Latest advances in atomic cluster collisions: structure and dynamics from the nuclear to the biological scale

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Latest Advances in

Atomic Cluster Collisions Structure and Dynamics from the Nuclear to the Biological Scale

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Latest Advances in

Atomic Cluster Collisions Structure and Dynamics from the Nuclear to the Biological Scale

editors

Jean-Patrick Connerade The Blackett Laboratory, Imperial College London, UK

Andrey Solov’yov Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe University, Germany

ICP

Imperial College Press

Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

LATEST ADVANCES IN ATOMIC CLUSTER COLLISIONS Structure and Dynamics from the Nuclear to the Biological Scale Copyright © 2008 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-1-84816-237-2 ISBN-10 1-84816-237-5

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

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PREFACE The Second International Symposium “Atomic Cluster Collisions: Structure and dynamics from the nuclear to the biological scale” (ISACC 2007) was organized as a satellite meeting of the XXVth International Conference on Photonic, Electronic, and Atomic Collisions (ICPEAC 2007, Freiburg, Germany) by the Frankfurt Institute for Advanced Studies (FIAS) and Gesellschaft f¨ ur Schwerionenforschung (GSI) and was held on July 19–23, 2007 at GSI, Darmstadt, Germany. ISACC was recognized by the European Physical Society (EPS) as a Europhysics Conference. ISACC started as the international symposium on atomic cluster collisions in St. Petersburg, Russia in 2003. ISACC 2007 promoted signiﬁcantly the growth and exchange of scientiﬁc information on the structure and properties of nuclear, atomic, molecular, biological and complex cluster systems studied by means of photonic, electronic, heavy particle and atomic collisions. In the symposium, particular attention was devoted to dynamical phenomena, many-body eﬀects taking place in the clusters, molecular and biological systems, which include problems of fusion and ﬁssion, fragmentation, collective electron excitations, phase transitions and many more. Both experimental and theoretical aspects of cluster physics, uniquely placed between nuclear physics on one hand and atomic, molecular, condensed matter and solid state physics on the other, were the subject of the symposium. The venue of the symposium was the Gesellschaft f¨ ur Schwerionenforschung (GSI) at Darmstadt, which is one of the world-renowned research centers in the ﬁeld of heavy ion physics. Several superheavy elements were synthesized here for the ﬁrst time. This was a very natural location for a symposium on cluster science, since cluster properties of the nuclei are very thoroughly investigated and often appear as a prototype for similar phenomena in other many-body systems of various degrees of complexity. The Symposium brought together more than 120 leading scientists in the ﬁeld of atomic cluster physics from around the world. The special emphasis of the Symposium was on the new methods of investigation of the structure and properties of atomic clusters, the collective excitations v

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in photoabsorption and photoionization processes of atomic clusters, ﬁssion and fusion dynamics of clusters, cluster dynamics in the laser ﬁeld, resonance processes in electron–cluster collisions, the interaction of ions, including multiply charged ions, with metal clusters and fullerenes and the processes of cluster deposition on a surface as well as of cluster collisions on a surface. The aim of the Symposium was to present the most recent achievements in all these ﬁelds of atomic cluster science. These proceedings, we hope, bear witness that this goal has been fulﬁlled. Part A of this book describes clustering phenomena at nuclear and subnuclear scales. It surveys recent advances in the synthesis of superheavy elements, clustering phenomena in ﬁssion and fusion processes of heavy nuclei and the properties of heavy and superheavy nuclei in supernova experiments. Part B is devoted to recent advances in the understanding of structure and essential properties, such as nanomagnetism, electronic and geometric shell eﬀects, of selected atomic cluster systems and conﬁned atoms. Both theoretical and experimental aspects of the ﬁeld are discussed. Part C describes recent advances in electron, photon and ion cluster collisions. These include the problem of molecular rotation in external ﬁelds, electron scattering on neon droplets, dynamical screening of atoms by fullerene cages, photoionization and fragmentation of fullerene ions, and collisions of molecules with cluster ions. Part D is devoted to the problem of clusters on a surface. In particular, it describes the recent eﬀorts in infrared spectroscopy and thermal desorption of clusters on self-assembled monolayers and the atomistic approach to the simulation of the nanoindentation process. Part E deals with the problems of phase transitions, ﬁssion, fusion and fragmentation in ﬁnite systems. In focus are quantum structuring of 4 H atoms around ionic dopants, phase transitions in polypeptides, and dissociation of charged rare gas clusters. Part F of the book contains a discussion of clusters in laser ﬁelds. These include the dynamics of metal clusters in laser ﬁelds: phase, amplitude and polarization shaping by interferometric pulse generation. Part G reports on recent advances in the understanding of clustering phenomenon in systems of various degrees of complexity. This chapter includes the description of structure and stability of novel objects: electron– positron quantum droplets. The new data on spectroscopy of chromophores

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in the gas phase is covered, and energy landscape analysis as a computational tool for analysis of complex molecular systems is discussed. Part H reports on recent advances in the understanding of structure and dynamics of biomolecules. These include a novel theoretical framework for the interpretation of NMR recidular dipolar couplings of unfolded proteins, computational simulation of antibody:antigen unbinding, mechanisms of magnetoreception in birds and biophysical modeling of fragment distributions of DNA plasmids after heavy ion irradiation. Part I includes the discussion of several important biological problems manifesting themselves at the mesoscopic scale, in which dynamics, clustering and other properties of matter at smaller scales play the essential role. Case studies presented in this chapter include molecular and nuclear mechanisms of ion cancer therapy and analysis of gene expression patterns in the Drosophila embryo. The subjects of the chapters in this book correspond to the sessions in the Symposium. The organizers of the ISACC 2007 wish to acknowledge generous support received from the European Physical Society, the Frankfurt Institute for Advanced Studies, the Gesellschaft f¨ ur Schwerionenforschung (GSI, Darmstadt), the European Commission within the Network of Excellence project EXCELL, which made this Symposium possible and a great success. The editors of this book would like to express their gratitude to Dr Andrey Lyalin, Dr Andriy Kostyuk, Dr Ilia Solov’yov, Mr Alexander Yakubovich and especially to Ms Stephanie Lo, for their great help in the preparation of the manuscript of this book for publication. Finally, we acknowledge our fruitful collaboration with Imperial College Press and the World Scientiﬁc Publishing Co. Jean-Patrick Connerade Imperial College, London, UK Andrey V. Solov’yov Frankfurt Institute for Advanced Studies, Germany Chairman The Second International Symposium “Atomic Cluster Collisions: Structure and dynamics from the nuclear to the biological scale” Frankfurt am Main and London, 2007

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LOCAL ORGANISING COMMITTEE

A. V. Solov’yov (Chairman) W. Greiner (Vice-Chair) S. Hofmann (Vice-Chair) T. Haberer H. Feldmeier C. Fournier T. Litvinova A. G. Lyalin O. I. Obolensky S. Raiss I. A. Solov’yov T. St¨ ohlker

— — — — — — — — — — — —

FIAS, Germany J.-W. Goethe University, Germany GSI, Germany GSI, Germany GSI, Germany GSI, Germany GSI, Germany FIAS, Germany FIAS, Germany GSI, Germany FIAS, Germany GSI, Germany

INTERNATIONAL ADVISORY COMMITTEE

U. Becker — Fritz Haber Institute, Germany C. Br´echignac — CNRS, France M. Broyer — LASIM, University Lyon 1, France E. Campbell — G¨ oteborg University, Sweden A. W. Castleman Jr. — Pennsylvania State University, USA J.-P. Connerade — Imperial College London, UK F. Gianturco — The University of Rome “La Sapienza”, Italy H. Haberland — University of Freiburg, Germany B. Huber — CIRIL, France V. K. Ivanov — St. Petersburg State Polytechnic University, Russia J. Jellinek — Argonne National Laboratory, USA T. Kondow — Toyota Technological Institute, Japan B. Zhang — Wuhan Institute of Physics and Mathematics, China ix

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CONTENTS

Preface

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Organizing Committees

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Conference Photo

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Part A. Clustering Phenomena at Nuclear and Subnuclear Scales

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Heaviest Nuclei from Yu. Oganessian

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Ca-Induced Reactions

Production and Decay of Superheavy Nuclei S. Hofmann Clustering Phenomena in Fission and Fusion Processes of Heavy Nuclei V. Zagrebaev and W. Greiner Nuclear Molecules G. G. Adamian, A. V. Andreev, N. V. Antonenko, S. P. Ivanova, R. V. Jolos, A. K. Nasirov, T. M. Shneidman, A. S. Zubov and W. Scheid

3

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23

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Properties of Heavy and Superheavy Nuclei in Supernova Environments T. J. B¨ urvenich, I. N. Mishustin and W. Greiner

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Part B. Structure and Properties of Atomic Clusters

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Clusters, Quantum Conﬁnement and Energy Storage J.-P. Connerade

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Site-Speciﬁc Analysis of Response Properties of Sodium Clusters K. Jackson, M. Yang and J. Jellinek

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Magnetism in Clusters A. Lyalin, A. V. Solov’yov and W. Greiner

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Strontium Clusters: Electronic and Geometry Shell Eﬀects A. Lyalin, I. A. Solov’yov, A. V. Solov’yov and W. Greiner

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New Deformed Single-Particle Shell Model D. N. Poenaru, R. A. Gherghescu, I. H. Plonski, A. V. Solov’yov and W. Greiner

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Part C. Electron, Photon and Ion Cluster Collisions

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Top Rotors in Electric Fields: Inﬂuence of the Asymmetry, the Flexibility and the Structure of the Molecules M. Abd el Rahim, R. Antoine, M. Broyer, P. Dugourd and D. Rayane Electron Scattering on Neon Droplets: Singly and Multiply Charged Neon Clusters S. Denifl, F. Zappa, I. M¨ ahr, P. Scheier, O. Echt and T. D. M¨ ark Dynamical Screening of an Atom Conﬁned Within a Finite-Width Fullerene S. Lo, A. V. Korol and A. V. Solov’yov Photoionization and Fragmentation of Fullerene Ions A. M¨ uller, S. Schippers, R. A. Phaneuf, S. Scully, E. D. Emmons, M. F. Gharaibeh, M. Habibi, A. L. D. Kilcoyne, A. Aguilar, A. S. Schlachter, L. Dunsch, S. Yang, H. S. Chakraborty, M. E. Madjet and J. M. Rost

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Collision of Transition Metal Cluster Ions with Simple Molecules M. Ichihashi, T. Hanmura and T. Kondow

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Part D. Clusters on a Surface

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Infrared Spectroscopy and Thermal Desorption Study of Vanadium–Mesitylene 1:2 Sandwich Clusters Soft-Landed onto a Long-Chain N-Alkanethiolate Self-Assembled Monolayer M. Mitsui, S. Doi, K. Ikemoto, S. Nagaoka and A. Nakajima Simulation of the Nanoindentation Procedure on Pure Nickel on the Smallest Length Scale: A Simple Atomistic Level Model P. Berke, M.-P. Delplancke-Ogletree, A. Lyalin, V. V. Semenikhina and A. V. Solov’yov Part E. Phase Transitions, Fusion, Fission and Fragmentation in Finite Systems Quantum Structuring of 4 He Atoms Around Ionic Dopants: Energetics of Li+ , Na+ and K+ from Stochastic Calculations E. Coccia, E. Bodo, F. Marinetti, F. A. Gianturco, E. Yurtsever, M. Yurtsever and E. Yildirim On the Theory of Phase Transitions in Polypeptides A. V. Yakubovich, I. A. Solov’yov, A. V. Solov’yov and W. Greiner Translational Kinetic Energy Released in the Dissociative Cascade of Charged Rare-Gas Clusters: Hints for Finite Size Phase Transitions? F. Calvo and P. Parneix

199

205

225

227

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Part F. Clusters in Laser Fields

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Dynamics of Metal Clusters: Free, Embedded and Deposited M. B¨ ar, F. Fehrer, P.-G. Reinhard, P. M. Dinh, E. Suraud, L. V. Moskaleva and N. R¨ osch

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Phase, Amplitude, and Polarization Shaping by Interferometric Pulse Generation A. Lindinger, S. M. Weber, F. Weise and M. Plewicki Part G. Clustering Phenomenon in System of Various Degrees of Complexity Electron–Positron Clusters: Structure and Stability V. K. Ivanov, R. G. Polozkov and A. V. Solov’yov Spectroscopy of Neutral Retinal and GFP Chromophores in the Gas Phase L. H. Andersen

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295 297

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The Energy Landscape as a Computational Tool J. M. Carr and D. J. Wales

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Part H. Structure and Dynamics of Biomolecules

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Theoretical Framework for the Interpretation of NMR Residual Dipolar Couplings of Unfolded Proteins O. I. Obolensky, A. V. Solov’yov, K. Schlepckow and H. Schwalbe

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Computational Simulations of Antibody: Antigen Unbinding E. S. Henriques and A. V. Solov’yov

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Iron Mineral Based Magnetoreception Mechanism in Birds I. A. Solov’yov and W. Greiner

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Biophysical Modeling of Fragment Distributions of DNA Plasmids After Heavy Ion Irradiation Th. Els¨ asser, M. Scholz, G. Taucher–Scholz, S. Brons, K. Psonka and E. Gudowska-Nowak

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Part I. From Biomolecules to Cells and System Biology Towards Monte Carlo Calculations of Biological Dose in Heavy-Ion Therapy: Modeling of Nuclear Fragmentation Reactions I. Pshenichnov, I. Mishustin and W. Greiner Mechanisms of Radiation Damage of Biomolecules E. Surdutovich, O. I. Obolensky, I. Pshenichnov, I. Mishustin, A. V. Solov’yov and W. Greiner On Modeling of Gene Expression Patterns in the Drosophila Embryo by the Gene Circuit Method A. M. Samsonov, V. V. Gursky, K. N. Kozlov and J. Reinitz Author Index

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Clustering Phenomena at Nuclear and Subnuclear Scales

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HEAVIEST NUCLEI FROM 48 Ca-INDUCED REACTIONS YURI OGANESSIAN Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, 141980 Dubna, Russia Relatively long half-lives of the new nuclides open possibilities for the study of the structure of superheavy atoms, in particular chemical properties of the new elements. Experiments on the observation of the socalled “relativistic eﬀect” in the electron structure of elements 112 and 114 using express radiochemical methods are now underway at the 48 Ca ion beam. The search for the most long-lived superheavy elements in nature is aimed at the registration of decay of element 108 isotopes. Preliminary results of measurements of spontaneous ﬁssion rare events in the underground laboratory at Modane (France) are discussed in context of the maximal nuclear half-lives at the top of the island of stability and in context of setting up new experiments. Keywords: Superheavy nuclei; long-lived isotopes.

A fundamental outcome of modern nuclear microscopic theory is the prediction of the “islands of stability” in the region of hypothetical superheavy elements. A signiﬁcant enhancement in nuclear stability when approaching the closed spherical shells with Z = 114 (possibly 120 and 122) and N = 184, which follow the doubly magic 208 Pb nucleus is expected for the nuclei with large neutron excess. Because of this, for the synthesis of nuclei with Z = 112–116 and 118 we chose the reactions 238 U, 242,244 Pu, 243 Am, 245,248 Cm and 249 Cf + 48 Ca, which are characterized by evaporation residues with a maximal number of neutrons. The formation and decay of the nuclei with Z = 112–116 and 118 were registered with the use of the Gas Filled Recoil Separator (Fig. 1), installed at the beam of the heavy ion accelerator.1 The mechanism of production of the heaviest nuclei induced by 48 Ca ions was studied separately. From the yield of the evaporation residues — products of the above-mentioned reactions (excitation functions), measured at diﬀerent ion-beam energies — it follows that they are formed in the 3

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Fig. 1.

Dubna Gas Filed Recoil Separator.

process involving the emission of two to ﬁve neutrons depending on the excitation energy (or temperature) of the compound nucleus.2 The maximal formation cross-section for the heaviest nuclei substantially depends on the neutron number in the compound nucleus and its position relative to the closed neutron shell N = 184. Figure 2 shows the formation cross-section of the evaporation products as a function of the ﬁssion barrier height of compound nuclei produced in the cold fusion (1n channel) and hot fusion (4n channel) reactions. In the cold fusion reactions (with a 208 Pb or 209 Bi target) a decrease in the cross-section with increasing ZCN is connected with dynamical prohibitions for the fusion of massive nuclei,3 whereas in the reactions of hot fusion (48 Ca + Act.) the major losses are determined by the strong ﬁssility of the heated nucleus. The survivability of compound nuclei grows with an increase in their ﬁssion barrier height in the vicinity of closed neutron shells N = 152, 162.4,5 An increase in the cross-section of the evaporation products of the heaviest nuclei, observed in the experiment, with the growing atomic number in the region of ZCN = 112–116 is connected with an increase in the neutron number and nuclear ﬁssion barrier as the neutron shell N = 184 is approached.

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Fig. 2. The upper panel shows cross-sections of the 1n and 4n evaporation channels of the cold and hot fusion reactions, correspondingly, as a function of the compound nucleus neutron number. The lower panel shows the fission barrier heights as a function of NCN .

The new nuclides undergo mainly sequential α-decays, which are terminated by spontaneous ﬁssion (SF). The total time of the decays ranges from 0.5 ms to ∼1 d, depending on the proton and neutron numbers in the synthesized nuclei. The experimental method used can be demonstrated with the example of the synthesis of elements 113 and 115 in the reaction 243 Am + 48 Ca.6,7 The evaporation of three neutrons and the emission of γ-rays by the compound nuclei of element 115, produced in the fusion reaction, lead to the formation in the ground state of the odd–odd nuclide with 115 protons and 173 neutrons. This nuclide is the parent of a radioactive “family” consisting of the Z = 115(α) → 113(α) → 111(α) → 109(α) → 107(α) → 105(SF) nuclei, formed as a result of the ﬁve consecutive emissions of α-particles, and terminated by spontaneous ﬁssion of the Db isotope (Z = 105).

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Fig. 3.

Synthesis of elements 113 and 115.

The properties of the long-lived Db isotope are themselves of special interest (Fig. 3). Due to the long lifetime, the atoms of element 105 can be separated by a classical oﬀ-line radiochemical method of ion-exchange chromatography, with the consequent measurement of their decay by spontaneous ﬁssion. In eight identical experiments 243 Am + 48 Ca, after the chemical separation, 15 spontaneous ﬁssion events were detected with T1/2 ∼ 1 d.8–10 It was shown that the spontaneous ﬁssion observed in the decay of the 288 115 nucleus comes from an element that is a chemical homologue of Nb and Ta, which are representatives of the ﬁfth group of Mendeleev’s Periodic Table of the Elements. The chemistry experiment gives an independent and unambiguous identiﬁcation of the atomic number of the ﬁnal nucleus (Z = 105) and at the same time the atomic numbers of all nuclides in the fully correlated chain of the decay of the parent nucleus 288 115. Another example is the study of a chemical behavior of the isotope 283 112 (T1/2 ∼ 3.8 s). To what extent element 112 is a homologue of Hg depends on the so-called “relativistic eﬀect” in the electronic structure of the superheavy atom. According to some relativistic calculations, the chemical behavior of element 112 (as well as of the other atoms with a higher atomic number) will somewhat diﬀer from that of its light homologue. Both characteristics — the high volatility and the ability to form inter-metallic

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compounds — of Hg/Au and (112)/Au independently conﬁrm the identiﬁcation of the atomic numbers of element 112.11 And ﬁnally, after a few attempts failed to synthesize the isotope 283 112 in the reaction 238 U + 48 Ca with diﬀerent set-ups,12,13 its decay properties were successfully reproduced in the recent experiments with the set-up SHIP (GSI).14 The majority of our experiments have been dedicated to the synthesis of even-Z elements. In the 238 U, 242,244 Pu, 245,248 Cm + 48 Ca reactions 15 new nuclides were synthesized for the ﬁrst time: these are the heaviest isotopes of elements 110, 112, 114 and 116. For the 291 116 isotope and its daughter nuclei 287 114 and 283 112, a rare branch of 6–4 consecutive αdecays was observed: 116(α) → 114(α) → 112(α) → 110(α, SF) → 108(α) → 106(α, SF) → 104(SF). It was terminated by spontaneous ﬁssion of the neutron-rich isotope 267 Rf (T1/2 ∼ 1.3 h). Simultaneously, ﬁve new odd-Z isotopes of elements 111, 113 and 115 were obtained in reactions 237 Np, 243 Am + 48 Ca. For the ﬁrst time, results are reported on the synthesis of the heaviest element with Z = 118. It has been shown that in the fusion of two massive nuclei 249 Cf + 48 Ca, after the emission of three neutrons, the even–even isotope 284 118, which undergoes α-decay (Eα = 11.65+0.06 MeV) with a halflife T1/2 = 0.9+1.1 −0.3 ms, is formed. Its further decay takes place within about 0.2 s via the chain 284 118(α) → 280 116(α) → 276 114(α, SF) → 272 112(SF). The properties of the daughter nuclei of 284 118 — the isotopes with Z = 116, 114 and 112 — were studied in an independent way in the 238 U, 242 Pu and 245 Cm + 48 Ca reactions (Fig. 4). A self-consistent picture of the decay of the isotope of element 118 was obtained.15 In the series of experiments using the 48 Ca-beam, performed during the past six years, in total 84 events, corresponding to the formation and decay of 34 new nuclides with Z = 104–118 and N = 161–177, were observed (Fig. 5). A comparison of the half-lives of the known nuclei with Z = 110–112 with those of the newly observed neutron-rich isotopes of the same elements shows that the increase of their mass by adding 6–8 neutrons brings forth an increase in nuclear stability by a factor of 104–105 (Fig. 6). The decay properties of the isotopes of the heaviest elements are now being compared with the predictions of microscopic nuclear models. This comparison gives evidence of the decisive inﬂuence of the nuclear structure of superheavy elements on their stability with respect to diﬀerent modes of radioactive decay. A more detailed analysis shows that experiments do not solely reproduce the theoretically expected decay scenarios, but are also consistent (within ∼5% accuracy) with the decay energies of all the synthesized 26 α-radioactive nuclei with Z = 106–118.17 From this point of view, the obtained results can be considered as the ﬁrst experimental

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Fig. 4. (a) Decay chains of the isotopes of element 116 synthesized in 2n- and 3n-evaporation channels of the reaction 245 Cm + 48 Ca (bottom). The daughter products of α-decay of these nuclides — the isotopes of elements 114 and 112 — were also produced in the reactions 242 Pu, 238 U + 48 Ca. (b) Decay of the nuclei of element 118 synthesized in the reaction 249 Cf + 48 Ca.15

evidence of the existence of “islands of stability” in the region of the heaviest elements, considerably extending the boundaries of the material world.16 Keeping in mind that the yield of the new nuclides in heavy ion reactions is extremely low, any further progress in the ﬁeld has to be connected primarily with the sensitivity of the experiment. The increase of the ion-beam intensity and the creation of new, more eﬃcient set-ups will make it possible to perform fundamental investigations in nuclear physics (determination of nuclear mass limits) and chemistry (relativistic eﬀect), aﬀecting at the same time inter-disciplinary ﬁelds: the models of nucleosynthesis, astrophysical aspects, studies of the structure of superheavy atoms and molecules, and so on. One of the new-generation set-ups, MASHA (Mass Analyzer of Super Heavy Atoms), is designed to be the ﬁrst step in achieving this aim. This new set-up, compared to the already existing kinematical separators, will be a few times more eﬃcient; it will have high selectivity and identiﬁcation ability for the mass numbers of the separated atoms.18 At the same time, it is a sophisticated detector that can be used in diﬀerent chemical techniques, for studying the chemical behaviors of the superheavy elements. After the

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Nuclide chart of elements with Z ≥ 104.16

determination of the physical and chemical properties of elements 112 and 114 in the 242,244 Pu+48 Ca reactions (such experiments have been performed in our laboratory in the last two years), further investigations will be carried out in conjunction with the MASHA set-up. Another problem is obtaining longer-lived superheavy nuclides. As the artiﬁcial synthesis of nuclei is limited, the possibility of searching for the most stable nuclei with Z = 106–110 and N ∼ 180 (the estimates are T1/2 ∼ 104 –106 y with high uncertainties) in nature is being considered. Among the possible candidates for the ﬁrst experiment, an isotope of element 108 (Hs) was chosen. The search for the long-lived Hs isotope in its chemical homologue — a sample of metallic Os (500 g) — is being performed in the underground laboratory in Modane (France). The observation of a single spontaneous ﬁssion event (measured as a neutron ﬂash, accompanying the ﬁssion process) over a one-year measuring period will correspond to a concentration amounting to 5 × 10−15 g/g of element 108 in the Os sample, assuming that its half-life is equal to 109 y. This small value is ∼10−16 times less than the concentration of uranium in the Earth’s crust.

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Fig. 6. Half-lives of the isotopes with Z ≥ 108 synthesized in cold fusion (open symbols) and 48 Ca-induced reactions (black symbols). The lines trace through calculated α-decay half-lives obtained in the MM-model.17

In spite of the high sensitivity of the experiment, the chances of ﬁnding surviving superheavy nuclei are small. However, the absence of any eﬀect will give an upper limit for the half-life of the long-lived nuclide at the level of T1/2 ≤ 5 × 107 y (shown in Fig. 6). Acknowledgments The experiments were carried out at the Flerov Laboratory of Nuclear Reactions (JINR, Dubna) in collaboration with the Analytical & Nuclear Chemistry Division of the Lawrence Livermore National Laboratory (USA). References 1. K. Subotic et al., Nucl. Instr. Meth. A 481, 71 (2002). 2. Y. Oganessian et al., Phys. Rev. C 70, 064609 (2004).

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3. V. Zagrebaev, M. Itkis and Y. Oganessian, Phys. At. Nucl. 66, 1033 (2003). 4. I. Muntian, Z. Patyk and A. Sobiczewski, Acta Phys. Pol. B 34, 2073 (2003). 5. A. Baran, Z. Lojewski, S. Kamila and M. Kowal, Phys. Rev. C 72, 044310 (2005). 6. Y. Oganessian et al., Phys. Rev. C 69, 021601 (2004). 7. Y. Oganessian et al., Phys. Rev. C 72, 034611 (2005). 8. S. Dmitriev et al., Mendeleev Commun. 15, 1 (2005). 9. D. Schumann et al., Radiochim. Acta 93, 727 (2005). 10. N. Stoyer et al., Proc. 9th Int. Conf. Nucleus–Nucleus Collisions, Rio de Janeiro, Brazil, 28 August–1 September, 2006. 11. R. Eichler et al., Nature 447, 72 (2007). 12. W. Loveland, K. E. Gregorich, J. B. Patin, D. Peterson, C. Rouki, P. M. Zielinski and K. Aleklett, Phys. Rev. C 66, 044617 (2002). 13. K. E. Gregorich et al., Phys. Rev. C 72, 014605 (2005). 14. S. Hofmann et al., Eur. Phys. J. A 32, 251 (2007). 15. Y. Oganessian et al., Phys. Rev. C 74, 044602 (2006). 16. Y. Oganessian, J. Phys. G: Nucl. Part. Phys. 34, R165 (2007). 17. A. Sobiczewski and K. Pomorski, Prog. Part. Nucl. Phys. 58, 292 (2007). 18. Y. Oganessian et al., Nucl. Instrum. Methods B 204, 606 (2003).

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PRODUCTION AND DECAY OF SUPERHEAVY NUCLEI S. HOFMANN∗ Gesellschaft f¨ ur Schwerionenforschung (GSI), D-64291 Darmstadt, Germany Johann Wolfgang Goethe-Universit¨ at, D–60438 Frankfurt, Germany www.gsi.de [email protected] An overview of present experimental investigation of superheavy elements is given. Using cold fusion reactions which are based on lead and bismuth targets, relatively neutron-deﬁcient isotopes of the elements from 107 to 113 were synthesized at GSI in Darmstadt, Germany, and/or at RIKEN in Wako, Japan. In hot fusion reactions of 48 Ca projectiles with actinide targets, more neutron-rich isotopes of the elements from 112 to 116 and even 118 were produced at FLNR in Dubna, Russia. Recently, part of these data, which represent the ﬁrst identiﬁcation of nuclei located on the predicted island of superheavy elements, were conﬁrmed in two independent experiments. The data are compared with theoretical descriptions. Keywords: Superheavy nuclei.

1. Introduction and status of experiments For the synthesis of heavy and superheavy elements (SHEs) fusionevaporation reactions are used. Two approaches have been successfully employed. Firstly, there are the reactions with medium mass ion beams impinging on targets of stable Pb and Bi isotopes (cold fusion). These reactions have been successfully used to produce elements up to Z = 112 at GSI1 and conﬁrmed at RIKEN2,3 and LBNL.4 Recently, a number of neutrondeﬁcient odd element isotopes were produced in a combination with 208 Pb target, and odd element projectiles5,6 at LBNL. Using a 209 Bi target, the isotope 278 113 was synthesized at RIKEN.2 Secondly, reactions between lighter ions, especially with beams of 48 Ca, and radioactive actinide targets ∗Josef

Buchmann Professor Laureatus. 12

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(hot fusion) have been used to produce more neutron-rich isotopes of elements from Z = 112 to 116 and 118 at FLNR.7 Recently, the results of two of these reactions, 48 Ca + 242 Pu and 48 Ca + 238 U, were conﬁrmed in independent experiments.8–10 Figure 1 summarizes the data as they are presently known or under investigation. Besides the discovery of the existence of these high-Z elements, two more important observations have emerged. Firstly, the expectation that halflives of the new isotopes should lengthen with increasing neutron number as one approaches the island of stability seems to be fulﬁlled. Secondly, the measured cross-sections for the relevant nuclear fusion processes reach values up to 5 pb, which is surprisingly high. Furthermore, they seem to be correlated with the variation of shell-correction energies as predicted by macroscopic–microscopic calculations.11,12

Fig. 1. Upper end of the chart of nuclei showing the presently (2007) known nuclei. For each known isotope the element name, mass number, and half-life are given. The magic numbers for the protons at elements 114 and 120 and for the neutrons at N = 184 are emphasized. The bold dashed lines mark proton number 108 and neutron numbers 152 and 162. Nuclei with that number of protons or neutrons have increased stability. However, they are deformed contrary to the spherical superheavy nuclei. The crossing at Z = 114 and N = 162 reflects the uncertainty, whether nuclei in that region are deformed or spherical. The background structure in gray shows the calculated shell correction energy according to the macroscopic–microscopic model.

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2. Nuclear structure and decay properties The calculation of the ground-state binding energy provides the basic step to determine the stability of SHEs. In macroscopic–microscopic models, the binding energy is calculated as the sum of a predominating macroscopic part (derived from the liquid-drop model of the atomic nucleus) and a microscopic part (derived from the nuclear shell model). This way, more accurate values for the binding energy are obtained than in the cases of using only the liquid drop model or the shell model. The shell correction energies of the ground-state of nuclei near closed shells are negative, which results in further decreased values of the negative binding energy from the liquid drop model — and thus increased stability. An experimental signature for the shell-correction energy is obtained by subtracting a calculated smooth macroscopic part from the measured total binding energy. The shell-correction energy is plotted in Fig. 2(a) using data from Ref. 13. Two equally deep minima are obtained, one at Z = 108 and N = 162 for deformed nuclei with deformation parameters β2 ≈ 0.22, β4 ≈ −0.07 and the other one at Z = 114 and N = 184 for spherical SHEs. Diﬀerent results are obtained from self-consistent Hartree– Fock–Bogoliubov calculations and relativistic mean-ﬁeld models.14–18 They

Fig. 2. (a) Shell-correction energy (b)–(d) and partial half-lives for SF, α, and β decay, respectively. The calculated values in (a)–(c) were taken from Refs. 11 and 13 and in (d) from Ref. 12. The squares in (a) mark the nuclei presently known; the filled squares in (d) mark the β stable nuclei.

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predict for spherical nuclei shells at Z = 114, 120 or 126 [indicated as dashed lines in Fig. 2(a)] and N = 172 or 184. The knowledge of ground-state binding energies, however, is not suﬃcient for the calculation of partial spontaneous ﬁssion (SF) half-lives. Here it is necessary to determine the size of the ﬁssion barrier over a wide range of deformation. The most accurate data were obtained for even– even nuclei using a macroscopic–microscopic model.11 Partial SF half-lives are plotted in Fig. 2(b). The landscape of ﬁssion half-lives reﬂects the landscape of shell-correction energies, because in the region of SHEs the height of the ﬁssion barrier is, ﬁrstly, mainly determined by the groundstate shell correction energy, while the contribution from the macroscopic liquid-drop part approaches zero for Z = 104 and above, and, secondly, the shell correction energy at the saddle point is small.19 Nevertheless, we see a signiﬁcant increase of SF half-life from 103 s for deformed nuclei to 1012 s for spherical SHEs. This diﬀerence originates from an increasing width of the ﬁssion barrier, which becomes wider in the case of spherical nuclei. Partial α half-lives decrease almost monotonically from 1012 s down to −9 10 s near Z = 126 [Fig. 2(c)]. The valley of β-stable nuclei passes through Z = 114, N = 184. At a distance of about 20 neutrons away from the bottom of this valley, β half-lives of isotopes have dropped down to values of one second12 [Fig. 2(d)]. Combining results from the individual decay modes, one obtains the dominating partial half-life as shown in Fig. 3(a) for even–even nuclei. The two regions of deformed heavy nuclei near N = 162 and spherical SHEs merge and form a region of α emitters surrounded by spontaneously ﬁssioning nuclei. Alpha decay becomes the dominant decay mode beyond darmstadtium with continuously decreasing half-lives. For nuclei at N = 184 and Z < 110 half-lives are determined by β − decay. For odd nuclei, Fig. 3(b), partial α and SF half-lives calculated in Ref. 13 have to be multiplied by a factor of 10 and 1000, respectively, thus making provisions for the odd particle hindrance factors. However, we have to keep in mind that ﬁssion hindrance factors show a wide distribution from 101 to 105 , which is mainly a result of the speciﬁc levels occupied by the odd nucleon.20 For odd–odd nuclei, the ﬁssion hindrance factors from both the odd proton and the odd neutron are multiplied. For odd and odd–odd nuclei, the island character of α emitters disappears and for nuclei with neutron numbers 150 to 160, α decay prevails down to rutherfordium and beyond. In the allegorical representation, where the stability of SHEs is seen as an island in a sea of instability, even–even nuclei portray the situation at high-tide and odd nuclei at low-tide, when the island is connected to the mainland.

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Fig. 3. Dominating partial half-lives for α, β + decay/EC, β − decay, and SF: (a) for even–even nuclei, and (b) for odd-A nuclei. Nuclei and decay chains known at present are marked; the known odd–odd nuclei are included in (b).

3. Nuclear reactions In fusion reactions towards SHEs the product Z1 Z2 reaches extremely large values and the ﬁssion barrier extremely small values. In addition, the ﬁssion barrier itself is fragile, because it is solely built up from shell eﬀects. For these reasons, the fusion of SHEs is hampered twofold: (i) in the entrance channel by a high probability for re-separation, and (ii) in the exit channel by a high probability for ﬁssion. In contrast, the fusion of lighter elements proceeds unhindered through the contracting eﬀect of the surface tension and the evaporation of neutrons instead of via ﬁssion. The mutual relation between measured cross-sections and calculated ﬁssion barriers is plotted in Fig. 4. Cross-sections and ﬁssion barriers decrease up to element 110; beyond, both values increase again. The reason that the cross-sections increase less than the ﬁssion barriers could be that the calculated latter values are less high in reality or, which seems more likely, that increasing Coulomb repulsion in the entrance channel leads to a reduced fusion probability.

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Fig. 4. Measured cross-sections of hot fusion reactions (upper panel) and calculated fission barriers11,21 (lower panel) revealing the mutual relation between these two quantities (see text for details). The figure was taken from Ref. 22.

A number of excitation functions were measured for the synthesis of elements from No to Ds using Pb and Bi targets.1 For the even elements, these data are shown together with the two data points measured for 278 112 in Fig. 5. The maximum evaporation residue cross-section (1n channel) was measured at beam energies well below a one-dimensional fusion barrier.23 At the optimum beam energy, projectile and target are just reaching the contact conﬁguration in a central collision. The relatively simple fusion barrier based on the Bass model23 is too high and a tunneling process through this barrier cannot explain the measured cross-section. Various processes are possible, and are discussed in the literature, which result in a lowering of the fusion barrier. Among these processes, the transfer of nucleons and an excitation of vibrational degrees of freedom are the most important.24–26

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Fig. 5. Left: Measured excitation functions of even elements from rutherfordium to element 112 produced in reactions with 208 Pb targets and beams from 50 Ti to 70 Zn. These data were measured in experiments at SHIP. For comparison, the excitation function for synthesis of element 114 in the reaction 48 Ca + 244 Pu is plotted in the bottom panel.22 The arrows mark the energy at the point where contact configuration is reached using the model by Bass.23 Right: Comparison of the cross-sections as functions of the excitation energy E ∗ for quasi-fission (σQF ), compound fission (σF ), and ERs (σER ) for the reaction 48 Ca + 238 U. The figure on the right was taken from Ref. 22.

Target nuclei of actinide targets are strongly deformed and the height of the Coulomb barrier depends on the orientation of the deformation axes. Two of the measured excitation functions for the production of 292 114 and 283 112 after evaporation of four and three neutrons,22 respectively, are shown in Fig. 5. A comparison with the cold fusion data reveals that the element 114 excitation function is located completely above the Bass contact conﬁguration, which was calculated for a mean radius of the deformed target nucleus. In addition, the curves for the hot fusion reactions are signiﬁcantly broader, 10.6 instead of 4.6 MeV (FWHM) as measured for 265 Hs.

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On the right of Fig. 5, the fusion evaporation excitation function is compared with the yield of quasi-ﬁssion and compound nucleus ﬁssion.27 At low projectile energies, nuclear reactions can occur only at polar orientation of the deformed target nucleus. In the upper part of the ﬁgure, the orientation at the touching point and the corresponding excitation energy is indicated. Although the excitation energy of the compound nucleus would be low, about 20–25 MeV, resulting in reduced ﬁssion of the compound nucleus, the nuclei do not fuse but re-separate with high probability at this elongated conﬁguration. Only the compact conﬁguration in the case of equatorial collisions results in fusion despite the fact that the compound nucleus is considerably more highly excited, about 35–40 MeV. It was pointed out in the literature28 that closed shell projectile and target nuclei are favorable synthesizing SHEs. The reason is not only a low (negative) reaction Q-value and thus a low excitation energy, but also that fusion of such systems is connected with a minimum energy dissipation. The fusion path proceeds along cold fusion valleys, where the reaction partners maintain kinetic energy up to the closest possible distance. In the case of cold fusion with spherical targets, the maximum fusion yield is obtained at projectile energies just high enough so that projectile and target nucleus come to rest when just the outer orbits are in contact. The conﬁguration at this point is plotted in Fig. 6. From there on, the fusion process occurs well ordered along paths of minimum dissipation of energy. Empty orbits above the closed shell nucleus 208 Pb favor a transfer of nucleons from the projectile to the target and thus initiate the fusion process. At ﬁrst glance, the situation seems to be diﬀerent in the case of hot fusion. The maximum of the excitation function is located at the higher energy side of the value needed to reach the contact conﬁguration according to the Bass model23 ; see Fig. 5. However, taking into account the deformation of the target nucleus and considering fusion at equatorial orientation, also then the projectile and target nuclei come to rest when just the outer orbits are in contact. This distance corresponds to the energy where the maximum yield is measured; see Fig. 5. It is, like in cold fusion, located on the left side of the “Bass” contact conﬁguration for deformed nuclei at equatorial collisions. Also, in this case the empty orbits in the equatorial plane of the prolate deformed target nucleus favor the transfer of nucleons.

4. Conclusion and outlook The experimental work of the last three decades has shown that crosssections for the synthesis of the heaviest elements do not decrease continuously as was measured up to the production of element 112 using a cold

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Fig. 6. Energy against distance diagram for the reaction of an almost spherical 64 Ni projectile with a spherical 208 Pb target nucleus resulting in the deformed fusion product 271 110 after emission of one neutron. At the center-of-mass energy of 236.2 MeV, the maximum cross-section was measured. In the top panel, the reaction partners are represented by their nuclear potentials (Woods–Saxon) at the contact configuration where the initial kinetic energy is exhausted by the Coulomb potential. At this configuration, projectile and target nuclei are 14 fm apart from each other. This distance is 2 fm larger than the Bass contact configuration, where the mean radii of projectile and target nucleus are in contact. In the bottom panel, the outermost proton orbitals are shown at the contact point. For the projectile 64 Ni, an occupied 1f7/2 orbit is drawn, and for the target 208 Pb an empty 1h9/2 orbit. The protons circulate in a plane perpendicular to the drawing. The Coulomb repulsion, and thus the probability for separation, is reduced by the transfer of protons. In this concept, the fusion is initiated by transfer (FIT, see also Refs. 24 and 25). The figure was taken from Ref. 29.

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fusion reaction. Recent data on the synthesis of elements 112 to 116 and 118 in Dubna using hot fusion break this trend when the region of spherical SHEs is reached. Some of the results originally obtained in Dubna were conﬁrmed in independent experiments and with diﬀerent methods, including the use of chemical, element-speciﬁc properties. We conclude that the region of the predicted spherical SHEs has ﬁnally been reached and the exploration of the “island” has started and can be performed even on a relatively high cross-section level. An opportunity for the continuation of experiments in the region of SHEs at low cross-sections aﬀord, among others, further accelerator developments. High current beams and radioactive beams are options for the future. A wide range of half-lives encourages the application of a wide variety of experimental methods in the investigation of SHEs, from the safe identiﬁcation of short-lived isotopes by recoil-separation techniques to atomic physics experiments on trapped ions, and to the investigation of chemical properties of SHEs using long-lived isotopes.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22.

S. Hofmann and G. M¨ unzenberg, Rev. Mod. Phys. 72, 733 (2000). K. Morita et al., J. Phys. Soc. Jpn. 73, 2593 (2004). K. Morita et al., J. Phys. Soc. Jpn. 76, 043201 (2007). T. N. Ginter et al., Phys. Rev. C 67, 064609 (2003). C. M. Folden III et al., Phys. Rev. Lett. 93, 212702 (2004). C. M. Folden III et al., Phys. Rev. C 73, 014611 (2006). Yu. Ts. Oganessian, J. Phys. G: Nucl. Part. Phys. 34, R165 (2007). R. Eichler et al., Nucl. Phys. A 787, 373c (2007). R. Eichler et al., Nature 447, 72 (2007). S. Hofmann et al., Eur. Phys. J. A 32, 251 (2007). R. Smolanczuk et al., Phys. Rev. C 52, 1871 (1995). P. M¨ oller et al., At. Data Nucl. Data Tables 66, 131 (1997). R. Smolanczuk and A. Sobiczewski, in Proc. of the XV. Nucl. Phys. Div. Conf. on Low Energy Nuclear Dynamics, eds. Yu. Ts. Oganessian et al. (World Scientiﬁc, Singapore, 1995), p. 313. S. Cwiok et al., Nucl. Phys. A 611, 211 (1996). K. Rutz et al., Phys. Rev. C 56, 238 (1997). A. T. Kruppa et al., Phys. Rev. C 61, 034313 (2000). M. Bender et al., Phys. Lett. B 515, 42 (2001). M. Bender et al., Nucl. Phys. A 723, 354 (2003). W. J. Swiatecki et al., Acta Phys. Pol. B 38, 1565 (2007). D. C. Hoﬀman, Nucl. Phys. A 502, 21c (1989). I. Muntian et al., Acta Phys. Pol. B 34, 2141 (2003). Yu. Ts. Oganessian et al., Phys. Rev. C 70, 064609 (2004).

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23. 24. 25. 26. 27. 28. 29.

R. Bass, Nucl. Phys. A 231, 45 (1974). W. von Oertzen, Z. Phys. A 342, 177 (1992). V. V. Volkov, Phys. Part. Nuclei 35, 425 (2004). V. Yu. Denisov and S. Hofmann, Phys. Rev. C 61, 034606 (2000). M. G. Itkis et al., Phys. Rev. C 65, 044602 (2002). R. K. Gupta et al., Z. Phys. A 283, 217 (1977). S. Hofmann, in Proc. of the XV. Nucl. Phys. Div. Conf. on Low Energy Nuclear Dynamics, St. Petersburg, Russia, 1995, eds. Yu. Ts. Oganessian et al. (World Scientiﬁc, Singapore, 1995), pp. 305–312.

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CLUSTERING PHENOMENA IN FISSION AND FUSION PROCESSES OF HEAVY NUCLEI VALERY ZAGREBAEV Flerov Laboratory of Nuclear Reactions, JINR, Dubna, Russia [email protected] WALTER GREINER Frankfurt Institute for Advanced Studies, Frankfurt am Main, Germany [email protected] Strongly coupled processes of deep inelastic scattering, quasi-ﬁssion and fusion are described concurrently with a common set of variables within the Langevin-type equations of motion. Shell eﬀects on the multidimensional potential energy surface play an extremely important role in these reactions. This leads to the two-body (shape isomeric states in ﬁssion) and three-body clustering phenomena in heavy nuclear systems. Enhanced yield of the nuclides far from the projectile and target masses was found in the multi-nucleon transfer reactions due to shell eﬀects. This suggests that the low-energy damped collisions of transactinide nuclei may be used as an alternative way for the production of surviving superheavy long-living neutron-rich nuclei. Keywords: Damped collisions; fusion; ﬁssion; clustering; superheavy elements.

1. Introduction A new approach was recently proposed1,2 for a uniﬁed and simultaneous description of strongly coupled deep inelastic (DI), quasi-ﬁssion (QF) and fusion–ﬁssion processes of low-energy heavy-ion collisions. The distance between the nuclear centers R (corresponding to the elongation of a mononucleus), dynamic spheroidal-type surface deformations δ1 and δ2 , mutual in-plane orientations of deformed nuclei ϕ1 and ϕ2 , and mass asymme1 −A2 try η = A A1 +A2 are used in this approach as the most relevant variables for description of fusion–ﬁssion dynamics. Note that we take into consideration all the degrees of freedom needed for description of all the reaction stages. 23

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Thus, in contrast with other models, we need not split artiﬁcially the whole reaction process into several stages, and we may consider concurrently the strongly coupled DI, QF and CN formation reaction channels within the same Langevin-type equations of motion. At low collision energies, the shell eﬀects play a very important role in the dynamics of heavy nuclear systems. Here, we concentrate mainly on the detailed study of these eﬀects. 2. Adiabatic potential energy The interaction potential of separated nuclei is calculated rather easily within the folding procedure with eﬀective nucleon–nucleon interaction or parameterized, e.g., by the proximity potential. Of course, some uncertainty remains here, but the height of the Coulomb barrier obtained in these models coincides with the empirical Bass parametrization3 within 1 or 2 MeV. After contact, the mechanism of interaction of two colliding nuclei becomes more complicated. For fast collisions (E/A ∼ εFermi or higher) the nucleus– nucleus potential, Vdiab , should reveal a strong repulsion at short distances protecting the “frozen” nuclei from penetrating each other and forming a nuclear matter with double density (diabatic conditions, sudden potential). For slow collisions (near-barrier energies), when nucleons have enough time to reach equilibrium distribution (adiabatic conditions), the nucleus– nucleus potential energy, Vadiab , is quite diﬀerent (Fig. 1). Thus, at energies well above the Coulomb barrier, we need to use a time-dependent potential

Fig. 1. Potential energy for 48 Ca + 248 Cm for diabatic (dashed curve) and adiabatic (solid curve) conditions (zero deformations of the fragments).

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Schematic view of nuclear system in the two-core approximation.

energy, which after contact gradually transforms from a diabatic potential energy into an adiabatic one: V = Vdiab [1 − f (t)] + Vadiab f (t). Here, t is the time of interaction and f (t) is a smoothing function with parameter τrelax ∼ 10−21 s, f (t = 0) = 0, f (t τrelax ) = 1. The calculation of the multi-dimensional adiabatic potential energy surface for a heavy nuclear system remains a very complicated physical problem, which has not yet been solved in full. The two-center shell model4 seems to be most appropriate for calculation of the adiabatic potential energy. However, the simplest version of this model with a restricted number of collective coordinates, using standard parametrization of the macroscopic (liquid drop) part of the total energy5,6 and overlapping oscillator potentials for a calculation of the single particle states and resulting shell correction, does not reproduce correctly values of the nucleus–nucleus interaction potential for well separated nuclei and at the contact point (depending on mass asymmetry). The same holds for the value of the Coulomb barrier and the depth of potential pocket at contact. These shortcomings are overcome in the extended version of the two-center shell model.7 The phenomenological two-core model8,9 (based on the two-center shell model idea) was also proposed for a calculation of the multi-dimensional adiabatic potential energy surface in which all the shell eﬀects are included by using experimental nuclear masses of the fragments a1 and a2 gradually dissolving with the increase in a number of the shared nucleons ∆A (see Fig. 2). 3. Clusterization and shape-isomeric states Within the two-center shell model, the processes of compound nucleus formation, ﬁssion and quasi-ﬁssion may be described both in the space of (R, η, δ1 , δ2 ) and in the space (a1 , δ1 ; a2 , δ2 ), because for a given nuclear conﬁguration (R, η, δ1 , δ2 ) we may unambiguously determine the two cores a1 and a2 . This is extremely important for interpretation of physical meaning of some deep minima on the potential energy surface.

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Fig. 3. Driving potential of nuclear system 296 116 ↔ 48 Ca + 248 Cm. (a) Potential energy in the “elongation — mass asymmetry” space. (b) Topographical landscape of the driving potential on the (z1 , z2 ) plane. Dashed, solid and dotted curves show most probable trajectories of fusion, quasi-ﬁssion and regular ﬁssion, respectively. The diagonal corresponds to the contact conﬁgurations (∆A = 0). (c) Three-humped barrier calculated along the ﬁssion path (dotted curve).

The adiabatic driving potential for formation and decay of superheavy nucleus 296 116 is shown in Fig. 3 as a function of z1 and z2 (minimized over n1 and n2 ) at R ≤ Rcont and also as a function of elongation and mass asymmetry at ﬁxed deformations of both fragments. It is easy to see that the shell structure, clearly revealing itself in the contact of two nuclei, is also retained at R < Rcont [see the deep minima in the regions of z1,2 ∼ 50 and z1,2 ∼ 82 in Fig. 3(b)]. Following the ﬁssion path [dotted curves in Figs. 3(a) and (b)] the nuclear system goes through the optimal conﬁgurations (with minimal potential energy) and overcomes the multi-humped ﬁssion barrier [Fig. 3(c)]. These intermediate minima correspond to the shape isomer states. From analysis of the driving potential, we may deﬁnitely conclude that these isomeric states are nothing but the two-cluster conﬁgurations with magic or semi-magic cores surrounded by a certain amount of shared nucleons.

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It is interesting to estimate the adiabatic potential energy for the threecenter conﬁguration. Such clusterization may play a role in the vicinity of the scission point, where the shared nucleons ∆A may form a third cluster located between the two heavy cores a1 and a2 . Because there are too many degrees of freedom, we calculated the potential energy of a threebody conﬁguration (shown in Fig. 4) only as a function of Z1 and Z3 at ﬁxed deformations (δ1 = δ2 = δ3 = 0.1) of the fragments being in contact. The corresponding potential energy (minimized over neutron numbers N1 and N3 ) is shown in Fig. 5 for the 248 Cm nucleus. One may see that the potential energy increases with increasing mass of the third fragment. This means that a ternary ﬁssion should be quite unfavorable for transactinide nuclei. However, the situation may change for heavier nuclear systems (see below).

Fig. 4.

Three-body clusterization of a heavy nuclear system.

Fig. 5. Landscape of potential energy of a three-body clusterization of the nucleus.

248 Cm

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4. Quasi-fission and SHE formation It is well known that in low-energy collisions of very heavy ions the quasiﬁssion process dominates, hindering formation of a compound nucleus. This process is also caused by the shell eﬀects, namely, by the deep valleys on the potential energy surface. Driving potentials for the 48 Ca + 248 Cm fusion reaction are shown in Fig. 6 for two diﬀerent initial orientations of the deformed 248 Cm nucleus. After overcoming the Coulomb barrier, the fragments ﬁrst become very deformed, then the mass asymmetry gradually decreases and the system ﬁnds itself in the quasi-ﬁssion valley with one of the fragments close to the doubly magic nucleus 208 Pb (see deep valley at η ≈ 0.4 in Fig. 6). Experimental10 and calculated energy–mass distributions of the primary reaction products at the near-barrier energy of Ec.m. = 203 MeV are shown in Fig. 7. The large yield of the fragments in the region of doubly magic nucleus 208 Pb (and complimentary light fragments) is the most pronounced feature of the TKE-mass distribution. Note that a reasonable quantitative description of the QF processes was attained for the ﬁrst time. The probability of CN formation in this reaction was found to be very small and depended greatly on the incident energy. Due to a strong dissipation of kinetic energy, just the ﬂuctuations (random forces) deﬁne the dynamics of the system after the contact of two nuclei. At near barrier collisions, the excitation energy (temperature) of the system is rather low, the ﬂuctuations are weak and the system chooses the most probable path to the

Fig. 6. Driving potentials for the nuclear system formed in the 48 Ca + 248 Cm collision at tip (left) and side (right) orientations of statically deformed 248 Cm. The solid lines with arrows show schematically (without ﬂuctuations) the projections of the QF trajectories (going to lead and tin valleys) and the path leading to formation of CN.

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Fig. 7. Experimental (a) and calculated (b) TKE-mass distributions of reaction products in collision of 48 Ca + 248 Cm at Ec.m. = 203 MeV. (c) Contributions of DI (1), QF (2,3) and fusion–ﬁssion (4) processes into inclusive mass distribution. (d) One of the trajectories in collision of 48 Ca + 248 Cm leading to QF channel (2).

exit channel along the quasi-ﬁssion valley (see Fig. 6). However, at non-zero excitation energy there is a chance for the nuclear system to overcome the multi-dimensional inner potential barriers and ﬁnd itself in the region of the CN conﬁguration. Within the Langevin calculations, a great number of events should be tested to ﬁnd this low probability. For the studied reaction, for example, only several fusion events have been found among more than 105 total tested events [see dark region 4 in Fig. 7(c)]. In our approach, we estimated the possibility of SH element production in the asymmetric fusion reactions of nuclei heavier than 48 Ca with transuranium targets. Such reactions can be used, in principle, for a synthesis of elements heavier than 118. Evaporation residue (EvR) cross-sections for the fusion reactions 50 Ti + 244 Pu, 50 Ti + 243 Am, 54 Cr + 248 Cm and 58 Fe + 244 Pu are shown in Fig. 8. SH elements beyond 118 may also be synthesized in the fusion reactions of symmetric nuclei (ﬁssion-like fragments). However, for such reactions an uncertainty in calculation of the cross-sections for CN formation is rather large. Dashed and solid curves in Fig. 8(d) reﬂect this uncertainty in our estimations of the EvR cross-sections

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Fig. 8. Evaporation residue cross-sections for the fusion reactions (a) 50 Ti + 244 Pu, (b) 50 Ti + 243 Am, (c) 54 Cr + 248 Cm, 58 Fe + 244 Pu (dashed curves) and (d) 136 Xe + 136 Xe.

in the 136 Xe + 136 Xe fusion reaction. If the experiment (performed in Dubna) will give the EvR cross-sections at the level of a few picobarns, then we may really dream about using neutron-rich accelerated ﬁssion fragments for the production of SH elements in the region of the “island of stability” (e.g., 132 Sn + 176 Yb → 308 120). 5. Low-energy damped collisions Low-energy damped collisions of very heavy transactinide nuclei (e.g., U + 248 Cm) were also used about thirty years ago for a synthesis of SH elements.11,12 The cross-sections were found to decrease very rapidly with increasing atomic number of surviving target-like fragments. However, Fm and Md neutron-rich isotopes were produced at the level of 0.1 µb. Recently, it was shown13 that the existence of a rather pronounced lead valley on the potential landscape of such giant nuclear systems may lead to the so-called “inverse” (anti-symmetrizing) quasi-ﬁssion process, in which one fragment transforms to the doubly magic nucleus 208 Pb, whereas another one transforms to complementary SH element. In spite of rather high excitation energy, this neutron-rich superheavy nucleus may survive in a neutron evaporation cascade giving us an alternative way of producing neutron-rich long-living SH elements (see Fig. 9). Good agreement of the experimental data with our predictions was obtained recently for near-barrier 238 U + 238 U damped collisions.14 Nevertheless, more detailed experiments have to be performed aimed at a study of the shell eﬀects in mass transfer in low-energy collisions of heavy nuclei. For this purpose, lighter nuclear systems may also be used. The mass distributions of the primary fragments in the 160 Gd + 186 W reaction calculated with and without the shell corrections to the potential energy are shown in Fig. 10. As can be seen at near barrier collision energies, the shell eﬀects may increase by two orders of magnitude the yield of the reaction products even for the transfer of twenty nucleons. 238

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Fig. 9. Yield of superheavy nuclei in collisions of 238 U + 238 U (dashed), 238 U + 248 Cm (dotted) and 232 Th + 250 Cf (solid lines) at 800 MeV center-of-mass energy. Solid curves in the upper part show isotopic distribution of primary fragments in the Th + Cf reaction.

Fig. 10. Primary fragment mass distribution in the 160 Gd + 186 W reaction at 460 MeV center-of-mass energy calculated with (solid) and without (dashed histogram) shell corrections in potential energy.

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We found also that, in low-energy collisions of transactinides, the shell eﬀects may lead to the formation of three-cluster conﬁgurations. In Fig. 11, the landscape of the potential energy surface is shown for a three-body clusterization of the nuclear system formed in the collision of U + U. As can be seen, the potential energy has a rather deep minimum corresponding to the Pb–Ca–Pb-like conﬁguration (or Hg–Cr–Hg) caused by the N = 126 and Z = 82 nuclear shells. The existence of this three-body clusterization

Fig. 11. Landscape of potential energy of three-body conﬁgurations formed in collision of 238 U + 238 U (see notations in Fig. 4).

Fig. 12. Radial dependence of potential energy for two-body (solid curve) and threebody (dashed curve) conﬁgurations of nuclear system formed in collision of 238 U + 238 U.

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can be proved experimentally by a coincident detection of the two Pblike fragments in collisions of transactinides. A more ﬂat radial dependence of the potential energy (as compared with a two-body system) is another feature of this three-body conﬁguration (see Fig. 12). Decay of a U + U-like nuclear system into the three-body conﬁguration may signiﬁcantly prolong reaction time, which could be important for spontaneous positron formation in a super-strong electric ﬁeld.15 References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

V. Zagrebaev and W. Greiner, J. Phys. G 31, 825 (2005). V. Zagrebaev and W. Greiner, J. Phys. G 34, 1 (2007). R. Bass, Nuclear Reactions with Heavy Ions (Springer Verlag, 1980), p. 326. U. Mosel, J. Maruhn and W. Greiner, Phys. Lett. B 34, 587 (1971). A. I. Sierk, Phys. Rev. C 33, 2039 (1986). P. M¨ oller et al., At. Data Nucl. Data Tables 59, 185 (1995). V. Zagrebaev, A. Karpov, Y. Aritomo, M. Naumenko and W. Greiner, Phys. Part. Nucl. 38, 469 (2007). V. I. Zagrebaev, Phys. Rev. C 64, 034606 (2001). V. I. Zagrebaev, AIP Conf. Proc. 704, 31 (2004). M. G. Itkis et al., in Fusion Dynamics at the Extremes, eds. Yu. Ts. Oganessian and V. I. Zagrebaev (World Scientiﬁc, Singapore, 2001), p. 93. M. Sch¨ adel et al., Phys. Rev. Lett. 41, 469 (1978). M. Sch¨ adel et al., Phys. Rev. Lett. 48, 852 (1982). V. I. Zagrebaev, M. G. Itkis, Yu. Ts. Oganessian and W. Greiner, Phys. Rev. C 73, 031602 (2006). A. C. C. Villari et al., AIP Conf. Proc. 891, 60 (2007). J. Reinhardt, U. M¨ uller and W. Greiner, Z. Phys. A 303, 173 (1981).

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NUCLEAR MOLECULES G. G. ADAMIAN, A. V. ANDREEV, N. V. ANTONENKO, S. P. IVANOVA, R. V. JOLOS, A. K. NASIROV, T. M. SHNEIDMAN and A. S. ZUBOV Joint Institute for Nuclear Research, 141980 Dubna, Russia W. SCHEID Institut f¨ ur Theoretische Physik der Justus-Liebig-Universit¨ at, 35392 Giessen, Germany [email protected] A nuclear molecule or a dinuclear system consists of two touching nuclei which carry out motion in the internuclear distance and exchange nucleons by transfer. The dinuclear system model can be applied to nuclear structure, fusion reactions leading to superheavy nuclei, multi-nucleon transfer and fission. Keywords: Dinuclear model; nuclear structure; fusion; quasifission.

1. Introduction Molecular structures appear in various ﬁelds of nuclear physics and are signatures for a high stability of nuclear sub-units, called clusters. Such clusters can be α-particles or heavy nuclei with closed shells like 40 Ca or 208 Pb. A well-studied example is the structure of 8 Be consisting of two α-particles. Nuclear molecular resonance structures were ﬁrst observed by Bromley et al.1 in the scattering of 12 C on 12 C and then seen up to the system Ni + Ni.2 The Frankfurt School of W. Greiner contributed much to the explanation of nuclear molecular eﬀects within molecular reaction theories, two-center shell model and fragmentation theory.3 In this contribution, we present more recent developments of the nuclear molecular idea on the basis of the dinuclear system concept proposed by V. V. Volkov.4 A nuclear molecule or a dinuclear system (DNS) is a conﬁguration of two touching clusters (nuclei) which maintain their individuality. 34

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The dynamics of such a system is ruled by two main degrees of freedom: (i) the relative motion between the nuclei describing molecular resonances in the internuclear potential, the capture of two heavy ions forming a DNS and the decay of the DNS which is denoted as quasi-ﬁssion, and (ii) the transfer of nucleons between the nuclei leading to a dependence of the dynamics on the mass and charge asymmetries in fusion and ﬁssion reactions. The latter processes are described by the mass and charge asymmetry coordinates η=

A1 − A2 A1 + A2

and ηZ =

Z1 − Z2 . Z1 + Z2

(1)

These coordinates can be assumed as continuous or discrete quantities. For η = ηZ = 0, we have a symmetric clusterization with two equal nuclei, and if η approaches the values ±1 or if A1 or A2 is equal to zero, a fused system has been formed. These coordinates were originally introduced for describing the fragmentation dynamics in heavy ion reactions5 and applied to mass distributions in ﬁssion.6 Their importance to nuclear reactions was pointed out by V. V. Volkov4 in the dinuclear system concept. Here, we give a short and concise review of several applications of the DNS model. We discuss hyperdeformed states, normal- and superdeformed bands, the multi-nucleon transfer between nuclei, leading to complete and incomplete fusion, especially the production of superheavy nuclei, and the decay of the dinuclear system, i.e. the quasi-ﬁssion. 2. Hyperdeformed states Nuclei have excited states with the properties of molecular (or cluster) states. Such states could be the hyperdeformed (HD) states which are caused by a third minimum in the potential energy surfaces (PES) of the corresponding nuclei. An interesting observation in shell model calculations was made by Cwiok et al.,7 that the third minimum of the PES of actinide nuclei belongs to a molecular conﬁguration of two touching nuclei (clusters), which is a DNS conﬁguration. Dinuclear systems have quadrupole moments and moments of inertia as those measured for superdeformed states and estimated for HD states.8 Under the assumption that hyperdeformed states can be considered as quasimolecular states in the internuclear potential, it should be possible to excite them in the scattering of heavy ions. In the following, we concentrate the discussion on the systems 48 Ca + 140 Ce and 90 Zr + 90 Zr as possible candidates for exploring the properties of hyperdeformed states.9 First, we calculated the potentials V (R, L) as a function of the relative distance for various angular momenta. These potentials are shown in Fig. 1. They have a minimum of around 11 fm at a distance Rm ≈ R1 + R2 + 0.5 fm, where R1

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140

V (MeV)

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Ca+

140

Ce

Bqf(L)

120 90

200

90

Zr+ Zr

180 160 10

11

12

13

14

15

R (fm) Fig. 1. The potential V (R, L) for the systems 48 Ca + 140 Ce (upper) and 90 Zr + 90 Zr (lower) as a function of R for L = 0, 20, 40, 60, 80 presented by solid, dashed, dotted, dash-dotted and dash-dot-dotted curves, respectively.

and R2 are the radii of the nuclei. The depth of this molecular minimum decreases with growing angular momentum and vanishes for L > 100 in the considered systems. The potential pocket has virtual and quasibound molecular resonance states situated above and below the barrier, respectively. Approximating the potential in the neighborhood of the minimum by a harmonic oscillator potential, we can easily estimate the positions of one to three quasibound states with an energy spacing of ω ≈ 2.2 MeV for L > 40. For example, in the 90 Zr + 90 Zr system, we ﬁnd the lowest quasibound state for L = 50 lying 1.1 MeV above the potential minimum. Quasibound states should be directly excited by tunneling through the potential barrier in R including the centrifugal potential. In this case, the DNS has no extra excitation energy. It stays in the potential minimum without changing the mass and charge asymmetries if spherical and stiﬀ nuclei (magic and double magic nuclei) are used in heavy ion reactions. The cross-section for penetrating the barrier and populating quasibound states can be estimated to be on the order of 1 µb. Only a range of partial waves contributes to the excitation of quasibound states and constitutes the so-called “molecular window” known in the theory of nuclear molecules with light clusters. In the reaction 48 Ca on 140 Ce, cold and long-living DNS states can be formed at an incident energy Ec.m. = 147 MeV and 90 < L < 100, and in the reaction 90 Zr on 90 Zr at Ec.m. = 180 MeV and 40 < L < 50. The

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spectroscopic investigation of the HD structures is diﬃcult because of the small formation cross-section and the high background produced by fusion– ﬁssion, quasi-ﬁssion and other reactions. However, the later processes have characteristic times much shorter than the lifetime of the HD states, which is of the order of 10−16 s. Therefore, the HD states should show up as sharp resonance lines as a function of the incident energy. 3. Nuclear structure with normal- and superdeformed bands The DNS model can be used to describe the normal-deformed (ND) and superdeformed (SD) bands of various nuclei. We applied this model to the structure of 60 Zn (Ref. 10) and of 190,192,194 Hg and 192,194,196 Pb.11 Also, alternating parity bands in rare earth nuclei and actinides can be explained with this model.12 The observed strong collective dipole transitions between the excited SD band and the lowest-energy SD band in 150 Gd, 152 Dy, 190,194 Hg, 196,198 Pb and between the SD and ND bands in 194 Hg and 194 Pb indicate a decay out of pronounced octupole deformed states. The measured properties of the excited SD bands in 152 Dy and 190,192,194 Hg have been interpreted in terms of rotational bands built on collective octupole vibrations.13 Conﬁgurations with large quadrupole and octupole deformation parameters and low-lying collective negative parity states are strongly related to a clustering describable with heavy and light clusters within the DNS model. The cluster picture of the above-mentioned ND and SD bands can be consistently treated by assuming a collective dynamics in the mass or charge asymmetry coordinates η or ηZ , respectively. To achieve this aim, we formulated a conventional collective Schr¨ odinger equation in ηZ (or η): 2 d 1 d − + U (ηZ , I) ψn (ηZ , I) = En (I)ψn (ηZ , I). (2) 2 dηZ BηZ dηZ The left-hand side of Fig. 2 shows the calculated potential (histogram) U of 194 Hg as a function of the charge number Z2 of the lighter cluster for two nuclear spins, I = 0 and 10. The potentials have minima for α-type clusterizations, namely, for Z2 = 2, 4, 6, 8, . . . . In addition, the probabilities |ψn (ηZ , I)|2 expressed with the intrinsic wave functions of the ND and SD states are presented. These probabilities are peaked around the minima of the potential indicating a corresponding cluster structure of the states. On the right-hand side of Fig. 2, we show the calculated level spectra of 194 Hg in comparison with the experimental data. We note that the shift of the negative parity states is reproducible with the dynamics in ηZ and is

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0

-

19 17 -

15 13 11 9 7 5 3 1

+

ND, exp.

ND, th.

Fig. 2. Left: Potential energy (histogram) U of 194 Hg for the spins I = 0 and 10. The curves are the absolute squares of the wave functions of the ground (solid) and first excited (dashed) ND bands and ground (dash-dotted) and first excited (dotted) SD bands. Right: Calculated and experimental levels of the ground state and superdeformed bands of 194 Hg.

related to the properties of the octupole degree of freedom. Also, electromagnetic transition probabilities can be evaluated10–12 with the intrinsic wave functions, which agree well with the experimental data. 4. Fusion to superheavy nuclei Heavy and superheavy nuclei can be produced by fusion reactions with heavy ions. We discriminate Pb or Bi based or cold fusion reactions, e.g., 70 Zn + 208 Pb → 278 112 → 277 112 + n with an evaporation residue crosssection of σ = 1 pb and an excitation energy of the 278 112 compound nucleus of about 11 MeV, and actinide based or hot fusion reactions, e.g., 48 Ca + 244 Pu → 288 114 + 4n, with the emission of more neutrons. The crosssections are small because of a strong competition between complete fusion and quasi-ﬁssion and small survival probabilities of the excited compound nucleus. 4.1. Models for production of superheavy nuclei The models for the production of superheavy nuclei depend sensitively on whether adiabatic or diabatic potentials in the internuclear coordinate R are assumed.

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Models using adiabatic potentials minimize the potential energy. In that case, the nuclei ﬁrst change their mass asymmetry in the direction of more symmetric clusters and then they fuse together by crossing a smaller fusion barrier in the relative coordinate around η = 0. The models tend to give larger probabilities for fusion if similar target and projectile nuclei are taken, which contradicts the exponential falling-oﬀ of the evaporation residue cross-section with increasing mass of the projectile nuclei in Pb-based reactions. Models using diabatic potentials describe the fusion process as a transfer of nucleons in a dinuclear conﬁguration. The diabatic potential has a a minimum in the touching range and a repulsive part towards smaller internuclear distances prohibiting the dinuclear system from amalgamating to the compound nucleus in the relative coordinate. Such a potential, achieved with a diabatic two-center shell model,14 evolves towards the adiabatic potential and has a survival time of about the reaction time of 10−20 s. It can also be justiﬁed with structure calculations based on group theoretical methods.15 The above distinction between diﬀerent models is based on a few degrees of freedom. If more collective coordinates like orientation angles and vibration coordinates of the fragments and the neck coordinate are included in the dynamical treatment, then the diﬀerences between the two approaches get diminished. Of similar importance to the diﬀerence between adiabatic and diabatic potentials is the coordinate-dependence of the various masses. As explored by Fink et al.16 for the case of 12 C+ 12 C scattering, a coordinate dependence of the mass of relative motion can be transformed into a constant mass and an energy-dependent repulsive potential which screens the outer range of the potential from the inner one and, therefore, has similar properties to the diabatic potential. 4.2. Evaporation residue cross-section The cross-section for the production of superheavy nuclei can be written J max σER (Ec.m. ) = σcap (Ec.m. , J)PCN (Ec.m. , J)Wsur (Ec.m. , J). (3) J=0

The three factors are the capture cross-section, the probability for complete fusion, and the survival probability. The maximal contributing angular momentum Jmax is on the order of 15–20. The capture cross-section σcap describes the formation of the dinuclear system at the initial stage of the reaction, when the kinetic energy of the relative motion is transferred

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into potential and excitation energies. The DNS can decay by crossing the quasi-ﬁssion barrier Bqf , which is on the order of 0.5–5 MeV. After its formation, the DNS evolves in the mass asymmetry coordinate. The center of the mass distribution moves towards more symmetric fragmentations and its width is broadened by diﬀusion processes. ∗ The part of the distribution that crosses the inner fusion barrier Bfus of the driving potential U (η) yields the probability PCN for complete fusion. The DNS also decays by quasi-ﬁssion during its evolution. Therefore, the fusion probability PCN and the mass and charge distributions of the quasi-ﬁssion have to be treated simultaneously. The fusion probability can be quantitatively estimated with the Kramers formula and results in ∗ − Bqf − Bsym )/T ), where the temperature T is related PCN ∼ exp(−(Bfus to the excitation energy of the DNS, and Bsym is the barrier in η to more symmetric conﬁgurations. Bsym is 4–5 MeV (>Bqf ) in cold fusion reactions and 0.5–1.5 MeV (
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-164 -166

72

Ge

280

-168

281

282 A

283

284

236

238

240

242

244

A

∗ Fig. 3. Left: Excitation energy ECN and evaporation residue cross-section σ1n for Ge+ 208 Pb → A 112. Right: Evaporation residue cross-section σ ∗ 3n,4n , excitation energy ECN and Q-value for 48 Ca + A Pu. The two experimental points are from Ref. 18.

5. Quasi-fission and incomplete fusion The process of quasi-ﬁssion is the decay of the DNS. Since quasi-ﬁssion leads to a large quantity of observable data like mass and charge distributions, distributions of total kinetic energies (TKE), variances of total kinetic energies and neutron multiplicities, a comparison of the theoretical description with experimental data provides sensitive information about the applicability and correctness of the DNS model. For this reason, we studied the dynamics of mass and charge transfer and the succeeding quasi-ﬁssion with master equations.19 As a starting point, we considered the shell model Hamiltonian of all dinuclear fragmentations of the nucleons. This Hamiltonian can be used to derive master equations for the probability PZ,N (t) to ﬁnd the dinuclear system in a fragmentation with Z1 = Z, N1 = N and Z2 = Ztot − Z, N2 = Ntot − N . The master equations contain the oneproton and one-neutron transfer rates which depend on the single-particle energies and the temperature of the DNS where the occupation of the singleparticle states is taken into account by a Fermi distribution. The simultaneous transfer of more nucleons is neglected. The master equations include a term describing the quasi-ﬁssion with a rate obtained with the Kramers

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formula. This rate causes a loss of the total probability PZ,N (t) ≤ 1. The primary yields Y (Z, N ) are calculated from the probabilities PZ,N (t). The DNS dynamics was also studied by Li et al.20 with similar master equations. We calculated quasi-ﬁssion distributions, TKEs, variances of TKE and neutron multiplicities for cold and hot fusion reactions19 and found satisfying agreement with the experimental data of Itkis et al.21 For heavier systems, e.g., 48 Ca + 248 Cm, the contribution of fusion–ﬁssion products to the mass distribution can be neglected since the probability of forming a compound nucleus is very small. The master equations also give probabilities for more asymmetric systems than the initial one. These systems can be used to describe incomplete fusion.22,23 In Fig. 4, we present evaporation residue crosssections for incomplete fusion in the reactions 48 Ca + 244,246,248 Cm. With these reactions, one can produce new isotopes of superheavy nuclei with 3

10

258

Md

2

σER (pb)

10

256

Md

257

No 260

Lr

259

262

No

Lr

264 261

No

1

10

264

Db Rf 268 Db 263 Rf 266 265 Db Rf

Lr

265

267

Sg 270

269

Sg

Bh

268 271

Bh

Sg

0

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Bh

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Hs 273

Hs

σER (pb)

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-5

10

-6

10

YZN

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10

-8

10

-9

10

102

104

Z

106

108

Fig. 4. Calculated primary yields (lower) and evaporation residue cross-sections (middle and upper) are shown by triangles, circles and squares for the reactions 48 Ca + 244,246,248 Cm (E c.m. = 207, 205.5 and 204 MeV), respectively. The heavy fragments after 1 n evaporation are indicated in the upper part of the figure. The closed and open symbols are calculated with the master equations and Kramers-type formula, respectively.

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Z = 104 − 108, which ﬁll the gap between the isotopes of heaviest nuclei obtained in cold and hot complete fusion reactions. Their measurement would be a proof for the fusion dynamics in the dinuclear system concept. Acknowledgments We thank DFG (Bonn), VW-Stiftung (Hannover) and RFBR (Moscow) for supporting this work. We thank Prof. Junqing Li (Lanzhou), Prof. Enguang Zhao (Beijing), Prof. Nikolay Minkov (Soﬁa), Prof. Shangui Zhou (Beijing), and Dr. Ning Wang (Giessen) for valuable discussions and help. References 1. D. A. Bromley, J. A. Kuehner and E. Almqvist, Phys. Rev. Lett. 4, 365 (1960). 2. L. Vannucci et al., Z. Phys. A 355, 41 (1996). 3. W. Greiner, J. Y. Park and W. Scheid, Nuclear Molecules (World Scientific, Singapore, 1995). 4. V. V. Volkov, Izv. AN SSSR ser. fiz. 50, 1879 (1986). 5. H. J. Fink et al., Z. Phys. 268, 321 (1974). 6. J. Maruhn and W. Greiner, Phys. Rev. Lett. 32, 548 (1974). 7. S. Cwiok et al., Phys. Lett. B 322, 304 (1994). 8. T. M. Shneidman et al., Nucl. Phys. A 671, 119 (2000). 9. G. G. Adamian et al., Phys. Rev. C 64, 014306 (2001). 10. G. G. Adamian et al., Phys. Rev. C 67, 054303 (2003). 11. G. G. Adamian et al., Phys. Rev. C 69, 054310 (2004). 12. T. M. Shneidman et al., Phys. Rev. C 74, 034316 (2006). 13. T. Nakatsukasa et al., Phys. Rev. C 53, 2213 (1996). 14. A. Diaz–Torres et al., Phys. Lett. B 481, 228 (2000). 15. G. G. Adamian et al., Phys. Lett. B 451, 289 (1999). 16. H. J. Fink et al., J. Phys. G (Nucl. Phys.) 1, 685 (1975). 17. G. G. Adamian et al., Phys. Rev. C 69, 011601 (2004); ibid. C 69, 014607 (2004). 18. Yu. Ts. Oganessian et al., Eur. Phys. J. A 13, 135 (2002); ibid. A 15, 201 (2002). 19. G. G. Adamian et al., Phys. Rev. C 68, 034601 (2003). 20. W. Li, N. Wang, J. F. Li, H. Xu, W. Zuo, E. Zhao, J. Q. Li and W. Scheid, Europhys. Lett. 64, 750 (2003). 21. M. G. Itkis et al., Nucl. Phys. A 734, 136 (2004). 22. G. G. Adamian and N. V. Antonenko, Phys. Rev. C 72, 064617 (2005). 23. G. G. Adamian et al., Phys. Rev. C 71, 034603 (2005).

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PROPERTIES OF HEAVY AND SUPERHEAVY NUCLEI IN SUPERNOVA ENVIRONMENTS ¨ T. J. BURVENICH, I. N. MISHUSTIN and W. GREINER Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe University, Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany The properties of nuclei embedded in an electron gas are studied within the relativistic mean-ﬁeld approach. These studies are relevant for nuclear properties in astrophysical environments such as neutronstar crusts and supernova explosions. The electron gas is treated as a constant background in the Wigner–Seitz cell approximation. We investigate the stability of nuclei with respect to α and β decay. We ﬁnd that the presence of the electrons leads to stabilizing eﬀects for α decay at high electron densities. Furthermore, the screening eﬀect shifts the proton drip-line to more proton-rich nuclei, and the stability line with respect to β decay is shifted to more neutron-rich nuclei. Implications for the creation and survival of very heavy nuclear systems are discussed. Keywords: Superheavy nuclei; supernova explosions; electron gas.

1. Introduction The properties of nuclei in astrophysical environments have enjoyed continuous interest for over more than 30 years. The seminal work by Negele and Vautherin1 laid the groundwork for microscopic studies of nuclei in stellar environments; see, e.g., Refs. 2–7. In this contribution, we focus on the physics of nuclei embedded in a dense electron gas8 at zero temperature. We consider nuclei across the periodic chart up to superheavy nuclei embedded in a Wigner–Seitz cell with constant electron density. The presence of the electrons will aﬀect the location of the β-stability line and the proton drip-line as well as decay modes such as α decay. The calculations are performed within the Wigner–Seitz (WS) approximation by dividing the system into (spherical) WS cells, each containing only one nucleus and the number of electrons equal to the nuclear charge Z. The construction of the WS cell is aimed at an approximate 44

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and convenient segmentation of an ensemble of nuclei surrounded by the uniform background of electrons. Its construction should ensure that the physics contained in the Wigner–Seitz cell is to an optimally large degree isolated from the surroundings. This requires, for example, charge neutrality of the cell. For the description of nuclei we employ the relativistic mean-ﬁeld (RMF) model.9,10 The eﬀective in-medium nucleon–nucleon interaction is parametrized via the exchange of several meson ﬁelds: scalar–isoscalar ρµ ). The meson ﬁelds (σ), vector–isovector (ωµ ) and vector–isovector ( are treated as classical potentials and the nucleons are represented by Dirac spinors. The isoscalar–scalar σ meson delivers the intermediate-range attraction, while the isoscalar–vector ω meson is responsible for the shortrange repulsion. The isovector–vector ρ meson couples to the isovector nucleon density and thus parametrizes the isovector properties of the model. The photon is coupled in the standard fashion. No (explicit) exchange terms are taken into account. In order to account for the electron background, we modify the source term of the photon ﬁeld by adding the electron density, i.e. (1) ∆φ = −e(ρp − ρe ) = −eρch, which is the Poisson equation for the electrostatic potential φ ≡ A0 . All other equations remain the same as in vacuum. We employ the mean-ﬁeld parametrization NL3 in these studies,11 which delivers accurate values for nuclear ground-state properties. 2. Periodic chart In this section, we analyze how the nuclear β-stability valley and the proton and neutron drip-lines change with increasing electron Fermi momentum. The proton single-particle potential experiences a constant downward shift. This downward shift of the proton potential and hence the gap between proton and neutron chemical potentials is the reason for the new β-stability line. In the presence of electrons, the equilibrium condition with respect to weak decays (n → p + e− + νe , p → n + e+ + ν¯e ) reads (2) µn = µp + µe , where µn and µp are neutron and proton chemical potentials calculated within the framework of the RMF model, i.e. (3) µN = (mN + gσ σ)2 + p2F N + gω ω 0 + gρ ρ0 τ3 , N = n, p. The electron chemical potential is simply given by µe = kF2 + m2e ,

(4)

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where the contribution of the electrostatic potential has been neglected since it cancels out with the corresponding contribution from the proton in Eq. (2). If µn > µp + µe holds, neutrons can decay into available proton states with the electron going on the top of the Fermi distribution. On the other hand, if µn < µp + µe , protons can capture electrons from the Fermi distribution and occupy neutron states. Only when the condition in Eq. (2) is fulﬁlled, are nuclear systems stable with respect to these decay modes. Note that this relation becomes the well-known condition for β-stability in vacuum, µn ≈ µp , for kF = 0. For zero temperature, the proton and neutron chemical potentials coincide with their Fermi energies. We note that this correspondence is no longer uniquely deﬁned for systems with pairing, where the eﬀective Fermi ˆ = A, but can lie somewhere between the energy is determined to yield N last level contributing with non-zero occupancy and the next one. Thus, in order to obtain the (N, Z) dependence of the stability line, instead of Eq. (2) we employ the criterion |µe − (µn − µp )| < ∆,

(5)

where ∆ = 1 MeV. In Fig. 1, the β-stability lines are plotted for various values of kF . In the same plot, we show also the proton and neutron drip-lines (see below). As kF increases, the line of β-stability is shifted more and more to the neutronrich side, i.e. the presence of electrons stabilizes neutron-rich nuclei. This shift is so strong already for kF = 0.1 fm−1 that β-stability is reached only in 180 160 140 120 100 Z

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the region of the two-neutron drip-line. This means that the β-equilibrium is reached only when free neutrons appear in the system. 3. α decay In this section, we investigate the inﬂuence of the electron background on α decay of nuclei. The Qα value of the reaction is deﬁned as Qα (N, Z) = B(N, Z) − B(N − 2, Z − 2) − B(2, 2),

(6)

and corresponds approximately, neglecting nuclear recoil, to the kinetic energy of the α-particle leaving the nucleus. In the considered environment, the binding energies of mother and daughter nuclei as well as the binding energy of the α-particle increase due to the attractive electromagnetic interaction of protons with the electron background. The calculation of the α decay lifetimes in the presence of electrons is not completely trivial. There are two competing eﬀects. The Qα value is lowered due to the increased nuclear binding caused by electrons, which alone (for the same barrier) could increase the lifetime. The presence of the electrons leads to screening and modiﬁcation of the Coulomb potential. Therefore, we also need to take into account the change of the barrier through which the α-particle has to penetrate. To quantify this eﬀect, we use the simple one-parameter model of Ref. 12, which expresses the α-particle potential Vα as Vα = 2Vp + 2Vn ,

(7)

where Vp and Vn are, respectively, proton and neutron single-particle potentials. The half-life is written as τ1/2 = ln 2/λ, where the decay constant λ is parametrized as λ = cP. Here, the pre-formation factor and the knocking frequency in the Gamov picture are absorbed into one parameter, c, which is adjusted to known data. The probability for transmission through the barrier, P = eS , is calculated within the WKB approximation R2 2µ[Vα (r) − Qα ] S = −2 dr , (8) 2 R1 where R1 and R2 are the turning points of the barrier. The α-particle potential is lowered as a function of increasing electron Fermi momentum, making it easier for the α-particle to escape the nucleus. Together with the Qα value, this leads to an overall decreasing of α half-lives; see Fig. 2. This holds true at low electron densities. However, this trend changes in the vicinity of the point where the Qα value turns negative at a certain electron Fermi momentum kFcrit . At kF > kFcrit , α decay becomes forbidden

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Fig. 2. The evolution of Qα values (left) and α decay half-lives (right) for the nucleus 218 U as a function of the electron Fermi momentum. In the right panel, the full squares correspond to calculations within the model of Ref. 12, multiplied with a prefactor taking into account tunneling near the threshold Qα = 0. The lines are drawn to guide the eye.

and the nucleus becomes stable with respect to this decay mode. In the uranium isotope discussed here, kFcrit = 0.24 fm−1 . However, the potential barrier experienced by the α-particle at kFcrit = 0.24 fm−1 is still ≈ 13 MeV high. It goes to zero, however, at the WS cell radius. 4. Fission barriers In ﬁssion studies, an important role is played by the energy of the system as a function of the deformation parameter β2 . Our analysis shows that the shape of the electron cloud has some eﬀect on the energy. While the spherical cloud does not produce any changes, the deformed cloud leads to visible eﬀects. Important key quantities related to ﬁssion are the width and the height of the ﬁssion barrier. The inﬂuence of the electrons on the ﬁssion barrier is a macroscopic eﬀect resulting from interaction of the protons with the Coulomb potential produced by the electrons. The electrostatic repulsion between the protons is weakened by the presence of the negative charge background. This change leads to the increase of barrier height and isomer energy. Physically speaking, the electron background tends to slightly stabilize the system with respect to symmetric spontaneous ﬁssion and to increase the excitation energy of the isomeric state. We expect that asymmetric ﬁssion will be altered in a similar way, although our consideration does not allow octupole shapes. We note that the potential energy surface

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is determined by the dependence of the energy on the deformation. Especially for superheavy systems, the liquid drop barrier vanishes, and their stabilization results from shell eﬀects. The trends discussed above are clearly seen in Fig. 3, which displays the ﬁssion barrier of plutonium for diﬀerent choices of the electron density in the spheroidal conﬁguration. The inner barrier increases with increasing kF leading to a stabilization eﬀect towards spontaneous ﬁssion. Further, the properties of the shape isomer (second minimum) are altered due to the electron background too. For kF = 0.5 fm, the inner ﬁssion barrier increases by approximately 0.3 MeV and the energy of the isomeric state increases by approximately 1 MeV. Note that the ground-state binding energy of 240 Pu with NL3 is 1814 MeV; the presence of the electrons increases it to the value of 2248 MeV (the latter does not include the kinetic energy of the electrons). 5. Conclusions and outlook We have studied the properties of atomic nuclei embedded in an electron gas as occuring in, e.g., neutron star crusts and supernova explosions. Nuclear structure calculations have been performed within the relativistic

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mean-ﬁeld approach employing the force NL3. The calculations have been performed within the Wigner–Seitz cell approximation in coordinate space. The presence of electrons leads to eﬀects of rather macroscopic character. Since the electron background is constant over the nuclear volume — by construction — it leads predominantly to a constant downward shift of the proton potential. The electron gas alters the stability condition for β decay and shifts the stability line to more neutron-rich nuclei. Electron capture becomes a relevant process: the transformation of protons into neutrons favors large isospin in nuclei. Furthermore, the two-proton drip-line is shifted to more proton-rich nuclei since the protons gain additional binding due to the attractive interaction with the electrons. The neutron drip-line is not altered by the presence of the electrons since the neutron single-particle potential remains (to a very good approximation) unaﬀected. We have found that in a dense electron background α decay is suppressed. Our calculations show that the α decay half-lives decrease as a function of the electron Fermi momentum kF until they increase again for larger kF . A general trend is that the Q values of those decay modes decrease with increasing electron density. In extreme astrophysical environments, the production of very exotic and superheavy nuclei could become possible. This might happen during the r-process (rapid neutron capture) and the rp-process (rapid proton capture) when the electron background prevents the heavy (and superheavy) nuclei from fast decay by spontaneous ﬁssion or alpha decay. As a consequence, the nuclei, which would otherwise be unstable, can provide a bridge to the island of superheavy elements. If long-lived or even stable superheavy nuclei exist, they could be created in such an environment and later ejected into space. Acknowledgments We gratefully acknowledge fruitful discussions with J. A. Maruhn, S. Reddy, P.-G. Reinhard, N. Sandulescu, L. Satarov, J. Schaﬀner-Bielich and S. Schramm. This work was supported in part by the Gesellschaft f¨ ur Schwerionenforschung (Germany) and by grants RFFR-02-04013 and NS-8756.2006.2 (Russia). References 1. J. W. Negele and D. Vautherin, Nucl. Phys. A 207, 298 (1973). 2. A. S. Botvina and I. N. Mishustin, Phys. Lett. B 584, 233 (2004). 3. P. Magierksi, A. Bulgac and P.-H. Heenen, nucl-th/0112003v1.

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P. Magierski and P.-H. Heenen, nucl-th/112018v1. N. Sandulescu, nucl-th/0403019v2. K. S. Cheng, C. C. Yao and Z. G. Dai, Phys. Rev. C 55, 2092 (1997). T. Maruyama, T. Tatsumi, D. N. Voskresenky, T. Tanigawa and S. Chiba, nucl-th/0402002v1. T. J. B¨ urvenich, I. N. Mishustin and W. Greiner, Phys. Rev. C 76, 034310 (2007). P.-G. Reinhard, Rep. Prog. Phys. 52, 439 (1989). M. Bender, P.-H. Heenen and P.-G. Reinhard, Rev. Mod. Phys. 75, 121 (2003). G. Lalazissis, J. K¨ onig and P. Ring, Phys. Rev. C 55, 540 (1997). Z. A. Dupre and T. J. B¨ urvenich, Nucl. Phys. A 767, 81 (2006).

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Structure and Properties of Atomic Clusters

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CLUSTERS, QUANTUM CONFINEMENT AND ENERGY STORAGE JEAN-PATRICK CONNERADE Quantum Optics and Laser Science Group, Imperial College London, UK One of the challenges posed by the demand for clean urban transportation is the compact and cyclically recoverable storage of energy in quantities suﬃcient for propulsion. Promising routes, such as the reversible insertion of Li+ ions inside solids for ‘rocking chair’ batteries, require a deformable host material with no irreversibility. Such ‘soft’ deformations are in general highly complex, but the compressibility of atoms or larger systems can be studied directly in situations with simpler symmetry. Thus, the search for ‘soft’ materials leads one to consider certain types of cluster, as well as linear or nearly-spherical structures (chains of metallofullerenes, for example) whose deformations can be computed from the Schrodinger equation. Extended or ‘giant’ atomic models allow one to construct compression-dilation cycles analogous in a rough sense to the Carnot cycle of classical thermodynamics. This simpliﬁed approach suggests that, even for idealised systems, there are constraints on the reversible storage and recovery of energy, and that (when applied to realistic structures) modelling based on such principles might help in the selection of appropriate materials. Keywords: Conﬁned atoms; atomic compressibility; energy storage.

