Methods of Experimental Physics (Part B)

Methods of Experimental Physics VOLUME 5

NUCLEAR PHYSICS PART B

METHODS OF

EXPERIMENTAL PHYSICS: 1. Marton, Editor-in-Chief Claire Marton, Assistant Editor

1. Classical Methods, 1959 Edited by lmmanuel Estermann

2. Electronic Methods, in preparation Edited by E. Bleuler and R. 0. Haxby 3. Molecular Physics, 1961 Edited by Dudley Williams

4. Atomic and Electron Physics, in preparation Edited by Vernon W. Hughes and Howard L. Schultz 5. Nuclear Physics (in two parts), 1961 and 1963 Edited by Luke C. L. Yuan and Chien-Shiung Wu

6. Solid State Physics (in fwo parts), 1959 Edited by K. Lark-Horovitz and Vivian A. Johnson

Volume 5

Nuclear Physics Edited by

LUKE C. 1. YUAN Brookhaven National Laboratory Upfon, New York

CHIEN-SHIUNG WU Columbia University New York, New York

PART B

1963

@

ACADEMIC PRESS New York and London

Copyright @ 1963, by

ACADEMIC PRESS INC. ALL RIGHTS RESERVED NO P A R T OF THIS BOOK MAY B E REPRODUCED I N ANY FORM B Y PHOTOSTAT, MICROFILM, OR A N Y O T H E R MEANS, WITHOUT WRITTEN PERMISSION FROM T H E PUBLISHERS

ACADEMIC PRESS INC. 111 FIFTHAVENUE NEW YORK3, N. Y.

United Kingdom Edilon Published by ACADEMIC PRESS INC. (LONDON) LTD.

BERKELEY SQUARE HOUSE,LONDON W. 1

Library of Congress Catalog Card Number 61-17860 PRINTED I N T H E UNITED STATES O F AMERICA

CONTRIBUTORS TO VOLUME 5, PART B E. AMBLER,Low Temperature Section, National Bureau of Standards, Washington, D. C.I Page 162. F. AJZENBERG-SELOVE, Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania. Page 339. M. BLAU,Institut f u r Radiumforschung, Vienna, Austria. Page 37. M. H. BLEWETT,Brookhaven National Laboratory, Upton, New York. Pages 580 and 623. 0. CHAMBERLAIN, Department of Physics, University of California, Berkeley, California. Page 485. B. CORK,Lawrence Radiation Laboratory, University of California, Berkeley, California. Page 747. H. DANIEL, M a x Planck Institute for Nuclear Physics, Heidelberg, Germany. Page 275. M. DEUTSCH, Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts. Page 303. H. E. DUCKWORTH, Department of Physics, McMaster University, Hamilton, Ontario, Canada. Page 1. R. D. EVANS,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts. Page 761. G. FEHER,Department of Solid State Physics, University of California, L a Jolla, California. Page 127. H. FRAUENFELDER, Department of Physics and Coordinated Science Laboratory, University of Illinois, Urbana, Illinois. Pages 129 and 214. W. B. FRETTER, Department ifPhysics, University of California, Berkeley, California. Page 35. W. GENTNER, M a x Planck Institute for Nuclear Physics, Heidelberg, Germany. Page 275. S. GESCHWIND, Bell Laboratories, Murray Hill, New Jersey. Page 20. J. G. HIRSCHBERG, Project Matterhorn, Princeton University, Princeton, New Jersey. Page 44. J. C. HUBBS,E-H Research Laboratories, Inc., Oakland, California. Page 58. 1

Present address: 1504 Summit Road, Berkeley, California. V

vi

CONTRIBUTORS TO VOLUME

5,

PART B

C. D. JEFFRIES,Department of Physics, University of California, Berkeley, California. Page 104. C. K. JEN, Applied Physics Laboratory, The Johns Hopkins University, Silver Spring, Maryland. Page 85. H. W. KOCH,Betatron Laboratories, National Bureau of Standards, Washington, D. C. Page 508. H. KOUTS,Brookhaven National Laboratory, Upton, New York. Page 590. D. W. MILLER,Department of Physics, Indiana University, Bloomington, Indiana. Page 366. W. A. NIERENBERG, Department of Physics, University of California, Berkeley, California. Page 58. G. D. O'KELLEY,Oak Ridge National Laboratory, Oak Ridge, Tennessee. Page 555. J. S. PRUITT, Betatron Laboratories, National Bureau of Standards, Washington, D. C. Page 508. S . REED,O$ce of Naval Research, Washington, D. C . Page 807. L. ROSEN,Department of Physics, Los Alamos Scientific Laboratory, Los Alamos, New Mexico. Page 366. A. ROSSI,Department of Physics, University of Illinois, Urbana, IElinois.2 Page 214. R. M. STERNHEIMER, Brookhaven National Laboratory, Upton, New York. Pages 691 and 821. A. H. WAPSTRA,Instztute voor Kernphysisch Onderzoek, Amsterdam, Holland. Page 152. W. WHALING, Department of Physics, California Institute of Technology, Pasadena, California. Page 9.

* Present address: Sorin, Saluggia (Vercelli), Italy.

CONTENTS, VOLUME 5, PART B CONTRIBUTORS TO VOLUME 5, PARTB . . . . . . . . . . . . .

v

CONTRIBUTORS TO VOLUME 5, PART A . . . . . . . . . . . . . xiii 5, PARTA . . . . . . . . . . . . . . . . xv CONTENTS, VOLUME

2. Methods for the Determination of Fundamental Physical Quantities (Continued from Vol. 5, Part A )

2.3. Determination of Mass of Nuclei and of Individual Particles. . . . . . . . . . . . . . . . . . . . . 2.3.1. Mass Spectroscopy. . . . . . . . . . . . . . . by H. E. DUCKWORTH

1 1

2.3.2. Determination of Atomic Masses from Nuclear Reaction and Nuclear Disintegration Energies . . . by WARDWHALING

9

2.3.3. Atomic Mass Determination from Microwave Spectra. . . . . . . . . . . . . . . . . . . 20 by S. GESCHWIND 2.3.4. Measurement of Mass with Cloud Chambers and Bubble Chambers . . . . . . . . . . . . . . 35 by W. B. FRETTER 2.3.5. Determination of Mass of Nucleons in Emulsions by MARIETTA BLAU 2.4. Determination of Spin, Parity, and Nuclear Moments . 2.4.1. Spectroscopic Methods . . . . . . . . . . . . 2.4.1.1. Optical and Ultraviolet Spectroscopy . . by JOSEPH G. HIRSCHBERG

.

37

.

44

. 44 . 44

2.4.1.2. The Investigation of Short-Lived Radionuclei by Atomic Beam Methods. . . . by JOHN C. HUBBS and WILLIAMA. NIERENBERG

58

2.4.1.3. Microwave Method. . . . . . . . . . . 85 by C. K. JEN 2.4.1.4. Nuclear Magnetic and Quadrupole Resonance . . . . . . . . . . . . . . . . 104 by C. D. JEFFRIES vii

...

CONTENTS. VOLUME

Vlll

5.

PART B

2.4.1.4.7.3. The Electron Nuclear Double Resonance (ENDOR) Technique . . . . . . . 127 by G . FEHER 2.4.2. Indirect Methods . . . . . . . . . . 2.4.2.1. Angular Correlation . . . . . by HANSFRAUENFELDER

. . . . . 129

. . . . . 129

2.4.2.2. Conversion Coefficients . . . . . . . . . 152 by A. H . WAPSTRA 2.4.2.3. Nuclear Orientat,ion. . . . . . . . . . . 162 by E . AMBLER 2.5. Determination of the Polarization of Electrons and Photons 214 by H . FRAUENFELDER and A. ROSSI 2.5.1. Introduction . . . . . . . . . . . . . . . . . 214 2.5.2. Description of Polarized Beams . . . . . . . . . 216 2.5.3. Motion of Electrons in Electromagnetic Fields . . 223 2.5.4. Polarization Transfer . . . . . . . . . . . . . . 233 2.5.5. Detection of Electron Polarization . . . . . . . . 239 2.5.6. Detection of Photon Polarization . . . . . . . . 264 2.6. Determination of Life-Time . . . . . . . . . . . . . . 275 2.6.1. Long Life-Time . . . . . . . . . . . . . . . . 275 by H . DANIEL and W . GENTNER 2.6.2. Short Lifetimes . . . . . . . . . . . . . . . . 303 by M . DEUTSCH 2.7. Determination of Nuclear Reactions . . . . . . . . . . 339 2.7.1. Determination of the Q Value for Nuclear Reactions 339 2.7.2. Determination of Nuclear Energy Levels from Reaction Energies . . . . . . . . . . . . . . . 352 by FAYAJZENBERB-SELOVE 2.7.3. 2.7.4. 2.7.5. 2.7.6. 2.7.7.

Total Interaction Cross Sections . . . . . Nonelastic Neutron Cross Sections . . . . . Differential Interaction Cross Sections . . . Differential Elastic-Scattering Cross Sections Supplementary Remarks . . . . . . . . . by LOUISROSENand DANW . MILLER

2.8. Determination of Flux and Densities . . . . . . . 2.8.1. Determination of Flux of Charged Particles . by 0. CHAMBERLAIN

. . . . . . . . .

. . . . . .

366 397 411 472 482

. . . 485 . . . 485

CONTENTSJ VOLUME

5,

ix

PART B

2.8.2. Determination of Differential X-ray Photon Flux and Total Beam Energy . . . . . . . . , . . 508 by J. S. PRUITT and H. W. KOCH

3. Sources of Nuclear Particles and Radiations 3.1. Radioactive Sources . . . . . . . . . . . . .

.

.

. .

by G. D. O J K ~ ~ ~ ~ ~ 3.2. Artificial Sources . . . . . . . . . . . . . . . . . . 3.2.1. Low-Energy Sources . . . . . . . . . . . . . . 3.2.1.1. Cascade Rectifiers . . , . . . . . . . . 3.2.1.2. The Electrostatic (Van de Graaff) Generator by M. H. BLEWETT 3.2.1.3. Nuclear Reactors. . . . . . . . . . . . by H. KOUTS 3.2.2. Medium- and High-Energy Sources . . . . . . . by M. H. BLEWETT

4. Beam Transport Systems 4.1. Introduction , . . . . . . 4.2.

. . . . . . . . . . . . . Beam Bending and Focusing Systems . . , . . . . . .

by R. M. STERNHEIMER 4.2.1. The Trajectories of Particles in a Strong Focusing (Quadrupole) Magnet . . . . . . . . . . . . 4.2.2. The Lens Equations for a Single Quadrupole Magnet and for a Two-Magnet System. Conditions of Double Focusing by a Two-Magnet System . . 4.2.3. The Matrix Method of Calculation for Strong Focusing Magnet Systems . . , . . . . . . . 4.2.4. The Focusing Equations for Deflecting WedgeShaped Magnets with Finite Field Index n . . . 4.3. Beam Separators . . . . . . . . . . . . . by BRUCECORK 4.3.1. Degrader Type Separation . . . . . . 4.3.2. Electromagnetic Separators . . . . . . 4.3.3. Radiofrequency Separators . . . . . . 4.3.4. Separation by Nuclear Interactions . .

555 580 580 581 584 590 623

691 692

694

696 719 731

. . .

747

. . . . . . . . . . . . . . . . . . . . . . . .

747 748 750 751

. . . .

4.4. Some Examples of Beam Transport Systems 752 by BRUCECORK 4.4.1. High Momentum Beams Using Counters as Detectors . . . . . . . . . . . . . . . . , . 752

.

CONTENTS. VOLUME

X

5.

PART B

4.4.2. Separated Beams for Bubble Chambers . 4.4.3. Special Beams . . . . . . . . . . . .

. . . . . 755 . . . . . 759

.

5 Statistical Fluctuations in Nuclear Processes by ROBLEY D . EVANS 5.1. Frequency Distributions . . . . . . 5.1.1. The Binomial Distribution . . 5.1.2. The Multinomial Distribution . 5.1.3. The Normal Distribution . . . 5.1.4. The Poisson Distribution . . . 5.1.5. The Interval Distribution . . . 5.2. Statistical Characterization of Data 5.2.1. Mean Value . . . . . . . . 5.2.2. Variance . . . . . . . . . 5.2.3. Sample Variance . . . . . . 5.2.4. Standard Deviation . . . . 5.2.5. Standard Error . . . . . . 5.2.6. Probable Error . . . . . . 5.2.7. Dimensions . . . . . . . . 5.2.8. Precision vs . Accuracy . . .

. . . . . . . . . 761 . . . . . . . . . 761 . . . . . . . . . 762 . . . . . . . . . 763 . . . . . . . . . 764 . . . . . . . . . 767

. . . . . . . . . . 771 . . . . . . . . . . 772 . . . . . . . . . . 773 . . . . . . . . . . 774 . . . . . . . . . . 774 . . . . . . . . . . 776 . . . . . . . . . . 778 . . . . . . . . . . 778 . . . . . . . . . . 779

5.3. Composite Distributions . . . . . . . . . . . . . . . 780 5.3.1. Combined Probabilities . . . . . . . . . . . . . 780 5.3.2, Superposition of Several Independent Random Processes . . . . . . . . . . . . . . . . . . 781 5.3.3. Propagation of Errors . . . . . . . . . . . . . 783 5.3.4. Difference of Two Mean Values . . . . . . . . . 784 5.3.5. Significance Levels and Confidence Intervals . . . 785 5.4. Tests for Goodness of Fit . . . . . . . . . . . . . . . 786 5.4.1. Pearson’s Chi-square Test . . . . . . . . . . . 786 5.4.2. Poisson Index of Dispersion . . . . . . . . . . . 789 5.4.3. Confidence Interval for the Standard Deviation of a Normal Distribution . . . . . . . . . . . . 791 5.5. Applications of Poisson Statistics to Some Instruments Used in Nuclear Physics . . . . . . . . . . . . . . 792 5.5.1. Effects of Resolving Time in Single-Channel Counting . . . . . . . . . . . . . . . . . . . . . 792 5.5.2. Effects of Resolving Time in Coincidence and Anticoincidence Circuits . . . . . . . . . . . . . 794 5.5.3. Scaling Circuits . . . . . . . . . . . . . . . . 795 5.5.4. Counting-Rate Meters . . . . . . . . . . . . . 799

CONTENTS. VOLUME

5.

xi

PART B

5.6. Useful Inefficient Statistics . . . . . . . . . 5.6.1. Estimate of the Mean Value . . . . 5.6.2. Estimate of Standard Deviation . . . . 5.6.3. Estimate of Standard Error . . . . . . 5.6.4. Estimate of x2. . . . . . . . . . . . 5.6.5. Examples . . . . . . . . . . . . . .

. . . . . 803 . . . . . 803 . . . . . 803 . . . . . 803 . . . . . 804 . . . . . 806

Appendix 1 by SIDNEYREED 1 . Evaluation of Measurement . . . . . . . . . . . . . . . 807 1.1. General Rules . . . . . . . . . . . . . . . . . . . 807 2 . Errors . . . . . . . . . . . . . . . . . . . . . . . . . 808 2.1. Systematic Errors, Accuracy . . . . . . . . . . . . . 808 2.2. Accidental Errors, Precision . . . . . . . . . . . . . 808 3. Statistical Methods . . . . . . . . . . . . . . . . . . . 809 3.1. Mean Value and Variance . . . . . . . . . . . . . . 809 3.2. Statistical Control of Measurements . . . . . . . . . . 810 4 . Direct Measurements . . . . . . 4.1. Errors of Direct Measurements 4.2. Rejection of Data . . . . . . 4.3. Significance of Results . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

811 812 813 814

5 . Indirect Measurement . . . . . . . . . . . . . . . . . . 814 5.1. Propagation of Errors . . . . . . . . . . . . . . . . 814 6. Prcliminary Estimation . . . . . . . . . . . . . . . . . 819

7. Errors of Computation . . . . . . . . . . . . . . . . . . 819

.

Appendix 2 Kinematics by R . M . STERNHEIMER 1 . Equations for the Lorentz Transformation from the Laboratory System to Center-of-Mass System; The Jacobian for Particle Reactions . . . . . . . . . . . . . . . . . . . . . . 821 2 . Maximum Angle Criterion for the Identification of Outcoming

Particles in Fundamental Collisions . . . . . . . . . . . 828 3 . Kinematics of Two-Body Decay . . . . . . . . . . . . . 831 4 . Threshold Energies for Associated Production of Unstable

Particles . . . . . . . . . . . . . . . . . . . . . . .

833

5 . Phase Space Factors for Two- and Three-Particle Final States 838

xii

CONTENTS. VOLUME

5.

PART B

.

Appendix 3 Properties of Elementary Particles and Particle Resonance States Table 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 845

Table 2 . . . . . . . . . . . . . . . . . . . . . . . . . .

846

Table3 . . . . . . . . . . . . . . . . . . . . . . . . . .

847

AUTHORINDEX. . . . . . . . . . . . . . . . . . . . . . .

849

SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . .

874

CONTRIBUTORS TO VOLUME 5, PART A D. E. ALBURGER, Brookhaven National Laboratory, Upton, New York M. BLAU,Institut f u r Radiumforschung, Vienna, Austria J. E. BROLLEY, JR., Los Alamos Scientijc Laboratory, Los Alamos, New

Mexico

B. CORK, Lawrence Radiation Laboratory, University of California, Berkeley, California J. W. M. DUMOND,Department of Physics, California Institute of Technology, Pasadena, California

R. D. EVANS,Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts H. FRAUENFELDER, Department of Physics, University of Illinois, Urbana, Illinois

W. B. FRETTER, Department of Physics, University of California, Berkeley, California S. S. FRIEDLAND, Solid State Radiations, Inc., Culver City, California

T. R. GERHOLM, Institute of Physics, University of Uppsala, Uppsala, Sweden

W. W. HAVENS,P u p i n Physics Laboratory, Columbia University, New York, New York R. HOFSTADTER, Physics Department, Stanford University, Stanford, California D. J. HUGHES,* Brookhaven National Laboratory, Upton, New York S. J. LINDENBAUM, Brookhaven National Laboratory, Upton, New York G. C. MORRISON, Atomic Energy Establishment, Harwelt, Berkshire, England G. D. 0’KELLEY,Oak Ridge National Laboratory, Oak Ridge, Tennessee F. REINES,Department of Physics, Case Institute of Technology, Cleveland, Ohio G. T. REYNOLDS, Princeton University, Princeton, New Jersey A. SILVERMAN, Department of Physics, Cornell University, Ithaca, New York

* Deceased. xiii

xi v

CONTRIBUTORS TO VOLUME

5,

PART A

R. M. STERNHEIMER, Brookhaven National Laboratory, Upton, New York R. W. WILLIAMS,Department of Physics, University of Washington, Seattle, Washington L. C. L. YUAN,Brookhaven National Laboratory, Upton, New York F. P. ZIEMBA, Solid State Radiations, Inc., Culver City, California

CONTENTS. VOLUME 5. PART A

.

1 Fundamental Principles and Methods of Particle Detection 1.1. Interaction of Radiation with Matter . . . . . . . by R . M . STERNHEIMER 1.1.1. Introduction . . . . . . . . . . . . . . . 1.1.2. The Ionization Loss d R / d x of Charged Particles 1.1.3. Range-Energy Relations . . . . . . . . . . 1.1.4. Scattering of Heavy Particles by Atoms . . . 1.1.5. Passage of Electrons through Matter . . . . . 1.1.6. Multiple Scattering of Charged Particles . . . 1.1.7. Penetration of Gamma Rays . . . . . . . .

. .

1

. . .

1 4 . . 44 . . 55 . . 56 . . 73 . . 76 .

I .2. Ionization Chambers . . . . . . . . . . . . . . . . . 89 by ROBERT W . WILLIAMS 1.2.1. General Considerations . . . . . . . . . . . . . 89 1.2.2. Pulse Formation . . . . . . . . . . . . . . . . 95 1.2.3. Quantitative Operation and Some Practical Considerations . . . . . . . . . . . . . . . . . 100 1.2.4. Amount of Ionization Liberated . . . . . . . . . 103 1.2.5. Noise: Practical Limit of Energy Loss Measurable . 105 1.2.6. Some Types of Pulse Ionization Chambers . . . . 107 1.2.7. Current Ionization Chambers and Integrating Chambers . . . . . . . . . . . . . . . . . 109 1.3. Gas-Filled Counters . . . . . . . . . . . . . . . . . 110 by ROBERTW. WILLIAMS 1.3.1. Gas Multiplication; Proportional Counters . . . . 110 1.3.2. Geiger Counters and Other Breakdown Counters . 118 I .4. Scintillation Counters and Luminescent Chambers . . . . 120 by GEORGE T. REYNOLDS and F. REINES 1.4.1. Scintillation Counters . . . . . . . . . . . . . 120 1.4.2. Solid Luminescent Chambers . . . . . . . . . . 159

1.5. Cerenkov Counters . . . . . . . . . . . . by S. J . LINDENBAUM and LUKEC . L. YUAN 1.5.1. Introduction . . . . . . . . . . . . 1.5.2. Focusing Cerenkov Counters . . . . . 1.5.3. Nonfocusing Counters . . . . . . . . XV

. . . . . 162

. . . . . 162 . . . . . 168 . . . . . 186

xvi

CONTENTS. VOLUME

5.

PART A

1.5.4. Total Shower Absorption eerenkov Counters for Photons and Electrons . . . . . . . . . . . . 189 1.5.5. Other Applications . . . . . . . . . . . . . . . 191 1.6. Cloud Chambers and Bubble Chambers . . . . . . . . 194 by W . B. FRETTER 1.6.1. Cloud Chambers . . . . . . . . . . . . . . . . 194 1.6.2. Bubble Chambers . . . . . . . . . . . . . . . 203 1.7. Photographic Emulsions . . . . . . . . . . . . . . . 208 by M . BLAU 1.7.1, Introduction . . . . . . . . . . . . . . . . . 208 1.7.2. Sensitivity of Nuclear Emulsions . . . . . . . . 210 1.7.3. Processing of Nuclear Emulsions . . . . . . . . . 216 1.7.4. Optical Equipment and Microscopes . . . . . . . 224 1.7.5. Range of Particles in Nuclear Emulsions . . . . . 226 1.7.6. Ionization Measurements in Emulsions . . . . . . 240 1.7.7. Ionization Parameters . . . . . . . . . . . . . 245 1.7.8. Photoelectric Method . . . . . . . . . . . . . 264 1.8. Special Detectors . . . . . . . . . . . . . 1.8.1. The Semiconductor Detector . . . . . and F. P. ZIEMBA by S. S. FRIEDLAND 1.8.2. Spark Chambers . . . . . . . . . . . by BRUCECORK

. . . . . 265 . . . . . 265

. . . . . 281

.

2 Methods for the Determination of Fundamental Physical Quantities 2.1. Determination of Charge and Size . . . . . . . . . . . 2.1.1. Charge of Atomic Nuclei and Particles . . . . . . 2.1.1.1 Rutherford Scattering . . . . . . . . . . . . . 2.1.1.2. Characteristic X-ray Spectra . . . . . . . . . by ROBLEY D . EVANS 2.1.1.3. Charge Determination of Particles in Photographic Emulsions . . . . . . . . . . . . . by M . BLAU 2.1.2. Principal Methods of Measuring Nuclear Size . . . by ROBERTHOFSTADTER

289 289 289 293

298 307

2.2. Determination of Momentum and Energy . . . . . . . 341 2.2.1. Charged Particles . . . . . . . . . . . . . . . 341 2.2.1.1. Measurement of Momentum . Electric and Magnetic Analysis . . . . . . . . . . . . . . . 341 by T. R . GERHOLM

CONTENTS, VOLUME

5,

PART A

xvii

2.2.1.1.4. Measurement of Momentum with Cloud Chambers or Bubble Chambers . . . . . . . . . 375 by W. B. FRETTER 2.2.1.1.5. Momentum Measurement in Nuclear Emulsions 388 by M. BLAU 2.2.1.2. Determination of Energy . . . . . . . . . . . 409 2.2.1.2.1. Energy Measurement with Ionization Chambers 409 by R. W. WXLLIAMS 2.2.1.2.2, Scintillation Spectrometry of Charged Particles 411 by G. D. O'KELLEY 2.2.1.2.3. Measurement of Range and Energy with Cloud Chambersand Bubble Chambers. . . . . . 436 by W. B. FRETTER 2.2.1.3. Determination of Velocity. . . . . . . . . . . 438 2.2.1.3.1. Time-of-Flight Method . . . . . . . . . . . 438 by LUKEG. L. YUANand S. J. LINDENBAUM 2.2.1.3.2. Measurement of Velocity . . . . . . . . . . 444 by W. B. FRETTER 2.2.1.3.3. Measurement of Velocity Using cerenkov Counters . . . . . . . . . . . . . . . . 454 by LUKEC. L. YUANand S. J. LINDENBAUM 2.2.2. Neutrons . . . . . . . . . . . . . . . . . . . 461 2.2.2.1. Recoil Techniques for the Measurement of Neutron Flux, Energy, Linear and Spin Angular Momentum . . . . . . . . . . . . . . . . 461 E. BROLLEY, JR. by JOHN 2.2.2.2. Time-of-Flight Method . . . . . . . . . . . . 495 by W. W. HAVENS,JR. 2.2.2.3. Crystal Diffraction. . . . . . . . . . . . . . 566 by D. J. HUGHES 2.2.2.4. Determination of Momentum and Energy of Neutrons with Her Neutron Spectrometer. . . . . 570 by G. C. MORRISON 2.2.3. Gamma-Rays . . . . . . . . . . . . . . . . . 582 2.2.3.1. Internal and External Conversion Lines. . . . . 582 by T. R. GERHOLM 2.2.3.2. Determination of Momentum and Energy of Gamma Rays with the Curved Crystal Spectrometer . . . . . . . . . . . . . . . 599 by J. W. M. DUMOND

xviii

CONTENTS, VOLUME

5,

PART A

2.2.3.3. Gamma-Ray Scintillation Spectrometry. . . . . 616 by G. D. O’KELLEY 2.2.3.4. Determination of the Momentum and Energy of Gamma Rays with Pair Spectrometers . . . . 641 by D. E. ALBURGER 2.2.3.5. Shower Detectors . . . . . . . . . . . . . . 652 by R. HOFSTADTER 2.2.3.6. Gamma-Ray Telescopes. . . . . . . . . . . . 668 by A. SILVERMAN 2.2.3.7. Measurement of ?-Ray Energy by Absorption. . 671 by ROBLEY D. EVANS 2.2.3.8. Detection and Measurement of Gamma Rays in Photographic Emulsions . . . . . . . . . . 676 by M. BLAU 2.2.4. Neutrino . . . . . . . . . . . . . . . . . . . 682 2.2.4.1. Neutrino Reactions. . . . . . . . . . . . . . 682 by F. REINES

AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . .

699

SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . . .

718

2. METHODS FOR THE DETERMINATION OF FUNDAMENTAL PHYSICAL QUANTITIES (Continued) 2.3. Determination of Mass of Nuclei and of Individual Particles 2.3.1. Mass Spectroscopy1 *

t

2.3.1 .l. Introduction. The first mass spectroscopes were invented by Thomsoq2 A ~ t o nand , ~ Dempster4 in the second decade of this century for the purpose of investigating isotopy among stable atoms. In these instruments, and in most of the mass spectroscopes of the present day, the specific charge ( e / M ) of a positive ion is deduced by observing the deflection experienced by the ion as it passes through electric and/or magnetic fields. The mass of the ion is readily calculated from this as the charge carried by the ion is generally known. Within the precision of the first experiments the masses of all atoms appeared to be integral multiples of that of HI. Small divergences from this “whole number rule” were discovered, however, in 1923 and these have since been the subject of intensive investigation by mass spectroscopists because of the information they yield concerning nuclear stability. The reader will recall in this connection that the mass of an atom is less than the combined masses of its constituent nucleons and electrons, and that it is this difference between deduced and observed masses, known as the binding energy, which accounts for the stability of the atom. Prior to September, 1960, atomic masses were expressed in atomic mass units (OIE = 16 amu), but since September, 1960 they have been expressed in relative nuclidic mass units, abbreviated u, so defined that C12 = 12u. On these scales the masses of all atoms are nearly integral, the nearest integer being in each case the mass number (number of

t See also Vol. 4, A, Section 4.1.1. K. T. Bainbridge, Charged particle dynamics and optics, relative isotopic abundances of the elements, atomic masses. I n “Experimental Nuclear Physics” (E. SegrB, ed.), Vol. 1. Wiley, New York, 1953;H. Ewald and H. Hintenberger, ‘‘ Methoden und Anwendungen der Massenspectroskopie.” Verlag Chemie, Weinheim, Germany, 1953; M. G. lnghram and R. J. Hayden, “ A Handbook on Mass Spectroscopy,” Nuclear Energy Series, Report No. 14. National Academy of Sciences, Washington, D.C., 1954;and H. E. Duckworth, “Mass Spectroscopy.” Cambridge Univ. Press, London and New York, 1958. * J. J. Thomson, “Rays of Positive Electricity.” Longmans, Green, London, 1913. F. W. Aston, Phil. Mag. [6]38, 709 (1919). ‘A. J. Dempster, P h y . Rev. 11, 316 (1918). 1

* Section 2.3.1 is by H. E.

Duckworth. 1

2

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

nucleons) of the atom in question, Aston and many since him have expressed the exact mass of an atom as zM*= A ( l f), where f is the “packing fraction” and A and Z are the mass number and atomic number (number of protons), respectively. Thus

+

packing fraction, f = ( M

- A)/A.

An algebraically small packing fraction indicates an atom whose mass is numerically small with respect to its mass number and whose stability is consequently high. It is more common to express this stability in terms of the “binding energy per nucleon,” defined as follows: b.e./nucleon =

ZH’

+ ( A - 2)n - zMA A

where H1and n are the masses of the hydrogen atom and neutron, respectively. The term b.e./nucleon implies that the binding energy is associated with the nucleus, as indeed moat of it is. The binding energies of the orbital electrons, here practically neglected, are not only small but increase with 2 in a gradual manner so that the b.e./nucleon gives a significant picture of the variations and trends in nuclear stability. Modern precision mass spectroscopy is concerned with the study of these variations and trends. 2.3.1.2. Mass Spectroscopes for the Determination of Atomic Mass. The mass spectroscopes employed in the determination of atomic masses are necessarily high resolution instruments. With a single exception those under construction or now in use are instruments in which the ions are deflected by a linear combination of magnetic and electric fields as, for example, in the instrument shown in Fig. 1. The essential parts of such an apparatus are (a) a source of positive ions, (b) a region in which the ions are accelerated, (c) a system of slits for collimating the ion beam, (d) an electromagnetic analyzer which separates the beam into its several specific charge groups, and (e) a device for detecting the ions after analysis. Depending upon whether photographic or electrical detection is employed, “lines” or “peaks” are recorded by the detector, each representing a particular group of ions present in the initial ion beam. The electrical detector of Fig. 1 could be replaced, if desired, by a photographic plate located at Saand lying in the direction SaS4. These instruments possess the property of focusing at the final detector ions which emerge from the collimating system with an angular spread in the plane of the paper and/or with a velocity spread. They are, therefore, said to be “double focusing,’’ a term which has unfortunately been

2.3.

