Advances in Atomic, Molecular, and Optical Physics, Volume 45

Advances in

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS VOLUME 45

Editors B B New York University New York, New York H W Max-Planck-Institut für Quantenoptik Garching bei Munchen Germany

Editorial Board P. R. B University of Michigan Ann Arbor, Michigan M. G F.O.M. Instituut voor Atoom-en Molecuulfysica Amsterdam The Netherlands M. I Argonne National Laboratory Argonne, Illinois W. D. P National Institute for Standards and Technology Gaithersburg, Maryland

Founding Editor S D R. B

Supplements 1. Atoms in Intense Laser Fields, Mihai Gavrila, Ed. 2. Cavity Quantum Electrodynamics, Paul R. Berman, Ed. 3. Cross Section Data, Mitio Inokuti, Ed.

ADVANCES IN

ATOMIC, MOLECULAR, AND OPTICAL PHYSICS Edited by

Benjamin Bederson        ,  

Herbert Walther     -- ¨   , 

Volume 45

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This book is printed on acid-free paper. Copyright  2001 by Academic Press All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. The appearance of code at the bottom of the first page of a chapter in this book indicates the Publisher’s consent that copies of the chapter may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per-copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts 01923), for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for chapters are as shown on the title pages; if no fee code appears on the chapter title page, the copy fee is the same for current chapters, 1049-250X/00 $35.00 ACADEMIC PRESS A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA http://www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK International Standard Book Number: 0-12-003845-5 International Standard Serial Number: 1049-250X

Printed in the United States of America 00 01 02 03 MB 9 8 7 6 5 4 3 2 1

Contents  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Comparing the Antiproton and Proton, and Opening the Way to Cold Antihydrogen G. Gabrielse I. II. III. IV. V. VI.

World’s Lowest Energy Antiprotons by a Factor of 10 . . . . . Million-Fold Improved Comparison of Antiproton and Proton Opening the Way to Cold Antihydrogen . . . . . . . . . . . . . . . . . Technological Spinoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Medical Imaging with Laser-Polarized Noble Gases Timothy Chupp and Scott Swanson I. II. III. IV. V. VI. VII. VIII.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Polarization Techniques . . . . . . . . . . . Basics of Magnetic Resonance Imaging (MRI) Imaging Polarized Xe and He Gas . . . . . . NMR and MRI of Dissolved Xe . . . . . . . . . Conclusions — Future Possibilities . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Polarization and Coherence Analysis of the Optical Two-Photon Radiation from the Metastable 22S1/2 State of Atomic Hydrogen Alan J. Duncan, Hans Kleinpoppen, and Marlan O. Scully I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. On the the Theory of the Two-Photon Decay of the Metastable State of Atomic Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. The Stirling Two-Photon Apparatus . . . . . . . . . . . . . . . . . . . . . . . . IV. Angular and Polarization Correlation Experiments . . . . . . . . . . . . . V. Coherence and Fourier Spectral Analysis — Experiment and Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Time Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Correlation Emission Spectroscopy of Metastable Hydrogen: How Real are Virtual States? . . . . . . . . . . . . . . . . . . . . . . VIII. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

100 101 108 111 127 133 133 144

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IX. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Laser Spectroscopy of Small Molecules W. Demtröder, M. Keil, and H. Wenz I. II. III. IV. V. VI. VII.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High Vibrational Levels in Electronic Ground States . . . . . . . . . . . Laser Spectroscopy of Electronically Excited Molecular States . . . . . Sub-Doppler Spectroscopy of Small Alkali Clusters . . . . . . . . . . . . . Time-Resolved Laser Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Coulomb Explosion Imaging of Molecules Z. Vager I. II. III. IV.

The Principle of Coulomb Explosion Imaging . . . . . . . . . . . Scientific Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of Recent Coulomb Explosion Imaging Studies and Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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S I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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C  V  T S . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contributors Numbers in parentheses indicate pages on which the authors’ contributions begin

T C (41), Departments of Physics and Radiology, University of Michigan, Ann Arbor, Michigan 48109 W. D¨  (149), Fachbereich Physik, Universita¨t Kaiserlautern, D67663 Kaiserlautern, Germany A J. D (99), Unit of Atomic and Molecular Physics, University of Stirling, Stirling FK9 4LA, Scotland G. G (1), Harvard University, Cambridge, Massachusetts 01238 M. K (149), Fachbereich Physik, Universita¨t Kaiserlautern, D-67663 Kaiserlautern, Germany H K (99), Unit of Atomic and Molecular Physics, University of Stirling, Stirling FK9 4LA, Scotland M O. S (99), Department of Physics, Texas A&M University, College Station, Texas 77843; and Max-Planck-Institut fu¨r Quantenoptik, D-85748 Garching, Germany S S (41), Departments of Physics and Radiology, University of Michigan, Ann Arbor, Michigan 48109 Z. V (203), Department of Particle Physics, Weizmann Institute of Science, 76100 Rehovot, Israel H. W (149), Fachbereich Physik, Universita¨t Kaiserlautern, D-67663 Kaislerlautern, Germany

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COMPARING THE ANTIPROTON AND PROTON, AND OPENING THE WAY TO COLD ANTIHYDROGEN G. GABRIELSE Harvard University, Cambridge, Massachusetts 02138 I. World’s Lowest Energy Antiprotons by a Factor of 10 . . . . . . . . . A. First Slowing and Trapping of Antiprotons . . . . . . . . . . . . . . . . B. Capturing and Cooling Antiprotons . . . . . . . . . . . . . . . . . . . . . . C. Vacuum Better than 5 ; 19\ Torr . . . . . . . . . . . . . . . . . . . . . . D. Stacking Antiprotons: Making the Antiproton Decelerator (AD) Possible. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Transporting Trapped Antiprotons . . . . . . . . . . . . . . . . . . . . . . . F. Later Duplication of TRAP Techniques by Others . . . . . . . . . . . G. Lower Temperature Antiprotons are Coming . . . . . . . . . . . . . . . II. Million-Fold Improved Comparison of Antiproton and Proton . . . . A. Testing PCT Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Comparing Cyclotron Frequencies . . . . . . . . . . . . . . . . . . . . . . . C. TRAP I: One Hundred Antiprotons Compared to One Hundred Protons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. TRAP II: Alternating One Antiproton and One Proton . . . . . . . E. TRAP III: Simultaneously Trapped Antiproton and H\ Ion . . . III. Opening the Way to Cold Antihydrogen . . . . . . . . . . . . . . . . . . . . . A. Cold Antihydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. 4.2 K Positrons in Extremely Good Vacuum . . . . . . . . . . . . . . . C. Demonstrating the Nested Penning Trap . . . . . . . . . . . . . . . . . . D. Closer to Cold Antihydrogen than Ever Before . . . . . . . . . . . . . . E. Recombination Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Selecting Processes within a Nested Penning Trap . . . . . . . . . 2. Other Formation Processes . . . . . . . . . . . . . . . . . . . . . . . . . . F. Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Technological Spinoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract: Our TRAP collaboration has developed and demonstrated slowing, trapping, and electron-cooling techniques that enable antiproton storage in thermal equilibrium at 4.2 K. This is an average energy that is more than 10 times lower than the energy of any previously available antiprotons. Months-long confinement of a single antiproton, at a background pressure 5 ; 10\ torr, and nondestructive detection of the radio signal from a single trapped antiproton, made it possible to show that the charge-to-mass ratios of the antiproton and proton differ in magnitude by 9 parts in

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Copyright  2001 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-003845-5/ISSN 1049-250X/01 $35.00

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G. Gabrielse 10. This 90 parts per trillion comparison is nearly a million times more accurate than previous comparisons, and is the most stringent test of PCT invariance with a baryon system by a similar amount. The availability of extremely cold antiprotons makes it possible to pursue the production of antihydrogen that is cold enough to trap for precise laser spectroscopy. The closest approach to cold antihydrogen to date is our simultaneous confinement of 4.2 K antiprotons and positrons. All cold antiproton experiments so far were carried out at the CERN Laboratory with antiprotons coming from its Low Energy Antiproton Ring (LEAR). This unique facility has now closed. Future antihydrogen experiments will be pursued at the new Antiproton Decelerator ring at CERN, which was constructed for this purpose. Using the techniques developed by TRAP, antiprotons will be accumulated within traps rather than in storage rings, thereby reducing the operating expenses to CERN.

I. World’s Lower Energy Antiprotons by a Factor of 1010 Stored antiprotons are now available in thermal equilibrium at 4.2 K. This is an energy that is 10 times lower (Fig. 1) than the lowest energy antiprotons available before our TRAP Collaboration (Table 1) developed and demonstrated new techniques over the last decade. Our extremely cold antiprotons made possible a series of three comparisons of the charge-tomass ratio of the antiproton and proton that improved this comparison by nearly a factor of 10, the most precise test of PCT invariance with baryons by approximately this factor. The extremely cold antiprotons, and a method to accumulate them in a trap, make it possible to pursue the production and study of cold antiprotons at the new Antiproton Decelerator facility at the CERN laboratory. Antiprotons are the antimatter counterparts of protons, the familiar constituents of ordinary matter (along with electrons and neutrons). Antiprotons occur naturally only as the very occasional products of a collision between a high energy cosmic ray and an atom in the atmosphere. Although believed to be stable particles, the naturally occurring antiprotons nonetheless live for only a short time. A single collision between an antiproton and any proton in ordinary matter can annihilate both particles. The antiproton and proton cease to be and a variety of lighter particles (mostly pions) are formed. Antiprotons can thus be stored only in a container that has no walls and essentially no ordinary matter within. Starting in 1955, in the Bevatron storage ring at Berkeley, usable numbers of antiproton have been produced and studied using giant storage rings within which antiprotons travel at speeds close to that of light. Protons of extremely high energy are made to collide with ordinary matter. In a small fraction of these collisions, antiprotons are formed, which can be directed into large circular storage rings. Technological improvements allowed the CERN Laboratory in Geneva to accumulate much larger

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F. 1. Our trapped antiprotons are the lowest energy antiprotons in the world by more than a factory of 10. The vast energy scale for charged particles is represented on a logarithmic ‘‘thermometer.’’

TABLE I TRAP C: CERN PS-196 Harvard University: Prof. G. Gabrielse?, Dr. S. Rolston, Dr. L. Orozco, Dr. W. Jhe, Dr. W. Quint, Dr. T. Roach, K. Helmerson, R. Tjoelker@, X. Fei@, D. Phillips@, A. Khabbaz@, D. Hall@, P. Yesley?, J. Estrada?. University of Bonn: Dr. H. Kalinowsky?, Dr. G. RouleauA, J. Haas@, J. GrobnerB, C. Heimann@. University of Washington: Prof. T. Trainor. Fermilab: Dr. W. Kells. ?Continuing on ATRAP. @Ph.D. earned at TRAP. AContinuing on ATHENA. BDiploma Thesis earned at TRAP.

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numbers of antiprotons that were collided with protons so that the shortlived W and Z particles could be observed and studied. At Fermilab in Illinois, higher energy collisions between antiprotons and protons are being investigated to learn about the top quark. Such high energies and velocities are desirable for experiments in which antiprotons are made to collide with other particles. However, new experiments became possible when the CERN Laboratory began operating the Low Energy Antiproton Ring (LEAR) in 1982. It has a modest circumference of only 79 m (much smaller than the 27 km circumference for the LEP ring at CERN). LEAR slowed and cooled antiprotons to an energy of 6 MeV, a speed that is approximately 0.1 of the speed of light, and sent them to various particle and nuclear physics experiments. The ‘‘Low’’ in Low Energy Antiproton Ring was initially appropriate insofar as at the time these antiprotons were much lower in energy than any other antiprotons available in the world. However, the energy of antiprotons in LEAR was nonetheless much higher than the average energy of particles in the sun. Compared to the much lower energy we needed to do precision mass spectroscopy of antiprotons, and to prepare for antihydrogen that is cold enough to be trapped for high precision laser spectroscopy, the LEAR energy was very high indeed — by at least a factor of 10. The new techniques developed by TRAP, listed in what follows, allow slowing and cooling of antiprotons to energies that are lower by the required factor of 10. The slowed and cooled antiprotons reside within a small volume (less than 1 mm) of an ion trap in a nearly perfect vacuum (better than 5 ; 10\ torr). Their average kinetic energy is so low, 1 MeV, that temperature units are often used. The energy ‘‘thermometer’’ of Fig. 1 contrasts the energy of antiprotons and protons in various giant storage rings at the top with LEAR energies in the middle. Towards the bottom, 10 times lower in energy than is possible in LEAR, is the new low energy frontier at only 4° above absolute zero (4 K). Even lower antiproton temperatures should be possible as illustrated by the 70 mK temperatures we recently realized with trapped electrons in a similar ion trap (Peil and Gabrielse, 1999). 1. Slowing in matter to very low energy. Some of the 6 MeV antiprotons from LEAR slowed below 3 keV while passing through a thin window of matter. The fraction was initially measured during a 24-h beam time allocation in 1986 (Gabrielse et al., 1986, 1989a). 2. Trapping antiprotons. Antiprotons slowed below 3 keV are captured whilew they are within the electrodes of a Penning trap (Brown and Gabrielse, 1986) by the sudden application of a 3-kV trapping potential (Gabrielse et al., 1986). This was first demonstrated in a 24-h beam time allocation in 1986.

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3. Electron cooling of trapped antiprotons. The trapped antiproton, with keV energies, cooled via collisions with simultaneously trapped electrons (Gabielse et al., 1989b). The electrons radiated synchrotron radiation to return to thermal equilibrium with their 4.2 K surroundings. 4. L ong-term storage of 4.2 K antiprotons monitored nondestructively. We held antiprotons for months, monitoring them via the currents induced in surrounding electrodes by their motion, without loss of antiprotons (Gabrielse et al., 1990). 5. A pressure 5;10\ torr. Antiprotons remained trapped for months without any detectable loss, establishing a 3.4 month limit for our antiproton storage time and for antiproton decay into all channels. Calculated cross sections allow us to convert this storage time limit to the pressure limit already mentioned here (Gabrielse et al., 1990). 6. Nondestructive monitoring of one trapped p (or a p and H\ together). Good control and detection sensitivity with one or two trapped particles allowed the comparisons of q/m for the antiproton and proton to be improved by almost a factor of one million (Gabrielse et al., 1990, 1995, 1999b). These techniques were reported in Physical Review L etters and in the Rapid Communications of Physical Review, as indicated. A semipopular account was presented in Scientific American (Gabrielse, 1992). A. F S  T  A The apparatus used to cool antiprotons to low temperatures and to measure accurately their charge-to-mass ratio falls in the realm of ‘‘table top’’ experiments in that its size would allow it to be mostly located on the top of a table. A big complication is that the table top must be located at a large particle physics facility capable of supplying antiprotons. My trip to Fermilab in 1981 did not succeed in generating much interest in low-energy antiproton experiments. An intense focus on the primary Fermilab mission of studying TeV collisions between protons and antiproton left no room for the low energy experiments envisioned, even though a small operating ring (used for cooling studies) might have been adapted for this purpose. In 1985, Bill Kells of Fermilab came to work with me for one year, bringing the news that the small storage ring was shut down and pieces were being shipped to other laboratories. We thus turned our attention to mounting an experiment in Geneva, since the unique LEAR facility at the CERN laboratory was the only place in the world that could slow antiprotons to the MeV energies that a table top apparatus could accept. Hartmut Kalinowsky of the University of Mainz in Germany joined

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forces with us, as did Tom Trainor of the University of Washington in Seattle. Initially there was some skepticism at CERN about our proposals to slow antiprotons in matter, capture them in an ion trap and cool them within the ion trap via collisions with cold electrons in the same trap. These unproven techniques were very different from the normal high-energy collision experiments done at CERN. Moreover, one of our physics goals was in direct competition with an experiment in which CERN had already invested a great deal. There was also concern because we had no financial support while making our proposal. Funding agencies in the United States, with limited resources in tough economic times, were cautious about a large new program to be carried out at CERN but which did not yet have CERN approval. CERN and the funding agencies were not reassured by the fact that none of us had regular academic positions with tenure. Fortunately, first the Atomic Physics Division of the National Science Foundation, then the Air Force Office of Scientific Research and the National Bureau of Standards (now National Institute of Standards and Technology (NIST)) decided jointly to fund the quest for low-energy antiprotons. Only after the experiments succeeded did we learn that an NSF consensus found the opportunity too good to pass up despite an estimated likelihood of success in trapping cold antiprotons of 20%. Eventually CERN allowed us 24 h access to LEAR antiprotons to demonstrate that it really was possible to slow antiprotons from MeV energies down to 3 keV. Figure 2a shows the gas cells and time-of-flight apparatus, and Fig. 2b shows the number of antiprotons and protons transmitted through the gas cells and degrader as the gas mixture was changed to tune the energy of these particles. When we were able to demonstrate that enough low-energy antiprotons were available for trapping in 1986, we were given a second 24-h access to antiprotons two months later, to demonstrate that we could capture the slowed antiprotons. This demonstration experiment took place before there was time to purchase very much modern equipment. An ancient superconducting magnet was loaned to us by the University of Mainz. An ion trap was constructed in one day from glass-to-metal seals of unknown origin that were found abandoned in glass blower’s drawer. After testing in our own laboratory, we shipped the apparatus by air owing to time pressure and the delicate nature of the helium Dewar we had constructed. After the Dewar arrived broken in Geneva, we learned that our ‘‘air’’ shipment actually traveled across Europe in trucks of unknown suspension quality. It is hard to forget aiming a misbehaving hand torch at ‘‘borrowed’’ hard solder within a Dewar that dangled from a rope slung over a beam in a CERN hallway.

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F. 2. (a) Gas cells and time-of-flight apparatus used to measure the number of antiprotons slowed to keV energies in a degrader. (b) Number of transmitted antiprotons as a function of the gas mixture varied to decrease the energy of the antiprotons incident on the degrader window. (Taken from Gabrielse et al., 1989a.)

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F. 3. (a) Scale outline of the apparatus that first trapped antiprotons. (b) Annihilation signals from the first trapped antiprotons when they were released from the trap after being held for 1000 s. (Taken from Gabrielse et al., 1986.)

The repaired apparatus (Fig. 3a) was ready several days before antiprotons were scheduled to arrive at noon on Friday. Feverish computer programming was continuing (‘‘just one half hour more of Basic’’) to read out information about attempted antiproton captures in real time. Then, late Thursday evening, disaster struck. Routine tests, done dozens of times before, revealed that we could no longer apply high voltages to our trap electrodes without causing an arc inside the part of the Dewar that was cooled to 4 K. It was 12 h before the antiprotons were scheduled to arrive and the apparatus had never been warmed to room temperature and then cooled back to 4 K in less than several days. Half of our team went to bed convinced that we had failed.

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Given CERN’s ambivalence about the feasibility of the proposed experiments, a failure in this test experiment would clearly be a major setback, so a repair had to be attempted. As the cold apparatus was prematurely opened, water condensed on it and streamed out, despite the hot air directed in from three industrial-strength hair dryers. The breakdown point was located and fresh copper cables installed to handle the high voltages. After much mopping of water, drying and cleaning, we reassembled the apparatus during the night and began cooling by 10 a.m., Friday. We informed the LEAR control room that we would indeed be ready for our antiproton test by shortly after noon. Our euphoria was short-lived. Their reply indicated that CERN was not yet ready to deliver antiproton to us. Such particles were indeed available in the large Antiproton Accumulator (AA) storage ring at CERN, but the ‘‘kicker’’ device used to extract antiprotons from this ring had broken. It was most likely that we would leave CERN without receiving antiprotons. I explained the urgency of the situation (how far we had come, the stakes involved etc.), then stumbled off to bed exhausted and discouraged. The test experiment appeared doomed for some time, since LEAR was soon scheduled to be shut down for more than 1 year. Several hours later I was awakened. An accelerator magician at CERN had managed to make a backup ‘‘kicker’’ work. Soon LEAR was ready to try to send us intense pulses of antiprotons (about 10) in short bursts (200 ns). This ‘‘fast extraction’’ mode of operation was new at LEAR. The operators counted ‘‘five, four, three, two, one,’’ in various versions of English, and then pushed a newly installed green button with a loud ‘‘go.’’ After several hours of adjusting the timing electronics, we started to see clear and unmistakable signals that indicated we were able to trap antiprotons in our ion trap at energies 3 keV. The emotional roller coaster that this antiproton test experiment had become ended on a pronounced high. The LEAR operators and physicists from other experiments crowded around during the countdown. Applause broke out any time the histogram (see Fig. 3b) on the computer monitor indicated that antiprotons had been trapped and stored. A few antiprotons were held as long as 20 min, thereby establishing the feasibility of the proposed measurements. B. C  C A This success turned CERN ambivalence into CERN enthusiasm. A connection to LEAR was constructed and dedicated to these experiments while LEAR was shut down for 1 year. Our new apparatus, installed in the fall of 1988, included a state-of-the-art superconducting solenoid and a carefully constructed ion trap suited to precise measurement of the antiproton mass.

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F. 4. (a) Outline of trap electrode surfaces and the location of cold trapped particles. (b) Typical potential well on axis.

The electrodes of the particle trap (Fig. 4a) were a stack of gold plated copper rings to which appropriate voltages could be applied. The electrodes resided within a 6-tesla magnetic field directed along the symmetry axis of the cylinders and produced by the superconducting solenoid. Charged particles make circular orbits around the direction of a magnetic field. Antiprotons and electrons in the trap thus orbit in circular trajectories rather than traveling radially outward to hit the trap electrodes. Before antiprotons are allowed into the trap, electrons are loaded into the small region indicated by the representation of the axial potential wells of the trap in Fig. 5. Internally generated electrons sent through the trap strike the flat plate at the left and dislodge adsorbed gas atoms. Other electrons collide with some of these atoms within the small region in such a way as to produce low-energy electrons in this location. Positive tens of volts on the electrodes at this location attract the negative electrons, keeping them from exiting at either end. The electrons in circular orbits about the magnetic field direction rapidly radiate their energy (typically in 0.1 s) and cool to the temperature of the surrounding electrodes near 4 K. The 6-MeV antiprotons in an intense pulse from LEAR crash through the flat plate electrode at the left of the trap, losing energy via collisions within the goldplated aluminum degrader. Approximately one in a few thousand of the antiprotons emerge with an energy (along the axis of the trap) of 3 keV. These can possibly be trapped using the voltages described here. Antiprotons that emerge from the aluminum with higher energy are not turned around by the modest 93 kV being applied to the electrode at the far right, so are annihilated upon hitting this electrode. Others lose too much energy by collisions within the aluminum, slow to a stop within the plate, and eventually annihilate. The flat degrader electode is kept slightly

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11

F. 5. (a) Cold trapped electrons in a small central trap well await the introduction of a pulse of hot antiprotons into the long ‘‘half well.’’ (b) When the maximum number of antiprotons are in the trap volume a high voltage is applied to complete the well. (c) On a much longer time scale hot antiprotons cool via collisions with cold electrons and eventually join the electrons in the small trap well.

positive while the antiprotons are passing through to keep secondary electrons from leaving the degrader and filling the trap. After the antiprotons enter the trap, but before they return to the entrance plate, this potential is changed to 93 kV to prevent them from striking the plate upon their return. To shut this door before the rapidly moving antiprotons can escape, this potential is changed in 20 ns. Figure 6 shows the relative number of antiprotons trapped as a function of the time that the door is shut and Fig. 7 shows how the number of antiprotons trapped is optimized by tuning the energy of the antiprotons incident on the degrader window (by changing the gas mixture). Captured antiprotons oscillate back and forth along the 12-cm length of the trap, with energies ranging between 0 and 3 keV, passing through the cold, trapped electrons. Just as a heavy bowling ball would eventually be slowed by collisions with light ping-pong balls, the antiprotons cool (in 100 s time we typically reserve for cooling) to thermal equilibrium with

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F. 6. Number of antiprotons trapped as a function of the time at which the trap is closed (below), with the injected antiproton pulse shown (above) on the same time scale.

F. 7. The energy of antiprotons that arrive at our apparatus at 6 MeV is tuned downward (by varying the amount of SF in the beam path) to maximize the number of trapped  antiprotons.

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F. 8. Energy spectrum for hot trapped antiprotons. (Taken from Gabrielse et al., 1989b.)

the trapped electrons at 4 K. Cooled antiprotons now reside in the same small region of the trap as do the electrons and their energy is 10 times lower than the energy of the antiprotons that came from LEAR. Figure 8 shows the trapped antiproton energy distribution before electron cooling. The number of annihilations is measured as the potential on one end of the long well is reduced. After electron cooling the greatly narrowed spectra of Fig. 9 are observed. The energy width of these spectra is only an upper limit insofar as the space charge of the low-energy antiprotons allows some of them to escape at a well depth that is higher than their kinetic energy. The cooling process is remarkably efficient in that upwards of 95% of the antiprotons in the long trap are so cooled, as illustrated in Fig. 10. As many p as will fit, limited by space charge to about 0.4 million, end up in the small inner, harmonic well with approximately 10 cooling electrons. We trap up to 0.6 million antiprotons from a single LEAR pulse in inner and outer traps together (Fig. 11), with an efficiency (compared to the number of antiprotons measured to leave LEAR) shown in Fig. 12. To selectively expel the cooling electrons they are heated by driving them at frequencies that correspond to their preferred oscillation frequencies. When the voltages on the neighboring electrodes are lowered for a very short time the hot electrons leak out of the trap. The cold, heavy antiprotons remain and are subsequently cooled using cold resistors connected between various nearby electrodes. Residual antiproton motions induce currents through these resistors. Power dissipated in the resistors is thereby extracted from the antiproton motion, cooling the antiprotons into thermal equilibrium with the resistors, which are near 4 K.

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F. 9. Energy spectra for decreasing numbers of cold antiprotons. (Taken from Gabrielse et al., 1989b.)

F. 10. Fraction of antiprotons cooled into the center harmonic well as a function of cooling time with 4 ; 10 cooling electrons.

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F. 11. Energy spectrum of 0.6 million antiprotons trapped from a single 250-ns pulse of antiprotons from LEAR. Approximately 0.4 million (below) were able to fit in the small central trap well with the electrons that cooled them. An additional 0.2 million were cooled somewhat but remained in the long well (above). (Taken from Gabrielse et al., 1999.)

F. 12. Number of trapped antiprotons versus the number of 6 MeV antiprotons sent down our beamline as measured by the LEAR beam monitor; gives an efficiency of 5 ; 10\.

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When only one antiproton is needed for q/m measurements we slowly reduce the well depth of the small trap to let all but one antiproton escape. We monitor the cyclotron signal (we will discuss this presently). When only a few antiprotons remain we can resolve the signals from each antiproton because of the relativistic shift in their cyclotron frequencies. C. V B  5 ; 19\  It seems that we can store the cold antiprotons indefinitely. In one trial we held approximately 10 antiprotons for about 2 months before deliberately ejecting them (Gabrielse et al., 1990). We actually observed no antiproton loss at all, but imprecision in our knowledge of the number of antiprotons initially loaded into the trap limits the lifetime we can set to p  3.4 months

(1)

Despite the much lower energies (and hence much higher annihilation cross sections), this lifetime limit is longer than directly observed for high-energy antiprotons in storage rings for decays into all channels. Based upon calculated cross sections (Morgan and Hughes, 1970), our containment lifetime limit given here requires a background gas density 100 atoms/cm. For an ideal gas at 4.2 K this corresponds to a pressure 5 ; 10\ torr. The low pressure is attained by cooling the trap and its sealed container to 4.2 K. D. S A: M  A D (AD) P We typically captured a pulse of antiprotons from LEAR and let it cool via its interaction with 4.2 K electrons over the following 100 s. For charge-tomass measurements we would then eject all but one antiproton. To facilitate the production and study of cold antihydrogen, however, the largest possible number of cold, trapped antiprotons is desired. To this end we demonstrated that once captured antiprotons had been cooled into an inner potential well it was possible to capture additional pulses of antiprotons from LEAR right over top of the first (Gabrielse et al., 1990). After the capture and cooling process represented in Fig. 5 is completed, one simply repeats it as many times as desired. The number of trapped antiprotons thus increases over what can be captured in a single pulse. The TRAP demonstration of ‘‘antiproton stacking’’ in a trap is the foundation upon which the new antiproton decelerator is based. Antiprotons delivered from LEAR were initially captured in the antiproton

COMPARING THE ANTIPROTON AND PROTON

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collector ring (ACOL), accumulated in the antiproton accumulator ring (AA), then slowed and cooled within the low energy antiproton ring (LEAR) and sent to our trap. When we demonstrated that antiprotons could instead be accumulated in a small trap, and much more inexpensively than in the large storage rings, the CERN management decided to shut down two of the three rings to save expenses. (The alternative was to entirely discontinue all low-energy antiproton physics.) The ACOL Ring was substantially modified to make the new antiproton decelerator (AD). The good news is that the experimental efforts to produce and precisely study cold antihydrogen can continue. The bad news is that the new facility brings additional challenges. To obtain the same number of cold, trapped antiprotons that we captured from LEAR in 250 ns will require us to accumulate (i.e., stack) antiprotons from the AD for an hour or more. E. T T A From our initial proposal at CERN, we stressed that our Penning trap was an intrinsically portable device. Trapped antiprotons could certainly be transported within our Penning trap if ever there was a good reason to do so. Although the compelling reason never emerged, we were nonetheless often asked about this possibility. Finally, in 1993 we used the opportunity of the delivery to our Harvard laboratory of a new superconducting solenoid constructed in California to make an experimental demonstration. In a Penning trap apparatus that was essentially identical to our antiproton apparatus, we transported stored electrons from California to Nebraska, then from Nebraska to Cambridge (see Fig. 13) (Tseng and Gabrielse, 1993). (The tale of an avoidable adventure in a Nebraska truck stop will not be retold here.) Electrons were used because they were much more readily available in California and Nebraska, but there is no doubt whatever that antiprotons could just as easily be transported if there had been a good reason. F. L D  TRAP T  O For many years, the techniques to obtain cold antiprotons, all of which were developed and demonstrated by TRAP, were used exclusively by TRAP. More recently, some of these techniques were duplicated by the PS-200 collaboration, which has since developed into ATHENA. The PS-200 motivation was to measure the gravitational acceleration of antiprotons using a time-of-flight technique. Most of a decade was spent puruing this goal but unfortunately this effort was not successful. As one would expect, the gravitational force of the earth on an antiproton is simply

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F. 13. Trapped electrons were transported from California to Nebraska, and then from Nebraska to Massachusetts. (Taken from Tseng and Gabrielse, 1993.)

too small compared to the electric force from stray charges in the apparatus. The conditional approval of PS-200 offered antiprotons only after the desired gravitational sensitivity in the time-of-flight measurements was demonstrated with negative ions. When the end of the LEAR program was in sight, however, permission was granted to try to duplicate the TRAP techniques for trapping and cooling antiprotons despite the absence of the promised demonstration. The trap used (Holzscheiter et al., 1996) was a larger version of the open endcap cylindrical trap used by TRAP (Gabrielse et al., 1989c). A higher trapping potential was also used. Up to a million antiprotons were trapped at the same time. The vacuum was not so good as that demonstrated by TRAP due to higher trap temperatures and the presence of warm surfaces in the vacuum system, and the antiproton storage time was of the order of 10 s rather than months. Antiprotons were eventually cooled with electrons — but not to 4 K. The only new feature of the PS-200 measurements was the surprising claim that antiprotons stored in poor vacuum annihilate much less rapidly than expected (Holzscheiter et al., 1996). However, very little quantitative study was done and the actual vacuum within the trapping volume was estimated rather than measured. Hopefully, this surprise will eventually be tested under more controlled conditions.

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G. L T A  C For some years single component plasmas of elementary particles have been studied at temperatures down to 4 K. We have now managed to cool stored electrons down to 70 mK and below. So far, only one trapped electron (at a time) has been studied in detail at this low temperature, though there is no reason to expect any difficulties with larger numbers. Quantum jumps between Fock states of a 1-electron oscillator reveal the quantum limit of a cyclotron (Peil and Gabrielse, 1999). With a surrounding cavity inhibiting synchrotron radiation 140-fold, the jumps show a 13 s Fock state lifetime, and a cyclotron in thermal equilibrium with 1.6—4.2 K blackbody photons. These disappear by 80 mK, a temperature 50; lower than previously achieved with an isolated elementary particle. The cyclotron stays in its ground state until a resonant photon is injected. A quantum cyclotron offers a new route to measuring the electron magnetic moment and the fine structure constant. Although the demonstration was done with trapped electrons, there is every reason to believe that the same apparatus would also cool antiprotons to the same 70 mK temperature. Our ATRAP collaboration, an expanded version of TRAP formed to study cold antihydrogen, plans to pursue this option.

II. Million-Fold Improved Comparison of Antiproton and Proton Our proposal to improve the comparison of the charge-to-mass ratio of the antiproton and proton to 1 part in 10 was a surprise at CERN. One reason was that the proposed techniques were very unfamiliar at CERN. Another was that CERN had already invested in an experimental program with similar goals (CERN PS-189), employing a large Smith-type mass spectrometer. (Unfortunately, the angular acceptance of the spectrometer was so small that it was never able to make any antiproton measurements.) Over several years we were able to achieve the accuracy we had proposed and even to do an order of magnitude better. Our series of three mass measurements (Gabrielse et al., 1990, 1995, 1999) began as soon as we produced 4.2 K antiprotons and eventually improved the comparison of antiproton and proton by approximately 10. Figure 14 shows how comparison of the antiproton and proton improved in time, starting with the first observation of the antiproton, and concluding with the three measurements by TRAP. A. T PCT I The P in PCT stands for a parity transformation. Suppose we do a certain experiment and measure a certain outcome. As we do the experiment, we

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F. 14. (a) Accuracy in comparisons of p and p. (b) The measured difference between q/m for p and p (TRAP III) is improved more than ten-fold. (Taken from Gabrielse et al., 1999.)

also watch what the experiment and outcome look like in a mirror. We then build apparatus and carry out a second experiment that is identical to the mirror image of the first. If our reality is invariant under parity transformations P then we should obtain the outcome seen in the mirror for the second experiment. Until 1956 it was universally believed that reality was invariant under parity transformations. Then Lee and Yang noted that this basic tenet of physicists’ faith had not been tested for weak interactions — those interactions between particles that are responsible for beta decay of nuclei. Shortly after, Wu and collaborators, and then several other experimental groups in rapid succession, showed in fact that experiments and mirror image experiments produced strikingly different results when weak interactions were involved. The widespread faith that reality was invariant under parity transformations had clearly been misplaced. A new faith, that our reality was invariant under PC transformations, rapidly replaced the discredited notion. The ‘‘C’’ stands for a charge conjugation transformation, which for our purposes is a transformation in which particles are turned into their antiparticles. To test whether reality is invariant under PC transformations, a mirror image experiment is constructed as described here but this time all the particles within it are also changed into antiparticles. It was widely believed that these two different experiments could not be distinguished by their outcomes until Cronin and Fitch surprised everyone by using kaon particles to explicitly demonstrate that our reality is not invariant under PC transformations. The experiment has been repeated by different groups in different locations and related measurements are still being pursued.

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F. 15. Comparison of the accuracy of baryon, lepton and meson PCT tests

Now most physicists believe that reality is instead invariant under PCT transformations, the ‘‘T’’ standing for a time reversal transformation. The PCT invariance seems more well-founded insofar as theorists find it virtually impossible to construct a reasonable theory that violates this invariance. To experimentally test for PCT invariance, one again compares the outcomes of two experiments. This time one makes a movie of the goings on in a mirror image experiment in which the particles are switched to antiparticles. The second experiment is constructed to mimic what one sees in the movie when the movie is run backwards (i.e., when ‘‘time is reversed’’). In practice, the cyclotron oscillation frequencies of a proton and an antiproton oscillating in the same magnetic field would be identically the same if reality is invariant under PCT transformations. The antiprotonproton frequency comparisons discussed in what follows thus test whether reality is PCT invariant and establish that any departures from this

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invariance must be smaller than the experimental error bars. This comparison is by far the most precise test of PCT invariance done with baryons, particles made of three quarks or three antiquarks. The antiproton-toproton charge-to-mass ratio comparison thus joins an experiment with kaons (made of a quark and an antiquark) and a comparison of the magnetic moment of an electron and positron as one of the most precise experimental tests of whether our reality is invariant under PCT transformations. The improved comparison of the antiproton and proton which we discuss next strengthens our belief in PCT invariance. The various tests of PCT made by comparing the measured properties of particles and antiparticles are represented in Fig. 15. The stable particles and antiparticles in these tests come in several varieties that are important to distinguish. The proton (antiproton) is a baryon (antibaryon). The proton (antiproton) is composed of three quarks (antiquarks) bound together. The K mesons, like all meson particles and antiparticles, are instead composed of a quark and an antiquark bound together. The third variety of particle is the lepton; the electron and the positron are one example of lepton particle and antiparticle. Leptons are not made of quarks. In fact, so far as we know, leptons are perfect point particles. No experiment has yet been devised that gives evidence of any internal structure at all. It seems crucial to test PCT invariance in a sensitive way for at least one meson system, one baryon system, and one lepton system. The comparison of q/m for the antiproton and protons, discussed next, is the most sensitive test of PCT invariance with a baryon system by approximately a factor of million. The proposed comparison of the hydrogen and antihydrogen, discussed later, is of great interest in that it promises to give a much more sensitive test of PCT invariance with leptons and baryons. B. C C F The first measurement with extremely cold antiprotons was a greatly improved comparison of the charge-to-mass ratios of the antiproton and the proton. Figure 14 represents previous comparisons (with different techniques) along with the series of three TRAP measurements. The basic ideas for TRAP comparisons are illustrated in Figs. 16 and 17. An antiproton, proton, or H\ ion makes a circular orbit in a plane perpendicular to the magnetic field direction as shown. The orbit angular frequency  is simply A related to the charge of the particle q, its mass m, and the strength of the magnetic field B, by q  : B A m

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F. 16. (a) For TRAP I and TRAP II, antiprotons and protons are alternated in the trap. (b) For TRAP III, a simultaneously confined antiproton and H\ ion were interchanged between larger and smaller concentric orbits.

In the strong magnetic field we use, antiprotons, protons and H\ ions make approximately 90 million revolutions. We detect the 90-MHz signal (Fig. 19) induced across the RLC circuit attached to the electrodes of the trap (Fig. 19) using a refined version of an FM radio receiver, and measure the oscillation frequency. The points in Fig. 14b indicate the amount that the ratio of measured antiproton and proton cyclotron frequencies differs from 1. If the magnetic field does not change between measurements of  for the A antiprotons and protons, the ratio of cyclotron frequencies can be interpreted as a ratio of q/m. In reality, an antiproton confined in a Penning trap follows the slightly more complicated orbits represented in Fig. 18. The small circular oscillation is the cyclotron motion discussed in the preceding except that the oscillation frequency is slightly modified by the trap to  . This cyclotron A motion is superimposed on another circular orbit perpendicular to the magnetic field, called magnetron motion, at a much lower frequency  . In K addition, the antiproton oscillates up and down along the direction of the

F. 17. The cyclotron motion induces a detectable voltage across an RLC circuit attached to a 4-segment ring. (Taken from Gabrielse et al., 1999.)

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F. 18. Superimposed cyclotron, axial and magnetron orbit of a particle in a perfect Penning trap. (Taken from Brown and Gabrielse 1986.)

magnetic field at the axial frequency  . The desired cyclotron frequency  X A is deduced from the three measurable frequencies  ,  , and  using an A X K invariance theorem which Lowell Brown and I discovered (Brown and Gabrielse, 1982),  :  ;  ;  (2) A A X K Much of the experimental effort goes into understanding and/or eliminating any imperfection in our apparatus that could change the measured frequencies even slightly. Nonetheless, each of the three measurable frequencies is slightly shifted from the ideal — by a misaligned magnetic field for example. Fortunately, the invariance theorem holds even when the three measurable frequencies are shifted by this misalignment and the other largest sources of frequency shifts. Depending on the accuracy of the measurements, approximations to this general expression can sometimes be used. It is essential that the magnetic field B not change between the time the proton frequencies are measured and the antiproton frequencies are measured. This is challenging in an accelerator environment in which magnetic fields in the accelerator rings are being changed dramatically as often as every couple of seconds. One important aid for all three of our measurements is a superconducting solenoid that not only makes the strong magnetic field but also senses when this field fluctuates and cancels the fluctuation at the location of our trapped particles. This invention (Gabrielse et al., 1988, 1991) is now patented (Gabrielse and Tan, 1990) because of applications in magnetic resonance imaging and ion cyclotron resonance. It illustrates the interplay between ‘‘pure science’’ and technology. Technology is pushed so hard in the pursuit of fundamental physics goals that practical applications with wider applicability can emerge.

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C. TRAP I: O H A C  O H P In our first measurement (Gabrielse et al., 1990), the cyclotron frequency of the center-of-mass of approximately 100 antiprotons was compared to that of protons. This measurement showed that the charge-to-mass ratios of the antiproton and proton are the same to within 4 ; 10\, which is 40 ppb. The self-shielding solenoid kept the magnetic field drift from being a major factor at this accuracy. The improvement over earlier comparisons of antiprotons and protons using exotic atoms was more than a factor of 1000. D. TRAP II: A O A  O P The second mass measurement compared a single trapped antiproton to a single trapped proton (Gabrielse et al., 1995). The radio signal of single antiprotons was detected nondestructively (Fig. 19a). Owing to our high resolution, this measurement provided a spectacular illustration of special relativity (Fig. 19b,c) at eV energies insofar as the antiproton’s cyclotron frequency qB  : A m

(3)

depends upon the familiar relativistic factor  : (1 9 v/c)\ : E/mc.

F. 19. Special relativity shifts the cyclotron frequency of a single trapped p as its cyclotron energy is slowly and exponentially dissipated in the detector. Cyclotron signals for three subsequent times in (a) have frequencies highlighted in the measured frequency versus time points in (b). A fit to the expected exponential has small residuals (c) and gives the cyclotron frequency for the limit of no cyclotron excitation. (Taken from Gabrielse et al., 1995.)

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This second measurement showed that the charge-to-mass ratios of the antiproton and proton differed by less than 1 ; 10\. The 1 ppb uncertainty arose almost entirely because the antiproton and proton have opposite sign of charge, and thus require externally applied trapping potentials of opposite sign. After the cyclotron frequency of one species was measured it would be ejected from the trap, the trapping potential would be reversed, and the second species loaded for measurement. Reversing the applied potential does not completely reverse the potential experienced by a trapped particle (e.g., due to the path effect on the inner surfaces of the trap electrodes). During the measurements of their respective cyclotron frequencies, the antiproton and proton thus reside at slightly different locations, separated by up to 45 m in this case. If the nearly homogeneous magnetic field differs slightly between the broad locations, the measured  for the different species A differs even if the charge-to-mass ratios do not.

E. TRAP III: S T A  H\ I The third and final measurement utilized a single antiproton and a single H\ ion trapped at the same time (Gabrielse et al., 1999b). Both had the same sign of charge and were confined simultaneously, eliminating the systematic effect that limited the previous measurement. To keep the two from interfering with each other, one particle was always ‘‘parked’’ in a large cyclotron orbit. Measurements were made of the cyclotron frequency of the other particle in a small orbit at the center of the large orbit. The electron-to-proton mass ratio, the hydrogen binding energy, and the H\ electron affinity were well enough known that no additional error was contributed by substituting an H\ ion for a proton. In the initial proposal to CERN I suggested that the most accurate q/m comparisons would come by comparing an antiproton and an H\ ion. During the TRAP I and TRAP II measurements we speculated occasionally about whether H\ ions might be formed during antiproton loading, but never got around to looking until we encountered the unavoidable disruption of an H\ ion loaded with a single antiproton. When we did look we found that we could always load negative ions with antiprotons. By reducing the number of cooling electrons we were able to typically load of order 500 H\ at the same time as antiprotons, presumably as hydrogen atoms liberated from the degrader picked up cooling electrons. The electrons then had to be ejected quickly to avoid collisional stripping of the H\. Loading a single antiproton and H\, and preparing them for measurement, typically required 8 h.

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F. 20. (a) Special relativity shifts the cyclotron frequency of an H\ as our detector slowly removes its energy. (b) Similar signals from a p kept simultaneously in a large orbit by three pulsed excitations. (Taken from Gabrielse et al., 1999.)

This mass measurement established that q q ( p ) ( p) : 90.999 999 999 91 (9) m m



(4)

The accuracy exceeds that of the second TRAP measurement by more than a factor of 10 (Fig. 14b), and improves upon the earlier exotic atom measurements by a factor of 6 ; 10. At a fractional accuracy f : 9 ; 10\ : 90 ppt there is thus no evidence for PCT violation in this baryon

F. 21. Alternating cyclotron decays of p and p (from H\) superimposed upon a slightly drifting magnetic field. (Taken from Gabrielse et al., 1999.)

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system. This is the most precise test of PCT invariance with a baryon system by many orders of magnitude as is illustrated in Fig. 15. The comparison of p and H\ also uniquely establishes the limit r&\4; SA 10\, where r&\ :  ( p ) f /(mc) quantifies extensions to the standard SA A model that violate Lorentz invariance, but not PCT (Bluhm et al., 1998). Such violations would make  ( p ) and  (H\) differ in addition to the A A familiar mass and binding energy corrections, without making q/m different for p and p. Our apparatus was clearly capable of a higher accuracy, perhaps even another factor of 10, but LEAR closed down before these measurements could be pushed to their limit.

III. Opening the Way to Cold Antihydrogen A. C A Antihydrogen is the simplest of antimatter atoms, being formed by a positron in orbit around an antiproton. The pursuit of cold antihydrogen began some time ago, long before a few antihydrogen atoms traveling at nearly the speed of light (Baur et al., 1996) generated great publicity. Unlike the extremely energetic antihydrogen, cold antihydrogen that can be confined in a magnetic trap for highly accurate laser spectroscopy offers the possibility of comparisons of antihydrogen and hydrogen at an important level of accuracy (Fig. 22). Gravitational measurements can also be contemplated (Gabrielse, 1988) because the antimatter atom is electrically neutral

F. 22. Relevant accuracies for the precise 1s—2s spectroscopy of antihydrogen are compared to the most stringent tests of PCT invariance carried out with the three types of particles — mesons, leptons, and baryons.

COMPARING THE ANTIPROTON AND PROTON

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and hence not very sensitive to electric and magnetic forces. In my 1986 Erice lecture (Gabrielse, 1987), shortly after we had trapped antiprotons for the first time (Gabrielse et al., 1986), I discussed the possibility of forming cold antihydrogen from cold, trapped antiprotons and positrons. I concluded: For me, the most attractive way . . . would be to capture the antihydrogen in a neutral particle trap. . . . The objective would be to then study the properties of a small number of [antihydrogen] atoms confined in the neutral trap for a long time.

I was inspired by the attempts to confine neutrons and the first trapping of atoms (Migdall et al., 1985). During the time that we were developing the techniques to make cold antiprotons and positrons available for the production of cold antihydrogen, the trapping of atoms including hydrogen (Cesar et al., 1996) also was becoming common. The formation of cold antihydrogen requires first that its ingredients, cold antiprotons and cold positrons, be available in the extremely high vacuum that is desirable for accumulating these particles and for storing cold antihydrogen. The first half of this chapter focused upon the techniques required to slow, trap, cool, and accumulate 4.2 K antiprotons. In the following section we summarize the availability of 4.2 K positrons. The next step on the path to cold antihydrogen is to bring the cold ingredients together. The ‘‘nested Penning trap’’ (Section III.C), proposed for this purpose, has since been demonstrated. Section III.D summarizes an experiment in which we stored cold antiproton and positrons for the first time in either a nested trap or at 4.2 K — the closest approach yet to cold antihydrogen. Several different recombination mechanisms (Section III.E) will be investigated whereby cold antiprotons and cold positrons could recombine to form cold antihydrogen. B. 4.2 K P  E G V Cold positrons are the other required ingredient for cold antihydrogen. Since only a few cold positrons had ever been confined in the extremely high vacuum that is desirable for cold antihydrogen experiments we set about capturing large numbers of cold positrons in this environment. We first used electronic damping (Haarsma et al., 1995). Then we discovered and developed a new technique in which we produced Rydberg positronium and ionized it within a trap. This approach yielded a vastly improved accumulation rate (Estrada et al., 2000), up to 10 positrons per hour for a 2.5 mCi Na source. Figure 23a shows the simplicity of the apparatus. A thin transmission moderator, a 2-m tungsten crystal W(110), is added to an open access

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F. 23. The electrodes of an open access Penning trap (a) are biased to produce an electric potential (b) and field (c) along the central axis that confines e> (solid curves) or e\ (dashed curves). A 5.3 T magnetic field parallel to this symmetry axis guides fast positrons entering from the left through the thin crystal and towards the thick crystal. (Taken from Estrada et al., 2000.)

Penning trap (Gabrielse et al., 1989c) at one end. A thick reflection moderator, a 2-mm tungsten crystal W(100), is added at the other. Positrons from the radioactive source, traveling along field lines of a strong magnetic field (5.3 T), pass through the transmission moderator to enter the trap from the left. The electric field of the trap ionizes the Rydberg positronium, which then accumulates in the location shown. Figure 24 shows the accumulation of more than a million positrons. We expect soon to increase the accumulation rate dramatically by simply increasing the size of the 2.5 mCi source to 150 mCi. The crucial time period for positron accumulation at the AD is of order of an hour, the amount of time it will take to stack a reasonable number of antiprotons in a trap. C. D  N P T The production of cold antihydrogen requires that antiprotons and positrons be allowed to interact. The nested Penning trap (Fig. 25) proposed for this purpose (Gabrielse et al., 1988) was demonstrated experimentally (Hall and Gabrielse, 1996) with protons and electrons. The demonstration shows that a nested Penning trap should allow antiprotons and positrons to interact with a low relative velocity, as illustrated in Fig. 26. Without cooling electrons in the central well, hotter antiprotons retain the higher energy distributions, illustrated on the right-hand side of the figure. With cooling electrons in the well the antiproton energy spectrum cools dramatically as shown on the left-hand side of the figure.