1. Introduction The purpose of this contribution is to draw attention to phenomena which are not usually thought about in the context of atomic and molecular physics, but which have potentially important applications. When one is studying new molecules, and especially large molecules and clusters, one is dealing with structures whose volume can change according to the environment. There are then two types of change to consider. The ﬁrst is heteromorphic, in the sense that the cluster or molecule changes its nature. This is standard chemistry and will not be discussed here. The second is isomorphic, meaning that the topology of the structure is preserved, but simply changes scale size. Since the atoms themselves account for much of the volume, the way their volume changes becomes important. Also, we 55

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may have a structure with open spaces and bond lengths, whose equilibrium distances can change. Both involve what I will call the flexibility of the atomic wave functions. The reasons why this subject is of practical importance are related to energy storage, ion storage, catalysis, valence changes, and so on. To introduce the subject, it is worth mentioning that systems larger and more complex than atoms, but restricted by their structure to nearly spherical symmetry, can be approximated as giant “quasi-atoms” for the description of some of their properties. So, a discussion of volume changes via central mean ﬁeld theory has a wider relevance than just for atoms. A number of systems fall into this category (nearly spherical molecules, spherical clusters, quantum dots, quantum “bubbles,” fullerenes and some metallofullerenes, to name but a few) and a number of papers have drawn analogies to a “quasi-atomic” description1 from which quite a few spectral features can be worked out. Such approximations, of course, employ nothing revolutionary. What makes them valuable is that (a) they are computationally “lighter” to consider, and (b) they can make it easier to understand some properties than more complete and accurate “ab initio” computations. They tell us at a glance how the level structure evolves for a system larger than an atom. The discrepancies between these simple models and either full calculations or experiment also tell us what is left out. As regards compressibility, note that such “quasi-atoms” occupy quite large volumes, and that these volumes can also change in response to the environment. I will mention the “quasi-atoms” brieﬂy at the outset, because they help as a ﬁrst step to generalize what I will be saying about atoms themselves. However, they are not my real subject. I refer to them only to generalize slightly the discussion of atomic compressibility to larger systems, whose interesting properties remain, in respect of compressibility, a terra incognita. Thus, a discussion of volume changes via central mean ﬁeld theory has a relevance wider than just to the properties of individual atoms. My argument is that the traditional “spectroscopic” approach (by which I mean obtaining spectral properties for an isolated quantum system, for example, by ab initio methods) does not exploit the full potential of quantum theory, because it involves no testing of the inﬂuence of changes in environment on the system. Even the most simpliﬁed quantum models are capable of telling us more. They can provide more than just an understanding of spectral structure, because changes in the environment can be introduced via changes in the boundary conditions. Computationally, this is a very minor modiﬁcation. In this way, one probes a property I will call the “ﬂexibility of wave functions.” The question is how to do it usefully and systematically. This “ﬂexibility” is not usually considered in atomic physics, but will be the real focus here. I emphasize again at the outset that we exclude chemical

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bonds, and discuss processes closer to what is termed physi-sorption in surface physics or topotactic insertion in solid state chemistry. Such processes occur without chemical bonding, and are therefore truly reversible in the thermodynamic sense. They are of special interest in a variety of situations, but especially for energy storage. Introducing the “ﬂexibility” of wave functions is best understood by considering the properties of atoms conﬁned under pressure, of which I will give a brief account. Then, I will pull the various threads together by considering the nature of microscopic energy storage cycles, whose properties still remain to be investigated both theoretically and experimentally. 2. Quasi-atoms The treatment of a number of spherically symmetric systems as “quasiatoms,” or systems of a size and complexity larger than ordinary atoms, but still described by the same basic Hartree–Fock equations, with somewhat modiﬁed potentials and subshell occupation numbers, has been described by a number of authors1,2 and therefore need not be repeated here. Sufﬁce it to say that the Hartree–Fock model is of course not the only one, that we can also use Thomas–Fermi or Density–Functional approaches, or indeed any quantum mechanical model with spherical symmetry. To reduce the complexity, it is even possible to adapt one-dimensional models of the atomic potential,3 which can be useful if the aim is to represent the properties of a linear chain. In fact, the full apparatus of atomic physics is adaptable to “quasi-atoms,” which allows predicted properties to be checked for consistency between diﬀerent models and for sensitivity to the various approximations involved. One of the best-known examples of a “quasi-atom” is the jellium model for metallic clusters (Fig. 1), which simply

Fig. 1.

A quasi–atom: the metallic cluster.26

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elaborates on a central ﬁeld model already tried and tested for atoms and for nuclei. When this approach is followed through for a confined system4–9 (and I will take here the example of a spherical metallofullerene with the conﬁned atom at the center), it is found that the excitation spectrum is essentially the same as that of the atom, augmented by the spectrum of the conﬁning shell.10,11 That is to say: in addition to the spectrum of the free atom (bound states, autoionizing resonances, continua and Auger spectra), we will ﬁnd the spectrum of the cavity (cavity resonances and continuum) within which the atom is held, as well as resonances arising by reﬂection of the electronic wave function between the inner and outer walls of the conﬁning shell. These diﬀerent features, being present within the same system, can now interact. Once this is recognized, we can classify the diﬀerent eﬀects present in the combined system spectrum,11 namely: (a) atomic resonances modiﬁed by the presence of the cavity; (b) cavity resonances altered by the presence of the atom; (c) resonances which only appear in the combined system and are not present in either of the two components considered separately. These spectral properties are sensitive in diﬀerent ways to the dimensions of the conﬁning shell. So, a change in the environment which changes the dimensions of the shell can cause migrations of the spectral features through each other, with all the usual eﬀects arising from the theory of avoided crossings. The simple “quasi-atom” models are useful to uncover such crossings and their properties when they involve (a) and (b). For example, in Fig. 2, we show such avoided crossings for hydrogen in a conﬁning shell10 as a function of the binding strength of the shell for the s-channel. An interesting point is that almost identical results are obtained for these curves if the full three-dimensional theory is used, or if the potential is replaced by the one-dimensional 1/(x2 + a2 )1/2 potential which possesses a Rydberg spectrum without conﬁnement.3 Shells containing many electrons are capable of screening a conﬁned atom from external inﬂuences. This property has attracted the attention of researchers seeking a compact system to realize the register of a quantum computer,12 and it has been suggested that endohedral fullerenes (i.e. a fullerene cavity containing an atom inside) could serve as building blocks, perhaps deposited on a surface. One should not forget, however, when dealing with shells of charge, that dynamical screening by a conducting shell of quasi-atomic size is not complete: plasmon properties arise from the collective behavior of many electrons in the conﬁning shell which couple to the spectrum of the atom. These can have a very important dynamical eﬀect on

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Avoided crossings for the atom confined in a shell.

the response of the system to excitation by external radiation.13 Even the pure spectroscopy of the complete system is modiﬁed by these dynamical properties. The microscopic theory of the Faraday cage must take plasmon eﬀects into account. Quasi-atomic modeling treats a wide range of phenomena which are a straightforward extension of atomic properties. Spherical shells may even be distorted from a spherical shape to describe conﬁnement in elongated cavities, which, for even a single conﬁned atom, gives rise to beautiful molecularlike properties without the actual formation of a molecule.8 In principle, all these properties can be explored, but this type of research faces the experimental challenge of producing a suﬃcient density of targets, or of trapping heavy systems long enough to observe their spectra. 3. High pressure eﬀects I turn now to high pressure eﬀects, which are closely connected to the problem of the spherical endohedral just described. Like conﬁned atoms, the system of the atom under a very high external pressure can also be described by modifying the external boundary conditions so as to introduce a rather high conﬁning step, with a radius equal to that of a conﬁning cavity of quantum size. This approach tests compressibility directly and leads to a number of interesting conclusions: (a) the atomic states and their ionization properties are altered by the conﬁnement; (b) the physical properties of the conﬁning cavity are no longer the same as those of an empty cavity;

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(c) the ﬂexibility of wave functions, which I will describe, becomes a consideration, whereas it has no importance for either of the two components considered separately. Let me comment on the “ﬂexibility” of wave functions. This terminology comes, of course, from the theory of springs and the formulation of Hooke’s law. We can consider the external step as a piston compressing a medium, which is the wave function. Notice that this property would disappear classically. It exists only because of quantization, which obliges the ground state of an atom to occupy a deﬁnite volume. We can calculate this volume from the expectation value of the operator r3 . Flexibility is a property akin to compressibility. It describes generically how a system copes with small changes in its environment without changing its fundamental properties. The fundamental property of a free atom is its shell structure. If we place an atom inside a closed spherical cavity with walls it cannot react chemically, we expect that it should, at least in the ﬁrst phases of compression, retain the integrity of its shells (the ground conﬁguration). However, the binding energy of the ground state is altered, and the expectation value of the radius is changed. By solving the Schr¨ odinger equation for the atom in the cavity, we can determine both observables. This procedure was applied by Sommerfeld early in the development of quantum mechanics,15 and set up for hydrogen in an impenetrable sphere. Thus, high pressure eﬀects were known at an early stage. What Sommerfeld did not consider was the natural compressibility of atoms (and, more generally, of quantum systems) treated as a quantum mechanical observable. The reason for this neglect was the kinetic theory of gases. In this conceptual framework, an ideal gas is a system of point particles, and the pressure P is due to momentum exchanges at the walls of its container. The volume of the particles is not introduced unless the van der Waals correction is applied to the ideal gas law. According to this correction, the volume of the atoms or molecules forming the gas is simply a ﬁxed parameter. In fact, this correction is clearly not ﬁxed. At very high pressures, the volume occupied by atoms or molecules of the gas changes, because the electronic wave functions are conﬁned by Pauli forces, according to rules dictated by quantum mechanics. This change of volume brings into play the ﬂexibility of atomic wave functions. Quantum mechanics tells us that there is a change in energy when a free atom is conﬁned, and that this change is accompanied by a change in radius, i.e. in the volume occupied by the conﬁned atom. We determine the volume change by working out the quantum mechanical expectation value of r3 before and after conﬁnement. If we relate the change in

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energy to the change in volume, we can extend a simple rule imported from thermodynamics16 : dE = P dV,

(1)

which deﬁnes a quantum or “Hellmann–Feynman pressure” P (see Ref. 17 for a discussion on this). This pressure P is a legitimate observable. It has nothing to do with thermodynamics, however. It is a quantum mechanical property of atoms, determined by solving the Schr¨ odinger equation. In essence, we are now treating the wave function of the atom as a ﬂuid, and studying how it responds to an externally driven compression obtained by driving the boundary conditions like a piston. Notice that an inﬁnitely high pressure cannot exist, because the electron would be conﬁned to a point, which would violate the Uncertainty Principle. In fact, there is no reason why the compression should only be driven externally. Imagine an atom which is being progressively ionized by removing the external electrons. The screening of the nuclear charge is reduced and produces a stronger force towards the center. This also results in compression, and in a change of volume. So, it again deﬁnes a quantum pressure P and a compressibility, driven this time internally.18 These two quantities (external and internal compressibility) are mathematically well-deﬁned and can be computed for the whole the Periodic Table. The result, of course, depends to some extent on the model used, but the principle is clear. From such calculations, we easily arrive at the conclusion that some atoms are hard and other soft. It is not diﬃcult to guess which ones. Helium is obviously hard, since it has a closed shell, a very high ionization potential and a very small radius. On the other hand, a large alkali atom is soft, because its ionization potential is very low; it has no external closed shell and a large atomic radius. For a many-electron atom, the question which now arises is that of the shell structure. Does the atom really preserve its shell structure when compressed? The answer, interestingly, is no. Even for hydrogen, there comes a point (as discovered by Sommerfeld and Welker15 ) where the electron is no longer attached to the nucleus but only to the cavity. More importantly, for many-electron atoms, the electronic structure can change under pressure, and the class of atoms for which this happens is particularly signiﬁcant for energy storage. It occurs for lanthanide and transition elements, i.e. the elements of Smith and Kmetko’s “Quasi-Periodic Table,”19 whose properties as hydrogen storage and superconducting elements are well documented. When these atoms are studied under pressure, it emerges that the Periodic Table itself is not the same for atoms under pressure as it is for free atoms.20 In fact, from this perspective, Smith and Kmetko’s table is a statement about

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how “soft” the transition elements and rare earths are, which is determined by their compressibility. This property is linked to the double-well potentials of these atoms, which determine their compressibility. We can even formulate a general law: for any many-electron atom, as the pressure is increased, the order of ﬁlling of the electronic shells will tend to return to the order predicted by the aufbau prinzip. That is to say, the chemistry of atoms is diﬀerent under high pressures. This change of electronic structure is observed experimentally by X-ray diﬀraction for Cs metal under pressure.17 It is responsible for the discontinuity in the curve for Cs of Figs. 3 and 4. 4. Nonlinear terms Now, I would like to turn to another quantum mechanical issue, which is the linearity of the stiﬀness constant for an atom. For a classical spring, it can be pretty good, but for an atom, such an assumption obviously does not work. Atomic electrons are subject to Coulomb’s law, and the compressibility therefore cannot be a constant.

Fig. 3. Atomic size against applied pressure plot for Cs metal: (a) by experiment, (b) by Dirac–Fock theory, and (c) in the non-relativistic limit of the Dirac–Fock calculation. The main discontinuity is discussed in the text. The small discontinuity in the experimental curve is due to a transition in the crystalline structure.

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Fig. 4. Showing plots of the dimensionless Energy against Volume compression parameters for Cs and He as discussed in the text.

What is needed is to ﬁnd a way to represent the scaling in which all atoms can be plotted onto a single graph,21 which represents the nonlinear contribution to the compressibility without worrying about the absolute value. The way we do this is to divide all energy diﬀerences by the corresponding energy for the free atom, i.e. to deﬁne a dimensionless reduced energy, and likewise to divide all volumes by the corresponding volumes for free atoms, so as to obtain a dimensionless reduced volume. Thus, we ﬁnd that pretty well all atoms exhibit the same nonlinearity, despite considerable diﬀerences in their structure. In essence, the dominant cause of this universal nonlinearity is the inverse square law of force. Such curves do tell us more, however: a change of electronic structure under pressure is signaled by a discontinuity, as shown in the case of Cs in Figs. 3 and 4. We may conclude that the quantum mechanical treatment of compressed wave functions is complex in detail, but, in principle, quite straightforward. Soft atoms are the large alkalis (like Cs) while hard atoms are small rare gases (like He), basically because closed shells are stable and therefore robust. More quantitatively, Dirac–Fock calculations yield the result that Cs is about 2000 times more compressible than He,17 so we see the eﬀectiveness of the rescaling in Fig. 4. We also conclude that “soft” atoms are generally located in the part of the Periodic Table where atomic structure can

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Fig. 5.

Changes of valence with cluster size.

reorganize itself at comparatively low pressures. Such transitions are known to occur also in clusters. For example, valence changes of atoms in clusters for some rare-earth elements as a function of size are well documented22 (see Fig. 5). A size change in a cluster is equivalent to a pressure change for the atoms inside it. In the case of Cs, the order of ﬁlling according to the aufbau prinzip would be 5d before 6s. However, in the free atom, 6s ﬁlls before 5d. Under pressure, the order of ﬁlling reverts to the aufbau ordering, and there is consequently a level crossing between the 6s and 5d in the (E − V ) curves. The discontinuity observed corresponds to this level crossing. We see that in its vicinity there are two possible compression regimes. If Cs is compressed slowly, then the reorganization has time to occur, so the atom remains in its ground state and the discontinuity is observed. If Cs were compressed very fast, then Cs would persist in the 6s state and subsequently the material would readjust under pressure by some energy loss process to drop down to the 5d ground state. Energy loss interferes with energy storage, so we see that, although soft atoms are highly desirable for energy storage, level crossings become a potential complication, because they impose longer time scales for full energy recovery. I return to the question of time scales below.

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5. Energy storage Let me now turn to energy storage, and in particular to the storage of electrical energy. A very serious and important issue is of course the generation of energy, but almost equal in importance is its storage, its transportation and its recovery at the point of use. This is particularly true for electrical energy and for the storage of hydrogen, which could provide clean sources for urban transportation in the future. In both cases, the problem is how to store large quantities of electricity (which basically means Li+ ions) or else hydrogen in a safe way, easily recoverable at the point of use. For Li+ ion batteries, storage by topotactic insertion in a host material (the electrode) is used.23 For hydrogen, again, storage is by adsorption on the surface of a solid material with a speciﬁc property (a good capacity to store hydrogen). In both cases, it has been found that materials containing elements not too far from the ridge in the Smith and Kmetko table (see Fig. 6) are very suitable. The reason for this is the ﬂexibility of their wave functions. There is in fact a double requirement, namely, the ﬂexibility of wave functions, allied to an open threedimensional lattice structure which allows high mobility of the inserted atoms. Why is this? And why do we seek soft materials? The underlying reason is that the storage process must be reversible, on energy scales of a few eV. Full reversibility in this context does not have quite the same meaning (again) as it would have in the theory of ideal gases. What it means here is that chemical bonds are not formed in the storage process, since otherwise they have to be broken on recovery. In other words, we need to involve materials which are soft at the quantum mechanical level and can easily bend their structures without breaking their intrinsic order (i.e. materials capable of polaronic deformations). The storage then tends to occur by a physical mechanism (like physisorption) rather than by a chemical mechanism (chemisorption). This is a requirement for good reversibility. Appropriate materials are shrouded in commercial secrecy, but the basic principles are clear. There may also be other classes of materials, based on “ﬂexible” molecular structures, whose properties would be suitable to act as hosts in reversible storage. That is why it is of practical importance to study compressibility, not just for atoms, but for other types of quantum system. Since both Li+ ions and protons are incompressible, it follows by conservation of energy that the storage of energy actually occurs in the deformation of the host material. In other words, it is by studying the compressibility of the host that one obtains information on the storage process and the storage cycle. The host material must expand or

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Fig. 6. The Smith and Kmetko Quasi-Periodic Table, which tracks the flexibility of wave functions through the diagonal boundary between localization (magnetism) and itinerant behavior (conductivity) of electrons in solids. The underlying cause is the soft atomic potential for transition elements and rare-earths.

contract without breaking. More explicitly: we can separate the stored energy into two parts. One part is energy stored in the compression or dilation of host atoms, and the other part is the energy contained in the stored “fuel.” The second depends on the number of stored atoms of fuel and the energy eﬃciency per atom of the fuel. The greater the volume change of the host atoms in the cycle, the greater the fuel storage capacity, which is why one seeks a highly compressible host to achieve eﬃcient storage. Energy loss can thus appear in two ways. First, it can be consumed in the compression–dilation of the host. Second, it can be lost because full recovery of the fuel is not achieved, i.e. the host material does not return to its original form, and some of the fuel remains trapped in the host lattice. In an ideal cycle, i.e. one which is reversible, this second process does not occur: the atoms all return to their original volume when the fuel is fully recovered. Even for such an ideal cycle, however, there can be an energy loss associated with the compression–dilation of the host.

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6. A new kind of cycle Thus, in connection with reversible compression, a new kind of cycle, which we might call the energy storage cycle, appears. It has some similarities with the Carnot cycle of thermodynamics, although, as I have stressed, the underlying physics is completely diﬀerent and rests on the Schr¨ odinger equation, without involving issues of temperature or entropy in its deﬁnition. In fact, as I will argue, heating is the enemy of eﬃcient storage, so temperature changes should really be avoided. Of course, it is possible to set up somewhat artiﬁcially a theory to introduce entropy and temperature into the argument, but this is not really useful. One way is to compress a gas of two-level atoms in the presence of a radiation ﬁeld of a given temperature. This simply elaborates on Albert Einstein’s famous argument to introduce the A and B coeﬃcients into physics. It does not, however, lead to a satisfactory energy storage theory, for three reasons. The ﬁrst is that two level atoms are not a good basis on which to discuss either temperature or entropy, since Boltzmann’s law cannot reasonably describe two-level atoms. The second is that a two-level atom cannot be “soft” because its compressibility is unknown and does not in any case correspond to a Coulombic force. The third is that the processes involved in energy storage do not depend on the radiation bath, so a radiation temperature is irrelevant anyway. It seems, therefore, that the best approach is to stick with the minimum, simplest theory to describe the basic features of this problem. Having made this point, I just note, in passing, that rapid level crossings produced by an intense acoustic wave result in inversions of population at each crossing, which do generate radiation — a mechanism that is perhaps of interest in connection with sono-luminescence. As noted above, in hydrogen and in ion storage, the material to be stored by insertion in a solid is hard (protons or Li+ ions) and essentially incompressible. The host material must therefore expand or contract without breaking (the “polaronic” distortion). It must therefore be built from atoms linked in a ﬂexible way. We must avoid breaking bonds and, since these are directional and involve angular properties of wave functions, we can drop the angular functions, which are associated with irreversible eﬀects. We are therefore led back quite naturally to the issue of atomic compressibility as deﬁned from the radial part of the wave function, which is not a bad measure of ﬂexibility to use for this problem. As just noted, energy is stored directly in the deformation of the host. So, we can simplify the process by considering the general behavior of “soft atoms” under compression. To keep things easy, I consider a case for which no level crossings occur. Cases with level crossings (as in Cs) produce more

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complex cycles, but the basic conclusion is the same. We can draw a graph of the change in energy against change in volume (an E–V graph). If compression of the system in the ground state occurs slowly along a branch of this graph (think of an isothermal process) then energy is gradually stored in the atoms of the host material as it is compressed. This is the path AB represented in Fig. 7. It is indeed a universal law that the energy as computed from the Schr¨ odinger equation always increases when a system is compressed. Otherwise, the ground state would be unstable, which is contrary to the very deﬁnition of a ground state (note, however, that the same restriction does not necessarily apply to excited states). If the compression is released at any point in the path AB, the energy stored in the system is recovered. However, this must again be done slowly (as in an isothermal process of thermodynamics). All of the stored energy can in principle be recovered, but only if the state of the system has not changed. If any excitation is produced in the cycle, then there is an energy loss, at a rate determined by the lifetime of the excited state against all the natural de-excitation processes. For a host made of soft atoms, there can be many of these. What does “slowly” mean in this case? It is clearly related to the fact that the system must remain in its ground state throughout the cycle, i.e. that no heating occurs. If we compress very fast, then we may generate a Landau–Zener transition. The system then jumps to an excited state. If this excited state can be sustained long enough, then at ﬁrst sight it seems that we can actually store much more energy than in slow compression, but then of course an excited state can survive only for a ﬁnite lifetime.

Fig. 7. The energy storage cycle discussed in the text. Note that energy losses arise when there is excitation.

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We can also conceive of short-time, or pulsed-excitation, curves. These, however, are not the analogs of “adiabatic” curves. In fact, short pulse compression gives rise, not to a single curve, but to a family of curves which starts out from a given energy point and fans out into a sequence of excited states. This is represented schematically as the section B to D in Fig. 7. Each one of these excited states gives rise to a new return path in the E–V graph. So, even in the simplest case (just one state excited) one cannot produce a closed cycle with a single path analogous to the Carnot cycle of thermodynamics. Rather, a family of cycles is always produced, each one with diﬀerent closure properties, because the lifetimes of all the excited states must be considered, and they are all diﬀerent from one another. There exists no time scale on which full energy recovery can be achieved once short pulse excitation is included as one of the compression steps. The only way to achieve full storage and recovery is therefore to use compression time scales long enough for the system to remain in its ground state throughout the cycle. The only conceivable exception to this rule would be if the state excited in fast compression happened to be metastable and if no other state were excited. Such a situation seems impossible to produce in nature, since metastable states usually lie close in energy to other excited states. Consequently, the slow compression and subsequent slow release of the “piston” remain the most eﬀective form of energy storage, exactly as in the ion storage processes of the reversible or “rocking chair” battery. We can compare this to the very simple classical process of pumping water up a hill to store energy and then letting it ﬂow down to recover it, except that the cycle must occur at the molecular level. Although the discussion was presented in terms of a “quantum pressure,” this term covers not only physical compression eﬀects induced, e.g., with a diamond anvil but also the so-called “chemical pressure,” due to the change in size of a cluster, or due to changes in concentration of the inserted ion in a lithiated host,24 or due to changes of concentration of impurities in mixed-valence materials,25 which thus are all equivalent to each other. We see that the quantum pressure is a very useful quantity which allows us to track many kinds of physical mechanism used for energy storage.

7. Conclusions To store and transport energy eﬀectively to the point of use, reversible mechanisms must be found which act at the molecular level. Wave function ﬂexibility is the key to picking out the optimum materials for energy storage (soft chemistry). To store energy with manageable pressures, the ﬂexibility must be high and atomic volume changes large,

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which means that the E–V curve should be nearly “ﬂat.” Rigid structures are unsuitable for energy storage. In order to model the process more accurately, we must of course take into account the fact that the real situation is not spherical, and does not involve atoms conﬁned by shells. Therefore, we need to study the linear stretch and compression in three dimensions of molecular building blocks which preserve their shape, or which undergo polaronic distortion without rupture. This is a challenge for molecular computations of a new type, in which the energy-distortion curves are plotted so as to identify the most favorable species. Naturally, this single criterion does not suﬃce: it is also desirable that the chosen materials should have open structures, through which the inserted ions can readily migrate. All the theorists engaged in computing new molecular structures should study their compressibility, which is anyway a by-product of stability calculations. That is the way to learn which ones could be useful in energy storage applications. Good materials may well be built from soft atoms as we described, but the building blocks should be such as to ﬁll space with a deformable structure, which is why spherical symmetry (which I have used to simplify the argument) is not really appropriate. Attention must therefore turn to more complicated molecular species. So, the message is clear: do not just calculate complex molecular structures and the spectra which are their signatures, but also squeeze them or distort them and look for cases which are “soft,” because these are the ones which might help in the energy-storage business. Take your favorite molecules and pull them around. How easy are they to distort without losing their structure? How much energy can you store inside and then recover without breaking them? If you ﬁnd good materials, make sure you get them patented quickly, because there are lots of sharks out there, waiting to see if you can ﬁnd the best one. Acknowledgments This work is partially supported by the European Commission within the Network of Excellence project EXCELL, and by INTAS under the grant 03-51-6170. References 1. J.-P. Connerade, Indian J. Phys. 76(4), 359 (2002), and references therein. 2. V. K. Dolmatov, J.-P. Connerade, P. A. Lakshmi and S. T. Manson, Surf. Rev. Lett. 9, 39 (2002).

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3. J.-P. Connerade, P. Kengkan and S. Kompichit, One-Dimensional Modeling of Atomic Confinement, to be published. 4. T. Sako and G. H. F. Diercksen, J. Phys. B 36, 1433 (2003). 5. V. K. Dolmatov, J.-P. Connerade, P. A. Lakshmi and S. T. Manson, Surf. Rev. Lett. 9, 39 (2002). 6. J.-I. Ahara, J. Phys. Chem. 106, 11371 (2002). 7. L. Forr´ o and L. Mih´ ay, Rep. Prog. Phys. 64, 649 (2001). 8. J.-P. Connerade, P. Kengkan and R. Semaoune, J. Chinese Chem. Soc. 48, 265 (2001). 9. J.-P. Connerade, V. K. Dolmatov and S. T. Manson, J. Phys. B 33, L395 (2000); ibid. 33, 2275 (2000). 10. J.-P. Connerade, V. K. Dolmatov, P. A. Lakshmi and S. T. Manson, J. Phys. B 32, L239 (1999). 11. J.-P. Connerade, V. K. Dolmatov and S. T. Manson, J. Phys. B 33, 2279 (2000). 12. W. Harneit, Phys. Rev. A 65, 032322 (2001). 13. J.-P. Connerade and A. V. Solov’yov, J. Phys. B 38, 807 (2005). 14. J.-P. Connerade, A. G. Lyalin, R. Semaoune and A. V. Solov’yov, J. Phys. B 34, 2505 (2001). 15. A. Sommerfeld and H. Welker, Ann. Phys. 32, 57 (1938). 16. J.-P. Connerade, J. Phys. C 15, L367 (1982). 17. J.-P. Connerade and R. Semaoune, J. Phys. B 33, 3467 (2000). 18. J.-P. Connerade and V. K. Dolmatov, J. Phys. B 31, 3557 (1998). 19. J. L. Smith and E. A. Kmetko, J. Less Common Metals 90, 83 (1983). 20. J.-P. Connerade, V. K. Dolmatov and P. A. Lakshmi, J. Phys. B 3, 251 (2000). 21. J.-P. Connerade, P. Kengkan, P. A. Lakshmi and R. Semaoune, J. Phys. B 33, L847 (2000). 22. C. Blancard, J. M. Esteva, R. C. Karnatak, J.-P. Connerade, U. Kuetgens and J. Hormes, J. Phys. B 22, L575 (1989). 23. J. Olivier-Fourcade, J.-C. Jumas and J.-P. Connerade, J. Solid State Chem. 147, 85 (1999). 24. J.-P. Connerade, J. Alloys Compounds 255, 79 (1996). 25. J.-P. Connerade, J. Alloys Compounds 227, 145 (1995). 26. W. Ekardt, Phys. Rev. B 29, 1558 (1984).

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SITE-SPECIFIC ANALYSIS OF RESPONSE PROPERTIES OF SODIUM CLUSTERS K. JACKSON∗,† and M. YANG Physics Department, Central Michigan University, Mt. Pleasant, Michigan 48859, USA † [email protected] J. JELLINEK Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439, USA [email protected] A scheme we have formulated recently for partitioning the total dipole moments and polarizabilities of finite systems into site-specific contributions is used to analyze the structure-/shape- and size-specific aspects of the dipole moments and polarizabilities of small sodium clusters. The procedure is based on dividing the system volume into cells associated with its atoms. The site-specific, or atomic, dipole moments and polarizabilities are computed from the charge densities within the individual cells (“atomic volumes”) and the changes in these densities in response to an external electric field. The atomic dipole moments and polarizabilities are further partitioned into local (or “dipole”) and “charge-transfer” components. It is shown that the polarizabilities associated with the individual Na atoms vary considerably with the structure/shape of the cluster and the location of the atom within a given structure. Surface atoms, especially those at edges, have larger polarizabilities than interior atoms. The contribution of the charge-transfer components to the total polarizability increases with the cluster size. Keywords: Clusters; polarizability; electric dipoles; size-dependence of properties.

1. Introduction The hallmark of atomic and molecular clusters is that for each size (number of atoms or molecules) they form many alternative structural forms ∗ Corresponding

author. 72

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(isomers) and that their properties vary both with the size and the isomeric form. A deeper understanding of these two dependencies can be achieved by examining the atomic (or molecular) level origin of the diﬀerent properties; one can also consider a coarser grained approach of site-speciﬁc origin of, or contributions to, these properties, where a site can be a submanifold of atoms or molecules. Metal clusters are of particular interest because of their relevance to and potential in many technological applications. Clusters of sodium were among the ﬁrst to be examined experimentally.1,2 It was found that many of the qualitative trends observed in measurements could be rationalized using the jellium model.3 Among these is the size-dependence of the magnitude αN of the (isotropic) electric polarizability of NaN , which can be approximated4 as αN = (RN + δ)3 ; here RN is the radius of the spherical positive jellium background and δ is the “spill-out” length, a measure of how far the electron charge density extends beyond the jellium background. The measured polarizabilities2,5,6 have also been compared with results of density functional theory calculations.7–11 The two are in good agreement when the calculations take into account the thermal expansion of the clusters at ﬁnite temperature. The focus of this contribution is on the atomic-level details of the response of NaN , N = 2–20, clusters to a small external electric ﬁeld. Among the issues addressed are the structure-/shape- and site-speciﬁc aspects of the atomic contributions to the total polarizability and the relative roles of its so-called local (or “dipole”) and charge-transfer components. The eﬀect of the cluster size is considered as well. We address these issues using a scheme we have formulated recently12 for partitioning the total dipole moments and polarizabilities of ﬁnite systems into contributions associated with their constituent atoms (or, more generally, sites or subsystems) and further decomposition of these into dipole and charge-transfer parts.12 The main elements of the scheme are recapped in the next section. The computational details are outlined in Sec. 3. The results are presented and analyzed in Sec. 4. A brief summary is given in Sec. 5. 2. Partitioning of the total dipole moment and polarizability into site-specific contributions The energy E of a ﬁnite system in a weak external electric ﬁeld F can be expressed as 1 E(F ) = E0 − µ · F − α · F · F , 2

(1)

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where µ is the vector of the electric dipole moment, and α is the polarizability tensor. The dipole moment and the polarizability can therefore be deﬁned as: dE µi = − , (2) dFi F=0

d2 E dµi αij = − = , dFi dFj F=0 dFj F=0

(3)

where the subscripts i and j denote the Cartesian components x , y and z . Alternatively, the electric dipole moment can be computed as µi = ri ρ(r)d3 r, (4) where ρ is the system total charge density, and the integration extends over the entire space. The charge density ρ incorporates contributions from both the electrons and the nuclei. The partitioning of the total dipole moment into site-speciﬁc contributions is accomplished by decomposing the integral in Eq. (4) into a sum of integrals over complementary, non-overlapping site volumes, where a site can be an atom or a collection of atoms. We deﬁne the volume ΩA associated with an atom A as the locus of those space points that are closer to its nucleus than to the nucleus of any other atom. The ΩA ’s are thus the Voronoi cells, or, in the nomenclature of solid state physics, the Wigner– Seitz cells. When a site is a collection of atoms, the site volume is the sum of the corresponding atomic volumes. Using this partitioning, Eq. (4) can be rewritten in the form µA (5) µi = i , A

where µA i are the Cartesian components of what we shall call the sitespeciﬁc, or atomic, dipole moment µA of atom A: = ri ρ(r)d3 r. (6) µA i ΩA

The electric dipole moment of a given charge distribution is independent of the location of the origin of the coordinate system, if the distribution corresponds to a zero total charge. Thus, the total dipole moment of a neutral system, as speciﬁed either by Eq. (4) or Eq. (5), is deﬁned uniquely. This is not necessarily the case for the individual atomic dipole moments µA deﬁned by Eq. (6), as the charge density in a given atomic volume ΩA may integrate to a nonzero value. The dependence of µA on the location of the

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origin of the system of coordinates can be made transparent by rewriting Eq. (6) as A,p µA + µA,q i = µi i ,

where

= µA,p i and = µA,q i

(ri − RiA )ρ(r)d3 r

(8)

RiA ρ(r)d3 r = q A RiA .

(9)

ΩA

ΩA

(7)

Here, RA deﬁnes the position of the nucleus of atom A and q A is the total charge in the volume ΩA , or, alternatively, the charge of atom A in the system. The component µA,p , Eq. (8), of the total atomic dipole moment µA represents the dipole moment of the charge distribution in ΩA with respect to an origin at RA , and we will refer to it as the local dipole moment of atom A or the local (“dipole”) component of µA . This local component is clearly independent of the choice of the origin of the global (i.e. common for all atoms) system of coordinates. The component µA,q , Eq. (9), however, does depend on this choice through RA . This component can be viewed as an eﬀective dipole moment of the atomic charge q A with respect to the origin of the global coordinate system, when q A is viewed as a point charge placed at RA . In the case of neutral systems, the local atomic charges are largely consequences of charge transfer between the atomic volumes caused by bonding. Therefore, we will refer to the component µA,q as the charge-transfer dipole moment of atom A or the charge-transfer part of µA . When summed over all the atoms A, µA,p and µA,q give, respectively, the local (or “dipole”) µp and the charge-transfer µq components of the total system dipole moment µ = µp +µq . For a neutral system, neither µp nor µq depends on the location of the origin of the global coordinate system. Using Eq. (7), one can rewrite Eq. (3) in the form A,p A,q αA ij = αij + αij ,

where αA,p ij

dµA,p i = dFj

,

(10)

(11)

F=0

and αA,q ij

dµA,q i = dFj

= RiA F=0

dqA . dFj F=0

(12)

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The αA,p component, Eq. (11), of the total site-speciﬁc, or atomic, polarizability αA characterizes the rate of change of the local atomic dipole moment in response to a very small external electric ﬁeld. It represents a dielectric (or “dipole”) type of a response as it reﬂects a local (conﬁned to ΩA ) ﬁeld-induced change in the distribution of the charge density; this change in the charge density may or may not be accompanied by a change in the value of q A . The αA,q component, Eq. (12), of αA constitutes a chargetransfer type of a response as it accounts for the ﬁeld-induced change, if any, in the atomic charge q A . Whereas αA,p does not depend on the choice of the origin of the global system of coordinates, αA,q does. The essence of the partition scheme described above is that it represents the charge distribution of a ﬁnite system by point charges and point dipole moments associated with the individual ΩA ’s or, alternatively, atoms. The changes in these charges and dipole moments caused by a small external electric ﬁeld determine, respectively, the atomic charge-transfer αA,q and dipole αA,p polarizabilities. These, when summed over all the atoms of the system, give the charge-transfer αq , A,q αij , (13) αqij = A

and dipole αp , αpij =

αA,p ij ,

(14)

A

components of the system total polarizability α, αij = αpij + αqij . p

(15) q

For a neutral system, both components α and α are independent of the choice of the origin of the global coordinate system. 3. Computational details We applied the scheme outlined above to analyze the polarizability properties of NaN clusters. The computations were performed using the gradientcorrected version of the density functional theory (DFT) with the Perdew, Burke, and Ernzerhof (PBE)13 exchange-correlation functional as implemented in the NRLMOL code.14,15 The all-electron basis set for the Na atoms was constructed from 16 primitive Gaussian functions contracted into six s-type, ﬁve p-type, and four d-type orbitals. The discretized version of the Eqs. (7)–(9) is (rsi − RiA )ρ(rs )ws + RiA ρ(rs )ws , (16) µA i = rs ∈ΩA

rs ∈ΩA

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where ws is the volume element associated with the grid point rs , and the sums extend over all the grid points lying in the volume ΩA . Because the grid volume elements are ﬁnite, for each atom they sum up to the true Voronoi cell volume only approximately. If a grid point lies near the boundary between two cells, its associated volume element may straddle the boundary, eﬀectively transferring a portion of the volume of one cell into the other. The eﬀect of this blurring of the boundaries on the computed values of αpij and αqij has been shown to be small.12 The derivatives in Eqs. (11) and (12) are computed using the ﬁnite difference method. An extra term, −e F · r, representing the potential energy of an electron (e denotes the absolute value of the electron charge) in an external electric ﬁeld F is added to the potential in the Kohn–Sham equations. The modiﬁed equations are then solved self-consistently to yield the electron charge density in the presence of the ﬁeld. The computations are performed for ﬁelds oriented along the “+” and “−” directions of the x, y, and z axes. The polarizability components are then evaluated as A,p(q)

αij

A,p(q)

=

µi

A,p(q)

(+Fj ) − µi 2Fj

(−Fj )

,

(17)

where Fj is the magnitude of the external ﬁeld applied in the jth coordinate direction. The small value of 0.005 a.u. (1 a.u. = 27.21 V/bohr = 5.14 × 1011 V/m) is used for Fj . This value has been found to yield results for the derivatives that are well converged to the zero-ﬁeld limit.16 In the analysis given below, we use the isotropic quantities 1 A,p(q) αA,p(q) = α (18) 3 i ii and

A A 2 A 2 A 2 dq dq dq dq + + . dF = dFx dFy dFz

(19)

4. Results and discussion 4.1. Na5 To illustrate the use of the partition scheme presented in the preceding section, we ﬁrst consider the case of Na5 . The lowest energy conﬁguration of this cluster is the symmetric, planar structure shown in panel a of Fig. 1. The HOMO–LUMO gap of this structure is 0.53 eV. The eﬀects of a small external electric ﬁeld F directed along two mutually perpendicular directions on the distribution of the charge density

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Fig. 1. Na5 . (a) The most stable structure of the cluster (the bond lengths are in ˚). (b) and (c) Isodensity contour plots of the total charge density redistribution δρ = A ρ(F ) − ρ(0) and changes in the atomic charges δq A = q A (F ) − q A (0) (in units of the magnitude e of the electron charge) induced by a weak (0.005 a.u.) external electric field F oriented as shown by the arrows. (d) and (e) The field-induced changes in the dipole parts of the total atomic dipole moments δµA,p = µA,p (F ) − µA,p (0) (the magnitudes of µA,p are in units of e ·bohr, their directions are shown by the arrows). (f) and (g) The same as (d) and (e), but for the charge-transfer parts of the total atomic dipole moments δµA,q = µA,q (F ) − µA,q (0). (h) and (i) The same as (d) and (e), but for the total atomic dipole moments δµA = µA (F ) − µA (0). See the text for details.

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in Na5 are shown in panels (b)–(i) of Fig. 1. Panels (b) and (c) depict δρ = ρ(F ) − ρ(0), the diﬀerence between the self-consistent cluster charge densities computed with and without the ﬁeld. The blue contours indicate depletion of the negative part of the total charge density, whereas the red ones represent its accumulation. Clearly, the electronic charge is driven in a direction antiparallel to the applied ﬁeld, and the more extensive charge separation occurs in the peripheral parts of the cluster. The panels also list the values of δq A = q A (F ) − q A (0), the ﬁeld-induced changes in the total charges of the individual atomic volumes. The ﬁeld-induced changes in the local and charge-transfer components of the atomic dipole moments are displayed in panels (d)–(g). Panels (d) and (e) show the directions and the magnitudes of δµA,p = µA,p (F ) − µA,p (0). The vectors δµA,p point largely in the direction of the applied ﬁeld. The magnitudes δµA,p depend on the direction of the ﬁeld and on the location (site) of the Na atom within the cluster structure. Panels (f) and (g) show the direction and the magnitude of δµA,q = µA,q (F ) − µA,q (0) for the case when the origin of the global system of coordinates is placed at the position of atom 1. Of course, the vectors δµA,q point in the same directions as the corresponding RA . The changes δµA = µA (F ) − µA (0) in the total atomic dipole moments in the same global coordinate system are shown in panels (h) and (i). 4.2. Na14 and Na20 The structures of the lowest energy isomers of Na14 and Na20 are shown in panels (a) and (b) of Fig. 2. Na14 is strongly oblate, as can be seen in the inset in the ﬁgure, and essentially all its atoms are on its surface. The structure of Na20 is more spherical (see the inset) and some of its atoms are in its interior. The HOMO–LUMO gaps of the clusters are nearly identical with values of 0.67 eV and 0.66 eV for Na14 and Na20 , respectively. Figure 2, panels (c)–(f) also depicts the changes δρ in the charge densities in each cluster caused by two small mutually orthogonal external electric ﬁelds. The blue and red isodensity surfaces represent δρ values of +0.0002 e/bohr3 and −0.0002 e/bohr3, respectively. The patterns of the ﬁeld-induced charge redistributions correlate with the directions of the ﬁelds. The overall eﬀect is a shift of charge, although interesting oscillations in δρ can also be noticed as one moves from the centers of the clusters to their surfaces along the directions of the ﬁelds. The atoms at the periphery along these directions experience the largest changes in their charges, whereas those closest to the center are aﬀected the least. Table 1 lists the values of αA,p , Eq. (18), and |dq A /dF |, Eq. (19), for every atom of Na14 and Na20 shown in Fig. 2. As discussed earlier, these

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Fig. 2. Na14 and Na20 . (a) and (b) The most stable isomers of Na14 and Na20 , respectively (the insets show a second view of the clusters). (c) and (d) Isodensity plots of the change in the total charge density δρ = ρ(F ) − ρ(0) induced by a weak (0.005 a.u.) external electric field F oriented horizontally. The red and blue surfaces represent δρ values of +0.0002 e/bohr3 and −0.0002 e/bohr3 , respectively. (e) and (f) The same as (c) and (d), but for an external field oriented vertically.

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Table 1. The dipole parts of the orientationally averaged atomic polarizabilities αA,p (in bohr3 ) and the magnitudes of the gradients of the atomic charges |dq A /dF | (in bohr2 ) for atoms of the Na14 and Na20 clusters shown in panels (a) and (b) of Fig. 2. The numbering of the atoms is as in the figures. Atom number

Na14

Na20

αA,p

|dq A /dF|

αA,p

|dq A /dF|

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

41.19 17.60 9.79 23.30 17.51 54.67 70.41 43.98 68.50 9.80 44.73 68.70 54.16 23.35

23.60 13.22 8.40 19.44 13.21 32.86 41.11 26.46 41.19 8.52 26.93 41.30 32.62 19.31

21.34 35.08 62.67 0.69 49.18 18.78 18.63 −0.44 56.42 49.51 42.37 45.32 33.80 27.16 64.87 6.14 46.90 41.66 21.09 28.01

14.14 23.22 40.33 1.81 31.38 9.89 10.05 1.80 34.56 31.50 27.19 28.30 22.56 17.62 41.37 4.50 28.84 26.76 14.14 18.07

Average

39.12

24.87

33.46

21.40

values do not depend on the choice of the origin of the global system of coordinates. They, however, change considerably with the location of the atoms. In the case of Na14 , the smallest value of αA,p , 9.79 bohr3 , is about seven times smaller than the largest value, 70.41 bohr3 . The smallest and largest values of αA,p for Na20 are −0.44 bohr3 and 64.87 bohr3 , respectively. The values of |dq A /dF | also exhibit strong site-dependence, which mirrors that in the values of αA,p . For the applied ﬁeld of 0.005 a.u., values of |dq A /dF | of about 20 a.u. (the approximate average of the values listed in Table 1) imply |δqA | values of about 0.10 of the magnitude of the electron charge. These are comparable to the |δqA | values for Na5 placed in an electric ﬁeld of the same strength [cf. panels (b) and (c) of Fig. 1]. The site-speciﬁcity of the αA,p and |dq A /dF | values is such that they are largest for atoms on the cluster surface, especially those that are the most remote from the cluster center. For example, in the case of Na14 atoms 7, 9 and 12 are the most peripheral (cf. Fig. 2) and they have the largest

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values of αA,p and |dq A /dF | — about 70 a.u. and 41 a.u., respectively. In Na20 , atoms 3 and 15 are the most peripheral, and their corresponding values are similarly large — about 65 a.u. and 41 a.u. In contrast, the values of αA,p and |dq A /dF | for the two interior atoms of Na20 (labeled in Fig. 2 as 4 and 8) are much smaller — 0.69 a.u. and 1.81 a.u. for atom 4, and −0.44 a.u. and 1.80 a.u. for atom 8. For an external ﬁeld of 0.005 a.u., these values imply changes of about (2–3)×10−3 a.u. in the magnitudes of the local parts of the atomic dipole moments and of 9 × 10−3 a.u. of the magnitude of the electron charge in the atomic charges of these atoms. In contrast, the values of αA,p and |dq A /dF | for the most peripheral surface atoms of Na20 imply changes of about 0.33 a.u. in the magnitudes of the local dipole moments and of 0.2 of the magnitude of the electron charge in the atomic charges of these atoms. The eﬀect of the external ﬁeld on the surface atoms is clearly considerably larger than on the interior ones. This indicates signiﬁcant screening of the interior of the cluster by its surface. The negative sign of the αA,p value for atom 8 of Na20 implies that the external ﬁeld has a net attenuating eﬀect on its local dipole moment. This eﬀect incorporates the consequences of the ﬁeld-induced redistribution of the charge density in all atomic regions of the cluster. 4.3. Size-dependence of the polarizabilities of NaN In this subsection, we present and analyze results on the size-dependence of the polarizability properties of the most stable structures of NaN , N = 2–20. Figure 3 displays graphs of the total orientationally averaged (i.e. isotropic) cluster polarizability α and its dipole αp and chargetransfer αq components [cf. Eq. (18)], all considered on a per-atom basis as functions of the cluster size N . The α/N graph shows an overall, albeit nonmonotonic, decrease with the size. As is clear from the ﬁgure, this decrease is primarily due to the decrease in the dipole component αp /N , which, however, displays an almost monotonic variation in its sizedependence. The charge-transfer part αq /N , whose relative contribution to α/N increases with the cluster size, changes nonmonotonically and shows almost no overall systematic trend as a function of size. As a consequence, the oscillations in the α/N graph closely follow those in the αq /N graph. These observations suggest that it is the structural details that are responsible for the oscillations in the total polarizability α/N , and it is the charge-transfer component αq /N that is sensitive to these details and imparts their eﬀect to α/N . The essentially monotonic variation of the αp /N graph indicates a dominant dependence of the dipole component on the cluster size and at most a weak sensitivity to the structural details.

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160 Polarizability (bohr3/atom)

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140 120

<α>/Ν

100 80

<α q>/Ν

60 40

<α p>/Ν

20 0 0

5

10

15

20

N Fig. 3. Isotropic polarizabilities per atom α/N, αp /N, and αq /N [cf. Eq. (18)] for the most stable structures of NaN as functions of the cluster size.

The smooth variation of αp /N with the cluster size N can be rationalized using the following qualitative considerations. αp is deﬁned by the rate of change of the sum µp of the local parts µA,p of the atomic dipole moments in response to an external electric ﬁeld. This change is caused by the redistribution of the original (ﬁeld-free) charge density within the local atomic volumes and the possible ﬂow of charge to or from these volumes, both triggered by the external ﬁeld. Because of the screening of the interior atoms and the larger degree of charge exchange in the periphery of the clusters, it is the surface atoms that can be expected to contribute to αp most. This means that αp roughly scales as the number of the surface atoms N 2/3 . αp /N then scales with the size N of the clusters as N −1/3 . A power-law ﬁt of the αp /N data in Fig. 3 yields an exponent of −0.475 in reasonable agreement with these qualitative considerations, which are based on the assumed spherical shape of the clusters; deviations from this shape, as in our case (e.g., the preferred structures of NaN , N ≤ 6, are planar), lead to the departure from the predictions of the spherical model. It is the surface atoms that play the dominant role in deﬁning αq as well. Here, however, an extra linear dependence on N 1/3 is introduced due to the fact that the charge-transfer parts of the individual atomic dipole moments and polarizabilities are referred to the origin of the global system of coordinates [cf. Eqs. (9) and (12)], a convenient choice for which is the “center” (e.g. center of mass) of the clusters. The distance RA of the surface atoms from the center of a spherical cluster scales with the number of

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atoms as N 1/3 . Hence, for spherical clusters the value of αq /N can be expected to be insensitive to the cluster size. The deviations, including the oscillations, in the αq /N graph of Fig. 3 from this expectation are largely due to the deviations of the real cluster structures from the spherical shape.

5. Summary In this contribution, we recapped the scheme we introduced recently for an atomic-level analysis of the dielectric properties of ﬁnite systems and applied it to study the size-, structure-/shape- and site-dependence of the dipole moments and polarizabilities of sodium clusters. The scheme is based on partitioning the volume of a system into atomic volumes and using the charge distributions within the atomic volumes and the changes in these distributions in response to an external electric ﬁeld to deﬁne what we refer to as atomic dipole moments and polarizabilities. These are further decomposed into local, or dipole, and charge-transfer components. When summed over all the atoms, these components give the local, or dipole, and charge-transfer parts of the total system dipole moment and polarizability. The scheme can straightforwardly be generalized to inhomogeneous systems and alternative partitionings into subsystems, or sites, that contain more than one atom. The atoms of the subsystems may or may not be related by proximity. The main ﬁndings for the response properties of sodium clusters with up to 20 atoms can be summarized as follows. The individual atomic polarizabilities, as well as their dipole and charge-transfer components, are strongly dependent on the location of the atom within the cluster. They are substantially larger for atoms at peripheral sites than for atoms at interior sites (the screening eﬀect), and the more peripheral an atom is, the larger its polarizability. As a consequence, the total polarizabilities of the clusters depend on their structure/shape. The charge-transfer contributions are responsible for the details, including the oscillations, in the size-evolution of the total polarizability evaluated on a per-atom basis. The dipole part of the total polarizability, when evaluated on a per-atom basis, decreases almost monotonically with the cluster size and its relative contribution to the total polarizability also decreases. The partition scheme used in this study represents a powerful new tool for exploration and detailed analyses of properties of ﬁnite systems and size-dependent phenomena. Our current work involves its applications to heterogeneous systems and to the phenomenon of size-induced transition to metallicity (cf., e.g., Refs. 17 and 18).

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Acknowledgments This work was supported by the Oﬃce of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, U.S. Department of Energy under grant number DE-FGO2-03ER15489 (KAJ and MY) and under contract number DE-AC-02-06CH11357 (JJ), by the National Science Foundation Grant PHY-0619407 (KAJ and MY), and by Research Excellence Funds from the State of Michigan (KAJ). The assistance of Mr. John Kamau and Mr. Chamila Dharmadharwana with Figs. 1 and 2 is appreciated. References 1. W. D. Knight, K. Clemenger, W. A. de Heer, W. A. Saunders, M. Y. Chou and M. L. Cohen, Phys. Rev. Lett. 52, 2141 (1984). 2. W. D. Knight, K. Clemenger, W. A. de Heer and W. A. Saunders, Phys. Rev. B 31, 2539 (1985). 3. M. Brack, Rev. Mod. Phys. 65, 677 (1993). 4. W. A. de Heer, W. D. Knight, M. Y. Chou and M. L. Cohen, in Solid State Physics, Vol. 40, eds. H. Ehrenreich and D. Turnbull (Academic Press, New York, 1987). 5. D. Rayane, A. R. Allouche, E. Benichou, R. Antoine, M. Aubert-Frecon, Ph. Dugourd, M. Broyer, C. Ristori, F. Chandezon, B. A. Huber and C. Guet, Eur. Phys. J. D 9, 243 (1999). 6. G. Tikhonov, V. Kasperovich, K. Wong and V. V. Kresin, Phys. Rev. A 64, 063202 (2001). 7. S. Kummel, J. Akola and M. Manninen, Phys. Rev. Lett. 84, 3827 (2000). 8. P. Calaminici, A. M. Koster and A. Vela, J. Chem. Phys. 113, 2199 (2000). 9. L. Kronik, I. Vasiliev and J. R. Chelikowsky, Phys. Rev. B 62, 9992 (2000). 10. L. Kronik, I. Vasiliev, M. Jain and J. R. Chelikowsky, J. Chem. Phys. 115, 4322 (2001). 11. K. R. S. Chandrakumar, T. K. Ghanty and S. K. Ghosh, J. Chem. Phys. 120, 6487 (2004). 12. K. A. Jackson, M. Yang and J. Jellinek, J. Phys. Chem. C 111, 17952 (2007). 13. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 14. M. R. Pederson and K. A. Jackson, Phys. Rev. B 41 7453 (1990). 15. K. A. Jackson and M. R. Pederson, Phys. Rev. B 42 3276 (1990). 16. K. A. Jackson, M. R. Pederson, K. M. Ho and C. Z. Wang, Phys. Rev. A 59, 3685 (1999). 17. J. Jellinek and P. H. Acioli, J. Phys. Chem. A 106, 10919 (2002); ibid. 107, 1610 (2003). 18. B. von Issendorf and O. Cheshnovsky, Ann. Rev. Phys. Chem. 56, 549 (2005).

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MAGNETISM IN CLUSTERS A. LYALIN∗,‡ , A. V. SOLOV’YOV† and W. GREINER† ∗

Imperial College London, Prince Consort Road, London SW7 2BW, UK †

Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe University, Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany ‡ [email protected]

The optimized structure, electronic and magnetic properties of La clusters consisting of up to 14 atoms have been investigated using ab initio theoretical methods based on density-functional theory. We show that increase in cluster symmetry can promote ferromagnetic instability in La clusters. A giant enhancement of magnetism in La4 , La6 and La13 clusters is reported. We also find that the ground states of La2 , La3 , La5 , La7 , La9 –La11 , and La14 clusters possess nonzero magnetic moments that ranged from ∼0.1–1.0 µB per atom. Strong dependence of the magnetic moment on temperature for T > 300 K is predicted. The results obtained are compared with the available experimental data and the results of other theoretical works. Keywords: Atomic cluster; magnetism; density-functional theory.

1. Introduction During the last two decades numerous theoretical and experimental eﬀorts have been devoted to studying structural, electronic, optical and magnetic properties of atomic clusters (see, e.g., Refs. 1–5). The properties of atomic clusters are very diﬀerent from those of the bulk materials and change drastically with increasing cluster size. This fact gives a unique opportunity to form new materials by properly assembling selected clusters. One of the most exciting developments in the physics of clusters relates to their magnetic behavior. Clusters exhibit novel magnetic properties essentially diﬀerent from those of the corresponding bulk solids (see, e.g., Refs. 6 and 7, and references therein). The study of evolution of magnetic 86

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properties from atoms to the bulk is important for the development of magnetic nanomaterials and understanding the fundamental principles of spin coupling in ﬁnite and low-dimensional systems. The most explored type of magnetic clusters are the clusters of the ferromagnetic transition-metal elements, such as Fe, Co, Ni (see, e.g., Refs. 6–11, and references therein). Theoretically, a strong enhancement of magnetism in clusters of elements that are ferromagnetic as bulk solids was predicted.6,8,10 The ﬁrst experimental measurements of the magnetic moments of small free Fe, Co and Ni clusters showed, however, that magnetization per atom was far less than the moment per atom in corresponding bulk material.12,13 This contradiction was explained by relaxations of the magnetization of superparamagnetic clusters in the magnetic ﬁeld.10,13 Over the past years, this model has been extensively used to interpret the experimental data.14 Compared to the amount of work done for transition-metal clusters, little eﬀort has been devoted to systematic investigation of magnetism in rare-earth metal clusters, in spite of the fact that rare-earth metals possess remarkable magnetic properties. The heavy rare-earth elements, such as Gd, Tb, Dy, Ho are ferromagnets in the bulk state. The magnetic moments of the rare-earth metals are dominated by the spin contribution from the highly localized 4f electrons, and therefore these metals are good examples of local-moment ferromagnets. The 4f electron shell can accommodate 14 electrons, and according to the empirical Hund’s rule, a half-ﬁlled shell has seven electrons with parallel spins. Thus, the 4f electrons contribute 7 µB to the total magnetic moment of Gd (∼7.6 µB /atom), and similarly make a large contribution to the total moments for the other magnetic rare-earth metals. The valence electrons contribute a small fraction of the overall magnetic moment per atom — in the case of bulk Gd, the 5d6s valence electrons contribute 0.6 µB of the total moment. It has been demonstrated experimentally that the Gd2 molecule is the diatomic molecule with the highest spin, which has in its ground state a magnetic moment 8.8 µB /atom (compared with 6.5 µB for the free Gd atom, and 7.6 µB /atom in the bulk metal).16 The recent Stern–Gerlach deﬂection experiments in Gd17 and Dy18,19 clusters showed enhancement of magnetism in small rare-earth metal clusters. In Ref. 20, the magnetic behavior of small rare-earth clusters with diﬀerent geometries is investigated within the quantum Heisenberg model. It is shown that magnetic properties of heavy rare-earth clusters are highly dependent on the symmetry and geometry of the cluster. It is important to note that even in a local-moment system such as rare-earth metals the behavior of the valence electrons is crucial because these electrons mediate the exchange interaction between neighboring magnetic moments localized on the parent atoms.15,23 Lanthanum is the ﬁrst

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element in the group of the rare-eath metals and its electronic conﬁguration is [Xe]4f 0 5d1 6s2 . Thus, La can be treated as a prototype of all rare-earth elements, sharing their properties that are not dependent on 4f electrons. As the magnetic structures of the rare-earth metals are dependent on the occupancy of the 4f electron shell, they are absent from bulk La. Therefore, the emergence of magnetism in small La clusters remains an open question. Systematic theoretical investigations of magnetic properties of La clusters are lacking. In Ref. 24, energetics and structural stability of La clusters with number of atoms N = 3–13 are investigated by performing molecular dynamics simulations with an empirical pair potential. The potential energy function was ﬁtted to the dimer potential energy proﬁle of La2 , which was calculated by the DFT method.24 The M¨ obius inversion pair potential in combination with genetic algorithm was used in Ref. 25 in order to predict the lowest energy structures of the La3 –La20 clusters. The ﬁrst principles DFT calculations have been performed to study the electronic structure and spectroscopic constants of the La2 dimer,26 as well as the structural − 27 and electronic properties of the icosahedral La13 , La+ 13 and La13 clusters. The stable structures and electronic properties of small La2 –La14 clusters were explored in Ref. 28. Recently, the magnetic properties of Sc, Y and La clusters containing 5–20 atoms have been investigated in a Stern–Gerlach molecular-beam deﬂection experiment.29 The results of experiment conﬁrm that all Sc clusters and most Y and La clusters in the size range N = 5–20 are elemental molecular magnets with particular strong enhancement of magnetism in Sc13 (6.0 ± 0.2 µB ), Y8 (5.5 ± 0.1 µB ), Y13 (8.8 ± 0.1 µB ) and La6 (4.8 ± 0.2 µB ) clusters.29 In this contribution, we report the results of a systematic theoretical investigation of optimized ionic structure, electronic and magnetic properties of La clusters within the size range N ≤ 14. We focus our study on emergence of magnetic properties in La clusters and ﬁnd a giant enhancement of magnetism in La4 , La6 and La13 clusters. We also ﬁnd that the ground states of La2 , La3 , La5 , La7 , La9 –La11 and La14 clusters possess nonzero magnetic moment that ranged from ∼0.1–1.0 µB per atom, clearly indicating that small La clusters display magnetic behavior, even though bulk La has no magnetic ordering. We show that magnetism in La clusters is governed by unpaired valence electrons, in contrast to the local-moment magnetism in clusters of heavy rare-earth elements. As already mentioned, the valence electrons are those that are responsible for the local-moment spin coupling in rare-earth metals. Therefore, an understanding of the origin of magnetism in La clusters is important in order to gain a better insight into the magnetism of the rare-earth metal clusters. In addition to the ground state isomers of La clusters, we found an ensemble of energetically

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low-lying spin isomers. We predict an increase in the average magnetic moments for ensembles of La2 , La3 , La5 , La8 , La9 , La11 , La12 , and La14 clusters with temperature due to the thermal population of the spin isomers. For ensembles of La4 , La7 , and La13 clusters, the average magnetic moment decreases with temperature. Such an anomalous behavior of the magnetic moment with temperature can be detected in Stern–Gerlach deﬂection experiments.

2. Theoretical methods Our calculations are based on ab initio theoretical methods invoking the density-functional theory. The standard LANL2DZ basis set of primitive Gaussians are used to expand the cluster orbitals formed by the 5s2 5p6 5d1 6s2 outer electrons of La (11 electrons per atom). The remaining 46 core electrons 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p6 4d10 of the La atom are represented by the Wadt–Hay eﬀective core potential (see, e.g., Refs. 30 and 31, and references therein). The computations are performed within the DFT method based on the gradient-corrected exchange-correlation functional of Perdew, Burke and Ernzerhof (PBEPBE).32,33 Such an approach has proved to be a reliable tool for the ab initio level studying of the structure and properties of clusters of the alkali, the alkaline earth and the transition metals. To check the applicability of this method to lanthanum, we performed a careful comparison of the results obtained with those of earlier ab initio and experimental studies. The cluster geometries have been determined by ﬁnding local minima on the multidimensional potential energy surface. We have applied an eﬃcient scheme of global optimization, called the Cluster Fusion Algorithm (CFA).34–36 The scheme has been designed within the context of determination of the most stable cluster geometries, and it is applicable for various types of clusters.36 We have used a similar approach to ﬁnd the optimized geometries for noble gas clusters, and Na, Mg and Sr metal clusters.34,38–42 While the global energy optimization for noble gas clusters is a relatively simple problem and optimization could easily be done for larger clusters, the calculations with metal clusters present a serious challenge and require signiﬁcant computational resources. For both types of calculations, the CFA has proven to be a reliable and eﬀective tool in multidimensional global optimization. The proposed algorithm belongs to the class of genetic (also called evolutionary) global optimization methods.44,45 Note that during the optimization process, the geometry of the cluster as well as its initial symmetry sometimes change dramatically. All the characteristics of the clusters, which we have calculated and present in

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the following section, are obtained for the clusters with the optimized geometry. With increasing cluster size, such calculations become demanding of computer time. In this work, we limit the calculations by using the cluster size N = 14. Calculations have been carried out with the use of the GAUSSIAN 03 software package.37

3. Numerical results and discussion We start our study with the lanthanum dimer, La2 . The investigation of the rare-earth metal dimers is still a challenge for both experimentalists and theoreticians.49 Although, several experimental43,46,47 and theoretical26,48–50 works on La2 have been done, their results are not consistent with each other. Our calculations show the ground state of the La2 dimer to be a triplet (µ = 1 µB ), with dissociation energy De = 2.44 eV, bond length d = 3.00 ˚ A and the harmonic vibrational frequency ωe = 163 cm−1 . The calculated dissociation energy is in an excellent agreement with the experimental results of Refs. 43 and 46, where the value of 2.52 ± 0.22 eV was reported. The calculated value of the bond length, d, slightly overestimates the bond distance of 2.80 ˚ A determined experimentally.43 The calculated vibrational frequency, ωe , underestimates signiﬁcantly the measured ground state vibrational constant of 230 cm−1 .43 We have also found several energetically close-lying spin isomers of La2 . The ﬁrst spin isomer is found to be a pentet (µ = 2 µB ), with the bond length of 3.10 ˚ A; its dissociation energy of 2.42 eV only 0.02 eV lower than the ground state. The second isomer state is a singlet, with d = 2.94 ˚ A and De = 2.40 eV. The existence of a large number of energetically close-lying states makes the ground state determination a very diﬃcult task. This fact can explain a large dispersion in theoretical results obtained by diﬀerent methods. Thus, the ﬁrst theoretical calculations of spectroscopic constants for La2 were performed within the conﬁguration interaction method with the inclusion of single and double excitations and size-consistency error corrections (CISD + SCC).50 The calculations performed in Ref. 50 predicted A, a pentet ground state for La2 and spectroscopic constants d = 3.25 ˚ ωe = 130 cm−1 , De = 1.17 eV in a strong disagreement with the current experimental data. In Ref. 24, spectroscopic constants of d = 3.28 ˚ A, ωe = 228 cm−1 , De = 3.47 eV were obtained for La2 within an all-electron relativistic density-functional (RDFT) method. The bond distance and dissociation energy calculated within the RDFT method for La2 considerably overestimate those measured in the experiment.

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The recent theoretical calculations for La2 were performed with the use of the coupled cluster method with single, double and perturbative triple excitations [CCSD(T)]49 and the complete active space self-consistent ﬁeld method (CASSCF).48 This calculation predicted the singlet to be a ground A, ωe = 218 cm−1 , state of La2 dimer. The theoretical values of d = 2.64 ˚ −1 De = 2.37 eV and d = 2.70 ˚ A, ωe = 186 cm , De = 2.31 eV were obtained within the CCSD(T) and CASSCF methods, respectively. The calculated values of dissociation energy and bond distance are in good agreement with experimental data; however, the vibrational frequency is underestimated. In Ref. 26, various DFT methods were used for calculation of the spectroscopic constants for La2 . The calculated values of dissociation energy vary considerably among diﬀerent DFT methods, from 1.09 eV (BP86/LANL2DZ) to 3.66 eV (BHLYP/LANL2DZ) (see Table I in Ref. 26 for details). Overall results reported in Ref. 26 are in a poor agreement with experiment. In Ref. 28, dissociation energy of 3.08 eV and bond length of 2.77 ˚ A were obtained for La2 within the BLYP/DNP method. In Table 1, we summarize results of our calculations of the spectroscopic constants for La2 performed within the PBEPBE/LANL2DZ method, as well as the results reported in previous theoretical24,26,28,48–50 and experimental43,46 works. As is seen from Table 1, the PBEPBE/LANL2DZ estimation for La2 dissociation energy is in excellent agreement with experimental data. The La2 bond length calculated with the use of the PBEPBE/LANL2DZ method only slightly overestimates the experimental value of 2.80 ˚ A.43 This comparison allows us to conclude that the PBEPBE/LANL2DZ method is a reliable tool for ab initio level studying of the structure and properties of La clusters. The results of the cluster geometry optimization for La clusters consisting of up to 14 atoms are shown in Fig. 1. La clusters possess various geometry and spin isomer forms whose numbers grow dramatically with increasing cluster size. In Fig. 1, we present only the lowest energy conﬁgurations optimized by the PBEPBE/LANL2DZ method. The interatomic distances are given in ˚ Angstroms for La2 –La7 clusters. The label above each cluster image indicates the point symmetry group of the cluster. Figure 1 shows that small La clusters form compact structures, maximizing the coordination number. The lowest energy state for La3 is an isosceles triangle, and for La4 a regular tetrahedron. La clusters are tridimensional already at N > 3. As we discuss below, the La4 cluster is relatively more stable and compact, as compared to the neighboring clusters. The La5 cluster has a structure of a slightly elongated triangular bipyramid, while La6 has a structure of a slightly ﬂattened octahedron, La7 is a pentagonal bipyramid. These geometrical structures are in a good agreement

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µ (µB )

d (˚ A)

ωe (cm−1 )

De (eV)

1 2 0 0 0 0 0 0

3.00 3.10 2.94 2.93 2.91 2.89 2.87 2.87 3.28 3.25 2.70 2.64 2.77 2.80

163 140 141 139 150 131 136 161 228 130 186 218

2.44 2.42 2.40 1.09 1.70 1.83 1.87 3.66 3.47 1.17 2.31 2.37 3.08 2.5 ± 0.2 2.52 ± 0.22

2 0 0 0

230

with the results of Ref. 28, except the case of La4 . We have found that the regular tetrahedron structure Td for La4 has a lower energy than that of the distorted tetrahedron structure Cs reported in Ref. 28. This discrepancy can be explained by the fact that the Td structure found in this work and Cs structure found in Ref. 28 are diﬀerent spin isomers. We have found that the spin-nonet (µ = 8 µB ) ground state becomes energetically more favorable for La4 in comparison to the spin-triplet state considered in Ref. 28. In the size region N ≥ 8, the structures of clusters optimized with the use of the PBEPBE/LANL2DZ method are diﬀerent from those derived on the basis of the BLYP/DNP approximation.28 Thus, the lowest energy state for La8 is a capped pentagonal bipyramid, Cs . The La9 cluster has a low symmetry structure C1 that is formed by fusion of a deformed triangular bipyramid with a strongly distorted rhombus, La10 has a C3v structure, and La11 is a distorted D4d structure. The La12 cluster is formed by fusion of a pentagonal bipyramid with a tetragonal pyramid, La13 is a regular icosahedron, and La14 has a Cs structure that is close to a slightly deformed C2v . The structural stability of La13 has already been studied in Ref. 27. It was found that the La13 cluster has its ground state of D2h symmetry, while Ih and D3d structures appear to be energetically close-lying isomers.27 Our calculations show that the Ih structure (with the total spin S = 13/2) is energetically more favorable in comparison with D2h and D3d structures. This disagreement is connected with the diﬀerence in total spins of the ground states considered in Ref. 27 and in our work.

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Fig. 1. Optimized geometries of lanthanum clusters La2 –La14 calculated in the PBEPBE/LANL2DZ approximation. Interatomic distances are given in ˚ Angstroms. The label above each cluster image indicates the point symmetry group of the cluster.

Figure 1 demonstrates a strong competition between icosahedral and octahedral growth motifs in the evolution of geometry structure of small La clusters with their size. The isocahedral packing is a typical growth motif for clusters of elements having a van der Waals type of bonding, such as, for example, clusters of rare gases (see, e.g., Refs. 1 and 34). A similar growth mode is also typical for clusters of alkaline earth metals, because the electronic shells in the divalent atoms are ﬁlled (the electronic conﬁguration of the valence electrons is ns2 ) and bonding between atoms is expected to have some features of the van der Waals type (see, e.g., Refs. 39, 41 and 42, and references therein). The appearance of the octahedral elements in cluster structures is typical for the d transition metal clusters. The d orbitals have a square symmetry, and they can be responsible for formation

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of elements of a cubic lattice. Since a lanthanum atom contains s and d valence electrons, one can expect a competition between compact structures maximizing the number of bonds and directional bonding compatible with the orientation and ﬁlling of the d orbitals.1 Figure 2 shows how the average bonding distance, Rav , evolves with increasing size of La clusters. The dependence of the average bonding distance on cluster size has non-monotonic oscillatory behavior atop its fast systematic growth towards the bulk limit. In Fig. 2, the bulk limit for the double hexagonal closest-packing (dhcp) lattice of lanthanum is indicated by the horizontal line. It is clearly seen in Fig. 2 that for N ≥ 11 the average bonding distance for La clusters closely approaches the bulk limit. The appearance of the maxima in the size dependence of the average bonding distance shows that La6 , La9 and La11 clusters are less compact than their neighbors. Exactly these structures possess elements of the cubic (or rhombic in the case of La9 ) lattice as is seen from Fig. 1. Therefore, the oscillatory behavior of the average bonding distance can be interpreted as the competition between icosahedral and octahedral growth modes. Figure 3 shows the dependence of the binding energy per atom for the most stable lanthanum clusters as a function of cluster size. The binding energy per atom for La clusters is deﬁned as follows: Eb /N = E1 − EN /N,

(1)

Fig. 2. Average bonding distance as a function of cluster size for La2 –La14 clusters. The horizontal dashed line indicates the bulk limit for the dhcp lattice of La.51

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Fig. 3. Binding energy per atom for the most stable La clusters as a function of cluster size. The open circle presents the binding energy per atom for the La2 dimer obtained from the experimental data by Verhaegen.43 Inset: secondary diﬀerences of total energy for La clusters.

where EN is the energy of a neutral N -particle atomic cluster, and E1 is the energy of a single La atom. For small La clusters, the binding energy per atom increases steadily with the cluster size. The local maxima of Eb /N at N = 7 and 13 correspond to the most stable conﬁgurations of the magic La clusters. The analysis of the secondary diﬀerences of the binding energy, ∆2 EN = EN +1 − 2EN + EN −1 , conﬁrms this conclusion and hints about the relative stability of the La4 , La5 and La9 clusters, in addition to the magic clusters La7 and La13 (see Fig. 3). The principal magic numbers 7 and 13 can be explained by atomic shell closing eﬀects. Indeed, the enhanced stability of La7 and La13 clusters arises when their ionic structure is highly symmetric and corresponds to the icosahedral type of packing. This icosahedral growth sequence for metal clusters has also been seen for clusters of alkaline earth metals such as Sr41 and Ba52 , which exhibit nonmetal-to-metal transitions with increasing size. It is important to note that for alkaline earth metal clusters there is a strong competition between atomic and electronic shells closure.39,41 However, for La clusters there is no direct inﬂuence of the electronic shell eﬀects on cluster stability. It is a typical feature of atomic clusters that they possess various isomer forms whose numbers grow dramatically with increasing cluster size. An important feature of La clusters is that, in addition to structural isomers,

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Fig. 4. Binding energy per atom for a variety of geometry and spin isomers of La clusters as a function of cluster size.

they possess a great variety of energetically low-lying spin isomers. Figure 4 demonstrates the binding energies per atom calculated for a variety of lowlying geometries and spin isomers of La clusters. The typical diﬀerence in binding energy between the ground state and the ﬁrst low-lying isomers is about 0.02–0.05 eV, as is seen from Fig. 4. This result suggests that an ensemble of energetically low-lying isomers will be thermally available already at relatively low temperatures. The multitude of spin multiplicities, 2S + 1, for La clusters results in considerable variations of the magnetic moment per atom µ = 2SµB /N . Figure 5 shows the dependence of the magnetic moment per atom for La clusters as a function of cluster size. The ﬁlled circles represent magnetic moments per atom calculated for the lowest energy isomers of La clusters. The calculations were performed within the PBEPBE/LANL2DZ method. The open circles in Fig. 5 show the results of the experiment by Knickelbein.29 The magnetic moments reported in Ref. 29 were obtained for clusters generated at 58 ± 2 K. Figure 5 shows that the dependence of the magnetic moment per atom, µ, on cluster size has a complex and non-monotonic behavior. For La2 and La3 clusters µ = 1 µB , that corresponds to ferromagnetic coupling of d electrons. Transition to 3D structure for N = 4 is accompanied by a giant enhancement of magnetism in La4 (µ = 2 µB ). The existence of an energetically favorable high-spin Td structure for La4 originates from the sd hybridization developed upon bonding. The results of our calculations

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Fig. 5. Magnetic moments per atom of La clusters as a function of cluster size. The ﬁlled circles represent the magnetic moments per atom calculated within the PBEPBE/LANL2DZ method. For La13 , the ground state structure Ih and the ﬁrst low-lying isomer C3 are presented. The open circles show the results of the experiment by Knickelbein.29

demonstrate that an anomalous enhancement of magnetism also occurs in the highly symmetric La6 (D4h ) and La13 (Ih ) clusters. It is known that anomalous magnetism in low-dimensional transition metals can be attributed to the enhancement of the densities of d states at the Fermi level resulting from spatial conﬁnement.53 For La clusters, the eﬀects of spatial conﬁnement on magnetic properties can be explained by the Stoner model of itinerant ferromagnetism (see, e.g., Ref. 6, and references therein). The Stoner model allows the prediction of the emergence of anomalous magnetic ordering in clusters of transition elements (see, e.g., Ref. 29, and references therein). The spatial conﬁnement leads to d-band narrowing53 which can in turn lead to ferromagnetic instability and the formation of spontaneous magnetic ordering in clusters. It is also important to note that narrowing of d states in clusters can result from symmetry eﬀects. Indeed, d states degenerate with increase in cluster symmetry, promoting ferromagnetic instability. It is clearly seen from Fig. 5 that the giant enhancement of magnetism occurs in the highly symmetric La4 (Td ), La6 (D4h or distorted Oh ) and La13 (Ih ) clusters. On the other hand, the structural changes are often accompanied by strong changes in magnetic behavior. Thus, the reduction in symmetry of La13 from the ground state icosahedral structure to the ﬁrst low-lying isomer with C3 symmetry results in a sharp decrease in magnetic moment per

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atom from 1.0 µB to 0.23 µB , respectively. Note that the diﬀerence in total binding energies between Ih and C3 structures of La13 is only 0.06 eV. The Mulliken analysis of atomic spin densities demonstrates ferromagnetic ordering in La4 and La6 clusters as is seen in Fig. 6. For La13 , there is a ferromagnetic coupling within the surface atoms. However, the spin of the highly coordinated central atom is antiferromagnetically coupled with the spins of the surface atoms. This leads to a total magnetic moment per atom of 1.0 µB for the ground state of La13 cluster. The enhancement of

Fig. 6. Mulliken atomic spin densities for La4 , La5 , La6 and La13 clusters. The green (lighter) and red (darker) colors denote the excess of spin up and spin down densities, respectively. The label above each cluster image indicates the total spin S of the cluster.

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the spin magnetization of La13 cluster is dominated by the surface contribution, even though the highest coordinated central atom has the highest moment, while the surface atoms have the lowest. Figure 5 demonstrates that the ground states of La5 , La7 , La9 –La11 , and La14 clusters possess nonzero magnetic moments, ranging from ∼ 0.1 to 0.5 µB per atom, clearly indicating that some La clusters display magnetic behavior, while clusters La8 and La12 are not magnetic. The total magnetic moment of La clusters is comprised of sizable local atomic magnetic moments [∼(0.3−1.0)µB ] that couple antiferromagnetically. For example, for La5 there is a ferromagnetic coupling within three base atoms of the triangular bipyramid, and an antiferromagnetic coupling between the base and its capping atoms, as is seen in Fig. 6. Overall, the calculated magnetic moments of La clusters are in good agreement with the experimental results of Ref. 29. With the exception of N = 13, the theoretical curve reproduces general features in the size dependence of the magnetic moment per atom obtained in experiment. The theory reproduces a giant enhancement of magnetism in La6 , discovered experimentally, as well as reproduces a sharp decrease in size dependence of µ for N = 7 and 8. We found that the magnetic moments per atom are relatively small for La clusters within the size range 8 ≤ N ≤ 12 in accord with experiment. The calculated value of the magnetic moment for La12 is 0. This result is also in full agreement with experiment, which shows no measurable deﬂection or broadening of the beam proﬁle for La12 .29 Unfortunately, there are no experimental data for La4 , for which we predict a giant enhancement in magnetism, with the magnetic moment per atom reaching 2.0 µB . The main disagreement between the theory and experiment arises for La13 . Figure 5 shows that the theoretical value of the magnetic moment for the icosahedral La13 cluster is considerably higher than that observed in the experiment. Such a diﬀerence can arise due to the existence of the energetically close-lying isomer structure of the C3 symmetry. The total binding energy, Eb , for the C3 isomer of the La13 cluster is only 0.06 eV smaller than that for the Ih ground state. The calculated magnetic moment per atom of 0.23 µB for the C3 isomer is in good agreement with experiment, as is seen from Fig. 5. We note that the size variation of the experimentally measured magnetic moment per atom of Sc and Y clusters displays pronounced maxima for N = 13.29 By contrast, the experimental data for La13 displays no giant enhancement of magnetism. This diﬀerence was explained in Ref. 29 by the possibility of diﬀerent structural motifs in the growth of Sc, Y and La clusters. As we have discussed above, La clusters possess various structural and spin isomer forms. The small energy diﬀerences between spin isomers result

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in the dependence of the thermally averaged magnetic moment on temperature due to the thermal population of low-lying spin isomers. Figure 7 shows the temperature dependence of the thermally averaged magnetic moment per atom, µ, which is deﬁned as i µi exp (−Ei /kT ) . (2) µ = i exp (−Ei /kT )

Fig. 7.

Thermally averaged magnetic moments per atom for La clusters.

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µi is the magnetic moment per atom for isomer i with the total energy Ei , k is the Boltzmann constant, and T is the ensemble temperature. The temperature dependence of µ is expected to be rather weak for low temperatures (T < 100–150 K), and only the ground state isomer will contribute to µ at a temperature of 58 K used in Ref. 29. However, for T > 200–300 K we found a strong dependence of the thermally averaged magnetic moment on temperature. Figure 7 demonstrates the increase of the average magnetic moments for ensembles of La2 , La3 , La5 , La8 , La9 , La11 , La12 and La14 clusters with temperature, which is due to the thermal population of the spin isomers. For the ensembles of La4 , La7 and La13 clusters, the average magnetic moment decreases with temperature. For the ensembles of La6 and La10 clusters, the average magnetic moment for practical purposes does not depend on temperature up to T = 1000 K. Such an anomalous behavior of the magnetic moment with temperature (T > 300 K) can be detected in Stern–Gerlach deﬂection experiments. We believe we can explain the thermal behavior of the La2 dimer. The increase in temperature depopulates the triplet state and populates the pentet and the singlet states. Therefore, we predict that for La2 a rise in temperature would lead ﬁrst to an increase in the thermally averaged magnetic moment per atom due to the increase in population of the pentet state. The subsequent decrease in µ (T > 300 K) would result from the higher population of the singlet state. Similar anomalous behavior of the magnetic moment has been predicted for small Pd clusters.54

4. Conclusion The optimized structure, electronic and magnetic properties of La clusters consisting of up to 14 atoms have been investigated using the ab initio DFT PBEPBE/LANL2DZ method. We found a giant enhancement of magnetism in La4 , La6 and La13 clusters. We also found that the ground states of La2 , La3 , La5 , La7 , La9 –La11 and La14 clusters possess nonzero magnetic moments, ranging from ∼0.1 to 1.0µB per atom, clearly indicating that small La clusters display magnetic behavior, even though bulk La has no magnetic ordering. The results of our calculations of the magnetic moment per atom for La clusters are in a good agreement with those derived from experiment. We show that the increase in cluster symmetry can promote ferromagnetic instability in La clusters. On the other hand, the structural changes can be accompanied by strong changes in magnetic behavior. A variety of structural and spin isomers were determined. We predict increases of the average magnetic moments for the ensembles of La2 ,

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La3 , La5 , La8 , La9 , La11 , La12 and La14 clusters with temperature for T > 300 K due to the thermal population of spin isomers. For the ensembles of La4 , La7 and La13 clusters, the average magnetic moment decreases with temperature. Our study suggests temperature-dependent Stern–Gerlach deﬂection measurements. Our results were obtained for free La clusters. Many interesting problems beyond the scope of the present work arise, however, when considering possible enhancement of magnetism in clusters deposited on a surface or embedded into a matrix. This problem can have important technological applications for the creation of new nano-structured materials with unique magnetic properties. Acknowledgments This work is partially supported by the European Commission within the Network of Excellence project EXCELL, and by INTAS under the grant 03-51-6170. The authors gratefully acknowledge support by the Frankfurt Center for Scientiﬁc Computing. References 1. J. A. Alonso, Structure and Properties of Atomic Nanoclusters (Imperial College Press, London, 2005). 2. C. Guet, P. Hobza, F. Spiegelman and F. David, Atomic Clusters and Nanoparticles, NATO Advanced Study Institute, les Houches Session LXXIII, les Houches, 2000 (EDP Sciences and Springer Verlag, Berlin, 2001). 3. J.-P. Connerade and A. V. Solov’yov (eds.), Latest Advances in Atomic Cluster Collisions: Fission, Fusion, Electron, Ion and Photon Impact (Imperial College Press, London, 2004). 4. P.-G. Reinhard and E. Suraud, Introduction to Cluster Dynamics (WileyVCH, Weinheim, 2004). 5. F. Baletto and R. Ferrando, Rev. Mod. Phys. 77, 371 (2005). 6. G. M. Pastor, in Atomic Clusters and Nanoparticles, eds. C. Guet et al. (EDP Sciences and Springer Verlag, Berlin, 2001). 7. N. Fujima and T. Yamaguchi, Phys. Rev. B 54, 26 (1996). 8. D. R. Salahab and R. P. Messmer, Surf. Sci. 106, 415 (1981). 9. K. Lee, J. Callaway and S. Dhar, Phys. Rev. B 30, 1724 (1985). 10. S. N. Khanna and S. Linderoth, Phys. Rev. Lett. 67, 742 (1991). 11. T. Oda, A. Pasquarello and R. Car, Phys. Rev. Lett. 80, 3622 (1998). 12. W. A. de Heer, P. Milani and A. Chatelain, Phys. Rev. Lett. 65, 488 (1990). 13. J. P. Bucher, D. C. Douglass and L. A. Bloomfield, Phys. Rev. Lett. 66, 3052 (1991). 14. M. B. Knickelbein, Phys. Rev. Lett. 86, 5255 (2001).

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15. C. Kittel, Introduction to Solid State Physics, 7th Edn. (John Wiley and Sons, New York, 1996). 16. R. J. Van Zee and W. Weltner, J. Chem. Phys. 100, 4010 (1994). 17. D. Gerion, A. Hirt and A. Chˆ atelain, Phys. Rev. Lett. 83, 532 (1999). 18. S. Pokrant, Phys. Rev. A 62, 051201(R) (2000). 19. S. Pokrant and J. A. Becker, J. Magn. Magn. Mater. 226–230, 1921 (2001). 20. F. L´ opez-Ur´ıas, A. D´ıaz-Ortiz and J. L. Mor´ an-L´ opez, Phys. Rev. B 66, 144406 (2002). 21. D. P. Pappas, A. P. Popov, A. N. Anisimov, B. V. Reddy and S. N. Khanna, Phys. Rev. Lett. 76, 4332 (1996). 22. V. Z. Cerovski, S. D. Mahanti and S. N. Khanna, Eur. Phys. J. D 10, 119 (2000). 23. S. D. Barrett and S. S. Dhesi, The Structure of Rare-Earth Metal Surfaces (Imperial College Press, London, 2001). 24. S. Erkoc, T. Bastug, M. Hirata and S. Tachimori, Chem. Phys. Lett. 314, 203 (1999). 25. Y.-H. Luo and Y. Wang, Phys. Rev. A 64, 015201 (2001). 26. Z.-J. Wu, J. S. Shi, S. Y. Zhang and H. J. Zhang, Phys. Rev. A 69, 064502 (2004). 27. D.-B. Zhang and J. Shen, J. Chem. Phys. 120, 5081 (2004). 28. D.-B. Zhang and J. Shen, J. Chem. Phys. 120, 5104 (2004). 29. M. B. Knickelbein, Phys. Rev. B 71, 184442 (2005). 30. J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods (Gaussian Inc., Pittsburgh, PA, 1996). 31. P. J. Hay and W. R. Wadt, J. Chem. Phys. 82, 299 (1985). 32. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 33. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 78, 1396 (1997). 34. I. A. Solov’yov, A. V. Solov’yov, W. Greiner, A. Koshelev and A. Shutovich, Phys. Rev. Lett. 90, 053401 (2003). 35. I. A. Solov’yov, A. V. Solov’yov and W. Greiner, Int. J. Mod. Phys. E 13, 697 (2004). 36. O. I. Obolensky, I. A. Solov’yov, A. V. Solov’yov and W. Greiner, Comput. Lett. 1, 313 (2005). 37. J. M. Frisch et al., Computer code GAUSSIAN 03, Rev. C.02, Wallingford CT: Gaussian Inc., 2004. 38. I. A. Solov’yov, A. V. Solov’yov and W. Greiner, Phys. Rev. A 65, 053203 (2002). 39. A. Lyalin, I. A. Solov’yov, A. V. Solov’yov and W. Greiner, Phys. Rev. A 67, 063203 (2003). 40. A. Lyalin, I. A. Solov’yov, A. V. Solov’yov and W. Greiner, Phys. Rev. A 75 053201 (2007). 41. A. Lyalin, A. V. Solov’yov, C. Br´echignac and W. Greiner, J. Phys. B: At. Mol. Opt. Phys. 38, L129 (2005). 42. I. A. Solov’yov, A. Lyalin, A. V. Solov’yov and W. Greiner, in Latest Advances in Atomic Cluster Collisions: Fission, Fusion, Electron, Ion and Photon

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Impact, eds. J.-P. Connerade and A. V. Solov’you (Imperial College Press, London, 2004). G. Verhaegen, S. Smoes and J. Drowart, J. Chem. Phys. 40, 239 (1964). D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, MA, 1989). Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, 3rd Edn. (Springer, Berlin, 1996). J. A. Connor, in Metal Clusters in Catalysis: Studies in Surface Science and Catalysis, Vol. 29, eds. B. A. Gates, L. Guczi and L. H. Knoezinger (Elsevier, Amsterdam, 1986). Y. Liu, Li Fang, X. Shen, X. Chen, J. R. Lombardi and D. M. Lindsay, Chem. Phys. 262, 25 (2000). X. Cao and M. Dolg, Theor. Chem. Acc. 108, 143 (2002). X. Cao and M. Dolg, Mol. Phys. 101, 1967 (2003). M. Dolg, H. Stroll and H. Preuss, J. Mol. Struct. (Theo. Chem.) 277, 239 (1992). B. J. Beaudry and K. A. Gschneidner, in Handbook on the Physics and Chemistry of Rare Earths, Vol. 1, eds. K. A. Gschneidner and L. Eyring (NorthHolland, Amsterdam, 1978). Q. Wang, Q. Sun, J.-Z. Yu, B.-L. Gu, Y. Kawazoe and Y. Hashi, Phys. Rev. A 62, 063203 (2000). F. J. Himpsel, J. E. Ortega, G. J. Mankey and R. F. Wills, Adv. Phys. 47, 511 (1998). M. Moseler, H. H¨ akkinen, R. N. Barnett and U. Landman, Phys. Rev. Lett. 86, 2545 (2001).

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STRONTIUM CLUSTERS: ELECTRONIC AND GEOMETRY SHELL EFFECTS A. LYALIN∗,‡ , I. A. SOLOV’YOV† , A. V. SOLOV’YOV† and W. GREINER† ∗

Imperial College London, Prince Consort Road, London SW7 2BW, UK †

Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe University, Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany ‡ [email protected] The optimized structure and electronic properties of neutral, singly and doubly charged strontium clusters have been investigated using ab initio theoretical methods based on density-functional theory. We have systematically calculated the optimized geometries of neutral, singly and doubly charged strontium clusters consisting of up to 14 atoms, average bonding distances, electronic shell closures, binding energies per atom, and spectra of the density of electronic states (DOS). It is demonstrated that the size-evolution of structural and electronic properties of strontium clusters is governed by an interplay of the electronic and geometry shell closures. The inﬂuence of the electronic shell eﬀects on structural rearrangements can lead to violation of the icosahedral growth motif of strontium clusters. It is shown that the excessive charge essentially aﬀects the optimized geometry of strontium clusters. Ionization of small strontium clusters results in the alteration of the magic numbers. The strong dependence of the DOS spectra on details of ionic structure allows one to perform a reliable geometry identiﬁcation of strontium clusters. Keywords: Metal cluster; density-functional theory; shell eﬀects.