MASS O F NUCLEI AND INDIVIDUAL PARTICLES

3

abrogated by beta-ray spectroscopists to describe instruments which achieve angular focusing in two directions. The original double focusing instruments of Dempster, Bainbridge and Jordan,B and Mattauch,7 and the majority of their lineal descendants, are based on first-order focusing theory. In these the final image

90" ELECTROSTATIC ELECTROS 90"

ELECTRON MULTIPLIER

FIG. 1. Schematic diagram of the Nier double focusing mass spectrometer (from reference 10).

breadth contains aberration terms involving second-order and higher powers of the angular and velocity spread in the transmitted ion beam. Notable exceptions are the mass spectroscope of Nier and his collaborators,*-1° shown in Fig. 1, which possesses second-order direction A. J. Dempster, Proc. Am. Phil. SOC.76, 755 (1935). K. T. Bainbridge and E. B. Jordan, Phys. Rev. 60,282 (1936). 7 J . Mattauch, Phys. Rev. 60, 617 (1936). * A. 0. Nier and T. R. Roberts, Phys. Rev. 81, 507 (1951). E. G. Johnson and A . 0. Nier, Phys. Rev. 91, 10 (1953). l o K. S. Quisenberry, T. T. Scolman, and A . 0. Nier, Phys. Rev. 102, 1071 (1956). 6

6

4

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

focusing plus first-order double focusing, and the new instrument a t the Max Planck Institute for Chemistry11v12which will possess first-order double focusing along the entire photographic plate and second-order direction focusing in the middle of the plate. The resolution of a mass spectroscope is written as A M / M , which implies that two ions of mass M and M AM, respectively, can be just resolved. In instruments of the type we have been discussing, the resolution improves linearly with the radius of curvature of the electrostatic analyzer and also as the widths of the defining slits (Sz, for photographic detection; Sz and 54 for electrical detection) are reduced. With mass spectroscopes employing photographic detection the grain size of the cm) sets the lower limit to the actual line photographic plate width with the result that the resolution cannot be reduced indefinitely by reducing the width of the object slit Sz. Because of this grain size limitation in the case of photographic detection instruments and also because of the difficulty, whatever the method of detection] of adjusting and aligning the diminutive slits required to secure high resolution in any small scale apparatus, the instruments currently (1960) under coiistruction for the purpose of precision mass determinations exceed their predecessors in size by about an order of magnitude. That is, high resolution is being sought by increasing the radius of curvature a, of the electrostatic analyzer (and, of course, the size of the magnetic analyzer, which is related to it). Included in this category are new instruments at Osaka University13*14(a,= 1.09 m), Harvard Universityl6 (ae = 2.14 m), the Max Planck Institute for Chemistry11v12(ae = 5.43m), and McMaster University16 (a,= 2.75 m). Whereas the first double focusing instruments6-’ possessed resolutions ranging from 1/2500 - 1/10,000, the values for these new instruments should be considerably better than 1/100,000.

+

11 H. H. Hintenberger, H. Wende, and L. A. Konig, 2.Naturfursch. lOa, 605 (1 055) ; laa, 88 (1957). H. Hintenberger, J. Mattauch, W. Miiller-Warmuth, H. Voshage, and H. Wende, in “Proceedings of the International Conference on Nuclidic Masses” (H. E. Duckworth, ed.). Univ. of Toronto Press, Toronto, 1960. 18 K. Ogata and H. Matsuda, 2.Naturforsch. 10a, 843 (1955). 14 K. Ogata and H. Matsuda, in “Nuclear Masses and Their Determination” (H. Hintenberger, ed.). Pergamon, London, 1957. 16X. T. Bainbridge, R. C. Barber, H. E. Duckworth, and P. E. Moreland, Jr., in “Proceedings of the International Conference on Nuclidic Masses” (H. E. Duckworth, ed.). Univ. of Toronto Press, Toronto, 1960. 16 N. R. Isenor, R. C. Barber, and H. E. Duckworth, in “Proceedings of the International Conference on Nuclidic Masses” (H. E. Duckworth, ed.). Univ. of Toronto Press, Toronto, 1960.

2.3.

MASS OF NUCLEI AND INDIVIDUAL

PARTICLES

5

Smith, 17-19 a t the Brookhaven National Laboratory, has employed a completely different type of mass spectroscope known as the “mass synchrometer.” Here the ions describe circular paths in a homogeneous magnetic field, the frequency of the motion being f = Be/27rM. This cyclotron frequency is independent of the velocity of the ion but linearly dependent upon its specific charge. Measurements are therefore made of the cyclotron frequencies of the ions under study. The resolution is adjustable electrically between 1/10,000 and 1/25,000. 2.3.1.3. T h e D o u b l e t M e t h o d of Mass Comparison. In theory, in a typical mass spectroscope, one could measure the potential through which an ion is accelerated, ascertain the deflection produced by an analyzer of known characteristics, and from these data calculate the ion mass. This procedure, however, involves a knowledge of certain absolute values which it is not yet possible to determine to the desired precision. Instead, in practice, the mass of an unknown atom is found by comparing it to one whose mass is known. As a rule the known and unknown ions have specific charges which are nearly equal, with the result that the two peaks or lines appear on the mass spectrum at the same mass number. For example, singly charged CL2 ( =12u) and doubly charged Mg24 (=23.9850446 f 19u) both appear a t mass number 12, where they are slightly displaced with respect to one another by virtue of the one part in 1600 mass difference between them. These two constitute a “doublet” and it is the concern of the mass spectroscopist to ascertain, in terms of mass, this doublet spacing. In the main, three factors determine the precision with which mass differences can be determined. These are (a) the accuracy with which the dispersion of the mass spectroscope is known, and (b) the accuracy with which the lines or peaks belonging to the atoms under study can be located, and (c) the extent to which the lack of perfect velocity and angular focusing in the mass spectroscope introduces aberrations in the image. OF DISPERSION LAW,The theoretical dispersion 2.3.1.3.1. UNIFORMITY law governing a mass spectroscope is never realized experimentally because of lack of homogeneity in the analyzer fields. Thus, although it is possible in principle to compare the masses of atoms even if their mass spectral lines are wide apart on a photographic plate, it is difficult in practice to secure a uniform dispersion law over the large region involved. It is much less difficult to do so over the region occupied by a doublet. This consideration has led to the almost exclusive use of the doublet method in mass spectroscopes that employ photographic detection. G. Smith, Phys. Rev. 81, 295 (1951). G. Smith and C . C . Damm, Phys. Rev. 90, 324 (1953). 19L.G. Smith and C. C . Demm, Rev. Sci. Instr. 27,638 (1956). 17L.

18L.

2.

6

DETERMINATION O F FUNDAMENTAL QUANTITIES

In the case of electrical detection the fact that the ions, a t the time of detection, have each traveled identical paths makes the achievement of a uniform dispersion law much easier than in the case of photographic detection. The doublet members are brought in turn to the collector by varying some circuit parameter as, for example, in the case of the mass spectroscope of Fig. 1, a resistance that controls the voltage across the electrostatic analyzer. The mass change is expected to be proportional to the resistance change and is so in practice to a very close approximation. This has been verified by direct determination of the H* mass as, for example, from the CaHs-CaH, mass difference. This, it will be realized, is an extraordinarily stringent test, as these two ions do not constitute a doublet but are spaced one mass unit apart. Twenty-four such determinationsZ0of the H1 mass yielded a mean of H’ = 1.0078239 13u, as compared with the best value from the same laboratory, based on doublet comparisons, of H’ = 1.0078247 jz 2u. The extensive linearity of this dispersion law makes feasible, therefore, the determination of mass-unit differences with a precision which is not much inferior to that associated with doublet comparisons. 2.3.1.3.2. LOCATION OF MASSSPECTRAL LINESOR PEAKS. The precision with which a given mass difference can be determined is dependent upon the accuracy with which the mass spectral lines or peaks may be located, Let us designate the mass width of a line or peak by AM, a quantity which is directly proportional to the resolution of the mass spectroscope. I n the case of photographic recording, the position of a mass spectral line may be determined to some fraction of its width, say, for an observer who is neither unduly optimistic nor unduly conservative, Thus, if a resolution of 1/20,000 be available, a mass difference can be determined with a precision of one part in a million. With electrical recording it has been demonstrated that it is possible to locate a peak with a precision of & of its width, and perhaps even more closely. At least a tenfold improvement over the photographic case is thus obtained. This is done by a “peak-matching” technique introduced by Smith and Damm17 and also employed, in a modified form, by Nierlo and several others. In this technique, by taking advantage of rapidly responding detector systems, the two doublet peaks are made to appear on an oscilloscope screen on alternate sweeps. These two peaks are then brought into coincidence by adjustment of some variable (in Smith’s case, a frequency; in Nier’s case, a resistance) whose value gives the doublet mass difference. The peaks are thus “matched” by the human eye, an organ which can discern lack of coincidence with exceptionally keen discrimination. Two peaks which have almost been brought into coincidence by this method are shown in Fig. 2.

*

10

K. S. Quisenberry, C. F. Giese, and J. L. Benson, Phys. Rev. 107, 1664 (1957).

2.3.

MASS

OF NUCLEI A N D INDIVIDUAL

PARTICLES

7

2.3.1.3.3. IMAGE ABERRATIONS RESULTING FROM IMPERFECT FOCUSING. It has been mentioned that at least two of the modern precision mass

spectroscopes7-~*provide second-order angular focusing. Actually, even for first-order angular focusing instruments, the lack of perfect angular focusing is often not serious for the reason that the ions are strongly collimated during their acceleration (25-50 kev). The lack of perfect velocity focusing, on the other hand, had led in the past t o serious systematic errors in cases where the velocity spread associated with one of the doublet members differs from that associated with the other. This will be true if one of the ions and not the other arises from the dissociation of a molecule. The one may then possess significant initial kinetic energy as, for example, the S32-ion (from HZS or SO,) in the

FIQ. 2. Peak matching of two doublet members. The degree of mismatch shown (as indicated by the lateral displacement of one peak with respect to the other) corresponds to a mass difference of approximately one part in 500,000. Resolution is 1/40,000. (Courtesy of Professor A. 0. C. Nier and Dr. K. S. Quisenberry.) 02-S32 doublet. As i t possesses no velocity focusing feature, the mass synchrometer of Smith1?-l9 is particularly vulnerable in this respect and Smith has, therefore, taken pains to study doublets in which both members are nondissociation or “parent” ions. I n this way, for example, masses for OX6and D2 can be obtainedz1from the two doublets C8Ds-C4D2 and CDd-D20. I n the usual deflection-type mass spectroscope the slit S3 in Fig. 1, located between the electrostatic and magnetic analyzers, serves the purpose of limiting the velocity band which enters the magnetic analyzer. This slit should be narrow enough to ensure that it is completely illuminated by both doublet members. 2.3.1.4. Accuracy of Mass Spectroscopic Mass Values. Although “best” values have a habit of changing, and any listed here will in all probability be out-of-date a t the time of publication of this text, it is 21

L. G. Smith, Bull. Am. Phys. SOC.[2] 2, 223 (1957).

8

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

nevertheless instructive to list a few, if for no other reason than to indicate the degree of precision of modern mass spectroscopy. The values for three key light atoms, shown in the tabulation, are the best available in the autumn of 1960. Value

Reference 20

H’ DZ

1.0078247 k 2u

0’8

15.9949142 -t 5u

Reference 21 2.01409964 f 3u 15.99491550 i 15u

Thus, the error associated with H1 is 200 ev, with D2, 30 ev and with 0 1 6 , 130-500 ev. In the case of D2 the accuracy approaches the point where one must take into account differences in the ionization potential of the two ions forming a doublet. The errors quoted above are statistical ones, based on the internal consistency of the experimental data, and therefore represent the actual errors only if systematic errors be absent. Systematic errors will manifest themselves as discrepancies between the values obtained in laboratories that employ different experimental methods; otherwise there is no basis for suspecting their existence. As mentioned above, the mass of 0l6has been determined to a high degree of precision both a t the University of Minnesota20 and a t Brookhaven National Laboratory, 2 1 using radically different types of mass spectroscopes. The discrepancy between the two values is -1.3 5- 0.6 micro-u, that is, somewhat greater than twice the probable error, suggesting that systematic errors are of more importance than statistical ones. This is a fairly typical example; if anything, the discrepancy is rather larger here than usual. On the whole it appears that the actual probable errors associated with mass spectroscopic mass values are 1.5-2 times the stated statistical errors, with the limits of error being larger again by an additional factor of two. Finally, a word will be said concerning the accuracy as a function of mass. At the time of writing, the actual probable errors in the various mass regions that have been studied with modern techniques are as follows: A = 32, -1.5 kev;20 A = 48, -2 kev;22 A = 64, -4 k e ~ ; ~ ~ A = 132, -5 kev;Z4 A = 150, -0.14 and A = 200, -25 kev.26 22

C. F. Giese and J. L. Benson, Bull. Am. Phys. SOC.[2] 2, 223 (1957).

K. S. Quisenberry, T. T. Scolman, and A. 0. Nier, Phys. Rev. 104, 461 (1956). R. R. Ries, R. A. Damerow, and W. H. Johnson, Jr., in “Proceedings of the International Conference on Nuclidic Masses” (H. E. Duckworth, ed.). Univ. of Toronto Press, Toronto, 1960. 26 V. B. Bhanot, W. H. Johnson, Jr., and A. 0. Nier, Phys. Rev. 120, 235 (1960). *a

24

2.3. MASS

OF NUCLEI AND INDIVIDUAL PARTICLES

9

Further details concerning these and other mass regions can be found in published tab~lations2~,~~,27 of mass spectroscopic atomic mass data.

2.3.2. Determination of Atomic Masses from Nuclear Reaction and Nuclear Disintegration Energies*

The energy balance in a nuclear reaction is expressed in terms of the

Q value of the reaction, the difference between the mass of the target and

+ + +

projectile atoms before the reaction Ma M1, and the mass of the atoms that are produced in the reaction M z M,. The Q value is customarily expressed in energy units, Q = c2(Mo M 1- M z- Ma). For decay processes the Q value is the difference between the mass of the radioactive atom and the daughter atom, in the case of @-decay, or the daughter atom plus a helium atom in the case of a-decay. The electron mass does not appear explicitly as an emitted particle in 6-decay because the daughter atom has just one more, or less, electron than the radioactive atom, and the use of atomic masses rather than nuclear masses automatically includes the emitted electron. However, in order that the Q value for positron decay include all of the energy released in the transition, one must add 2m0c2,the energy which eventually appears as annihilation radiation, to the beta endpoint energy. The atomic masses are customarily expressed in atomic mass units. Prior to 1960, the mass unit universally used in precise studies of atomic masses was OL6/16.In September 1960, the International Union of Pure and Applied Physics adopted C12 as the mass standard and C12/12 as the atomic mass unit in order to take advantage of the fact that mass spectroscopic doublets containing carbon provide the most convenient measurement of the mass of heavy atoms. The new unit CI2/12 is larger than 016/16 by a factor 1.000 317 917 k 17,' and all atomic masses expressed in the new units are thereby decreased, but the mass to energy conversion factor is increased to (C12/12) c2 = 931.441 Mev so that any mass difference expressed in energy units is independent of the mass unit chosen. * B H. E. Duckworth, B. G . Hogg, and E. M. Pennington, Revs. Modern Phys. 26, 463 (1954). 27 H. E. Duckworth, Revs. Modern Phys. 29, 767 (1957). F. Everling, L. A. Konig, J. H. E. Mattauch, and A. H. Wapstra, Nuclear Phys. 16, 342 (1960).

* Section 2.3.2 is by Ward Whaling.

10

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

The following discussion will deal with the mass of atoms in their ground state, and the Q values will always refer to ground-state transitions. The experimental measurement of a Q value is based on the fact that the kinetic energy of the particles produced in the reaction exceeds the kinetic energy of the reacting atoms by just the amount of the Q value, and the experimental measurement consists of the determination of the kinetic energy of all of the atoms involved.* If the particles produced in the reaction carry away some of the energy released in the reaction as internal energy of excitation, either nuclear or atomic, the kinetic energy of the outgoing particles is thereby reduced, and the measured Q value does not represent the true mass difference. Nuclear excitation energy is usually so large that it is not easily overlooked, but it is much more difficult to detect, or otherwise make allowance for, atomic excitation of the residual atom. An estimate2 of the amount of energy that might be expected to appear as electronic excitation indicates that it is of the order of a kilovolt for reactions of atoms with atomic number in the neighborhood of ten. For reactions among nuclei with large atomic number, this atomic excitation energy may constitute a significant correction to the measured Q value3 and must be taken into account. From the mathematical point of view the experimental Q value is equivalent to a linear equation relating two or more masses. By proper choice of the &-value equations, one obtains a set of simultaneous equations which may be solved to find the unknown masses. It will be recognized that this procedure is identical with that used to determine masses from mass-spectroscopic doublet measurements. Indeed, a mass spectroscopic doublet is also a linear relation between unknown masses and hence is exactly equivalent to a Q-value measurement. I n selecting the set of simultaneous equations for solution, one may include mass spectroscopic doublets as well as reaction Q values. The mass tables published in the past few years (1957) have been based either on mass spectrographic doublets or Q values, and the two types of measurements have not been combined. The purpose behind this artificial separation of the data from the two sources is to discover possible inconsistencies between the different methods of measurement. As more accurate measurements remove the sources of disagreement, the information provided by both * A wide variety of techniques have been developed for measuring Q values, of which the most precise are based on electric and magnetic analysis (see Sections 2.2.1.1.1 and 2.7.1). ¶ A . B. Brown,C. W. Snyder, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 82, 159 (1951).

'R. Serber and H. Snyder, Phys. Rev. 87, 152 (1952).

2.3.

MASS OF NUCLEI AND INDIVIDUAL

PARTICLES

11

types of experiment will be combined to provide the most accurate information about atomic masses. The procedure used in determining masses from Q values differs from that used with mass doublets only in that a system of equations involving Q values is necessarily large, much larger than the minimum system of equations that can be used when treating mass doublets. 2.3.2.1. Formulation of Input Data. The large number of equations introduces difficulties, in addition to the obvious mathematical complexity, since it becomes necessary to combine values measured in different laboratories by different methods, values which may be based on inconsistent energy calibration standards. All of the input data must first be examined and, if necessary, corrected to correspond to best values of the energy calibration standards. It is convenient to consider the mass excess M , expressed in energy units, rather than the atomic mass M. If A is the mass number, then M = (M - A)$, and the Q value of a reaction is a simple combination of the mass excesses of the atoms taking part in the reaction. The set of input Q value equations would have the form

2

AikMk =

Qi

AQt

i

=

1, 2, 3,

. . . ,n

(2.3.2.1)

k=l

where A i k is a n integer, usually i 1 or 0, the Jfk are the unknown mass excesses which one wishes to find, Qi is an experimental Q value, and AQ, is the standard deviation in the experimental measurement. It will be assumed throughout the following discussion that AQiare standard errors.* This set of equations, called conditional equations in the mathematical literature, may be solved to find Mk if the following conditions are satisfied. (1) The determinant of the Aik must not be zero. If the set of equations (2.3.2.1)is large, evaluation of the determinant may be quite tedious, and the following graphical method determines immediately whether this condition is satisfied. Let us consider, for example, the set of Q value equations listed in Table I. This set of equations is represented graphically in Fig. 1, in which the nuclides are located as a function of A - 2 and 2. 4 link connecting two nuclides represents a mass difference equation relating the two nuclides. The mass difference equation given by a

* In practice, the errors quoted by various experimental physicists are sometimes standard errors, sometimes probable errors, and are frequently not identified at all. Statistical analyses of experimental data from many laboratories have indicated that the internal consistency of the measurements is such that, on the average, the quoted error is usually greater than the probable error and smaller than the standard error.

2.

12

DETERMINATION O F FUNDAMENTAL QUANTITIES

TABLEI. A Set of 40 Experimental Q Values Expressed as Linear Equations Which Relate the Masses of 20 Nuclides between n and N16 In these equations the symbol for a nuclide represents the mass excess of the atom, expressed in units of MeV.&

(1) n - HI = 0.783 f 0.013 Mev (2) n HI - HZ = 2.227 L- 0.002 = 6.251 f 0.008 (3) n t Ha - H3 (4) 2Ha - n - He3 = 3.267 f 0.007 = 0.0185 f 0.0002 (5) H 3 - He3 (6) He3 n - H1 - H3 = 0.7637 f 0.001 (7) Lie H' - He4 - He3 = 4.023 f 0.002 (8) Lie Ha - Li7 - H1 = 5.027 f 0.003 (9) 2 H a - H' - HS = 4.038 f 0.005 = 1.6449 f 0.0004 (10) Be7 n - H1 - Li7 HI - 2He4 = 17.346 0.010 (11) Li7 (12) Be8 - 2He4 = 0.0941 rt 0.007 (13) Be8 n - Be9 = 1.665 f 0.0014 = 6.816 f 0.006 (14) Be9 n - Bela (15) Be9 H' - HZ - Be8 = 0.559 f 0.001 (16) Be9 H' - He4 - Lie = 2.126 f 0.002 (17) Be9 HZ - H1 - Belo = 4.587 f 0.005 (18) Be9 H Z - H a - Be8 = 4.598 f 0.012 (19) Be9 Ha - He4 - Li7 = 7.153 f 0.003 = 0.557 f 0.004 (20) Belo - BIO (21) BO ' n - He4 - Li7 = 2.786 f 0.008 (22) BIO H' - He4 - Be7 = 1.148 f 0.002 (23) B'O Ha - H' - B" = 9.229 f 0.005 (24) BI1 H1 - He4 - Be8 = 8.585 rt 0.006 (25) B11 Ha - He4 - Be9 = 8.024 rt 0.004 = 4.949 f 0.006 (26) CI2 n - CI3 (27) N13 + n - H2 - C" = 0.281 f 0.003 (28) C'a H* - H' - CIa = 2.721 f 0.002 (29) NIS + n - HI - CI3 = 3.003 f 0.003 (30) C" + H a - H' - CI4 = 5.943 f 0.003 (31) C" Ha - H3 - CIa = 1.310 f 0.003 (32) C13 + H a - He4 - B1' = 5.164 f 0.004 (33) C'4 - N'4 = 0.156 0.001 0.005 (34) N14 + n - HI - CI4 = 0.6264 = 2.23 k 0.010 (35) N'3 - C'* (36) N14 n - Nx6 = 10.832 f 0.008 (37) N" + H a - H' - N" = 8.614 f 0.007 (38) N16 H1 - He4 - CI2 = 4.961 f 0.003 He - He4 - CIa = 7.681 f 0.009 (39) N16 = 3.115 f 0.0025 (40) 0 1 6 + H e - He4 - "4

+

+ + +

+

+

*

+ + + + + + + + + + + + + + +

+

*

+ + +

o This particular set of equations is taken from reference 8. The experimental Q values are t a b n from D. M. Van Patter and W. Whaling, Revs. Modem Phys. 26,

402 (1954)'.

2.3.

MASS OF NUCLEI AND INDIVIDUAL

13

PARTICLES

nuclear reaction Q value is represented by a link connecting the target and residual nuclei in the reaction; e.g., the line connecting Li6 and Li7 might represent the Li6(d,p)Li7reaction, or the equation Li6

+ H* - Li7 - H1 = &.

The legend in the corner of the figure indicates the links established by several common types of reactions.

a 7

6 5 z

4

3 2

I

0

0

I

2

3

4

5

6

7

8

A-Z FIG.1. Graphical representation of the set of &-value equations in Table I. A link connecting two nuclides represents a nuclear reaction or decay process in which one of the nuclides is the target, the other is the residual nucleus. The links established by common reactions are indicated in the lower corner of t h e figure.

In order that the determinant of the set of equations represented by the figure have a nonzero determinant, it is necessary that the links in the figure form a continuous path connecting C12, or some other atom of known mass, with every other nuclide that appears in the set of equations. From the figure it is apparent that the set of equations in Table I satisfies this requirement. However, if the C13(d,a)B" reaction were removed

14

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

from Table I, the remaining set could not be solved. Furthermore, the subset formed by reactions (28)-(40) in Table I can be solved for this group of masses connected to CI2 only if the p , n, d, I, and a masses, the bombarding and emitted particles in these reactions, were furnished as auxiliary information. (2) The mass standard C12,or some other atom of known mass, must appear in a t least one of the equations in the set. This requirement follows from the fact that the conditional equations are restricted to those in which the mass number, had it been included in the expression for the mass, would cancel out of the equation. For example, suppose the first equation in set (2.3.2.1) were n - H’ = QI (we may neglect the +AQi in this discussion). Since the difference n - H’ is the only combination of these masses which cancels the mass numbers, any other equation containing n and H1 must also contain at least one additional mass (e.g., n H1 - H2 = Q 2 ) . Each additional equation adds at least one additional unknown mass, and any system of s equations must contain therefore at least s 1 unknown masses. However, if to this system we add the (a 1)th equation, in which the only new mass appearing is C12,the set of equations now becomes soluble. If our second equation had been 6n 6 H - C12= Q2, the two equations would determine the n and H1 masses. Mattauch has shown that certain linear combinations of the masses, such as the nuclear binding energy, may be determined from a set of equations which does not contain the mass standard.4 2.3.2.2. Reaction Cycles. Since the set of equations (2.3.2.1) usually contains many more equations than unknowns, n > rn, there exist n - rn independent equations relating the Q values in the set. These equations, usually called reaction cycles, are of the form Z, a,&, = 0, where a. is a n integer, usually ? 1. The reaction cycles are useful in screening the input data for experimental Q values that contain gross errors. From the experimental values Q, k AQ, one computes the value of the cycle sum, and the uncertainty in the sum. If the value differs from zero by several times the standard deviation of the sum, it is likely that a t least one of the experimental Q values in the cycle is wrong. One then examines one at a time each Q value in the original cycle by summing other cycles which contain one of the original Q values. I n this way, it is possible to uncover experimental Q values which appear to be in error and eliminate them from the original set before proceeding with the solution.

+

+ +

+

J. Mattauch, The masses of light nuclides according to an adjustment of nuclear data and a comparison with masses based on mass spectroscopic doublets. I n “NUdear Masses and Their Determination’’ (H. Hintenberger, ed.), p. 123. Pergamon, New York, 1957.

2.3.

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

15

TABLE 11. Reaction Cycles These are linear relationships which must be satisfied by the Q values in Table I by virtue of the fact that there are more equations in Table I than unknown masses.'

Table I1 contains 20 reaction cycles that relate the Q values of Table I. The first cycle, Q 1 6 - Q 2 4 &26 may be written out and evaluated

+

+ p - Be8 - d -& - p + Be8 + a Qza B11 + d - Be9 - a QI6 = Be9

-B11

24

=

0

= = =

0.559 k 0.002 -8.570 k 0.009 8.018 k 0.007 0.007 k 0.012

I n this example the experimental Q values are consistent with a cycle sum of zero. Cycles (6)-(11) all contain Q1= n - H1. If the cycle sum is set equal to zero, each cycle (6)-(11) determines a value of the mass difference n - HI, and the values determined in this way may have errors smaller than the direct determinations of the mass difference. For example, cycle (6) gives the value n - H1 = 0.7822 5 0.001 MeV, whereas the direct measurement of the beta-decay endpoint energy gives the experimental value listed in Table I, 0.783 jz 0.013 MeV. The weighted average of the several values of n - H1 obtained from the cycles and the directly measured value, should be substituted in the cycles (6)-(11) to reduce

16

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the error in the cycle sum. Similarly, cycles (12)-(16) determine values of n H1 - H2 = Q2, cycles (18)-(21) contain the mass difference 2 H Z - He4. In principle, the reaction cycles could be discovered by eliminating the unknown masses from the original set of n equations. However, in practice, it is much easier to construct the cycles by examining the closed figures formed by the links in Fig. 1 . Since each mass must appear at least twice in a reaction cycle, the links representing the reactions in the cycle form a closed polygon in the graphical representation. The sum or difference of any pair of cycles is also a cycle, and the choice of the cycles is therefore somewhat arbitrary. For the purpose of screening input data, the shortest cycles and those with the smallest errors in the sum are most useful. Mattauchs discusses limitations that should be observed in order to obtain independent cycles; he also considers the significance of the deviation from zero relative to the probable error of the sum. 2.3.2.3. Solution of the Equations. The method of least squares is the common method of solving the set of equations (2.3.2.1) to find the most probable values of t'he mass excesses, defined as those values M k for which the sum

+

m

has its minimum value, or those masses M k for which d S / d M k = 0. It is evident from the form of the sum S that an experimental observation Qi f AQi is to be given a weight ( l / A Q i ) 2 .This weighting of the observations may be accomplished conveniently by multiplying the ith equation of the set (2.3.2.1) by a factor (l/AQi). If aik = (l/A&i)Aik, and qi = (l/AQi)Qi, after multiplication the set of equations (2.3.2.1) becomes

and the least squares condition is n

m

for all Mk. Setting the derivative with respect to each of the m d u e s 6

J. Mattauch, L. Waldmann, R. Bieri, and F. Everling, Ann. Rev. Nuclear Sci. 6,

179 (1956).

2.3.

of

Mk

17

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

equal to zero leads to the m equations of the form n

m

C2

n

astaaiMi

s=l i

-

C

astq8= 0,

t = 1, 2, 3,

. . . , m.

(2.3.2.3)

8

The equations (2.3.2.3) are called the normal equations of the least squares solutions. I n vector form Eqs. (2.3.2.3)may be written aaa =

aq

(2.3.2.4)

where a is the matrix whose elements are aik, and 6 is the transpose of 6,&k = ski. The solution of Eq. (2.3.2.4) is =

(2.3.2.5)

(6a)-l&j = bQ

where (iia)-' is the inverse of the matrix &a,and the matrix b is defined by Eq. (2.3.2.5).Inversion of the square matrix 6a may be a very extensive computation if the number of unknown masses is large. Table 111 TABLE 111. Atomic Masses of the First Few Nuclides Appearing in Table I These are most probable values obtained by a least squmes solution of the set of equations in Table I. The values below were computed using the 0l6mass standard; the results are also expressed in terms of the C1*/12 mass unit.= 0 1 8 Mass standard atomic mass unit = 016/16

Nuclide

Mass excess (Mev)

n H1 H* Ha Hea He4

8.3622 f 11 7.5811 f 14 13.7189 f 20 15.8228 f 37 15.8044 f 37 3.6047 f 15

Atomic mass (am4 1.008 1.008 2.014 3.016 3.016 4.003

9806 1417 7334 993 973 8713

f 12 f 15 f 21 f 4

f 4 f 16

C12 Mass standard atomic mass unit = CL*/12

Atomic-mass (&mu) 1.008 1.007 2.014 3.016 3.016 4.002

6599 8213 0931 034 014 5988

Mass excess (MeV)

f 11 8.0662 f 12 f 14 7.2851 f 15 f 20 13.1269 L- 21 f 37 14.935 f 4 f 37 14.916 L- 4 5~ 15 2.4206 f I6

~~

For mass values computed from more recent and accurate Q values, see reference 1.

lists the first few mass values obtained by the least squares method of solution of the observational equations listed in Table I. 2.3.2.4. Accuracy of the Solution. According to Eq. (2.3.2.5), M i is a linear combination of the experimental Q values,

2.

18

DETERMINATION O F FUNDAMENTAL QUANTITIES

and the standard deviation AMi may be computed from the experimental AQ's n

(AMi)2 =

2 n

[(5) AQ.]'

=

aQ8

b:8 =

(bb)ii =

(6~);~.

(2.3.2.6)

8

8

Equation (2.3.2.6) follows from the assumption that all of the AQ's are random errors, independent of each other. Although this assumption is not true in practice, since many Q-value measurements contain a common source of error in the energy calibration standards used, no one has yet attempted to trace the truly independent errors though the vast amount of data involved in the input equations, and the assumption leading to Eq. (2.3.2.6) is usually made. One attractive feature of the least-squares method of solution is that the matrix (&a)-l which contains the AIMi information is computed during the course of the solution. Table IV lists the first few elements of this matrix (&a)-' obtained in the solution of the observational equations in Table I. The square root of the diagonal elements are the errors which appear in Table 111. The off-diagonal elements of the matrix (&a)-l are used to find the error AC in a linear combination of masses, C = ZlclMl, such as a Q value that one might compute from the masses: (AC)z

=

(2.3.2.7)

c~c~(&u),'.

Since the sum in Eq. (2.3.2.7) may contain negative terms, the value of AC may be much smaller than any of the AMt appearing in the expression for C. For example, from Table I11 we find that the value of H3 - He3 is 15.8228 - 15.8044 = 0.0184 MeV. The error in this value of the beta decay energy can be found by Eq. (2.3.2.7) with the matrix elements in Table IV.

A2(H3 - HeS) = aEs,H8

+

U E ~ S , H~ S2

=

13.76

~ ~ 8 , ~ ~ :

- 2(13.77)

+ 13.82

= 0.04; AC = 0.2 kev.

The error in this mass difference is far smaller than the error in H8 or Hea, and reflects the small error in the fourth observational equation in Table I. In the least squares solutions that have been published, Mattauchs and Drummonda have also published the matrix from which the errors can be computed. 8

J. E. Drummond, Phys. Rev. 97,

1004 (1955).

2.3.