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31

F. 24. (a) Accumulation of 1.1 million positrons and (b) their electrical signal. (Taken from Estrada et al., 2000.)

D. C  C A  E B In the last week of LEAR’s operation we got closer to cold antihydrogen than anyone has ever been before (Gabrielse et al., 1999a). Figure 27 shows the first simultaneous confinement of 4 K antiprotons and positrons, and Fig. 28 shows trapped positrons heated by trapped antiprotons.

F. 25. Scale outline of the inner surface of the electrodes (a), and the potential wells (b), for the nested Penning trap. (Taken from Hall and Gabrielse, 1996.)

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F. 26. Energy spectrum of the hot protons (right-hand side) and the cooled protons (left-hand side), obtained by ramping the potential on electrode K downward and counting the protons that spill out to the channel plate. The hot and cooled spectra for 4 initial proton energies are summed. (Taken from Hall and Gabrielse, 1996.)

F. 27. (a) Electrode cross sections and the initial position of the simultaneously trapped p and e>. (b) Trap potential on the symmetry axis. Fits (solid curves) to the electrical signals from simultaneously trapped e> (c) and p (d) establish the number of trapped particles. (Taken from Gabrielse et al., 1999a.)

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F. 28. The signal from cold trapped positrons (below) changes dramatically (above) when heated antiprotons pass through cold positrons in a nested Penning trap, showing the interaction of antiprotons and positrons. (Taken from Gabrielse et al., 1999a.)

E. R M To form antihydrogen, a positron and antiproton must have kinetic energy to approach each other, and this energy must be removed to form an atomic bound state. Energy and momentum cannot be conserved unless a third particle is involved. Different antihydrogen formation processes provide different ways to conserve energy and momentum. Of course, this recombination must occur within the extremely good vacuum demonstrated with antiprotons or else the antihydrogen will not live long enough to be studied to an interesting accuracy. Long ago we compared different mechanisms by which cold antihydrogen might be formed in a Penning trap (Gabrielse et al., 1988), suggesting that a ‘‘nested Penning trap’’ might provide the most useful environment. Three of the antihydrogen formation processes that have been studied (and possibly one hybrid) are attractive candidates. A very nice feature of the nested Penning trap we have demonstrated is that it gives a very easy way to select one process from another, and to rapidly switch between them within the same apparatus. A shallow central trap will select a three-body recombination process. A deep central trap will select radiative recombination, whose rate can be enhanced by switching on an appropriate laser. We will first consider the processes that look most attractive (Section III.E.1), then briefly look at other processes (Section III.E.2). 1. Selecting Processes Within a Nested Penning Trap Within a shallow nested Penning trap, the electric field is very low. Antihydrogen initially formed in a high Rydberg state will be less easily

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ionized when the electric field is low. (The positron plasma will screen the electric field along the axis of the magnetic field, but not in the radial direction.) The dominant process should then be the three-body processes p ; e> ; e> ; H ; e>

(5)

because this promises to have a much higher rate than any other process (Gabrielse et al., 1988). De-excitation to the ground state takes some time (Glinsky and O’Neill, 1991; Fedichev, 1997), but not nearly so long as was originally thought (Menshikov and Fedichev, 1995). In a shallow central well it should be possible to use 10 antiprotons at 4 K, submerged within an extended plasma of 4 K positrons at a density of 10/cm. (This density is lower by 10 than what we have already achieved in a deeper well.) Under these conditions, except with no magnetic field, the calculated recombination rate is an astounding 10/s (Gabrielse et al., 1988). A strong magnetic field (e.g., from the trap containing antiprotons and positrons) would reduce this high rate (Glinsky and O’Neill, 1991) by approximately a factor of 10, and an electric field (also part of the trap) also has some effect (Menshikov and Fedichev, 1995), but the rate is still higher than for any other recombination processes. The related three-body recombination process p ; e> ; e\ ; H ; e\

(6)

has also been mentioned (Gabrielse et al., 1988). Within a deeper nested Penning trap, there will be a much stronger electric field in the central region where antiprotons and positrons interact. The rapid three-body mechanism (Eq. 5) should thus be essentially turned of since antihydrogen initially formed in a high Rydberg state will be ionized before de-excitation occurs. The slower radiative recombination process would then be selected, and it could be enhanced by the illumination of an appropriate laser. Radiative recombination can be thought of as producing an excited hydrogen atom that radiates a photon to conserve energy and momentum, p ; e> ; H ; h

(7)

This process suffers from a lower rate because it takes approximately a nanosecond to radiate a photon, much longer than the duration of the interaction between a p and an e>, even when these particles have an energy low enough to correspond to 4 K. For 10 antiprotons at 4 K within a 4 K positron plasma of density n : 10/cm (a positron density that we have C already realized), the estimated recombination rate (Gabrielse et al., 1988)

COMPARING THE ANTIPROTON AND PROTON

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is 3 ; 10/s. The radiative recombination process has the attractive feature that most of the antihydrogen is formed in the ground state. If we succeed in increasing the positron density to 10/cm, this rate would be 100 times higher. The radiative recombination rate can be increased by stimulating the process with a laser, p ; e> ; h ; H ; 2h

(8)

Laser-stimulated, radiative recombination has been observed in merged beams (Schramm et al., 1991; Yousif et al., 1991) but has yet to be observed in a trap. It has the useful diagnostic feature that the formation rate will increase sharply as the laser is tuned through resonance. For cold positrons and antiprotons, the two transitions that are easily accessible with relatively high power lasers are to n : 3 at 820.6 nm, and to n : 11 at 11,032 nm (Gabrielse et al., 1988). Stimulating to n : 11 has the higher rate, and subsequent radiation will take 99% of the population to the ground state. A N O laser or a CO   laser with a modest and manageable power of 10 W/mm will nearly saturate the transition. Nearly 2% of the antiproton should be converted to antihydrogen at this power for a positron density of 10/cm. We use this low positron density because here a hybrid process could be even more attractive. The rate would be much higher if a three-body recombination to approximately 4 kT below the ionization limit was followed by a laserstimulated recombination to n : 11. 2. Other Formation Processes Another antihydrogen formation process has been extensively discussed. The process uses positronium, the bound state of an electron and a positron, with the electron carrying off the excess (Humbertson et al., 1987). p ; e>e\ ; H ; e\

(9)

One advantage is that the antihydrogen is produced preferentially in the lowest states. When antiprotons are not available, the charge-conjugate process can be studied (recombining protons and positronium to form hydrogen and positrons). This was recently observed (Merrison et al., 1997) using a beam of protons. Unfortunately, comparable quantities of antiprotons are very difficult to arrange. It should be possible to increase the recombination rate by initially exciting the positronium atom with a laser

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(Charlton, 1990). Nonetheless, the disadvantage of the recombination using positronium is that the projected rate is still extremely low for both realizable numbers of antiprotons and existing positronium beams. Estimates place the production rate at perhaps 100 atoms per day for continuous production at the AD, or perhaps 2 antihydrogen atoms per pulse of 10 positrons if this could be arranged (Surko et al., 1997). With such a low rate, and an apparatus somewhat different from what is required for the other formation processes, we do not plan to pursue this approach. However, we are intrigued by the possibility that the high Rydberg positronium we have already realized may allow us to resurrect it. Finally, there are more recent suggestion. Charge exchange processes may provide a route to the formation of cold antihydrogen (Hessels et al., 1997). Recombination of electrons and positive ions aided by an electric field, has recently been observed (Wesdorp et al., 2000); this technique may be applicable to antihydrogen formation as well. We are investigating both routes. F. F A substantial Antiproton Decelerator (AD) facility is nearing completion at CERN to carry forward experiments with low energy antiproton. It looks like the AD will be delivering useful numbers of antiprotons in fall 2000, and meeting (or even exceeding) its design specifications in summer of 2001. Two large collaborations have formed to produce and study cold antihydrogen. Our TRAP collaboration expanded to become ATRAP (ATRAP, 2000). Our competition is ATHENA (ATHENA, 2000), which grew out of the attempt mentioned earlier to measure the gravitational acceleration of antiprotons. In July, our ATRAP colloboration announced the first trapping of antiprotons from the AD, the first electron-cooling of these trapped antiprotons, and began stacking antiprotons in a trap. ATHENA apparatus should be ready soon. Experiments to build upon the TRAP foundation have just begun. Despite some claims to the contrary, antihydrogen production is a very difficult undertaking that will take some time. Estimated production rates are dautingly low even though it should be possible to detect single antihydrogen atoms, by detecting the pions from antiproton annihilation and the gammas from positron annihilation. New techniques must be devised to cool antihydrogen to the low energies required for trapping, since current hydrogen cooling techniques involve collisions with cold surfaces that would cause the antihydrogen to be annihilated. It also remains to be demonstrated that useful spectroscopic measurements can be done with only a few antihydrogen atoms in a trap. The cause is worthy but many challenges remain.

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IV. Technological Spinoffs In the pursuit of fundamental measurements it is not uncommon to push technology very hard with the result that unpredicted new techniques and devices emerge. I mention two examples from our antiproton studies: ( The charge-to-mass measurements required a very stable magnetic field, but were performed in an accelerator hall where the cycle of the Proton Synchrotron (PS) caused the magnetic field in our experiment to make a large change every 2.4 s. Our solution was to invent a self-shielding superconducting solenoid (Gabrielse and Tan, 1988). We demonstrated (Gabrielse et al., 1991) that this addition reduced the size of the fluctuations in the magnetic field by a factor of 150 or more. Without this invention the extremely accurate q/m measurements would not have been possible. The same invention makes it possible to do more accurate ion cyclotron resonance (ICR) measurements (to analyze the composition of potential drugs, etc.) and nuclear magnetic resonance. The self-shielding solenoid is now patented (Gabrielse and Tan, 1990) and available commercially. ( The open endcap Penning trap (Gabrielse et al., 1989c) we developed to allow antiprotons to enter our trap will also provide ready access for any other charged particle, laser beams, etc. This design is increasingly being used as the cell design of choice for ICR measurements

V. Acknowledgments It was an honor and pleasure to lead the TRAP collaboration (Table I); a succession of gifted students and postdocs immersed themselves in this work. I am especially grateful to early collaborators, Kells and Trainor, and my long time collaborator, Kalinowsky, for their courage in embarking upon an adventure few thought would succeed or even be supported. Without the unique LEAR facility at the CERN laboratory, the antiproton experiments would not have been possible. We profited from the help and personal encouragement of the LEAR staff, the SPSLC, the research directors and the directors general of CERN. Most of the support for the antiproton experiments came from the NSF and AFOSR of the USA, with an initial contribution from NIST. The German contribution to these experiments came from the BMFT. The positron experiments were supported by the ONR of the USA.

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VI. References ATHENA (2000). httpp://athena.web.cern.ch/athena/. ATRAP (2000). http://hussle.harvard.edu/:atrap. Baur, G. et al. (1996). Production of antihydrogen. Phys. Rev. B 368:251. Bluhm, R., Kostelecky, V. A., and Russell, N. (1998). CPT and Lorentz tests in Penning traps. Phys. Rev. D57, 3932—3943. Brown, L. S. and Gabrielse, G. (1982). Precision spectroscopy of a charged particle in an imperfect Penning trap. Phys. Rev. A 25:2423—2425. Brown, L. S. and Gabrielse, G. (1986). Geonium. Theory: Single Electrons and Ions in a Penning trap. Rev. Mod. Phys. 58:233—311. Cesar, C. L., Fried, D. G., Killian, T. C., Polcyn, A. D., Sandberg, J. C., Yu, I. A., Greytak, T. J., Kleppner, D., and Doyle, J. (1996). Two-photon spectroscopy of trapped atomic hydrogen. Phys. Rev. L ett. 77:255. Charlton, M. (1990). Antihydrogen production in collisions of antiprotons with excited states of positronium. Phys. Rev. A 143:143. Estrada, J., Roach, T., Tan, J. N., Yesley, P., Hall, D. S., and Gabrielse, G. (2000). Field ionization of strongly magnetized Rydberg positronium: A new physical mechanism for positron accumulation. Phys. Rev. L ett. 84:859—862. Fedichev, P. O. (1997). Formation of antihydrogen atoms in an ultra-cold positron-antiproton plasma. Phys. Rev. A 226:289—292. Gabrielse, G. (1987). Penning traps, masses and antiprotons, in Fundamental Symmetries, P. Bloch, P. Paulopoulos, and R. Klapisch, eds., New York: Plenum, p. 59—75. Gabrielse, G. (1988). Trapped antihydrogen for spectroscopy and gravitation studies: Is it possible? Hyper. Int. 44:349—356. Gabrielse, G. (1992). Extremely cold antiprotons. Sci. Amer., December, 78—89. Gabrielse, G., Fei, X., Helmerson, K., Rolston, S. L., Tjoelker, R. L., Trainor, T. A., Kalinowsky, H., Haas, J., and Kells, W. (1986). First capture of antiprotons in a penning trap: A keV source. Phys. Rev. L ett. 57:2504—2507. Gabrielse, G., Fei, X., Orozco, L. A., Rolston, S. L., Tjoelker, R. L., Trainor, T. A., Haas, J., Kalinowsky, H., and Kells, W. (1989a). Barkas effect observed with antiprotons and protons. Phys. Rev. A 40:481—484. Gabrielse, G., Fei, X., Orozco, L. A., Tjoelker, R. L., Haas, J., Kalinowsky H., Trainor, T. A., and Kells, W. (1989b). Cooling and slowing of trapped antiprotons below 100 meV. Phys. Rev. L ett. 63:1360—1363. Gabrielse, G., Fei, X., Orozco, L. A., Tjoelker, R. L., Haas, J., Kalinowsky, H. Trainor, T. A., and Kells, W. (1990). Thousand-fold improvement in the measured antiproton mass. Phys. Rev. L ett. 65:1317—1320. Gabrielse, G., Haarsma, L., and Rolston, S. L. (1989c.). Open-endcap Penning traps for high precision experiments. Intl. J. Mass Spec. and Ion Phys. 88:319—332. Gabrielse, G., Hall, D. S., Roach, T., Yesley, P., Khabbaz, A., Estrada, J., Heimann, C., and Kalinowsky, H. (1999a). The ingredients of cold antihydrogen: Simultaneous confinement of antiprotons and positrons at 4 K. Phys. L ett B 455:311—315. Gabrielse, G., Khabbaz, A., Hall, D. S., Heimann, C., Kalinowsky, H., and Jhe, W. (1999b). Precision mass spectroscopy of the antiproton and proton using simultaneously trapped particles. Phys. Rev. L ett. 82:3198—3201. Gabrielse, G., Tan, J. N., Clateman, P., Orozco, L. A., Rolston, S. L., Tseng, C. H., and Tjoelker, R. L. (1991). A superconducting solenoid system which cancels fluctuations in the ambient magnetic field. J. Mag. Res. 91:564—572.

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Gabrielse, G., Phillips, D., Quint, W., Kalinowsky, H., Rouleau, G., and Jhe, W. (1995). Special relativity and the single antiproton: forty-fold improved comparison of P and P charge-tomass ratios. Phys. Rev. L ett. 74:3544—3547. Gabrielse, G., Rolston, S. L., Haarsma, L., and Kells, W. (1988). Antihydrogen production using trapped plasmas. Phys. L ett. A 129:38—42. Gabrielse, G. and Tan, J. N. (1988). Self-shielding superconducting solenoid systems. J. Appl. Phys. 63:5143—5148. Gabrielse, G. and Tan, J. (1990). US patent 4974113, issued 27 Nov. 1990. Glinsky, M. E. and O’Neil, T. M. (1991). Guiding center atoms: Three-body recombination in a strongly magnetized plasma. Phys. Fluids B 3:1279. Haarsma, L. H., Abdullah, K., and Gabrielse, G. (1995). Extremely cold positrons accumulated electronically in ultrahigh vacuum. Phys. Rev. L ett. 75:806—809. Hall, D. S. and Gabrielse, G. (1996). Electron-cooling of protons in a nested Penning trap. Phys. Rev. L ett. 77:1962—1965. Hessels, E. A., Homan, D. M., and Cavagnero, M. J. (1997). Two-stage rydberg charge exchange: An efficient method for production of antihydrogen. Phys. Rev. A 57:1668. Holzscheiter, M. H., Feng, X., Goldman, T., King, N. S. P., Nieto, M. M., and Smith, G. A. (1996). Are antiprotons forever? Phys. L ett. A 214:279. Men’shikov, L. I. and Fedichev, P. O. (1995). Theory of elementary atomic processes in an ultracold plasma. Zh. Eksp. Teor. Fiz. 108:144. (JETP 81, 78). Merrison, J. P., Bluhme, H., Chevallier, J., Deutch, B. I., Hvelplund, P., Jorgensen, L. V. Knudsen, H., Poulsen, M. R., and Charlton, M. (1997). Hydrogen formation by proton impact on positronium. Phys. Rev. L ett. 78:2728—2731. Migdall, A. L., Prodan, J. V., Phillips, W. D., Bergeman, T. H., and Metcalf, H. J. (1985). First observation of magnetically trapped neutral atoms. Phys. Rev. L ett. 54:2596. Morgan, D. L., Jr. and Hughes, V. W. (1970). Atomic processes involved in matter-antimatter annihilation. Phys. Rev. D 2:1389. Peil, S. and Gabrielse, G. (1999). Observing the quantum limit of an electron cyclotron: QND measurements of quantum jumps between Fock states. Phys. Rev. L ett. 83:1287—1290. Schramm, U., Berger, J., Grieser, M., Habs, D., Jaeschke, E., Kilgus, G., Schwalm, D., Wolf, A., Neumann, R., and Schuch, R. (1991). Observation of laser-induced recombination in merged electron and proton beams. Phys. Rev. L ett. 67:22. Surko, C. M., Greaves, R. G., and Charlton, M. (1997). Stored positrons for antihydrogen production. Hyper. Int. 109:181—188. Tseng, C. and Gabrielse, G. (1993). Portable trap carries particles 5000 kilometers. Hyper. Int. 76:381—386. Wesdorp, C., Robicheaux, F., and Noordam, L. D. (2000). Field-induced electron-ion recombination, a novel route towards neural (anti-) matter. Phys. Rev. L ett. 84:3799—3802. Yousif, F. B., Van der Donk, P., Kucherovsky, Z., Reis, J., Brannen, E., Mitchell, J. B. A., and Morgan, T. J. (1991). Experimental observation of laser-stimulated radiative recombination. Phys. Rev. L ett. 67:26.

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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 45

MEDICAL IMAGING WITH LASER-POLARIZED NOBLE GASES TIMOTHY CHUPP and SCOTT SWANSON Departments of Physics and Radiology, University of Michigan Ann Arbor, Michigan 48109 I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Historial Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Nuclear Polarization Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Optical Pumping and Spin Exchange . . . . . . . . . . . . . . . . . . . . . 1. He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Xe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Lasers for Spin Exchange Pumping . . . . . . . . . . . . . . . . . . . . 4. Optical Pumping with Laser Diode Arrays . . . . . . . . . . . . . . B. Metastability Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Lasers for Metastability Exchange . . . . . . . . . . . . . . . . . . . . . C. Polarization and Delivery Systems . . . . . . . . . . . . . . . . . . . . . . . III. Basics of Magnetic Resonance Imaging (MRI) . . . . . . . . . . . . . . . . A. Nuclear Magnetic Resonance (NMR) . . . . . . . . . . . . . . . . . . . . B. One-Dimensional Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Magnetic Resonance Imaging and k-Space . . . . . . . . . . . . . . . . D. Imaging Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Selective Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Back Projection Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Gradient Echo Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Chemical Shift Imaging (CSI) . . . . . . . . . . . . . . . . . . . . . . . . E. Contrast in Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . F. Low Field Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Imaging Polarized Xe and He Gas . . . . . . . . . . . . . . . . . . . . . . A. Magnetic Resonance Imaging of Polarized Gas: General Concerns 1. Sampling of the Magnetization . . . . . . . . . . . . . . . . . . . . . . . 2. Diffusion and k-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Airspace Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Injection of He and Xe Carriers . . . . . . . . . . . . . . . . . . . . . . V. NMR and MRI of Dissolved Xe . . . . . . . . . . . . . . . . . . . . . . . . . A. Spectroscopy of Xe in Vivo . . . . . . . . . . . . . . . . . . . . . . . . . . B. Xe Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Lung Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Time Dependence and Magnetic Tracer Techniques . . . . . . . . . 1. Dynamics of Laser-Polarized Xe in Vivo . . . . . . . . . . . . . . VI. Conclusions — Future Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Copyright  2001 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-003845-5/ISSN 1049-250X/01 $35.00

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Timothy Chupp and Scott Swanson Abstract: The field of medical imaging with polarized rare gases, just five years old, has brought optical scientists together with medical researchers to perfect techniques and pursue new opportunities for biomedical research. This review, written for the likely reader of these volumes, aims to present the field from several perspectives. The historical perspective shows how applications of nuclear polarization for experiments in nuclear and particle physics led to techniques for production of large quantities of highly polarized He that are increasingly reliable and economical. The atomic/optical physics perspective details the underlying processes of optical pumping, polarization, and relaxation of the rare gases. The biomedical perspective describes work to date and the potential applications of imaging in medicine and research.

I. Introduction Five years ago, a short article was published in the journal Nature showing magnetic resonance images (MRI) of Xe gas that had filled the airways of an excised mouse lung (Albert et al., 1994). The images were acquired at SUNY, Stony Brook, on Long Island, NY. But the gas, prepared by laser optical pumping methods, in Princeton, New Jersey, was transported over 100 km by car in a small glass cell immersed in a cup of liquid nitrogen. (The gas was ‘‘polarized’’ in Princeton to provide 10,000 times greater NMR signal per atom than produced by ‘‘brute force.’’ This compensated for the 10,000 times lower concentration of gas.) Reading the Nature article led many in the field of laser optical pumping to turn their attention to the new possibilities, and many radiologists sought out laser physicists as collaborators to help develop potential biomedical applications. Today, physicists, radiologists, neuroscientists, medical researchers, and clinicians are working together in teams around the world. The promise of entirely new ways to use NMR and MRI information from He and Xe images of gas in the lungs and of xenon dissolved in lung, heart, and brain tissues has attracted the attention of scientists and physicians, as well as the pharmaceutical industry. The promise is that this marriage of laser/optical physics and medical imaging will provide new ways to study and map brain function, measure physiological parameters, and diagnose diseases of the lungs, heart, and brain that depend on the flow of gas and blood through the vital organs. In Figs. 1 and 2 we show magnetic resonance images produced with laser polarized He and Xe. In Fig. 1, a series of consecutive images of a slice through the human lungs shows the flow of gas into the air spaces after a breath is taken and exhaled (Saam et al., 1999). This moving picture of gas flow is called a ventilation image. Ventilation images made with scintigraphy of radioactive gas (usually Xe) are used to assess lung function and find nonfunctioning portions of the lung. Combined with measures of blood flow through the lung, ventilation scans help diagnose a variety of lung diseases, such as pulmonary embolism, with moderate specificity. (The efficacy of a diagnostic technique is characterized by its sensitivity and

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F. 1. A He ventilation image: a series of images taken at 0.8-s intervals after a breath of laser polarized He is inhaled while the human subject is in the MRI scanner. The gas starts in the trachea, then moves out and down to the lower parts of the lung. The last frame shows the beginning of exhalation as the upward motion of the diaphragm expels gas from the lungs. Images courtesy of Brain Saam. Used with permission.

specificity. Sensitivity is essential to discover a malady while specificity is required to determine the exact problem and the course of treatment.) In Fig. 2 (see also Color Plate 1), we show images of Xe gas in the lung and dissolved in tissue and blood of a rat that had been breathing a polarized xenon-oxygen mixture. In contrast to helium, xenon crosses the blood-gas barrier in the lungs, dissolves in blood, and is carried to distal organs where it diffuses into tissue as the blood flows through capillaries from artery to vein. The NMR frequencies of Xe differ by about 200 ppm for gas and dissolved phases, and vary by several ppm among different types of tissue and blood. The development of techniques of laser-enhanced nuclear polarization (or hyperpolarization), has been most strongly motivated by nuclear and particle physics. Targets of polarized He are used in accelerator experiments such as those that probe the elementary particle, short-range structure of the neutron (Chupp et al., 1994a). Polarized He is also used to polarize neutrons for nuclear physics and neutron scattering research (Coulter et al., 1988). These driving motivations and applications along with other historical developments are described in Section I.A. The requirements

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F. 2. Magnetic resonance images of Xe in the lungs and dissolved in the blood and tissue of a rat. The gray-scale images are conventional proton MRI (spin-echo) images that show the animal’s anatomy. The false-color images show the concentration of Xe magnetization for each of three spectral features corresponding to xenon in the gas phase (C and F), dissolved in tissue (B and E) and dissolved in blood (A and D). Panels (A through C) are called axial images across the body, and (D through F) are coronal images through the body. (See also Color Plate 1).

of these experiments have pushed us to understand the physics and technical limitations of optical pumping at high densities. We can now produce liters (at STP) of He, polarized to 50% or more, and similar quantities of Xe. Optical pumping, polarization techniques, lasers, and other technical details are discussed in Section II, and the basics of NMR and MRI are described in Section III. The exciting new possible applications to medical imaging described in Sections IV and V deal with air space imaging and dissolved phase imaging, respectively. We conclude in Section VI by emphasizing some of the potential applications and future promise of this new technique — it gives a wonderful example of transfer of technology motivated by fundamental physics research. A. H P The atomic nucleus of an odd-A or odd-Z isotope in general has nonzero nuclear spin and nonzero magnetic moment. These nuclear spins and

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moments have long been important in the development of nuclear physics through the comparison of experiment with the nuclear shell model theory (see, e.g., Ramsey’s book on nuclear moments, 1953). Nuclear spin has also been an important variable for a range of approaches to studying nuclear interactions. Perhaps the best example is He. The stable A : 3 isotope He has been extremely important in nuclear physics. Calculation of the magnetic moment has clarified the role of meson exchange corrections. Nuclear reactions induced by He and H allow study of isospin symmetry and isospin dependence in a unique way because the A : 1 isodoublet is much more difficult experimentally — accelerated neutron beams are not feasible. Perhaps most important is the fact that the neutron polarization in a polarized He nucleus is 87%. This has allowed important short-range properties of the neutron to be measured including the neutron electric charge distribution — the electric form factor GL — and the neutron deep # inelastic scattering spin structure functions g (x). Therefore, beams and L targets of polarized He have been sought since at least the 1950s. It was the late 1980s before the basic physical processes and technology came together to foresee practical polarized He targets. The He isotope was accidently discovered by Alvarez (1987) at the Berkeley cyclotron in a test run intended to use the cyclotron as a mass spectrometer to detect He produced in nuclear reactions. (It had been assumed, following the suggestion of Bethe, that He was unstable.) After the experiment, the magnet was ramped down with the RF on, revealing an ion with Z/A : 2/3. Once the cyclotron magnet was shimmed properly for this mass, the discovery of He was confirmed. One surprising consequence of that discovery was that He and not H is the stable A : 3 isotope. With two protons and one neutron, the He nucleus must have half-integer spin, and naive consideration of the Pauli principle suggests that the protons’ spins pair in a singlet l : 0 state. In that case the entire angular momentum and magnetic moment of the He nucleus would be due to the neutron. In fact He as well as H are spin 1/2, but the tensor component of the nuclear force and isospin breaking lead to a complicated many-component wave function with l : 0, 1 and 2, and mixed isospin states (Afnan and Birrell, 1977). The total angular momentum of the nucleus has contributions from the D-state and a small proton polarization opposite the He polarization. Even this is not enough to account for the measured magnetic moments: isovector meson exchange currents apparently contribute opposite amounts to the He and H magnetic moments. Therefore the total magnetic moment of He can be written

 (He) : PL ; PN ;  ( L ; A I ) +#! X X L N , X

(1)

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Timothy Chupp and Scott Swanson

where PL : 2 sL : 0.87 and PN : 90.027 are, respectively (Friar et al., X 1990), the neutron and proton polarizations, and L : 0.061. The meson X exchange correction is A I : 90.35 for the isospin projection +#! X I : 91/2. The nuclear magneton is  : e /2m c. X , N The He isotope is rare, and this is a problem. Primordial abundance of He produced in Big Bang nucleosynthesis is [He]/[He] : 0.00004 (Arnett and Turan, 1985). Additional He has been produced in stellar burning (Trimble, 1982), in the atmosphere due to cosmic ray interactions, and underground due to natural radioactivity. Cosmic ray production of He on the moon, which does not have atmospheric shielding from cosmic rays, has left much greater abundances embedded in lunar rocks (Wittenberg et al., 1986), although mining the moon may remain so expensive as to be impractical. Most of the stored He reserve, 1000 kg, has come from the decay of tritium (H) produced for thermonuclear weapons. Another feature of He has motivated work to develop polarization techniques. It turns out to be a potentially perfect spin filter for polarization of neutrons. The strong neutron absorption reaction n;He ; p ; H is nearly 100% polarization dependent, due to an unbound J : 0 resonance in He. With the neutron and He spins opposite, the absorption cross section is  (v) : 5230v /v barns (v : 2200 m/s is the rms velocity of ?   thermal neutrons). Other n ; He interactions are negligible. Passing a neutron beam through a filter of polarized He produces a beam polarized parallel to the He, though reduced in flux (Williams, 1980; Coulter et al., 1988; Coulter et al., 1990). The polarization and transmission for a filter with He thickness [He]t and polarization P are given by  P : tanh( [He]tP )P : cosh( [He]t)P exp( [He]tP ) L ?  R ?  ? 

(2)

Polarized neutron beams are widely sought for condensed matter and materials science research (Fitzsimmons and Sass, 1989) and for studies of the nuclear interactions of scattered (Heckel et al., 1982), absorbed (Mitchell et al., 1999), and decaying neutrons (Abele et al., 1997). Scattering of neutrons from materials reveals structure and momentum distributions, and the spin dependence is used to study magnetism, for example of multiple thin layers sought for magnetic recording media (Kubler, 1981). In contrast to synchrotron x-rays, the magnetic interactions of neutrons are comparable in strength to electric interactions; for photons, magnetic interactions are suppressed by 300. The decay of polarized neutrons provides the opportunity to study the weak interactions (Jackson et al., 1957), and weaker interactions (Herczeg, 1998), such as those that emerge from extensions of the Standard Model of elementary particle interactions including SuperSymmetry.

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The first attempts to produce usable polarized He targets were not successful in spite of heroic efforts. Most notable was the effort by Timsit et al. (1971a,b). Employing optical pumping of metastable helium atoms with lamps (described in Section II), they developed a mercury Toeppler pump (later adapted by Becker and co-workers, 1994) and provided an important study of He polarization relaxation in the presence of many materials (Timsit et al., 1971b). One particularly crucial conclusion was the importance of a predominantly glass system. Timsit, Daniels, and their co-workers presented a theory for predicting He polarization relaxation rate dependence on helium permeability and glass iron content that confirmed that one should use alumino-silicate glasses such as Corning 1720 (Fitzsimmons et al., 1969), Schott Supermax (Becker et al., 1994), and Corning 7056 (Smith, 1998). Quartz and fused silica, though relatively porous to helium, can be produced with extremely low iron content, and are useful particularly for neutron spin filters since the B in most glasses strongly absorbs low-energy neutrons. Timsit and Daniel’s efforts fell short of the goals of 0.5 l-atm of He with 25% polarization. A decade later, the availability of lasers led to success with the first He neutron spin filter (Coulter et al., 1990) and the first targets for electron scattering for study of the neutron charge form factor GL by quasielastic scattering of # polarized electrons from the polarized He (Woodward et al., 1990; Thompson et al., 1992; Chupp et al., 1992; Becker et al., 1994). (Quasielastic scattering breaks up the nucleus by momentum transfer to a single nucleon. The spin dependence is generally dominated by the neutron.) The next generation of polarized He targets was used for electron scattering at SLAC in a program that revealed the spin content of the neutron’s quarks in deep inelastic scattering (Anthony et al., 1993; Middleton et al., 1993; Abe et al., 1997). These targets employed spin exchange with laser-polarized Rb vapor, a technique that had been considered less favorable for several reasons including the extremely weak hyperfine spin exchange interaction (Walker and Happer, 1997) and problems of radiation trapping — depolarization by multiple scattering of photons in the dense alkali-metal vapor. However, it had been shown that 60—100 torr of N is  sufficient to suppress radiation trapping (Hrycyshyn and Krause, 1970) and that optical pumping with lasers was effective at extremely high optical density with [Rb]  10 cm\ (Chupp and Coulter, 1985). More detailed studies of optical pumping at high alkali-metal density (Chupp et al., 1987; Wagshul and Chupp, 1994) showed that laser intensity was the primary limitation and that He pressures of greater than 10 bar in volumes limited only by laser power to 200 cm became possible with CW standing wave titanium::sapphire lasers (Larson et al., 1991).

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The two methods for polarizing He, discovered in the early 1960s, became competing techniques in the 1980s. Metastability exchange was pursued by Becker et al. (1994) and Bohler et al. (1988). This led to the neutron spin filter program at ILL, Grenoble (Surkau et al., 1997; Heil et al., 1998), and quasielastic scattering measurements at Mainz (Becker et al., 1994, 1998), both using a two-stage, titanium pump compressor to increase the He pressure from 1 torr to 1 bar. The metastability exchange technique has also been used to pump He into a cooled cell in a quasielastic scattering experiment (Woodward et al., 1990) and to fill a ‘‘storage cell’’ that is coaxial with the circulating 30 GeV positron beam in the HERA ring at DESY, Hamburg, Germany (Ackerstaff et al., 1997). Spin exchange has been most successful in producing high-density polarized He that is essential for targets used in extracted beam experiments such as the SLAC End Station A deep inelastic scattering program (Abe et al., 1997; Anthony et al., 1996) and recent efforts at TJNAF in Newport News, Virginia (Gao, 1998). The possibility of nuclear spin gyroscopes also emerged as optical pumping techniques were developed (Colgrove et al., 1963; Grover, 1983). A nuclear spin gyroscope does not require the large quantities of highly polarized He demanded by applications of polarized nuclear targets and polarized neutrons. However, the concept does rely on the longest possible spin-relaxation and spin-coherence times. Long spin-relaxation times are also important for high polarization, and the development of gyroscopes at industry laboratories helped advance the study of surface relaxation mechanisms. While the technical advances in polarized He have been largely motivated by work on polarized targets for nuclear and high energy physics, Xe polarization was advanced in optical pumping studies (Zeng et al., 1985). Early in the 1980s they began extensive investigations of spin exchange between noble gases and optically pumped alkali-metal vapors (Happer et al., 1984). They studied many processes involved in optical pumping of alkali-metal vapors in the presence of buffer gases, providing extensive data on the xenon-Rb system (Zeng et al., 1985). The Xe polarization of nearly 100% in small cm volumes was produced; experiments included studies of I : 3/2 Xe as well as radioactive isotopes (XeK, Xe, and XeK) (Calaprice et al., 1985). The work of Cates and Happer with co-workers (Cates et al., 1990; Gatzke et al., 1993) on polarized frozen Xe as a means for accumulating large quantities of polarized gas may have been the initial inspiration for the development of MRI with laser-polarized xenon. The first experiment at Stony Brook with gas polarized in Princeton relied on freezing the xenon for transport by car. The magnetization lifetime of frozen Xe is generally much longer than in the gas phase (Gatzke et al., 1993).

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Studies of spin exchange between Rb and lighter noble gases Ne (Grover, 1983) and He (Chupp and Coulter, 1985; Chupp et al., 1987) were motivated by nuclear physics applications, in particular the use of symmetry violations such as parity (P) and time reversal (T) to study weak interactions in the presence of the dominant strong and electromagnetic interactions (Chupp et al., 1988). Several experiments used I : 3/2 Ne and He simultaneously to search for quadrupolar interactions such as a possible dependence of nuclear binding energy on orientation with respect to an assumed absolute rest frame of the Universe (Chupp et al., 1989). These pulsed NMR experiments were probably the first applications that specifically used laser-polarized rare gases to enhance rare gas NMR signals by many orders of magnitude. A variety of experimenters have since used laser-polarized He and other noble gases to enhance NMR measurements. The low-temperature work at Ecole Normal Supe´rieur has used NMR to measure polarization and probe such phenomena as spin waves (Tastevin et al., 1985) and the properties of Fermi liquids (Leduc et al., 1987), and He-He mixtures (Himbert et al., 1989; Nacher and Stolz, 1995). Geometric phases have been measured with Xe (Appelt et al., 1995). Measurement of the NMR splittings of Xe and He in the presence of an electric field is used to search for T-violation (Rosenberry, 2000). This experiment used a spin exchange pumped Zeeman maser (Chupp et al., 1994b; Stoner et al., 1996) that exploits cavity-spin coupling and the population inversion pumped into the nuclear spins (Robinson and Myint, 1964; Richards et al., 1988). Conventional NMR research with Xe (i.e., not laser enhanced) has focused on a variety of problems including cross polarization, molecular dynamics, xenon molecules (e.g., XeF ), diffusion in porous media, polymers,  and liquid crystals. The Xe isotope has been used to study quadrupolar relaxation on surfaces, in macromolecules, and porous media. Xenon has been extremely important because it is normally gaseous, can be easily frozen or liquified, is relatively soluble, and is characterized by large NMR chemical shifts of up to 500 ppm between the gas and dissolved phases. It was recognized that many of these applications of NMR research could be enhanced with laser polarized Xe (Raftery et al., 1991), and this inspired the original pursuit of MRI with laser-polarized noble gases (Song et al., 1999).

II. Nuclear Polarization Techniques Polarization of He and Xe can be contemplated by brute-force, SternGerlach, or optical-pumping techniques. Brute-force polarization uses high magnetic fields and low temperatures to create an imbalance of nuclear spin

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state populations. At low temperatures He atoms in the liquid phase are indistinguishable, obeying Fermi-Dirac statistics with the consequence that neglible polarization can be achieved at reasonable magnetic fields. (The effective spin temperature does not drop below the Fermi temperature of T : 0.18 K.) For solid He, the lattice positions, not the momentum states, $ distinguish atoms and Boltzmann statistics describe the polarization. The result is that solid He can be polarized, achieving the equilibrium value P  tanh(B/kT ) 

(3)

which at 10 mK and 10 T gives P : 91.5%. Low temperature alone is not  sufficient to produce solid He — high pressures are also needed. The Pomeranchuk method involves cooling the liquid in an applied magnetic field under pressure. For T  0.32 K, the liquid’s entropy is less than that of the solid and the sample cools itself in a process similar to cooling by evaportation (Lounasmaa, 1974). Frossati (1998) has proposed producing a thousand liters of highly polarized He per day using this method, followed by rapid warming of the polarized He through the liquid phase. It is not known whether Xe can be polarized in this way, though measured spin diffusion times seem favorable. Stern-Gerlach techniques have been used to produce beams of highly polarized He. However, the tradeoffs of acceptance and polarization have limited fluxes to 10/s with P : 0.9 for a hexapole magnet. This is not  sufficient for accumulation of useful quantities, though it is useful for applications where a trace amount of highly polarized He is required (Golub and Lamoreaux, 1994). Optical polarization employing either hyperfine spin exchange with an alkali-metal vapor (Bouchiat et al., 1960) for He and Xe or optical pumping of metastable helium atoms for He (Colgrove et al., 1963) both emerged as promising techniques with the availability of lasers. Both techniques are now used to produce liter quantities (at STP) with polarization P  50% that are used for neutron polarization, polarized targets,  and medical imaging. In all cases, relaxation of nuclear spin must be balanced by polarization rates. (Note that nuclear spin relaxation in a biological environment in vivo or in vitro is completely different from relaxation in a carefully prepared polarization system as discussed in Sections IV and V.) Rare gas nuclear spin relaxation occurs by bulk collisions with impurities, dipole-dipole interactions in the bulk, magnetic field gradients, and surface wall interactions. The most important impurity is paramagnetic O . Relaxation  rates are proportional to the oxygen impurity level with rate constants k(O -Xe)  0.3 s\/amagat (Jameson et al., 1988) and k(O -He)   

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0.45 s\/amagat (Saam et al., 1996) at 14.1 kG and at room temperature with temperature dependence T \. (One amagat is the number density of a gas at STP.) In order to achieve high He polarizations, O impurity  levels must be below parts per million. Relaxation due to dipole-dipole interactions have rate constants k(Xe-Xe) : 5 ; 10\ s\/amagat (Hunt and Carr, 1963) and k(He-He) : 4 ; 10\ (Mullin et al., 1990; Newbury et al., 1993). Magnetic field gradient contributions to nuclear spin relaxation arise due to nonadiabatic evolution of the nuclear spin as the atoms move in the gradients between collisions. For the high densities encountered in most applications  D

 B  ;  B  V W B

(4)

Typically, magnetic field gradients of 0.3—1%/cm are sufficient for He and Xe polarization, respectively. Wall interactions are moderately well understood. For He, the work of Timsit and Daniels already described here shows that paramagnetic impurities and sticking time, dominated by diffusion of helium into the surface, are most important. Coating the surfaces of glass or fused silica with cesium has proved effective for attaining He relaxation times of 2 d or more (Cheron et al., 1995; Surkau et al., 1997). For highly polarizable xenon atoms, the sticking times are much shorter, but relaxation rates can be reduced with silane wall coatings (Zeng et al., 1985; Oteiza, 1992; Sauer et al., 1999). Sauer has shown evidence that Xeproton dipolar interactions dominate relaxation in coated cells. With coatings, relaxation times for Xe of 10 min or more are common at low magnetic fields. At 2T, T  2 h has been observed for Xe, indicating  decoupling of the wall relaxation mechanisms. A. O P  S E Optical pumping (Kastler, 1950) is the means by which the internal degrees of freedom of a sample of atoms can be manipulated with light, and the angular momentum of the photons can be transferred with high efficiency to the atoms (see Harper, 1972). The most effective way to understand optical pumping and spin exchange is by derivation of rate equations describing these processes. For optical pumping, we begin by considering an atom with J : 1/2, such as an alkali-metal atom with nuclear spin I : 0. The polarization, P, is given by P: 9   \\

and



 

; :1 \\

(5)

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Both polarization and spin destruction processes must be considered. For polarization, we assume that the atoms are illuminated with right circularly polarized ( ) light, and we define the total for the rate per atom of pumping > out of the m : 91/2 state and into the m : ;1/2 state as  (1/2). For  H H atoms with resonant frequency    (r) : k 



d (r, )( 9  ) 

(6)

The laser intensity per unit frequency is (r, ) : dI(r)/d, which is, in general, position dependent. The cross section for absorption of unpolarized light is (), and k is a constant that accounts for the relative probability that an atom, after absorbing a photon, also absorbs its angular momentum. For alkali-metal atoms in the presence of sufficient buffer gas pressure to collisionally mix and randomize the spin projections in the p states, k : 1. The optical pumping rate equations for the two-state system are   d(<1/2) : < 1" ;  (91/2) h 1" (;1/2)  2 2 dt





(7)

We have included possible relaxation of electron spin polarization in the term  . 1" For spin exchange-pumped He, Rb spin relaxation is dominated by collisions with Rb atoms, He and N , and to a lesser degree by wall inter actions (Wagshul and Chupp, 1994). A surprising magnetic field dependence to the Rb-Rb relaxation process that decouples at relatively low fields of a few hundred gauss was discovered recently (Kadlecek et al., 1998). This suggests a time scale much longer than characteristic of binary collisions between Rb atoms. Although the mechanism is not yet understood, it is clear that optical pumping at magnetic fields of a few kG can turn off the Rb-Rb collisions with the advantages of potentially higher Rb polarization or less laser power. These Rb-Rb collisions are generally less important than Rb-He or Rb-Xe collisions. For He polarization, it is most effective to use high He density so that Rb-He collisions dominate the Rb spin destruction rate. For Xe, spin destruction is so strong that the same is effectively true, though xenon densities are much lower. Therefore the laser intensity requirements are determined by the Rb-noble gas spin destruction rate  . 1" There are good spin destruction collisions and there are bad ones. A good one, of course, results in a spin exchange to the noble gas nucleus. In a bad collision, the Rb atom loses its electron spin polarization to rotational

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angular momentum. It turns out that the ratio of spin exchange to spin rotation varies significantly among the alkali-metal atoms. Rubidium is worse than K, by nearly an order of magnitude, though the spin exchange rate constants are apparently approximately equal (Romalis et al., 1998). This has become very important recently with the availability of highpowered LDA at 770 nm, the D1 wavelength for K. As long as it is practical to operate at temperatures of 250 °C, at which the density of K is sufficient for spin exchange to balance He relaxation, we can expect increased use of K as the spin exchange partner. Radiation trapping is also a potential limitation to optical pumping in polarized targets (Holstein, 1947). This occurs when the mean free path for the unpolarized photons is much less than the dimensions of the pumping vessel. The incident  photons can be reemitted (i.e., resonantly scattered) > and depolarized. Each unpolarized photon can multiply scatter and depolarize many atoms and therefore radiation trapping can be thought of as an additional relaxation mechanism that is a function of incident laser power. Radiation trapping would limit the density of the mediating alkali-metal species in spin exchange pumped He targets. However, molecular N (Zeng  et al., 1985; Chupp and Coulter, 1985) (for He) have been shown to mitigate radiation trapping effects effectively. At high magnetic field the Zeeman splitting of the S and P states causes the scattered photons to   be off resonance and only very weakly absorbed in depolarizing transitions. The presence of N or other molecular species quenches the P states   nonradiatively, thereby reducing the branching ratio for radiative decay (i.e., resonant scattering) (Wagshul and Chupp, 1989). Assuming that the complication of radiation trapping has been practically eliminated, the steady-state solution to the rate equations predicts electron spin polarization P : 1 

 (r )  (r ) ;   1"

(8)

and a time constant ( (r ) ;  )\ that is typically milliseconds.  1" For atoms with nuclear spin, including alkali-metal atoms and metastable He atoms, the hyperfine coupling results in total angular momentum F. Laser optical pumping must provide the angular momentum for complete atomic polarization, the time dependence becomes more complicated than the single exponentials that describe the two state system, the transients become longer, and the nuclear spin serves as a reservoir of angular momentum (Bhaskar et al., 1982; Nacher and Leduc, 1985; Wagshul and Chupp, 1994; Appelt et al., 1998). However, the levels rapidly reach a

54

Timothy Chupp and Scott Swanson

spin-temperature equilibrium mediated by electron spin exchange (Anderson and Ramsey, 1961) and it is sufficient to consider only the evolution of electron spin S. For the metastability exchange, the discharge itself also leads to relaxation. The spin exchange rate equations, including relaxation, can be written P :  (P 9 P ) 9 P ' 1# 1 ' '

(9)

where P : 2 I (for I : 1/2) is the rare gas nuclear polarization and P : ' X 1 2 S is the alkali-metal electron polarization. The steady-state solution is X  1# P :P ' 1 ; 1#

(10)

The goal is therefore to maximize alkali-metal electron spin polarization and effect long relaxation times so that    . 1# For He polarized by spin exchange with Rb, 1/ is typically many 1# hours and relaxation times of days have been achieved, resulting in high polarizations 50%. Relaxation seems to be limited by many factors including wall relaxation, interactions with impurity gases (probably paramagnetic O ), and dipolar relaxation in He-He collisions.  For Xe, 1/ is typically several minutes but can be a short as 10 s. 1# Relaxation times in silane-coated cells seem to be 10—30 min at low magnetic fields, and several hours at 2T (Zeng et al., 1985; Oteiza, 1992). Deuterated coatings have been suggested to reduce relaxation at low field (Sauer et al., 1999). Relaxation is often dominated by wall collisions, though impurities and dipolar relaxation are also important. In a collision between an alkali-metal atom with electron spin polarization and a rare gas atom with I : 1/2, the electron spin couples to the nuclear spin and to the rotational angular momentum of the pair (Happer et al., 1984). The dominant contributions to the spin dependent Hamiltonian are H : N · S ; AK · S ; A K · S 1#

(11)

where A is the alkali-metal hyperfine interaction and A is the spin 1# exchange hyperfine interaction, both of which are, in general, position dependent. However, the long-range contributions vanish for spherically symmetric collisions, and only the Fermi-contact term acts, so that 8 A : 2 2 (R) ' 1# 3

(12)

MEDICAL IMAGING WITH LASER-POLARIZED NOBLE GASES

55

Where (r 9 R) : (R) is the probability that the alkali-metal valence electron (coordinate r) is at the position of the noble gas nucleus (coordinate R). Herman has shown that (R) is in fact enhanced due to the electron exchange interactions as the electron is attracted by the positive charge of the nucleus (Herman, 1965). An enhancement factor is defined in terms of the free alkali-metal electron wave function ( ) by (R) :   (R). The     varies from about 10 for Rb-He to 50 for Rb-Xe (Walker, 1989). The hyperfine interaction is, of course, time dependent as an alkalimetal atom and rare gas atom move past each other. For He, the time scale is about 10\ s because the collisions are always binary — in contrast to Xe, which can form a Van der Waals molecule with an alkali-metal atom in a three-body collision (Bouchiat et al., 1972). The lifetime of this molecule can be 10\ s or longer, limited in fact by the break up of the molecule in a collision with another buffer gas molecule. One consequence is that the rate constants for spin exchange are much different for He polarization and Xe polarization: k (Rb-He) : 6—12;10\ cm/s and k (Rb-Xe  1# 1# 4 ; 10\ cm/s, with this lower limit set by binary spin exchange in the absence of three-body formation of Van der Waals molecules (Cates et al., 1992). Spin rotation is a sink of angular momentum resulting from the coupling of the electron spin to the rotation of the alkali-metal-noble-gas pair, and is generally dominated by the heavier partner as discussed by Walker and Happer (1997). If we neglect wall interactions, alkali-alkali collisions, and alkali-N  collisions, the alkali-metal electron spin destruction rate reduces to the sum of spin exchange and spin rotation:  D  k E[I] ; k R[I] 1 1 1

(13)

for a rare gas of number density [I]. As the incident, circularly polarized laser photons must balance this spin destruction rate, it is useful to consider the spin exchange efficiency k E 1

: 1# k E ; k R 1 1

(14)