1. Introduction During the last two decades numerous theoretical and experimental works have been devoted to the study of stability, ionic structure and electronic properties of metal clusters. A comprehensive survey of the ﬁeld can be found in review papers and books; see, e.g., Refs. 1–12. The most explored type of metal clusters are the clusters of the alkali metals. It is worth mentioning that the electronic shell structure of metal 105

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clusters was discovered through the observation of the strong peaks in the mass spectra of sodium clusters.13 The enhanced stability of some clusters, the so-called magic clusters, has been explained by the closure of shells of delocalized electrons. A simple physical model describing the electronic shell structure of metal clusters has been developed within the jellium approximation (see, e.g., Refs. 6, 15 and 16) by analogy with the shell model of atomic nuclei (see, e.g., Ref. 14). The jellium model is very successful for the simple alkali metals, for which one electron per atom is delocalized (see, e.g., Refs. 1, 2 and 15–23, and references therein). The jellium model electronic shell closures for alkali metal clusters deﬁne the magic numbers N = 8, 20, 34, 40, 58 and 92 that are in good agreement with experiment. However, clusters of alkali metals are not representative of the whole class of metal clusters. Clusters of alkalineearth metals of the second group of the periodic table are expected to diﬀer from the jellium model predictions at least at small cluster sizes. In this case, bonding between atoms is expected to have some features of the van der Waals type of bonding, because the electronic shells in the divalent atoms are ﬁlled. Therefore, the evolution of the alkaline-earth metal cluster properties is governed by an interplay between the electronic and geometry shell closures. That fact implies using direct ab initio molecular dynamics simulations methods rather than simple jellium model approaches when exploring electronic properties and structure of alkaline-earth metal clusters. The structural behavior of alkaline-earth metals is peculiar. With increasing atomic number, the crystalline structure of the bulk alkalineearth metals alters from hexagonal closed packed (hcp) for Be and Mg to faced-centered cubic (fcc) for Ca and Sr and ﬁnally to body-centered cubic (bcc) for Ba.24 Thus, clusters of alkaline-earth metals are very appropriate for studying structural transformations, non-metal to metal transitions, testing diﬀerent theoretical methodologies, and conceptual developments of atomic cluster physics. However, relatively little work has been done so far on the exploration of the alkaline-earth metal clusters in comparison with that for the alkali metal clusters (see, e.g., Refs. 6 and 25, and references therein). Among clusters of the alkaline-earth metals, the most signiﬁcant attention has been paid to the berillium and magnesium clusters (see, e.g, Refs. 12 and 26–35, and references therein). The geometrical structure and bonding nature of MgN clusters with N up to 13 was studied in Ref. 29 using the density-functional molecular-dynamics method. The size evolution of bonding in magnesium clusters MgN with N = 8–13, 16, 20 was studied in Ref. 33 using the local-density approximation, which accounts for gradient corrections. Structural and electronic properties of small magnesium clusters (N ≤ 13) were studied in Ref. 30 using a ﬁrst-principles

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simulation method in conjunction with the DFT and the generalized gradient correction approximation for the exchange-correlation functional. It was shown30 that the metallization in magnesium clusters has a slow and non-monotonic evolution, although jellium-type magic clusters were also observed.29,33 In order to extend such calculations to larger systems, symmetry restricted methods have been developed. The spherically-averagedpseudo-potential scheme with the local and non-local pseudopotentials has been used for the investigation of the electronic structure and shell closures of spherical MgN clusters up to N = 46.36 Various properties of magnesium clusters have been investigated theoretically, such as their structure and formation of the basic elements of the hcp lattice, the binding energy, ionization potentials, the energy gap between the highest occupied and the lowest unoccupied molecular orbitals (HOMO–LUMO gap), average distances, and their evolution with the cluster size (see Refs. 26–28, and references therein). All-electron DFT calculations of the energetic and structural properties of neutral magnesium clusters MgN (N = 2 to 22 and selected clusters up to 309) were performed in Ref. 64. The mass spectrum of magnesium clusters was recorded37,38 and the sequence of magic numbers was determined. Not many works have been devoted to strontium clusters. Initially, the structures and stability of strontium clusters with the number of atoms N up to 20 were investigated using an empirical atomistic potential of the Murrell–Mottram type.39 The pairwise additive Morse potential was utilized in Ref. 40. Both of these potentials have been used to predict properties of bulk strontium and can also be employed to describe strontium clusters. In the case of the Morse potential, clusters with polytetrahedral components dominate the growth process,40 whereas in the case of the Murrell–Mottram potential, the icosahedral growth dominates, with local regions of enhanced stability at N = 4, 7, 13 and 19.39 At the ab initio level, calculations have been performed within the local-density approximation for the Sr2 molecule41 and small strontium clusters up to 13 atoms.42 The ab initio molecular-dynamics method with a plane-wave basis and ultrasoft pseudopotentials have been used for the study of the evolution of electronic states and multishell relaxations in strontium clusters SrN with number of atoms N = 2–35, 55 and 147.43 In Ref. 44, a many-body potential for strontium clusters was developed with parameters ﬁtted to the energy surface of strontium clusters containing up to ten atoms calculated within densityfunctional theory (DFT) in the generalized gradient approximation. Structure and energetics of the most stable cluster isomers with up to 63 atoms have been obtained with genetic algorithms.44 It has been shown that the sequence of magic clusters possessing enhanced stability with respect to its neighboring sizes changes signiﬁcantly with temperature.44 This behavior

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is due to structural transitions of the strontium clusters that occur at ﬁnite temperatures. Experimentally obtained mass spectra of the strontium clusters show that the magic numbers are substantially diﬀerent to those of Lennard–Jones clusters.44–46 Stability and ﬁssion of the positively charged strontium clusters have been studied theoretically within the simple liquid-drop model47 and experimentally.47–50 It has been found experimentally that the internal thermal excitation can inﬂuence ﬁssion channels and promote Coulombic ﬁssion.48 On the other hand, ﬁssion into two charged fragments can stimulate an additional ejection of a neutral atom during or immediately after the system overcomes the ﬁssion barrier. Such an interplay between the Coulombic ﬁssion and the evaporation processes was observed recently in Ref. 50, and analyzed theoretically in Ref. 51. The thermal promotion has been predicted 51 . Stability towards of Coulombic ﬁssion of the Sr2+ 7 monomer evaporation and ﬁssion of small neutral and positively charged strontium clusters was studied by means of ab initio DFT methods in Ref. 51. In this contribution, we investigate the optimized ionic structure and the electronic properties of neutral, singly and doubly charged strontium clusters within the size range N ≤ 14. We calculate optimized geometries, average bonding distance, electronic shell closures, binding energies per atom, and spectra of the electronic density of states. We demonstrate that the sizeevolution of structural and electronic properties of strontium clusters is governed by an interplay between the electronic and geometry shell closures. We show that the inﬂuence of the electronic shell eﬀects on structural rearrangements can lead to violation of the icosahedral growth motif of strontium clusters. We study how the excessive charge aﬀects the optimized geometry and other properties of the strontium clusters and demonstrate that ionization of the small strontium clusters results in the alteration of the sequence of magic numbers. We demonstrate that the strong dependence of the DOS spectra on details of ionic structure allows one to perform a reliable identiﬁcation of the geometry of strontium clusters. Our calculations are based on ab initio theoretical methods invoking density-functional theory and molecular dynamics simulations. The results obtained are compared with the available experimental data and the results of other theoretical works. The atomic system of units, |e| = me = = 1, has been used throughout, unless other units are indicated.

2. Theoretical methods Our calculations are based on ab initio theoretical methods invoking density-functional theory. The standard SDD(6D,10F) basis set of primitive

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Gaussians has been used to expand the cluster orbitals formed by the 4s2 4p6 5s2 outer electrons of Sr (ten electrons per atom). The remaining 28 core electrons 1s2 2s2 2p6 3s2 3p6 3d10 of the Sr atom are represented by a core-polarization potential (see, e.g., Ref. 52, and references therein). The computations are performed within the DFT method based on the hybrid Becke-type three-parameter exchange functional53 paired with the gradientcorrected Perdew–Wang 91 correlation functional (B3PW91).54,55 Such an approach has proven to be a reliable tool for ab initio level study of the structure and properties of strontium clusters.44 The cluster geometries have been determined by ﬁnding local minima on the multi-dimensional potential energy surface. We have applied an eﬃcient scheme of global optimization, called the Cluster Fusion Algorithm (CFA).56–58 The scheme has been designed within the context of determination of the most stable cluster geometries and it is applicable for various types of clusters.58 We have used a similar approach to ﬁnd the optimized geometries for noble gas clusters, and Na, Mg and La metal clusters.25,26,56,59 While the global energy optimization for noble gas clusters is a relatively simple problem and optimization could easily be done for larger clusters, the calculations with metal clusters present a serious challenge and require signiﬁcant computational resources. For both types of calculations, the CFA has proven to be a reliable and eﬀective tool in multi-dimensional global optimization. The proposed algorithm belongs to the class of genetic (also called evolutionary) global optimization methods.60,61 In applications to clusters, the genetic methods are based on the idea that the larger clusters evolve to low energy states by mutation and/or by mating smaller structures with low potential energy. Our method uses the strategy of adding one atom to a cluster of size N − 1. There are, however, two important improvements which allow for a much faster convergence in comparison with standard genetic methods. The ﬁrst is the fact that the atom is not added at a random place on the surface of the initial cluster. Rather, we use the deterministic approach and the new atom is added to certain places on the cluster surface, such as the midpoint of a face. The second important feature of our method is that we add the new atom not only to the ground state isomer of size N − 1, but also to the other, energetically less favorable, isomers. This ensures that we do not miss sizes at which smooth evolution within one family of clusters (say, with the same type of lattice) is interrupted and the global energy minimum of the next cluster size lies within another cluster growth branch. Note that during the optimization process, the geometry of the cluster as well as its initial symmetry sometimes change dramatically. All the characteristics of the clusters, which we have calculated and present in the following section, are obtained for the clusters with the optimized geometry. With

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increasing cluster size, such calculations become demanding of computer time. In this work, we limit the calculations to the cluster size N = 14. Calculations have been carried out with the use of the GAUSSIAN 03 software package.62 3. Numerical results and discussion 2+ 3.1. Geometry optimization of SrN , Sr+ N and SrN clusters

The results of the cluster geometry optimization for neutral, singly and doubly charged strontium clusters consisting of up to 14 atoms are shown in Figs. 1–3, respectively. Strontium clusters possess various isomer forms whose number grows dramatically with cluster size. In Figs. 1–3, we present the lowest energy conﬁgurations optimized with the B3PW91 functional. The interatomic distances are given in ˚ Angstroms. The label above each cluster image indicates the point symmetry group of the cluster. For the determination of symmetry of diﬀerent clusters, certain constraints were used. The interatomic distances and the angles were considered as equal within the tolerance of 0.07 ˚ A and 1◦ , respectively. The constraints on the distances and on the angles roughly correspond to the error in the total energy of the system, which in our case is about 10−5 a.u. If the deviation of some distances (or angles) was greater than the chosen constraint value, and the cluster is topologically close to a structure with higher symmetry, we indicate the higher symmetry in brackets (see Sr+ 12 , 2+ Sr2+ 12 and Sr13 in Figs. 2 and 3). Figure 1 shows that neutral strontium clusters form compact structures, maximizing the coordination number. The Sr2 dimer is weakly bound, possessing the dissociation energy 0.139 eV, bond length 4.659 ˚ A, and the harmonic vibrational frequency 45.33 cm−1 , which is in good agreement with the experimental results of Ref. 63, where the values 0.131 ± 0.004 eV for the dissociation energy, 4.45 ˚ A for the bond length and 40.32 ± 0.02 cm−1 for the vibrational frequency have been reported. The lowest energy state for Sr3 is an equilateral triangle, and for Sr4 a regular tetrahedron. As we discuss below, the Sr4 cluster is relatively more stable and compact, as compared to the neighboring clusters. The Sr5 cluster has the structure of a slightly elongated triangular bipyramid. These structures are in good agreement with the results obtained with the use of an empirical potential of the Murrell–Mottram type.39 For larger strontium clusters, in particular with N = 6, 9, 10, we found that the lowest-energy isomers optimized within the B3PW91/SDD(6D,10F) DFT method are considerably diﬀerent from the Murrell–Mottram structures. Although the Murrell–Mottram

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Fig. 1. Optimized geometries of neutral strontium clusters Sr2 –Sr14 calculated in the B3PW91/SDD(6D,10F) approximation. The interatomic distances are given in ˚ Angstroms. The label above each cluster image indicates the point symmetry group of the cluster.

method reproduces most stable structures with the closed geometry shells, it obviously fails to take into account the inﬂuence of electronic shell eﬀects on cluster geometry. The Sr6 consists of three pyramids connected by their faces, Sr7 is a pentagonal bipyramid, and Sr8 is a capped pentagonal bipyramid. These geometrical structures are in good agreement with the results of Ref. 44. For N ≤ 8, the lowest-energy strontium isomers are the same as for magnesium clusters (see, e.g., Refs. 26–28 and 64). Figure 1 demonstrates that the bonds between the highest coordinated atoms for small neutral strontium clusters are the shortest, and therefore the delocalization of valence electrons is more pronounced in the vicinity of such bonds.43 Examples of such bonds are the short bond between the base atoms in Sr5 , the bond sharing the pyramids in Sr6 , and the bond joining the apex atoms in Sr7 and Sr8 . A similar behavior has been observed for magnesium clusters.26,29 It is worth noting that the optimized geometry structures for small neutral strontium and magnesium clusters diﬀer signiﬁcantly from those obtained for sodium clusters (see, e.g., Refs. 25, 65 and 66, and references therein). Thus, the optimized sodium clusters with N ≤ 6 have a

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plane structure. For N a6 , both plane and spatial isomers with very close total energies exist. The planar behavior of small sodium clusters has been explained as a result of the successive ﬁlling of the 1σ and 1π symmetry orbitals by delocalized valence electrons,65 in accord with deformed jellium model calculations.22 The geometry structure of small alkali metal clusters is mainly deﬁned by the closure of electronic shells of the valence electrons. In contrast to the small sodium clusters, the strontium and magnesium clusters are tri-dimensional already at N = 4, forming structures nearly the same as the van der Waals bonded clusters. It has been shown that the evolution of the alkaline-earth metal clusters properties is governed by an interplay between the electronic and geometry shell closures.26,30 In the size region N ≥ 9, the optimized geometry structures for strontium and magnesium clusters become diﬀerent. Thus, the Mg9 cluster has the structure of a tricapped trigonal prism. The formation of the trigonal prism core plays an important role in the magnesium cluster growth process and is closely connected with the future formation of elements of the hexagonal closest-packing (hcp) lattice of bulk magnesium.26 For small strontium clusters, however, a motif based on the icosahedral structure dominates the cluster growth. It was shown in Ref. 43 that the icosahedral growth of strontium clusters is induced by the sp–d hybridization. Icosahedral growth in the cluster size region 8 ≤ N ≤ 13 results in the successive capping of the pentagonal bipyramid core (structure of the Sr7 cluster). Thus, the Sr8 cluster is a capped pentagonal bipyramid and Sr9 has the structure of a bicapped pentagonal bipyramid. We have found, however, that for N = 10 the icosahedral isomer structure is not the lowest-enegry one. The most stable structure of Sr10 is a tricapped trigonal prism with an additional atom in the center. This structure appears to be new and more bounded compared to those of Refs. 39 and 42–44. As we show below, icosahedral growth violation for Sr10 is a result of the strong inﬂuence of electronic shell eﬀects on structural rearrangements. The conﬁguration of Sr11 is a derivative of the structure of Sr10 . It can be obtained by the addition of an atom to one of the convex quadrangular faces of Sr10 and by allowing for relaxation. Starting from Sr12 , an icosahedral growth mode is restored, which leads to a regular icosahedron structure for Sr13 . The lowest-energy structure for Sr14 is a capped icosahedron. Figures 2 and 3 show the optimized geometries of singly and doubly 2+ charged cationic strontium clusters, respectively. The Sr+ 2 and Sr2 cationic dimers are more bound and compact as compared to the neutral dimer. For + the singly charged strontium dimer Sr+ 2 , the dissociation energy D2 = −1 1.104 eV and the harmonic vibrational frequency ω = 75 cm are in good agreement with the experimental values D2+ = 1.092 ± 0.016 eV and ω = 86 ± 3 cm−1 reported in Ref. 45.

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+ The same as in Fig. 1, but for singly charged strontium clusters Sr+ 2 –Sr14 .

The ground state geometries of the cationic strontium clusters are not very diﬀerent from those obtained for neutral parent clusters. Exceptions are observed for small cationic strontium clusters with N ≤ 4, where the 2+ 2+ equilibrium geometries of Sr+ 3 , Sr3 and Sr4 are linear chains due to the Coulombic repulsion. A single ionization of the Sr4 cluster with a structure of a regular tetrahedron Td lowers its symmetry to D2d for Sr+ 4. An interesting situation occurs for N = 5. A single and double ionization of the Sr5 cluster does not change its D3h symmetry, while changing the cluster’s shape. It has already been noticed that the bonds between the highest coordinated atoms for small neutral strontium clusters are the shortest. Thus, the bond length between the base atoms of Sr5 is considerably shorter than that between the base and the apex atoms. Ionization of the Sr5 cluster leads to a gradual increase in the bond length between the base atoms. Therefore, an elongated triangular bipyramid structure for the Sr5 isomer transfers to a regular triangular bipyramid structure for Sr+ 5 and ﬁnally to an oblate triangular bipyramid structure for Sr2+ 5 . Single and double ionization of the Sr7 cluster lowers its point symmetry group from D5h to C2v . Double ionization of the Sr8 cluster raises its symmetry from Cs to the D6h point symmetry group for the Sr2+ 8 cluster. The formation of

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Fig. 3.

2+ The same as in Fig. 1, but for doubly charged strontium clusters Sr2+ 2 –Sr14 .

a highly symmetric hexagonal bipyramid structure for the Sr2+ 8 cluster is a result of electronic shell closure. It has already been noticed that closure of electronic shells for Sr10 results in violation of the icosahedral growth of 2+ small neutral strontium clusters. However, charged Sr+ 10 and Sr10 clusters possess open electronic shells and as a result there is no violation of icosahedral growth. The doubly charged Sr2+ 11 cluster possesses closed electronic shell structure and higher symmetry D4d when compared with the open shell neutral Sr11 cluster (C4v ). In Fig. 4, we present the average bonding distance, Rav , calculated within the B3PW91 functional for neutral, singly and doubly charged Sr clusters. When calculating the average bonding distance in a cluster, only interatomic distances smaller than 5.0 ˚ A have been taken into account. The bulk limit for the strontium fcc lattice24 is indicated in the ﬁgure by the horizontal dashed line. Figure 4 demonstrates that the dependence of the average bonding distance on cluster size has an essentially non-monotonous oscillatory behavior. For the weakly bounded Sr2 dimer, the bonding distance is equal to 4.659 ˚ A, which is in good agreement with the experimental result 4.45 ˚ A of Ref. 63.

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Fig. 4. Average bonding distance as a function of cluster size for neutral, singly and doubly charged strontium clusters. The open squares present the results of the work by Wang et al.44

The appearance of the minima in the size dependence of the average bonding distance shows that Sr4 , Sr7 , and Sr10 clusters (8, 14 and 20 valence electrons, respectively) are more tightly packed than their neighbors. This behavior can be interpreted by the inﬂuence of electronic shell eﬀects on the geometrical structure of strontium clusters. It supports the conclusion of Ref. 67 that electronic shell eﬀects can enhance the stability of geometric structures resulting from dense ionic packing. Figure 4 demonstrates the good agreement of our results with the dependence of Rav on N calculated in Ref. 44 for neutral Sr clusters, except for the case of Sr10 . It has already been noted that the Sr10 cluster violates icosahedral growth due to the strong inﬂuence of electronic shell eﬀects. Therefore, this structure is more bounded when compared with that of Ref. 44. Evolution of the average bonding distance with cluster size diﬀers for alkaline-earth clusters from that for clusters of alkali metals. For neutral alkali metal clusters, one can see odd–even oscillations of Rav atop its systematic growth and approaching the bulk limit.25 These features have a quantum origin and arise due to the spin coupling of the delocalized valence electrons. For alkaline-earth metal clusters, the average bonding distance depends on their size non-monotonically. Such an irregular behavior is induced both by the closure of electronic shells of the delocalized electrons and by the structural rearrangements.26

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The manifestation of the magic numbers in the dependence of the average bonding distance on cluster size coinciding with the deformed jellium model magic numbers does not imply, however, the rapid metallization of strontium clusters. To investigate the transition of van der Waals to metal bonding in strontium clusters, it is necessary to explore the evolution of their electronic properties. Below we perform such analysis in detail. The ﬁlled circles and triangles in Fig. 4 represent the average bonding distance as a function of cluster size calculated for singly and doubly charged strontium clusters, respectively. Figure 4 demonstrates the essential diﬀerence in the cluster size dependence of Rav for the ionized and neutral strontium clusters with N ≤ 8. The small singly charged strontium clusters are more compact in comparison with the corresponding neutral clusters. For example, for Sr+ 2 the bonding distance is equal to 4.247 ˚ A, which is much less than in the case of Sr2 . This phenomenon has a simple physical explanation: the removed electron is taken from the antibonding orbital. The fact that cationic strontium clusters are more stable than the parent neutral and anionic clusters was already noted in Refs. 26 and 31. Within the size range N ≥ 7, the average bonding distances for singly charged and neutral strontium clusters behave similarly. The absolute value of Rav for singly charged clusters is slightly smaller in this region of N . Figure 4 demonstrates that singly charged Sr+ 5, + Sr+ and Sr clusters are more tightly packed than their neighbors. Such an 7 9 alteration in packing after ionization occurs due to the manifestation of electronic shell eﬀects and has already been observed for cationic magnesium26 and strontium51 clusters. and Sr2+ clusters are Figure 4 shows that small doubly charged Sr2+ 2 3 more compact in comparison with the corresponding neutral and singly ˚ charged clusters. Thus, for Sr2+ 2 the bonding distance is equal to 4.106 A. However, in the size range 4 ≤ N ≤ 10 doubly charged clusters are less compact. This fact can be explained by the inﬂuence of Coulombic repulsion forces on cluster structure. It is known that small doubly charged strontium clusters are metastable within the size range N ≤ 7 and can decay via Coulombic ﬁssion if they have enough energy to overcome the ﬁssion barrier.51 3.2. Binding energy per atom for SrN , Sr+ N and Sr2+ N clusters The binding energy per atom for neutral, singly and doubly charged atomic clusters is deﬁned as follows: Eb /N = E1 − EN /N,

(1)

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+ /N, Eb+ /N = (N − 1)E1 + E1+ − EN 2+ + 2+ Eb /N = (N − 2)E1 + 2E1 − EN /N,

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(2) (3)

+ 2+ where EN , EN and EN are the energies of a neutral, singly and doubly charged N -particle atomic cluster, respectively. E1 and E1+ are the energies of a single atom and an ion. Figure 5 shows the dependence of the binding energy per atom for the most stable neutral, singly and doubly charged Sr clusters as a function of cluster size. The ﬁlled squares, circles and triangles represent binding energies Eb /N , Eb+ /N and Eb2+ /N obtained within the B3PW91/SDD method. The crosses in Fig. 5 present the results of the work by Wang et al.44 For small Sr clusters, the binding energy per atom increases steadily with the cluster size. The peculiarity in the size dependence of Eb /N at N = 4, 7 and 10 corresponds to the most stable conﬁgurations of neutral Sr clusters. The same magic numbers have also been obtained from the analysis of binding energies of small neutral Mg26 clusters. The analysis of the second diﬀerences of the binding energy, ∆2 EN = EN +1 − 2EN + EN −1 , points to a relative stability of the Sr13 cluster, in addition to the magic clusters Sr4 , Sr7 and Sr10 (see Fig. 6). The principal magic numbers 7 and 13 can be explained by atomic shell closing

Fig. 5. Binding energy per atom for the most stable neutral (ﬁlled squares), singly charged (ﬁlled circles) and doubly charged (ﬁlled triangles) strontium clusters as a function of cluster size. The crosses present the results of the work by Wang et al.44

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Fig. 6. Second diﬀerences of total energy for neutral (ﬁlled squares), ∆2 EN = EN+1 − + + + + = EN+1 − 2EN + EN−1 , stron2EN + EN−1 , and singly charged (ﬁlled circles), ∆2 EN tium clusters.

eﬀects. Indeed, the enhanced stability of Sr7 and Sr13 clusters arises when their ionic structure is highly symmetric and corresponds to the icosahedral type of packing. This icosahedral growth sequence for metal clusters has also been seen for clusters of Ba,68 which exhibit the nonmetalto-metal transition with increasing cluster size. It is important to note that for alkaline-earth metal clusters there is strong competition between geometrical and electronic shell closures.26,51 The electronic conﬁguration of the Sr atom is [Kr]5s2 , which means that there are two valence electrons per atom. Accounting for the semi-core 4p electrons of strontium increases the absolute value of the binding energy by about 10–20% although it does not change the general qualitative trend in the evolution of properties of small strontium clusters.43 The most stable magic clusters Sr4 , Sr7 and Sr10 possess Nel = 8, 14 and 20 valence electrons respectively, which is in agreement with the deformed jellium model (see, e.g., Refs. 20–23, and references therein as well as the discussion in Refs. 26 and 51). The ﬁlled circles in Fig. 5 show the binding energy for singly charged strontium clusters. The singly charged small strontium clusters are more stable towards decay in comparison with neutral clusters. This phenomenon

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has a simple physical explanation: the removed electron is taken from the antibonding orbital, and thus small cationic strontium clusters are more strongly bounded. A similar eﬀect was discussed for cationic magnesium clusters in Ref. 26. The local maxima in the size dependence of the binding energy Eb+ /N + + for the Sr+ 5 , Sr7 and Sr11 clusters indicate their enhanced stability. This result is in very good agreement with experimental data on cation clusters, which show 5, 7, 11 to be magic numbers.46 The analysis of the second diﬀerences of the binding energy (see Fig. 6) also suggests relative stability + of the Sr+ 9 and Sr13 clusters. Figures 5 and 6 clearly demonstrate that the single ionization of small strontium clusters results in alteration of the “electronic” magic numbers. A similar change of the magic number from N = 4 for neutral to N = 5 for cationic magnesium clusters was noticed in our recent work.26 This fact can be explained by the manifestation of shell eﬀects. The singly charged alkaline-earth metal clusters always possess an odd number of valence electrons and, thus, always contain open electronic shells. In this case, the enhanced stability of a singly charged alkaline-earth metal cluster ion arises, when the electronic conﬁguration of the ion has one hole in or an extra electron above the ﬁlled shells.26 Thus, the electronic conﬁguration containing + an extra electron becomes more favorable for Sr+ 5 and Sr11 . Note that magic numbers N = 7 and 13 correspond to the geometry shell closures, and therefore they do not alter due to single ionization. This eﬀect is clearly seen in Fig. 6. The ﬁlled triangles in Fig. 5 present the binding energy per atom, Eb2+ /N , for the doubly charged strontium clusters. Figure 5 demonstrates the fast increase in the value of Eb2+ /N with cluster size. Binding energy 2+ Eb2+ /N possesses a negative value for Sr2+ 2 , and is neglible for Sr3 . This 2+ 2+ fact means that Sr2 and Sr3 clusters are intrinsically unstable towards complete fragmentation, because the ﬁnal state of the system is energetically more favorable in comparison with the initial state of the parent cluster. However, in order to decay the parent cluster has to overcome the Coulombic ﬁssion barrier. The doubly charged strontium clusters remains metastable for N ≤ 7.51 For larger sizes N ≥ 8 doubly charged strontium clusters are intrinsically stable towards the Coulombic ﬁssion. Therefore, the critical appearance size for the doubly charged strontium clusters is equal to Napp = 8.51 It is important to note that we found a drastic alteration of the cluster geometry upon ionization for N = 8. The eﬀect of the alteration of the ionic structure for Sr2+ 8 arises due to the inﬂuence of the electronic shell eﬀects on the cluster geometry. It is very important to take into account such an alteration when calculating the critical size of stability of doubly charged strontium clusters.

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3.3. Electronic structure Let us now consider the density of electronic states (DOS) for Sr-clusters. Figures 7, 8 and 9 demonstrate the evolution of the DOS for neutral, singly and doubly charged strontium clusters, respectively. For the sake of comparison, for each spectrum the zero level of energy is chosen equal to the Fermi level Ef of the corresponding cluster. For the strontium atom, the energy level of the 5s valence electron lies at 1.52 eV below the Fermi level, while unoccupied 5p and 4d levels lie at 1.52 eV and 1.89 eV above the Fermi level, respectively. For the dimer Sr2 valence electrons form one bonding and one antibonding σ-orbital. The HOMO–LUMO gap is relatively high (2.11 eV), and therefore there is no interaction of valence electrons with

Fig. 7. Density of states for neutral strontium clusters. Gaussian broadening of halfwidth 0.15 eV has been used. The zero level of energy is chosen equal to the Fermi level Ef (vertical line).

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Fig. 8.

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The same as in Fig. 7, but for singly charged strontium clusters.

the 4d and 5p states. Figure 7 clearly demonstrates the formation of the electronic shell structure of small strontium clusters in accord with the jellium model. Sharp peaks in the energy spectrum of Sr4 and Sr10 correspond to the closed electronic shells 1s2 1p6 and 1s2 1p6 1d10 2s2 , respectively. The DOS spectra for small neutral strontium clusters are in good agreement with the results of Ref. 43 with the exception of Sr10 . As we discussed above, we found that the most stable Sr10 structure violates icosahedral growth and is more bounded compared to those of Refs. 42–44. The icosahedral growth violation for Sr10 is a result of the strong inﬂuence of electronic shell eﬀects on structural rearrangements. It is seen from Fig. 7 that the DOS spectrum for the Sr10 cluster contains well separated maxima below the Fermi level. These maxima correspond to the closed electronic shells 1s2 1p6 1d10 2s2 .

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Fig. 9.

The same as in Fig. 7, but for doubly charged strontium clusters.

As cluster size increases, the HOMO–LUMO gap decreases, the unoccupied 5p and 4d states shift to the higher binding energy, while the highest occupied molecular orbital shifts towards the lower binding energy. This results in an increase in interaction between valence and unoccupied states and, hence, in an increase in sp–d hybridization. In turn, the sp–d hybridization signiﬁcantly changes electronic states near the HOMO level. The most stable clusters possess local maxima in the size dependence of the HOMO–LUMO gap and therefore for such clusters sp–d hybridization decreases. Therefore, the electronic states of magic clusters are similar to those expected from a jellium model as can be seen from Fig. 7. For charged strontium clusters, the DOS spectra of low-lying valence states are quite similar to those obtained for neutral clusters. The diﬀerence

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can be observed for the states near the HOMO as well as for the unoccupied states. Single and double ionization of the strontium dimer leads to removing valence electrons from the antibonding HOMO orbital, which can be seen in Figs. 8 and 9. The removal of an electron from the antibonding 2+ orbital results in increased stability of Sr+ 2 and Sr2 ions. The most stable + 2+ 2+ conﬁguration of Sr3 , Sr3 and Sr4 is a linear chain. For linear geometry, the successive ﬁlling of σ-orbitals is more favorable energetically. It is seen from Figs. 8 and 9 that this fact has an eﬀect on the DOS spectra of 2+ 2+ Sr+ 3 , Sr3 and Sr4 clusters. As we discussed above, the ionization of the Sr10 magic cluster results in the alteration of the magic number from 10 to 11 due to electronic shell eﬀects. Figure 9 conﬁrms the conclusion about the origin of such an alteration. Indeed, energetically well separated maxima below the Fermi level correspond to the closed electronic conﬁguration 1s2 1p6 1d10 2s2 of the Sr2+ 11 cluster. In conclusion, it is necessary to note that DOS spectra of free electronic states obtained for cationic clusters are more complicated when compared with those calculated for neutral clusters due to the additional Coulombic ﬁeld. The DOS spectra are very sensitive to the cluster isomer structure, and thus can be used for the determination of the geometry of strontium clusters. Recently, a similar approach was used for the identiﬁcation of a speciﬁc icosahedral growth motif of medium-sized sodium clusters.69 Figure 10 presents the DOS spectra calculated for three diﬀerent isomers of the Ih , Cs , and C2v point symmetry groups of the Sr13 cluster. The energies of the two lowest lying states are not aﬀected by the cluster geometry alterations. These states can be associated with the 1s2 and 1p6 levels in accord with the spherical jellium model. However, from Fig. 10 one can see considerable rearrangement and splitting of the states lying in the vicinity and above the Fermi level. The level splitting due to the geometry distortion of the cluster was explored in detail in Refs. 20–22 within the deformed jellium

Fig. 10. Density of states for three diﬀerent isomers of Sr13 . From left to right: isomers of the Ih , Cs , and C2v point symmetry groups, respectively. Gaussian broadening of half-width 0.15 eV has been used. The zero level of energy is chosen equal to the Fermi level Ef (vertical line).

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model. A similar behavior of energy levels is well known for deformed nuclei (see Ref. 70). 4. Conclusion The optimized structures and electronic properties of neutral, singly and doubly charged strontium clusters have been investigated using ab initio theoretical methods based on density-functional theory. We have systematically calculated the optimized geometries of neutral, singly and doubly charged strontium clusters consisting of up to 14 atoms, average bonding distances, electronic shell closures, binding energies per atom, the gap between the highest occupied and the lowest unoccupied molecular orbitals, and spectra of the electronic density of states. We have shown that the sizeevolution of the structural and electronic properties of strontium clusters is governed by an interplay between the electronic and geometry shell closures. The inﬂuence of the electronic shell eﬀects on structural rearrangements can lead to the violation of the icosahedral growth motif of strontium clusters. We have demonstrated that the electronic shell structure of small strontium clusters forms in accord with the jellium model. It is shown that the excessive charge essentially aﬀects the optimized geometry of strontium clusters. Ionization of small strontium clusters results in the alteration of the magic numbers. We have demonstrated that the strong dependence of the DOS spectra on the details of ionic structure allows one to perform a reliable identiﬁcation of the geometry of the strontium clusters. Acknowledgments This work is partially supported by the European Commission within the Network of Excellence project EXCELL, and by INTAS under the grant 03-51-6170. The authors gratefully acknowledge support by the Frankfurt Center for Scientiﬁc Computing. References 1. W. A. de Heer, Rev. Mod. Phys. 65, 611 (1993). 2. M. Brack, Rev. Mod. Phys. 65, 677 (1993). 3. C. Br´echignac and J.-P. Connerade, J. Phys. B: At. Mol. Opt. Phys. 27, 3795 (1994). 4. H. Haberland (ed.), Clusters of Atoms and Molecules, Theory, Experiment and Clusters of Atoms, Springer Series in Chemical Physics, Vol. 52 (Springer Verlag, Berlin, 1994).

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5. U. N¨ aher, S. Bjørnholm, S. Frauendorf, F. Garcias and C. Guet, Phys. Rep. 285, 245 (1997). 6. W. Ekardt (ed.), Metal Clusters (Wiley, New York, 1999). 7. C. Guet, P. Hobza, F. Spiegelman and F. David, (eds.), Atomic Clusters and Nanoparticles, NATO Advanced Study Institute, les Houches Session LXXIII, les Houches, 2000 (EDP Sciences and Springer Verlag, Berlin, 2001). 8. J. Jellinek (ed.), Theory of Atomic and Molecular Clusters. With a Glimpse at Experiments, Springer Series in Cluster Physics (Springer Verlag, Berlin, 1999). 9. K-H. Meiwes-Broer (ed.), Meal Clusters at Surfaces: Structure, Quantum Properties, Physical Chemistry, Springer Series in Cluster Physics (Springer Verlag, Berlin, 1999). 10. J.-P. Connerade and A. V. Solov’yov (eds.), Latest Advances in Atomic Cluster Collisions: Fission, Fusion, Electron, Ion and Photon Impact (Imperial College Press, London, 2004). 11. F. Baletto and R. Ferrando, Rev. Mod. Phys. 77, 371 (2005). 12. J. A. Alonso, Structure and Properties of Atomic Nanoclusters (Imperial College Press, London, 2005). 13. W. D. Knight, K. Clemenger, W. A. de Heer, W. A. Saunders, M. Y. Chou and M. L. Cohen, Phys. Rev. Lett. 52, 2141 (1984). 14. J. M. Eisenberg and W. Greiner, Nuclear Theory. Vol. 1: Collective and Particle Models (North Holland, Amsterdam, 1985). 15. W. Ekardt, Phys. Rev. 29, 1558 (1984). 16. W. Ekardt, Phys. Rev. B 32, 1961 (1985). 17. W. Ekardt and Z. Penzar, Phys. Rev. B 38, 4273 (1988). 18. W. Ekardt and Z. Penzar, Phys. Rev. B 43, 1322 (1991). 19. B. Montag, Th. Hirschmann, J. Meyer, P.-G. Reinhard and M. Brack, Phys. Rev. B 52, 4775 (1995). 20. A. G. Lyalin, S. K. Semenov, A. V. Solov’yov, N. A. Cherepkov and W. Greiner, J. Phys. B 33, 3653 (2000). 21. A. G. Lyalin, S. K. Semenov, A. V. Solov’yov, N. A. Cherepkov, J.-P. Connerade and W. Greiner, J. Chin. Chem. Soc. (Taipei) 48, 419 (2001). 22. A. Matveentzev, A. Lyalin, I. A. Solov’yov, A. V. Solov’yov and W. Greiner, Int. J. Mod. Phys. E 12, 81 (2003). 23. A. G. Lyalin, A. Matveentzev, I. A. Solov’yov, A. V. Solov’yov and W. Greiner, Eur. Phys. J. D 24, 15 (2003). 24. N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, New York, 1976). 25. I. A. Solov’yov, A. V. Solov’yov and W. Greiner, Phys. Rev. A 65, 053203 (2002). 26. A. Lyalin, I. A. Solov’yov, A. V. Solov’yov and W. Greiner, Phys. Rev. A 67, 063203 (2003). 27. P. H. Acioli and J. Jellinek, Phys. Rev. Lett. 89, 213402 (2002). 28. J. Jellinek and P. H. Acioli, J. Phys. Chem. A 106, 10919 (2002). 29. V. Kumar and R. Car, Phys. Rev. B 44, 8243 (1991). 30. J. Akola, K. Rytk¨ onen and M. Manninen, Eur. Phys. J. D 16, 21 (2001).

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NEW DEFORMED SINGLE-PARTICLE SHELL MODEL D. N. POENARU∗ , R. A. GHERGHESCU and I. H. PLONSKI Horia Hulubei National Institute of Physics and Nuclear Engineering (IFIN-HH), P.O. Box MG-6, RO-077125 Bucharest-Magurele, Romania www.theory.nipne.ro ∗ [email protected] A. V. SOLOV’YOV† and W. GREINER Frankfurt Institute for Advanced Studies, Max-von-Laue Str. 1, 60438 Frankfurt am Main, Germany † [email protected] We present a new single-particle shell model, derived by solving the Schr¨ odinger equation for a semi-spheroidal potential well. It may be used to study atomic clusters deposited on a planar surface. By assuming opacity of the surface only the negative parity states of the Z(z) component of the wave function are allowed, so that new magic numbers are obtained. The maximum degeneracy is reached at a superdeformed semi-spheroidal prolate shape whose magic numbers are identical to those obtained at the spherical shape of the spheroidal harmonic oscillator. Keywords: Atomic clusters on surface; solutions of wave equations; single-particle levels; theory of electronic structure.

1. Introduction The nanostructured coating of surfaces by cluster deposition is a rapidly growing ﬁeld.1 The shapes can be observed by using scanning probe microscopy. The ﬁnal shape of some of them2,3 may be approximated by a semi-spheroid. In this way, we are motivated to investigate how the known properties of the spheroidal harmonic oscillator are changed when we consider just one half of the spheroidal shape. In all studies using a harmonic oscillator published since 1955, the maximum degeneracy of the quantum 128

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states was reached for a spherical shape, explaining the high stability of the doubly magic nuclei or of the metal clusters with spherical closed shells. One would expect to arrive at a similar conclusion for the semi-spheroidal shape: maximum degeneracy at zero deformation (semi-sphere). However, we arrived at an unexpected result: the maximum stability of the semispheroidal quantum harmonic oscillator occurs at a superdeformed prolate shape (semiaxes ratio a/c = 1/2). For several decades, the spheroidal harmonic oscillator has been used in various branches of physics, e.g., the Nilsson model,4 which is very successful in nuclear physics. Its variants for free atomic clusters5–7 are also widespread. Major spherical-shells N = 2, 8, 20, 40, 58 and 92 have been found5 in the mass spectra of sodium clusters of N atoms per cluster, and Clemenger’s shell model6 was able to explain this sequence of spherical magic numbers. It is not clear how these numbers for clusters deposited on diﬀerent surfaces change. We think that a new single-particle shell model based on the semi-spheroidal harmonic oscillator may be useful in this respect. In order to make a comparison with a well-known model, we shall ﬁrst consider the analytical relationships for the energy levels of the spheroidal harmonic oscillator. 2. Surface parametrization K. L. Clemenger introduced the deformation δ by expressing the dimensionless two semiaxes, in units of the radius of a sphere with the same volume, A for Na,8,9 as R0 = rs N 1/3 , where rs is the Wigner–Seitz radius, 2.117 ˚ 1/3 2/3 2+δ 2−δ and c = . (1) a= 2+δ 2−δ We use dimensionless cylindrical coordinates ρ and z. The volume of the spheroid is Vol = 4πa2 cR03 /3. Inspired by experimental shapes, we consider a semi-spheroidal cluster deposited on a surface with the z-axis perpendicular to the surface and the ρ-axis in the surface plane (see Fig. 1). The surface equation is given by (a/c)2 (c2 − z 2 ), z ≥ 0, 2 (2) ρ = 0, z < 0. The radius of the semi-sphere obtained for the deformation δ = 0 is Rs , given by volume conservation, leading to Rs = 21/3 R0 . For the semispheroidal shape, we shall give ρ, z, a, c in units of Rs instead of R0 . According to volume conservation, a2 cRs3 /2 = R03 so that a2 c = 1. Other kinds of shapes obtained from a spheroid by removing less or more than half (as in the liquid drop calculations10 ) will be considered in the future.

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Fig. 1. Semi-spheroidal superdeformed shape of a prolate atomic cluster. Semiaxes ratio a/c = 1/2.

3. Wave equation and the eigenvectors For spheroidal shapes, generated by a potential with cylindrical symmetry, M R02 2 2 (ω⊥ ρ + ωz2 z 2 ). (3) 2 S. G. Nilsson4 introduced the deformation ε by expressing the two deformation dependent frequencies using ε 2ε ω⊥ = ω0 (ε) 1 + , ωz = ω0 (ε) 1 − . (4) 3 3 V =

One can separate the variables in the Schr¨ odinger equation HΨ = EΨ. The eigenvalues of the three-dimensional Hamiltonian are 1 E = ω⊥ (n⊥ + 1) + ωz nz + 2

(5)

(6)

or11 in units of ω00

E ω0 ε 1 2ε + nz + = = 0 (n⊥ + 1) 1 + 1− , ω00 ω0 3 2 3

(7)

where n⊥ + nz = n, the main quantum number, which is an integer, so that the levels can be labeled by two quantum numbers n⊥ , n in which for every n = 0, 1, 2, . . . , one has n⊥ = 0, 1, 2, . . . , n.

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For the same kind of spheroidal equipotential surfaces, the states of the valence electrons were found6 by using an eﬀective single-particle Hamiltonian with a potential 2/3 4/3 2−δ M ω02 R02 2 2 + δ 2 ρ +z . (8) V = 2 2−δ 2+δ In order to get analytical solutions, we shall neglect an additional term proportional to (l2 − l2 n ). Its component along the symmetry axis at δ = 0 is plotted in Fig. 2. The new potential well we have to consider for the semi-spheroidal equipotential surface is shown in Fig. 3. The potential along the symmetry

z5=5.5 z4=4.5 z3=3.5 z2=2.5 z1=1.5 z0=0.5

0 Fig. 2. Harmonic oscillator potential V = V (ξ), the wave functions Znz = Znz (ξ) and the corresponding contributions to theptotal energy levels z nz = Enz /ωz = (nz + 1/2) for spherical shapes, δ = 0. ξ = zR0 / /M ωz .

z5=5.5

nz=5

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nz=1 0

Fig. 3. Semi-spherical harmonic oscillator potential V = V (ξ), the wave functions Znz = Znz (ξ) and the corresponding contributions to the total energyplevels z nz = Enz /ωz = (nz + 1/2) for semi-spherical shapes, δ = 0. ξ = zR0 / /M ωz . Only negative parity states are retained.

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axis, Vz (z), has a wall of an inﬁnitely large height at z = 0, and concerns only positive values of z, implying opacity of the surface: ∞, z ≤ 0, (9) Vz = M Rs2 ωz2 z 2 /2, z > 0. The energy spectrum {i } is discrete, since we consider inﬁnite-depth potential wells. For the spheroidal harmonic oscillator (SHO), the wave function12,13 may be written as Ψ(η, ξ, ϕ) = ψnmr (η)Φm (ϕ)Znz (ξ),

(10)

where each component of the wave function is orthonormalized, so that √ (11) Φm (ϕ) = eimϕ / 2π, ψ(η) =

Nnmr η |m|/2 e−η/2 L|m| nr (η),

Nnmr

=

2nr ! α⊥ (nr + |m|)!

1/2 ,

(12)

with η = R02 ρ2 /α2⊥ and the quantum numbers m = (n⊥ − 2i) with i = 0, 1, . . . up to (n⊥ − 1)/2 for an odd n⊥ or to (n⊥ − 2)/2 for an m even n

⊥ . Ln (x) is the associated Laguerre polynomial and the constant α⊥ = /M ω⊥ has the dimension of a length: 2 1 Znz (ξ) = Nnz e−ξ /2 Hnz (ξ), Nnz = , (13) √ n 1/2 (αz π2 z nz !)

/M ωz , and the main quantum number where ξ = R0 z/αz , αz = n = n⊥ + nz has integer values 0, 1, 2, . . . . The parity of the Hermite polynomials Hnz (ξ) is given by (−1)nz , meaning that the even order Hermite polynomials are even functions H2nz (−ξ) = H2nz (ξ) and the odd order Hermite polynomials are odd functions H2nz +1 (−ξ) = −H2nz +1 (ξ). For the semi-spheroidal case, the wave functions should vanish at the origin, where the potential wall is inﬁnitely high [see Eq. (9)], so only negative parity Hermite polynomials (nz odd) should be taken into consideration. The orthonormalization condition of the Zns z component of the wave function becomes +∞ Zns z (z)Zns z (z)dz = δnz nz (14) 0

with nz = 1, 3, 5, . . . , n for odd n and nz = 1, 3, 5, . . . , n − 1 for even n. Consequently, √ 2 1 , (15) Zns z (ξ) = 2Nnz e−ξ /2 Hnz (ξ), Nnz = √ n 1/2 (αz π2 z nz !) √ that is, the normalization factor is 2 times the preceding one.

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4. Eigenvalues and magic numbers The eigenvalues are given by Eq. (6). Hence, for a SHO in units of ω0 , = E/(ω0 ), they can be expressed as a function of the deformation δ as 2 n 1 3 = + δ n + − n + . (16) ⊥ 2 2 4 (2 − δ)1/3 (2 + δ)2/3 In Fig. 4, for a prolate spheroid, δ > 0, at n⊥ = 0 the energy level decreases with deformation except for n = 0, but when n⊥ = n it increases. For a semi-spheroidal harmonic oscillator (SSHO), Eq. (16) from the SHO, in units of ω0 is still valid, but one should only allow the values of n and n⊥ for which nz = n − n⊥ ≥ 1 are odd numbers. The shell gap at δ = 0 for an atomic cluster14 is given by −2 2 13.72 eV ˚ A t ω0 (N ) = . (17) 1+ rs Rs rs N 1/3

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(spheroidal deformation) Fig. 4. Energy levels of a spheroidal harmonic oscillator in units of ω0 versus the deformation δ. Each level is labeled by n, n⊥ quantum numbers shown on the right-hand side. Six major shells are considered.

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Since we consider solely monovalent elements, N in this equation is the number of atoms, and t denotes the electronic spill-out for the neutral cluster.14 We start with the SHO. Each energy level, labeled n⊥ , n, may accommo date 2n⊥ + 2 particles. One has 2 nn⊥ =0 (n⊥ + 1) = (n + 1)(n + 2) atoms in a completely ﬁlled shell characterized by n, and the total number of states n of the low-lying n + 1 shells is n=0 (n + 1)(n + 2) = (n + 1)(n + 2)(n + 3)/3, leading to the magic numbers 2, 8, 20, 40, 70, 112, 168, . . . for a spherical shape. As shown in Fig. 4, besides the important degeneracy at a spherical shape (δ = 0), one also has degeneracies at some superdeformed shapes, e.g., for prolate shapes at the ratio c/a = (2 + δ)/(2 − δ) = 2, i.e. δ = 2/3. More details may be found in Fig. 6. The ﬁrst four shells can reproduce the experimental magic numbers mentioned above; in order to describe the other shells, Clemenger introduced the term proportional to (l2 − l2 n ).

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2 1 -0.6

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(spheroidal deformation)

Fig. 5. Energy levels of the semi-spheroidal harmonic oscillator in units of ω0 versus the deformation δ. Each level is labeled by n, n⊥ quantum numbers shown on the righthand side. Eight major shells are considered. Only odd values of nz = n−n⊥ = 1, 3, 5, . . . are allowed.

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Fig. 6. Variation of shell corrections with N for spheroidal Na clusters. Minima for magic numbers. Top: δ = 0. SHO spherical magic numbers: (n + 1)(n + 2)(n + 3)/3 = 2, 8, 20, 40, 70, 112, 168, . . . . Oblate and prolate shapes are considered on the left-hand and right-hand sides, respectively.

Now in the case of SSHO, from the energy levels given in Fig. 4 we have to select only those corresponding to the condition of odd nz = n − n⊥ = 1, 3, 5, . . . . In this way, the former lowest level with n = 0, n⊥ = 0 should be excluded. From the two levels with n = 1, we can retain the level with n⊥ = 0, i.e. nz = 1. This will be the lowest level for the semi-spherical harmonic oscillator and will accommodate 2n⊥ + 2 = 2 atoms. From the three levels with n = 2 only the one with nz = n⊥ = 1 and 2n⊥ + 2 = 4 degeneracy is retained such that the ﬁrst two magic numbers at spherical shape (δ = 0) are now 2 followed by 6, etc. Some deformed magic numbers may be found at the positions of minima in Fig. 7.

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20 1.5

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Fig. 7. Variation of shell corrections with N for semi-spheroidal Na clusters. Top: δ = 0. The semi-spherical magic numbers are identical to those obtained at the oblate spheroidal superdeformed shape: 2, 6, 14, 26, 44, 68, 100, 140, . . . . For a prolate superdeformed (δ = 2/3) shape the magic numbers are identical to those obtained at the spherical shape.

Each level, labeled n⊥ , n, may accommodate 2n⊥ + 2 particles. When n is an odd number, one should only have even n⊥ in order to select the odd nz = n−n⊥ . The contribution of the shells with odd n to the semi-spherical magic numbers will be n−1

(n + 1)2 , (18) (2n⊥ + 2) = 2 neven =0 ⊥

leading to the sequence 2, 8, 18 for n = 1, 3, 5. The contribution of the shells with even n to the semi-spherical magic numbers will be n−1

n(n + 2) , (19) (2n⊥ + 2) = 2 odd n⊥ =1

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which gives the sequence 4, 12, 24 for n = 2, 4, 6. This should be interlaced with the preceding one so that the magic numbers will be 2, 2 + 4 = 6, 6 + 8 = 14, 14 + 12 = 26, 26 + 18 = 44, 44 + 24 = 68, as shown in Fig. 5. The shell correction energy, δU ,15 in Fig. 7 shows minima at the oblate and prolate magic numbers. The striking result is that the maximum degeneracy is obtained at a superdeformed prolate shape (δs = 2/3). The magic numbers are those of spherical shape (δ = 0). This is a consequence of the fact that the dependence on n and n⊥ of eigenvalues in Eq. (16) for δ = 0 is given simply by n with all possible combinations of n⊥ and nz , while for δs = 2/3 one has n + n⊥ with only odd quantum numbers nz = 1, 3, . . . . In both cases, the degeneracy associated to an energy level labeled n, n⊥ is 2n⊥ + 2. In a similar way, at δ = −1 we compare 3n − 2n⊥ with the semispheroidal 6n − 2n⊥ at δs = −0.4, and at δ = −2/3 the term 2n − n⊥ with n for the semi-sphere (δs = 0). In conclusion, we expect that the increased stability of the prolate superdeformed semi-spheroidal shapes will be experimentally observed. Acknowledgments This work was partly supported by Deutsche Forschungsgemeinschaft, Bonn, and by the Ministry of Education and Research, Bucharest. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

J. V. Barth, G. Costantini and K. Kern, Nature 437, 671 (2005). K. Seeger and R. E. Palmer, Appl. Phys. Lett. 74, 1627 (1999). B. Bonanni and S. Cannistraro, J. Nanotech. Online 1, 1 (2005). S. G. Nilsson, Dan. Mat. Fys. Medd. 29 (1955). W. D. Knight, K. Clemenger, W. A. de Heer, W. A. Saunders, M. Y. Chou and M. L. Cohen, Phys. Rev. Lett. 52, 2141 (1984). K. L. Clemenger, Phys. Rev. B 32, 1359 (1985). S. M. Reimann, M. Brack and K. Hansen, Z. Phys. D 28, 235 (1993). M. Brack, Phys. Rev. B 39, 3533 (1989). C. Yannouleas and U. Landman, Phys. Rev. B 51, 1902 (1995). V. V. Semenikhina, A. G. Lyalin, A. V. Solov’yov and W. Greiner, Zh. Eksp. Teor. Fiz. 133, 781 (2008) (in Russian); JETP 106, 678 (2008). D. N. Poenaru and I. H. Plonski, in Nuclear Decay Modes (Institute of Physics Publishing, Bristol, UK, 1996), pp. 433–486. A. J. Rassey, Phys. Rev. 109, 949 (1958). D. Vautherin, Phys. Rev. C 7, 296 (1973). K. L. Clemenger, Ph.D. thesis, University of California, Berkeley, 1985. V. M. Strutinsky, Nucl. Phys. A 95, 420 (1967).

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Electron, Photon and Ion Cluster Collisions

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TOP ROTORS IN ELECTRIC FIELDS: INFLUENCE OF THE ASYMMETRY, THE FLEXIBILITY AND THE STRUCTURE OF THE MOLECULES M. ABD EL RAHIM, R. ANTOINE, M. BROYER, P. DUGOURD and D. RAYANE Universit´e de Lyon, Universit´e Lyon 1; CNRS; LASIM UMR 5579, bˆ at. A. Kastler, 43 Bvd. du 11 novembre 1918, 69622 Villeurbanne, France We report on electric deﬂection experiments of di-substituted benzene derivative molecules [para, meta and ortho amino-benzonitrile (PABN, MABN and OABN) and para and meta dimethyl-amino-benzonitrile (PDMABN and MDMABN), and para-ﬂuoroaniline (PFAN)]. They are used as prototypes to study the inﬂuence of the asymmetry and rotation– vibration couplings in deﬂection experiments. Experimental deﬂection proﬁles are compared to results of ab initio calculations in the framework of the rigid rotor Stark eﬀect and of the statistical linear response. Keywords: Molecules; symmetry; ab initio calculations.

1. Introduction Molecular beam deﬂections or manipulations of polar molecules with static ﬁelds are widely used. The response of a molecule to electric or magnetic ﬁelds strongly depends on its rotation motion and on its possible couplings with other molecular motion or potential terms.1–8 In electric deﬂection experiments, two limiting cases are well described: (i) For a symmetric top like rigid rotor, the deﬂection of the molecule is due to the ﬁrst order Stark eﬀect and is deduced from calculation of the energy of the rotational levels of the molecule in an electric ﬁeld. The electric ﬁeld induces a broadening of the molecular beam. (ii) In the limiting case of molecules with strong couplings between rotation and vibration, a statistical orientation of the molecules in the electric ﬁeld, similar to paraelectricity, is observed. In particular, this statistical orientation has been observed for hot molecules1,9 and for ﬂexible systems.10–12 In this case, all the molecules are deﬂected by the same amount. The average dipole does not depend on the initial rotation of the molecule and is described by Debye linear response theory.13,14 141

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However, the transition from the broadening observed for rigid symmetric rotors to the global deﬂection predicted by the linear response is not fully understood. A factor which plays a crucial role in the interaction of a polar molecule in an electric ﬁeld is its degree of asymmetry. Indeed, from a symmetric molecule to an asymmetric molecule in electric ﬁeld, the system becomes non-integrable (due to the loss of invariants in the rotational motion).15 A chaotic motion can then occur on classical trajectories. This behavior may have some consequences for the memory of the rotation in the electric ﬁeld. The eﬀect of chaotic rotational motion can be enhanced by other perturbations such as rotation–vibration couplings. In this contribution, we present electric deﬂections of di-substituted benzene derivative molecules. We chose the structural isomers of aminobenzonitrile (ABN) molecules. These molecules have strong permanent dipole moments. By varying the position of substituents, from para to ortho, the asymmetry of the molecule can be changed. By varying the nature of substituents, from amino to dimethyl-amino groups, the ﬂexibility of the molecule can be changed. Thus, they represent models to study the inﬂuence of both the asymmetry and the ﬂexibility of a molecule in electric deﬂection experiments (as summarized in Fig. 1). Furthermore, in the last part

Fig. 1. Amino-benzonitrile and dimethyl-amino-benzonitrile molecules used in this study to probe the inﬂuence of the molecular asymmetry and ﬂexibility. Molecules are para, meta and ortho amino-benzonitrile (PABN, MABN and OABN) and para and meta dimethyl-amino-benzonitrile (PDMABN and MDMABN).

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of this contribution, we use the di-substituted benzene derivative molecules to explore the sensitivity of electric deﬂection experiments, to determine structures for near-symmetric top molecules. 2. Experiments and calculations 2.1. Experimental set-up Our experimental apparatus consists of a molecular beam source coupled to an electric deﬂector and a time-of-ﬂight mass spectrometer. The molecular beam is produced by a laser vaporization source as has been previously described.16,17 The third harmonic of a Nd3+ :YAG laser was used for the ablation of the sample, which was rotated and translated in a screw motion. A short pulse of helium gas, synchronized with the ablation laser, was supplied by a piezo-electric valve. At the exit of the source, the molecules were thermalized in a 5-cm long and 3-mm diameter chamber maintained at room temperature. A thermal molecular beam was produced without supersonic expansion. After collimation, the beam traveled through a 15-cm long electric deﬂector. In the deﬂector, a molecule with an electric dipole moment µ is submitted to an instantaneous force along the Z-axis (Z is the direction of the electric ﬁeld and of its gradient): f = µ

∂ F . ∂Z

(1)

A deﬂection is produced that is proportional to the time-averaged value (µZ ) in the deﬂector of the projection on the Z-axis of the dipole. Beam proﬁles are analyzed 1.025 m after the deﬂector, with a time-of-ﬂight mass spectrometer coupled to a position sensitive detector.18 2.2. Simulations of electric deﬂection proﬁles As described in previous works,10,19 electric deﬂection proﬁles can be simulated in two limiting cases: (i) rigid rotor, and (ii) linear response theory. 2.2.1. Rigid rotors The Hamiltonian for a rigid rotor in an electric ﬁeld may be written [the eﬀect of the electronic polarizability is small and for simplicity is not included in Eq. (2)] H = Hrot + HStark = AJa2 + BJb2 + CJc2 − FZ µg φZg , (2) g=a,b,c

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where a, b, c are the principal axes of inertia of the rotor; Ja , Jb , and Jc are the corresponding components of the angular momentum and A, B, and C are the rotational constants. Z is chosen as the ﬁeld direction in a laboratory ﬁxed coordinate frame, µg are the components of the dipole moment along the principal inertial axes, and φZg is the direction cosine, or projection of the various molecular axes onto the ﬁeld direction Z. For an asymmetric rigid rotor, the eigenvalues of the Hamiltonian H are obtained by numerical diagonalization of the corresponding matrix in the basis of the eigenvectors of the prolate symmetric rotor. Proﬁles of deﬂection are calculated numerically from the derivatives of the energy eigenvalues as a function of the electric ﬁeld, and assuming that the molecules adiabatically enter the electric ﬁeld. The result is a broadening of the molecular beam. This quantum mechanical approach is described in detail in Ref. 17. Simulations are done using rotational constants and dipole moment components given by MP2 calculations.16,17 2.2.2. Linear response When couplings to rotation cannot be neglected, the motion of the molecule is more complicated and no longer described by Eq. (2). In general, the calculation of the average value of the projection of the dipole on the axis of the electric ﬁeld is not possible. However, when the correlation of the dipole on the axis of the electric ﬁeld tends toward zero as the molecules travel t → ∞ (i.e. a loss of memory of the through the deﬂector, µZ (t = 0)µZ (t)− −−→ orientation of the rotational motion) and assuming a canonical distribution before the molecules enter the deﬂector, the average value of the dipole on the axis of the electric ﬁeld µZ is the same for all the molecules and is given by linear response theory13,14 : µ2 F =0 F, (3) µZ = α + 3kT where µ2 F =0 is the average value of the square dipole of the molecule calculated at equilibrium without the electric ﬁeld (at temperature T = 300 K) and α is the static electronic polarizability. In this case, beam proﬁles are globally deﬂected towards the high ﬁeld region with a deﬂection d given by d=

∂F K , µZ 2 mv ∂Z

(4)

where K is a geometrical constant, and m and v are the mass and velocity of the molecule.

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3. Results and discussion 3.1. Asymmetric top molecules in an electric ﬁeld: The eﬀect of collisions on deﬂection proﬁles OABN is an asymmetric top molecule. Asymmetric top rotors in electric ﬁelds are non-integrable systems (due to the loss of invariants in the rotational motion). A chaotic motion can then occur on classical trajectories. A way to probe this chaotic behavior is to look at the evolution of the rotational levels as a function of the electric ﬁeld. Figure 2 shows the electric ﬁeld dependence of the 108th to the 125th energy levels for the OABN molecule for M = 1. This corresponds to 8 ≤ J ≤ 13. This ﬁgure illustrates the typical behavior of asymmetric top molecules in an electric ﬁeld. All the crossings are avoided with diﬀerent energy gaps. In Fig. 2, we see that for F = 0 V/m, the energy spacing between rotational lines is very irregular and that the series of levels are nearly degenerated. On the other hand, for F = 1.2 × 107 V/m, due to the repulsion between the diﬀerent rotational levels, the spacing is much more regular. This sample of levels illustrates the typical pattern of spectra of classically integrable systems (F = 0) as opposed to non-integrable systems (F = 0). In a static electric ﬁeld, the rotational motion of an asymmetric molecule is non-integrable, which may display chaotic behavior. The chaotic behavior can be quantiﬁed by a statistical analysis. From these energy levels, we performed a statistical analysis leading to nearest-neighbor spacing histograms (Brody analysis).20 The Brody analysis shows that OABN is chaotic. A chaotic motion means that any perturbation on the rotational motion will induce

Fig. 2. Electric ﬁeld dependence of the 108th to the 125th energy levels for the OABN molecule (for M = 1).

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a loss of memory of the initial rotation. Perturbations can be induced in particular by interaction between a helium carrier gas or a residual gas and OABN molecules. We performed experiments where we probe this hypothesis by changing the number of collisions between OABN molecules and rare gas atoms in the electric deﬂector. Electric deﬂection experiments in the presence of a large number of collisions were performed on OABN molecules. Figure 3 shows the variation in the intensity of the signal at the maximum of the peak (I/I0 ) as a function of the electric ﬁeld, for OABN molecules, in the presence of collisions. Clearly, a strong dependence on the collision conditions is observed for OABN. In the experiments with the neon leak in the deﬂector or with the helium leak in the source, the broadening of the beam induced by the electric ﬁeld is reduced. In this case, there is a clear disagreement between the experiments and the simulation performed for an isolated rigid rotor. For high electric ﬁelds (F > 6 × 106 V/m), electric deﬂection proﬁles are no longer symmetric; a global deﬂection of the proﬁle toward high electric ﬁeld is observed. The quasi-absence of broadening is due to the fact that with the collisions and the strong chaotic behavior of the

Fig. 3. Relative intensity of the beam proﬁle on the beam axis as a function of the electric ﬁeld F measured for diﬀerent conditions of collisions. The ﬁlled squares correspond to experiments with a pressure inside the electric deﬂector chamber of 2.3 × 10−8 mBar. The open squares correspond to experiments with a neon leak in the electric deﬂector chamber (3.6 × 10−6 mBar). The open circles correspond to experiments with a small constant ﬂow of carrier helium gas in the molecular beam, and the line to the rigid rotor quantum simulations.

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rotational trajectories, the molecules have lost the memory of their initial rotational level. 3.2. Inﬂuence of internal torsions in molecular beam deﬂection experiments In the absence of collisions, the eﬀect of chaotic rotational motion can be enhanced, for example, by rotation–vibration couplings. We performed experiments where we probe the inﬂuence of rotation–vibration couplings on molecular deﬂections, to demonstrate the importance of the eﬀect, and to provide a basis for a better understanding of electric and magnetic deﬂections of complex systems. Figure 4 shows the experimental beam proﬁles recorded for PABN and MDMABN molecules at F = 0 V/m and F = 0.61 × 107 V/m in the electric deﬂector. The beam proﬁle of PABN in the presence of the electric ﬁeld is diﬀerent from that of MDMABN. While an almost symmetric broadening is observed for PABN, a global deﬂection with almost no broadening is observed for MDMABN. Experimental results are compared to simulations for rigid rotors (Sec. 2.2.1) and using the linear response (Sec. 2.2.2). For PABN, a good agreement between experiment and rigid rotor simulations is observed. The deﬂection observed for MDMABN is not reproduced by the Stark eﬀect calculated for the rigid rotor. We interpret this as the signature of rotation–vibration couplings. The molecules are almost all deﬂected by the same amount, and deﬂection proﬁles can be simulated using the linear response voltage across the deﬂector. To summarize, deﬂections of PABN molecules are described with rigid rotor simulations while the deﬂections of MDMABN molecules are described by the linear response theory. The diﬀerences between these molecules are the symmetry and the addition of methyl groups. The chaotic behavior can be quantiﬁed using a Brody statistical analysis.17,20 Results of this analysis are shown in Fig. 4. They indicate that PABN symmetric molecules are a little chaotic, while MDMABN asymmetric molecules are much more chaotic. The introduction of methyl groups in ABN molecules induces some soft vibration modes which are populated at room temperature.16 These modes could induce a strong coupling between the torsion of the N(Me)2 group and the rotation of the whole molecule and therefore could aﬀect the deﬂection. Even if the number of populated vibrational states is low, this coupling may be particularly enhanced in the MDMABN molecules due to the repartition of rotation levels and the strong chaotic behavior of its rotational motion in electric ﬁelds.

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Fig. 4. Top: Experimental beam proﬁles of PABN and MDMABN molecules obtained at F = 0 V/m and F = 0.61 × 107 V/m in the deﬂector. Middle: Simulated beam proﬁles of PABN and MDMABN molecules obtained at F = 0 V/m and F = 0.61 × 107 V/m using the rigid rotor (solid line) and the linear response (dashed line) approaches. Bottom: Nearest-neighbor spacing histograms obtained with the procedure described in Ref. 21, at F = 1.21 × 107 V/m for PABN and MDMABN. S is the spacing between two adjacent levels. The histograms are for the ﬁrst 700 levels (with M = 1).16 A non-chaotic “Poisson type” distribution is obtained for PABN. A chaotic “Wigner type” distribution is obtained for MDMABN.

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3.3. Near-symmetric top molecules: Inﬂuence of the structure in molecular beam deﬂection experiments As shown in the previous section, electric deﬂections of near-symmetric top molecules are well described by rigid rotor simulations. In the last part of this contribution, we explore the sensitivity of electric deﬂection experiments to determine the structure of di-substituted benzene derivative molecules. As an illustration, we performed deﬂection measurements on para-fuoro-aniline PFAN molecules (NH2 –C6 H4 –F), which are nearsymmetric top molecules. In parallel, we calculated optimized geometries for planar (–NH2 in the same plane as ﬂuoro-benzene) and non-planar (–NH2 out of plane of ﬂuoro-benzene) structures at the MP2 level of theory. The non-planar geometry is calculated to be the lowest energy structure (see relative energies in Table 1). Both structures have similar rotational constants; however, strong diﬀerences are obtained for the dipole components. For the planar structure, the dipole moment lies on the molecular axis of the molecule and has only one component (µa = 0). For the non-planar structure, the dipole moment has two components (µa = 0 and µc = 0) due to the out-of-plane position of the polar amine group. The variation in the intensity of the signal at the maximum of the peak (I/I0 ) as a function of the electric ﬁeld, was simulated in the framework of rigid rotor approach (Sec. 2.2.1), for the planar and the non-planar structures, using the dipole values and rotational constants in Table 1. Results of simulations are displayed in Fig. 5 and compared to the experimental data for the PFAN molecule. A good agreement is obtained between the experimental results and the simulation for the lowest-energy structure of Table 1. Relative energies, rotational constants, and calculated electric dipole of PFAN in planar and non-planar geometries obtained at the MP2 level of theory. The calculations were performed with Gaussian 98.21 Rotational constants (cm−1 ) Basis set (Hartree)

A

B

C

Energy

Dipole components (Debye) µa

µb

µc

µ total

PFAN Planar 6–311++G∗∗ −385.888509 0.1862 0.04800 00.3897 −3.34 0.00 0.00 3.34 MP2 PFAN NonPlanar 6–311++G∗∗ −385.8925274 0.1858 0.04802 0.03802 MP2

2.60 0.00 1.14 2.83

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Fig. 5. Relative intensity of the beam proﬁle on the beam axis as a function of the electric ﬁeld F for PFAN molecules. Squares correspond to experimental data and the lines to the rigid rotor quantum simulations performed using dipole values and rotational constants for planar and non-planar structures (see Table 1).

the PFAN molecule (non-planar geometry). The diﬀerence between simulated proﬁles for non-planar and planar structures is signiﬁcant for electric ﬁelds above 3×106 V/m. The planar and non-planar structures have similar components and the diﬀerence mainly comes from the diﬀerent values of µc . The non-zero value of µc for the non-planar structure induces a stronger dispersion of the molecular beam, leading to lower I/I0 ratios in particular for electric ﬁelds higher than 3 × 106 V/m. Thus, due to this sensitivity to out-of-plane dipole components, electric deﬂection experiments may be used to discriminate such structures. 4. Conclusion The electric deﬂection experiments performed on di-substituted benzene derivative molecules have brought out the key factors that inﬂuence the interaction of an isolated polar molecule in an electric ﬁeld. The concerted eﬀects of the chaotic motion of the asymmetric top molecules in an electric ﬁeld and the intermolecular interactions (through collisions, for example) and/or the intramolecular interactions (through rotation–vibration coupling, for example) induce a response for the molecule which is diﬀerent to that expected for a rigid top rotor. This chaotic behavior explains the loss of memory of the initial rotation motion and is a major ingredient for understanding the observed deﬂections

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in many complex systems. These results point out the diﬃculty of measuring a dipole moment by electric deﬂection of a molecular beam for molecules without symmetry axes and also the diﬃculty of manipulating such molecules. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

F. W. Farley and G. M. McClelland, Science 247, 1572 (1990). B. Friedrich and D. R. Herschbach, Comm. At. Mol. Phys. 32, 47 (1995). H. J. Loesch, Annu. Rev. Phys. Chem. 46, 555 (1995). D. H. Parker and R. B. Bernstein, Annu. Rev. Phys. Chem. 40, 561 (1989). M. Broyer et al., C. R. Physique 3, 301 (2002). A. Amirav and G. Navon, Phys. Rev. Lett. 47, 906 (1981). M. B. Knickelbein, J. Chem. Phys. 121, 5281 (2004). X. Xu et al., Phys. Rev. Lett. 95, 237209/1 (2005). F. W. Farley et al., J. Chem. Phys. 88, 1460 (1988). R. Antoine et al., J. Am. Chem. Soc. 124, 6737 (2002). R. Antoine et al., Eur. Phys. J. D 12, 147 (2000). R. Moro et al., Phys. Rev. Lett. 97, 123401 (2006). P. Debye, Polar Molecules (Dover, New York, 1929). J. H. Van Vleck, Phys. Rev. 30, 31 (1927). M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990). R. Antoine et al., J. Phys. Chem. A 110, 10006 (2006). M. Abd El Rahim et al., J. Phys. Chem. A 109, 8507 (2005). M. Abd El Rahim et al., Rev. Sci. Instrum. 75, 5221 (2004). M. Broyer et al., C. R. Physique 3, 301 (2002). T. A. Brody et al., Rev. Mod. Phys. 53, 385 (1981). M. J. Frisch et al., Gaussian, Inc., Pittsburgh PA, 1998.

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ELECTRON SCATTERING ON NEON DROPLETS: SINGLY AND MULTIPLY CHARGED NEON CLUSTERS ∗ ¨ S. DENIFL∗ , F. ZAPPA∗,† , I. MAHR , P. SCHEIER∗ , ∗,§ ¨ O. ECHT and T. D. MARK∗,‡,¶ ∗

Institut f¨ ur Ionenphysik und Angewandte Physik, Leopold Franzens Universit¨ at, Technikerstr. 25, A-6020 Innsbruck, Austria

†

Universidade Est´ acio de S´ a, Rio de Janeiro, Brazil Department of Plasma Physics, Comenius University, SK-84248 Bratislava, Slovak Republic ¶ [email protected]

‡

We have analyzed the stability and fission dynamics of singly and multiply charged neon cluster ions. The critical sizes for the observation of long-lived ions are for charge states 2 and 3 with a measured n2 = 284 and n3 = 656, respectively, a factor 3 to 4 below the predictions of a previously successful liquid-drop model. The preferred fragment ions of fission reactions are surprisingly small and their kinetic energy distributions peak at 200 meV or below. The size of these fragments and their average kinetic energies are much less than those predicted by the liquiddrop model. In contrast, magic numbers observed here for singly charged cluster ions are in accordance with previous observations for the other rare gases. Keywords: Neon; cluster; Coulomb explosion.

1. Introduction The extensive number of spectroscopic studies with doped helium droplets shows impressively the ability of superﬂuid helium droplets to be a perfect matrix for the preparation and study of cold targets.1 In contrast, inelastic electron interaction with doped helium droplets has been a much less studied subject. This is all the more surprising as clusters of biomolecules can be formed from the gas phase by successively embedding single biomolecules §Permanent address: Department of Physics, University of New Hampshire, Durham, NH 03824, USA. ¶Corresponding author.

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into a cold droplet. Moreover, it has been known since recently that electrons can induce eﬃcient DNA damage. This is important because in cells secondary electrons are produced in high abundance by ionizing radiation. Thus, the underlying chemical and physical processes of the inelastic electron interaction with isolated and also solvated biomolecules is of relevance for the investigation of DNA damage by ionizing radiation. We have recently constructed a helium cluster source which was initially used to study in detail the properties of electron impact ionization of pure helium clusters,2 as well as metastable decays of the helium cluster ions produced.3 Recently, we modiﬁed our set-up by adding a pick-up chamber including molecular beam ovens, a pick-up cell and external gas inlets which allow the embedding of various molecules in cold neon and cold superﬂuid helium droplets. Initial studies of the pick-up process have been performed with DNA nucleobases adenine and thymine, both of which are well studied in the gas phase. Several interesting phenomena could be observed, e.g., in contrast to the gas phase situation electron attachment in this environment leads to the production of parent anions for adenine and thymine.4 Moreover, site selectivity in the electron attachment process recently discovered in our laboratory for isolated nucleobases5,6 is preserved in this complex environment and in addition a novel two-step reaction scheme has been proposed to explain characteristic diﬀerences in the attachment spectra. These pick-up experiments have recently been extended by embedding other systems into the helium (e.g., chloroform, valine and fullerenes) and in some cases even in the additional presence of water molecules. In addition to these experiments with helium droplets, we have recently carried out similar experiments with neon droplets. In the course of these studies, we also analyzed the stability and ﬁssion dynamics of singly and multiply charged neon cluster ions,7 including the determination of magic numbers and the determination of critical sizes for the observation of longlived doubly- and triply-charged ions. As will be shown here, the critical size, the size of the ﬁssion fragments, and their average kinetic energies are much less than those predicted by the previously accepted liquid-drop model. Multiply charged atomic clusters are prone to charge separation by Coulomb explosion.8,9 Coulomb explosion occurs after electron or photon ionization of clusters,10–12 collisions with high energy transfer,13 or collisions with highly charged ions.14–17 A novel Auger-like mechanism has been suggested to explain the surprisingly eﬃcient formation of multiply charged van der Waals clusters by electron or photon ionization.18–20 The weak binding between the neutral constituents in van der Waals clusters implies instability for charge states as small as z = 2 unless the cluster contains tens or even hundreds of monomers. A large body of experimental data

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has been collected for these systems; they have been explained satisfactorily within a liquid-drop model.12,21 However, the van der Waals systems investigated so far feature relatively large binding energies and correspondingly small critical sizes, n2 < 100, for doubly charged clusters. Critical sizes have not yet been reported for the most weakly bound systems, namely helium, neon and hydrogen.

2. Experimental details The apparatus used for the present measurements consists of a cluster source, an electron impact source, and a high resolution mass spectrometer. The cluster source, described in detail in Refs. 2 and 3 is mounted on a closed cycle helium cryostat that enables temperatures of the cluster source down to 9.5 K at operating conditions of 20 bar. At a distance of about 1 cm from the nozzle, the cluster beam passes a skimmer of 1 mm diameter before it crosses downstream the electron beam of an ion source. Neutral neon clusters are produced by expanding neat neon from a stagnation chamber at T0 = 40 K and a pressure of typically 8 bar through a pin-hole nozzle of 5 µm diameter into vacuum. Either neon with natural isotopic abundance (purity 99.999%) or 20 Ne enriched to 99.95% are used. Neutral clusters are ionized by electron impact. The ions are extracted by an electric ﬁeld from the ion source and accelerated into a high resolution double focusing mass spectrometer of reversed Nier–Johnson type geometry.22 They pass through the ﬁrst ﬁeld free region, are momentum-analyzed by a magnetic sector ﬁeld, enter a second ﬁeld-free region, pass through a 90◦ electric sector ﬁeld, and are detected by a channeltron type secondary electron multiplier. Metastable (spontaneous) reactions of ions and the kinetic energy distribution of fragment ions (KED) may be recorded by mass-analyzed ion kinetic energy (MIKE) scans.22 However, MIKE scans probe reactions that happen a long time after ionization (within 89 ≤ t ≤ 114 µs for Ne2+ 284 , the smallest observed doubly charged neon cluster). We did not observe here any spontaneous ﬁssion of doubly charged neon cluster ions on this time scale, in agreement with previous studies of doubly charged van der Waals clusters.12,21 We have chosen here an alternative technique recently developed in our laboratory, namely, analysis of the z-deﬂection proﬁle of the ion beam in the ion source.23 The kinetic energy of an ion in the z-direction is proportional to the square of the deﬂection voltage in that direction. The ﬁrst derivative of the ion signal as a function of the deﬂection voltage yields the kinetic energy distribution (KED) after transformation of the energy scale. The ion deﬂection method covers a much earlier time window than the MIKE

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technique and it includes prompt reactions. For example, for Ne2+ 284 the time window spans 0 ≤ t ≤ 6 µs. However, with this technique it is not possible to determine the mass-to-charge ratio of the parent ion. 3. Results Figure 1 shows a mass spectrum of neon cluster ions formed from isotopically natural neon. This mass spectrum is an averaged spectrum over 5 independent, high statistics, mass scans (each overnight). Hence, inﬂuences of ﬂuctuations of the cryostat onto the mass spectrum can be eliminated. The large number of isotopomers for natural neon makes it impossible to resolve individual cluster ions in this mass range. Nevertheless, a series of maxima and minima can be seen and the respective cluster size is determined by division of the mass with 20.26, which corresponds to the isotope enrichment to 13% of 22 Ne. With the exception of only a few, in general, the presently deduced magic numbers (see Table 1) are the same as observed for the other rare gas clusters, where icosahedral shell and subshell closure (see references given in Table 1) are responsible for these abundance anomalies. A high resolution mass spectrum of neon cluster ions, formed by expansion of 20 Ne and ionization at Ee = 120 eV, is shown in Fig. 2. The dominant ion peaks correspond to Ne+ n ; the abscissa has been labeled by the size-to-charge ratio n/z. A series of ions is observed between the main peaks

Fig. 1. Mass spectrum of neon cluster ions (averaged over five runs; see text) formed from isotopically natural neon.

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S. Denifl et al. Table 1. Magic numbers (minima and maxima) of neon cluster ions (present results) and of argon, krypton and xenon cluster ions (previous results). Neon

Argon

Krypton

Xenon

Neon

Maxima 130 146 158 164 177 182 195 202 207

130a 147d 157c 164c 178c 182c 196a,c 202a 207a

130b 147b,c 157b,c 163b,c 182c 196c 202c

Note: a P. Scheier24 ; et al.27

b M.

Argon

Krypton

Xenon

Minima 130c 147c 157c 163c 178c 196c 202c 207c

133 137 149 162 166 180 185 199 204

133a,c 149a,d 162c 166a 179c 183c

132b,c 137b,c 149b,c 160b 166c 179c 185c 199c 204c

Lezius et al.25 ; c O. Echt et al.26 ;

d I.

132c 137c 148c 160c 166c 179c 184c 200c 204c A. Harris

Fig. 2. Mass spectrum of isotopically pure 20 Ne cluster ions. The arrows indicate the smallest observable doubly charged cluster, Ne2+ 287 .

starting at n/z = 143.5; these ions arise from odd-sized Ne2+ n n ≥ 287 (see also inset showing a blow up of the mass spectrum around n/z = 143.5). In contrast, cluster ions formed in an expansion of neon with natural isotopic composition ionized at 120 eV show a large number of isotopomers, which makes it impossible to resolve individual cluster ions. Still, at a mass-tocharge ratio of m/z = 2875 ± 80 (with 20 Ne set to 20 u) we see a stepwise increase in the ion intensity; another step occurs at m/z = 4425 ± 120. The steps have been determined by ﬁtting a sum of power laws shown as a solid gray line. As argued previously,12 these steps are characteristic of

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multiply charged clusters beyond their critical size. They are not observed at Ee = 40 eV, which is an electron energy not suﬃcient to form multiply charged cluster ions. Assigning charge states z = 2 and 3 to the values of m/z given above, we calculate critical sizes n2 = 284 ± 8 and n3 = 656 ± 12, respectively, using an average monomer mass of 20.25 u, which takes into account the enrichment of 22 Ne in the cluster ions.28,29 The n2 value thus deduced agrees with the value determined in the resolved spectrum of isotopically pure 20 Ne. However, the observed critical sizes are much smaller than the values n2,theo = 868 and n3,theo = 2950 computed from a liquiddrop model for neon clusters.12 Although the model is fairly crude, it successfully explains the critical sizes of many atomic and molecular van der Waals and hydrogen-bound clusters and the observed size distributions of their ﬁssion fragments. To gain more insight into the source of this discrepancy, we have measured the z-proﬁles of the ions; they reﬂect the kinetic energy distributions (KED) in the ion source as given in Fig. 3 deduced from the z-proﬁles. The narrow proﬁle in Fig. 3 (top panel) represents, Ne+ formed by ionizing the

Fig. 3.

Kinetic energy distributions (KEDs) of ions derived from the spatial profiles.

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collimated cluster beam (with a 6 meV KED on average, limited by the instrumental resolution) at 40 eV. The KED value does not depend on the electron energy because monomer ions are ionization products of atoms. When neon is present as background gas, the KED is broad because of the random momentum directions of the neutrals. + The low energy KEDs of Ne+ 2 and Ne3 formed at 40 eV by ionizing the collimated cluster beam are much broader than that of the respective monomer ion. The broadening of the low energy peak at 40 eV, below the threshold for eﬃcient formation of multiply charged ions, is due to several factors. We have previously measured the total kinetic energy release (KER) for Ne evaporation from Ne+ n . A value of 2 meV was found for n ≥ 10 for events that occur in the metastable time regime of ≈ 22 µs after ionization.22 The much larger kinetic energies observed in the present work stem from several factors. Firstly, more than one evaporation is likely to occur during the experimental time window which includes t = 0. Secondly, during early times, non-statistical processes including relaxation of long-lived electronic excitations (excitons)30 may occur. The KED becomes bimodal at Ee ≥ 42 eV when formation of multiply charged cluster ions becomes possible. The average KED values of the second peak shown in Fig. 3 are 160 (30), 200 (15), 180, 105, 85, 66, 26 and 5 meV for n = 2, 3, 5, 10, 20, 30, 50 and 100, respectively, where values in parentheses refer to Ee = 40 eV, i.e. for 2 ≤ n ≤ 5 we observe a second peak near 200 meV, whereas for larger ions the peak becomes gradually less pronounced and shifts toward lower energies. For n ≥ 50, the second maximum in the KED is barely visible. It is clear this large, narrowly deﬁned second component is the signature of Coulomb explosion into two charged fragments. In contrast, multifragmentation processes would not lead to such a narrow KED. Moreover, an additional analysis reveals that ∼ 50% of all + + light ﬁssion fragments end up as Ne+ 2 or Ne3 ; another ∼ 20% end up as Ne4 + or Ne5 . Some of these fragments may stem from very small precursor ions by more or less symmetric ﬁssion. On the other hand, the high intensity of large cluster ions in mass spectra recorded at 40 eV and the instability of doubly charged cluster ions below n2 = 284 implies that the ﬁssion fragments arise from a broad range of precursor sizes. Thus, we have to conclude from these results that ﬁssion of these doubly charged Ne cluster ions is extremely asymmetric.

4. Discussion Figure 4 shows the prediction using the liquid-drop model from Ref. 12 for the ﬁssion barrier, the reverse ﬁssion barrier (the total reaction energy in the

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Fig. 4. Predictions calculated from the liquid-drop model12 for Ne2+ n . Left ordinate, solid and dashed lines: Height of the fission barrier and reverse fission barrier, respectively. Right ordinate, dash-dotted line: Size nf of the (smaller) fragment ion.

absence of non-adiabatic eﬀects) and, plotted along the right ordinate, the size of the lighter of the preferred fragment ions, i.e. the one that minimizes the ﬁssion barrier. The vertical arrows in Fig. 4 highlight the discrepancy between the observed and calculated critical size for Ne2+ n (n2 = 284 experimentally observed and n2 = 868 predicted). Two other discrepancies are immediately apparent from Fig. 4. The model predicts a reverse ﬁssion barrier exceeding 0.9 eV for fragment ions from Ne2+ n , n ≤ 284, and a preferred fragment size of 70 for doubly charged parent clusters around 284. Despite incorporating discrete charges (in contrast to the less realistic assumption of the Rayleigh model of a continuous charge distribution) the liquid-drop model used here from Ref. 12 is rather simplistic. The total energy of a multiply charged cluster is written as the sum of a volume, a surface, and a Coulomb term. The ﬁssion barrier is estimated by considering the reverse reaction and determining the point at which the two spherical fragments make contact. Moreover, the model does not consider the formation of a neck which will lower the ﬁssion barrier and the predicted critical size. However, a more reﬁned liquid-drop model,21 and a molecular 31 dynamics simulation of Xe2+ yield nearly the same values for the critn , ical sizes of doubly charged Xe van der Waals clusters in good agreement with experimental values. The model also ignores the dynamics and thermally activated processes. The initial vibrational excitation may be quite large as a result of dimer ion formation, but most of this excess energy will be released by monomer ejection, which proceeds signiﬁcantly faster than

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ﬁssion.31 Furthermore, thermally activated ﬁssion does not signiﬁcantly change the critical size because the ﬁssion barrier increases rapidly with cluster size.21 Another shortcoming of the model is the neglect of coupling of the recoiling fragments with internal (vibrational) modes. Nevertheless, 31 also indicates very ineﬃcient couthe molecular dynamics study of Xe2+ 51 pling with intracluster modes in elemental clusters. 5. Conclusion The critical sizes of doubly and triply charged neon cluster ions observed experimentally here are much smaller than those predicted by a liquiddrop model.12 Furthermore, the small size and low kinetic energy of ﬁssion fragments are at variance with predictions from the liquid-drop model. Although the model is crude and ignores the dynamics, it has successfully explained previous experiments on atomic clusters of heavier inert gases and many molecular clusters. Furthermore, its predictions were found to agree with more reﬁned models applied to xenon and carbon dioxide clusters.21,31 Possible reasons for the failure of the liquid-drop model invoked for neon are quantum eﬀects, and the eﬀect of a solvation shell, which will reduce the separation between the holes over the simpliﬁed estimate that places the holes on the surface of the cluster. Acknowledgments This work was supported in part by the FWF, Wien and the European Commission, Brussels. F.Z. acknowledges support from Brazilian agency CNPq. References 1. J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. 43, 2622 (2004). 2. S. Denifl, M. Stano, A. Stamatovic, P. Scheier and T. M¨ ark, J. Chem. Phys. 124, 054320 (2006). 3. S. Feil, K. Gluch, S. Denifl, F. Zappa, O. Echt, P. Scheier and T. M¨ ark, Int. J. Mass Spectr. 252, 166 (2006). 4. S. Denifl, F. Zappa, I. M¨ ahr, J. Lecointre, M. Probst, T. D. M¨ ark and P. Scheier, Phys. Rev. Lett. 97, 043201 (2006). 5. S. Ptasinska, S. Denifl, P. Scheier, E. Illenberger and T. M¨ ark, Angew. Chem. Int. Ed. 44, 6941 (2005). 6. S. Ptasinska, S. Denifl, V. Grill, T. M¨ ark, E. Illenberger and P. Scheier, Phys. Rev. Lett. 95, 093201 (2005).

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7. I. M¨ ahr, F. Zappa, S. Denifl, D. Kubala, O. Echt, T. M¨ ark and P. Scheier, Phys. Rev. Lett. 98, 023401 (2007). 8. U. N¨ aher, S. Bjornholm, S. Frauendorf, F. Garcias and C. Guet, Phys. Rep. 285, 245 (1997). 9. O. Echt, P. Scheier and T. M¨ ark, C. R. Physique 3, 353 (2002). 10. U. N¨ aher, S. Frank, N. Malinowski, U. Zimmermann and T. P. Martin, Z. Phys. D 31, 191 (1994). 11. K. Sattler, J. M¨ uhlbach, O. Echt, P. Pfau and E. Recknagel, Phys. Rev. Lett. 47, 160 (1981). 12. O. Echt, D. Kreisle, E. Recknagel, J. Saenz, R. Casero and J. Soler, Phys. Rev. A 38, 3236 (1988). 13. Y.-N. Wang, H.-T. Qiu and Z. Miskovic, Phys. Rev. Lett. 85, 1448 (2000). 14. L. Chen, S. Martin, R. Bredy, J. Bernard and J. Desesquelles, Europhys. Lett. 58, 375 (2002). 15. O. Hadjar, P. F¨ oldi, R. Hoekstra, R. Morgenstern and T. Schlath¨ olter, Phys. Rev. Lett. 84, 4076 (2000). 16. W. Tappe, R. Flesch, E. R¨ uhl, R. Hoekstra and T. Schlath¨ olter, Phys. Rev. Lett. 88, 143401 (2002). 17. F. Chandezon, S. Tomita, D. Cormier, P. Grbling, C. Guet, H. Lebius, A. Pesnelle and B. A. Huber, Phys. Rev. Lett. 87, 153402 (2001). 18. R. Santra, J. Zobeley, L. Cederbaum and N. Moiseyev, Phys. Rev. Lett. 85, 4490 (2000). 19. T. J. et al., Phys. Rev. Lett. 93, 163401 (2004). 20. Y. M. et al., Phys. Rev. Lett. 96, 243402 (2006). 21. J. S. R. Casero and J. Soler, Phys. Rev. A 37, 1401 (1988). 22. K. Gluch, S. Matt-Leubner, L. Michalak, O. Echt, A. Stamatovic, P. Scheier and T. M¨ ark, J. Chem. Phys. 120, 2686 (2004). 23. S. Feil, A. Bacher, M. Zangerl, W. Schustereder, K. Gluch and P. Scheier, Int. J. Mass Spectr. 233, 325 (2004). 24. P. Scheier, PhD thesis, Innsbruck, 1988. 25. M. Lezius, P. Scheier, A. Stamatovic and T. M¨ ark, J. Chem. Phys. 91, 3240 (1989). 26. O. Echt, O. Kandler, T. Leisner, W. Miehle and E. Rechnagel, J. Chem. Soc. Faraday Trans. 86, 2411 (1990). 27. I. Harris, R. Kidwell and J. Northby, Phys. Rev. Lett. 53, 2390 (1984). 28. P. Scheier and T. M¨ ark, J. Chem. Phys. 87, 5238 (1987). 29. W. C. M. J. DeLuca, D. M. Cyr and M. Johnson, J. Chem. Phys. 92, 7349 (1990). 30. M. Foltin and T. M¨ ark, Chem. Phys. Lett. 180, 317 (1991). 31. J. Gay and B. Berne, Phys. Rev. Lett. 49, 194 (1982).

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DYNAMICAL SCREENING OF AN ATOM CONFINED WITHIN A FINITE-WIDTH FULLERENE S. LO∗ , A. V. KOROL∗,† and A. V. SOLOV’YOV∗,‡ ∗

Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe Universit¨ at, Ruth-Moufang-Str. 1, 60438, Frankfurt am Main, Germany †

Department of Physics, St. Petersburg State Maritime Technical University, Leninskii prospect 101, St. Petersburg 198262, Russia This is an investigation of the dynamical screening of an atom conﬁned within a fullerene of ﬁnite width. The two surfaces of the fullerene lead to the presence of two surface plasmon eigenmodes. In the vicinity of these two frequencies, there is a large enhancement of the conﬁned atom’s photoabsorption rate. The dynamical screening factor is given for two cases. In the ﬁrst case, the atom is ﬁxed at the center of the fullerene. The eﬀect of the interaction between the dipole moments of the shell and the atom on the dynamical screening factor is investigated. In the second case, it is calculated for the atom placed at an arbitrary position inside the fullerene. A method for obtaining the spatially averaged screening factor is outlined. Keywords: Fullerene; photoabsorption; plasmon.