19

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

TABLE IV. The Matrix (&z)-~ in Units of (kev)2 The standard error in the neutron mass is ( ( E U ) ; ; ) ~ / * = (1.277 kev*)”a = 1.13 kev. These matrix elements were computed using the 0 1 6 ma68 standard.

n H‘ HZ

Hs Hea He4

n

H’

H2

Ha

Hea

He

1.277 1.191 1.860 2.927 2.924 1.170

1.191 1.903 2.381 3.845 3.874 1.549

1.860 2.381 3.960 5.943 5.964 2.431

2.927 3.845 5.943 13.76 13.77 3.597

2.924 3.874 5.964 13.77 13.82 3.612

1.170 1.549 2.431 3.597 3.612 2.152

2.3.2.5. Other Methods of Solution. From the original set of n equations (2.3.2.1) containing m unknowns, one may choose a particular subset of m equations and solve them to find the m unknown masses. The mass values obtained in this way will, of course, depend on the particular subset chosen, unless the original set of equations (2.3.2.1)is self-consistent, i.e., unless the original equations satisfy the auxiliary conditions imposed by the reaction cycles. This follows from the fact that any particular subset can be converted into any other subset by adding or subtracting reaction cycles; if the cycle sums are zero, the solutions to the modified subset will be the same as the solutions to the initial subset. Hence another method of solution, employed by Tollestrup, Fowler, and Lauritsen17is to adjust the experimental Q values so that one obtains a set of Q values which satisfy all of the reaction cycles. The increment SQi = Q:di - Qf””, the amount which is added to the ith experimental Q value to produce a Q;dj in the self-consistent set, is made directly proportional to the weight of the experimental Q value, SQi a (l/AQi)z. Li et aZ.* describe a solution of this type for the equations listed in Table I in which the adjustment process was iterated until each cycle sum was within 1 kev of zero. Once the set of self-consistent Q values is determined, the solution of any m equations of the original set for the m unknowns is straightforward, as is the calculation of AM from the errors in the adjusted Q values. Whittaker and Robinsong describe the method of computing the error in the adjusted Q values. I n order to compute the error in a linear combination of the masses, it is necessary to express the desired combination as a sum of the adjusted Q values, then compute A. V. Tollestrup, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 78, 372 (1950). 8 C . W. Li, W. Whaling, W. A. Fowler, and C. C. Lauritsen, Phys. Rev. 83, 512 (1951). 9 E. Whittaker and G. Robinson, “The Calculus of Observt~tions,”p. 252. Blackie, Glmgow, 1924.

2.

20

DETERMINATION OF FUNDAMENTAL QUANTITIES

the error in the sum from the errors in the adjusted Q values. This method of solutions has the advantage, a t least for hand computation, that in the iterative procedure of adjusting the Q values, one sees a t each stage of the solution which of the input experimental values requires the largest adjustment in order to bring it into consistency with the remaining data. The examination of the input data by means of the reaction cycles is thus continued throughout the solution, and one has the opportunity of altering the choice of input data if that should appear to be desirable.

2.3.3. Atomic Mass Determination from Microwave Spectra* 2.3.3.1. Introduction. In microwave spectroscopy, t information on relative atomic masses is obtained by correlating the shift in frequency of a pure rotational absorption line of a molecule with isotopic mass substitution for one of the atoms in the molecule. The rotational frequencies of molecules fall in the microwave range (1 to 300 mm in wavelength), a region of the electromagnetic spectrum which only became accessible in the last two decades. Prior to the advent of microwave techniques the rotational spectra of molecules were observed as a fine structure on their vibrational spectra which fall in the infrared region. Historically there was one very notable infrared mass measurement of this type at its time. In 1932 Hardy et al.‘ measured the deuterium mass t o an accuracy of 0.1 mmu by comparing the spacing of the vibrational lines in the DCl vibrational spectrum with those of HC1. The most general class of molecule suitable for mass measurements is the symmetric-top molecule shown in Fig. 1 whose rotational energy levels are given byz-* Wg where B

=

==

hBJ(J

+ 1) + h ( A - B ) K z

(2.3.3.1)

h/87r21B and A = h/8?r21A.Here I B and I A are the moments

t See also Vol. 2, Chapter 10.6.1; Vol. 3, Chapter 2.1; and Vol. 4, Chapter 4.1.2. J. D . Hardy, E. F. Barker, and D. M. Dennison, Phys. Rev. 42, 279 (1932). Diatomic Molecules.” Van Nostrand, Princeton, New Jersey, 1950. * G . Herzberg, “Infrared and Raman Spectra.” Van Nostrand, Princeton, New Jersey, 1945. 4 C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy.” McGraw-Hill, New York, 1955. 1

* G. Hemberg, “Spectra of

* Section 2.3.3 is by S. Geschwind.

2.3.

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

21

of inertria of the molecule perpendicular and parallel respectively to the symmetry axis of the molecule; [ J ( J 1)]*% is the total angular momentum of the molecule and K h is its projection along the symmetry axis. If the molecule possesses a permanent electric dipole moment it will have a pure rotational absorption spectrum corresponding to the transition J - 1 -+J with AK = 0. The pure rotational absorption frequencies are then v = 2BJ. (2.3.3.2)

+

In many cases the rotational energy levels of a molecule are split by interactions of the rotation with the nuclear electric quadrupole moment

L-SYMMETRY AXIS 1

OF MOLECULE

I

H

FIQ.1. Symmetric-top molecule.

and nuclear magnetic moment. The theory of these effects are thoroughly explored in reference 4,for example. It has been worked out in sufficient detail so that the value of the rotational frequency as it would be in the absence of these effects can be precisely determined, and will be assumed to have been corrected for in all that follows. For a diatomic molecule B in Eq. (2.3.3.2) is inversely proportional t o the reduced mass which for DC1 is about twice that of HC1 so that the isotopic shift in this case is quite large, i.e., a factor of two in the rotational spacings. However, for heavier atoms such as C1 these shifts are, of course, considerably smaller. For example, in IC1 the shift in rotational frequency for the two isotopes C136and C13’ is only about 5% and, because of the

22

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

limited resolution of infrared techniques, mass measurements in the infrared region are completely unfavorable. However, in the microwave range, due t o the monochromatic oscillator sources available along with refined techniques of frequency measurement, the situation for accurate mass determination becomes extremely favorable. For example, the above-mentioned shift of 5% in the IC1 rotational spectrum corresponds t o a frequency shift of 1200 Mc in the 24,000 Mc region which in turn was measured to an accuracy of 0.01 Mc so that this change of two mass units in C1 can be experimentally measured to an accuracy of 0.05 mmu, and even more accurately (to 0.02 mmu) in other alkali halides. Even more striking is the relative measurement of the carbon and oxygen isotopes in the CO molecule to 2.5 micromass units by Rosenblum et aLs I n general, isotopic frequency shifts may vary from a few Mc to a few thousand Mc per mass unit in the 24 kMc region. In the middle mass region around Se, typical shifts are 100 Mc/mass unit in the 24 kMc region, which measured to an accuracy of 0.002 Mc correspond to a relative mass measurement 0.02 mmu, which compares favorably with other techniques of measurement. The major contributions t o this technique of mass determination have been made by Professor C. H. Townes and his students and reference to their work may be found in the review articles cited which contain tables of all masses measured by this technique.&* The accuracy of experimental measurement of isotopic frequency shift will be determined by absorption signal-to-noise ratio combined with absorption linewidth. These considerations will be outlined in the next section dealing with the various spectrometers. Secondly, the accuracy of mass determination will depend on small molecular corrections needed t o correlate experimentally determined isotopic frequency shifts of rotational spectra with mass change and this will be treated in Section 2.3.3.3. The general requirements for relative mass determination of isotopes of the same atom from microwave spectra may be summarized as follows. (1) The atom should be contained in a molecule with a permanent electric dipole moment (requirement for pure rotational spectrum). (2) The molecule should be in a gaseous state (lo-$ to mm of Hg), or in a molecular beam. B. Rosenblum, A. H. Nethercot, Jr., and C. H. Townes, Phys. Rev. 109,400 (1958). 5. Geschwind, G. R. Gunther-Mohr, and C. H. Townes, Revs. Modern Phys. 26, 444 (1954). 7 S. Geschwind, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flilgge, ed.), Vol. 38, Part 1, p. 38. Springer, Berlin, 1958. 8B. Rosenblum, C. H. Townes, and 5. Geschwind, Revs. Modern Phys. SO, 409 (1958). 5

f)

2.3.

MASS OF NUCLEI A N D INDIVIDUAL PARTICLES

23

(3) The molecule should have a reasonable isotopic shift of its rotational frequency for the atom in question (generally realizable and almost characteristic of microwave spectra). (4) The molecule should have a symmetry no lower than a symmetric top, as the spectra of asymmetric rotors are too complicated for mass determination.

Requirement 2 implies the need for only a very small amount of substance and only gm of S3swas present in the form of OCS in the gas absorption cell in the S3‘jmass determination by W. A. Hardy (private communication). In the case of radioactive isotopes much larger amounts may be needed, however, only to do the chemistry. The list of radioactive isotopes whose masses have been measured by microwave spectra may be found in earlier 2.3.3.2. Experimental Methods for Measuring Microwave Absorption Lines. Detailed accounts of the techniques of microwave spectroscopy may be found in the l i t e r a t ~ r e , ~ so s ~that - ~ ~only the salient features of the different types of spectrometers which have been used in mass determinations will be outlined here, It is characteristic of microwave gas absorption spectra that over many decades of pressure peak absorption intensity is constant, as long as the linewidths are determined by collision broadening. Thus, while the total number of molecules is proportional to pressure the peak absorption of an individual molecule is inversely proportional to the mean time between collisions or pressure, so that the peak absorption intensity of the gas is constant. As narrower lines are obviously desirable for more accurate frequency measurement, one therefore tries to work at as low a pressure as possible to lo-* mm of Hg) before other sources of line broadening set in, after which any further reduction in pressure would mean a loss in intensity. The different sources of line broadening are listed in Table I and are evaluated for the case of the linear molecule OCS. This table is pertinent to an understanding of the factors that lead to the selection of a particular spectrometer as described below. The factors contributing to the peak absorption coefficient, a! (the fractional absorption per centimeter), such as the dipole moment, matrix element for the transition, etc., are fully discussed in Chapter 13 of the work by Townes and S c h a ~ l o wThese . ~ authors also include various tables in the appendices which are helpful in calculating a! for the different molecules. The value of a! may range from to cm-’ or less in the M. W. P. Strandberg, “Microwave Spectroscopy.” Wiley, New York, 1954. W. Gordy, W. V. Smith, and R. F. Trambarulo, “Microwave Spectroscopy.” Wiley, New York, 1953. l1 “Microwave Spectroscopy.” Ann. N . Y . Acad. Sci. 66 (November, 1952). 9

lo

24

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

TABLE I. Contributions to Line Width of Microwave Gas Spectra Evaluated 2(Av) for OCS at room temperature

Half line width, AV (half-half width)d Doppler effect

A V / V = 3.58 X l O - T z / T l i i ?

Collision broadening (10-1 mm Hg)

Avo0ll

Power saturation and collision broadening.

120 kc

= 1 / 2 ~

” ‘v AvcO1l

38 kc at 24 kMc

-

4’

~?FPI~.&,,, 280 kc for P = 200pw -I- hic A ~ ; ~ , , and Avcall = 120 kc

Stark-modulation broadeningb

120 kc for f = 120 kc

dT

-

Wall collisionsc

Av

‘v

Av ,

0.6 X lo9

M

17 kc for 24 kMc wavcguide

R. Karplus and J. Schwinger, Phys. Rev. 73, 1020 (1948). R. Karplus, Phys. Rev. 73, 1027 (1948). M. Danos and 5. Geschwind, Phys. Rev. 91, 1159 (1953). KEY: T = temperature in degrees Kelvin; M = molecular weight; T = mean time between collisions; P = microwave power flow in ergs/sec cma; Ip.t,lmsx = largest dipole moment matrix element between the rotational levels a and b; f = Stark modulation frequency; A / V = wall area to volume ratio of gas absorption cell in cm-1. a

b

24 kMc region. A discussion of the limiting sensitivity of gas absorption cell spectrometers may be found in many place^.^*^^^^ 2.3.3.2.1. GAS ABSORPTION CELLS. 2.3.3.2.1.1 Basic Gas Absorption Cell (Crystal Video Detection). An outline of the waveguide gas absorption cell is shown in Fig. 2. The molecule under examination is contained in the hollow waveguide cell, in gaseous form at a pressure of approximately to mm. The microwave klystron oscillator is swept in frequency by application of a sawtooth voltage t o the repeller electrode. This same sawtJoothvoltage is simultaneously applied t o the horizontal plates of an oscilloscope to provide a time base proportional to frequency. As the frequency of the oscillator sweeps through a molecular absorption in the gas cell, the resultant reduction in the power reaching the microwave crystal detector, after passage through a video amplifier, is displayed on the vertical plates of the oscilloscope. The strength of these absorptions expressed in terms of the fractional absorption per centimeter, a, ranges from lo-* t o cm-l or less in the 24,000 Mc region. These absorption coefficients vary with the third power of frequency so 12

C. H. Townes and 6. Geschwind, J . Appl. Phys. 19, 795L.

2.3. MASS

25

OF NUCLEI AND INDIVIDUAL PARTICLES

that for very light molecules such as CO whose first rotational transition is in the 100 kMc region, the absorption becomes as large as 50% in a 6-foot length of waveguide. The frequency of an absorption line is measured very precisely by comparing the signal oscillator frequency with the harmonics of a standard quartz crystal oscillator which have been obtained by multiplication up to microwave f r e q ~ e n c i e s . ~ * ' ~ WAVEGUIDE GAS. ABSORPTION CELL

GAS INLET

TO VACUUM PUMP

\

\

M SAW TOOTH VOLTAGE GENERATOR

AMPLIFIER CROSS SECTION

FIG.2. Basic gas absorption cell microwave spectrometer.

2.3.3.2.1.2. Stark-Modulation Spectrometer. In practice these very small absorptions are masked by spurious signals from frequency sensitive guide reflections and discontinuities in the mode pattern of the microwave source klystron. To separate the true absorption from these spurious effects, the characteristic gas absorption frequencies are Stark modulated by applying a time varying (100 kc) square wave of voltage between the guide proper and a metal septum insulated from the guide as shown in Fig. 3. This technique was first introduced by Hughes and Wilson.14 The 100 kc component of the absorption is detected by a crystal rectifier, amplified in a tuned 100 kc amplifier and then detected in a phasesensitive detector. This scheme of detection greatly reduces spurious signals, since the amplitude of noise or spurious signals which fall in the pass band of the amplifier is small. Widths of absorption lines, as measl*S. Geschwind, Ann. N . Y . Acad. Sci. 66, 751 (1952). I4 R. H, Hughes and E. B. Wilson, Jr., Phys. Rev. 71, 562 (1947).

26

2.

DETERMINATION OR FUNDAMENTAL QUANTITIES

ured between points of half-maximum intensity, will vary from about 150 to 300 kc/sec in this type of spectrometer. This width results from the combined effects of Doppler broadening, pressure broadening, collisions with walls, Stark modulation, and power saturation (see Table I). Power saturation and collision broadening are major contributors to this width. A minimum of several hundred microwatts of power must impinge upon the silicon crystal detector for optimum detection sensitivity. However, this amount of power will saturate the absorption lines

SIGNAL OSCl LLATOR

I d l

-i

I

- VARIABLE -ATTENUATOR

I 1

I

1

INSULATING ,*STRIPS STARK WAVE GU IDE SECT I ON

i00 KC SQUARE WAVE GENERATOR

'I '

CRYSTAL DETECTOR

TUNED 100 KC AMPLIFIER

I

1 SAWTOOTH GEN ERATOR

-

FIG.3. Stark-Modulation spectrometer.

if the gas pressure is reduced too far, and so one is constrained by this power requirement for good detection sensitivity to work at pressures where collision broadening is a factor. 2.3.3.2.1.3. Balanced Microwave Bridge with Superheterodyne Detection.'* If one could use lower power levels without a reduction in sensitivity and consequently reduce the gas pressure, then linewidths would be limited by Doppler effect and wall collisions. This can be done by using a balanced microwave bridge with superheterodyne detection (shown in Fig. 4). The optimum power necessary for sensitive detection is furnished by the beat oscillator rather than the signal oscillator whose power can now be kept low enough (a few microwatts) t o avoid saturation with

2.3.

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

27

the reduced gas sample pressure of 10-3 to 10-4 mm. In the initial work with this spectrometer, the Stark modulation was dispensed with t o avoid distortion or displacement of the absorption line by Stark effects. Even without Stark modulation, amplitude fluctuations and mode discontinuities in the signal klystron and reflections in the arms of the bridge were canceled in first order by use of a symmetrical bridge. Linewidth of 60 kc (for OCS) and a sensitivity of a 3X cm-' were achieved with this spectrometer. The limit in sensitivity was here imposed by uncancelled second-order effects of reflections in the bridge arms.

-

SAW TOOTH

FIG.4. Balanced microwave bridge spectrometer with superheterodyne detection.

An improvement of sensitivity of a factor of ten in the balanced bridge scheme was realized by adding low frequency, 1 kc Stark modulation.16 However, great care must be exercised to accurately clamp the Stark voltage on zero to avoid displacement of the absorption line. 2.3.3.2.1.4. High Temperature Spectrometer. A noteworthy achievement was the extension of microwave gas absorption spectroscopy t o very Such a high temperahigh temperatures (25Oo-930"C) by Stitch et aZ.16v17 ture is necessary to obtain a sufficient vapor pressure for the alkali halide diatomic molecules, many of whose pure rotational absorption spectrum were studied by this technique yielding accurate mass information for the C1, Br, Li, K, and Rb isotope^.'^ A standard 100 kc Stark modulation spectrometer was used in which formidable difficulties were overcome in extending it to this high temperature range. l a G . R. Gunther-Mob, R. L. White, A. L. Schawlow, W. E. Good, and D. K. Coles, Phys. Rev. 94, 1184 (1954). 16 M. L. Stitch, A. Honig, and C. H. Townes, Rev. Sci. Instr. 26, 759 (1954). A. Honig, M. Mmdel, M. L. Stitch, and C. H. Townes, 86, 629 (1954).

28

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

2.3.3.2.1.5. Millimeter-Wave Spectrometers. Since the absorption coefficient, a,varies as the third power of the frequency, a in the wavelength region of 1 to 10 mm is often high enough so that absorptions can be seen directly without the need for Stark-modulation or superheterodyne techniques. Furthermore, Stark modulation cells are quite lossy a t millimeter wavelengths so that instead one uses frequency modulation of the signal klystron. This source modulation was used in a millimeter-wave spectrometer by Rosenblum et al.6,1sin their very high precision determination of the carbon, oxygen, and hydrogen mass ratios. 2.3.3.2.2. MOLECULAR BEAMELECTRIC RESONANCE.19s2o The ultimate limit upon linewidth imposed by the Doppler broadening in the gas absorption cell spectrometers, described in the preceding section, is '

MICROWAVE RADIATION

0

A-FIELD

C-FIELD

0-FIELD

\

FLOPPED-, BEAM

HOT WIRE DETECTOR

>

FIG.5. Molecular beam electric resonance apparatus.

circumvented in the electric resonance molecular beam type spectrometer shown in Fig. 5. The beam of molecules traverses successive regions of oppositely directed inhomogeneous electric fields separated by an intermediate region (the c-field) which for the study of rotational transitions has no static field. The effective electric dipole moment, peff,of a molecule is, among other things, a function of the rotational state, J. In addition, the force on a molecule is proportional to p e f f ( ~ E / d 2 Thus, ). by a suitable arrangement of stops a particular rotational state is focused at the detector. By irradiating the molecules in the c-region with microwaves of the proper frequency, transitions are induced from the J to J 1 rotational state, The resultant change in the J for some of the molecules changes the magnitude of their deflection in the B-field so that they are not refocused at the detector, causing a decrease in beam intensity.

+

B. Rosenblum, A. H. Nethercot, Jr., Phya. Rev. 97, 84 (1955). H. K. Hughes, Phye. Rev. 72, 614 (1948). zo J. W. Trischka, Phys. Rev. 74, 718 (1948). l*

lo

2.3.

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

29

Doppler broadening is absent as the molecules travel in essentially one direction relative to the microwave field. The linewidth is determined by the amount of time T , the molecules spend in the transition region 1. Typical linewidths that have been as is given by the relation Av * T achieved in mass measurements with this type of spectrometer range from 10 to 20 kc.21 By use of split RF field techniques,22linewidths of 1 kc are possible although no mass measurements have been made by this technique. 2.3.3.2.3. MASER-TYPESPECTROMETER.23A third general type Of microwave gas spectrometer is the gas-type maser (Fig. 6). While no mass measurements have been made with this device, it represents such an important conceptual and practical advance in the state of the art that it is worthy of mention here. I t is a combination of the above methods in that a molecular beam is used but a change in microwave

-

! U -d OUTPUT GUIDE

INPUT GUIDE

I,=_=L,_,_,_,---'----------_--_______________--__----J

GAS SOURCE

CAVITY

FOCUSER

FIG. 6. Gas maser spectrometer.

power absorbed by the beam is measured. The beam of molecules passes through a system of electrodes which produce a field gradient which focuses only those molecules in a particular energy state above the ground state into a microwave cavity. Microwave radiation of the proper frequency will stimulate emission to the ground state so that the output radiation is enhanced. A linewidth of 4 kc has been obtained for NHI at room t e m p e r a t ~ r e . ~ ~ , ~ ~ The maser-type spectrometer should also be capable of far greater sensitivity than the conventional ones.26 2.3.3.3. Correlation between Isotopic Shift of Rotational Frequency MOLECULES. The correlation and Mass Change. 2.3.3.3.1. DIATOMIC between frequency shift in the rotational spectrum and mass change is C. A. Lee, B. P. Fabricand, R. 0. Carlson, and I. I. Rabi, Phys. Rev. 91,1395 (1953) . N. F. Ramsey, Phys. Rev. 76, 996 (1949). 23 J. P. Gordon, H. Zeiger, and C. H . Townes, Phys. Rev. 95, 282 (1954). Z 4 J. P . Gordon, Phys. Rev. 99, 1253 (1955). M. W. P. Strandberg and H . Dreicer, Phys. Rev. 94, 1393 (1954). Z 8 C. H. Townes, Phys. Rev. Letters 6, 428 (1960). *I

22

30

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

most easily made for a diatomic molecule. Equation (2.3.3.2)is only a first approximation which assumes a rigid rotator. The effect of the vibration of the molecule is in a first approximation represented by v = 2J[B, -

(Y(V

+ 4)]

(2.3.3.4)

where Be = h/8r21, and I , is the equilibrium moment of inertia assuming the nuclei are stationary at their equilibrium separation, r,; v is the vibration quantum number and a is a correction which measures the rotation-vibration interaction; I , = pr.% where p is the reduced mass of the molecule. By measuring v for two different vibrational states, a can be found and hence Be. The equilibrium internuclear distance, r,, is determined to a very high approximation only by the electronic structure of a molecule, so that it is the same for different isotopic species. Thus by measuring the equilibrium B values first with masses M and mo and then with M and ml, where ml and mo are isotopes of the same atom, the relative masses of ml and mo will be given by (2.3.3.5)

Note that knowledge of re or constants such as h is not required. Assuming (ml - mo) is small, an uncertainty in M/mo of A produces an uncertainty in ml/mowhich is given by (2.3.3.6)

In practice this is usually negligible. In RbC1, for example, an uncertainty of 1 mmp in both Rb and C1 produces errors of only 1 part in lo7in the C1 mass ratio or Rb mass ratio. There are a set of approximations or uncertainties connected with Eq. (2.3.3.5)most of which can usually be very accurately corrected for or are negligible, which will now be briefly described. (a) Centrifugal distortion of the molecule due to rotation. This correction to Be is of the order of B,3/u82where ws is the vibrational frequency of the molecule. In all but the very lightest molecules this correction is usually negligible, When it is important, however, we can usually be determined with sufficient accuracy from optical band spectra to make this correction. 4sb (b) Anharmonicity of the potential function.27 Here corrections are also of the order of B,3/wz and are usually negligible except in lighter molecules, where they can also be corrected for as in *'J. 4. Duaham, Phys. Rev. 41, 721 (1932).

2.3.

MASS OF NUCLEI A N D INDIVIDUAL PARTICLES

31

(c) Contribution of molecular electrons to moment of inertia. Implicit in the writing of Eq. (2.3.3.5) is that the mass of an entire neutral atom, nucleus plus electrons is concentrated at a point. In reality, of course, the electrons have an extension in space about the nuclei and one must consider their extra contribution to the moment of inertia of the molecule due to this spatial extension. If the electrons are assumed to be almost spherically distributed about their respective nuclei and if indeed they rotated rigidly with the molecule, then one might expect the moment of inertia to be greater than the point mass assumption by approximately an amount equal to the moment of inertia of the electrons about their respective nuclei. If really present, this contribution to the moment VALENCE ELECTRONS NUCLEUS I

/’ ./

CLOSED

FIG.7. Illustration of the slipping of closed shell electrons in a rotating molecule.

of inertia would be quite large; but fortunately it is absent because the orientation of a closed shell of electrons remains fixed in space as the molecule r0tates.28,2~This slipping of the closed shells as the molecule rotates is illustrated in Fig. 7 and is akin t o the motion of a chair on a ferris wheel. On the other hand, the valence electrons may be visualized as rotating rigidly with the molecule and will give an extra contribution t o the moment of inertia, A single rigidly jotating valence electron at an average distance of approximately 1 A from its nucleus would give an error of less than one part per million in the C136/C137ratio in the alkali halides and is in general negligible in the heavier molecules. However, a similar valence electron in a lighter molecule such as CO results in an uncertainty of one part in lo6in the C12/C1aratio, which is one hundred times greater than the experimental error in this case. 2 8 G. C. Wick, Phys. Rev. 75, 51 (1948). 29See C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy,” p. 213. McGraw-Ell, New York, 1955.

32

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

However, the nonslipping valence electrons contribute proportionately t o the magnetic moment of the molecule, so that by measuring the magnetic moment of the molecule, one can very accurately correct for this extra contribution, A I of the valence electrons t o the moment of inertia above the idealized situation. This connection between the two derives from nothing more than Larmor’s theorem for rotating charges and is given by4J-g (2.3.3.7) where B:‘Om is the idealized Be value assuming all the mass for the neutral atom concentrated at a point. Here M p is the proton mass; m, the electron mass; p2&,the nuclear magneton; pJ, the magnetic moment of the molecule in the state J . As magnetic moments of molecules are of the order of a nuclear magneton for lighter molecules, this fractional correction t o the B value will be of the order of a part in several thousand and corresponded t o a correction of one part in loEin the C13/C12ratio in CO, which was one hundred times greater than the experimental error.6 (d) “Wobble” stretching: Rosenblum et aL6 have pointed out that in a Z molecule, even though the projection of the electronic orbital angular momentum along the internuclear axis is zero, it may still have perpendicular components which precess about the internuclear axis causing it t o “wobble” and hence produce a type of centrifugal distortion. The effect of wobble stretching can be taken into account only by using one accurately known mass ratio t o compare with the microwave value of the same mass ratio. In this sense it represents a basic limit on the accuracy of mass determination by the microwave method. This correction in CO (which was ten times the experimental error) amounted t o about 1 part in lo6 in the C and 0 mass ratios and was applied to the C12/C1sand 018/01E mass ratio determination using the known C12/C14 ratio to calibrate for this correction. The very detailed examination of the afore-mentioned electronic corrections t o mass determinations in the molecule CO by Rosenblum et aLs has confirmed the validity of the molecular theory involved t o very high accuracy. This is illustrated in Table I1 by noting the consistency of their results for the Cla/C12ratio in the molecules C 0 l Eand CO1*and the good agreement of these results with other determinations. In the higher mass region these electronic corrections are far less important. This is borne out by R b mass ratios in Table I1 which have not been corrected for elect,ronic effects. The microwave result in RbCl which was obtained with a molecular beam apparatus is seen t o be consistent with the determinations in RbI and RbBr which were made in a

2.3.

33

MASS O F NUCLEI AND INDIVIDUAL PARTICLES

TABLE11. Illustrative Mass Determinations in Diatomic Molecules Mass ratio Molecule

From microwave spectra

Rbsa/Rb87 RbC136 RbI RbBr C13/C1* C016 COls

0.9770163 k 45" 0.9770177 f 45d 0.9770146 f 55d 1.08361283 f 27" 1.08361290 f 28"