This quantity in principle sets an upper limit on the ‘‘photon efficiency’’ (defined by Bhaskar et al., 1982), with which optical pumping can balance noble gas relaxation. In general, however, photon efficiency is much lower than because of other alkali-metal spin destruction mechanisms, neces1# sarily inefficient optical transport of laser light into the cell, and the fact that lasers used in practical situations are broadband (as discussed in what

56

Timothy Chupp and Scott Swanson

follows). The magnetic field dependence of spin exchange, spin rotation, and relaxation mechanisms are, of course, important, particularly in magnetic imaging applications at fields of 2T and greater (Happer et al., 1984). 1. He Spin exchange cross sections for Rb-He have been estimated by Walker (1989) and measured by several groups. Measurements of He nuclear spin relaxation rates in the presence of Rb (Bouchiat et al., 1960; Gamblin and Carver, 1965; Coulter et al., 1988; Cummings et al., 1995) show that  v : 4—8 ; 10\ cm/s. Measurement of the frequency shifts of He 1# NMR and Rb EPR frequencies are consistent with this range (Baranga et al., 1998). The frequency shift measurements also allowed comparison of the Rb-He and K-He spin exchange interaction, showing that they are within 10% of each other. Since He nuclear spin relaxation times are generally tens of hours, polarization times must be only a few hours. This requires alkali-metal density 10/cm. As cell volumes are 100 cm, or greater, the total number of alkali-metal atoms can be 10, the incident photon flux must balance the loss of angular momentum by the alkali-metal atoms. The dominant processes relevant to Rb-He spin exchange can be summarized by the Rb spin destruction rate (Wagshul and Chupp, 1994; Walker and Happer, 1997; Appelt et al., 1998) SD : kRb-He [He] ; kRb-N [N ] ; kRb-Rb [Rb]  

(15)

where the k are rate constants for spin destruction due to collisions with each of the species in the optical pumping cell. For a typical application,   500 Hz and 10 photon/s/cm or 100 mW/cm are necessary. 1" 2. Xe There are crucial distinctions for Xe polarization: The xenon-alkali-metal spin exchange and spin rotation rate constants are many orders of magnitude larger than for helium (Cates et al., 1992), and long-lived Van der Waals molecules, formed in three-body collisions with lifetimes comparable to the hyperfine mixing time, may dominate spin exchange and spinrotation. As a result, polarization rates have characteristic time constants in the range of 10 s to several minutes in practical situations (Zeng et al., 1985). These times are comparable to and shorter than Xe nuclear spin relaxation times in the polarization apparatus so the Xe polarization is generally limited by the Rb or K polarization, not relaxation mechanisms, as in the case of He. The situation can be quite different, however, in

MEDICAL IMAGING WITH LASER-POLARIZED NOBLE GASES

57

systems designed to collect xenon gas that has flowed through a polarization chamber, such as that developed by Driehuys et al. (1996). In this case, the Rb density is probably not well controlled, and the xenon atoms may not uniformly sample the Rb polarization in the pumping chamber. This may be the reason that the observed Xe polarization is generally much lower than the Rb polarization (Hasson et al., 1999b). 3. Lasers for Spin Exchange Pumping Lasers have been the essential light source for successfully polarized He and Xe experiments. Originally dye lasers were used, producing up to 1 W near 795 nm with linewidths less than or comparable to the pressurebroadened Rb absorption linewidth (Chupp et al., 1987). (Typical standing wave dye laser linewidths are 30 GHz; the Rb D1 line is broadened by about 18 GHz per amagat of He.) In the late 1980s, high-powered arrays of laser diodes (LDA) became available, and their suitability for spinexchange pumped He polarization was of immediate interest (Wagshul and Chupp, 1989). Simultaneously the titanium::sapphire laser was developed for high-power applications and soon became commercially available. By about 1990, the cost per useful watt of LDA and Ti::sapphire lasers was comparable, but a single Ti::sapphire set up could produce 5 W whereas the most powerful available LDA was 2 W. Further, 795 nm was at the edge of reliable LDA production. Several experiments were undertaken, each using one or more Ti::sapphire lasers. Experiment E142 at SLAC ran with up to five (Middleton et al., 1993). By 1994, bars of LDA had become available with a price per watt of $500 and falling rapidly. This has been the single most important technology advance driving this field. By comparison, a Ti::sapphire laser pumped by a large frame argon ion laser has a price per watt of $15—$20 K. Current LDA prices are $100—$200 per W. The LDA will dominate future experiments and make polarized He and Xe much more widely accessible. 4. Optical Pumping with Laser Diode Arrays Laser diodes are widely recognized as work horses in atomic and optical physics. For example, near-IR lasers used in cooling and trapping of K, Rb and Cs are generally single-mode (linewidths on the order of MHz) and low-powered (50—100 mW with 500-mW amplifiers commonly in use). High-powered LDA are produced for a variety of commercial, industrial, and communications applications (including stripping the paint from battleships). Currently available LDA packages utilized for Rb optical pumping consist of bars of individual LDA. Bars with 20—50 W of nominal output

58

Timothy Chupp and Scott Swanson

F. 3. Profile of intensity vs wavelength for a typical GaAlVAs\V LDA (from Optopower Corp.), indicated by a solid line. The total power output per laser is about 15 W. The dotted and dashed lines show the laser profile 5- and 10-cm into the cell, respectively, for cell parameters of: 10 amagat of He and 0.1 amagat N at 180°C (left); 0.1 amagat of Xe, 0.2  amagat N , and 2.7 amagat He at 110°C (center); and 2.2 amagat of Xe, 0.2 amagat N , and   0.5 amagat He at 110°C (right).

consisting of about 20 1—3 W elements with GaAlAs and InGaAsP can be purchased for a few thousand dollars each. The injection current and temperature of the device are used to tune the arrays to 794.7 nm, the Rb DI wavelength, and typical bandwidth is 2—4 nm. Recently, 20-W bars at 770 nm with K D1 wavelength have become commercially available. Though the broadband light from the LDA is spread over 1—2 nm, much greater than the 0.1—0.2 nm typical homogeneously broadened absorption linewidth of Rb, the convolution of the light intensity and the absorption cross section provides a sufficiently high photon absorption rate that light 1 nm or more off resonance can effectively polarize Rb. The photon absorption rate of laser light by Rb atoms is defined in Eq. 6. In the case of LDA, () is spread over 2 nm or more, as shown in Fig. 3. The total power output per laser is about 15 W. As the light propagates through the cell (along z), it is absorbed by the Rb at a rate d () : 9()[S] (z)(1 9 P (z)) 1 dz

where

 (z)  P : (16) 1  (z) ;   1"

Computer modeling based on numerical integration of these equations is generally reported by several authors to predict results for He and Xe polarization that are within 10% of that measured (Wagshul and Chupp, 1994; Walker and Happer, 1997; Smith, 1998; Appelt et al., 1998). The requirement for significant Rb polarization is    . For He, a  1" large portion of the initial laser spectral profile is useful. In Fig. 3, we show

MEDICAL IMAGING WITH LASER-POLARIZED NOBLE GASES

59

the spectral profile at three positions along the axis of the cell for He density of 10 amagat with 0.1 amagat N . As light burns its way into the  cell, the central portion of the spectral profile is absorbed more strongly than the wings. Therefore, the front of the cell is essentially polarized by the near-resonance light. The more off-resonance light polarizes a greater portion of the cell’s length and is more important in the back of the cell. The large optical thickness of Rb typically used for He polarization ([Rb] : 10\/cm) is the main reason polarization with LDA can be so effective. The pressure broadening of the Rb absorption line is of secondary importance in most cases; in fact, the gains due to pressure broadening tend to saturate above 4—5 amagat of He. The situation is quite different for Xe. The spin destruction rate of Rb due to Xe is so much greater than that due to He that much greater laser intensity or spectral density (or both) is required to satisfy    .  1" Consequently, only a much narrower part of the LDA spectrum is useful for Xe, even at very low xenon concentration, as illustrated in Fig. 3. Broadening the absorption line with a buffer gas such as helium, which does not appreciably increase  , is helpful, but it is only practical to increase 1" the absorption line to approximately 0.5 nm with 10 amagat of buffer gas (Driehuys et al., 1996). The problem of balancing trade offs of noble gas polarization, production rates, volumes, and/or magnetization involves exploring a large parameter space. For example, increasing the total density of gas produces pressure broadening of the Rb absorption line, increasing the integral  , but also  increasing  . Greater Rb density increases  but also increases  and 1" 1# 1" the absorption of the light as it propagates through the pumping cell, reducing  further into the cell. For example, one can produce 60% Xe  polarization in 7.5 torr-liters per hour per watt of standard LDA laser power. The actual photon efficiency is less than 0.5%, compared to the 4% efficiency for Rb-Xe prediction (Walker and Happer, 1997). A standard liter would require about 100 W. For He, over 50% polarization of more than 1 l with 30 W of laser power has been achieved. Significant improvement of Xe polarization is possible if the LDA light is spectrally narrowed. In Fig. 4 we show a calculation of the expected Rb and Xe polarizations for different combinations of xenon density, temperature, that is, Rb density, etc. for 15 W of laser power. The total pressure is held constant at 2000 torr; for example, with 500-torr xenon, we use 100-torr N , and 1400-torr helium. We show results for two cases: low  xenon density, that is, 100-torr xenon and high helium buffer gas density as suggested by Dreihys et al. (1996); and high xenon density 1500-torr xenon used by Rosen et al. (1999). Narrowing LDA spectra provide significant gains in either case. The width parameter for the LDA is essentially a

60

Timothy Chupp and Scott Swanson

F. 4. Calculated Xe polarization as a function of the LDA linewidth (FWHM). For each temperature, curves for 0.13 amagat, 0.65 amagat, 1.3 amagat, and 2 amagat of Xe are shown from top to bottom, respectively. A constant total density of 2.8 amagat is maintained by loading with helium buffer gas. The solid dot on the 110° shows the measured Xe polarization reported by Rosen et al., 1999.

measure of full width half maximum (FWHM) of the spectrum. We emphasize that narrowing in this case does NOT mean that the lasers need to be single mode as in the case of cooling/trapping/BEC. Recent progress on narrowing off the shelf LDA in external cavities (MacAdam et al., 1992) has been reported (Nelson et al., 1999; Zerger et al., 1999). For example, the Littman Metcalf configuration has been used with 2-W off-the-shelf LDA. The spectral profiles for 1.0—1.5 W output have FWHM 20—30 GHz, and the central frequency could be tuned over several nm. Simulations of the expected performance show that a single 15-W LDA could be replaced by a 3-W external-cavity LDA. With the recent commercial availability of 4-W broad area LDA, 3 W may be possible with a single device. The 2-W LDA is similar to a single facet of a typical multiarray bar. For most commercially available CW bars, the filling factor is only 30%, and efficient optical feedback from the grating would be difficult. However, bars that are intended for pulsed use are available with filling factors of up to 90%. Thermal management problems limit the duty factor of these in normal operation, but reliable operation might be feasible for 10 W or more. B. M E In the metastability exchange scheme, a sample of He atoms is excited by a weak electric discharge so that a fraction of the atoms (:10\) is in the metastable 2S state. This long-lived state can be optically pumped to the  2P and 2P states by 1.083 m circularly polarized light. For example,   the 2S state is split into hyperfine levels with F : 1/2 and 3/2. Pumping 

MEDICAL IMAGING WITH LASER-POLARIZED NOBLE GASES

61

into the F : 3/2, m : ;3/2 state (the C9 line) produces high atomic and $ nuclear polarization of the metastable fraction. Resonant exchange of the excitation energy in metastability exchange collisions does not affect the nuclear spin, because the collision duration is short compared to the hyperfine mixing time. Thus, the ground state population attains high nuclear polarization (Colgrove et al., 1963). In general, the same principles of optical pumping apply to metastability exchange and spin exchange. There are, however, some crucial distinctions. One distinction is the ratio of widths of the atomic absorption line and the Doppler profile. For spin exchange, the high density of He or Xe and N lead to homogeneous collisional broadening of the Rb absorption line  of 18 GHz/amagat for He and 14 GHz/amagat for N (Che’en and  Takeo, 1957). This greatly exceeds the natural (5.7 MHz) and Doppler widths. Under these conditions, broadband laser light, from standing wave lasers or laser diode arrays, is effective for optical pumping (Wagshul and Chupp, 1989; Cummings et al., 1995). For metastability exchange polarization of He, the densities are hundreds of times less and Doppler broadening is dominant. Effective optical absorption by all of the atoms requires careful matching of the laser frequency distribution to the Doppler distribution. Another distinction between spin exchange pumping and metastable pumping is optical thickness. We can define an absorption length for polarized resonant photons with m : ;1 J  : 2([m] (91/2))\  

(17)

where [m] is the number density of metastable atoms or the alkali-metal vapor, and  is the resonant absorption cross section for unpolarized light.  For spin exchange pumping the absorption length is less than the dimension of the optical pumping vessel, which leads to the radiation trapping problems discussed earlier. The quantity  is generally more than 1 m for  metastability pumping, and radiation trapping does not present any limitations. Under optimum conditions, samples of He gas at a density 1.5 ; 10/cm can be pumped to an equilibrium polarization of over 80% with polarization rates of 10 atoms/s. The dependence of the equilibrium polarization and polarization rate on gas pressure, discharge level, and frequency has been studied in detail by Lorenzon et al. (1993). Metastability exchange polarization of Xe in a discharge has been studied by a few groups with little success (Schaerer, 1969; Lefevre-Seguin and Leduc, 1977). Although electron polarization in the metastable states indicates effective optical pumping, the discharge may induce excessive nuclear spin relaxation. As an alternative, the metastable 5p6s J : 2 state may be populated by two-photon laser excitation with ( : 317 nm), or a

62

Timothy Chupp and Scott Swanson

metastable atomic beam used to separate the discharge from the optical pumping region. These methods will probably not become practical for producing large quantities of polarized Xe, but may be useful for studying the physical processes at work. 1. Lasers for Metastability Exchange The success of metastability exchange-based He applications has also been strongly supported by laser developments. The first lasers for 1083 nm were color center F> ; NaF. Two Nd-based laser materials, Nd:Yap (Schearer and Leduc, 1986; Bohler et al., 1988) and Nd:LNA (Hamel et al., 1987) are now available. Five W of laser power at the helium transition is routinely obtained by pumping a crystal of Nd:LNA with a cw, krypton arc-lamp in a commercial Nd:YAG cavity. The laser can be tuned to the different pumping lines by use of a solid uncoated etalon in the cavity (Aminoff et al., 1989). The LDA-pumped LNA lasers have also been used (Hamel et al., 1987). The most recent laser development for metastability pumping of He is the diode-pumped fiber laser and fiber laser amplifier (Goldberg et al., 1998; Lee et al., 1999). C. P  D S Several devices combine optical pumping and polarization with delivery of the polarized gas to a subject or a storage container. For He, the basic designs used for polarized targets are applied for both metastability exchange and spin-exchange pumped systems. The metastability exchange systems have a valved port that connects to a transport container. Gentile and co-workers presented in 1999 a relatively compact and inexpensive compressor that may see wide use. For spin exchange systems an additional valve of the appropriate material is straightforward. With He polarization relaxation times of several days typical in glass containers, transport almost anywhere can be contemplated. For Xe, the high rate of Rb electron spin depolarization in spinrotation collisions limits the rate of Xe production, and a method of accumulation is essential. Cates, Happer, and co-workers have shown that frozen and liquid xenon provide very long nuclear spin relaxation times for Xe (Cates et al., 1990; Sauer et al., 1999), and that freezing is an ideal accumulation method (Driehuys et al., 1996). Relaxation times are on the order of an hour at liquid N temperatures and days at liquid He  temperatures (Gatzke et al., 1993). For human studies, it is sufficient to collect the polarized gas in a plastic bag, where it is held for several minutes before inhalation and breath-hold.

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For animal studies, voluntary breathing is not possible, and delivery to the animal requires a polarized gas ventilator. There are many technical difficulties, and very few such ventilators have been constructed (Hedlund et al., 1999; Rosen et al., 1999). Delivery of polarized gas by shipping from a geographically centralized production facility is one possible operating procedure for future medical imaging. In the case of He, relaxation times of several days are routine in clean, uncoated, glass containers (Middleton et al., 1993; Chupp et al., 1996), and all that is needed is a portable holding field magnet. Magnetic fields of 10—20 gauss are sufficient to dominate the magnetic field gradients expected in normal commercial shipping. Both battery-operated, wire-wound coils (Hasson et al., 1999a) and permanent magnet systems (Surkau et al., 1999) have been developed. If liquid He transport of polarized Xe becomes practical, its shipment would also be feasible.

III. Basics of Magnetic Resonance Imaging (MRI) Conventional magnetic resonance imaging (MRI) creates a map of the distribution of water protons in the body and has become one of the most versatile and powerful imaging methods in clinical medicine (Wehrli, 1995). The MRI system uses static, RF, and gradient magnetic fields to create images. A large, static magnetic field B , generally between 0.5—1.5 tesla,  creates an axis of quantization, energy level separation, and energy level population difference for the spin states. A radio frequency field, B (t),  oscillating at the proton larmor frequency causes transitions between the spin states and converts longitudinal magnetization into detectable transverse magnetization. Finally, pulsed magnetic field gradients, B /x(t), X B /y(t), or B /z(t), are used to both localize and spatially encode the X X nuclear spin magnetization in order to create an image. Here we present a synopsis of conventional MRI. A complete treatment can be found elsewhere (Callaghan, 1991). In addition, we review specific aspects of MRI related to imaging laser-polarized noble gases. A. N M R (NMR) MRI is an application of NMR (Abragam, 1961) with the fundamental relationship given by the larmor equation  : B  

(18)

The precessing spins are detected by tipping the magnetization by an angle

64

Timothy Chupp and Scott Swanson

 with a radio frequency pulse B (t) applied orthogonal to the B field. The   signal recorded as a function of time in a pick-up coil is s(t) .  M sin e\GSŠR 

(19)

where M is the total magnetization of the system and  is the ‘‘tip angle’’ of the magnetization relative to the axis defined by B .  B. O-D I Lauterbur (1973) realized that a map of the spatial distribution of the magnetization could be obtained by acquiring the NMR signal in the presence of a magnetic field gradient. The frequency of the nuclear spin is then proportional to the position of the spin and given by (x) : (B ; xG )  V

(20)

where x is the position of the spin and G is the gradient of B along the x V  axis, B G : X V x

(21)

The time evolution of the transverse magnetization is given by s(t) : M(x)e\GA Š>V%VR : M(x)e\GA ŠRe\GAV%VR

(22) (23)

Where  is a calibration constant that depends on  , , and electronic and  geometric factors. The only interesting component of s(t), from an imaging point of view, is the additional frequency due to the magnetic field gradient. Moving into a reference frame rotating at  , the time evolution of the  magnetization is given by s (t) : M(x)e\GAV%VR M

(24)

The signal s (t) is detected by mixing signals from an oscillator at  M  (64 MHz for protons at 1.5 T) with s(t). One practical consequence of detection in the rotating frame is that the signals can be sampled at audio frequencies rather than RF frequencies. Again, see Callaghan (1991) for a complete description.

MEDICAL IMAGING WITH LASER-POLARIZED NOBLE GASES

65

We now consider a one-dimensional (1D) distribution of spins along the x-axis. The time evolution of the magnetization is given by

 

s (t) :  M

V

:

M(x)e\GAV% VR dx

(25)

M()e\GSVR dx

(26)

V

This is the Fourier transform of M(x). Mansfield and Grannell (1973) showed that a 1D image could therefore be created by taking the Fourier transform of the NMR signal in the presence of a magnetic field gradient. 1 M() : F(s (t)) M 

(27)

C. M R I  k-S For imaging, the goal is to create a plot of the intensity of magnetization as a function of a spatial coordinate. A more appropriate representation for MRI is a coordinate system with spatial dimensions x and inverse spatial dimensions k where V 2 k : G t V V

(28)

Equation (26) can then be rewritten s(k ) : V



V

M(x)e\GLI VV dx

(29)

In this formulation, k and x are the conjugate Fourier variables. The V Fourier transform with respect to k provides a 1D map of the magnetizV ation. The applied gradient, and hence k , may be time dependent, V 2 k (t) :  V s(k ) : V



V



R O

G () d V

M(x)e\GLI VRV dx

(30) (31)

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Timothy Chupp and Scott Swanson

Generalizing to two dimensions, we then have s(k , k ) : V W



V W

M(x, y)e\GLI VRV>IWRW dx dy

(32)

Much of the progress in MRI over the last decade has been made by controlling the amplitudes and durations of gradients to appropriately sample k-space. These advances have been made possible by improvements in the hardware that produce the magnetic field gradients. It is important to realize that each point acquired in k-space is spread throughout real space. The point at k : 0 represents the dc component of V the magnetization and is proportional to the magnitude. As k  increases, V we measure the Fourier coefficients of higher frequency terms. By summing together all of the Fourier components in real space, one obtains an image of the magnetization. Artifacts in MRI arise because some of the terms in k-space are not sampled correctly or are lost. For example, a beating heart introduces time dependence not due to G (t). The artifact does not appear V at one location in real space, rather it is spread according to the sampling error in k-space. A solution to such an artifact is cardiac gating of the signal, triggered by heart monitors. Another important concept in k-space is prephasing and rephasing of transverse magnetization. Applying a gradient adds a phase to the spins that depends on their position in the sample. (x) : 



xG (t) dt V

(33)

If the direction of the gradient is reversed, the spins at each position acquire an opposite phase. When the  G (t) dt of the two gradients is of equal V magnitude, all transverse magnetization is in phase and a gradient echo occurs. In the language of k-space, we first move to a point where k is V negative. Changing the sign of the gradient changes the direction we move in k-space. The gradient echo occurs when we traverse the point where k : 0. Most pulse sequences are designed to symmetrically sample k-space V in order to maximize signal-to-noise. D. I S In most cases an MRI tomograph is a two-dimensional (2D) image of a slice of the body. The slice is isolated by selective excitation of spins along the third dimension. The spatial information is encoded by either frequency

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dispersion or phase dispersion, as discussed in the sections that follow. 1. Selective Excitation Slice selection is typically accomplished by simultaneous application of a magnetic field gradient and a shaped RF pulse with relatively long duration (1—10 ms) and correspondingly narrow bandwidth. This gradient creates a frequency ramp along its direction and the shaped RF pulse excites spins only within a relatively narrow slice. The sinc pulse, sin(t)/t, is the most common because its Fourier transform is a rectangle. In practice, the sinc shape does provide a reasonable approximation of a rectangular pulse in space coordinates. The combination of a gradient and a frequency selective pulse only excites spins within a region defined by z 

2 G  V

(34)

where G is the strength of the magnetic field gradient and  is the duration V between the first zero crossings of the sinc pulse. 2. Back Projection Imaging Back projection imaging in MRI detects the NMR signal in the presence of a magnetic field gradient, applied immediately after the slice selective RF pulse. This was the first type of imaging to be performed (Lauterbur, 1973) and is most directly related to other imaging methods such as computed tomography (CT) or positron emission tomography (PET). For the most part, back projection imaging has been replaced by Fourier imaging. However, it still maintains a niche in studies of tissues with a short transverse relaxation time T . In laser-polarized noble gas imaging, back  projection imaging is useful because all views acquired contain the dc component of k-space, which is proportional to the total intensity of the image. Therefore, if image intensity changes from pulse to pulse due to a different amount of gas magnetization, it is possible to normalize the acquired signals for proper reconstruction. This is not possible in Fourier imaging sequences such as gradient echo imaging. The pulse sequence needed for 2D back projection imaging is shown in Fig. 5. The frequency selective RF excitation pulse only excites spins in a slice of magnetization along the z-axis in the magnet. Signal acquisition commences immediately after the RF pulse is applied, and the NMR signal is recorded in a constant magnetic field gradient. The direction of the

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F. 5. Pulse sequence for back projection imaging in two dimensions. The slice-selective gradient and the frequency-selective RF pulse only excite spins in a slice or slab along the z-axis. The half sinc slice-selective pulse does not require that the transverse magnetization be refocused. The two projection gradients are varied in a sinusoidal pattern.

applied gradient is varied by changing the magnitude of both the x and the y gradients according to G : G cos( ) V G G : G sin( ) W G

(35) (36)

The different amplitudes in the x-projection- and y-projection-gradients are represented in Fig. 5 by the lines of different heights. Each radial step in k-space corresponds to a different value of  . For each step, a slice selective G pulse is followed by application of the gradients during which the MRI signal is acquired. Typically  is varied from 0 to 2 in 128 steps. The G sampling of k-space is shown in Fig. 6. Sampling of k-space is radial. Back projection images are reconstructed with a specialized algorithm and not by a 2D Fourier transform. 3. Gradient Echo Imaging All the elements of 2D Fourier MRI are contained in the gradient echo imaging sequence shown in Fig. 7. Slice selection and read-out gradients are applied as in back projection. The main difference is phase-encoding, first proposed by Kumar et al. (1975) and later modified by Edelstein et al. (1980). Phase encoding now forms the basis of many MRI pulse sequences. In phase encoding, phase dispersion occurs during an interval t before the  signal is acquired during the interval t . The duration of t or the phase  encode gradient can be varied to step through k -space, with t fixed. W 

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F. 6. Sampling of k-space by back projection imaging. In this example, we illustrate just the first few sampled rays obtained by adjusting the gradients according to Eq. (35—36).

Discrete samples are acquired during the interval t to form a 2D dataset.  The 2D Fourier transformation yields a correlation spectrum in f — f space   or real space. The slice selective pulse in the back-projection imaging sequence of Fig. 5 is a self-refocusing pulse, allowing the magnetization to be sampled immediately following the RF pulse (Green and Freeman, 1991). In general one needs to apply a slice refocusing gradient of opposite magnitude after the RF pulse so that the spins are in phase at the beginning of acquisition. This is shown in Fig. 7. The area of the negative gradient must be one-half the area of the slice selection gradient pulse. At the same time, the read-out dimension is prephased and the phase encoding gradient is applied. Prephasing in the read-out dimension k is done to allow symmetric V sampling of k-space by first moving in the negative k direction before the V read-out gradient moves in the positive k direction. Phase encoding V gradients are applied along the y-dimension. Part of the trajectory through k-space during the gradient-echo sequence is shown in Fig. 8. Starting in the middle of k-space particular values of k and k are determined by the W V phase-encode and read-out prephase gradients. The amplitude of the phaseencode gradient is changed for the next step to move to a different point in k . By continuing to raster across k for the different values of k , a complete W V W and even sampling of k-space is achieved. In typical imaging sequences, k V is acquired with 256 datapoints and k with either 128 or 256 datapoints. W

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F. 7. Gradient echo imaging (GRAS or FLASH). Magnetization is sampled in k and k V W as shown in Fig. 8. A 2D Fourier transform of uniformly sampled k-space creates the image.

Each value of k requires repeating the sequence. This is not true for k , W V which is called the free dimension in MRI. The number of k points is V typically determined by the desired resolution and the transverse relaxation time T .  4. Chemical Shift Imaging (CSI) Chemical shift imaging (CSI), a hybrid application of imaging and spectroscopy, is used to obtain spatially resolved spectral information or images of specific spectral components. Since gradients, which would disperse frequency across spatial dimensions, cannot be applied during acquisition, phase encode gradients are applied along either one, two, or three dimen-

F. 8. Sampling of k-space by the gradient echo pulse sequence. The phase-encode gradient varies from scan to scan and allows complete sampling of k . The readout gradient is prephased W to 9k  and runs to ;k . The resolution of the image is determined by the value of k  V V V and the field-of-view of the image is determined by the step size in k-space.

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F. 9. Two-dimensional chemical shift imaging (CSI) sequence. After slice selection, phase encode gradients are simultaneously applied along k and k . After the gradients are applied, V W the magnetization precesses freely in the B field so that a frequency spectrum can be measured. 

sions. A 2D chemical shift imaging pulse sequence is shown in Fig. 9. After the gradients are applied, the magnetization freely precesses in the B field  so that a frequency spectrum can be measured. This pulse sequence has been used to separate the different components of xenon magnetization in both the rat brain and body (Swanson et al., 1997; Swanson et al., 1999b) and it will be described in Section V. The CSI sequence requires discrete steps through each dimension of k-space, and is much slower than back projection and gradient echo sequences, which step through only one dimension in k-space. To collect a 16 ; 16 image, 256 different acquisitions are required. E. C  M R I Proton density varies only slightly in tissue, and MRI contrast therefore depends on changes in the magnetization characterized by relaxation times. The longitudinal or spin-lattice relaxation time T determines the time  required for the spin polarization to return to equilibrium following excitation by a radio-frequency (RF) pulse. If the spin magnetization is flipped by /2, the longitudinal magnetization recovers according to M (t) : M(1 9 e\R2) X X

(37)

The transverse or spin-spin relaxation time T is the time constant for  decay of magnetization in the transverse plane M (t) : M (e\R2 ‚) VW VW

(38)

Both T and T weightings require the spin-echo sequence. The spin-echo   sequence is similar to the gradient echo sequence, but a pulse refocuses spins that dephase in the intrinsic magnetic field inhomogeneities of the sample. The pulse is typically applied 10 and 50 ms after the initial

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Timothy Chupp and Scott Swanson TABLE I T R T  P  B T †

Gray matter White matter †

T 

T 

1000 ms 650 ms

110 ms 70 ms

Bottomley et al., 1984.

RF pulse for T and T weighting, respectively. In brain imaging, for   example, proton concentrations in white matter and gray matter are nearly equal, in contrast to the relaxation times given in Table I. For cerebral spinal fluid (CFS), motion effectively increases T . The relaxation time  differences are exploited to produce images such as those shown in Fig. 10. F. L F I Nuclear magnetic resonance with nuclei polarized by laser optical pumping is less dependent on large magnetic fields than is conventional NMR, and the potential of low-field imaging has emerged. The signal to noise ratio

F. 10. Conventional proton MRI tomographic images of the human brain. The images were acquired using spin-echo sequences. The detected magnetization depends on T or T ,   depending on the echo time. This provides the contrast. Both white and gray matter in each lobe of the cerebrum are distinguished in the T weighted image on the left. Cerebral spinal  fluid and tissue are distinguished in the T weighted image on the right. 

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(SNR) is the important parameter, and we therefore consider both signal and noise. For conventional NMR, the signal S due to a nuclear spin I : /2 (for H, H, and Xe) with concentration [I] is proportional to the product of precession frequency () and magnetization: S .  [I]P ' '

(39)

where I :  B and the brute-force polarization is P   B/kT. Thus ' ' ' S




. B[I] '

(40)

In contrast, for NMR with laser-polarized nuclei, P is independent of field, ' and S . B[I]  '

(41)

The most important MRI noise sources are Johnson noise due to the pick-up coil resistance, R , amplifier noise, and dissipation in the sample due A to loading characterized by R . Skin depth effects generally increase the coil Q resistance so that R . (B. The SNR for brute force and laser polarization A for fixed bandwidth are B SNR .
 (1 ; B\

1 SNR .  (1 ; B\

(42)

where   (0.2 T)\ (Edelstein et al., 1986). This shows that above 0.2 T, the SNR for laser-polarized NMR and MRI increases very little, that is, it is approximately independent of B. There are many advantages that may be gained from NMR and MRI at lower fields. The cost of magnets is less, open geometry permanent and conventional magnets may provide friendlier NMR scanners (important for pediatrics), and high-field effects such as susceptibility dependence may be less. Low-field work has been most effectively pursued by Darrasse et al. (1998). They have shown that the combination of 0.1 T magnet and a low-polarization metastability pumped He polarizer can produce lung images with resolution comparable to standard Xe nuclear medicine techniques such as shown in Fig. 11. One-dimensional images of polarized He have been used to study diffusion effects (Saam et al., 1996). Very low-field imaging at 0.003 T has been demonstrated by Tseng et al. (1998).

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F. 11. Lung images of a healthy volunteer produced with laser-polarized He at 0.1 T using a multispin-echo sequence. The slice thickness is 5 cm. Less than 30 cm of He with initial polarization 15% was mixed with a buffer gas just prior to inhalation. Image courtesy of Laboratoire Kastler-Brossel of ENS, Paris. Used with permission.

IV. Imaging Polarized

129

Xe and 3He Gas

Although either He or Xe may be used for gas imaging, the majority of lung ventilation imaging studies have used He. Helium has a number of advantages over xenon for creation of high-resolution gas images: The magnetic moment of He is nearly three times larger than that of xenon, and it has generally been easier to create high magnetization with He. The He polarizations are generally of 20—50% whereas typical Xe polarizations used for imaging are currently at 5%. A recent study imaging both gases concluded that in general helium is approximately 10 times more sensitive than xenon for MRI studies (Moller et al., 1999a). Helium also has fewer biological effects than xenon. Helium is biologically inert and the only consequence of helium inhalation (apart from the well-known change in voice pitch) is the risk of lowering the blood oxygen content due to oxygen being removed from the inhaled gas. Xenon on the other hand is anesthetic at concentrations of 35%. These effects are well known and have been addressed in CT studies where xenon is used to measure regional cerebral blood flow (rCBF) by monitoring the spatial and temporal attenuation of x-rays. Although helium provides greater signal strength and fewer medical complications, a major concern for widespread clinical studies with helium

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F. 12. Three lung images. Left: A Xe nuclear medicine scan of a patient with chronic obstructive pulmonary disease (COPD). Center: A laser-polarized He MRI of the same patient. Right: Laser-polarized He MRI of a healthy volunteer. Image courtesy of the University of Virginia. Used with permission.

will be the limited supply of He discussed earlier. Most studies are performed with He gas with an isotopic concentration of approximately 99% at a cost of approximately 100—150 USD/liter. Xenon is present in the air at a concentration of approximately 0.04%. The abundance of the spin 1/2 isotope, Xe, is 26.44%. Naturally abundant xenon can be purchased for approximately 10 USD/liter. The Xe enriched to approximately 75% can be purchased for about 300 USD/liter. This price is determined primarily by demand and could drop dramatically if specific clinical uses are identified. A. M R I  P G: G C 1. Sampling of the Magnetization In conventional MRI, longitudinal magnetization is sampled by an RF pulse and then replenished by relaxation to thermal equilibrium with time constant T . For laser-polarized gases, the longitudinal magnetization in the  body must be replenished by a fresh supply of polarized gas. With each sampling of the magnetization, the RF pulse destroys a portion of the longitudinal magnetization. The nonequilibrium polarization created by optical pumping would be entirely lost if sampled by a /2 RF pulse. Since MRI requires many excitations in order to appropriately sample k-space, /2 pulses cannot be used. The gradient echo sequence shown in Fig. 7 with a small tip angle is the most widely used approach. As the gas is sampled,

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the longitudinal magnetization decays, with magnetization after n pulses given by M (n, ) : M cosL() X X

(43)

where  is the tip angle. Thus the sampled magnetization in the initial pulses is larger than that in the later pulses if the tip angle is constant. For instance, if the tip angle is 10°, the value of the magnetization at the end will be only 20% of the initial value for the 128 pulses typically used to collect an image. This will cause different Fourier components of k-space to have intensities modified by an exponential decay. This leads to blurring of the real space image as each pixel is the convolution of the true magnetization with a Lorentzian (the Fourier transform of the exponential loss of magnetization to pulsing). Variable tip angle series have been applied to economically use laser-pumped magnetization in two-species experiments that probe fundamental principles (Chupp et al., 1989; Oteiza, 1992). An MRI sequence with variable pulse angle that produces the proper intensity of the Fourier coefficients in k-space has been proposed (Zhao et al., 1996). In principle, the variable flip angle sequence has better SNR because all of the magnetization is sampled. In practice, it is difficult to program this sequence on clinical MRI systems and most studies use a constant flip angle. 2. Diffusion and k-Space The basic description of MRI in Section III neglected effects due to the diffusion of spins during acquisition. For gas imaging, these effects are large and present many problems, as well as a few opportunities. The main problem stems from the fact the positions and therefore the frequencies of the spins change due to diffusion as k-space is sampled during the read-out gradient. As k-space is sampled along one dimension, the mean path length for 1D self-diffusion is d : (2Dt where D is the diffusion constant and t is the time. At 1 atm xenon has a self-diffusion constant of approximately 0.06 cm/s and helium approximately 2.0 cm/s. Therefore, during a typical MRI experiment with a sampling time of about 6 ms, the resolution for He is limited to about 1.5 mm. This assumes that the spins are free to diffuse. In lung alveoli and other porous media free diffusion is restricted. This allows measurement of pore size, which has recently been applied to lung imaging. A full treatment of diffusion and restricted diffusion can be found in Callaghan (1991). A number of studies have investigated this phenomenon. Edge enhancement of the signal intensity near the walls of rectangular glass cells in 1D

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images of polarized He has been observed (Saam et al., 1996). These studies were extended to demonstrate image distortion by molecular diffusion during the read-out gradient (Song et al., 1998). In this study, the investigators varied the strength of the gradient to follow the images from the strong diffusion regime to the weak diffusion regime. In another study using thermally polarized xenon, gas diffusion was used to measure both tortuosity and surface-to-volume ratio in a system of glass beads (Maier et al., 1999). Work from the same group also showed that the gas diffusion constant can be measured in a single experiment (Peled et al., 1999). B. A I Lung ventilation imaging is currently based on nuclear medicine scintigraphy of either Xe or aerosol sprays with Tc. Laser-polarized noble-gas imaging research with animals and human subjects has already shown that tomographic (slice-selected) high resolution images can be produced. A comparison of Xe scintigraphy and laser-polarized He images shown in Fig. 12. The first human ventilation studies with He were performed in Mainz (Ebert et al., 1996) and at Duke. The group at Mainz has continued with more clinical studies of volunteers with diagnosed lung diseases (Bachert et al., 1996; Ebert et al., 1996; Kauczor et al., 1997) (see Fig. 13). Other studies have looked at helium images of the lungs of smokers (de Lange et al., 1999) and ventilation defects have been found in a few cases.

F. 13. Laser polarized He lung image. The patient is suffering from pulmonary artery obstruction. The image shows a large ventilation defect that surprisingly corresponds to an obstruction of the pulmonary arterial branch. Image courtesy of Radiologie Klinik at Mainz University. Used with permission.

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In fact even apparently healthy, active, volunteers have ventilation defects that are revealed in high-resolution laser-polarized He MRI (Mugler et al., 1997). A study of subjects with chronic asthma suggests that ventilation defects may allow a measure of the progression and treatment of the disease (Altes et al., 1999). Although it will be some time before the utility of high-resolution lung images is clarified, it is clear that they provide new information and raise new questions: for example, what are the mechanisms of signal destruction in diseased lungs (Kauczor et al., 1998). The lungs are not the only organ amenable to gas imaging. The sinus cavities (Rizi et al., 1998) and bowel (Hagspiel et al. 1999) can also be imaged with laserpolarized He or Xe. Animal studies provide, appropriate disease models for eventual clinical studies. An advantage of using a small animal model is that the amount of polarized gas needed to create an image is significantly reduced compared to an equivalent human study. Impressive results using specialized small pick-up coils to attain high resolution images of He in animal models have been obtained by the group at Duke University. They showed the first in vivo images of helium in the lungs using 2D and 3D gradient echo imaging (Middleton et al., 1995). They also have demonstrated that the back projection imaging sequence can be used to reduce problems associated with changes in signal amplitude as k-space is sampled. Figure 14 shows images from a guinea pig model. These studies also show that one can vary the tip angle to capture either the early or later phases of inhalation. More recent work has concentrated on the magnetic behavior of both He and Xe gas in the lungs. One study finds that the effective transverse relaxation time (T *) for He is approximately 14 ms in the trachea but 8 ms in the  intrapulmonary airspaces. For Xe, T * is 40 ms in the trachea and 18 ms  in the intrapulmonary airspaces. This indicates that Xe interacts more strongly with the tissue of the infra pulmonary airspaces as it crosses the blood gas barrier. The regional variation of the diffusion constant was measured in vivo in guinea pigs (Chen et al., 1998). A study from a group in Lyon examined combining an MRI of He gas with proton-based methods to measure lung perfusion (Cremillieux et al., 1999). The goal is to provide a regional assessment of lung function. Methods in nuclear medicine typically provide only low-resolution images that are projections through the entire lungs and are not tomographic. A combination of conventional and laser-polarized gas MRI has the potential to provide very high resolution images for diagnosis of certain lung diseases, such as pulmonary emboli. A collaboration between the Duke and Lyon groups has presented images of guinea pig lungs with 2D resolution of 100  (Viallon et al., 1999).

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F. 14. Ventilation images of a guinea pig lung showing the exceptional spatial resolution possible with polarized He and the specialized techniques of in vivo microscopy. Image courtesy of the Center for in vivo Microscopy at Duke University. Used with permission.

C. I  He  Xe C Laser-polarized gas dissolved or encapsulated in injectable carriers is also under study (Goodson, 1999). Since xenon is highly soluble in nonpolar liquids, it is possible that images of xenon can be obtained in vivo by injection of xenon dissolved in an appropriate carrier. Work at Pines’s laboratory at the University of California, Berkeley, has shown that xenon dissolved in different carriers may have a significantly greater SNR than can be created by inhalation of xenon gas (Goodson et al., 1997). At Duke, laser-polarized He was trapped in microbubbles and introduced into the tail vein and arterial blood of a rat (Chawla et al., 1998). This new form of angiography provided high-resolution images. Also at Duke, laser-polarized Xe was dissolved in biologically compatible lipid emulsions (Intralipid 30% (Moller et al., 1999b). Measured relaxation times were T : 25.3 < 

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2.1 s, and T * : 37 < 5 ms. Analysis of magnetization inflow was used to  deduce the mean blood flow velocity in several organs. Several other potential carriers have been investigated including perfluorooctyl bromide (PFOB), which is a blood substitute (Wolber et al., 1998).

V. NMR and MRI of Dissolved

129

Xe

In contrast to He, which is most useful for imaging air spaces such as the lungs and colon, Xe is soluble in blood with 17% solubility and tissue with varying solubility (Chen et al., 1980). Many of the biological properties of xenon have been established through research with radioactive isotopes, particularly Xe. Xenon freely diffuses across biological membranes including the blood gas barrier in the lungs and capillary walls between blood and tissue. Xenon is metabolically inert, and is carried to distant organs where it accumulates in tissue. The size of the Xe magnetization signal in a specific region of interest can be a measure of the rate of blood flow or perfusion through the tissue. Studies using radioactive Xe have shown that xenon can be used in diagnosis and research to measure kidney perfusion (Cosgrove and Mowat, 1974), and cardiac perfusion (Marcus et al., 1987). Most exciting may be the study of regional brain activation. A variety of techniques has enormously enriched our understanding of the functional organization of the nervous system. The methods of Kety and Schmidt (1945) for measuring total blood flow following administration of a metabolically inert gas have been combined with radiotracer imaging techniques to measure changes in regional cerebral blood flow (rCBF) correlated with sensory stimulation, motor activity and inferred information processing in the brain. Early experiments used inhaled or injected gamma-emitting gases such as Xe (Lassen, 1980) or Kr (Lassen and Ingvar, 1961) to measure altered blood flow in the cerebral cortices. More recently, PET methods, most notably those employing O-H O, have been used to measure rCBF  (Phelps, 1991). However PET techniques have intrinsic resolution limited to 2—4 mm due to the range of positrons in tissue and often require a complementary imaging technique such as MRI or CT for accurate anatomical mapping of the PET functional information. The MRI methods are not subject to these intrinsic limitations and can provide functional information and anatomical registration with a single modality and apparatus. Several methods for measuring brain function with MRI have been explored (Shulman et al., 1993), and techniques based on blood oxygen level dependence of proton NMR have demonstrated high spatial resolution (Ogawa et al., 1990), although the physiological basis for the detected changes in signal is not well understood (Shulman et al., 1993).

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F. 15. Spectra from the body and head of a rat breathing a Xe-O mixture. Left: the  body spectrum showing signals from gas in the lungs (at 0 ppm), and dissolved in the blood (210 ppm) and tissue (192 ppm and 199 ppm). Right: the spectrum from the head showing several tissue peaks and possibly the blood peak near 210 ppm.

A. S  Xe in Vivo Figure 15 shows an NMR spectrum of  Xe from the body and head of a rat that had been breathing a mixture of Xe and oxygen gas (Swanson et al., 1999b). Similar spectra have been observed in humans after a single breath-hold of laser polarized Xe (Brookeman, 1998). The peaks in the rat body-spectrum (Fig. 15a) as well as the time dependence of the peaks have been identified on the basis of work by several authors (Wagshul et al., 1996; Sakai et al., 1996; Swanson et al., 1999b), the location of each resonance determined by imaging (see Fig. 2), and the chemical shifts revealed in in vitro experiments (Wolber et al., 1999a). The chemical shift may also depend on the oxygenation level of the blood (Wolber et al., 1999b) and varies with tissue type. The spectrum from the head (Fig. 15b) reveals at least four peaks in addition to the apparent blood peak at 210 ppm. Although there is not yet a definitive identification of the separate tissue types, this does show that several kinds of brain tissue are highly perfused and/or have large partition coefficients for dissolved xenon. An exciting direction for future research is the identification of each chemical shift component and functional study of the differences. It may become possible to identify the kinds of brain tissue involved in specific neurological functions. B. Xe I As Xe is carried throughout the body by the flow of blood, it is deposited in tissue with time dependent concentration that depends on several factors including the rate of blood flow, that is, perfusion. Perfusion measurement

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F. 16. Illustration of the data provided by the CSI imaging sequence. For each pixel, a frequency spectrum is produced. Spectra for four pixels are shown. The background gray-scale image is a proton MRI acquired with the spin-echo sequence. The oval surrounds the heart region. (See also Color Plate 2).

has many applications, ranging from rCBF measurement and research in cognitive neuroscience to assessment of pulmonary, renal, and cardiac health. One key goal of laser polarized Xe MRI is the development of perfusion measurement techniques (i.e., Xe as a magnetic tracer that uses chemical shifts to isolate each tissue type). The development of such techniques is discussed in Section V.D. Images of each chemical shift component of Xe can be created using the CSI sequence (described in Section III.D.4) and possibly frequency selective excitation. The CSI sequence produces frequency spectra for each pixel as illustrated in Fig. 16 (see also Color Plate 2), where we show spectra acquired for each of four adjacent pixels. The pixel map is superimposed on proton images acquired with the spin-echo sequence described in Section III.5. In Fig. 2, actual images of Xe in gas, blood, and tissue are shown. These images are magnetization maps of the signal in each of the peaks

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F. 17. A CSI image of Xe dissolved in tissue of the rat brain. On the left is the gray-scale Xe image and on the right is that image false-colored and superimposed on a proton spin-echo image.

indicated in Fig. 15a. An image of Xe dissolved in tissue in the rat head (Swanson et al., 1997) is shown in Fig. 17. The images of Xe in dissolved phases shown in Fig. 2 demonstrate some potential medical applications that may emerge in the coming years. Images of the lungs in the gas phase (Fig. 2A,D) show the region of ventilation. In a healthy lung, xenon crosses the blood-gas barrier, appearing also in tissue (Fig. 2B,E) and blood phase images (Fig. 2C,F). We discuss further analysis of lung function in the next section. The blood carries the Xe magnetization from the lungs to the left side of the heart. In the heart, the blood phase signal is dominated by pooled blood in the left heart chambers. Perfusion in the healthy heart is indicated by the appearance of Xe magnetization in the dissolved tissue and fat phases in the heart are also shown in Fig. 2B,E. Restricted blood flow (ischemia) and unperfused regions (infarction) would be revealed by the absence of the dissolved tissue phase in that region. C. L F The main functions of the lung are ventilation and perfusion. Many problems in the lungs result when there is a ventilation-perfusion mismatch. For example, regions of the lung that are ventilated but not perfused characterize about 70% of pulmonary embolism cases. Tomographic measurement of ventilation and perfusion, combining gas phase imaging in the lungs and Xe-dissolved phase imaging of the blood and tissue provide a

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F. 18. Images of ratios of Xe resonances. Left: gas, tissue; middle: gas, blood; right: blood, tissue.

new way to study lung function and may assist in appropriate treatment of lung disease. The data of Fig. 2 can be analyzed to extract ratios of blood and gas Xe concentrations. The image in Fig. 18a shows that the gas tissue ratio is relatively uniform except near the trachea and in the peripheral regions of the lung. The gas blood ratio image (Fig. 18b) shows a similar mismatch in the trachea but also more variation throughout the lungs. Some of this variation may be normal. Other possible pulmonary MRI methods using polarized Xe are venous injection of dissolved gas (see Section IV), followed by simultaneous imaging of the blood and gas components and study of the spatial variation in the frequency of the blood resonance, likely related to the oxygen content of the blood. The rich information content of Xe spectra and images provides interesting opportunities for pulmonary applications. D. T D  M T T The time dependence of the different chemical shift components of Xe is important in several applications. As we show here, a laser polarized Xe magnetic tracer can measure blood flow and the dynamics of exchange across blood gas and blood tissue barriers. In general, the time dependence of a chemical shift magnetization component depends on the rate of delivery to the tissue in the region of interest (perfusion) and on the local magnetization relaxation time T . This relaxation time is also, in general, time  dependent as oxygen concentration changes. Several authors have developed multicompartment models of Xe magnetization time dependence (Peled et al., 1996; Martin et al., 1997; Welsh et al., 1998). The goal is to measure the time dependence and use the model to extract quantities of interest, in particular T and blood flow independently.  Although nuclear medicine methods based on PET are highly developed, MRI-based methods of tissue perfusion measurement may have advantages:

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F. 19. Schematic of the magnetic tracer technique described in the text.