1. Introduction Currently, there is much interest in plasmon excitations and their eﬀects on the properties of the object (see Refs. 1–6, and references therein). Plasmons are the collective excitation of electrons and are found in strongly correlated systems such as nanotubes, fullerenes and metallic clusters. The endohedral atom, i.e. an atom conﬁned within a fullerene, is another area uller et al.10 focused on of interest.5,7–9 Recent experimental work by M¨ the photoionization of endohedral fullerene ions and considered the mutual ‡ On

leave from the Ioﬀe Physical-Technical Institute, Russian Academy of Sciences, Polytechnicheskaya 26, St. Petersburg 194021, Russia. 162

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inﬂuence of the fullerene cage and the encapsulated species on the photoionization process. Connerade and Solov’yov studied the eﬀect of electromagnetic screening on an atom conﬁned within a C60 molecule.5 They found that the screening is strongly dependent on the frequency of the light, hence the term dynamical screening. Near the plasmon frequency of the fullerene, there is a large ampliﬁcation of the ﬁeld within the fullerene, leading to an enhancement of the photoabsorption cross-section of the endohedral atom. oster et al.12 demonstrate Recent works by Scully et al.11 and by Reink¨ the existence of a second giant resonance in the single photoionization crosssection of multiply ionized C60 molecules. The ﬁrst resonance is known to correspond to the surface plasmon of the fullerene. The nature of the second resonance has been discussed by Scully et al. in the original paper and also by Korol and Solov’yov in a comment.13 Its presence indicates that the volume or thickness of the fullerene should be taken into account. The problem of dynamical screening of an atom conﬁned at the center of a fullerene cage of ﬁnite thickness (neglecting the eﬀect of the polarized atom on the cage) was studied using a semi-classical method in Ref. 14. Ab initio calculations were performed by Madjet et al.15 on argon centrally conﬁned within C60 and showed good agreement with the results presented in Ref. 14. Here the eﬀect of the polarized atom on the fullerene cage is investigated. The semi-classical method of Ref. 14 is generalized for the case where the atom is conﬁned at an arbitrary position inside the fullerene. 2. Central position of the atom: Theoretical framework When this system of an atom within C60 is exposed to an external electromagnetic ﬁeld, the fullerene shell dynamically screens the conﬁned atom. The atom experiences a ﬁeld that is enhanced or suppressed depending on the frequency of the light, ω. The result is that, for the same external ﬁeld, the photoabsorption rate of the conﬁned atom diﬀers from that of the free atom. A dynamical screening factor F ≡ F(ω) may be deﬁned to relate the photoabsorption cross-sections of these two atoms: σconf = F σfree .

(1)

This problem of dynamical screening may be studied using quantum mechanical methods with many-body theory, or with classical methods such as the hydrodynamic approach detailed in Ref. 6. Here, a classical electrostatic approach is taken, where the fullerene is modeled as a dielectric shell.16

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Fig. 1. The atom, indicated by the solid circle, is conﬁned at the center on the fullerene molecule of average radius R and thickness ∆R. The inner and the outer radii of the fullerene are given by R1 and R2 respectively. This system is exposed to an external ﬁeld Eext .

The fullerene is treated as a spherical shell of dielectric function ≡ (ω). It has a thickness ∆R. It has an inner and an outer radius of R1 and R2 respectively, with R being their average. The point-like atom, located at r = 0 is characterized by the polarizability αa ≡ αa (ω). This system is exposed to a monochromatic electromagnetic wave. Using the dipole approximation and neglecting the magnetic part, this is treated as an external uniform electric ﬁeld Eext (t) = E0 eiωt . An object’s photo absorption rate is proportional to its photoabsorption cross-section. One may therefore calculate the dynamical screening factor F (ω) by comparing the absorption rate of the conﬁned atom to that of the free atom. In the dipole limit, an object exposed to an electromagnetic wave will absorb its energy with a rate Q given by17 ω Q = Im E∗ (r) dD(r). (2) 2 Here, the integration is carried out over the volume of the considered system. E(r) is the electric ﬁeld intensity at the position r and dD(r) is the dipole moment of the integration element at that position. For a point-like atom located at r, Eq. (2) becomes Qa = ω Im(E∗ (r)D(r))/2 = ωE02 Im(αa (ω))/2,

(3)

with D(r) being the dipole moment of the atom. Thus, the dynamical screening factor becomes F (ω) =

Im(E∗tot (r) Dtot (r)) , E02 Im(αa (ω))

(4)

where Etot (r) and Dtot (r) are the total electric ﬁeld and total dipole moment at r, the position of the atom.

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2.1. The field at the atom When the endrohedral atom is exposed to a uniform electric ﬁeld, both the shell and the atom are polarized. The induced dipole moments of these two objects act on each other, producing further polarization. One may account for this inﬁnite cycle of mutual polarization of the fullerene and the atom by considering the electric ﬁeld of the following two situations. Case 1: The dielectric shell in the external ﬁeld E0 . The resulting uniform ﬁeld inside the fullerene gives the zeroth order approximation, where the eﬀect of the atom on the shell is neglected. Case 2: A point-like dipole D at the center of the shell in the absence of the external ﬁeld. This allows one to account for each cycle of mutual polarization. The dipole D polarizes the shell. The resulting electric ﬁeld is the one that further polarizes the atom, beginning the next iterative cycle. Both problems are easily solved using standard methods, such as the scheme detailed in Ref. 18. Equations (5) and (6) give the electric ﬁeld at r = 0, the position of the atom, for the ﬁrst and second case, respectively: Ecase1 (r = 0) = wE0 ,

(5)

˜ Ecase2 (r = 0) = wD,

(6)

where w=

9 , ∆

(7)

w ˜=

2 (2 + )( − 1) + (1 − )(2 + )ξ 3 R13 ∆

(8)

with ∆ = (2 + )(2 + 1) − 2(1 − )2 ξ 3

(9)

and ξ, the ratio of the inner to outer shell radii, given by ξ = R1 /R2 ≤ 1. The total ﬁeld at the atom may be constructed by combining these results. Let Ei refer to the electric ﬁeld at r = 0 due to the polarization of the shell and Di refer to the induced moment of the atom in the ith iteration. Then the iteration scheme is as follows: 1st iteration: The external ﬁeld E0 polarizes the shell, as described by case 1, creating the polarization ﬁeld E1 = (w−1)E0 . The atom is also polarized, gaining an induced dipole moment of D1 = αa wE0 . 2nd iteration: As described by case 2, the dipole moment D1 brings about ˜ 1 . This ﬁeld induces D2 = αa E2 in the atom. the polarization ﬁeld E2 = wD ith iteration: The creation of a dipole moment of the atom causes further polarization of the shell, which results in an additional dipole moment of

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the atom, thus starting the cycle anew. It follows that for the iterations where i > 2, the relations for Ei and Di are given by Ei = wD ˜ i−1 ,

Di = αa Ei .

(10)

Therefore, the total electric ﬁeld at the atom and its dipole moment are Etot = F E0 , where F =

Dtot = αa F E0 , w . 1 − αa w ˜

(11)

(12)

2.2. Dynamical screening factor Using Eq. (11) in Eq. (2), one obtains that the dynamical screening factor F (ω) of an atom conﬁned at the center of the fullerene has the value |F |2 . Further analysis of this factor requires the dielectric function of the shell to be expressed in terms of ω, which is dependent on the material nature of the fullerene. Henceforth, the fullerene shall be treated as being metallic on account of the delocalization of the valence electrons. Following the Drude model,19 the dielectric function is expressed as (ω) = 1 − The plasma frequency ωp is

ωp2 . ω2

(13)

4πN , (14) V with N being the number of valence electrons and V being the volume of the shell. The dynamical screening factor is therefore given by 2 w 2 , F (ω) = |F | = (15) 1 − αa (ω)w ˜ ωp2 =

and in the case where the eﬀect of the polarized atom on the fullerene is small, it may be approximated as F (ω) ≈ |w|2 .

(16)

The components of the screening factor w, Eq. (7), and w, ˜ Eq. (8), are transformed into 1 1 2(N1 − N2 ) − w = 1+ , (17) R23 ω 2 − ω12 + iΓ1 ω ω 2 − ω22 + iΓ2 ω N2 N1 2 + w ˜=− 3 3 , (18) R1 R2 ω 2 − ω12 + iΓ1 ω ω 2 − ω22 + iΓ2 ω

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where ωp2 ωp2 (3 − p), ω22 = 3 + p , p = 1 + 8ξ 3 . (19) 6 6 The frequencies ω1 and ω2 are the eigenfrequencies of the surface plasmon modes of the fullerene.4 The plasmons at the inner and the outer surfaces of the fullerene are coupled oscillators and have two normal modes. In the symmetric mode, characterized by ω1 , the charge densities of the two surfaces oscillate in phase, whereas in the antisymmetric mode of frequency ω2 they are in antiphase. Both N1 and N2 may be thought of as the number of oscillators in the symmetric and the antisymmetric mode: ω12 =

N1 = N

p+1 , 2p

N2 = N

p−1 . 2p

(20)

The widths of the two surface plasmon modes have been introduced via Γ1 and Γ2 . These describe the decay rate from the collective excitation to the incoherent sum of single-electron excitations. 3. Results The dynamical screening factor was calculated for C60 . It is known to have a a mean radius of 3.5 ˚ A and a thickness ∆R of 1.5 ˚ A, which were obtained by R¨ udel et al.20 Thus, the two resonant frequencies are given by this model as 16.9 and 33.5 eV, with the ﬁrst characterizing the symmetric plasmon mode. It must be noted that the plasmon frequencies for C60 are experimentally measured to be approximately 20 and 40 eV.11 The discrepancy between the calculated values and the known experimental values may be explained by the fact that the model is purely classical and also by the choice of ∆R. In these calculations, it is necessary to quantify the surface plasmon widths. In this contribution, we do not attempt to calculate these widths. Various methods for calculating the widths are described in Refs. 3 and 21–23. Due to the lack of experimental data, these are parametrized by the resonant frequency in the following manner: Γ1,2 = γω1,2 , where γ is a constant. 3.1. Neglecting the interaction between the shell and the atom When the interaction between the shell and the atom is neglected, the dynamical screening factor is given by Eq. (16). Below, we present the results obtained for the fullerene C60 . Here, the interaction between the polarized

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200 Γi = 0.1 ωi

180 Dynamical screening factor

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160

Γ = 0.5 ω i

i

140 120 100 80 60 40 20 0 0

10

20 30 40 photon energy ω (eV)

50

60

Fig. 2. The dynamical screening factor of C60 for various plasmon widths. The pair of thin vertical lines indicate the two resonant frequencies at 16.9 and 33.5 eV.

atom and the polarized fullerene shell is neglected. Figure 2 gives the dynamical screening factor for various plasmon widths — the values of γ used here are 0.1, 0.25 and 0.5. In the static limit, the atom is completely shielded from the external ﬁeld. At high frequencies the fullerene is transparent to the light and there is no enhancement. This is as expected for a metallic fullerene. The proﬁle of the screening factor shows a strong dependence on the value of γ. There is a peak in the vicinity of the ﬁrst resonant frequency and another feature near the second resonant frequency due to the symmetric and the antisymmetric plasmon modes, respectively, as demonstrated in Fig. 3. For low values of γ, γ ≈ 0.1, there is a well-deﬁned peak at each plasmon frequency. For larger values of γ, the individual contribution of the second plasmon mode is strongly suppressed by the damping. Here, the interference terms of the screening factor become more signiﬁcant and the feature at the antisymmetric eigenfrequency is reduced to an extension of the ﬁrst peak. 3.2. Accounting for the interaction between the fullerene and the atom In the regimes where the dynamical polarizability of the atom is large, the interaction between the polarized atom and the polarized fullerene shell may

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35 screening factor symmetric mode antisymmetric mode

30 Dynamical screening factor

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25 20 15 10 5 0 0

10

20 30 40 photon energy ω (eV)

50

60

Fig. 3. The screening factor for C60 with Γ1,2 = 0.25ω1,2 . The individual contributions of the symmetric and the antisymmetric plasmon modes (dashed lines) are also shown. The dotted black lines indicate the two plasmon resonant frequencies.

become signiﬁcant. (These occur primarily for photon energies in the vicinity of giant resonances in the photoionization cross-section of the conﬁned atom.) Accounting for this interaction could result in changes in both the proﬁle and the magnitude of the dynamical screening factor. The complete expression of the dynamical screening factor, given in Eq. (15), includes this eﬀect. The numerical results for argon conﬁned with C60 are presented here. The widths of the surface plasmons are ﬁxed at Γ1,2 = 0.25ω1,2. The dynamic dipole polarizability of argon, αa (ω), calculated within the RPAE approximation is given in Fig. 4. For photon energies below 17 eV (just above the ﬁrst ionization potential of argon), Re(αa (ω)) is smoothed out. −2 Figure 5 shows the correction factor of 1−αa (ω)w ˜ from Eq. (15). The corrected dynamical screening factor is given in Fig. 6. In the vicinity of the ﬁrst plasmon resonance, there is a marked change caused by the inclusion of the interaction between the polarized argon atom and the shell. The discontinuity at 17 eV is a direct consequence of the threshold of Im(αa (ω)) at that energy. This also results in the apparent “shift” of the peak to lower energies. For ω > 17 eV, there is a marked decrease in the height of the peak.

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60 Re(αa(ω)) Im(αa(ω))

Dipole dynamic polarizability of Ar

50 40 30 20 10 0 −10 −20

0

10

20

30

40

photon energy ω, eV

50

60

Fig. 4. The real and imaginary parts of the dynamic dipole polarizability of argon. The threshold for Im(αa (ω)) is at 17 eV.

5 4.5 4 The correction factor

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0

10

20

30 40 photon energy ω (eV)

50

60

Fig. 5. The correction factor, which results from the accounting of the interaction between the polarized atom and fullerene. The dashed line marks 17 eV, the threshold of Im(αa (ω)).

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35 F(0,ω), disregarding αa(ω) F(0,ω), including α (ω)

30

Dynamical screening factor

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a

25

20

15

10

5

0

0

10

20

30

40

photon energy, ω, eV

50

60

Fig. 6. The eﬀect of including the interaction between the dipole moments of the fullerene and of the atom.

4. Non-central position of the atom In a physical system, the atom is not ﬁxed at the center of the fullerene. Instead, it can occupy an arbitrary position inside the shell as illustrated by Fig. 7. One must derive a general expression F (ρ, ω) for the dynamical screening factor, which accounts for a non-central position of the atom. It is then necessary to average over all possible positions.

Fig. 7. The atom, indicated by the ﬁlled circle, is conﬁned at ρ inside the fullerene. The angle between ρ and the external ﬁeld E0 (ω) is denoted θ. The ﬁeld at the position of the fullerene is given by Etot (ρ, ω).

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The dynamical screening factor F (ρ, ω) is calculated using the procedure outlined in Sec. 2. The total electric ﬁeld Etot (ρ, ω) for an arbitrary position ρ inside the fullerene, given in Eq. (21), is found by combining the ﬁeld of the two situations in an iterative manner. The crucial diﬀerence is that now the ﬁeld in case 2 contains non-dipole terms. Thus, Etot (ρ, ω), which contains multipole series, is expressed in the following form: w s2 αa (ω)/R13 Etot (ρ, ω) = (nE0 )n . E0 − 1 + s1 αa (ω)/R13 1 + (s1 + s2 )αa (ω)/R13 (21) The quantity n is the unit vector in the direction of ρ. The product wE0 is the electric ﬁeld of case 1 and is as deﬁned in Eq. (7) and Eq. (17). Both s1 and s2 are related to the multipole expansion of the electric ﬁeld of case 2 at ρ. The component of this ﬁeld in the direction of D is proportional to s1 while s2 is proportional to the component in the direction of ρ. The explicit formulas for s1,2 are

∞ 2l−2 s1 ρ βl (ω) 1 − ξ 2l+1 3 l(l + 1)2 . (22) = 2R23 R1 2l + 1 1 − ξ 3 s2 l(l2 − 1) l=1

Here, l is the multipole moment. Information on the l-pole surface plasmon modes of the fullerene shell is contained within βl (ω), which deﬁnes the multipole dynamical response of the fullerene shell. Within the plasmon resonance approximation, this function reads as βl (ω) =

ω2

Nl1 Nl2 + 2 2 2 + iΓ ω , − ω1l + iΓ1l ω ω − ω2l 2l

(23)

where (ωjl , Γjl , Nlj )j=1,2 are the eigen-frequency, the width, and the oscillator strength participating in the l-pole asymmetric/symmetric surface plasmon mode. These quantities are now deﬁned as:   ωp2 pl + 1  2   (2l + 1 − p ω = ),   l Nl1 = N 2p ,  1l 2(2l + 1) l (24) 2   pl − 1 ωp   2   = N , N ω2l = l2 (2l + 1 + pl ), 2pl 2(2l + 1) pl = 1 + 4l(l + 1)ξ 2l+1 . (25) The widths of these l-pole surface plasmon modes are treated in the same way as before: Γjl = 0.25ωjl It follows, from Eq. (4) and from the results of this section, that the dynamical screening factor for an atom placed at an arbitrary position ρ

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within the fullerene is

2 s2 αa (ω)/R13 |w(ω)|2 F (ρ, ω) = 1 + 3 3 2 |1 + s1 αa (ω)/R1 | 1 + (s1 + s2 )αa (ω)/R1 s2 αa (ω)/R13 2 − 2Re cos θ , 1 + (s1 + s2 )αa (ω)/R13

(26)

where θ is the angle between ρ and E0 . Figure 8 gives the spatial dependences of the dynamical screening factor for various values of ω. A spatial average of the dynamical screening factor F(ρ, ω)ρ , based on the interaction between the atom and the shell, can be carried out as follows: F(ρ, ω)ρ = C

V

F (ρ, ω)WT (ρ)dρ.

(27)

This integration is carried out over the volume V where the atom may reside — a sphere of radius R1 − a, where a is related to the size of the

Fig. 8.

Plots of F (ρ, ω) for Ar@C60 . Here Γjl = 0.25ωjl , j = 1, 2.

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conﬁned atom. If the atom were closer to the fullerene, quantum eﬀects such as hybridization come into play and this classical model is no longer suitable. The Boltzmann distribution WT (ρ) governing the probability of ﬁnding the atom at ρ and temperature T is given by Utot (ρ) − Utot (0) WT (ρ) = C exp − , (28) kT where k is the Boltzmann factor and C is the normalization constant, Utot (ρ) − Utot (0) exp − dρ. (29) C −1 = kT V The total interaction between the fullerene and the conﬁned atom can be presented as a sum of two terms: Utot (ρ) = UvdW (ρ) + U (ρ).

(30)

Here, UvdW (ρ) stands for the van der Waals type interaction between the shell and the atom. The second term, U (ρ), is the potential energy of the interaction between the dipole of the atom D(ρ) and the total electric ﬁeld Etot (ρ) at position ρ is deﬁned as 1 (31) U (ρ) = − Re(Etot (ρ)∗ D(ρ)). 2 Numerical analysis of this function is currently being performed. The results will be presented elsewhere. 5. Conclusion In this contribution, the dynamical screening factor obtained by Connerade and Solov’yov5 for an atom conﬁned within an inﬁnitely thin fullerene is generalized to that of an atom conﬁned by a fullerene of ﬁnite width. A classical dielectrical approach is taken to study the screening of the endohedral atom. We have focused on two cases. In the ﬁrst case, the atom is located at the center of the fullerene. The eﬀect of the interaction between the dipole moments of the shell and the atom on the dynamical screening factor is investigated. In the second case, the atom is no longer restricted to the center of the fullerene. We give a qualitative description of the dynamical screening factor with this classical approach. This thick fullerene has two surface plasmon eigenmodes. They result in two frequency ranges where there are large enhancements of the conﬁned atom’s photoabsorption cross-section. This is in contrast to the single frequency range for the inﬁnitely thin fullerene.

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The frequency proﬁle of the dynamical screening factor depends strongly on the damping of the surface plasmons. For cases of minor damping of the plasmons, there are two distinct peaks of the screening factor in the vicinity of the plasmon resonant frequencies. When damping is large, the second peak is suppressed and becomes an extension of the ﬁrst peak. The eﬀect of including the interaction between the dipole moment of the atom and the fullerene is two-fold. Firstly, the proﬁle and magnitude of the dynamical screening factor are changed depending on the dynamical dipole polarizability of the atom. Secondly, the screening factor becomes dependent on the position of the conﬁned atom. As the atom cannot be physically ﬁxed to any position within the fullerene, a method of spatial averaging, based on the Boltzmann distribution, is outlined. Despite the simplicity of this model, it provides an instructive description of the phenomena discussed here. This model may be further applied to study the dynamic responses of more complicated situations for various processes. Acknowledgments We are grateful for the helpful discussions with Prof. Dr. W. Greiner. This work was supported by INTAS under the grant 03-51-6170, by the European Commission within the Network of Excellence project EXCELL (project number 515703) and by PECU under the grant 004916(NEST). References 1. I. A. Solov’yov, A. V. Solov’yov and W. Greiner, J. Phys. B: At. Mol. Opt. Phys. 37, L137 (2004). 2. A. V. Solov’yov, Int. J. Mod. Phys. B 19, 4143 (2005). 3. L. G. Gerchikov, A. N. Ipatov, R. G. Polozkov and A. V. Solov’yov, Phys. Rev. A 62, 043201 (2000). ¨ 4. D. Ostling, P. Apell and A. Ros´en, Z. Phys. D Suppl. 26, 1794 (1993). 5. J.-P. Connerade and A. V. Solov’yov, J. Phys. B: At. Mol. Opt. Phys. 38, 807 (2005). 6. J.-P. Connerade and A. V. Solov’yov, Phys. Rev. A 66, 013207 (2002). 7. Y. M. Amusia and A. S. Baltenkov, Phys. Rev. A 73, 063206 (2006). 8. Y. M. Amusia and A. S. Baltenkov, Phys. Rev. A 73, 062723 (2006). 9. V. K. Dolmatov and S. T. Manson, Phys. Rev A 73, 013201 (2006). 10. A. M¨ uller et al., J. of Phys. Conf. Ser. 88, 012038 (2007). 11. S. W. J. Scully et al., Phys. Rev. Lett. 94, 065503 (2005). 12. A. Reink¨ oster et al., J. Phys. B: At. Mol. Opt. Phys. 37, 2135 (2004). 13. A. V. Korol and A. V. Solov’yov, Phys. Rev. Lett. 98, 179601 (2007).

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14. S. Lo, A. V. Korol and A. V. Solov’yov, J. Phys. B: At. Mol. Opt. Phys. 40, 3973 (2007). 15. E. Madjet, H. S. Chakraborty and S. T. Manson, Phys. Rev. Lett. 99, 243003 (2007). 16. P. H. Lambin, A. A. Lucas and J.-P. Vigneron, Phys. Rev. B 46, 1794 (1992). 17. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media (Pergamon Press, UK, 1960). 18. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, USA, 1965). 19. C. F. Bohren and D. R. Huﬀman, Absorption and Scattering of Light by Small Particles (Wiley, USA, 1983), pp. 148–150 and 251–257. 20. A. R¨ udel, R. Hentges, U. Becker, H. S. Chakraborty, M. E. Madjet and J. M. Rost, Phys. Rev. Lett. 89, 125503 (2002). 21. A. A. Lushnikov and A. J. Siminov, Z. Physik 270, 17 (1974). 22. C. Yannouleas and R. A. Broglia, Ann. Phys., 217 105 (1992). 23. C. Yannouleas, Phys. Rev. B 58, 6748 (1998).

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PHOTOIONIZATION AND FRAGMENTATION OF FULLERENE IONS ∗ ¨ A. MULLER and S. SCHIPPERS

Institut f¨ ur Atom- und Molek¨ ulphysik, Universit¨ at Giessen, D-35392 Giessen, Germany ∗ [email protected] R. A. PHANEUF, S. SCULLY† , E. D. EMMONS, M. F. GHARAIBEH‡ and M. HABIBI Department of Physics, MS 220, University of Nevada, Reno, NV 89557-0058, USA A. L. D. KILCOYNE, A. AGUILAR and A. S. SCHLACHTER Advanced Light Source, MS 7-100, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA L. DUNSCH and S. YANG Leibniz-Institut f¨ ur Festk¨ orper-und Werkstoﬀforschung Dresden, D-01171 Dresden, Germany H. S. CHAKRABORTY§ , M. E. MADJET¶ and J. M. ROST Max-Planck-Institut f¨ ur Physik komplexer Systeme, D-01187 Dresden, Germany † Present

address: Kerr Henderson Bacon & Woodrow Limited, 29–31 College Gardens, Belfast BT9 6BT, UK. ‡ Present address: Department of Applied Physical Sciences, Jordan University of Science and Technology, Irbid 22110, Jordan. § Present address: Department of Chemistry and Physics, Northwest Missouri State University, Maryville, MO 64468, USA. ¶ Present address: Institut f¨ ur Chemie, Freie Universita¨ at Berlin, D-14195 Berlin, Germany. 177

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Mass selected fullerene ions are exposed to synchrotron radiation in the 17–300 eV energy range. Selected absolute cross-sections for single and multiple ionization as well as fragmentation were measured for ions of C60 , C70 , C80 , C82 , and C84 . More recently, the first ever experiments + with endohedral Sc3 N@C+ 80 and Ce@C82 fullerene ions have been conducted. Keywords: Fullerene; photoionization; fragmentation.

1. Introduction Understanding electron dynamics of mesoscopic systems is a fundamental goal for science at the threshold between the microscopic world of atomic and molecular species and the macroscopic world of solid bodies. Such understanding is not only of fundamental interest but also a requirement for progress in nanoscience and technology. Quantitative measurements on excitation and decay processes of clusters and complex molecules are essential for the development of the required understanding of complex systems. A particularly interesting class of mesoscopic particles are the fullerenes. Because of their size, structure and symmetry, their properties and behavior are intermediate between a free molecule and a solid body. In studies on neutral fullerenes or clusters in the gas phase, two requirements complicate the determination of absolute cross-sections: (i) the material has to be of high chemical purity so that background signals are negligible, and (ii) the vapor density and the target thickness have to be experimentally determined with suﬃcient accuracy. Since it is diﬃcult to satisfy both requirements simultaneously, most of the existing data are on a relative scale only. In the present study, ionic fullerene molecules including Cq+ 60 (q = 1, 2, 3) as well as higher order fullerenes are investigated, and most recently, the research program was extended to ions of endohedral fullerenes. Experiments with charged particles provide a number of advantages. Even with samples that are not chemically pure, one can produce ion beams with well-deﬁned energies that are then puriﬁed by magnetic mass and charge analysis. The electric charge on the accelerated ions also facilitates the quantitative measurement of ion beam density proﬁles which are required for quantifying the overlap of the ion beam with an interacting beam of photons. This in turn is a condition for the measurement of absolute cross-sections. Moreover, the charge state on a particle provides an additional, tunable parameter for investigating interactions under diﬀerent, well-controlled physical conditions. In addition to the potential of

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obtaining absolute cross-sections in the present experiments, the use of intense wavelength-resolved VUV photon beams provides access to ﬁne details in the photo-ion spectra. 2. Experiment Cross-sections for photoionization and fragmentation are measured in the range 17–340 eV of photon energies available at Beamline 10.0.1.2. of the Advanced Light Source (ALS) in Berkeley. For the measurements, a beam of monochromatized synchrotron radiation from an undulator is merged with accelerated and mass/charge analyzed fullerene ion beams, and the yield of product ions is measured as the photon energy is stepped. Descriptions of the ALS merged-beams endstation1 and its application to absolute photoionization measurements with fullerene ions2 have been presented previously. In the present experiments, the energy resolution of the photon beam is typically between 20 and 500 meV depending on photon energy and cross-section to be measured. Beams of fullerene ions are produced by evaporating chemically pure or enriched powder samples at low pressure (∼10−6 hPa) into the discharge of a permanent-magnet 10-GHz electron– cyclotron–resonance (ECR) ion source operated at RF power levels of typically only a few mW to minimize fragmentation. Atoms, molecules and clusters are eﬃciently ionized when entering the ion source plasma. Ions extracted from the plasma are accelerated by 4 kV and a beam of mixed constituents is formed. By using a 60◦ magnetic-dipole mass spectrometer with a resolution of 1%, the ion beam is mass/charge selected and thus chemically puriﬁed. Ion beam currents in the merged-beams interaction region vary between 100 nA when vaporizing puriﬁed C60 powder and only about 1 pA when using enriched endohedral fullerene samples. Such low ion currents correspond to less than 100 ions in the photon-ion interaction region at any given time. Photon and ion beams are merged over a length of about 30 cm for absolute measurements at selected photon energies and about 1 m in spectroscopy mode where the photon energy is scanned in ﬁne steps while the relative ion yield is monitored. For the measurement of absolute crosssections the overlapping beams are probed in three dimensions to determine the density distributions in space and, thereby, the beam overlap. Product ions in a higher charge state, or ionic fragments, are collected by a highly sensitive low-background detector behind a second magnet spectrometer that deﬁnes the mass, charge, and energy of the detected particles. The cross-sections obtained for the fullerenes have an estimated uncertainty of ± 30%.

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3. Studies on C+ 60 The neutral C60 molecule is known to exhibit a giant resonance in its singlephoton absorption spectrum at energies near 20 eV.3,4 This resonance is associated with the collective motion of the 240 delocalized valence electrons that form a negatively charged cloud around the positively charged ion cage of the C60 molecule and can oscillate relative to the positive charge of the carbon ion cores that form the C60 molecular cage. In single ionization of C+ 60 ions, the well-known surface plasmon resonance near 20 eV is observed. Figure 1 shows absolute cross-section mea2 surements for single photoionization of C+ 60 ions. In addition to the dominant peak centered at 22 eV, a higher order resonance is seen near 40 eV. Two Lorentzian curves centered near 22 eV (surface plasmon) and 38 eV (volume plasmon) were ﬁtted to the experimental cross-section to visualize and quantify the contributions of these resonances. These broad resonance features are also evident in photoionization measurements of neuoster et al.5 and Kou et al.6 Similar plasmon contributions tral C60 by Reink¨ at almost identical energies are found in the photoionization of all other fullerene ions studied within the present research program. Based on ab initio quantum-mechanical calculations within the framework of the time-dependent local-density approximation (TDLDA), the collective nature of the resonance features was conﬁrmed. A classical picture7 was used to interpret the two resonances as in-phase and out-of-phase

Fig. 1. Absolute cross-section for single photoionization of C+ 60 ions. The thin solid line results from a fit to the measured data of a linear background (not shown) plus two separately displayed Lorentzian curves which represent the plasmon resonances.

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oscillations of charges on the inner and outer surfaces of the spherical C60 shell. The dynamics of the second plasmon lead to its interpretation as a volume plasmon. It is associated with a spatial modulation of the electron density, in contrast to a collective excitation with the constant electron density of the traditional surface plasmon observed at 22 eV. This reﬂects the diﬀerence between the in-phase-motion of the two surface charges with lower resonance frequency at constant electron density and the out-of-phase motion at higher resonance frequency with a modulated electron density.8 Moreover, this higher frequency is close to the plasma frequency ωp which describes the modulation in a plasma and is identical to the volume plasmon frequency. Figure 2 shows cross-sections for photoabsorption and photoionization normalized to their maximum at about 20 eV. The abscissa is the frequency ω scaled to the plasma frequency ωp . The TDLDA calculation for photoabsorption by C+ 60 clearly shows the higher order plasmon at about 38 eV, which corresponds to the volume plasmon shown in Fig. 1. A simpler LDA (local density approximation) calculation normalized by the same factor as the TDLDA result fails to reproduce the volume plasmon feature. Also a TDLDA calculation for a full-sphere Na198 cluster only shows a strongly suppressed resonance feature in the vicinity of the plasma frequency of the

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Fig. 2. Photoionization cross-section of C+ 60 ions as a function of the scaled frequency ω/ωp , where ωp is the plasma frequency of the delocalized electrons. For C+ 60 , the plasma frequency ωp = 36.77 eV was used and for Na198 ωp = 5.89 eV. The cross-sections are normalized to their maximum near 20 eV. The LDA results have been multiplied with the same factor as the TDLDA results. All calculated spectra are convoluted with a width of 0.1 eV. The photoionization data for C+ 60 ions are represented by the open circles, the TDLDA result for C+ 60 is the solid line, the associated LDA calculation is the dotted line, and the TDLDA result for Na198 is the dashed line.

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delocalized electrons. Obviously, it is the hollow-sphere nature of C60 that supports a dipole-excited volume plasmon in addition to the well-known surface plasmon. For a solid conducting sphere, such a volume plasmon is classically dipole forbidden. Figure 3 shows the cross-section for photoionization of C+ 60 over a wide range of photon energies. The experimental data are remarkably well represented by the semiempirical cross-section for photoabsorption of neutral C60 calculated by using the prescription of Henke et al.9 This method is based on the approximation that the photoabsorption coeﬃcients of compound materials can be estimated from the coeﬃcients of their individual atoms. It is expected to provide useful data for photon energies beyond 30 eV and far away from absorption edges. The interactions between the individual atoms in the material are neglected. Accordingly, the absorption cross-section of a molecule can be approximated by xi σa,i , (1) σa = i

where xi is the number of atoms of element i in the molecule and σa,i is the absorption cross-section for an individual atom of type i. The elementspeciﬁc cross-sections σa,i follow from scattering factors calculated and tabulated by Henke et al.9 The size of the cross-section step at the carbon K-edge is unexpectedly well represented by the Henke method; however, theory and experiment diverge towards higher photon energies. In fact,

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Fig. 3. Overview of the absolute cross-sections measured for photoionization of C+ 60 ions in the energy range from 17 eV to about 300 eV. The scan measurement from Fig. 1 is also shown (thin solid line). The thick solid line is the total photoabsorption cross-section for neutral C60 obtained by using the method of Henke et al.9

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Fig. 4. Absolute cross-section measurements σ1,2 (C+ 80 ) for single photoionization of C+ 80 (open circles). Typical total uncertainties are indicated by solid black error bars. Also shown is the photoabsorption cross-sections for C80 (dotted line) constructed by using the method of Henke et al.9 [see Eq. 1]. For comparison, the scaled cross-section σ1,2 (C+ 60 )×80/60 is also shown (gray dotted line) together with error bars (solid gray) at selected energies.

the experiment shows detailed resonance structures in the vicinity of the K-edge which are not included in the Henke treatment. One should also keep in mind that, in addition to single photoionization, photoabsorption may lead to excitation and fragmentation, as well as multiple ionization. 4. Scaling behavior of cross-sections The scaling of cross-sections with the number of carbon atoms in a fullerene that is inherent in the method of Henke et al. should bring the cross+ + section σ1,2 (C+ 60 ) for photoionization of C60 up to σ1,2 (C80 ) by multiplying + σ1,2 (C60 ) by a factor 80/60. Figure 4 investigates this issue. Within the experimental error bars, the cross-sections of fullerene ions are compatible with the scaling rule predicting a linear relation between the cross-section and the number of C atoms in the molecule. 5. Endohedral Sc3 N@C80 Endohedral fullerenes, characterized by atoms or molecules encapsulated within a cage of carbon atoms, have stimulated increasing research activity

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within the last several years, partly because of their intriguing structures but also because they provide new possibilities for applications in nanostructure science and technology.10 In order to make use of such applications, detailed knowledge about the production, handling and structure of endohedral fullerenes is required. For the present research, milligram quantities of endohedral Sc3 N@C80 were synthesized at the Leibniz-Institute for Solid State and Materials Research in Dresden. The material being of Ih symmetry was puriﬁed to 99%. From the miniscule amounts produced in a time-consuming process, samples were taken to (i) produce microscopic crystals for photoabsorption measurements, and (ii) supply the ion source plasma with vapors from which Sc3 N@C+ 80 ions were produced for mergedbeams photoionization experiments. Clearly, photoabsorption by endohedral fullerenes depends on whether the carbon cage is empty or ﬁlled. The question is how the cage and the atoms inside are mutually inﬂuenced in their response to ionizing radiation. As an example, Fig. 5 shows an absorption spectrum of endohedral Sc3 N@C80 recorded on the Scanning Transmission X-ray Microscope11 at beamline 5.3.2 of the Advanced Light Source (ALS) using a µm-sized crystal structure of the Sc3 N@C80 Ih isomer. The relative absorption coeﬃcient resulting from the experiment was normalized at high photon energies to

Fig. 5. Measured near edge X-ray absorption fine structure spectrum (open circles) for Sc3 N@C80 normalized at high energies to the photoabsorption cross-section of Sc3 N@C80 resulting from the method of Henke et al.9 (dotted line). The X-ray energy was calibrated to within ±0.05 eV; the data were taken at a resolution of 100 meV.

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the photoabsorption cross-section constructed by employing the method of Henke et al.9 The resulting cross-section σa for the Sc3 N@C80 fullerene is shown as a dotted line in Fig. 5. It reproduces the gross features of the measured absorption spectrum but deviates in the details. Diﬀerent from the prediction of a simple scandium L-edge, strong resonances are observed at about 401.5 eV and 405.7 eV that are associated with excitation from the L3 (2p3/2 ) and the L2 (2p1/2 ) sub-shells of Sc to the lowest unoccupied molecular orbitals of the fullerene. The energy shift of these resonances in the measured spectrum from the L3 and L2 ionization thresholds of neutral Sc at 398.7 eV and 403.6 eV, respectively, is due to the valency of about 2.4 of the scandium atoms bound in the Sc3 N@C80 fullerene.12 The Sc3 N cluster gives stability to the C80 cage structure due to transfer of six electrons to the fullerene cage.12 The N atom draws about 0.4e from each Sc atom, which thus remain as Sc2.4+ ions. The photoionization cross-section of the endohedral fullerene ion Sc3 N@C+ 80 was measured by employing the merged photon-ion beam technique.13 The cross-section as a function of photon energy exhibits an enhanced resonance structure in the vicinity of the expected volume plasmon. By comparison with the cross-section for photoionization of hollow C+ 80 , the excess cross-section caused by the caged Sc3 N cluster can be extracted. A model calculation assuming independent atoms in the endohedral molecule is in quantitative agreement with the experimental ﬁndings. Despite the extreme diluteness of the ion beam, the experiment was feasible because of the large magnitude of the cross-section and the absence of signiﬁcant background in the ionization measurement. A similar study on endohedral Ce@C+ 82 is in progress. Acknowledgments The authors acknowledge support by the Oﬃce of Basic Energy Sciences of the U.S. Department of Energy and the Deutsche Forschungsgemeinschaft.

References 1. A. M. Covington, A. Aguilar, I. R. Covington, M. F. Gharaibeh, G. Hinojosa, ´ C. A. Shirley, R. A. Phaneuf, I. Alvarez, C. Cisneros, I. Dominguez-Lopez, M. M. Sant’Anna, A. S. Schlachter, B. M. McLaughlin and A. Dalgarno, Phys. Rev. A 66, 062710 (2002). 2. S. W. J. Scully, E. D. Emmons, M. F. Gharaibeh, R. A. Phaneuf, A. L. D. Kilcoyne, A. S. Schlachter, S. Schippers, A. M¨ uller, H. S. Chakraborty, M. E. Madjet and J. M. Rost, Phys. Rev. Lett. 94, 065503 (2005).

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3. I. V. Hertel, H. Steger, J. de Vries, B. Weisser, C. Menzel, B. Kamke and W. Kamke, Phys. Rev. Lett. 68, 784 (1992). 4. T. Mori, J. Kou, Y. Haruyama, Y. Kubozono and K. Mitsuke, Proceedings of the 14th International Conference on Vaccum Ultraviolet Radiation Physics, in J. Elec. Spectrosc. Rel. Phenom. 144–147, 243 (2005). 5. A. Reink¨ oster, S. Korica, G. Pr¨ umper, J. Viefhaus, K. Godehusen, O. Schwarzkopf, M. Mast and U. Becker, J. Phys. B 37, 2135 (2004). 6. J. Kou, T. Mori, S. V. K. Kumar, Y. Haruyama, Y. Kubozono and K. Mitsuke, J. Chem. Phys. 120, 6005 (2004). 7. P. Lambin, A. A. Lucas and J.-P. Vigneron, Phys. Rev. B 46, 1794 (1992). 8. S. W. J. Scully, E. D. Emmons, M. F. Gharaibeh, R. A. Phaneuf, A. L. D. Kilcoyne, A. S. Schlachter, S. Schippers, A. M¨ uller, H. S. Chakraborty, M. E. Madjet and J. M. Rost, Phys. Rev. Lett. 98, 179602 (2007). 9. B. L. Henke, E. M. Gullikson and J. C. Davis, At. Data Nucl. Data Tables 54, 181 (1993). 10. L. Dunsch and S. Yang, Small 3, 8, 1298 (2007). 11. A. L. D. Kilcoyne, T. Tyliszczak, W. F. Steele, S. Fakra, P. Hitchcock, K. Franck, E. Anderson, B. Harteneck, E. G. Rightor, G. E. Mitchell, A. P. Hitchcock, L. Yang, T. Warwick and H. Ade, J. Synchr. Rad. 10, 125 (2003). 12. L. Alvarez, T. Pichler, P. Georgi, T. Schwieger, H. Peisert, L. Dunsch, Z. Hu, M. Knupfer, J. Fink, P. Bressler, M. Mast and M. S. Golden, Phys. Rev. B 66, 035107 (2002). 13. A. M¨ uller, S. Schippers, M. Habibi, R. A. Phaneuf, A. L. D. Kilcoyne, S. Yang and L. F. Dunsch, submitted.

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COLLISION OF TRANSITION METAL CLUSTER IONS WITH SIMPLE MOLECULES MASAHIKO ICHIHASHI∗,‡ , TETSU HANMURA†,§ and TAMOTSU KONDOW∗,¶ ∗

Cluster Research Laboratory, Toyota Technological Institute, 717-86 Futamata, Ichikawa, Chiba 272-0001, Japan www.clusterlab.jp ‡ [email protected] † East Tokyo Laboratory, Genesis Research Institute, Inc., 717-86 Futamata, Ichikawa, Chiba 272-0001, Japan § [email protected] ¶ [email protected] Absolute cross-sections for dehydrogenation of an ethylene molecule onto + + + Ti+ n (n = 3−25), Tin O (n = 2−25), Fen (n = 2−28), Con (n = 8−29) + and Nin (n = 3 − 30) were measured as a function of the cluster size, n, in a gas-beam geometry at a collision energy of 0.4 eV in an apparatus equipped with a tandem-type mass spectrometer. It is found that (i) + the di-dehydrogenation proceeds on Ti+ n and Tin O , and the mono+ + + dehydrogenation on Fen , Con and Nin , (ii) the di-dehydrogenation cross+ section increases gradually with the cluster size of Ti+ n and Tin O , (iii) the mono-dehydrogenation cross-section increases rapidly above a cluster + + size of ∼18 for Fe+ n , ∼13 and ∼18 for Con , and ∼10 for Nin , and (iv) the + rapid increase of the cross-section for Mn (M = Fe, Co and Ni) occurs at a cluster size where the 3d-electrons start to contribute to the highest occupied levels of M+ n . These findings lead us to conclude that the 3d+ electrons of M+ n play a central role in the dehydrogenation on Mn . Keywords: Transition metal cluster; ethylene; reaction; electronic structure.

1. Introduction Size-dependent reactivity in the collisional processes of transition metal clusters has been one of the most intriguing and challenging problems both

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experimentally and theoretically in the last twenty years. Experimental studies of the reaction of transition metal clusters in the gas phase originate from the epoch-making studies of Fen + H2 .1–3 They have revealed that the reactivity of Fen with hydrogen molecules starts to increase at n ≈ 10, and then decreases before increasing again n ≈ 20. The size-dependence of the reactivity has a strong correlation with the ionization potential of Fen . This strong correlation implies that chemical bonds involved in the reaction are formed via electron exchange between Fen and H2 . Subsequent studies of transition metal clusters have elucidated the reactivity in relation to their electronic structures.4,5 Reactivity studies of corresponding metal cluster ions have also been undertaken extensively,6–10 and give complementary results to those of the neutral metal clusters. These studies have discovered that the reactivity exhibits close correlation with the ionization potential, HOMO–LUMO gap, d-vacancy, and so on. Metal clusters are one of the ultimate catalysts of the reactions, and adsorption of unsaturated hydrocarbons on a metal surface has an essential importance for extending our understanding about the mechanism of catalytic reactions. For instance, transition metals are prominent components of the catalysts in Fischer–Tropsch synthesis, that is, conversion of carbon monoxide and hydrogen to hydrocarbons. Ethylene is one of the simplest unsaturated hydrocarbon products in Fischer–Tropsch synthesis, and the chemisorption of ethylene molecules onto iron has been extensively studied. A study of the ethylene chemisorption onto Fe(100) by use of HREELS12 suggests that ethylene chemisorbs undissociatively in the di-σ bonding conﬁguration at 100 K, and that it decomposes to methylidyne (CH) or ethynyl (CCH) at 253 K. In this dissociative chemisorption, a hydrogen molecule is a unique desorbing product from the surface. Nakajima et al. have studied reactions of cobalt cluster ions, Co+ n (n = 2–22), with ethylene molecules in a fast ﬂow reactor,6 and have found that Co+ n (n = 4, 5 and 10–15) have particularly high reaction rates. Irion and coworkers have studied reactions 7,8 of iron cluster ions, Fe+ n (n = 2–13), with ethylene molecules in FT–ICR. + They have found that only Fen (n = 4 and 5) are reactive for ethylene, and that three ethylene molecules are adsorbed on them and dehydrogenated into a benzene molecule. In the present study, we systematically measured the absolute reaction cross-section for the collision of transition metal cluster ions, Ti+ n (n = + + 3–25), Tin O+ (n = 2–25), Fe+ n (n = 2–28), Con (n = 8–29) and Nin (n = 3–30), with an ethylene molecule as a function of the cluster size. Speciﬁc size dependencies of the cross-sections imply that 3d-electrons of the cluster ion contribute greatly to the dehydrogenation of an ethylene molecule on the cluster ion.

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2. Experimental details The apparatus employed is described brieﬂy in this article (see Ref. 11 for a detailed description). Transition metal cluster ions were produced by a laser-ablation technique using the second harmonic of an Nd:YAG laser (∼20 mJ/pulse and 25 Hz) and helium gas. The metal cluster ions produced were mass-selected by a quadrupole mass ﬁlter, and were allowed to enter a reaction cell ﬁlled with ethylene (C2 H4 ) gas. To avoid the scattering loss of the product ions, the reaction cell was equipped with an octapole ion beam guide. The size-selected cluster ions collide with an ethylene molecule under single collision conditions. Product and unreacted parent ions were accelerated up to the translational energy of 6 keV and mass-analyzed by a double-focusing mass spectrometer. The ions were detected by a secondary electron multiplier equipped with an ion-conversion dynode. 3. Results Figure 1 shows a typical mass spectrum of ions produced by the collision of a titanium cluster ion, Ti+ 13 , with an ethylene molecule, C2 H4 . The product , was observed, and this indicates that the di-dehydrogenation ion, Ti13 C+ 2 of C2 H4 proceeds on Ti+ exclusively as n + Ti+ n + C2 H4 → Tin C2 + 2H2 .

(1)

Similar di-dehydrogenation proceeds on Tin O+ as well.

Fig. 1. (a) Mass spectrum of product ions in Ti+ 13 + C2 H4 at the collision energy of + 0.4 eV, and (b) the distribution of the isotopomers (Ti+ 13 and Ti13 C2 ) calculated from the natural abundance of Ti isotopes.

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A reaction cross-section, σ, was derived from Ir + Ip kB T log σ= , P Ir

(2)

where Ir and Ip represent the intensity of an intact parent ion and that of the product ions, respectively. P and T are the pressure and the temperature of ethylene gas, respectively. (= 120 mm) is the eﬀective path length of the interaction region of the metal-cluster ion beam in the ethylene gas, and kB is Boltzmann’s constant. Figure 2 shows the cross-sections for the ethylene-dehydrogenation on + Ti+ n and Tin O as a function of the cluster size, n. The dehydrogenation cross-section of Ti+ n increases gradually with the cluster size, and that of Tin O+ is as large as that of Ti+ n . This implies that a single oxygen atom does not have a signiﬁcant inﬂuence on the reactivity of Ti+ n for an ethylene molecule. On the other hand, the mono-dehydrogenation proceeds on M+ n (M = Fe, Co and Ni), whenever any product ion is observed: + M+ n + C2 H4 → Mn C2 H2 + H2 .

(3)

Figure 3 shows the cross-sections for the ethylene-dehydrogenation on + + Fe+ n , Con and Nin as a function of the cluster size, n. As Irion and coworkers 7,8 have reported, C2 H4 chemisorbs onto Fe+ 4,5 , and dehydrogenation of the chemisorbed ethylene proceeds. In our measurements, the dehydrogenation ˚2 cross-section for Fe+ n is No. 3 A at n = 4 and 5, and increases sharply at + n ≈ 18, while Nin exhibits a similar size dependence, but the threshold size is about 6. Similarly, Co+ n exhibits sharp cross-section peaks at n = 15

+ Fig. 2. Cross-sections for ethylene dehydrogenation on Ti+ n (solid circles) and Tin O (open circles) at the collision energy of 0.4 eV. Dashed and dotted lines show the simple and modified geometrical cross-sections, respectively.

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+ Fig. 3. Cross-sections for ethylene dehydrogenation on Fe+ n (open circles), Con (solid (open squares) at the collision energy of 0.4 eV. circles) and Ni+ n

and 26. The maximum of n = 15 in the relative eﬃciency of chemisorption of C2 H4 on Co+ n (n = 3–22) has been measured in a fast ﬂow reactor experiment by Nakajima et al.6 4. Discussion The reaction cross-section of Ti+ n increases monotonically with the increase in the cluster size, n. A collisional cross-section of Ti+ n with C2 H4 is estimated approximately from the geometrical radius of Ti+ n and the chargeinduced dipole interaction between Ti+ n and C2 H4 in the same way as mentioned in Ref. 13. As shown in Fig. 2, this shows poor agreement with the measured reaction cross-section of Ti+ n with C2 H4 , while the simple geometrical cross-sections reproduce the reaction cross-sections. This coincidence suggests that the collision of Ti+ n with C2 H4 at a large impact parameter does not contribute to the reaction. In that case, scattering is the dominant process. The dehydrogenation reaction may proceed not in a grazing collision but in a nearly head-on collision. On the other hand, it is more complicated to interpret the size-speciﬁc reactivity of M+ n (M = Fe, Co and Ni) with C2 H4 . In this collisional process, it is conceivable that the reaction proceeds via the following steps: + M+ n + C2 H4 → Mn (C2 H4 ),

M+ n (C2 H4 ) M+ n (C2 H4 )

→ →

M+ n (C2 H2 )(H)2 →

M+ n + C2 H4 , M+ n (C2 H3 )(H) → M+ n (C2 H2 ) + H2 .

(4) (5) M+ n (C2 H2 )(H)2 ,

(6) (7)

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An ethylene molecule is chemisorbed as M+ n (C2 H4 ) with the formation + of a di-σ bond between M+ n and C2 H4 [step (4)], and then Mn (C2 H4 ) is dehydrogenated into M+ (C H ) leaving H [steps (6) and (7)] by 2 2 2 n analogy with ethylene chemisorption on a metal surface. The branching ratio of step (5) to step (6) depends critically on the cluster size. Step (6) occurs slowly for small clusters, but more rapidly for larger clusters probably because of a lower activation energy barrier. Step (5) takes place only for small clusters. In step (6), both the cleavage of a C–H bond and the formation of an M–H bond occur simultaneously, which are initiated by valence electrons of M+ n . Evidently, this bond rearrangement is closely related to the character of the electrons. Let us compare the cross-section for ethylene dehydrogenation on a cluster ion with the feature of the photoelectron spectrum of the corresponding cluster anion, on the premise that the electronic structure of a transition metal cluster cation can be compared with that of the corresponding neutral cluster. 4.1. Dehydrogenation cross-section versus separation of 4s and 3d peaks in photoelectron spectra − − − Photoelectron spectra of Ti− n , Fen , Con and Nin measured by Wang and 14–16 exhibit peaks assignable to 4s- and 3d-electrons; in general, coworkers the peak due to 4s-electrons (4s peaks) tends to merge with those of 3delectrons, as the cluster size increases. The 4s peak of Ni− n which is clearly distinguishable from the 3d peak at n ≈ 9, merges with the 3d peak above this size, while the 4s peak of Co− n merges with the 3d peak at n ≈ 13 except for 15 ≤ n ≤ 20 for which the 4s and the 3d peaks are separated. On the other hand, the 4s peak of Fe− n is distinguishable from the 3d peak even at n ≈ 23, but seems to merge above this size. The comparison of the spectral features with the size dependencies of the reactivities of the iron-group cluster cations reveals that the dehydrogenation cross-section becomes signiﬁcantly large at the sizes where the 4s and the 3d peaks are merged, but very small at the sizes where these peaks are separated. This ﬁnding implies that 3d-electrons in the valence bands of a cluster cation, M+ n , contribute greatly to ethylene dehydrogenation on it. The 3d-electrons of M+ n are considered to participate in the C–H bond rupture of the ethylene molecule chemisorbed onto M+ n , while the 4s-electrons favor weakly bound ethylene chemisorption rather than the bond rupture. This notion is consistent with the ﬁnding of Armentrout and coworkers that Fe+ (6 D, 3d6 4s1 ) has lower reactivity for D2 than Fe+ (4 F, 3d7 ) because of the existence of the 4s-electron in Fe+ (6 D, 3d6 4s1 ).9

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The electronic structure of Ti+ n is signiﬁcantly diﬀerent from those of the + + iron-group cluster ions, Fe+ n , Con and Nin , because the 3d-electrons of Ti are delocalized. Even for a small Ti− , it has been reported that the energy n levels of the 4s- and 3d-electrons are already mixed.14 Then, the electronic structures of small titanium clusters are acceptable for the reaction, and the geometrical size should be critical. + + 4.2. Calculated electronic structures of Fe+ n , Con and Nin

As described in the previous sections, the cross-section for dehydrogenation of an ethylene molecule on an iron-group cluster ion is closely related to the electronic structure of the valence electrons of the cluster ion. This experimental ﬁnding is supported computationally as follows. The densities of states (DOS) of Fe+ n (n = 13, 19 and 25) were calculated using density functional theory (BPW91) with the 6–311G basis set. The calculation was performed by employing the Gaussian 03 suite of programs.17 Figure 4 + + shows DOS’s of Fe+ 13 (2S + 1 = 36), Fe19 (2S + 1 = 58) and Fe25 (2S + 1 = 70) having icosahedral and fused icosahedral structures. As shown in the

Fig. 4. Densities of states of Fe+ n (n = 13, 19 and 25). The shaded parts show the contribution from the 4s-electrons, and the clear parts show the contribution from 3d-electrons. Fermi energies are indicated by dashed lines.

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DOS of Fe+ 13 , the peak around −7 eV [labeled (a)] is contributed mainly from the 4s-electrons, while the peak around −8 eV [labeled (b)] comes from the 3d-electrons. As the cluster size increases, the 3d peak is broadened while the 4s peak shifts toward the 3d peak. This trend is consistent with the size-dependent proﬁle of the photoelectron spectrum, and can explain the size-dependent dehydrogenation observed. 5. Conclusion + In the collision of transition metal cluster ions, Ti+ n (n = 3–25), Tin O + + + (n = 2–25), Fen (n = 2–28), Con (n = 8–29) and Nin (n = 3–30), with an ethylene molecule, dehydrogenation of the ethylene molecule was found to proceed dominantly. It was found that the size dependence of the dehydrogenation is related to the energy separation of the 4s- and 3d-electrons in the highest occupied molecular orbitals. These ﬁndings indicate that 3d-electrons play a central role in the dehydrogenation. The reactivity of a transition metal cluster is controlled by changing the density of the delectrons in the valence states of the cluster.

Acknowledgments The calculations were performed on SGI2800 at the Research Center for Computational Science, Okazaki Research Facilities, National Institutes of Natural Sciences. References 1. S. C. Richtsmeier, E. K. Parks, K. Liu, L. G. Pobo and S. J. Riley, J. Chem. Phys. 82, 3659 (1985). 2. R. L. Whetten, D. M. Cox, D. J. Trevor and A. Kaldor, Phys. Rev. Lett. 54, 1494 (1985). 3. M. D. Morse, M. E. Geusic, J. R. Heath and R. E. Smalley, J. Chem. Phys. 83, 2293 (1985). 4. J. Concei¸ca ˜o, R. T. Laaksonen, L.-S. Wang, T. Guo, P. Nordlander and R. E. Smalley, Phys. Rev. B 51, 4668 (1995). 5. H. Kietzmann, J. Morenzin, P. S. Bechthold, G. Gantef¨ or and W. Eberhardt, J. Chem. Phys. 109, 2275 (1998). 6. A. Nakajima, T. Kishi, Y. Sone, S. Nonose and K. Kaya, Z. Phys. D 19, 385 (1991). 7. P. Schnabel, M. P. Irion and K. G. Weil, J. Phys. Chem. 95, 9688 (1991). 8. P. Schnabel and M. P. Irion, Ber. Bunsenges. Phys. Chem. 96, 1101 (1992).

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9. J. Concei¸ca ˜o, S. K. Loh, L. Lian and P. B. Armentrout, J. Chem. Phys. 104, 3976 (1996). 10. M. Ichihashi, T. Hanmura, R. T. Yadav and T. Kondow, J. Phys. Chem. A 104, 11885 (2000); R. T. Yadav, M. Ichihashi and T. Kondow, J. Phys. Chem. A 108, 7188 (2004). 11. M. Ichihashi, T. Hanmura and T. Kondow, J. Chem. Phys. 125, 133404 (2006). 12. W.-H. Hung and S. L. Bernasek, Surf. Sci. 339, 272 (1995). 13. T. Orii, Y. Okada, K. Takeuchi, M. Ichihashi and T. Kondow, J. Chem. Phys. 113, 8026 (2000). 14. S.-R. Liu, H.-J. Zhai, M. Castro and L.-S. Wang, J. Chem. Phys. 118, 2108 (2003); M. Castro, S.-R. Liu, H.-J. Zhai and L.-S. Wang, J. Chem. Phys. 118, 2116 (2003). 15. L.-S. Wang, H.-S. Cheng and J. Fan, J. Chem. Phys. 102, 9480 (1995); L.-S. Wang, X. Li and H.-F. Zhang, Chem. Phys. 262, 53 (2000). 16. S.-R. Liu, H.-J. Zhai and L.-S. Wang, Phys. Rev. B 65, 113401 (2002). 17. M. J. Frisch et al., Gaussian 03, Revision C.02 (Gaussian, Inc., Wallingford CT, 2004).

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INFRARED SPECTROSCOPY AND THERMAL DESORPTION STUDY OF VANADIUM–MESITYLENE 1:2 SANDWICH CLUSTERS SOFT-LANDED ONTO A LONG-CHAIN N-ALKANETHIOLATE SELF-ASSEMBLED MONOLAYER MASAAKI MITSUI∗ , SHINGO DOI∗ , KAORI IKEMOTO∗, SHUHEI NAGAOKA∗ and ATSUSHI NAKAJIMA∗,†,‡ ∗

Department of Chemistry, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan †

JST-CREST, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan ‡ [email protected] Gas-phase synthesized vanadium–mesitylene 1:2 V(mes)2 cluster cations were size-selectively deposited onto a long-chain alkanethiolate selfassembled monolayer-coated gold substrate under ultrahigh vacuum conditions. Examination of the resultant clusters was conducted by infrared reflection absorption spectroscopy (IRAS) and temperature programmed desorption (TPD), showing the clusters molecularly adsorbed and maintaining a sandwich structure on the substrate. Keywords: Soft-landing; cluster; self-assembled monolayer.

1. Introduction Soft-landing of mass-selected cluster ions on suitable substrates oﬀers a promising route to realizing cluster-based nanomaterials.1,2 Since isolated clusters possess unique properties that diﬀer from those of the bulk material, it is important to establish a technique of preserving the clusters’ structures and properties intact on solid substrates. Toward this end, we have recently used long-chain alkanethiolate self-assembled monolayers (SAMs) as “soft” cluster-trapping matrixes and succeeded in the non-destructive isolation of the gas-phase synthesized open-shell organometallic sandwich clusters V(benzene)2 (Refs. 3–5) and V2 (benzene)3 (Ref. 6), at around room temperature. The room-temperature cluster isolation can be well explained 199

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Fig. 1.

Mesitylene (mes).

by cluster penetration into the long-chain alkanethiolate self-assembled monolayer (SAM).3 –5 In the current work, we applied this organic-matrix isolation technique to another open-shell organometallic sandwich complex that is composed of a vanadium (V) atom and more bulky organic ligand of mesitylene (mes) (see Fig. 1). The vanadium–mes 1:2 V(mes)2 cluster cations were size-selectively deposited onto a long-chain alkanethiolate SAM, and their adsorption state and thermal stability were investigated by infrared reﬂection absorption spectroscopy (IRAS) and temperature programmed desorption (TPD). With the aid of density functional theory (DFT) calculations, it is conﬁrmed that the V(mes)2 clusters are physisorbed and maintain the sandwich structure intact on the SAM substrate. 2. Experimental and computational methods The soft-landing apparatus has been described elsewhere.4,5,7 Brieﬂy, it consists of a cluster source, a mass selection stage, and a deposition chamber where the IRAS and TPD measurements are conducted. V(mes)2 cations were produced in the expansion from a piezo-driven pulsed valve under a He stagnation pressure of 2 atm by the reaction between laser-vaporized vanadium and mesitylene vapors. The cations thus produced were guided by a series of ion optics, separated from the neutrals and anions by a quadrupole deﬂector, and size-selected by a quadrupole spectrometer; subsequently, the clusters were deposited onto a substrate with 20 ± 10 eV collision energy under UHV conditions (∼10−10 Torr). The substrate was cooled to 200 K, and the total amount of the deposited cluster ions was determined by monitoring the ion current on the substrate during the deposition time. In the IRAS measurements, an FT-IR spectrometer (Bruker IFS66v/S) operating at ∼0.1 Torr was used. The detector and the lens system were mounted in a separate vacuum chamber pumped to a pressure of about 0.1 Torr to remove the spectral background due to atmospheric gases. All spectra were recorded with a spectral resolution of 2 cm−1 . Two hundred scans were accumulated for background and sample spectra, which were

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recorded before and the cluster deposition, respectively. In the TPD experiments, which have been described elsewhere,5 the substrate is positioned with the surface normal parallel to the quadrupole axis of the mass spectrometer and heated at 1 K/s. The desorbed species were detected by a mass spectrometer. In this study, a commercially available 10 × 10 mm2 gold substrate, Au/Ti/Silica (Auro Sheet, Tanaka Precious Metal Co. Ltd.), was used. To remove organic contaminants from the surface, the substrate was immersed in a piranha solution. Solution-phase chemisorption was then conducted by immersing the chemically polished gold substrate in a 2 mM ethanolic solution of 1-hexadecanethiol (C16 , Wako) for one day. The formation of C16 –SAM on the gold surface was conﬁrmed by IRAS and contact angle measurements.5 A geometrical optimization and harmonic frequency analysis of the neutral V(mes)2 in the doublet electronic state was carried out with the Gaussian 98 program8 using DFT with the BLYP functional. The 6–31G(d,p) basis set was employed for mesitylene, while the TZVP basis was used for the V atom. The selection of the BLYP functional had already provided excellent results for the V–benzene complex.9,10 In the comparisons between the observed and computed IR spectra, no scaling factor was used. 3. Results and discussion 3.1. IRAS of soft-landed V(mes)2 clusters Figure 2 shows the 600–1600 cm−1 region of the IRAS spectrum for V(mes)2 clusters deposited onto C16 –SAM-coated gold substrates at 180 K, along with the calculated IR spectrum for the minimum-energy structure of the V(mes)2 neutral cluster in doublet electronic state (as indicated in Fig. 2). Note that the apparent total amounts of the deposited clusters were set to provide a monolayer coverage (∼2 × 1014 cluster/cm2 ) and that the calculated spectrum in Fig. 2 was obtained by convoluting each stick peak with a 8 cm−1 (FWHM; full width half maximum) Gaussian line shape. As the number of the deposited clusters increases, ﬁve IR absorption bands at 816, 988, 1032, 1377, and 1454 cm−1 (within ±1 cm−1 ) start to appear and gradually increase in absorption intensity. These bands can compare well with the main absorption bands reported in IR data of V(mes)2 neutral complex in nitrogen matrix at 20 K.11 Although many other weak bands were also reported in the literature,11 the close similarity of the main IR features indicates that the soft-landed V(mes)2 cluster cations lose their charge and the resulting neutral clusters adsorb on the C16 –SAM substrate, while maintaining their native sandwich structure. As can be seen in Fig. 2,

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Fig. 2. IRAS spectrum of V(mes)2 clusters soft-landed onto C16 –SAM substrate (upper trace), compared with calculated IR absorption spectrum of the minimum-energy structure of the neutral V(mes)2 sandwich complex (as indicated).

excellent agreement of the observed spectrum with the calculated one further supports these facts. Thus, we can assign the bands as follows: that at 816 cm−1 to a ring C–H out-of-plane bending mode δs (CH)ring , 988 cm−1 to the symmetric ring-breathing mode ν2 (CC)ring , 1032 cm−1 to the methyl C–H rocking mode ρ(CH)methyl , and that at 1377 and 1454 cm−1 to the symmetric and asymmetric CH3 bending mode, δs (CH)methyl and δa (CH)methyl , respectively. These assignments are summarized in Table 3.1. Table 1. Measured IR absorption bands of the IRAS spectrum of adsorbed V(mes)2 clusters and their assignments. Support Modea δs (CH)ring νs (CC)ring ρs (CH)methyl δs (CH)methyl δa (CH)methyl

C16 –SAM

N2 Matrixb

816 988 1032 1377 1454

818 987, or 990 1034 1378 1450, 1454, or 1458

Note: a Vibrational modes: δs (CH)ring , ring CH out-ofplane bending; νs (CC)ring , ring-breathing; ρs (CH)methyl , methyl CH rocking; δs (CH)methyl , symmetric CH3 bending; δa (CH)methyl , asymmetric CH3 bending. b Taken from Ref. 11 in which IR band assignments were not presented.

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3.2. TPD measurements of soft-landed V(mes)2 clusters To perform mass analysis of the species adsorbed on the substrate, the TPD spectrum of the parent mass 291 [V(mes)2 ] following deposition of 3 × 1013 V(mes)2 cations onto C16 –SAM/gold substrate at 180 K were recorded [Fig. 3(a)]. For comparison, the TPD spectrum of V(benzene)2 recorded under similar experimental conditions is also shown in Fig. 3(b), which has already been reported in detail in Refs. 3 and 5. In TPD measurement, the deposition-number of the cluster ions was set to provide a relatively low coverage of ∼0.2 ML (i.e. 3–4 × 1013 ions/cm2 ) to make the level of aggregations and interactions between the adsorbed clusters themselves on the substrate negligible. The other main desorbed species detected is composed + of three kinds of ions: V(mes)+ 2 (m/z = 291), V(mes)1 (m/z = 171), and + mes (m/z = 120). These ion signals were not observed without electron impact ionization in the mass spectrometer, a result demonstrating that the soft-landed V(mes)2 cations are indeed neutralized on the substrate. The TPD spectra of these three ions yield a peak shape identical to that of V(mes)+ 2 ; the temperatures of the threshold and maximum desorption are the same (data not shown). The close similarity between the peak shapes of parent and daughter ion species indicates that these ions were obtained from an ionization process of neutral V(mes)2 . As can be seen in Fig. 3, the temperatures of the threshold and maximum desorption rate of V(mes)2 are almost the same as those of the

Fig. 3. Typical TPD spectra of (a) V(mes)2 (m/z = 291) and (b) V(benzene)2 (m/z = 207) for the deposition of 3–4 × 1013 cluster cations onto C16 –SAM substrate at 180 K. The arrows indicate the threshold desorption temperature of the adsorbed clusters. Note that other mass fragment signals of V(mes)2 exhibit TPD spectra identical to mass 291 and are not shown here.

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V(benzene)2 clusters that are incorporated inside the C16 –SAM.3–5 This agreement is very interesting because larger polarizability of V(mes)2 compared to V(benzene)2 should result in stronger intermolecular interactions with the C16 –SAM than the case of V(benzene)2 . Indeed, we have observed that the temperature of the threshold desorption from a bare gold substrate increases about 40 K from V(benzene)2 to V(mes)2 (data not shown). As described elsewhere,12 thermal desorption of the SAM-incorporated clusters has a close connection with the melting behavior (i.e. latent heat of melting) of the SAM. Hence, the similar thermal chemistry of the softlanded V(mes)2 and the SAM-incorporated V(benzene)2 clusters suggests that the V(mes)2 clusters are also trapped inside the SAM accompanied by substantial structural rearrangements/re-assembling of the neighboring alkanethiolate molecules on gold. Acknowledgments This work is partly supported by the 21st Century COE program “KEIO LCC” from the MEXT. S. N. expresses his gratitude for the research fellowship from the JSPS for Young Scientists. References 1. W. Harbich, Phil. Mag. B 79, 1307 (1999). 2. A. M. Bittner, Surf. Sci. Rep. 61, 383 (2006). 3. S. Nagaoka, E. Okada, S. Doi, M. Mitsui and A. Nakajima, Eur. Phys. J. D 34, 239 (2005). 4. M. Mitsui, S. Nagaoka, T. Matsumoto and A. Nakajima, J. Phys. Chem. B 110, 2968 (2006). 5. S. Nagaoka, T. Matsumoto, E. Okada, M. Mitsui and A. Nakajima, J. Phys. Chem. B 110, 16008 (2006). 6. S. Nagaoka, T. Matsumoto, K. Ikemoto, M. Mitsui and A. Nakajima, J. Am. Chem. Soc. 129, 1528 (2007). 7. K. Judai, K. Sera, S. Amatsutsumi, K. Yagi, T. Yasuike, S. Yabushita, A. Nakajima and K. Kaya, Chem. Phys. Lett. 334, 277 (2001). 8. M. J. Frisch et al., Gaussian 98, Revision A.11.2 (Gaussian, Inc., Pittsburgh PA, 2001). 9. P. Weis, P. R. Kemper and M. T. Bowers, J. Phys. Chem. A 101, 8207 (1997). 10. J. Wang, P. H. Acioli and J. Jellinek, J. Am. Chem. Soc. 127, 2812 (2005). 11. A. McCamley and R. N. Perutz, J. Phys. Chem. 95, 2738 (1991). 12. S. Nagaoka, T. Matsumoto, K. Ikemoto, M. Mitsui and A. Nakajima, J. Am. Chem. Soc. 129, 6, 1528 (2007).

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SIMULATION OF THE NANOINDENTATION PROCEDURE ON PURE NICKEL ON THE SMALLEST LENGTH SCALE: A SIMPLE ATOMISTIC LEVEL MODEL PETER BERKE∗,§ , MARIE-PAULE DELPLANCKE-OGLETREE†,¶ , ANDREY LYALIN‡ , VERONIKA V. SEMENIKHINA‡ and ANDREY V. SOLOV’YOV‡, ∗

BATir Department CP 194/2, Universit´e Libre de Bruxelles (ULB), av. Roosevelt, 50, 1050 Bruxelles, Belgium † Chemicals and Materials Department CP 165/63, Universit´e Libre de Bruxelles (ULB), av. Roosevelt, 50, 1050 Bruxelles, Belgium ‡

Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany § [email protected] ¶ [email protected] [email protected]

An atomic scale model has been developed to study the response of the pure nickel material during nanoindentation with shallow indentation depths (necessarily used for thin ﬁlm applications) as an alternative to the frequently used continuum methods. Keywords: Nanoindentation; atomistic approach.

1. Introduction Nanoindentation is one of the testing procedures for characterization of the mechanical behavior of bulk materials and thin ﬁlms at nanoscales.1,2 In typical nanoindentation experiments, an indenter is pushed into the sample material deforming a very small volume of the sample. In order to understand elastic and plastic deformations of the material and the contact between the indenter and the sample surface, numerical models are often used. An atomic scale model is set up to study the response of the considered pure nickel material during nanoindentation with shallow indentation depths (necessarily used for thin ﬁlm applications) as an alternative to the frequently used continuum methods.

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2. Nanoindentation procedure The prime advantage of the nanoindentation procedure is the possibility to test very small material volumes and its applicability to thin ﬁlm characterization. The principle of the nanoindentation experiment is similar to the “classical” hardness measurement of materials: a hard indenter with a deﬁned geometry is pushed into the sample material causing the elastic– plastic deformation of a small volume of the sample. Generally, the indenter is made of diamond in order to avoid spurious eﬀects of indenter tip deformation. Many indenter shapes are commercially available, the most frequently used one is the Berkovich indenter type corresponding to a pyramidal shape with a tip rounding of a few hundred nanometers (for the sharpest tips). The residual imprint on the sample surface after nanoindentation is on the order of a micrometer diameter even with high force levels and a blunt indenter type with a conical geometry (with a nominal radius of curvature of 2 µm) leaving a rather large imprint. In the case of annealed pure nickel, the average grain size of the nickel, sample can be much larger than the imprint size. As a consequence, indentations can be performed in the middle of the grains, and the measured average reponse can thus be considered to be that of a pure nickel single crystal with arbritary orientation. In contrast to traditional hardness testers, the nanoindentation system allows the application of a speciﬁed force on the indenter (on the order of µN) while simultaneously monitoring the indenter displacement (on the order of nm) to obtain the applied force–indenter displacement curve as the output (Fig. 1). The experiment follows a predeﬁned loading sequence where the applied force or the indenter displacement as a function of time is speciﬁed. Generally, three parts of the loading sequence are distinguished: the loading part where the applied force is increased until a peak value, the holding part where this peak load is maintained for a prescribed amount of time, and ﬁnally the unloading part where the applied force is decreased gradually to zero. A nanoindentation experiment lasts generally only for some tens of seconds in order to remain in the high precision domain of the measuring equipment. The indenter velocity varies between tens of nm/s to µm/s in the experiment. In thin ﬁlm–substrate sandwich nanoindentations if the indentation depth is too large the substrate inﬂuences the response of the ﬁlm, thus very small indentation depths are required to characterize the mechanical properties of thin ﬁlms. As a rule of thumb, to avoid the eﬀect of the substrate, 1/10 of the ﬁlm thickness (under 1 µm for thin ﬁlms) is usually taken as the maximum indentation depth. In the case of such shallow indents,

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Fig. 1. Nanoindentation load–displacement curve for pure nickel with a conical indenter of 2 µm nominal tip radius obtained from numerical simulations based on the finite element method.

various artifacts of the load–displacement curve have been observed that cannot be explained by the continuum theory. The size dependence of the materials is frequently observed, that is, the yield strength of the material depends on an internal length scale, an intrinsic material parameter.3 When reaching this material length scale in the experiments, the contribution of the size eﬀects is more and more pronounced. Another phenomenon is the pop-in, i.e. a sudden displacement burst (as shown in Fig. 2) in the load– displacement curve, most frequently observed at low load levels. Pop-ins are explained in the literature by sudden dislocation nucleations.4,5 The characteristic length scale and the deformed material volume in nanoindentation are on a mesoscopic scale; the material response in such an experiment is the sum of behaviors that can be described by continuum models and of individual atoms. The shallower the indent, the more the atomic level behavior of the material becomes dominant, witnessed by increasing size eﬀects and pop-ins. As mentioned before, the result of a nanoindentation experiment is the continuously monitored applied force–indenter displacement curve also known as the load–displacement curve as presented in Figs. 1 and 2.

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Fig. 2. Schema of the displacement bursts on the loading part of the load–displacement curve, also known as pop-ins.

The load–displacement curve obtained is considered to be the mechanical ﬁngerprint of a material’s response to deformation. In fact, it is composed of various contributions: the behavior of the material (elastic–plastic, time dependence, size dependence, the eﬀect of residual stresses, etc.), the geometry of the contact (surface topology and the real indenter geometry), and the eﬀects of the contact interface behavior. As a consequence, this measuring method is not straightforward to interpret and shows a strictly nonlinear behavior (the sum of each nonlinear physical phenomenon involved). Specially adapted numerical simulation tools are thus very helpful in the understanding of nanoindentation experiments. Considering the multiple scales involved, the developed numerical models are also working on diﬀerent levels. A brief review of the numerical models developed for nanoindentation and related problems are presented in the next chapter. 3. Nanoindentation numerical models Depending on the purpose of the simulation, diﬀerent aspects of the nanoindentation procedure are addressed in diﬀerent numerical models. Every

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numerical model is adapted to speciﬁc requirements; it is rarely possible to directly compare the simulation results issued from diﬀerent models. The authors refer readers to the review paper by Gouldstone,2 which considers diﬀerent numerical models and techniques applied at diﬀerent length scales of nanoindentation. When the numerical framework has to be coupled directly to experiments, most frequently the continuum models are preferred because of their computational eﬃciency. These models are well adapted for reproducing the overall average response of the material in nanoindentation and for conducting parametric studies concerning parameters that are experimentally diﬃcult to access (material properties and diﬀerent behaviors, surface roughness6 and contact geometry,7 etc.). The authors would like to mention that a ﬁnite element code is being developed for the simulation of the nanoindentation procedure in the large and moderate indentation depths domain.8 The authors refer to the work of Cheng,9 which considers the identiﬁcation of the key parameters of the nanoindentation problem and the eﬀects of various material parameters using dimensional analysis together with continuum numerical models. In order to describe the more complex physics of shallow nanoindentations where the size-dependent response of the material has to be taken into account, the continuum methods are specially adapted to microscale simulations, using higher order theories in a ﬁnite element framework (Refs. 10 and 11, for example). However, when the purpose of the numerical simulation is the understanding of plastic deformation, models usually work on smaller scales. The descent to scales where continuum mechanics is no longer applicable implies the use of more computationally expensive simulations as more information is included in the model itself. One of the relatively computationally cheap numerical methods uses the discrete dislocation plasticity model, which focuses on the plastic deformation of the crystalline material considering only the slip planes without details concerning the positions of the atoms.12–14 Experimental eﬀorts have been made to investigate the deformation mechanisms on the corresponding nanoscale by specially adapted experimental techniques. The authors refer readers to the work of Ashok,15 which considers nanoindentation with sharp AFM tips at very small indentation depth (down to 25 nm). Nibur 16 studied experimentally the activated slip systems of FCC crystals in a microindentation setting. In the experimental work of Tanaka,17 the plastic zones and slip band formation around a crack tip in silicon were observed together with the produced dislocation structure. Finally, the recent experimental work of Fujikane5 gives insight into the elastic–plastic transition in GaN crystals.

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When the details of the onset of the plastic deformation and the dislocation activity are in the focus of interest, the numerical models have to descend to the lowest scale: atomistic simulations are conducted. Generally, atomistic models are used for system sizes not larger than 100 nm (around the upper limit of the range of thin ﬁlm nanoindentations) beyond which scales of other numerical methods are more adequate to describe the behavior of the material.2 Atomic level models have the indisputable advantage of identifying the trends concerning the main physical variables of a problem and give precious qualitative information for the understanding of experimentally observed complex phenomenon (such as the size eﬀects and pop-ins) that cannot be reproduced by coarser models. The main handicap of purely atomistic models for mechanical applications is that they are computationally expensive and hence, generally, they handle length (on the order of tens of nm) and time scales (some ps) many orders of magnitude smaller than in the experiments, extending the latter being the biggest challenge of numerical modeling on the atomic level. This results, for example, in molecular dynamics simulations where the prescribed indenter speed can reach 100 m/s18–22 due to computational limitations. Even though atomic level calculations have been computationally expensive in the past years, the dynamic increase in computational power has allowed the adaptation of atomic level models to interesting mechanical applications.23 Many numerical works using molecular dynamics have considered the problem of nanoscratching24 or nanoscale machining25 and have been used for the identiﬁcation of diﬀerent wear regimes26 and the explication of tribological phenomena, like the nanoscale stick-slip27 and adhesion.28 The possible qualitative comparison of the nanoscratch behavior of Au and Pt using the numerical model and the experimental data22 also encourages the use of atomic level simulations for problems concerning nanocontact and friction.29 Considering the modeling of the nanoindentation procedure on the atomic scale, the authors refer readers to a review paper by Szlufarska30 for a general overview. The prime concern of such calculations is the understanding of the early stages of plastic deformation in the sample material (the determination of a suitable criterion deﬁning the onset of defect nucleation),31 the study of the dislocation nucleation,32–34 the observation of the indenter–sample contact,35 and the induced wear mechanism in the tip.36 From the point of view of this work, one of the most interesting papers presents the results of modeling the nanoindentation of pure nickel.37 In recent years, it has been possible to model very large systems (several million atoms38,39 ) using parallel computational techniques, but

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the computational eﬀort, directly related to the enormous quantity of information of atomic level models still remains very large. Finally, to complete our overview, the complicated but very promising hybrid methods have to be mentioned. These methods build a bridge between low and high scales using suitable diﬀerent numerical methods on each scale. The higher scales are informed by the lower scale behavior allowing the seamless treatment of multiple scales. One member of this family of methods is the quasi-continuum method, which considers atomic and structural scales (using an adaptive FEM mesh40 ) simultaneously41 for the analysis of fracture and plasticity.42,43 Atomic level models are used to precisely describe the local behavior of the material. Other methods e.g. discrete dislocation12,32 quasi-continuum39 and advanced ﬁnite element models31 prescribe more realistic boundary conditions to the atomic scale model. Coupling these two types of methods allows problems with large geometrical size to be addressed. It has to be noted, however, that ﬁnding the right boundary conditions for each considered model (working potentially at diﬀerent length scales) in the numerical multi-model assembly to assure the transition between the diﬀerent scales is far from being evident. The main drawback of these models is their advanced complexity compared to single-model methods. Taking into consideration the diﬀerent aspects of the presented numerical methods, a single-level purely atomistic numerical model has been chosen (described in detail in the next section) to simulate the nanoindentation procedure, because of its relative simplicity compared to a multi-level model. Some further simpliﬁcations of the chosen model allowed us to calculate a reasonable system size. A quantitative comparison with experimental results is beyond the scope of this study considering the extremely large computational eﬀort necessary to model nanoindentation even with very sharp AFM tips (with a nominal tip radius as small as 30 nm)15 at indentation depths where the experimental scattering is small enough to obtain results with a reasonable accuracy for the comparison. The chosen numerical model, the boundary conditions and the numerical results are presented in the next section.

4. The numerical model In nanoindentation, even for shallow indentation depths, the deformation of quite a large volume of material has to be considered with respect to the atomic scale. On the other hand, from the experimental point of view indents are so small that they can be made inside a grain and the material properties corresponding to those of single crystals (with defects) can be

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measured. For the sake of simplicity and to be able to handle large system size calculations in the most eﬃcient way, an atomic level numerical model using empirical potentials has been chosen for the simulation task. The choice of a quasi-static simulation of the nanoindentation procedure was taken as in Refs. 32 and 36. The speed of the indenter is considered to be negligible with respect to the speed of the atom vibrations, reducing the numerical problem to a structural minimization with quasi-static increments calculated using the NAMD44 software. A common practice in atomic level simulations of the nanoindentation procedure is to model the indenter using a purely repulsive potential as in Refs. 32 and 38. Even though the main purpose of this work is to study the behavior of the pure nickel sample material during shallow nanoindentation, a diﬀerent choice is made in this model, and the diamond lattice of the indenter is represented and the interaction between the tip and the sample is taken into account as in Refs. 33 and 35, etc. The interaction potentials of both the Ni–Ni and Ni–C interactions are modeled with Lennard–Jones type (Eq. 1) interaction potentials: φLJ (r) = [(r0 /r)12 − 2(r0 /r)6 ],

(1)

A] the equilibrium distance where [eV] is the energy well depth, r0 [˚ between two atoms and r [˚ A] the distance measured between two atoms. The two sets of Lennard–Jones potential parameters have been calculated to ﬁt Ni–Ni and Ni–C interactions obtained from ab initio density functional theory (DFT) calculations.25,45 The parameter sets used for the Ni–Ni and A Ni–C interactions are respectively: N i−N i = 0.4245 eV, r0N i−N i = 2.56 ˚ and N i−C = 0.1 eV, r0N i−C = 2.4 ˚ A. The indenter is constructed from 3308 carbon atoms in a diamond latA), the same material as tice (with the lattice parameter46 adiam = 3.56 ˚ in the experimental setting. Even though in reality the nanoindenter has a tip radius of Rmin = [100–800 nm] for the sharpest tips, in this study a cono-spherical tip of 2 nm radius has been considered due to computational limitations. The chosen tip geometry is in good agreement with other works using atomistic models of nanoindentation, where the indenter tip radius ranges generally from 2 nm21,33 to 18 nm.36 The deformability of the indenter is not taken into account in this work; it is modeled merely as a rigid body. Note that some works have pointed out the necessity of taking into account the tip deformation7,35 (and the tip lattice orientation) for hard sample materials. In the case of the pure nickel sample, however, the assumption that the indenter can be modeled as a rigid body holds because its hardness (64HV47 ) is well under that of diamond. A defect-free slab of FCC Ni lattice (the lattice parameter aNi = 3.52 ˚ A46 ) with dimensions 10 nm × 10 nm × 6.5 nm built from roughly

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65,600 atoms is considered for the nickel sample. The sample is constructed from unit cells with [001] orientation. Although the variation of the lattice orientation of the nickel structure is not the subject of this study, we keep in mind the dependence of the material’s response19 on the lattice orientation. The size of the lattice has been found to be suﬃciently large with respect to the size of the diamond indenter with 2 nm tip radius and the imposed displacement of 0.8 nm. The system size and the indentation depth are comparable to those used in other works.25,35 The displacement of the atoms on the lateral and bottom sides of the nickel lattice are prescribed to be zero during the whole simulation, leaving only the upper contact surface of the lattice and the enveloped volume to deform (Fig. 3). The nanoindentation simulation is divided into quasi-static increments. At the beginning of each increment, the position of the rigid diamond indenter is updated and then a structural optimization step using the structural optimization feature of the NAMD numerical code is made, recalculating the position of the free nickel atoms in order to minimize the total energy of the system. During the simulation, the indenter moves downwards and comes into contact with the sample. The rigid indenter then deforms the sample volume and is ﬁnally retracted upwards to its ﬁnal position. This methodology corresponds to a displacement controlled simulation. A program has been created which drives the whole calculation as an external frame to the structural optimization step. An increment is considered converged when a total energy tolerance condition has been satisﬁed. The average displacement step of the

Fig. 3. Schema of the applied boundary conditions in the simulation; the atoms in the indenter are frozen.

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indenter per increment in the simulation is 0.064 ˚ A, with a ﬁxed minimum step size of 0.001 ˚ A. 5. The numerical results The results of the above-mentioned quasi-static atomic level numerical model are presented in this section. The main outputs of the numerical simulation are the positions of the atoms in the deformed conﬁgurations (Fig. 4) and the corresponding total energy of the structure as a function of the indenter displacement (Fig. 5), from which the atomic scale reaction force–displacement or load–displacement curve can be computed (Fig. 6). Since the carbon atoms in the indenter are frozen, the change in the total energy of the structure is related to two contributions: the deformation of the sample volume and the contact interaction between the nickel and the carbon atoms on the indenter–sample interface. The zero reference energy level corresponds to the initial conﬁguration with the defect-free undeformed and relaxed perfect nickel lattice

Fig. 4. Snapshot in cut view of the deformed configuration during indentation at an indenter penetration of 4.38 nm. The atoms on the left, right and bottom sides are blocked (in green). The indenter carbon atoms are represented in black.

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Fig. 5. The variation of the total energy of the system as a function of the indenter displacement from the numerical model.

Fig. 6. The reaction force–indenter displacement curve or load–displacement curve from the numerical model.

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and the indenter suﬃciently far so that there is no contribution of the sample–indenter interaction to the energy balance. The initial separation distance between the indenter and the sample’s contact surface is 0.8 nm. Three domains have to be distinguished on the obtained total energy– displacement and load–displacement curves corresponding to three phases in the nanoindentation procedure on the atomic scale: the approach phase followed by the elastic then the elastic–plastic deformation of the sample volume, and ﬁnally, after reaching the peak indenter penetration value, the unloading phase. The nature of the obtained total energy–displacement curve is in very good agreement with other works considering the indentation in materials with the FCC lattice.31,33,37,38 In the approach phase, the indenter moves toward the sample until contact is established. The convention is used that the zero value on the axis of the indenter displacement on all ﬁgures corresponds to the position of the sample’s contact surface. The negative values of the indenter displacement stand for the separation between the contacting bodies (in a geometrical sense) whereas the positive values correspond to the penetration of the indenter in the sample. With diminishing separation distance (from 0.8 nm to 0.1 nm) the total energy of the system ﬁrst decreases (Fig. 5); the indenter and the sample volumes attract each other bulging the nickel contact surface. In forcecontrolled simulations, the above-mentioned attraction between the two surfaces may result in a jump-to-surface phenomenon as mentioned in Refs. 30 and 36. Further approach of the indenter to the sample (from a separation distance of around 0.1 nm onwards) the interaction force between the indenter and the sample changes to repulsion; the energy of the system increases. The authors would like to mention that the jump-to-surface feature of the numerical model could only be observed experimentally with the condition of having to manipulate atomically clean surfaces. Such clean surfaces are obviously extremely hard to prepare and are, in practice, never produced in a nanoindentation setting. Even in the case of experiments with controlled environment oxide layers, impurities are installed rapidly on the surfaces after surface preparation. The next phase of the nanoindentation, the indentation itself, starts when the repulsive interaction force between the indenter and the sample deforms the nickel lattice. This deformation is ﬁrst elastic and then followed by an elastic–plastic regime. The forced atomic displacements in the sample volume, i.e. the deformation process, results in a globally increasing energy level of the system during the indentation phase. The elastic–plastic transition in an atomic level model corresponds to the start of the nucleation of dislocations in the sample volume,31 causing the ﬁrst large energy jump on the total energy–displacement curve at around 0.24 nm (Fig. 5) of

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Fig. 7. Comparison between the results of the numerical model (dots) and the Hertzian elastic contact theory with Esample = 5000 GPa (solid line).

indenter penetration. Until this point, the assumption is made that all the sample deformation is reversible, elastic. In this case, the elastic contact model of Hertz,48 considering a deformable body with a ﬂat surface and a rigid spherical body with 2 nm of contact radius, is applicable to the problem (as in Ref. 33). Indeed, a rather good agreement between the Hertzian contact model and the numerical results has been observed (Fig. 7) by taking the sample’s Young’s modulus as Esample = 5000 GPa. This value is an order of magnitude higher than the macroscopically measured average of EN i = 207 GPa.47 The indenter penetration value corresponding to the start of major dislocation activity where the numerical load–displacement curve deviates from the Hertzian elastic contact model is near that obtained by Saraev,37 considering a nickel sample and a diamond indenter with a tip radius of 3 nm. At this indenter penetration, considering the theory of Hertz a peak contact pressure of 1300 GPa is generated. This value is two orders of magnitude larger than the theoretical shear strength of the nickel crystal deﬁned by τmax = G/2π = 12 GPa.31 Deviations of the numerical results from the theoretical predictions are often observed but generally with considerably smaller magnitude.31 Using molecular dynamic simulations, Vliet31 found that the maximum contact pressure (of some tens of GPa) corresponding to the defect nucleation is larger than the theoretical value of the shear strength of Cu crystals explained by the complicated

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stress condition under the indenter. In a paper by Fang,21 overestimated Young’s modulus and hardness values (up to 300% of the expected value) from the molecular dynamics simulation of Cu are reported. The plastic yielding seems to start oﬀ the symmetry axis (as represented in Fig. 4) of the indenter, as observed in Ref. 38, contrary to the continuum model’s and the Hertzian analytical model’s predictions (Fig. 8). With larger penetration of the indenter, up to the peak value of 0.8 nm, the system response is composed of subsequent energy and force jumps. The spacing of these energy jumps on the total energy–indenter displacement curve (Fig. 5) is closer at larger indentation depths (around 0.11 nm to 0.07 nm) and their magnitude also increases at large penetration values (up to 62 eV), showing the increasing plastic deformation of the nickel lattice. The tangent of the loading curve portions between the energy jumps increases with larger extent of sample deformation at higher indenter penetration. The sharp falls in the total energy of the system indicate the plastic relaxation of the stresses in the lattice corresponding to the reorganization of the position of a larger number of atoms via dislocation activity.

Fig. 8. The position of the plasitc yielding is under the symmetry axis of the indenter in the axisymmetric finite element simulation. The shades of gray represent the accumulated plastic deformation in the indentation direction.