From nuclear reactions ~~~

~~~~

From mass spectra ~~

~

0.9770191 f 20 1.08361308 f 20a 1,08361278 k 200

1.08361309

3*

See reference 5. T. T. Scolman, K. S. Quisenberry, and A. 0. Nier, Phys. Rev. 102, 1076 (1956). J. W. Trischka and R. Braunstein, Phys. Rev. 06, 968 (1954). See reference 17. a T. L. Collins, W. H. Johnson, Jr., and A. 0. Nier, Phys. Rev. 94, 398 (1954).

0

high temperature gas absorption cell, and all three in good agreement with the mass spectrographic result. 2.3.3.3.2. MOLECULES MORECOMPLEX THANDIATOMIC MOLECULES. In molecules other than diatomic, zero point vibrations prevent the determination of accurate mas6 ratios but nonetheless an accurate determination of atomic masses can be made in terms of mass difference ratios. In analogy to the representation of the rotation-vibration interaction for a diatomic molecule, the effective B value, Bo for a polyatomic molecule may be written as2-4

B O = Ba -

2

ai(Vi

+ di/2)

(2.3.3.8)

i

where Be = equilibrium B value, ai is the rotation-vibration interaction, vi is the vibrational quantum number, and di the degree of degeneracy of the ith vibrational mode. In order to determine Be, one must measure BOfor the same rotational transition for each vibrational mode in both the ground vibrational state, vi = 0, and the first excited state, vi = 1. This is, however, exceedingly difficult as the Boltzmann population of the excited vibrational states of the less abundant isotopic species will be too low so that to measure all the a's is in general beyond the reach of the sensitivity of present-day microwave techniques. In addition, the a's in a polyatomic molecule depend in a rather complex way upon anharmonic potential constants and mass, so that knowing their values for a more abundant isotopic species does not allow one to theoretically predict their values for the less abundant species as one can do in a simple diatomic molecule. By a straightforward application of the parallel axis theorem of mechanics, it can be shown that the following relationship obtains for

34

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

isotopic substitutions of an atom lying on the symmetry axis of a symmetric-top or linear molecule : 4 * 6

m l - mo - BLl)) - M1B!2)(BLo) M~B:~)(B:o’ -~12))‘ m2 - mo

(2.3.3.9)

Here mo,m land m2 are the isotopic masses, M1 and MO are the molecular masses with isotopes ml and m2, respectively, and the B,‘s are the equilibrium B values corresponding t o the three different isotopic species. While the equilibrium B values appear in Eq. (2.3.3.8), it can be shown that the error introduced by using instead the effective or measured B value, Bo (determined from the measured rotational frequency by vrot = 2BoJ), is small. For example, a detailed estimate* of the error introduced by this neglect of a in the (P3- S a 2 ) / ( S 3 *- S12) mass ratio as determined from the linear molecule OCS shows that it is less than 1 part in 15,000. This corresponds to an uncertainty of 0.03 mmp in the determination of S33mass taking the and V4masses as known, and is representative of the accuracy obtainable in the middle mass region. As a matter of fact the best evidence for the smallness of the error introduced by neglecting zero point vibrations is furnished by the very good agreement between mass difference ratios determined by the microwave method, as outlined, neglecting zero point vibrations, and other methods. Such comparisons have been tabulated in several p l a ~ e s For . ~ exam~ ~ ~ ~ ~ ple, in OCS the oxygen atom, being light, undergoes large displacements from its equilibrium position so that one might expect the effect of zero point vibrations to show up in this case; yet the microwave value of (0’’ - 016)/(018 - OI6) = 0.501042 f 8 6 is in good agreement with the most recent mass spectroscopic values of 0.501045 & 3 3 1 and 0.5010446 jz 9 . 3 2 In addition, it can be shown that the typical uncertainty in M1/Mz is also generally unimportant.6 Thus if one measures the rotational frequencies corresponding to three isotopic substitutions for an atom on the symmetry axis of a symmetric top molecule and takes the masses of two of the isotopes as known, one can accurately determine the mass of the third. Taking the masses of two of the isotopes as known can be regarded as a calibration of the isotope shift and the effect of zero point vibrations. Note that no information concerning the structural parameters of the molecule, such as bond angles and internuclear distances is required. 30 S.Geschwind, in “Nuclear Masses and Their Determination,,’ p. 163. Pergamon, New York, 1947. 31 H. Ewald, Z. Naturjorsch. 6a, 293 (1951). 31 T. T. Scolman, K. S.Quisenberry, and A. 0. Nier, Phys. Rev. 102, 107 (1956).

2.3.

MASS OF NUCLEI AND INDIVIDUAL

PARTICLES

35

Further increase of the experimental accuracy, would bring one into the region where electronic effects and zero point vibrations would undoubtedly become important. It would be very difficult to correct for these effects in polyatomic molecules as one is able to do for diatomic molecules and so one must regard the present state of the theory of the electronic structure of polyatomic molecules as setting the limit of accuracy attainable in mass determinations in polyatomic molecules.

2.3.4. Measurement of Mass with Cloud Chambers and Bubble Chambers*

Cloud chambers have proved very useful in the determination of the mass of the unstable nuclear particles produced in nuclear disintegrations, and with the rapid development of bubble chamber technique, there is every hope that mass measurements will be commonly made with those instruments. The mass of a charged particle detected in a cloud or bubble chamber may be determined in various ways. The most important of these are the following: 1. 2. 3. 4. 5.

Momentum-ionization Momentum-range (energy) Momentum-change in momentum Ionization-range Scattering-range

Momentum-ionization. If the momentum p = mopyc and the velocity @c are measured, the mass mo can be found, since y depends only on p. The momentum can be measured either prior to the entry of the particle into the chamber, or in the chamber, if the chamber is immersed in a magnetic field. Likewise the velocity may be determined either within the chamber by measurement of ionization, or outside the chamber by time-of-flight, cerenkov, or ionization techniques. Since these two measurements are independent, the error in the mass is obtained from

where A p and AI are the errors in p and I , and m ois taken to be a function of p and I . It is found that the error in ionization usually contributes

* Section 2.3.4 is by W.

6. Fretter.

36

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

more to the error in mass than does the error in momentum, when both are measured' in the same chamber. Under favorable circumstances the error in mass for an individual particle may be as small as 5%. I n the relativistic region, where the slope of the rise is rather slow, the error in mass determination is much larger, but it is still possible to distinguish mesons from nucleons in the momentum range of 5-20 Bevlc. Momentum-range. If the particle can be stopped and its range determined, and a range-energy relation is available, the mass can be found from a combination of the equations p = moPycand kinetic energy

= moc2(y -

1).

There are two general procedures used, depending on the energy of the particle: either the particle is stopped in the medium of the chamber, or in heavy material as in a multiplate chamber. In either case a separate measurement of the momentum is usually necessary, since momentum measurements are virtually impossible in a multiplate chamber and inaccurate in a chamber when a particle approaches the end of its range. In a large bubble chamber, however, momentum and range measurements are simultaneously possible. Where the range-energy relation is accurately established, and accurate momentum measurements can be made, the momentum-range method is the most accurate for mass measurements. The range straggling error on an individual measurement is only one or two per cent, and momentum measurements can usually be made to a per cent or so under good conditions. Depending on the region in which the measurement is done, the error in mass on an individual determination is also of the order of one or two per cent. Momentum-change in momentum. When it is inconvenient to bring a particle to rest, but momentum measurements may be made, the momentum loss of a particle in passing through a given thickness of material may be used together with the momentum itself, to determine the mass. This method was discussed by Wheeler and Ladenburg.* It is not very accurate because it involves differences in momentum measurements which are ordinarily fairly small, and is not often used. Ionization-range. If momentum measurements cannot be made, the mass can be determined if the velocity is found from the ionization, and the energy from the range of the particle. Because of the inaccuracies in the ionization measurements, and the limited region over which they can be made, this method has not been used very much, except for fairly rough measurements. In a large bubble chamber, where simultaneous range and velocity measurements should be possible, this should provide

* W. B. Fretter and E. W. Friesen, Rev. Sci. Innstr. 26, 703 (1965). * J. A. Wheeler and R. Ladenburg, Phys. Rev.

60, 754 (1941).

2.3.

MASS

OF NUCLEI A N D INDIVIDUAL

PARTICLES

37

a method for mass measurements in the absence of a magnetic field. If the medium is so heavy as to give large multiple scattering making an ordinary magnetic field useless, this method should give useful results, as i t does in emulsions. Scattering-range. Rarely applied to chamber conditions, this method gives mo-yp2cand moc2(r- 1) which can be combined to determine the mass. It has been applied to multiplate chambers3 and to high-pressure cloud chamber^.^

2.3.5. Determination of Mass of Nucleons in Emulsions* I n previous sections (2.1.1.3, 2.2.1.1, an d 2.2.1.2) the method of charge, momentum, and energy determination of particles have been discussed. I n all calculations, connected with these problems, the knowledge of the particle mass or the simultaneous mass determination was implicitly involved. Therefore we will discuss below only a few cases of mass determination which are interesting from a methodical standpoint. The discussion will be limited to particles of single charge and mainly treat particles ending in the emulsion. 2.3.5.1. Constant Sagitta M e t h ~ d . l J .I~n the case of singly charged, nonrelativistic particles (pat = 2E), ending in the emulsion, the rangeenergy relation together with the measured mean scattering angle give sufficient information for the determination of the particle mass. The range-energy relation can be written in the form

E

= aRnm1-n

(2.3.5.1)

where a is a constant and n the power factor which actually is not a constant, but varies slightly with particle ~ e l o c i t yThe . ~ mean scattering angle is given by6

(2.3.5.2)

K being the scattering constant and t the cell length. Using relation C. Peyrou, Nuovo cimento [9]11, Suppl. No. 2, 322 (1954). P. Baxter and F. R. Stannard, Proc. Phys. Soc. (London) A70, 19 (1957). 1 S. Biswas, E. C. George, and B. Peters, Proc. Indian Acad. Sci. S A Y418 (1953). 2 C. Dilworth, S. J. Goldsack, and L. Hirschberg, Nuovo cimento 191 11, 113 (1954). 3 R. G . Glasser, Phys. Rev. 98, 174 (1955). 4 See Section 1.7.5.2. 6 See Section 2.2.1.2. 3 4

~

* Section 2.3.5 is by

Marietta Blau.

38

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

(2.3.5.1) with the approximation that the product momentum times velocity is equal to twice the particle energy, (2.3.5.2) can be written in the form 8 = -tllZR-nmn-l K

2a

(2.3.5.3)

where m is the only unknown magnitude, which therefore can be calculated. The method of constant sagitta, which has been briefly mentioned in connection with other scattering methods, was developed independently by the laboratories of Brussels and Bombay and was presented in a joint communication at the Congress on Cosmic Rays at Bagnere de Bigore, July, 1953. The most common procedure in scattering measurements of fast particles is the use of constant cell lengths along the whole track or limited track segments within which the particle energy can be considered constant. The rate of energy loss per unit length, however, increases with decreasing residual range and toward the end of a track the sections of nearly constant energy become exceedingly small. I n this region, measurements made with cell lengths, appropriate for the more energetic part of the track, lead by necessity to results of low statistical accuracy. In the constant sagitta method variable cell lengths are chosen in such a way that the mean sagitta remains constant along the whole track. With the aid of the range-energy relation one can calculate for particles of given mass the cell length required such that at a given residual range the mean value of the second difference I) has a predetermined value A, (2.3.5.4)

The relation between t, R, and m for a given value A can be written: t3/2

-

ml-nRn.

(2.3.5.5)

The general procedure in constructing a “scattering scheme” is the following. One starts at a point of the track with the residual range Ro and chooses for the first cell a value t o ; the subsequent cells tl, t2, . . . tn can be easily found from a log-log plot, describing the relation (2.3.5.5) for a particle of given mass m. A particle of mass m2 # ml, measured with the same scattering scheme will again lead to a constant second difference D,, which, however, is different from D,. The ratio of the two second differences is related to the ratio of the masses by (2.3.5.6)

2.3.

MASS OF NUCLEI AND INDIVIDUAL PARTICLES

39

Equations (2.3.5.5) and (2.3.5.6) are not strictly correct for various reasons. (1) The power factor n is not constant, but varies with particle energy. (2) The relation p@c = 2E is not valid for higher energies. (3) The scattering constant varies slightly with particle velocity and cell length. (4) Corrections for “noise,” so far neglected, have to be introduced. The effects due to factors 1-3 can be accounted for in various ways. Dilworth et aL2 have shown that an adequate procedure consists in first calculating the scattering scheme with a single power factor (the authors use n = 0.58), assuming a constant value for the scattering constant and neglecting deviations due to the assumption ppc = 2E, and afterward using correction terms to adjust the calculated value D,,,‘to Bobs,the value of the mean second difference observed. Biswas et al.’ use values for the power factor n which were found in experiments with protons with energies up to 35 MeV, and, for higher energies, values derived from a n empirical formula; for the scattering constant theoretical values, depending on cell length and particle energy, were used. Glasser3 determines the constants b and n

ID1 = bR-n

(2.3.5.4a)

from measurements on proton tracks of 57 Mev from a cyclotron beam by fitting the experimental data to a power law and found a power factor n, = 0.607. Equation (2.3.5.4a) is derived from (2.3.5.4) and the measurements were started a t a point RO= to of the track, where to is the initial cell length: the following cell lengths ti are then related to the first by ti = t o ( R / t ~ ) 2 n ’ [compare 3 (2.3.5.5)]. According to Glasser’s experiments the mean absolute second differences as a function of particle mass are then given

ID1 = 0.053

e)””” (to)0.893.

(2.3.5.7)

where m, is the proton mass. The noise problem (point 4) is less complicated in the constant sagitta than in the constant cell method. The cells are smaller and therefore the troublesome stage noise is either completely absent or has a t least a constant value. The most important source of noise is the grain noise, arising from the fact that the grain centers are not completely aligned along the trajectories: the error is independent of cell length and of the order of half a grain diameter. The independence of the error from cell length is a great advantage of the constant sagitta method, because of the possibility of choosing cell lengths in such a way that the signal-tonoise ratio assumes an optimum value for the whole trajectory.

2.

40

DETERMINATION OF FUNDAMENTAL QUANTITIES

According to Biswas et al.’ the optimum conditions for the signal-tonoise ratio can be expressed by the relation A = 2.42

(2.3.5.8)

where A, the mean absolute value of the true second differences, is given by A = dDZ - p2, and a the mean value of the noise. Relation (2.3.5.8) suggests that the scattering scheme should be chosen in accordance with particle mass, since A is a function of mass [Eq. (2.3.5.6)]. RESIDUAL RANGE R in mm,SCALE

IT

-

500

200 -SCALE

II

i

I

100

I I-

(3

z w

-1

50-

-I -I W

0

2 0 - SCALE

10

-

I

20

50 100 200 RESIDUAL RANGE IN p ,SCALE

500

1000

I

FIQ.1. Plot of cell length versus residual range for protons and pions

Figure I, taken from the paper by Biswas et aLli is a plot of cell length versus residual range for protons and pions, chosen in such a way that the mean absolute second differences after noise subtraction assume the constant values of 1, and 1.6 p, respectively. Glasser’sa experiments indicate that the optimum signal-to-noise ratio is obtained for D = 2.4Z where D is the observed value due to scattering noise. He finds that the optimum initial cell t o for R = t o is 4.4 p for protons, 1.9 for pions, and 3.9 p for a particle of mass 1000 me, where m, is the electron mms.

+

41

2 . 3 . MASS OF NUCLEI AND INDIVIDUAL PARTICLES

The error in mass measurements is relatively large, even if optimum conditions are used since: am 1 aA - = -_. m 1-nA [See Eqs. (2.3.5.4) and (2.3.5.5).] In Table I (Glasser) the percentage statistical error in mass measurements for particles of 1000 m, measured a t optimum conditions is given for tracks of various lengths. The error becomes considerably less severe, of course, if many tracks are available. TABLE I. Statistical Error in Mass Measurements Available length,

(I0 1000 3000 6000 10,000 50,000

Number of cells

45 81 131 178 350

Errors on 1 track

(%I

Errors on 25 tracks (%)

45 33 26 23 14

8.9 6.7 5.3 4.5 3

Tables with scattering schemes for various particles were published by different authors. 2.3.5.2. Photometric Method. Mass measurements on singly charged particles, ending in the emulsion, can be made with any one of the ionization methods described in earlier sections. However, for dense tracks near the end of the range, the blob counting or mean gap length methods are not suitable. In this case the method of gap or blob length per unit path length should be used, or a photometric method based on opacity or profile measurements of the track. In the following the Lund photometric method, the so-called mean track width method, will be discussed. The general measuring arrangement and the measuring procedure has already been described in the section on charge determination. Details of the multiplier equipment, the electronic part, and the stabilized high-voltage supply of the measuring instrument are published by von Friesen and Kristiansson.' The objective used in the measurements is a K.S. X 100, and the dimensions of the slit referred to the objective plane are 2 . 4 X 30 1.1. The optimum slit dimensions depend on the density of the track, the track inclination, and the background conditions. Each measurement with the track H. Fay, K. Gottstein, and K. Hain, Nuovo cimento Suppl. [9]11, 234 (1954). S.von Friesen and K. Kristiansson, Arkiv Fysik 4, 505 (1952).

42

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

segment below the slit was followed by two background measurements to each side of the track; the three measurements are used for the determination of the true opacity, due only to grains activated by the particle along its path through the emulsion. The method of “mass measurement” is based upon the following considerations, The mean track width, or MTW(Rm), of a particle of mass m a t a point with residual range R is for singly charged particles a function of the particle velocity, v , and hence of the ratio R/m. (2.3.5.9)

Because MTW depends also on development and emulsion conditions in a complicated way, it was found impossible to obtain a simple analytic expression for f(R/m). found it expedient to write the function MTW (Rm) The in the form of a power series: MTW(Z2m) = MTWo

+

(2.3.5.10)

where A, B . . . in the expansion depend on emulsion and development condition, but are independent of particle mass; MTWo, the mean track width a t zero range, is normalized to the value 100. The number of terms in the expansion depends on the length of the track. The constants A, B . . . can be determined for a special emulsion stack and development conditions if particles of known mass, ending in the emulsion are available. The MTW of two particles with masses ml and mz a t the same residual range can be written in the form MTW,

=

100

+ A I R + BIRZ+ -

. and MTWZ = 100

+ AzR + BzR2 +

*

.

*

where (Al/Az) = (mz/ml) and (Bl/B2) = (mz/m1)2. It is advantageous to check this ratio on tracks of particles with known mass in order to ascertain the absence of systematic errors. The mass of an unknown particle is determined by the method of least squares with the condition, that Eq. (2.3.5.10)should fit the experimental results as well as possible, or that

0=

2

{MTWL- MTW(Rm)) aMTW(Rm) am

n 8

S.von Friesen and K. Kristiansson, Arkiv Fysik 8, 121 (1954). K. Kristiansson, Arkiu Fysik 8, 311 (1954).

(2.3.5.11)

2.3.

MASS OF NUCLEI AND INDIVIDUAL

PARTICLES

43

where MTW1, MTWz . . . MTWL represent measured values of the unknown track a t residual ranges R I , Rz, . . . RL. The mass of the unknown particle m, is related to the mass m of the known particle by (m/m,) = (RL/RL,) if R I , R2, . . . RL and R1,,R2, . . . RL, are the residual ranges where the ionization and hence the MTW of the two particles are equal. Particles of unknown mass should be compared with protons or other known particle tracks found a t nearly the same part of the emulsion. The value of the MTW of a track depends on the depth in the emulsion; dept,h corrections can be applied, but may introduce additional errors. Furthermore the inclination of tracks used for comparison should be the same, and the best results can be obtained only if the dip angle is small. The error in mass measurements depends on the total track length measured of the particle used for comparison, on the residual range of the unknown particle, on the mass ratio, and finally on the resolving power of the emulsion. The errors are relatively small if optimum conditions prevail, because the method is based on a great number of single measurements. Kristianssons estimates the error in mass to 5 3 % ’ if tracks of a t least 1 cm length are available. 2.3.5.3. M a s s Measurement of Particles Which Do Not End in the Emulsion. I n the case of particles not ending in the emulsion, the conditions are more complicated, because in addition to the mass the particle velocity must be determined. It is difficult to treat this problem in a general way, since it depends strongly on the velocity of the particle, length of track, dip angle, and emulsion conditions. For particle tracks near minimum ionization, where the grain density does not change perceptibly over an appreciable length in the emulsion, the only possible method in determining the mass consists in a combination of ionization and scattering measurements (constant cell method). In this case it is helpful to use sets of curves-scattering angle or p p versus blob density-for particles of various masses. The accuracy of the method depends nearly entirely on the error made in the scattering measurement; in general this error will be large unless the tracks are very long and flat and the emulsion is free from distortion. I n gray tracks with more than 1.5 minimum ionization, the gradient of the blob density along the track gives additional information on the particle mass, provided that the track length is sufficient. Johansson and Kristianssonlo have applied the photometric method to gray tracks which do not end in the emulsion. They approximate the MTW versus range relation by a straight line, and the slope of this line represents the rate lo

F. Johansson and K. Kristiansson, Arkiv Fysik 11, 467 (1956).

44

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

of change, d(MTW),/dR, which is a measure of the particle mass. They again use normalization procedures and comparison with tracks of particles of known mass and velocity. The MTW and d(MTW)/dR, the latter in units of MTW, of the unknown track, are calculated by the least square method. They find that identification is possible even without additional scattering measurements if very long and flat tracks are available; the best results were obtained with blob densities between 2-5 times minimum value, but even with tracks as light as 1.75 times, minimum ionization mass measurements can be made if the track length exceeds 10 mm. Yekutielli and Rosendorf" have recommended blob density measurements along the track, plotted as a function of the distance from the entrance of the track to its final disappearance. They discuss the problem in a mathematical way and give the solution of the problem in an analytical form.

2.4. Determination of Spin, Parity, and Nuclear Moments 2.4.1. Spectroscopic Methods 2.4.1.1. Optical and Ultraviolet Spectroscopy.* t 2.4.1.1.1. HISTORICAL BACKGROUND. Hyperfine structure in optical spectra was discovered in 1891 by A. A. Michelson using his celebrated interferometer. Fabry and Perot, and Lummer and Gehreke also brought their interferometers to bear, and by 1910 a considerable number of observations had been made. At length it became clear that two causes of this structure were present, one due to the different isotopes of an element, now usually called isotope shift, and the other due to some effect within a single isotope. I t was not until 1924 that Pauli' suggested that the hfs in the case of a single isotope was due to nuclear moments. Subsequent experiment rapidly disclosed that many nuclei possess spin, and a number of magnetic and electric quadrupole moments were 11

R. Rosendorf and C. Yekutielli, Nuovo eimento Suppt. [9] 12, 416 (1954).

t See also Vol. 3, Part 2; Vol. 4, A, Chapter 4.2. W. Pauli, Naiurwissenschaften 12, 741 (1924).

J. E. Mack, Revs. Modern Phys. 22, 64 (1950). G. Laukien, Handbuch der Physik, XXXVIII/l, 338 (1958). G. H. Fuller, 1960 Nuclear Data Tables, National Academy of Sciences, National Research Council Part IV, Table 4, 72 (1961). J

4

~

* Section 2.4.1.1is by Joseph G.

Hirschberg.

2.4.

DETERMINATION OF SPIN,

PARITY,

A N D NUCLEAR MOMENTS

45

In the case of elements with several isotopes the measurement of nuclear moments is complicated by the fact that the structure of each of the various isotopes is displaced by the isotope shift. Tin, for example, has ten stable isotopes, several of which have additional hfs caused by nuclear spin. With the advent, since the Second World War, of the availability of separated isotopes, the evaluation of isotope shifts and nuclear moments has been greatly facilitated. 2.4.1.1.2. ISOTOPE SHIFT.A short description of isotope shift is included here for two reasons. In any evaluation of nuclear moments of an element with several isotopes, the isotope shift structure must be disentangled from the structure due to the moments of the nucleus in order to measure the latter. Also, the isotope shifts themselves, in the case of heavy elements may directly shed light on the structure of the nucleus. The isotope shift can be divided into three categories: that for light elements, that for heavy elements, and that for intermediate elements. For light elements, the effect is due to the difference in mass of the nucleus for the different isotopes, In its simplest form this occurs in the spectrum of hydrogen. In one-electron spectra the effect of a finite nuclear mass is taken into consideration by the use of the reduced mass p :

Mm ' " W m

(2.4.1.1.1)

where M is the nuclear mass and m is the electron mass. The energy T , of a particular level will depend on M , the nuclear mass according to (2.4.1.1.2) where T , is the value which would obtain if M were infinite. The energy levels of a lighter isotope are shifted more toward the series limit than those of a heavier one. This simple situation obtains only with the two-body case; hydrogen and hydrogen-like spectra. When more than one electron is involved, corrections to the simple reduced mass solution must be made. In this case, detailed knowledge of the wave functions is necessary to apply the corrections, and the measured shifts do not agree well with calculation, except in the simplest cases. Beyond mass 50 or 60, the mass effect becomes very small, and is finally obscured by the nuclear volume isotope shift which may be thought of as being caused by the finite size of the nucleus. Since the nucleus is not an infinitely small point, but has a radius of the order of All3 X 10-13 cm, the

46

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Coulomb field breaks down in its neighborhood. Orbital electrons whose wave functions have an appreciable value at the nucleus, such as configurations involving s and to a smaller extent p electrons, are most affected. This effect becomes important for 2 > 20 and increases toward heavier nuclei approximately as Z2.The effect on the energy levels is just opposite to the mass effect, the energy levels of the lighter isotopes lying lowest in the energy level diagram. For details in the evaluation of both shifts the reader is referred to references 5, 6, and especially 7. 2.4.1.1.3. EFFECTSOF NUCLEAR MOMENTSAND SPIN. Besides the isotope shift another cause of hyperfine structure was soon required, since such structure was often found to occur in nuclei with only a single isotope. As indicated above, this was ascribed in 1924 by Pauli to the existence of a nuclear spin, I, and to the interaction between the resulting nuclear moments and those of the optical electrons. The most important of the nuclear moments is the magnetic moment pr. The change in the energy of the atom due to the magnetic energy between this moment and the orbital electrons is given by: AW,

=

-

-prH(O)cos prH(0)

(2.4.1.1.3)

where H(O)is the average value of the magnetic field of the orbital electrons in the neighborhood of the nucleus; p~ is the magnitude of the nuclear magnetic moment; H(0) and pr are the vector field and nuclear moment respectively. This expression becomes : AW,

=

A I J cos(1,J)

(2.4.1.1.4)

when it is remembered that H(0) and J are antiparallel (due to the negative sign of the electronic charge), and pr is parallel to I (because the proton has a positive magneticlmoment). A = -prH(O)/IJ and is evaluated by a method considered below. We may form a vector diagram for the composition of two angular moments (see references 6 and 7), as in Fig. 1, where I, the spin of the nucleus, and J, the total angular momentum of the electrons, are combined to form F. We can apply the law of cosines and write cos(1,J) =

FZ - 1 2 - J 2 21J

(2.4.1.1.5)

5E. U. Condon and G . H. Shortley, “Atomic Spectra.” Cambridge Univ. Press, London and New York (1935). 6 H. E. White, “Introduction to Atomic Spectra.‘’ McGraw-Hill, New York, 1934. 7 H. Kopfermann, “Nuclear Moments.” Academic Press, New York, 1958.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

47

Thus :

W,

=

A - (F2 - I 2 - J')

(2.4.1.1.6)

2

A rigorous solution on the quantum mechanics6 necessitates the following changes: F2, 12,and J 2 are replaced by F ( F l), 1(1 l), and J ( J 1) respectively. The magnitudes pr and H ( 0 ) are replaced by the z components of and H(0) if I and J are taken parallel to the z axis. Thus we obtain: A (2.4.1.1.7) W , = 2, F ( P 1) - 1 ( 1 1) - J ( J 1)

+

+

+

+

+

+

where F can assume the values

I+J,

I + J - l ,

I + J - 2 * . . ( I - J ) . (2.4.1.1.8)

This expression for the splitting AW, results in the characteristic hyperfine flag patterns, in which the same interval rule occurs as in

FIG. 1 . Vector diagram for combining I and J.

FIG.2. Hyperhe structure levels showing interval rule.

ordinary fine structure: The energy difference between two adjacent hfs levels, for instance one of F and one of F - 1, is just proportional to F. For example, if I = and J = 2 then F can have the values 4, $, +, 4 by Eq. (2.4.1.1.8)and the levels will appear as in Fig. 2. In many cases, the nuclear spin can be determined simply from the number of levels, or from the intervals, but in more difficult cases where some of the structure cannot be resolved, intensities must be determined. Detailed methods are discussed by Kopfermann.7 Sometimes it is noticed that the interval rule is not exactly followed. This effect is attributed to an electrostatic interaction with the orbital electrons by the nuclear quadrupole moment. The only other nuclear moment which normally can be measured by

2.

48

DETERMINATION OF FUNDAMENTAL QUANTITIES

means of hyperfine structure, is the electric quadrupole moment, Q. The additional energy due t o the quadrupole moment AWQ is given by: (2.4.1.1.9)

where e is the elect,ronic charge, the electric gradient a t the nucleus in the z direction and 0 is the angle between the direction of the quadrupole moment and the electric field gradient. The quadrupole moment may be thought of as a measure of the departure of the nucleus from spherical shape and has positive or negative sign according t o whether the nucleus is prolate or oblate, respectively. Casimirs has expressed the formula for AWQ as follows:

where

c = F ( F + 1) - 1(1+ 1) - J(J + 1)

and

B

= eQ@JJ(O)

where ipJJ(0)is the average gradient in the J direction of the electric field due t o the orbital electrons at the nucleus. We have now obtained expressions for the effect on the atomic energy levels of the magnetic dipole and electric quadrupole moments in terms of two constants A and B. If these formulas are combined we obtain for the total effect: AWF = AWm AWQ. (2.4.1.1.11)

+

As has been suggested above, much the largest effect is due t o th e magnetic moment ; the AWQ is often negligible and when present can be treated as a perturbation on AW, resulting in a departure from the intervals predicted. The hyperfine structure levels are affected by external magnetic fields in a similar way t o the ordinary fine-structure levels, resulting in Zeeman and Paschen-Back effects. A discussion of these will be found in Kopfermann’ and White,6 and is beyond the scope of this treatment. I n the study of spectra, transitions between levels are observed, not the levels themselves. This requires a consideration of the combination rules, telling us which levels combine with each other. The general rule is that values of a given F will combine so that AF = 0 or k 1. The intensities depend on I and J as well as F , and tables may be found in standard texts. 8

H. B. G. Casimir, Teyler’s Tweede Genootshap, Verhandel. (Haadem) (1936).

2.4.

DETERMINATION

OF SPIN, PARITY,

AND NUCLEAR MOMENTS

49

As an example of the transitions to be expected, consider a simple example: I = $, and J = 0 and 1. Here (see Fig. 3) there are just as many lines as F levels in the lower J state since there is only one hyperfine upper state. Other simple cases arise when one of the states arises from a configuration not split by the nuclear moments, the result then being similar t o the example.

1/2

FIG.3. Hyperfine structure transitions.

2.4.1.1.4. EVALUATION OF MOMENTS. The nuclear moments which have been determined from the study of hfs are the mechanical or spin moment I , the magnetic dipole moment p, and the electric quadrupole moment Q. As mentioned above, F , the total angular momentum, can assume J - 2 * - ( I - J ) , (or if J is the values I J , 1 J - 1, I larger than I , two I 1 levels) and for each value of F there is a hyperfine structure level. I can be determined if the spectrum is sufficiently well resolved by counting the maximum number of hfs components. If this cannot be done, for example because of insufficient resolution, the interval rule, intensities, or Paschen-Back effect may be used. By the use of Eqs. (2.4.1.1.7) and (2.4.1.1.10), the constants A and B can be deduced from the measured structure. As indicated above, A is related t o the nuclear magnetic dipole moment and B t o the electric quadrupole moment, and these latter could be determined exactly if the values of the mean magnitude field H ( 0 ) and the mean electric gradient in the J direction in the neighborhood of the nucleus were known. Goudsmitg has deduced formulae for A in the single electron case in terms of the nuclear g-factor gr, where pr = (e/2MP)hIgr, e is the electron charge, h is the Planck-Dirac constant, M , is the proton mass, c is the velocity of light, and I is the nuclear spin; for electrons with I > 0:

+

+ +

A. =

+

MPhcR,a2Z3 mn3(Z

9

-

+ + ) j ( j + 1) gz

S. Goudsmit, Phys. Rev. 48, 636 (1933).

(2.4.1.1.12)

50

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

and for s electrons

8 M,hcR,a2ZS A, = 3

mn3

BI

(2.4.1.1.13)

+,

Here R, is the Rydberg, a the fine structure constant, 1 and j the orbital and total electronic momentum quantum numbers respectively. For alkali-like and alkaline earth-like spectra similar formulas exist, semiempirical in nature, and indeed these calculations can be extrapolated to complex spectra with good accuracy.I Casimirs has obtained an expression for the B factor involving the quadrupole moment Q : 1 2j B = -@&(2.4.1.1.14) < l / r 3 > Rr(l,j) H. 2j+ 2 Unfortunately, the factor < l / r > , the average reciprocal of the magnitude of the electron radius vector, appears in this equation, but. this can be eliminated by consideration of the A factor in the derivation of which < l / r > also appears. The following relation7 can then be obtained:

Q

+

B PB' 41(1 1) P I m 1 F, A e2 j(2j - 1) pn M, I R,

= EOPO- -

(2.4.1.1.15)

where eo and P O are the electric and magnetic fundamental constants, 8.85 X 10-l2 farads per meter and 41 X lo-' henrys per meter respectively, p n is the nuclear magneton, pB the Bohr magneton, p r the nuclear magnetic moment and P, and R, are relativity corrections tabulated by Kopfermann. 'I 2.4.1.1.5. LIGHTSOURCES. The hfs splitting is smaller than the ordinary fine structure of atomic spectra by about the ratio of the nuclear magneton p n t o the electron magneton, or roughly %&. Thus, the evaluation of nuclear properties by optical spectroscopic methods ordinarily involves the measurement of very small energy differences, often of the order of thousandths of a wave number unit, To measure energy differences of the order of a few thousandths of a wave number unit, only a few light sources can be used. The principal causes of broadening that are encountered are: Doppler effect, Zeeman effect, and Stark broadening. Doppler broadening resulting from random thermal motion is given by:

where r D is the half-intensity breadth of the Doppler-broadened line, T is the absolute temperature of the source, and M the molecular weight

2.4.

DETERMINATION

OF SPIN, PARITY, AND NUCLEAR MOMENTS

61

of the radiating atom. From this we see that for narrow lines a cooled source is necessary; for light elements cooling by liquid air is commonly used. An atomic beam, where the effective temperature may be very small indeed, has been used t o obtain some of the narrowest spectral lines yet attained. The Zeeman effect is proportional t o the magnetic field strength, and in most sources is not a troublesome source of broadening. The Stark effect is of two types. A relatively large effect proportional t o the electric field occurs in hydrogen, and hydrogen-like atoms. In other cases, a much smaller quadratic effect obtains, proportional to the square of the field. Because of random intermolecular and interatomic electric fields, the linear Stark effect can contribute t o the broadening of spectral lines, the Stark broadening, r8,being approximately proportional to the density. For hydrogen and hydrogen-like spectra subject to the linear Stark effect, rdis especially large and the spectral lines are greatly broadened, especially at high pressures or if large externally applied electric fields are present. Because of the factors discussed above, we see that sources for high resolution must exhibit low temperature and low density, and should have as small magnetic and electric fields as possible. Other requirements, in some cases, are at least moderate intensity and sometimes high excitation. The accompanying tabulation summarizes the most commonly used sources with their advantages. ~~~~~~~~~

Type of source Arc Spark Vacuum spark Hollow cathode

~

Excitation energy Medium High Very high Medium

Electrodeless Low (radio frequency) Atomic beam in Medium emission Atomic beam in Resonance absorption lines only

Doppler temperature High High High May easily be cooled May be cooled May be very low May be lowest

Brightness

Pressure

Fields

High High High Medium

High High Low Low

Medium High High Very low

Medium

Very low

Low

Low

Lowest

Very low

Low

Lowest

Very low

From this tabulation we see that three sources appear to be of greatest use for the study of hfs: the hollow cathode, the electrodeless discharge tube, and the atomic beam. The hollow cathode (see Fig. 4), first used by Paschen, has been brought to a high state of development by many workers, especially

52

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Schiiler, Arroe, and Mack. It combines simplicity, ease of cooling, and very low electric fields with relatively low pressures, & to 10 torr. The electrodeless tube, excited by radio frequency, has the great advantage t ha t no electrodes are necessary, so that the tube can simply be of glass or quartz. It can also be cooled, but not quite so easily, and has the complication that suitable radio-frequency excitation must be provided. It also operates at low pressure, in the neighborhood of torr. An example is the well-known Meggers lamp, containing Hg198.

FIG.4. Hollow cathode discharge tube.

FIG.5. Atomic beam.

For ultimate resolution, where the line breadth must be reduced as far as possible, the atomic beam (Fig. 5 ) is used. Here the effective temperature is dependent only on the component of the atomic velocity in the line of observation. If the average velocity of the atoms emerging from the oven at temperature To is Do, where mijo2/2 = kTo, then the component V8 in the line of sight is 2fio tan 8, so that T,, the effective temperature, is given by T , = 4To tan2 0. For a well-collimated beam, 0 may be as small Thus even for a n oven a t 1000"K, the as lo, so that tan2 0 = 3 X effective temperature in the beam can well be reduced t o the order of l0K! The atomic beam is, however, much more complicated than the other light sources mentioned, requiring an oven and some means, for example an electron gun, for exciting the spectrum (in the case of the

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

53

emission beam) : it is usually not especially bright. A discussion of atomic beams has been given by Meissner.10 2.4.1.1.6. SPECTROGRAPHIC APPARATUS.* As noted above, the problem of determining nuclear properties by measuring the hyperfine structure of atomic spectra entails high resolving power. Of the many possible high-resolution spectroscopic instruments, only a few can be selected for discussion here and we will confine our remarks to the two most commonly used-the diffraction grating and the Fabry-Perot interferometer. As developed by Rowland in the nineteenth century and especially by Harrison in recent years, the diffraction grating (plane and concave) is a practical instrument of high resolution, suitable from the vacuum ultraviolet t o the far infrared. The theoretical resolving power R of the grating is given by the simple relation : (2.4.1.1.17) R = A/Ah = nN where n is the order number of the spectrum, and N is the total number of grooves in the grating. With modern ruling techniques it has been possible t o produce highly efficient gratings 25 cm wide with upwards of 15,000 grooves per centimeter, or more than 375,000 grooves. This affords a resolving power of over a million in the third order; such diffraction gratings can be very useful in the measurement of hfs, especially in the ultraviolet where high orders can be employed. The Fabry-Perot interferometer, devised toward the close of the nineteenth century, has emerged during the last decade with the advent of mnltilayer coatings and the photomultiplier as a n instrument t o be used for highest resolution. The resolving power of the Fabry-Perot is virtually unlimited, since it is just proportional to the separation bet ween the plates. Separations of more than a meter have been achieved, corresponding t o resolution widths smaller than any presently-available spectral lines. The Fabry-Perot, operating as it does at extremely high order, ordinarily requires a predispersion monochromator to eliminate unwantrd orders. Despite this complicating factor, the use of the Fabry-Perot, both photographically and photoelectrically, has become very widespread. This interferometer, as is well known, consists of a pair of partially reflecting mirrors, accurately flat and parallel (see Fig. 6). The light passed by the Fabry-Perot is a series of ring-shaped fringes a t infinity, the intensity of which, for perfect plates, is given by the Airy forrnula:

I * See also lo

=

T,,,[l

+ F Z sin2 6/2]-'

Vol. I , Part 7.

K. W. Meissner, Revs. Modern Phys. 14, 68 (1942).

X IO

(2.4.1.1.18)

54

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

where T,,, is the maximum transmittance, T,,, = T 2 / ( 1- R)2, where T and R are the transmittance and reflectance of the partially reflecting mirrors; F is proportional to the sharpness of the fringes and is given F = 2 di?/(l- R ) ; 6 is the phase difference of adjacent rays, and 10is the incident intensity. When the absorption of the partially-reflecting films is very low, T,,, is very nearly unity, and as Jacquinot” pointed out, the Fabry-Perot is, under these conditions, the most luminous of spectrometers ; its transmittance is highest for a given spectral range. Multilayer films achieve extremely high reflection, upwards of 95 %, with very low absorption, so that the sharpness of the fringes is limited not by the reflectivity through the function F, but by the available flatness of the plates.

Fro. 6. The Fabry-Perot interferometer.

The traditional method of using the Fabry-Perot is photographic and fully described elsewhere.10p12The photographic method has a number of disadvantages, however-among them nonlinearity in both wavelength and intensity, the first being quadratic and the latter logarithmic. A photoelectric method, developed by Meissner,I3 Jacquinot, and others, eliminates the two difficulties mentioned, and in addition has a considerable advantage in sensitivity. Here a diaphragm is placed at the center of the fringe pattern, and the light which passes through falls on a photomultiplier or other suitable detector. It can be shown that 6 = (%/X)2pt cos 8, so that I can be made to vary by changing 1/X, p , t, or cos 8. In the photoelectric method p is usually changed by varying the gas pressure between the interferometer plates, although sometimes t and 8 are varied. As 6 is changed, by whatever method, the spectrum is spread out repeatedly, and a monochromatic line appears as in Fig. 7. The condition P. Jacquinot, J . Opt. Sot. Am. 44, 761 (1954). S. Tolansky, “High Resolution Spectroscopy.” Pitman, New York, 1947. 18 K. W. Meissner, J . Opt. SOC.Am. 31, 405 (1941); ibid. 32, 185 (1942).

11

12

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTG

55

for the maxima is evidently 6 = 2 m , whence

nX = 2pt cos

e

(2.4.1.1.19)

often quoted as the fundamental formula for the Fabry-Perot. The halfintensity breadth, A61/z, for the perfect (exactly plane plates of infinite extent) interferometer is given to a first approximation by Meissner as 2F/r or 0.64F. Figure 8, which shows the logarithm of the transmittance TO,where To = Z/Zo [see Eq. (2.4.1.1.18)lof the interferometer plotted against the logarithm of the phase angle 6, indicates the strong dependance of the half-intensity width on the reflectance of the mirrors. In practical interferometers, a flatness of about & wavelength (green) can be expected, which would allow a reflectance of about 97% to be used. A 63

I I

2

6-

I

'

-I

FIG.7. Schematic appearance of a monochromatic line in a photoelectric Fabry-Perot.

higher R than this would merely lead to loss of light; for poorer plates, correspondingly lower reflectances must be used. As can be seen from Eq. (2.4.1.1.19), the wavelength free spectral range, @A, between successive peaks (n changing by l ) , is just X2/2pt, or, in terms of wave number, 6iv = l / 2 ~ t . In general, the predispersion instrument must have a wavelength band pass less than or equal to 26i to prevent the confusion of orders alluded to earlier. 2.4.1.1.7. RESULTSFROM SPECTROSCOPIC OBSERVATIONS. Of the many possibilities, two examples are given of the phenomena and methods described above. The first is a measurement of the spin of U 2 3 5 by Blaise et al.14Figure 9 shows two orders of a tracing by a photoelectric FabryPerot interferometer of a uranium hollow-cathode discharge cooled by is separated from the liquid nitrogen. The very strong line due to UZ3* U236and U233structure by the isotope shift. The U236components are not l4

J. Blaise, S. Gerstenkorn, and M. Louvegnies, J . phys. radium 18, 318 (1957).

8

FIG.8. The Airy formula.

-1

235

i- 233-l

FIQ.9. Hyperfine structure of Ua36. 56

235 :-233-1

.

100

50

0

Tb159 I 3/2 FIG.10. Hyperfine structure of

TbI59.

2.

58

DETERMINATION OF FUNDAMENTAL QUANTITIES

completely resolved after the first three, a, b, and c, but by the use of the intensity rules, the spin, I, of 4 was confirmed. The other example,16 Fig. 10, very clearly shows the hfs flag pattern in Tb169.Two orders of a photoelectric Fabry-Perot trace are presented, clearly showing I = $ and exemplifying the excellent resolution and intensity measurement possible with this method. 2.4.1.1.8. SUMMARY. The optical spectroscopic method provided the first knowledge of nuclear moments, and still is the source for much of present-day knowledge. Although superseded in some cases by more accurate methods, such as molecular beams and nuclear resonance, it is still indispensable in many cases where the other methods are not applicable or are applicable only with difficulty. Two examples are: (i) shortlived or rare isotopes where the quantities required for a beam are out of the question; and (ii) elements whose electronic ground state has no magnetic moment such as the rare gases. Several lists of moments have been p u b l i ~ h e d . ~ - ~ ACKNO w LEDGMENTS It is a pleasure for the author to acknowledge the kind help of many in preparing this section, especially of Dr. Jean Blaise, Bellevue, France, who put original photographs of hfs at the author's disposal, and of Professor Julian Ellis Mack of the University of Wisconsin, who read the section and made several valuable comments.

2.4.1.2. The Investigation of Short-lived Radionuclei by Atomic Beam Methods.* 2.4.1.2.1. INTRODUCTION.Atomic and molecular beams pro-

vide one of the oldest methods of experimental physics. Their earliest applications were, as to be expected, to the kinetic theory of gases, because study of the effusion of gas molecules from a source was of fundamental importance in understanding the Maxwellian theory of gases. The technique was soon recognized, however, as especially valuable for the study of molecular, atomic, and, later, nuclear structure. I n a beam, the atomic or molecular system is in effect isolated from any other molecule or atom. Thus, in a study of atomic, molecular, or nuclear properties, all experimental complications arising from the effects of the solid, liquid, or gaseous state can be avoided. If the experiment utilizes a resonance method, the lower limit to the width of the resonance line depends only upon the radiation lifetime and the duration of 0bservations.t lr

J. Blaise and F. Tomkins, private communication (1961).

t For further details of the atomic beam method see also Vol. 4, A, Chapters and 2.2; Vol. 4, B, Sections 7.1.1, 7.2.1, and 9.1.2.

* Section 2.4.1.2 is by John C.

Hubbs and William A. Nierenberg.

1.3

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

59

Advantages of the reaction-free state become apparent from consideration of the numbers involved in a typical experiment with a nonradioactive beam. Such a beam consists of 1O1O or lo1' atoms per second at the detector in an apparatus whose length is of the order of 1 meter. The distribution of velocities is the Maxwellian distribution multiplied by the velocity v. The mean velocity may be 50,000 cm/sec, giving a lineal density of -loB atoms/cm; for a beam 0.01 cm wide and 1 cm high, this is a volume density of -lo8 atoms/cm3. The result is a much smaller collision rate than in a standard gas density of 10lg atoms/cm3. Furthermore, if the cross section for collision of the beam atoms with themselves is spherically symmetric, the probability that products of collisions will remain in the beam is only a little greater than the apparatus transmission. Since this is about or less, the effect of collisions is negligible. Scattering due to residual gas in the apparatus is scarcely significant: the pressure of the residual gas is 5 X lo-' mm of Hg, or less, which corresponds to a mean free path ten times the length of the apparatus. The molecular (or atomic) beam became an important device for modern physical research when 0. Stern and his collaborators used it for studying such diverse phenomena as molecular scattering and space quanti~ation.l-~I n these experiments he studied the deflection of the atomic system in very strong inhomogeneous magnetic f i e l d ~ . ~The J specific conformation of magnet pole tips required to produce this inhomogeneity still goes by the name Stern-Gerlach field. The next major step in the technique, based on an interesting theoretical observation by Breit and Rabi,6 was put into experimental practice by Rabi and his collaborators.~-9 This concerned the strong magnetic interaction between the currents produced by the electron angular momentum of the atom and the nuclear magnetic moment. The nuclear spin is rather strongly coupled to the electronic angular momentum, and the two systems would not be decoupled in a magnetic field of less than 1 L. Dunoyer, Compt. rend. acad. sci. 162, 594 (1911); 167, 1068 (1913); Le radium 8, 142 (1911); 10, 400 (1913). 9 W. Gerlach and 0. Stern, Ann. Physik [4] 74, 673 (1924); [4] 76, 163 (1925).

0. Stern, 2.Physik 7, 249 (1921). R. D. Frisch and 0. Stern, 2.Physik 86, 4 (1933). I. Estermann and 0. Stern, Z. Physik 86, 17 (1933); 86, 132 (1933); I. Estermann, 0. C. Simpson and 0. Stern, Phys. Rev. 62, 555 (1937). 'G. Breit and I. I. Rabi, Phys. Rev. 38, 2082 (1931). I. I. Rabi, Phys. Rev. 49, 324 (1936). *I. I. Rabi and V. W. Cohen, Phys. Rev. 48, 582 (1933); D. R. Hamilton, Phys. Rev. 66, 30 (1939). SS. Millman, I. I. Rabi, and J. R. Zacharias, Phys. Rev. 63, 384 (1938); N. A. Renzetti, Phys. Rev. 67, 753 (1940). 4

60

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

several hundred gauss.* I n a weak field, therefore, the two systems behave as one, with an angular momentum that is the vector sum of the two angular momenta, but with a magnetic moment that, to a very good approximation, is due to the electronic structure alone. Thus the behavior of the system as a whole in a n external field is primarily influenced by the spin of the nucleus, and not by its magnetic moment, except through the hyperfine coupling. This was the basis for the zero-moment method used by Rabi and his collaborators for the measurement of the spins, magnetic moments, and quadrupole moments of the stable isotopes of the alkalies, gallium, and indium. (The Breit-Rabi diagram is later explained in some detail.) The next step in the development of the subject was the discovery and application by Rabi of the nuclear radiofrequency resonance method to molecular beams. This was an advance of many orders of magnitude in the precision of measurement of nuclear moments, and the application of the method to HD and D2led to the discovery and measurement of the deuteron quadrupole moment. lo Immediately afterward, the method was applied to the radiofrequency spectroscopy of alkali atoms by Millman et al.,” and because of its enormous precision, virtually supplanted the zero-moment method. Zacharias measured the spin of K40, the rare radioactive isotope of potassium, by an ingenious application of the method, and thereby demonstrated its applicability to trace amounts of material in the presence of large numbers of background atoms. l2 The resonance method was extended to measurements

* Throughout this paper all magnetic moments are in units of

the Bohr magneton,

PO, about 0.927 X lO-*O erg/gauss. The vector magnetic moment of the electron

system is gJpoJ, where J is the vector angular momentum in units of h, and gJ is known as the gyromagnetic ratio of the system and depends upon the details of the e‘ectronic motion. For a single electron in an s state, we have gJ E - 2, and, in general, g J - 1. For a nucleus, however, the magnetic moment is gIp& where I is the vector nuclear mlm. As a n example, for a single s spin in units of h, and, what is important, gr electron, the magnetic field at the nucleus due t o the spin oi’ the electron is roughly 2p0/r3. If r is taken as ~ 0 . X 5 10-8 cm, this is equivalent to about lo6 gauss. A unit nuclear moment in this field of 106 gauss is, therefore, equivalent to about 5 X erg. When this is divided by Planck’s constant, one gets 108 cps or 100 Mc/ sec, if frequency is used as a unit of energy. This represents a rather strong coupling of the nuclear spin t o the electronic angular momentum, and, since the electron precession frequency in a magnetic field, if uncoupled, is gJpoH/h, or about 1.4 Mc/ sec/gauss, for gr = 1, it would take a field of several hundred gauss to completely decouple the two systems. l o J. M. B. Kellogg, I. I. Rabi, N. F. Ramsey, and J. R. Zacharias, Phys. Rev. 66, 728 (1939); H. G. Koski, T. E. Phipps, N. F. Ramsey, and H. B. Silsbee, Phys. Rev. 87, 395 (1952); N. J. Harrick, R. G. Barnes, P. J. Bray, and N. F . Ramsey, Phys. Rev. 90, 260 (1953). I I S . Millman, P. Kusch, and I. I. Rabi, Phys. Rev.66, 165 (1939). l2 J. R. Zacharias, Phys. Rev. 60, 168 (1941); 61, 270 (1942).

-

-

2.4.

DETERMINATION

OF SPIN,

PARITY,

AND NUCLEAR MOMENTS

61

of quadrupole interaction in nuclei, including what is known as “pure quadrupole resonance.”13 The Kusch-Foley measurements established the anomalous magnetic moment of the electron14 and set the stage for the rapid development of the renormalized-electron-field theory and Lamb’s experiment leading to the discovery of the “Lamb shift.”16 For a broad discussion of procedures and techniques of atomic beams, the reader should refer to several articles that cover all the above applications. l 6 * This discussion is primarily concerned with the applications to the measurement of the spins, magnetic moments, and quadrupole moments of unstable nuclei of short half-lives. These quantities are of primary importance in the study of nuclear structure, but until recently there has been very little done in the field, considering the large number of known isotopes. As a result, although much has been known about the nuclei t ha t lie along the “stabie line” of the Seer&chart and longlived isotopes that may be produced in abundance from these in reactors, very little was known about those isotopes that are neutron-deficient by two, three, four, or more neutrons. The primary reason is that these are cyclotron-produced isotopes and, in general, are not available in quantities greater than about 1013 atoms. By a n amalgamation of the techniques of nuclear physics with atomic beams, measurements have been successfully carried out on 1 O l o atoms and on species whose halflives are as short as 10 minutes. Within the last four years, the nuclear spins of about one hundred isotopes have been determined by this extension of the method of atomic beams. The half-lives of these isotopes have ranged from 10 minutes to 24,000 years. It should be emphasized that these measurements have been made with no loss of the great resolution available and thus have a very high reliability compared with other methods. The usefulness of the results can be appreciated from the fact that nuclear spins are now known for ten isotopes of Cs, ranging from a magic number of neutrons for 65Csi23’ to ten less than the magic number of neutrons for 5sCs:227. I n addition, long-lived isomeric states have been investigated. Much useful information has been obtained on Au and on Ag isotopes. Because these atoms are in an electronic state with J = $, it is not feasible to measure the quadrupole moments of these nuclei.

* See also the Vol. 4 references cited on page

58. W. A. Nierenherg and N. F. Ramsey, Phys. Rev. 72, 1075 (1947). 1 4 P. Kusch and H. H. Foley, Phys. Rev. 74, 250 (1948); 72, 1256 (1947). 16 W. E. Lamb, Jr., R. C. Retherford, S. Treibwasser, and E. S. Dayhoff, Phys. Rev. 73, 241 (1947); 79, 549 (1950); 81,222 (1951); 86, 259 (1952); 86, 1014 (1952); 89, 98 (1953); 89, 106 (1953). 16 J. G. King and J. R. Zacharias, Advances in Electronics 8 , l (1956); N. F. Ramsey, “Molecular Beams.” Oxford Univ. Press, London and New York, 1956. 13

62

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Instead, the hyperfine-structure anomalies for these nuclei are observed. This term is also nuclear-shape-dependent. The advantage to be gained is that measurements can be made on a J = $ state, and a quadrupole (as well as an octupole) term can be measured. If this is possible, it will result in the knowledge of the quadrupole moments for eight or nine isotopes of the same element, and should be extremely useful in the interpretation of the unified nuclear model and similar theories. 2.4.1.2.2. GENERAL PRINCIPLE.* The fundamental method for the spin and magnetic moment measurements is based on space quantization. Because of its spin 1, the nucleus alone has a spatial degeneracy of 21 1. Because of the electronic angular momentum J , the electronic system has a degeneracy 2J 1 . In the absence of any external fields or interactions between the electrons and the nucleus, the total degeneracy is (21 1)(2J 1 ) . Because of the magnetic interaction between the nucleus and the electrons, the degeneracy between levels of the total angular momentum, F = I J, is removed. There are either 21 1 or 2J 1 of these hyperfine levels, whichever number is the smaller. F takes on the values I J, I J - 1 , . . . \I - JI,and each of these levels is 2F 1 degenerate. On the application of a weak magnetic field H , this degeneracy is removed, and the energy dependence of the levels is a constant plus a term proportional to H . The difference between any two energy levels is a possible transition frequency under the influence of an applied radiofrequency magnetic field, and this transition is observed, provided that it is allowed by the selection rules, and provided that the apparatus is designed to detect the particular transition involved. There are many different types of transitions, and a t least three different ways of designing the atomic beam magnets to observe them. As an example, the class of transitions that involve AF = _ + I is particularly suitable for precision measurements of hyperfine intervals. If no previous value of this separation is available, however, a tedious search is involved because of the high resolution of an atomic beam apparatus. A procedure that operates in stages and requires only a minimum amount of isotopes has been devised for determination of the various nuclear constants of interest. The quantities to be determined are, in order, the spin, the magnetic dipole term in the hyperfine interaction, the electric quadrupole term, the sign of each, and, if possible, the magnetic octupole term and the hyperfine-structure anomaly. From these terms, the values of the nuclear magnetic moment and electric quadrupole moment can be determined by ratios. The sequence as described requires monotonically increasing precision. As an example, the alkalies have a 2S1,2ground state; therefore, the quadrupole and octupole terms vanish identically, and

+

+

+

+

+

+

+

* See also Vol. 3, Part 6.

+

+

+

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

63

there is no hope of measuring these moments in this atomic state. Since the hyperfine-structure anomaly may be as high as several per cent, there is no point in measuring the dipole interaction with extreme accuracy for the purpose of determining the nuclear magnetic moment b y comparison with a known isotope. However, in order to determine the sign of the interaction] it is generally necessary to measure the hyperfine-structure constant to a greater accuracy to check the consistency of the assignment. As a n example of some aspects of this technique, we consider C S ' ~ ~ . The spin of this isotope is 2. Figure 1 is the plot of the energy levels versus the magnetic field. The energy is in units of a, the hyperfinestructure constant, and the abscissa is the dimensionless quantity,

+

X = (-gJ gr)poHha(I $1 +

(9.7

< 0 for electrons)

(2.4.1.2.1)

The Hamiltonian representing the interaction between nuclear spin and electronic angular momentum, and the interaction with the external applied field, is X

=

a1

- J - gJpoJzH - ____ grpoIpH h h ~

(2.4.1.2.2)

where X and a are in frequency units. The Land6 factor g.7 is known very accurately from atomic beam experiments on the stable Cs isotopes and is very nearly equal to -2; g I is small, mrnlm; a is approximately several kilomegacycles per second. Initially, in the weak field or Zeeman region, the levels vary linearly with H . The corresponding frequency differences between adjacent upper levels, F = 8, AF = 0, and AM = f1, are equal to one another and to approximately Q poH/h, or about 0.5 Mc/sec/gauss. The field is too weak to decouple the nucleus from the electrons, but it does cause a precession of the entire system due to the torque on the total dipole, which is why all the upper frequencies are the same for low field. For a free electron ( I = 0), this frequency is gJpoH/h or 2.8 Mc/sec/ gauss. The nucleus adds inertia to the system because of its angular momentum, but makes a negligible contribution to the torque; therefore the precession frequency is reduced. The exact expression for 2S1/2states to first order in the field for the upper levels is y o = - -

g.rpoH/h 21 1

+

2IgrpoH/h. 21 1

+

(2.4.1.2.3)

The last term is negligible for initial measurements in the determination of the spin. If a resonance can be observed a t low enough values of H , the spin is determined by the factor (21 l)-l. The advantage is the discreteness of the search. The resonance of the carrier or known isotope

+

64

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

2 .O

M

1.0

F.5

2

w -

AW

F

=% -1.c

-2.0

I

I

I

.

1.0

2.0

30 3.0

X

ENERGY LEVELS

J . 12

1.2,

FIQ.1. The Breit-Rabi diagram for I

=

2.

is used to calibrate the field, and the position of possible resonances for different spins can, in general, be discretely predicted and compared. Once a spin resonance is located and verified, an estimate can easily be made of the hyperfine interval Av, the difference in energy between the upper and lower F levels: A v = [(2I 1)/2]a. This is done by following the resonance to a field that will introduce a small deviation from the linearity expressed by Eq. (2.4.1.2.3).This term is, in general, quadratic, and is due to the incipient deeoupling of I and J by the external field,

+

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

65

and is 2Iv02 Av

(2.4.1.2.4)

V ~ V O + - -

to second order in the field. The shift 21v02/Av gives a rough estimate of Av, and this shift can be increased by increasing the frequency until Av is determined to any reasonable desired accuracy. Since this procedure is capable of determining Av to 0.1% or better if necessary, one must use the exact solutions for the Hamiltonian, including the terms in g r ; gr itself is estimated from Av by comparison with a known isotope and the relation” Avi/Avz = (211 l ) g r 1 / ( 2 I 2 l)grE. (2.4.1.2.5)

+

+

For alkalies and other 2S1/2 states, this formula is good to about 1%, which is more than enough for the small correction it affords. The deviation of this relation from equality is a measure of the finite distribution of the nuclear magnetism and is known as the hyperfine anomaly. However, the sign of gr is not determined, and the usual method is to check the constancy of the calculated A v versus H for assumed gr > 0 or gr < 0. If the resolution of the apparatus is sufficiently good, there is enough discrimination to tell the sign. So far, nothing has been said about the apparatus needed to observe this resonance. A particular arrangement of magnets in an atomic beam apparatus is used that results in what is inelegantly called a “flop-in)’ trajectory rather than the original “flop-out” design. Reference to Fig. 1 shows the rather interesting phenomenon that the particular level corresponding to the most negative value of M = - I - J (e.g., the level -$ in the diagram) varies exactly linearly with H and “crosses the diagram” in the Paschen-Back region. to take its place with the levels WLJ = The level immediately above, M = - I - J 1 (e.g., in the diagroup. The slope of gram) eventually winds up with the WLJ = these lines at a particular magnetic field is a measure of the force on an atom in that state in an inhomogeneous magnetic field. The two levels, M = - I - J and M = - I - J 1, therefore have equal but opposite forces exerted on them in the same inhomogeneous field, providing only that the field is sufficient to decouple the angular momenta completely as in the Paschen-Back region. Figure 2 is a schematic of an actual apparatus. The atoms leave the source and pass between the pole tips of two successive magnets that supply the field necessary to cause the decoupling and the inhomogeneity to deflect the atoms. These magnets are called the “A” and “B” magnets respectively, and are so arranged that the fields of each and the field gradients of each are in the same

-+

+

17E.Fermi,

Z . Physik 60, 320 (1930).

+ +&

-+

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

I

FIG.2. A cross section of an atomic beam apparatus.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

67

direction. The apparatus then acts like a Stern-Gerlach system in the first instance. The atoms are deflected either to the left or to the right, depending upon the sign of mJ, and there is no signal a t the detector. Between the two deflecting magnets there is a uniform field, the “C” field, which is normally at just a few gauss for spin determinations. The resonance takes place in this field via a small perturbing rf field induced by a pair of wires, generally called a “hairpin,” running parallel to and near the beam. If the resonance condition is met, transitions are caused between the two states,

F=I+J -I - J

MF =

and M F =

-I - J

+ 1.

The result is that the moments of the atoms initially in these states may be reversed in sign in the B field with respect to the A field, and these atoms will go in a sigmoid path and be refocused onto the detector yielding a signal. The result is a positive signal compared to no signal, which gives far better statistics than the flop-out method in which the indication is a reduction in beam intensity. This is an essential feature in experiments involving trace amounts of material. I t is important to note that the matching and stability of the A and B fields can be quite crude without too much effect on the efficiency of the apparatus. The C field must be stable within a fraction of the line width, and since this may be as low as 20 kc/sec, this is often a stringent requirement. Since only two 1) are involved, the best fractional inlevels out of a total of 2 ( 2 1 tensity obtainable is ( 2 1 l ) - I , and because of various losses, about one-quarter of this is actually realized. The final identification of the radioisotope is usually made certain by a determination of the half-life of a collected resonance sample. Figure 3 shows the results of the decay of a Cs beam produced by a-particles on Xe. The undeflected-beam decay curve is composite, with a t least three components. Three of the spin buttons collected showed counts that were above background and each decayed with a half-life ~ ~ and , CsI3* characteristic of a known species. Thus the spins of C S ~CsI3l, were measured. Cs129 and Cs131 had been measured earlier after having been produced by different methods. The procedure for atomic and nuclear levels where both I and J are greater than one-half is more complicated because of the existence of the quadrupole term. However the spirit of the method is the same, but two different transitions must be observed for sufficient accuracy, this is

+

+

68

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

always possible because there are always two or more “flop-in” transitions in this case. Further accuracy gives a direct measurement of gr. 2.4.1.2.3. CONSIDERATIONS IN APPARATUS DESIGN. The design of an atomic beam apparatus for the study of radioactive species should clearly involve the optimization of the transmission which is the total number of atoms from the source that reach the detector. This is a different condition from the many that are usually employed, so a brief description is required here. The apparatus designed and constructed a t

-

0

5

I5

10

20

25

DAYS

FIQ.3. The decay of three resonances corresponding to three different Cs isotopes.

Berkeley will be discussed as an example. I t is more fully described in the thesis of R. A. Sunderland, and the calculations that follow are abstracted from more complete accounts in his thesis. The apparatus being considered is to be designed so that the largest possible fraction of the manufactured atoms is available a t the detector. The C field will, therefore, be made as short as possible, thus sacrificing ultimate precision, and dead spaces between the source, the magnets, and the detector will be held to a minimum. In order to obtain a first idea, it will be assumed that the C-field length and the dead spaces are negligible. The deflection

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

69

of an atom from a straight-line path is then (2.4.1.2.6)

where 6y is the deflection perpendicular to the axis of the apparatus, p is the effective moment of the atom, a H / a y is the perpendicular magnetic field gradient, 1 is the length of the deflecting field, M is the mass of the atom, and v is the velocity of the atom. For l 2 = 1000 cm2,

aH/ay

=

6000 gauss/cm

T = 1000"K, and p = 10Vo erg/gauss, the deflection is 0.3 cm. In what follows, numerical factors of the order of unity will be discarded. If T is the source temperature, Mu2 = ICT for a n average atom. The field inhomogeneities are developed by pole tips which are cylindrical arcs. If the radius of one of the pole tips is a, we can assume ( l / H ) ( a H / a Y ) = U/a> therefore