(1) chemical shift information allows blood and various tissue types to be isolated; (2) with an entirely MRI-based technique, the perfusion map can be anatomically registered with conventional proton images; (3) the resolution is not inherently limited, in the way PET is limited to several millimeters by the range of high energy positrons in tissue; and (4) radioactive dose restrictions that limit repeated PET studies do not have an impact on MRI techniques. In Fig. 19 we schematically illustrate how MRI of laser-polarized Xe can be used as a magnetic tracer to measure perfusion. Once inhaled, Xe is carried from the lungs to the heart, brain, and other distal organs. The signal produced at the frequency of the tissue resonance in a given organ (or pixel in an organ) is a measure of the total Xe magnetic moment in the measured volume of tissue. Tissue magnetization M calibrated in units of 2 the arterial magnetization M depends on blood flow F and the local  magnetization relaxation rate 1/T in different ways. If M is uncalibrated,  2 data can be used to determine relative blood flow. As the blood carries Xe with magnetization M into tissue, the NMR  signal size of the tissue resonance in each volume element of the tomographic image changes with time. The differential equation describing the tissue magnetization (M ) in a voxel is 2 F 1 dM 2 : FM 9 ; M  2  T dt 2 






(44)

where F is the rate of blood flow in units of ml /minute/ml , and   2  2 is the blood-tissue partition coefficient — the ratio of concentrations of xenon in blood to that in tissue. The time constant for relaxation of Xe

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magnetization to thermal equilibrium is T . This differential equation is  quite similar to that for the standard nuclear medicine formulation describing wash-in of a radioactive tracer (e.g., O-H O for PET or Xe for  SPECT). However, there is an extremely important difference — the relaxation time constant T is not uniform, rather it is generally different in  different tissues and blood, and it depends on the blood’s oxygenation level (Wilson et al., 1999; Wolber et al., 1999a). Measurement of dynamics of Xe tissue resonance in the rat brain is consistent with T  30 s (Welsh  et al., 1998). Techniques have been proposed for separating F and T  (Swanson et al., 1999a). Absolute measurement of F in units of ml/min/ml requires calibration of M in units of M . This requires measuring the magnetization signal from 2  known volumes of tissue and blood, respectively. For a quantitative measure of rCBF, it may be possible to image the blood in the carotid artery. For cardiac perfusion, imaging of the pulmonary veins and left heart chambers is possible (see Fig. 16). One important caveat follows from the small separation of the blood and tissue peaks, 150 Hz at 1.5 T. With the observed T * varying from 2 ms in blood to 20 ms in brain tissue, any NMR pulse that  tips the magnetization of Xe in tissue will perturb the blood magnetization. Thus M will come to an equilibrium value that is, in general, less  than the unperturbed M . However, the perturbation can be relatively small  with proper design of the pulse shape and phasing and because the rate of blood flow to the region of interest is high compared to the pulse rate 1/ (Geen and Freeman, 1991). Another possible complication is that the blood and tissue concentrations may not equilibrate rapidly on the time scale of the imaging experiments (about 1 s), resulting in an apparent variation of  . 2 1. Dynamics of Laser-Polarized Xe in Vivo Features of the dynamics of laser-polarized Xe in the lungs, body and brain of rats in vivo are shown in Fig. 20 (Swanson et al., 1999b). Frequency spectra collected as a function of time were used to study the dynamics of laser-polarized Xe. Qualitative interpretation suggests that the blood component builds up more quickly and saturates with respect to the lung input function, whereas the tissue component builds up more slowly due to greater tissue capacity for xenon, and falls off more slowly due to the longer intrinsic T in tissue and the relatively slow wash out of xenon. The  amplitude of the blood resonance closely follows the amplitude of the scaled gas resonance. The blood resonance plateaus after about 13 s of xenon delivery, but the tissue peak and the fat peak continue to grow and do not level off, even when xenon delivery is stopped at about 25 s.

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F. 20. Dynamics of Xe gas, blood, and tissue resonances.

VI. Conclusions — Future Possibilities The future is exceptionally bright for research in biomedicine, neuroscience, and materials science using laser-polarized rare gas imaging. The scientific problems relating to polarization techniques and the delivery of polarized gas with devices and in solutions are challenging, but progress continues. Ventilation images of animals and humans in the United States and Europe provide unprecedented resolution and are likely to provide new information, as is often the case when we can look at something with greater sensitivity, precision, and resolution. Figure 21 provides a stunning example. The new techniques possible with Xe provide resolution in chemical shift frequency and time that promise to develop into methods to measure perfusion of specific tissues as well as organs, thereby serving to complement PET. The potential for a complexity quantitative measure of perfusion promises broad application. All of these possibilities have been discussed in this chapter. However, research with a new imaging modality does not ensure its application as a medical diagnostic procedure. Among the potential applications of high-resolution lung ventilation imaging, colonscopy, lung function assessment, and perfusion measurement, MRI with laser-polarized gases must pass the tests of: 1. sensitivity to disease or injury; 2. specificity for a unique diagnosis; and 3. effectiveness based on cost and risk. For example, high-resolution lung imaging with He has been shown to be clearly sensitive to small ventilation defects — regions of the lung that do not effectively fill with gas in a normal breath. However the question of which specific malady this indicates is currently open. On the other hand,

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F. 21. A surface rendering of the human lung constructed with laser-polarized He MRI. Image courtesy of the University of Virginia. Used with permission.

lower resolution He or Xe lung images, produced with less gas and lower polarization (see Fig. 13) provide the same ventilation information as a Xe nuclear medicine scintigraphy, but without the radiation dose of nearly 1 rad from a single study. Such lower resolution scans would therefore provide the demonstrated sensitivity and specificity of the widely used nuclear medicine techniques. However, the cost of an MRI is currently many times greater than a Xe nuclear medicine study, and the additional cost of laser-polarized gas would significantly increase the cost of an MRI. Low-magnetic-field imaging systems may bring the cost down. Early diagnosis procedures and repeated studies that would be limited by radiation dose may be developed by physicians with these new tools. Pediatric pulmonary medicine may be an important application of the combination of diagnosis without radiation dose and low-field, open-geometry magnets. With the promise of these and a host of other potential applications, clinical efforts are underway in the United States and Europe. In the United States efforts are organized by commercial interests, which would produce the polarized gas in regional centers and ship it, overnight, to medical facilities. In Europe, a collaboration of industry, academic, and

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hospital-based researchers is developing the clinical program. The goals of both these groups include regulatory approval for administration of polarized gas as a contrast agent and its use for medical diagnosis. Interestingly, the final step in regulatory approval, following demonstration of safety and other issues, is a demonstration of efficacy — the sensitivity and specificity for diagnosis of specific maladies that would prove useful to clinicians/ physicians.

VII. Acknowledgments The authors are grateful to several colleagues for discussions and advice regarding this chapter and for scientific inspiration and guidance. They are Bernie Agranoff, Jim Brookeman, Gordon Cates, Kevin Coulter, Tom Chenevert, Will Happer, Bob Koeppe, Pierre-Jean Nacher, Eduardo Oteiza, Matt Rosen, Brian Saam, Ron Walsworth, Robert Welsh, and Jon Zerger. Images were provided by Brian Saam, Jim Brookeman, Tom Chenevert, Hans-Ulrich Kauczor, Pierre-Jean Nacher, and Al Johnson.

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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 45

POLARIZATION AND COHERENCE ANALYSIS OF THE OPTICAL TWOPHOTON RADIATION FROM THE METASTABLE 2S STATE OF  ATOMIC HYDROGEN ALAN J. DUNCAN and HANS KLEINPOPPEN Unit of Atomic and Molecular Physics, University of Stirling, Stirling FK9 4LA, Scotland

MARLAN O. SCULLY Department of Physics, Texas A&M University, College Station, Texas 77843; and Max-Planck-Institut fu¨ r Quantenoptik, D-85748 Garching, Germany I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. On the the Theory of the Two-Photon Decay of the Metastable State of Atomic Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. The Stirling Two-Photon Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . IV. Angular and Polarization Correlation Experiments . . . . . . . . . . . . . . A. Two-Polarizer Experiments: Polarization Correlation and Einstein-Podolsky-Rosen Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Tests of Garuccio-Selleri Enhancement Effects . . . . . . . . . . . . . . . C. Three-Polarizer Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Breit-Teller Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Coherence and Fourier Spectral Analysis — Experiment and Theory . VI. Time Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Correlation Emission Spectroscopy of Metastable Hydrogen: How Real are Virtual States? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VIII. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract: This chapter first summarizes fundamental aspects and results of the quantum electrodynamical theory of the two-photon radiation from the decay of the metastable 2S atomic hydrogen state. After a brief description of the second improved Stirling  two-photon coincidence experiment polarization correlations of the two-photon decay are described in which both two or three linear polarizers are applied in order to test predictions of such correlations based upon quantum mechanics and local realistic theories (i.e., Einstein-Podolsky-Rosen type experiments). It is particularly noticeable that the three-polarizer coincidence measurement provided the largest difference (about

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I. Introduction It is generally recognized that the spectroscopy of atomic hydrogen has provided crucial tests of the foundation of basic quantum physics, quantum electrodynamics, and even areas of elementary particle physics (Series, 1988; Selleri, 1988; Greenberger and Zeilinger, 1995; and Scully and Zubairy, 1997). As early as 1887 American physicists Michelson and Morley observed that the first spectral line H of the Balmer series of atomic hydrogen was ? split into two components, which Sommerfeld subsequently interpreted as a relativistic effect, afterwards called spin-orbit coupling and described by introducing a further quantum number referred to as electron spin. The experimental detection of the fine structure of the hydrogen Balmer line was the beginning of the precision spectroscopy of atoms. The 2S and 2P   states were predicted by Dirac’s quantum mechanics to be degenerate but this was proved incorrect by the sensational detection of an energy difference between these states by Lamb and Retherford (1947) (Lamb shift

E(P 9 S ) 5 1050 MHz). The question as to whether the 2S -state    of atomic hydrogen would be metastable in practice was a source of controversy during the first part of this century as discussed in detail by Novick (1969). However, the successful radio-frequency experiment of Lamb and Retherford (1947), which detected transition between the 2S and  2P states, depended on the metastability of the 2S state. The first   experiments for the direct detection of two-photon radiation from metastable states was reported for the decay of He>(2S ) by Lipeles et al.  (1965) and of H(2S ) by O’Connell et al. (1975) and also by Kru¨ger and  Oed (1975). The possibility of a spontaneous two-photon transition in general had been predicted by Go¨ppert-Mayer (1931) based upon her pioneering theory of multiphoton processes of atomic systems. Following this theory, Breit and Teller (1940) estimated that the dominant decay mode of the atomic hydrogen 2S state should be the two-photon emission and  much subsequent theoretical work has been carried out on the subject (Series, 1988; Drake, 1988, in the work edited by Series, 1988). In this chapter the data from the Stirling two-photon experiment is summarized and evaluated. This summary will include reports of measure-

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ments of the geometric angular correlations of the two photons and their polarization correlation observed in coincidence (including confirmation of the Breit-Teller hypothesis). As will be discussed, the two-photon metastable atomic hydrogen source also provides a means of carrying out fundamental experiments of the Einstein-Podolsky-Rosen (EPR) type (Einstein et al., 1935). In addition to experiments involving two linear polarizers, following proposals by Garuccio and Selleri (1989), experiments with three linear polarizers are described that provide particularly sensitive tests to distinguish between the predictions of quantum mechanics and local realism. A Stokes parameter analysis of the coincident two photons, which proves the coherent nature of the two-photon transition of metastable hydrogen, is also discussed. An experiment based on a delay of one of the orthogonal polarization components of one photon of the two-photon pair by a multiwave plate leads to the measurement of the coherence length of a single photon of the two-photon pair, which is shown to be extremely short. A novel Fourier-transform spectroscopic method using a Stokes parameter analysis of the two-photon polarization to determine the spectral distribution of the two photons emitted in the spontaneous decay of metastable atomic hydrogen is described. The theoretical analysis (Biermann et al., 1997) of the two-photon correlation spectroscopy of metastable atomic hydrogen in comparison to two-photon cascade emission from a three-level atom will be discussed. Attention is also drawn to more general summaries on the physics of atomic hydrogen (including collision processes) by Series (1988), Basassani et al. (1988) Friedrich (1998) and McCarthy and Weigold (1995).

II. On the Theory of the Two-Photon Decay of the Metastable State of Atomic Hydrogen As already mentioned here, the theory of the two-photon emission of the metastable 2S state of atomic hydrogen was initiated by a paper  from Breit and Teller (1940), which was based upon Maria Goeppert’s (1929) and Goeppert-Mayer’s (1931) quantum theory of multiple-photon processes in atomic spectroscopy. Since then substantial progress has been made both theoretically and experimentally with regard to the physics of the two-photon emission of metastable atomic hydrogen. The exact quantum mechanical description of the two-photon process is based upon the four-component Dirac equation. (see, e.g., Drake, 1988). This theoretical approach is well discussed in terms of the matrix formalism of quantum electrodynamics (Akhiezer and Berestetskii, 1965).

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Before we describe some results of the theoretical formalism for the two-photon decay of the metastable 2S state the competing electromag netic transitions in the absence of any perturbation such as an electric field or atomic collisions will be considered. Due to the Lamb shift the 2P  state is the only one of the n : 2 states lower than the 2S state. However,  as the energy difference is so small the possible spontaneous electric dipole transition from the 2S to the 2P state has a negligibly small   transition probability (corresponding lifetime of about 20 yr). Magnetic dipole and electric quadrupole transitions of the 2S state are forbidden  in the Pauli approximation but magnetic dipole M transitions are allowed  if exact Dirac theory with Dirac wave functions are allowed with a decay rate of 2.496 ; 10\s\, corresponding to a medium lifetime of about 2 days. The two-photon transition probability is much greater and has been estimated to be about 14 s\ for the two-photon transition of metastable atomic hydrogen (corresponding to a mean lifetime of 1/7 s). Figure 1 illustrates spontaneous and field-induced radiative transition modes of the metastable state of atomic hydrogen. While interest here is exclusively related to the two-photon decay, the study of field-induced quenching radiation reveals measurable interference effects and quantum beat phenomena applied in atomic spectroscopy (Andra¨, 1974 and 1979; van Wijngaarden et al., 1974; Drake et al. 1979). The quantum electrodynamical theory of the simultaneous emission of the two photons with vector potentials A (x) and A (x) can be illustrated   by the second-order Feynman diagrams shown in Fig. 2. The relevant second-order S-matrix element for these transitions is S : (e/ ) GD



; (e/ )

 (x )A *(x )SC (x , x )A (x ) (x )dx dx D D   A       



 (x )A *(x )SC (x , x )A *(x ) (x )dx dx   D    A      

where SC is the electron propagator in the external field of the nucleus. With A substitutions of  (r ) (r ) 1  dweGUR\R‚  L  L  SC(x , x ) : A    (1 9 i ) ;  2 i \ L L



one obtains S : (92 i/ )U ( ;  9  ;  ) GD GD   G D

(1)

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F. 1. Spontaneous and field-induced radiative transitions of the metastable 2S state of  atomic hydrogen. The 2E1 denotes the two-photon decay mode, M1 and M2 the magnetic dipole, magnetic quadrupole and E1 the electric dipole decay modes. The states between the metastable and the ground state represent ‘‘virtual’’ states (dotted lines) associated with the emission of the two photons of energies h and h . Single photon decay modes may lead to   cross terms to produce quantum beats and interference effects and also contributions to the two-photon decay rates (Drake, 1988). The dashed-dotted lines indicate the mixing of the metastable state with the 2P and 2P states by an external perturbation such as electric   fields or atomic collisions. Note that the energy differences are not scaled.

where  and  are the angular frequencies of the two emitted photons;   the positive frequency  for the electron and 9 the negative frequency D G for the positron follow from the solutions of the Dirac equation. The preceding equation includes the second-order interaction energy expressed by the formula: f  · A*( )n n · A*( )i

e     U : 9  GD  ; 9

L L  G f  · A*( )n n · A*( )i

    (2) ;  ; 9 L  G





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F. 2. Feynman diagrams for the two-photon decay of metastable atomic hydrogen: (a)  (x ) and  (x ) are the wave functions of the final ground state and the metastable state;     A (x ) and A (x ) are the vector potentials of the two photons, and SC is the electron A     propagator in the external field of the nucleus (Eq. 1); x are the relativistic four-component coordinates (after Akheizer and Berestetskii, 1965). (b) In nonrelativistic electric dipole approximation the energies of the ground, final, intermediate P state and metastable (initial) states, W , W , and W , are connected by the products p A and p A of the canonical momenta D N G     p and p and the vector potential A and A to result in the photon energies h and h . It       demonstrates that the emission of the two photons can be considered in either order as shown in the Feynman diagram (2b).

The summation in this relativistic expression is taken over both positive and negative frequency states i ,  f and n denote the initial (2S ), the final  (1S ) and the intermediate state for the two-photon emission. The symbol   is the usual Dirac matrix. The spectral distribution of the two-photon radiation only requires  ;  :  9  which means that only one of   G D the two-photon frequencies is independent. The so-called triply differential emission rate in the energy interval dE between the two energy states E Q and E is given by Q 2 w( ,  )d d dE : U ( ) ( )dE GD D  D       

:

a    Q( ,  )d d dE      (2 )

(3)

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where Q( ,  ) : 9   L



f a · e e\GI Pn na · e e\GI‚ Pi

   ; 9 L  G f a · e e\GI‚ Pn na · e eGI Pi

  ;  ; 9 L  G



(4)

where e and e are unit vectors in the polarization directions of the two   photons and an average will be taken over the directions of emission and polarization. The interference of the two terms in Eq. (4) results from the fact that complementary pairs of photons of energies hv and hv (or hv ,    hv ) are indistinguishable. It should be emphasized that the picture of  simultaneous emission of the two photons assumed here has some limitations as it follows from the analysis of coherence effects as measured and reported in the chapter by Z. Vager (see p. 203, this volume). In the nonrelativistic electric dipole approximation the following expression is replaced by  · e e\GIP ; p · e /mc and the sum in Eq. (4) is restricted to positive frequency intermediate states. The central problem is to evaluate numerically the expressions of Eq. (3). The summation is to be taken over all intermediate states n and their related energy states E :  . There are L L several important consequences of the results of the theory: 1. Based upon the preceding replacement of the Dirac matrix , the Feynman diagram of Fig. 2a can be drawn as shown in Fig. 2b with the energies of the states involved. 2. The energies of a complementary pair of two photons add up to the energy difference between the 2S and 1S states. Figure 3 shows   shapes of such energy distributions for Z : 1 and Z : 92. 3. According to Breit and Teller (1940), the electric dipole operators in the interaction Hamiltonian are diagonal in nuclear and electron spins. As a result the presence of hyperfine and fine structure effects can be neglected in the two-photon emission process (see Section V). 4. If e and e are the unit vectors of the linear polarizations for the two   photons with energies hv and hv the transition probability has a   cosine square dependence I(v ) . e · e  : cos where  is the angle    between e and e (Fig. 4).   5. Averaging over these polarizations results in an angular correlation for the coincident detection of the two photons: I(v ) . e 9 e  . (1 ; cos)    ?T

(5)

where  is the angle between the directions of coincident detection of the

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F. 3. Energy distribution of the 2S ; IS two-photon continuum radiation of   hydrogen-like atomic systems with nuclear charges Z : 1 and Z : 92 (uranium). Here (y, z) is the spectral distribution function and y is the fraction of the total transition energy transported by one of the two photons; correspondingly 1 9 y is the fraction of the energy of the other photon. The areas under the curves are normalized to unity (from Goldman and Drake, 1981).

two photons. In the chapter by Demtro¨der, Keil, and Wenz such polarization and angular correlations have been confirmed experimentally. In most of the experiments, however, the two photons are normally observed in diametrically opposite directions, that is, with  : . A relevant and interesting form to describe the polarization for two photons can be based upon the formulation of a two-photon state vector for  : . Conservation of angular momentum and parity implies the following arguments. In the transition H(2S) ; H(1S) no orbital angular momentum change occurs. Therefore the two photons with energies hv and hv must have  

F. 4. Diagram illustrating the angular correlation  and the correlation angle  between the polarization unit vectors e and e involved in the detection of two-photon emission.  

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equal helicities and the photon pairs are either right-right R R or   left-left L L handed circularly polarised or in a superposition of both   kinds, that is, 1 R R < L L   : ! (2     The parity operator P transforms the handedness of right- and left-circularly polarized light and the propagation of the two photons in opposite directions: PR : L and PL : R . The two states of 2S and 1S     have definite even parity (l : 0) so the two photons should also have even parity (otherwise the resulting parity could be a superposition of even and odd parity), that is, the plus signs should be valid in the preceding equation, viz, 1 R R ; L L   :     > (2

(6)

Because of the usual relations between circular and linear optical polarizations R : 2\(x ; iy ),    L : 2\(x 9 iy ),   

R : 2\(x 9 iy )    L : 2\(x ; iy )   

(7)

the two-photon state vector can be written alternatively as  : 2\(x x ; y y ).    

(8)

The following implications of these state vectors should be noted. a. They are invariant with respect to rotation about the detection z-axis (see Fig. 5). b. The state vector represents a pure quantum mechanical state, not a mixture of the R R and L L states (or alternatively not a     mixture of the x x and y y states).     c. Consider a typical ideal coincidence experiment for measuring linear or circular polarization correlations in the opposite directions of z and 9z (Fig. 5). With such measurements, the state vector  * ‘‘collapses’’ into R R or L L or alternatively into x x or y y ,         each possibility occuring with a probability of one-half. This collapse of the state vector implies that detection of a photon, say in the

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F. 5. Coordinate system with reference to the emission and polarization correlations of the two-photon emission (h and h ) of metastable atomic hydrogen H(2S) detected in the z   and 9z directions by the two detectors and D and D . The transmission axes of the two   polarizers are at angles  and  with respect to the directions (after Perrie et al., 1985). 

z-direction, with special choices for linear or circular polarizers determines the result of the measurement of the polarization in the other detector, say in the 9z direction, irrespective of the distance between them. The result obtained on one side of the source depends on the choice made for the setting of the polarizer on the other side of the source. This situation clearly violates the ‘‘principle of locality’’ in classical and relativistic physics, according to which the value obtained for a physical quantity at point A cannot be dependent on the choice of measurement made at point B as long as the physical quantities at point A are not correlated with the ones at point B. This discussion already in essence leads to the Bohm-Aharonov (1957) version of the Einstein-Podolsky-Rosen-Paradox concerning the incompleteness of quantum mechanics (see Section IV).

III. Stirling Two-Photon Apparatus Figure 6 illustrates schematically the second improved two-photon apparatus built at Stirling University (Perrie, 1985). Metastable D (2S) atoms were produced as a result of the capture reaction d ; Cs ; D(2S) ; Cs*, which is favored by a high resonance cross section (10\ cm). The deuterium ions (d) were extracted from a radio frequency ion source and passed through a cesium vapor cell constructed after a design by Bacal et al. (1974) and Bacal and Reichelt (1974). Deuterium was used rather than hydrogen since the radiation noise generated by interaction of the deuterium beam with the background gas was less than with hydrogen. Best statistical data of the two-photon coincidences could be achieved at an energy of 1 keV for deuterium. The D(2S) beam leaving the charge exchange cell passed

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F. 6. Schematic diagram of the second Stirling two-photon apparatus.

through a set of electric field prequench plates, allowing the metastables of the beam to be switched on and off by the effect of Stark mixing of the 2S  and 2P states.  At the end of the beam apparatus the metastables were completely quenched by the electric field of another set of quench plates; the resulting Lyman9(L ) radiation was used to normalize the two-photon coincidence ? signal from the metastables. The L —signal was registered by a solar blind ? UV photomultiplier together with an oxygen filter cell with LiF windows through which dry oxygen flowed continuously. Tests were made regularly to confirm that the two-photon coincidence signal was proportional to the L —signal, which depends linearly on the density of the metastables. ?

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The two-photon radiation was collimated and detected at right angles to the D(2S)-beam. Two lenses of 50-mm diameter were used, each with a focal length of 43 mm at a wavelength of 243 nm. For the linear polarization correlation measurements two high-transmission UV polarizers were used, each consisting of 12 amorphous silicon plates polished flat to at 2 at 243 nm and set nearly at Brewster’s angle as shown in Fig. 6. Additional optical elements such as quarter-wave, half-wave and multiwave plates could be inserted as required for other polarization correlation measurements. The material of the various lenses and plates was high-quality fused amorphous silica with a short-wavelength cut-off at 160 nm. This in turn corresponds to a complementary long wavelength cut-off at 355 nm. Accordingly, all photons in the wavelength range from 185 to 355 nm can contribute to possible two-photon coincidence signals. The quantum efficiency of the photomultipliers was about 20% over this range. The transmission efficiencies and and of the polarizers, for light polarized + K parallel and perpendicular, respectively, to the transmission axes of the polarizer, were measured by making use of the 253.7-nm optical line from a mercury lamp. Two of the polarizers used had transmission efficiencies of

: 0.908 < 0.013 and : 0.0299 < 0.0020; a third polarizer with plates + K from a different manufacturer had values of : 0.938 < 0.010 and +

: 0.040 < 0.002. The pulses detected by fast-rise-time photomultipliers K were fed into a coincidence circuit described by O’Connell et al. (1975). It consisted of the common combination of constant-fraction discriminators, a time-to-amplitude converter, and a multichannel analyzer operating in the pulse-height analysis mode. A typical run for acquiring coincidence signals lasted at least 20 h. Spurious coincidence signals due to cosmic rays and residual radioactivity in the apparatus occurred at a rate of one every 100 s, and decreased as the distance between the photomultipliers increased. The density of the metastable atomic deuterium D(2S) for realistic coincidence measurements was about 10 cm\ (equivalent to a partial gas pressure of about 0.3 · 10\ torr). A typical coincidence signal of the two-photon radiation is shown in Fig. 7. As can be seen, the shape of the coincidence peak is symmetrical, which is expected for the effectively simultaneous emission process of the two photons from the decay of the metastables. The background signal results mainly from the single count rates of the order of 10s\, which were due mainly to radiation produced by the metastable atomic beam interacting with the background gas of the vacuum system at a pressure of 2 ; 10\ torr; uncorrelated photons from the two-photon decay contribute only about 0. 01% to the background coincidence signal.

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F. 7. Typical coincidence spectrum for the two-photon emission from metastable atomic deuterium. The time-correlation spectrum is that which was obtained after subtraction of the spectrum produced when the metastable component of the atomic beam was quenched. Polarizers were removed for this example. Time delay differences between relative channel numbers are 0.8 ns, total collection time 21.5 h. Singles count rate with metastables present (quenched) is about 1.15 · 10 s\ (0.85 · 10 s\). The true two-photon coincidence rate from the decay of the metstable is 490 h\ for this example.

IV. Angular and Polarization Correlation Experiments A. T-P E: P C  E-P-R T The first coincidence measurement of the two-photon decay of atomic hydrogen was made by O’Connell et al. (1975) for three different angles between directions of the detected photons (Fig. 8). While the accuracy of these early experiments was limited, the data approached the shape of the theoretical prediction given in Eq. (5) and clearly demonstrated a disagreement with a circularly symmetric angular correlation in the detection plane defined by the directions of the two coincident photons given by the equation proportional to (1 ; cos), which is expected by the theoretical prediction.

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F. 8. Angular correlation data for two-photon coincidences from H(2S) with detectors D  and D at the three correlation angles  equals 90°, 135°, and 180°. The circle represents a  symmetrical angular correlation with the other curve representing the theoretical prediction, which has the form (1 ; cos).

In a subsequent improved experiment involving two linear polarizers (Fig. 5) for the detection of photons in diametrically opposite directions ( : ), the coincidence rate ratio R()/R was measured as a function of M the angle  between the transmission axes of the polarizers; R() is the coincidence count rate with the two polarizers inserted while R is the M coincidence count rate with the two polarizers removed. For this case quantum mechanics predicts (Clauser et al., 1969) in the ideal case, a (1 ; cos) dependence of the coincidence signal. In practice, quantum mechanics predicts 1 R() 1 : ( ; ) ; ( 9 )F( ) cos K K 4 + 4 + R M

(9)

where in the current case the transmission efficiency : 0.908 < 0.013, +

: 0.0299 < 0.0020, the half-angle subtended by the lenses near the source K of the two-photon radiation is : 23° and F( ) : 0.996. As can be seen in

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F. 9. Linear polarization correlation of the two photons emitted in decay of metastable D(2S) as measured and compared to Q.M. (quantum mechanics) and two local realistic models (I) and (II). The theoretical curves take account of the finite transmission efficiencies and the angles of acceptances of the lenses. The transmission axes of the polarizers are rotated by the angles  and  with  :  9  .  

Fig. 9 the quantum mechanical curve fits the data of our coincidence measurements of the two-photon radiation very well while the predictions of the two local realistic theories discussed in what follows fail to do so. The horizontal straight line (curve I) for R()/R in Fig. 9 is based upon the M following local realistic model. In this model it is assumed that the source emits R pairs of photons per second in the ;z and 9z directions with the M photons emitted on either side independently of each other possessing an isotropic distribution of polarization vector directions. Because, in the ideal case each polarizer transmits only one-half of the photons, the coincidence rate R() observed is reduced to R /4 independently of angle  and hence M

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F. 10. Coincidence signal R() divided by the coincidence signal R with the plates of M both linear polarizers removed for the circular polarization experiment as a function of the angle  made by the fast axis of the quarter-wave plate in one detection arm of the two-photon coincidence apparatus with respect to the fast axis of the quarter-wave plate in the other detection arm. The solid line represents the theoretical quantum mechanical curve for comparison.

with corrections for the transmission efficiencies and of the polarizers, + K R( )/R : 0.22 (see Fig. 10). Another specific example of a local realistic M model (curve II in Fig. 9) originally due to Holt (1973) can be described as follows. As in the preceding, we assume an isotropic source emitting pairs of photons in the ;z and 9z directions but, in this case, each with the same polarization vector at an angle  to the x-axis. These angles  have values from 0 to with equal probabilities. Taking R decays per second and M detectors D and D with 100% efficiency the coincidence signal would be  dS : R cos( 9 ) cos( 9 )d/ M  for photons with polarization angles of between  and  ; d. Integrating

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over all angles  from 0° to gives R() 1 1 : 1 ; cos 2 R 4 2 M





with  :  9  

This result compares to the quantum mechanical prediction of R() 1 : [1 ; cos 2] 4 R M in the ideal case, which does not have the factor  in front of the cosine term  found in the local realistic model of Holt (1973). This discussion leads to the Einstein-Podolsky-Rosen (EPR) debate about the completeness of quantum mechanics. The literature on this topic has increased dramatically over the past 15 years so that we refer only to some relevant reviews (Selleri, 1988; Duncan and Kleinpoppen, 1988; Greenberger and Zeilinger, 1995). The controversy between local realistic theories and quantum mechanics may be characterized as follows. Bell (1964) and later Clauser, Horne, Shimony, and Holt (1969) showed that quantum mechanics predicts strong correlations in ideal two-photon experiments of which local realistic theories are incapable. In local theory a measurement of a physical quantity at some point A in a space-time representation is not influenced by a measurement made at another point B spatially separated from A in a relativistic sense. A realistic theory assumes that the world is made up of objects with physical properties, which exist independently of any observation made on them. Contrary to both local and realistic theory quantum mechanics is neither local nor realistic. Bell (1964) showed in the form of his famous inequality that all local deterministic hidden variable theories (which are a subclass of local realistic theories) predict a weaker correlation between photons than that given by quantum mechanics. Freedman’s (1972) form of Bell’s inequality showed that, for local realistic theories, the quantity must satisfy the following inequality: :

R(22.5°) 9 R(67.5°)  0.250 R M

where R(22.5°) is the coincidence rate for  9  : 22.5°, R(67.5°) the  coincidence rate for  9  : 67.5°, and R the coincidence rate with the  M two polarizers removed. Contrary to these limits quantum mechanics predicts, in the ideal case, a variation as cos( 9  ), which results in the  value : 0.354. Taking account of the solid angle of detection, the

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transmission efficiencies of the polarizers and applying Eq. (9), quantum mechanically : 0.273 < 0.011 for this experiment. Experimentally the two-photon data from the results in Fig. 9 provide a value of : 0.275 < 0.016(1) in agreement with quantum mechanics and beyond the limits of Bell’s inequality. A large number of experiments using two-photon radiation from atomic cascades (e.g., Fry and Thomson, 1976; Aspect et al., 1982), from positronium annihilation (Paramanande and Butt, 1987), as well as from interference experiments (Brendel et al., 1992) with laser photon pairs have also clearly confirmed the agreement with quantum mechanics. Positronium annihilation experiments can be criticized on the grounds that quantum mechanics itself must be assumed in order to analyze the experimental data. Doubts about atomic cascade experiments have been expressed (Garuccio and Selleri, 1984; Kleinpoppen et al., 1997), concerning the correctness of the results where rescattering effects may not be completely negligible. The metastable D(2S) experiment is free of these objections but the low efficiency of the photon detectors here and in cascade experiments leaves the possibility that the results could be interpreted in terms of local realistic theories if the assumptions of Clauser et al. (1969) are questioned. In a further experiment the circular polarization correlation of the two photons from D(2S) was measured by inserting a quarter-wave plate between the collimating lens and the linear polarizer in each detection arm of the apparatus. These quarter-wave plates were achromatic with a retardation that varied by only about <10% in the wavelength range from 180 to 300 nm. The presence of the plates reduced the solid angle and the overall sensitivity of the system so that considerably longer recording times were necessary to achieve satisfactory statistical accuracy. In this experiment the transmission axes of the two linear polarizers were fixed perpendicular to each other and the fast axis of the quarter-wave plate in one detection arm was oriented at 45° to the transmission axis of the linear polarizer to form an analyzer for light of right-handed helicity. The fast axis of the other quarter-wave plate was then rotated from a position where it acted as an analyzer of light of right-handed helicity (i.e., fast axes parallel,  : 0°) to a position for analyzing light of left-handed helicity (i.e., fast axes perpendicular,  : 90°). In this way, and on the basis of the collapse of the state vector to R R the measured coincidence ratio R()/R was expected to vary   M in the ideal case as (1 ; cos 2)/4. However, due to the various imperfections of the linear polarizers and in particular of the quarter-wave plates, a satisfactory test of Bell’s inequality using circular polarization was not possible, but the required left-left and right-right circular polarization correlation as required from the state vector of the two-photon radiation

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F. 11. Comparison of linearly and circularly polarized - and -components for onephoton (a)- and two-photon (b) transitions. Note the higher nP states, which ‘‘contribute’’ to the two-photon transition as states represented in the dipole matrix elements of the secondorder quantum mechanical approximation theory.

process could certainly be verified (Fig. 10). However, in a recent experiment using a down conversion source (Torgerson et al., 1995) it has been shown that Bell’s inequality has been violated by about 40 standard deviations using circularly polarized light. This confirmation of the linear and circular polarization correlations of the two-photon transition of metastable atomic hydrogen suggests comparison to traditional one-photon transitions with their linear ( ) and circular (> and \) polarization components as illustrated in Fig. 11 for a normal transition from a n P state to a n S state. Analogous to this picture, the    M two-photon transitions can be associated with two linearly polarized and correlated -transitions   or   or two right-right > >

      and left-left \ \ circularly polarized components. In comparing these  

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one- and two-photon dipole transitions one has, however, to keep in mind that the two emitted photons are detected in coincidence. B. T  G-S E E While the preceding linear and circular polarization experiments of the two-photon radiation appear to agree well with the predictions of quantum mechanics, further critical considerations and suggestions have been made for testing all possible assumptions with regard to quantum mechanics versus local realistic theories. Important proposals were made by Garuccio and Selleri (1984) involving single-photon physics and two-polarizer type experiments to be interpreted in local realistic terms. They assumed that in addition to a polarization vector l, a photon possessed a detection vector  and, on this basis, were able to explain experimental results in local realistic terms. However, Haji-Hassan et al. (1987) tested this concept in an extension of the previously described linear polarization correlation experiment. They inserted a half-wave plate in one detection arm between the linear polarizer P and the photomultiplier D . By adding this half-wave plate to the system   the plane of polarization of the photons incident on photomultiplier D  could be varied independently of the polarization axis of P . At first the  transmission axes of the polarizers P and P in both arms of the apparatus   and the fast axis of the half-wave plate were set parallel to each other. When polarizer P was rotated by an angle , the fast axis of the half-wave plate  was rotated by an angle  : /2. With this procedure it was arranged that the orientation of the plane of polarization of the photon incident on the photomultiplier D did not change as the rotation of the polarizer and  half-wave plate took place. As the relative angle between the planes of polarization of the photons impinging on the detectors did not change significantly the results could not be distorted by enhancement effects due to the detection vector . Within the limits of experimental error the results were once again in agreement with quantum mechanics as shown in Fig. 12. In a second experiment with the half-wave plate the polarization axes of the two linear polarizers were fixed parallel to each other and the fast axis of the half-wave plate was rotated by an angle  relative to the axes of the polarizers. In this way the relative angle between the planes of polarization of the photons impinging on the photomultipliers could be varied continuously. It was also verified that the singles count rates did not vary as the half-wave plate was rotated. The two-photon coincidence measurements with the half-wave plate clearly establish the assumption that the relative angle between the planes of linear polarization of the two photons prior to detection plays no role in establishing experimental observation of polarization correlations. This

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F. 12. Ratios R()/R as a function of the orientation angle  of the fast axis of the M half-wave plate inserted between the linear polarizer P and the photomultiplier D . The   coincidence signal R() results from the observations with the transmission axes of the two linear polarizers parallel to each other ( : 0°) and the coincidence signal R is obtained with M the linear polarizers removed.

statement confirms the assumption of Clauser et al. (1969) that the probability of the joint detection of a pair of photons that emerge from two polarizers is independent of the relative angle between the polarization planes of the two photons just prior to their detection at the photomultipliers. There is no experimental evidence to support the fore mentioned idea of Garuccio and Selleri (1984) of introducing a detection vector and considering an enhanced or modified photon detection depending on the combined action of the detection vector  and polarization vector l. C. T-P E The further proposal by Garuccio and Selleri (1984) to introduce a second linear polarizer in one of the detection arms of the two-photon coincidence apparatus provided an opportunity to test quantum mechanics versus local realistic theories in a hitherto unexplored and novel procedure. The experiment and the relevant geometries for the polarization directions of the three polarizers (Haji-Hassan et al., 1987) are shown in Figs. 13 and 14. The orientation of the polarization plane of polarizer a was held fixed while that for polarizer b was rotated through an angle  in a clockwise sense and polarizer a through an angle  in an anticlockwise sense. The

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F. 13. Schematic arrangement of the three-polarizer experiment (Haji-Hassan et al., 1987). The orientation of polarizer a is fixed with its polarizer transmission axis parallel to the x axis. The transmission axes of polarizers b and a are rotated, respectively, through angles  and  relative to the x-axis.

ratio R(, )/R(, -) was measured where R(, ) is the coincidence rate with all three polarizers in place and R(, -) the rate with polarizer a removed. Results of these ratios are shown in Figs. 15 and 16 along with quantum mechanical predictions and the limits of the Garuccio-Selleri local realistic model (1975). The quantum mechanical prediction for  : 0 is close to the form cos ; sin with , the transmission efficiencies of + K + K polarizer a. These data confirm the validity of Malus’ cosine-squared law for the transmission of polarized light from a very weak source through polarizer a.

F. 14. Geometry and angles of the transmission axes of the three polarizers used in the Stirling three-polarizer experiment.

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F. 15. Ratios R(, )/R(, -) as a function of  and  in the experiment by Haji-Hassan et al. (1987). The solid lines represent quantum mechanical predictions for  : 0°, 33°, and 6.75°.

On the other hand the model of Garuccio and Selleri (1984) showed that, for any angle   90° (note that  is not the angle between the polarizers in this case), arranging the angles of the transmission axes of the three polarizers to satisfy the relations  : 3 and  ;  :  the ratio of the quantum mechanical prediction to that of their local realistic model must always be greater than some minimum value  as shown in Fig. 17. The * range 58°    80° where   1 can be used in particular as a test between * quantum mechanics and the local realistic theories of Garuccio and Selleri. For the maximum of  : 1.447,  : 71° and the corresponding value for  * is 33° and for  is 38°. For these values the approach of Garuccio and Selleri sets an upper limit on the predictions of the local realistic theories of 0.413 for the ratio R(, )/R(, -) while the experimental value according to

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F. 16. Variation of the ratio R(, )/R(, -) as a function of the angle  in the experiment of Haji-Hassan et al. (1987) for  : 0°, 15°, 30°, 45°, and 60°. The points marked (*) correspond to the results for  : 0°, the point marked (●) to  : 33°. The solid curve represents the quantum mechanical prediction for  : 0°, while the broken curve shows the upper limit for the ratio set by the local realistic model of Garuccio and Selleri (1984) for various angles .

Fig. 17 is 0.585 < 0.029, thereby violating the Garuccio-Selleri model by about six standard deviations or a difference of more than 40%. Even with some modification of the maximum value of  to 0.162 suggested by Selleri * (1985), the preceding ratio will be only 0.514, which is still violated by the experimental data by almost three standard deviations or about 15%. Accordingly the three-polarizer experiment appears to rule out the class of local realistic theories of Garuccio and Selleri in a more convincing way than do the currently published two-polarizer coincidence experiments. The quantum-mechanical prediction for  : 0° is close to the form cos + ; sin with , the transmission efficiencies of polarizer a. These K + K results thus confirm the validity of Malus’ cosine-squared law for the transmission of polarized photons from a very weak light source passing through polarizer a.

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F. 17. Lower limit  of the ratio of the quantum mechanical prediction to that of local * realistic theories with enhancement of the type proposed by Garuccio and Selleri (1984).

As can be seen in Fig. 16 for  : 33° and  5 40° there is a relative difference of more than about 40% between the prediction of quantum mechanics and the local realistic theory of Garrucio and Selleri, which is so much larger than from experiments with photons from cascades (see Fry and Thomson, 1976; Aspect et al., 1982), positronium annihilation (Paramannada and Butt, 1987), and in interference experiments with laser photon pairs (Brendel et al. 1992; Tittel et al., 1998; Weihs et al., 1998). We note that the three-polarizer method, which gives a much more sensitive test result as a large difference of at least 40% between quantum mechanics and Bell’s limit for local realistic theories (Fig. 16), is not simply based on a kind of gray filtering effect by the additional polarizer, but is based on an interference effect of the observed polarization correlation. This can be seen as follows (see Fig. 16). With directions of polarization of polarizer a parallel to the x-direction, that of polarizer b at angle  and of polarizer a at angle , the polarization correlation of the coincident two-photon radiation becomes P(, ) 5 cos cos

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with  :  ;  and  :  9    P(, ) : cos( 9 ) cos : [cos  ; sin( 9 ) sin ] : cos ; sin( 9 ) sin ; 2 cos  sin( 9 ) sin 

(10)

The last part of this equation represents kinds of interference terms. Accordingly, this type of the three-polarizer system for the detection of the two-photon radiation in opposite directions leads to a different polarization correlation and, in a way, a surprisingly larger difference between quantum mechanical and local realistic predictions for certain combinations of the directions of polarization of the polarizer as demonstrated in Fig. 16. D. B-T E It follows from considerations of parity and angular momentum conservation (Section II) that the two-photon state vector can be written as described by Eqs. (6) and (8) for circularly or linearly polarized components. In a coincidence experiment the ‘‘collapsed’’ components of the state vector are the right-right-hand R R or left-left-hand L L circularly polarized     components or, alternatively, the linearly polarized components x x

  and y y , respectively. These collapsed components of the state vector   are compatible with the hypothesis of Breit and Teller (1940) that the fine and hyperfine interaction due to electron and nuclear spin should not affect the two-photon polarization correlation since the Hamiltonian describing the two-photon emission has no off-diagonal components for fine and hyperfine interactions in second order. Figure 18a—c demonstrates the various cases of the polarized components of the collapsed state vectors: with electron and nuclear spin neglected (Fig. 18a); including the electron spin (Fig. 18b); and including both nuclear and electron spin (Fig. 18c). For the case of zero nuclear and electron spin (I : 0, S : 0), only the collapsed state vectors R R , L L , x x and y y are compatible         with the two-photon transitions through the intermediate virtual P states (Fig. 18a). For the case S : 1/2, I : 0 (Fig. 18b), possible ‘‘collapsed’’ state vectors would be R x , R x , L x or L x with m : <1 as a         H selection rule, which is compatible with one photon carrying one unit of angular momentum away. A state vector of the corresponding polarization correlation for m : ;1 would be H  :

1 (2

(R x ; L x )    

(11)

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125

F. 18. The ‘‘collapsed’’ state vectors for the two-photon transitions of metastable atomic hydrogen with (a) fine and hyperfine structure interactions neglected (m : 0), (b) fine  structure interaction included but hyperfine structure interaction neglected (m : 0, <1), and H (c) fine and hyperfine structure interactions included (m : 0, <1, <2). D

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F. 19. Experimental arrangement for the coherence and Fourier analysis of the twophoton radiation of metastable atomic hydrogen. While the detection arm B contains only the linear polarizer B the detection arm A includes a retardation plate of given thickness, a/4 wave plate, and the linear polarizer A in order to measure Stokes parameters.

which predicts, for example, that with a linear polarizer on one side of the two-photon source no variation in the coincidence signal would be observed when the transmission axis of a linear polarizer on the other side was rotated. However, the measured variation with rotation angle is described well by the state vectors in Eqs. (6) and (7), which rules out to a high degree of accuracy state vectors of the type given in Eq. (11). In other words, the electron spin plays no role in the polarization correlation of the two-photon decay. Including both the nuclear and electron spins of atomic hydrogen will again result in the collapsed state vectors as shown in the preceding. R x with m : <1, which are excluded by the experimental observa  $ tions. However, as can be seen from Fig. (19c), possible m : <2 transi$ tions are to be associated with the collapsed state vectors R L and   L R and this would require a state vector of the type    :

1 (2

(R L ; R L )    

(12)

The experiments of Perrie et al. (1985), however, give effectively zero coincidence count rates for the two photons with opposite helicity. In conclusion it can be stated that the polarization correlation experiments confirm the statement of Breit and Teller (1940) that the influence of the nuclear and electron spins can be neglected in the two-photon emission of metastable hydrogen. The correct state vector model of the two-photon emission in diametrically opposite directions is well represented by the state vectors of Eqs. (6) and (7) and not by those of Eqs. (11) and (12). Based

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upon the ‘‘simultaneous’’ process of the two-photon emission of the metastable hydrogen atom (i.e., within a time interval of 10\s, (see also the chapter by Demtro¨der, Keil, and Wenz, p. 149, this volume) the available time during which the two-photon emission takes place is too short for setting up the coupling mechanisms of the fine (10\—10\s) and hyperfine interactions (10\—10\s).

V. Coherence and Fourier Spectral Analysis — Experiment and Theory The coherence properties and spectral distribution of one-photon of the two-photon pair emitted by metastable atomic hydrogen were investigated in an experiment in which a series of birefingent retardation plates were placed in one arm of the detection system as shown in Fig. 19. The two photons emitted by the source are detected in coincidence by detectors D  and D . A linear polarizer with its transmission axis orientated in the  x-direction is placed on the left-hand side of the source (polarizer B). Thus, by virtue of the entangled nature of the two-photon state vector, Eq. (8) upon detection of a photon on the left, the complementary photon on the right, and before being detected itself, can to all intents and purposes also be regarded as polarized in the x direction. In the absence of any wavelength filter on the left the frequency of the photon on the right is indeterminate, and the radiation on the right can be considered to consist of a sequence of single frequency photons with spectral distribution determined by the spectrum of the source or, alternatively, as a sequence of minimum uncertainty wavepackets with the spectrum and coherence length of each wavepacket determined by the spectral characteristics of the two-photon source. If this radiation on the right is now passed through a uniaxial birefringent retardation plate with its axis at an angle of 45° to the x axis as shown in Fig. 19, its state of polarization will be changed and it will also be depolarized to an extent that depends upon the thickness of the retardation plate. The state of polarization of the radiation emerging from the retardation plate may then be monitored by measuring the Stokes parameters P , P ,   and P of the radiation on the right detected in coincidence with the  radiation on the left. The Stokes parameters are defined (Born and Wolf, 1965) as: I(0°)9I(90°) , P :  I(0°);I(90°)

I(45°)9I(945°) P :  I(45°);I(945°)

I(RHC)9I(L HC) P :  I(RHC);I(L HC)

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where I(0°) is the strength of the coincidence signal when the transmission axis of polarizer A is set at angle  : 0° to the axis with corresponding definitions for I(90°), I(45°) and I(945°). Similarly, I(RHC) and I(L HC) refer to the strength of the coincidence signal with the achromatic quarterwave plate shown in Fig. 19 in place and orientated with its x axis at <45° to the x axis so as to detect, respectively, right-hand-circularly (RHC) polarized light on left-hand-circularly (L HC) polarized light. The degree of the total polarization P is then given by P : (P ; P ; P   

(13)

which is also equal to the degree of coherence of the orthogonally polarized components if these components have equal intensity (Born and Wolf, 1965). If the retardation of the retardation plate of thickness d is  then (n 9 n )d M : C c where n and n are, respectively, the extraordinary and ordinary refractive C M indices of the material of the retardation plate. It is then easy to show (Sheikh, 1993) that, for monochromatic radiation incident upon the retardation plate, the Stokes parameters of the emerging radiation are given by P : cos , 

P : 0, 

P : 9sin  

However as the two-photon radiation has a continuous spectral distribution A(), the expected values of P , P , and P of the emerging radiation are    P : 





cos ()A()d







A()



P :0  P : 







sin ()d





A()d



Examination of the preceding expression reveals that, if the birefringence (n 9 n ) were frequency independent, P and P would be precisely the C M   Fourier cosine and Fourier sine transforms of the spectral distribution A() with the quantity (n 9 n ) d/c acting as the ‘‘time’’ variable. C M

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129

It has been shown (Duncan et al., 1997) that the frequency dependence of the birefringence can be allowed for by defining an effective spectral distribution F() in terms of P : 





F() cos td



P :0 



P :9 



F() sin td



which allows the easy calculation of P and P , and also allows an inverse   transformation analysis to the performed, which permits the easy calculation of the spectral distribution F() and hence A() from knowledge of P  and/or P .  In the the experiment, a series of retardation plates were placed one at a time on the right-hand side of the source. These plates were in the form of zero-order half-wave plates at wavelengths of 200, 300, and 486 nm. They consisted of two flat pieces of crystal quartz of slightly different thicknesses cut parallel to the optical axis and placed in contact with their axes perpendicular to give ‘‘effective’’ thicknesses d : 7.69, 18.84, 14.56, and 26.27 m deduced from the known birefringence properties of quartz. Additional effective thicknesses of d : 3.15 and 18.53 m were obtained by placing the 200- and 243-nm plates in series with optical axes parallel. An effective thickness of d : 37.11 m was obtained by placing the 243- and 486-nm plates in series with their axes parallel. The experimental results with allowance made for the imperfections of the polarizers are shown in Fig. 20 and demonstrate good agreement with the theoretical calculations based on the preceding Fourier analysis, taking into account that the spectrum is cut off above an angular frequency of 1.02;10 rad s\ ( : 185 nm) and below at 5.31;10 rad s\ (:355 nm). The problem of determining the spectral distribution from the Stokes parameter measurements is straightforward in principle but, in the case here, difficult in practice, due to the small number of measured points. However, by making a reasonable interpolation between the points and performing an inverse Fourier transform, it has been possible to show (Duncan et al., 1997) that the spectral distribution has the expected broadband form as show in Fig. 20. A bandwidth extending between angular frequencies of 1.02 ; 10 rad s\ and 5.31 ; 10 rad s\, on the basis of the usual Fourier bandwidth-time relationship, implies a coherence time of about 1.3 fs and

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F. 20. The Stokes parameters P , P and the degree of total polarization P as a function   of the t : d/c or the relative displacement of the photon wavepacket envelopes,

: d(n 9 n ) 9  C  A A

(n 9 n ) A  d A 





where d is the effective thickness of the retardation plate. The solid lines are calculated from Fourier transforms of the theoretical spectral distribution of the two-photon radiation.