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After reaching the maximum prescribed value of the indenter penetration, the indenter is retracted, corresponding to the beginning of the unloading phase. The initial decrease of the total energy during unloading can be explained by the elastic relaxation (to some extent) of the accumulated stresses in the nickel lattice. There are no signiﬁcant energy and force jumps at the beginning of the unloading up to an indentation depth of around 0.6 nm (Fig. 5), suggesting the mainly elastic nature of this nanoindentation phase. This conﬁrms the assumption of elastic unloading of the classical post-treatment method of nanoindentation developed by Oliver and Pharr49 for the extraction of the elastic properties of the material (Young’s modulus) based solely on the unloading part of the load–displacement curve. This method has been applied to the part of the numerical unloading curve where the sample material response is assumed to be purely elastic (penetration depth from 0.8 nm to 0.6 nm; Fig. 9) and the following Ol−Ph = 646 GPa, material properties have been obtained: a hardness HNi two orders of magnitude larger than the documented values for bulk nickel Ol−Ph bulk = 4 GPa50 and a Young’s modulus ENi = 4580 GPa that matches HNi well the value found by Hertzian elastic contact analysis of the loading curve. Note the similar multiplicative factor of around two orders of magnitude of the initial yield strength and of the hardness of the material in the numerical model with respect to the average material properties of pure nickel. The good agreement between the Young’s modulus calculated by the Hertzian elastic contact approximation using the numerical loading curve and the value obtained by this post-treatment method conﬁrms the simplifying assumptions of the Oliver–Pharr method based on observations on the continuum level. On the other hand, the shape of the atomic-scale unloading curve is different from those obtained with continuum models observed ones Fig. 1 and Fig. 2, i.e. the variation of the unloading tangent with decreasing indentation depth in the numerical model is more important probably due to the sample–tip adhesive interaction (having a negligible eﬀect in real scale experiments). This same interaction is mainly responsible for the energy jumps during the retraction of the indenter from penetration depths of 0.55 nm onwards (Fig. 5) due to the rearrangement of a large number of nickel atoms on the sample surface by these adhesive forces during contact separation. During the retraction of the indenter at shallow penetration depths between 0.4 nm and 0.1 nm, the determinant feature is again the interaction between the indenter and the sample surface witnessed on the load–displacement curve (Fig. 6) by the very similar shape of this portion of the curve and the approach phase. Contact separation between the indenter and the sample occurs at around 0.1 nm penetration depth, which gives an indication of the residual imprint depth. Unlike in some adhesion,28 wear29

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Fig. 9. Left: The unloading part of the total energy–displacement curve assumed to represent the elastic relaxation of stresses in the sample material. Right: The derived load–displacement curve for unloading. The dots are the numerical results and the thick solid line shows the fitted unloading curve using the Oliver–Pharr post-treatment method.

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and nanoindentation simulations,2,30 no neck formation was observed during contact separation in this study. As expected, the ﬁnal total energy of the system suﬀering permanent deformation after indentation is higher than that corresponding to the initial defect-free conﬁguration. The results obtained and the trends of this very simple atomic level model can be explained qualitatively from a physical point of view, and they are in agreement with observations in other works in the domain. Even though the response of the system is undoubtably much stiﬀer than expected (the reaction force levels, the hardness and elastic modulus are orders of magnitude larger than in experiments) the shape of both the total energy–indenter displacement and load–indenter displacement curves are similar to those documented in other works.18,19,37 6. Discussion The numerical results from this simple atomic level model set-up for nanoindentation at very shallow indentation depths are coherent with the physics of the problem and the derived trends are in good agreement with other works. For very shallow nanoindentations, an atomic scale numerical model reveals interesting features of the nanoindentation experiment concerning the plastic deformation of the sample. As demonstrated in other works,33,37,38 a very eﬀective way to observe the dislocation activity in atomic level nanoindentation simulations is the use of the centrosymmetry parameter (Eq. 2) based on the property of central symmetry of the defect-free, undeformed FCC lattice. In a future work, the authors plan to implement a post-treatment method identifying the nucleation and the propagation of dislocations in the sample material using the centrosymmetry parameter: |Ri + Ri+1 |2 , (2) P = i=1,6

where P is the centrosymmetry parameter, and Ri and Ri+1 are the vectors corresponding to six pairs of opposite nearest neighbors in the lattice. Even though the results obtained are rather satisfying, there is, however, no doubt that some aspects of the developed qualitative model are to be revisited. The stiﬀ response of the system is a result of the use of Lennard–Jones potentials. In order to describe nanoindentation at the atomic level more precisely, we are going to use the Embedded Atom Model (EAM) type of potentials which are more sutable for nickel.32,33,37,38 Note that Lee pointed out38 that diﬀerent potential types result in different dislocation mechanisms in the same simulation. The outlook of this

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study that emerges is to use more realistic potentials to describe the material behavior.

7. Conclusion A simple atomic level quasi-static model for the nanoindentation procedure has been set up and applied to the theoretical problem of a very shallow indentation (8 ˚ A deep) in a nickel single crystal with a diamond conical indenter with 2 nm contact radius modeled as a rigid body. The purpose of this work is to help to understand some features of the nanoindentation procedure linked to atomic scale mechanisms like the plastic deformation of the sample at very small indentation depths. The contact interaction between the indenter tip and the sample material is taken into account in the model. The main assumptions of the model are the use of Lennard– Jones potentials for the atomic interactions. The proposed simple model performs overall rather well: the results of this qualitative numerical model considering all the simpliﬁcations are in rather good agreement with other works. The response of the modeled system is, however, considerably stiﬀer than expected, with a Young’s modulus an order of magnitude and the hardness and yield strength two orders of magnitude higher than experimentally observed. The high stiﬀness of the system has been conﬁrmed by comparing the elastic part of the loading curve with the predictions of the Hertzian theory of elastic contact and by the Oliver–Pharr post-treatment method of the elastic unloading curve. The elastic parameters obtained from these two methods are in good agreement, conﬁrming the initially elastic unloading assumption of the Oliver–Pharr post-treatment method of nanoindentation. The main reason for this stiﬀ behavior is believed to be related to the Lennard–Jones approximation of the interaction potentials. The onset of plastic deformation in the sample material is witnessed by energy jumps on the total energy–displacement curve. The unloading phase of the nanoindentation has been identiﬁed to be initially mainly of elastic nature, which supports the use of post-treatment methods based on the unloading part of the load–displacement curves for the identiﬁcation of elastic material properties. Due to all the simpliﬁcations, the quasi-static nanoindentation can be rapidly calculated even for quite large system sizes and ﬁne displacement steps. In a future work, the system size can be increased to model a system with a realistic size. A parametric study is also planned, considering diﬀerent tip geometries. The model behavior can certainly be improved by using more realistic model potentials for the Ni–Ni and Ni–C interactions. In order to study the plastic deformation mechanism in the nickel lattice,

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a post-treatment method using the centrosymmetry parameter is planned. Atomic level models give an insight into the physics of the nanoindentation procedure. Even such simple models as the one considered have the advantage of providing important qualitative results. In a future work, the atomistic model used is planned to be applied to the problem of delamination of thin ﬁlms in a nanoindentation setting.

Acknowledgments This work was carried out through the cooperation between the Universit´e Libre de Bruxelles and the Frankfurt Institute of Advanced Studies within the EXCELL European Network of Excellence. The authors acknowledge the support of the EXCELL program.

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18. S.-R. Jian, T.-H. Fang, D.-S. Chuu and L.-W. Ji, Appl. Surf. Sci. 253, 833 (2006). 19. A. Richter, R. Ries, R. Smith, M. Henkel and B. Wolf, Diam. and Relat. Mat. 9, 170 (2000). 20. A. Noreyan, J. G. Amar and I. Marinescu, Mat. Sci. Eng. B 117, 235 (2005). 21. T.-H. Fang, C.-I. Weng and J.-G. Chang, Mat. Sci. Eng. A 357, 7 (2003). 22. T.-H. Fang, W.-J. Chang and C.-I. Weng, Mat. Sci. Eng. A 430, 332 (2006). 23. H. Raﬁi-Tabar, Phys. Rep. 325, 239 (2000). 24. R. Komanduri, N. Chandrasekaran and L. M. Raﬀ, Wear 240, 113 (2000). 25. Z.-C. Lin, J.-C. Huang and Y.-R. Jeng, J. Mat. Process. Tech. 192, 27 (2007). 26. L. Zhang and H. Tanaka, Wear 211, 44 (1997). 27. M. H. Cho, S. J. Kim, D. S. Lim and H. Jang, Wear 259, 1392 (2005). 28. J. Song and D. J. Srolovitz, Acta Mat. 54, 5305 (2006). 29. T. Harano, A Fundamental Study on Friction and Wear Behavior between Metals, JSME Symposium on Motion and Power Transmissions, The Japan Society of Mechanical Engineers (2004), p. 319. 30. I. Szlufarska, Mat. Today 9, 42 (2006). 31. K. J. Van Vliet, J. L. T. Zhu, S. Yip and S. Suresh, Phys. Rev. B 67, 104105 (2003). 32. R. E. Miller and D. Rodney, J. Mech. Phys. Sol. 56, 1203 (2008). 33. E. T. Lilleodden, J. A. Zimmerman, S. M. Foiles and W. D. Nix, J. Mech. Phys. Sol. 51, 901 (2003). 34. C. L. Kelchner, S. J. Plimpton and J. C. Hamilton, Phys. Rev. B 58, 11085 (1998). 35. D. Christopher, R. Smith and A. Richter, Nucl. Instrum. Meth. Phys. Res. B 180, 117 (2001). 36. J. H. A. Hagelaar, E. Bitzek, C. F. J. Flipse and P. Gumbsch, Phys. Rev. B 73, 045425 (2006). 37. D. Saraev and R. E. Miller, Acta Mat. 54, 33 (2006). 38. Y. Lee, J. Y. Park, S. Y. Kim, S. Jun and S. Inn, Mech. Mat. 37, 1035 (2005). 39. P. Vashishta, R. K. Kalia and A. Nakano, J. Phys. Chem. B 110, 3727 (2006). 40. V. B. Shenoy, R. Miller, E. B. Tadmor, D. Rodney, R. Phillips and M. Oritz, J. Mech. Phys. Solids 47, 611 (1999). 41. J. Knap and M. Oritz, J. Mech. Phys. Solids 49, 1899 (2001). 42. R. Miller, M. Oritz, R. Phillips, V. Shenoy and E. B. Tadmor, Eng. Frac. Mech. 61, 427 (1998). 43. R. E. Miller, L. E. Shilkrot and W. A. Curtin, Acta Mat. 52, 271 (2004). 44. NAMD Scalable Molecular Dynamics, www.ks.uiuc.edu/Research/namd/. 45. Y. Shibuta and S. Maruyama, Comp. Mat. Sci. 39, 842 (2007). 46. C. Kittel, Introduction to Solid State Physics, 7th Edn. (John Wiley and Sons, New York, 1996). 47. Metals Handbook, Vol. 2, 10th edn. (ASM International, 1990). 48. H. Hertz, J. Reine Angew. Math. 92, 156 (1882). 49. W. C. Oliver and G. M. Pharr, J. Mat. Res. 7, 1564 (1992). 50. Y. Pauleau, S. Kukielka, W. Gulbinski, L. Ortega and S. N. Dub, J. Phys. D: Appl. Phys. 39, 2803 (2006).

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QUANTUM STRUCTURING OF 4 He ATOMS AROUND IONIC DOPANTS: ENERGETICS OF Li+ , Na+ AND K+ FROM STOCHASTIC CALCULATIONS E. COCCIA, E. BODO, F. MARINETTI and F. A. GIANTURCO∗ Department of Chemistry and CNISM, University of Rome “La Sapienza”, Piazzale A. Moro 5, 00185 Rome, Italy www.uniroma1.it ∗ [email protected] E. YURTSEVER Department of Chemistry, Ko¸cUniversity, Rumeligeneri Yolu, 34450 Sariyer, Istanbul, Turkey M. YURTSEVER and E. YILDIRIM Department of Chemistry, Istanbul Technical University, 34469 Maslak, Istanbul, Turkey Quantum variational and diﬀusion Monte Carlo (VMC, DMC) calculations are carried out using accurate, ab initio interaction potentials for alkali metal ions doping small 4 He clusters. The results for Li+ , Na+ and K+ with clusters up to n = 16 reveal an interesting interplay between ionic forces and He–He interactions when driving the spatial delocalization of the solvating helium adatoms which form the initial snowball structures within the larger droplets. Keywords: He droplets; Monte Carlo calculations; ionic dopants.

1. Introduction Our knowledge and understanding of He droplets has witnessed tremendous growth in recent years and it has contributed, to a large extent, to maintaining the physics and the chemical physics of liquid helium, and of quantum ﬂuids in general, as a lively research ﬁeld attracting several, 227

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diverse researchers from many-body theory, molecular and atomic spectroscopy and quantum molecular methods.1–3 He droplets share various features with other helium systems in restricted geometries, such as ﬁlms on weakly interacting substrates, and they are liquid at low pressures and zero temperature, a property not present in any other physical system. From the standpoint of picking up possible impurities, He droplets are able to do so with any species with which they collide,3 the latter then residing either in the bulk of the droplet or away from it, depending on the strength of the helium-dopant interaction with respect to that between He–He solvent atoms (see, for example, Ref. 4). One question about 4 He clusters frequently discussed in the literature is the possible existence of magic numbers which are usually associated with particular conﬁgurations that are energetically more stable than neighboring ones and that contain a speciﬁc number of adatoms.5–7 The theoretical estimates for He droplets, in fact, indicate that in pure aggregates well-deﬁned structures are not expected,5 and even recent calculations and experiments8 have connected the presence of such magic numbers observed in pure 4 He clusters of small sizes to surface excitation processes rather than to special stability eﬀects. The situation, however, is drastically modiﬁed when a cationic impurity is being picked up by the droplet: the charged dopant is expected to modify the local environment of the liquid and to give rise to a region of increased density and to electrostriction eﬀects that can generate regular structures around the charged impurity (snowball eﬀects),9,10 which appear for speciﬁc numbers of adatoms.

2. Computational tools: An outline It is the purpose of the present work to show that quantum stochastic calculations in small 4 He clusters containing alkali atom cations (Li+ , Na+ and K+ ) can provide evidence for speciﬁc collocational features of the solvent atoms around the impurity (electrostriction) and for the appearance of regular structures of largely equivalent adatoms (snowballs). The interaction potentials between the cationic impurities and the single He atom were obtained from correlated, ab initio calculations using the coupled-clusters-single-and-double [CCSD(T)] method with the augcc-pv5Z expansion as we have discussed in more detail elsewhere.11 Our Fig. 1 reports the three potentials and compares their relative strengths: we clearly see that the interaction strength decreases in going from Li+ to K+ , a feature that will be important in the following discussion, and that the location of the potential minimum moves further out when we go from Li+ to K+ as dopants.

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Fig. 1. Computed potential energy curves for the cation–He interactions, taken from Ref. 11. Distances are in ˚ A; energies are in cm−1 .

The stochastic calculations were carried out using ﬁrst a variational Monte Carlo (VMC) approach to generate the trial functions with an increasing number of He atoms,12,13 which were in turn employed within a diﬀusion Monte Carlo (DMC) calculation for all clusters.14,15 The overall interaction was described within the sum-of-potentials (SOP) approximation: Vtot (R, r) =

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E

∆E

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−520.007 ± 0.002 −1041.35 ± 0.05 −1556.30 ± 0.10 −2059.54 ± 0.10 −2477.65 ± 0.14 −2890.79 ± 0.21 −2998.90 ± 0.27 −3107.69 ± 0.47 −3142.22 ± 1.13 −3187.59 ± 2.34 −3239.51 ± 7.15 −3261.82 ± 6.32 −3289.24 ± 4.37 −3320.55 ± 4.25 −3354.92 ± 4.48 —

— 521.35 ± 0.05 514.95 ± 0.11 503.24 ± 0.14 418.11 ± 0.17 413.14 ± 0.25 108.11 ± 0.34 108.79 ± 0.54 34.53 ± 1.22 45.37 ± 2.60 51.92 ± 7.52 22.31 ± 9.54 27.42 ± 7.68 31.31 ± 6.10 34.37 ± 6.18 —

−257.2515 ± 0.0006 −515.01 ± 0.04 −772.09 ± 0.08 −1024.84 ± 0.13 −1279.36 ± 0.08 −1521.33 ± 0.12 −1717.89 ± 0.13 −1892.97 ± 0.17 −1997.32 ± 0.34 −2051.30 ± 0.42 −2073.62 ± 1.03 −2093.87 ± 1.85 −2132.51 ± 3.49 −2154.25 ± 6.04 −2175.20 ± 7.78 —

— 257.76 ± 0.04 257.08 ± 0.09 252.75 ± 0.15 254.52 ± 0.15 241.97 ± 0.14 196.56 ± 0.18 175.08 ± 0.21 104.35 ± 0.38 53.98 ± 0.47 22.32 ± 1.11 20.25 ± 2.12 38.64 ± 3.95 21.74 ± 6.98 20.95 ± 9.85 —

−137.0076 ± 0.0005 −274.691 ± 0.004 −411.97 ± 0.05 −543.85 ± 0.18 −687.72 ± 0.06 −824.25 ± 0.08 −954.27 ± 0.05 −1083.03 ± 0.08 −1202.77 ± 0.15 −1303.50 ± 0.19 −1377.00 ± 0.36 −1472.28 ± 0.45 −1474.24 ± 0.61 −1489.62 ± 0.31 −1491.10 ± 1.43 −1494.00 ± 2.73

— 137.683 ± 0.004 137.28 ± 0.05 131.88 ± 0.19 143.87 ± 0.19 136.53 ± 0.10 130.02 ± 0.09 128.76 ± 0.09 119.74 ± 0.17 100.73 ± 0.24 73.50 ± 0.41 95.28 ± 0.58 1.96 ± 0.76 15.38 ± 0.68 1.48 ± 1.46 2.90 ± 3.08

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system. We clearly see the high level of accuracy attained in all cases and the relatively small errors associated with computed ∆E values, even for the larger clusters. In the following section, we shall further analyze the energetics and the spatial arrangements extracted from present calculations. 3. Energetics and density distributions As mentioned in the previous section, the diﬀerences existing among the cationic interactions with the adatoms allow one to immediately relate to them the variations found in the relative energetics between diﬀerent doped clusters. This is explicitly shown by the data we report in Figs. 2–4, which describe several energy features of the smaller clusters. In particular, the total DMC energies are presented in Fig. 2, where clusters up to n = 15 are computed [n = 16 for K+ (He)n ]. One clearly sees there that all three systems exhibit marked changes in energy gradient as the clusters grow larger, this being clearly seen for n = 6 in Li+ -doped clusters, for n = 10 for Na+ -containing clusters, and for n = 12 in K+ -doped clusters. The spatial diﬀerences between 0

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cation–solvent interactions are thus important in deﬁning the smallest size of a cluster beyond which that interaction is largely screened and markedly weakened with respect to the smaller aggregates. Such diﬀerences are even more visible when we plot the single-particle evaporative energies as given by the data of Fig. 3. The ﬁlled circles, representing the Li+ -doped aggregates, show a marked equivalence of the ﬁrst four He adatoms (forming a tetrahedron around Li+ ; see Ref. 18) and only a small variation for the next two (leading to the octahedral conﬁguration), while a dramatic drop in binding energy is occurring for clusters with n > 6. The latter value remains essentially the same up to the largest clusters we have considered. The situation is, however, very diﬀerent when we look at the Na+ -doped cluster energies, given by the open squares in Fig. 3. The initial binding energies are smaller than for the Li+ (He)n aggregates as is expected from the potential diﬀerences shown in Fig. 1. They also remain largely equivalent for the ﬁrst six adatoms but only change by small amounts as n increases up to nine, after which the ∆E values are very similar in behavior to those shown by Li+ -doped clusters thereby indicating an essential independence of the binding energies from the nature of the central cationic dopant (see, for example, Ref. 19).

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0 2 4 6 8 10 12 14 16 number of helium atoms

Fig. 4. Computed zero-point-energy values of the three sets of doped clusters as a function of cluster size. The values are given as percentages of the total classical energy minima computed by Ref. 11.

The behavior of the clusters which contain K+ as an impurity (open triangles in the ﬁgure) is diﬀerent yet again: the equivalent adatoms now extend up to n = 12 and the energy values necessary to evaporate them are much smaller than for the previous cations. Furthermore, the energetic equivalence remains, albeit with small oscillations, up to that n value, beyond which the binding energy of a single 4 He atom drops below 20 cm−1 and hovers around such a value until n = 16. In other words, the ﬁrst regular structure of the adatoms surrounding the dopant cation seems to reach a microsnowball conﬁguration by n = 12, i.e. the dodecahedral regular structure we discussed in Refs. 18 and 20. Another interesting indicator of the energetics of the doped clusters is provided by an estimate of their zero-point-energy (ZPE) values as a function of cluster size. The data reported in Fig. 4 show, therefore, the ZPE% values along the clusters examined in the present work: they were obtained by combining knowledge of total potential minima found in our earlier study11 with the present quantum ground states from stochastic calculations. It is interesting to note that the lighter Li+ cation, because of its stronger interaction with the adatoms, provides smaller values of ZPE

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and therefore a behavior closer to classical aggregates compared with both Na+ (He)n and K+ (He)n . The latter, in fact, turns out to be the one with the largest quantum eﬀect on the energy and with the most marked increase in ZPE as n increases. The diﬀerences in the quantum behavior of these aggregates can also be seen when the spatial conﬁgurations are analyzed along the series: the data reported in Figs. 5–7, in fact, help us to better understand the detailed features of the doped clusters as their size varies and as one dopant cation is replaced by another. The computed radial densities (in units of ˚ A−1 ) are measured either from the center-of-mass (CoM) of the clusters (given by solid lines) or from their geometric center17 (given by dashes): the latter frame of reference simply includes all distances without the mass weighting present in the CoM deﬁnition. The comparison between the three diﬀerent cations in Fig. 5 tells us the following: (i) The Li+ impurity, small and strongly interacting with the solvent, gets located at the center of the microaggregate, thereby making the two distributions essentially coincide with one another, with the same also

Fig. 5. Computed radial density distributions of the cationic dopants (inner regions) and of the 4 He adatoms (outer regions) measured from the CoM (solid line) or from the geometric center (dashes).

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Fig. 6. The same quantities as in Fig. 5 but for the larger clusters containing 15 4 He adatoms.

occurring for the 4 He adatom distribution: the complex therefore corresponds to an octahedral arrangement of the solvent atoms around a central Li+ dopant.18 (ii) In the case of the Na+ , on the other hand, the cations’ distributions diﬀer from each other depending on whether one considers the reference frame to be placed in the CoM of the system or at its geometric center (GC): this feature tells us that the heavy impurity is not placed at the center of the cluster and that the solvent atoms are clustered asymmetrically in space. This shifted location feature is even more prominent in the case of the potassium cation (bottom panel of Fig. 5), where the K+ radial distribution is more markedly shifted and where the 4 He distribution is also more spread out when the geometric center is considered. As we move to larger clusters, as shown by the density distributions reported in Fig. 6, the features discussed above remain present along the series of cationic dopants and further reveal the spatial diﬀerences between their distributions of solvent atoms. In particular, we see conﬁrmed the Li+

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Fig. 7. Computed angular distributions for three diﬀerent doped clusters. The solid lines refer to an angle centered on the cationic dopant while the dashes describe the data for angles centered on one of 4 He adatoms.

collocation at the center of its aggregates and the lopsided placement of Na+ and K+ within their respective clustering structures. It is interesting to note, in fact, that the cluster containing Li+ as a dopant has now completed its ﬁrst microsnowball with n = 6, where that number of adatoms forms a regular, highly localized structure,18 while the remaining solvent atoms are still not enough to complete a second shell and therefore provide 4 He distributions which are very diﬀuse: the cationic dopant, on the other hand, still sits at the center of the cluster. In the examples of doped clusters containing Na+ and K+ we see, on the other hand, that both cationic impurities do not sit, as yet, at the center of the clusters although the structures are now more symmetric than those in Fig. 5. Furthermore, the density distributions associated with the 4 He adatoms show an interesting diﬀerence: the Na+ aggregate completes its ﬁrst shell around n = 10–12 and therefore the additional adatoms cover a less localized range of distances outside the inner shell. On the other hand, the K+ -containing cluster is just completing the ﬁrst shell, thereby showing a more localized radial distribution of the solvent atoms present in that cluster.20

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The formation of regular structures around the cationic dopants together with the existence of more liquid layers beyond those structures is one of the characteristic features of the charged impurity behavior in 4 He droplets and is pictorially described as a snowball. The data shown in Fig. 7 indeed provide another way of detecting the above behavior. We report there the probability distributions of two sets of angles between partners in each cluster. Those given by a solid line refer to angles for which the ionic dopant is at the center of the three partners deﬁning that angle, while those given by dashes report the distributions of angles which are centered on one of the solvent atoms. (i) The ﬁrst shell for Li+ gets completed around n = 8 and therefore the remaining solvent atoms in that cluster behave more like in a liquid state, showing a marked delocalization. As a result, the angular distribution centered on the dopant exhibits a bimodal behavior: the smaller angles referring to the octet structure and the large angles to the more liquid component. The fact that the cationic dopant sits at the center of the cluster further produces a broad distribution over smaller angular values for those angles that have 4 He as their central partner (dashes in the top panel). (ii) The Na+ -containing cluster, on the other hand, is completing its snowball structure and therefore all angular distributions are seen to be very regular and markedly localized in speciﬁc angular regions: no solvent atoms are available to produce delocalized, liquid-like behavior outside that primary shell. (iii) Finally, the K+ -doped cluster follows the behavior of the sodium-doped aggregate because here its ﬁrst shell is not, as yet, completed and therefore its solvent atoms attempt the formation of a regular spatial conﬁguration18 while being on the way to completing their ﬁrst shell. The diﬀerent stages of formation of possible, regular structures in the different clusters can also be seen by looking at the radial density distributions along the series of aggregates we have analyzed in the present work. The presence of an electrostriction eﬀect18 implies that such densities are always higher than those of bulk 4 He because the stronger ionic attractions drive the 4 He solvent to crowding the spatial regions adjacent to the dopant cations (see, for example, Ref. 19; see also Ref. 20). Our calculations reported in Fig. 8 indeed conﬁrm the above expectation. The Li+ -containing clusters clearly show full shell completion at n = 8 and also a tremendous density increase near the dopant (taken to be at the origin of the radial axis) with respect to the bulk value (∼0.02 ˚ A−3 ),1 when n = 8 is reached. In this case, in fact, the electrostriction eﬀects also extend to the outer region, where a second shell begins to form from n = 10.

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Fig. 8. Computed DMC densities for the three diﬀerent doped clusters. The number of adatoms is indicated for each distribution.

In the case of the Na+ dopant, instead, we see shell completion by the time n = 12 is reached, with an intermediate shell contraction for n = 10 and with still larger solvent densities beyond ten with respect to the bulk value. Finally, the results for the K+ -containing clusters indicate full shell completion for n = 15 and electrostriction features already clearly seen for n = 12. In fact, our data for K+ do not as yet show the further formation of the second shell beyond the ﬁrst solid-like structure. 4. Summary and conclusions In the present study, we have analyzed via a stochastic approach, i.e. by employing a VMC treatment followed by a DMC calculation, the quantum energetics and quantum spatial structures of small 4 He clusters which contain cationic dopants like Li+ , Na+ and K+ . The interaction potential has been put together within the SOP approximation, i.e. disregarding at this level the additional contributions from MB forces. One of our earlier studies11 has already addressed this aspect and found that the SOP treatment yields the correct structures with respect to those that include MB corrections and that the latter modify only marginally the overall

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energetics.4 Thus, we have used ab initio potentials to describe ion–He interactions11 and one of the most realistic empirical potentials to describe the He–He interaction.16 The present computational analysis has conﬁrmed the dominance of the ionic forces in driving the cluster spatial structures to their energy minima and further conﬁrmed the weaker and spatially more diﬀuse nature of the clusters doped by Na+ and K+ with respect to Li+ . Thus, the latter was shown to have smaller contributions from ZPE eﬀects and was seen to complete the ﬁrst shell with the smallest number of adatoms (eight 4 He atoms) while Na+ and K+ have 12 and 15 solvent atoms in their ﬁrst solvation shell, respectively. The energy calculations also show the formation of microsnowball structures around the dopants, occurring with 6, 8 and 10 adatoms for Li+ , with 12 solvent particles for Na+ and K+ . Such solid-like structures are also conﬁrmed by an analysis of radial densities in various clusters, where the comparison between CoM-originating distributions and geometric-centered distributions reveals that both Na+ and K+ start growing their clusters with asymmetric collocations of the solvent atoms around an oﬀ-center cation. On the other hand, the more compact Li+ dopant locates itself at the center of the droplet and thus the 4 He adatoms are more symmetrically located around that impurity. The present calculations also conﬁrm the existence of marked electrostriction eﬀects in the systems under study, i.e. that the solvent atom placement in the smaller clusters is driven by the strong interactions between the solvated cations and the quantum adatoms, where the relevant forces generate marked crowding of such partners in the regions immediately surrounding the ionic dopant with a sizeable increase of 4 He density values with respect to the bulk estimates of this quantity. Acknowledgments The ﬁnancial support of the Research Committee of the University of Rome “La Sapienza” and the computational help from the Supercomputing Consortium CASPUR are gratefully acknowledged. We also acknowledge the support of the EU Network COMOL (contract number HPRN-CT-200200290). References 1. J. P. Toennies and A. F. Vilesov, Ann. Rev. Phys. Chem. 49, 1 (1998). 2. F. Stienekemeier and A. F. Vilesov, J. Chem. Phys. 115, 10119 (2001).

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3. J. P. Toennies and A. F. Vilesov, Angew. Chem. Int. Ed. 43, 2622 (2004). 4. C. Di Paola, F. Sebastianelli, E. Bodo, I. Baccarelli, F. A. Gianturco and M. Yurtsever, J. Comp. Theor. Chem. 1, 1045 (2005). 5. R. Melzer and J. G. Zabalitzky, J. Phys. A 17, L565 (1984). 6. J. P. K. Doyl and F. Calvo, J. Chem. Phys. 116, 8307 (2002) 7. M. Casas and S. Stringari, J. Low Temp. Phys. 79, 135 (1990). 8. R. Guardiola, O. Kornilov, J. Navarro and J. P. Toennies, J. Chem. Phys. 124, 084307 (2006). 9. F. Stienkemeier and S. Mende, Rev. Sci. Instrum. 74, 4071 (2003). 10. K. P. Atkins, Phys. Rev. 116, 1339 (1959). 11. F. Marinetti, E. Coccia, E. Bodo, F. A. Gianturco, E. Yurtsever, M. Yurtsever and E. Yildrim, Theor. Chem. Acc. 118, 53 (2007). 12. D. Bressanini, G. Morosi and M. Mella, J. Chem. Phys. 116, 5345 (2005). 13. A. Mushinski and M. Nightingale, J. Chem. Phys. 101, 8831 (1994). 14. D. M. Ceperly and B. Alder, Science 231, 555 (1986). 15. F. Paesani and F. A. Gianturco, J. Chem. Phys. 116, 10170 (2002). 16. K. T. Tang and J. P. Toennies, J. Chem. Phys. 118, 4976 (2003). 17. F. Sebastianelli, E. Bodo, I. Baccarelli, C. Di Paola, F. A. Gianturco and M. Yurtsever, Comp. Mat. Sci. 35, 261 (2006). 18. E. Coccia, F. Marinetti, E. Bodo, F. A. Gianturco, E. Yurtsever, M. Yurtsever and E. Yildirim, J. Chem. Phys. 126, 124319 (2007). 19. M. Rossi, M. Verona, D. E. Galli and L. Reatto, Phys Rev. B 69, 212510 (2004). 20. E. Coccia, E. Bodo and F. A. Gianturco, submitted to Phys. Rev. Lett.

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ON THE THEORY OF PHASE TRANSITIONS IN POLYPEPTIDES ALEXANDER V. YAKUBOVICH∗, ILIA A. SOLOV’YOV, ANDREY V. SOLOV’YOV and WALTER GREINER Frankfurt Institute for Advanced Studies, J. W. Goethe University, Frankfurt am Main 60438, Germany ∗ [email protected] We suggest a theoretical method based on statistical mechanics for treating the α-helix ↔ random coil transition in polypeptides. This process is considered to be a ﬁrst-order-like phase transition. The developed theory is free of model parameters and is based solely on fundamental physical principles. We apply the developed formalism to the description of thermodynamical properties of alanine polypeptides of diﬀerent lengths. We analyze the essential thermodynamical properties of the system, such as heat capacity, phase transition temperature, and latent heat of the phase transition. Also, we obtain the same thermodynamical characteristics from molecular dynamics simulations and compare the results with those of statistical mechanics calculations. The comparison proves the validity of the statistical mechanics approach and establishes its accuracy. Keywords: Dynamics and conformational changes in polypeptides; structural transitions in nanoscale materials; general studies of phase transitions.

1. Introduction The phase transitions in ﬁnite complex molecular systems, i.e. the transition from a stable 3D molecular structure to a random coil state or vice versa [also known as the (un)folding process], has a long history of investigation (for review see, e.g., Refs. 1–4). The phase transitions of this or a similar nature occur or can be expected in many diﬀerent complex molecular systems and in nano objects, such as polypeptides, proteins, polymers, DNA, fullerenes and nanotubes.5 They can be understood as ﬁrst-order phase transitions, which are characterized by the rapid growth of the system’s

241

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internal energy at a certain temperature. As a result, the heat capacity of a system as a function of temperature acquires a sharp maximum at the temperature of phase transition. In our recent paper,6 a novel ab initio theoretical method for the description of phase transitions in the above-mentioned molecular systems was suggested. In particular, it was demonstrated that in polypeptides (chains of amino acids) one can identify speciﬁc, so-called twisting degrees of freedom responsible for the folding dynamics of amino acid chains, i.e. for the transition from a random coil state of the chain to its α-helix structure. The twisting degrees of freedom are also referred to as the torsion degrees of freedom. The essential domain of the potential energy surface of polypeptides with respect to these twisting degrees of freedom can be calculated and thoroughly analyzed on the basis of ab initio methods such as density functional theory (DFT) or the Hartree–Fock method. It was shown6,7 that this knowledge is suﬃcient for the construction of the partition function of a polypeptide chain and thus for the development of its complete thermodynamic description, which includes the calculation of all essential thermodynamic variables and characteristics, e.g., free energy, heat capacity and phase transition temperature. The method has been proved to be applicable for the description of phase transitions in polyalanine chains of diﬀerent lengths via the comparison of theory predictions with the results of several independent experiments and of molecular dynamics simulations. Similar descriptions can be developed for a large variety of complex molecular systems. Earlier studies of the folding process based on statistical mechanics principles8–11 always contained some empirical parameters and thus could hardly be used for ab initio predictions of essential characteristics of the phase transitions. Since then, a very large number of published papers have been devoted to this problem. Here, we do not intend to review all of them, but refer only to those which are related directly to our work (for a review, see also Refs. 1, 3, 4, 7 and 12, and references therein). In the present contribution, we describe a theoretical approach based on statistical mechanics for treating the α-helix ↔ random coil phase transition in alanine polypeptides. The suggested method is a further development of the method suggested in Refs. 5 and 6, which is based on the construction of a parameter-free partition function for a system experiencing a phase transition. All the necessary information for the construction of such a partition function can be calculated on the basis of ab initio DFT, combined with molecular mechanics theories. Comparison of the results of this method with the results of molecular dynamics simulations allows one to establish the accuracy of the new approach for suﬃciently large molecular systems and then to extend the description to larger

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molecular objects, which is especially essential in those cases when molecular dynamics simulations are almost impossible because of computer power limitations. We note that the suggested method is considered to be an eﬃcient, novel alternative to the existing theoretical approaches for the study of helix–coil transitions in polypeptides since it does not contain any model parameters and gives a universal recipe for constructing the partition function in complex molecular systems. The partition function of the polypeptide is constructed based on a minimal number of assumptions about the system, which are diﬀerent from those used in earlier theories. It includes all essential physical contributions needed for the description of the helix–coil transition in polypeptides. Therefore, the ﬁnal expression for the partition function obtained within the framework of our theory is diﬀerent from those suggested earlier. The developed formalism is applied to the description of the α-helix ↔ random coil phase transition in alanine polypeptides of diﬀerent lengths. We have chosen this system because it has been widely investigated both theoretically8–11,13–25 and experimentally26–29 during the last ﬁve decades (for a review see, e.g., Refs. 1–4, 7 and 12) and thus is a perfect system for testing a novel theoretical approach. We have calculated the potential energy surface (PES) of polyalanines of diﬀerent lengths with respect to their twisting degrees of freedom. This was done within the framework of classical molecular mechanics. The calculated PES was used to construct the parameter-free partition function of a polypeptide, which was then in turn used to derive various thermodynamical characteristics of alanine polypeptides as a function of temperature and polypeptide length. Namely, we have calculated and analyzed the temperature dependence of the heat capacity and the latent heat of alanine polypeptides consisting of 21, 30, 40, 50 and 100 amino acids.

2. Statistical mechanics model for the α-helix ↔ random coil phase transition Let us consider a polypeptide, consisting of n amino acids. The polypeptide can be found in one of its numerous isomeric states that have diﬀerent energies. A group of isomeric states with similar characteristic physical properties is called a phase state of the polypeptide. Thus, a regular bounded α-helix state corresponds to one phase state of the polypeptide, while all possible unbounded random conformations can be denoted as the random coil phase state.

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2.1. Hamiltonian of a polypeptide chain To study thermodynamic properties of the system, one needs to investigate its potential energy surface with respect to all degrees of freedom. For the description of macromolecular systems, such as polypeptides and proteins, eﬃcient model approaches are necessary. One of the most common tools for the description of macromolecules is based on the so-called molecular mechanics potential.30–32 Here, we do not discuss it, but refer to our recent paper, where this was done in detail.7 The most important twisting degrees of freedom for the description of the helix–coil transition in polypeptides are the twisting degrees of freedom along the backbone of the polypeptide.5–7,33–38 These degrees of freedom are deﬁned for each amino acid of the polypeptide except for the boundary ones and are described by two dihedral angles ϕi and ψi (see Fig. 1). Both angles are deﬁned by four neighboring atoms in the polypeptide chain. The angle ϕi is deﬁned as the dihedral angle between the planes formed by the atoms (Ci−1 − Ni − Ciα ) and (Ni − Ciα − Ci ). The angle ψi is deﬁned as the dihedral angle between the (Ni − Ciα − Ci ) and (Ciα − Ci − Ni+1 ) planes. A Hamiltonian function of a polypeptide chain is constructed as a sum of the potential, kinetic and vibrational energy terms. For a polypeptide chain in a particular conformational state j consisting of n amino acids and N atoms, we obtain Hj =

3N −6 1 (j) p2i P2 (j) (j) + (I1 Ω21 + I2 Ω22 + I3 Ω23 ) + + U ({x}), 2M 2 2mi i=1

(1)

Fig. 1. Dihedral angles ϕ and ψ used for characterization of the secondary structure of a polypeptide chain. The dihedral angle χi characterizes the rotation of the side radical along the Ciα –Ciβ bond.

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(j)

where P, M , I1,2,3 , Ω1,2,3 are, respectively, the momentum of the whole polypeptide, its mass, its three main momenta of inertia, and its rotational frequencies. pi , xi and mi are the momentum, the coordinate and the generalized mass describing the motion of the system along the ith degree of freedom, respectively. U ({x}) is the potential energy of the system, a function of all atomic coordinates in the system. One can group all the degrees of freedom in a polypeptide into the two classes: “stiﬀ” and “soft” degrees of freedom. We refer to the degrees of freedom corresponding to the variation of bond lengths, angles and improper dihedral angles (see Fig. 1) as “stiﬀ”, while degrees of freedom corresponding to the angles ϕi and ψi are classiﬁed as “soft” degrees of freedom. The “stiﬀ” degrees of freedom can be treated within the harmonic approximation because the energies needed for a noticeable change in the system structure with respect to these degrees of freedom are about several eV, signiﬁcantly larger than the characteristic thermal energy of the system at room temperature, which is on the order of 0.026 eV.30–32,37–39 The Hamiltonian of the polypeptide can be rewritten in terms of the “soft” and “stiﬀ” degrees of freedom. Transforming the set of Cartesian coordinates {x} to a set of generalized coordinates {q} corresponding to the “soft” and “stiﬀ” degrees of freedom, one obtains Hj =

ls ls psi psj 1 (j) P2 (j) (j) + (I1 Ω21 + I2 Ω22 + I3 Ω23 ) + gij 2M 2 2 i=1 j=l

+

ls ls +lh i=1 j=ls +1

gij psi phj

+

ls +lh

ls +lh

i=ls +1 j=ls +1

gij

phi phj + U ({q s }, {q h }), 2

(2)

where q s and q h are, respectively, the generalized coordinates corresponding to the “soft” and “stiﬀ” degrees of freedom, and ps and ph are the corresponding generalized momenta. ls and lh are the numbers of the “soft” and “stiﬀ” degrees of freedom in the system, satisfying the relation 3N − 6 = ls + lh . U ({q s }, {q h }) in Eq. (2) is the potential energy of the system as a function of the “soft” and “stiﬀ” degrees of freedom. 1/gij has the meaning of generalized mass. The motion of the system with respect to its “soft” and “hard” degrees of freedom occurs on the diﬀerent time scales, as discussed in Ref. 40. The typical oscillation frequency corresponding to the “soft” degrees of freedom is on the order of 100 cm−1 , while for the “stiﬀ” degrees of freedom it is more than 1000 cm−1 .40 Thus, the motion of the system with respect to the “soft” degrees of freedom is uncoupled from the motion of the system with respect to the “stiﬀ” degrees of freedom. Therefore, the ﬁfth term

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in Eq. (2), which describes the kinetic energy of the “stiﬀ” motions in the polypeptide can be diagonalized. The corresponding set of coordinates {˜ qh} describes the normal vibration modes in the “stiﬀ” subsystem: 2 2 lh µhi ωi2 q˜ih p˜hi 1 (j) 2 P2 (j) 2 (j) 2 + (I1 Ω1 + I2 Ω2 + I3 Ω3 ) + + Hj = 2M 2 2 2µhi i=1 +

ls ls

gij

i=1 j=1

psi psj + U ({χ}) + U ({ϕ, ψ}). 2

(3)

Here, ωi and µhi are the frequency of the ith “stiﬀ” normal vibrational mode and the corresponding generalized mass. Note that the fourth term in Eq. (2) vanishes if the “soft” and the “stiﬀ” degrees of freedom are uncoupled. The last two terms in Eq. (3) describe the potential energy of the system with respect to the “soft” degrees of freedom. For every amino acid, there are at least two “soft” degrees of freedom, corresponding to the angles ϕi and ψi (see Fig. 1). Some additional “soft” degrees of freedom involve the rotation of the side radicals in amino acids. A typical example is the angle χi , which describes the twisting of the side chain radical along the Ciα –Ciβ bond (see Fig. 1). The angle χi is deﬁned as the dihedral angle between the planes formed by the atoms (Ci –Ciα –Ciβ ) and by the bonds β Ciα –Ciβ and Ciβ –Hi1 . Note that the notations χ, ϕ and ψ are used for simplicity and for further explanation of our theory. The set of these dihedral angles builds up the set of “soft” degrees of freedom of the polypeptide: {q s } ≡ {χ, ϕ, ψ}. 2.2. Partition function If one assumes a harmonic potential for the “stiﬀ” degrees of freedom, it is possible to derive the following expression for the partition function of a polypeptide in a particular conformational state j 7 :   (j) (j) (j) ls s Vj M 3/2 I1 I2 I3 µi l i=1  B(kT )3N −3− 2s Zj =  lh ls −l 3N χ (2π) 2 π i=1 ωi

π

π U ({ϕ,ψ}) × ··· e− kT dϕ1 · · · dϕn dψ1 · · · dψn −π

−π

3N −3− l2s

= Aj B(kT )

Usr ({ϕ, ψ}) + Ulr ({ϕ, ψ}) ··· exp − kT −π −π π

× dϕ1 · · · dϕn dψ1 · · · dψn ,

π

(4)

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where Aj denotes the factor in the square brackets. Vj is the speciﬁc volume of the polypeptide in conformational state j, lχ is the number of the χ degrees of freedom in the system, and k and T are the Boltzmann constant and the temperature, respectively. Note that generalized masses µhi are reduced during the integration and do not enter into the expression of the partition function. lχ kχ kχ B = exp − , I0 kT kT where kχi is the stiﬀness parameter of the potential describing the rotation of the side radical, and I0 is the Bessel function. Usr in Eq. (4) describes the potential energy of the system corresponding to the short-range interactions of individual amino acids in the polypeptide. Ulr accounts for the long-range interactions, such as the Coulomb and van der Waals interactions between distant amino acids. Since a polypeptide exists in diﬀerent conformational states, one needs to sum over the contributions of all possible conformations Zj in order to calculate the complete partition function of the polypeptide. For an ensemble of N non-interacting polypeptides, the partition function reads as N  ξ Zj  Z= j=1

 l

s = B(kT )3N −3− 2

ξ j=1

Aj

π

−π

Usr ({ϕ, ψ}) + Ulr ({ϕ, ψ}) exp − kT −π

···

π

N × dϕ1 · · · dϕn dψ1 · · · dψn  ,

(5)

where Zj is deﬁned in Eq. (4) and ξ is the total number of possible conformations in a polypeptide. Equation (5) has been derived with a minimum number of assumptions about the system. In general, however, its use for a particular molecular system is not so straightforward. Expression (5) can be further simpliﬁed, if one makes additional assumptions about the structure of the system. For the sake of simplicity, we write further equations for only one polypeptide instead of N . Generalization for the case of N statistically independent polypeptides can always be done according to (5). One can expect that the factors Aj in (5) depend on the chosen conformation of the polypeptide. However, since the values of speciﬁc volumes, momenta

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of inertia and frequencies of normal vibration modes of the polypeptide in diﬀerent conformations are expected to be close,6,41 the values of Aj in all these conformations can be considered to be equal, at least in the zero-order approximation. Thus, Aj ≡ A. It is important to note that for neutral, non-polar amino acids the long-range term Ulr ({ϕ, ψ}) in the partition function (5) can be neglected because the long-range interactions in this case are small. With the assumptions made, the partition function of the polypeptide reduces to ξ n π π (j) i (ϕ, ψ) 3N −3− l2s dϕdψ, (6) exp − Z = AB(kT ) kT j=1 i=1 −π −π (j)

where i (ϕ, ψ) is the potential energy of the ith amino acid in the polypeptide, in one of its ξ conformations denoted by j. The potential energy of the amino acid is calculated as a function of its twisting degrees of freedom ϕ and ψ. Further simpliﬁcations of the partition function Eq. (6) for a polypeptide consisting of identical non-polar, neutral amino acids can be achieved if one assumes that each amino acid in the polypeptide can occupy two states only, below referred to as the bounded and unbounded states. The amino acid is considered to be in the bounded state when it forms one hydrogen bond with the neighboring amino acids. In the unbounded state, amino acids do not have hydrogen bonds. When the α-helix is formed, all amino acids are in the bounded state, while in the case of the random coil state all amino acids occupy the unbounded states. With the assumptions outlined above and assuming the polypeptide consists of n identical amino acids, the partition function (6) of the system can be rewritten as n−2 3N −3− l2s (n − i)Zbi Zun−i + Zun βZbn−1 Zu + β Z = AB(kT ) i=4

(n−3)/2

+

i=2

βi

n−i−3 k=i

 (k − 1)!(n − k − 3)! Z k+3i Zun−k−3i  . i!(i − 1)!(k − i)!(n − k − i − 3)! b (7)

Here, the ﬁrst and the third terms in the square brackets describe the partition function of the polypeptide in the α-helix and in the random coil phases, respectively, while the second term in the square brackets accounts for the phase co-existence. The summation in the second term in (7) is performed from i = 4, because the shortest α-helix consists of four amino

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acids. The last term in the square brackets accounts for the polypeptide conformations in which a number of amino acids in the helix conformation are separated by amino acids in the random coil conformation. The ﬁrst summation in this term goes over the separated helical fragments of the polypeptide, while the second summation goes over individual amino acids in the corresponding fragment. Polypeptide conformations with two or more helical fragments are energetically unfavorable. This fact was discussed in our recent paper.7 Therefore, the fourth term in Eq. (7) can be omitted in the construction of the partition function. Zb and Zu are the contributions to the partition function from a single amino acid being in the bounded or unbounded states respectively: (b) (ϕ, ψ) exp − dϕdψ kT −π −π (u)

π π (ϕ, ψ) exp − dϕdψ kT −π −π (b) 3 π π 1 (ϕ, ψ) + (u) (ϕ, ψ) exp − dϕdψ kT Zu3 −π −π 3EHB exp − , kT

Zb = Zu = β= ≈

π

π

(8) (9)

(10)

where (b) (ϕ, ψ) and (u) (ϕ, ψ) are the potential energies of a single amino acid in the bounded or in the unbounded states, respectively, calculated versus the twisting degrees of freedom ϕ and ψ. β is a factor accounting for the entropy loss of the helix initiation. EHB is the energy of a single hydrogen bond. Substituting (8)–(10) into (7), one obtains the ﬁnal expression for the partition function of a polypeptide undergoing an α-helix ↔ random coil phase transition. This result can be used for the evaluation of all thermodynamical characteristics of the system. 3. Molecular dynamics Molecular dynamics (MD) is an alternative approach that can be used for the study of phase transitions in macromolecular systems. Within the framework of MD, one tries to solve the equations of motion for all particles in the system interacting via a given potential. Since the MD technique is well known and described in numerous textbooks,42–44 we will only present the basic equations and underlying ideas.

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MD simulations usually imply the numerical solution of the Langevin equation44–46 : mi ai = mi r¨i = −

∂U (R) − βi vi + η(t). ∂ri

(11)

Here, mi , ri , vi and ai are, respectively, the mass, radius vector, velocity and acceleration of the atom i. U (R) is the potential energy of the system. The second term describes the viscous force, which is proportional to the particle velocity. The proportionality constant βi = mi γ, where γ is the damping coeﬃcient. The third term is the noise term, which represents the eﬀect of a continuous series of collisions of the molecule with the atoms in the medium. In the MD formalism, the system of Langevin equations for all particles is being integrated with respect to time. In this contribution, we use the MD approach to study the α-helix ↔ random coil phase transition in alanine polypeptides and compare the results with those obtained using the statistical mechanics approach. For the simulations, we use the CHARMM27 force ﬁeld32 to describe the interactions between atoms. This is a common empirical force ﬁeld for treating polypeptides, proteins and lipids.32,47–49 We perform MD simulations of alanine polypeptides consisting of 21, 30, 40, 50 and 100 amino acids. For this study, it is necessary to specify the initial conditions for the system, i.e. to deﬁne the initial positions of all atoms and set their initial velocities. We assume the initial structure of the polypeptide to be an ideal α-helix2,50 and assign the particle velocities randomly according to the Maxwell distribution at a given temperature. The set of parameters used in our simulations can be found in Refs. 42–44. 4. Results and discussion 4.1. Potential energy surface of alanine polypeptide To construct the partition function (7), one needs to calculate the PES of a single amino acid in the bounded, (b) (ϕ, ψ), and unbounded, (u) (ϕ, ψ), conformations versus the twisting degrees of freedom ϕ and ψ (see Fig. 1). The PES of an alanine depends both on the conformation of the polypeptide and on the amino acid index in the chain. The PESs for different amino acids of the 21-residue alanine polypeptide calculated as a function of twisting dihedral angles ϕ and ψ are shown in Fig. 2. The PESs (a)–(e) in Fig. 2 correspond to the variation of the twisting angles in the second, third, fourth, ﬁfth and tenth amino acids of the polypeptide, respectively. Amino acids are numbered starting from the NH2 terminal of

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Fig. 2. PESs for diﬀerent amino acids of alanine polypeptide consisting of 21 amino acids calculated as a function of twisting dihedral angles ϕ and ψ in (a) second alanine, (b) third alanine, (c) fourth alanine, (d) ﬁfth alanine, and (e) tenth alanine. Amino acids are numbered starting from the NH2 terminal of the polypeptide. Energies are given with respect to the lowest energy minimum of the PES in eV.

the polypeptide. We do not present the PES for the amino acids at the boundary because the angle ϕ is not deﬁned here. On the PES corresponding to the tenth amino acid in the polypeptide [see Fig. 2(e)], one can identify a prominent minimum at ϕ = −81◦ and ψ = −71◦ . This minimum corresponds to the α-helix conformation of the corresponding amino acid, and energetically, the most favorable amino acid conﬁguration. In the α-helix conformation, the tenth amino acid is stabilized by two hydrogen bonds (see Fig. 3). With the change of the twisting angles ϕ and ψ, these hydrogen bonds become broken and the energy of the system increases. The tenth alanine can form hydrogen bonds with the neighboring amino acids only in the α-helix conformation, because all other amino acids in the polypeptide are in this particular conformation. This fact is clearly seen from the corresponding PES [Fig. 2(e)], where all local minima have energies signiﬁcantly higher than the energy of the global minima. (The energy diﬀerence between the global minimum and a

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Fig. 3. Alanine polypeptide in the α-helix conformation. The dashed lines show the hydrogen bonds in the system. The ﬁgure shows that the second alanine forms only one hydrogen bond, while the ﬁfth alanine forms two hydrogen bonds with the neighboring amino acids.

local minimum with the closest energy is ∆E = 0.736 eV, which is found at ϕ = 44◦ and ψ = −124◦.) The PES depends on the amino acid index in the polypeptide. This fact is clearly seen from Fig. 2. The three boundary amino acids in the polypeptide form a single hydrogen bond with their neighbors (see Fig. 3) and therefore are more weakly bounded than the amino acids inside the polypeptide. The change in the twisting angles ϕ and ψ in the corresponding amino acids leads to the breaking of these bonds, thereby increasing the energy of the system. However, the boundary amino acids are more ﬂexible than those inside the polypeptide chain, and therefore their PES is smoother. Figure 2 shows that the PESs calculated for the fourth, ﬁfth and the tenth amino acids are very close and have minor deviations from each other. Therefore, the PESs for all amino acids in the polypeptide, except the boundary ones, can be considered identical. Each amino acid inside the polypeptide forms two hydrogen bonds. However, since these bonds are shared by two amino acids, there is only eﬀectively one hydrogen bond per amino acid (see Fig. 3). Therefore, to determine the potential energy surface of a single amino acid in the bounded, (b) (ϕ, ψ), and unbounded, (u) (ϕ, ψ), conformations, we use the potential energy surface calculated for the second amino acid of the alanine polypeptide [see Fig. 2(a)], because only this amino acid forms a single hydrogen bond with its neighbors (see Fig. 3). The PES of the second amino acid Fig. 2(a) has global minima at ϕ = −81◦ and ψ = −66◦ and corresponds to the bounded conformation of the alanine. Therefore, the part of the PES in the vicinity of these minima corresponds to the PES of the bounded state of the polypeptide, (b) (ϕ, ψ).

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The potential energy of the bounded state is determined by the energy of the hydrogen bond, which for an alanine is equal to EHB = 0.142 eV. This value is obtained from the diﬀerence between the energy of the global minima and the energy of the plateaus at ϕ ∈ (−90◦ . . . −100◦) and ψ ∈ (0◦ . . . 60◦ ) [see Fig. 2(a)]. Thus, the part of the potential energy surface with the energy lower than EHB corresponds to the bounded state of alanine, while the part with the energy larger than EHB corresponds to the unbounded state. 4.2. Internal energy of the alanine polypeptide Knowing the PES for all amino acids in the polypeptide, one can construct the partition function of the system. The expressions for Zb and Zu [see Eqs. (8)–(9)] are integrated numerically and the partition function of the polypeptide is evaluated according to Eq. (7). The partition function deﬁnes all essential thermodynamical characteristics of the system as discussed in Ref. 7. Figure 4 shows the dependencies of the internal energy on temperature calculated for alanine polypeptides consisting of 21, 30, 40, 50 and 100

Fig. 4. Dependencies of the internal energy on temperature calculated for the alanine polypeptide chains consisting of 21, 30, 40, 50 and 100 amino acids. The thick solid lines correspond to the results obtained within the framework of the statistical model. The dots correspond to the results of MD simulations, which are ﬁtted by thin lines.

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amino acids. The thick solid lines correspond to the results obtained using the statistical approach, while the dots show the results of MD simulations. From Fig. 4, it is seen that the internal energy of alanine polypeptide rapidly increases in the vicinity of a certain temperature corresponding to the temperature of the ﬁrst-order phase transition. The value of the steplike increase of the internal energy is usually referred to as the latent heat of the phase transition, denoted as Q. The latent heat is the energy which the system absorbs in the course of the phase transition. Figure 4 shows that the latent heat increases with the growth of the polypeptide length. This happens because the number of hydrogen bonds in longer α-helices is larger than in shorter ones and thus more energy is required for the formation of the random coil state. The characteristic temperature region of the abrupt change in the internal energy (half-weight of the heat capacity peak) characterizes the temperature range of the phase transition. We denote this quantity as ∆T . With the increase of the polypeptide length, the dependence of the internal energy on temperature becomes steeper and ∆T decreases. Therefore, the phase transition in longer polypeptides is more pronounced. In the following subsection, we discuss in detail the dependence of ∆T on the polypeptide length. With the molecular dynamics, one can evaluate the dependence of the total energy of the system on temperature, which is the sum of the potential, kinetic and vibrational energies. Then, the heat capacity can be factorized into two terms: one corresponding to the internal dynamics of the polypeptide and the other to the potential energy of the polypeptide conformation. The conformation of the polypeptide inﬂuences only the term related to the potential energy, and the term corresponding to the internal dynamics is assumed to be independent of the polypeptide’s conformation. This factorization allows one to distinguish from the total energy the potential energy term corresponding to the structural changes of the polypeptide. The formalism of this factorization is discussed in detail in Ref. 7. The term corresponding to the potential energy of the polypeptide conformation has a step-like dependence on temperature that occurs at the temperature of the phase transition. Since we are interested in the manifestation of the phase transition, we have subtracted the linear term from the total energy of the system and consider only its nonlinear part. The slope of the linear term was obtained from the dependence of the total energy on temperature in the range of 300–450 K, which is far beyond the phase transition temperature (see Fig. 4). Note that the dependence shown in Fig. 4 corresponds only to the nonlinear potential energy terms.

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Figure 4 shows that the results obtained using the MD approach are in reasonable agreement with the results obtained from the statistical mechanics formalism. With increasing polypeptide length, the temperature width of the phase transition decreases, while the latent heat increases (see Fig. 4). These features are correctly reproduced both in MD and in our statistical mechanics approach. Furthermore, MD simulations demonstrate that with increasing polypeptide length, the temperature of the phase transition shifts towards higher temperatures (see Fig. 4). Note also that the increase in the phase transition temperature is reproduced correctly within the framework of the statistical mechanics approach, as seen from Fig. 4. Nevertheless, the results of MD simulations and the results obtained using the statistical mechanics formalism have several discrepancies. As seen from Fig. 4, the latent heat of the phase transition for the Ala100 polypeptide obtained within the framework of the statistical mechanics approach is higher than that obtained in MD simulations. This happens because within the statistical mechanics approach, the potential energy of the polypeptide is underestimated. Indeed, long polypeptides (consisting of more than 50 amino acids) tend to form short-lived hydrogen bonds in the random coil conformation. These hydrogen bonds lower the potential energy of the polypeptide in the random coil conformation. However, the “dynamic” hydrogen bonds are neglected in the present formalism of the partition function construction. 4.3. Heat capacity of alanine polypeptide The dependence of the heat capacity on temperature for alanine polypeptides of diﬀerent lengths is shown in Fig. 5. The results obtained using the statistical mechanics approach are shown by the thick solid lines, while the results of MD simulations are shown by the thin solid lines. Since the classical heat capacity is constant at low temperatures, for the purposes of plotting it is convenient to subtract this constant contribution. Figure 6 shows the dependence of the α-helix ↔ random coil phase transition characteristics on the length of the alanine polypeptide. The maximum heat capacity C0 and the temperature range of the phase transition ∆T are plotted against the number of amino acids squared (n2 ) and the inverse of the number of amino acids (1/n), respectively, while the temperature of the phase transition T0 and the latent heat of the phase transition Q are plotted against the number of amino acids (n). The squares and triangles represent the phase transition parameters calculated using the statistical approach and those obtained from the MD simulations, respectively.

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Fig. 5. Dependence of the heat capacity on temperature calculated for the alanine polypeptides consisting of 21, 30, 40, 50 and 100 amino acids. Results obtained using the statistical mechanics approach are shown by thick solid lines, while results of MD simulations are shown by thin solid lines. C300 denotes the heat capacity at 300 K.

Fig. 6. Phase transition parameters C0 , ∆T , T0 and Q calculated as a function of polypeptide length. The squares and triangles represent the phase transition parameters calculated using the statistical approach and those obtained from the MD simulations, respectively.

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The results obtained within the framework of the statistical mechanics model are in good agreement with the results obtained on the basis of MD simulations. However, since the MD simulations are computationally demanding of time, it is diﬃcult to simulate phase transitions in large polypeptides. The diﬃculties arise due to the large ﬂuctuations which appear in the system at the phase transition temperature and to the large time scale of the phase transition process. The relative error in the phase transition temperature obtained on the basis of MD approach is on the order of 3–5%, while the relative error in the heat capacity in the vicinity of the phase transition is signiﬁcantly larger.

5. Conclusion In the present contribution, a novel ab initio theoretical method for treating the α-helix ↔ random coil phase transition in polypeptide chains is discussed. The suggested method is based on the construction of a parameter-free partition function for a system undergoing the ﬁrst-order phase transition. All the necessary information for the construction of such a partition function can be calculated on the basis of ab initio DFT combined with molecular mechanics theories. The suggested method is considered to be an eﬃcient alternative to the existing theoretical approaches for the study of the helix–coil transition in polypeptides since it does not contain any model parameters. It gives a universal recipe for a statistical mechanics description of complex molecular systems. However, the formalism should be further advanced in order to account for long-range interactions. The partition function of the polypeptide is written with a minimum number of assumptions about the system which makes our method more general and universal in comparison with other theoretical approaches. Within the suggested theoretical framework, we derived and analyzed the temperature dependence of the heat capacity, latent heat and helicity of alanine polypeptides consisting of 21, 30, 40, 50 and 100 amino acids. In addition, we have obtained the same thermodynamical characteristics using molecular dynamics simulations and compared them with the results of our statistical mechanics approach. The comparison proved the validity of our method and established its accuracy. It was demonstrated that the heat capacity of alanine polypeptides has a peak at a certain temperature. The parameters of this peak (i.e. the maximal value of the heat capacity, the peak temperature, the width at half maximum, and the area under the peak) were analyzed as a function of polypeptide length.

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In this contribution, we demonstrated that the new statistical mechanics approach is applicable to the description of the α-helix ↔ random coil phase transition in alanine polypeptides. However, this method is general and can be used in studies of similar processes involving other complex molecular systems. For example, it is interesting to apply the formalism to the description of the β-sheet ↔ random coil phase transition, nonhomogeneous polypeptides (i.e. consisting of diﬀerent amino acids), or even small proteins. With some modiﬁcations and advances of the method, it could also be applied to the description of the protein folding process — an important direction for future work. Acknowledgments We acknowledge support of this work by the NoE EXCELL. The possibility to perform complex computer simulations at the Frankfurt Center for Scientiﬁc Computing is also gratefully acknowledged. References 1. E. Shakhnovich, Chem. Rev. 106, 1559 (2006). 2. A. Finkelstein and O. Ptitsyn, Protein Physics: A Course of Lectures (Elsevier, Oxford, 2002). 3. J.-E. Shea and C. L. Brooks, Ann. Rev. Phys. Chem. 52, 499 (2001). 4. N. V. Prabhu and K. A. Sharp, Ann. Rev. Phys. Chem. 56, 521 (2005). 5. A. V. Yakubovich, I. A. Solov’yov, A. V. Solov’yov and W. Greiner, Europhys. News 38, 10 (2007). 6. A. V. Yakubovich, I. A. Solov’yov, A. V. Solov’yov and W. Greiner, Eur. Phys. J. D 40, 363 (2006). 7. A. V. Yakubovich, I. A. Solov’yov, A. V. Solov’yov and W. Greiner, Eur. Phys. J. D 46, 215 (2008); I. A. Solov’yov, A. V. Yakubovich, A. V. Solov’yov and W. Greiner, ibid. 46, 227 (2008). 8. B. Zimm and J. Bragg, J. Chem. Phys. 31, 526 (1959). 9. J. Gibbs and E. DiMarzio, J. Phys. Chem. 30, 271 (1959). 10. S. Lifson and A. Roig, J. Chem. Phys. 34, 1963 (1961). 11. J. A. Schellman, J. Phys. Chem. 62, 1485 (1958). 12. I. A. Solov’yov, A. V. Yakubovich, A. V. Solov’yov and W. Greiner, Eur. Phys. J. D (2007), arXiv:0704.3085v1 [physics.bio-ph], 23 Apr 2007. 13. S. Lifson, J. Chem. Phys. 40, 3705 (1964). 14. D. Poland and H. A. Scheraga, J. Chem. Phys. 45, 1456 (1966). 15. T. Ooi and M. Oobatake, Proc. Natl. Acad. Sci. USA 88, 2859 (1991). 16. J. Gomez, V. J. Hilser, D. Xie and E. Freire, Proteins Struct. Func. Gen. 22, 404 (1995). 17. D. J. Tobias and C. L. Brooks, Biochem. 30, 6059 (1991).

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18. A. E. Garcia and K. Y. Sanbonmatsu, Proc. Natl. Acad. Sci. USA 99, 2781 (2002). 19. H. Nymeyer and A. E. Garcia, Proc. Natl. Acad. Sci. USA 100, 13934 (2003). 20. A. Irb¨ ack and F. Sjunnesson, Proteins Struct. Func. Gen. 56, 110 (2004). 21. D. Shental-Bechor, S. Kirca, N. Ben-Tal and T. Haliloglu, Biophys. J. 88, 2391 (2005). 22. R. A. Kromhout and B. Linder, J. Phys. Chem. B 105, 4987 (2001). 23. A. Chakrabartty, T. Kortemme and R. L. Baldwin, Prot. Sci. 3, 843 (1994). 24. M. Go, N. Go and H. A. Scheraga, J. Chem. Phys. 52, 2060 (1970). 25. H. A. Scheraga, J. A. Villa and D. R. Ripoll, Biophys. Chem. 101-102, 255 (2002). 26. J. M. Scholtz, S. Marqusee, R. L. Baldwin, E. J. York, J. M. Stewart, M. Santoro and D. W. Bolen, Proc. Natl. Acad. Sci. USA 88, 2854 (1991). 27. I. K. Lednev, A. S. Karnoup, M. C. Sparrow and S. A. Asher, J. Am. Chem. Soc. 123, 2388 (2001). 28. P. A. Thompson, W. A. Eaton and J. Hofrichter, Biochem. 36, 9200 (1997). 29. S. Williams, R. G. Thimothy, P. Causgrove, K. S. Fang, R. H. Callender, W. H. Woodruﬀ and R. B. Dyer, Biochem. 35, 691 (1996). 30. W. Scott and W. van Gunsteren, The GROMOS software package for biomolecular simulations, in Methods and Techniques in Computational Chemistry: METECC-95 , eds. E. Clementi and G. Corongiu (STEF, Cagliari, Italy, 1995). 31. W. Cornell et al., J. Am. Chem. Soc. 117, 5179 (1995). 32. A. MacKerell et al., J. Phys. Chem. B 102, 3586 (1998). 33. S. He and H. A. Scheraga, J. Chem. Phys. 108, 271 (1998). 34. S. He and H. A. Scheraga, J. Chem. Phys. 108, 287 (1998). 35. A. Yakubovich, I. Solov’yov, A. Solov’yov and W. Greiner, Eur. Phys. J. D 39, 23 (2006). 36. A. Yakubovich, I. Solov’yov, A. V. Solov’yov and W. Greiner, Khimicheskaya Fizika (Chemical Physics) 25, 11 (2006) [in Russian]. 37. I. A. Solov’yov, A. V. Yakubovich, A. V. Solov’yov and W. Greiner, Phys. Rev. E 73, 021916 (2006). 38. I. A. Solov’yov, A. V. Yakubovich, A. V. Solov’yov and W. Greiner, J. Exp. Theor. Phys. 102, 314 (2006). 39. I. A. Solov’yov, A. V. Yakubovich, A. V. Solov’yov and W. Greiner, J. Exp. Theor. Phys. 103, 463 (2006). 40. N. Go and H. A. Scheraga, J. Chem. Phys. 51, 4751 (1969). 41. S. Krimm and J. Bandekar, Biopolymers 19, 1 (1980). 42. D. Rapaport, The Art of Molecular Dynamics Simulation (Cambridge University Press, UK, 2004). 43. J. C. Phillips et al., J. Comp. Chem. 26, 1781 (2005). 44. D. Frenkel and B. J. Smit, Understanding Molecular Simulation (Academic Press, 2001). 45. W. Coﬀey, Y. Kalmykov and J. Waldron, The Langevin Equation, World Scientiﬁc Series in Contemporary Chemical Physics, Vol. 14 (World Scientiﬁc, Singapore, 2004).

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46. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw Hill, New York, 1965). 47. E. Henriques and A. Solov’yov, Eur. Phys. J. D 46, 471 (2008). 48. M. Sotomayor, D. P. Corey and K. Schulten, Science 13, 669 (2005). 49. J. Gullingsrud and K. Schulten, Biophys. J. 86, 3496 (2004). 50. D. L. Nelson and M. M. Cox, Principles of Biochemistry (W. H. Freeman and Company, New York, 2005).

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TRANSLATIONAL KINETIC ENERGY RELEASED IN THE DISSOCIATIVE CASCADE OF CHARGED RARE-GAS CLUSTERS: HINTS FOR FINITE SIZE PHASE TRANSITIONS? F. CALVO∗ LASIM, Universit´e Claude Bernard Lyon 1, 43 Bld du 11 Novembre 1918, 69622 Villeurbanne Cedex, France ∗ [email protected] P. PARNEIX Laboratoire de Photophysique Mol´eculaire, CNRS, Bˆ at 210, Universit´e Paris-Sud, F91405 Orsay Cedex, France The evaporative cascade of neutral atoms from Ar+ n clusters is investigated using phase space theory and kinetic Monte Carlo simulations. We focus here on the kinematics of sequential dissociation, by recording the translational kinetic energy distributions of the fragments for thermal excitations at ﬁxed excitation energy or at ﬁxed canonical temperature. The average kinetic energy of the monomers exhibits non-monotonic variations with excess energy, backbendings being found as the consequence of successive cooling processes. These eﬀects are signiﬁcantly washed out by the thermal broadening of the initial excitation. However, these backbendings do not necessarily reﬂect the intrinsic caloric curves of the clusters. Keywords: Atomic clusters; statistical fragmentation; phase transitions.

1. Introduction As is the common practice in nuclear physics, many physical and chemical properties of atomic and molecular clusters are experimentally determined from their fragmentation behavior.1–6 These properties are often inferred from measurements through simple theoretical models, especially those developed for unimolecular dissociation. Recent experimental progress has improved our ability to accurately characterize the internal energy of 261

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the clusters from their fragmentation pattern, thus allowing calorimetric measurements to be undertaken.7–10 The ﬁnding of a backbending in the microcanonical caloric curve of sodium,11 strontium,9 and hydrogen8 clusters conﬁrmed early theoretical predictions by the Berry group.12 Most experiments on isolated clusters deal with ionized species that are suitable for mass selection. The abundance spectrum is then one of the most direct observables that can be used to locate the special sizes often known as “magic numbers.” The kinetic energy distribution of the ionized fragments can be subsequently determined using speciﬁc techniques such as velocity ion imaging. In a few cases, a complete characterization of the dissociation event can be achieved, including neutral fragments.13 In a pioneering eﬀort, Weerasinghe and Amar14 showed that phase space theory (PST), in its orbiting transition state version highly developed by Chesnavich and Bowers,15 was able to quantitatively describe the statistics of cluster evaporation. By extending this work to the case of rotating,16 molecular,17 and non-spherical18 clusters, we have recently been able to incorporate PST into a uniﬁed kinetic Monte Carlo (kMC) framework for the dissociation cascade.19 Similar ideas have been applied to the fragmentation of ﬁnite systems in the past,20,21 but they were limited by the diﬃculty of calculating realistic potential energy surfaces or densities of states. Such diﬃculties have essentially been alleviated with the advent of powerful sampling algorithms such as parallel tempering Monte Carlo, and the PST/kMC combination turns out to be accurate when compared to numerically exact molecular dynamics (MD) simulations.19 In the present work, we further extend the kinetic Monte Carlo scheme to calculate translational kinetic energies of the various fragments emitted in the dissociative cooling of Ar+ n clusters. Cationic systems are more readily accessible by mass spectrometry measurements. As they undergo consecutive dissociations, their kinetic energy varies as the combined result of total energy and momentum conservation. The present framework is not only useful for monitoring the properties of the main cluster experiencing dissociative cooling, but also the kinetic energies of the evaporating atoms. In particular, as will be shown below, the kinetic energy of the monomers shows intricate variations with excess energy or temperature, which we are tempted to interpret in terms of ﬁngerprints of solid-like–liquid-like phase changes, whereas the larger charged fragments do not show such complex behavior. In the next section, we brieﬂy describe the methods used to describe unimolecular dissociation, paying particular attention to the kinematics of the dissociation cascade and the partitioning between internal, translational, and rotational energies. Our application to thermally excited Ar+ n clusters is presented and discussed in Sec. 3, before we summarize and conclude in Sec. 4.

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2. Methods The ground electronic state of cationic argon clusters is modeled using a diatomic-in-molecules quantum Hamiltonian already discussed previously.22 Since a detailed description of the PST/kMC framework is available in our recent work,19 here we only give the details that are speciﬁc to Ar+ n clusters. 2.1. Phase Space Theory approach to cluster fragmentation For a parent cluster having n atoms, a total energy E and angular momentum J, phase space theory provides the rate constant kn and total kinetic energy released εtr upon dissociation,15,16 as well as its translational εt and rotational εr contributions. The angular momentum of the product cluster can also be calculated.15,16 The main ingredients of such calculations are the microcanonical densities of vibrational states Ωn and Ωn−1 of the parent and product clusters, respectively, the rotational constants Bn and Bn−1 , and the dissociation energy ∆En = En − En−1 corresponding to the loss of a single atom from Ar+ n . The constraints on energy and angular momentum conservation are accounted for by assuming a loose transition state, the interaction between the fragments here being modeled by an explicit A3 is the atomic radial potential V (r) = −αe2 /8πε0 r4 , where α = 1.586 ˚ polarizability of argon and e the electronic charge. In this form, phase space theory predicts kinetic energy and angular momentum distributions in quantitative agreement with molecular dynamics results, provided that the densities of states incorporate anharmonic eﬀects.14,16 However, the rate constants can diﬀer by two orders of magnitude, and it is then useful to calibrate them by performing additional MD trajectories at high energies. We have followed this procedure here, repeating the calculations for all parent sizes in the range 4 ≤ n ≤ 30 with a sample of 1000 evaporative trajectories for each size. The densities of states were obtained from exchange Monte Carlo simulations with 50 replicas, with a geometric progression in the allocated temperatures in the range 2 K ≤ T ≤ 100 K for n > 10, and 5 K ≤ T ≤ 150 K for n ≤ 10. For each cluster size, 106 MC steps were performed for each replica. 2.2. The dissociative cascade For a given initial parent size N , the excitation energy is either ﬁxed (microcanonical excitation), or drawn from a Boltzmann distribution at ﬁxed temperature, obtained from the available density of states. The kinetic Monte Carlo scheme proceeds according to the following rules.19

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For each cluster with size n, total energy En , and angular momentum Jn , the rate constant kn is ﬁrst calculated, then the probability distribution P (εtr ) of total kinetic energy release. A random value εtr is drawn from this distribution, and the conditional probability P (Jr |εtr ) of the product angular momentum Jr at ﬁxed εtr is calculated, from which a random value Jr is selected. The rotational energy carried by the product cluster is obtained using εr = Bn−1 Jr2 , which yields the translational energy lost as εt = εtr − εr . The cluster Ar+ n−1 is then created after a time lapse 1/kn with excess energy En−1 = En − ∆En − εt and angular momentum Jn−1 = Jr , ∆En being the diﬀerence in binding energies between the parent and fragment clusters. This process is repeated until the required waiting time is reached. The absolute kinetic energies of the fragments are calculated in the reference frame of the initial cluster, by following the sequence of events illustrated in Fig. 1. Each time a fragmentation event occurs, the translational kinetic energy εt is converted into the velocities vg and vi for the main cluster and the neutral ith neutral atom leaving, respectively: vg2 =

2εt 2(n − 1)εt and vi2 = , n(n − 1)m nm

with n being the size of the parent cluster and m the argon mass. A random orientation for the dissociation is chosen by drawing a unit vector on the surface of a sphere, yielding the vectors vg and vi relative to the parent reference frame. Changing to the reference frame of the initial cluster provides

Fig. 1. Schematic representation of the sequential dissociation. The cluster is initially at the origin of the reference frame indicated by the (x, y, z) axes, and undergoes a first v1 . At time t2 , a second dissociation at time t1 leading to a monomer with velocity dissociation leaves a monomer with velocity v2 . At the time of the third dissociation, the main cluster has velocity vg in the initial reference frame.

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the absolute velocities and kinetic energies of the various fragments, including the main cluster. 3. Results and discussion We ﬁrst consider the case of clusters with ﬁxed excess energies. The average size n of the remaining fragment and the average kinetic energy of the neutral monomers, obtained 10 µs after heating, are represented in Fig. 2 as a function of excitation energy, for initial cluster sizes of N = 15, 20, 25

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and 25. All simulations have been performed with 104 kMC trajectories for each excess energy or temperature. The trends observed for the three parent clusters are the same. As the excess energy increases, the average cluster size decreases in a monotonic fashion, but through successive plateaus. As was already found for Lennard–Jones clusters,19 these patterns are related to the correlation between excess microcanonical energy and observation time: increasing the excitation energy at a given waiting time is equivalent to increasing — exponentially — the waiting time at ﬁxed energy. This is a consequence of evaporative cooling, the rate constant being multiplied by a nearly constant factor after each dissociation. An indication of the specially stable or unstable cluster sizes can be inferred from the apparent lengths of the various plateaus of Fig. 2(a). However, in contrast to neutral rare-gas clusters which have well-deﬁned magic numbers,19,23 the present cationic clusters do not exhibit strong special stabilities, except maybe around 14 atoms, size 19 being slightly unstable with respect to its neighbors. It is worth noting that at ﬁxed excitation energy per atom, larger clusters tend to dissociate more atoms than small clusters. Cationic rare-gas clusters have a structure similar to a covalent core of 2–4 atoms, solvated by neutral atoms.24 The neutral atoms bind to the charged core through polarization forces, and to the remaining neutral atoms via dispersion interactions (as well as interactions between induced dipoles). On average, small clusters are thus more bound than large clusters, in agreement with the higher number of dissociating atoms found in Fig. 2(a). The kinetic energy of the monomers globally increases with excitation energy, but these variations are non-monotonic. Each onset of a new atom loss is accompanied by a signiﬁcant drop in the kinetic energy, resulting in several backbendings. The magnitude of these backbendings is higher for smaller clusters, and slightly decreases after each new evaporation due to the broadening of the energy distribution for each new oﬀspring cluster. As was ﬁrst found by Weerasinghe and Amar,14 the variations of the total kinetic energy released with excess energy mimic the caloric curves of the product cluster, because they are intimately connected with its densities of vibrational states. It would seem that the same holds here for the purely translational energy released, each backbending in Fig. 2(b) being then related to the phase change in each successive product cluster. However, the microcanonical temperatures of Ar+ n clusters, shown as an inset in Fig. 2(b) for the three parent sizes, do not exhibit backbendings as a function of total energy. Thus, the backbendings observed in the translational kinetic energy released are likely to be the consequence of the sudden cooling of the cluster following an extra dissociation, rather than the signature of the caloric curve itself.

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Assuming now that the clusters are thermalized within a heat bath at ﬁxed temperature T , the average size and monomer kinetic energies are now obtained by performing an additional sampling of the initial excess energy. The results of this canonical sampling are represented in Fig. 3. The average cluster size smoothly decreases with increasing temperature, without exhibiting any plateau associated with possible special stabilities. Smoother variations due to thermal broadening of the initial energy distribution are also found for the monomer kinetic energies, as seen in Fig. 3(b). However, a residual backbending is found at low temperatures, corresponding to the ﬁrst evaporation from the initial cluster. After this evaporation, the energy distribution of the product cluster becomes excessively broad. 25

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Fig. 4. Average kinetic energy of Ar+ n fragments (n = 16–18) obtained 10 µs after sequential loss of Ar monomers from Ar+ 20 , at (a) constant excitation energy, and (b) constant temperature.

The kinetic energy of the main fragment is also a useful observable that could be experimentally measured. Figure 4 represents the variations of the average kinetic energy of selected fragment sizes 16 ≤ n ≤ 18 obtained after the loss of several monomers from Ar+ 20 excited in the microcanonical or canonical ensembles. Well-deﬁned excess energies lead to narrow size distributions, and each cluster size found after 10 µs exists only in a limited energy range, higher energies leading to smaller clusters. As expected, the kinetic energy of the main fragment increases with initial energy, but it does not depend signiﬁcantly on the fragment size. No backbending is found in the variations of this observable. Thermally broadened excitations yield two main diﬀerences with the microcanonical case, namely, the existence range (extended to high temperatures), and the generally increasing kinetic energy of smaller fragments. The former diﬀerence comes from the occasional selection of low initial energies for Ar+ 20 , even at high temperatures, leading to fragments larger than Ar+ 15 after 10 µs. Conversely, the initial distribution is narrow at low temperatures, hence the lower bound for the appearance of a given fragment size seen in Fig. 4(b). The higher fragment kinetic energy found here for smaller clusters reﬂects the contribution of evaporation events starting with a high excess energy, these events leading to more energetic fragments according to Fig. 4(a). Apart from these diﬀerences and ﬂuctuations at low temperatures, no signiﬁcant backbending is found in the variations of the fragment kinetic energy with increasing temperature.

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4. Conclusion Previous experimental works have suggested that phase changes in clusters may be detected on suitable dissociation observables, most notably the fragments’ kinetic energies. In the present contribution, we have theoretically modeled the evaporative cascade of neutral atoms from initially hot Ar+ n clusters having a few tens of atoms. Using phase space theory for the individual dissociation steps within a kinetic Monte Carlo framework, the translational kinetic energies of the various fragments have been characterized for well-deﬁned or thermally sampled initial excitation energies. The kinetic energies of the monomers carry important information about each dissociation event, especially for speciﬁc initial excitations, through as many backbendings. However, these backbendings should not be misinterpreted as the signature of underlying ﬁnite-size phase transitions in the clusters, but rather as the consequence of the signiﬁcant cooling occuring upon evaporation. The kinetic energies of the larger, ionized fragments were not found to be particularly enlightening, especially in the canonical ensemble where most features are generally smoothened if not washed out. This somewhat deceptive result is likely to be due to the rather short range of the interaction in rare-gas clusters, which tend to evaporate as soon as they melt. The situation is worse in molecular clusters such as C60 due to the near-contact intermolecular forces25 enhancing sublimation. It would then be more interesting to focus on metallic clusters, for which the interactions are long-ranged, and the liquid state expected to be stable over a wide range of energies or temperatures. However, an additional diﬃculty for metal clusters is the competition between several evaporation channels, mostly the monomer and the dimer, which would require separate PST treatments. Provided that these channels are correctly accounted for, the present method could be very useful for getting insight into the relation between the fragment kinetic energies and the liquid–vapor transition in ﬁnite-size systems.

References 1. U. Ray, M. F. Jarrold, J. E. Bower and J. S. Kraus, J. Chem. Phys. 91, 2912 (1989). 2. C. Br´echignac, Ph. Cahuzac, J. Leygnier and J. Weiner, J. Chem. Phys. 90, 1492 (1989). 3. S. Wei, B. Tzeng and A. W. Castleman, Jr., J. Chem. Phys. 92, 332 (1990). 4. C. X. Xu, D. A. Hales and P. B. Armentrout, J. Chem. Phys. 99, 6613 (1993). 5. P. Brockhaus, K. Wong, K. Hansen, V. Kasperovich, G. Tikhonov and V. V. Kresin, Phys. Rev. A 59, 495 (1999).

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6. L. Chen, S. Martin, J. Bernard and R. Br´edy, Phys. Rev. Lett. 98, 193401 (2007). 7. M. Schmidt, R. Kusche, W. Kronm¨ uller, B. von Issendorﬀ and H. Haberland, Phys. Rev. Lett. 79, 99 (1997). 8. F. Gobet, B. Farizon, M. Farizon, M. J. Gaillard, J. P. Buchet, M. Carr´e, P. Scheier and T. D. M¨ ark, Phys. Rev. Lett. 89, 183403 (2002). 9. C. Br´echignac, Ph. Cahuzac, B. Concina and J. Leygnier, Phys. Rev. Lett. 89, 203401 (2002). 10. G. A. Breaux, C. M. Neal, B. Cao and M. F. Jarrold, Phys. Rev. Lett. 94, 173401 (2005). 11. M. Schmidt, R. Kusche, T. Hippler, W. Kronm¨ uller, B. von Issendorﬀ and H. Haberland, Phys. Rev. Lett. 86, 1191 (2001). 12. R. S. Berry, J. Jellinek and G. Natanson, Chem. Phys. Lett. 107, 227 (1984). 13. G. Martinet et al., Phys. Rev. Lett. 93, 063401 (2004). 14. S. Weerasinghe and F. G. Amar, J. Chem. Phys. 98, 4967 (1993). 15. W. J. Chesnavich and M. T. Bowers, J. Chem. Phys. 66, 2306 (1977). 16. F. Calvo and P. Parneix, J. Chem. Phys. 119, 256 (2003); P. Parneix and F. Calvo, ibid. 119, 9469 (2003). 17. F. Calvo and P. Parneix, J. Chem. Phys. 120, 2780 (2004). 18. P. Parneix and F. Calvo, J. Chem. Phys. 121, 11088 (2004). 19. F. Calvo and P. Parneix, J. Chem. Phys. 126, 034309 (2007). 20. J. M. Soler and N. Garc´ıa, Phys. Rev. A 27, 3300 (1983). 21. C. Barbagallo, J. Richert and P. Wagner, Nucl. Phys. A 324, 97 (1986). 22. F. Calvo, P. Parneix and F. X. Gad´ea, J. Phys. Chem. A 110, 1561 (2005). 23. J. A. Northby, J. Chem. Phys. 87, 6166 (1987). 24. J. Galindez, F. Calvo, P. Paska, D. Hrivnak, R. Kalus and F. X. Gad´ea, Comput. Phys. Commun. 145, 126 (2002). 25. F. Calvo, J. Phys. Chem. B 105, 2183 (2001).

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DYNAMICS OF METAL CLUSTERS: FREE, EMBEDDED AND DEPOSITED ¨ F. FEHRER and P.-G. REINHARD M. BAR, Institut f¨ ur Theoretische Physik, Universit¨ at Erlangen, Staudtstrasse 7, D-91058 Erlangen, Germany P. M. DINH and E. SURAUD Laboratoire de Physique Th´eorique, UMR CNRS – Universit´e Paul Sabatier, 118, route de Narbonne F-31062 Toulouse C´edex, France ¨ L. V. MOSKALEVA and N. ROSCH Department Chemie, Theoretische Chemie, Technische Universit¨ at M¨ unchen, D-85748 Garching, Germany We present a recently introduced hierarchical model for the description of clusters in contact with an environment (embedded or deposited cluster). We brieﬂy outline the ingredients of the model and show the relevance of a proper treatment of the degrees of freedom of the environment. We then discuss the eﬀects of cluster–substrate interaction in three diﬀerent dynamical scenarios: optical response of a small Na cluster deposited on an MgO surface, a comparison of the dynamics of deposition of an Na cluster on an MgO surface versus an Ar surface, and strong laser excitation of an Na cluster embedded in an Ar matrix. Keywords: Metal clusters; dynamics; cluster-substrate interaction.

1. Introduction Physics at the nanometer scale is very promising from a technological perspective as it allows one to control material at the nanometer scale.1 Equally, it is a very interesting ﬁeld of basic research. Metal clusters in particular have attracted much attention over the past decades.2–7 They serve as a generic laboratory for studying a ﬁnite electron cloud with its pronounced quantum and many-body eﬀects.3,4 Their dynamics is 273

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distinguished by a pronounced optical resonance, the Mie surface plasmon, which provides a strong coupling to photons in a very narrow frequency window. That makes metal clusters an ideal laboratory for laser-induced nonlinear dynamics.8,9 The plasmon also serves as a versatile handle for various scenarios in pump and probe experiments,10–12 and it provides the route to driving the large Coulomb explosion of large clusters.13,14 An even richer variety of phenomena emerges when considering metal clusters in contact with other materials (embedded in a matrix or deposited on a surface), both in terms of fundamental questions and in terms of potential applications, such as the dedicated shaping of clusters.15,16 As the environment simpliﬁes handling, many interesting experiments can only be done with clusters in contact with a carrier material.17–20 Again, the plasmon plays a dominant role in this context. It is a key ingredient in the polarization interaction with the substrates, and clusters in contact with insulators are also versatile model systems for chromophores, which can be used for studies of radiation damage in materials21,22 or as indicators in biological tissues.23,24 From the theoretical side, however, embedded/deposited clusters are plagued by the complications brought up by the cluster–matrix interface and because of the huge number of atoms in the matrix. However, the broad range of applications has forced the development of a robust and eﬃcient hierarchical scheme for the dynamics of metal clusters in contact with rare gas materials.25–30 They treat the diﬀerent constituents at diﬀerent levels of reﬁnement, mixing quantum mechanical with molecular mechanical (QM/MM) descriptions. In this contribution, we will present recent results from QM/MM calculations for optical response, deposition and laser-induced dynamics of Na clusters in/on insulating substrates (Ar and MgO).

2. A brief summary of the QM/MM model The Na cluster is treated in full microscopic detail. We use quantummechanical single-particle wave functions for the valence electrons and classical molecular dynamics (MD) for the Na ions, both coupled by soft, local pseudo-potentials.31 The electronic wave functions are propagated by the time-dependent local-density approximation (TDLDA) augmented by a self interaction correction (SIC) through the averaged density SIC (ADSIC) scheme.32 Their motion is coupled non-adiabatically to MD for the ions. The thus emerging TDLDA-MD model has been widely validated for linear and nonlinear dynamics in the case of free metal clusters.7,8 The substrate is treated at a fully classical level, associating two degrees-of-freedom to each atom in an Ar substrate, position and electrical dipole moment.

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For MgO, the O++ anion carries also a dipole moment while Mg++ does not. The atomic dipoles allow one to explicitly treat the dynamical polarizability of the atoms through polarization potentials.33 The Coulomb ﬁeld of the dipoles provides the polarization potentials which are the dominant agents at long range. The dipole parameters are adjusted to reproduce the dynamical polarizability of the material at low frequencies. The Na+ ions of the metal cluster predominantly interact with the Ar dipoles through their monopole moment. The small dipole polarizability of the Na+ core is neglected. The cluster electrons also naturally couple to the Coulomb ﬁeld generated by the atoms (cores and electron clouds). The short-range repulsion between the constituents is obtained by adding specially constructed local core potentials. The ﬁne-tuning to known system properties is detailed in Ref. 26 for Ar substrate and Ref. 34 for MgO substrate. In the case of Ar, the Van-der-Waals interaction is known to be crucial in determining the Na–Ar binding, and it is also accounted for in an approximate way.26 The various ingredients in the actual QM/MM models have been tested in several separate applications. Concerning TDLDA(-MD) for Na clusters see, e.g., Refs. 7, 8 and 35–37. The modeling for the case of Ar substrate was developed and tested ﬁrst in Ref. 38 and its TDLDA extension in Ref. 25. The QM/MM model for MgO substrate was developed ﬁrst for noble metal adatoms and clusters39–42 and was extended to alkalines in Ref. 34. One has to keep in mind that applications to large-amplitude dynamics in the highly excited regime, as those reported later in that contribution, are still at an exploratory stage.

3. Optical response Optical absorption spectra are among the most important observables in cluster physics. For metal clusters, they are dominated by the Mie surface plasmon whose position depends on the overall extension (radius), and its splitting of the x, y, and z modes are related to the global quadrupole deformation while ﬁner details of the Landau fragmentation emerge from the interference with energetically close one-electron-one-hole states.7 Figure 1 compares the spectra of free and deposited Na clusters. In general, the overall average peak position changes very little. This conﬁrms previous ﬁndings for Na clusters in contact with Ar (Refs. 26 and 43) and other rare gases.44 Hence, the overall size of the cluster remains robust under deposition. The trends in the collective splitting diﬀer notably among the various clusters. Free Na8 is the most symmetric system. Its spectrum (upper-right panel of Fig. 1) is distinguished by near degeneracy of the three resonance modes. This changes somewhat for deposited Na8 (lower-right panel). The main

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Fig. 1. Optical response strength for the alkaline clusters Na6 (left) and Na8 (right). The upper panels show the results for free clusters, the lower panels for clusters deposited on MgO substrate. The three modes along the principle axes are shown as indicated. The z-axis coincides with the symmetry axis of the cluster and is orthogonal to the MgO surface while the x- and y-axes lie parallel to the surface.

eﬀect is an enhanced spectral fragmentation due to strong changes of the underlying single particle spectrum. The eﬀect is particularly dramatic in the z-direction. The core repulsion from the surface is fully explored here, and it leads to a huge Landau broadening, practically a dissolution, of the Mie resonance. The case of Na6 is shown in the left-hand panels of Fig. 1. The free cluster starts with a fragmented spectrum with signiﬁcant collective splitting (cf. z versus x mode). The nearly axial symmetry leaves x and y modes still perfectly degenerate. Deposition has much stronger eﬀects on the geometry,34 and this leads to more signiﬁcant changes in the spectrum (lower-left) where particularly the x-y degeneracy is broken. Core repulsion leads again to the near dissolution of the z mode and further fragmentation of the x mode, but surprisingly to a re-concentration of the y mode. The latter feature shows that Landau fragmentation emerges from a delicate balance of eﬀects and can hardly be predicted by simple rule of thumb. 4. Dynamics of cluster deposition In this section, we study the deposition dynamics of an Na6 cluster at moderate collisional energies comparing the Ar(100) and MgO(100) surfaces.