6y = PHp/kaT.

(2.4.1.2.7)

If the exit slit is of height h and width w, we may assume, dimensionally, that the source slit and width are proportional to h and w (and, in fact, h, w, and 6y must be proportional to a in a n analysis of this type). The total number of atoms that reach the detector is then (remembering that 1 a h a w 0~ 6y a a) (2.4.1.2.8)

Here p is the number of atoms per cm3 in the vapor phase in the oven. For an experiment with an unlimited supply of material, the maximum permissible p 00 l / w by the Knudsen condition. This is the condition that the slit width should not exceed the mean free path in the oven, so that the condition for pure molecular effusion can be maintained. I n this case, N varies as the length of the apparatus. The only limit to the increase in 1 is the scattering of the beam atoms by the residual gas and the expense. In practice, it has seldom been necessary to consider an apparatus of length greater than 2 meters, and most of this length is due to the C field needed for high precision. In the situation in which trace amounts of material are involved, the total amount of material is fixed and all of it is needed. This is the case a t hand. The proper condition is that phw equal a constant. Then we find

N

a

Hp/T.

(2.4.1.2.9)

70

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

This means that the efficiency of the apparatus is independent of its length. The apparatus can be made quite short until the dead spaces and the finite C field introduce other limiting factors. A convenient over-all length is about 75 cm. Of course, the half-life of the material also enters these considerations. As the apparatus is shortened, the rate of efflux is decreased and the carrier needed must be decreased. This can reach the point a t which the amount of carrier is too small to be successfully handled. The foregoing analysis was carried through on the assumption that the ratios of ail lengths were held constant. There are other considerations, however. Provision must be made for a collimating slit and for a stop wire. The latter is an obstacle that just blocks the direct beam from the detector. It prevents the fast, relatively undeflected tail of the Maxwellian distribution from reaching the detector. Furthermore, the ratio of the A to B magnet lengths need not be unity. I n fact, if the A magnet has all its dimensions reduced, it remains equivalent in focusing power and transmission, as the above analysis showed (including the necessary reduction in source dimensions). If carried to the limit, this means a factor of two in length for the total apparatus and, therefore, a factor of four in transmission. In practice, a factor of three can be obtained, A factor of this size is not trivial when viewed against cyclotron time and counting time. One pays for this improvement with a higher source pressure. For the alkalies, this represents a reasonable temperature rise. For low-vapor-pressure substances, such as U, Pu, Am, Np, Th, etc., the necessary rise in oven temperature is barely tolerable, and the improvement in intensity probably is not worth it. Under these conditions, the stop wire allows only about one-half the atoms through that undergo a change of effective moment of 1 Bohr magneton. The transmission of such an apparatus is about when a cosine distribution is assumed. Channeling the source never seems to produce the improvements expected from the simple theory and, further, is less effective for material of short half-life. The exit slit is 0.1 by 0.5 in., and the source is 0.002 by 0.080 in. The one remaining question of importance is the length and value of the rf field. A typical beam velocity is 50,000 cm/sec, and if the hairpin is 1 cm in length, the transit time is 1/50,000 sec, given a resonance band with 50,000 cps. At a field of 10 gauss, a nucleus of spin I = 0 in an alkali atom resonates at 28 Mc/sec, and this bandwidth will just resolve spins in the neighborhood of I = 50. Such resolution is far more than necessary, but usually the C field is not sufficiently uniform to attain ideal resolution, and resolutions of 200 kc/sec are more common. This is still more than adequate. The strength of the rf field is determined as follows.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

71

The system precession a t resonance about the C-field direction can be viewed from a rotating coordinate system that has the same velocity and sense of rotation. In the rotating system, the angular-momentum vector appears stationary, and the applied magnetic field is taken as zero. Such is the essence of Larmor’s theorem. The rf field, which appeared as a rotating vector perpendicular to the C field in the laboratory system, is now stationary if the oscillator is tuned to resonance. In this coordinate system, the spin will precess about the perturbing field H’ with the Larmor frequency corresponding to H’. If the strength is just right, it will make just one-half revolution in the time the atom spends in the rf field. Denoting this time by At, the condition for a precession of ?r is (2.4.1.2.10)

For spin 0, At = 1/50,000 sec-I, this yields a field of 20 mgauss, a field that is very easily obtained up to quite high frequencies. For a monoenergetic beam, the resonance will decrease if this field is exceeded. I n practice, the Maxwellian velocity distribution will smooth out the oscillations of the resonance versus perturbing field. 2.4.1.2.4. P R O B L E M S INVOLVED I N P A R T I C L E DETECTION OF ATOMIC BEAMS.2.4.1.2.4.1.General Considerations. Only experimental methods that take advantage of particle detection can now give the high sensitivities required for the investigation of short-lived radioisotopes. Two essentially complementary techniques now being used meet this requirement; one is the detection of individual beam atoms by ionization, massspectrometric analysis, and subsequent detection with electron multipliers; the other is radioactive collection and detection. For the first scheme, the following limiting situation is typical: at least lo3 positive ions must be observed a t the electron multiplier to obtain a statistically significant sample of the beam; mass-spectrometer transmission is about 10%; and the fraction of beam atoms that is ionized ranges from a lower limit of (with the universal electron-bombardment detector first introduced by Wessel and Lewls) to unity for special elements, such as the alkalis and Group IIIb series, in which Langmuir-Taylor surface ionixationlg is part,icularly applicable. For radioactive collection and detection this situation is typical; a beam sample is required to have a decay rate in excess of 1 cpm and is observed in detectors with a n efficiency of 25%. Under these simple assumptions the crossover point, where one method becomes comparable to the other in sensitivity, occurs G. Wessel and H. Lew, Phys. Rev. 92, 641 (1953). I. Langmuir and K. H. Kingdon, Proc. Roy. SOC.A107, 61 (1925); J. B. Taylor, 2.Physik 67, 242 (1929); Phgs. Rev. 86, 375 (1930). ‘8

l9

72

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

for materials with half-lives between one week and 100 years, depending upon the ionization efficiency attainable. There is, however, another consideration, difficult to evaluate, but nonetheless important. Even when there is no common stable isotope having the same mass as a nuclide to be investigated by the ion method, there is inevitably some spillover of carrier, beam contamination, and apparatus background, especially oils, into the mass channel of interest. The effect is to make the ion method somewhat less sensitive than would first appear. The present cross-over, therefore, occurs somewhere in the neighborhood of 1- to 100-year activities. An excellent r6sum6 of the ion system is contained in an article by King and Zacharias16 and the publication of Lew,20therefore, this discussion is generally limited to research on materials with lives of one year or less. The central procedures of the radioactive method are the collection of beam material for a suitable integration time and the detection of the sample decay by a n appropriate counter. Let us first consider the conditions under which the material available may be most effectively utilized. The central fact is that if reliable counters are available, the relative uncertainty t in a beam sampling is (I, t =

+

c)l'2 ct"2

+

(2.4.1.2.11)

where c is the sample decay rate, b G is the observed decay rate, and t the counting time. It is assumed that the background b is well known. We dispense with the b term by noting that in most instances an efficient counting system with background of about 1 cpm or less can be found (more on this point later), and that 1 cpm is also about the lower limit of decay rate set by the time required t o count down the results of a run and by the investigator's patience. Thus any discussion of radioactive detection rather naturally separates into two categories; into the first class fall all materials with half-lives of several days or more, and into the second go isotopes with shorter lifetimes. For investigations in the first class, the rate of progression of research is strictly proportional to the rate a t which the beam exposures can be counted, since the exposures themselves can be taken a t essentially any rate, and since the investigator will have sufficient time to analyze the data and decide on a n optimum search procedure. The rate a t which exposures can be counted is, from Eq. (2.4.1.2.11), ne2c2 (2.4.1.2.12) d N / d t = -= nt2c c+b za

H. Lew, Phys. Rev. 74, 1550 (1948); 76, 1086 (1949).

2.4.

DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS

73

where n is the number of counters used, c is a prescribed relative uncertainty in the beam sampling, c is the net counting rate of beam samples, and b the counter background. The total activity available for the experiment will invariably be limited by availability, expense, or health precautions, and the counting rate c will be some appropriate apparatus constant times the total activity divided by the number N of exposures required to execute the experiment. Thus perhaps a more illustrative form of Eq. (2.4.1.2.12) is 1/r = 2Qne2/N2 (2.4.1.2.13) where r is the counting time required for the entire research project, and Q is proportional to the total activity. Note that there is a large premium connected with optimum search procedures, which reduce N, and with increase of the allowable relative uncertainty in the individual resonance points. In addition, the counter effect occurs as the first power of the number of counters. We now consider investigations falling into the second class, where one deals with materials in which the lifetime itself is a major factor in the control of exposure and counting procedures. Here the progress of the research is taken to be proportional to the number of resonance points that can be obtained per run, although other considerations, such as the fact that no extensive data analysis is possible during the course of the run, also play an important role. The maximum counting time per resonance point is T n / N , where N is now the number of resonance points taken during the run, T is a characteristic time which is essentially the half-life, and n is the number of counters. But, as before, the individual counting rates are & / N ; so for this case, assuming b << c, we find N = e(Qnr)1/2. (2.4.1.2.14) Although this result is formally like Eq. (2.4.1.2.13),the physical situation is quite different, for T is now fixed. First of all, under these circumstances, there is little to be gained from large numbers of counters, quite aside from the fact that, in the interest of reliability and uniformity, cach beam exposure should be counted in each counter, so that the dead time involved in switching samples would become high (particularly for half-lives of less than a few hours). Also interesting is the fact that N goes only as the square root of Qr, which essentially measures the total radiation h k e n by the investigator. Because the run generally extends over tbe ent,ire lifetime of the material and little time is left for calculations, the principal profit is in precalculation of situations that may be of interest during the course of the search procedure. We now consider a question pertinent to either class of investigationindeed, to any search procedure-namely, what method of resonance

74

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

coverage, i.e., spacing of resonance points, will lead to the fastest convergence from an initial uncertainty u in a parameter such as gJ or the hyperfine structure, whose effect on the resonance frequency is controllable (e.g., by the strength of the static magnetic field), to a final uncertainty u’. Assuming that the assigned uncertainty in each resonance center is the interval between resonance points, and that a t each stage the effect of the desired parameter on the resonance frequency is multiplied by a factor n, the total number N of points needed for the procedure will be approximately given by u/u’ =

n”n.

(2.4.1.2.15)

Minimizing N with respect to n, we find that n is ideally e = 2.7 with a flat maximum between n = 2 and n = 5. The effect of this consideration, in combination with the practical ones of field drift and time spent in adjusting the field, is to make an optimum procedure one in which the effect of the parameter is increased by a factor of 3 to 6 per stage, and in which resonance points are separated by intervals of 4 to 8 the line width. Once the material has been produced, chemistry must, in general, be performed to separate the product from the few grams of target material and to put the product in a suitable state for production of an atomic or molecular beam. The range of such “chemistry” procedures is limited only by the ingenuity and resources of the investigator. The methods are thus best illustrated by examples. The central theme of all such processes is that between 1Olo and loLe atoms must be transferred under conditions of high radiation level to the apparatus in which the beam is to be produced under controlled conditions. The natural procedure for n - y reactions is to bombard the pure element, which is then placed in bulk quantities into an appropriate source. Investigations using this procedure have principally been carried out by Goodman and WexleP and by Srnith.z2 For cyclotron bombardments or secondary reactions in piles involving a product that is not the same element as the parent, the opportunity for controlling the quantity of the carrier is a very considerable advantage, as is discussed below. Therefore, in the investigations of cesium and rubidiurn,23~24the target *l L. S. Goodman and 9. Wexler, Phys. Rev. 95, 570 (1954); 97, 242 (1955); 90, 192 (1955); 100, 1245 (1955); 100, 1796 (1955). z* J. Smith, Thesis, Harvard University, Cambridge, Massachusetts, 1951. ** J. P. Hobson, J. C. Hubbs, W. A. Nierenberg, and H. B. Silsbee, Phys. Rev. 96, 1450 (1954); 104, 101 (1956). 24 W. A. Nierenberg, J. C. Hubbs, H. A. Shugart, H. B. Silsbee, and P. 0. Strom Bull. Am. Phys. SOC.[2] 1,343 (1956); W. A. Nierenberg, H. A. Shugart, H. B. Silsbee, and R. J. Sunderland, Phys. Rev. 104, 1380 (1956).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

75

is not a n alkali bromide but an alkaline earth bromide, and chemical separation of the target and the product is made, with a suitable quantity of carrier whose primary purpose is to prevent losses of the minute quantities of the radioisotope. The final product form is rubidium bromide or cesium iodide. The alkali halide is then reduced, in the oven, b y calcium a t a temperature of about 400°C. The chemistry of these two alkalis might have been done, however, in any one of the following ways.

‘I

It

.

I

II 1 ,

::

I s

I

I !I:

It

(I It

\

,GLASS _+HIGH

WATER IN-

TO METAL SEAL

VOLTAGE

EVAPORATOR

FIG.4. Detail of part of an evaporator for separating thallium from gold.

The halide nitrate could be produced in the chemistry process, the nitrate put in the oven, and then successively decomposed first to the oxide and then to the alkali metal a t high temperatures; or the azide of the alkali could be produced in the chemistry process and then decomposed directly to the metal in the oven. The gross target material, barium bromide or iodide plus carrier plus radiobromide o r iodide, might be placed in the oven and the entire load reduced by calcium a t a n appropriate temperature. I n investigation of the neutron-deficient thallium25 isotopes, it was found that chemistry procedures took entirely too long for the investigation of the 1- and 2-hour isotopes, therefore a different procedure was used. The thallium was evaporated (Fig. 4) from the gold target into a small cup with a n appropriate amount of carrier thallium. This material was then placed in the oven and a beam made of thallium metal. Again, t 6 G. 0. Brink, J. C. Hubbs, W. A. Nierenberg, and J. L. Worcester, Phys. Rev. 107, 189 (1957).

76

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

however, one alternatively might have put the gold target directly into the oven and evaporated out the thallium. In gallium investigationslZ6it was discovered that the activity coefficient of gallium in copper metal is so small that one is unable to separate gallium and copper by vapor pressures, even though the ratio for the free metals a t the melting point for copper is 103/1, and as a result, a long and involved chemistry process is required. 2.4.1.2.4.2. Oven Channeling. A possibility not heretofore mentioned is that the fraction of beam atoms that reach the detector, the effective transmission, may be materially increased by the use of long thin tubes or channels a t the source orifice to reduce the fraction of beam atoms issuing a t large angles to the beam direction. This technique has been extensively used by the MIT group in their investigations of the rare and long-lived alkalies and halides. l 6 They report typical forward effusion gains as large as several hundred from the channeling process. This technique is not of particular importance for the investigation of short-lived (less than 1 day) isotopes for the following reasons. First, in this case, one must effuse the entire activity in a period of an hour to a day. With a standard oven geometry, only a few hundred milligrams may be effused in this time. If, now, the source is channeled by a factor K , the total effusion rate is decreased by a factor K , and the source pressure must also be reduced by about a factor K . Therefore, the total quantity of material that can be effused per unit time is decreased by a factor K 2 . Absorption of the beam material by the walls of the source and inevitable sample contamination resulting from hurried chemistry set an effective lower limit of about 1 mg on the carrier; therefore, in this case, the maximum allowable channeling gain is about 20. But, from Eq. (2.4.1.2.14), the effect on the over-all research picture will be a factor of only 4 or 5. An effect this small does not justify the additional line-up problems and increased probability of failure of a run. On the other hand, for moderately long-lived materials (greater than 1 day), channeling is desirable, for the chemistry procedure can be made arbitrarily clean, ovens can be properly outgassed before use, and furthermore, the effect on the research goes as the first power of the channeling gain. [SeeEq. (2.4.1.2.13).] 2.4.1.2.4.3. Beam Collection. A general problem uniquely associated with the radioactive method is the collection of the beam on some surface in such a manner as to be useful later for counting. While this process need not necessarily be 100 % efficient, reproducibility is highly desirable. Bellamy and Smith, in their parent investigations of alkalies by the 18 J. C. Hubbs, W. A. Nierenberg, H. A. Shugart, and J. L. Worcester, Phys. Rev. 106, 1928 (1957); J. L. Worcester, J. C. Hubbs, and W. A. Nierenberg, Bull. Am. Phys. Soc. [2] 2, 310 (1957)$ J. L. Worcester, private communication.

2.4.

DETERMINATION O F SPIN, PARITY, AND NUCLEAR MOMENTS

77

radioactive method, have used a scheme where the beam is first ionized and then attracted t o a brass surface by a potential of several hundred volts.22It is found by Hobson and Hubbs that the retention by a brass surface for rubidium ions, but at ion currents lo2lower than that reported by Bellamy and Smith, is about 1% for ion energies between 10 and 1000 ev.27,28The retention rises a t either end of this range. It is known from other sources that the retention of aluminum for 3-kev ions in this mass region is essentially It is also found by Hobson and Hubbs that the retention of surfaces for thermal rubidium atoms is similarly complex. A tabulation of their data, some of which were taken only once, is given in Table I. The data were taken with a tracer technique, whereby TABLE I. Retentivity Coefficients 1. Brass 2. Silver

3. Platinum 4. Tungsten 5. Sulfur

0.10 f 0.01 0.07 f 0.01 0.24 f 0.02 = 0.21 f 0.03 = 1.0

a = a = a = LY

a

the counting rate of samples was compared with the integrated current to a surface ionization detector, from a beam whose ratio of carrier to active rubidium is known. Because of the well-known difficulties in the precision measurement of the latter ratio, the results are normalized to unity for sulfur. (The value obtained from the measured activity ratio was about 125% for the retentivity of sulfur.) On the other hand, Goodman and Wexler find large and reproducible retention for thermal cesium and indium atoms falling on a brass surface at liquid nitrogen temperature,21and Hamilton et al. find a similar situation for beams of copper and gold on surfaces of The Berkeley group finds that sulfur surfaces give reproducible and probably unit retention for beams of cesium, rubidium, potassium, thallium, gallium, indium, copper, silver, gold, and neptunium. They also find flamed platinum surfaces to be equally effective for beams of thallium, plutonium, uranium, thorium, americium, protactinium, and curium. Difficulties are found by this group in the collection of bismuth, iodine, and bromine, and by Cohen in the collection of astatine.31Lipworth et ~ 1 . 27 J. P. Hobson, Thesis, University of California, unpuhlished; J. C. Hubbs, Thesis, University of California, unpublished. 28 H. Lew and G. Wessel, Phys. Rev. 90, 1 (1953). 2gL.Davis, D.E. Nagle, and J. R. Zacharias, Phys. Rev. 76, 1068 (1949). 30 D. R. Hamilton, A. Ternonick, F. M. Pipkin, and J. B. Reynolds, Phys. Rev. 96, 1356 (1954). 31 V. W. Cohen, private communication. 31 E. Lipworth, private communication.

~ 2

78

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

have tried several surfaces, including platinum and sulfur, and find a silver surface to be the only acceptable one for the collection of iodine and bromine and to give retention that is less than unity and not highly reproducible, even under conditions where the silver surface is deposited by vacuum evaporation, removed from the evaporator, and stored under mechanical pump vacuum until ready for use. In summary, only sulfur is found to give uniformly acceptable retentivities for all the electropositive elements tried. This is presumably the result of a negative bonding energy for these materials and of the fact that sulfur forms in air a surface layer of acid which is rapidly pumped off in vacuum. The effective vapor pressure of solid sulfur a t room temperature is about lo-$ mm Hg; therefore, the lifetime of a surface atom is more than an hour. A corresponding “universal” surface for electronegative elements has not yet been found. 2.4.1.2.4.4. Beam Detection. The cross-sectional area of an atomic or molecular beam is typically 0.1 cm2 or less. This immediately allows the investigator who takes advantage of this fact to design counting systems having backgrounds far below the average levels encountered in nuclear physics research. While each nuclide presents unique opportunities for the design of low-background and high-counting-efficiency systems, the attention required by other phases of the research is such that we have standardized on three counter types (described below), which detect essentially any decay with reasonable efficiency and with background less than 5 cpm. Counter for X-rays, low-energy-y-rays, and 0-particles. The of Rbsl first prompted the development of high-efficiency, low-background counters to detect K X-rays (15 kev) emitted when the material K-captures or internally converts. The product of the development is equally useful when decay goes largely through @-emissionor when there is a prominent gamma (with energy below 150 kev) in the delay scheme. The counter and appropriate collecting surface are shown in Fig. 5. The counting head is a brass cup that fits over the face of the photomultiplier, and into which are cemented the y-window of 0.5-mil 2-5 aluminum, the crystal, 0.25 by 0.5 inch by 1 mm, and the quartz light window. Care is taken to prevent a film of the cement, Myva Wax, from covering either face of the scintillator. The y-window transmission is approximately 90 % for rubidium K X-rays; thin-film techniques might well be used to extend the useful lower limit of operation to 1 kev. Sodium iodide crystals are obtained in large blocks and split to size in a dry box with relative humidity less than 5% a t 30°C. Cleavages are 3 3 J. B. Hobson, J. C. Hubbs, W. A. Nierenberg, and H. B. Silsbee, Phys. Rev. 99, 612 (1955).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

79

made by delivering a light blow to a razor blade held against the surface in the direction of a crystal plane, and acceptable fragments with no sign of fracture or discoloration are immediately sealed between the two windows to prevent surface decomposition. Photomultiplier tubes are selected from the large number of tubes based on the requirements of less than one dark-current pulse per minute

BH 11 I

1

EDGE

I

I I

SIDE

TOP

BOTTOM

FIQ.5. Drawing of the phototube assembly for the X-ray counters. The lower part of the drawing is the sketch of the sample carriers or buttons.

above a pulse height equivalent to 7-kev gamma rays in NaI. The common denominator of tubes so selected is a very large amplification for a given dynode voltage and presumably a high photocathode conversion efficiency. The relative dark-current rate is very nearly independent of dynode voltage over the normal region of operation of the tubes. Highvoltage curing of the photomultipliers leads to no improvement of the noise figure. Operation. Counting is performed with conventional preamplifiers and single-channel differential analyzers. The only stabilization used is that of the analyzer and a Sorenson regulator, which is used to supply

80

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

110 voltsac. The counter window is customarily set to accept 90% of the X-ray line, which for 15-kev X-rays results in a 100% window, is., from one-half to three-halves of the peak counting-position. Only a 50 % window is required to achieve the same results above 50 kev. Under such adjustments as this, small drifts in window width and height contribute a negligible change in counter efficiency and background, a factor of paramount importance for counting periods which sometimes are as long as three months. The counters are shut down only after a counter electronics failure. The counter head is shielded with a 2.541. thick lead cylinder. For window settings as given above, the counter background in the shield is one count per minute or less for all energy settings between 5 kev and 100 kev. Minimum background occurs in the neighborhood of 40 kev, where it is typically 0.35 cpm. Counter background outside the shield is approximately 5 cpm. Integral settings of the counter in the shield, with a threshold corresponding to 3 to 5 kev, result in approximately 8 cpm background for a situation in which all p mesons, low-energy gamma rays, and medium-energy electrons are detected with unit efficiency. The integral contribution of dark-current pulses to counter background is, for a typical unit, approximately lo4 exp(-EE/0.5 kev) cpm between 1 and 10 kev. The scintillation counter is ineffective for low-energy p- and &-emitters because of the relatively thick (7 mg/cm2) aluminum window. Alpha emitters are detected in flow proportional counters large enough that the a is stopped in the counting gas. Discrimination against pulses corresponding to less than 3 or 4 Mev lost in the gas effectively eliminates all background except that arising from large showers and from a’s from contaminants in the chamber walls. This latter contribution is minimized by coating the chamber walls with a heavy layer of common aquadag which has been found to contain a-emitters in much lower concentration than metals. Such chambers have backgrounds of three to ten counts per hour and a detection efficiency of about 50%. p emitters are detected in similar large-volume flow counters, or in special units with extremely small interior wall area (about 20 cmz). With large chambers it is again found that a layer of aquadag substantially reduces chamber background. Large units have been found to have a detection efficiency of 50% for p’s above 5 kev and a background of about 3 counts per minute. The special small-volume chambers built of stainless steel typically have a background of 1 count per minute with 50% detection efficiency for p’s above 30 kev. 2.4.1.2.5. ISOTOPE PRODUCTION AND PREPARATION. In this section, the problem of producing sufficient radioactive nuclei for detection and

2.4.

DETERMINATION

OF SPIN,

PARITY, AND NUCLEAR MOMENTS

81

measurement is analyzed, and some numerical examples are given. For illustration, the examples of Rbsl, Rbs2, RbS3,and RbS4are discussed. The stable isotopes of Rb are Rbss and RbS7,with spins of Q and Q respectively. Their nuclear magnetic moments are 1.35 and 2.75 in units erg/gauss. The most convenient of the nuclear magneton, 5.05 X way to produce the neutron-deficient isotopes is via a-bombardment of Br. Since Br has two stable isotopes in nearly equal abundance, B P and BrS1,there may be more than one way to produce the same isotope. If 50-Mev a-particles are available, reactions of the type (a,kn) are possible, where k = I, 2 , 3, or 4.Other reactions are possible, but the yield of

ALPHA

CALCULATED

LAB ENERGY

-

MEV

EXCITATION FUNCTIONS

Br(a, hn) Rb

FIG.6. The differential cross section for a-particles on bromine to produce rubidium.

radioisotope is so low that they may be neglected. Furthermore, we will see that the identification, by means of chemistry, vapor pressure, electronic structure, and characteristic X-rays, is sufficiently rigid to remove any possibility that the finally detected beam is not rubidium. Figure 6 is a production curve for a-particle reactions, estimated from the evaporation model. These curves and the initial energy of the a-particles can be used in a numerical integration to estimate the yield of the radioisotopes. The numbers used as a n example are based on an actual run a t the Berkeley 60-in. cyclotron. The overriding problem in working with the neutron-deficient isotopes of short half-life is one of intensity. It is important to obtain yields as great as possible from a given cyclotron bombardment. This means that the high-intensity beam should be uied. I n working with isotopes of Au

82

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

that can be made by a-particles on Ir, this is no problem because the heat developed on a conventional water-cooled target can never be enough to melt or vaporize the Ir or Au under reasonable circumstances (up to about 100 pa of beam at -45 Mev). However, for most substances, such as the bromides in this example, a beam of only a few microamperes a t this energy vaporizes and decomposes the salt in a conventional holder. Therefore, a special target assembly (Fig. 7) was designed to permit bombardment by currents up to 40 pa without destruction of the salt. For estimation of the number of atoms involved, reference is made to Fig. 6, which gives the differential cross section for each of the four reactions (a,kn) on bromine, for k = 1, 2, 3, 4. Knowing the density of BaBrz and interpolating a n appropriate range-energy relation for a-particles, one can reach the estimates given in Table 11. The yield for an TABLE11. Results of Br

(a&) Rb Reactions from Bombardment of Br with a Cyclotron Beam of 45-Mev a-Particles at 24 ha (Average)

Total effective production cros8 section (emz X 3.19 3.06 1.50 0.310 Number of atoms produced per minute (XlO'Z) 1 . 0 0.97 0.47 0 . 1 Equivalent number of disintegrations per minute after 5.75-hr bombardment ( X lo8) 570 420 0.94 0.49 Reaction not completely realized because energy available (45 MeV) takes in only half the ((rI4n)peak.

average cyclotron beam current of 24 pa and 45-Mev a-particles is also given in the table, as well as the number of disintegrations per minute immediately after a 5.75-hr bombardment. These numbers can be most useful if they are translated to counts per minute. The counters used are K X-ray counters (more fully described later). We use here only those numbers necessary for intensity estimates. First, we have the factors common to all four isotopes: (a) Counter solid-angle factor of 0.5. (b) Window efficiency-the effect of the finite resolution of the counter discriminator. This is about 0.85. (c) A fluorescent yield of 1/1.62. for K versus L captures. (d) A factor of Combined, these factors give a counting eficiency of 0.24. Now there are factors that are different for each isotope: Rb8': 0.87 for the electron capture versus @+ emission. RbS2:0.94 for the electron capture versus /3+ emission. In addition, there is a guessed factor of for competition in production with the metastable RbS2that does not decay to the ground state.

+

FIG.7. High-yield cyclotron target for production of the radio-alkalies from halide salts. The cross ribbing is for the more effective removal of the heat.

2.

84

DETERMINATION OF FUNDAMENTAL QUANTITIES

Rbs3: The low-energy y-rays of Kr*lm,the daughter, which augments the counting by a factor 1.4. Rbs4:A factor of 0.76 because of electron branching. The total predicted counting rate immediately after bombardment is Rb81: 1.2 X RbE2:0.5 X Rbs3: 3.0 X RbE4:0.9 X

10” cpm 10” cpm lo8 cpm lo8 cpm.

This can be compared with the results obtained from the analysis of a sample from the “straight-through” or undeflected beam. The ratio of the observed counting rates for the Rb83 Rb84 components t o

+

t

I I

Rbe’

i

Rb

Rbe2

f

Rbs3

5 b

8 8

8

e

I0

%

, .60

F I ~8.. The normalized peak resonance intensity of the Rb isotopes for a number of runs. The intensities do vary approximately as 2 / ( 2 I

+

+ 1).

predicted and 1.9 X Rb81 Rb82 is 2.5 X observed. A rough absolute check also can be undertaken. The sample used for beam analysis was 0.02 of the total beam determined by monitoring the beam carrier. The transmission of the apparatus is 3.3 X 10-6/?r,and the decay rate (extrapolated) for the sample was 13,700 cpm, giving 0.7 X 10” cpm total, or about one-half that predicted. This is very good agreement,,

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

85

considering the nature of the estimates. There is still another check on the intensities, and th at involves the intensities of the observed resonances. The spins of Rbsl, RbE2,Rbs3, and Rb84 are $, 5, Q, and 2, respectively ; their resonances were observed under many magnetic field and production conditions. If each of the counting rates a t the spin resonances is normalized to the particular component present in the beam and l), they do indeed roughly scatter about a plotted versus 1/(21 straight line through zero (Fig. 8). The theory predicts just such a n intensity dependence. 2.4.1.2.6. CONCLUSION. The experimental methods outlined here are being used to materially extend our knowledge of the ground-state properties of nuclei. Since the radioactive technique is essentially universal, the detection limitations formerly imposed on atomic and molecular beam investigations have essentially disappeared for a class of isotopes considerably larger than th at which encompasses the stable and verylong-lived ( > 10-yr) species. Perhaps the most significant contribution to knowledge of nuclear structure has come from the enumeration of ground-state spins and magnetic moments for a large number of isotopes of the same element; most theoretical treatments of the nuclear system are considerably simplified by the fact that the “core” remains the same across such a series. I n addition to this general feature of the research, results are obtained having particular and specific interest, such a s the spin of Cs127 and Cs129, which showed conclusively the breakdown of the shell model in this region,2‘ and the spin 0 of Gass that, besides being interesting in its own right, has had some import in parity experiment^.^^

+

2.4.1.3. Microwave Method.* t During the last decade, the microwave method of measuring nuclear spin and moments has offered a n additional tool for determining nuclear properties. The principal feature of this method lies in the observation of the hyperfine structure which may be present in the microwave spectra of atoms or molecules. The existence of a n observable hyperfine interaction is the necessary condition for the derivation of any nuclear information from microwave spectroscopy. 3 4 H. Frauenfelder, A. 0. Hanson, N. Levine, A. Rossi, and G. DePasqualli, Phys. Rev. 107, 910 (1957). t See also Vol. 2, Part 10; Vol. 3, Chapter 3.1; Vol. 4, A, Sections 4.1.2 and 4.2.3.

* Section 2.4.1.3 is by C.