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F. 20. Continued.

coherence length of about 385 nm for a single photon of the two-photon pair. The coherence length is, therefore, less than two wavelengths of the predicted 243-nm center wavelength of the spectral distribution. The coherence length and the coherence time of single photons of the two-photon pair are thus very short and, in the ideal case, in the absence of any filtering, are determined by the lifetime of the virtual intermediate state of the decay rather than having the long lifetime of the metastable state determine both the coherence time and length of the two-photon pair. In fact an analogous situation arises in the case of certain experiments involving the two photons produced by parametric down conversion where single-photon, secondorder interference effects occur over distances determined by the short coherence lengths of the photons after conversion, whereas two-photon, fourth-order interference effects take place over distances related to the usually much longer coherence length of the pump radiation (Franson, 1989; Ou et al., 1990; Kwiat et al., 1990; Brendel et al., 1992). It should be pointed out, of course, that although the lifetime of an excited atomic state is related to the coherence time of the resulting radiation produced when it decays, it is not equal to it. If the excited state has, say an energy width E, then by Heisenberg’s uncertainty principle, the lifetime  5 /2E, whereas on the basis of the Fourier bandwidth-time  relationship  5 1, the coherence time  5 1/, where  is the freA A

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quency bandwidth. Hence 1 h 2

 : : : : 4  A  E E 

(14)

and the coherence time  is greater than the lifetime  by a factor 4 . A  An alternative interpretation of the results can be made in terms of photon wavepackets, which can be considered to exist on the right following detection of complementary photons on the left. As a result of their different group velocities, the two orthogonally polarized wavepackets passing through the retardation plate are separated by an amount where

: (n 9 n ) d 9  C M SC A



d(n 9 n ) C M d d SC



(15)

 is the center angular frequency of the wavepacket. For quartz from A knowledge of its birefringence : 1.667 ; 10\ d. When is large enough the polarization of the radiation emerging from the retardation plate is essentially zero, and the distance over which the degree of polarization or degree of coherence P reduces to near zero gives a measure of coherence length of the wavepackets. The degree of polarization P is plotted against

in Fig. 20, from which it can be seen that the coherence length is approximately 350 nm with a corresponding coherence time of 1.2 fs, in agreement with the previous observations. Theoretical predictions of P , P and P are calculated based upon the   preceding Fourier analysis and the theoretical spectral distribution for hydrogen cut off above an angular frequency of 1.02 · 10 rad s\ (corresponding to  : 185 nm) and below at the complementary frequency of 5.31 · 10 rad s\ (corresponding to  : 355 nm). Allowance has been made for imperfections of the polarizers in the two arms for coincident photon detection. The results of Fig. 20 show good agreement between the calculated values and experimental data. The modulus P of the total polarization vector falls rapidly to a low value at 5 350 nm. The minimum length of the wave packets of the photons is thus approximately 350 nm, which can be considered as the experimentally determined coherence length l : ct   of a single photon of the coincident two-photon pair for which the corresponding coherence time t is thus approximately 1.2 · 10\ s and the  bandwidth 0.8 · 10 Hz. The experimentally determined coherence length of 350 nm is approximately one-and-a-half times the wavelength of the theoretically predicted center and maximum of the spectral distribution of the two-photon radiation at 243 nm.

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133

Some caution, of course, must be applied in using the wavepacket concept in the current case. Such a short, wide bandwidth classical wavepacket would undergo a significant amount of spreading and chirping in propagating through a dispersive medium. However, the maximum of the packet envelope would travel at the group velocity and the degree of overlap and hence the degree of polarization or coherence would be essentially unaffected. (See the discussion by Steinberg et al., 1992.)

VI. Time Correlation The existing theory of the two-photon decay implies that the spontaneous emission of the two photons takes place essentially simultaneously. However, the angular frequency bandwidth  sets a lower limit on the correlation time T (as opposed to coherence time) between the detection of A the two photons given by (Huang and Mandel, 1985) 1 T : A 

(16)

In the current case for an unfiltered signal the full bandwidth  : 1.546 ; 10 rad s\ and hence T : 6.47 ; 10\ s; these values for the correlation A time are very much less than the electronic resolving time of a few nanoseconds in the current experiment (Fig. 19), and much less than could be achieved with the most modern photon detectors and counting electronics. Biermann et al. (1997) tackled the problem of the time correlation between detection of the photons in the decay of metastable atomic hydrogen from the more sophisticated viewpoint of quantum electrodynamics and found a value T 5 3.33 ; 10\ s, which is not too different from the value A found in the preceding. Their theory also compares the nature of the two-photon decay process with a cascade through a three-level atom. It successfully predicts the asymmetrical nature of the time-correlation spectrum for a cascade source and the symmetrical nature of the time-correlation spectrum (Fig. 21) for the two-photon source.

VII. Correlated Emission Spectroscopy of Metastable Hydrogen: How Real are Virtual States? Over the past few years it has been shown that atomic spectroscopic (Rathe and Scully, 1995; Rathe et al., 1995; Scully and Zubairy, 1997) properties such as the energies and width of atomic states involved in the production

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F. 21. Spectral distribution deduced by performing a Fourier analysis of the P and P   data (both from Fig. 20), and as compared to the theoretical prediction with an upper cut-off at 0.531 · 10 rad s\ (after Duncan et al., 1997).

of correlated cascade radiation can be probed via intensity correlation interferometry (Hanbury-Brown and Twiss, 1954; Scully and Dru¨hl, 1982; Ou and Mandel, 1988; Shih and Alley, 1998; Rubin et al. to be published; Franson, 1989; Rarity and Tapster, 1900; Kwait et al., 1992; Herzog et al., 1995). This correlated emission spectroscopy was heretofore studied in the context of real states. However, recent intriguing experiments by Kleinpoppen et al. (1997) on the 2s  1s transition in atomic hydrogen extend these considerations to the realm of virtual states. These experiments raise many questions, to wit: Can we interpret these studies as yielding a measure of a delay between emissions? And is there a virtual state between the 2s and 1s states, which is ‘‘like’’ a rapidly decaying real state? Motivated by the preceding considerations, we here calculate the twophoton correlation function for the radiation spontaneously emitted by metastable hydrogen. These studies taken together with the cascade emission problem (Fig. 22); we yield insight into what we mean by the ‘‘delay time’’ between emissions and the extent to which the virtual state concept is a useful one in the current context. We proceed to sketch the derivation of the two-photon correlation

ANALYSIS OF THE OPTICAL TWO-PHOTON RADIATION

135

F. 22. Figure illustrating Gedanken coincidence measurement of radiation from atoms, which are of either the hydrogen metastable (upper part) or the cascade type (lower part). In both cases the two photons emitted travel to detectors located at distances r and r from the   emitting atom. Both detectors are equipped with shutters such that the detection time is defined within narrow time windows about the times t and t . The near detector at about r    establishes the ‘‘start fate’’ and a coincidence is observed in detector 2 at a time that is determined by the position of detector 2 indicated by x. When one considers the cascade problem (see text) one finds that when the first photon goes to detector 1 coincidences are observed for x  0 and when the second photon goes to detector 1 coincidences occur for x  0. If the photons are indistinguishable then these two possibilities are added and this yields the ‘‘cusp’’ correlation function. In the case of hydrogen decay a similar logic prevails in which the functional form now changes from a cusp function to the smoother ‘‘Bessel function’’ result of Eq. (30).

function, which describes the joint count probability for registering a count in both a detector located at r at time t and one at r at time t . This is     given by the Glauber second-order correlation function G(1, 2) : E\(I)E\(2)E *(2)E *(I)

(17)

where 1 and 2 stand for r t and r t and the positive frequency (annihila  tion operator) part of the field is given by E*(1) :  ak ei( k 9I ) I  k

(18)

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in which : (  / V ), where is the vacuum permittivity and V the I I   quantization volume. ak is the annihilation operator for the k th mode and the polarization of the field is neglected as an unessential complication. The state of the radiation field  will be derived here. However, it is useful first to note that we may insert a complete set of states of the field between the positive and negative operator parts of Eq. (17) and take advantage of the fact that only the vacuum term survives to write G(1, 2) : (1, 2)

(19)

(1, 2) : 0E*(2)E *(I)

(20)

where

The two-photon decay of the hydrogenic 2s state proceeds via energy nonconserving transitions to intermediate p states depicted in Fig. 22 (upper part). The Hamiltonian for the process is given by H :   m m ;   k a k*ak K  k K ;  (g (k )a n ; adj.)(a *k e9ik ·r ; adj.) ?L  k

(21)

;  (g (q )b n ; adj.)(a * e9i g ·r ; adj.) @L  q The notation is largely explained in Fig. 22 where  is the energy of the K state m , g (k ) is the usual atom-field coupling constant which goes as (k, ?L and a*k (a k ) are the usual creation and annihilation operators for the kth mode with similar coupling constants and radiation operators associated with the qth mode. It is noted that energy nonconserving operations of the type a *k n a dominate the 2s ; 1s decay process. We proceed by removing the intermediate states via a canonical transformation. The result is the two-photon, two-level Hamiltonian H :  a a ;   k a*k a k ;  (G k,q b aa*k a*g ei(k;g ) · r ; adj) (22)    k k,  q where the effective coupling constant is given by g (k)g (q) g (q)g (k) L@ L@ G :  ?L ; ?L IO  9 9  9 9 L  L I  L O






(23)

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ANALYSIS OF THE OPTICAL TWO-PHOTON RADIATION

The sum over states can be carried out (Breit and Teller, 1940) to yield a Ak  9 B (kq G : 4 & IO e kq





(24)

where a : /me, e is the electronic charge, ck :  , and A and B are the &   Breit-Teller constants, (A  9, B  15). In order to calculate the two-photon correlation function we must first specify the two-photon quantum state. The atom-field state is given by (t) : (t), 0 ;   k , q (t)1 k , 1q

(25)

 k,  q

and in view of the Hamiltonian equation (17) we have i a : 9i  9  Gk,q e9i(k;g ) · r  k ,q 

k , q

(26)

i  k ,q : 9( ;  ) k ,q 9 G ei(k;q ) · r  I O

IO

(27)

If we proceed in the spirit of Weisskopf and Wigner, we note that the excited state amplitude is (t) : exp(9(i ; )t) and upon insertion of this in Eq.  (27) and performance of the integration we find the state of the field in the long time limit to be G / i(k;q ) · r IO e 1k , 1q i 9  k , q IE

 : 

(28)

where  :  9  9  . Using Eqs. (20) and (28) we find that IO  I O (1, 2) :

I r M  t 9  e\AE\EAe\GSŠE\EA  r r c  r r ;C (t 9 t ) 9  9  e\GSŠE\EA ; (1  2) (29)    c c












where we have taken the atom to be at origin, that is, r 9 0.I is an overall M constant and j (k /2x) j (k /2x) C (x) : A   9B    (k /2x) (k /2x)  

(30)

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in which j (y) is the spherical Bessel function of order n. The quantity C (x) L  is sharply peaked at about x : 0, as is seen in Fig. 22. This is the main result of this chapter. The two-photon detection amplitude Eq. (29) has several interesting features that can best be brought out against the backdrop of cascade radiation. As indicated in Fig. 22, cascade radiation involves the emission of a photon associated with the transition between real states a and b

followed by another emission in the passage from b to c . Thus cascade radiation can be said to consist of two single photon events. The state of the radiation field associated with such a double emission cascade is given by Scully (1995) and Huang and Eberly (1993): g (k)g (q) ?@ @A ei(k;q ) · r 1k , 1q

(i 9  )(i 9 ( 9  ))  k, q ? IO @ @C O

 : 

(31)

where  :  9  9  . IO ?A I O The probability amplitude associated with state equation (31) is then found to be r I (1, 2) :   t 9  e\A?R\EAeGS?@R\EA  c r r  r r ;C (t 9 t ) 9  9  e\GS@AE\EA ; (1  2)    c c












(32)

where C () : ()e\A @K 

(33)

The 1  2 interchange in Eq. (32) is discussed later and is a manifestation of overall symmetry. In fact, if the intermediate state b in Fig. 22 is long lived then it is possible to write (1, 2) :  (1) (2) ;  (1) (2) ? @ @ ?

(34)

where  (1) goes as (t 9 r /c) exp(9 (t 9 r /c)) and  (2) is a similar ?   ?   @ expression with t , r ; t , r and  ;  . In this sense the 1  2 inter    ? @ change term in Eq. (32) is a reflection of the boson character of (1, 2). With the photon-photon correlation function for the two-photon- and cascade decay processes in hand, we turn to the question of experimental implications. In order to probe the very short coherence time between the two 2s ; 1s photons we consider the experimental setup as sketched in

ANALYSIS OF THE OPTICAL TWO-PHOTON RADIATION

139

F. 23. Figure illustrating experimental configuration in which atom located symmetrically between detectors 1 and 2 (D and D ) has the possibility of two paths to detector 2   (corresponding to a long and short path) whereas there is only one path to detector 1. No shutters are involved. The experimental arrangement is contained in references Rathe and Scully (1995), Rathe et al. (1995), and Scully and Zubairy (1997) and is conceptually equivalent to the experiment of Kleinpoppen et al. (1997).

Fig. 23. There we see that there are two amplitudes contributing to (1, 2), the short-short case in which r : r : S, and the short-long situation in   which r : S and r : L . The total amplitude is now given by   (1, 2) : (S, S) ; (S, L )

(35)

The joint photodetection probability is given by P : 

  



dt 





dt (12) 

(36)

For cascade radiation (S, S) is obtained from Eq. (32) with r : S, r : S   and (S, L ) are obtained by taking r : S, r : L . Carrying out the   integrations over t and t we find   2I 1 [1 ; e\*\1‚/l cos( (L 9 S)/c)] P :  @A  r   ? @

(37)

Thus we see that correlated emission interferometry can yield spectroscopic information, that is, the linewidth  even when  is much larger than the @ @ bandwidth of detectors 1 and 2. The potential utility of such ‘‘correlated emission spectroscopy’’ (CES) was discussed in Rathe and Scully (1995),

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Rathe et al. (1995), and Scully and Zubaiy (1997); for an analysis of the current ‘‘long-short’’ path CES set-up, see Scully and Zubairy (1997) and Meyer et al. (199?). The point is that even very short delay times \  \ @ ? may be measured by simply ‘‘moving mirrors’’ so as to vary the path length difference L 9 S. Returning now to the 2s ; 1s two-photon problem and taking r : S,  r : S or L in Eq. (29), we again integrate over t and t to find a good    approximation  2I 1 1 ; e\*\1‚J‚ cos  (L 9 S)/l P :   2 r c/l








(38)

In obtaining Eq. (38) we have made use of that fact that we may, to a good approximation, replace the ‘‘spherical Bessel function’’ expression for C ,  Eq. (30), by a Gaussian. The coherence length l can be ‘‘read-off’’ from Fig. 24 by noting that at the 1/e point k l/2 5 3 and so l 5 6/2 . That is,  the coherence length is of the order of the Lyman- wavelength, which is in reasonable agreement with the results of Kleinpoppen et al. (1997).

F. 24. Plot of the conditional detection amplitudes C for the metastable hydrogen, (solid  line describes Eq. (29), short dashed line describes Eq. (40), long dashed line describes Eq. (41)) as a function of position of detector 2. The key point is that all three curves are remarkably similar even though the atomic configurations are vastly different. This supports a ‘‘pulse overlap’’ paradigm as opposed to a ‘‘time delay’’ interpretation of Eq. (29).

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141

Having considered experimental implications of our main result in Eq. (29), we return to the somewhat philosophical questions: time delay, yes, or no? Virtual states: physical or unphysical? To answer these questions we focus on the conditional amplitudes equations (29) and (32). Consider the first term in Eq. (32): It describes the cascade situation in which the first photon is detected at r , t and the second at r , t . Let us fix     t , t , and r and vary r . The (t 9 t 9 (r /e 9 r /e)), function in C then          tells us that we will get a count in detector 2 only if ct 9 r  ct 9 r or     r  ct 9 ct ; r ; set r : R at the equality point, that is, define R :        c(t 9 t ) ; r . Thus R is the location of the second photon at time t given      that the first photon was detected at time t and location r .   Thus, a count in detector 2 satisfying these conditions had to be from a second photon emitted simultaneously with the first. Now let us move detector 2 such that r : R ; x, while still accepting counts only at time   t . The C function in Eq. (32) now reads C (x) : (9x) exp  x, which    @ describes events in which the second photon was emitted at a time x/c after the first. To sum up: the x  0 region depicted in Fig. 22 describes the case in which the first photon goes to detector 1 and the second to detector 2 with a possible dwell time in state b of order 1/ . @ The second (1 ; 2) term in Eq. (32) describes the case in which the second photon goes to detector 1 and the first goes to detector 2. Now the C function reads C (x) : (x) exp 9  x and such events account for the   @ x  0 region in the C (x) vs the x curve depicted in Fig. 22.  By comparing the preceding with the two-photon case equation (29), we see that if we take  :  :  /2 in Eq. (32) the two-photon and cascade ?@ @A  coincidence results are very similar. The upper part of Fig. 22 depicting the two-photon correlation function for metastable hydrogen is also symmetric in the first and second photons. This would seem to say that the 2S ; 1S photons are not so much simultaneous (delay time of order of an optical period) as symmetric (can not tell which photon was emitted first). The message is clear: There is no way to tell which photon is emitted first in the two-photon case involving transitions from a to the virtual level at  /2  and from the virtual level on to c . This is in contrast to the cascade case where we could have used the frequency of the first (a ; b) photon to distinguish it from the second (b ; c) photon (but did not). But where ‘‘are’’ the atoms during this delay? And how did the virtual level at  /2 come about? That is, in the cascade problem the propagators  arose from the poles in Eq. (31). However, in the two-photon state equation (28) there are no poles at  /2, so where do the exp 9i( /2)(t 9 r /c)     propagators come from?

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Let us address the virtual state issue first. By inserting Eq. (28) into Eq. (20) and doing the implied integrals, expressions such as exp 9 (t 9 r/c) 9 1  arise. This is rewritten as exp 9i( /2)(t 9 r/c) sin( /2)(t 9 r/c) in order to   arrive at Eq. (28) with the spherical Bessel functions (which, after all, are just combinations of sines and cosines) appearing in Eq. (30). This is to be compared with the propagator terms in the cascade problem, for example, exp(9i (t 9 r /c)) in Eq. (31). If we take  :  :  /2 these terms ?@   ?@ @A  are the same in Eqs. (28) and (31). However, their origin is very different. In the cascade problem the propagator terms can be traced to the poles of Eq. (31), which reflect real physical states. Even in the limit of very rapid decay of intermediate state b, it still carries quantum numbers such as n, l, m, that is, it is a ‘‘real’’ state. This is not the case in the ‘‘virtual’’ level at  /2 in the  2s ; 1s decay problem. In some sense the virtual level at  /2 would seem  to be virtually not in the cards. It is therefore interesting to see it arriving in such a ‘‘backdoor’’ fashion. Let us now return to the time delay question. We have already noted that the similarity between the cascade problem (in which there is a delay between emissions) and the two-photon case would seem to suggest the conclusion that there is a short delay between the two-photon 2s ; 1s emission. (See Fig. 23.) If there is a short delay, where is the atom during this time? One might think that perhaps it is ‘‘in’’ the virtual state, that is, it is residing in the n

states for a delay time L  ( 9  9  )\ and the sum of all such times B ? L I yields  (total)  1/  1/ . B  In order to check the preceding hypothesis, consider the problem in which we replace all the levels in Eq. (23) by only one state. For simplicity let us concentrate on two special cases, in which the first case corresponds to the 2P level at energy  , and the second to a continuum p-level having  energy   . L  In the first example the effective coupling constant becomes 1 1 ; (kq G :  IO  k q




(39)

where  is an interesting constant.  Now when Eq. (38) is inserted in Eq. (27) and the result is used in Eq. (20) to calculate (1, 2), we pick up another factor of (kq from the electric

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field operators: G (kq :  (k\ ; q\)kq :  (k ; q) :  k . In calcu@O     lating (12) we again find Eq. (29), but now we have k C (x) : sin  x  2

k x 2





(40)

similarly for the second example, in which we consider a very high energy intermediate state, we now find k x C (x) : j   2

k x 2





(41)

These results are summarized in Fig. 24. Both examples yield correlation amplitudes, which are essentially the same as the full result given in Eq. (30). Thus we see that the ‘‘dwell time,’’ if there is one, is independent of the energy of the intermediate state n. This contrasts with the L  ( 9  9  )\ hypothesis and bodes ill B ? L I for the delay time paradigm. These considerations favor the picture of simultaneous emission of two short pulses, with the C (x) correlation factor  representing the overlap between the pulses. In conclusion, we have calculated the two-photon correlation function for the decay of metastable hydrogen and analyzed an experimental arrangement for measuring the photon-photon correlation length; the results are found to be in reasonable agreement with the measurements of Kleinpoppen et al. (1997). A comparison between the current results and those obtained from a cascade emission scheme shows that they are substantially similar. It was found that if, in the cascade problem, we set  :  :  /2 then compari?@ @A  son with the 2s ; 1s two-photon correlation function suggests a virtual level at  /2 and, as seen in Fig. 22, the correlation functions are essentially the  same. This would tend to support a ‘‘delay time’’ paradigm to explain the photon correlation results in the decay of metastable hydrogen. However, upon further reflection, it is found that the ‘‘dwell-time’’ (if one exists) in the intermediate state is essentially independent of the position of this state. This suggests a ‘‘pulse-overlap’’ paradigm, in which the metastable state decays by emitting two simultaneous ‘‘pulses’’ or wavepackets, each having one photon energy. From this point of view the form of C (x) is  understood as arising from the pulse overlap area. From this perspective it is misleading to think of a virtual state at  /c that is similar to a rapidly  decaying real state.

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VIII. Conclusions This chapter summarizes and highlights the activities of research on the two-photon coincidences of metastable atomic hydrogen with a look at both the theoretical and experimental aspects. In comparison to similar two-photon investigations with one-electron hydrogen-like ions (see, e.g., Gould and Marrus, 1983) the atomic hydrogen case has the disadvantage of low intensity of the two-photon radiation; however, the advantage is the possibility that coincident optical polarization techniques and Fourier analysis in the near ultraviolet spectral region can be applied. The experiment with three polarizers, two polarizers in one, and one polarizer in opposite direction for the detection of the two-photon emission appears to be the one of the many EPR-type measurements that has, for certain observational conditions, the largest difference (40%) between predictions of nonlocal quantum mechanics and local classical theories. This effect results from an interference effect of the optical polarization analysis. One would have to judge future three-or-more particle EPR experiments based on the work by Greenberger et al. (1989) and other possible techniques in comparison to the results of the threepolarizer EPR experiments for the two-photon radiation of metastable atomic hydrogen. The various physical characteristics of two-photon radiation are linked to the properties of the energy scheme and the spectroscopic transitions of atomic hydrogen and their correlation effects. While static properties such as the energies of the states and their various one-photon transitions of atomic hydrogen are well understood theoretically to a highest degree of accuracy, the newly measured coherence effects of the two-photon radiation require further, more detailed developments of theory in order to get a better agreement with the results of the recent polarization correlation and coherence experiments. It is obvious that a modern representation of the hydrogen energy states and spectrum should take into account the influence of the ‘‘virtual states’’ between its real states (e.g., the 1S and 2S states but also any combination of higher states). The first recent approach to this representation by Biermann et al. (1997) for considering coherence effects from cascade transitions of real higher states leads to a program to extend and consider different distributions of intensities and amplitudes for these transitions. This appears to be another important fundamental problem of quantum mechanical applications for the structure of the physical description of static and dynamical properties of atomic hydrogen.

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IX. Acknowledgments M. O. Scully would like to thank S. Biermann and A. Toor for their help in preparing parts of the manuscript and R. Arnowitt, D. Church, E.-R. Hu, E. Fry, G. Welch, M. Hillery, M. Fleischhauer, H. Schuessler, and S. Yelin for stimulating discussions. The support of The ONR, The Welch Foundation, and the Texas Advanced Research Project are also gratefully acknowledged. H. K. Kleinpoppen would also like to acknowledge his appreciation to The Leverhulme Trust (London) for their award of an Emeritus Fellowship. A. J. Duncan tragically died during the joint preparation of this manuscript and, in recognition of his outstanding contribution, we wish to dedicate this chapter to his memory.

X. References Akheiser, A. I. and Berestetskii (1965). Quantum Electrodynamics, New York: Wiley. Andra¨, H. J. (1974). Phys. Scr. 9:257. Andra¨, H. J. (1979). Progress in Atomic Spectroscopy, W. Hanle and H. Kleinpoppen eds., part B, New York: Plenum Press, pp. 829—953. Aspect, S., Dalibard, J., and Roger, G. (1982). Phys. Rev. L ett. 49:1804. Bacal, M., Truce, A., Doucet, H. J., Lamain, H., and Chretian, M. (1974). Nucl. Instr. Meth. 114:407. Bassani, G. F., Inguscio, M. and Ha¨nsch, T. W. (eds.). (1989). T he Hydrogen Atom, Berlin: Springer Verlag. Bell, J. F. (1964). Physics 1:195. Biermann, S., Scully, M. O., and Toor, A. H. (1997). Phys. Scr. T72:45. Bohm, D. and Aharonov, Y. (1957). Phys. Rev. 108:1070. Born, M. and Wolf, E. (1965). Principles of Optics, Edinburgh, Scotland: Pergamon Press. Breit, G. and Teller, E. (1940). Astrophys. J. 91:215. Brendel, J., Mohler, E., and Martienssen, W. (1992). Europhysics L ett. 20:575. Clauser, J. F., Horne, M. A., Shimony, A., and Holt, R. A. (1969). Phys. Rev. L ett. 23:880. Drake, G. W. F. (1988). T he Spectrum of Atomic Hydrogen, G. W. Series, ed., Singapore: World Scientific, p. 137. Drake, G. W. F., Goldman, S. P., and van Wijngaarden, A. (1979). A. Phys. Rev. A20:1299. Duncan A. J. and Kleinpoppen, H. (1988). Quantum Mechanics Versus L ocal Realism — The Einstein-Podolsky-Rosen Paradox, F. Selleri, ed., New York: Plenum Press, p. 175. Duncan, A. J., Sheikh, Z. A., Beyer, H. J., and Kleinpoppen, H. (1997). J. Phys. B30:1347. Einstein, A., Podolsky, B., and Rosen, N. (1935). Phys. Rev. 47:779. Franson, J. D. (1989). Phys. Rev. L ett. 62:2205. Freedman, S. J. (1972). PhD thesis, University of California, Berkeley. Friedrich, H. (1998). T heoretical Atomic Physics, 2nd edition, Berlin: Springer Verlag. Fry, E. S. and Thompson, R. C. (1976). Phys. Rev. L ett. 37:465. Garrucio, A. and Selleri, F. (1984). Phys. L ett. 103A:99.

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Goeppert, M. (1929). Naturwiss. 17:932. Goeppert-Mayer, M. (1931). Ann. Phys. 9:173. Goldman, S. P. and Drake, G. W. F. (1981). Phys. Rev. A24:183. Gould, H. and Marrus, R. (1983). Phys. Rev. 28:2001. Greenberger, D. M. and Zeilinger, A. (1995). Fundamental Problems in Quantum T heory, New York: New York Academy of Sciences. Greenberger, D. M. and Zeilinger A. (eds.). (1995). Fundamental problems in quantum theory: A conference held in honour of Professor J. A. Wheeler, Annals of the New York Academy of Sciences, 755. Greenberger, D. M., Horne, M., and Zeilinger, A. (1989), in: Bell’s T heorem, Quantum T heory and Conceptions of the Universe, Ed. M. Kafatos, Boston: Kluwer Academic, p. 73; see also Greenberger, Horne, Shimony and Zeilinger. (1990). Am. J. Phys. 58:1131. Haji-Hassan, T. et al. (1987), in Book of Abstracts. Fifteenth Int. Conf. Physics of Electronic and Atomic Collisions. J. Geddes, H. B. Gilbody, A. E. Kingston, G. J. Latimer, H. J. R. Walters, eds., p. 74. Hanbury-Brown, H. and Twiss, R. (1954). Phil. Mag. 45:633; see also Bayin, G. (1955). L ectures on Quantum Mechanics, New York: Benjamin Press, p. 431. Herzog, T., Kwait, P., Weinfurter, H., and Zeilinger, A. (1995). Phys. Rev. L ett. 75:3034. Holt, R. A. (1973). PhD thesis, Harvard University, Cambridge. Hong, C. K. and Mandel, L. (1985). Phys. Rev. 31:2409. Huang, H. and Eberly, J. H. (1993). J. Mol. Opt. 40:915. Kleinpoppen, H., Duncan, A. J., Beyer, H. J., and Sheikh, Z. A. (1997). Phys. Scr. T72:7. Kru¨ger, H. and Oed, A. (1975). Phys. L ett. 54A:251. Kwait, P., Steinberg, A., and Chiao, R. (1992). Phys. Rev. A45:7729. Kwiat, P. G., Vareka, V. A., Hong, K. C., Nathel, H., and Chiao, R. Y. (1990). Phys. Rev. A 41:2910. Lamb, W. E. and Rutherford, R. C. (1947). Phys. Rev. 72:241. Lipeles, M., Novick, R., and Tolk, N. (1996). Phys. Rev. L ett. 15:690. McCarthy, I. E. and Weigold, E. (1995). Electron-Atom Collisions, Cambridge: Cambridge University Press. Meyer, G. et al., (I997). Electron T heory and Quantum Electrodynamics: 100 Years Later. J. P. Dowling, ed., New York: Plenum Press. Novick, R. (1963). Physics of the One- and Two-Electron Atoms, F. Bopp, and H. Kleinpoppen, eds., Amsterdam: North Holland, pp. 296—325. O’Connell, D., Kollath, K. J., Duncan, A. J., and Kleinpoppen, H. (1975). J. Phys. B8:214. Ou, Z. and Mandel, L. (1998). Phys. Rev. L ett. 61:50. Ou, Z. Y., Zou, X. Y., Wang, L. J., and Mandel, L. (1990). Phys. Rev. L ett. 65:321. Paramananda, V. and Butt, D. K. (1987). J. Phys. G13:449. Perrie, W. (1985). PhD thesis, Stirling; Duncan, A. J., Perrie, W., Beyer, H. J. and Kleinpoppen, H. (1985), in Fundamental Processes in Atomic Physics, H. Kleinpoppen, J. S. Briggs, and H. O. Lutz, eds., New York/London: Plenum Press, pp. 555—572; and Perrie, W., Duncan, A. J., Beyer, H. J. and Kleinpoppen, H. (1985). Phys. Rev. L ett. 54:1970. Rarity, G. and Tapster, P. (1990). Phys. Rev. L ett. 64:2495. Rathe, U. and Scully, M. (1995). Math. Phys. 34:297. Julian Schwinger Memorial Lecture. Rathe, U., Scully, M., and Yellin, S. (1995). Fundamental problems in quantum theory, Annals N.Y. Acad. of Sci. Scully, M. and Dru¨hl, K. (1982). Phys. Rev. A25:2208; Eichmann, U. et al. (1993). Phys. Rev. L ett. 70:2359. Scully, M. O. (1995), in Advances in Quantum Phenomena, E. Beltrametti, and J. M. LevyLeblond, eds., New York: Plenum Press.

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Scully, M. O. and Zubairy, M. S. (1997). Quantum Optics, Cambridge: Cambridge University Press. Selleri, F. (1985). Private communication. Selleri, F. (1988). Quantum Mechanics Versus L ocal Realism — the Einstein-Podolsky-Rosen Paradox, New York: Plenum Press. Series, G. W. (ed.). (1998). T he Spectrum of Atomic Hydrogen. Advances, Singapore: World Scientific Publishing Company. Sheikh, Z. A. (1993). PhD thesis, University of Stirling. Shih, Y. and Alley, C. (1998). Phys. Rev. L ett. 61:2921; see also Rubin, M., Klyshoko, D. . Shih, Y. and Sergienk, A. (to be published). Steinberg, A. M., Kwiat, P. G., and Chiao, R. Y. (1992). Phys. Rev. A45:6659. Targerson, J. R., Branning, D., Monken, C. H., and Mandol, L. (1995). Phys. Rev. A 51:4400. Tittel, W., Brendel, J., Zbinden, H., and Gisin, N. (1998). Phys. Rev. L ett. 81:3563. Rathe, U. and Scully, M. (1995). L ett. Math. Phys. 34:297—307, Julian Schwinger Memorial Issue. Weihs, G., Jenneweinn, T., Simon, C., Weinfurter, H., and Zeilinger, A. (1998). Phys. Rev. L ett. 81:5039.

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ADVANCES IN ATOMIC, MOLECULAR, AND OPTICAL PHYSICS, VOL. 45

LASER SPECTROSCOPY OF SMALL MOLECULES W. DEMTRÖDER, M. KEIL, and H. WENZ Fachbereich Physik Universität Kaiserslautern, D-67663 Kaiserslautern, Germany I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. High Vibrational Levels in Electronic Ground States . . . . . . . . . . . . . A. Overtone Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Modulated Absorption Spectroscopy . . . . . . . . . . . . . . . . . . . . . 2. Optothermal Spectroscopy in Molecular Beams . . . . . . . . . . . . . B. Fluorescence Spectroscopy of High Vibrational Levels . . . . . . . . . . C. Stimulated Emission Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Laser Spectroscopy of Electronically Excited Molecular States . . . . . . A. Laser Induced Fluorescence and Resonant Two-Photon Ionization. B. Optical Double Resonance Techniques . . . . . . . . . . . . . . . . . . . . . . IV. Sub-Doppler Spectroscopy of Small Alkali Clusters . . . . . . . . . . . . . . . V. Time-Resolved Laser Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Lifetime Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Time-Resolved Spectroscopy of Collision Processes . . . . . . . . . . . . C. Time-Resolved Measurements of Photodissociation Processes . . . . D. Coherent Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 152 152 153 158 160 167 171 172 173 180 180 187 190 192 194 196 196

I. Introduction During the last thirty years remarkable progress has been achieved in our understanding of structure and dynamics of small molecules. This is due mainly to the development of new experimental techniques with improved sensitivity and spectral resolution (Demtro¨der, 1998), but to a large extent also to the impressive advances in computational methods, which allow abinitio calculations of small molecules with astonishing accuracy (Hehre, et al., 1986; Hinchcliffe, 1987; Ha¨ser and Ahlrichs, 1989; Kutzelnigg, 1991; Lee and Head-Gardon, 1999). To date, most of our knowledge on the structure of molecules in their electronic ground states stems from microwave spectroscopy (Gordy and Cook, 1970), NMR spectroscopy (Shaw, 1987; Friebolin, 1988), and infrared and Raman spectroscopy (Griffith and de Hareth, 1986). These techniques 149

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have brought detailed and accurate information on bond lengths and bond angles, force constants, electric quadrupole moments and spin effects, such as spin-rotation splittings or hyperfine structures of stable molecules and small radicals. The application of laser spectroscopy has extended the range of accessible molecular energy levels to high lying vibrational levels in the electronic ground state and to electronically excited states. This range of excitation energies is of great interest for chemical reactions, which are generally substantially enhanced by internal excitation of the reactant molecules because the excitation energy facilitates their penetration through possible reaction barriers (Wyatt and Zhang, 1996). The goals of modern molecular spectroscopy are therefore not only the elucidation of molecular structures in excited states and the interactions between different states but, in particular gains in information on both the dynamics and the possible reaction paths of selectively excited molecules. These goals can be met by spectroscopic techniques with high spectral or temporal resolution. A breakthrough in spectral resolution was achieved with the invention of sub-Doppler techniques, such as nonlinear saturation spectroscopy (Shimoda, 1976; Field et al., 1998), polarization spectroscopy (Demtro¨der, 1998), Doppler-free multiphoton spectroscopy (Sieber et al., 1988; Grynberg and Cagnac, 1977), or linear laser spectroscopy in collimated molecular beams (Scoles, 1992), where the Doppler width is reduced by collimating the transverse molecular velocity components. The spectroscopy of molecules under nearly collision-free conditions in molecular beams allows investigations of long-living excited molecular states, such as metastable states or Rydberg levels closely below or above the ionization limit, which may decay by competitive processes, such as radiative decay, auto-ionization, or predissociation. Classical spectroscopy was restricted mainly to ‘‘allowed’’ transitions in molecules. Perturbations of excited levels by other interacting states showed up as deviations of line positions or intensities from their expected regular values (Lefebvre and Field, 1986). The development of more sensitive laser spectroscopic techniques allows direct observation of ‘‘forbidden’’ transitions, thus opening the way to direct access to many more excited molecular states which had not been found before. In the case of strong nonlinear couplings between different highly excited levels even chaotic behavior of molecular dynamics might be observed, thereby altering the statistical features of the distribution of energy separations between neighboring vibrational levels in polyatomic molecules. In the classical model of a vibrating molecule such chaotic behavior implies

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that the movements of the nuclei can no longer be described by the superposition of normal vibrational modes instead but follow chaotic paths in phase space (Casati, 1990). With the development of pico- and femtosecond lasers, time resolution down to 10\s has been reached (Rullie`re, 1998). This allows direct observations of molecular dynamics on a femtosecond time scale, as for example the wave packet dynamics of vibrating molecules or the microscopic view on the breaking and forming of molecular bonds during dissociation or recombination processes (Zewail, 1994; Assion et al., 1996). This chapter is restricted to laser spectroscopy of small molecules with two to four atoms, with the major focus on diatomic and triatomic molecules. Although most examples are taken from our own work in Kaiserslautern, it should be emphasized that many groups in different laboratories around the world have contributed to this field. Only a small fraction of their work can be mentioned here due to limited space. The references should help the reader to become familiar with at least part of the international research in this field. This chapter is organized in the following way. Investigations of highly excited vibrational-rotational levels in the electronic ground states of polyatomic molecules by different sensitive techniques of overtone spectroscopy or by laser-induced fluorescence and stimulated emission spectroscopy are first presented. The interesting question arises as to whether selective excitation might result in the enhancement of wanted decay channels and reaction paths or if fast intramolecular vibrational redistribution (IVR) due to strong couplings of the selectively excited level with a manifold of other levels might destroy any selectivity (Butler, 1998). The next section covers methods for measuring electronically excited states of diatomic and triatomic molecules and illustrates the results with several examples. Particularly interesting research objects are small alkali clusters, which show a complex dynamics and may serve as model examples for floppy molecules with no rigid geometrical structure. They are discussed in Section IV. The last section deals with time-resolved laser spectroscopy and some of the recent results obtained on lifetime measurements and investigations of collision processes. The fascinating new field of femtosecond dynamics, where coherent control of the excitation process may finally lead (in favorable cases) to real control of chemical reactions, will be only briefly covered here, as it has already been partly discussed in a review (Baumert and Gerber 1995; Baumert et al., 1997).

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II. High Vibrational Levels in Electronic Ground States Highly excited vibrational levels v* in the electronic ground state of molecules can be either reached by one-photon absorption from the lowest thermally populated vibrational levels (overtone-spectroscopy) or by fluorescence or stimulated emission from a selectively populated ro-vibronic level v in an excited electronic state (Fig. 1). Since with the first method the levels are reached by one photon, and in the second by two photons, the parities of the final levels are opposite. The two techniques therefore reach different levels and supplement each other. We will discuss both techniques and illustrate them with several examples.

A. O S In the harmonic oscillator model electric dipole transitions between vibrational levels v and v of polar molecules are allowed only for v : v 9 v : <1. In real molecules the potential becomes more and more anharmonic with increasing energy. This softens the selection rule v : <1 and transitions with v : <2, 3, 4, . . . also appear in the spectra, although at much lower intensities. These overtone transitions can be explained by two effects. First, for larger vibrational amplitudes x the molecular dipole

F. 1. Level diagram for the population of high vibrational levels in the electronic ground state: (a) by overtone transitions; and (b) by fluorescence or stimulated emission pumping.

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 M(x) : M ;  M xG  G G

(1)

moment

depends in a nonlinear way on the displacement amplitudes x of the vibrating nuclei. Therefore, the matrix element R : TYTYY



* M(x) dx TY TYY

(2)

also gives small contributions for overtone transitions with v  1, even if the wave functions (x) are taken as Hermitian polynomials, that is, eigenfunctions of the harmonic oscillator. Second, the anharmonicity of the potential changes the eigenfunctions in Eq. (2). This gives nonvanishing contributions for overtone transitions even if M(x) contains only linear terms. The second effect generally contributes the major part to the intensity of overtone transitions (Scholz, 1932; Lehmann and Smith, 1990). The anharmonicity of the potential has another essential effect. It couples different vibrational modes. The anharmonicity and therefore also the coupling strength both increase with increasing energy. The total energy of highly excited vibrational levels therefore no longer can be represented by the sum of normal mode energies because the mutual interaction energy cannot be neglected. For sufficiently high vibrational excitation energy in a polyatomic molecule it is therefore no longer meaningful to characterize these vibrational states by normal mode quantum numbers. Even if initially a specific vibrational mode has been excited (e.g., the bending vibration), the strong coupling causes a rapid intramolecular vibrational redistribution (IVR) (Troe, 1997) and the stationary state of the excited molecule must be represented by a linear combination of all vibrational modes involved, including the coupling terms (Boyarkin and Rizzo, 1996). Accurate measurements of the term energies and widths of highly excited vibrational levels yield not only the potential surfaces but also the coupling strength of vibronic interactions. Overtone spectroscopy provides a powerful tool to investigate these couplings and the IVR. In the following we will discuss some sensitive experimental techniques of overtone spectroscopy, all of which use tunable lasers in the visible or near infrared spectral region. 1. Modulated Absorption Spectroscopy Semiconductor lasers with external cavities are very convenient widely tunable narrowband radiation sources for overtone absorption spectros-

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F. 2. Widely tunable single-mode semiconductor laser with external cavity.

copy. A typical setup, shown in Fig. 2, consists of the laser diode with an antireflection coating on one end face inside an external Littmann resonator (Duarte, 1995) with a grazing incidence diffraction grating and a plane end mirror. Tilting the mirror results in continuous mode-hop-free tuning of the laser wavelength, if the tilting axis is correctly chosen (Wenz et al., 1996). Single-mode continuous tuning ranges of up to 400 GHz are possible without mode hops. The zero diffraction order reflected from the grating is used as the output beam, which passes through the absorption cell inside a multipass nearly confocal mirror configuration. This arrangement allows up to 100 passes through the absorption cell without serious overlap of the multiple beams at the mirror surfaces. Such overlaps cause interference fringes when the laser wavelength is scanned and therefore should be avoided, as they deteriorate the signal-to-noise ratio (SNR) and therefore the sensitivity. As in microwave spectroscopy the sensitivity can be greatly enhanced by modulation techniques. Several of such techniques have been reported in the literature (Wong and Hall, 1989; Silver, 1992). One example is the phase modulation of the laser beam when passing through an electro-optic modulator driven at an rf frequency . The frequency spectrum of the laser beam then consists of the carrier frequency  and two sidebands  < , which have opposite phases (Fig. 3). The transmitted laser intensity after the absorption cell is detected by a fast photodiode followed by a lock-in amplifier that is tuned to the modulation frequency . Without absorption the two heterodyne contributions

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F. 3. Carrier and sideband frequencies of a phase-modulated single-mode laser: (a) without; and (b) with absorption of a sideband.

S ( 9 ( ; )) and S ( 9 ( 9 )) of the signal due to the superposition   of carrier and sidebands are equal in amplitude but opposite in phase and therefore exactly cancel. This implies that every intensity fluctuation of the laser beam incident onto the absorption cell is completely canceled. If one of the sidebands coincides with an absorption line, the balance is perturbed and a signal is detected (Großkloß et al., 1994a). This method has been used by several authors for high sensitivity overtone spectroscopy (Ye et al., 1998; Werle and Lechner, 1999). An optimum compromise between signal amplitude and spectral resolution is achieved when the modulation frequency  equals the linewidth of the absorbing transitions. Since lock-in amplifiers cannot directly handle frequencies of 1 GHz, down conversion techniques have to be applied in order to detect the signal at lower frequencies. Another sensitive modulation technique is based on two-tone modulation, where the rf voltage that drives the EOM for frequency modulation is additionally amplitude-modulated at a lower frequency. Instead of using the EOM modulator the laser diode current can be modulated, which affects both laser frequency and intensity. The basic principle is explained in Fig. 4. In the upper portion of Fig. 4 the unmodulated laser frequency is shown, as well as the corresponding absorption profile (which is obtained when the laser frequency is tuned over the absorption line). In the lower portion the frequency spectrum of the modulated laser is depicted with the corresponding absorption profiles due to carrier and sidebands. Assume that the rf modulation is periodically switched on and off. The middle portion of Fig. 4 illustrates the signal, which is then obtained as the difference (a)—(b), when the lock-in is synchronized with the switching frequency (two-tone modulation technique). The increase in sensitivity obtained by the modulation techniques is demonstrated by Fig. 5, which shows a rotational line in the overtone band (1, 2, 1) < (0, 0, 0) of the H O molecule without modulation (insert) and  with two-tone modulation. The gain in sensitivity amounts to two orders of magnitude.

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F. 4. Basic principle of two-tone modulation spectroscopy and the resulting absorption line profiles.

With this technique overtone spectra of H S (Großkloß et al., 1994b,  Flaud et al., 1995) and C H , C D (Großkloß et al., 1995) have been     measured. In H S a local vibrational mode can be excited, where one H  atom vibrates against the S atom. Because the mass of the S atom is much larger than that of the H atom, its vibrational amplitude is correspondingly

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F. 5. The rotational line in the overtone band (1, 2, 1) < (0, 0, 0) of the H O molecule  measured without modulation (insert) and with two-tone modulation.

smaller, diminishing the coupling of the excited local mode to the vibration of the other H atom. The time scale of IVR in H S is therefore longer than  in C H . This is corroborated by the overtone spectrum, which shows only   weak perturbations. Instead of high-frequency phase modulation, slow modulation of the laser wavelength can be used by modulating the length of the laser cavity. In this case the laser wavelength is modulated across the absorption line profile while it is continuously tuned across the spectrum. In combination with a beamsplitter in front of the absorption cell and a balanced detector (Houser and Granire, 1994; Hobbs, 1997) to compare transmitted and reference intensities, the high sensitivity that can be achieved allows measurements of relative absorptions 10\. With an absorption cell inside a multipass configuration, a total absorption length of 70 m was realized in our laboratory. This gives a sensitivity limit for the absorption coefficient of as   10\cm\. For illustration, Fig. 6 shows a section of the overtone spectrum of ozone O around  : 1.5 m, which has been measured at an O pressure of   35 mbar. Note the extremely high line density, due to the high level density and the strong couplings between the different vibrational levels that are slightly below the dissociation limit of the electronic groundstate.

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F. 6. Section of the O -overtone absorption spectrum around  : 1.5 m measured with  the wavelength-modulated diode laser in a multipath cell at an O pressure of 35 mbar. 

2. Optothermal Spectroscopy in Molecular Beams The spectral resolution of modulated absorption spectroscopy in a multipath cell is, in principle, limited by the Doppler width of the absorption lines. At higher pressures pressure broadening also may further increase linewidth. In contrast to these limitations, spectroscopy in a collimated molecular beam allows sub-Doppler spectral resolution. For investigation of long-lived highly excited vibrational level, optothermal spectroscopy, as first proposed and applied by Scoles and coworkers (Scoles, 1991; Bassi, 1992; Miller, 1992) is a very suitable and sensitive technique. Its basic principle is illustrated in Fig. 7. Molecules seeded in a supersonic cold helium atomic beam are excited into higher rotational-vibrational levels by a laser beam that crosses the molecular beam exactly perpendicularly, either in a multiple reflection arrangement or inside a high-finesse enhancement resonator. The excited molecules, with excitation energy h ·  and a radiative lifetime exceeding their flight time, travel to the detector, which consists of a doped silicon bolometer cooled to 1.6 K and a low-noise preamplifier. The impinging molecules stick at the cold surface and transfer their energy

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F. 7. Schematic illustration of optothermal laser spectroscopy in a collimated molecular beam.