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A broader range of energies and conﬁgurations for Ar substrates is discussed in Ref. 28. Na6 consists of ﬁve Na ions in a ring plus a sixth ion topping the ring. The top ion sits on the symmetry axis. The Na6 centerof-mass starts from 15a0 above the surface facing the ring side towards 0 of 0.06 Ry per the surface. The initial boost gives it a kinetic energy Ekin Na ion. The time evolution of atomic and ionic z-coordinates is shown in the left-hand panels of Fig. 2. In both cases, there is a very fast stopping and quick capture of the Na cluster. The other aspects of the process develop very diﬀerently for the two diﬀerent substrates. For Ar(100), stopping is achieved by the substrate, which is highly excited and leaves little energy for the cluster, just suﬃcient to allow gentler oscillations of the top ion through the ring back and forth. The Mg(100) surface, however, is hardly excited at all and the sticking of Na6 is achieved by conversion of the initial center-of-mass energy into huge internal cluster excitation. The complementary information on kinetic energies is shown in the right-hand panels of Fig. 2. They conﬁrm the ﬁndings: the soft substrate

Fig. 2. Temporal evolution of the deposition of Na6 on Ar(100) substrate (upper) or MgO(100) substrate (lower) at an initial kinetic energy of 0.06 Ry. The left-hand panels show the z-coordinates of Na cluster and substrate. The right-hand panels show the corresponding kinetic energies.

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Ar(100) achieves the most gentle and eﬃcient stopping by absorbing nearly all the impact energy while the rather hard MgO(100) substrate remains almost unaﬀected. Both cases agree in that the main energy transfer proceeds on a very short time scale, within a few tens of femtoseconds. Further thermal equilibration takes much longer, on a scale of nanoseconds, far beyond simulation time.43

5. Hindered Coulomb explosion We study here an Na8 cluster embedded in a large Ar434 cluster, serving as a ﬁnite model of an Ar matrix. This is a generic example of a chromophore inside an inert environment. We consider low initial temperatures, safely below the Ar melting point of 84 K, where spatial ﬂuctuations of the Ar atoms and Na ions can be safely neglected. The small Na cluster at low temperature has a very clean excitation spectrum with rather narrow excitation peaks in the plasmon range around 2.5 eV.7 The surrounding Ar cluster is suﬃciently large to absorb the energy from the highly excited chromophore without destruction. The Na8 cluster with a radius of 7a0 (r.m.s. radius of 5.5a0 ) resides in a cavity of about 10a0 radius. The outermost Ar shell has a radius of 31a0 , which provides a substantial coverage of the embedded Na8 . The dynamics in the linear regime was studied in Ref. 26. A ﬁrst case of moderate laser excitations was discussed in Ref. 43. Here, we aim at a more violent process. To that end, we irradiate the Na8 @Ar434 system with a laser of frequency 1.9 eV, intensity 5 × 1012 W/cm2 , and a cos2 pulse envelope with total pulse length of 100 fs (FWHM = 33 fs). The laser is polarized along the symmetry axis of the system, henceforth called the z-axis. The rather short and intense laser excitation very quickly produces a net ionization stage 4+ and deposits a large amount of energy in the Na8 cluster. The initial reaction to laser excitation is exactly the same as we had already observed previously for the more moderate cases.43 There is a fast direct emission of electrons much similar than for free clusters. The ﬁnite Ar environment does not impose a hindrance to the electrons. The subsequent dynamics of ions and atoms is shown in Fig. 3. The cavity inside the Ar system is spacious, leaving a large initial separation of the Na ions from the ﬁrst shell of Ar atoms. Thus, we see in the lower part of Fig. 3 for the initial 200 fs the beginning of a fast Coulomb explosion of the cluster almost as in case of a free Na8 cluster. The explosion is stopped abruptly if the ions go against the repulsive Ar cores and the Na cluster goes into bound oscillations for a while. The Ar atoms instantaneously take up the momentum from the stopped ions. That momentum propagates like a sound wave

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Fig. 3. Temporal evolution of the hindered Coulomb explosion for a Na8 cluster embedded in an Ar434 matrix and excited with a short laser pulse to a net ionization of Q = 4. The lower panel shows the z-coordinates of cluster ions and substrate atoms. The upper panel shows the corresponding kinetic energies.

through the Ar medium. The perturbation is ultimately strong enough to produce a direct emission of Ar atoms when the wave reaches the outermost shell. On the way out, the wave traveling at velocity of sound distributes the energy over all the shells, exciting them and causing large ﬂuctuations. The Ar matrix remains surprisingly robust because the initial energy is distributed over so many atoms. The stabilization of the highly charged cluster lasts only for a limited, transient time. At about 4 ps, the cluster expansion revives, however, on a slower time scale rather like a Coulomb driven diﬀusion. The upper part of Fig. 3 shows the evolution of kinetic energies. The sudden stopping of the Na ions is associated with an instantaneous and almost exhaustive energy transfer to the Ar atoms. Although ultimately the Coulomb pressure regains the lead, we conclude that the Ar environment changes the cluster dynamics dramatically. The initial Coulomb explosion is eﬃciently stopped and charge stability is established for a rather long transient time of several picoseconds, more than suﬃcient for performing pump and probe analysis of such an exotic compound. The reconstituted Coulomb drift is, again, very diﬀerent to the straightforward Coulomb explosion of a free cluster.

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6. Conclusion In this contribution, we have used a hierarchical description within a combined quantum-mechanical–molecular-mechanical (QM/MM) modeling to study the eﬀects of cluster–substrate interactions on the dynamics of metal clusters. We considered three scenarios — optical response, deposition, and strong laser excitation — for a combination of Na clusters with inert substrates, Ar or MgO. The gross features of optical response (average Mie resonance position) are only weakly aﬀected by the substrate while the details of spectral fragmentation depend sensitively on the actual set-up and combinations. In the deposition processes, the energy balance depends sensitively on the substrate. Ar is a very soft material serving as a perfect shock absorber which takes away almost all the collisional energy, while MgO constitutes a rather hard surface which remains little aﬀected and achieves capture by converting the impact energy into strong internal excitations of the cluster. Intense laser pulses on embedded clusters couple ﬁrst to the metal cluster which acts as chromophore in that set-up. The energy absorbed from the laser is used mostly for fast ionization of the cluster. That creates an immense Coulomb pressure which initiates an explosion of the cluster. But that motion is abruptly stopped by the ﬁrst atomic shell. Almost all ionic momentum is transferred at once to the innermost Ar shell and distributed over the whole Ar substrate at the velocity of sound. The Ar environment stabilizes the for a transient time of 4 ps until a slow very highly charged state Na4+ 8 Coulomb diﬀusion leads ﬁnally to the separation of four Na+ ions. The three examples when compared show that dynamics in the domain of linear response is only slightly aﬀected by the presence of the substrate. This is totally diﬀerent for highly excited and large-amplitude dynamics, where the substrate is a decisive partner determining the overall features of the whole process. Acknowledgments This work was supported by the Deutsche Forschungsgemeinschaft (RE 322/10-1, RO 293/27-2), Fonds der Chemischen Industrie (Germany), a Bessel–Humboldt prize, and a Gay–Lussac prize. References 1. W. Eberhardt, Surf. Sci. 500, 242 (2002). 2. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters, Springer Series in Materials Science (Springer-Verlag, Berlin, 1995).

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3. M. Brack, Rev. Mod. Phys. 65, 677 (1993). 4. W. A. de Heer, Rev. Mod. Phys. 65, 611 (1993). 5. H. Haberland (ed.), Clusters of Atoms and Molecules 2 — Solvation and Chemistry of Free Clusters, and Embedded, Supported and Compressed Clusters, Springer Series in Chemical Physics (Springer Verlag, Berlin, 1994). 6. W. Ekardt (ed.), Metal Clusters (Wiley, New York, 1999). 7. P.-G. Reinhard and E. Suraud, Introduction to Cluster Dynamics (Wiley, New York, 2003). 8. F. Calvayrac, P.-G. Reinhard, E. Suraud and C. A. Ullrich, Phys. Rep. 337, 493 (2000). 9. S. Teuber, T. D¨ oppner, T. Fennel, J. Tiggesb¨ aumker and K. H. Meiwes-Broer, Eur. Phys. J. D 16, 59 (2001). 10. B. Bescos, B. Lang, J. Weiner, V. Weiss, E. Wiedemann and G. Gerber, Eur. Phys. J. D 9, 399 (1999). 11. K. Andrae, P.-G. Reinhard and E. Suraud, J. Phys. B 35, 1 (2002). 12. T. D¨ oppner, T. Fennel, T. Diederich and J. Tiggesb¨ aumker and K. H. MeiwesBroer, Phys. Rev. Lett. 94, 013401 (2005). 13. T. Ditmire, T. Donnelly, A. M. Rubenchik, R. W. Falcone and M. D. Perry, Phys. Rev. A 53, 3379 (1996). 14. S. A. Buzza, E. M. Snyder, D. A. Card, D. E. Folmer and A. W. C. Castleman Jr, J. Chem. Phys. 105, 7425 (1996). 15. G. Seifert, M. Kaempfe, K.-J. Berg and H. Graener, Appl. Phys. B 71, 795 (2000). 16. H. Ouacha, C. Hendrich, F. Hubenthal and F. Trger, Appl. Phys. B 81, 663 (2005). 17. N. Nilius, N. Ernst and H.-J. Freund, Phys. Rev. Lett. 84, 3994 (2000). 18. J. Lehmann, M. Merschdorf, W. Pfeiﬀer, A. Thon, S. Voll and G. Gerber, Phys. Rev. Lett. 85, 2921 (2000). 19. M. Gaudry, J. Lerm´e, E. Cottancin, M. Pellarin, J.-L. Vialle, M. Broyer, B. Pr´evel, M. Treilleux and P. M´elinon, Phys. Rev. B 64, 085407 (2001). 20. T. Diederich, J. Tiggesb¨ aumker and K. H. Meiwes-Broer, J. Chem. Phys. 116, 3263 (2002). 21. M. G. M. Bargheer and N. Schwentner, J. Chem. Phys. 117, 5 (2002). 22. M. B. M. Y. Niv and R. B. Gerber, J. Chem. Phys. 113, 6660 (2000). 23. C. Mayer, R. Palkovits, G. Bauer and T. Schalkhammer, J. Nanopar. Res. 3, 361 (2001). 24. B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou and A. Libchaber, Science 298, 1759 (2002). 25. B. Gervais, E. Giglio, E. Jaquet, A. Ipatov, P.-G. Reinhard and E. Suraud, J. Chem. Phys. 121, 8466 (2004). 26. F. Fehrer, P.-G. Reinhard, E. Suraud, E. Giglio, B. Gervais and A. Ipatov, Appl. Phys. A 82, 151 (2005). 27. P. M. Dinh, F. Fehrer, G. Bousquet, P.-G. Reinhard and E. Suraud, to appear Phys. Rev. B 75, 235418 (2007). 28. P. Dinh, F. Fehrer, P.-G. Reinhard and E. Suraud, Eur. Phys. J. D 45, 415 (2007).

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29. F. Fehrer, P. Dinh, M. B¨ ar, P.-G. Reinhard and E. Suraud, Eur. Phys. J. D 45, 447 (2007). 30. F. Fehrer, P. M. Dinh, P.-G. Reinhard and E. Suraud, Comp. Mat. Sci. 42, 203 (2008). 31. S. K¨ ummel, M. Brack and P.-G. Reinhard, Euro. Phys. J. D 9, 149 (1999). 32. C. Legrand, E. Suraud and P.-G. Reinhard, J. Phys. B 35, 1115 (2002). 33. B. G. Dick and A. W. Overhauser, Phys. Rev. 112, 90 (1958). 34. M. B¨ ar, L. V. Moskaleva, M. Winkler, P.-G. Reinhard, N. R¨ osch and E. Suraud, Eur. Phys. J. D 45, 507 (2007). 35. K. Yabana and G. F. Bertsch, Phys. Rev. B 54, 4484 (1996). 36. F. Calvayrac, P.-G. Reinhard and E. Suraud, Ann. Phys. (NY) 255, 125 (1997). 37. K. Yabana and G. F. Bertsch, Phys. Rev. A 58, 2604 (1998). 38. F. Dupl` ae and F. Spiegelmann, J. Chem. Phys. 105, 1492 (1996). 39. A. V. Matveev, K. M. Neyman, G. Pacchioni and N. R¨ osch, Chem. Phys. Lett. 299, 603 (1999). 40. A. M. Ferrari, C. Xiao, K. M. Neyman, G. Pacchioni and N. R¨ osch, Phys. Chem. Chem. Phys. 1, 4655 (1999). 41. C. Inntam, L. V. Moskaleva, K. M. Neyman, V. A. Nasluzov and N. R¨ osch, Appl. Phys. A 82, 181 (2006). 42. N. R¨ osch, V. A. Nasluzov, K. M. Neyman, G. Pacchioni and G. N. Vayssilov, in Computational Material Science, ed. J. Leszczynski, Theoretical and Computational Chemistry Series, Vol. 15 (Elsevier, Amsterdam, 2004), p. 367. 43. F. Fehrer, P.-G. Reinhard and E. Suraud, Appl. Phys. A 82, 145 (2006). 44. F. Fehrer, P. Dinh, P.-G. Reinhard and E. Suraud, Phys. Rev. B 75, 235418 (2007).

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PHASE, AMPLITUDE, AND POLARIZATION SHAPING BY INTERFEROMETRIC PULSE GENERATION A. LINDINGER∗ , S. M. WEBER, F. WEISE and M. PLEWICKI Institut f¨ ur Experimentalphysik, Freie Universit¨ at Berlin, Arnimallee 14, 14195 Berlin, Germany ∗ [email protected] We present an interferometric pulse shaper set up for unrestricted phase, amplitude, and polarization pulse control. It is realized by integrating a 4f-shaper set up in both arms of a Mach–Zehnder interferometer and rotating the polarization by 90◦ in one arm before overlaying the phase and amplitude modulated beams. We demonstrate the capabilities of this set-up by introducing a method for generating parametrically tailored three-dimensional electrical ﬁelds of femtosecond laser pulses. Keywords: Pulse shaping; polarization; laser pulse parametrization.

1. Introduction The control of femtosecond pulse shapes has gained considerable interest in the past years. Particularly in the ﬁeld of coherent control, there have been many achievements using this versatile tool.1 However, most experiments were performed with phase and/or amplitude shaping of linearly polarized light. Recently, a new degree of pulse shaping was achieved by manipulating the polarization state of the laser ﬁeld, as such pulses are best suited for three-dimensional quantum objects and give rise to new fundamental research. The ﬁeld of coherent control beneﬁts from the vectorial character of the light ﬁeld since it opens up new and more ﬂexible optimized paths where, e.g., the speciﬁc directions of the molecular transition dipole moments can be utilized. Polarization pulse shaping was ﬁrst described in a contribution by Wefers and Nelson,2 where they used two liquid crystal arrays with optical axes at ±45◦ to the horizontal. With this now standard pulse former device, they were able to produce polarization ellipses with restricted major axis angles, but with considerably diﬀering intensities for the horizontal and vertical light components due to polarization dependent grating 283

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diﬀraction. Later, Brixner et al.3 addressed this problem by placing a stack of glass plates at the Brewster angle in the beam to diminish the more intense component. They used this set-up in the ﬁrst polarization coherent control experiments, where they optimized the photo-induced ionization process in K2 and found an improved optimization factor compared to phase-only modulation.4 Other groups utilized the same type of setup on aligned iodine molecules5 or for controlling the angular momentum distribution in atoms.6 Recently, volume phase holographic gratings, which have a reduced reﬂectivity dependence on polarization, were utilized to generate shaped pulses for controlling nano-structures.7 A major improvement in the degree of polarization pulse shaping was made by Polachek et al.8 by sequentially combining the mentioned double array with a single array, having a horizontal optical axis which allows arbitrary major ellipses angles in the spectral domain. None of these former studies enable one to modulate the amplitude simultaneously, which is a major restriction regarding full controllability of the light ﬁeld. Recently, we developed a serial set-up employing two round-trips9 through one double array shaper, which allows restricted phase, amplitude, and polarization shaping. In this contribution, we present a parallel pulse shaper scheme which is capable of simultaneous unrestricted phase, amplitude, and polarization shaping. This is achieved by placing the modulator in a Mach–Zehnder interferometer to shape the ﬁeld’s x- and y-components independently. We will further demonstrate its pulse shaping capabilities in the time-domain using tailored, parametric pulses. Parametric pulse shaping was introduced by Bardeeen et al.10 and later further developed for independent sub-pulse encoding by Motzkus et al.11 ,12 Here, we extend this method to polarization states, and hence full parametric sub-pulse control of the light ﬁeld.

2. Shaper set-up In this section, we report our approach to simultaneous polarization, phase, and amplitude control. The concept is to spatially overlay two ﬁelds Ex and Ey , having perpendicular linear polarizations, and then choose the proper phases and amplitudes of these ﬁelds. The mathematical description of such an electrical ﬁeld by light ﬁeld modulation with the parallel set-up is given by (ω) out (ω) = Ein √ E 2

1

(φax (ω) 21 cos 2 (φay (ω)

cos

i − φbx (ω)) e 2 (φax (ω)+φbx (ω)) , i − φby (ω)) e 2 (φay (ω)+φby (ω))

(1)

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with φax , φbx and φay , φby being the retardances of two regions of the liquid crystal arrays. This is experimentally realized by incorporating a shaper in both paths of a Mach–Zehnder interferometer and rotating the polarization by 90◦ in one of the paths before combining. In order to keep the same polarization, it is crucial to maintain a stable relative phase between the vertical and horizontal polarization components. Employing a single shaper set-up for both paths of the interferometer, as shown in Fig. 1, provides more stability compared to two independent 4f-shapers. Any possible vibration or thermal deformation from these components will have the same impact on both paths, and consequently, have minimal inﬂuence on the relative phase stability. This is done by splitting the laser beam and then directing both beams on the ﬁrst grating (600 lines/mm) at the same position. After propagating through one cylindrical lens (f = 250 mm), the modulator (SLM–640, CRi), and the second cylindrical lens they reach the other grating. Since the beams are spatially separated after reﬂection from the second grating, a waveplate is used to turn the polarization of one of the Modulator Cylindrical lens

Cylindrical lens

Grating

Grating

Beamsplitter

Polarizing beam-splitter

Waveplate λ/2 Delay

Fig. 1. Interferometric parallel pulse shaper conﬁguration. The laser pulses are split by the beamsplitter and hit the grating at the same position, but at diﬀerent incidence angles. The spectral components of the pulses travel parallel to each other through the shaper and reach the second grating. Afterwards, the polarization of one of the components is rotated with a waveplate by 90◦ and passes through a delay stage. Both pulses are spatially and temporally overlapped by a polarization beam cube.

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beams by 90◦ and a delay line is included in one of the optical paths. To combine the two beams, we employ a polarizing beam cube with almost no beam intensity loss. We used a waveplate instead of a periscope to rotate the beam proﬁle, regardless of the introduced chirp, since for unsymmetric proﬁles this leads to a better spatial overlap with the other beam. For reliable pulse control, a precise alignment of the polarization component overlap in time and space is required. This can be reached by also taking advantage of light interference properties. A polarizer oriented at 45◦ is placed in the outgoing beam and creates two equally polarized pulses originating from the x- and y-components. When these pulses are delayed with respect to each other, spectral fringes can be observed and the contrast can be used as a feedback signal, since there exists a coupling between the spatial overlap and the maximum–minimum contrast of the fringes. The interference spectrum is also very helpful to ﬁnd the exact temporal overlap, as the interval between spectral interference peaks is inversely proportional to the pulse delay. For zero delay, one peak is observed for constructive interference and no peak for destructive interference. Moreover, the liquid crystal modulator allows one to ﬁne-tune the spectral and temporal overlap. The spectra are compared and modiﬁed by an amplitude ﬁlter in order to match the intensities, and the temporal features of the polarization components are tuned by phase factors with the help of sum frequency generation (SFG) cross-correlation traces. The stability of the relative phase between polarization components was tested by rotating the polarization of the outgoing light by 45◦ with a waveplate and transmitting the beam through a horizontally oriented polarizer. To determine the stability, the SFG signal of shaped pulses and reference pulses was recorded as a function of time. This signal was almost constant for several minutes, which allowed precise pulse generation and detection. An active phase stabilization should be considered for problems demanding a higher stability over extended periods of time.

3. Polarization ellipse control By applying certain voltages to the liquid crystals of the above-mentioned polarization shaper set-up, arbitrarily formed polarization ellipses are generated within the technical limits of the set-up, but no desired ellipse form for each frequency component is achieved. Hence, in the following a determination of the ellipse parameters major axis orientation γ, minor to major axes ratio Hba = Hb /Ha , and relative intensity I/I0 is outlined. The above introduced parameters together with the sign of the relative phase form a complete and intuitive description of the polarization state.

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Thus, a transformation between the ellipse and electric ﬁeld parameters will be performed. Thereto, we calculated the transmissions for both components, Tx and Ty and the relative phase shift as a function of the ellipse parameters. This was obtained by comparison between general ellipse and Lissajous curve equations, leading to 2 (Hba 1 I − 1) cos(2γ) Tx = 1− , (2) 2 + 1) 2 I0 2(Hba 2 − 1) cos(2γ) 1 I (Hba Ty = 1+ , (3) 2 + 1) 2 I0 2(Hba 1/2 2 Hba −1 sin(2γ) arccos . (4) =± 4 + H 2 (cot2 (γ) + tan2 (γ)) | sin(2γ)| 1 + Hba ba The change of sign in Eq. (4) is due either to the clockwise (negative ) or counterclockwise (positive ) rotating vector of the electric ﬁeld. The observable is the transmitted power after a rotatable output polarizer. The recorded power P (I, Hba , γ, θ) is dependent on the intensity I, the ratio Hba , the orientation of the polarization ellipse γ, and the orientation of the output polarizer θ: H2 − 1 1 cos(2(γ − θ)) . (5) P (I, Hba , γ, θ) = I 1 − ba 2 +1 2 Hba To measure the example pulses, we use a simpliﬁed version of the TRE (time resolved ellipsometry) technique,13 which is suﬃcient for our purposes, as the helicities can be pre-selected by the pulse form calculation. We detect a multitude of SFG-CCs (sum-frequency-generated cross-correlations) by rotating the polarization of the shaped beam with a λ/2-plate, and then focus and overlap it with the reference pulse in a BBO crystal, which is oriented to inherently select the vertical polarization direction. Next, we will show some experimental examples of changing one of the introduced parameters, while keeping the others constant. The laser pulses with a bandwidth (FWHM) of 25 nm were provided by an oscillator (Mira, Coherent). As an initial demonstration, we rotated the polarization of the pulse. This was done by varying the orientation γ for a few selected axes ratios with the output polarizer aligned to θ = 45◦ . As the simulation in Fig. 2(a) shows, the maximum modulation of the function should be observed for linear pulses where ratios are close to zero (or inﬁnity), and a ﬂat function for circularly polarized light, where Hba = 1. The agreement between our experimental scans in Fig. 2(b) and the theoretical curves conﬁrms that our set-up is able to perform a controlled major axes rotation. Without changing other parameters, the same scans

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(a)

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Fig. 2. (a) Theoretical and (b) experimental scans of the major axis orientation at a few constant ratios. The outgoing beam passes a 45◦ oriented polarizer and the intensity behind it is recorded in the experiment and calculated for the theoretical values. The curves correspond to diﬀerent axis ratios.

were experimentally conducted for a polarizer oriented at θ = 0◦ , where the signal is independent of the relative phase since the vertical component is not relevant. The experimental curves show minimal deviation from theory, and hence we believe that the noise observed in the scans for θ = 45◦ is only due to the unstable relative phase. Next, the ratio was scanned keeping the diﬀerent polarization orientations constant. Equation 5 indicates that the change of Hba has no inﬂuence

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in the case of cos(2(γ − θ)) = 0. This condition implies that the signal remains constant for γ − θ = 45◦ ±n × 90◦ (with integer n) while changing the ratio. Since the simulations and measurements are executed for the polarizer oriented at θ = 45◦ , the scans are ﬂat for γ = 0◦ ±n × 90◦ . The simulation and experimental graphs are shown in Fig. 3.

Fig. 3. Theoretical and experimental scans of the axes ratio of the polarization ellipse. The outgoing beam passes a 45◦ oriented polarizer and the intensity is calculated or measured behind it. The curves correspond to diﬀerent orientations of the polarization ellipse.

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The presented experimental scans agree well with the calculations. They ﬁnish at almost the same value for Hba = 1. For γ = 0, the results diﬀer from the simulation and this could be explained by an imperfect or changing relative phase alignment.

4. Parametric pulse shaping method With the above outlined transformations and the modulation functions, still only simple desired temporal pulse forms are possible. In order to generate user-deﬁned parametric pulse shapes, we use a sub-pulse encoding that consists of the parameters: intensity I, position in time, zero order phase, chirps, and polarization states (major axis angle β/2, minor-to-major axis ratio Hba , and helicity), which are kept constant for the sub-pulse duration. One could alternatively choose these parameters arbitrarily in time, if fundamental limitations of the temporal derivatives of the respective parameters are observed.14 By parameter transformation, these ellipse parameters are converted to the electric ﬁeld parameters: phase φN , amplitudes Ax,N , Ay,N , and phase diﬀerence N between the x- and y-components. The conversion between the time and frequency domains is via a Fourier transform. To assemble the complex ﬁeld, ﬁrst, a spectral phase ﬁlter φN (ω) is constructed for every sub-pulse N , using spectral Taylor terms bn (where the linear term b1 holds the distance from time zero, the quadratic term b2 the linear chirp etc.) with equal Taylor components for both x- and y-directions, except for the y-component of the zero order phase, which has to be set to b0,y = b0 + ε (whereas b0,x = b0 ), to implement the polarization. The determined electric ﬁeld parameters φN , Ax,N , Ay,N , and N can then be entered for every sub-pulse. out , the N sub-pulses are superposed, In order to yield the target ﬁeld E still in the frequency domain, as Ax,N eiφN,x (ω) ˜ out (ω) = E √in (ω) · E , (6) Ay,N eiφN,y (ω) 2 N

H(ω)

˜in (ω) is the linear input pulse. where E x |2 and |H y |2 represent the components of the now vectorial, Then, |H spectral transmission T (ω) to be written on the two modulator regions. The arguments constitute the phase ﬁlter φ(ω) for the two parallel paths, respectively, whereby energy conservation demands 1 ˜in (ω)|2 dω, (7) |E˜x,y (ω)|2 dω = Tx,y (ω) · |E 2 independently for x and y.

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The generated pulse shapes can be detected and visualized in threedimensional plots. The user-friendly, simpliﬁed TRE method records crosscorrelations of the shaped pulses which can then be ascribed to ellipse parameters by ﬁtting them for every time step to the projection formula, Eq. (5). This yields a three-dimensional plot without phase and helicity information. As the displayed information originates from crosscorrelations, the temporal proﬁles still have to be deconvoluted, which for this case would only cause a sub-pulse narrowing. To receive the total pulse shape including phase and helicity information, pulse spectra also need to be recorded for the diﬀerent polarization directions, and a retrieval procedure favorably using a multi-goal evolutionary algorithm has to be employed, or possibly dual channel interferometry like POLLIWOG.

4.1. Parametric polarization pulse shapes Figure 4 depicts three interferometrically generated double pulses with distances of 400 fs, calculated using the above formulas, showing the time dependent temporal intensity I, major axis angle γ, and minor-to-major axis ratio Ha,b . In Fig. 4(a), a double pulse with orthogonal linear polarization is generated with major axis angles of 0◦ and 90◦ . There is only a minimal deviation from the desired values for the ranges where there is suﬃcient intensity to provide a reliable pulse reconstruction. In the 3D representation of the ﬁeld amplitudes, the noise originates from the employed method of detection due to the ratio of major/minor axis involved; it is also responsible for the frayed edges for the vertical sub-pulse. The next example pulse in Fig. 4(b) shows an intended double linear/circular combination whereby a minor-tomajor axis ratio of 0.89 could be achieved for the circular sub-pulse. The last pulse form, depicted in Fig. 4(c), consists of an elliptical sub-pulse (at 30◦ , with a minor-to-major axis ratio of 0.27; 0.33 was intended), and a consecutive linear pulse. The example pulses oﬀer good agreement with the expected values. Creating temporally expanded pulse forms seems not to impair the interferometric overlap quality which is apparent from the low ellipticities of the quasi-linear pulses, and the small intensities between the sub-pulses.

5. Conclusion and outlook We have presented the ﬁrst interferometrically generated, parametrically tailored polarization pulse forms using a novel parallel set-up. The experimental example pulses lack the restrictions of other set-ups, like limited

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Fig. 4. Experimental parametric test pulses: (a) orthogonal 0◦ /90◦ linear/linear subpulses, (b) linear/circular sub-pulse, (c) elliptical/linear sub-pulses (axis ratio 0.3, and major axis angle 30◦ /90◦ angle, respectively). The dashed and dotted lines represent Hba and the major axis angle β/2, respectively. The corresponding 3D plots of the pulses are shown in (d), (e), and (f), where the shadows are the projections onto the respective planes.

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control over the amplitude or limitations to the polarization transients. Such pulses could be favorably employed to systematically investigate complex molecular systems with non-orthogonal dipole moments, highlighting the signiﬁcance of parametric tailoring without inherent restrictions. The parametric encoding allows parameter scans, and could be applied to research ﬁelds like femtochemistry, polarization compensation in ﬁbers, for super-continuum-generation, ﬁlamentation in air, nano-optical manipulation eﬀorts, or periodic electron circulation in molecules. Acknowledgments The authors acknowledge Ludger W¨oste, and the Deutsche Forschungsgemeinschaft (SFB 450) for ﬁnancial support. References 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

D. Goswami, Phys. Rep. 374, 385 (2003). M. M. Wefers and K. A. Nelson, Opt. Lett. 20, 1047 (1995). T. Brixner and G. Gerber, Opt. Lett. 26, 557 (2001). T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe, C. Horn, D. Liese and T. Baumert, Phys. Rev. Lett. 92, 208301 (2004). T. Suzuki, S. Minemoto, T. Kanai and H. Sakai, Phys. Rev. Lett. 92, 133005 (2004). N. Dudovich, D. Oron and Y. Silberberg, Phys. Rev. Lett. 92, 103003 (2004). M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiﬀer, M. Rohmer, C. Spindler and F. Steeb, Nature 446, 301 (2007). L. Polachek, D. Oron and Y. Silberberg, Opt. Lett. 31, 631 (2006). M. Plewicki, S. M. Weber, F. Weise and A. Lindinger, Appl. Phys. B 86, 259 (2007). C. J. Bardeen, V. Yakovlev, K. R. Wilson, S. D. Carpenter, P. M. Weber and W. S. Warren, Chem. Phys. Lett. 280, 151 (1997). T. Hornung, R. Meier and M. Motzkus, Chem. Phys. Lett. 326, 445 (2000). T. Hornung, R. Meier, D. Zeidler, K.-L. Kompa, D. Proch and M. Motzkus, Appl. Phys. B 71, 277 (2000). G. E. Jellison and D. H. Lowndes, Appl. Phys. Lett. 47, 718 (1985). T. Brixner, Appl. Phys. B 76, 531 (2003).

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Clustering Phenomenon in System of Various Degrees of Complexity

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ELECTRON–POSITRON CLUSTERS: STRUCTURE AND STABILITY V. K. IVANOV∗ and R. G. POLOZKOV St. Petersburg State Polytechnic University, Politekhnicheskaya 29, 195251, St. Petersburg, Russia www.spbstu.ru ∗ [email protected] A. V. SOLOV’YOV Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, 60438, Frankfurt am Main, Germany [email protected] Structure, stability and properties of a new type of cluster, electronpositron quantum cluster, are discussed. The analysis is based on both the non-relativistic Hartree–Fock and self-consistent local-density approximations. An essential role of many-body eﬀects in the formation of the droplets is demonstrated. Their properties are compared with the known physical objects such as metal clusters and clusters of excitons in a solid. Keywords: Cluster; electron–positron droplet; stability.

1. Introduction In this contribution, we discuss a new quantum object of ﬁnite size — electron–positron cluster or electron–positron droplet (EPD).1 These systems consist of a number of electrons and positrons which are held together by the attractive Coulomb and mutual polarization forces and form a shell structure similar to metal clusters.2 In contrast to metal clusters, in electron–positron clusters the motion of negatively and positively charged subsystems is identical and quantized. The number of particles in EPD can be varied from several up to inﬁnity. In the limit of large numbers, such a system envolves into an electron–positron gas or plasma.3 At small numbers, the system is essentially ﬁnite and its quantum features manifest 297

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themselves more prominently. The main properties of these clusters of diﬀerent size, such as the structure, stability and dynamics, are under consideration. The calculated structure is compared with the properties of metal clusters, which were obtained in particular within the ordinary2 and optimized jellium model (OJM).4,5 From an experimental point of view, electron–positron clusters can be created during the condensation or density collapse of a number of electrons and positrons initially localized in a certain volume. The necessary conditions for such a process can be met in experiments with electron and positron aligned beams of high density and of equal energy. When the beams are brought together the mutual attraction of electrons and positrons should result in the growth of particle density, the formation of electron–positron plasma, and the subsequent creation of electron–positron clusters. A similar many-body phenomenon occurs during electron cooling of ionic beams.6 In this case, the cooling of ions and subsequent growth of the density of the ionic beam is achieved by alignment of the ionic beam with the beam of electrons. The density of high energy electron–positron bunches in modern colliders can be as high as 10−21 cm−3 ,7 which is only three orders of magnitude lower than the characteristic value of the electronic density in solids. Therefore, it is plausible to expect that it will be suﬃcient to initiate the collapse of the electron–positron density. An alternative idea for creation of electron–positron clusters concerns the formation of these objects in Penning traps. We expect that the condensation of electron–positron densities into clusters of ﬁnite size can take place in external electric and magnetic ﬁelds holding particles together in a trap, providing the temperature of the electron–positron gas is suﬃciently low. A proof of the existence of EPD might have important consequences for various ﬁelds of science. Thus, these objects can be relevant to astrophysical problems in connection with the possible presence of antimatter in the universe. In nuclear physics, similar kinds of objects arise when antibaryons become bound in nuclear matter.8 In solid state physics, the condensation of electron–hole pairs or bound states of several excitons in the form of a cluster has been experimentally observed under certain conditions.9 There has been no self-consistent many-body theory developed for these systems so far. The EPD structure has many features common to atomic clusters of simple metals, like Na or K, in which strong delocalization of valence electrons takes place. However, there are important diﬀerences between atomic clusters and EPD. In metallic clusters the motion of valence electrons is quantized and creates the shell structure while the positively charged ions form the background. In the simplest case, the jellium model for metal clusters suggests a homogeneous distribution of the ionic density within

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the cluster volume.2 In more advanced jellium-type models for metal clusters, like the optimized jellium model,4,5 the ionic background is no longer homogeneous. On the contrary, in the EPD case, no assumption about the background in the systems is made either for electronic or for positronic subsystems. Moreover, both subsystems are considered dynamical, i.e. the kinetic energy for both electrons and positrons is taken into account. Our analysis of the electron–positron cluster properties is based on both the Hartree–Fock (HF) approximation and the formalism of non-relativistic density functional theory and the self-consistent solution of the Kohn–Sham equations. We do that for a neutral bound system consisting of N -electrons and N -positrons. Another constraint that we impose on the system is related to its sphericity. Here, we focus on spherical systems only, thus restricting ourselves to electron–positron clusters with close electronic and positronic shells. 2. HF and LDA approaches to electron–positron clusters Within the ordinary jellium model, a cluster consists of two subsystems: a positively charged core and valence electrons. The positively charged core is considered to be a ball with radius R and uniform charge-density distribution2 ( = me = |e| = 1). This core creates the potential  r 2 N   , r ≤ R, 3−  2R R U (r) = (1)   N , r > R, r where R = rs N 1/3 with rs standing for the average distance between atoms in a bulk material and N for the number of atoms in a cluster. Valence electrons in this potential are considered within the Hartree–Fock (HF) or local density (LDA) approximations. 2.1. The Hartree–Fock approximation The system of HF equations for single-electron wave functions and energies is   ∆ i − − U (r) + ϕj (r )Ve−e (|r − r |)ϕj (r )dr  ϕi (r) 2 j −

j

ϕj (r)

ϕj (r )Ve−e (|r − r |)ϕi (r )dr = εi ϕi (r),

(2)

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where Ve−e is the Coulomb interaction between electrons. Characterizing electron states by principle quantum number n, orbital quantum number l and its projection m, one obtains the order of ﬁlling states similar to that in a short-range potential well: 1s2 2p6 3d10 2s2 4f 14 3p6 5g 18 4d10 6h22 3s2 · · · . The total energy of cluster indicates its stability. The total cluster energy is equal to the sum of a positive jellium core and an electronic system: Etot = Ecore + Ee .

(3)

The positive core energy is deﬁned by classical formulas for electrostatic interaction: 1 ρcore (r)U (r)dr, Ecore = (4) 2 where ρcore is the positive-charge density distribution. The energy of the electronic system is determined by (5) Ee = Ψ|Hˆe |Ψ. Here, Ψ is the total wave function of the electronic system, and the Hamiltonian is ∆i ˆ He = − U (ri ) + Ve (ri , rj ). (6) − 2 i i=j

Regarding the wave function Ψ as a Slater determinant of single-electron HF wave functions ϕi (r) = |i, one may obtain the following expression for Ee : 1 Ee = εi − (ij|Ve (r, r )|ij − ji|Ve (r, r )|ij). (7) 2 i i,j In the case of the electron–positron cluster, the electronic and positronic subsystems are identical and quantized, so they should be considered equal. Because of this equality of electronic and positronic subsystems, one has the condition of local neutrality: ρe (r) = −ρp (r).

(8)

This result is also proved within the optimized jellium model (OJM)4,5 in which the positive-charge distribution of the classical core is chosen in accordance with the absolute minimum of the total energy Etot . This OJM model does not include any free parameters. Using the double variation procedure, one can obtain a precise result: the total electronic density is equal to the “optimal” positive-charge density at each space point. Within the Hartree–Fock approximation due to the condition of local electro-neutrality

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and compensation for both the Hartree term and the positive core contributions, we obtain the following equation4 : ∆ 1 dr = εi ϕi (r). − ϕi (r) − ϕj (r) ϕj (r )ϕi (r ) (9) | 2 |r − r j So such a cluster system within the HF is bound exceptionally due to the non-local exchange interaction between free electrons, including selfinteraction. There is no diﬀerence between an electron–positron cluster and a metallic cluster when one calculates only the electronic subsystems within the HF + OJM approximation. The essential diﬀerence must appear because two subsystems in EPD are quantized and when one calculates the total energy of the electron–positron cluster. The total energy of the electron– positron cluster within the HF must be calculated using the following equation: p e , Etot Etot

p e + Etot − E e−p , Etot = Etot

(10)

are the total energies of each quantum subsystem, which where are calculated using Eq. (5). E e−p is energy of the electron–positron interaction. More diﬀerences between EPD and metal clusters appear when one takes into account many-electron and many-positron correlations. 2.2. The local density approximation (LDA) The Kohn–Sham equations for electron and positron single-particle wave functions may be written 2 ˆ p e,p e,p e,p + Veﬀ (r) ϕe,p (11) i (r) = εi ϕi (r). 2 Here, the eﬀective interaction for each subsystem includes three terms: ρp,e (r ) ρe,p (r ) e,p e,p + dr + Vxc Veﬀ (r) = dr (r). (12) |r − r | |r − r | The ﬁrst two terms in Eq. (12) produce the potential created by the charge density distributions of the electron and positron subsystems, ρe,p (r ) =

N i=1

2 |ϕe,p i (r)| ,

(13)

e,p where N is the number of electron–positron pairs. The third term Vxc (r) in Eq. (12) represents the exchange-correlation potential, which includes three parts: the local exchange interaction between equivalent particles [Vxe−e (r) for electrons and Vxp−p (r) for positrons] and the correlation interaction between equivalent and non-equivalent particles [Vce−e (r) + Vce−p (r) for

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electrons and Vcp−p (r) + Vcp−e (r) for positrons]1 : e (r) = Vxe−e (r) + Vce−e (r) + Vce−p (r), Vxc

(14)

p Vxc (r)

(15)

=

Vxp−p (r)

+

Vcp−p (r)

+

Vcp−e (r).

Due to the condition of local electro-neutrality, Eq. (8), the ﬁrst two terms in Eq. (12) compensate-each other, so the electron–positron cluster is maintained due to the exchange-correlation potential. In order to test the selfconsistency of the applied approach, we used an iterative procedure to solve Eqs. (11)–(14). These calculations show that the electron and positron densities are equal to each other, ρe (r) = −ρp (r), as expected from the charge symmetry of the two subsystems. The total energy of the cluster is calculated using Eq. (10). Within the LDA, one can obtain the following expression for the total electron (positron) energy: N 1 e−p,p−e e,p e,p e−e,p−p E = εe,p − + E − drρe,p (r)Vxc (r), (16) Etot xc i 2 i=1 are the single particle energies deﬁned from the solution of where εe,p i Eq. (11). The energy of the Coulomb interaction between electron and positron subsystems is equal: e,p ρ (r)ρp,e (r) drdr . (17) E e−p = E p−e = |r − r | The exchange-correlation energy within the LDA is determined by e,p Exc = ρe,p (r)εe,p (18) xc (r)dr, with the density of exchange-correlation energy εxc being equal to e,p e,p e−p,p−e (r). εe,p xc (r) = εx (r) + εc (r) + εc

(19)

Note that the Coulomb repulsion energy of electron and positron subsystems in (16) reads as e,p ρ (r)ρe,p (r) 1 drdr . (20) E e−e,p−p = 2 |r − r | The exchange-correlation potential is taken as a sum of exchange and correlation parts10,11 : 1 9 3 1 LDA + Vcorr , (21) Vxc (r) = Vx + Vcorr = − 4π 2 rs (r) where 13 3 rs (r) = . (22) 4πρ(r)

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The correlation part of the potential within the Gunnarson–Lundqist approximation10 reads as 11.4 Vcorr = −0.0333 ln 1 + . (23) rs (r) The correlational part of the exchange-correlation potential within the Perdew–Wang (PW91) parameterization11 is given by √ 1 + 76 1.1581 rs + 43 0.3446rs Vcorr (rs ) = εcorr (rs ) , (24) √ 1 + 1.1581 rs + 0.3446rs εcorr (rs ) = −

0.1471 . √ 1 + 1.1581 rs + 0.3446rs

(25)

The calculations were performed within both the Gunnarson–Lundqvist approximation and Perdew and Wang parameterization for the exchangecorrelation potential and the results obtained are compared with each other and with HF calculations. 3. Potential and density distribution In order to prove the existence of electron–positron clusters of a ﬁnite size, we calculate the electron–positron energy level structure and analyze the stability of these objects. The electron–positron droplet structure has been calculated for clusters with numbers of electron–positron pairs N = 2, 8, 18 and 20. These numbers of electrons (positrons) correspond to the clusters with the closed 1s2 , 2p6 , 3d10 and 2s2 shells and spherically symmetric density distributions: N = 2 (1s2 ), N = 8 (1s2 2p6 ), N = 18 (1s2 2p6 3d10 ), N = 20 (1s2 2p6 3d10 2s2 ). The calculated potentials within the LDA with the Gunnarson–Lundqvist approximation and Perdew and Wang (PW91) parameterization for these clusters are presented in Figs. 1 and 2. One can see a potential well with a minimum at the center of the electron–positron cluster. The depth of the well increases with cluster size. A comparison between the Gunnarson– Lundqvist approximation and the Perdew and Wang (PW91) parameterization shows that the latter takes into account more correlation eﬀects and makes the potential well deeper.

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0,0

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1 - N=2 2 - N=8 3 - N=18

-1,5 -2,0 -2,5 -3,0

0

20

40

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r (a.u.) Fig. 1. Potentials of small electron–positron clusters calculated with the Gunnarson– Lundqvist approximation for the correlation term (23).

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1 - N=2 2 - N=8 3 - N=18 4 - N=20

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40

60

80

r, a.u. Fig. 2. Potentials of small electron–positron clusters calculated with the PW91 parameterization for the correlation term.

The radial dependence of the electron–positron cluster equilibrium densities obtained within the OJM model and the LDA are presented in Figs. 3–5. The ﬁgures show that the electron and positron densities are strongly dispersed in the EP cluster outer region. The dispersion of the electron density in the vicinity of the cluster surface is known for metal clusters as the spill-out eﬀect. In EP clusters, spilling out of the density is

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r (a.u.) Fig. 3. Radial distribution of equilibrium densities calculated for cluster N = 8 within the OJM and LDA with the PW91 parameterization.

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r (a.u.) Fig. 4. Radial distribution of equilibrium densities calculated for cluster N = 18 within the OJM and LDA with the PW91 parameterization.

much stronger than in metal clusters, because the EP cluster does not have a ﬁxed core determining the system geometry and the density of delocalized electrons or positrons. Thus, electron–positron clusters are much more compact objects than the corresponding metallic clusters whose densities were obtained within the OJM. This is a consequence of the quantization of the positively charged positron subsystem.

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4

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r (a.u.) Fig. 5. Electron–positron equilibrium densities for small EPD (N = 2, 8, 18, 20) within the LDA with the PW91 parameterization.

The calculated total energies per particle and the average distance between particles for the EPDs with N = 2, 8, 18 and 20 are presented in Tables 1 and 2 and compared with the corresponding characteristics obtained for Na clusters and with the positronium binding energy. The following deﬁnitions have been used: HF JM is the ordinary Hartree–Fock jellium model for metal clusters2,12 ; LDA JM is a similar model developed within the LDA2 ; OJM is the optimized jellium model for metal clusters4,5 ; EPD 1 and EPD 2 are the results of this work obtained without and without accounting for the correlations between electronic and positronic subsystems,1 respectively. Our calcuations demonstrate that the EPDs have higher densities (smaller Wigner–Seitz radius rs ) than those of sodium clusters. The correlation interaction between electron and positron subsystems plays a very important role. As one can see from Table 1, it reduces the average distance between particles, being in the range 2.6–2.8 a.u. for the EPDs considered. Table 1. The average distances between particles rs in sodium clusters and electron–positron droplets (in a.u.). Theory HF JM LDA JM OJM EPD1 EPD2

N =2

N =8

N = 18

N = 20

3.29 4.0 3.33 3.33 2.96

3.15 4.0 3.35 3.12 2.79

3.10 4.0 3.50 3.20 2.86

3.10 4.0 3.50 3.05 2.73

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Table 2. The total energies of clusters per particle E/2N for diﬀerent numbers of pairs (in a.u.). Theory

N=2

N =8

N = 18

N = 20

HF JM LDA JM OJM EPD1 EPD2 Ps

−0.049

−0.047 −0.072 −0.051 −0.07 −0.10 −0.125

−0.045 −0.074 −0.049 −0.071 −0.101 −0.125

−0.450 −0.0740 −0.0490 −0.072 −0.102 −0.125

−0.054 −0.065 −0.094 −0.125

Also, the results presented in Table 2 show that the EPDs have larger binding energy E/2N than those for Na clusters calculated both in HF JM12 and LDA JM.2 The correlation between electron and positron subsystems signiﬁcantly lowers the EPD total energy and makes it very close to the positronium (Ps) binding energy. To estimate the EPD stability against the decay into Ps, we use our analogy between this system and metal clusters. In neutral metal clusters, the evaporation of a single atom usually means overcoming a barrier, which is typically of about 1 eV. It can be very roughly estimated from the double value of the binding energy per atom in a cluster. Applying this argument to the EPD, one can state that the evaporation energy of a single Ps from the EPD can be roughly estimated as 0.22 eV (see Table 2). The emission of Ps from the EPD occurs via the ﬁssion barrier originating due to the restructuring of both electronic and positronic energy levels in the system. A similar situation takes place in the ﬁssion of multiply charged metal clusters.13,14 The height of this barrier can be estimated from the binding energy of two Ps. The Ps dimer has recently been described and the binding energy 0.4 eV for this system has been reported; see Ref. 3 and references therein. This value should give a very rough estimate for the value of the barrier to be overcome during the Ps emission process. The fact that the binding energy of electrons and positrons in the EPD turns out to be smaller than the Ps binding energy (see Table 1) implies the possibility of Mott phase transition in the system and the formation of a cluster of Ps with a well-deﬁned lattice structure. For a medium of ortho-positronium this also means the possibility of transition into a Bose– Einstein condensate state at suﬃciently low temperatures. We evaluate the probability of the EPD annihilation from the following equation: (26) |ϕen (r1 )|2 |ϕpm (r2 )|2 δ(r1 − r2 )dr1 dr2 , W = 4πr02 c mn

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where m, n are the sets of quantum numbers for positron and electron 2 states, respectively, and r0 = mee c2 . Integrating over the angular variables, one derives: (p)2 (e)2 N l Nl Pn l (r)Pnl (r)dr. (27) W = r02 c n l

(p)

nl

(e)

Here, Pn ,l (r) and Pn,l (r) are the radial parts of the positron and the electron wave functions; Nl = 2(2l + 1), Nl = 2(2l + 1) are the occupation numbers for the electron and positron shells with orbital momentum l and l , respectively. This expression is proportional to the probability of twophoton annihilation of para-positronium: WPs =

r02 c . 2

(28)

A comparison of the probability of the two-photon annihilation in EPD with the probability of the two-photon annihilation in a system of equivalent number of positroniums may be made using the following ratio: R=

W . N WPs

(29)

Integrating (24) one derives the following ratios: RN =2 = 0.68;

RN =8 = 2.27;

RN =18 = 4.79;

RN =20 = 4.52.

These estimates demonstrate that the EPD lifetime against annihilation is comparable with the Ps lifetime, which means it is very large at the atomic scale of units, and it grows quickly with increasing EPD size: τPs =

2 = 1.23 × 10−10 c. mc2 α5

(30)

Accounting for the annihilation in EPD of a suﬃciently large size should lead to the additional polarization of the medium within the EPD and as a result to the additional attraction between electrons and positrons. This eﬀect should reduce the EPD total energy. For an EPD of larger size, one may expect that accounting for the additional attraction in the system caused by the annihilation could make it energetically more stable than the Ps condensate. Investigation of this interesting phenomenon could be a subject for further work. Another interesting question concerns the calculation of the probability of immediate annihilation of several electrons and positrons into one high energy photon as well as all other possible annihilation channels in the system, including its explosion.

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4. Concluding remarks The calculations we have performed show that electrons and positrons are able to form a bound system like a cluster or a droplet with properties similar to those of metal clusters. Electron–positron clusters or droplets are purely quantum many-body systems where the electron and positron subsystems are identical. An EPD has a more compact structure than a possibly existing positronium molecule. We have performed our calculations for closed shell electron–positron clusters. The extension of this consideration for open shell systems implies accounting for deformations similarly to how it was done for clusters in the Hartree–Fock deformed jellium model.15 Our theory is based on the nonrelativistic density functional formalism. Treatment of the EPDs on the basis of the Hartree–Fock–Dirac equation and consistent many-body perturbation theory is another interesting problem to be solved in the future. In this contribution, we have not discussed collective dynamics of particles in the EPD, which is another interesting topic for further consideration. Indeed, one can extend our model and add to the Hamiltonian of the system a part responsible for the collective dynamics of the EPD in a similar way as was done for metal clusters within the dynamical jellium model,16 accounting for collective breathing, dipole and quadrupole deformation modes. An EPD at ﬁnite temperatures can be treated using the technique developed in Ref. 16. In conclusion, we have described a new physical object, the electron– positron quantum cluster or droplet, which opens up a broad spectrum of problems for further investigation. Acknowledgments This work is supported by INTAS (grant number 03-51-6170) and the Swiss National Scientiﬁc Foundation (SNSF IB7420-111116). References 1. A. V. Solov’yov, V. K. Ivanov and R. G. Polozkov, Eur. Phys. J. D 40, 313 (2006). 2. W. Ekardt (ed.), Metal Clusters (Wiley, New York, 1999). 3. H. Yabu, Nucl. Instr. Meth. Phys. Res. B 221, 144 (2004). 4. V. K. Ivanov, V. A. Kharchenko, A. N. Ipatov and M. L. Zhizhin, Pis’ma Zh. Eksp. Teor. Fiz. 60, 345 (1994). 5. V. K. Ivanov, A. N. Ipatov and V. A. Kharchenko, Zh. Eksp. Teor. Fiz. 109, 902 (1996).

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14.

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G. I. Budker, Atomic Energy USSR 22, 346 (1967). S. Eidelman et al., Phys. Lett. B 592, 1 (2004). I. N. Mishustin et al., Phys. Lett B 542, 206 (2002). A. A. Rogachov, Progr. Quant. Electron. 6, 141 (1980). O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976). J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992). V. A. Kharchenko, V. K. Ivanov, A. N. Ipatov and M. L. Zhizhin, Phys. Rev. A 50(2), 1459 (1994). J.-P. Connerade and A. V. Solov’yov (eds.), Latest Advances in Atomic Clusters Collision: Fission, Fusion, Electron, Ion and Photon Impact (Imperial College Press, London, 2004), pp. 1–398. A. G. Lyalin, O. I. Obolensky, A. V. Solov’yov and W. Greiner, Phys. Rev. A 65, 043202 (2002); Phys. Rev. B 72, 085433 (2005); J. Phys. B: At. Mol. Opt. Phys. 37, L7 (2004). A. G. Lyalin et al., J. Phys. B: At. Mol. Opt. Phys. 33, 3653 (2000). L. G. Gerchikov, A. V. Solov’yov and W. Greiner, Int. J. Mod. Phys. E 8(3), 289 (1999); J. Phys. B: At. Mol. Opt. Phys. 33, 4905 (2000).

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SPECTROSCOPY OF NEUTRAL RETINAL AND GFP CHROMOPHORES IN THE GAS PHASE L. H. ANDERSEN Department of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark www.phys.au.dk/molecular [email protected] In this contribution, we discuss aspects of spectral tuning of molecular chromophores at the atomic level. The basis for the discussion is a series of measurements performed at the electrostatic ion-storage ring ELISA at the University of Aarhus. The experimental technique, which is a type of “action spectroscopy” based on photo dissociation, will brieﬂy be discussed. We will present examples of spectral tuning due to external charges and proton exchange with the hosting medium. Keywords: Chromophore ions; storage ring; absorption spectroscopy; gas phase.

1. Introduction Molecular chromophores are found in many places in nature, where they serve a variety of purposes such as energy harvesting and signaling. They are found in so-called photoactive proteins like the green ﬂuorescent protein (GFP),1 the photoactive yellow protein (PYP)2 and the large family of retinal containing proteins.3 Other important areas include heme-containing proteins and porphyrins4 due to their potential application in photonic devices5 and in photodynamic therapy.6 Organic chromophore molecules normally consist of about 20–50 atoms and are of such a size that the energy diﬀerence between the electronic ground state and the ﬁrst excited state is a couple of electronvolts, corresponding to a transition in the visible or near visible region of the spectrum. The charge state of the chromophore is determined by its state of protonation (cations are protonated and anions are de-protonated). The state of protonation proves to be a very important parameter as will be discussed later in the present contribution. However, other factors also play a role.

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The environment may distort the chromophore by twisting it and hence shift the absorption maximum. Besides the interaction due to the overall electrostatic ﬁeld of the surrounding medium, there may be local charged groups in the vicinity of the chromophore. Counterions may be present when the chromophore itself is charged. There may also be hydrogen bonding to, for example, water molecules hidden in the protein structure, and the situation becomes even more complicated when external interactions (inter-molecular) create new intra-molecular interactions. The so-called color tuning that is of contemporary interest is related to environmental perturbations of the electronic levels in the chromophores. To obtain knowledge hereof at the atomic scale, a variety of methods is being used. By a biological approach, site-speciﬁc mutations are made and the consequences for the spectral absorption are being studied.7 For many years, the spectral properties have been compared to those in solutions. But the problem of deﬁning and controlling the interactions in liquids is not a trivial matter. Quantum calculations have also been performed, but with the size of the chromophores of interest such calculations are not trivial. The best reference for theory and for absorption in other media seems to be the absorption proﬁle of chromophores in vacuo, where there are no external perturbations. With this as a reference one may have an important benchmark value for calculations and a measure of perturbations occurring in speciﬁc hosting media. Many biological species use proteins which have retinal as the chromophore, one example being bacteriorhodopsin8 (λmax = 568 nm) found in the membrane of halobacteria.9 Other examples of opsin proteins are rhodopsin (λmax = 498 nm) and the visual pigments [λmax = 425 nm (blue), 530 nm (green), and 560 nm (red)]10 in the human eye enabling night vision and color vision, respectively. The opsins share a very similar structure and features of their photocycle. The chromophore is in all cases retinal, in the all-trans form in the halobacterial opsins and in the 11-cis form in the rhodopsins. In both cases, retinal is linked to the protein via a Schiﬀ-base linkage to a speciﬁc lysine residue. The primary event in vision is the absorption of a photon by the chromophore which is here retinal in the protonated Schiﬀ-base form, i.e. a charged species. The absorption of the photon results in a conformation change of the chromophore, which then starts a cascade of processes that eventually leads to a recognizable signal for the brain. Granit, Hartline and Wald were awarded the Nobel Prize in Medicine in 1967 for their discoveries concerning the primary physiological and chemical visual processes in the eye.10 Yet, today — almost 40 years later — the very mechanisms that are responsible for the shifts of the electronic energy levels in retinal that allow for color vision are still the subject of intense research.

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Even though the Schiﬀ base is protonated in the ground state in a large majority of the opsins, a neutral Schiﬀ base is also encountered. One example is the intermediate produced in the photocycle of most opsin proteins. Light absorption triggers a very fast isomerization of the retinal chromophore (all-trans to 13-cis in the case of halobacterial proteins, and 11-cis to all-trans in the case of rhodopsin). The isomerization is followed by a proton transfer from the Schiﬀ base to a nearby amino acid residue making the Schiﬀ base neutral. In rhodopsin, this intermediate is known as metarhodopsin II (λmax = 380 nm). The neutral Schiﬀ-base intermediate in bacteriorhodopsin is termed the M intermediate (λmax = 410 nm). The above-mentioned opsins with unprotonated Schiﬀ bases all absorb in the UV, whereas the ground states of the halobacterial opsins and the visual pigments have maximum absorption wavelengths in the visible part of the spectrum. The mere protonation of the chromophore represents in itself a potential factor for color tuning that we will discuss here. Two examples are considered: Schiﬀ-base retinal and the chromophore of the green ﬂuorescent protein.

2. Experimental technique The experimental technique that we use to obtain the absorption spectra was ﬁrst applied in 2001, when we recorded for the ﬁrst time the gas-phase absorption spectrum of a deprotonated GFP-model chromophore.11 Since then, it has been applied in a number of measurements.12–24 Brieﬂy, we use the ELectrostatic Ion Storage ring in Aarhus, Denmark, (ELISA)25–27 (Fig. 1). The chromophore molecules are ﬁrst electronically excited by light from a pulsed laser. The internal energy is quickly converted into nuclear vibrations (internal conversion), and as a result fragmentation occurs, typically on the sub-millisecond to millisecond time scale. The storage-ring technique is essential to the work, since it allows us to record the delayed action (dissociation) of the photoexcitation of the chromophore. Freshly prepared chromophore samples are brought into the gas phase by an electrospray-ion source.26 The ions are accumulated for 100 ms in an ion trap with a helium buﬀer gas which cools them down to room temperature. After extraction and acceleration to a kinetic energy of 22 keV, the desired ions are selected by a bending magnet, and about 10–30 µs long bunches of typically 104 ions are injected into the storage ring at a repetition rate of 10 Hz. The ions circulate in the ring with a revolution time of about 50–100 µs until they change their mass-to-charge ratio, either by unimolecular dissociation or by collisions with residual gas (pressure ∼10−11 mbar).

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PAL101 Laser power meter

Ion bunch

Laser pulse

Raman cell

ELISA

Magnet

OPPO Nd:YAG

Injection 1m Accelerator with electrospray ion source

Detector for neutrals Injection

Heated Octupole Ion trap capillary

Laser

N0

Nsignal

needle Pumps spray solution

Ion bunch

Time after injection [ms]

Fig. 1. The electrostatic ion storage ring ELISA. The lower part of the ring is the injection and detection section, and the upper part is the laser interaction section. The electrospray-ion source is seen in the inset to the left. In the inset to the right, the counts of neutrals as a function of time is seen. The windows are used for calculation of the absorption cross-section [Eq. (1)].

Neutral particles are not deﬂected by the electric ﬁelds in ELISA and continue on straight trajectories. Those formed in the ﬁrst section of the ring hit a micro-channel plate detector located at the end of the section (see Fig. 1) and constitute the signal of absorption. The chromophore ions are laser irradiated after about 50 ms of storage. Tuning of the laser wavelength to an absorption band results in bond dissociation and hence in the formation of detectable neutral products. Figure 1 also shows a typical decay spectrum where an increased count rate of neutrals are recorded after the laser is ﬁred. Initially, the chromophore is in the electronic ground state at room temperature where only a few of the 3N − 6 vibrational modes are excited (N is the number of atoms in the molecule). Photoabsorption brings the molecule into the ﬁrst excited state and radiation-less internal conversion brings the molecule back to the electronic ground state with about 2–4 eV of energy distributed over the vibrational modes. This eventually leads to a statistical dissociation on the sub-millisecond time scale (see Fig. 1). Such decays have been modeled successfully with an Arrhenius expression for the decay rate.19,28

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The photo-absorption cross-section σ(λ) is given as N (λ) − N0 Epulse (λ) σ(λ) ∝ , N0 hc/λ

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(1)

where N (λ) is the total counts of neutral particles in the signal peak (see Fig. 1). N0 is a small contribution of neutrals from residual-gas collisions, which is subtracted after proper scaling. N0 is proportional to the number of ions in the ring and is therefore also used for normalization. Epulse (λ) is the average laser-pulse energy and Epulse (λ)/(hc/λ) the number of photons in the laser pulse at wavelength λ. We do not know the exact overlap between the ion beam and the laser beam, and hence we do not obtain absolute cross-sections. 3. Results 3.1. The retinal chromophore with a neutral Schiﬀ base We consider here the three retinal chromophores shown in Fig. 2. The ﬁrst one is the standard protonated Schiﬀ-base retinal, the second has a neutral Schiﬀ base but a positive charge adjacent to the Schiﬀ base, and the third

(1) Protonated Schiff base

N

H+

N

N

+ NMe3

+ NMe3

(2) Neutral Schiff base with close charge

(3) Neutral Schiff base with distant charge

Fig. 2. Three retinal chromophores. (1) In the protonated Schiﬀ-base form, (2) with a neutral Schiﬀ base and a positively charged group nearby, and (3) with a neutral Schiﬀ base and a positively charged group further away.

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one has its positive charge further away (as described above, only charged species may be studied in ELISA). Let us discuss the eﬀect of the approaching positive charge on the neutral retinal chromophore. In Fig. 3, we show the absorption spectra in the gas phase of the three retinal chromophores. The protonated all-trans n-butyl retinal Schiﬀ base (1) has its absorption maximum at 610 nm.14 The two model chromophores with neutral Schiﬀ bases both experience an immense blue shift compared to the protonated retinal Schiﬀ base, as seen in Fig. 3. The absorption maximum of the chromophore with the most distant positive charge (3) is at 388 nm, and the shift between the protonated and this neutral Schiﬀ-base chromophore is thus more than 200 nm! The compound with a positive charge adjacent to the Schiﬀ base (2) has its absorption maximum at 480 nm, which is between the other maxima. We clearly observe that the location of the positive spectator charge has an inﬂuence on the position of the absorption maximum for the two neutral Schiﬀ-base models. One may ask: What is the origin of the almost 100 nm diﬀerence in absorption maxima of the two neutral Schiﬀ-base models (2) and (3)? When retinal is excited from S0 to S1 , a charge transfer from the ionone ring to the Schiﬀ base is taking place,29 i.e. negative charge is transferred from the ring end to the Schiﬀ base containing the nitrogen atom. This negative

(1) Protonated Schiff base (2) Neutral Schiff base (close) (3) Neutral Schiff base (distant)

1.2

Absorption cross-section (arb. units)

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0.8

0.6

0.4

0.2

0.0 300

350

400

450

500

550

600

650

700

λ (nm)

Fig. 3.

Absorption spectra of three retinal chromophores of Fig. 2.

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charge transfer can qualitatively explain the shift between the two chromophores when we consider what happens as the positive spectator charge is approaching the Schiﬀ base. With an almost even charge distribution, the S0 state of the neutral chromophore is not disturbed much. The excited state (S1 ) will, on the other hand, be stabilized more in model (2), where the spectator charge is closer to the Schiﬀ base than in model (3), resulting in an absorption spectrum of (2) which is red-shifted compared to the spectrum of (3), in complete agreement with the observation. Moving the positive charge to inﬁnity should simulate the absorption of the true neutral chromophore, which may indeed be expected to be even further to the blue (364 nm30 or 357 nm31 ). The gas phase value of 388 nm is important for comparison to theory in order to optimize calculational methods. Sekharan et al.,31 for example, recently reported a CASPT2 calculational study on the chromophore (2) with the -N(CH3 )3 + group reduced to -NH3 + . They obtained an S0 to S1 excitation wavelength of 484 nm, which is in good agreement with the experimental value at 480 nm. 3.2. The neutral GFP chromophore The green ﬂuorescent protein (GFP) has a chromophore which may be either neutral or protonated. As a result, the protein has two absorption bands, one at about 400 nm and another at about 480 nm for the neutral and anion chromophores, respectively. It is estimated that in wild-type GFP there is about six times more neutral than deprotonated chromophores.32 In our initial study of the deprotonated GFP chromophore,11 we found that the absorption of the bare chromophore in vacuum essentially coincides with the corresponding absorption band of the protein, which indicates that the environmental tuning is indeed very gentle in the protein. An alternative explanation is that S0 and S1 are perturbed by an equal amount. We have addressed experimentally the absorption of an isolated neutral GFP model chromophore.13 To this we have synthesized33 a novel molecule carrying a positive charge well separated from a neutral chromophore, which is akin to that in GFP. Indeed, by studying two chromophores of GFP with diﬀerent protonation stages, we have much ﬁrmer ground when it comes to conclusions about the degree of vacuum-like conditions that possibly may exist inside the GFP cavity. As seen in Fig. 4, it is evident that the wavelength for maximum absorption of the anionic as well as neutral chromophores in vacuum is very close to the wavelengths of the corresponding peaks in the protein (479 versus 476 nm and 396 versus 415 nm, respectively). The role of hydrogen bonding in the neutral GFP chromophore between NH+ 3 and =O is currently being investigated.

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Fig. 4. Absorption spectra of the neutral (left) and anionic (right) GFP model chromophores. The spectra are both normalized to 100 at their peak values.

The results are in full agreement with the recent theoretical ﬁnding of Olivucci and collaborators,34 who found that the protein cavity provides an environment for the anionic GFP chromophore where diﬀerent perturbations, like the presence of counterions, balance each other both in the S0 and S1 states, and cause very little disturbance of the electronic and molecular structure of the chromophore. We conclude that the absorption properties of the green ﬂuorescent protein to a high degree are determined by the intrinsic chromophore properties. One of the important roles of the protein environment is to allow for protonation and deprotonation of the chromophore, which indeed shifts the absorption, as shown in the works discussed here. For both chromophores (retinal and the GFP chromophore), the state of protonation seems to be of major importance for the location of the absorption maximum. Interestingly, the proton-transfer process is also vital to the functioning of the signaling proteins. Acknowledgments This work has been supported by the Danish Natural Science Research Council (grant no. 272-06-0427), the Carlsberg foundation (grant no. 200601-0229) and by the Lundbeck foundation.

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References 1. R. Y. Tsien, Annu. Rev. Biochem. 67, 509 (1998). 2. D. E. McRee, J. A. Tainer, T. E. Meyer, J. van Beeumen, M. A. Cusanovich and E. D. Getzoﬀ, Proc. Natl. Acad. Sci. USA 86, 6533 (1989). 3. K. Palczewski, T. Kumasaka, T. Hori, C. A. Behnke, H. Motoshima, B. A. Fox, I. Le Trong, D. C. Teller, T. Okada, R. E. Stenkamp, M. Yamamoto and M. Miyano, Science 289, 739 (2000). 4. J. A. Shelnutt, X. Z. Song, J. G. Ma, S. L. Jia, W. Jentzen and C. J. Medforth, Chem. Soc. Rev. 27, 31 (1998). 5. A. P. Desilva, H. Q. N. Gunaratne and C. P. Mccoy, Nature 364, 42 (1993). 6. Z. Huang, Tech. Cancer Res. Treatment 4, 283 (2005). 7. S. Yokoyama, Prog. Retinal Eye Res. 19, 385 (2000). 8. R. A. Mathies, S. W. Lin, J. B. Ames and W. T. Pollard, Ann. Rev. Biophys. Biophys. Chem. 20, 491 (1991). 9. W. Stoeckenius and R. A. Bogomolni, Ann. Rev. Biochem. 51, 587 (1982). 10. G. Wald, Science 162, 230 (1968). 11. S. B. Nielsen, A. Lapierre, J. U. Andersen, U. V. Pedersen, S. Tomita and L. H. Andersen, Phys. Rev. Lett. 87, 228102 (2001). 12. L. Lammich, I. B. Nielsen, H. Sand, A. Svendsen and L. H. Andersen, J. Phys. Chem. A 111, 4567 (2007). 13. L. Lammich, M. ˚ A. Petersen, M. B. Nielsen and L. H. Andersen, Biophys. J. 92, 201 (2007). 14. I. B. Nielsen, M. ˚ A. Petersen, L. Lammich, M. B. Nielsen and L. H. Andersen, J. Phys. Chem. A 110, 12592 (2006). 15. I. B. Nielsen, L. Lammich and L. H. Andersen, Phys. Rev. Lett. 96, 018304 (2006). 16. M. A. Petersen, I. B. Nielsen, M. B. Kristensen, A. Kadziola, L. Lammich, L. H. Andersen and M. B. Nielsen, Organ. Biomol. Chem. 4, 1546 (2006). 17. L. H. Andersen, I. B. Nielsen, M. B. Kristensen, M. O. A. E. Ghazaly, S. Haacke, M. B. Nielsen and M. ˚ A. Petersen, J. Am. Chem. Soc. 127, 12347 (2005). 18. I. B. Nielsen, S. Boy´e-P´eronne, M. O. A. E. Ghazaly, M. B. Kristensen, S. B. Nielsen and L. H. Andersen, Biophys. J. 89, 2597 (2005). 19. L. H. Andersen, H. Bluhme, S. Boy´e, T. J. D. Jørgensen, H. Krogh, I. B. Nielsen, S. B. Nielsen and A. Svendsen, Phys. Chem. Chem. Phys. 6, 2617 (2004). 20. S. Boy´e, H. Krogh, I. B. Nielsen, S. B. Nielsen, S. U. Pedersen, U. V. Pedersen, L. H. Andersen, A. F. Bell, X. He and P. J. Tonge, Phys. Rev. Lett. 90, 118103 (2003). 21. S. Boy´e, I. B. Nielsen, S. B. Nielsen, H. Krogh, A. Lapierre, H. B. Pedersen, S. U. Pedersen, U. V. Pedersen and L. H. Andersen, J. Chem. Phys. 119, 338 (2003). 22. S. Boy´e, S. B. Nielsen, H. Krogh, I. B. Nielsen, U. V. Pedersen, A. F. Bell, X. He, P. J. Tonge and L. H. Andersen, Phys. Chem. Chem. Phys. 5, 3021 (2003).

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23. L. H. Andersen, A. Lapierre, I. B. Nielsen, S. B. Nielsen, U. V. Pedersen, S. U. Pedersen and S. Tomita, Nucl. Phys. Rev. 19, 176 (2002). 24. L. H. Andersen, A. Lapierre, S. B. Nielsen, I. B. Nielsen, S. U. Pedersen, U. V. Pedersen and S. Tomita, Eur. Phys. J. D 20, 597 (2002). 25. S. P. Møller, Nucl. Instrum. Meth. Phys. Res. A 394, 281 (1997). 26. J. U. Andersen, P. Hvelplund, S. B. Nielsen, S. Tomita, H. Wahlgreen, S. P. Moller, U. V. Pedersen, J. S. Forster and T. J. D. Jorgensen, Rev. Sci. Instrum. 73, 1284 (2002). 27. L. H. Andersen, O. Heber and D. Zajfman, J. Phys. B 37, R57 (2004). 28. S. B. Nielsen, J. U. Andersen, J. S. Forster, P. Hvelplund, B. Liu, U. V. Pedersen and S. Tomita, Phys. Rev. Lett. 91, 048302 (2003). 29. R. Mathies and L. Stryer, Proc. Natl. Acad. Sci. USA 73, 2169 (1976). 30. R. S. Becker and K. Freedman, Proc. Natl. Acad. Sci. USA 107, 1477 (1985). 31. S. Sekharan, O. Weingart and V. Buss, Biophys. J. 91, L7 (2006). 32. K. Brejc, T. Sixma, P. Kitts, S. Kain, R. Tsien, M. Orm¨ o and S. Remington, Proc. Natl. Acad. Sci. USA 94, 2306 (1997). 33. M. ˚ A. Petersen, P. Riber, L. H. Andersen and M. B. Nielsen, Synthesis 23, 3635 (2007). 34. A. Sinicropi, T. Andruniow, N. Ferre, R. Basosi and M. Olivucci, J. Am. Chem. Soc. 127, 11534 (2005).

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THE ENERGY LANDSCAPE AS A COMPUTATIONAL TOOL J. M. CARR and D. J. WALES∗ University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, UK www-wales.ch.cam.ac.uk ∗ [email protected] Coarse-graining the potential energy surface in terms of its local minima, and the transition states that connect them, provides a framework for global optimization, and the calculation of global thermodynamic and kinetic properties. Here, we provide an overview of the computational tools used in such analyses, and present some new results for kinetic analysis of pathways obtained from a stationary point database for the GB1 hairpin peptide. Keywords: Energy landscapes; global optimization; superposition approach; discrete path sampling.

1. Introduction The potential energy surface (PES) determines all the observed structures and calculated thermodynamic and kinetic properties in computer simulations of a molecular or condensed matter system.1 For example, the stable equilibrium geometries at zero Kelvin correspond to local minima of the PES, while the energy and gradient used in Monte Carlo and molecular dynamics simulations are directly calculated from the potential energy function or its ﬁrst derivatives.2 A convenient coarse-graining scheme can also be formulated using the local minima and transition states of the PES, which both correspond to stationary points where the gradient vanishes. Here we deﬁne transition states geometrically, as stationary points with a single imaginary normal mode frequency,3 while minima have only real frequencies. Focusing on local minima leads to the basinhopping approach to global optimization4–6 (Sec. 2) and the basin-sampling approach to thermodynamics7 (Sec. 3). To obtain kinetic properties, we must also sample the transition states that link local minima together. The discrete path sampling approach8,9 encompasses several schemes for 321

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growing stationary point databases that are geared towards analysis of global kinetics (Sec. 4). Some selected examples of calculations based upon analysis of stationary points are presented in the following sections. In particular, a new approach to extracting mean ﬁrst-passage times and rate constants from stationary point databases is discussed in Sec. 5, with applications to the folding of a β hairpin peptide. 2. Basin-Hopping global optimization Basin-hopping global optimization involves stepping between local minima of the PES.1,4–6 Steps may be accepted and rejected using a simple Monte Carlo scheme4,5 or proposed on the basis of evolutionary algorithms.13 Large moves are possible because the potential energy that is used to assess structures is not the instantaneous value, but the minimized quantity. Figures 1 and 2 illustrate structures recently investigated using basinhopping for the Thomson problem14 and a cluster of discoids interacting via the potential suggested by Paramonov and Yaliraki (PY),15 respectively. The Thomson problem consists of N unit charges constrained to the

Fig. 1. Defect migrations linking the global minimum to a low-lying local minimum for N = 732 (top) and two low-lying minima for N = 972 (bottom). The forward and reverse barriers in atomic units are 6.60 × 10−4 and 4.22 × 10−4 for N = 732 and 4.24 × 10−3 and 1.04 × 10−3 for N = 972.

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286,0 V1

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287,2

Fig. 2. Disconnectivity graph10 ,11 for the potential energy of local minima in a 20-particle PY cluster composed of particles with a11 = 0.05, a21 = 0.4, and σ0 = 18. The lowest 100 minima are included, four of which are also illustrated.12

surface of a sphere with potential energy i<j 1/|ri − rj |, with |ri | = 1 for all particles i and j. It was introduced in 1904 in an attempt to study atomic structure,14 but has since become a model for constrained systems with spherical topologies.16 Such systems generally exhibit defects as an unavoidable consequence of geometry; for example, twelve ﬁve-coordinate particles (ﬁvefold disclinations) provide one way to achieve the required

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disclination charge. However, for larger systems the increased strain energy makes diﬀerent sorts of defects more favorable. Preliminary results also suggest that low-barrier rearrangements may generally exist between lowlying minima with alternative arrangements of defects.17 Figure 1 illustrates two such processes for migration of an extended dislocation in N = 732 and interconversion of twinned and conventional grain boundaries in N = 972. Global optimization of clusters of discoids has revealed that helices are favored for a signiﬁcant region of parameter space.12 In particular, helical global minima are generally found when the dimer geometry is a shifted stacked conﬁguration. Furthermore, analysis of the PES reveals the “palm tree” structure illustrated in Fig. 2, which is known to produce eﬃcient self-assembly because there is a well-deﬁned, kinetically accessible free energy minimum.1,18,19 This observation suggests a design principle for producing self-assembling supramolecular polymers with well-deﬁned ﬁber morphologies.20 3. Global thermodynamics from the superposition approach The superposition approach to global thermodynamics1,21–27 is based on the formally exact decomposition of the density of states, Ω(E), or canonical partition function, Z(T ), into a sum over contributions from the basins of attraction of the local minima: Ωα (E) and Z(T ) = Zα (T ). (1) Ω(E) = α

α

To evaluate these expressions exactly would require calculation of the anharmonic density of states for all minima. In practice, suitable samples of local minima can be used so long as the correct weighting can be obtained.26,28,29 Harmonic densities of states, which can readily be obtained from normal mode analysis, are also commonly used as a ﬁrst approximation. The superposition formulation does not assume that the system only samples conﬁgurations “close” to local minima. However, if the harmonic approximation is used for local densities of states then the resulting thermodynamic properties will exhibit systematic errors at higher temperatures. Harmonic superposition calculations have proved to be particularly useful and accurate in treating low-temperature equilibria between free energy minima separated by high barriers.30,31 For example, in solid–solid transitions that involve a change in morphology, a reasonably accurate and naturally ergodic partition function is obtained when a few key minima from the corresponding regions of conﬁguration space are included.

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Although reweighting schemes have been developed26,28,29 for superposition calculations involving high-entropy phases, there are still problems in sampling correctly. The basin-sampling approach7 provides a systematic way to deal with this problem. It is closely related to basin-hopping, but employs moves that retain detailed balance. In the ﬁrst reported applications to cluster thermodynamics, anharmonic densities of states were estimated using data obtained from the local minimizations.7

4. Global kinetics from discrete path sampling To obtain insight into global kinetics, we must characterize not only local minima, but also the transition states that connect them. The results reported in Sec. 5 employed hybrid eigenvector-following32,33 and the doubly-nudged34 elastic band35,36 approach for this purpose. We also require some way to construct a database of stationary points that will be relevant for describing global kinetics. Dijkstra’s shortest-path algorithm37 applied to a network with missing connections can be used to generate an initial path for local minima that are separated by many transition states.38 The discrete path sampling approach8,9 can be thought of as a framework for growing a stationary point database that optimizes the pathways between two end points. The coarse-graining of global kinetics into transitions between local minima, or groups of minima, is usually implemented within a linear master equation approach.39,40 The resulting rates are formally exact within a Markov approximation if we have suﬃcient sampling and the correct densities of states and rate constants for transitions between local minima. In practice, the databases of stationary points are ﬁnite, and we usually employ a harmonic approximation for the densities of states, along with the corresponding formulation of transition state theory.41 The principal assumption involved in transition state theory is that recrossings of the transition surface are neglected.42 In particular, it is not assumed that the true dynamical trajectories lie close to the geometrical transition state or the corresponding steepest-descent minimum energy paths. However, the harmonic approximation will introduce greater systematic errors for high temperature trajectories. A number of applications of the DPS approach have been presented to atomic and molecular clusters, and to biomolecules.8,9,43–46 In each case, we must also address the problem of extracting phenomenological two-state rate constants from databases that potentially contain hundreds of thousands of local minima. A graph transformation method has recently been developed for this purpose, which can provide mean ﬁrst-passage times

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equivalent to a kinetic Monte Carlo simulation, but at signiﬁcantly less computational cost.47,48 However, it is also useful to extract paths that make the most signiﬁcant contributions to the rate constants for the purposes of analysis and visualization. The following section describes a general way to extract such paths, with application to the GB1 peptide. 5. Kinetic analysis and the k shortest paths algorithm Once a connected database of stationary points has been generated using DPS, we can also extract from it a set of discrete paths that make signiﬁcant contributions to the overall rate constants within an approximate formulation. The database is described as a network in exactly the same way as for a kinetics-based shortest-path analysis.45 However, instead of just using Dijkstra’s shortest-path algorithm, the recursive enumeration algorithm50 (REA) of Jim´enez and Marzal is employed to obtain the k shortest paths between two end points, i.e. the k discrete paths, in rank order, that make the largest contributions to the DPS phenomenological rate constant using a steady-state formulation for intervening minima.8,51 5.1. A k shortest paths algorithm The recursive enumeration algorithm starts from a single-source shortestpath algorithm (such as Dijkstra’s) to compute the shortest path from the source (or start) node to every other node in the network. The REA then recursively solves a set of equations that generalize the Bellman equations52 for the single shortest-path problem to multiple paths. What this means in practice is that the algorithm recursively ﬁnds the kth shortest path by considering (at most all of) the nodes in the (k −1)th path and selecting from a set of candidates for the next shortest paths from the source to each of the relevant nodes. Not including Dijkstra’s algorithm, the computational complexity is at worst O(m + k n log[m/n]), where m and n are the number of edges and nodes, respectively. The worst case requires the k shortest paths to all nodes in the network to be calculated, and is seldom encountered in practice. However, we found that the memory requirements can become prohibitive for substantial databases of stationary points (tens of thousands of minima and transition states) and large values of k (thousands). For each of the paths generated with the REA, we calculate the mean ﬁrst-passage time (MFPT) for that unbranched path, in the absence of any oﬀ-path minima, using the graph transformation method.47 In this way, we can obtain the rate constant for a particular path (as the reciprocal of the MFPT) without assuming that the intervening minima are in steady

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state. The set of k paths are then ranked according to their non-steadystate (NSS) rate constants and analyzed. We are, therefore, assuming that our chosen value of k is suﬃciently large that the set of paths that make the largest contribution to the DPS steady-state (SS) rate constant will contain the most important NSS paths, even if the rank orders diﬀer. Paths on which some barrier heights are very small compared to the thermal energy kB T often give rise to sets of fast paths that only diﬀer in the number of times the low-barrier transition states are recrossed. Such sets of paths all have the same NSS rate constant, and are therefore treated as indistinguishable. 5.2. Folding of the GB1 hairpin peptide The GB1 hairpin comprises residues 41–56 from the C-terminal fragment of the B1 domain of protein G. It forms a β hairpin both in the intact protein,53 and for the isolated fragment in solution.54 The peptide was modeled in the current work using the CHARMM19 united-atom potential55 and the EEF1 implicit solvation model,56 as in previous studies.45,57 The two end points for the DPS were, after local minimization, a fully extended structure and the lowest-energy structure from a constanttemperature molecular dynamics trajectory started from the experimental structure53 (Protein Data Bank58 code 2GB1). After constructing an initial discrete path and DPS database as described in Sec. 4, the 500 paths with the largest contributions to the SS rate constants were extracted using the REA. For this database, the rank order of the 500 fastest paths is diﬀerent depending on whether SS or NSS rate constants are considered. The rate constants for the k = 1 and k = 120 paths are compared in Table 1, and snapshots from the two paths are presented in Fig. 3. There are some points of congruency in the two paths, but appreciable deviations in between, as illustrated in Fig. 3. Both paths do, however, belong to the same mechanistic family, showing early formation of one of the β strands followed by formation of the turn and ﬁnally docking of the second strand to complete the hairpin. A full analysis of this DPS database will be presented Table 1. Comparison of two of the fastest paths for the GB1 hairpin DPS database. k = 1 path /s−1

SS rate constant contribution NSS rate constant /s−1 Rank order based on NSS rate constant

10−29

2.4 × 4.1 × 108 1

k = 120 path 9.3 × 10−30 9.8 × 107 475 equal

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Fig. 3. Snapshots from the k = 1 and k = 120 paths from a DPS database representing the folding of the GB1 hairpin from an extended structure (top left) to a hairpin-like conformation (bottom right). The figure was prepared using the NewCartoon representation in VMD.49

elsewhere. Here, we simply note that the numerical values in Table 1 are very diﬀerent because within the SS formulation we only obtain the contribution of a particular path to the overall rate constant. The actual value would require a sum over all the combinatorial possibilities for alternative discrete paths, allowing for all possible branches. In contrast, the NSS value is itself an estimate of the overall rate constant because we do not allow branches oﬀ the path, only recrossings. References 1. D. J. Wales, Energy Landscapes (Cambridge University Press, Cambridge, 2003). 2. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987). 3. J. N. Murrell and K. J. Laidler, J. Chem. Soc. Faraday Trans. 64, 371 (1968). 4. Z. Li and H. A. Scheraga, Proc. Natl. Acad. Sci. USA 84, 6611 (1987). 5. D. J. Wales and J. P. K. Doye, J. Phys. Chem. A. 101, 5111 (1997). 6. D. J. Wales and H. A. Scheraga, Science 285, 1368 (1999). 7. T. V. Bogdan, D. J. Wales and F. Calvo, J. Chem. Phys. 124, 044102 (2006). 8. D. J. Wales, Mol. Phys. 100, 3285 (2002). 9. D. J. Wales, Mol. Phys. 102, 891 (2004). 10. O. M. Becker and M. Karplus, J. Chem. Phys. 106, 1495 (1997). 11. D. J. Wales, M. A. Miller and T. R. Walsh, Nature 394, 758 (1998). 12. S. Fejer and D. J. Wales, Phys. Rev. Lett. in press (2007).

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49. W. Humphrey, A. Dalke and K. Schulten, J. Molec. Graphics 14, 33 (1996). 50. V. M. Jim´enez and A. Marzal, Computing the k shortest paths: A new algorithm and an experimental comparison, in Algorithm Engineering: 3rd International Workshop (WAE ’99), London, UK, July 1999, eds. J. S. Vitter and C. D. Zaroliagis (Springer, Berlin, 1999). 51. D. J. Wales, Int. Rev. Phys. Chem. 25, 237 (2006). 52. R. Bellman, Q. Appl. Math. 16, 87 (1958). 53. A. M. Gronenborn, D. R. Filpula, N. Z. Essig, A. Acharia, M. Whitlow, P. T. Wingfield and G. M. Clore, Science 253, 657 (1991). 54. F. J. Blanco, G. Rivas and L. Serrano, Nat. Struct. Biol. 1, 584 (1994). 55. B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan and M. Karplus, J. Comp. Chem. 4, 187 (1983). 56. T. Lazaridis and M. Karplus, Proteins Struct. Func. Gen. 35, 133 (1999). 57. S. V. Krivov and M. Karplus, Proc. Natl. Acad. Sci. USA 101, 14766 (2004). 58. H. M. Berman, J. Westbrook, Z. Feng, G. Gilliland, T. N. Bhat, H. Weissig, I. N. Shindyalov and P. E. Bourne, Nucleic Acids Res. 28, 235 (2000) [the Protein Data Bank home-page is www.rcsb.org/pdb].

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THEORETICAL FRAMEWORK FOR THE INTERPRETATION OF NMR RESIDUAL DIPOLAR COUPLINGS OF UNFOLDED PROTEINS O. I. OBOLENSKY† and A. V. SOLOV’YOV∗,† Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str.1, 60438 Frankfurt am Main, Germany ∗ [email protected] K. SCHLEPCKOW Institute for Organic Chemistry and Chemical Biology, Center for Biomolecular Magnetic Resonance, Johann Wolfgang Goethe University, 60438 Frankfurt am Main, Germany Frankfurt International Graduate School for Science, Johann Wolfgang Goethe University, 60438 Frankfurt am Main, Germany H. SCHWALBE Institute for Organic Chemistry and Chemical Biology, Center for Biomolecular Magnetic Resonance, Johann Wolfgang Goethe University, 60438 Frankfurt am Main, Germany A theoretical framework for the prediction of nuclear magnetic resonance (NMR) residual dipolar couplings (RDCs) in unfolded proteins under weakly aligning conditions is presented. The unfolded polypeptide chain is modeled as a random ﬂight chain while the alignment medium is represented by a set of regularly arranged obstacles. For the case of bicelles oriented perpendicular to the magnetic ﬁeld, a closed-form analytical result is derived. With the analytical expression obtained, the RDCs are readily accessible for any locus along the chain, for chains of diﬀering length, and for varying bicelle concentrations. The two general features predicted by the model are (i) RDCs in the center segments of a polypeptide chain are larger than RDCs in the end segments, resulting in a bell-shaped sequential distribution of RDCs, and (ii) couplings are larger for shorter chains than for longer chains at a given bicelle concentration. The presented framework is an important step towards a solid †On

leave from: A. F. Ioﬀe Institute, St. Petersburg 194021, Russia. 333

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theoretical foundation for the analysis of experimentally measured RDCs in unfolded proteins in the case of alignment media such as polyacrylamide gels and neutral bicelle systems which align biomacromolecules by a steric mechanism. Various improvements and generalizations are possible within the suggested approach. Keywords: Liquid crystal media; NMR residular dipolar couplings; random ﬂight model; unfolded polypeptide chains.