K. Jen.

86

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Since the energy splitting due to nuclear interactions is in general rather small, the high resolving power inherent in microwave and low-frequency techniques proves to be particularly effective in such investigations. There are essentially two areas where the microwave method has been used in the measurement of nuclear properties. One is the microwave spectroscopy in gases, and the other is the paramagnetic resonance in solids. The former, dealing principally with free molecules in electronically nonparamagnetic states, has led to a number of new determinations on nuclear spins and electric quadrupole moments and, to a lesser extent, on nuclear magnetic moments. The latter, dealing principally with paramagnetic ions in solids under the influence of external magnetic fields, has yielded another set of new determinations on nuclear spins and moments. Although the two approaches just mentioned differ somewhat in their detailed experimental execution, they are basically similar in applying microwave techniques to the observation of absorption spectral lines. 2.4.1.3.1. EXPERIMENTAL METHODSI N MICROWAVE SPECTROSCOPY. * 2.4.1.3.1.1. General Considerations for Microwave Spectroscopic Observation. a. General Description. When reduced to its minimum essentials, a spectroscopic system of the absorption type should contain a monochromatic source of radiation with a continuously variable frequency, a material whose absorption of radiation is under investigation and a detector of the radiation following absorption. These conditions are readily met by a microwave system. A monochromatic microwave source for the spectroscopic work is usually supplied by a klystron tube whose frequency can be varied continuously over a range. The wave propagates from the source through an absorption cell containing the material to be studied. The absorption cell may be a section of the waveguide which when containing a gaseous material can be blocked off at the two ends by sheets of material (usually mica) transparent to the microwave, or it may be a microwave resonant cavity which contains a small solid substance for the study of paramagnetic resonance. In any case, the outgoing wave is allowed to reach a detector for which a semiconductor (usually silicon) crystal is commonly used. Thus it is, in principle, possible to observe an absorption spectrum by noting the change of the crystal

* Most of the material presented in Section 2.4.1.3.1 can be found in the general references given below.laJb,lo la C. H. Townes and A. L. Schawlow, “Microwave Spectroscopy.” McGraw-Hill, New York, 1955. lb N. F. Ramsey, “Molecular Beams.” Oxford Univ. Press, London and New York, 1956. lo B. Bleaney and K. W. H. Stevens, Paramagnetic resonance. Repts. Progr. in Phya. 10, 108 (1953).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

87

current with the frequency (or alternatively the magnetic field as in the case of paramagnetic resonance). The absorption of a material is usually described by the imaginary part (x”) of the complex quantity x denoting the susceptibility of the material as

x = XI - i f .

(2.4.1.3.1)

The susceptibility may be due to electric dipoles as in most cases in gaseous microwave spectroscopy, or it may be due to magnetic dipoles as in paramagnetic resonance. I n practice it is seldom necessary to consider the absorption due to both kinds of susceptibility at the same time. b. Absorption in a Gas-Filled Waveguide. It is customary to use the absorption coefficient a instead of XI‘ for the description of the gaseous absorption. The relationship between the two quantities is (2.4.1.3.2)

where X is the free space wavelength and X I ‘ is the susceptibility per unit volume. Since X I ’ is dimensionless, a has the dimension of cm-l. If we let y denote the absorption of an empty waveguide, then we have

pl

=

Poe-(~+Y)~

(2.4.1.3.3)

where 1 is the length of the waveguide absorption cell, PO and PI are, respectively, the powers a t the input and output ends of the cell. Since in general a << y, we have for the power change due to gaseous absorption (i.e., signal power) P, = alPoe-Y1. (2.4.1.3.4) Equation (2.4.1.3.4) indicates that P, reaches a maximum if the cell length is chosen to be 1 = y-1. But this optimum condition is somewhat altered when ultimate sensitivity is considered in relation to the noise power (considered in a later section). The absorption coefficient a for a spectral line is given by the Van Vleck-Weisskopf equation: (2.4.1.3.5)

where A = constant of a gas which varies inversely as the absolute temperature, n = number of molecules per unit volume, v o = resonant frequency of a spectral line, and Av = linewidth parameter. At the pressure range most commonly used mm Hg), the linewidth is almost all due to “pressure broadening” and is directly proportional to

88

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

pressure. Hence, the peak absorption is independent of pressure [when there is no “intensity saturation” which Eq. (2.4.1.3.5) does not consider]. c. Absorption in a Cavity Containing a Paramagnetic Sample. A cavity can be characterized by a resonant frequency and a quality factor Q. The fact that a cavity is equivalent to a waveguide of length (X/27r)& and is physically concentrated in a small volume makes it ideally suited to a paramagnetic resonance experiment, where a sample is usually a small crystal placed in a strong homogeneous magnetic field. Since the absorption by magnetic dipoles is the only point of interest in this connection, the sample should be placed in the region of strongest microwave magnetic field, which in a cavity is fortunately also the location of weakest electric field. In a resonance experiment, the microwave and steady magnetic fields should be mutually perpendicular. Assuming that the microwave magnetic field is represented by H I cos at, we have for the power absorbed by the sample

P,

$wx:’H~~

=

(2.4.1.3.6)

where xi‘ is the imaginary part of the magnetic susceptibility for the whole sample, having the dimension of a volume. The absorption by a n empty cavity can be similarly expressed as w

P “ --o-- Q

HI2V 87r

(2.4.1.3.7)

where &I, is the “unloaded” quality factor, and V is effective volume in the expression < J H 2 do>.. = H l 2 V . For maximum sensitivity, QO should be made twice as large as the “loaded” Q , leading to the result P, = P0/2,where Po is the input power. Using the above relations, we have for the power absorbed by the sample (2.4.1.3.8) The expression for xi’ of a paramagnetic substance is often rather similar to that in Eq. (2.4.1.3.5) and can be generally written as

xi’

= CN*J(u)

(2.4.1.3.9)

where C is a constant of the substance and varies inversely as the absolute temperature, N , is the number of paramagnetic’ions in the sample, and f(v) is the spectral form factor. At the center of the resonance line, f(v) = v o / A v , where Au is the half-width.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

89

d. Noise Power. We shall not give an analysis of the various sources of the noise power except to mention that the noise generated at the crystal is often the largest among them. The crystal noise power can be described by the expression

P (crystal noise)

=

(k T + Y2) ~

Af

(2.4.1.3.10)

where k = Boltzmann’s constant, I’ = absolute temperature, C = constant of a crystal, I = crystal current, f = output frequency, Af = output frequency bandwidth. The first term in Eq. (2.4.1.3.10) denotes the thermal agitation or the “Johnson” noise, and the second term the specific noise of a crystal. Equation (2.4.1.3.10) suggests that, for minimization of the crystal noise, f should be as high and Af as low as possible. The lowest limit of the noise power is set by the thermal term kT At. It is a common practice to describe the over-all noise power as N k T Af, with N designated as a noise figure not only for the crystal, but for the whole system. e. Sensitivity of Detection. The sensitivity of detection in a microwave spectroscopic system can be expressed by the minimum detectable a in gaseous spectroscopy or x:’ in paramagnetic resonance. This can be done by equating the signal voltage to the effective thermal noise voltage. For a gas-filled waveguide, the result is (2.4.1.3.11)

where e = Naperian base, I, = 2/7 (“optimized” length of waveguide). Similarly, for the case of a cavity, the result is (2.4.1.3.12)

Equations (2.4.1.3.11) and (2.4.1.3.12) point to the disadvantage of observing a signal at a very low output frequency, because the noise figure N with reference to Eq. (2.4.1.3.10) would be an enormously large number. Thus a spectrograph operating on the principle of detecting dc or very low frequency signals is apt to be extremely insensitive. The situation is somewhat improved if the microwave frequency is periodically swept (e.g., by a saw-tooth voltage) at a chosen rate across the frequency of a spectral line so that a band-pass receiver at a high frequency can be used. But the method that really gives a major improvement in sensitivity lies in the use of a modulation frequency such that the inverse frequency noise contribution in Eq. (2.4.1.3.10) is greatly reduced and the bandwidth Af can be made very narrow.

90

2.

DETERMINATION OF FUNDAMENTAL QUANTlTIES

2.4.1.3.1.2. Methods of Modulation and Detection. a. Stark Modulation. The method of modulation by employing the Stark effect is most extensively used in microwave spectroscopy of gases.* The modulation frequencies used by various workers are in the range of 6-100 kc. Consider first that the microwave frequency is tuned to an absorption line. If we apply an electric field, the absorption at this frequency can decrease considerably and often to zero because of the displacements in frequency for the Stark components of the original line. In the case of a periodically varying electric field, the fluctuating absorption would give a signal a t the modulation frequency. The variation of the amplitude of this signal with a slowly swept microwave frequency would yield a spectral presentation of the absorption line. The use of a square-wave (so biased that the field is zero half of the time) is especially rewarding in that the resulting spectrum will not only show the zero-field line, but also the Stark components at various positions as determined by the constant field during half of the cycle. However, there are also cases where we rather want to avoid seeing the Stark components when, for instance, we observe the Zeeman splitting of a spectral line. This is exactly our situation in the study of nuclear magnetic moments. Under these circumstances, we can use, for example, a coaxial waveguide for the absorption cell in which the electric field is very inhomogeneous, and hence will cause the Stark components to smear out. Such an arrangement is actually used in the experimental layout shown in Fig. 1. b. Zeeman Modu1ation.t Zeeman or magnetic field modulation is extensively used in paramagnetic resonance experiments. The modulation frequencies used by various workers are in the range of 10-1000 cps. In a few cases the modulation frequency has been raised to 100 kc or higher. A small sinusoidally varying magnetic field is usually placed along the direction of a high steady magnetic field. If the amplitude of the modulation field is much smaller than the spectral line width, then a signal at the modulation frequency would appear after the detection of the microwaves with a magnitude proportional to the derivative of the absorption coefficient (i.e., da/dH). c. Detection of Modulation Signals. Where a modulation scheme is used, only the information carried by the modulation frequency component in the detector output needs to be preserved. This signal is usually fed in succession to a preamplifier, a narrow-band amplifier, and a phase detector. In the phase detector, the signal is mixed with a modulation frequency reference voltage to produce essentially a dc output which can be delivered to an oscilloscope or a recorder. If it happens that there

* See for example, Vol. 3, Section 2.1.4.

t See also Vol. 4, A, Section 4.2.3.3.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

91

is more than one modulation frequency being used for special reasons, then each frequency component can be separately handled through its own channel as in the above description. The spectroscopic sensitivity of a relatively high frequency modulation system has been experimentally found to be considerably higher than that of a nearly dc system. But, even here, the noise figure N in typical setups is far greater than the ideal ( N = 1).Referring to Eq. (2.4.1.3.11), under the conditions of a good waveguide spectrometer, e.g., I,, lo3cm, T 300"K, A! 30 cps, P o 1 mw, the experimental value for amin is

-

-

MICA WINDOW ROTATABLE POLARIZER

1

-

-

/,,\ \ LONGITUDINAL FIELD COIL

TRANSVERSE FIELD ELECTROMAGNET STARK ELECTRODE

I

ROTATABLE lDETECTOR

AMPLIFIER

T147] OSClLLOSCOPE

1OOkC

PHASE

Fro. 1. Schematic diagram of a microwave Zeeman spectrograph.

roughly in the range lo-* to cm-l. The corresponding value for N is of the order of lo3. The application of field modulation method to paramagnetic resonance in solids calls for some special consideration. Here the modulation frequency for the magnetic field is, with a few exceptions, usually restricted below 1 kc because of the eddy current limitations of the cavity used as the absorption cell. I n order that full advantage can be taken for the lower noise figure a t higher frequencies, a superheterodyne scheme can be adopted with increased sensitivity. A local microwave oscillator is set a t a frequency differing by a constant amount, for example, 30 Mc/sec, from the signal microwave frequency. The difference frequency due to the mixing of the local oscillator output and the signal wave after absorption can be handled by an I.F. amplifier, and eventually by a phase-sensitive detector. If a field modulation is used, then the output contains the

92

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

signal at the modulation frequency which can be handled as discussed before. The sensitivity of a paramagnetic resonance spectrometer can be conveniently rated according to the minimum number of unpaired electron spins N , in Eq. (2.4.1.3.9) that the apparatus can detect when the sample is a t a given temperature. By the use of Eq. (2.4.1.3.12),it can be shown that the ultimate sensitivity of a cavity spectrometer is of the order of 1011 spins at room temperature. Practical spectrometers are considered to be good if their capability of detecting spins covers the range 1 0 1 2 to 1013. 2.4.1.3.1.3. High-Resolution Spectroscopic Methods.* High-resolution spectrometers are occasionally needed to resolve very closely spaced lines (hyperfine structure or Zeeman splitting) , or to enable more accurate determination of line frequencies. Increase in resolution is mostly accomplished by seeking ways to reduce the linewidth. When a modulation method is used for the sake of sensitivity, it also broadens the line in the same process. This means that when high resolution is important one must choose a sufficiently low modulation frequency and/or low degree of modulation at the expense of sensitivity. In gaseous spectroscopy, the linewidth can often be reduced by using lower pressures when pressure broadening is the dominating feature. For further reduction of linewidth, methods for decreasing Doppler broadening have been successfully worked out. But the systems used are often rather complex. For paramagnetic resonance in solids, the linewidth can be reduced by decreasing the concentration of the paramagnetic species, or sometimes by lowering the temperature. A rather different form of high-resolution spectroscopy has been introduced by FeherId with his double resonance method. Here, a resonance condition is first established for the electrons in a paramagnetic species. The resonance is actually between the hyperfine levels of two magnetic states of the electron. A second resonance is then introduced between one of the participating hyperfine levels and another hyperfine level of the same electronic state. The second resonance is essentially of the nuclear magnetic type, being in the range of radio frequencies. If the electronic resonance is initially intensity saturated with a marked reduction of signal, then the application of the nuclear resonance can cause a visible change in the electronic resonance. There is thus a natural selection of only such nuclear transitions that can affect a given electronic transition. This method has the advantage of combining the high resolution of a nuclear resonance experiment with the high sensitivity of an

* See also Vol. 4, A, Section 4.2.3.2. Id

G. Feher, C. S. Fuller, and E. A. Gere, Phgx. Rev. 107, 1462 (1957).

2.4.

DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS

93

electron resonance experiment. But, more important for our present subject, the method has been demonstrated to give a direct determination of the nuclear g factor from the separation between two lines in the radiofrequency spectrum. 2.4.1.3.1.4. Measurement of Frequency, Magnetic Field, and Spectral Intensity. 2.4.1.3.1.4.1. Frequency measurement.* Measurement of the signal oscillator frequency is mostly done with reference to an absolute frequency standard like the transmissions of the station WWV. In gaseous spectroscopy, the frequencies a t the centers of all the spectral lines are to be measured. However, for the measurements involving the hyperfine structure or Zeeman spectrum, only frequency differences between lines are essential. This less stringent requirement can often be met by a simpler operation. In paramagnetic resonance, the oscillator frequency is usually held at a fixed value while the resonance condition is fulfilled by the variation of the magnetic field. It is comparatively a simple matter to make an absolute measurement of a constant frequency. The accuracy of frequency determination depends upon the stability of the signal frequency and the spectral linewidth. Under typical conditions, it is possible to determine the line frequency within about one-tenth of the linewidth. Since a typical linewidth in gaseous spectroscopy is of the order of 1 Mc/sec, the accuracy of the line frequency determinations is usually within the range 0.01 to 0.1 Mc/sec. 2.4.1.3.1.4.2.Magnetic jield measurement.t The measurement of magnetic field is mostly done through the use of a proton resonance magnetometer. Under typical conditions, it is usually practicable to measure the field with an accuracy within 0.1 oersted for a field of several thousand oersteds. This accuracy is more than adequate for the Zeeman measurements in gaseous spectroscopy and is quite satisfactory for most of the paramagnetic resonance work. 2.4.1.3.1.4.3.Intensity measurement.$ Unless extreme care is taken, intensity measurement is subject to many types of variability. Fortunately, only relative intensities are of interest for all measurements concerning the hyperfine structure from which nuclear information is derived. The average accuracy of relative intensity determination is of the order of 1%. 2.4.1.3.1.5. Illustrative Microwave Spectrographs and Spectral Presentations. 2.4.1.3.1.5.1. Spectroscopy in gases. As an illustrative example, the schematic diagram of a microwave spectrographs specially designed

* See also Vol. 2, Chapter 9.2.

t See also Vol. 1, Chapter 9.3.

1See Vol. 3, Section 2.1.7.5. 5 The specific description given here refers to

a spectrograph of this type at the Applied Physics Laboratory of The Johns Hopkins University.

94

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

for measuring the Zeeman effect in gaseous spectroscopy is given in Fig. 1. The absorption cell is formed by a cylindrical waveguide 0.62 in. inside diameter and 53 in. long, with mica windows a t the two ends. A center conductor of 0.125 in. outside diameter serves as the Stark electrode, being supported by perforated mica discs in the waveguide. A zero-biased square-wave voltage a t 500 volts maximum and 100 kc is used for the Stark modulation. The absorption cell is placed in the gap of a 4-fOOt long electromagnet, which can supply a transverse magnetic field up to about 8000 oersteds. An auxiliary coil wound around the absorption cell can supply a longitudinal magnetic field up to about

24,019.6 Mc/ssc

FIG.2. Observed hyperfine spectrum of O I @ C W 3for J

= 1 + 2.

1000 oersteds. The longitudinal field is used only for the determination of the sign of the magnetic moment. For the latter operation, a circular polarization of the microwaves is produced by a “polarizer” (mica phase changer) and detected by an “analyzer” (rotable rectangular wave with crystal detector). Using a spectrograph as described in Fig. 1, we can observe the hyperfine spectrum of a molecule in the absence of a magnetic field. Such a spectrum for 016C12S33 is shown in Fig. 2, for the rotational transition J = 1 4 2. A very good fit between the observed pattern and the theoretical predictions can be obtained only when the following assumptions are made: I ( S 3 3 ) = +, en& = -29.1 Mc/sec [see (Eq. 2.4.1.3.13)l. When a magnetic field is applied, each line in the hyperfine spectrum is split in manner characteristic of the magnetic properties of the par-

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

95

ticipating states. The Zeeman pattern2 is the simplest for the transition F = + 4, where the magnetic splitting gives a doublet shown in Fig. 3. The nuclear g factor for S33can be calculated with the known value for the molecular g factor of 0cs3*( -0.026).

+

F=

v2

-bZ

I

t

24,019.6 Mc/sec

H = 3912 oc. -2

-I

0

I

2

Mc/sec

FIG. 3. Zeeman splitting (after Esbach et ~ 1 . 2 ) .

(r components)

for F =

-

8 4 (J

= 1 + 2) of O‘~C”Sa8

CAVITY AND SAMPLE

OSCl LLATOR

1

FIG.4. Schematic diagram of a microwave paramagnetic resonance spectrograph. 2.4.1.3.1.5.2. Paramagnetic resonance. A moderately simple paramagnetic resonance spectrograph is given schematically in Fig. 4. A field modulation of several hundred cycles per second can be used, even with a thick cavity wall. If a thin metallic wall for the cavity is used, 2 J. R. Esbach, R. E. Hillger, and M. W. P, Strandberg, Phys. Rev. 86, 532 (1952).

96

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

then the modulation frequency can be raised to the range of several hundred kilocycles with the attendant advantage of high sensitivity. The source modulation and its related circuitry are used to stabilize the klystron frequency to the sample cavity resonance frequency. This frequency should be kept at least several times higher than the field modulation frequency. There should be a provision, not indicated in the figure, for rotating the sample in the cavity if a single crystal is used. A sample spectrum of paramagnetic resonance is given in Fig. 5 which shows the magnetic hyperfine structure of Ce141 obtained by Kedzie et aLs with an irradiated fission product of U236 in a L a ~ M g ~ ( N 0 , ) ~ ~ - 2 4 H ~ 0 single crystal. * The eight-line spectrum is clearly associated with a

t---

1060 o e r s t e d s

t

3725 oersteds

FIQ.5. Hyperfine structure due to Ce"' in the paramagnetic resonance of Ce3+ ions in LazMgs(N03)~ 2 4 H ~ This 0 . spectrum was observed at 4.2"K and 9380 Mc/sec (after Kedzie et aZ.3).

nuclear spin of 4. The magnetic moment of Ce141is evaluated from the measured hyperfine coupling constant to be about 0.89 nm. BACKGROUND AND RESULTS OF MEASUREMENT 2.4.1.3.2. THEORETICAL FOR SPIN AND NUCLEARMOMENTS.2.4.1.3.2.1. Spins and Nuclear Moments by Microwave Spectroscopy in Gases. Microwave spectroscopy in gases has been largely concerned with the study of the rotational transitions in various molecules with negligible mutual interaction except for the effect of molecular collisions on line broadening. For the most part, the molecules studied do not possess a resultant electron spin or orbital angular momentum. This is the natural situation for almost all stable gaseous molecules in their ground electronic states with an extremely small number of exceptions. To be eligible for the present consideration of nuclear interaction, the molecule in question must have a t

* In their original work, the authors got several other types of spectral lines including those for Nd"7 from the same sample. We have reproduced only the part for Ce141for the benefit of a simple presentation. R. W. Kedzie, M. Abraham, and C. D. Jeffries, Phys. Rev. 108, 54 (1957).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

97

least one nucleus with a spin. Generally speaking, both the electric and magnetic hyperfine interactions are possible under these conditions. It turns out, however, that magnetic hyperfine interactions are usually negligible in view of the fact that the magnetic fields, due to paired electrons, tend to cancel out almost completely. The few molecules which do show a detectable magnetic hyperfine structure have not yet figured significantly in giving new nuclear moments. On the contrary, the electric quadrupole hyperfine interaction in a molecule is often very prominent except for a nuclear spin of +, which has no quadrupole moment, or, in some cases, a hyperfine coupling much too small for detection. Assume now a molecule answering the above description with just one nuclear spin and also being, for simplicity, a symmetric top variety (which includes a linear molecule as a special case). The restriction to a single nuclear interaction and a simple molecular configuration is fortunately not too stringent because most of the pertinent information to be presented has been obtained in just this way. Also assume that no magnetic field is being applied externally for the moment. Then the nuclear electric quadrupole interaction energy is, by a standard formula (reference la, page 154)

where

+

+

+

c = F ( F 1) - 1(1 1) - J ( J 1) F = J + I , J + I - 1 , . . . [J-I1 eqQ = quadrupole coupling constant K = component of J on molecular symmetry axis in a symmetric top, being zero for a linear molecule in the ground vibrational state. The quadrupole coupling constant is the product of the proton charge e and the nuclear quadrupole moment Q and the second derivative of V , q = r32V/r3z2,evaluated along the molecular axis z for the potential V a t the nucleus due to all charges outside of the nucleus. The quantity WQ is the hyperfine energy for each state characterized by the quantum number F , with all the F states associated with the same rotational state characterized by J and K . The rotational energy for a symmetric top is W R= B J ( J 1) - ( A - B ) K 2 , where A and B are the usual rotational constants. Any given transition is a result of the relation

+

A(WR

+ WQ)

=

hv

where Y is the microwave frequency and h is Planck’s constant. The selection rules in F , J, and K are: AF = 0, 1; A J = 0 , +_ 1; AK = 0 .

+

98

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

The intensity distribution for the transitions is well known and will not be given here. It is clear from the above discussion that, with the knowledge of J and K ,the nuclear spin I can be determined by matching the pattern of a calculated hyperfine spectrum (including intensity distribution) based upon an assumed I with that of the observed spectrum. The adjustable scale factor for perfect matching gives directly the constant eqQ. As a result of such measurements over a large number of molecules, the known spin values of very many nuclei have been confirmed; but, more significantly, more than ten new nuclear spins have been established by the microwave spectroscopic method for the first time. Table I gives a listing TABLE11 Atom Mass number I

B

0 S

c1 Ge Se

I

10 11 17 33 35 36 73 75 77 79 125 129 131

dPN)

3

Q

$ +0.633(10)

8 2

i

+l.OO(lO) +1.32(8)

H

8 8 4

-1.018(15) 2.74(14) 2.56(12)

cma) Reference

+

1.80081 (49)] [+2.68852(4)1 [-1.89370(9)] [O.64342(13)]

+0,06(4) +0.0355(2) -0.005(2) -0,050 +O ,035 [ + I .28538(6)] 0.0172(4) [ -0.8791 4 (12)] -0.21 (10) 1.1 [+0.534058(14)] I < 2 x 10-31 0.7 -0.66 [+2.617266(12)] - 0 . 4 3 0 5 )

[

i3

&(lo-24

Ib lb lb lb,2,4 lb lb,5 lb 6 lb lb 7,8 lb,9 8

This table lists only those atoms whose nuclear spins Z were first determined by microwave spectroscopy in gases. The other quantities p and & were also first obtained from the same method, but, where they are improved by another method, the more accurate results are given in square brackets. The numbers in parentheses indicate experimental errors in the last figures of the associated values.

of the nuclei and their new spin values, together with other items to be d i s c ~ s s e d . ~ ~Notable ~ ~ ~ 4 -among ~ these new spins are the ones such as I(B1O) and I(C136)which ran counter to the then current predictions. It is just this type of information that is most helpful to the theories of nuclear structure. We come now to the other result of the hyperfine structure observation, J. R. Esbach, R. E. Hillger, and C. K. Jen, Phys. Rev. 80, 1106 (1950). C. Aamodt and P. C. Fletcher, Phys. Rev. 98, 1317 (1955). EL. C. Aamodt and P. C. Fletcher, Phys. Rev. 98, 1224 (1955). 7 P. C. Fletcher and E. Amble, Bull. Am. Phys. SOC.2, 30 (1957). * P. C. Fletcher and E. Amble, Phys. Rev. 110, 536 (1958). W. Gordy, 0. R. Gilliam, and R. Livingston, Phye. Reu. 76, 443 (1949). 6L.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

99

namely, eqQ. Unlike the direct determination of nuclear spin, the nuclear quadrupole moment Q is as yet concealed in its product with the quantity q which needs to be ascertained. Of course, if the electronic wavefunction is accurately known, there should be no difficulty in calculating q. As this is practically not the case, some methods of approximation have to be used. The problem is greatly simplified by the observation of Townes and Schawlow (see reference la, page 232) that, for a heavy atom participating in a covalent bond in a molecule through a p orbital, the contribution to q is nearly the same as in an atom. In the case of an atom, q can be calculated readily from the fine structure separations or the magnetic hyperfine structure. However, measurements of eqQ for molecules containing the same atom show rather large variations, which can only mean that q varies from molecule to molecule on account of local conditions. The local conditions are construed to consist mainly of the ionic character of the bond and the degree of s or d hybridization. I n spite of some inherent uncertainties in this manner of analysis, it has been demonstrated that reasonable values for the nuclear quadrupole moments (with a specification of their signs) can be obtained. Such values for Q are listed in Table I only for those nuclei whose spins were first determined by the microwave method. There have been, in fact, many other nuclear quadrupole moments estimated from microwave measurements even though the corresponding nuclear spins were established by other methods earlier. We shall now take up the subject of nuclear magnetic moments in microwave spectra of gas molecules. I n the absence of an external magnetic field, nuclear magnetic moment can only reveal itself in a magnetic hyperfine interaction. However, except for an exceedingly few molecules, each of which has an electronic angular momentum, most molecules very rarely show any detectable magnetic hyperfine structure, which is rather difficult for interpretation in any case. Therefore, the information on nuclear magnetic moment has to be sought in the Zeeman splitting of the quadrupolar hyperfine lines previously discussed. Assume now we apply an external magnetic field H . For an electronically nonparamagnetic molecule, the only quantities which could be influenced by the magnetic field are the nuclear magnetic moment and the rotational moment of the molecule. These moments are usually of the same order of magnitude and are sufficiently small that the magnetic energy involved is often considerably smaller than the quadrupolar interaction for the fields commonly applied. Under these conditions, the first order magnetic energy (reference l a , page 290)

WH = -MpArH(ar.rg,

+ wgz)

(2.4.1.3.14)

2.

100

DETERMINATION O F FUNDAMENTAL QUANTITIES

where

M

magnetic quantum number associated with F nuclear magneton g J = rotational g factor = p(rotation)/J qr = nuclear g factor = ~ I / I (YJ = [F(F 1) J(J 1) - I ( I 1)1/2F(F 1) ffI = 1 - ( Y J . As a consequence of this situation, and subject to the well-known selection rule of AM = 0 for a ?r transition and AM = 1 for a u transition, there results a number of Zeeman components, with a defined intensity distribution, for each of the hyperfine lines. From the measured splittings for two or more hyperfine lines, it is possible to determine 1gJJ and 1911 and their relative sign, assuming that F , J , and I values are all known. The absolute sign of the g-factors can be experimentally determined by using circularly polarized microwaves. The aforementioned method has offered a few new determinations of nuclear magnetic moments before many of these moments were redetermined by some other methods (chiefly nuclear resonance) with much higher accuracy. In the interest of the present discussion, the first determinations of nuclear magnetic moments by the microwave spectroscopic method are listed in Table I, with the more accurate values by some other methods placed in square brackets. 2.4.1.3.2.2. Spins and Nuclear Moments by Paramagnetic Resonance. Paramagnetic resonance means, by usage, the electron spin magnetic resonance rather than any other paramagnetic resonance such as nuclear magnetic resonance. The presence of an unpaired electron spin, then, is an essential feature of this study. When a spinning electron, as a part of an atom, is placed in a magnetic field, a resonant absorption of radiation energy occurs when the radiation frequency is related to the magnetic field in a specific manner. It so happens that, with a convenient laboratory field of a few kilogauss, the resonant frequency falls directly in the microwave region. This is the only visible connection that paramagnetic resonance has with the microwaves. Paramagnetic resonance can be and has been done in gases, liquids, or solids. More productive work, however, has been done with the solid phase than any other, partly because most paramagnetic compounds are solids. The most extensively studied paramagnetic atoms or ions are those in the transition groups, the iron and the rare earth groups of elements. Paramagnetic resonance studies in recent years, mainly led by Bleaney at Oxford, have supplied the much needed experimental basis for many of the deductions and speculations in the quantum theory of paramagnetic solids. Among the significant contributions on the subject is a clear =

~ L N=

+ +

+

+

+

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

101

demonstration of the effect of the crystalline field on the splitting of the orbital degenerate energy levels, “quenching” of the orbital angular momentum, and magnetic anisotropy. Though somewhat disparaging to our present discussion, it must be said that the nuclear effects come in somewhat as by-products of the principal development in paramagnetic resonance in much the same way as they do in gaseous microwave spectroscopy. The nucleus of a paramagnetic ion can make its presence felt only through the channel of hyperfine interaction. Whenever the nucleus possesses a resultant angular momentum, and hence a nuclear magnetic moment, it can have a prominent magnetic hyperfine interaction with the electronic magnetic moment. Also, there may be a nuclear electric quadrupolar interaction, although generally very much smaller. The dominating role of the crystalline field in paramagnetic resonance of solids, and the variety of field symmetries involved, make it difficult to describe the situation in most general terms. For simplicity of presentation as well as to cover practically the most important case, let us assume that the crystalline field has an axial symmetry and the “spin Hamiltonian” takes the formga X = DSz2

+ P[gllHzSz + gI(HzSz + HuSg)l + AIzSz + B(1zSz(2.4.1.3.15) +

where the coordinate system is chosen with z along the symmetry axis and Sz,Su,S,= components of the electron spin S H,,H,,H, = components of magnetic field H 911 = electronic g factor parallel to symmetry axis gI = electronic g factor perpendicular to symmetry axis p = Bohr magneton and A , B, and D are constants. I n Eq. (2.4.1.3.15),it is assumed th a t the lowest orbital state is a singlet and the nuclear electric quadrupole energies are negligible. The complete solution of Eq. (2.4.1.3.15) is too complicated to be given here, but a simple description of the results can be made in sufficiently strong magnetic fields. If the field-independent terms are all negligible (i.e., A B D A 0), then there is just one resonance with the relation hv = gPH witheb

--

92 =

gI12COS*

8

+ gL2 sin2 8

((2.4.1.3.16)

where e = angle between magnetic field and the symmetry axis. If the term DSz2 is small but not negligible, there are 2 s equally spaced lines with sa

B. Bleaney and K. W. H. Stevens, Repts. ProgT. ,in Phps. 16,134 (1953). reference 9a, page 139.

@hSee

102

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

the intensity decreasing from the center. This spectrum is known as the fine structure, since it is completely independent of the nuclear effects. If, in addition, A and B are also not negligible, there is a hyperfine structure of 21 1 equally spaced lines for each fine-structure line present. All the hyperfine lines in one group are of equal intensity, since all the discrete nuclear orientations relative to the magnetic field are equally probable under normal temperature conditions. The spacing K between any two adjacent hyperfine lines is given assb