(translational plus excitation energy) to the bolometer, thus increasing its temperature by T : (dn/dt) · A · h · /G where dn/dt is the flux of molecules with excitation energy h · , impinging per second onto the detector with surface area A, and G is the thermal heat conductivity for the heat transfer from the bolometer to its surroundings. This decreases the electric resistance R by R :

dR T dT

with dR/dT  0

(3)

When a small current I is sent through the bolometer and the excitation laser with intensity I is chopped at a frequency f  1 KHz, the bolometer * delivers an ac signal with amplitude U( f ) : I · R( f ) . (dN/dt) · h · 

(4)

which is proportional to the rate dN/dt . (dn/dt) · I of absorbed photons * with energy h ·  on an overtone transition. The minimum detectable absorbed power is about 10\ W, which corresponds at a photon energy of 2 eV to dN/dt : 3 · 10s\ absorbed photons/s. With an enhancement cavity an intracavity laser power of 100 W is readily achieved. Then a relative absorption of I/I  10\ still can be detected! The technique is therefore well suited for measuring even weak transitions. The achievable sensitivity is demonstrated in Fig. 8, which shows the optothermal signal of

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F. 8. Sub-Doppler optothermal signal of the P(3) line in the overtone band (1030°0°) < (0, 0, 0,0°0°) of C H .  

the P(3) line in the overtone band (1030°0° < 0000°0°) of acetylene C H at    : 789 nm. In order to emphasize the large SNR, the background is also shown 50; magnified. The linewidth of 5.5 MHz is about 8 · 10\ of the Doppler width in a gas at T : 300 K and is limited by the residual Doppler width due to the small divergence of the collimated molecular beam. The advantages of higher spectral resolution and SNR are illustrated by Fig. 9, where a comparison of the same spectra of ethylene C H at around    : 1.5 m is given, measured with Doppler-limited Fourier spectroscopy (upper trace), optoacoustic spectroscopy (middle), and Doppler-free optothermal spectroscopy (Platz, 1998). The higher spectral resolution allowed the successful analysis of the C H -overtone spectrum corresponding to the  ;  combination band     and the identification of perturber levels that interact by vibronic and Coriolis couplings with the rotational levels, selectively excited by a single mode color center laser (Platz et al., 1998). B. F S  H V L Another method to reach and analyze high vibrational levels in the electronic groundstate is the selective population of a ro-vibronic level i in an excited electronic state by the pump laser and the observation of the

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F. 9. Comparison of spectral resolution and signal to noise ratio (SNR) for a section of the ethylene overtone spectrum around  : 6150 cm\ measured by Fourier-, optoacousticand optothermal spectroscopy.

spectrally dispersed laser induced fluorescence. With the term value of the initial level T , the wavenumber of the laser  and that of the fluorescence * line  , the term value of the terminating vibrational level m in the $J electronic groundstate (see Fig. 1b) is T : T  ;  9  K * $J

(5)

Measurements of the spectrally resolved fluorescence lines yield the term energies of all vibrational-rotational levels that are accessible by fluorescence transitions from the excited level v with sufficiently large FranckCondon factors. The experimental arrangement used in our laboratory is shown in Fig. 10. The molecules in a collimated molecular beam are excited into a selected ro-vibrational level i in an upper electronic state by a cw single-mode laser. For many molecules the necessary excitation wavelength lies in the ultraviolet (UV) region. Therefore, the laser frequency is doubled in an optical nonlinear crystal inside an enhancement resonator. The laser-induced fluorescence is collected by a mirror-lens system and imaged onto an optical fiber bundle with circular entrance cross section. The exit cross section of this bundle is formed to a rectangular shape adapted to the entrance slit of the

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F. 10. Experimental arrangement for sensitive measurements of spectrally dispersed laser-induced fluorescence spectra of molecules in a molecular beam.

spectrometer. A liquid nitrogen-cooled CCD chip with 1752 ; 532 pixels and total length L at the exit of the spectrometer collects fluorescence photons simultaneously within a spectral range  : L · d/dx, which is determined by the inverse dispersion d/dx of the spectrometer and by the length L of the CCD chip. The spectral resolution  : (d/dx) · x is limited by the width x of the entrance slit, which is set to five times the width of a pixel on the CCD chip. A cooled CCD system with integration times of several minutes allows detection of even very weak fluorescence lines with only a few fluorescence photons per second. The selective excitation of a single upper level results in fluorescence spectra that are much simpler than the emission spectra from the manifold of thermally populated levels and therefore easier to analyze. In case of diatomic molecules each vibrational fluorescence band consists of only two or three rotational lines (P- and R-lines for — transitions and additional Q-lines for other electronic transitions).

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F. 11. Two sections of LIF spectra in the same spectral range emitted from two different selectively excited upper levels of CS . The wavenumber scale gives the energy of the  terminating vibrational levels in the X ground state.

For polyatomic symmetric top molecules, where each level is characterized by its vibrational quantum numbers v , the rotational quantum number G J and projection quantum number K, there are six lines per vibrational band, according to the selection rules J : 0 < % < 1, K : <1 for perpendicular transitions or three lines for parallel transitions with K : 0 (Herzberg, 1966). This is in contrast to the manifold of lines in absorption spectra of electronic transitions starting from many thermally populated lower rotational-vibrational levels. The structure of the spectra depends strongly on the vibrational wave functions of the emitting levels. This is illustrated by Fig. 11, which depicts the same spectral section of two fluorescence spectra of carbondisulfide CS  molecules in a cold molecular beam, excited by a frequency-doubled, single-mode dye laser at  : 31777.371 cm\ (left spectrum) and   : 31974.614 cm\ (right spectrum). Each line corresponds to a single  vibrational fluorescence band of the electronic  ;  transition consisting of two (here not resolved) rotational lines (P and R lines). Although in both spectra the fluorescence lines terminate at vibrational levels in the electronic ground state with the same term values around 16,500 cm\, the two spectra look completely different. This is due to different Franck-Condon factors for the transitions from two different upper vibrational levels (Brasen and Demtro¨der, 1999). Analysis of such spectra gives direct information on the term energies of the vibrational levels in the  ground state of CS , Fermi resonances, and  Coriolis interactions up to high vibrational excitation energies. At term energies 10,000 cm\, however, the density of vibrational levels becomes

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F. 12. (a) Schematic illustration of vibronic coupling and vibrational levels in two different electronic states; and (b) section of the Doppler-free absorption spectrum of CS  around  : 32.000 cm\.

so high that an unambiguous assignment is no longer possible without the support by accurate ab initio calculations, which have to take into account all possible vibrational-rotational interactions and the resulting level shifts. Such calculations have been performed for the CS molecule (Meyer and  Rosmus, 1998). For a certain class of triatomic molecules (e.g., NO , SO , and CS ) the    dissociation energy of the electronic ground state is much higher than the bottom of the potential surface of excited electronic states. This means that many ro-vibronic levels of an excited state are below the dissociation limit of the electronic ground state. They can therefore strongly interact with energetically close ro-vibronic levels of the electronic ground state. Such an interaction implies an exchange of electronic and vibrational energy and represents a breakdown of the Born-Oppenheimer approximation. Since the vibrational level density of the ground state levels at this high vibrational energy is much higher than that of the excited state levels at the same total energy (Fig. 12), each vibronic level in the excited state may interact with many levels in the ground state. Because the matrix elements for overtone transitions from the vibrational ground state into these high vibrational levels * within the same electronic state are vanishingly small, they are called ‘‘dark states’’ and the levels in the electronically excited state accessible by allowed dipole transitions are the

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‘‘bright states.’’ Each bright state can interact with many dark states, with the result that the oscillator strength of the allowed transition is diluted into many weak transitions. This may prolong the radiative lifetimes of the perturbed levels by several orders of magnitude (Douglas, 1966). In the absorption spectrum these perturbations lead to an irregular pattern of many weak lines around the position of the allowed transition. This is illustrated by Fig. 12b, which shows a section of the Doppler-free absorption spectrum of CS around  : 32,000 cm\ measured in a cold  molecular beam, where mostly the lowest vibrational level in the  ground E state was populated. One can clearly recognize the clustering of several weak lines around strong allowed transitions. The question now is, whether the nonlinear interaction among the many vibronic levels is sufficiently strong to mutually couple these levels in such a way that they can no longer be characterized by vibrational quantum numbers of normal modes. This can be checked by statistical means, even if an assignment of quantum numbers is no longer possible, as was proved much earlier in nuclear physics for the vibrational levels of large nuclei (Dyson and Mehta, 1963). The procedure is as follows. The energies E of all measured vibrational levels are arranged according G to increasing energy. Then the energy separations E : E 9 E between G G G\ neighboring levels are determined and the numbers N (E ) of level pairs L G with energy separations within the interval n · E  E  (n ; 1) · E

G (n : integer) are plotted as a function of n. The energy interval E can be chosen arbitrarily but should be wide enough to have a sufficiently large and statistically relevant number N (E ) within each interval. Generally, E is L G chosen as a fraction of the average spacing S : E . Since the density of G vibrational levels increases with EK where m is the number of vibrational degrees of freedom (for example m : 3 for a nonlinear triatomic molecule), one has to normalize N(E ) in order to obtain equal statistical conditions G for all energy ranges (Brody et al., 1981; Karrlein, 1991). Instead of N(E ) G often the probability



P(E ) : N(E )  N(n · E ) G G G L

(6)

is plotted. Berry showed in 1987 that in level systems with no (or only weak) interactions the nearest neighbor distribution can be described by the Poisson distribution 1 P(E ) : · e\ #G1 G S

(7a)

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F. 13. Plots of the nearest neighboring distributions in (a) NO and (b) CS .  

while for strong nonlinear interactions it is represented by the Wignerdistribution · E G e\L #‚G1 P(E ) : G 2·S

(7b)

The Poisson distribution has its maximum for E : 0 while the Wigner G distribution is zero for E : 0 and shows a maximum for E : S. The G G physical reason is the level repulsion for strongly interacting levels. In the classical model of vibrating nuclei the Wigner distribution implies a chaotic movement of the vibrating nuclei in phase space. In Fig. 13 a comparison is given between the nearest neighbor distributions for the vibrational levels of NO and CS . While the distribution   for NO shows chaotic behavior, it clearly indicates a Poisson distribution  and therefore nonchaotic behavior for CS . This difference is probably due  to the weaker couplings between high vibronic levels in the linear ground state of CS and those in the bent excited electronic state. This means  smaller overlaps of the vibrational wave functions in the two electronic states, while for NO the molecular structures in both electronic states are  bent. Several groups have studied the statistical behavior of NO (Persch et  al., 1988; Delon and Jost, 1991) and CS (Michaille et al., 1997; Brasen and  Demtro¨der, 1999). Although the statistical analysis of molecular spectra cannot provide an assignment in the sense that quantum numbers are attributed to vibrational energy levels, it does give a measure of the interaction strength and the amount of its nonlinearity for mutually interacting vibrational levels within the same electronic state or within different states.

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C. S E P While the spectral resolution of fluorescence spectroscopy (discussed in the previous section) is limited by the fluorescence dispersing instrument (spectrograph or interferometer), stimulated emission pumping provides a resolution that is limited only by the bandwidth of the lasers used for pumping and probing. With cw single mode lasers sub-Doppler resolution can be achieved. The basic principle of stimulated emission pumping, which may be regarded as stimulated resonance Raman scattering, is depicted in Fig. 14. The pump laser L1 is kept on the transitions 1 ; 2 , while the probe laser is tuned across the transition 2 ; 3 . For the resonance condition hc(



9  ) : E 9 E   

(8)

the population N is depleted by stimulated emission while N increases.   The maximum population transfer to the final level 3 is 30% (He et al., 1990). The depletion of N can be monitored either by the corresponding  decrease of the fluorescence intensity I (2) or by ionization of the molecules $J in level 2 with a third laser L3 and the observation of the decrease in the ion rate, when the probe L2 laser is tuned across the resonance (ion-dip spectroscopy) (Weber et al., 1990). The advantages of this technique as compared to laser-induced fluorescence are high sensitivity and high spectral resolution. This allows

F. 14. Level scheme for stimulated emission pumping.

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observation of downward transitions even with small Franck-Condon factors. Of particular interest are high vibrational levels closely below the dissociation limit of the electronic ground state, where the level density is high and the spectral resolution of laser-induced fluorescence may not be sufficient to resolve individual lines (Dai and Field, 1995) The method was first applied with pulsed lasers to study high vibrational levels of the iodine molecule I (Kitrell et al., 1981). Another example is the  investigation of the Renner-Teller effect in NCO by Wu and Sears, (1995) using stimulated emission pumping of the molecules in a supersonic jet. The Renner-Teller effect (Jungen and Meerer, 1980) describes the coupling between the electronic orbital angular momentum and the nuclear vibrational angular momentum of the bending vibration in a linear triatomic molecule. The SEP allows access to higher vibrational levels than those that can be reached with fluorescence spectroscopy. Furthermore, forbidden transitions also can be seen in the spectra. If the laser bandwidths are small compared to all level widths  , the G linewidth  of the stimulated emission signal is (Letokhov and Chebotayev, 1977):  :  ; [(k /k ) ; (1 < k /k ) ](1 ; S       

(9)

where k , k are the wavenumbers of lasers 1 and 2,  the level width of level   G i and S the saturation parameter. For counterpropagating pump and probe beams the ; sign in Eq. (9) applies, while for copropagating beams the minus sign holds. The latter yields the minimum width , because the influence of the level width  is reduced by the factor (1 9 k /k ). For    k /k  1 and    the signal width  becomes even smaller than the     natural linewidth  :  ;  of the optical transition 1 ; 2 (Hackel and   Ezekiel, 1979; Weickenmeier, 1986). We will illustrate here the spectral resolution achieved with cw lasers using the diatomic Cs molecule as our example. With the pump laser  tuned to a single rotational line in the D> (v, J) < X> (v, J) system, S E upward transitions into selected high vibrational levels v  50 of the D S state are induced. The excited state is pumped at the inner turning point of the potential (Fig. 15), where the FC factor is maximum. Downward transitions are stimulated by a tunable second dye laser from the outer turning point at large internuclear distances into high vibrational levels of the electronic ground state. At these large internuclear distances the energy separation between the > and > potentials (which is due to the E S exchange energy) becomes comparable to the hyperfine splittings of the Cs atoms. Therefore the hyperfine interaction can break the g-u symmetry and mix the electronic even-odd states. The stimulated downward transitions

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F. 15. Population of high rotational-vibrational levels in the coupled X> 9 a E S ground states of Cs by stimulated emission pumping. 

therefore terminate on mixed singlet-triplet levels. This is demonstrated in Fig. 16, which shows the stimulated emission spectrum from the upper level D (v : 53, J : 48) into the lower hyperfine components of the J : 47 S  and J : 49 rotational levels (Weickenmeier et al., 1986).  Accurate measurements of such high ro-vibronic levels give detailed information on the long range part of the potential (Weickenmeier et al., 1985). This subject has recently gained increasing interest after the realization of Bose-Einstein condensation of cold atoms (Anderson et al., 1995) confined in a magnetic trap. Since spin-flip collisions that correspond in the molecular picture to transitions from a triplet to a singlet state would lead to a loss of atoms from the trap, the hyperfine interactions (which cause singlet-triplet mixing) are of essential importance for such a condensate. Recent investigations of ro-vibronic levels in the Na molecule up to  the last bound vibrational states closely below the dissociation limit have been performed by Elbs et al. (1999). The measurements used a triple resonance scheme with three cw lasers (Fig. 17). Starting from level (v:0, J:J) in the X> ground state, the first laser L 1 pumped a level E v : 15, J : J ; 1 in the A> state, which decays by fluorescence  S mainly into high vibrational levels v of the X> state (Franck-Condon E pumping). From these levels v the second laser selectively pumps very high levels v : 100—140 in the A state and a third laser L 3 finally  populates the wanted levels v : 61—65 by stimulated emission pumping. V

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F. 16. Hyperfine structure of P- and R-transitions observed for v : 136 with stimulated  emission pumping from v : 53, J : 48.

From the accurately measured level energies the vibrational wavefunctions could be calculated and from their asymptotic phase (k) the scattering length, defined as 1 L : lim 9 tan (k)  k I





(10)

could be deduced. The scattering length, which is an important quantity to describe low energy atom-atom collisions, depends on the kinetic energy E : k/2 of the colliding atoms with reduced mass  : (m m )/     (m ; m ) and on the dissociation limit approaching the different hyperfine   states of the two dissociating atoms (Crubellier et al., 1999). The results are L : 55a for F : F : 1, F : 2 and L : 50a for F : F : 1, F : 0. A       Bose-Einstein condensate is only stable (i.e., it does not collapse into a solid) for positive values of L .

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F. 17. Level system for the investigation of vibrational levels close to the dissociation limit by triple resonance stimulated emission spectroscopy.

Once the scattering length is known, it is possible to determine the energy of the last bound level. For the > state of Na , for example, the value of S  L confirms that v : 15 is the last vibrational level in the shallow potential of the a> triplet state (Verhaar, 1999; Elbs et al., 1999). S

III. Laser Spectroscopy of Electronically Excited Molecular States The most commonly used spectroscopic method for obtaining information on excited electronic states is based on various techniques of absorption spectroscopy. The modulation techniques discussed in Section II.A on overtone spectroscopy also can be of course applied in the visible and UV region to look for electronic transitions. However, detection sensitivity is generally much higher for electronic transitions than for overtone transitions, because in the former case laserinduced fluorescence detection or selective photoionization of the electronically excited molecules allows (in favorable cases) the detection of even single absorbed photons. Since the line density in electronic spectra is often much higher than in infrared spectra, the spectral resolution, limited by Doppler width and pressure broadening, might not be sufficient in many cases to resolve the rotational structure or even finer details, such as fine and hyperfine splittings. In such cases Doppler-free techniques are demanded.

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A sensitive sub-Doppler technique is linear laser spectroscopy of molecules in a collimated molecular beam, where the molecules travel in the z-direction and the laser beam pointing into the x-direction crosses the molecular beam perpendicularly. If a tunable single mode cw laser is used, the linewidth of the absorption lines is generally limited by the residual Doppler-width

 5 (v /v ) ·  0" V X "

(11)

which is smaller by a factor 5 10\ than the Doppler width  : " ( /c) · (8kT · ln 2/m of the corresponding transitions for molecules with  mass m in a thermal gas at a temperature T. The factor equals the collimation ratio (v /v ) of the molecular beam. V X In cold supersonic beams, where the molecules are seeded (typically with a concentration of a few percent) in a cold noble gas atomic beam, the population distribution N(v, J) of rotational-vibrational levels is compressed into the lowest vibrational and rotational levels. Although N(v, J) does not follow a Boltzmann distribution, it is often described by a rotational temperature T (typical values are T : 1—10 K) and a vibrational tem  perature (T : 10—100 K) (Demtro¨der, 1992). The drastic reduction of 
thermally populated levels reduces the line density in the absorption spectrum and facilitates its assignment considerably. A. L-I F  R T-P I Because of the low molecular densities and the short absorption path lengths across the collimated beam, the absorption of the laser intensity is generally very small and hard to detect. A widely used detection technique is based on laser-induced fluorescence, where special optical collection systems have been created, consisting of spherical or elliptical mirrors, to collect a maximum percentage of the fluorescence onto the detector (see, e.g., Fig. 10). An even more sensitive detection technique is the resonant two-photon ionization (RTPI), where the first laser L 1 is tuned through the absorption spectrum, corresponding to transitions i ; (k) and the second laser L 2 (generally at a fixed wavelength) ionizes the excited molecules in levels (k . The photo-ions can be extracted by an electric field, accelerated and detected with 100% efficiency. The ion rate dN /dt is  dN P dNa P dN * ·  · x · I' : I'  : N · G dt GI P ;R dt P ; R dt I' I I' I

(12)

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where N is the density of molecules in level (i , dN /dt the incident photon G * flux,  the absorption cross section for the transition i ; k , P the GI I' ionization probability of molecules in level k , R the relaxation probability I of level k by fluorescence or radiationless transitions, and dNa/dt the absorbed photon rate of L 1. With pulsed lasers P generally is large compared to R . This means that I' I the ion rate equals the rate of absorbed photons. Every excited molecule is then converted by L 2 into a molecular ion. With cw lasers the intensities are much lower. Therefore, the ionizing transition will not be saturated, that is, P  R . However, the measured total ion rate may still be higher than I' I obtained with pulsed lasers with their low duty cycle (at a pulsewidth of 10 ns and a repetition rate of 100 Hz it is only 10\!). In order to increase the intensity of cw lasers but still allow all molecules in the molecular beam to pass through the two superimposed ‘‘sheets of light’’ for excitation and ionization, both laser beams have to be focused by a cylindrical lens into the molecular beam. In Fig. 18 the comparison between two identical sections of the absorption spectrum of cold Cs -dimers in a supersonic beam (T : 5 K), detected   by laser-induced fluorescence (a) and resonant two-photon ionization (b) with a cw dye laser (L 1) and a cw argon ion laser (L 2) demonstrates the superior sensitivity of RTPI. The SNR of the two spectra is comparable, but the RTPI-spectrum had been measured at a 10-fold lower density of Cs  molecules, which shows that RTPI for this example is about 10 times more sensitive than LIF. B. O D R T For perturbed electronic transitions (which is generally the rule and not the exception) the assignment of the spectrum is by no means trivial, even if all lines are resolved. Here the technique of optical double resonance is very helpful. It is based on the simultaneous interaction of the molecules with two laser waves, tuned to two molecular transitions that share a common level (Fig. 19). The first laser L 1 is tuned to a wanted transition 1 ; 2 and is stabilized at its peak wavelength. Due to saturation the population N is  decreased and N is increased. If the intensity I of L 1 is chopped the   populations of levels 1 and 2 are both modulated at the chopping frequency but with opposite phases. The probe laser L 2 is now tuned across the spectral region of interest. If its wavelength coincides with a transition 1 9 m the absorbed laser power will be modulated according to the modulated population N (t) of  the absorbing molecules. This can be monitored by either the modulation of the transmitted laser power of L 2, the fluorescence induced by L 2, or the

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F. 18. Comparison of two identical sections of the (0 < 0) band in the C < X> E S system of Cs measured (a) with LIF and (b) with RTPI at a 10-fold lower molecular density. 

rate of ions produced by photoionization of molecules in levels m by the first laser L 1 or a third laser L 3. The optical-optical double resonance (OODR) spectrum consists of all allowed transitions starting from the labeled levels 1 or 2 , and is therefore much simpler than the normal absorption spectrum, which contains all allowed transitions from all thermally populated lower levels. Since the modulation phases for transitions from 1 are opposite to those from 2 , the two classes can be readily distinguished when using lock-in detection. The separations of all OODR-transitions starting from the same lower level correspond just to level separations in the upper state.

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F. 19. Level scheme for optical-optical and optical-microwave double-resonance spectroscopy in a molecular beam.

Taking into account the selection rules for allowed transitions the assignment of the upper levels is much facilitated because the lower level is known. This drastic simplification is evident in Fig. 20, where in the upper part a small section of the dense absorption spectrum of the Na trimer (parallel  band with (K : 0 of the AE(0, 0, 0) < XE(0, 0, 0) electronic transition measured with RTPI) is shown. Even at a spectral resolution of 30 MHz, the absorption spectrum is not completely resolved because the hyperfine structure splittings, pseudorotation, and spin-rotation splittings of neighboring rotational transitions overlap each other. This should be compared to the OODR-spectrum in the lower part of Fig. 20 where the pump laser L 1 was stabilized on the R(3) transition. The OODR-spectrum essentially consists of three rotational transitions P(3), Q(3) and R(3), with each of them split into several not completely resolved subcomponents, which are caused by hyperfine structure and pseudorotation splittings (see next section). Only the OODR technique or optical-microwave double resonance are able to resolve all structures. In the optical-microwave-double resonance the microwave frequency is tuned to a transition between two rotational levels, where one of them is depopulated by the laser L 1, kept at optical transition. When the microwave is in resonance the level populations are altered. This is detected by a corresponding increase or decrease of the optical signal. This opticalmicrowave DR directly determines the splittings of the ground state levels, while all optical transitions show the difference of the splittings in the upper and lower levels of the electronic transitions. The OODR and OMDR

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F. 20. Comparison of a section of the sub-Doppler absorption spectrum of the zero-zero band (AE(0, 0, 0) < XE(0, 0, 0)) of the sodium trimer Na detected with RTPI (upper  spectrum) and the OODR-spectrum with the pump laser stabilized onto the R(3) line (lower spectrum).

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F. 21. Schematic experimental set-up for optical-microwave double-resonance (OMDR) in Na (Ernst, 1999). 

therefore supplement each other. The level scheme and the experimental realization of OMDR in Na used by Ernst (1999) are shown in Fig. 21.  Another level scheme of optical double resonance spectroscopy is stepwise excitation of high lying molecular levels. The first laser L 1 excites a selected level v, J in an intermediate state and the second laser L 2 is tuned to all transitions into higher levels, accessible from v, J . Examples are high lying Rydberg levels, which can be populated by two-step excitation (Bernheim et al., 1983). Since all transitions induced by L start from a  single selectively populated level, the excitation spectrum of L is relatively  simple, even if perturbations by couplings between different Rydberg levels shift the line positions. A second illustrative example of stepwise excitation is the investigation of high lying triplet states by perturbation-facilitated OODR (Li and Field, 1987; Li and Lyyra, 1999). Here L 1 excites a level v, J in the A state S of Na or Li , which is strongly perturbed by a nearby level of the  state.   S The wave function of this perturbed level is a linear combination of singlet and triplet state functions. For the second, step transitions are therefore allowed into higher even singlet as well as into triplet levels. (Fig. 22). A further example of Doppler-free stepwise excitation spectroscopy with single-mode cw lasers is the selective population of high lying ungerade Rydberg levels of Li molecules in a cold molecular beam. Three steps are 

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F. 22. Level scheme of perturbation-facilitated OODR in alkali dimers.

required in order to reach with visible lasers the odd Rydberg levels from the X> ground state of Li . Therefore, three independently tunable lasers E  were used to select specific Rydberg levels Ry(n*, v*, J*) with principal quantum number n*, vibrational quantum number v*, and rotational quantum number J*. The population of Ry is monitored by detecting the ions, which are produced either by field ionization for levels marginally below the field-free ionization limit IP or by auto-ionization of levels above IP. Auto-ionization of molecular Rydberg levels is caused by energy transfer from the rotating or vibrating molecular core to the Rydberg electron (Fig. 23a), which takes place during the time the Rydberg electron penetrates the core. It represents a breakdown of the Born-Oppenheimer approximation. In the level scheme of Fig. 23b the auto-ionization can be regarded as the interaction of a bound level at energy E of the neutral molecule with the 0 energy continuum of the molecular ion. The excitation probability P . A eGP ; A eGP‚  

(13)

of the auto-ionizing level is proportional to the square of the sum of two probability amplitudes A for the bound-bound transition and A for the   bound-free transition with the corresponding phases  ,  . The interference   of these two excitation amplitudes, which may be constructive or destruc-

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F. 23. Auto-ionization of molecular energy levels: (a) schematic diagram of the core penetrating Rydberg electron; and (b) level scheme.

tive, depending on the phase difference  — , results in the Fano profile   (E 9 E ; q(/2)) 0 ; () :  @ ? (E 9 E ) ; (/2) 0

(14)

of the total absorption cross section () around the resonant energy E 0 for transitions starting from the bound level k . The line profile parameter q equals the ratio A /A of the two probability amplitudes and  is the   halfwidth (FWHM) of the auto-ionizing level (Connerade, 1998). In Fig. 24 measured Fano profiles of Li -transitions from  >(v : 2, J : 1) E into Rydberg levels with n*, v*, N> : J or J < 1 with the angular momentum l : 1 of the Rydberg electron are shown. From the halfwidth  the effective lifetime of the level can be deduced (Baig et al., 2000). At sufficiently high principle quantum numbers n* the coupling of the electronic angular momentum to the rotational angular momentum of the molecular ion core becomes stronger than its coupling strength to the molecular axis (transition from Hund’s coupling case b to case d) (Schwarz et al., 1988). Because of the large average spatial separation of the two valence electrons (i.e., the Rydberg electron and the remaining 2s-electron) the exchange energy may become comparable or even smaller than the hyperfine interaction of the remaining core valence electron with the nuclei. This means that the hyperfine splitting due to the Fermi-interaction of the

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F. 24. Fano profiles of transitions into different autoionizing Rydberg levels of the Li  molecule.

core electron spin with the nuclear spins becomes larger than the singlettriplet separation of the Rydberg levels, which are therefore mixed by hyperfine-coupling (Beigang, 1988). The total electron spin is no longer defined and the Rydberg levels lose their singlet or triplet character. This results in a hyperfine structure of the Rydberg levels, which depends on the parity of the rotational levels N of the ion core, the angular momentum l of the escaping Rydberg electron, and the total quantum number F, which includes the total nuclear spin (Merkt, 1998).

IV. Sub-Doppler Spectroscopy of Small Alkali Clusters Over the last two decades, clusters have been viewed with increasing interest (Haberland, 1994; Jellinek, 1999). They represent a transition regime between free molecules and solid microparticles or liquid droplets. Investigations of their properties, such as geometric structure, electronic level energies and their dependence on cluster size might aid in understanding microscopic scale condensation processes, the formation of solids or liquids from the gaseous phase, and catalytic effects at surfaces.

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Alkali clusters A can be considered prototypes of metal clusters because L of their freely mobile valence electrons. The atomic coordination of small clusters differs considerably from that of the crystalline bulk (Koutecky and Fantucci, 1986; Bonacic-Koutecky et al., 1994) and their size dependent ionization energy does not smoothly converge with increasing size towards the work function of the bulk but shows maxima and minima at certain values of the atomic number n (Schumacher et al., 1984; de Heer, 1993). Alkali clusters can be produced by expanding a mixture of alkali vapor and noble gas (density ratio about 1:100) from a hot oven through a narrow nozzle (50—100-m diameter) into a vacuum. The resulting adiabatic cooling reduces the relative velocities and facilitates recombination of atoms and molecules by three body collisions. The noble gas atoms act as the third collision partner, which removes the binding energy. The combination of high-resolution laser spectroscopy and mass spectrometry allows isotope-specific spectroscopy of molecules and clusters. This often facilitates the assignment of spectra considerably because the comparison of calculated and measured isotope shifts of vibrational or rotational levels allows determination of vibrational and rotational quantum numbers. When using cw lasers a quadrupole mass spectrometer is a good choice; however, for time-resolved spectroscopy with pulsed lasers a time-of-flight spectrometer has advantages because here the spectra of different isotopes can be measured in parallel. For illustration a typical mass spectrum of Li L clusters, shown in Fig. 25, was obtained by two-photon ionization with pulsed lasers and detected with a time-of-flight mass spectrometer. We will restrict the following discussion to triatomic alkali clusters. The alkali trimers represent small systems, which show typical properties of loosely bound clusters, such as floppy geometrical structure. In particular, the lithium trimer Li with only nine electrons and three nuclei, which is still  theoretically treatable with sufficient accuracy, may well serve as a model example, as it shows all kinds of interactions between rotational, vibrational and electronic motions, which is typical for systems with large amplitude vibrations in a shallow potential. At first sight, one might expect an equilateral triangle-shaped structure for such trimers of three equal atoms. However, for all homonuclear alkali trimers the electronic ground state and several excited electronic states are double degenerate in the D configuration (E symmetry). Therefore, accordF ing to the Jahn-Teller theorem (Jahn and Teller, 1937; Bersuker, 1984) vibrational modes of lower symmetry lift the electronic degeneracy and this results in stabilization of the lower Jahn-Teller components of the splitted states at an equilibrium configuration of lower symmetry. For the Li  ground state this leads to an obtuse triangle-shaped structure with an apex angle of  : 71.6°. A cut through the potential energy surface of the ground

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F. 25. Mass spectrum of Li alkali clusters produced by adiabatic expansion of lithiumL vapor seeded in argon.

state of Li according to ab initio calculations by Meyer is shown in Fig. 26  (Keil et al., 2000a). The coordinates give the nuclear displacements from the equilateral geometry in atomic units a  0.5 Å.  The energy surface resembles a Mexican hat with three potential minima (obtuse triangle geometry) separated by shallow barriers (acute triangle geometry) and a central high maximum (equilateral triangle geometry) at the conical intersection of the two Jahn-Teller potential energy sheets. The energy difference between conical intersection and the minima is called stabilization energy. Even in its lowest vibrational level (0, 0, 0) the system undergoes continuous motion along the dashed curve in Fig. 26, where it changes its structure periodically from an obtuse through an acute to an obtuse triangle again. During this motion the apex angle switches from one atom to the next one. This motion can be described by synchronous but 120°-phase shifted movements of the three nuclei on nearly circular paths around the position they would occupy in the D geometry (see Fig. 26). Therefore it is called F ‘‘pseudorotation,’’ differing from the rotation of the whole molecular frame

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F. 26. Cut through the potential energy surface of the Li ground state XE. The  coordinates give the nuclear displacements (in atomic units a ) from the equilateral configur ation. The energy separation of the equipotential curves is 50 cm\ (calculated by W. Meyer, Kaiserslautern).

around its center of mass (Mayer et al., 1996). Similar to the well-known tunneling splitting in the NH molecule the pseudorotation through three  barriers and three minima of the potential results in a splitting of all vibrational levels into a nondegenerate component with vibronic A symmetry and a twofold degenerate component with vibronic E symmetry. For the vibrational ground state (0, 0, 0) of Li this splitting amounts to 36.3 cm\,  corresponding to a tunneling frequency of about 10 s\. This is large compared to the rotational energies. For Na , on the other hand, the potential barriers and the nuclear masses  are both higher, resulting in a stronger localization of the system within the minima and a pseudorotation splitting of only 0.003 cm\ in the lowest vibrational state. This results in small splittings of the rotational lines into A and E ro-vibronic components, while for Li the large splittings cause the  appearance of two well-separated sub-bands that overlap only for high rotational quantum numbers. The pseudorotation moves the system periodically around the conical intersection of the two Jahn-Teller potential sheets at the D configuration, F where the electronic states are twofold degenerate. In this case the phase of the electronic wave function in the B.-O. approximation changes by (not by 2 !) during the motion along the closed path in Fig. 26 (geometrical phase, often called Berry phase) (Berry, 1984; Kendrick, 1997). Since the total wave function which is written as product of electronic and nuclear part, must have unique values, the nuclear wavefunction must also show a

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F. 27. Small section of the (0, 0, 0) < (0, 0, 0) band in the AE < XE transition of Li ,  measured with resonant two photon ionization, and compared with calculations by W. Meyer.

phase change of < . This results in a change of the energetic sequence of vibronic levels with E or A symmetry. As has been proven by Ham (1987) the vibronic E component of the lowest vibrational level has to lie below the A component if the Berry phase is present. This is exactly what we find experimentally for Li as well as for Na .   How can these facts, obtained from theory, be deduced from the experimental spectra of Li or Na ?   The excitation of Li in a cold collimated molecular beam by a tunable  single-mode laser using RTPI as the detection method results in isotopeselective sub-Doppler spectra with a spectral resolution of about 30 MHz. This allows for complete resolution of the rotational structure (Fig. 27). The ionization of the excited levels by a cw argon laser in the ion chamber of a quadrupole mass spectrometer is a very sensitive and mass selective detection technique. Because of strong perturbation of the Li levels by couplings  of the electronic vibrational and rotational motions, the assignment of the lines is not always unambiguous in spite of the complete rotational resolution. Here the optical-optical double resonance technique which selects

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those transitions starting from a single lower level, that is labeled by the pump laser, helps to identify those lines covered by the probe laser (see Section III). Together with the predictions of line positions by ab initio calculations, the spectrum of several bands in the (AE < XE) transition could be well assigned. In Fig. 27 a small section of the measured (0, 0, 0) < (0, 0, 0) band is compared with the theoretical calculations of Meyer (1998), where the calculated absolute wavenumber of the band origin had been adapted to the experimental value. The agreement is very satisfactory. The vibronic selection rules of parallel electronic transitions (AE<XE) are E < E, A < A and A < A . Because of the large pseudorotation     splitting of 36.3 cm\ in the (0, 0, 0) vibrational level of the XE groundstate of Li and the low vibrational temperature in the supersonic beam the  upper pseudorotation component is only weakly populated. The most intense sub-bands consist of vibronic E < E transitions while the A < A transitions are observed as much weaker ‘‘hot bands.’’ The two types of sub-bands can be readily distinguished by their different rotational structures and the different hyperfine structures of the rotational transitions. This allows us to identify unambiguously the energetic order of the vibronic E and A components (Keil et al., 2000a) and proves that the nuclear wavefunctions require a phase of during one cycle of the pseudorotation. In case of Na the measured rotational patterns of several vibronic transitions are  compared with spectra calculated with and without the Berry phase. Here the comparison also unambiguously proves the existence of the Berry phase. This is to our knowledge the first experimental proof of the existence of the Berry phase in free molecules (Busch et al., 1998). For the sodium trimer Na the absorption spectrum (recorded with LIF  as well as with RTPI) is much more complex than for Li (Fig. 20). This is  due to the larger mass (smaller rotational and vibrational constants causing smaller spacings of rotational lines) as well as the larger hyperfine splittings, which cause an overlap of the hyperfine components of adjacent rotational lines. The analysis of the spectrum is only possible with the help of OODR (see Fig. 20b) and guidance that relies on ab initio calculations. For the lowest vibrational level (0, 0, 0) the Na system is well localized  in the wells of the obtuse triangle configuration with C symmetry. The T tunneling splitting, which is only 90 MHz in the (0, 0, 0) level of the X state and 510 MHz in the excited A state, increases rapidly however with increasing vibrational energy. This allows detailed investigation of the transition regime from a strongly hindered to a nearly free pseudorotating system (Meyer zur Heyde et al., 1992; Vituccio et al., 1997; Busch et al., 2000). The hyperfine structure of Li and Na , which is caused mainly by Fermi   contact interaction of the unpaired core electron spin with the nuclei,

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F. 28. Hyperfine pattern of the rotational Q-line in the (0, 0, 0) < (0, 0, 0) band of Li  (AE < XE) starting from level (H : 4, K : 4) with A ro-vibronic symmetry. The solid  A curve represents the experimental pattern, the dashed curve the calculated pattern convoluted with the experimental linewidth.

provides information on the spatial distribution of the electronic wavefunction of the valence electron. The measured hyperfine splittings are nearly independent of the rotational quantum number N but they do depend strongly on the symmetry of the rotational levels. This is due to the different nuclear spin statistics for different symmetries. The results of such measurements (Keil et al., 2000b) show that the measured hfs patterns can be very well fitted by the expression A E(F, I, S) : E ; [G(G ; 1) 9 I(I ; 1) 9 S(S ; 1)]  2

(15)

for the energies of the hfs components with quantum numbers I for the total nuclear spin, S for the electron spin, and G for the total angular momentum :I ; S . A is the hyperfine coupling constant. In Fig. 28 one calculated G hfs pattern (bars) convoluted with the experimental line width of 27 MHz is compared to the measured curves. The coupling constant A turns out to be A : 34 MHz. This proves that the probability density of the unpaired electron at the three nuclei is considerably smaller than that in the 2S  ground state of Li atoms.

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V. Time-Resolved Laser Spectroscopy In the previous sections we have discussed the role of stationary spectroscopy with cw lasers in creating sub-Doppler spectral resolution. Any information on time-dependent processes could only be deduced from the measured homogeneous linewidths of molecular transitions. This restricts the time scale to T  1/(2 ) where  is the spectral resolution of the experiment. In this section we will briefly discuss some recent experiments on time-resolved spectroscopy of small molecules, within the time domain from microseconds to femtoseconds.

A. L M The first example is the CS molecule, illustrating the amount of informa tion that can be obtained from spectroscopy with a time resolution in the microsecond range but with high spectral resolution. The experimental equipment consists of a single mode cw dye laser, an electro-optic modulator, and the nonlinear optical crystal LiIO for optical frequency doubling.  A small voltage applied to the electro-optic crystal turns the plane of polarization sufficiently to destroy the optical phase-matching condition in LiIO . This diminishes the frequency doubling efficiency by at least two  orders of magnitude. Therefore, only small rf amplitudes are required for a complete intensity modulation of the UV laser beam. Rectangular pulses as well as sinusoidal modulation at frequencies of up to 10 s\ can be generated, thus realizing a spectrally narrowband-pulsed UV source with variable pulsewidth. This technique has been applied to lifetime measurements of ro-vibronic levels of CS molecules in a collimated molecular beam selectively excited  by the UV pulses. The fluorescence photons are detected by a photomultiplier, followed by a pulse-forming discriminator and a time-to pulse height converter, which measures the delay time between laser pulse and emission of the fluorescence photon (Weyh and Demtro¨der, 1996). The excited singlet levels J, v are perturbed by vibronic and spin-orbit interaction with nearby triplet levels with the same rotational quantum number J. The perturbation strength depends on the energy separation of the interacting levels. When an external magnetic field is applied, the triplet levels are shifted against the singlet levels due to their magnetic spin moment. This affects the coupling efficiency and the mixing coefficients in the linear combination of the wavefunctions describing the interacting levels. In Fig. 29a the Doppler-free absorption spectrum of CS around  : 

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F. 29. (a) Doppler-free excitation spectrum of CS at different magnetic fields; (b) lifetimes  of singlet and triplet levels as a function of the magnetic field strength.

31344.20 cm\ is shown. Without magnetic field only the singlet transition can be seen, because the forbidden transitions from the X> ground state E into the triplet levels are too weak to be detected. With increasing magnetic field B the singlet line splits into (2J ; 1) Zeeman components due to the

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F. 29. Continued

admixture of triplet wave functions, and the weak triplet transitions also appear, due to the admixture of the singlet wave function, which increases the transition probability. The magnetic sublevels M : 93 of the triplet level are shifted towards the singlet level until they overlap the M : 93 component of the singlet level. At zero field the interaction between the singlet and triplet levels is weak and the lifetime T : 3.61 s of the singlet level is much shorter than  : 33.5 s of the triplet level. With increasing field the mixture of the wavefunctions becomes stronger and the lifetimes of the two levels approach each other, as can be seen in Fig. 29b. The lifetimes of the levels are therefore dependent on the magnetic field. From these measurements, the mixing coefficients and the magnetic moments of the levels can be deduced. These results demonstrate that the combination of sub-Doppler spectroscopy with time-resolved measurements allows detailed insight into the characteristic features of perturbed molecular levels.

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B. T-R S  C P Time-resolved spectroscopy can provide absolute integral cross sections of inelastic collision processes. Molecules in a selectively excited level k may suffer collisions during their lifetimes, which cause in addition to their radiative decay a depopulation of this level, following the equation dN I : 9A N 9  · v · N · N I I I I dt

(16)

where A :  A is the total radiative transition probability to all lower I G IG levels i , accessible from k ,  is the total collision quenching cross I section of level k , v the mean relative velocity between the excited molecule and collision partners B with density N . Integration of Eq. (16) yields the exponential decay N R : N (0) · e\RO I I

(17)

1 1 : ·A ;  v N : ;  · (8/ K T ) · p I I I    ) 

(18)

with

where  : (m · m )/(m ; m ) is the reduced mass of the collision partners, ) ) K the Boltzmann constant and T the temperature of the gas. Measuring the inverse effective lifetime versus the density N or the pressure p of the collision partners yields a Stern-Volmer plot from which the radiative lifetime   : 1/A I I

(19)

and the quenching cross section  can be obtained. I Individual cross sections for selected collision-induced transitions from level k to another level m (this may be another rotational-vibrational level within the same electronic state or in another electronic state) can be measured, if the time-dependent population N (t) can be monitored. It is K described by the rate equation dN K :  v N N 9 (A ;  v · N )N : c N 9 N /  K IK  K K IK I K K dt

(20)

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The integration of Eq. (20) gives  · I K (e\ROI 9 e\ROK) N (t) : N (0) · C K I IK  9  K I

(21)

where  ,  are the effective lifetimes. The population N (t) can be K I K monitored either through the spectrally resolved fluorescence intensity I (t) . N (t) on a transition m ; i or through the ion rate N>(t), when KG K the ions are produced by selective photoionization of level m . For instance, this can be achieved by exciting m into a higher level n , which is then photoionized. The method is well suited to measure collision-induced transitions between a selectively exited singlet Rydberg level and other singlet and triplet levels. The difference in the cross sections of singlet-singlet transitions and singlet-triplet transitions gives information on the spin-orbit coupling of the collision pair. For illustration the measured time-dependent population of the 2 state of Na is shown in Fig. 30. This state is populated by E  collision-induced transitions from (v, J) levels in the C>-state of Na , S  selectively excited by a frequency doubled mode-locked dye laser pulse. The rising part of the curve depends on the transfer cross section  , the decay IK part on the effective lifetime . A fit of the measured curve to Eq. (21) K

F. 30. Time dependent fluorescence from the 2 state of Na , that had been populated E  by collisions from the optically excited C state. S

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yields all relevant parameters, after the effective lifetime  of the feeding ) level had been determined separately. C. T-R M  P P A further example of successful applications of time-resolved spectroscopy is the investigation of the photodissociating dynamics of excited states in Na . In the absorption spectrum of Na around  : 725—757 nm a broad   continuous feature was found. It was not clear whether it is only an unresolved dense line spectrum or a true continuum corresponding to transitions from the Na XE ground state into a dissociative excited state.  Because such dissociative states generally have very short lifetimes, their fluorescence is very weak and often hard to detect. In such cases photodepletion spectroscopy is a good choice to detect even weak absorption features (Wang et al., 1990). Here another laser, tuned to an absorbing transition from level i into a bound level k with high fluorescence efficiency, or into an ionizing state where the ions can be monitored, is used to detect the depletion of i (Fig. 31). However, when Na clusters in a molecular beam are illuminated  simultaneously with a visible laser and a UV laser, one observes not only Na> ion but also fragments of Na> and Na> and the question arises about   the possible fragmentation channels, that is, whether the fragment ions are produced through the process

F. 31. Level scheme for photodissociation of Na with subsequent state selective detection  of the fragments Na . 

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Na ; h ; Na>    Na> ; h ; (Na>)* (22)     ; Na c Na>  (Na>)*  Na ; Na>  or by photodissociation of the neutral Na with subsequent ionization of  the fragments Na ; h ; Na* ; Na ; Na      Na ; h ; Na> (23)    Na ; h ; Na>  The distinction between the two possible reactions (22) and (23) could be achieved with time-resolved spectroscopy using a pulsed XeCl laser at  : 308 nm and a tunable dye laser, pumped by the same XeCl laser. A variable optical delay line between the two laser pulses was set up, where either the XeCl or the dye laser pulse could be delayed up to 20 ns against the other. When the XeCl pulse comes first, it could not detect any depletion of the Na ground state by the dye laser pulse. Fragments Na> or Na> can then   only be produced by channel (23); on the other hand, only channel (22) can work if the dye laser pulse comes first. The experimental results clearly showed (Wang et al., 1999) that the main part of the absorption continuum around  : 740 nm is due to transitions Na X(E) ; A(A ) to an excited state of the neutral Na . Although this state    cannot directly dissociate, the predissociation channel involves a three-step process: The A state is vibronically coupled to two other excited states,  which form a pseudo-Jahn-Teller pair. These states are in turn coupled to high vibronic levels of the electronic ground state above its dissociation energy. At higher resolution vibrational structures can be seen in the continuum and they can be attributed to higher vibronic levels of the pseudo-Jahn-Teller pair. A minor part of the continuum can be traced to transitions in the Na> ion, which  excite a directly dissociating state. Detailed studies on higher dissociative states of Na have been performed by Broyer and his group (Broyer et al.,  1986), who measured vibrational structures in the depletion spectrum of Na  and simultaneously the appearance of Na molecules in an excited state, which  were photoionized for detection. In this way the dissociation channels into specific vibrational levels of the fragment could be identified. These examples demonstrate that detailed information on fragmentation can be extracted from properly designed experiments in which the necessary selectivity is guaranteed.

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D. C C For many years the well-defined breaking of selected bonds in a molecule by optical excitation had been dreamed about for achieving controlled initiation of chemical reactions. Although selective excitation of individual vibrational levels can be achieved with narrowband lasers the rapid intramolecular transfer of the excitation energy among many ro-vibronic levels due to strong mutual couplings often prevents controlled bond breaking (see Section IIA). Recently a new approach based on the coherent properties of specially prepared femtosecond laser pulses has been attempted (Brunner and Shapiro, 1986; Baumert et al., 1997; deVivie-Riedle et al., 1999). The basic principle can be understood as follows: If an excited state of a molecule (this might be a nonstationary state) that decays into the wanted reaction channel can be simultaneously excited via two different coherent excitation pathways, the wavefunction of the excited state is formed by coherent superposition of two contributions. Depending on the phase relations between these two coherent laser pulses the interference between these two contributions may be constructive or destructive, thus enhancing or suppressing the population at this level and with it the reaction channel. The crucial point of this scheme is the correct amplitude- and phase shaping of the excitation pulses. This can be achieved by passing the spatially widened laser pulses through an electro-optic phase plate that consists of many pixels. These pixels can be separately controlled and will superimpose phase retardations onto the spatial parts of the laser beam. This affects the time-and-frequency profile of the laser pulses, focused into the reaction zone. The signal produced by the wanted reaction products is amplified and fed back to the phase plate. Using a computer algorithm, an iteration procedure is started that changes the individual phase retardations of the pixels until the wanted reaction signal becomes maximum. This technique has been applied by deVivie-Riedle et al. to the collision complex Na-H , which is unbound in the electronic ground state but bound in an  excited state. The collision pair was excited by an optimal laser pulse to form a molecular wave packet focused toward the conical intersection of the two potential surfaces. Here the phase of the wave packet is essential for the two possible reaction channels: formation of the stable NaH* or decay into the  separated components Na ; H .  Another approach to coherent control is based on the pump-and-probe technique with two femtosecond pulses from the same laser with a variable time delay (Baumert et al., 1997). The method was applied first to ionization and fragmentation of Na molecules. The first laser pulse excites via two photon transitions a vibrational wavepacket consisting of the coherent superposition of vibrational wavefunctions in the 2 state of Na (Fig. 32), E 

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F. 32. Potential curve diagram for the excitation three-photon ionization and fragmentation of Na with 620-nm laser photons (Baumert et al.) 

which moves periodically with a period of about 800 fs back and forth between the two turning points of the 2 potential. The second delayed E pulse excites the molecule further, either into a Na> state (ionization) or  into the dissociation channel Na> ; Na. The selection between the two possible channels depends on the spatial location of the wave packet in the 2 state. If the second excitation occurs at the inner turning point, the E ionization channel Na* ; h ; Na> ; e\(E )     is opened while at the outer turning point the dissociative channel Na* ; h ; Na> ; Na  becomes accessible.

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The time delay between the pump-and-probe pulse therefore controls which of the two reaction channels is opened. There are many more examples of coherent control applications. While at first small molecules were only used meanwhile also medium-sized molecules have been studied with this technique.