1. Introduction It is now widely recognized that a thorough understanding of the structure and dynamics of unfolded proteins is of great importance for understanding protein folding, namely, how a highly dynamic polypeptide chain devoid of any signiﬁcant structure attains its biologically active three-dimensional conformation.1 Unfolded proteins are, however, not only important from the viewpoint of protein folding. It became evident only recently that intrinsically disordered proteins comprise a large part of the proteins being encoded in eukaryotic genomes.2 Despite being (partially) unfolded they are able to fulﬁll biological functions, and as such they are involved in important regulatory processes in the cell such as cell cycle control and transcriptional and translational regulation, respectively.3 Moreover, (partially) unfolded forms of proteins seem to play an important role in the onset of neurodegenerative diseases as is the case in Alzheimer’s4 or Parkinson’s disease.5 High-resolution, liquid-state nuclear magnetic resonance (NMR) spectroscopy has proven to be invaluable in the investigation of the structure and dynamics of unfolded proteins. Several experimental observables can be used to gain detailed information about the heterogeneous ensemble of conformations at the atomic level.6 Chemical shifts deviating from random coil values may indicate local conformational propensities7,8 whereas 15 N transverse magnetization relaxation measurements yield information about long-range interactions9 and conformational exchange within the ensemble.10 Elements of residual structure can be inferred from nuclear Overhauser eﬀect (NOE11 ) data.12 Other approaches such as the use of spin labels and 2D photo-CIDNP (chemically induced dynamic nuclear polarization) enable the determination of residual long-range order13–16 and of diﬀerential accessibility of aromatic side-chains involved in hydrophobic clustering.17 Another especially valuable NMR observable is the direct dipole–dipole interaction between nuclear spins (the so-called residual dipolar couplings, RDCs), e.g., between the spins of a 15 N and a 1 H nuclei. RDCs directly report on the average orientation of internuclear vectors with respect to

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the magnetic ﬁeld. Net alignment is achieved by dissolving the protein in a liquid crystal medium, which creates an anisotropic environment. As RDCs are correlated to each other via the alignment tensor, they provide unique long-range structural information. Therefore, they can be used in structure validation and reﬁnement processes.18 Recently, RDCs have been measured on a variety of unfolded proteins and small peptides.15,19–37 The interpretation of the RDC data is nontrivial due to the highly heterogeneous character of the unfolded ensemble. The earliest reports provided evidence for a native-like topology within the unfolded ensemble.19–21 In a more recent study, however, it was argued in favor of simple local conformational propensities.27 Due to this controversy in interpreting RDCs collected on unfolded proteins, it is obvious that a theoretical treatment is needed in order to develop a solid foundation for data analysis. A few theoretical approaches have been proposed for folded proteins. Reasonable predictions were obtained within the models including only the steric eﬀects.38–41 Upon inclusion of electrostatic contributions, it was also possible to predict RDCs of folded proteins being dissolved in charged alignment media.42–44 However, these models fail in the case of unfolded proteins since not only one predominant conformation — as for a folded protein — but an ensemble of conformations, which might diﬀer signiﬁcantly from each other, has to be considered. Recently, two diﬀerent groups reported similar approaches to provide a basis for the prediction of RDCs of unfolded proteins.45,46 These studies employed the concept of a statistical coil earlier introduced47–49 in which ensembles of unfolded conformations are constructed from amino acid-speciﬁc distributions of Ramachandran angles φ/ψ taken from the loop regions of high-resolution X-ray structures in the protein database. In Ref. 46, steric overlap was avoided by residue-speciﬁc volume exclusion and reasonable agreement with experimental data was demonstrated. In Ref. 45, nearest neighbor eﬀects were additionally taken into account, which led to a signiﬁcant improvement in the agreement between calculated and experimental RDCs. The method proposed in Refs. 45 and 46 relies on the numerical sampling of the conformational space of the unfolded polypeptide. An attractive alternative to such a computational approach is the derivation of an analytic expression which can be used (i) to reveal and analyze the general trends of the behavior of the system, and (ii) to obtain quantitative predictions not having to perform the numerical sampling for each particular polypeptide. Unfortunately, this is not always possible. In the present work, we show that a closed-form analytic expression for the RDCs in an unfolded polypeptide can be derived within the random

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ﬂight chain model with the alignment medium represented by a set of planar obstacles. A random ﬂight chain is a rather crude approximation for an unfolded polypeptide. However, this idealized model captures many aspects of the behavior of unfolded polypeptides. The random ﬂight chain model allows an analytic description of many characteristics of unfolded polypeptides such as the radius of gyration or the mean end-to-end distance.50 The ﬁrst attempt to obtain RDCs within the random ﬂight chain model was made by Annila and co-workers,51 who were able to derive a representation of the RDCs as a threefold integral that had to be calculated numerically. This approach was later extended to the valence chain model to take into account steric hindrance between residues.52 Unfortunately, the formalism presented in the initial study51 contains three shortcomings. The most serious one is, essentially, adding the (weighed) probabilities of statistically independent events rather than multiplying them [see Eq. (1) in Ref. 51]. Secondly, only one obstacle has been accounted for, while, according to the model, the polypeptide chain is conﬁned between two obstacles. And ﬁnally, the one-dimensional random walk formalism has been employed for describing the probabilities of the possible chain conformations while, as we discuss below, the three-dimensional formalism must be used. The last shortcoming can easily be eliminated by a suitable re-scaling of the length of the chain segments. For the ranges of experimental parameters for which the polypeptide does not “feel” two obstacles simultaneously (this is a rather mild approximation; see below), the second shortcoming can also be eliminated by re-scaling. In this case, the distance between the obstacles is actually twice as large as assumed in the original paper. The ﬁrst shortcoming, however, cannot be remedied within the formalism presented in Ref. 51. In this contribution, we present a general mathematical framework suitable for describing various characteristics of random ﬂight chains, including RDCs in various alignment media. We use this framework to derive values of RDCs within the model suggested by Annila et al. We show that the RDCs √ can be represented in a form of an expansion over a small parameter 1/ N (where N is the number of segments in the chain). Only the few ﬁrst terms of this expansion are necessary for an accurate approximation of the exact formula. The series representation makes it possible to obtain analytical dependencies of the RDCs on obstacle concentration, chain length, and on position of a segment within the chain. Two general features of RDCs in unfolded polypeptide chains are predicted by the random ﬂight chain model. The ﬁrst is that RDCs are larger for segments in the middle of the chains as compared to the end segments.

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The second general feature predicted by the random ﬂight chain model is that shorter chains exhibit larger RDCs. 2. Theory of residual dipolar couplings in random flight chains 2.1. General approach Let us start with formulating the random ﬂight chain model. A random ﬂight chain is constructed as a set of line segments connecting subsequent steps of a three-dimensional random walk. The unfolded polypeptide is, therefore, represented in the model by a sequence of inﬁnitely thin rods of equal, ﬁxed length attached to one another at the tips. Each rod/segment represents a structural subunit of the polypeptide. The segments are randomly oriented: there is no interaction, including steric hindrance, with the other segments. The dipolar coupling between nuclei P and Q depends on the angle β between the internuclear vector and the magnetic ﬁeld53 : µ0 γP γQ 3 cos2 β − 1 DP Q = . (1) 4π 2 RP3 Q 2 Here, γP and γQ are the gyromagnetic ratios of the nuclei, and RP Q is the internuclear distance. In the case of axially symmetric segments, the dipolar couplings can be expressed via the angle θ between the axis of the segment to which the nuclei belong and the magnetic ﬁeld (see Fig. 1): µ0 γP γQ 3 cos2 αP Q − 1 3 cos2 θ − 1 DP Q = , (2) 4π 2 RP3 Q 2 2 where αP Q is the angle between the internuclear vector and the axis of the segment. Since the angle θ = θ(t) changes with time due to ﬂuctuations of the chain, one has to average the coupling over the time of the measurements; that is, denoted by the angular brackets: 2 1 τ cos2 θ dt. (3) cos θ = τ 0 For residual dipolar couplings, the experimental parameter is averaged over a much longer time than the characteristic times of thermal motion of segments of the polypeptide chain. If no special aligning conditions are imposed (alignment due to the presence of the magnetic ﬁeld is negligible in the case of diamagnetic proteins), the chain will freely ﬂuctuate in solution and the segment Si will spend equal times at each possible value of θ and

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Fig. 1. (a) Representation of an unfolded polypeptide chain by a random ﬂight chain. The segments of the random ﬂight chain are built by connecting subsequent Cα atoms. The Rn indicate residue-speciﬁc side-chains. (b) The angles between the vector of magnetic ﬁeld B0 , the unit vector S along the segment of interest, and the internuclear vector connecting nuclei N and H, respectively.

averaging over a long time τ will result in zero coupling, cos2 θ = 1/3, DP Q = 0. If, however, segments of the chain have a preferred orientation in space, the dipolar couplings become an observable quantity and can serve as an important source of information about the structural properties of the polypeptide. The optimal conditions correspond to a very experimental small degree of alignment, cos2 θ − 1/3 ∼ 10−3 −10−4 , hence the name residual dipolar couplings. The random ﬂight chain model is suitable for the NMR experiments in which alignment is caused by a spatial obstruction of chain motions due to various kinds of oriented media (bicelles in an external magnetic ﬁeld, strained gels, etc.). Statistical mechanics postulates54 that averaging over inﬁnite time is equivalent to averaging over the phase space of the system with a probability distribution function describing how frequently the system visits each point of its phase space. Since τ is much longer than the characteristic times of ﬂuctuations and, additionally, since large ensembles of the polypeptide chains contribute to the measured signal, one can replace the averaging over time in Eq. (2) by averaging over the conﬁgurational space of the polypeptide. In this section, we show how the results of the theory of random walks allow one to construct a probability distribution function which depends on

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the angle θ only. To obtain such a function P (θ), one has to integrate over all other variables characterizing the conﬁgurational space of a polypeptide. The distributions P (θ) are diﬀerent for each chain segment i, and they also (N ) depend on the length of the chain N : P (θ) = Pi (θ); we will omit these indices as their values are always clear from the context. In the next section, we will apply this general recipe of ﬁnding P (θ) in calculating the RDCs for a particular experimental set-up. The desired distribution P (θ) gives the probability dP of ﬁnding a given segment Si of the chain oriented at an angle in an inﬁnitesimally small vicinity of θ, dP = P (θ)dθ. Then, π 1 2 cos θ = P (θ) cos2 θ sin θdθ = P (θ) cos2 θd(cos θ). (4) 0

−1

The sine of θ in (4) appears as a remnant of the integration over the complete phase space of the chain in which the integration over the polar angle dΩ = sin θdθdφ is done. The probability distribution must be properly normalized: 1 P (θ)d(cos θ) = 1. (5) −1

Ifno alignment is imposed, all values of θ are equally probable, P (θ) = 1/2, cos2 θ = 1/3, and the RDC is zero. The function P (θ) can generally be thought of as the ratio of the number of polypeptide conformations in which the given segment Si is at angle θ with respect to the magnetic ﬁeld to the total number of polypeptide conformations (cf. generic deﬁnition of probability, the number of successful events divided by the total number of events). In continuous space, it is impossible to explicitly enumerate all the conformations of the polypeptide chain. Therefore, one has to operate with continuous distributions characterizing the density of conformations in a space of continuous variables. In order to ﬁnd the density of conformations P (θ), we use the results of the theory of random walks which connect the starting point of a walk and the number of random steps with a probability to arrive at some ending point. The segment Si , for which P (θ) is sought, divides the chain into two parts (before and after Si ) of lengths N1 = i − 1 and N2 = N − i. Each subchain corresponds to a random walk characterized by the starting point (deﬁned by the position and orientation of the segment Si ), number of steps (N1 and N2 , respectively), and by the ending point (ending points of the chain). Therefore, the function P (θ) can be expressed via the random walk distribution functions integrated over the positions of the end points of the chain, and over the position and azimuthal orientation of the given segment Si . In this probability distribution, all other variables of the

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conﬁgurational space of the polypeptide chain (i.e. the coordinates of the rest of the segments) have already been integrated over. We introduce a function W (r0 , r1 , r2 , si ) proportional to the density of polypeptide conformations such that the given segment Si is oriented along the unit vector si = {cos φ sin θ, sin φ sin θ, cos θ} (here and throughout this work the magnetic ﬁeld is taken to be along the z-axis; θ and φ are the polar angles), while the ends of the polypeptide chain are located at the points r1 and r2 . Vector r0 points to the middle of Si (see Fig. 2). Then, the expression for P (θ) can be written as P (θ) = 1 −1

where

N (θ) =

dr0

2π 0

N (θ) N (θ)d(cos θ)

dφ

,

(6)

dr1

dr2 W (r0 , r1 , r2 , si ).

(7)

Roughly speaking, in Eq. (7) we ﬁnd the number of conformations of the chain for a given position r0 and orientation si of the segment Si by integrating over the number of conformations with positions of the end points r1 and r2 . Then, we integrate over the position (r0 ) and orientation (φ) of Si . Integration over r0 , r1 , and r2 is carried out over the whole accessible space. This results in the total number of conformations in which Si is at angle θ with respect to the z-axis. The total number of conformations is obtained by integrating N (θ) over the interval (0, π). The densities of conformations for the subchains before and after the segment Si are given by a probability distribution of a random walk with the corresponding length, starting and ending points. Let us denote this distribution by G(a, r). For the ﬁrst subchain, the ending point of the random

Si r2

θ

φ

r1

r0 B0 , z

x

y

Fig. 2. Deﬁnition of the angle φ and vectors r0 , r1 , r2 . Segment Si is enlarged for illustrative purposes.

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walk is r1 ; the starting point is one of the tips of segment Si , a1 = r0 + si /2. The segment length is taken to be 1, thereby deﬁning the scale for all length related quantities. For the second subchain, the random walk ends in r2 and it starts from the other tip of the segment Si , a2 = r0 − si /2. The subchains are independent, and hence the probability distributions are multiplied, that is, the number of conformations of the whole chain is a product of numbers of conformations of the subchains: W (r0 , r1 , r2 , si ) = G(r0 + si /2, r1 )G(r0 − si /2, r2 ).

(8)

The random walk probability distribution G(a, r), which is to be substituted into Eq. (8), should be determined based on the particular geometry of obstructing media, but it can also accommodate further improvements in the model (self-avoiding random walk, non-Markovian processes, etc.). In the simplest case — no alignment, no interactions between the segments of the chain — G(a, r) was ﬁrst obtained by Lord Rayleigh.55 It reads as56 1 3(r − a)2 G(a, r) = exp − . (9) 2n (2πn/3)3/2 Here, n is the number of the steps in the random walk. Substitution of (9) into (8), (7) and (6) gives a constant (1/2), which does not depend on θ, as expected. We note that, in principle, the binominal distributions should be used in (9) instead of the Gaussian functions. In the limit of an inﬁnitely long chain, n → ∞, the binominal distributions coincide with the Gaussians. Therefore, strictly speaking, the analysis based on the use of the Gaussian functions is applicable for suﬃciently long subchains, i.e. for segments which are far from the ends of a long chain. However, already for n > 2 the Gaussians are a rather good approximation. Alignment media in the random ﬂight chain model are represented by a set of barriers obstructing the random walk, and the function G(a, r) has to be modiﬁed correspondingly. The role of the obstructing media is to exclude from the conﬁgurational space all the conformations of the chain which cross any of the barriers. This is achieved by modeling the obstructing media as a set of absorbing barriers. Indeed, the probability distribution of a random walk in the presence of absorbing barriers disregards the pathways connecting the starting and the ending points, which cross any of the barriers. Representing the obstructing media as a set of reﬂecting barriers would contradict the physical picture behind the random ﬂight chain model, since in this case some random trajectories (or conformations) are counted twice; see, e.g., Ref. 56. It is instructive to make the following general remark. The integral over the probability distribution G(a, r) in the presence of obstructing media is

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less than 1:

I(a) =

G(a, r)dr < 1.

(10)

In the language of the random walk, the diﬀerence 1 − I reﬂects the probability of absorption. In chain conformations language, the integral I corresponds to the fraction of possible polypeptide conformations in the obstructed space as compared to the number of conformations of a free chain. After integrating over r1 and r2 in (7), one arrives at 2π dφI(r0 + si /2)I(r0 − si /2). (11) N (θ) = dr0 0

2.2. Residual dipolar couplings in the presence of bicelles Let us consider a particular type of experiment for the measurement of RDCs in which alignment of the polypeptide chains is achieved by placing bicelles into the solution. Bicelles are large, disk-shaped bodies which weakly interact with polypeptides and orient in the presence of an externally applied magnetic ﬁeld.57 We assume that the bicelles are aligned perpendicular to the magnetic ﬁeld. Neglecting the ﬁnite size of the bicelles, we approximate them by a set of parallel inﬁnite planes. Typical interplanar distances encountered in experiments are 400–600 ˚ A in the case of phospholipid bicelles58 and a few nanometers in the case of n-alkyl-poly(ethylene glycol)/n-alkyl alcohol bicelles,59 respectively. We have taken the length of one amino acid residue (Cα –Cα distance ≈ 3.8 ˚ A) to be the unit of length in our model. Hence, an L value of 150 corresponding to an interplanar distance of about 570 ˚ A seems to be a reasonable choice for the calculations. We note that the model — random walk between two inﬁnite planes oriented perpendicular to the z-axis while there are no obstructions in the x- and y-directions — implies a one-dimensional character of the random walk problem. However, the random walk probability distribution has to be taken from the 3D random walk formalism rather than from the one-dimensional one. The reason is that in the 3D case the characteristic width of the random walk probability distribution (which translates into √ the radius of gyration, end-to-end distance, etc.) is a factor of 3 smaller than in the 1D case. When applying the 1D formalism in this model, one eﬀectively replaces the steps of equal length with the steps of equal projection onto the z-axis. The probability distribution of the 3D random walk between two parallel inﬁnite absorbing barriers is easily obtained by generalization (based on

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symmetry considerations) of a one-dimensional walk in the presence of one barrier56: 3(r − a)2 1 3(r − a)2 exp − G(a, r) = − exp − 2n 2n (2πn/3)3/2

3(r − a)2 − exp − . (12) 2n Here, a is the starting point of an n-steps walk, r is the ending point, r is the point symmetric to r with respect to mirror reﬂection in the plane of the ﬁrst barrier, and r = r − 2D n (where D is the distance between r and the ﬁrst barrier; n is the unit normal to the plane of the ﬁrst barrier). Analogously, r = r − 2D n is the image of the point r in the plane of the second barrier (Fig. 3). Equation (13) is valid for n < L, when the chain cannot “feel” two barriers simultaneously. However, completely or almost completely extended conformations for which the size of the chain approximately equals N are extremely improbable. Rather, the typical size of the chain is on the order √ of N . Therefore, in fact Eq. (13) works well until N ∼ L2 , i.e. it is satisﬁed for most of the experimental conditions.

D´´

r ´´ L 2

n´´

D´´

B0 , z

a

r

n´

D´

L − 2

r´

D´

Fig. 3. Deﬁnition of the vectors r and r . The two ellipsoids represent two bicelle particles oriented perpendicular with respect to B0 . The bicelles are approximated by inﬁnite planes in the calculations. The segment which divides the chain into two subchains thickened for illustrative purposes.

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The second and third terms in the square brackets in (13) correspond to exclusion of the trajectories/conformations which cross the ﬁrst and the second barriers, respectively. These terms are responsible for the integrals (10) being less than 1. Let us now calculate this integral. For convenience, we will assume that the barriers are perpendicular to the z-axis, crossing it at the points −L/2 and L/2 (see Fig. 3). Then, integrations over x and y result in the factor (2πn)/3. The integration over z from −L/2 to L/2 results in a set of error functions Erf(x): I(a) = G(a, r)dr = Erf

3 2n

L − az 2

+ Erf

3 2n

L + az 2

− 1.

(13)

Substituting (13) into (11), one obtains 2π dφ N (θ) = dr0 0

Erf

× Erf

L L 3 − z0 − c/2 + Erf + z0 + c/2 − 1 2 2N1 2

L L 3 3 − z0 + c/2 + Erf + z0 − c/2 − 1 , 2N2 2 2N2 2 (14)

3 2N1

where N1 = i − 1 and N2 = N − i are the lengths of the parts of the chain before and after the given segment Si , (N1 + N2 + 1 = N ), z0 is the z-component of the vector r0 , and c ≡ cos θ

(15)

is the z-component of the vector si . In (14), due to the symmetry of the problem, there are no dependencies on x0 , y0 , and the azimuthal angle φ. The integrals over these variables produce the inﬁnitely large factor 2πσ, where σ is the (inﬁnite) surface of the barriers. However, exactly the same factor is produced in the denominator of Eq. (6), so the value of P (θ) remains ﬁnite. Therefore, the probability distribution is expressed via the one-dimensional integral over the position of the middle of the given segment Si , z0 : P (c) = 1 −1

E(c) E(c)d(c)

,

(16)

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where

L − z0 − c/2 2 −L/2+|c|/2

3 L + Erf + z0 + c/2 − 1 2N1 2

L 3 × Erf − z0 + c/2 2N2 2

L 3 + Erf + z0 − c/2 − 1 dz0 . 2N2 2

E(c) =

L/2−|c|/2

Erf

3 2N1

(17)

At this stage of the derivation, the reason for the nonzero RDCs can be clearly seen. Indeed, E(c) is just a renormalized number of chain conformations for which the segment of interest is at a particular angle θ (c ≡ cos θ). In (17) there are terms |c|/2 in the limits of the integration which are due to the fact that the center of the given segment cannot approach the barriers closer than |c|/2. The conformations in which the segment of interest is parallel to the barriers (c = 0) have larger space available to them and, therefore, they are better populated. Expression (17) can be further simpliﬁed by expanding the error functions in a series over |c|/2. Using the standard Taylor series, 2 2 Erf(x + δ) = Erf(x) + √ exp(−x2 )δ − √ x exp(−x2 )δ 2 π π 2 2x2 − 1 2 2x2 − 3 exp(−x2 )δ 3 − √ x exp(−x2 )δ 4 + · · · , +√ π 3 π 6 (18) one obtains after some algebra 2 2 2(N1 + N2 ) 3 1 1 2 E(c) = L − √ −√ c + |c|3 3 π N1 N2 π π 2(N1 + N2 ) 3 1 1 3 4 (N1 + N2 ) + √ c − |c|5 + · · · . 3 3 2 π 2(N1 + N2 ) 20π N1 N23 (19) As shown in Fig. 4, one has to include terms up to |c|5 in order to achieve the accuracy (10−5 ) needed for the calculations of the RDCs.

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Fig. 4. Comparison of P (θ) for the center segment of a 21-mer chain calculated with the series representation (19) [dash-dotted green (including terms up to |c3 |), dotted blue (including terms up to c4 ), and dashed purple (including terms up to |c5 |) curves] and with the exact expression (17) (red). The bottom graph is an enlargement of the bottom-right part of the top graph.

The denominator in (16) is calculated, then, as follows:

1

4 2(N1 + N2 ) 3 − √ 3 3 π 2(N1 + N2 ) 1 1 3 1 + + √ 2π N1 N2 5 π 2(N1 + N2 )3 1 1 (N1 + N2 ) − + ··· . 20π N13 N23

4 E(c)dc = 2L − √ π −1

(20)

Equations (16), (19) and (20) deﬁne the probability distribution for a given segment to be found at a particular angle θ. With the given probability

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distribution, we can now ﬁnd the average angular distribution in (2): 8 3 µ0 γP γQ 3 cos2 αP Q − 1 × − √ DP Q ≈ 4π 2 RP3 Q 2 15 π 2(N1 + N2 ) 1 1 3 1 4 N1 + N2 + + √ − 4π N1 N2 32π N13 N23 35 π 2(N1 + N2 )3 4 4 1 2(N1 + N2 ) 3 1 2L − √ − √ + π 3 3 π 2(N1 + N2 ) 2π N1 N2 3 1 1 N1 + N2 + √ − . (21) 20π N13 N23 5 π 2(N1 + N2 )3 Equation (21) is the ﬁnal result of our derivation. 3. Results and discussion In order to reveal the two general patterns in the behavior of the RDCs predicted by the random ﬂight chain model, √ expres√ let us further simplify sion (21) by expanding it over 1/L and 1/ N provided that L N 1. This condition is fulﬁlled for typical ranges of experimental parameters. We keep only the ﬁrst terms in the expansion: µ0 γP γQ 3 cos2 αP Q − 1 4π 2 RP3 Q 2

1 4 1 3 1 √ × − , L 15 π 2(N1 + N2 ) 8π N1 N2

DP Q ≈ −

N = N1 + N2 + 1.

(22)

From Eq. (22) it is evident that shorter chains have larger (in absolute value) RDCs at ﬁxed distance L. At ﬁrst glance, this is a counterintuitive result. Indeed, one would expect that the lengthier chains, which are “bigger,” should be more aﬀected by the presence of the obstacles compared to the “smaller” shorter chains. However, within the same theoretical framework, one can show that the chain clews almost never approach the √ obstacles closer than their characteristic radius ∼ N ; the conformations in which the chain is thinly spread along the obstacle are statistically insigniﬁcant. In the vicinity of the obstacle, the average “shape” of the chain resembles a sphere ﬂattened on one side, in contrast to the unperturbed spherical shape characteristic of non-obstructed chains. The deformation is maximum

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at the surface of the chain clew while the inner region is less aﬀected. From this point of view, it is evident that the inﬂuence of the obstacle is greater for clews with greater surface-to-volume ratios. That is, the short chains are more aﬀected by the obstacles and therefore have larger RDCs. The second general feature predicted by the random ﬂight chain model is that the RDCs are larger for the segments in the middle of the chain compared to the end segments,a leading to the bell-like shape of the RDC proﬁle, as illustrated in Fig. 5. This feature can also be explained in terms of average shapes of the chain clews. The reason why the middle segments are more aﬀected by the obstacles is the following. As we have just discussed, the chain clews resemble ﬂattened spheres with maximum deformation on one side of the clew, at the contact with the obstacle. In this situation the end segments (located mostly on the surface of the clew) are aﬀected most, while the middle segments are aﬀected less. However, when the clew rotates (keeping the distance to the obstacle constant) other end segments become aﬀected by the obstacle while the middle segments are still constrained. This rotational degree of freedom leads to the fact that on average the middle segments are more strongly aligned and the RDCs in such segments are larger. These two features are illustrated in Figs. 5 and 6. The presented data correspond to 1 DNH couplings using a value of −21,700 Hz for the constant term in Eq. (2). The angle αNH [αP Q , see Eq. (2)] was taken to be 90◦ , a value close 0

DNH (Hz)

−1

1

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0

20

40 60 Segment number

80

100

Fig. 5. NH residual dipolar couplings as a function of the segment number for random ﬂight chains of diﬀerent lengths: 11 segments (blue), 21 segments (green), 51 segments (yellow), and 101 segments (red). a It is assumed that the internuclear vectors throughout the chain are on average oriented at the same angle αP Q with respect to the symmetry axis of the corresponding segment.

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0

DNH (Hz)

−1

1

−2 −3 −4

0

20

40

60

80

100

N

Fig. 6. NH residual dipolar couplings as a function of the chain length N for the segment next to the end segment (upper curve) and the center segment (lower curve).

to ∼85◦ dictated by the rigid part of geometry of the chain. One can easily check that deviations from the 90◦ value do not exceed 10–15◦ (0.26 radian) in all sensible conformations (α-helix, β-sheet, polyproline II) being populated in unfolded polypeptides. Such deviations are insigniﬁcant for the RDC values, since cos2 (90◦ ± 15◦ ) ≈ 0.07 1 [see Eqs. (21) and (22)]. Let us furthermore compare our results with those obtained in Ref. 51. In Fig. 7, we present the RDC proﬁles calculated with the expressions from Ref. 51 (two upper curves) and with the use of the present formalism (two lower curves). It is seen from the ﬁgure that the shortcomings mentioned in the introduction lead to three times smaller RDCs (in absolute values), and also to the less pronounced bell shape. 0

(Hz)

− 0.5

D

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20 30 Segment number

40

50

Fig. 7. Comparison of NH residual dipolar couplings as a function of the segment number for random ﬂight chains consisting of 21 and 51 segments (blue and red, correspondingly). The upper proﬁles are obtained with the expression from Ref. 51; the lower proﬁles are the present results reproduced from Fig. 5.

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4. Conclusions In this work, we present a theoretical framework for the prediction of residual dipolar couplings in unfolded proteins. The framework is rather general and can serve as a basis for determining RDCs in unfolded polypeptide chains under a wide spectrum of experimental conditions. The framework allows one to employ various models for the polypeptide chains and for aligning media in order to ﬁnd RDCs with the desired degree of accuracy. Using the framework, we showed that within a simple model which approximates bicelles as inﬁnite planes and in which unfolded polypeptides are simulated by random ﬂight chains, it is possible to obtain a closed-form analytical result for the RDCs. In this case, RDCs are readily accessible for chains, of diﬀering length, for diﬀerent loci along the chain, and for varying bicelle concentrations. The two general features predicted by the model are (i) a sequentially symmetric bell-shaped distribution of RDCs with center segments showing larger alignment than segments at the ends of the chain, and (ii) larger alignment for shorter chains than for longer chains at a given bicelle concentration (interplanar distance L). We note that the employed model represents the ideal case of fully unfolded proteins, and it additionally contains signiﬁcant simpliﬁcations. Advancing the model will account for the excluded volume, the actual geometry of the alignment medium, long-range interactions, and so on. Unfortunately, accounting for these factors will probably prevent one from deriving a closed-form analytical solution for the RDCs but will allow one to obtain better agreement with experimental data. Such developments can be implemented within the general framework of random walk dynamics formulated in the present work. Acknowledgments This work is partially supported by the European Commission within the Network of Excellence project EXCELL and by the EU project UPMAN. References 1. M. Vendruscolo and C. M. Dobson, Philos. Transact. A Math. Phys. Eng. Sci. A 363, 433 (2005). 2. A. L. Fink, Curr. Opin. Struct. Biol. 15, 35 (2005). 3. H. J. Dyson and P. E. Wright, Nat. Rev. Mol. Cell Biol. 6, 197 (2005). 4. E.-M. Mandelkow and E. Mandelkow, Trends Cell Biol. 8, 425 (1998). 5. V. N. Uversky, J. Li and A. L. Fink, J. Biol. Chem. 276, 10737 (2001).

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6. J. Wirmer, C. Schlorb and H. Schwalbe, Conformation and dynamics of nonnative states of proteins studied by NMR spectroscopy, in Protein Folding Handbook, Part I, 1st edn., eds. J. Buchner and T. Kiefhaber (Wiley-VCH, Weinheim, 2005), pp. 737–794. 7. A. Bundi and K. Wuthrich, Biopolymers 18, 285 (1979). 8. S. Schwarzinger, G. J. Kroon, T. R. Foss, J. Chung, P. E. Wright and H. J. Dyson, J. Am. Chem. Soc. 123, 2970 (2001). 9. J. Klein-Seetharaman, M. Oikawa, S. B. Grimshaw, J. Wirmer, E. Duchardt, T. Ueda, T. Imoto, L. J. Smith, C. M. Dobson and H. Schwalbe, Science 295, 1719 (2002). 10. M. Tollinger, N. R. Skrynnikov, F. A. Mulder, J. D. Forman-Kay and L. E. Kay, J. Am. Chem. Soc. 123, 11341 (2001). 11. D. Neuhaus and M. P. Williamson, The Nuclear Overhauser Eﬀect in Structural and Conformational Analysis, 2nd edn. (Wiley-VCH, New York, 2000). 12. Y. K. Mok, C. M. Kay, L. E. Kay and J. Forman-Kay, J. Mol. Biol. 289, 619 (1999). 13. J. R. Gillespie and D. Shortle, J. Mol. Biol. 268, 158 (1997). 14. J. R. Gillespie and D. Shortle, J. Mol. Biol. 268, 170 (1997). 15. C. W. Bertoncini, Y. S. Jung, C. O. Fernandez, W. Hoyer, C. Griesinger, T. M. Jovin and M. Zweckstetter, Proc. Natl. Acad. Sci. USA 102, 1430 (2005). 16. S. Kristjansdottir, K. Lindorﬀ-Larsen, W. Fieber, C. M. Dobson, M. Vendruscolo and F. M. Poulsen, J. Mol. Biol. 347, 1053 (2005). 17. C. Schlorb, S. Mensch, C. Richter and H. Schwalbe, J. Am. Chem. Soc. 128, 1802 (2006). 18. M. Blackledge, Progr. Nucl. Magn. Reson. Spectrosc. 46, 23 (2005). 19. D. Shortle and M. S. Ackerman, Science 293, 487 (2001). 20. M. S. Ackerman and D. Shortle, Biochemistry 41, 3089 (2002). 21. M. S. Ackerman and D. Shortle, Biochemistry 41, 13791 (2002). 22. S. Ohnishi and D. Shortle, Proteins 50, 546 (2003). 23. A. T. Alexandrescu and R. A. Kammerer, Prot. Sci. 12, 2132 (2003). 24. K. Ding, J. M. Louis and A. M. Gronenborn, J. Mol. Biol. 335, 1299 (2004). 25. S. Ohnishi, A. L. Lee, M. H. Edgell and D. Shortle, Biochemistry 43, 4064 (2004). 26. W. Fieber, S. Kristjansdottir and F. M. Poulsen, J. Mol. Biol. 339, 1191 (2004). 27. R. Mohana-Borges, N. K. Goto, G. J. Kroon, H. J. Dyson and P. E. Wright, J. Mol. Biol. 340, 1131 (2004). 28. K. Fredriksson, M. Louhivuori, P. Permi and A. Annila, J. Am. Chem. Soc. 126, 12646 (2004). 29. S. Meier, S. Guthe, T. Kiefhaber and S. Grzesiek, J. Mol. Biol. 344, 1051 (2004). 30. C. O. Sallum, D. M. Martel, R. S. Fournier, W. M. Matousek and A. T. Alexandrescu, Biochemistry 44, 6392 (2005). 31. C. W. Bertoncini, C. O. Fernandez, C. Griesinger, T. M. Jovin and M. Zweckstetter, J. Biol. Chem. 280, 30649 (2005).

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32. E. B. Gebel, K. Ruan, J. R. Tolman and D. Shortle, J. Am. Chem. Soc. 128, 9310 (2006). 33. A. Binolﬁ, R. M. Rasia, C. W. Bertoncini, M. Ceolin, M. Zweckstetter, C. Griesinger, T. M. Jovin and C. O. Fernandez, J. Am. Chem. Soc. 128, 9893 (2006). 34. N. Sibille, A. Sillen, A. Leroy, J. M. Wieruszeski, B. Mulloy, I. Landrieu and G. Lippens, Biochemistry 45, 12560 (2006). 35. S. A. Dames, R. Aregger, N. Vajpai, P. Bernado, M. Blackledge and S. Grzesiek, J. Am. Chem. Soc. 128, 13508 (2006). 36. S. Ohnishi, H. Kamikubo, M. Onitsuka, M. Kataoka and D. Shortle, J. Am. Chem. Soc. 128, 16338 (2006). 37. S. Meier, M. Strohmeier, M. Blackledge and S. Grzesiek, J. Am. Chem. Soc. 129, 754 (2007). 38. M. Zweckstetter and A. Bax, J. Am. Chem. Soc. 122, 3791 (2000). 39. M. X. Fernandez, P. Bernado, M. Pons and J. Garcia de la Torre, J. Am. Chem. Soc. 123, 12037 (2001). 40. H. F. Azurmendi and C. A. Bush, J. Am. Chem. Soc. 124, 2426 (2002). 41. A. Almond and J. B. Axelsen, J. Am. Chem. Soc. 124, 9986 (2002). 42. A. Ferrarini, J. Phys. Chem. B 107, 7923 (2003). 43. M. Zweckstetter, G. Hummer and A. Bax, Biophys. J. 86, 3444 (2004). 44. M. Zweckstetter, Eur. Biophys. J. 35, 170 (2006). 45. A. K. Jha, A. Colubri, K. F. Freed and T. R. Sosnick, Proc. Natl. Acad. Sci. USA 102, 13099 (2005). 46. P. Bernado, L. Blanchard, P. Timmins, D. Marion, R. W. Ruigrok and M. Blackledge, Proc. Natl. Acad. Sci. USA 102, 17002 (2005). 47. L. J. Smith, K. A. Bolin, H. Schwalbe, M. W. MacArthur, J. M. Thornton and C. M. Dobson, J. Mol. Biol. 255, 494 (1996). 48. K. M. Fiebig, H. Schwalbe, M. Buck, L. J. Smith and C. M. Dobson, J. Phys. Chem. 100, 2661 (1996). 49. H. Schwalbe, K. M. Fiebig, M. Buck, J. A. Jones, S. B. Grimshaw, A. Spencer, S. J. Glaser, L. J. Smith and C. M. Dobson, Biochemistry 36, 8977 (1997). 50. P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, New York, 1953). 51. M. Louhivuori, K. Paakkonen, K. Fredriksson, P. Permi, J. Lounila and A. Annila, J. Am. Chem. Soc. 125, 15647 (2003). 52. M. Louhivuori, K. Fredriksson, K. Paakkonen, P. Permi and A. Annila, J. Biomol. NMR 29, 517 (2004). 53. R. R. Ernst, G. Bodenhausen and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford University Press, New York, 1987). 54. L. D. Landau and E. M. Lifschitz, Course of Theoretical Physics, Vol. 5 (Pergamon Press, 1958). 55. Lord Rayleigh, Collected Papers, Vol. 6, 2nd edn. (Dover, New York, 1964). 56. S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). 57. C. R. Sanders, II. and J. P. Schwonek, Biochemistry 31, 8898 (1992). 58. A. Bax and N. Tjandra, J. Biomol. NMR 10, 289 (1997). 59. M. Ruckert and G. Otting, J. Am. Chem. Soc. 122, 7793 (2000).

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COMPUTATIONAL SIMULATIONS OF ANTIBODY: ANTIGEN UNBINDING E. S. HENRIQUES and A. V. SOLOV’YOV∗ Frankfurt Institute for Advanced Studies, Goethe University, Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany http://fias.uni-frankfurt.de/mbn ∗ [email protected] The unbinding process of the mAb4–4–20:ﬂuorescein complex was investigated by means of a computational approach at the atomistic classical mechanical level, probing only a relevant set of generalized coordinates in order to determine the putative dissociation paths of the system. The complex problem was reduced to a low-dimensional scanning along a selected distance between the protein and the escaping ligand. The unbinding was further characterized by assessing the relative positional and orientational coordinates of the ligand. Solvent eﬀects were accounted for by means of the Poisson–Boltzmann continuum model. The complex’s dissociation time was derived from the calculated barrier height, in compliance with the experimentally reported Arrhenius-like behavior. The computed results are in good agreement with the available experimental data. Keywords: Antibody:antigen; dissociation; koﬀ .

1. Introduction Biological processes are essentially driven by interactions between proteins and their target ligands, in particular the association/dissociation events these cellular components undergo. The dissociation events are, in essence, a fragmentation of complex multiatomic aggregates. Many-body aggregates have been the object of extensive experimental and theoretical studies in a wide range of natural science research ﬁelds ranging from atomic and cluster physics to cellular biology. However, despite the vast amount of data that has now accumulated, there is still a need for an eﬃcient and physically sound theoretical approach that will rationalize these data and make insightful predictions. A ﬁrst step is to try and identify the common features underlying dissociation events of diﬀerent nature. 353

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One emerging key idea arising from atomic cluster physics is that a typical fragmentation process can be successfully described in terms of a few generalized coordinates that determine the overall conﬁguration of the escaping and parent fragments.1,2 This concept also holds for analogous events in more complex systems,3 and it prompted us to now address the dissociation process of an aggregate of higher complexity, a biological protein–ligand adduct (often referred to as a complex). A most remarkable protein–ligand system is the antibody–antigen one, which is involved in a fundamental recognition process during the body immune response. This response is triggered by foreigner molecules — the antigens (antibody generator, AG). One key mechanism whereby the immune system recognizes and targets them for destruction is by releasing antibodies (anti-foreign body, AB).4 ABs are very large proteins, and the human body has a potential repertoire of 2.5 × 1011 diﬀerent ones. Yet, they all feature a basic scaﬀold: they consist of two identical “light” (L) and “heavy” (H) chains of amino acids entangled in a Y-shape fold as shown in Fig. 1. Each tip of the Y branches displays a distinctive variable region, i.e. the speciﬁc “lock” for which the target AG has the “key” (see the schematic inset in Fig. (1); the two tips are identical for each AB. The “key” region of the AG can be a small protein fragment or a hapten. A hapten is a low molecular weight compound originally attached to some carrier protein, which will also trigger the release of ABs. Upon exposure to a particular AG, a set of ABs is reﬁned to target it, via a mutation process.5,6 The mutations occur in the referred variable region (hence it is called “variable”). Along a maturation series, the increase in aﬃnity strongly correlates with an increase in the corresponding AB–AG dissociation times, τ .4,7–9 Usually, τ is expressed in terms of the rate of spontaneous dissociation, koﬀ = 1/τ . Not surprisingly, much eﬀort has been devoted to the determination of those koﬀ values, with some of the most innovative experiments involving sensitive micromanipulation techniques like atomic force microscopy (AFM) and other force probe procedures to measure AB–AG binding forces.4,10–13 Some further insight into the molecular structure, interactions and unbinding pathways underlying such single molecule experiments has been gained from computer simulations using “force probe” molecular dynamics (FPMD).14 However, the question arises as to what extent the measured unbinding force in the mechanically sped up process of pulling out the ligand relates to the thermodynamic or kinetic parameters describing the spontaneous dissociation. The latter arises in the minute time scale8 in contrast to the time scales of AFM (millisecond) and FPMD (nanosecond). There is also the matter of across which pathway is unbinding being forced.

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L-chain

355

L-chain

H-chains Antigen key va

H-chains

ria

bl

e

lock L-chain

L-chain Fab

Antibody

Fig. 1. Overall ribbon representation of a complete AB structure. The two pairs of heavy chains are depicted in red and blue, and the corresponding light chains in yellow and gray. The dashed ellipse highlights one of the branches that binds to the antigen (the so-called Fab, after fragment binding antigen), in an all-atom representation; the trapezoidal region highlights the Fab variable domains (with added hydrogens), and the dashed arc illustrates the chains’ cleavage sections for these variable domains to be detached. A simpliﬁed scheme of AB–AG binding is presented in the inset.

In the absence of a pulling force, one regains the spontaneous (natural) mode of AB–AG dissociation, a thermally activated barrier-crossing along a preferential path in a multidimensional energy landscape. The contributing activated states (which determine koﬀ ) may well be described in terms of a few collective coordinates, by close analogy with other studied fragmentation processes.1–3 Within this context, it is reasonable to constrain the many other degrees of freedom that only contribute to the negligible ﬁne structure of the energy landscape. This is a rational approach to probing the unbinding of a complex biological system like the AB–AG one, in order to calculate the corresponding energetic barrier and derive koﬀ from it. Starting with an experimentally well studied AB–AG complex, an antiﬂuorescein one (vide infra), here we describe a computational approach at the molecular (atomistic) level to explore its preferential unbinding pathways by probing only a few relevant degrees of freedom. A detailed analysis

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of its dissociation pathway and dependence on the distance and relative orientation of the molecules in question is presented. The introduction of solvent eﬀects is also discussed along with its implications on the results, and the dissociation rate (koﬀ ) is derived from the calculated energy barriers. Following this introduction, the selection of the AB–AG system is described in detail. Next, a brief overview of the theoretical methods adopted in this study is given, in particular the computational level, the force ﬁeld and the extent to which the solvent eﬀects have been introduced. In Sec. 4, the results are presented, compared with the available experimental data, and discussed. The last section is devoted to the conclusions. 2. The test case Fluorescein (Flu) is a synthetic hapten. It is extensively used in ﬂuorescence-based kinetic measurements of oﬀ-rates (koﬀ ),15 and a valuable reference system for the understanding of important immunological issues. Anti-ﬂuorescein AB–AG complexes are also clear-cut models in the sense that Flu is a small inert and rigid ligand (see Fig. 2) and the oﬀ-rates of a number of anti-Flu complexes have been found to display an Arrhenius-like behavior.7 The current study has been carried out for the anti-ﬂuorescein IgG monoclonal antibody 4–4–20 (mAb4–4–20), one of the most extensively studied by thermodynamic, kinetic, structural, spectroscopic, and mutational methods (see Ref. 16 and references therein), and for which two crystallographic structures of its Fab region (highlighted in Fig. 1) have already been reported.17,18 A complete IgG mAb4–4–20 molecule has two identical Fab fragments, each consisting of two constant and two variable domains. The two variable domains (labeled VL and VH ) constitute the so-called Fv fragment (also highlighted in Fig. 1), which is the minimal antigen-binding fragment. In fact, there are many genetically engineered ABs that feature only the VL and VH domains.19 This practice further endorses the idea of a system with a restricted number of binding-determinant degrees of freedom. It also makes it realistic (and computationally less demanding) to consider just the mAb4–4–20 variable domains: VL with 112 amino acids and VH with 118. 3. Methods 3.1. Force field Even reducing the system to the mAb4–4–20 two variable domains plus Flu, it amounts to ca. 3600 atoms. It is, thus, too big to be computationally

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H15 H16

O5

C17

C18

C16 C19

C20

H6

C15

C14 H13

O4

H7

C8 C9

O3

H9

C1

C6 C7

H14

C5

C2

C4

C3

H10 C10 C11 C12 C13

O1

O2 H8 Name C1 C6, C10 C7, C11 C9, C13 C12 C2, C5 C3, C4 C8 C14 C15, C18 C16, C17 C19 C20

Charge Type Name xanthenone ring 0.215 0.000 H6, H9 −0.400 CA H7, H10 −0.450 H8, H11 0.415 H12 −0.250 O1 C66 0.450 O2 0.730 C O3 carboxyphenyl ring −0.050 −0.200 H13, H16 CA −0.145 H14, H15 0.140 0.450 CC O4, O5

H12

H11 Charge Type

0.140 0.160 0.220 0.400 −0.550 −0.350 −0.600

HP H OH1 OS O

0.125 0.100

HP

−0.650

OC

Fig. 2. Structural formula and assigned atom labels for Fluorescein {2-(6-hydroxy-3oxo-(3H)-xanthen-9-yl) benzoic acid}. The force-ﬁeld atom types (see Sec. 3.3) and the partial charges (units of e) are listed in the table. The dashed line highlights the two aromatic (ring) fragments labeled and grouped in the table.

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addressed at any level of quantum mechanics. A realistic simpliﬁcation is to assume that the nuclei move in the average ﬁeld created by all particles, and use an empirical ﬁt to this ﬁeld — an eﬀective potential commonly known as a force ﬁeld. One then uses the computationally less demanding classical mechanics formalism to calculate both static properties (equilibrium structures, relative energies, etc.) and the time evolution of the system. Much eﬀort has been devoted to developing force ﬁelds suitable for studying proteins, the CHARMM force ﬁeld20 (the one used in the present work) being one of the most widely used nowadays. Its potential energy function reads E=

Nr

kir (ri − ri0 )2 +

i=1

i=1

+

kiθ (θi − θi0 )2 +

i=1

Nχ

+

Nθ

kiχ (χi − χ0i )2 +

N i, j=1 i<j

ij

Rij rij

12

Nφ i=1

NS

kiφ [1 + cos(ni φi + δi )]

kiS (Si − Si0 )2

i=1

−2

Rij rij

6 +

N qi qj . εrij

(1)

i, j=1 i<j

The energy E is a function of the positions of all atoms in the system. The ﬁrst four summations are known as bonding terms: they extend to the topologically deﬁned Nr covalent bonds “r,” Nθ bond angles “θ,” Nφ dihedral angles “φ,” and Nχ improper torsion angles “χ,” respectively. For some speciﬁc bond angles, an additional bonding term may be required as a function of the distance “S” between the ﬁrst and third atoms.20 The term for the bonds describes the energy required to deform a bond from its equilibrium value (denoted by the subscript “0”) within a harmonic approximation, and an analogous description holds for the remaining harmonic terms; the dihedral term is chosen diﬀerently to satisfy the dihedral periodicity. The last two summations of Eq. (1) are extended to all N non-bonding atom pairs ij separated by three or more covalent bonds. The Lennard–Jones 6–12 potential term accounts for the van der Waals (vdW) interactions: for each atom type (say i), there is a Ri distance corresponding to a well depth i , with Ri = 21/6 σi and σi the distance for which the Lennard–Jones potential equals zero; the Lennard–Jones parameters between pairs of diﬀerent atoms are obtained from combination rules, the ij values based on the geometric mean of i and j , and Rij values form the arithmetic mean between Ri and Rj . The Coulombic potential is deﬁned for the pairs of charges qi and qj separated by a distance rij , and for a given dielectric constant ε (the vacuum one by default). The equilibrium values in the harmonic terms and the Ri and i values are parameters derived from experimental data (e.g.,

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crystallographic structures) and ab initio quantum mechanical calculations on small reference molecules, presented and discussed in Ref. 20. Partial charges (qi , qj ) are also derived from such ab initio calculations. 3.2. Implicit solvent Proteins operate in aqueous solution. Solvation, stability and dissociation of molecules in water are largely governed by electrostatic interactions. This is particularly pertinent in proteins: more than 20% of all amino acids in globular proteins are ionized under physiological conditions and polar side-chains occur in over another 25% amino acids.21 However, introducing explicit water molecules in a computational simulation to account for solvent eﬀects dramatically increases the calculation time. Moreover, when the calculations involve any energy minimization-based technique like calculating minimum energy reaction paths, the explicit water molecules will arrange in a single conformation matrix, exerting forces on the solute that are very diﬀerent from the solvent mean force. Alternatively, a continuum treatment of the solvent as a uniform dielectric may provide an accurate enough description of such interactions, as long as one accounts for the fact that a protein in aqueous solution (the physiological medium) yields a system with two very diﬀerent dielectric media.22 The most physically correct implicit solvent model arises from solving the so-called Poisson–Boltzmann (PB) equation (see Refs. 22 and 23, and references therein). The protein (macromolecule) is treated as a lowdielectric cavity bounded by the molecular surface and containing partial atomic charges — typically taken from the classical molecular mechanics force ﬁeld. The solvent (water) is implicitly introduced by assuming a highdielectric surrounding of the protein. And since under physiological conditions macromolecules are dissolved in dilute saline solutions (water with a dissolved electrolyte), a term for the average charge density due to the mobile ions is also included. This classical continuum electrostatics treatment relies on the (reasonable) assumption that it is possible to replace the ionic potential of mean force with the mean electrostatic potential, neglecting non-Coulombic interactions (e.g., vdW) and ion correlations. The actual PB equation reads: ∇ · [ε(r)∇ϕ(r)] = −4πρ(r) − 4π

N

eqi ni (r)λ(r),

(2)

i=1

with ni (r) = n0i exp(eqi ϕ(r)/kB T ).

(3)

Equation (2) relates the electrostatic potential ϕ to the distribution of the protein atomic partial charges [charge density ρ], the dielectric properties

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of both the protein and solvent [ε, position dependent (r)], and the charge density due to the mobile ions given by the summation term; qi is the charge of ion type i, ni (r) its local concentration, e the elementary charge, and λ(r) a parameter that describes the ions’ accessibility at position r. The boundary condition is ϕ(∞) = 0. For each ion type, ni (r) is described by a Boltzmann distribution (3) where n0i is the ion concentration in bulk solution, kB the Boltzmann constant, and T the absolute temperature. As for the accessibility parameter, a general consensus is that any point within one ionic radius from the macromolecule is inaccessible [i.e. λ(r) = 0], and it is implicit that the region inside the macromolecular surface is inaccessible. The remaining region outside has λ(r) = 1. A typical value for the ionic A), given that Na+ Cl− (sodium chloride) is a very raidus is that of Cl− (2 ˚ frequently chosen electrolyte. A description of the several possible approximations and numerical techniques used to solve Eq. (2) is beyond the scope of the present contribution, the reader being referred to the supporting literature of the software used in this work, APBS (Adaptative Poisson–Boltzmann Solver).23,24 Brieﬂy, the solute’s charges are mapped onto a mesh and the electrostatic potential in the presence of the dielectric continuum solvent is determined at each point, via a ﬁnite diﬀerence numerical solution of the PB equation. The mesh being a ﬁnite one, it is necessary to set up the boundary potentials (at the lattice edge) accordingly. For the present work, they are approximated by the sum of the Debye–H¨ uckel potentials of all the charges, meaning ϕ=

N

eqi

i=1

exp(−ri /λD ) , εwater ri

(4)

where λD , the Debye length, reads λD =

εwater kB T . N 4πNA i=1 n0i e2 qi2

(5)

3.3. Fluorescein parameters The available CHARMM parametrization already has parameters for all amino acids but not for ﬂuorescein, so one has ﬁrst to describe the latter consistently with the force ﬁeld. In the present work, the required bonding and Lennard–Jones parameters where derived by analogy with similar ones existing in CHARMM. Partial atomic charges were ﬁtted to reproduce the molecular electrostatic potential (MEP) at selected points around the molecule according to the Merz–Singh–Kollman scheme25,26 implemented in the Gaussian03 program.27 The points are located in layers around the molecule, the ﬁrst layer corresponding to the van der Waals

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molecular surface scaled by a factor of 1.4; the default scheme then adds three more layers with scaling factors 1.6, 1.8 and 2.0. The MEP was generated at the DFT/B3LYP level with the 6-31G(d) basis set. DFT (density functional theory) is now a widely used and computationally convenient quantum mechanical method well documented in many reference books (e.g., Ref. 28): it makes use of exchange correlation functionals dependent on the electron density and its gradient to tackle electron correlation eﬀects. B3LYP was the functional of choice and stands for the three-parameter Becke functional combined with the Lee–Yang–Parr correlation functional.28 For quantum mechanical calculations, the coordinates for the starting Flu conformation were taken from the complex crystal structure with the best resolution (1.85 ˚ A18 ), which has the coordinates deposited in the RCSB 29 Protein Data Bank with entry name 1FLR. Only the acidic deprotonated form of Flu was considered, since this is the active form in the experiments underlying the current study. The structure was energy optimized before charge ﬁtting. The charges were then further reﬁned by their similarity to the set of already deﬁned ones in the CHARMM force ﬁeld (for details on the approach, see Refs. 30–32). The ensuing Flu set of parameters was then used as an extension of the CHARMM parametrization, the corresponding CHARMM atom types and partial charges being indicated in Fig. 2. 3.4. Reference geometry The variable domains (VL and VH ) were extracted from the L and H segments of the anti-Flu 4–4–20Fab 1FLR crystal structure.18,29 Crystallographic waters were stripped from the structure and the C-terminal amino acids were capped with –NH2 functional groups. A representation of the system is shown in Fig. 3. The positions of the protein’s missing hydrogens were initially guessed. Next, any latent close contacts or anomalous bonding positions were cleared out by relaxing the structure to an energy gradient tolerance of 0.05 eV·˚ A−1 , at the classical mechanics level using the NAMD 33 program with the extended CHARMM parametrization. This relaxed structure fully retains the experimental X-ray conformational features and was used as the starting conformation for the scanning. A full structure optimization (i.e. using a tighter energy gradient tolerance) was also carried out but it introduced many small errors at the protein’s secondary structure level. This is because secondary structure relies on a network of backbone hydrogen bonds, which are less accurately described in the framework of the simpliﬁed molecular mechanics force ﬁeld theories. The CHARMM energy diﬀerence between the relaxed and fully minimized geometries is ∼ 100 eV.

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Fig. 3. All-atom representation of the Fv-fragment of the mAb4–4–20-Flu complex structure. The ribbon representation highlights the backbone of the two chains, the H-chain in blue and the L-chain in gray. The Flu molecule is depicted in ball-and-stick and colored according to element (CPK space-ﬁlling).

The crystallographic structure itself has been resolved at a temperature of ∼290 K,18 thus an estimate of the corresponding average thermal energy (considering kB T per degree of freedom) amounts to ∼270 eV for our simulation system. This indicates that the full minimization only reaches some local minima. Considering the above limitations of the force ﬁeld, it is judicious to take the minimally relaxed structure (closer to the X-ray one) as the reference structure for the subsequent simulations. Flu is a particularly rigid molecule. Its essential degree of freedom is the torsion around the bond between the xanthenone and the carboxyphenyl aromatic rings (see Fig. 2), deﬁning the angle between the planes of these two rings. This angle has the value of −63◦ in both the crystal complexed form18 and the crystalline free Flu.34 For the above-mentioned CHARMM energy relaxed structure, the value of this angle is −67◦ . An energy optimization was also carried out for the hapten alone (without the AB), the value for the angle in question being −62◦ . Moreover, the RMSD (rootmean-square deviation) between the crystallographic and relaxed Flu bound structures is 0.201 ˚ A. These results are a good indication of the validity of the derived set of CHARMM parameters.

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3.5. Distance scanning In the pursuit of suitable reaction coordinates to describe the system’s unbinding, the distance between the protein and Flu mass centers could be a ﬁrst option, by close analogy with some cluster ﬁssion processes.1 Yet for reasons that will become clear next, a distance between two rationally selected atoms has been considered instead. The shape of the binding pocket hosting Flu is most complementary to this hapten, with a few amino acids at the rim of the pocket gating the entrance. Superimposing the two available crystal structures results in an overall RMSD of 0.419 ˚ A for the Flu atoms and 1.854 ˚ A for the protein, with a few of those rim amino acids exhibiting some of the larger individual RMSD values (up to 3.7 ˚ A). Out of those, ﬁve amino acids — His31L , Asn33L , Tyr56H, Tyr102H and Tyr103H — strategically “frame” amino acid Arg39L at the bottom of the pocket, as depicted in Fig. 4. Amino acids are labeled (in the text and Fig. 4) according to the standard amino acid 3-letter code (see Ref. 6 and also www.chem.qmul.ac.uk/iupac/AminoAcid/), the

(a)

(b)

His31L

Asn33L

Tyr103H Tyr102H

Arg39L

Arg52H Tyr56H Glu59H

His31L

Tyr103H

Tyr102H

Asn33L Tyr56H

Fig. 4. Fluorescein (van der Waals spheres in shades of gray) docked in the antibody’s binding cavity. The backbone of the antibody is depicted by black sticks; the framing amino acid acids (labeled as described in the text) have their side-chains displayed in colored ball-and-stick. (a) Front view of the entrance of the cavity. (b) Top view showing residue Arg39L at the bottom of the cavity, with its atom Cζ (zeta-carbon, standard nomenclature) highlighted in yellow; the red dotted lines represent hydrogen bonding between Arg39L and the hapten.

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number of the amino acid in the 1FLR ﬁle and the chain identiﬁer in subscript (e.g., Arg39L , refers to an Arginine that is numbered 39 in the Lchain). As highlighted in Fig. 4(b), Arg39L has its +1-ionized group (centered on atom Cζ ) directly involved in hydrogen bonding to the hydroxyl group of Flu (atoms O1–H12 in Fig. 2). Arg39L is also a mutation introduced during the maturation process of mAb4–4–209 : the original residue was a neutral, weakly polar Histidine. Upon this His-to-Arg mutation a slowing in the unbinding of Flu by a 1.5-fold was experimentally observed,9 no doubt as a consequence of the increased attraction between the mutated amino acid 39L and Flu. Thus, it was only logical to consider the distance between the groups of Arg39L and Flu engaged in that driving hydrogen bonding as the most likely unbinding coordinate. The distance between the Cζ atom of Arg39L and the Flu’s hydroxyl oxygen (O1) was then set as the appropriate coordinate for scanning. The scanning started from the distance in the reference conformation and progressed in increments of 0.25 ˚ A up to ∼40 ˚ A. At this distance and for the set cut-oﬀ, the interaction energy between the hapten and the AB becomes zero. The scanning was also performed for a few decreasing steps, i.e. for distances smaller than the one in the reference structure. The distance value at each scanning step was imposed by means of a strong harmonic constraint (force constant = 26 eV·˚ A−2 ) between the referred oxygen and a dummy atom placed at the same coordinates of the Cζ atom. The need for a dummy atom arises from the fact that, in CHARMM, the non-bonding energy of all atom pairs separated by less than three covalent bonds is excluded20 ; the introduced harmonic constraint is an “artiﬁcial bond” and therefore should not be directly set between the Cζ and OH atoms, otherwise several pair interactions between Arg39L and Flu would be wrongly excluded. For each scanning step, the system was energy minimized with NAMD A−1 . A 12 ˚ A cut-oﬀ on longto an energy gradient tolerance ≤ 4 × 10−4 eV·˚ range interactions with a switch smoothing function between 10 and 12 ˚ A was used. During minimization, the hapten was free to move (subject only to the scanning harmonic constraint) while the AB was kept frozen for all but the side-chain atoms of a few key amino acids gating the passage of the hapten. The unconstrained side-chains belong to His31L , Asn33L , Arg52H , Tyr56H, Glu59H , Tyr102H and Tyr103H. The reported energy of each minimized structure was calculated after removing the afore-mentioned harmonic constraint. In NAMD, it is not possible to set two diﬀerent dielectrics within the same system, so minimizations were performed for ε = 1, and solvent eﬀects were introduced as corrections a posteriori, as described next. For the ﬁnal conformation of each scanning step, the electrostatic energy was recalculated using the APBS program (refer to Sec. 3.2). The

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conformation of the last scanning step roughly occupies a 70 ˚ A-side cubic box. The side was extended by an extra 20 ˚ A for solvent media, resulting in a 90 × 90 × 90 ˚ A3 box that was set equal for all scanning steps. Calculations were performed using the APBS’ adaptive reﬁnement.35 A low dielectric constant of = 2 was set for the macromolecule cavity22,36 and the typical water value of ≈ 80 was set for the continuum solvent medium. The eﬀect of a dilute electrolyte in solution was assessed with a second run of calculations, for a salt bulk concentration of 0.150 mol·dm−3 as in a typical physiological media,25 and a temperature of 298 K. 3.6. Exploring relative orientations The distance scanning scheme described above does not enforce an escaping channel along a straight line, nor does it restrain the AB–hapten relative orientations. For the sake of completeness, a comprehensive overlook of the two molecules relative position and mutual orientation should be performed. The designated appropriate descriptors are the spherical coordinates (r, θ, φ) and three Euler angles (α, β, γ), for which two coordinate frames are required. The referential frame, set as the protein’s principal axes of moment of inertia given that the protein is ﬁxed in space, and the movingbody local frame, i.e. the Flu frame. Care was taken to select this later, given that during the scanning Flu does not evolve in space as a completely rigid body. Its center was set in Flu’s atom C1 since during the scanning the position of Flu’s mass center is approximately coincident to this atom (0.2– 0.4 ˚ A RMSD). The xy-plane was made coincident to the rigid xanthenone ring, with the x-axis pointing in the direction of atom O1, as shown in Fig. 5.

Fig. 5.

System of internal axes for Flu.

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3.7. Calculation of koﬀ In compliance with the experimentally reported Arrhenius-like behavior7 and within the context of the reaction-rate theory,37 the oﬀ-rate constant along the scanned pathway was computed using the expression, koﬀ = ω exp(−∆E ‡ /kB T ),

(6)

where ω is the pre-exponential factor determining how frequently the system approaches the barrier, and ∆E ‡ is the activation energy (i.e. the barrier height). The harmonic approximation was used to estimate the Arrhenius-like pre-factor. It reads k 1 , (7) ω= 2π µ where µ is the reduced mass of the system and k the harmonic force constant. The latter was obtained from a parabolic ﬁt of the data (the bounding R region of the well in the energy proﬁles) using the Mathematica software package. For systems similar to the one presented here (with reduced masses in the 200–500 range and binding pocket length within 3–7 ˚ A), an estimation of the frequency ω falls in the 1011 –1012 s−1 range. 4. Results and discussion 4.1. Energetic and structural analysis The energy proﬁles resulting from the scanning runs with and without solvent correction are plotted in Figs. 6 and 7. Figure 6 displays the in vacuo results for four diﬀerent scanning runs, corresponding to diﬀerent constraining schemes on the protein atoms. The scheme referring to the seven unconstrained side-chains enumerated in Sec. 3.5 has been labeled “c1” in Fig. 6. To better assess the inﬂuence of those seven amino acids on the escaping proﬁle, they were subject to successive constraining procedures, exempliﬁed in Fig. 6 for three representative cases, labeled “c2,” “c3,” and “c4,” that correspond to six, ﬁve and four unconstrained side-chains (out of the initial seven). The plotted “c3” curve, for instance, results from moving only the side-chains of the ﬁve gating residues signaled out in the second paragraph of Sec. 3.5 and visible in Fig. 4(b). The constraining limit is the set of amino acids Asn33L , Tyr56H, Tyr102H and Tyr103H corresponding to the “c4” curve. Within this limit, no general signiﬁcant diﬀerences on the energy proﬁle arise from the explored diﬀerent schemes. These four amino acids always experience significant conformational changes upon the hapten’s passage, in comparison to

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c1 c2 c3 c4

-120

-122 Total energy in eV

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-126

-128

-130 0

5

10

15

20

25

30

35

40

Distance in Ångstroms

Fig. 6. Energy proﬁles (in vacuo) for diﬀerent distance scanning runs, corresponding to diﬀerent constraining schemes on the protein atoms. Each curve corresponds to a diﬀerent number of gating amino acid side-chains that were allowed to move during each scanning, namely, seven (c1), six (c2), ﬁve (c3) and four (c4) side-chains (details in the text).

the remaining moving ones which just slightly adjust positioning. The plane deﬁned by the side-chain oxygens of the four amino acids in question can be taken as the outermost limit of the protein’s pocket, and it is intersected at a ∼15 ˚ A scanning distance. Below this separation distance, the total energy plots in Fig. 6 depict the expected proﬁle for an activated process. For the diﬀerent curves, the height and shape of the energetic barrier at ∼7 ˚ A is essentially the same: 1.029, 1.027, 1.060 and 1.026 eV, respectively, for seven, six, ﬁve and four moving side-chains. Past the 15 ˚ A distance, the in vacuo proﬁles depict an asymptotic increase to a ﬁnal plateau above the aforementioned energetic barrier, making unbinding unfeasible. Predictably, the inclusion of solvent eﬀects rectify the asymptotic behavior depicted in the in vacuo proﬁles, as exempliﬁed in Fig. 7 for two scanning runs. At larger separation distances, the energy proﬁle has been signiﬁcantly ﬂattened, and it is also for the larger distances that the eﬀect of the dissolved electrolyte becomes perceptible. In solution, the electrostatic interactions between the protein and the escaping hapten are eﬀectively screened by the high-dielectric, allowing for unbinding to happen. Of relevance is also the decrease in the height of the energetic barrier at ∼7 ˚ A: with implicit solvent eﬀects, this barrier value is 0.863 and 0.871 eV, respectively for the “c2” and “c3” schemes. The jagged contour emerging at ∼20 ˚ A also deserves some attention. A detailed analysis of the energetic components was carried out, with

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-118 in vacuo APBS correction, no salt (+10.9 offset) APBS correction, 0.150M NaCl (+ 5.5 offset)

c2

Total energy in eV

-120

-122

-124

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5

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Distance in Ångstroms

in vacuo APBS correction, no salt (+ 8.6 offset) APBS correction, 0.150M NaCl (+ 3.3 offset)

c3

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-122 Total energy in eV

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5

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25

30

35

40

Distance in Ångstroms

Fig. 7. Comparison of the distance scanning energy proﬁles in vacuo and with implicit solvent corrections (with and without dissolved electrolyte), for the “c2” and “c3” constraining schemes.

particular emphasis on the Coulombic component, as exempliﬁed in Fig. 8 for scanning run “c2.” A previous work had already shown that, at larger distances, electrostatics play a major role in fragmentation.3 By comparing Figs. 7 and 8, one perceives the jagged pattern similarities between the total energy and its electrostatic component. Moreover, the electrostatic energy and its intermolecular component, i.e. the electrostatic interaction energy between the protein and the hapten, run parallel. To this intermolecular energy, a major contribution comes from the COO− group of Flu. This group gets anchored via hydrogen bonds at the protein’s surface

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c2 0 Electrostatic energy in eV

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-4

-6

Total (+ 141 offset) Intermolecular Inter FluCOO-

-8 0

5

10

15

20

25

30

35

40

Fig. 8. Electrostatic energy proﬁles (in vacuo) for the overall system, its intermolecular component and the interaction energy between the protein and the COO− group (atoms C20, O4, O5 of the Flu’s carboxyphenyl ring; see Fig. 2). The picture on top portrays the path of that ring during the scanning: the dashed yellow straight line highlights the coordinate being scanned while the circle emphasizes the anchor point of the COO− group at the protein’s surface (see details in the text). The gray area shows a jagged region in the proﬁle.

as the hapten leaves the pocket: the anchor point is circled yellow in the top image of Fig. 8. The hapten rotates around this point as the scanning distance is further increased, until it ﬁnally detaches from the surface. The detachment features a somewhat irregular trajectory of the escaping hapten: it is the region in the top image of Fig. 8, right above the gray area highlighting the jagged contour in the plot. A better perception of the hapten’s rotation as it leaves the binding pocket can be gained from the plotting of the Euler angles during the scanning, presented in Fig. 9. The steeper variation of the angles in the 15–20 ˚ A region corresponds to the anchoring track of the COO− group at the protein’s surface. As the hapten detaches, a swift change in its orientation is observed, made evident by the plots for the Euler angles from ∼20 ˚ A onwards. At this stage, one cannot ascertain whether or not this pronounced “trapping” of the hapten to the

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100

Angle in Degrees

50 0 -50 -100 -150 -200

β 160

Angle in Degrees

140 120 100 80 60 40 20 0 350

γ

300 Angle in Degrees

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100 10

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Distance in Ångstroms

Fig. 9.

Euler angles as a function of the scanned distance coordinate.

40

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protein’s surface is a genuine feature of the unbinding. That would at least require the other known mutations of the anti-ﬂuorescein mAb4–4–20 to be subject to an analogous study, which is beyond the scope of the present contribution. The fact remains that, overall, the total energy proﬁles are smooth (without discontinuities), as is clear from the top plot in Fig. 10 of the

-123 scanning distance radial distance r

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Total energy in eV

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Angle in Degrees

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90

80

70

60 0

5

10

15

20

25

30

35

40

Distance in Ångstroms

Fig. 10. Spherical coordinates along the “c2” scanning run. The total energy of the system is plotted as a function of both the radial distance (r) and the scanned distance coordinate. The angles θ and φ are plotted only as functions of the scanning distance coordinate.

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energy as a function of both the scanning coordinate and the radial distance (r). They portray a plausible unbinding channel, provided solvent eﬀects are included, though one cannot claim that they correspond to the minimum energy pathway. One possible step to assess that would be to perform a more comprehensive probing of the positional/orientational space of the hapten — beyond the points determined by the presently selected reaction coordinate. However, such a study involves a substantial computational eﬀort, even if conﬁned to some plausible escaping window in space. On the other hand, the presently computed proﬁles can be used to derive koﬀ , and by comparison to the corresponding experimental values(s), a ﬁrst evaluation of the scanning approach introduced here can be made. 4.2. koﬀ determination Table 1 presents the calculated values of koﬀ , with and without solvent correction, resulting from parabolic ﬁts to the proﬁles (0.99 ≤ R2 ≥ 0.87), given the energy barrier at ∼7 ˚ A (vide supra). Experimentally available koﬀ values are also presented for comparison. It becomes immediately evident that, even for an extensively studied system like mAb4–4–20–ﬂuorescein, experimental koﬀ values may diﬀer by an order of magnitude, depending on set-up conditions and techniques.9,38,39 As for our estimated values, while the in vacuo results are oﬀ-range, the solvent-corrected ones are comparable with the experimental results. The equilibrium distance between the antibody and the hapten (the well minimum) also compares better to the experimental value in the case of the solvent-corrected simulations. Finally,

Table 1. Kinetic and equilibrium parameters obtained from calculations based on the computational scanning. Available experimental values are also presented for comparison: (a) and (c) determined in solution (Refs. 38 and 39, respectively), and (b) at a surface by SPR.9 Parameter

Simulations In vacuo c2

koﬀ (s−1 )

Equilibrium distance (˚ A)

Experimental

Solvent corrected c3

c2

c3

3.4 × 10−6 6.8 × 10−6 4.1 × 10−3 5.4 × 10−3 1.9 × 10−3 6.8 × 10−3 4.3 × 10−3 2.5 × 10−2 3.50

T (K)

3.55

3.60

3.65

3.65

(a) (b) (c) (c)

291 298 298 298 291

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remark that the diﬀerent constraining schemes have little inﬂuence on the order of magnitude of the koﬀ values. 5. Concluding remarks We have presented our ﬁrst attempt to describe the unbinding of a complex biomolecular system in terms of a reduced set of relevant generalized coordinates while restricting most of its conformational internal degrees of freedom. The reported results reveal a practical and physically sound procedure to compute energy proﬁles along the selected reaction coordinate(s). This was demonstrated in the present work for an experimentally well studied complex of the biologically relevant antigen–antibody system. For the example in question, it was actually possible to ﬁnd a distance dependent escaping channel in the multidimensional potential energy landscape, thus reducing the unbinding to a low-dimensional problem: the system seems to be eﬃciently bound by this one distance coordinate. The eﬀect of the solvent was also accounted for, and despite the fact that it was introduced as a correction a posteriori, it allowed us to ascertain that this is one eﬀect that needs to be included, for it has a signiﬁcant inﬂuence on the overall energetic proﬁle and subsequent parameters derived from it. With solvent eﬀects, the derived oﬀ-rates are in reasonable agreement with the experimentally determined ones, a result that can be regarded as an indicator that ours is indeed a realistic approach. The proposed approach would no doubt beneﬁt from further reﬁnements, namely, in the way solvent eﬀects are introduced (viz. include them during scanning, both implicitly and explicitly) and in the kinetic model, which we intend to address in the near future. We also have in mind to apply this same approach to the maturation series and engineered mutants of the 4–4–20–ﬂuorescein complex. That would allow us to further test our strategy, in particular its sensitiveness to the energetic diﬀerences arising from antibody’s single-point mutations. In addition, computed dissociation rates would be valuable parameters in diﬀerent areas of immunological research, namely, theoretical immunology.40 Remark also that calculated koﬀ values could be used to determine the related association rate kon using the relation kon = koﬀ /Kd ,9 for those systems where only the equilibrium dissociation constant Kd has been experimentally measured. And by identifying and rationalizing the key structural features and interactions determining the unbinding involved, one could make insightful predictions and propose, for instance, aﬃnity-enhancing mutations. Our long-term goal is to extend our research to other molecular recognition processes besides the antigen–antibody one and to test the applicability and universality of our approach.

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Acknowledgments Financial support from the NoE EXCELL EU project is gratefully acknowledged. The authors also thank Ilia A. Solov’yov for his valuable assistance in the implementation of the Euler angle analysis, and Dr. Michael MeyerHermann for drawing our attention to the importance of the problem considered for theoretical immunology. Finally, we are grateful to Prof. Walter Greiner for useful discussions.

References 1. A. V. Solov’yov, J.-P. Connerade and W. Greiner, Latest Advances in Atomic Cluster Collisions (Imperial College Press, London, 2004). 2. O. I. Obolensky, A. Lyalin, A. V. Solov’yov and W. Greiner, Phys. Rev. B 72, 085433 (2005). 3. I. A. Solov’yov, A. V. Yakubovich, A. V. Solov’yov and W. Greiner, J. Exp. Theo. Phys. 103, 463 (2006). 4. G. J. Wedemayer, P. A. Patten, L. H. Wang, P. G. Schultz and R. C. Stevens, Science 276, 1665 (1997). 5. T. Manser, J. Immunology 172, 3369 (2004). 6. D. L. Nelson and M. M. Cox, Lehninger Principles of Biochemistry (W. H. Freeman and Company, New York, 2005). 7. F. Schwesinger, R. Ros, T. Strunz, D. Anselmetti, H.-J. Gntherodt, A. Honegger, L. Jermutus, L. Tiefenauer and A. Pl¨ uckthun, Proc. Natl. Acad. Sci. USA 97, 9967 (2000). 8. J. Foote and H. N. Eisen, Proc. Natl. Acad. Sci. USA 92, 1254 (1995). 9. R. Jimenez, G. Salazar, T. J. J. Yin and F. E. Romesberg, Proc. Natl. Acad. Sci. USA 101, 3803 (2004). 10. P. Hinterdorfer, W. Baumgartner, H. J. Gruber, K. Schlicher and H. Schindler, Proc. Natl. Acad. Sci. USA 93, 3477 (1996). 11. U. Dammer, M. Hegner, D. Anselmetti, P. Wagner, M. Dreier, W. Huber and H.-J. G¨ untherodt, Biophys. J. 70, 2437 (1996). 12. S. Allen, X. Chen, J. Davies, M. Davies, A. C. Dawkes, J. C. Edwards, C. J. Roberts, J. Sefton, S. J. B. Tendler and P. M. Williams, Biochemistry 36, 7457 (1997). 13. T. A. Sulchek, R. W. Friddle, E. Y. L. K. Langry, H. Albrecht, T. V. Ratto, S. J. DeNardo, M. E. Colvin and A. Noy, Proc. Natl. Acad. Sci. USA 102, 16638 (2005). 14. H. Grubm¨ uller, Protein-Ligand Interactions (The Human Press Inc., New Jersey, 2005), Chap. “Force probe molecular dynamics simulations,” pp. 493–515. 15. E. W. Voss Jr., J. Mol. Recog. 6, 51 (1993). 16. K. S. Midelfort, H. H. Hernandez, S. M. Lippow, B. Tidor, C. L. Drennan and K. D. Wittrup, J. Mol. Biol. 343, 685 (2004).

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17. J. N. Herron, X. M. He, M. L. Mason, E. W. Voss Jr. and A. B. Edmundson, Proteins 5, 271 (1989). 18. M. Whitlow, A. J. Howard, J. F. Wood, E. W. Voss Jr. and K. D. Hardman, Protein Eng. 8, 749 (1995). 19. S. Jung and A. Pluckthun, Protein Eng. 10, 959 (1997). 20. A. D. MacKerell Jr., D. Bashford, M. Bellott, R. L. Dunbrack Jr., J. Evanseck, M. J. Field, S. Fischer, J. Gao, H. Guo, S. Ha, D. Joseph, L. Kuchnir, K. Kuczera, F. T. K. Lau, C. Mattos, S. Michnick, T. Ngo, D. T. Nguyen, B. Prodhom, W. E. Reiher III, B. Roux, M. Schlenkrich, J. Smith, R. Stote, J. Straub, M. Watanabe, J. Wiorkiewicz-Kuczera, D. Yin and M. Karplus, J. Phys. Chem. B 102, 3586 (1998). 21. F. Fogolari, A. Brigo and H. Molinari, J. Mol. Recogn. 15, 377 (2002). 22. F. Fogolari, P. Zuccato, G. Esposito and P. Viglino, Biophys. J. 76, 1 (1999). 23. N. A. Baker, D. Sept, S. Joseph, M. J. Holst and J. A. MacCammon, Proc. Natl. Acad. Sci. USA 98, 10037 (2001). 24. U. C. Singh and P. A. Kollman, J. Comput. Chem. 5, 129 (1984). 25. B. P. Singh, H. B. Bohidar and S. Chopra, Biopolymers 31, 1387 (1991). 26. B. H. Besler, K. M. Merz Jr. and P. A. Kollman, J. Comput. Chem. 11, 431 (1990). 27. Gaussian 03: Revision C.02, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery Jr., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Cliﬀord, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez and J. A. Pople, Gaussian Inc., Wallingford CT (2004). 28. R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1994). 29. H. Berman, K. Henrick, H. Nakamura and J. L. Markley, Nucleic Acids Res. Database issue D1–D3 00 (2006) [doi:10.1093/nar/gkl971]. 30. E. Paci, A. Caﬂisch, A. Pl¨ uckthun and M. Karplus, J. Mol. Biol. 314, 589 (2001). 31. E. S. Henriques, M. Bastos, C. F. G. C. Geraldes and M. J. Ramos, Int. J. Quant. Chem. 73, 237 (1999). 32. E. S. Henriques, M. A. C. Nascimento and M. J. Ramos, Int. J. Quant. Chem. 106, 2107 (2006).

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33. L. Kale, R. Skeel, M. Bhandarkar, R. Brunner, A. Gursoy, N. Krawetz, J. Phillips, A. Shinozaki, K. Varadarajan and K. Schulten, J. Comp. Phys. 151, 283 (1999). 34. J. P. DuBost, J. M. Leger, J. C. Colleter, P. Levillain and D. Fompeydie, C. R. Acad. Sci. Paris S´er. II 292, 965 (1981). 35. M. J. Holst, N. A. Baker and F. Wang, J. Comput. Chem. 21, 1343 (2000). 36. M. T. Neves-Petersen and S. B. Petersen, Biotech. Annu. Rev. 9, 315 (2003). 37. P. H¨ anggi, P. Talkner and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990). 38. M. E. Mummert and E. W. Voss Jr., Biochemistry 35, 8187 (1996). 39. E. T. Boder, K. S. Midelfort and K. D. Wittrup, Proc. Natl. Acad. Sci. USA 97, 10701 (2000). 40. M. E. Meyer-Hermann, P. K. Maini and D. Iber, Math. Med. Biol. 23, 255 (2006).

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IRON MINERAL BASED MAGNETORECEPTION MECHANISM IN BIRDS I. A. SOLOV’YOV∗ and W. GREINER Frankfurt Institute for Advanced Studies, J. W. Goethe University, Frankfurt am Main 60438, Germany ∗ [email protected] In an earlier work, it was suggested that the “magnetic sense” in birds may be mediated by the blue light receptor protein, cryptochrome, which is known to be localized in the retinas of migratory birds. The present contribution discusses an alternative mechanism of avian magnetoreception based on the interaction of two iron minerals (magnetite and maghemite), recently found in subcellular compartments within the sensory dendrites of the upper beak of several bird species. The analysis of forces acting between the iron particles shows that the orientation of the external geomagnetic ﬁeld can signiﬁcantly change the probability of the mechanosensitive ion channels opening and closing. Our theoretical analysis shows that the suggested magnetoreceptor system might be a sensitive biological magnetometer providing an essential part of the magnetic map for navigation. Keywords: Sensory perceptions; ferrimagnetics; ﬁne-particle systems; magnetic ﬁeld eﬀect; radical pair mechanism; avian magnetoreception.

1. Introduction In an earlier paper,1 it was already mentioned that during the last ﬁve decades it has been demonstrated that a wide variety of living organisms have the ability to perceive magnetic ﬁelds and can use spatial information from Earth’s magnetic ﬁeld in orientation behavior. For example, migratory birds use magnetic cues (in addition to light polarization, star position, and the position of the sun) to ﬁnd their way south in fall and north in spring.2,3 Salamanders, frogs and sea turtles use the magnetic ﬁeld for orientation when they have to ﬁnd the direction of the nearest shore quickly, e.g., to evade a predator.4,5 Magnetoreception by honeybees (Apis mellifera) is demonstrated by such activities as comb building and homing

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orientation, which are aﬀected by the geomagnetic ﬁeld.6–9 Magnetotaxis has been shown in bacteria,10–12 whose directional movement is oriented by the local geomagnetic ﬁeld. The best studied example is the use of the geomagnetic ﬁeld by migratory birds for orientation and navigation during migration. Reviews of these studies are given in Refs. 3 and 13–18. Two, likely co-existing, mechanisms underlying animal magnetoreception are widely discussed in the literature (for review, see Refs. 13, 18 and 19). In a preceding paper,1 it was suggested that the “magnetic sense” in birds may be mediated by the blue light receptor protein, cryptochrome, which is known to be localized in the retinas of migratory birds. The present contribution discusses an alternative mechanism of avian magnetoreception which is based on the interaction of two iron minerals [magnetite (Fe3 O4 ) and maghemite (γ−Fe2 O3 )], recently found in subcellular compartments within the sensory dendrites of the upper beak of several bird species.18,20–22 From the analysis of forces which act between these particles, we show that the iron-mineral system considered can serve as a magnetoreceptor with distinct orientational properties. We demonstrate that, depending on the orientation of the external magnetic ﬁeld, the probability of opening of mechanosensitive ion channels signiﬁcantly changes, thus leading to diﬀerent nerve signals. The suggested mechanism is diﬀerent from the magnetite-based magnetoreception mechanism suggested earlier because it involves two diﬀerent types of iron minerals (see, e.g., Refs. 23–28). In these papers, only magnetite was considered and it was suggested that the magnetite clusters depending on the orientation of the external magnetic ﬁeld will attract or repel each other, deforming the membrane and possibly opening (closing) the ion channels. This mechanism was suggested about ﬁfteen years ago,29 long before maghemite platelets were discovered in the beaks of birds. Since maghemite has a ferrimagnetic nature30 and possesses pronounced magnetic properties, the maghemite platelets could potentially play an important role in the magnetoreception mechanism.

2. The magnetoreceptor system Based on histology studies of the upper beak of homing pigeons, it has been shown21,22 that there are six patches in the beak where iron minerals are concentrated. The iron minerals have been found in symmetrical spots near the lateral margin of the skin of the upper beak inside the dendrites of nerve cells. The structure of a typical bird nerve cell (neuron) is schematically shown in Fig. 1(a).

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Fig. 1. (a) Schematic illustration of a nerve cell (neuron). The dendrites form a tree-like structure and constitute the nerve signal input interface of the neuron; the axon connects the dendrites with the axon terminal, the signal output interface of the neuron. The cell nucleus is located in the soma. The characteristic size of the cell components is indicated. (b) The characterization and subcellular localization of iron minerals within a dendrite. The drawing shows schematically the structure of a single dendrite as derived from serial ultrathin sections with the three subcellular components containing iron: chains of maghemite crystals (1 × 0.1 × 1 µm) magnetite clusters of nanoparticles (diameter about 1 µm), and the iron-coated vesicle (diameter 3 to 5 µm). Figure courtesy of Gerta and G¨ unther Fleissner, Universit¨ at Frankfurt am Main.

The neuron consists of several dendrites, which induce the nerve signal; an axon, a long ﬁber used to transmit the signal; a soma with the cell nucleus; and an axon terminal, which transmits the nervous signal. The dendrites build the nerve signal input interface, while the axon terminal is the signal output interface [see Fig. 1(a)]. The size of the iron mineral patches in dendrites was found to always be about the same, 350 µm long and 200 µm in diameter.21,22 In every patch, the iron minerals were found parallel to the axon bundles with a certain spatial orientation. It was shown22 that the dendrites in the frontal, middle and caudal parts of the beak are aligned in three perpendicular directions: the frontal ones have a preferred dorsal-to-ventral direction, the middle

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ones a median-to-lateral direction, and the caudal ones a caudal-to-rostral direction. In addition, it was demonstrated that dendrites containing iron also form a regular pattern. Several of them may align side by side but longitudinally the distance between them is about 100 µm. This fact was independently conﬁrmed in the µ-SXRF-measurements of histologically undisturbed and unstrained material.22 The construction of a primary magnetoreceptor unit, which is discussed in the present work, is motivated by the experimental ﬁndings of Fleissner et al.21 ,22 In order to explain the choice of our model in Fig. 1(b), we show schematically the structure of a single dendrite containing iron minerals. The ﬁgure is based on the experimental results discussed in Refs. 21 and 22. Within the dendrite there are three diﬀerent subcellular compartments containing iron minerals: several magnetite clusters, an iron-coated vesicle, and maghemite platelets, which are marked in Fig. 1(b). Each dendrite contains 10–15 clusters of magnetite nanocrystals of average size 5 nm.22,31 The clusters have an average diameter of 1 µm21,22 and adhere to the cell membrane. The maghemite platelets form bands that extend through the entire dendrite. The magnetite clusters are usually found at the edges of maghemite bands, which include about ten platelets. Each platelet was found to be 1 µm wide and long and less than 0.1 µm thick.21,22 The vesicle is most often located in the center of the dendrite and its composition is still not well understood. In Refs. 21 and 22, it was demonstrated that the vesicle is covered by some non-crystalline iron substance and has a diameter of about 5 µm. If the suggested magnetoreception mechanism works, then the forces which act on the magnetite clusters in the dendrite should be diﬀerent at diﬀerent orientations of the external magnetic ﬁeld. Here, this problem is considered theoretically and it is shown that the iron mineral based magnetoreception mechanism is indeed possible. To do so, we have deﬁned a primary magnetoreceptor unit which is the smallest structure possessing the magnetoreception properties of the whole dendrite. This magnetoreceptor unit consists of ten maghemite platelets and one magnetite cluster is shown in Fig. 1(b). We chose to study the forces acting on a single magnetite cluster. In a dendrite there are about 10–15 of such units [see Fig. 1(b)], which should have similar behaviors in the external magnetic ﬁeld. Therefore, if the entire dendrite is subject to the external magnetic ﬁeld, the repetition of the magnetoreceptory units increases the functional safety of the dendrite magnetoreception. The geometry of the magnetoreceptor unit is determined from experimental observations.22 Thus, the maghemite platelets have the dimensions 1 × 0.1 × 1 µm and the magnetite cluster has a diameter of 1 µm. The

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Fig. 2. Magnetoreceptor unit consisting of ten maghemite platelets (boxes) and a magnetite cluster (sphere). The coordinate frame shown here is used in the computations. is characterized by two polar The direction of the external magnetic induction vector B angles Φ and Θ. The magnetic moments of the maghemite platelet i, m i , and of the are indicated. magnetite cluster M

maghemite platelets are located in the xz-plane, being aligned along the x-axis (see Fig. 2). The distance between two neighboring platelets is equal to 0.1 µm. 3. Theoretical model The size of a single maghemite platelet is suﬃcient for the formation of magnetic domains in the xz-plane of the platelet26 (see Fig. 2). Thus, the maghemite platelets have a magnetic moment in this plane even in the absence of the external magnetic ﬁeld. The direction of the magnetic moment of a platelet is governed by the projection of the total magnetic ﬁeld on the xz-plane. The total magnetic ﬁeld at the site of the platelet i , is the sum of the external magnetic ﬁeld, the magnetic with index i, H ﬁeld created by other platelets, and by the magnetic ﬁeld created by the magnetite cluster. Thus, the magnetic moment of a platelet is given by   Hix 1   (1) m i = Mv  0 , 2 2 Hix + Hiz Hi z where M is the remnant magnetization of maghemite. v ≡ lx ly lz , where lx , ly and lz are the dimensions of a platelet along the x-, y- and z-axes, respectively. With M = 50 emu/cm3 ,32 lx = lz = 1 µm and ly = 0.1 µm, one obtains mi ≈ 3.12 eV/G.

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The magnetite cluster consists of nanoparticles which are 5 nm in diameter.22 In the case of ﬁnite temperature and ﬁnite magnetic ﬁeld, the is mean total moment of the magnetite cluster, M, 2 ≈ nµ H = χH, M (2) 3kT where n is the number of nanomagnets in the cluster, µ is the magnetic is the magnetic ﬁeld strength moment of an individual nanomagnet, H at the site of the magnetite cluster, T is the temperature, and k is the Bolzmann constant. With R0 = 0.5 µm and r0 = 2.5 nm being the radii of the magnetite cluster and of the nanoparticle respectively, one obtains n ≈ R03 /r03 ≈ 8 × 106 . The proportionality constant between the magnetic moment and the ﬁeld strength, χ, is the magnetic susceptibility. Dividing it by the volume of the magnetite cluster, one obtains the volume susceptibility of the magnetite cluster, χv , which at 300 K is equal to 0.12 CGS units. With H = 10 Oe, which is a typical value of the total ﬁeld at the site of the magnetite cluster, one obtains M ≈ 0.392 eV/G. Note that this value is about an order of magnitude smaller than the magnetic moment of a single maghemite platelet. The potential energy of the magnetite cluster reads as 2 N B 4 3 Hj (R) , + (3) E(R) = −χv πR0 3 µmed j=1

deﬁnes the position of the magnetite cluster, B is the induction where R vector of the external magnetic ﬁeld, µmed ≈ 1 is the permeability of the is the mag i (R) medium, N is the number of maghemite platelets, and H netic ﬁeld created by the ith maghemite platelet at the site of the magnetite cluster, which is known to be33 − rj )) − m − rj |2 j (R j |R j (R) = 3(R − rj )(m H . (4) − rj |5 |R j is its magnetic Here, rj describes the position of the jth platelet and m moment deﬁned in Eq. (1). The total magnetic ﬁeld at the site of the ith maghemite platelet is i = H

B µmed

+

N

j (ri ). H

(5)

j=1 j=i

The ﬁrst term describes the external magnetic ﬁeld while the second describes the magnetic ﬁeld created by all maghemite platelets except the ith one.

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i is determined by It follows from Eq. (5) that the total magnetic ﬁeld H the magnetic moments of the platelets. Thus, Eqs. (1) and (5) have to be treated iteratively. To a zeroth-order approximation, m i are aligned along the x-axis, which is energetically the most favorable conﬁguration of the system without the external magnetic ﬁeld applied. The magnetic moment (0) of a platelet is then m i = M vi, where i is the unit vector along the x-axis. The total magnetic ﬁeld to a ﬁrst-order approximation at the site of the ith maghemite platelet reads as N 1 i = B (1) = B + 2M v + 2M vξii, H (6) i |xi − xj |3 j=1 j=i

1/|xi − xj |3 . where xi is the x-coordinate of the ith platelet and ξi = Substituting Eq. (6) into Eq. (1), one yields the ﬁrst-order approximation for m i: M v(Bx + 2M vξi ) (1) , (7) mix = (Bx + 2M vξi )2 + By2 + Bz2 (1)

miy = 0,

(8)

M vBz (1) miz = . (Bx + 2M vξi )2 + By2 + Bz2

(9)

Here, Bx and Bz are the x- and z-components of the external magnetic induction vector, respectively. Thus, the potential energy E(R) of the magnetite cluster reads as = −χv 4 πR3 B E (1) (R) 0 3 2 (1) (1) N − rj )(m j (R − rj )) m j 3(R . − (10) + − rj |5 − rj |3 |R |R j=1

From Eq. (10), one calculates the force which acts on the magnetite cluster according to F = −∇E(R). (11) 4. Results and discussion To illustrate the eﬀect of the external magnetic ﬁeld on the magnetoreceptor system, we have calculated the diﬀerences in forces acting on the magnetite cluster due to the 90◦ change in direction of the external magnetic ﬁeld. The

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Fig. 3. Diﬀerence in force acting on the magnetite cluster and arising due to the 90◦ change in direction of the external magnetic ﬁeld. The thin line shows the force diﬀerence arising due to the change in external magnetic ﬁeld direction from x to z, and the thick line shows the force diﬀerence arising due to the change in external magnetic ﬁeld direction from x to y. The force diﬀerences were calculated as a function of the x-coordinate of the magnetite cluster, while y = 0.8 µm and z = 0 µm.

force diﬀerences are shown in Fig. 3. The thin line shows the force diﬀerence arising due to the change in external magnetic ﬁeld direction from x to z, and the thick line shows the force diﬀerence arising due to the change in external magnetic ﬁeld direction from x to y. The force diﬀerences were calculated as a function of the x-coordinate of the magnetite cluster, while y = 0.8 µm and z = 0 µm. Figure 3 shows that the force change, caused by the 90◦ change in direction of the external ﬁeld, is 0.1–0.2 pN in both cases. It was experimentally demonstrated21,22 that the magnetite clusters are connected to the nerve cell membrane. Depending on the magnetic ﬁeld strength, the magnetite cluster exerts forces on the membrane and activates mechanosensitive ion channels, increasing the ﬂux of ions into the cell. The ions change the membrane potential. If the potential is reduced to the threshold voltage,34 an action potential is generated in the cell, which opens up hundreds of voltage-gated ion channels in the membrane. During the millisecond that the channels remain open, thousands of ions rush into the cell,34 producing a nerve signal that is sent to the brain. The mechanosensitive ion channels inﬂuence the time needed for the membrane potential to reach the threshold value, and thus inﬂuence the bird’s behavior. A typical example of a mechanosensitive ion channel is the transduction channel of a hair cell (for review, see Refs. 35–37). The opening/closing of the mechanosensitive ion channel is regulated by the so-called gate, which is a large biological complex (protein or complex of proteins) at the edge of the ion channel.35–38 The gate is connected to an elastic element, the gating spring,35–38 transmitting the force to the gate.