VI. Conclusion The examples given here have shown that the combination of different spectroscopic techniques for improving spectral or time resolution and detection sensitivity has brought essential new insights into the structure and dynamics of excited small molecules. Several methods that facilitate the assignment of spectra, such as cooling of molecules in supersonic beams or double resonance methods for simplifying complex spectra, have allowed us to assign even complex spectra and to extract from such spectra information which otherwise would have been masked by overlapping lines in dense spectra.

VII. References Anderson, M. H., Enscher, J. R., Matthews, M. R., Wiemann, C. E., and Cornell, E. A. (1995). Observation of Bose-Einstein condensation in a dilute atomic vapor. Science 269:198. Andrews, D. L. and Demidov, A. A. (1995). An Introduction to L aser Spectroscopy, New York: Plenum. Ashfold, M. M. R. and Baghott, J. E. (eds). (1987). Molecular Photodissociation Dynamics. London: Royal Soc. of Chemistry. Assion, A., Baumert T., Seyfried, V., Weiss, V., Wiedemann, E., and Gerber, G. (1996). Femtosecond spectroscopy of the (2)> double minimum state of Na . Z. Phys. D S  36:265—271. Baig, A., Bylicki, F., Zhu, J., and Demtro¨der, W. (2000). Doppler-free laser spectroscopy of autoionizing Rydberg levels in Li (to be published).  Bassi, D. (1992). Detection principles, in Atomic and Molecular Beam Methods, G. Scoles, ed., Oxford: Oxford Univ. Press. Baumert, T. and Gerber, G. (1995). Femtosecond spectroscopy of molecules and clusters, Adv. At. Mol. and Opt. Phys., vol. 35, New York: Academic Press. Baumert, T., Helbing, J., and Gerber, G. (1997). Coherent control with femtosecond laser pulses. Advances in Chemical Physics 101:47—77. Beigang, R. (1988). High resolution Rydberg-state spectroscopy of stable alkaline-earth elements. J. Opt. Soc. Am. B 5:2423. Bernheim, R. A., Gold, L. P., and Tipton, F. (1983). Rydberg states of Li by pulsed  optical-optical double resonance spectroscopy. J. Chem. Phys. 78:3635. Berry, M. V. (1984). Quantal phase factors accompanying adiabatic changes. Proc. Royal Soc. L ondon A 392:45.

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COULOMB EXPLOSION IMAGING OF MOLECULES Z. VAGER Department of Particle Physics, Weizmann Institute of Science, 76100 Tehovot, Israel I. The Principle of Coulomb Explosion Imaging . . . . . . . . . . . . . . . . . . . II. Scientific Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The Study of Molecular Structure . . . . . . . . . . . . . . . . . . . . . . . . . . B. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III. Experimental Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Preparation of Fast Molecules for Coulomb Explosion Imaging . . . 1. The ANL Old Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Weizmann Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Heidelberg Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. A Realizable Future Cooling Method for Coulomb Explosion Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Stripping of Binding Electrons in Coulomb Explosion Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Detectors for Coulomb Explosion Imaging . . . . . . . . . . . . . . . . . . . 1. Detectors Used at ANL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Weizmann Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Detectors for Coulomb Explosion Imaging in Heidelberg . . . . . . 4. Outlook for Future Detectors for Coulomb Explosion Imaging . IV. Example of Recent Coulomb Explosion Imaging Studies and Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. The XH> Quasilinear Molecules, Experimental . . . . . . . . . . . . . . .  1. Conversion of Measured Parameters to V-Space . . . . . . . . . . . . 2. Bending Information in V-Space . . . . . . . . . . . . . . . . . . . . . . . . . 3. Study of Bond Angle Distributions Including Wake Effects . . . . 4. A Paradigm: The Study of the NH> Species . . . . . . . . . . . . . . . .  5. Digression: Coulomb Explosion Imaging of Diatomic Molecules with Wake Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Reorientation Replaced by Initial Condition . . . . . . . . . . . . . . . . 7. Bending Distributions from Coulomb Explosion Imaging and Comparison with the Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Systematics of the Differences between the Data and the Theory . B. Accuracy of Adiabatic Wavefunctions for Linear Molecules . . . . . . C. Comparison with Coulomb Explosion Imaging Experiments . . . . . V. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Z. Vager Abstract: Recent ideas and developments in coulomb explosion imaging (CEI) — observation of three-dimensional (3D) positions of nuclei within single molecules — are reviewed. Both experimental and theoretical concepts are covered. In particular, modern studies are highlighted by surveying new experimental progress that includes thermodynamically defined molecular beam preparation, suitable detector improvements, and other changes in experimental notions. Reconstruction of conformational distributions of molecular ensembles is given using examples, and discussion absent to date on the influence of wake phenomena on such reconstructions is detailed. Special care is devoted to the meaning of the experimental results in terms of quantum mechanics, spectroscopy, and other structural concepts dogma. Understanding the seemingly conflicting results from different dogmata is rationalized and scientific consequences are drawn. This review is not intended to be inclusive; instead, relatively new developments are summarized.

I. The Principle of Coulomb Explosion Imaging Coulomb explosion imaging (CEI) is an experimental method that provides instantaneous three-dimensional images (3D) of the nuclear positions within individual small molecules. In an ideal CEI measurement, a high-velocity (2—3% of light velocity) beam of molecules impinges upon an extremely thin foil. Within the passage time of the first few atomic layers (10\s), molecular electrons are scattered away with cross sections of approximately Å, leaving positive ions in the positions of the molecular nuclei. Typically, at least all binding electrons are stripped. As stripping time is shorter than representative periods of nuclear motion within the molecule, the molecular Hamiltonian suddenly changes into a purely repulsive, essentially Coulomb potential Hamiltonian of the ions’ degree of freedom. It can be shown that when at least one of the fragments charge is two or more, classical trajectories approximation is sufficient for predicting the final fragments velocity vectors from an initially assumed conformation of the atoms within the molecule. Thus, simultaneous determination of the final fragments velocity vectors from each single molecule, which is what CEI detectors do, resolves the 3D initial conformation. Collecting conformations from an ensemble of molecules unravels the density function of conformations of that ensemble, conveying the detailed correlated variations within the molecular structure. This experimental scheme of obtaining detailed structure of molecules is relatively direct and much less model dependent than other methods for obtaining the density of structures of isolated molecules. The advent of ultrashort powerful lasers opened up the possibility of stripping many electrons in a very short time, apparently in a way similar to that of the CEI The CEI quantum calculations, which also prove the aforementioned assertion, were fully carried out for only two fragments and used for the interpretation of CEI for hydrogen species.

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method. Yet, the foil stripping time is shorter by more than an order of magnitude compared with the most ideal future fast lasers. Thus, the sudden approximation condition mentioned in the preceding, and which is needed for simplicity of interpretation, is much better fulfilled for the original thin foil CEI and marginally fulfilled for the fastest laser pulses. Of course, there are many other advantages of dissociation by intense and fast laser pulses, but here, only thin foil CEI will be reviewed and hereafter the acronym CEI will be used to refer to thin foil stripping.

II. Scientific Considerations A. T S  M S The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws lead to equations much too complicated to be soluble. P. A. M. Dirac 1929

In spite of the vast improvements in modern computers, this well-known quotation readily applies to the equations describing molecules and small molecules are not exceptional. Nevertheless, there exists a theoretical framework that supplies, at various levels of approximations, satisfactory solutions and predictions to phenomena in molecular gas phase spectroscopy. This is particularly true for eigenenergies of isolated molecules. However, when radiative transition probabilities or chemical reactions are concerned, the existing theory does not do as well. In view of the quotation from Dirac, this is not too surprising. Although it seems that the needed approximations for extraction of molecular structure by CEI are very simple and transparent, this quotation applies also to the complete quantum mechanical treatment of a swift molecule hitting a thin foil, which is the relevant theory for CEI. This raises the following question: Are the spectroscopically accepted methods for obtaining structural features of molecules compatible with the experimentally obtained CEI structural results, and if not, are the CEI results better suited for explaining chemical reactions? In this review, some effort is devoted to answering this question. Some background for explaining the scientific importance of this question is given in what follows. B. B Knowledge of molecular structure is key to the comprehension of their functionality. This is a guiding concept in biology, chemistry, and material

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sciences. The definition, understanding, and characterization of molecular structure have been major challenges in science. But, as with many other general scientific concepts, the definition of molecular structure differs depending on the scientific application stratum. For example, in many biological applications, the results of detailed analysis of the crystal form of the studied molecule are sufficient for defining and understanding a molecular structure. For other biologists, the characteristics of a biological molecule must be tested within its natural environment; thus, it cannot be uncoupled from the vicinity properties. This is also the approach of modern physics, in which the exact quantum theory of matter, though believed to be valid but impractical for large molecules, demands strong correlations between the complex molecular structure and its surroundings. This is connected to the well-known entanglement phenomenon, which claims that the wavefunction of a molecule cannot be separable from its environment. In spite of this, the approach of biochemistry where a large molecule is concerned, considers the influence of certain molecular groups and radicals to be very dominant and, to a large extent, characteristics of large molecules are given by the weighted properties of such groups. This restrictive assumption that small molecules and radicals with their unique chemical and physical properties are building blocks of organic materials is very successful in explaining many properties of matter over a range of temperatures that includes living environmental temperatures. Noticeably, a most important quality of those molecules and radicals is their chemical reactivity and not their spectral features. As was mentioned before, this quality is believed to be strongly connected to their structural characteristics. The results of CEI measurements are an important facet of those structural characteristics, therefore their relation to current structural theories is of scientific importance.

III. Experimental Details In Section I the principle of an ideal CEI measurement was sketched. Over the years, the approach to an ideal experiment has been accompanied by better comprehension of the weak experimental points and vast improvements in several directions, some of which are outlined in what follows. An essential missing section is the description of the CEI simulation computer code (Zajfman et al., 1990, 1992), which is necessary for reduction of coulomb explosion images into the physical world. It is hoped that the physical principles that are used in the code are sufficiently covered here for appreciation of its scientific level of accuracy. Regretfully, the CEI experiments reviewed here do not include the initial period of one- and two-dimensional (2D) CEI experiments.

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A. P  F M  C E I The study of molecular structure by CEI requires careful preparation of a beam of fast molecules with a speed of v/c < 2—3%. Low-energy accelerators provide this speed to molecular ions. Extremely important care is given to the thermodynamic properties of the prepared molecules, either before or/and after acceleration. This step is essential for meaningful comparison with existing theories. What follows are several descriptions of preparation methods used in the past and today, and some preliminary future ideas in CEI experiments are also provided. 1. The ANL Old Set-Up The pioneering CEI set-up used to be at the Argonne National Laboratory (ANL) (Graber et al., 1992). Positive molecular ions were prepared at the terminal of a 4-MV dynamitron by expansion of seeded molecules in an inert gas through a pulsed nozzle. The molecules were ionized by an electron gun right at the edge of the expansion cone. The length of the expansion cone (2 cm) was expected to be large enough for vibrational cooling. Indeed cooling effects on He> were observed for CEI measurements of the 2>  charge states by noticing the narrowing of the CEI relative velocity distribution as well as the larger mean relative velocity as a function of cooling conditions. Good agreement with theoretical prediction for the vibrational ground state was achieved. Many molecular ions were studied at the ANL set-up and only some of those were published. Among the favorites of this author are a realistic demonstration of the Jahn-Teller effect in CH> (Vager et al., 1993a) and the structure of C H> (Vager et al.,    1993b), which conflicted with theoretical predictions and created an endless controversy (Marx and Parrinello, 1996) and even doubts about reliability of CEI as a method (Oka, 1993). Generally, these pioneering results provide direct confirmation of the structure of molecules that had already been predicted by theory and are compatible with indirect structural evidence gained from spectroscopy. When detailed comparison with theory showed significant deviation, doubts were raised regarding the quality and control of the cooling at the accelerator terminal. The adequate preparation of fast molecules for CEI became a pressing issue. 2. The Weizmann Set-Up As far as preparation of a molecular beam, the innovation of the WeizmannInstitute set-up involved introduction of a laser controlled accelerated neutral molecular beam (Kovner et al., 1988; Algranati et al., 1989). In the first step a pulsed negative molecular beam was prepared by pulsed nozzle

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technique followed by an electron gun or by expansion of laser-ablated negative molecular ions. This pulsed beam was injected into the 14 UD pelletron accelerator for acceleration to 10—12 MeV toward the positive terminal. When the pulse of negative molecules arrived at the terminal (the 3-m length of the pulse were matched to the length of the terminal), a triggered pulse from a tuned laser passing through the accelerating tube photodetached the electrons from a significant fraction of the ions. The produced neutral fraction drifted through the rest of the accelerator into the CEI stripper and detectors. Thus, this system allowed the study of neutral species using the CEI method. Probably the most significant measurement carried out by this set-up was the structure of neutral vinylidene (Levin, 1997; Levin et al., 1998) — an energetic isomer of acetylene — 6 s after production. Before this experiment, the species was expected to disappear within picoseconds into acetylenic structures. The main issue in this section is that the controlled laser detachment allowed preparation of very special states of neutral molecules, with a known narrow distribution of eigenenergies. 3. The Heidelberg Set-Up A novel CEI set-up is located at the heavy ion storage ring TSR of the MPI in Heidelberg, Germany (Wester et al., 1998). Accelerated positive molecular ions are prepared in regular heavy ions sources and accelerators, such as a 12-MV tandem or a 2-MV van-de-Graaff. Those beams are injected into the TSR and extracted after a variety of storage times into the CEI installation for structural measurements. While in the TSR, the molecular ions cool radiatively to room temperature and the cooling process is monitored by the structural changes of the CEI results as a function of the storage time. Except for the interesting study of structure variations while cooling, a beam of well-defined temperature is obtained after a verifiable long enough storage time. A detailed example of the measurements of bond angle > > distributions in NH>  , ND  , and CH  with this method will be discussed later. 4. A Realizable Future Cooling Method for Coulomb Explosion Imaging The inelastic scattering of almost zero energy electrons from molecular ions is bound to result in ions with even smaller internal excitation. Pictorially, the molecular ions can be considered as immersed in a cold bath where electrons exchange momenta between the bath and the excited molecules. For H>  , it was found that this process is by far more probable than the dissociative recombination (Tanabe et al., 1999; Krohn et al., 2000). If a

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dense electron beam with very small energy spread is prepared parallel to a stored molecular ion beam with almost the same velocity, then the internal excitations of the molecules are reduced by inelastic scattering of electrons. This is different from the usual ‘‘electron cooler’’ task for cooling atomic ion beams. The process was recently found (Tanabe et al., 1999; Krohn et al., 2000) by observing the internal energy of dissociative recombination products of H>  ; e, where anomalously large contribution of the ground state of H> was found. Of course, the H>   species is special because its radiative transition rates are extremely small. Nevertheless, trying to cool molecular ions by such electron coolers to below room temperature should always work to a certain degree due to the competition between radiative heating and electron inelastic scattering. The only controllable external parameter is the density of ‘‘cold’’ electrons in the common path of the ion-beam and the electrons. The internal parameters are the dipole radiative transition rates and the cross sections for inelastic transitions of thermally excited states. It is quite conceivable that reducing the molecular beam temperature significantly below room temperature is not only realistic, but very useful in resolving structural issues. B. T S  B E  C E I The cross sections for ionization and capture of electrons from and to atomic ions when swiftly passing through solid material are well known. Less known are these cross sections for the molecular species. In CEI it is essential to remove, at the very least, enough electrons such that all the involved atoms are ionized. Moreover, if an initially binding electron is not removed, there is a fair chance that the dissociating potential is significantly different from purely the coulombic. When dealing only with the ionization of binding electrons within very thin targets, the following empirical rule is useful. Consider the impinging molecule as a fictitious atom, or atomic ion, with all of the charges of the nuclei lumped together at one point. The ionization properties of such species are well known. If the probability of removal of all, previously binding electrons from such a concoction is high, then a pure coulomb Hamiltonian is expected to be a valid approximation after stripping. For small molecules with a speed of 2—3% of light velocity, the atomic ion fragments with popular charge states are found to behave accordingly. As mentioned before, if the stripping process is efficient and fast enough, the sudden approximation between the free molecular Hamiltonian and coulomb interacting ions Hamiltonian is applicable and facilitates a direct and simple interpretation of the CEI results. For this reason the stripper target must have a high density of electrons. On the other hand, the passing

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ions coulomb scatter while being in the stripper target by every target atom along their path, a process which is called multiple scattering (MS). To minimize MS effects, the target has to be as thin as possible with the smallest target atomic number. The most modern CEI target that optimizes those somewhat conflicting demands is made out of diamond-like-carbon (DLC) (Levin et al., 2000) with a density of 2 g/cm and as thin as 30 Å. Those previously popular targets in CEI that are still in use are made of formvar (Both et al., 1987) with a density of 1 g/cm and as thin as 80 Å. The high density of electrons supports high-frequency plasmons, which interact with the passing ions in a nontrivial way as can be observed in all CEI results. These, generally called wake effects (Gemmell and Vager, 1985), are known only qualitatively. Though precise bond length distributions from CEI can be analyzed with almost no regard to wake effects, it has been shown recently (Baer, 2000) that their influence on bond angle distributions is appreciable. Discussion of the nature of wake effects and an example of how to handle such bond angle distributions with correction for wake effects is detailed in what follows. C. D  C E I A short description of various detectors is given here. (Early detectors for CEI that did not cover all orientations of the studied molecules or did not have 3D capabilities are not reviewed in this chapter.) First, prototype detector characteristics are sketched. 1. The resolution for a single event measurement is intended to be compatible with zero-point fluctuation of a bond length. Invariably, the demand on time resolution governs the design of the whole detector. Assume that a 2% bond length resolution per event is demanded for an HX species where X is very heavy and having a final charge of ;4. For :1 Å bond length, the final center of mass (CM) kinetic energy of the proton is about 60 eV. The corresponding CM velocity is v  3.6 · 10\ c, which is supposed to be measured with an accuracy of 1% on top of the beam velocity of, say, V : 0.03 c. Thus

v/V : 1.2 · 10\ : /T where  and T are the time resolution of the detector and the time of flight between the stripper foil and the detector, respectively. Assume an excellent electronic time resolution of  : 50 ps, then, by the simple relation between length, velocity and time, all other dimensions and sizes are determined. For instance, the time of flight is T : 420 ns and the flight path length is 3.8 m. The radius of the CEI sphere is :4.5 cm, which demands a position resolution of better than half a millimeter. The latter is easily fulfilled by all CEI detectors. Of course, there could be a trade-off between

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time resolution, flight-path, and beam velocity. With the modern version of detectors (which use microchannel plates), the choice of detector diameter is limited only by cost. Otherwise, a longer flight path allows moderation in the time resolution. 2. An important demand is the possibility of simultaneous detection of several particles with high efficiency. 3. It was found very useful in all CEI setups to separate the masses of the CEI fragments by either electrostatic or magnetic means. The effect of such separation on the CEI spheres is corrected numerically. Such separation allows identification of charge states as well as optimization of time resolution. 1. Detectors Used at ANL The very first detector with almost all the features of an ideal CEI detector was developed at ANL (Koenig et al., 1985; Faibis et al., 1986). The detection principle was based on the multistep avalanche chamber scheme (Breskin and Chechik, 1985) with three sets of parallel detection wires, oriented with their delay-line readout, 120° from each other in the detection plane. This three-fold detection allowed the multiparticle detection feature. The time resolution was better than 200 ps and the flight path was almost 6 m. The effective radius of the detector was as large as 10 cm. At a later stage, a rectangular (Belkacem et al., 1990) detector was added and allowed for a comfortable separation of fragment masses. In retrospect, these pioneering detectors demanded quite huge installations due to limited time resolution and gas handling systems. As for data analysis, the cumbersome position calibration process, which was accompanied by highly complex software, demanded a lot of attention and extra care by the experimentalists. 2. Weizmann Detectors A new type of detector was developed at the Weizmann Institute (Kella et al., 1993) in which a CsI-coated thin foil was used to convert from CEI fragments to clusters of electrons. These electrons were then accelerated and multiplied by a 2-stage MCP. The resulting bunches of electrons were further accelerated into a phosphor screen. Attached to the phosphor screen are parallel wires, with a density of 1/mm, which pick up the signals of the multiplied electrons for timing information. The phosphor image, read by a CCD camera, records the transverse 2D information, which is correlated with the timing information for each fragment. The largest MCP used in this manner had a diameter of 12 cm combined with a ‘‘heavy ion detector’’ MCP with a 10 ; 8 cm rectangular shape. The best achieved resolution is 70 ps full width half minimum (FWHM) and the flight path length is 2.2 m.

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The monitoring of the arrival of single molecules on the detector by simple inspection of a TV monitor was a revelation and is still very helpful in monitoring and setting up CEI experiments. Compared with the pioneering version at ANL, the 2D position readout is by far more transparent, simple, and reliable. Although the timing still needs much testing and calibration, the operation has been very much improved. 3. Detectors for Coulomb Explosion Imaging in Heidelberg Detectors for CEI at the TSR (MPI) were developed at the Weizmann Institute (Baer, 2000) and later transferred and used at the CEI setup in Heidelberg (Wester et al., 1998). They are not very different from the WI version except that the anode wires have been replaced by a multistrip assembly on thin kapton foil coated with a phosphor layer. The performance of such detectors has been improved over the first WI version, but the main concern in operating such detectors remains calibration of timing information and overall homogeneity. 4. Outlook for Future Detectors for Coulomb Explosion Imaging In a recent article (Strasser et al., 2000) an innovative approach to multiparticle 3D imaging is presented. The ratio of intensities of a pair of 2D images is used to extract timing information. For example, these images could have been two CCD camera frames of the traces of CEI fragments arriving on the detector at ever slightly different times. The optical path of one of the images is time distorted with an optical switch such that late arrivals are chopped. Thus, at one image the late arrivals receive less light and the early arrivals receive more light in proportions that are linear (or another well-defined monotonic function) with time. This novel idea had been patented (Heber et al., 1999) for 3D vision application, but in Strasser et al. (2000) an actual demonstration of the method already proved to have sub-nanosecond resolution. It is not farfetched to assume that such compact and simple devices could be used in the near future for CEI, eliminating the existing cumbersome electronics for accurate timing.

IV. Example of Recent Coulomb Explosion Imaging Studies and Consequences A. T XH>  Q M, E Coulomb explosion imaging (CEI) measurements on the XH>  species were first studied at ANL and a full report was published in Graber et al. (1997).

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TABLE I B P

Ion

Source

NH>  ND>  CH> 

Duoplasmatron Duoplasmatron Sputter

Source molecule

Precursor

NH  ND  CH OH 

NH\  ND\  CH O\ 

Beam energy (keV) 5860 (366)? 5190 (288)? 6690 (478)?

? Energy per nucleon in parenthesis.

As mentioned earlier, doubts were raised as to the quality of the cooling of the supersonic expansion ion source at ANL and thus new improved measurements were carried out at the TSR setup in Heidelberg, where the molecular ions can be cooled to 300 °K. > Most of the experimental details concerning the NH>  and ND  CEI measurements are similar to those already described for the measurement of CH>  in Baer et al. (1999). More details on this particular CEI setup are given in Wester et al. (1998). For completeness, the accelerated molecular > > ion preparation and storage for the NH>  and ND  as well as the CH  species are summarized here. The molecular ions were initially produced from stable gas vapors from which a precursor negative ion was created by either a duoplasmatron or a Cs-sputter source. The negative precursor was injected into a 12-MV tandem accelerator and accelerated to a few MeV towards a gas target located in the tandem high-voltage terminal, where charge exchange occurred by collision with gas. Positive ion products were further accelerated and the desired molecular ion was mass and charge selected. The final beam velocity was chosen to be about 0.03 c. Once stored in the ring, a small part of the molecular ion beam was extracted toward the CEI beamline where the structure measurements were performed. For all practical purposes the sampled storage times are continuous up to several seconds. A summary of beam parameters is given in Table I. 1. Conversion of Measured Parameters to V-Space Each measured molecule leaves a ‘‘fingerprint’’ of positions and times of all the fragments on the CEI detectors. Their interpretation as 3D asymptotic images in velocity space includes some ensemble averages since the different charge states and masses ‘‘spheres’’ are measured at different parts of the detectors. Consequently, corrections due to the charge-to-mass separator distortion are applied. There are many useful visual tests for inspection of

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these conversions and critical tests of complete control involve comparison of the same views by simulations. Many times, the almost trivial parts of such tests are never revealed in publications in order not to confuse the reader with details. Here, for later clarification, it is preferred to show some of these visual comparisons. In particular, Figs. 1 and 3 (see also Color Plates 3 and 5) display 2D histograms of projections of the N-H velocity vectors from a CEI experiment on the NH> species, shown on the XY  detector plane (perpendicular to the beam) and XZ plane. The corresponding simulated figures are found in Figs. 2 and 4 (see also Color Plates 4 and 6). All of the acquired parameterized information is already built into these simulations. The overall view manifests a projection of a sphere having a skin thickness with some deviations from isotropy of the N-H vectors. Some of the reasons for anisotropy involve well-calibrated, detector inefficiencies. However, at the upstream pole along the beam V  (0, 0, 90.05) AU there ,& is an increased density due to the wake phenomenon, which will be explained and investigated in a forthcoming section. 2. Bending Information in V-Space As explained in Baer et al. (1999), for each coulomb exploding molecule the storage time as well as the 3D velocities and charge of each emerging ion are recorded. Of particular interest here are the histograms of the cosine of the H-C-H, H-N-H, and D-N-D angles at the measured asymptotic velocity space, called V-space, selected for the heavy ion’s most populated ;4 charge state. Figures 5, 6, and 7 represent the storage time dependence of the bending distributions of the three studied species. As storage time increases, the structural changes damp, indicating an approach to equilibrium at room temperature. Notice (Figs. 5 and 6) that for both nitrogen isotopomers the peak of the experimental cosine distributions at higher excitations (short storage time) shifts away from the linear conformation at cos  : 91 whose population declines. This trend is the opposite of what was found for CH> (Baer et al., 1999)  (see Fig. 7), where probability of linear conformation increases at higher excitations. This systematics is compared later to theoretical systematics. By choosing storage time where the curves labeled a in Figs. 5, 6, and 7 flattened, cosine probability histograms of CEI measurements of 300 °K ensembles are achieved. 3. Study of Bond Angle Distributions Including Wake Effects The redistribution of coulomb exploding particles in matter has been measured and interpreted in Gemmell and Vager (1985). The physics of this

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F. 1. The CEI of NH>; 2D histogram of x and y components of measured V in atomic  ,& units. (See also Color Plate 3).

F. 2. The CEI of NH>; 2D histogram of x and y components of simulated V in atomic  ,& units. (See also Color Plate 4).

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F. 3. The CEI of NH>; 2D histogram of x and y components of measured V in atomic  ,& units. (See also Color Plate 5).

F. 4. The CEI of NH>; 2D histogram of x and z components of simulated V in atomic  ,& units. (See also Color Plate 6).

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F. 5. (a) Storage time dependence of the NH> cosine density at x : 91. (b) The cosine  density at different storage time windows.

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F. 6. (a) Storage time dependence of the ND> cosine density at x : 91. (b) The cosine  density at different storage time windows.

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F. 7. (a) Storage time dependence of the CH> density ratio : P(x : 91)/P(x ) where  K P(x ) is the maximum density. (b) The cosine density at different storage time windows. K

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effect are found already in Bohr’s ideas (Bohr, 1948) on a wake of electrons that trail swift charge particles while passing through solid targets. This heap of negative charge behind the moving external positive charge is responsible for the ‘‘electronic stopping power,’’ which is the dominant stopping mechanism at 3% of the speed of light. Although accurate parameterization of the stopping power of ions in matter is quite well known, the exact shape of the wake of electrons is only approximately understood. Moreover, the response of target electrons to a cluster of ions of molecular dimension is yet unknown. To appreciate the nature of the effect, the following approximate picture is helpful (Vager and Gemmell, 1976): When the plasma frequency of the target electrons is well defined, the wake of electrons can be described by a stationary line of charge behind the moving charged particle describing sinusoidal amplitude starting negative. The wavelength is given by the plasma frequency and the velocity of the moving particle. The charge is created by a small deviation of the electrons from their normal distribution towards the external moving charge. For carbon targets the plasma frequency is :25 eV thus, for molecular velocities of :0.03 c the wake wavelength is :15 Å. When a cluster of positively charged particles is moving in the target, then in a linear response approximation each charged particle produces its own stopping electrons in the form of a wake. However, those wakes also influence the motion of neighboring ions. This is the basis for the notion of reorientation effects. So far, measured effects of the reorientation of coulomb exploded diatomics have been only qualitatively explained by wake models (Gemmell and Vager, 1985; Vager and Gemmell, 1976). In CEI experiments employing extra thin targets with a molecular center of mass velocity of :0.03 c, the internuclear distances within the target are a small fraction of the wake wavelength and, therefore, a linear approximation of the wake potential as a function of these distances is reasonable. Opposite to the beam direction there is a ‘‘stopping’’ field component due to the total effective charge carried by the ionized molecule. For example, a 6-MeV beam of NH > molecules produces in the thin foil a field that  corresponds to the stopping power of a 6-MeV fluorine ion with the same velocity (see Ben-Hamu et al., 1997). However, this monopole stopping effect is uninteresting for CEI, as only the relative differentials of these wake forces are of importance. 4. A Paradigm: The Study of the NH> Species  Assume for beginning this discussion that the magnetic sublevels of the total angular momenta of the studied molecules are equally populated. An experimental support of this physically reasonable isotropy assumption is

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given later. In a dream world of no multiple scattering or wake forces in the target, the assumed isotropy in R-space is also retained in V-space, after the coulomb explosion transpires. This ideal does not happen for even the thinnest real solid target and therefore a comparison between theory and experiment must be carried out with the aid of a computer simulation code that takes into account such effects. The physics of multiple scattering in the XY plan and its interplay with screened coulomb interaction within the target is supposed to be taken into account correctly within the CEI simulation code. This code also includes accurately measured detector efficiencies, which also contribute to the overall anticipated anisotropy in the measured V-space. However, the existing code does not include wake forces, simply because so far there had not been quantitative reproduction of the experimental wake features, such as in Figs. 1, 3 and in Gemmell and Vager, 1985 and Vager and Gemmell, 1976, by existing wake force theories. Although the existing simulation code is sufficient for extraction of bond length distributions, for experimental extraction of H-N-H bond angle density, all reorientations of N-H vector directions have to be accounted for quantitatively. This is essentially true for any other species’ bond angle distribution extraction from CEI. The simplest indication of reorientation is given by Fig. 8, in which the measured density is shown as a function of X , the cosine of the measured ,& N-H vectors with respect to the beam direction, and compared with the expected cosine density as given by the simulation code. Clearly, beside the sharp peaks at the edges, there is significant deviation from isotropy that is not explained by the existing simulation. This type of azimuthal reorientation is compatible with the notion of wake forces. It seems that this fact must have devastating effects on the bond angle distribution in V-space due to the distortions of the H-N-H angles, depending on the orientation of each of the two N-H arms. To see this, a 2D histogram is created from the observed data, in which one dimension is x : cos(H-N-H) and the other dimension is X , the cosine of the azimuthal angle of one of the N-H arms. ,& Each event is counted twice for each of the two arms of the H-N-H angle. The proper statistical error can be easily accounted for by noticing the number of double events within each bin. Indeed, as anticipated, the cosine distributions for different cuts in the azimuthal angle X vary significantly ,& (Fig. 9). Not only the number of events in each cut vary, but also the characteristics of the shape differ from one cut to the next. On the other hand, the existing simulation code produces cosine distributions for different The usual practice is to cut away the pole regions where distortions of the bond length by the wake are found. The simulated X distribution (dashed line) is not entirely flat due to the multiple,& scattering process as well as inclusions of detectors efficiencies.

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F. 8. Experimental and simulations histograms of the cosine of NH vectors with respect to the beam direction.

cuts in X , which hardly differ from each other (not shown). This seems to ,& rule out the use of the existing simulation code (which does not consider wake effects) for accurate determination of bond angle distributions. In an ideal simulation code, realistic wake forces should have been taken into account and if the simulation results would have been compatible with each of the Fig. 9 histograms and with the reorientation histogram (Fig. 8), only then should it be assumed that (spatially isotropic) R-space with its bond angle distribution is to be trusted to represent the measured ensemble bond angle distribution. Fortunately, for thin enough stripper foils, it is possible to reach this ideal situation without detailed knowledge of the wake forces and, as a spinoff, gain empirical knowledge of these forces. 5. Digression: Coulomb Explosion Imaging of Diatomic Molecules with Wake Forces To understand how this is done, a short introduction on the action of wake forces on diatomic species within thin foils is given. Because the characteristic wavelength of the wake of electrons is by far larger than the molecular

F. 9. Ten cuts in X from the 2D histogram of X vs x : cos . The cut windows are marked in the figure. ,& ,&

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size, the interaction of the induced potential with the molecular constituents can be expanded by multipoles. The monopole interaction is responsible for the stopping power of the cluster as a whole (see Ben-Hamu et al., 1997)  and is irrelevant to CEI interpretation. The next term is a vector field F wherein its scalar product with the diatomic bond vector affects the relative motion of the nuclear constituents within the foil. Due to symmetry along the beam direction, only the spherical coordinates F and F are different 0 F than zero. The force F generates changes in the final radial distances in V 0 space as seen in Figs. 1 and 3 and F generates the reorientations as reflected F by the cosine of the N-H vectors — X — in Fig. 8. Consider a two-body ,& interaction after electron stripping. The only tangential momenta are coming from initial rotational states with rotational periods of the order of 10\ s. As the characteristic time in the coulomb dissociation process is 10\ s, negligible reorientation effects are expected due to realistically anticipated angular momenta within molecules of the beam. Thus, the measured reorientation is only due to wake forces and can be simply interpreted as a dissociation processes with initial angular momenta L : RF t, where t is the dwell time within the stripper foil. A coulomb F scattering with an initial angular momentum yields a unique and wellknown scattering angle. In other words, to the first order in the dwell time within the stripper foil, the azimuthal angle shift by the wake can be simulated with the existing code (which does not contain wake forces), providing that the initial isotropic distribution will be replaced by a proper anisotropic distribution reflecting half the coulomb scattering angle. This allows for reorientation effects to be corrected empirically by finding initial momenta that reproduce histograms such as Fig. 8. 6. Reorientation Replaced by Initial Condition Coming back to the interpretation of the CEI results of NH>, the reorien tation distribution in Fig. 8 provides a pretty good clue to the needed initial distortion of the distribution in R-space. The aim is to find a shift function S( G ) of the initial isotropic azimuthal angles G (meaning random cos G ,& ,& ,& distribution) such that the simulated distribution of X : cos  assumes ,& the experimental distribution. If the simulation would have yielded a one-to-one correspondence between the initial orientation and the final orientation, then the function S(G ) is simply given by the measured ,& distribution P(X) (Fig. 8) as follows:  : G ; S(G ) ,& ,& ,& XG : cos G ,& X : cos  ,&

(1) (2) (3)

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and by the isotropy of the incoming molecules P(X) dX : Q(XG) dXG :  dXG 

(4)

The shift function can be found by integration of P(x) XG(X) : 2



6

P(x) dx 9 1

(5)

\

As the simulation code includes effects of multiple scattering within the stripper foil, there is no one-to-one correspondence between the initial and the final angles. However, it is a good starting point for finding the correct shift function by trial and error. The orientations that are most sensitive to multiple scattering are those wherein the NH vector is almost parallel to the beam direction. A modified shift function in which those pole regions are sharpened (by trial and error) does the job adequately. A test of shift function is whether it is compatible with the reorientation histogram (Fig. 8, the histogram with error bars). A background for the crucial test is set. If indeed the physics of reorientation distortion is given simply by the shift function (due to orientation-dependent torque in the target) then there must be an isotropic density function of NH > structure, which upon applying the prescribed  reorientation on each initial NH vector (distorting H-N-H angles) and sampling it through the CEI simulation code, should make it compatible with every cut shown in Fig. 9. As can be seen by the full-line curves in the figures, indeed there is an R-space distribution that, following the forementioned prescription, fits nicely the variations in the different cuts. There is no extra normalization for each cut, only one normalizing parameter, which matches the number of simulated events to the number of measured events, and is used for the entire 2D histogram. The fit parameters are all in R-space, related to the structural feature of the species. This satisfying compatibility of all the x : cos(H-N-H) cuts histograms, together with compatibility of the reorientation histogram (which includes a very large number of measured single molecule events) is an indication of the accuracy and reliability of such bond angle distribution measurements as well as the empirical treatment of the wake effects. Although it might be redundant to mention, the shift corrections are physically equivalent to nonzero initial (tangential) momenta conditions. Those initial conditions are not related to the initial density matrix of the studied molecules before the stripper foil. Therefore compatibility with all V-space cuts in the cosine distribution (Fig. 9) is not at all trivial and strongly supports the empirical physics behind this process. Moreover, data

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from nonthermally equilibrated molecules at short storage time extraction (see Fig. 5) have, within statistics, the same reorientation curve as in Fig. 8. Thus, although there is a clear indication of structural changes with time, the so-called reorientation process of the NH vectors is time independent, and therefore structurally independent. Any assumption on storage-ringrelated anisotropies is most likely to be storage time dependent. This supports the basic reasonable assumption that the orientations of the molecules in the incoming beam are randomly distributed and the observed anisotropy is related only to subsequent target effects. In summary, a procedure has been devised to examine both in great detail and directly the bond angle distribution of some simple molecules by CEI and quantitative information on reorientation effects due to wake forces. Applications of this procedure to molecular structure studies are given in the next section. So far, this procedure has not been applied to the study of wake phenomena.

7. Bending Distributions from Coulomb Explosion Imaging and Comparison with the Theory The lesson from the previous section is that accurate bending distributions can be obtained from CEI data provided that the wake effect is taken into account, either empirically or theoretically. The use of the 2D histogram of X and x (measured cosines of X-H and H-X-H angles) for comparison with simulated theories seems mandatory. The total number of degrees of freedom (bins) in Fig. 9 histograms is 450 and a  test (the sum of all cuts , each cut has 45 bins) should be compared to that number. For convenience, only projections of such 2D histograms along x will be shown from this point. However, comparison to a given theory will be carried out on all the bins of the histograms. For example, the measured cosine histogram for 300 °K NH> species is shown in Fig. 10. The  quoted in the  figure should be compared with 450 degrees of freedom. It is very clear from the values of the  that the so-called best fit (full-line curve) is significantly better than the theory (dashed-line curve). The source for theoretical calculations here was provided by Jensen and Bunker (1999) and based on several published studies (Osmann et al., 1999; Kraemer et al., 1994; Jensen et al., 1995a; Osmann et al., 1997a; Jensen et al., 1995b; Osmann et al., 1997b). These are conformational density ensembles averaged for 300 °K. Although it may seem pedantic to make the small difference between the theory and the data important, there is a good reason for noticing it and this will be elucidated soon. Figure 11 shows clearly the statistical signif-

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F. 10. The measured x : cos  histogram for NH> (independent of X ) and comparison  ,& to theory.

icance of this difference. Similarly to NH>, measured cosine distributions for  ND> and CH> are shown in Figs. 12 and 14 together with proper   (including empirical wake effects as for NH>) simulations of theoretical  predictions. Corresponding relative difference plots are shown in Figs. 13 and 15.

F. 11. The relative difference of the measured histogram (P ) and the theory (P ) for " 2 NH>. 

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F. 12. The measured x : cos  histogram for ND> (independent of X ) and comparison  ," to theory.

Before analyzing the differences between experiment and theory, similarity should be emphasized. Although these species have been spectroscopically measured and structural information had been retrieved from parameterized calculations, in the case of quasilinear molecules, it is especially difficult to extract the molecular structure from such measurements. A quotation from Okumura et al. (1992) concerning the structure of NH > is  clear on this point:

F. 13. The relative difference of the measured histogram (P ) and the theory (P ) for " 2 ND>. 

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229

F. 14. The measured x : cos  histogram for CH> (independent of X ) and comparison  !& to theory.

Although the spectrum can be fit to either a linear or an asymmetric rotor Hamiltonian, the large amplitude of the bending vibration prevents us from determining the molecular structure from the observed constants.

For the case of CH >, Ro¨sslein et al. (1992) can be quoted:  The spectrum could be fitted equally well to the linear molecule (or a bent one).

F. 15. The relative difference of the measured histogram (P ) and the theory (P ) for " 2 CH>. 

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It is evident that in both cases spectroscopy could not determine the structure of these quasilinear molecules and therefore the findings of CEI and their similarity to theoretical predictions should be taken as some confirmation of the theoretical concepts for such close-to-linear structures. 8. Systematics of the Differences between the Data and the Theory The comparison between the best fitted density functions and the theoretical predictions at both 300 and 0 °K are plotted in R-space and shown in Figs. 16, 17, and 18. The statistical error at large bending angles, where the probability approaches zero, is quite large (see Figs. 11, 13, and 15). Therefore, more attention is given to the region near the linear conformation. For all of the studied molecules, the predictions at linearity are significantly lower than experimentally found. For the two nitrogen isotopomers, the predictions at 0 °K have more linear probability, overshooting the trend of the data. On the other hand, for CH > the probability prediction at 0 °K is  strictly zero and thus opposite of the nitrogen isotopomers’ theoretical trend. The experimental probability distributions also have trends that can be seen in the cooling plots, the b part of Figs. 5, 6, and 7. The trends here are exactly the same: When the excitations are 300 °K (shorter storage times), the nitrogen isotopomer linear conformations are less populated while for

F. 16. Theoretical and experimental (marked ‘‘Best R’’) probability distributions in R-space for NH>. 

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F. 17. Theoretical and experimental (marked ‘‘Best R’’) probability distribution in Rspace for ND>. 

F. 18. Theoretical and experimental (marked ‘‘Best R’’) probability distributions in R-space for CH>. 

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the CH> species at higher excitations the linear conformation is more  populated. If theoretical predictions are taken seriously and used roughly to estimate the measured ensemble temperatures, one obtains for the nitrogen isotopomers ensembles a significantly lower temperature than 300 °K; for the CH> ensemble a significantly higher temperature than 300 °K is ob tained. This absurd situation is a manifestation of the incompatibility of theory with CEI data. For all cases, the experimental linear conformation is more probable than expected with a much larger deviation for CH> species.  Thus, the spectroscopically geared latest theoretical predictions for quasilinear molecules are at odds with the CEI results. In the following, a theoretical rationalization of this situation is proposed.

B. A  A W  L M Traditionally, the theory for linear molecules is only slightly different than the adiabatic theory for nonlinear species. In the first step, potential energy surfaces (PES) are calculated for different fixed point positions of the molecular nuclei. If the ground state electronic eigenenergy near the minimum ‘‘equilibrium’’ conformation is too close to the next electronic eigenstate, then in the next ‘‘dynamic’’ step the two low lying states and the coupling through the nuclear motion have to be taken into account for calculations of full molecular wavefunctions of low lying states. In particular, the electronic projection of the angular momentum at linear conformations takes the values , , . . . . Therefore, except for the  case, at linearity at least two degenerate states with a different sign of the electronic angular momentum projection are taken into account. For the species here, the NH>  and ND> have the  electronic ground state at linearity while for CH > the   lowest electronic states are of the  nature. If the theory of the lowest states of these species deals only with the foregoing states, such as in Osmann et al. (1999) then the treatment is defined as an adiabatic approximation. Nonadiabatic contributions are then calculations that take into account higher excited states. For CH> this is justified in Osmann et al. (1999)  because other electronic states at ‘‘equilibrium’’ have energies in excess of 6 eV. In the following it is argued that the usual adiabatic separation of electrons and nuclei in molecular treatment fails to predict the correct wavefunctions near linear conformations. For the sake of simplicity, spins are ignored and the isolated molecular Hamiltonian is employed. In the isolated molecule approximation, the total angular momentum and its projection along one axis are exactly conserved for stationary states. It will be shown that the last exact requirement, chosen along an inertial

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axis, is inconsistent with the existence of a small parameter for the traditional adiabatic expansion near linear conformations. The center of mass translational motion is irrelevant and so the effective number of particles is reduced by one. For instance, Jacobi coordinates are chosen in the order of decreasing masses of the participating particles. Discarding the kinetic energy of the center of mass operator, the remaining Hamiltonian deals with 3(N 9 1) nuclear coordinates and 3N electronic L C coordinates with their appropriate reduced masses. For example, in an atom, only electronic coordinates are relevant and their reduced masses are approximately equal to their free mass. For diatomic molecules there is only one effective nuclear vector, the vector difference between the two nuclei, with its usual reduced mass (and so on for larger molecules). All of the effective particles are well defined in a Cartesian inertial frame and electrons and nuclei are still easily distinguished by the very different sizes of their reduced masses. Choose arbitrarily an inertial frame z axis, which passes through the center of mass and cylindrical coordinates z ,  ,  . For each I I I effective particle (nuclei separately from electrons) the z component of the angular momentum operator is j : 9i (/ ). Notice that these operators I I and their partial or total sum are independent of the masses of the particles, thus, angular momentum conservation rules are independent of the ratio of the electron mass to the nuclear masses. The sum of such operators for N C electrons is defined as J and similarly J for N 9 1 nuclear coordinates. C L L The conjugate coordinates to J and J are defined as  and  . C L C L It can be shown that the kinetic energy operator includes the following term: T:9 where the dynamic variables

 

  9 2I  2I  L L C C

(6)

I and I are C

L

, I : C CG CG

(7)

C

G ,L\ I :  LH LH L H

(8)

For stationary states, the angular momentum along the (inertial frame) z axis M is conserved exactly. It is the eigenvalue of the operator   J : 9i

; X   C L






(9)

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The conserved quantity M is the sum of electronic and nuclear contributions. A few simple remarks are needed. All expectation values of the moments of inertia (or their inverse) are positive. For states of small molecules, the expectation of /2I is of the order of a few eV. C It is possible to transform to new coordinates where the conserved quantity belongs to only one coordinate, conjugate to J (with equivalent X mathematics to the transformation of the two body system into the center of mass coordinate, which is conjugate to the center of mass momentum, and the relative coordinate):

I  ;I  L L

: C C

: 9 C L

I

(10)

then T :9

 

  9 2J  2I  

(11)

where

J\ : I \ ; I \ C L I:I ;I

(12)

(13) C L are the z axis reduced moment of inertia and the z axis total moment of inertia dynamic variables. The angular momentum operator is transformed into:  J : 9i

X 

(14)

In general, the collective coordinate (Eq. (10)), which carries the angular momentum along the z axis, depends on nuclear and electronic coordinates. For stationary states, the total angular momentum J and its projection M are good quantum numbers. Consider a multiplet of degenerate states, J " 0, with M : 9J, . . . , J. The last term in Eq. (11) can be written as

M/2I and could be included in the potential energy. The corresponding eigenstates have a multiplying factor of the form eG+. For a given J, the M states are degenerate and the M-dependent expectation value of the last term of Eq. (11)

M

 I 1 2

(15)

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must be compensated by the expectation value of the rest of the Hamiltonian, which is independent of M. This ought to be expressed in the state itself by more than a phase factor. As for the first term in Eq. (11), the Fourier components of  :  9  A L are not restricted by angular momentum conservation rules. Thus far, no approximations were assumed and there is no constraint requiring that the amplitude of any state should vanish when the nuclei are situated on the z axis. The first step in the adiabatic approximation is to find the electronic eigenenergies for different conformations — except that here the exact quantum number M is already set as a demand. Start with linear conformations where the nuclei are fixed along the z axis. The last term of Eq. (11) forces the electronic state of this conformation to be of , , , . . . nature, depending only on the value of M : 0, 1, 2. . . . . The corresponding electronic eigenvalues increase rapidly with M in the scale of electronic energies. The argument can be extended to slightly bent conformations as long as the now constant /I is larger than a few eV. It is very clear now L that the adiabatic approximation where highly excited electronic states are ignored is erroneous because it should take the entire series of electronic excited states to obtain exactly degenerate eigenstates for all of these M states. That alone already makes the adiabatic approximation inconsistent with linear structures when angular momentum is considered properly, as should be the case for gas phase molecules. It might be argued that the choice of M classically corresponds to a choice of orientation; therefore, fixing the nuclei on the z axis is incompatible with that choice. Indeed, the problem lies in treating a linear conformation as a classical entity and trying to fix it later by projections on quantum mechanically allowed states. However, in most of the phase space, I  I . Therefore, the coordinate L C

is of almost a pure nuclear nature and the moments of inertia J and I are of almost pure electronic and nuclear natures, respectively. Thus, energy matrix elements can be well approximated by disregarding the remaining small phase space where the adiabaticity fails. The consequence is that for spectroscopy a model that result in reasonably accurate eigenenergies is sufficient and it does not matter whether the model wavefunctions are inaccurate. The expectation value of the last term in Eq. (11) is given by

M



1 2( I ; I ) C L



: M



1 2I · (1 ; I /I ) C L C



As long as I is very small the expectation value is of the electronic energies scale. L

(16)

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A question may be asked: What makes this formulation better than the traditional adiabatic formulation? The answer is that here an exact conservation law was taken into account explicitly and it prevents the wrong assumption to be made regarding the existence of a small parameter (for certain conformations) that can be used for perturbative expansion. C. C  C E I E If the consequence of not knowing the behavior of the wavefunction near linearity is accepted, it is still possible to attempt to guess its trend by ignoring the last term in Eq. (11), and to disregard its important influence on the form of the states (as already discussed). Assume that an adiabatic procedure is reapplied and the wavefunction is expanded in Fourier terms A of the now electronic variable . Each A is a function of the K K conformations of the nuclei. At linearity only A differs from zero. The other  components may build up for bent conformations starting at a rate slower than I . Even this is essentially different from the traditional adiabatic state L behavior, such as the Renner-Teller treatment where terms such as A are  discarded for linear conformations (as in the CH> species) due to energy  considerations. Ignoring the last term in Eq. (11) as rotational and then applying the adiabatic approximation on the rest of the Hamiltonian requires dealing with linear conformations with a pure A electronic state and developing a  solution for the bent region where traditional solutions normally apply. This has not yet been done by quantum chemistry. As a simple example, consider the CH> species at a very low temperature. In the regular Renner-Teller  treatment, the electronic wavefunctions are of a  nature near linearity. The  state at that conformation is at least 6 eV higher. This means that for fixed nuclei along the z axis the only nonzero Fourier components of  are with Fourier indices m : <1. This restriction forces a certain behavior on the nuclear function: It must be exactly zero at linearity and rise away from that point not faster than I such that the expectation value of L





 : ; 2I 2J 2I L C

(17)

remains finite. As was already mentioned here, no such strict condition is ever demanded before fixing the nuclear positions. In fact, as the z component of the total angular momentum is already taken care of by the conjugate of the coordinate in the second term of Eq. (11), the first term of Eq. (11)

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allows for a superposition of several Fourier components — m — of the  variable in the electronic functions. For example, an additional  component with a finite nuclear amplitude multiplier which falls rapidly enough in order to minimize the total energy a bit better than the solely  components. Such an additional term contributes to the relaxation of the traditional zero nuclear amplitude at linearity, as is supported by CEI measurements. At very low temperatures, the density of linear conformations of nonRenner-Teller species may acquire a maximum. This is consistent with the  nature of the traditional solution. Many times, a single PES with a minimum at linearity is a sufficient approximation for both the (single ground state) wavefunction and eigenenergy. However, when the temperature is increased, nonzero vibrational angular momenta are excited. Again, in a single PES those states must have a strictly zero nuclear amplitude at linearity due to an effective centrifugal barrier. The first term in Eq. (11) nonetheless allows any state to be a superposition of m components where pure m : 0 must be correlated with linear conformations. Therefore, at elevated temperatures (by the approximated formulation given here) less inhibition is expected at linearity as compared with the traditional theory. This is confirmed by CEI measurements of NH> and ND> species at   300 °K. Moreover, it is predicted that the discrepancy between the traditional theory and experiment will vanish at low temperatures for these species and will be enhanced for the CH> species.  Without ignoring the last term in Eq. (11), however, it is clear that the adiabatic wavefunction predictions are invalid for linear conformations and, therefore, the CEI results for that region may differ from the traditional theoretical scenario.