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Fig. 4. Schematic illustration of the transducer mechanism of the geomagnetic ﬁeld. The opening/closing of the mechanosensitive ion channel is regulated by the gate, which is connected to an elastic element, the gating spring. The channel has two conformations: (a) closed and (b) open, in thermal equilibrium.

The ion channel has two conformations, closed [see Fig. 4(a)] and open [see Fig. 4(b)]. Because the gate swings through a distance λ upon opening, an external force f changes the energy diﬀerence between open and closed states and can bias the channel to spend more time in its open state. The gating springs are connected to the magnetite cluster, which produces an external pull on the gates. As follows from our calculations, the magnitude of this pull is about 0.2 pN, when the direction of the external magnetic ﬁeld is changed by 90◦ (see Fig. 3). The work done in gating the channel is38 ∆E = ∆ε − f λ, where the ﬁrst term represents the change in the intrinsic energy between the open and the closed states of the channel and the second term shows the work of external force required for opening the channel. λ is the displacement of the gate. For the mechanosensitive ion channels in hair cells λ ≈ 4 nm.35,38 The probability of the ion channel being open in the presence of external force is p=

1

∆ε−f λ . 1 + exp kT

(12)

If no external force is applied then f = 0 and the corresponding probability of the channel being open is p˜0 . Thus, the change in channel opening probability due to the applied force is

fλ

exp kT −1 exp ∆ε p − p˜0 kT = (13) η=

.

fλ p˜0 + exp ∆ε exp kT kT

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Fig. 5. Change in the mechanosensitive ion channel opening probability calculated as a function of change in the intrinsic energy between the open and the closed states of the channel.

The value of ∆ε is not known. Usually,38 it is assumed that ∆ε = 0, but in general it is not because the gate can form hydrogen bonds with the membrane, which break when the gate is opened. Thus, ∆ε > 0. Figure 5 shows the dependence of the change in channel opening probability, η, on ∆ε (thick line). From Fig. 5 and from Eq. (13), it follows that the change in channel opening probability saturates at large values of ∆ε. The limiting value is ηmax = exp(f λ/kT ) − 1. For the given f , λ and T , ηmax = 0.21, the maximal change in channel opening probability possible in the suggested mechanism. If ∆ε = 0 then η0 = 0.096. If ∆ε is positive then η is somewhere between η0 and ηmax . 5. Conclusion In the present contribution, a possible mechanism of avian orientation in a magnetic ﬁeld is discussed. It was shown that in an external magnetic ﬁeld the magnetite clusters experience an attractive (repulsive) force leading to their displacement, which induces a primary receptor potential via mechanosensitive membrane channels, leading to a certain orientation eﬀect in the bird. We believe that the suggested magnetoreception mechanism is a realistic candidate for the magnetoreception mechanism. It might also be responsible for magnetosensation in other animals, like ﬁsh, salamanders, and bees. Unfortunately, a lack of suﬃcient information about magnetic particles in these species prevents us from drawing conclusions about the precise magnetoreception mechanism.

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The analysis performed in this work and in the preceding paper1 supports the hypothesis that birds possibly have two magnetic ﬁeld receptors: one located in the eye and the second located in the beak. Both receptors may supplement each other and enhance the eﬀect of the weak geomagnetic ﬁeld on birds. One possibility might be that the nervous signals induced in the beak and in the eye modulate each other, leading to an enhanced eﬀective signal being sent to the brain of the animal. To understand if this is the case, it is necessary to study the nervous system of the bird and clarify which nervous signals can be produced due to the described magnetoreception systems. Another possibility for the two mechanisms to operate together is inspired by recent electrophysiological and histological studies, which suggest that, in birds, a radical pair mechanism located in the right eye provides directional information, while a magnetite-based mechanism located in the upper beak records magnetic intensity, thus providing positional information.17 These hypotheses should be studied both experimentally and theoretically. Acknowledgments We thank Professors Gerta and G¨ unther Fleissner for many helpful discussions and valuable comments on data presented in this work. We are also grateful to Prof. Andrey V. Solov’yov and Prof. Klaus Schulten for many insightful comments. The possibility to perform complex computer simulations at the Frankfurt Center for Scientiﬁc Computing is gratefully acknowledged. This work is partially supported by the European Commission within the Network of Excellence project EXCELL. References 1. I. A. Solov’yov, D. Chandler and K. Schulten, Cryptochrome–1 magnetoreception, in Latest Advances in Atomic Cluster Collisions: Structure and Dynamics from the Nuclear to the Biological Scale, eds. A. V. Solov’yov and J.-P. Connerade (Imperial College Press, London, 2008). 2. W. Wiltschko and F. Merkel, Verh. dt. Zool. Ges. 59, 362 (1966). 3. W. Wiltschko and R. Wiltschko, J. Comp. Physiol. A 191, 675 (2005). 4. K. Lohmann, C. Lohmann, L. Erhart, D. Bagley and T. Swing, Nature 428, 909 (2004). 5. J. Phillips and S. Borland, Nature 359, 142 (1992). 6. J. Kirschvink, Trends Neurosci. 5, 160 (1982). 7. C.-Y. Hsu and C.-W. Li, Science 265, 95 (1994). 8. H. Nichol, M. Locke, J. L. Kirschvink, M. M. Walker, M. H. Nesson, C.-Y. Hsu and C.-W. Li, Science 269, 1888 (1995).

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H. Schiﬀ and G. Canal, Biol. Cyber. 69, 7 (1992). R. Blakemore, Science 190, 377 (1975). R. B. Frankel, R. P. Blakemore and R. S. Wolfe, Science 203, 1355 (1979). Y. A. Gorby, T. J. Beveridge and R. P. Blakmore, J. Bacteriol. 170, 834 (1988). I. A. Solov’yov, D. Chandler and K. Schulten, Biophys. J. 92, 2711 (2007). W. Wiltschko and R. Wiltschko, Naturwissenschaften 89, 445 (2002). R. C. Beason, Integr. Comp. Biol. 45, 565 (2005). H. Mouritsen and T. Ritz, Curr. Opin. Neurobiol. 15, 406 (2005). R. Wiltschko and W. Wiltschko, BioEssays 28, 157 (2006). I. A. Solov’yov and W. Greiner, Biophys. J. 93, 1493 (2007). I. A. Solov’yov, Magnetoreception Mechanisms in Birds — Towards the Discovery of the Sixth Sense, Dissertation, J.W. Goethe Universit¨ at Frankfurt am Main, Germany (2007). M. Hanzlik, C. Heunemann, E. Holtkamp-R¨ otzler, M. Winklhofer, N. Petersen and G. Fleisner, BioMetals 13, 325 (2000). G. Fleissner, E. Holtkamp-R¨ otzler, M. Hanzlik, M. Winklhofer, G. Fleissner, N. Petersen and W. Wiltschko, J. Comparat. Neurol. 458, 350 (2003). G. Fleissner, B. Stahl, P. Thalau, G. Falkenberg and G. Fleissner, Naturwissenschaften 94, 631 (2007). A. F. Davila, M. Winklhofer, V. P. Shcherbakov and N. Petersen, Biophys. J. 89, 56 (2005). M. Winklhofer, Physik unserer Zeit 35, 120 (2004). J. L. Kirschvink, M. M. Walker and C. E. Diebel, Curr. Opin. Neurobiol. 11, 462 (2001). J. Kirschvink and J. Gould, Biosystems 13, 181 (1981). S. Johnsen and K. J. Lohmann, Neuroscience 6, 703 (2005). M. Walker, T. Dennis and J. Kirschvink, Curr. Opin. Neurobiol. 12, 735 (2002). J. Kirschvink, Phys. Rev. A 46, 2178 (1992). C. Hunt, B. Moskowitz and S. Banerjee, Rock Physics and Phase Relations, A Handbook of Physical Constants, AGU Reference Shelf 3, 189 (1995). M. Winklhofer, Modelle hypotethischer Magnetfeldrezeptoren auf der Grundlage biogenen Magnetits (Verlag Merie Leidorf GmbH, Rahden/Westfallen, 1999). V. Math´e and F. L´eveque, Eur. J. Soil Sci. 56, 737 (2005). L. Landau and E. Lifshitz, The Classical Theory of Fields (Pergamon, London, 1965). D. L. Nelson and M. M. Cox, Principles of Biochemistry (W.H. Freeman and Company, New York, 2005). A. J. Hudspeth, Y. Choe, A. Mehta and P. Martin, Proc. Natl. Acad. Sci. USA 97, 11765 (2000). V. S. Markin and A. Hudspeth, Ann. Rev. 24, 59 (1995). O. P. Hamill and B. Martinac, Physiol. Rev. 81, 685 (2001). D. P. Corey and J. Howard, Biophys. J. 66, 1254 (1994).

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BIOPHYSICAL MODELING OF FRAGMENT DISTRIBUTIONS OF DNA PLASMIDS AFTER HEAVY ION IRRADIATION ∗ ¨ TH. ELSASSER , M. SCHOLZ and G. TAUCHER–SCHOLZ

Gesellschaft f¨ ur Schwerionenforschung, Biophysics, Planckstr. 1, 64291 Darmstadt, Germany www.gsi.de ∗ [email protected] S. BRONS Heidelberger Ionen Therapie, Im Neuenheimer Feld 450, 69120 Heidelberg, Germany K. PSONKA and E. GUDOWSKA-NOWAK Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, Krakow, Poland The investigation of fragment length distributions of plasmid DNA following ion irradiation leads to a better understanding of the induction of DNA damage, particularly the clustering of double strand breaks. We present a model that calculates the fragment distributions of plasmid DNA following heavy ion irradiation by combining the Local Eﬀect Model with a statistical model initially developed for X-rays. The integration of experimental constraints into the model calculations changes the resulting distributions strongly. We ﬁnd a good agreement of our simulations with experimental fragment distributions based on atomic force microscopy studies. The model provides the means to rapidly predict the results for all ions. It may thus help to ﬁnd the best ion species in order to demonstrate the impact of the localized energy distribution of particles. Keywords: Biophysical modeling; DNA damage; atomic force microscopy; ion irradiation.

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1. Introduction The production of double strand breaks (DSB) is considered to be the most severe single radiation-induced damage for numerous types of biological endpoints like cell inactivation or chromosomal aberrations.1 However, it has become apparent that the mere number of DSB(s) does not fully account for the observed radiation damage. Instead, the spatial distribution of DSB(s) seems to be crucial for understanding the radiation-induced damage as well. Whereas for low-LET radiation a homogeneous energy deposition results in a randomly distributed pattern of lesions, for high-LET particle radiation the localized dose deposition along individual tracks leads to a non-random clustering of radiation events and damage induction.2 In order to investigate the impact of localized energy deposition on biological endpoints, it is desirable to exploit a model system allowing one to deduce the inﬂuence of track structure on the induction of radiation damage. In cellular systems, it has been diﬃcult to separate the inﬂuence of track structure from the eﬀects arising due to the three-dimensional organization of chromatin. Therefore, several groups have used plasmid DNA as a suitable experimental system to investigate the inﬂuence of diﬀerent radiation qualities on strand break production using gel electrophoresis3,4 or atomic force microscopy (AFM).5 –7 In this contribution, we present a simple model which predicts fragment length distributions of linear and supercoiled plasmid DNA after irradiation with heavy ions in order to highlight the impact of local energy distribution on the experimental results. For this purpose, we combine the widely used broken stick approach for X-rays8 with the Local Eﬀect Model,9,10 which is being used in the context of heavy ion tumor therapy. Additionally, we include experimental constraints which are speciﬁc for the fragmention analysis by AFM. With this concept, it is possible to rapidly calculate fragment distributions for a broad range of ions and energies.

2. Broken stick model Our numerical simulations of fragment distributions after heavy ion irradiation partly exploit the model presented in Ref. 8 to describe the fragment length pattern of plasmids following X-irradiation. In this section, we brieﬂy summarize the derivation and its results with respect to fragment distributions of plasmid DNA in two diﬀerent conformations, namely, the linear and the supercoiled conformation. In the case of DNA initially in the supercoiled form, the ﬁrst DSB produced by irradiation breaks the supercoiled shape of the plasmid and converts it into a linear string of full plasmid length.

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In the case of initially linear plasmids, which are experimentally produced using enzymes speciﬁcally cutting the molecule once, a single DSB due to irradiation already produces two fragments. In either case, we need to know the biological yield expressed in terms of the average number of DBS(s), which is given by a linear-quadratic relation4 n ¯ DSB = αD + βD2 ,

(1)

with α and β being parameters determined by experiments. For X-rays, D is considered to be the macroscopic dose delivered to the plasmid sample, whereas for ion irradiation, D is equivalent to the local dose deposited on each plasmid in the suspending medium, as will be explained in the next section. The fragment distribution FD (l) of plasmid DNA in its most general form can be expressed by FD (l) =

∞

f (l, m)p(m, n ¯ DSB ),

(2)

m=0

where l denotes the fragment length, m is the actual number of DSB(s) ¯ DSB as the mean value derived from the Poissonian p(m, n ¯ DSB ) with n depending on the macroscopic dose D. The fragment distribution comprises the sum of all possible conditional fragment densities f (l, m) depending on the number of induced DSB(s) m, weighted by the probability of actually inducing m DSB(s) according to the Poisson distribution with a mean value of n ¯ DSB . The conditional fragment density f (l, m) diﬀers for linear and supercoiled DNA. For the determination of the fragment distribution using linear plasmids, we can simply state the result of Ref. 8:  −¯nDSB e , l = L,   FD,lin (l) = (3) n ¯ n ¯ DSB l  e−l DSB L 2+n ¯ DSB 1 − , l < L. L L In the case of supercoiled DNA, we can easily transfer the considerations to calculate the fragment distribution:  ¯ DSB e−¯nDSB , l = L,  n (4) FD,sc (l) = n ¯ n ¯ 2DSB  e−l DSB L , l < L. L 3. Local Eﬀect Model The simulation of fragment distributions following heavy ion irradiation relies on the Local Eﬀect Model (LEM),9 which exploits the knowledge about the dose-response to photon irradiation. In particular, it states

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that the quantitative diﬀerence between photons and ions can be solely attributed to the diﬀerent dose distribution in the target. For a certain local dose inside the sensitive target, the response to ion irradiation is assumed to be the same as it would be for the same photon dose. In the case of plasmid fragmentation, the local dose distribution at the plasmid site is determined according to the random spatial distribution of ion tracks and the radial dose distribution around the trajectory of each single ion. From the average local dose deposited at the plasmid site, the average number of DSB(s) n ¯ DSB can be calculated [Eq. (1)] and used as an input parameter of the broken stick model [Eq. (2)] to determine the fragment length distribution. In the following, we will focus on the input parameters of the LEM, which is applied to simulate the plasmid fragmentation pattern. 3.1. Input from experimental data The three main ingredients of the LEM consist of the geometrical size of the sensitive target, the radial dose distribution of the traversing particles, and the DSB yield after X-irradiation. 3.1.1. Geometric target size For simplicity, the plasmid DNA is assumed to be densely packed within a sphere. Its volume (cross-sectional area of the DNA double strand times the plasmid length) is calculated and distributed within a sphere. In the calculations, the radius of this sphere (12 nm) is used as the radius of the sensitive area. 3.1.2. Radial dose distribution Experiments in gas chambers and Monte Carlo simulations show a 1/r2 dependence of the radial dose distribution. Although these experiments are not performed for liquid water, we can assume that this dependence also holds for the plasmid environment. We use the following representation of the physical radial dose proﬁle:  2  λ LET/rmin , r < rmin , rmin ≤ r ≤ rmax , D(r) = λ LET/r2 , (5)   0, rmax , where LET denotes the linear energy transfer, λ is a normalization constant to assure that the two-dimensional integral over the area perpendicular to the beam reproduces the LET. The parameter rmin = 0.3 nm describes the transition from the dose plateau in the track center to the

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1/r2 -behavior.10 rmax is the maximum radius determined by the range of the fastest electrons ejected due to primary interaction with the traversing ion. It has been experimentally found11 that rmax only depends on the energy E of the primary ion and can be parametrized by rmax = γ E δ ; γ = 0.062 µm(MeV/u)−1.7 , δ = 1.7, where rmax is in µm, and E in MeV/u. Additionally, indirect DNA damage due to radical diﬀusion must also be taken into account, since after dose deposition diﬀerent radicals (e.g., OH− ) are produced, which travel a certain distance and may ﬁnally react with the DNA molecule. We consider this eﬀect by convolving the physical dose distribution (5) with a two-dimensional Gaussian with a width σ (9 nm) according to the radical diﬀusion length in the considered buﬀer solution. 3.1.3. DSB yield for X-rays We already introduced Eq. (1) as the model for the DSB yield after photon irradiation. The α and β values are determined by measurements which are performed under the same conditions as the fragmentation experiments. The nonlinear β-term is the result of additional DSB(s) due to clustering of single strand breaks (SSB) and is important for high doses and buﬀer solutions with relatively low scavenging capacities.12

4. Experimental constraints for AFM For the comparison with measurements, we have to take several experimental constraints into account. Firstly, despite the high sensitivity of AFM, there is a certain detection limit l0 below which DNA fragments cannot be distinguished from background noise. Secondly, the determination of the fragment length is subject to a resolution limit σl determined by the image quality and the technique of length measurement. Therefore, we convolve the frequency distributions FD (l) with a Gaussian function with σl according to the spread at l = L for fragment lengths between L − 3σl < L < L + 3σl . Since the length is estimated by measuring the bent polymer fragments on the AFM image, not all twists of the molecules can be detected precisely, leading to an additional uncertainty. On the basis of the limited resolution, it is reasonable to collect the fragments of a certain size l + ∆l in bins, with ∆l being the bin size. Moreover, in the experimental data only fragment distributions normalized to the total number of detected fragments are accessible. For the theoretical frequency distributions, the same normalization needs to be applied, i.e. the integration of FD (l) for all detectable fragments from l0 to L taking into account the lower detection limit and the bin size.

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5. Results We applied our theoretical model to a realistic scenario of Φ × 174 plasmid DNA (supercoiled and linear) irradiated in aqueous solution with 4 MeV/u nickel ions (LET = 3950 keV/µm) accelerated at the UNILAC at GSI. For all calculations, the radius of the plasmid was set to 12 nm, and we use a radical diﬀusion length of 9 nm according to the lifetime of OH radicals in the 20 mM Hepes buﬀer solution used in the experiments.33,34 In Figs. 1(a) and (b) we show the frequency distributions after irradiation with nickel ions of four diﬀerent doses of 3, 300, 3000 and 30,000 Gy (0.0047, 0.47, 4.7, 47 × 108 particles/cm−2). The probabilities for l = 1 bp and l = 5386 bp are depicted with broader bars for better readability. For all doses, the probability of ﬁnding small fragments below 100 bp is greatly enhanced, this being the key feature of the distributions induced by heavy ion irradiation. Additionally, Fig. 1 shows that the amount of very small fragments

Fig. 1. (a) and (b) Theoretical fragment distributions for supercoiled and linear plasmids for four diﬀerent doses of 3, 300, 3000 and 30,000 Gy after irradiation with 3.9 MeV/u Ni. The bars at l = 0 and l = L are artiﬁcially broadened for better readability. The vertical line shows the lower detection limit below which fragments are not detectable. (c) and (d) Model prediction of fragment distributions including experimental conditions for supercoiled and linear DNA for the same doses as in (a) and (b). Model parameters: ∆l = 130 bp, l0 = 150 bp, σl = 400 bp.

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increases with dose. For supercoiled plasmids, the contribution of full-size fragments at l = L is highest for 3000 Gy due to the fact that one DSB is necessary to produce a linear plasmid. This diﬀers from the situation of initially linear plasmids shown in Fig. 1(b), where the probability of ﬁnding full-size plasmids decreases steadily with dose. However, for doses below 3000 Gy, it is rather close to unity. Comparing this large fraction of full-size plasmids to the high proportion of small fragments, we can assume that many plasmids remain intact and that the damage is produced by single tracks. This is a clear indication of the inﬂuence of the heterogeneous local dose distribution on the induction of clustered damage.7 5.1. Experimental constraints The situation is clearly diﬀerent when we take experimental constraints such as the lower detection limit [indicated by the vertical line in Figs. 1(a) and (b)], the limited resolution, the binning size and the diﬀerent normalization into consideration [see Figs. 1(c) and (d)]. Not only the quantitative values but also the qualitative shape of the distributions change drastically. Unfortunately, also the detectable diﬀerences in the distributions partly vanish, especially for the supercoiled form of DNA. Nevertheless, also in the experimentally adapted case, the enhanced probability of small fragments for large doses is still clearly visible for linear as well as for supercoiled DNA. For supercoiled plasmids, the distributions for 300 and 3000 Gy are almost the same, whereas for initially linear plasmids, the distribution can be nicely distinguished especially in the full length part and for small fragment sizes. The reason for the diﬀerence between the distributions with and without experimental constraints is mainly due to the two constraints of minimum detection size and the normalization. As can be seen from Fig. 1, the largest values of FD occur at fragment sizes below the detection limit of about 150 bp. Therefore, the largest fraction of fragments remains experimentally undetectable. Additionally, the diﬃculty to assess absolute values for FD,sc , which was accounted for by proper normalization, leads to similar results for supercoiled plasmids over a large range of dose values. The reason is that the shapes of the fragment distributions at diﬀerent doses are very similar despite a considerable variability of the absolute values. 5.2. Comparison with experimental data In order to investigate how well the Local Eﬀect Model including experimental conditions agrees with measured data, we compare the results for Φ × 174 plasmid DNA with a full length of 5386 bp irradiated with 3.9 MeV/u Ni ions at a dose of 2510 Gy. The samples were irradiated in a

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Fig. 2. Comparison of experimental and simulated fragment distributions of supercoiled plasmid DNA after irradiation with 3.9 MeV/u Ni ions (∆l = 150 bp, l0 = 150 bp, σl = 376 bp).

moderately radioprotective Hepes buﬀer (20 mM, pH 7.5) at a concentration of 100 ng µl−1 , and further diluted for AFM imaging. Usually, about 200 fragments are measured and distributed into typically 40 fragmentation bins. Therefore, frequencies much below 0.01 are experimentally not accessible and we decided to use a linear representation of the normalized frequency for the theoretical calculations with experimental constraints and the experimental data. As depicted in Fig. 2, we ﬁnd a good agreement with respect to the key characteristics of the high fraction of small fragments compared to a low number of full-size DNA. Also, the strong increase in frequency with decreasing fragment length correlates with experimental ﬁndings. However, the detected number of full-size linear plasmids is smaller than in the simulations, which might be attributed to additional damage due to the handling procedure. 6. Conclusions We adapted the broken stick model for the description of fragment distributions of plasmid DNA after X-irradiation and combined it with the

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ideas of the Local Eﬀect Model to obtain theoretical predictions for ion irradiation. The calculations show that the majority of fragments produced by ions exhibit a short length even for rather low doses. The theoretical distributions change signiﬁcantly if we consider the experimental constraints of the detection method, in our case, atomic force microscopy. We revealed that part of the diﬀerence due to radiation quality is disguised by the lower detection limit for the fragment length and the ﬁnite resolution. Our comparison of theoretical calculations including experimental constraints with measurements of fragment distributions after irradiation with Ni ions shows a reasonable agreement between our model and experimental data. In summary, the developed model provides a fast and easy means to simulate the fragmentation of plasmid DNA given the inevitable experimental constraints. The model can be applied to predict the distributions for the entire range of ions and energies. Due to its simplicity and low computational eﬀort, it would facilitate the investigation of the predicted outcome to ﬁnd the best ion species in order to elucidate the impact of track structure. In future work, we will concentrate on improving the geometrical structure of the plasmid molecule to get a more realistic representation of the target.13 Then, also an explicit modeling and distribution of DSB(s) onto the molecule will be important to account for the non-uniform breakage probability. Furthermore, we plan to include additional experimental constraints like the possible conversion of heat-labile sites of damage into additional double strand breaks during handling to further improve the agreement between experiment and model.

Acknowledgments This research was supported by the Marie Curie TOK COCOS grant (6th EU Framework Program under Contract No. MTKD-CT-2004-517186).

References 1. C. von Sonntag, in The Chemical Basis of Radiation Biology (Taylor & Francis, London/New York/Philadelphia, 1987). 2. K. M. Prise, M. Pinto, H. C. Newman and B. D. Michael, Radiat. Res 156, 572 (2001). 3. J. R. Milligan, J. A. Aguilera and J. F. Ward, Radiat. Res. 133, 151 (1993). 4. G. Taucher-Scholz and G. Kraft, Radiat. Res. 151, 595 (1999). 5. D. Pang, J. E. Rodgers, B. L. Berman, S. Chasovskikh and A. Dritschilo, Radiat. Res. 164, 755 (2005).

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6. K. Psonka, S. Brons, M. Heiss, E. Gudowska-Nowak and G. Taucher-Scholz, J. Phys. Cond. Matt. 17, S1443 (2005). 7. K. Psonka, E. Gudowska-Nowak, S. Brons, Th. Els¨ asser, M. Heiss and G. Taucher-Scholz, Adv. Space. Res. 39, 1043 (2007). 8. T. Radivoyevitch and B. Cedervall, Electrophoresis 17, 1087 (1996). 9. M. Scholz and G. Kraft, Radiat. Prot. Dosim. 52, 29 (1994). 10. Th. Els¨ asser and M. Scholz, Radiat. Res. 167, 319 (2007). 11. J. Kiefer and H. Straaten, Phys. Med. Biol. 31, 1201 (1986). 12. R. E. Krisch, M. B. Flick and C. N. Trumbore, Radiat. Res. 126, 251 (1991). 13. E. A. K¨ ummerle and E. Pomplun, Eur. Biophys. J. 34, 13 (2005).

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PART I

From Biomolecules to Cells and System Biology

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TOWARDS MONTE CARLO CALCULATIONS OF BIOLOGICAL DOSE IN HEAVY-ION THERAPY: MODELING OF NUCLEAR FRAGMENTATION REACTIONS IGOR PSHENICHNOV∗,†,§ , IGOR MISHUSTIN∗,‡ and WALTER GREINER∗ ∗

Frankfurt Institute for Advanced Studies, J.-W. Goethe University, 60438 Frankfurt am Main, Germany †

Institute for Nuclear Research, Russian Academy of Science, 117312 Moscow, Russia ‡ Kurchatov Institute, Russian Research Center, 123182 Moscow, Russia § [email protected]

We study the propagation of carbon nuclei in tissue-like media within a Monte Carlo Model for Heavy-Ion Therapy (MCHIT) based on the GEANT4 toolkit. The results of the calculations are compared with experimental data on depth–dose distributions and fragment ﬂuences obtained with water phantoms irradiated by 12 C beams. The MCHIT model describes well the depth–dose distributions and the attenuation of the primary beam due to nuclear fragmentation reactions, as well as the production of boron, beryllium and lithium fragments. However, the model should be improved with respect to the description of helium and hydrogen yields at diﬀerent depths. We also made predictions for the doses at large distances from the beam axis, which are expected mostly from secondary neutrons, protons and alpha particles. The MCHIT capability of calculating three-dimensional dose distributions is demonstrated. The distributions obtained separately for each kind of secondary fragments provide information for calculating biological dose distributions, as the biological eﬀects of ions depend on their charge, mass and energy. Keywords: Theory and algorithms in physics of heavy-ion therapy; Monte Carlo applications; simulations.

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1. Introduction A conformal irradiation of the tumor volume while sparing surrounding healthy tissues and organs at risk is possible with beams of protons and light nuclei.1 The positive clinical experience obtained with this kind of cancer therapy initiated the construction of several new centers which will provide therapeutic proton, helium and carbon-ion beams in Germany, France, Austria and Japan.1 Carbon ions have an additional advantage close to the Bragg peak, i.e. at the end of their range in tissues, due to their increased biological eﬀectiveness in killing tumor cells. Some carbon nuclei decay into fragments in nuclear reactions on their way to a deeply-seated tumor. Therefore, the amount of primary beam attenuation and the characteristics of secondary fragments should be properly taken into account in calculating therapeutic dose, which is delivered by mixed radiation ﬁelds from primary nuclei and secondary fragments. Several approaches for modeling the transport of light-ion beams in tissuelike materials were proposed, from a one-dimensional model based on semi-empirical approximations of fragmentation cross-sections2 to threedimensional Monte Carlo simulations with SHIELD-HIT,3 PHITS4 and FLUKA codes.5 In the present work, we calculate three-dimensional distributions of radiation ﬁelds from carbon nuclei and all secondary nuclear fragments in water. We use our Monte Carlo model for Heavy-Ion Therapy (MCHIT),6–8 which is based on version 8.2 of the GEANT4 toolkit.9

2. MCHIT model The electromagnetic interaction of charged particles is described within a set of GEANT4 models called “standard electromagnetic physics.”10,11 At each simulation step, the energy loss of a charged particle due to interaction with atomic electrons (ionization) is calculated according to the Bethe– Bloch formula with the mean excitation potential for water molecules set to 77 eV. Multiple Coulomb scattering of charged particles on atomic nuclei is also taken into account. The binary cascade model10,11 describes inelastic interactions of energetic hadrons and nuclei with nuclei of the phantom material. Nuclear remnants of various size, charge and excitation energy are produced in nucleus–nucleus collisions. The decay of excited nuclear remnants is simulated according to their excitation energy. Evaporation of nucleons, deutrons, tritons, 3 He and 4 He is a dominant de-excitation process at excitation energies of nuclear remnants below 3 MeV per nucleon.12 The Fermi

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break-up model is used to describe the explosive decay of light hot fragments up to 16 O with excitation energies above 3 MeV per nucleon.12 The decay of hot nuclear systems is characterized by a large number of open decay channels. This makes it diﬃcult to perform calculations on the basis of analytical formulas and approximations for the fragmentation cross-sections which can be based only on scarce experimental data. Alternatively, Monte Carlo models for nuclear fragmentation reactions are employed in MCHIT. More details on the physical models employed in MCHIT and their veriﬁcation with available experimental data on depth–dose distributions and yields of secondary fragments can be found in Refs. 6–8.

3. Comparison with experimental data on fragmentation of therapeutic beams The depth proﬁles of average lineal energy deposition per beam particle calculated with MCHIT for 200 and 400 A MeV 12 C in water are shown in Fig. 1. A water phantom of 40 × 15 × 15 cm was divided into 0.25 mm slices and energy deposition was calculated in each of the slices. The calculations were performed for a Gaussian beam proﬁle of 5 mm FWHM in the transverse plane to the beam direction. The energy spread of the beam was assumed to be Gaussian with the FWHM of 0.2% of the reported beam energy. The experimental data on relative ionization, which were recently obtained by Haettner et al.13 at GSI, are also shown in Fig. 1 for these two beam energies. The data were rescaled to the calculated absolute dose proﬁles at the ﬁrst data points located close to the beam entrance into the phantom. The MCHIT model successfully describes both the positions and the peak-to-plateau ratios of the depth–dose proﬁles for 200 and 400 A MeV 12 C ions in water. The normalized ﬂuences of primary carbon nuclei and secondary nuclear fragments were also calculated with MCHIT and presented in Fig. 1. The same characteristics measured in the experiment13 are plotted in Fig. 1 for comparison with the results of the calculations. As the ﬂuence distributions were measured for fragments emitted within 10o with respect to the beam axis,13 corresponding acceptance corrections were introduced in the calculations. The model successfully describes the observed attenuation of the carbon beam in water both at 200 and 400 A MeV. One can note that since only nuclear charge was detected in experiment, the measured ﬂuence distribution for carbon nuclei characterizes the mixture of primary 12 C and other carbon isotopes, mostly 9 C, 10 C and 11 C. Therefore, it is proper to say that more than 40% of primary 200 A MeV 12 C nuclei undergo fragmentation

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before they reach the Bragg peak position, while this fraction exceeds 70% for 400 A MeV 12 C beam. The yields of 10 C and 11 C suitable for PET monitoring in carbon-ion therapy were also calculated with MCHIT and compared with available experimental data.7 The calculated ﬂuences of secondary hydrogen, helium, lithium, beryllium and boron fragments produced by 400 A MeV 12 C in water are presented in Fig. 1 together with the corresponding experimental data.13 The production of secondary fragments is characterized by slowly increasing yields until primary ions reach the depth where the Bragg peak is located, and where they stop. Secondary fragments propagate further beyond the distal edge of the Bragg peak, and their yields after this depth are decreasing due to secondary fragmentation reactions on target nuclei. As boron fragments have the largest charge and inelastic cross-section among other produced elements, the boron ﬂuence drops faster. In contrast, H and He fragments are less attenuated and most of them leave the water phantom of 40 cm depth (see Fig. 1). It is found that B and Li ﬂuences are well described by the model, while Be ﬂuence is overestimated by MCHIT by ∼30%. The agreement with hydrogen and helium data is better, within ∼20% accuracy, and the calculated values strongly depend on the angular acceptance introduced in calculations.

4. Calculations of radiation fields from projectile nuclei and secondary fragments One can use the MCHIT model to calculate the contributions to the total depth–dose distribution from each kind of secondary fragments.6,8 Here, we consider three-dimensional dose distributions for fragments of each element. Such distributions for C, B, Be, Li, He and H are shown in Fig. 2 for a 300 A MeV 12 C beam of 3 mm FWHM in water. In such calculations, a water phantom of 30×30×30 cm was divided into 0.25 mm-thick slices, and volume energy deposition in MeV/mm3 per beam particle was calculated inside each of the slices as a function of the distance from the beam axis. Due to the multiple scattering of the primary carbon beam, it will be slightly broadened at the depth where the Bragg peak is located. However, the carbon beam is still well focused and there is no dose from beam nuclei at depths of more than 173 mm, i.e. beyond the distal edge of the Bragg peak. Only a very small dose from carbon nuclei is found well beyond the Bragg peak. This dose stems from recoil carbon nuclei produced by secondary fragments from oxygen nuclei in water. This small dose can be safely neglected.

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Fig. 2. Volume energy deposition in water from 300 A MeV 12 C with 3 mm FWHM as a function of the depth in phantom and the distance from beam axis. The contributions from carbon, boron, beryllium, lithium, helium and hydrogen fragments are shown in separate panels with diﬀerent dose scales used for diﬀerent elements.

The radial dose distributions for B, Be, Li, He and H fragments are very diﬀerent in shape from the dose distribution for the primary beam. Indeed, the secondary fragments, which are produced in nuclear fragmentation reactions of 12 C, have much wider distributions in angle Θ between their momentum and the beam axis. As a result, secondary fragments deliver doses at larger distances from the beam axis, especially close to the Bragg peak of 12 C beam. The shape of dose distributions from the heaviest fragments, e.g., B and Be, is diﬀerent from one for H fragments for several reasons. First, the Θ-distribution of H fragments produced in nuclear reactions is wider compared to B and Be. Second, light fragments

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deviate more from their initial trajectories due to the multiple Coulomb scattering on target nuclei compared to heavy fragments. This results in smearing of the dose peak for H fragments both in depth and in radius. 5. Doses at large distances from the beam axis

energy deposition (MeV/mm3)

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a 300 A MeV 12 C beam with 3 mm FWHM. As discussed in Sec. 4, these doses are delivered by secondary H fragments, mostly by protons. The doses from ternary recoil protons emitted in (n, p) reactions are also included. In contrast to highly ionizing charged particles, neutrons can travel large distances without losing their energy. The contribution from neutrons provides a dominant part of the total dose very far from the beam. The total dose in water phantoms due to secondary neutrons emitted by carbon ions has been calculated by several groups,6,14 and good agreement with data15 has been found. Here, we consider the distributions of such dose in depth and radius. As shown in Fig. 3(a), the dose at large radii of ∼100 mm is only 10−4 –10−5 of the dose at the Bragg peak. At large radii and depths, the dose weakly depends on the depth in the phantom, as it is similar at the Bragg peak and beyond it at 200 mm. As shown in Fig. 3(b), the doses calculated at 50 mm and 100 mm from the beam axis are almost independent of the depth if they are measured beyond the Bragg peak. These results can be directly veriﬁed with future microdosimetry measurements in water phantoms irradiated with carbon beams. 6. On calculations of the biological impact of mixed radiation fields The ultimate goal of radiation therapy is to kill most of the tumor cells while sparing healthy cells. Modern heavy-ion therapy planning procedures16–18 are based on the calculations of the surviving fraction of cells S(D)/S0 in each voxel after being irradiated with a certain total dose D: S(D) ¯ 2 ), = exp(−α ¯ D − βD S0

(1)

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and 6 1 ¯ β= βi D i . D i=1

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The coeﬃcients αi and βi depend on the linear energy transfer of particles of the kind i, and this dependence is speciﬁc for each kind of nuclear fragments and each kind of cells.16–18 As argued by Kase et al., who used GEANT4 for calculating physical and biological doses from carbon beams with the spread-out Bragg peak,18 their Monte Carlo simulations should be, in principle, more precise than their analytical one-dimensional calculations with the HIBRAC code. However, since the coeﬃcients αi and βi were not measured for each kind of fragment, it was assumed18 that they are all equal to those for the primary carbon beam: αi = αC , ¯ = αC , β¯ = βC . As follows from Secs. 4 βi = βC , and, therefore, α and 5, this simpliﬁcation is acceptable for the cells located in the beam sport, where the dose from carbon nuclei dominates. However, it would be more appropriate to assume α ¯ = αH and β¯ = βH beyond the distal edge of the Bragg peak and at large distances from the beam axis. We believe that future measurements of the radial dependence of the biological eﬀects of monoenergetic carbon ion beams will help to evaluate αi and βi and to validate further Monte Carlo models based on the GEANT4 toolkit with respect to the production of secondary fragments.

7. Conclusions The MCHIT model has been validated with experimental data13 for attenuation of 200 and 400 A MeV 12 C beams in water and for ﬂuences of secondary fragments at 400 A MeV. The ﬂuences of carbon and boron nuclei, which provide the major contribution to the total dose before and at the Bragg peak, are well described. The calculated total depth–dose distribution agrees well with the data13 before and after the Bragg peak. In the latter case, the contribution of Be, Li, He and H dominates. However, there is still room for improvement of the model in calculating the ﬂuences of these light fragments. It is demonstrated that the MCHIT model is capable of calculating three-dimensional dose distributions for the primary beam and secondary nuclear fragments. Therefore, the cell survival rates in mixed radiation ﬁelds can also be calculated providing that the coeﬃcients of the linear-quadratic model are known for nuclear fragments of each element.

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Acknowledgments This work was partly supported by Siemens Medical Solutions. We are grateful to Prof. Hermann Requardt and Dr. Thomas Haberer for discussions which stimulated the present study. We are indebted to Dr. Dieter Schardt for useful discussions and for providing us tables of experimental data. We thank Dr. Alexander Botvina for discussions concerning fragmentation models. References 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

U. Amaldi and G. Kraft, Rep. Prog. Phys. 68, 1861 (2005). L. Sihver, D. Schardt and T. Kanai, Japan. J. Med. Phys. 18, 1 (1998). I. Gudowska and N. Sobolevsky, Radiat. Prot. Dosim. 116, 301 (2005). K. Niita, T. Sato, H. Iwase, H. Nose, H. Nakashima and L. Sihver, Radiat. Meas. 41, 1080 (2006). F. Sommerer, K. Parodi, A. Ferrari, K. Poljanc, W. Enghardt and H. Aiginger, Phys. Med. Biol. 51, 4385 (2006). I. Pshenichnov, I. Mishustin and W. Greiner, Phys. Med. Biol. 50, 5493 (2005). I. Pshenichnov, I. Mishustin and W. Greiner, Phys. Med. Biol. 51, 6099 (2006). I. Pshenichnov, I. Mishustin and W. Greiner, in Proc. Int. Conf. on Nuclear Data for Science and Technology, Nice, France, April 23–27, 2007 arXiv:0705.0248 [physics.med-ph]. GEANT4, http://geant4.web.cern.ch/geant4/. S. Agostinelli et al. (GEANT4 Collaboration), Nucl. Instrum. Meth. A 506, 250 (2003). J. Allison et al. (GEANT4 Collaboration), IEEE Trans. Nucl. Sci. 53, 270 (2006). J. P. Bondorf, A. S. Botvina, A. S. Iljinov, I. N. Mishustin and K. Sneppen, Phys. Rept. 257, 133 (1995). E. Haettner, H. Iwase and D. Schardt, Rad. Prot. Dosim. 122, 48 (2006). I. Gudowska, M. Kope´c and N. Sobolevsky, Rad. Prot. Dosim. 126, 652 (2007). K. Gunzert-Marx, D. Schardt and R. S. Simon, Rad. Prot. Dosim. 110, 595 (2004). J. J. Wilkens and U. Oelfke, Phys. Med. Biol. 51, 3127 (2006). M. Kr¨ amer and M. Scholz, Phys. Med. Biol. 51, 1959 (2006). Y. Kase, N. Kanematsu, T. Kanai and N. Matsufuji, Phys. Med. Biol. 51, N467 (2006). A. M. Kellerer and H. H. Rossi, Rad. Res. 75, 471 (1978). M. Zaider and H. H. Rossi, Rad. Res. 83, 732 (1980).

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MECHANISMS OF RADIATION DAMAGE OF BIOMOLECULES E. SURDUTOVICH Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48202, USA [email protected] O. I. OBOLENSKY∗ , I. PSHENICHNOV, I. MISHUSTIN, A. V. SOLOV’YOV∗,† and W. GREINER Frankfurt Institute for Advanced Studies, Ruth-Moufang Str. 1, 60438, Frankfurt am Main, Germany † [email protected] We discuss diﬀerent physical processes which may be responsible for biological damage induced by an ion beam propogating through living tissue. The creation of free radicals and an electron plasma, dissociative attachment of low-energy electrons, and local heating mechanisms are considered. Numerical estimates are performed for the case of carbon ion beams. We conclude that all three mechanisms are capable of inﬂicting single and double strand breaks in DNA molecules and thus can be responsible for the observed enhanced biological eﬀectiveness of ion beams in cancer therapy. Keywords: Ionizing radiation; heavy ion beam; cancer therapy; DNA damage; energy loss; dosimetry assessment.

1. Introduction In this contribution, we give a brief overview of physical processes which may be responsible for inﬂicting biological damage when an ion beam propogates through a living tissue. Namely, we consider creation of free radicals and an electron plasma, dissociative attachment of low-energy electrons, and local heating mechanisms of DNA damage. We also perform numerical estimates characterizing the eﬃciency of these mechanisms. Such an ∗ On

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analysis represents a ﬁrst step in reaching our ﬁnal goal: establishing a quantitative connection between the amount of energy deposited by an ion into the tissue and the induced biological consequences. There are three basic approaches to treating malignant tumors. The most eﬃcient and widely used method is surgery, i.e. direct removal of the tissues aﬀected by the cancer. It is, however, an invasive procedure, which is moreover not always possible. Administering drugs which prevent mitosis (cell division) or cause apoptosis (cell death), known as chemotherapy, frequently leads to severe side eﬀects inherent to non-speciﬁc actions of the drugs on the cells in the organism. The third approach is radiation therapy — killing the cancer cells by depositing enough energy into the tumor. The side eﬀects of this method are associated with damaging healthy tissues due to poor focusing of the deposited energy. Commonly, radiation therapy is associated with irradiation by MeVphotons, conventionally called “X-rays” by medical doctors. The weak point of photons is that their energy deposition proﬁle has a wide smooth maximum near the surface of the tissue. So, if a deeply embedded tumor is to be irradiated, one has to resort to sophisticated techniques of using many beam directions (Intensity Modulated Radiation Therapy, IMRT) in order to avoid severe damage to healthy tissues in the entrance channels. Another way to deliver energy into a precisely determined location inside a medium is to employ the phenomenon discovered in 1903 by W. H. Bragg who observed a peak in the energy loss of alpha particles at the very end of their trajectory. This peak is nowadays referred to as a Bragg peak. In 1946, R. Williams suggested that ion beams could be used to kill cells in malignant tumors. The ﬁrst clinical trials began in 1954 at the Berkeley cyclotron; however, the ﬁrst hospital-based center for proton therapy became operational only in 1992. Currently, there are about a dozen such centers around the world. Simultaneously, other ions were investigated. It has eventually been concluded that carbon ions are generally better suited than protons for treating cancers. The advantages of carbon ions are associated with a larger relative dose of energy deposited at the Bragg peak, smaller divergence of the beam, ability to control the position of the Bragg peak with positron-emission tomography (PET) and an enhanced biological eﬃciency, which refers to the fact that carbon ions kill tumor cells with a greater probability than protons, even at equal energy deposition rates. More details on the history, current status and comparison of proton and heavy ion therapies can be found in the review articles.1–3 To increase the eﬃciency and to reduce the side eﬀects of ion therapy, one needs to be able to very accurately predict the distribution of the damages in the tissue and the minimum biologically eﬀective dose. One

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of the most diﬃcult unsolved problems in this study concerns establishing a quantitative connection between the amount of energy deposited by an ion into the tissue and the induced biological consequences. In order to approach this problem, one needs a clear understanding and a quantitative description of a chain of events at the atomic and molecular level leading to irreparable damage of biological systems. When an energetic charged projectile enters a medium, it experiences a number of atomic and nuclear interactions. Nuclear processes, mainly fragmentation reactions, result in the transformation of the beam particles to new species. The nuclear processes can be very reliably modeled with the use of Monte-Carlo simulations.4,5 Atomic processes, such as elastic scattering, scattering with excitation, ionization, and multiple Coulomb scattering, are much more probable but less destructive events, occuring very frequently along the projectile’s trajectory. Ionization of host atoms is the main channel of energy loss by the projectile. This process is actually even more important for biological applications since it results in a shower of secondary electrons, which may inﬂict more damage than the projectile itself. In the next section, we consider the behavior of the ionization crosssection as a function of the energy of the projectile and the kinetic energy of the ejected electron. The data obtained for the parameters typical for cancer therapy are analyzed and numerical estimates characterizing eﬃciency of the mechanisms possibly responsible for DNA damage are carried out in Sec. 3.

2. Ionization cross-section The ﬁrst goal of our work is a description of the passage of a single ion through the biological medium. The medium is approximated by liquid water, which is the major component of all living tissues. The ion typically enters the medium with a sub-relativistic speed and then it gradually slows down because of energy losses due to inelastic processes, such as ionization and excitation of the medium. Inelastic cross-sections are typically small at the beginning of the trajectory (at higher velocities) but then increase and reach a maximum at the Bragg peak. In the vicinity of this maximum, the ion loses its energy at the highest rate. For even smaller velocities, the cross-section drops because of kinematical constraints. For simplicity, we include in our analysis only impact ionization, which is the dominant process contributing to the overall energy loss and most importantly resulting in the production of secondary electrons. In this work, we also limit our consideration to non-relativistic velocities of the

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projectiles; this is done for simplicity. However, the relativistic eﬀects should be taken into account for 400 Mev/n ions and at this point it is hard to estimate the errors caused by this limitation. An experiment studying such a process measures the energy and momentum of secondary electrons produced in this interaction. Therefore, the quantity of interest is the probability of observing N secondary electrons with kinetic energy in the interval dW around W , emitted from a segment ∆x of the trajectory of a single ion at the depth x corresponding to the kinetic energy of the ion, T , within the solid angle dΩ. This quantity is proportional to the doubly-diﬀerentiated cross-section (DDCS): d2 σ(W, T ) d2 N (W, T ) = n∆x , dW dΩ dW dΩ

(1)

where n is the number density of water molecules (at standard conditions n ≈ 3.3 × 1022 cm−3 ). The DDCS, d2 σ(W, T )/dW dΩ, carries all necessary physical information about the interaction of the ion with water molecules causing ionization of the latter and producing secondary electrons with the energy of interest. For our current analysis, the angular distribution is not needed. Therefore, we integrate over the solid angle and DDCS is replaced with the singly-diﬀerentiated cross-section (SDCS). Thus, the total impact ionization cross-section, diﬀerentiated in secondary electron kinetic energy, dσ(W, T )/dW , becomes the basic quantity in our analysis. Besides the kinetic energy of secondary electrons and the properties of water molecules, the SDCS depends on the kinetic energy of the projectile T and its charge z. We use a semi-empirical Rudd’s expression6 for the SDCS, which is a parametric adjustment combining the experimental data, the plane wave Born approximation, and other theoretical models. It can be written in the following form: 4πa0 Ni R 2 F1 (vi ) + F2 (vi )ωi dσ(W, T ) 2 , =z 3 dW I I i i (1 + ωi ) (1 + exp(α(ωi − ωimax )/vi )) i (2) where the sum is taken over the electron shells of the water molecule, a0 = 0.0529 nm is the Bohr radius, R = 13.6 eV is Rydberg, Ni is the shell occupancy, Ii is the binding energy of the shell, ωi = W/Ii is the dimensionless normalized kinetic energy of the ejected electron, and vi is the dimensionless normalized velocity, given by mT vi = . (3) M Ii

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m The combination M T reveals that vi is actually constructed from the kinetic energy of an electron having the same velocity as the projectile (M is the projectile’s mass, m is the mass of electron). Furthermore, F1 and F2 in Eq. (2) are given by C1 v D1 ln(1 + v 2 ) + (4) F1 (v) = A1 B1 /v 2 + v 2 1 + E1 v D1 +4 and A2 v 2 + B2 F2 (v) = C2 v D2 . (5) D C2 v 2 +4 + A2 v 2 + B2 The cut-oﬀ energy ω max is given by R ωimax = 4v 2 − 2v − . (6) 4Ii The ﬁrst term on the right-hand side represents the free electron limit, the second term represents a correction due to electron binding, and the third term gives the correct dependence of the SDCS for v 1.6 The SDCS for water as a function of projectile’s and secondary electron’s energy is plotted in Fig. 1. The corresponding ﬁtting parameters A1 , . . . , E1 , A2 , . . . , D2 , α are given in Tables 1 and 2. Integration of the singly-diﬀerentiated cross-section over secondary electron energy W gives the total cross-section of impact ionization by the ion energy T : W max dσ(W, T ) dW. (7) σ(T ) = dW 0

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Table 1. Fitting parameters for the three outer shells of a water molecule with the ionization potentials I1 = 12.61 eV, I2 = 14.73 eV and I3 = 18.55 eV. A1

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The results of this integration for 12 C6+ ions are shown in Fig. 2. The next important characteristic that we can obtain from the SDCS is the average energy of the secondary electrons. It is given by max 1 W dσ(W, T ) W dW. (8) Wave (T ) = σ 0 dW

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Fig. 3. Average energy of secondary electrons produced as the result of impact ionization as a function of kinetic energy of 12 C6+ ions.

The results of this integration are shown in Fig. 3. It is worth noting that on average, the secondary electrons are not very energetic, about 50–60 eV, and the average energy levels out as the energy of the projectile increases. This fact is important for estimating the avalanche due to secondary electrons. The stopping cross-section is deﬁned as Wimax dσi (W, T ) dW. (9) (W + Ii ) σst (T ) = dW 0 i This gives the average energy lost by a projectile in a single collision, which can be further translated into energy loss within an ion’s trajectory segment, dx: dT (T ) = −nσst (T ). (10) dx This quantity is also known as the linear energy transfer (LET), one of the most important quantities in radiology. Why did we not use the Bethe–Bloch formula to calculate LET in one step? The reason is that the average ionization energy used in the Bethe– Bloch formula is not very well deﬁned for water. At the same time, the parameters used in our approach are more reliable and, in principle, can be modiﬁed, given diﬀerent media. The LET found from Eq. (10) depends on the energy of the ion rather than the ion’s penetration depth in the medium. The dependence of LET

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(and other quantities) on the penetration depth, however, is much more convenient for cancer therapy applications. To obtain such a dependence, one can integrate (10): T0 dT . (11) x(T ) = |dT /dx| T Here T0 is the initial energy of the projectile. The dependence x(T ) for carbon ions is shown in Fig. 4. This ﬁgure clearly shows the Bragg peak, a sharp maximum of energy loss at the end of the range. The energy loss is very high at the top of the peak (not shown to maintain the clarity of the representation of the overall behavior). In fact, the LET reaches 3600 MeV/mm which is 36 times that of the commonly accepted value of about 100 MeV/mm. This factor is exactly the squared charge of the carbon ion which enters as a scaling factor in the ionization cross-section. Therefore, the discrepancy is explained by the fact that as the ion approaches the Bragg peak, it captures electrons and the eﬀective charge at the end of the trajectrory is close to 1. We also note that despite seemingly excellent agreement between the theoretical curve and the experimental points in Fig. 4, the depth of the Bragg peak predicted within the Rudd’s model signiﬁcantly exceeds the observed values. The good agreement between the theory and experiment in Fig. 4 is due to the fact that the Rudd’s model

Fig. 4. Linear energy transfer of 12 C6+ ions versus the depth of their penetration into the bulk of water. T0 = 280 MeV per nucleon. The dots represent the data from Ref. 1.

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works well in the middle and at the end of the trajectory, at lower ion energies, while the experimental values of the LET are actually given in Ref. 1 versus distances remaining till the Bragg peak. Rudd’s model significantly underestimates the ionization cross-sections at higher ion energies, which leads to greater penetration length. We attribute this inaccuracy to the neglect of multiple ionization processes, loss of the projectile’s energy in inelastic scattering on nuclei, and to the relativistic eﬀects in the beginning of the ion trajectories. This inaccuracy, however, does not inﬂuence our analysis since we are interested in the Bragg peak area where the main biological damage takes place. Using the relationship (11), all quantities of interest can be mapped along the trajectory of the ion. One of these quantities is the number of secondary electrons with kinetic energy W , produced by a single ion on the interval ∆x at the depth x corresponding to a certain kinetic energy T of the ion. This quantity is a product of the remapping of Eq. (1) integrated over the angles; it can be written as follows: dσ(W, x) dN (W, x) = n∆x . dW dW

(12)

Four curves shown in Fig. 5 show this dependence at diﬀerent energies or depths of the ion. This number, however, represents only the secondary electrons produced by the projectile. In order to determine the total number of electrons in the avalanche, one has to analyze the ionizing capabilities of these secondary electrons. To address this issue, we calculate the fraction of secondary electrons with energies higher than the threshold of ionization produced in a single collision: 1 φi = σ

Wmax

Ii

dσ(W , T ) dW . dW

(13)

The dependencies for diﬀerent electronic shells are shown in Fig. 6. These results are somewhat surprising since they indicate that the fraction of secondary electrons “capable of further ionization” decreases with increasing energy of the ion. This is probably a result of averaging: even though higher energetic secondary electrons are produced, there are many more lower energy ones. The fraction of secondary electrons capable of further ionization is substantial for the electrons created by the projectile, even at relatively low projectile energies. However, the further secondary electrons are still less energetic, so that further ionizations are much less probable. This is supported by calculations using the Rudd’s model for incident

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Fig. 5. Number of secondary electrons with kinetic energy W , produced by a single ion in a 1-nm interval versus kinetic energy of the secondary electrons.

Fig. 6. Fraction of secondary electrons produced by 12 C6+ capable of further ionization. Diﬀerent lines indicate diﬀerent electronic shells of the water molecule.

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electrons. However, it can also be explained by a simple estimate: Figure 3 indicates that the average energy of the secondary electrons in the vicinity of the Bragg peak is about 35 eV. The maximum average energy that can be transferred to the further secondary electrons is just (35 − Ii )/2, which is just over 12 eV for the outermost electron, an energy barely enough to cause further ionization. Therefore, we conclude that there is no avalanche of secondary electrons and that Eq. (12) gives a reasonable estimate of the number of secondary electrons, within a factor of two.

3. Numerical estimates of eﬃciency 3.1. Free radicals and electron plasma Let us now estimate the number density of the electron plasma produced by an ion. The crucial quantity for this estimate is the penetration length of secondary electrons after they have been created by the projectile. Direct experimental observations of penetration depths of electrons in liquid water are scarce. There are values recommended by the International Commission on Radiation Units and Measurements for 10–1000 keV electrons,7,8 and there are data for thermalization lengths of “subexcitation electrons” for lower energies, 0–4 eV.9 However, there is a gap in experimental data for electron energies between few eV and several hundred eV, the most interesting ones from the viewpoint of cancer therapy applications. The average energy of secondary electrons in the Bragg peak area (T ∼ 0.1 MeV per nucleon) is about 30 eV (see Fig. 3). Even though there are no direct experimental values for the penetration lengths of electrons in this energy range, interpolation of existing experimental data points10 and theoretical predictions based on Monte-Carlo simulations11 suggest that the penetration lengths of such electrons are on the order of 10 nm or less. The next important question which needs to be addressed before any estimates can be done is the following: Is there any collective action of ions in the beam or the secondary electrons produced by the ions which leads to biological damage? Or does every ion act independently and the total biological eﬀect equals the sum of the damages from individual ions? Comparison of electron thermalization lengths and typical distances between the ions in the beam leads us to conclude that the latter situation takes place under realistic conditions and ion tracks do not interact, as schematically shown in Fig. 7. In order to get realistic “order of magnitude” estimates we will assume that the energy deposited by an ion is thermalized within a tube of a radius r ∼ 10 nm; see Fig. 8 for a schematic view.

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Fig. 7. Schematic view of an ion beam cross-section. The red dots represents ion tracks. The rings surrounding the ion tracks represent the areas in which the energy deposited by the ion is distributed.

Fig. 8. Schematic view of an ion track. The ion’s trajectory is represented by a solid line. The volume in which host atoms are ionized and in which the ion’s energy is deposited is schematically shown as a cylinder of radius r. Schematic trajectories of secondary electrons are also shown.

The total ionization cross-section for carbon ions 12 C6+ of kinetic energy T ∼ 0.1 MeV per nucleon is about σ ≈ 180 × 10−16 cm2 = 1.8 nm2 . The majority of secondary electrons are produced in the process 12

C6+ + H2 O →12 C6+ + H2 O+ + e− .

(14)

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The number of electrons produced per one nanometer of the ion’s trajectory is then nσ = 33 nm−3 · 1.8 nm2 ≈ 60 nm−1 . This quantity is easily translated into the number density of the electron plasma: nσ 60 nm−1 ≈ ≈ 0.2 nm−3 = 2 × 1020 cm−3 . (15) 2 πr π100 nm2 Both secondary electrons and ionized water molecules react to form the radical ions ne =

H2 O+ → H+ + OH·, −

−

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(16) (17)

Our estimate, therefore, shows that passage of a single ion can create a high density of free radicals. It goes beyond the scope of the present work to discuss the further chemical reactions between the free radicals and the DNA molecules. For such a discussion, we refer to a comprehensive review.12 3.2. Dissociative recombination by low energy electrons Traditionally, it was believed that non-thermalized secondary electrons cannot cause any signiﬁcant genotoxic damage, unless they are already solvated in water and can participate in chemical reactions of the type in Eq. (17) or unless they have enough kinetic energy. The singly-diﬀerentiated crosssection, however, falls rapidly with energy of the ejected electron and therefore energetic secondary electrons are not abundant, cf. Figs. 1 and 3. In 2000, Leon Sanche and co-workers published a paper13 in which they argued that “abundant free secondary electrons with ballistic energies between 1 and 20 electron volts . . . induce substantial yields of singleand double-strand breaks in DNA, which are caused by rapid decays of transient molecular resonances localized on the DNA’s basic components.” In their experiments a monoenergetic electron beam at various incident energies was used to irradiate solid DNA samples. They observed that one electron out of 1000–2000 (depending on the electron energy) induced a single strand break and one electron out of 5000–10,000 induced a double strand break of a DNA molecule.14 This type of experiments, however, cannot be directly related to the problem of DNA damage induced by ions. The diﬀerence is in the density of free electrons. In Sanche’s experiments, the electron density was on the order of 104 cm−3 , 16 orders of magnitude lower than is created by an ion in the Bragg peak area, as is estimated in the previous section. One can nevertheless estimate the average number of electrons created in a volume occupied by one convolution of the DNA molecule. The DNA

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radius is about 1 nm, while the convolution (linear) length is about 4 nm, so that the volume of interest is about VDNA = 10−20 cm3 . Assuming that at least two electrons are needed in the volume of one DNA convolution to produce a double strand break, one can estimate that the required electron density is 2 = ≈ 2 × 1020 cm−3 . (18) ncrit e VDNA It is remarkable that exactly such a density has been predicted by our simple estimates in the previous section. 3.3. Local heating Finally, let us estimate the local heating produced in a tube of 10 nm radius by a carbon ion of 0.1 MeV per nucleon. The simplest way to do that is to use the thermodynamic equation, Q = µc∆T, (19) relating the heat Q transferred to a system with the system’s mass µ, speciﬁc heat capacity c (4.2 J/g K for water) and the increase of the system’s temperature ∆T . ∆T can then be found from the linear energy transfer Q/∆x (a typical value for 12 C ions is 100 MeV/mm = 1.61010 J/cm in the Bragg peak region), the tube’s radius r and the density of water ρ = 1 g/cm−3 : (20) Q = ρπr2 ∆xc∆T, Q/∆x . (21) ρπr2 c Substituting the numerical values, one obtains ∆T ∼ 10 K for r = 10 nm and ∆T ∼ 100 K for the lower estimate of the tube’s radius, r = 3 nm. This estimate shows that the local heating within the electron thermalization radius along the ion track can be quite substantial. The melting temperature of DNA molecules, is about 85◦ C. Therefore, in our opinion, there is a need for a more thorough study of the local heating mechanism, which would include investigation of such problems as the detailed description of thermalization of secondary electrons in water, heat transfer in water, and modeling of DNA dynamics under local heating. To the best of our knowledge, this local heating mechanism has not yet received suﬃcient attention. ∆T =

Acknowledgments This work is partially supported by the European Commission within the Network of Excellence project EXCELL and by the EU project PECU.

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References 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14.

A. Amaldi and G. Kraft, Rep. Prog. Phys. 68, 1861 (2005). E. Pedroni, Europhys. News 31, 18 (2000). M. Goitein, A. Lomax and E. Pedroni, Phys. Today 55, 45 (2002). I. Pshenichnov, I. Mishustin and W. Greiner, Phys. Med. Biol. 50, 5493 (2005). I. Pshenichnov, I. Mishustin and W. Greiner, Phys. Med. Biol. 51, 6099 (2006). M. E. Rudd, Y.-K. Kim, D. H. Madison and T. Gay, Rev. Mod. Phys. 64, 441 (1992). International Commission on Radiation Units and Measurements, Bethesda, MD, Report 37 (1984); [www.icru.org]. M. J. Berger, J. S. Coursey, M. A. Zucker and J. Chang, National Institute of Standards and Technology, ESTAR database, http://physics.nist.gov/PhysRefData/Star/Text/contents.html. V. V. Konovalov, A. M. Raitsimring and Yu. D. Tsvetkov, Radiat. Phys. Chem. 32, 623 (1988). D. E. Watt, Quantities for Dosimetry of Ionizing Radiations in Liquid Water (Taylor & Francis, London, 1996). J. Meesungnoen, J.-P. Jay-Gerin, A. Filali-Mouhim and S. Mankhetkorn, Rad. Res. 158, 657 (2002). C. Chatgilialoglu and P. O’Neill, Exp. Gerontol. 36, 1459 (2001). B. Boudaiﬀa, P. Cloutier, D. Hunting, M. A. Huels and L. Sanche, Science 287, 1658 (2000). M. A. Huels, B. Boudaiﬀa, P. Cloutier, D. Hunting and L. Sanche, J. Amer. Chem. Soc. 125, 4467 (2003).

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ON MODELING OF GENE EXPRESSION PATTERNS IN THE DROSOPHILA EMBRYO BY THE GENE CIRCUIT METHOD ALEXANDER M. SAMSONOV∗ and V. V. GURSKY, The Ioﬀe Institute of the Russian Academy of Sciences, St. Petersburg 194021, Russia ∗ [email protected]ﬀe.ru K. N. KOZLOV The State Polytechnical University, St. Petersburg 195251, Russia J. REINITZ University of Stony Brook, NY 11794, USA We review some recent results of modeling the pattern formation by segmentation genes during the early development of the Drosophila embryo. The study of gene expression patterns is based on the “gene circuit” method consisting of four steps: obtaining gene expression experimental data, formulating a model, ﬁtting the model to the data, and inferring new biology from the analysis of results. The biological data has the form of processed images of immunostained embryos and is adopted in the form of concentration curves for proteins coded by various segmentation genes averaged over many embryos. The model is formulated as deterministic reaction-diﬀusion equations with protein concentrations in many cell nuclei as state variables. The values of parameters in the model are calculated by ﬁtting the solution of model equations to the experimental concentration curves. We also describe how the gene circuit approach allows one to elucidate a role in the pattern formation played by nuclear cleavages in the developing embryo. Keywords: Mathematical biology; systems biology; pattern formation; reaction-diﬀusion; Drosophila.

1. Introduction In modern genetics, the regulatory interactions between genes in multicellular organisms are inferred by comparisons of mutant and wild type 426

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phenotypes. An important example of use of this method is the deduction that virtually all of the pair-rule class of segmentation genes in the fruit ﬂy Drosophila melanogaster are regulated by members of gap gene class. The validity of a model of the regulatory mechanism can be established only by keeping track of all regulatory inputs to a speciﬁc gene (cf. Ref. 1). Current experimental approaches, however, are limited in their ability to monitor regulatory contributions simultaneously, as such interactions are inferred from mutant expression patterns and it is rarely possible to obtain mutants in more than three genes. Moreover, mutant regulatory systems by deﬁnition have an incomplete or otherwise defective set of regulatory interactions. For these reasons, the regulatory structure of the wild-type network must be assembled on the basis of evidence from diﬀerent experiments. The consistency of such an inferred mechanism can be established conclusively only by testing it in the intact and complete developmental system. Finally, there is a fundamental issue concerning completeness of a proposed regulatory mechanism, which cannot be addressed by experimental approaches alone. The fact that all maternal and gap genes are necessary for correct gap gene expression does not prove that they are also suﬃcient. It is impossible to prove suﬃciency of the inferred mechanism without reconstituting the system ab initio, using only well-deﬁned ingredients. Such a reconstitution is not possible by contemporary experimental methods and hence has to be attempted by using mathematical modeling and simulations. The problems illustrated above show that to establish consistency, uniqueness, and completeness of a regulatory mechanism, we need a method that allows us to reconstitute wild-type gene expression patterns in silico, infer underlying regulatory interactions from these wild type patterns, and keep track of all regulatory interactions in all nuclei at all times. Among the current methods of mathematical modeling of biological systems, we have chosen one in which concentrations of products of gene expression are assumed to be continuous functions of time and space in spatially extended systems. The corresponding models are based on systems of nonlinear ordinary or partial diﬀerential equations (PDEs). The gene circuit method1–4 aims to extract information about dynamical regulatory interactions between transcription factors from given gene expression patterns. This is achieved in four steps: (i) formulation of a mathematical modeling framework, (ii) image processing and collection of gene expression data, (iii) ﬁtting of the model to expression data to obtain regulatory parameters, and (iv) biological analysis of the resulting gene network.

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The ability of the gene circuit method to interpret and predict regulatory mechanisms has been used as an example of the segment determination system in Drosophila. We have selected this process because of its biological importance, and also because computational investigations of gene regulation can be done in an exceptionally accurate way in this system. In this short review, we aim to demonstrate the ability of the method to interpret and predict regulatory mechanisms using as an example the segment determination system in the fruit ﬂy Drosophila.

2. Formulation of the problem The Drosophila blastoderm permits exceptionally precise modeling, since pattern formation results from regulatory interactions. As a rule, well characterized genetics alone does not provide enough information to model and understand the system behavior. The blastoderm is a very important exception to this rule. Segmentation gene mutations do not cause any morphological defects before the onset of gastrulation.5,6 Thus, the internal state of each nucleus can be described by concentration levels of transcription factors encoded by segmentation genes. The segmentation genes have been cloned, and, hence, their level of expression can be recorded by antibody methods. In addition, there is no tissue growth, therefore, no consideration of intercellular signaling since nuclei are not yet surrounded by membranes during the syncytial blastoderm stage.7 These properties allow one to understand important aspects of developmental genetics in detail.

2.1. Model and method description The model has been described in detail in Refs. 1 and 2. In the presumptive segmented germ band, segmentation gene expression may be described as a function of A–P position to a good level of approximation, and therefore the model considers the one-dimensional line of nuclei moving laterally along the A–P axis. In most cases, each nucleus was explicitly represented and numbered with an index i from anterior to posterior. The model has three rules governing the behavior of nuclei in time t: (i) interphase, (ii) mitosis, and (iii) division. Rules (i) and (ii) describe the dynamics of protein synthesis and decay within a nucleus and protein diﬀusion between nuclei. Rule (iii) is discrete and describes how each nucleus is replaced by its two daughter nuclei upon division. The schedule8 for these rules is summarized in Fig. 1. The internal state of nucleus i is described by concentrations via of transcription factors encoded by segmentation genes denoted by index a.

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Fig. 1. The gene circuit method. (A) The basic principle. Regulatory interactions are inferred from wild-type expression patterns by fitting gene circuit models to quantitative data. (B) Time schedule for gap gene networks. The model spans the time from the onset of cycle 13 (0.0 min) to the onset of gastrulation at the end of cycle 14A (71.1 min). The three rules of the model (interphase, mitosis and nuclear division) are shown on the right. There is one time class in cycle 13 (C13), and eight time classes (T1–T8) in cycle 14A. Time points used for comparison of model output to data for time classes C13 and T1–T8 are indicated. (C) The regulation-expression function g(u). Total regulatory input u is shown on the horizontal axis. Corresponding relative activation of protein synthesis g(u) is shown on the vertical axis. g(u) rapidly approaches saturation for values of u above 1.5, and rapidly approaches zero for values of u below −1.5 (dashed lines). (D) Regulatory interactions within a gene network are represented by the genetic interconnection matrix T (shown here for interactions of hb, Kr, gt and kni). More details are in text.

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The change in transcription factor concentration over time dvia /dt depends on three processes during interphase: (i) protein synthesis, (ii) protein diffusion, and (iii) protein decay, represented by the summation terms on the right-hand side of the following equation: N dvia = Ra g T ab vib + ma viBcd + ha dt b=1

a

a a − via ) + (vi+1 − via )] − λa via , + D (n)[(vi−1

(1)

where a = 1, . . . , N , and N is the total number of zygotic genes in the model. Here, T ab represents an interaction matrix, i.e. the matrix of regulatory coeﬃcients T ab which characterize the regulatory eﬀect of the product of gene b on the expression of gene a. This matrix is independent of i, reﬂecting the fact that each nucleus contains a copy of the same genome. The value viBcd is the time-independent concentration of Bcd maternal protein in nucleus i. The regulatory eﬀect of Bcd on gene a is represented by the parameter ma . Parameter ha is a threshold representing regulatory contributions of uniformly expressed maternal transcription factors. The relative rate of protein synthesis is given by the sigmoid regulation-expression function g(ua ) = 12 [(ua / (ua )2 + 1)+1], where N ua = b=1 T ab vib + ma viBcd + ha is the total regulatory input on gene a. The graph of g is given in Fig. 2. The maximum synthesis rate for the product of gene a is given by Ra . The diﬀusion coeﬃcient Da (n) depends on the number of nuclear divisions n that have taken place before the current time moment t by the law Da (n) = 4Da (n − 1), reﬂecting that 1.0 0.8 0.6 g 0.4 0.2 0.0 − 10

Fig. 2.

−5 0 5 Total regulatory input

10

The graph of regulation-expression function g(·).

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the diﬀusion coeﬃcient is assumed to vary inversely with the square of the distance between neighboring nuclei and this distance is halved upon nuclear division. The value λa is the decay rate of the product of gene a a and is related to the protein half-life τ1/2 of the product of gene a by a a τ1/2 = ln 2/λ . Equation (1) is written in a form corresponding to the interphase stage. During the mitosis stage, the equations have the same form but without the synthesis term (Ra are set to zero). Finally, the nuclear divisions are modeled as an instantaneous doubling of the number of nuclei at the end of each mitotic stage. Proteins are the only regulatory molecules considered in (1) synthesized by segmentation genes; there is no evidence for a direct role of RNA in the regulation of zygotic segmentation genes. The nuclear structure in (1) can be mathematically abolished by rewriting the equations for a spatial continuum, giving the PDE form of (1), which is written ∂ 2 v a (x, t) ∂v a (x, t) = Ra (t)g(ua ) − λa v a (x, t) + Da , ∂t ∂x2

a = 1, . . . , N,

(2)

where x is the spatial coordinate along the A–P axis of the embryo. The maximum synthesis rate Ra (t) now depends on time to capture the eﬀects of mitosis. Possible speciﬁc forms of this dependence are discussed below. The PDEs can be used in two ways. First, with PDEs it is possible to probe the role of nuclear structure in the segment determination (see Sec. 4.1). Second, sometimes PDEs can be solved analytically and exact solutions can be compared with numerical results. Before solving any PDE for quantitative expression data we need the initial/boundary conditions and some test points in time and space to validate the calculations. This means we need concentration curves produced from the images of expression patterns of segmentation genes. These patterns in the fruit ﬂy were obtained by immunostaining whole mount embryos using a confocal scanning microscope.9 The datasets were generated from these images. The expression levels were normalized per gene to a relative ﬂuorescence intensity range of 0–255 based on the most intensely ﬂuorescent pattern on each slide with multiple embryos. This dataset was generated from 2862 images of expression patterns by applying a ﬁve-step data pipeline.9–15 The dataset contains quantitative data on expression of gap and pair-rule genes, as well as maternal genes caudal (cad) and bcd at nuclear resolution and for a period spanning 71 min of development (cleavage cycles 13 and 14). The temporal resolution of data at cleavage cycle 14 is about 6.5 min of development.

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2.2. Fitting of model to data: Parallel Lam Simulated Annealing (PLSA) and Optimal Steepest Descent Algorithm (OSDA) The phenomenological parameters in the model equations (1) or (2) are found by ﬁtting solutions of the equations to the gene expression data. There are two methods of optimization in the parameter space used in the method for diﬀerent gene networks. PLSA was used as described in Refs. 1, 16 and 17. The set of equations (1) was solved numerically using either the forward Euler method, with 10–20% residual error (in numerical experiments with an old dataset) or a Bulirsch–Stoer adaptive step size solver.18 In the second case, equations were solved with a relative accuracy of 0.1%, and solutions were tested for numerical stability. The parameters Ra , T ab , ma , ha , Da and λa were adjusted to minimize the following cost function, representing the diﬀerence between a model solution and data: E= (via (t)model − via (t)data )2 . (3) Summation is performed over the total number of data points Nd , i.e. the number of protein measurements across all genes a, nuclei i and time classes t. Parameter search spaces were deﬁned by explicit search limits for Ra , Da and λa and a collective penalty function for T ab , ma , ha .1 Parameters ha for Kr, kni, gt and hb were ﬁxed to negative values representing a constitutive “oﬀ” state of the gene. This accelerated the annealing process considerably and slightly improved annealing results while not altering the overall quality of the resulting gene networks. The root mean square (r.m.s) score, E , r.m.s = Nd was used as a measure of the quality of a gene network. The r.m.s. represents the average absolute diﬀerence between protein concentrations in model and data. PLSA is a stochastic optimization method yielding gap gene networks of varying quality. Gene networks most faithfully reproducing gap gene expression were selected as follows. First, only networks with an r.m.s. < 12.0 were considered. All gap gene networks with an r.m.s. > 12.0 showed obvious pattern defects, some of them severe, such as displaced or missing expression boundaries. Second, each of the selected networks was carefully tested for patterning defects by visual inspection and plotting of squared diﬀerences between model and data for each protein and time class.

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OSDA is described in Refs. 15 and 19. This algorithm is based on a Lagrangian approach, in which the functional (3) is minimized subject to the constraints that (1) is satisﬁed and that the parameters lie within their search space. The search space constraints, initially expressed as inequalities, are transformed into additional constraint equations. An expanded cost function (the Lagrangian) is constructed by adding each constraint equation multiplied by its Lagrange multiplier to (3), and minimized by steepest descent approach. The algorithm was used in a study20 of the model (2) for a ﬁve-gene network. 3. Analysis of regulatory mechanisms in the system of gap genes Using a high resolution dataset, a new eﬀect was reported17 : Spatial positions of domains of gap gene expression on the anterior–posterior axis of the embryo (“gap domains”), which was expected to be almost stationary during the domain formation, are in fact substantially shifting towards anterior pole during the cleavage cycle 14A. The gap gene network comprising six zygotically expressed genes: Kr, gt, kni, hb, cad, and tailless (tll), was modeled as (1).17,21 Concentrations of proteins produced by maternally expressed hb and cad (“maternal gradients”) were used as initial conditions at cleavage cycle 13 for related zygotic gene products. Another maternal gradient, produced by bcd, came into model equations as an external input. All parameter values were found by ﬁtting the solution of the model to the expression data at eight consecutive time points from cycle 13 to the end of cycle 14A. Optimal parameter values found by PLSA predicted the activation of gap genes by maternal factors and gap–gap cross-repression, which was consistent with results of qualitative studies of mutant gene expression patterns. A solution related to the optimal parameter values found mimic data at high accuracy and temporal resolution, including the described shifts of gap domain boundaries. The main results of the analysis can be formulated as the following four points: (i) The gap gene system is reconstitution in silico by means of the novel PLSA technique. The patterns in the model are in excellent agreement with the data. (ii) The feed-forward interactions by maternal genes merely set initial patterns. Later the domain shifts and, hence, their positions at the later cycle 14 are determined by gap–gap cross-regulation. There are no such things as a stable border and co-moving coordinate system.

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(iii) The domain shifts result from a shift between production/decay areas and expression domains. Diﬀusion plays just an auxiliary role. (iv) The main regulatory mechanism in the system is an asymmetric feedback: within two nuclei system the posterior one represses its anterior neighbor more strongly than the other way around. Another important mechanism is relieved the repression of anterior products of gt and hb. 4. The role of nuclear divisions in the segmenting embryo The pattern formation by segmentation genes in the developing Drosophila embryo is accompanied by the process of nuclear divisions. We used the gene circuit method to answer the fundamental question about possible role of mitosis in the pattern formation.20,22 4.1. Pattern formation and nuclear divisions are uncoupled in Drosophila segmentation The PDE model (2) was used to study the correlation between the nuclear structure of the embryo and expression pattern formation by segmentation genes. Nuclei are replaced in this model by continuum, and we try to ﬁnd out if such a system may provide correct pattern formation. The network of four gap genes (Kr, hb, gt, and kni) and one pair-rule gene eve was considered. The model is formulated in the spatial domain on the A–P axis which covers a central part of the embryo including eve stripes 2–5, and in the time interval from cleavage cycle 11 to cycle 14A. The choice of time-dependence of Ra (t) in the equations is related to the way that mitosis is represented in the model, which is a non-trivial theoretical issue. Actual nuclei in the embryo divide at the end of each cleavage cycle; therefore, at the subsequent cleavage cycle we have doubled the number of gene copies, and, hence, we might expect double the potential for protein synthesis. If the nuclei were smaller in size than any spatial scale in which gene expression changes, and if macromolecular synthesis took place in a region of inﬁnitesimal spatial extent, writing (2) would be a straightforward exercise in taking concentrations, and Ra (t) would double in each successive cleavage cycle. In fact, this approximation does not hold. This is both because nuclei are large compared to the scale of spatial variation and because the actual process of protein synthesis takes place in a volume larger than a nucleus, since RNA must be transcribed and processed in the nucleus and transported to the cytoplasm for translation. Newly translated protein returns to the nucleus to bind to chromatin and regulate other genes.

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Three possible formulations of Ra (t) were considered, representing different approximations of this dependence, and in order of increasing complexity they are as follows: (A) Ra (t) = Ra , a constant. (B) Ra (t) = 0 during mitosis and has a positive value Ra (t) = Ra during interphase. This means that there is no synthesis during a short time period right before a cleavage takes place. (C) Ra = 0 during mitosis and Ra (t) = 2C−14 Ra during interphase, where C is the number of the cleavage cycle and Ra is the cycle 14 synthesis rate. This is the same as B, but Ra (t) → 2Ra (t) in each successive cleavage cycle. The schemes A–C reﬂect diﬀerent ways of incorporating mitosis into the model. Their ability to reproduce expression patterns (or lack thereof) allows us to draw conclusions about possible inﬂuence of mitosis on the pattern formation process. Parameter values were found for all three continuum models by means of OSDA.19 We obtained the correct pattern dynamics from all of the models A–C, as well as from the model with explicit nuclear structure (Fig. 3).20 The sets of parameter values in all the models considered were quantitatively diﬀerent from each other, but were qualitatively equivalent. Therefore, they represented the same genetic regulatory system (or the same gene network topology) which was independent of the representation of subcellular structure and the implementation of mitosis in the model. This led us to conclude that nuclear divisions were not coupled to pattern formation and serve only to populate the blastoderm with nuclei. More precisely, patterns of the form shown in Fig. 3 can be reproduced by the same gene network almost independently of the arrangement form of the nuclear structure in the embryo. 4.2. Nuclear divisions as a mechanism for selection in stable steady states We studied attractors in a two-gene model of the form (1) which mimicked cross-repressing gap genes in the segmentation system of the early Drosophila embryo.22 Diﬀerent numbers of nuclear cleavages were considered in the model. It turned out that the fraction that actual attractors in such a model form in the set of all stable steady states essentially depends on the number of divisions. Namely, the more nuclear cleavages the system undergoes, the fewer percent of steady states are reachable as attractors by the dynamics in the system. In this way, mitosis plays a speciﬁc dynamical

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 A−P Position Fig. 3. The expression patterns in the five-gene network (hb, Kr, gt, kni, and eve): comparison between data and models (A)–(C). Protein concentration profiles are shown at early (a)–(d) and late (e)–(h) cleavage cycle 14A: (a) and (e) data; (b) and (f) model (A); (c) and (g) model (B); (d) and (h) model C. The horizontal axis represents the rescaled spatial domain covering the middle 32% of the A–P axis of the embryo, the vertical axis represents protein concentrations in conventional units.

role, providing a mechanism for selection in stable states of multi-stationary gene networks. This eﬀect originates from the fact that the phase space of model (1) has a time-dependent dimension, which doubles after each division event. The steady states possible in the system are calculated in the ﬁnal phase space of a large dimension, while initial conditions are varied in the initial phase space of a smaller dimension. The larger number of divisions leads to the larger dimension of the ﬁnal phase space and, evidently, to a larger complexity of possible steady states. At the same time, the computations show that the number of actual attractors essentially depends only on the dimension of the initial phase space, which is ﬁxed. As an example, it was shown22 that if we consider the two-gene systems ﬁnishing in the same phase space of dimension 32 (16 nuclei in the last cleavage cycle) but having diﬀerent number d of nuclear divisions, then, for d varying from 1 to 3, the fraction of attractors in the set of all stable steady states decreases from, at maximum, 4% to, at maximum, 0.07%.

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There are two factors responsible for this selective mechanism of nuclear divisions. These factors provide two diﬀerent constraints on the dynamics on its way to stable steady states. The ﬁrst one relates to the assumption that the protein concentrations in two daughter nuclei are equal to that in their mother nucleus. This forbids certain states in certain neighboring nuclei right after a division occurs. The second constraint follows from the fact that the solution trajectories come into the next phase space already having been arranged in basins of attraction of pseudo-attractors from the previous phase space. As the previous phase space has a smaller number of degrees of freedom, again some states are forbidden for the system in the next phase space. The described role that cell divisions play in multi-stationary gene network models can be important in the evolutionary context. Having this property on hand, we can ask which combination of two numbers, the number of initial cells and the number of divisions to occur in the system, is the most suitable for selecting the appropriate number of degrees of freedom in the dynamical system and, consequently, the appropriate number of steady states. The system should deﬁnitely have enough steady states for evolution to be able to easily pick up the required ones. On the other hand, too many attractors would lead to narrow basins of attraction. This means an unstable situation for pattern formation, because there is a high probability of biological noise pushing the system from a correct to an incorrect basin. The actual protein concentrations from gap genes start to appear in the Drosophila embryo at early cleavage cycle 13, when there are about 50 nuclei along the A–P axis of the embryo. The “ﬁnal” patterns of segmentation gene expression are set at the end of cycle 14A; hence, one division occurs in the embryo in this period of transforming “initial” 50 to 100 nuclei. Taking into account the above arguments, we may hypothesize that this speciﬁc timing of gap pattern appearance is not casual but is the result of evolutionary selection. The purpose of this selection in mathematical terms is to construct a phase space for protein concentrations suitable for robust pattern formation.

5. Conclusions How can gene networks be used to solve a number of problems concerning the mechanisms of gene regulation? Being in silico methods, gene networks enable us to predict the results of experiments that have not been done, or which are quite diﬃcult to carry out, as well as to prove the suﬃciency of the inferred mechanisms without reconstructing the system ab initio. Simulations probing the stripe

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forming architecture of the gap gene system, as well as the necessity of nuclear divisions for pattern formation, would be examples of the former; the analysis of regulatory interactions in the gap gene system would be an example of the latter. Another important property of gene networks is their ability to keep track of all regulatory inputs to a speciﬁc gene in the intact and complete developmental system. This cannot be done in genetics, where functional information comes from removing genes one at a time from a complete system via mutation, and the regulatory structure of the wild-type network must be assembled on the basis of evidence from diﬀerent experiments. An example of the gene network obtained is presented in Fig. 4, as in Ref. 21. Regulatory loops of mutual repression create positive regulatory feedback between complementary gap genes, providing a straightforward mechanism for their mutually exclusive expression patterns. This mechanism is complemented by repression among overlapping gap genes. Overlap in expression patterns of two repressors imposes a limit on the strength of repressive interactions between them. Accordingly, repression between neighboring gap genes is generally weaker than between complementary ones (Fig. 4). The additional power of the gene network resides in its support of quantitative reasoning about the dynamics of living systems. As applied to the

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Drosophila segmentation system, this property allows one to reveal the role of autoactivation in sharpening the gap domain boundaries, as well as to explain the mechanism governing the posterior domain shifts during cycle 14A.17 It is also indispensable for answering fundamental questions, such as the question about the role of cell divisions, which are hard to answer with the help of experiment alone.20,22 The theoretical method requires a mathematical description of a network as a starting point and enables one to deduce the qualitative properties of genetic networks. The gene network approach is able to infer the regulatory net directly from data, as well as to understand the dynamical behavior of the genetic network. Merging the two methodologies into a general purpose tool is a worthwhile task for the future.

Acknowledgments We would like to thank our colleagues who participated in the work described in this overview: J. Jaeger, H. Janssens, D. Kosman, Manu, E. M. Myasnikova, A. Pisarev, E. Pustel’nikova, D. Sharp, M. Samsonova and S. Surkova. The support of this work by NIH Grants RR07801 and the CRDF GAP Awards RUB1-1578, and by RFBR-NWO grant 047.011.2004.013 is gratefully acknowledged.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11.

J. Reinitz and D. H. Sharp, Mech. Dev. 49, 133–158 (1995). E. Mjolsness, D. H. Sharp and J. Reinitz, J. Theor. Biol. 152, 429–453 (1991). J. Reinitz, E. Mjolsness and D. H. Sharp, J. Exp. Zool. 271, 47–56 (1995). J. Reinitz, D. Kosman, C. E. Vanario-Alonso and D. H. Sharp, Dev. Genet. 23, 11–27 (1998). P. T. Merrill, D. Sweeton and E. Wieschaus, Development 104, 495 (1988). E. Wieschaus and D. Sweeton, Development 104, 483 (1988). J. A. Campos-Ortega and V. Hartenstein, The Embryonic Development of Drosophila melanogaster (Springer, Germany, 1985). V. E. Foe and B. M. Alberts, J. Cell Sci. 61, 31 (1983). D. Kosman, J. Reinitz and D. H. Sharp, in Proceedings of the 1998 Pacific Symposium on Biocomputing, eds. R. Altman, K. Dunker, L. Hunter and T. Klein (World Scientiﬁc Press, Singapore, 1997), p. 6 [also available at www.smi.stanford.edu/projects/helix/psb98/kosman.pdf]. H. Janssens, D. Kosman, C. E. Vanario-Alonso, J. Jaeger, M. Samsonova and J. Reinitz, Dev. Genes Evol. 215, 374–381 (2005). D. Kosman, S. Small and J. Reinitz, Dev. Genes Evol. 208, 290 (1998).

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12. E. Myasnikova, M. Samsonova, D. Kosman and J. Reinitz, Dev. Genes Evol. 215, 320–326 (2005). 13. I. Aizenberg, E. M. Myasnikova, M. G. Samsonova and J. Reinitz, Math. Biosci. 159, 145 (2002). 14. K. N. Kozlov, E. M. Myasnikova, A. S. Pisarev, M. G. Samsonova and J. Reinitz, In Silico Biol. 2, 125 (2002). 15. K. N. Kozlov, E. M. Myasnikova, M. G. Samsonova, J. Reinitz and D. Kosman, Comp. Technol. 5, 112 (2000). 16. K. W. Chu, Y. Deng and J. Reinitz, J. Comput. Phys. 148, 646 (1999). 17. J. Jaeger et al., Nature 430, 368 (2004). 18. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, UK, 1992). 19. K. N. Kozlov and A. M. Samsonov, Tech. Phys. 48, 6–14 (2003). 20. V. V. Gursky, J. Jaeger, K. N. Kozlov, J. Reinitz and A. M. Samsonov, Physica D 197, 286–302 (2004). 21. J. Jaeger et al., Genetics 167, 1721 (2004). 22. V. V. Gursky, K. N. Kozlov, A. M. Samsonov and J. Reinitz, Physica D 218, 70–76 (2006).

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AUTHOR INDEX

Gharaibeh, M. F., 177 Gherghescu, R. A., 128 Gianturco, F. A., 227 Greiner, W., 23, 44, 86, 105, 128, 241, 377, 401, 411 Gudowska-Nowak, E., 389 Gursky, V. V., 426

Abd el Rahim, M., 141 Adamian, G. G., 34 Aguilar, A., 177 Andersen, L. H., 311 Andreev, A. V., 34 Antoine, R., 141 Antonenko, N. V., 34

Habibi, M., 177 Hanmura, T., 187 Henriques, E. S., 353 Hofmann, S., 12

B¨ ar, M., 273 Berke, P., 205 Bodo, E., 227 Brons, S., 389 Broyer, M., 141 B¨ urvenich, T. J., 44

Ichihashi, M., 187 Ikemoto, K., 199 Ivanov, V. K., 297 Ivanova, S. P., 34

Calvo, F., 261 Carr, J. M., 321 Chakraborty, H. S., 177 Coccia, E., 227 Connerade, J.-P., 55

Jackson, K., 72 Jellinek, J., 72 Jolos, R. V., 34 Kilcoyne, A. L. D., 177 Kondow, T., 187 Korol, A. V., 162 Kozlov, K. N., 426

Delplancke-Ogletree, M.-P., 205 Denifl, S., 152 Dinh, P. M., 273 Doi, S., 199 Dugourd, P., 141 Dunsch, L., 177

Lindinger, A., 283 Lo, S., 162 Lyalin, A., 205, 86, 105

Echt, O., 152 Els¨ asser, Th., 389 Emmons, E. D., 177

Madjet, M. E., 177 M¨ ahr, I., 152 Marinetti, F., 227 M¨ ark, T. D., 152

Fehrer, F., 273 441

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Author Index

Mishustin, I., 401, 411 Mishustin, I. N., 44 Mitsui, M., 199 Moskaleva, L. V., 273 M¨ uller, A., 177 Nagaoka, S., 199 Nakajima, A., 199 Nasirov, A. K., 34 Obolensky, O. I., 333, 411 Oganessian, Yu., 3 Parneix, P., 261 Phaneuf, R. A., 177 Plewicki, M., 283 Plonski, I. H., 128 Poenaru, D. N., 128 Polozkov, R. G., 297 Pshenichnov, I., 401, 411 Psonka, K., 389 Rayane, D., 141 Reinhard, P.-G., 273 Reinitz, J., 426 R¨ osch, N., 273 Rost, J. M., 177 Samsonov, A. M., 426 Scheid, W., 34 Scheier, P., 152

Schippers, S., 177 Schlachter, A. S., 177 Schlepckow, K., 333 Scholz, M., 389 Schwalbe, H., 333 Scully, S., 177 Semenikhina, V. V., 205 Shneidman, T. M., 34 Solov’yov, A. V., 86, 105, 128, 162, 205, 241, 297, 333, 353, 411 Solov’yov, I. A., 105, 241, 377 Suraud, E., 273 Surdutovich, E., 411 Taucher-Scholz, G., 389 Wales, D. J., 321 Weber, S. M., 283 Weise, F., 283 Yakubovich, A. V., 241 Yang, M., 72 Yang, S., 177 Yildirim, E., 227 Yurtsever, E., 227 Yurtsever, M., 227 Zagrebaev, V., 23 Zappa, F., 152 Zubov, A. S., 34