V. Conclusions It was shown that for three triatomic species the probability for finding linear conformations is significantly larger than was theoretically expected. It is suspected that the strict singularity of the nuclear centrifugal barrier in the theoretically used PES is the source of the problem. Theoretical arguments to support this are conveyed. As concerns the chemical reactivity of such species, it is not overly difficult to imagine a chemical reaction wherein the reaction rate depends exponentially on amplitude at linearity. For species such as CH>, large  discrepancies are expected in reaction rate estimates due to he difference between the results of CEI and the traditional adiabatic approximation. The unique advantages of direct observation on molecular structures that is permitted by the CEI method is clarified by new developments in the

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experimental technique. It is foreseen that, in the near future, new concepts will greatly simplify and add reliability to such experiments. Such results are of importance in understanding better the structure-reactivity relation of molecules and radicals.

VI. References Algranati, M., Feldman, H., Kella, D., Malkin, E., Miklazky, E., Naaman, R., Vager, Z., and Zajfman, J. (1989). J. Chem. Phys. 90:4617. Baer, A. (2000). PhD thesis, Weizmann Institute. Baer, A. et al. (1999). Phys. Rev. A 59:1865. Belkacem, A., Faibis, A., Kanter, E. P., Koenig, W., Mitchell, R. E., Vager, Z., and Zabransky, B. J. (1990). Rev. Sci. Instrum. 61:945. Ben-Hamu, D., Baer, A., Feldman, H., Levin, J., Heber, O., Amitay, Z., Vager, Z., and Zajfman, D. (1997). Phys. Rev. A 56:4786. Bohr, N. (1948). K. Dan V idensk. Selsk. Math.-Fys. Medd. 18:8. Both, G., Kanter, E. P., Vager, Z., Zabransky, B. J., and Zajfman, D. (1987). Rev. Sci. Instrum. 58:424. Breskin, A. and Chechik, R. (1985). IEEE Trans. Nucl. Sci. N2-32:504 and references therein. Faibis, A., Koenig, W., Kanter, E. P., and Vager, Z. (1986). Nucl. Instrum. Methods B 13:673. Gemmell, D. S. and Vager, Z. (1985). The electronic polarization induced in solids traversed by fast ions, in T reatise on Heavy-Ion Science, vol. 6, D. A. Bromley, ed., New York: Plenum, p. 243 and references therein. Graber, T., Kanter, E., Levin, J., Zajfman, D., Vager, Z., and Naaman, R. (1997). Phys. Rev. A 56:2600. Graber, T., Zajfman, D., Kanter, E. P., Vager, Z., Naaman, R., and Zabransky, B. J. (1992). Rev. Instr. 63:3569. Heber, O., Zajfman, D., and Vager, Z. (1999). Israeli patent 123747 approved 1999. Jensen, P., Brumm, M., Kraemer, W. P., and Bunker, P. R. (1995a). J. Mol. Spectrosc. 171:31. Jensen, P., Brumm, M., Kraemer, W. P., and Bunker, P. R. (1995b). J. Mol. Spectrosc. 172:194. Jensen, P. and Bunker, P. R. (1999). Private communication. Kella, D., Algranati, M., Feldman, H., Heber, O., Kovner, H., Malkin, E., Miklazky, E., Naaman, R., Zajfman, D., Zajfman, J., and Vager, Z. (1993). Nuclear Instrum. Method A 329:440. Koenig, W., Faibis, A., Kanter, E. P., Vager, Z., and Zabransky, B. (1985). Nucl. Instrum. Methods B 10:259. Kovner, H., Faibis, A., Vager, Z., and Naaman, R. (1988). Proc. Int. Workshop on the Structure of Small Molecules and Ions, R. Naaman and Z. Vager, eds., New York: Plenum Press, p. 113. Kraemer, W. P., Jensen, P., and Bunker, P. R. (1994). Can. J. Phys. 72:871. Krohn, S., Amitay, Z., Baer, A., Levin, J., Zajfman, D., Lange, M., Knoll, L., Schwalm, D., Wester, R., and Wolf, A. (2000). Accepted for publication in Phys. Rev. A. Levin, J. (1997). PhD thesis, Weizmann Inst. of Sci. Levin, J., Baer, A., Knoll, L., Scheffel, M., Schwalm, D., Vager, Z., Wester, R., Wolf, A., and Zajfman, D. (2000). Nucl. Instrum. Meth. B. 268:168. Levin, J., Feldman, H., Baer, A., Ben-Hamu, D., Heber, O., Zajfman, D., and Vager, Z. (1998). Phys. Rev. L ett. 81:3347.

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Marx, D. and Parrinello, M. (1996). Science 271:179. Oka, T. (1993). Private communication. Okumura, M., Gabrys, B. D., Jagod, M. F., and Oka, T. (1972). J. Mol. Spectrosc. 153:738. Osmann, G., Bunker, P. R., Jensen, P., and Kramer, W. P. (1997a). Chem. Phys. 225:33. Osmann, G., Bunker, P. R., Jensen, P., and Kramer, W. P. (1997b). J. Mol. Spectrosc. 186:319. Osmann, G., Bunker, P. R., Kraemer, W. P., and Jensen, P. (1999). Chem. Phys. L ett. 309:299. Ro¨sslein, M., Gabrys, C. M., Jagod, M.-F., and Oka, T. (1992). J. Mol. Spectrosc. 153:738. See, for example, Schiff, (1949). Quantum Mechanics, McGraw-Hill. Strasser, D., Urbain, X., Pedersen, H. B., Altstein, N., Heber, O., Wester, R., Bhushan, K. G., and Zajfman, D. (2000). Rev. Sci. Instruments. 71: (August). Tanabe, T. et al. (1999). Phys. Rev. L ett. 83:2163. Vager, Z. and Gemmell, D. S. (1976). Phys. Rev. L ett. 37:1352. Vager, Z., Graber, T., Kanter, E. P., and Zajfman, D. (1993). Phys. Rev. L ett. 70:3549. Vager, Z., Naaman, R., and Kanter, E. P. (1989). Science 244:426. Vager, Z., Zajfman, D., Graber, T., and Kanter, E. P. (1993). Phys. Rev. L ett. 71:4319. Wester, R., Albrecht, F., Baer, A., Grieser, M., Knoll, L., Levin, J., Repnow, R., Schwalm, D., Vager, Z., Wolf, A., and Zajfman, D. (1998). Nucl. Instrum. Methods Phys. Res A 413:739. Zajfman, D., Both, G., Kanter, E. P., and Vager, Z. (1990). Phys. Rev. A 41:2482. Zajfman, D., Kanter, E. P., Graber, T., and Vager, Z. (1992). Phys. Rev. A 46:194.

a This Page Intentionally Left Blank

Index

B

A Airspace imaging, 77—78 Alkali clusters, 180—186 Angular correlation experiments Breit-Teller effects, 123—127 Einstein-Podolsky-Rosen tests, 111— 118 Garuccio-Selleri effects, 118—119 Antihydrogen advances, 31—32 description, 28—29 4.2 K positrons, 29—30 nested penning trap, 30—31 recombination mechanisms formation processes, 35 future studies, 36 selecting processes, 30—31 Antiprotons capturing, 9—16 cooling, 9—16 decelerator, 16—17 description, 2 energy lowering, 3—4 LEAR, 7—13 lower temperature, 19 protons, comparisons cyclotron frequencies, 23—24 PCT invariance, 19—22 spinoffs, 36—37 TRAP I, 25 TRAP II, 25—26 TRAP III, 26—28 slowing, 5—9 stacking, 16 transporting, 17 TRAP duplications, 17—18 trapping, 5—9 vacuum, 16 Argonne National Laboratory set-up, 207, 211

Back projection imaging, 67—68 Bending distributions, 226—230 Binding electrons, 209—210 Bond angle distributions, 216—220 Breit-Teller effects, 123—127 C Chemical shift imaging, 70—71 Collision processes, 190—192 Coulomb explosion imaging background, 205—206 comparisons, 236—237 dectectors ANL, 211 description, 210—211 future, 212 TSR, 2112 Weizmann, 211—212 fast molecules ANL set-up, 206 future cooling method, 208 Heidelberg set-up, 208 linear molecules, 232—236 quasilinear bending distributions, 226—230 binding electron stripping, 209— 210 bond angle distributions, 216— 220 data/theory, comparison, 230— 232 description, 204—205 diatomic molecules, 222—224 framework, 212—213 molecular structure, 205 reorientation, 224—226 species, 220—222 V-space, 213—215 241

242

INDEX

Coulomb explosion imaging (Contd.) Weizmann set-up, 207—208 Cyclotron frequencies, 23—24

D Diatomic molecules, 222—224 Distributions bending, 226—230 bond angle, 216—220

E Einstein-Podolsky-Rosen tests, 111—118 Electronic states excited laser spectroscopy description, 171—172 fluorescence, 172—173 ionization, 172—173 optical double resonance, 173— 180 ground, high vibrational laser spectroscopy fluorescence, 160—167 stimulated emission pumping, 160—167 overtone spectroscopy description, 152—153 modulated absorption, 153—157 optothermal, 153—157 Electrons. see Binding electrons Emission pumping, stimulated, 167—171 Excitation, selective, 67

F Fast molecules coulomb explosion imaging ANL set-up, 206 future cooling method, 208 Heidelberg set-up, 208 Weizmann set-up, 207—208 Fluorescence, electronic states excited, 172—173 high vibrational, 160—167 Fourier spectral analysis, 127—133

G Gradient echo imaging, 68—70 Gyroscopes, nuclear spin, 48 H Heidelberg set-up, 207—208 Helium, He discovery, 45 polarization deliver systems, 62—63 description, 49—51 imaging carrier injection, 79—80 considerations, 62—63 future studies, 87—89 metastability exchange description, 60 laser, 62 methods, 48—49 next generation, 47 optical pumping description, 50—56 LDA, 57—60 spin exchange description, 50—56 LDA, 57 problems, 46 Hydrogen, atomic. see also Antihydrogen H\ ion, 26—28 role, 100 two-photon decay coherence, 127—133 correlated emissions, 133—143 fourier spectral analysis, 127—133 metastable state, 101—108 polarization correlation Breit-Teller effects, 123—127 Einstein-Podolsky-Rosen test, 101—108 three polarizers, 119—123 two polarizers, 111—118 time correlation, 133 I Invariance, PCT, 19—22 Ionization, two-photon, 172—173

243

INDEX L Laser spectroscopy advantages, 150 excited molecular states description, 171—172 fluorescence, 172—173 ionization, 172—173 optical double resonance, 173—180 high vibrational ground states fluorescence, 160—167 overtone description, 152—153 modulated absorption, 153—157 optothermal, 158—160 stimulated emission pumping, 167— 171 history, 150—151 metastability exchange, 62 optical pumping, 57—60 polarization in vivo, 86—87 sub-doppler, 180—186 time-resolved coherent control, 194—196 collision processes, 190—192 lifetime measurements, 187—190 photodissociation processes, 192— 193 LEAR, 7—13 Lifetime measurements, 187—190 Linear molecules, 232—236 Lung function, 83—84

M Magnetic resonance images basic techniques contrast, 71—72 -space, 65—66 low field, 72—73 NMR, 63—64 one-dimensional, 64—65 sequences back projection, 67—68 chemical shift, 70—71 gradient echo, 68—70 selective excitation, 67 description, 42—43 development, 43—44

future possibilities, 87—89 history, 44—49 polarized gas, 75—77 Xe framework, 80 lung function, 83—84 time dependence, 84—86 tracer techniques, 84—86 in vivo, 81—83 Magnetic tracer techniques, 84—86 Magnetization, sampling, 75—76 MRI. see Magnetic resonance images N Nested penning trap demonstrating, 30—31 selecting process, 32—34 NMR. see Nuclear magnetic resonance Nuclear magnetic resonance description, 63—64 Xe framework, 80 lung function, 83—84 time dependence, 84—86 tracer techniques, 84—86 in vivo, 81—83 Nuclear polarization airspace imaging, 77—78 He deliver systems, 62—63 description, 49—51 imaging carrier injection, 79—80 considerations, 74—75 future studies, 87—89 metastability exchange description, 60—62 laser, 62 methods, 48—49 next generation, 47 optical pumping description, 50—56 LDA, 57—60 spin exchange description, 50—56 LDA, 57 MRI imaging, 75—76 Xe deliver systems, 62—63

244

INDEX

Nuclear polarization (Contd.) imaging carrier injection, 79—80 considerations, 74—75 future studies, 87—89 in vivo, 86 metastability exchange description, 60 laser, 62 methods, 48—49 optical pumping description, 50—56 LDA, 57—60 spin exchange description, 50—56 LDA, 57 Nuclear spin gyroscopes, 48 O One-dimensional imaging, 64—65 Optical double resonance techniques, 173—180 Optical pumping description, 50—56 LDA, 57—60 Overtone spectroscopy, 152—153 P PCT invariance, 19—22 Penning trap, 37 Photodissociation processes, 192—193 Polarization correlation experiments Breit-Teller effects, 123—127 Einstein-Podolsky-Rosen tests, 111—118 Garuccio-Selleri effects, 118—119 three polarizers, 119—123 two polarizers, 111—118 nuclear (see Nuclear polarization) Positrons, 4.2K, 29—30 Protons antiprotons, comparisons cyclotron frequencies, 23—24 PCT invariance, 19—22 spinoffs, 36—37 TRAP I, 25

TRAP II, 25—26 TRAP III, 26—28 Q Quasilinear molecules, XH>  bond angle distributions, 216—220 production, 212—213 species, 220—222 V-space bending information, 214—215 conversion, 212—213 R Radiation, two-photon atomic hydrogen, 101—108 coherence, 127—133 correlation experiments Breit-Teller effects, 123—127 Einstein-Podolsky-Rosen tests, 111—118 emissions, 133—143 Garuccio-Selleri effects, 118—119 three polarizers, 119—123 two polarizers, 111—118 fourier spectral analysis, 127—133 function, 100—101 stirling apparatus, 109—110 time correlation, 133 Resonance optical double, 173—180 two-photon, 173—180 S Small molecules alkali clusters, 180—186 electronically excited description, 171—172 fluorescence, 172—173 ionization, 172—173 optical double resonance, 173—180 properties, 149—150 high vibrational states fluorescence spectroscopy, 160—167 overtone spectroscopy description, 152—153 modulated absorption, 153—157

245

INDEX optothermal, 158—160 sub-doppler spectroscopy, 180—186 time-resolved coherent control, 194—196 collision processes, 190—192 lifetime measurements, 187—190 photodissociation processes, 192— 193 Spin exchange description, 50—56 laser, 57 Sub-doppler spectroscopy, 180—186 T Time depencence, 84—86 TRAP duplications, 17—18 proton/antiproton, 25—28 type I, 25 type II, 25—26 type III, 26—28 W Wake effects diatomic molecules, 222—224 quasilinear molecules, 216—220 Wave functions, 232—236 Weizmann set-up, 207—208, 212

X Xenon, Xe dissolved imaging framework, 80 lung function, 83—84 time dependence, 84—86 tracer techniques, 84—86 in vivo, 81—83 polarization delivery systems, 62—63 description, 49—51 imaging carrier injection, 79—80 considerations, 62—63 future studies, 87—89 in vivo, 86—87 metastability exchange description, 60 laser, 62 optical pumping description, 49—51 LDA, 57—60 spin exchange description, 50—56 LDA, 57

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Contents of Volumes in This Serial Volume 1 Molecular Orbital Theory of the Spin Properties of Conjugated Molecules, G. G. Hall and A. T. Amos

Volume 3 The Quantal Calculation of Photoionization Cross Sections, A. L . Stewart Radiofrequency Spectroscopy of Stored Ions I: Storage, H. G. Dehmelt

Electron Affinities of Atoms and Molecules, B. L. Moiseiwitsch

Optical Pumping Methods in Atomic Spectroscopy, B. Budick

Atomic Rearrangement Collisions, B. H. Bransden

Energy Transfer in Organic Molecular Crystals: A Survey of Experiments, H. C. Wolf

The Production of Rotational and Vibrational Transitions in Encounters between Molecules, K. Takayanagi

Atomic and Molecular Scattering from Solid Surfaces, Robert E. Stickney

The Study of Intermolecular Potentials with Molecular Beams at Thermal Energies, H. Pauly and J. P. Toennies

Quantum Mechanics in Gas Crystal-Surface van der Waals Scattering, E. Chanoch Beder

High-Intensity and High-Energy Molecular Beams, J. B. Anderson, R. P. Andres, and J. B. Fen

Reactive Collisions between Gas and Surface Atoms, Henry Wise and Bernard J. Wood

Volume 2 The Calculation of van der Waals Interactions, A. Dalgarno and W. D. Davison Thermal Diffusion in Gases, E. A. Mason, R. J. Munn, and Francis J. Smith Spectroscopy in the Vacuum Ultraviolet, W. R. S. Garton The Measurement of the Photoionization Cross Sections of the Atomic Gases, James A. R. Samson The Theory of Electron-Atom Collisions, R. Peterkop and V. Veldre Experimental Studies of Excitation in Collisions between Atomic and Ionic Systems, F. J. de Heer Mass Spectrometry of Free Radicals, S. N. Foner

Volume 4 H. S. W. Massey — A Sixtieth Birthday Tribute, E. H. S. Burhop Electronic Eigenenergies of the Hydrogen Molecular Ion, D. R. Bates and R. H. G. Reid Applications of Quantum Theory to the Viscosity of Dilute Gases, R. A. Buckingham and E. Gal Positrons and Positronium in Gases, P. A. Fraser Classical Theory of Atomic Scattering, A. Burgess and I. C. Percival Born Expansions, A. R. Holt and B. L . Moiselwitsch Resonances in Electron Scattering by Atoms and Molecules, P. G. Burke Relativistic Inner Shell Ionizations, C. B. O. Mohr 247

248

CONTENTS OF VOLUMES IN THIS SERIAL

Recent Measurements on Charge Transfer, J. B. Hasted Measurements of Electron Excitation Functions, D. W. O. Heddle and R. G. W. Keesing Some New Experimental Methods in Collision Physics, R. F. Stebbings Atomic Collision Processes in Gaseous Nebulae, M. J. Seaton

The Diffusion of Atoms and Molecules, E. A. Mason and T. R. Marrero Theory and Application of Sturmian Functions, Manuel Rotenberg Use of Classical Mechanics in the Treatment of Collisions between Massive Systems, D. R. Bates and A. E. Kingston

Collisions in the Ionosphere, A. Dalgarno

Volume 7

The Direct Study of lonization in Space, R. L. F. Boyd

Physics of the Hydrogen Master, C. Audoin, J. P. Schermann, and P. Grivet Molecular Wave Functions: Calculations and Use in Atomic and Molecular Processes, J. C. Browne Localized Molecular Orbitals, Harel Weinstein, Ruben Pauncz, and Maurice Cohen General Theory of Spin-Coupled Wave Functions for Atoms and Molecules, J. Gerratt Diabatic States of Molecules — QuasiStationary Electronic States, Thomas F. O’Malley Selection Rules within Atomic Shells, B. R. Judd Green’s Function Technique in Atomic and Molecular Physics, Gy. Csanak, H. S. Taylor, and Robert Yaris A Review of Pseudo-Potentials with Emphasis on Their Application to Liquid Metals, Nathan Wiser and A. J. Greenfield

Volume 5 Flowing Afterglow Measurements of IonNeutral Reactions, E. E. Ferguson, F. C. Fehsenfeld, and A. L. Schmeltekopf Experiments with Merging Beams, Roy H. Neynaber Radiofrequency Spectroscopy of Stored Ions II: Spectroscopy, H. G. Dehmelt The Spectra of Molecular Solids, O. Schnepp The Meaning of Collision Broadening of Spectral Lines: The Classical Oscillator Analog, A. Ben-Reuven The Calculation of Atomic Transition Probabilities, R. J. S. Crossley Tables of One- and Two-Particle Coefficients of Fractional Parentage for Configurations s s p , C. D. H. Chisholm, A. Dalgarno, N  and E. R. Innes Relativistic Z-Dependent Corrections to Atomic Energy Levels, Holly Thomis Doyle

Volume 6 Dissociative Recombination, J. N. Bardsley and M. A. Biondi Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines, A. S. Kaufman The Rotational Excitation of Molecules by Slow Electrons, Kazuo Takayanagi and Yukikazu Itikawa

Volume 8 Interstellar Molecules: Their Formation and Destruction, D. McNally Monte Carlo Trajectory Calculations of Atomic and Molecular Excitation in Thermal Systems, James C. Keck Nonrelativistic Off-Shell Two-Body Coulomb Amplitudes, Joseph C. Y. Chen and Augustine C. Chen Photoionization with Molecular Beams, R. B. Cairns, Halstead Harrison, and R. I. Schoen

CONTENTS OF VOLUMES IN THIS SERIAL The Auger Effect, E. H. S. Burhop and W. N. Asaad

Volume 9 Correlation in Excited States of Atoms, A. W. Weiss The Calculation of Electron—Atom Excitation Cross Sections, M. R. H. Rudge Collision-Induced Transitions between Rotational Levels, Takeshi Oka The Differential Cross Section of Low-Energy Electron—Atom Collisions, D. Andrick Molecular Beam Electric Resonance Spectroscopy, Jens C. Zorn and Thomas C. English Atomic and Molecular Processes in the Martian Atmosphere, Michael B. McElroy

Volume 10 Relativistic Effects in the Many-Electron Atom, Lloyd Annstrong, Jr. and Serge Feneuille The First Born Approximation, K. L. Bell and A. K. Kingston Photoelectron Spectroscopy, W. C. Price Dye Lasers in Atomic Spectroscopy, W. Lange, J. Luther, and A. Steudel Recent Progress in the Classification of the Spectra of Highly Ionized Atoms, B. C. Fawcett A Review of Jovian Ionospheric Chemistry, Wesley T. Huntress, Jr.

Volume 11 The Theory of Collisions between Charged Particles and Highly Excited Atoms, I. C. Percival and D. Richards Electron Impact Excitation of Positive Ions, M. J. Seaton The R-Matrix Theory of Atomic Process, P. G. Burke and W. D. Robb Role of Energy in Reactive Molecular

249

Scattering: An Information-Theoretic Approach, R. B. Bernstein and R. D. Levine Inner Shell Ionization by Incident Nuclei, Johannes M. Hansteen Stark Broadening, Hans R. Griem Chemiluminescence in Gases, M. F. Golde and B. A. Thrush

Volume 12 Nonadiabatic Transitions between Ionic and Covalent States, R. K. Janev Recent Progress in the Theory of Atomic Isotope Shift, J. Bauche and R.-J. Champeau Topics on Multiphoton Processes in Atoms, P. Lambropoulos Optical Pumping of Molecules, M. Broyer, G. Goudedard, J. C. Lehmann, and J. Vigué Highly Ionized Ions, Ivan A. Sellin Time-of-Flight Scattering Spectroscopy, Wilhelm Raith Ion Chemistry in the D Region, George C. Reid

Volume 13 Atomic and Molecular Polarizabilities — A Review of Recent Advances, Thomas M. Miller and Benjamin Bederson Study of Collisions by Laser Spectroscopy, Paul R. Berman Collision Experiments with Laser-Excited Atoms in Crossed Beams, I.V. Hertel and W. Stoll Scattering Studies of Rotational and Vibrational Excitation of Molecules, Manfred Faubel and J. Peter Toennies Low-Energy Electron Scattering by Complex Atoms: Theory and Calculations, R. K. Nesbet Microwave Transitions of Interstellar Atoms and Molecules, W. B. Somerville

250

CONTENTS OF VOLUMES IN THIS SERIAL

Volume 14 Resonances in Electron Atom and Molecule Scattering, D. E. Golden The Accurate Calculation of Atomic Properties by Numerical Methods, Brian C. Webster, Michael J. Jamieson, and Ronald E. Stewart (e, 2e) Collisions, Erich Weigold and Ian E. McCarthy Forbidden Transitions in One- and TwoElectron Atoms, Richard Marrus and Peter J. Mohr Semiclassical Effects in Heavy-Particle Collisions, M. S. Child Atomic Physics Tests of the Basic Concepts in Quantum Mechanics, Francis M. Pipkin Quasi-Molecular Interference Effects in Ion— Atom Collisions, S. V. Bobashev Rydberg Atoms, S. A. Edelstein and T. F. Gallagher UV and X-Ray Spectroscopy in Astrophysics, A. K. Dupree

Volume 15 Negative Ions, H. S. W. Massey

Inner-Shell Ionization, E. H. S. Burhop Excitation of Atoms by Electron Impact, D. W. O. Heddle Coherence and Correlation in Atomic Collisions, H. Kleinpoppen Theory of Low Energy Electron-Molecule Collisions, P. G. Burke

Volume 16 Atomic Hartree—Fock Theory, M. Cohen and R. P. McEachran Experiments and Model Calculations to Determine Interatomic Potentials, R. Düren Sources of Polarized Electrons, R. J. Celotta and D. T. Pierce Theory of Atomic Processes in Strong Resonant Electromagnetic Fields, S. Swain Spectroscopy of Laser-Produced Plasmas, M. H. Key and R. J. Hutcheon Relativistic Effects in Atomic Collisions Theory, B. L. Moiseiwitsch Parity Nonconservation in Atoms: Status of Theory and Experiment, E. N. Fortson and L. Wilets

Atomic Physics from Atmospheric and Astrophysical Studies, A. Dalgarno Collisions of Highly Excited Atoms, R. F. Stebbings

Volume 17

Theoretical Aspects of Positron Collisions in Gases, J. W. Humberston

Collective Effects in Photoionization of Atoms, M. Ya. Amusia

Experimental Aspects of Positron Collisions in Gases, T. C. Griffith

Nonadiabatic Charge Transfer, D. S. F. Crothers

Reactive Scattering: Recent Advances in Theory and Experiment, Richard B. Bernstein

Atomic Rydberg States, Serge Feneuille and Pierre Jacquinot

Ion—Atom Charge Transfer Collisions at Low Energies, J. B. Hasted Aspects of Recombination, D. R. Bates The Theory of Fast Heavy Particle Collisions, B. H. Bransden Atomic Collision Processes in Controlled Thermonuclear Fusion Research, H. B. Gilbody

Superfluorescence, M. F. H. Schuurmans, Q. H. F. Vrehen, D. Polder, and H. M. Gibbs Applications of Resonance Ionization Spectroscopy in Atomic and Molecular Physics, M. G. Payne, C. H. Chen, G. S. Hurst, and G. W. Foltz Inner-Shell Vacancy Production in Ion— Atom Collisions, C. D. Lin and Patrick Richard

CONTENTS OF VOLUMES IN THIS SERIAL Atomic Processes in the Sun, P. L. Dufton and A. E. Kingston

Volume 18 Theory of Electron—Atom Scattering in a Radiation Field, Leonard Rosenberg Positron—Gas Scattering Experiments, Talbert S. Stein and Walter E. Kauppila Nonresonant Multiphoton Ionization of Atoms, J. Morellec, D. Normand, and G. Petite Classical and Semiclassical Methods in Inelastic Heavy-Particle Collisions, A. S. Dickinson and D. Richards Recent Computational Developments in the Use of Complex Scaling in Resonance Phenomena, B. R. Junker Direct Excitation in Atomic Collisions: Studies of Quasi-One-Electron Systems, N. Anderson and S. E. Nielsen Model Potentials in Atomic Structure, A. Hibbert Recent Developments in the Theory of Electron Scattering by Highly Polar Molecules, D. W. Norcross and L. A. Collins Quantum Electrodynamic Effects in FewElectron Atomic Systems, G. W. F. Drake

Volume 19 Electron Capture in Collisions of Hydrogen Atoms with Fully Stripped Ions, B. H. Bransden and R. K. Janev Interactions of Simple Ion—Atom Systems, J. T. Park High-Resolution Spectroscopy of Stored Ions, D. J. Wineland, Wayne M. Itano, and R. S. Van Dyck, Jr. Spin-Dependent Phenomena in Inelastic Electron—Atom Collisions, K. Blum and H. Kleinpoppen The Reduced Potential Curve Method for Diatomic Molecules and Its Applications, E. Jenc

251

The Vibrational Excitation of Molecules by Electron Impact, D. G. Thompson Vibrational and Rotational Excitation in Molecular Collisions, Manfred Faubel Spin Polarization of Atomic and Molecular Photoelectrons, N. A. Cherepkov

Volume 20 Ion—Ion Recombination in an Ambient Gas, D. R. Bates Atomic Charges within Molecules, G. G. Hall Experimental Studies on Cluster Ions, T. D. Mark and A. W. Castleman, Jr. Nuclear Reaction Effects on Atomic InnerShell Ionization, W. E. Meyerhof and J.-F. Chemin Numerical Calculations on Electron-Impact Ionization, Christopher Bottcher Electron and Ion Mobilities, Gordon R. Freeman and David A. Armstrong On the Problem of Extreme UV and X-Ray Lasers, I. L. Sobel’man and A. V. Vinogradov Radiative Properties of Rydberg State, in Resonant Cavities, S. Haroche and J. M. Ralmond Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction — Rydberg Molecules, J. A. C. Gallas, G. Leuchs, H. Walther, and H. Figger

Volume 21 Subnatural Linewidths in Atomic Spectroscopy, Dennis P. O’Brien, Pierre Meystre, and Herbert Walther Molecular Applications of Quantum Defect Theory, Chris H. Greene and Ch. Jungen Theory of Dielectronic Recombination, Yukap Hahn Recent Developments in Semiclassical Floquet Theories for Intense-Field Multiphoton Processes, Shih-I Chu

252

CONTENTS OF VOLUMES IN THIS SERIAL

Scattering in Strong Magnetic Fields, M. R. C. McDowell and M. Zarcone

Volume 24

Pressure Ionization, Resonances, and the Continuity of Bound and Free States, R. M. More

The Selected Ion Flow Tube (SIDT): Studies of Ion—Neutral Reactions, D. Smith and N. G. Adams Near-Threshold Electron—Molecule Scattering, Michael A. Morrison

Volume 22 Positronium — Its Formation and Interaction with Simple Systems, J. W. Humberston Experimental Aspects of Positron and Positronium Physics, T. C. Griffith

Angular Correlation in Multiphoton Ionization of Atoms, S. J. Smith and G. Leuchs Optical Pumping and Spin Exchange in Gas Cells, R. J. Knize, Z. Wu, and W. Happer Correlations in Electron—Atom Scattering, A. Crowe

Doubly Excited States, Including New Classification Schemes, C. D. Lin Measurements of Charge Transfer and Ionization in Collisions Involving Hydrogen Atoms, H. B. Gilbody Electron—Ion and Ion—Ion Collisions with Intersecting Beams, K. Dolder and B. Pearl Electron Capture by Simple Ions, Edward Pollack and Yukap Hahn Relativistic Heavy-Ion—Atom Collisions, R. Anholt and Harvey Gould Continued-Fraction Methods in Atomic Physics, S. Swain

Volume 23 Vacuum Ultraviolet Laser Spectroscopy of Small Molecules, C. R. Vidal Foundations of the Relativistic Theory of Atomic and Molecular Structure, Ian P. Grant and Harry M. Quiney Point-Charge Models for Molecules Derived from Least-Squares Fitting of the Electric Potential, D. E. Williams and Ji-Min Yan Transition Arrays in the Spectra of Ionized Atoms, J. Bauche, C. Bauche-Arnoult, and M. Klapisch Photoionization and Collisional Ionization of Excited Atoms Using Synchroton and Laser Radiation, E. J. Wuilleumier; D. L. Ederer, and J. L. Picqué

Volume 25 Alexander Dalgarno: Life and Personality, David R. Bates and George A. Victor Alexander Dalgarno: Contributions to Atomic and Molecular Physics, Neal Lane Alexander Dalgarno: Contributions to Aeronomy, Michael B. McElroy Alexander Dalgarno: Contributions to Astrophysics, David A. Williams Dipole Polarizability Measurements, Thomas M. Miller and Benjamin Bederson Flow Tube Studies of Ion—Molecule Reactions, Eldon Ferguson Differential Scattering in He—He and He>— He Collisions at KeV Energies, R. F. Stebbings Atomic Excitation in Dense Plasmas, Jon C. Weisheit Pressure Broadening and Laser-Induced Spectral Line Shapes, Kenneth M. Sando and Shih-I Chu Model-Potential Methods, G. Laughlin and G. A. Victor Z-Expansion Methods, M. Cohen Schwinger Variational Methods, Deborah Kay Watson Fine-Structure Transitions in Proton-Ion Collisions, R. H. G. Reid

CONTENTS OF VOLUMES IN THIS SERIAL

253

Electron Impact Excitation, R. J. W. Henry and A. E. Kingston

Volume 27

Recent Advances in the Numerical Calculation of Ionization Amplitudes, Christopher Bottcher

Negative Ions: Structure and Spectra, David R. Bates

The Numerical Solution of the Equations of Molecular Scattering, A. C. Allison High Energy Charge Transfer, B. H. Bransden and D. P. Dewangan Relativistic Random-Phase Approximation, W. R. Johnson Relativistic Sturmian and Finite Basis Set Methods in Atomic Physics, G. W. F. Drake and S. P. Goldman Dissociation Dynamics of Polyatomic Molecules, T. Uzer

Electron Polarization Phenomena in Electron—Atom Collisions, Joachim Kessler Electron—Atom Scattering, I. E. McCarthy and E. Weigold Electron—Atom Ionization, I. E. McCarthy and E. Weigold Role of Autoionizing States in Multiphoton Ionization of Complex Atoms, V. I. Lengyel and M. I. Haysak Multiphoton Ionization of Atomic Hydrogen Using Perturbation Theory, E. Karule

Photodissociation Processes in Diatomic Molecules of Astrophysical Interest, Kate P. Kirby and Ewine F. van Dishoeck

Volume 28

The Abundances and Excitation of Interstellar Molecules, John. H. Black

The Theory of Fast Ion-Atom Collisions, J. S. Briggs and J. H. Macek

Volume 26 Comparisons of Positrons and Electron Scattering by Gases, Walter E. Kauppila and Talbert S. Stein

Some Recent Developments in the Fundamental Theory of Light, Peter W. Milonni and Surendra Singh Squeezed States of the Radiation Field, Khalid Zaheer and M. Suhail Zubairy Cavity Quantum, Electrodynamics, E. A. Hinds

Electron Capture at Relativistic Energies, B. L. Moiseiwitsch The Low-Energy, Heavy Particle Collisions — A Close-Coupling Treatment, Mineo Kimura and Neal F. Lane Vibronic Phenomena in Collisions of Atomic and Molecular Species, V. Sidis Associative Ionization: Experiments, Potentials, and Dynamics, John Weiner, Françoise Masnou-Sweeuws, and Annick Giusti-Suzor On the  Decay of Re: An Interface of Atomic and Nuclear Physics and Cosmochronology, Zonghau Chen, Leonard Rosenberg, and Larry Spruch Progress in Low Pressure Mercury-Rare Gas Discharge Research, J. Maya and R. Lagushenko

Volume 29 Studies of Electron Excitation of Rare-Gas Atoms into and out of Metastable Levels Using Optical and Laser Techniques, Chun C. Lin and L. W. Anderson Cross Sections for Direct Multiphoton Ionization of Atoms, M. V. Ammosov, N. B. Delone, M. Yu. Ivanov, I. I. Bondar, and A. V. Masalov Collision-Induced Coherences in Optical Physics, G. S. Agarwal Muon-Catalyzed Fusion, Johann Rafelski and Helga E Rafelski Cooperative Effects in Atomic Physics, J. P. Connerade

254

CONTENTS OF VOLUMES IN THIS SERIAL

Multiple Electron Excitation, Ionization, and Transfer in High-Velocity Atomic and Molecular Collisions, J. H. McGuire

Volume 30 Differential Cross Sections for Excitation of Helium Atoms and Helium-Like Ions by Electron Impact, Shinobu Nakazaki Cross-Section Measurements for Electron Impact on Excited Atomic Species, S. Trajmar and J. C. Nickel The Dissociative Ionization of Simple Molecules by Fast Ions, Colin J. Latimer Theory of Collisions between Laser Cooled Atoms, P. S. Julienne, A. M. Smith, and K. Burnett Light-Induced Drift, E. R. Eliel Continuum Distorted Wave Methods in Ion— Atom Collisions, Derrick S. F. Crothers and Louis J. Dubé

Volume 31 Energies and Asymptotic Analysis for Helium Rydberg States, G. W. F. Drake Spectroscopy of Trapped Ions, R. C. Thompson Phase Transitions of Stored Laser-Cooled Ions, H. Walther Selection of Electronic States in Atomic Beams with Lasers, Jacques Baudon, Rudolf Düren, and Jacques Robert Atomic Physics and Non-Maxwellian Plasmas, Michèle Lamoureux

Volume 32 Photoionization of Atomic Oxygen and Atomic Nitrogen, K. L. Bell and A. E. Kingston Positronium Formation by Positron Impact on Atoms at Intermediate Energies, B. H. Bransden and C. J. Noble

Electron—Atom Scattering Theory and Calculations, P. G. Burke Terrestrial and Extraterrestrial H >,  Alexander Dalgarno Indirect Ionization of Positive Atomic Ions, K Dolder Quantum Defect Theory and Analysis of High-Precision Helium Term Energies, G. W. F. Drake Electron—lon and Ion—Ion Recombination Processes, M. R. Flannery Studies of State-Selective Electron Capture in Atomic Hydrogen by Translational Energy Spectroscopy, H. B. Gilbody Relativistic Electronic Structure of Atoms and Molecules, I. P. Grant The Chemistry of Stellar Environments, D. A. Howe, J. M. C. Rawlings, and D. A. Williams Positron and Positronium Scattering at Low Energies, J. W. Humberston How Perfect are Complete Atomic Collision Experiments?, H. Kleinpoppen and H. Handy Adiabatic Expansions and Nonadiabatic Effects, R. McCarroll and D. S. F. Crothers Electron Capture to the Continuum, B. L Moiseiwitsch How Opaque Is a Star? M. J. Seaton Studies of Electron Attachment at Thermal Energies Using the Flowing Afterglow— Langmuir Technique, David Smith and Patrik Spanel Exact and Approximate Rate Equations in Atom-Field Interactions, S. Swain Atoms in Cavities and Traps, H. Walther Some Recent Advances in Electron-Impact Excitation of n : 3 States of Atomic Hydrogen and Helium, J. F. Williams and J. B. Wang

Volume 33 Principles and Methods for Measurement of Electron Impact Excitation Cross Sections for Atoms and Molecules by Optical

CONTENTS OF VOLUMES IN THIS SERIAL Techniques, A. R. Filippelli, Chun C. Lin, L. W. Andersen, and J. W. McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Analysis of Scattered Electrons, S. Trajmar and J. W. McConkey Benchmark Measurements of Cross Sections for Electron Collisions: Electron Swarm Methods, R. W. Crompton

255

Polarization and Orientation Phenomena in Photoionization of Molecules, N. A. Cherepkov Role of Two-Center Electron—Electron Interaction in Projectile Electron Excitation and Loss, E. C. Montenegro, W. E. Meyerhof, and J. H. McGuire Indirect Processes in Electron Impact Ionization of Positive Ions, D. L. Moores and K. J. Reed

Some Benchmark Measurements of Cross Sections for Collisions of Simple Heavy Particles, H. B. Gilbody

Dissociative Recombination: Crossing and Tunneling Modes, David R. Bates

The Role of Theory in the Evaluation and Interpretation of Cross-Section Data, Barry I. Schneider

Volume 35

Analytic Representation of Cross-Section Data, Mitio Inokuti, Mineo Kimura, M. A. Dillon, and Isao Shimamura

Laser Manipulation of Atoms, K. Sengstock and W. Ertmer

Electron Collisions with N , O and O: What   We Do and Do Not Know, Yukikazu Itikawa

Advances in Ultracold Collisions: Experiment and Theory, J. Weiner Ionization Dynamics in Strong Laser Fields, L. F. DiMauro and P. Agostini

Need for Cross Sections in Fusion Plasma Research, Hugh P. Summers

Infrared Spectroscopy of Size Selected Molecular Clusters, U. Buck

Need for Cross Sections in Plasma Chemistry, M. Capitelli, R. Celiberto, and M. Cacciatore

Femtosecond Spectroscopy of Molecules and Clusters, T. Baumer and G. Gerber

Guide for Users of Data Resources, Jean W. Gallagher

Calculation of Electron Scattering on Hydrogenic Targets, I. Bray and A. T. Stelbovics

Guide to Bibliographies, Books, Reviews, and Compendia of Data on Atomic Collisions, E. W. McDaniel and E. J. Mansky

Relativistic Calculations of Transition Amplitudes in the Helium Isoelectronic Sequence, W. R. Johnson, D. R. Plante, and J. Sapirstein

Volume 34 Atom Interferometry, C. S. Adams, O. Carnal, and J. Mlynek

Rotational Energy Transfer in Small Polyatomic Molecules, H. O. Everitt and F. C. De Lucia

Optical Tests of Quantum Mechanics, R. Y. Chiao, P G. Kwiat, and A. M. Steinberg

Volume 36

Classical and Quantum Chaos in Atomic Systems, Dominique Delande and Andreas Buchleitner

Complete Experiments in Electron—Atom Collisions, Nils Overgaard Andersen and Klaus Bartschat

Measurements of Collisions between LaserCooled Atoms, Thad Walker and Paul Feng

Stimulated Rayleigh Resonances and RecoilInduced Effects, J.-Y. Courtois and G. Grynberg

The Measurement and Analysis of Electric Fields in Glow Discharge Plasmas, J. E. Lawler and D. A. Doughty

Precision Laser Spectroscopy Using AcoustoOptic Modulators, W. A. van Wijngaarden

256

CONTENTS OF VOLUMES IN THIS SERIAL

Highly Parallel Computational Techniques for Electron—Molecule Collisions, Carl Winstead and Vincent McKoy Quantum Field Theory of Atoms and Photons, Maciej Lewenstein and Li You

Volume 37 Evanescent Light-Wave Atom Mirrors, Resonators, Waveguides, and Traps, Jonathan P. Dowling and Julio GeaBanacloche Optical Lattices, P. S. Jessen and I. H. Deutsch Channeling Heavy Ions through Crystalline Lattices, Herbert F. Krause and Sheldon Datz Evaporative Cooling of Trapped Atoms, Wolfgang Ketterle and N. J. van Druten Nonclassical States of Motion in Ion Traps, J. I. Cirac, A. S. Parkins, R. Blatt, and P. Zoller The Physics of Highly-Charged Heavy Ions Revealed by Storage/Cooler Rings, P. H. Mokler and Th. Stöhlker

Volume 38 Electronic Wavepackets, Robert R. Jones and L. D. Noordam Chiral Effects in Electron Scattering by Molecules, K. Blum and D. G. Thompson Optical and Magneto-Optical Spectroscopy of Point Defects in Condensed Helium, Serguei I. Kanorsky and Antoine Weis Rydberg Ionization: From Field to Photon, G. M. Lankhuijzen and L. D. Noordam Studies of Negative Ions in Storage Rings, L. H. Andersen, T. Andersen, and P. Hvelplund Single-Molecule Spectroscopy and Quantum Optics in Solids, W. E. Moerner, R. M. Dickson, and D. J. Norris

Volume 39 Author and Subject Cumulative Index Volumes 1—38 Author Index Subject Index Appendix: Tables of Contents of Volumes 1—38 and Supplements

Volume 40 Electric Dipole Moments of Leptons, Eugene D. Commins High-Precision Calculations for the Ground and Excited States of the Lithium Atom, Frederick W. King Storage Ring Laser Spectroscopy, Thomas U. Kühl Laser Cooling of Solids, Carl E. Mungan and Timothy R. Gosnell Optical Pattern Formation, L. A. Lugiato, M. Brambilla, and A. Gatti

Volume 41 Two-Photon Entanglement and Quantum Reality, Yanhua Shih Quantum Chaos with Cold Atoms, Mark G. Raizen Study of the Spatial and Temporal Coherence of High-Order Harmonics, Pascal Salie`res, Ann L’Huiller Philippe Antoine, and Maciej Lewenstein Atom Optics in Quantized Light Fields, Matthias Freyburger, Alois M. Herkommer, Daniel S. Krähmer, Erwin Mayr, and Wolfgang P. Schleich Atom Waveguides, Victor I. Balykin Atomic Matter Wave Amplification by Optical Pumping, Ulf Janicke and Martin Wilkens

Volume 42 Fundamental Tests of Quantum Mechanics, Edward S. Fry and Thomas Walther

CONTENTS OF VOLUMES IN THIS SERIAL Wave-Particle Duality in an Atom Interferometer, Stephan Dürr and Gerhard Rempe Atom Holography, Fujio Shimizu Optical Dipole Traps for Neutral Atoms, Rudolf Grimm, Matthias Weidemüller, and Yurii B. Ovchinnikov Formation of Cold (T1K) Molecules, J. T. Bahns, P. L. Gould, and W. C. Stwalley High-Intensity Laser-Atom Physics, C. J. Joachain, M. Dorr, and N. J. Kylstra Coherent Control of Atomic, Molecular, and Electronic Processes, Moshe Shapiro and Paul Brumer

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Molecular Processes in Reactive Plasmas, Yoshihiko Hatano Ion—Molecule Reactions, Werner L indinger, Armin Hansel, and Zdenek Herman Uses of High-Sensitivity White-Light Absorption Spectroscopy in Chemical Vapor Deposition and Plasma Processing, L. W. Anderson, A. N. Goyette, and J. E. L awler Fundamental Processes of Plasma—Surface Interactions, Rainer Hippler Recent Applications of Gaseous Discharges: Dusty Plasmas and Upward-Directed Lightning, Ara Chutjian

Resonant Nonlinear Optics in Phase Coherent Media, M. D. Lukin, P. Hemmer, and M. O. Scully

Opportunities and Challenges for Atomic, Molecular, and Optical Physics in Plasma Chemistry, Kurt Becker, Hans Deutsch, and Mitio Inokuti

The Characterization of Liquid and Solid Surfaces with Metastable Helium Atoms, H. Morgner

Volume 44

Quantum Communication with Entangled Photons, Harald Weinfurter

Volume 43 Plasma Processing of Materials and Atomic, Molecular, and Optical Physics: An Introduction, Hiroshi Tanaka and Mitio Inokuti The Boltzmann Equation and Transport Coefficients of Electrons in Weakly Ionized Plasmas, R. W inkler

Mechanisms of Electron Transport in Electrical Discharges and Electron Collision Cross Sections, Hiroshi Tanaka and Osamu Sueoka Theoretical Consideration of PlasmaProcessing Processes, Mineo Kimura Electron Collision Data for PlasmaProcessing Gases, L oucas G. Christophorou and James K. Olthoff Radical Measurements in Plasma Processing, Toshio Goto

Electron Collision Data for Plasma Chemistry Modeling, W. L . Morgan

Radio-Frequency Plasma Modeling for Low-Temperature Processing, Toshiaki Makabe

Electron—Molecule Collisions in LowTemperature Plasmas: The Role of Theory, Carl W instead and V incent McKoy

Electron Interactions with Excited Atoms and Molecules, L oucas G. Christophorou and James K. Olthoff

Electron Impact Ionization of Organic Silicon Compounds, Ralf Basner, Kurt Becker, Hans Deutsch, and Martin Schmidt Kinetic Energy Dependence of Ion—Molecule Reactions Related to Plasma Chemistry, P. B. Armentrout Physicochemical Aspects of Atomic and

Volume 45 Comparing the Antiproton and Proton, and Opening the Way to Cold Antihydrogen, G. Gabrielse

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CONTENTS OF VOLUMES IN THIS SERIAL

Medical Imaging with Laser-Polarized Noble Gases, T imothy Chupp and Scott Swanson Polarization and Coherence Analysis of the Optical Two-Photon Radiation from the Metastable 2S State of Atomic  Hydrogen, Alan J. Duncan, Hans Kleinpoppen, and Marlan O. Scully

Laser Spectroscopy of Small Molecules, W. Demtro¨der, M. Keil, and H. Wenz Coulomb Explosion Imaging of Molecules, Z. Vager