+

K2g2= A2g$ cos2 9

+ B2gL2sin2 0.

(2.4.1.3.17)

It is clear from the preceding discussion that the nuclear spin I can be directly determined from the number of hyperfine components. The nuclear magnetic moment can be easily determined if the moment of an isotopic nucleus is known from a previous direct measurement (e.g., nuclear magnetic resonance). Then the ratio of the nuclear magnetic moments between the two isotopes is simply equated to that of the total splittings of the two corresponding hyperfine spectra observed in the paramagnetic resonance experiment. When such information is not available, then one has to calculate A or B from the observed spacing between the hyperfine lines. Either A or B is directly proportional to the nuclear magnetic moment p with a proportionality factor which, for a non-s electron, depends principally upon the factor < F ~ >which , is the inverse cubic distance between the electron and the nucleus averaged over the spatial function of the electron. Without going into the detailed processes of evaluation, it suffices to say for our present purpose that the nuclear magnetic moments have been estimated for a number of nuclei from results of paramagnetic resonance. Table I1 tabulates the nuclear spins and magnetic moments, along with some rough values for the nuclear quadrupole moments, which represent new results first obtained by paramagnetic r e s o n a n ~ e . ~ ~ ~ l ~As , ~in ~ ~Table 0 - 1 9I, more prelo

M. M. Weiss, R. 1. Walter, 0. R. Gilliam, and V. W. Cohen, Bull. Am,. Phys.

SOC.2, 31 (1957).

Dobrowolski, R. V. Jones, and C. D. Jeffries, Phys. Rev. 104, 1378 (1956). R. V. Jones, W. Dobrowolski, and C. D. Jeffries, Phys. Rev. 102, 738 (1956). 13 W. Dobrowolski, R. V. Jones, and C. D. Jeffries, Phys. Rev. 101, 1001 (1956). l4 J. Owen and I. M. Ward, Phys. Rev. 102, 591 (1956). 16 M. Abraham, R. Kedsie, and C. D . Jeffries, Phys. Rev. 108, 58 (1957). 16 J. G . Park, PTOC. Roy. SOC.A246, 118 (1958). P. B. Dorain, C. A. Hutchison, Jr., and E. Wong, Phys. Rev. 105, 1307 (1957). B. Bleaney, P. M. Llewelyn, M. H. L. Pryce, and G . R. Hall, Phil. Mag. [7] 46, l1 W. l*

991 (1954). J. C. Hubbs, R. Marrus, W. A. Nierenberg, and J. L. Worcester, Phys. Rev. 109, 390 (1958).

2.4.

DETERMINATION

OF SPIN, PARITY,

AND NUCLEAR MOMENTS

103

cise values later derived from some other methods are placed in square brackets. Not listed in Table 11, but which can be found elsewhere, are the cases where paramagnetic resonance confirmed previous spin assignments but offered new values for nuclear magnetic and, sometimes, quadrupolar moments. TABLEI I e Atom Mass number I

-P

v Mn

co Mo Ru Ce Nd Sm

Eu DY

Er U P

PbN)

&(lo-24 cm*) Reference

-

32 49 50 53 56 57 60 95 97 99 101 141 143 145 147 147 149 152 154 161 163 167 233 235 239 24 1

1

5

-0.2523(3) 4.46(5)

6

4

4

4

5

[+3.34702(94)] 5.050(7) 3.855(7) 4.65(20) 3.800(7)

4

[ -0.93270(18)] [ -0.95229(10))

Q

= 1.09(3)

3 & 4 4 4 4

0.89(9) --1.0(2) -0.62(9) 0.56(6) -0.78(37) -0.65(23)

3

2.1 -0.37(4) +0.51(6) 0.50(12) 0.48-0.54 0.32-0.35 0.4(2) 1.4(6)

4 4 4 8 4

$

.’”” = 0.965(10) &I64

+

1.1(4) +1.3(4) 10.2(30) 3.3-3.5 3.9-4.1

[O ,021

Id 10 lb 11 12 lb 13 lb, 4 lb,14 Ib lb 3 Ib Ib 3 lb lb 15 15 16 16 lb 17 17 18,19 18

This table lists only those atoms whose nuclear spins Z were f i s t determined by paramagnetic resonance in solids. The other quantities p and Q were also first obtained from t h e same method, but, where they are improved by another method, the more accurate results are given in square brackets. The numbers in parentheses indicate experimental errors in the last figures of the associated values.

In concluding the subject on the microwave method for measuring spins and nuclear moments, it may be said that this method has offered a number of new assignments for the spins and many reliable values for the magnetic and quadrupolar moments of nuclei. This type of measurement has the advantage of requiring only small samples, a factor of great importance, particularly in connection with radioactive nuclei.

104

2.

DETERMINATION O F FUNDA4MENTAL QUANTITIES

2.4.1.4. Nuclear Magnetic and Quadrupole Resonance. 2.4.1.4.1.*

t

PRINCIPLES OF THE METHOD. 2.4.1.4.1.1. Introduction. Since their introduction by Purcell, et al.1 and by Bloch et a1.,2the techniques of nuclear magnetic resonance and nuclear induction have been successfully used in the precision determination of the magnetic moments of over 100 nuclei and the spins of about 25. The subsequent observation of nuclear quadrupole resonance absorption in solids by Dehmelta and by Pound4 has resulted in the measurement of nuclear electric quadrupole moments for some 20 nuclei. These experiments are all so closely interrelated that they will be collectively treated. The method is a modified extension of Rabi's principle of magnetic resonance (see Section 2.4.1.2) to nuclei contained in matter of normal density; here the resonance condition is observed by a macroscopic electromagnetic absorption or dispersion in contrast to the deflection and subsequent detection of free atoms. It should also be noted that the method is the nuclear analog of microwave paramagnetic resonance absorption (see Section 2.4.1.3). Consider a nucleus characterized by its angular momentum Ih and total magnetic moment p = yhI. Here p represents the component along I of the resultant magnetic moment of the individual nucleons, which are so tightly coupled by nuclear forces in comparison to the energies involved in magnetic resonance that the assumed proportionality between p and I is completely justified. The maximum possible components of I and t) in any given direction are called the spin I and the magnetic moment The proportionality factor y = p / I h is here called the gyromagnetic ratio; for protons, y E 2.67 X lo4 sec-l oersted-'. The interaction of the nucleus with a n applied dc magnetic field H will be given by the usual magnetic dipole term X M = - y . H = -yhI * H. The 21 1 equally spaced energy levels (see Fig. 1) are given by Em = -ykHom where I 2 rn 2 - I , if we assume H is directed along t See also Vol. 2, Chapter 9.7;Vol. 3, Chapter 4.1;Vol. 4, A, Chapter 4.4. E. M. Purcell, H. C. Torrey, and R. V. Pound, Phys. Rev. 69,37 (1946);N.Bloem-

+

bergen, E. M. Purcell and R. V. Pound, Phys. Rev. 73, 679 (1948). *F. Bloch, W. W. Hansen, and M. Packard, Phys. Rev. 89, 127 (1946); F. Bloch, Phys. Rev. 70, 460 (1946);70, 474 (1946). a H. G. Dehmelt, Am. J . Phys. 22, 110 (1954),which see for earlier references. 4R.V. Pound, Phys. Rev. 79, 685 (1950). 6 I.( is in Gaussian units, which are used throughout this section; the unit of magnetic field H i s the oersted. Sometimes nuclear magnetic moments are given in units of t h e nuclear magneton 10 = eh/2mc = 5.05 X erg/oersted. Usually experiments are performed in magnet air gaps in which H (oersteds) = B (gauss) to high precision; for this reason, the literature of magnetic resonance often specifies t h e magnetic field in units of t h e gauss.

* Sections 2.4.1.4.1 through 2.4.1.4.7.2 are by C. D. Jeffries. Section 2.4.1.4.7.3 is by G. Feher.

2.4.

DETERMINATION OF SPIN,

PARITY,

AND NUCLEAR MOMENTS

105

the z axis with magnitude Ha. Consider now a large number of such nuclei which are immersed in a lattice, i.e., a reservoir of thermal energy, such as the protons in a sample of water or a crystal of ice. Because of thermal agitation, there will be relaxation processes which tend to bring the system of proton spins into thermal equilibrium with the lattice.

m =+A 2

(b)

(a)

FIQ.1. (a) Energy levels of a magnetic dipole with spin I = in a magnetic field. (b) Additional splitting due to nuclear electrical quadrupole interaction with electric field gradient in a crystal.

We can describe the system by assigning Boltzmann population factors to the energy levels. Between an adjacent pair of levels there will exist the population ratio: N(m)/N(m

+ 1)

=

exp[-E,/kT

+ E,+l/lcT]

E 1 - rhHo/kT

1-

lo-'

(typical value a t room temperature). Now if electromagnetic radiation at the Larmor frequency wo = [Em- Em+J/h = ~ H radians/sec o is applied to the system, stimulated magnetic dipole transitions may occur according t o the selection rule Am = fl. However, there will be more stimulated absorption (Am = - 1) than stimulated emission (Am = 1) because of the slight excess population in the lower state of any pair of levels. Thus there will be a net resonant absorption of electromagnetic energy by the system, resulting in a slight heating of the lattice. For HD fields of a few thousand oersteds, Y O = w 8 / 2 r is of the order of 10 Mc/sec, and the resonance absorption can be detected by straightforward radio frequency techniques, e.g., b y wrapping a coil around the sample and observing by means of a bridge the loss factor of the coil for frequencies in the neighborhood of va. As in related phenomena, the absorption is accompanied by a dispersions which can be observed by adjusting the

+

R. d e L. Nronig, J . Opt. SOC.Am. 12, 547 (1926); H. A. Kramers, Atti del congr. J. Gorter and R. de L. Kronig, Physica 3,1009 (1936); A. R. Von Hipple, "Dielectrics and Waves." Wiley, New York, 1954. 6

inst. jk., Como 2, 515 (1927); C.

106

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

bridge to detect the change in reactance of the coil. Thus by observing the resonance frequency v o in a known magnetic field, one obtains in a relatively simple experimental arrangement the nuclear gyromagnetic ratio p / I h , from which the magnetic moment p may be determined if the spin I is known. Furthermore, as shown later, the spin may be determined from the amplitude of the resonance absorption. If I > 1, the nucleus may be further characterized by an electric quadrupole moment and there may be in addition to an interaction of the form XQ = Q * AE, which represents the dyadic product of the nuclear quadrupole moment operator Q with the electric field gradient tensor AE a t the nuclear site. The nuclear electric quadrupole moment, defined explicitly in reference 44, is a measure of the deviation from spherical symmetry of the nuclear charge distribution. Although the quadrupole interaction may be averaged to zero in the rapid tumbling of liquids, we find in single crystals that the magnetic energy levels are shifted unequally (see Fig. 1) by this additional term, so that resonance absorption between pairs of levels may be observed a t several different frequencies. I n the case of zero applied magnetic field, resonance absorption between pure quadrupole levels may be observed. The measurement of the resonance frequencies determines the product qQ of a principal value q of the electric field gradient tensor with the scalar quadrupole moment Q . Calculated values of q must be used to obtain Q from the data; consequently the precision in the determination of Q, in this or any other method, is about two or more orders of magnitudes lower than for p. However the precise ratio of the quadrupole moments for two isotopes can be easily measured. Furthermore the spin I is explicitly given by the number of resonance lines. From the broad field of nuclear magnetic and quadrupole resonance absorption, we discuss here only those aspects of direct interest to nuclear physics. For details we refer to more general and extensive treatments.’-’l 2.4.1.4.1.2. The Bloch Phenomenological Theory. The simple optical model of magnetic resonance absorption given above does not take into consideration the interactions of the spin system with itself or with the lattice. Detailed treatments of these relaxation processes have been N. F. Ramsey, “Nuclear Moments.” McGraw-Hill, New York, 1953. G. E. Pake, Solid State Phys. 2, 1 (1956). E. R. Andrew, “Nuclear Magnetic Resonance.” Cambridge Univ. Press, London and New York, 1955. lo A. Abragam, “Principles of Nuclear Magnetism.’’ Oxford Univ. Press, London and New York, 1961. 11 T. P. Das and E. L. Hahn, Nuclear quadrupole resonance spectroscopy. Solid Stale Phys. Suppl. 1 (1958).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

107

given,lJ2-14 but from the viewpoint of nuclear physics it is sufficient to consider the original phenomenological treatment of Bloch in which these effects are represented by two empirical relaxation times.2 Consider n identical nuclei per unit volume of spin 1 and magnetic moment p under the influence of an applied field H with the components: H , = 2H1 COB wt, H , = 0, H , = H ; here H is a large and uniform field; HI is a small oscillating field applied by means of a coil around the sample to induce magnetic dipole transitions. Following Ehrenfest’s theorem one assumes that M, the resultant magnetization per unit volume of all the nuclei, obeys the classical equation of motion dM/dt = M = rM X H in the absence of relaxation effects. These are next taken into account as follows: owing to thermal processes in which the spin system exchanges energy with the lattice, it is assumed that the longitudinal component of the magnetization M , approaches its thermal equilibrium value Mo in a characteristic thermal relaxation time T1 according to the relation: k?,= - ( M z - Mo)/T1. Here M ais given by the familiar Curie formula

+

M~ = I--HO1 npz I 3kT

= (I -I- 1 ) l - nyh2 wo

3kT

(2.4.1.4.1)

where T is the absolute temperature of the sample and k is Boltzmann’s constant. It is also assumed that the transverse components M,, M , will vanish exponentially in a characteristic time T 2according to the relations M , = -M,/T2; a, = -M,/TZ. The time T, 5 TI and is called the transverse or total relaxation time, and is essentially the characteristic time for incoherence of M,, M , due to processes in which the total energy of the spin system does not change; more specifically, there may be mutual spin flips, or variations of the locally effective value of H throughout the sample due to the dipole fields of neighboring nuclei. By the Fourier theorem the resonance linewidth Aw 0 1/T2, unless there is appreciable inhomogeneous broadeningl4” due, e.g., to magnetic field nonuniformities. I n the following we assume that there is no appreciable inhomogeneous broadening, as is often the case for liquids but not for solids. The total effect of both applied fields and internal interactions is then approximately given by the phenomenological equations :

M,

=

kf,= Ma =

-MZ/TS - r M , H -MM,/T2 r(M,2HI cos w t - M,H) - ( M z - Mo)/TI - rM,2H1 cos wt,

+

R. K.Wangsness and F. Bloeh, Phys. Rev. 89, 728 (1953). F. Bloch, Phys. Rev. 102, 104 (1956); 106, 1206 (1957). l 4 A. G. Redfield, ZBM J . Research Develop. 1, 1 (1957). 148 A. M. Portis, Phys. Rev. 91, 1071 (1953). 12

(2.4.1.4.2)

108

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

In the Purcell technique’ the experimentally observed quantity is essentiaIIy a bridge unbalance voltage V , proportional to A$z. I n the Bloch technique2 of nuclear induction, an additional coil is wound around the sample along the y axis (see Fig. 3), and one observes the induced voltage V , a Mu. I n either technique, resonance is usually observed by noting the periodic change in V , or V , as the applied field H is periodically swept through the resonance value H O = wo/y by superposing on the dc field a small modulation field H , cos Qt. From the nature of Eqs. (2.4.1.4.2), it is seen that the exact shape of the resonance curves V,(H) and V,(H) will depend upon the rate a t which the resonance is swept through, i.e., upon whether or not the passage through the resonance line in a time T is adiabatic in the Ehrenfest sense (7 >> l/yHl), and also upon the ratios r/T1 and ~ / T zPerhaps . the simplest and most common case is that of adiabatic slow passage: resonance is swept through so slowly that M , continuously reaches its steady-state value under the influence of HI and the relaxation processes. This requires T >> T l , T z , l/yHl. Another interesting case is that of adiabatic fast passage in which T >> l/yH1, T << T1,Tz; this has the property of reversing the algebraic sign of M,, i.e., turning over the magnetization during the passage. Still another case of experimental interest is that of nonadiabatic passage (very small H I ) with T 5 Tz, which gives rise to transient oscillation or “wiggles.”1,’6These and other modulation effects have been treated, 16--20 but here we will be chiefly concerned with the adiabatic slow passage case for which Eqs. (2.4.1.4.2) yield these solutions, if H is considered to be the variable and AH = ( H o - H ) is the deviation from the resonance value :

M,

= u cos w t

- v sin w t

(2.4.1.4.3a) (2.4.1.4.3b)

M u = T (u sin w t + v cos w t )

(2.4.1.4.3~)

ly)2HiTz2 AH

(dispersion mode) + ( T z Y A W ’ + ( Y H I ) ~ T I ABO TZ (2.4.1.4.3d) lrlH1Tz M (absorption mode). u(H) = 1 + (Tzr AH)’ + (YHI)~TITZ (2.4.1.4.3e)

u(H) =

1

o

In Eq. (2.4.1.4.3b) the (-) sign is to be taken for positive y, the (+) sign B. A. Jacobsohn and R. K. Wangsness, Phys. Rev. 73, 946 (1948). E.E.Salpeter, Proc. Phys. Soc. (London) A68, 337 (1949). 17 K. Halbach, Helv. Phys. Acta 27, 259 (1954);29, 37 (1956). R. Karplus, Phys. Rev. 73, 1027 (1948). 19 B. Smaller, Phys. Rev. 83, 812 (1957). 50 J. Burgess and R. Brown, Rev. Sci. Znstr. 23, 334 (1952). 16

2.4.

DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS

109

for negative y. In the nuclear induction technique, e.g., there will be induced into the receiver coil along the y axis a voltage

V,

a

M,

=

T (u, cos w t -

ow

sin w t ) .

Because of incomplete decoupling with the transmitter coil along the z axis, there will also be induced into the receiver coil a leakage voltage V e = A sin w t B cos w t , which may generally be considerably larger than V,. By adjusting the phase of this leakage voltage by means of flux steering paddles and/or phase shifting networks, one can make either A / B << 1 or A / B >> 1. Thus the total receiver coil voltage V , V , can be made essentially proportional to either v or u;i.e., one may observe either the absorption or the dispersion (or any combination). Similar considerations hold for the Purcell single coil technique : by adjusting the balance of an rf bridge containing the coil as a n element, one can V e at nuclear resonance make the total bridge unbalance voltage V , proportional to either the imaginary or the real part of the complex nuclear susceptibility corresponding to the absorption v or the dispersion u,respectively. Note that whereas the algebraic sign of the nuclear induction signals are dependent upon the sign of the gyromagnetic ratio y, those of t>hesingle coil signals are not. This is due to the circumstance that two (crossed) coils are needed to establish the sense of a rotating field. The shape of experimentally observed homogeneously broadened resonance curves for protons in HzO under adiabatic slow passage conditions are shown in Figs. 2a and 2b. They follow closely the form expected from Eqs. (2.4.1.4.3): the absorption v has a maximum at resonance ( A H = 0) and a half-width at half-maximum H1,2 = l/yTz, provided the rf field H 1 is small enough so that ( ~ H I ) * T I
+

+

+

(yH1)2TlT, = 1 after which it decreases while the line width increases. This behavior is called saturation: the transition probability between two levels due to a sufficiently large H can considerably exceed the transition probability due to thermal relaxation processes, resulting in a population difference, and hence a signal, considerably less than that at thermal equilibrium. The dispersion u, on the other hand, does not display this saturation effect. Bot,h u and v have the same maximum value: ( M o / 2 ) . ( T Z / T I ) ~ / ~ . The magnitude of the received signal Vs, which is proportional to uu or W U , will thus have the optimum value:

110

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Equation (2.4.1.4.4) shows clearly that to make the signals as large as possible the highest available static magnetic field (and hence frequency) should be used. It is possible in the case of liquids to shorten Ti by the addition of paramagnetic ions until TIc Tz.This gives optimum signal heights which, for equal numbers of nuclei, are proportional to p(I 1)

+

a

b

d

C

FIG.2. Oscilloscope traces of signals for protons in 1 cc of HpO as observed at I f o s 3300 oersteds with apparatus of Fig. 3. A 40-cps modulation field is used, sweeping through resonance twice in each cycle, the phase being shifted to separate the two traces. In (a) and (b) are shown the adiabatic-slow-passage dispersion and absorption modes, respectively, observed in a 1.0 molar aqueous solution of Fe(N03)3. In (c) and (d) are shown the nonadiabatic-passage dispersion and absorption modes, respectively, observed in a 0.01 2M aqueous solution of FeCla; note the characteristic “wiggles.”16

+

at constant frequency and to p s ( l 1)/Pa t constant field. Tables are availablezoEgiving these relative sensitivity figures for all nuclei for which p and I are known. Figures 2c and 2d show observed signals for protons in H 2 0 under nonadiabatic conditions. Such signals are frequently observed in prac*0*

Varian Associates, Palo Alto, California.

2.4.

DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS

111

tice; for the absorption mode, the resonance field H o lies on the line of symmetry of the figure.16 The phenomenological theory given above yields the so-called Lorentz line shapes of Eq. (2.4.1.4.3);these are usually observed for liquids under slow passage conditions. However, in viscous fluids or solids the assumed model is not valid, and the observed line shapes are often nearly Gaussian; we refer to Pake and Purcel12' for a general discussion of line shapes. 2.4.1.4.2. EXPERIMENTAL APPARATUS. Since the early experiments, there have been many changes and improvements in the apparatus used to observe nuclear magnetic resonance resulting in greatly improved sensitivity and resolution. For example, with present-day techniques, the resonance of the deuterons in 1 cc of ordinary water may be observed several times above the noise background; relative line widths as small as H 1 p / H o = lo-* have been measured for protons. The details of various experimental apparatus have been given adequately in the literature to which we refer. The two coil nuclear induction arrangement of Bloch et aL2 has been modified for various applications by ProctorlZ2Weaver,23 Redfield,24ArnoldlZ6and others. The single-coil bridge method of Purcell et a1.l has been adapted to a more readily tunable twin-tee bridge by Torrey,26 AndersonlZ7Collins,28and others. Another notable single-coil method is the regenerative detector of Pound and Knight,29Roberts130 and others, in which the absorption is detected by its reaction on an oscillator. Other related techniques have also been d e ~ c r i b e dI.n~ experi~ ments in which the nuclear magnetic resonance is split by quadrupole interaction, any of the preceding methods may be used. In zero magnetic field experiments, where there is no preferred axis of orientation, only the single-coil methods will work. Circuits developed particularly for this case have been described by Dehmeltla2W ~ n gL,i ~ i~n g s t o nand , ~ ~others. G. E. Pake and E. M. Purcell, Phys. Rev. 74, 1184 (1948). W. G . Proctor, Phys. Rev. 74, 35 (1950). 23 H. E . Weaver, Phys. Rev. 89, 923 (1953). p* A. G. Redfield, Rev. Sci. Znstr. 27, 230 (1956). 26 J. T. Arnold, Phys. Rev. 102, 136 (1956). * 6 H. Torrey, Phys. Rev. 76, 1059 (1949). $7 H. L. Anderson, Phys. Rev. 76, 1462 (1949). 2 8 R. L. Collins, Rev. Sci. Znstr. 28, 502 (1957). 29 R. V. Pound and W. D. Knight, Rev. Sci. Znstr. 21, 219 (1950). 30 A. Roberts, Rev. Sci. Znstr. 18, 845 (1947). 3 1 J. R. Zimmerman and D . Williams, Phys. Rev. 76, 350 (19491; H. S. Gutowsky, L. H. Meyers, and R. E. McClure, Rev. Sci. fnstr. 24, 644 (1953); H. W. Knoebel and E . L. Hahn, ibid. 22, 904 (1951). 82 H. G. Dehmelt and H. Kriiger, Z.Physik 129, 27 (1951). 33 T. C. Wang, Phys. Rev. 99, 566 (1955). 3 4 R. Livingston, Ann. N . Y . Acad. Sci. 66, 800 (1952). 21 22

112

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Well-engineered apparatus of both the single- and double-coil types are now commercially a ~ a i l a b l e . ~ ~ ~ ~ ~ All techniques essentially measure the same thing: the components of the nuclear magnetization given by u and v in Eq. (2.4.1.4.3).The exception is that in the regeneration detectors only the absorption mode may be detected; also the range of operation of the applied rf field H I is somewhat restricted so that optimum sensitivity may not be achieved for very short or very long relaxation times. On the other hand, the regeneration detector is basically the simplest, since it does not involve a separate transmitter and receiver and is easily tunable with a single variable condenser; no frequency-sensitive balance conditions are involved as in the single-coil bridge method and in nuclear induction. These latter methods have the advantage, however, that they operate over wide ranges of H1 and may be preset to observe the dispersion under optimum signal conditions: (7H1)2T1Tz> 1. This feature is particularly useful in searching for weak signals. Current practice in most nuclear induction arrangements is to gang and track the transmitter and receiver so that single knob frequency control is available. Independent in-phase and out-of-phase leakage controls facilitate a rapid adjustment of the desired mode of operation. The sign of nuclear magnetic moments can be determined only with the nuclear induction arrangement. Permanent magnets may be used to provide a highly stable moderate field (-5 to 8 kilo-oersteds) with a minimum of upkeep. Greater flexibility is usually desired, however, and electronically regulated precision electromagnets of the Stanford type" have been widely used. They provide a continuously variable field from 0 to 15 kilo-oersteds with a stability of 1 in los or better. To prevent excessive broadening of the resonance lines by field nonuniformities within the sample, large diameter poles (-12 in.) and/or shimming by thin ferromagnetic materials or auxilliary coils is usually resorted to. For nuclear moment determinations, an inhomogeneity 6H = l / y T z 5 0.1 oersted over a typical sample volume of 1 cc is desirable. Otherwise it is difficult to shorten TI by the addition of paramagnetic ions to make TI = T z . An upper limit of the signal-to-noise ratio to be expected with a nuclear induction apparatus, for example, can be calculated approximately as follows: the voltage induced at resonance into a cylindrical receiver coil of N turns and diameter d by a sample completely filling the coil will be V , = Nn2d2i@,X 10-8 volt. If the sample does not comVarian Associates, Palo Alto, California. Perkin-Elmer Corporation, Norwalk, Connecticut. 37 Available from Varian Associates (reference 35) ; Pacific Electric Motors, Oakland, California; Harvey-Wells Corp., Framingham, Massachusetts. 86

36

2.4.

DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS

113

pletely fill the coil, the voltage will be correspondingly less because of incomplete flux linkage. For slow passage homogeneously broadened lines, the maximum value of A$, = (MOw/2) l/Tz/Tl. If the coil is resonated with a capacitor in a parallel resonant circuit of quality factor Q, the available rms signal voltage across this resonant circuit then will be

V , = QV,

=

+

[ n ~ 2 N d ~ & H o ~ 7 ~1)Ih2/6 (1 4

3 kT] d m X lo-'

volt. (2.4.1.4.5)

Taking the typical values d = 1 cm; N = 10 turns; Q = 100; H o = 10,000 oersteds; y = 4 X lo3rad/oersted for deuterons; I = 1; n = 6.6 X 1OZ2/cc for D20; T = 300°K; TI = T2, one finds V , = 2 X volt. T h e rms Johnson noise voltage across the same circuit will be Vn = 44kTRAv volts, where Av is the bandwidth of the receiver in cps, and R = QwL is the shunt resistance of the resonant circuit. For cylindrical coils of length = diameter d, L = Q.7N2dX lo-* henry. The signal-to-noise ratio then becomes

where the factor f 2 1 takes into account all additional noise, e.g., in the amplifier, and is here called the noise figure of the apparatus. It is possible to achieve f = 2 in practice. From Eq. (2.4.1.4.6)one finds that a n over-all working rule for the actual observable signal-to-noise ratio in a typical apparatus a t room temperature under optimum condi= ~13Tis:~ tions [TI = T z ; ( ~ H I ) ~ T

(2.4.1.4.7) where ~.l/~.lo is the nuclear magnetic moment in units of the nuclear magneton, Ho is the field in oersteds, n = number of nuclei/cc in sample, Av = apparatus bandwidth in cps. Similar considerations for the single coil method have been given b y Pound.' The general conclusion is that for all the methods, about 102O nuclei are required to give a visible signal 1, p PO, H lo4 oersteds. for the typical values I A block diagram for a nuclear induction spectrometer is shown in Fig. 3. Strong signals are simply displayed on a n oscilloscope by using a modulation field H , several times the linewidth, requiring a bandwidth Vv lo3 cps. For weak signals, a modulation field approximately 4 the linewidth is often used in conjunction with a narrow band (Av s 0.1 cps) lock-in amplifier to record the derivative of the absorption, as shown.

- - -

-

114

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

m

oscil lotor

frequency

FIG.3. Block diagram of nuclear induction spectrometer.

FIG.4. Recorded derivative of the dispersion signal of

0 1 7 in 1 ec of natural HzO; 10,000 oersteds; H I ir. 0.2 oersted; H,,, E O . 3 oersted; lock-in bandwidth AV = 0.1 cps. [H. E. Weaver, Phys. Rev. 89, 923 (1953).]

Ho

sz

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

115

This narrow-banding technique usually reduces the noise by a factor of 100. The derivative signal from 0'' in natural HzO recorded under optimum conditions is shown in Fig. 4. Figure 5 is a block diagram of an apparatus for observing pure quadrupole resonances.

FIG.5. Block diagram of pure quadrupole resonance spectrometer. The resonance is modulated by applying periodically a magnetic field which causes Zeeman splitting.

2.4.1.4.3. TYPICAL PROCEDURES IN NUCLEAR MAGNETIC RESONANCE. The linewidth = l / r T z is usually considerably narrower in nonviscous liquids than in solids because of the averaging out of the local dipole fields by the rapid tumbling motion in the former case. For this reason, and also because T I may be controlled by the addition of paramagnetic ions, liquid samples are usually used in the determination of nuclear moments by magnetic reaonaiice. The nuclei in question are often contained in ions or complexes in aqueous solution. One usually chooses the most soluble compounds to maximize n. If the nuclei have an appreciable quadrupole moment, this will usually prove to be effective in providing sufficient thermal relaxation so that TI = T z ; in this case no paramagnetic ions need to be added. In fact, the quadrupole relaxation is sometimes so effective that the relaxation times are too short and the

116

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

line widths consequently too broad for the resonance to be seen. I n these cases it is necessary to use a complex ion that is highly symmetrical so that the resultant electric field gradient is minimized. For example, the resonance of As16 can only be seen in the AsOT- or ASST- complexes. Similarly, the resonance of Zr91 has been observed only in the compound (NH4)zZrFa.When it is desired to make TI = Ta by the addition of paramagnetic ions (e.g., Fe++ or Cu++), one may roughly assume that T z is determined by the known magnetic field inhomogeneity, and then use the approximate formula:' l/T1 y2p;JVi X (10q/lcT) sec-I, where y = gyromagnetic ratio of nucleus in rad/oersted; peff= magnetic moment in units of the Bohr magneton of the paramagnetic ion and has the typical values 2-6; Ni = number of paramagnetic ions per cc; 1 = viscosity of liquid; k = Boltzmann's constant; and T = absolute temperature. The rare earth and transuranic elements are known to form only paramagnetic compounds and are not amenable to nuclear magnetic resonance studies in liquids because of the relatively huge (-lo6 gauss) and randomly oriented magnetic field existing a t the nuclear sites due to the hyperfine coupling with the unpaired electrons. Nuclear moments for these particular elements have been measured, e.g., by observing the hyperfine structure of the microwave paramagnetic resonance in single crystals (see Section 2.4.1.3). The other transition groups ( 3 4 4d, 5d) can fortunately be made to form diamagnetic complexes and are amenable to both nuclear magnetic resonance and paramagnetic resonance studies. I n some cases, one can also observe nuclear resonances in paramagnetic materials, as discussed in Section 2.4.1.4.7.2. 2.4.1.4.3.1. Magnetic Moment Measurements. The usual operation in a nuclear magnetic resonance experiment is to measure the ratio y/yp = p/2ppI, where y = p/Ih is the unknown gyromagnetic ratio and y p = 2pp/h is the proton gyromagnetic ratio. It is sometimes more convenient to measure y / y d , where Y d is the deuteron gyromagnetic ratio, and then use the precisely known ratio y d / y p . The unknown p in units of the nuclear magneton is then given by P/PO =

2I(Y/YP)

*

(PP/VO).

(2.4.1.4.8)

The proton magnetic moment in units of the nuclear magneton, pP/po, has been by comparing the magnetic resonance frequency to the cyclotron frequency of protons in the same magnetic field. The

** H. Sommer, H. A. Thomas, and J. A. Hipple, Phys. Rev. 82, 697 (1951). 99 F. Bloch and C. D. Jeffries, Phys. Rev. 80, 305 (1950); C. D. Jeffries, ibid. 81, 1040 (1951). '0 D. J. Collington, A. N. Dellis, J. H. Sanders, and K. C. Tuberfield, Phys. Rev. 99, 1622 (1955);J. H. Sanders, Nuovo cimento Suppl. [lo]6, 242 (1957). 'l K . R. Trigger, Bull. Am. Phys. SOC.[ 2 ] 1, 220 (1956).

2.4.

DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS

117

best value, as compiled by Cohen et ~ l . is, pp/pa ~ ~ = 2.79275 k 0.00003, including the diamagnetic correction. The ratio y / y p is usually measured as follows. A suitable sample is prepared containing both the unknown nuclei and the reference nuclei, H or D, as HzO or DzO, for example. If the signals are large, they can be quickly found on the oscilloscope and, holding the dc magnetic field constant, the ratio of the resonance frequencies w / w , = y/yp is measured by tuning the spectrometer to first one resonance and then the other, being careful in each case to adjust to a pure absorption mode so that the oscilloscope pattern is exactly symmetrical about its center point. A weak unknown signal is best found through searching a t constant frequency with a narrow band recording spectrometer by slowly varying the field H with a motor-driven current control. For optimum sensitivity, the dispersion mode is used under the following conditions: T1 = Tz; After the signal is located, the ratio (yH1)2T1Tz= 2; H , = u / w p = y / y p may be measured in constant field by sweeping through the two resonances consecutively with a slow motor drive on the frequency control, and putting frequency markers on the recorder tapes. If the lines are reasonably sharp (Hip -0.1 gauss) the ratio w / w , can be easily measured to 1 part in lo5 and with care to 1 in los, Such precision is not always warranted, because diamagnetic effects may cause the magnetic field at the unknown nucleus to differ slightly from that a t the reference nucleus, thus violating slightly the assumed equality w / w p = y / y p . Necessary corrections are discussed in Section 2.4.1.4.5.2. The sign of p is determined relative to p, by simply noting, with a nuclear induction spectrometer, whether the two signals have the same or opposite polarity when consecutively examined by varying the H field, leaving the spectrometer frequency and the leakage controls fixed. 2.4.1.4.3.2. S p i n Measurements in L i q u i d s . Under conditions of adiabatic slow passage and negligible saturation, we obtain from Eq. (2.4.1.4.3)for the signal voltage from the absorption mode at resonance V 8 = yH~Tzii?ow= H1wz[Tzy2n(1 1)1]. Now Tz = l/yHllz and hence [ y n ( I l)I] = V,Hl,z, which, being the product of the signal height and the half-width is proportional to the area A under the absorption curve if displayed as a function of the magnetic field. Thus, if we have two nuclei with identical resonance line shapes, and the two resonances are observed in turn by varying H while keeping W , H I , and the apparatus sensitivity all constant, we can write;

+

+

2 ):( ):( =

(11 (12

+ 1)Il + 1)Iz'

(2.4.1.4.9)

4 2 E. R. Cohen, J. W. M. Dumond, T. W. Layton, and J. S. Rollett, Revs. Modern Phys. 27, 363 (1955); E. R. Cohen, Nuovo cimento Suppl. [lo] 6 410 (1957).

118

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Thus we can determine the spin I1 relative to I z by comparing the signal areas, knowing the gyromagnetic ratios and the relative nuclear concentrations in the sample. This signal area comparison can be made directly on the oscilloscope for strong signals, after making certain that both are observed under unsaturated adiabatic slow passage conditions. Weak signals are usually examined by a narrow band differentiating type of apparatus which records the derivative of the resonance lines. To consider this case and to allow for other than Lorentzian line shapes, we rewrite the undifferentiated signal for unsaturated adiabatic slow passage conditions as V , a nw21ylH1(I 1)1 - f ( H ) , where f ( H ) is a resonance line shape function with a maximum valuef(H = H o ) = l/Hlp and so normalized that f ( H ) dH = 1. Now the differentiated signal recorded by the lock-in apparatus is proportional to

+

lom

G(H)

=

(aVa/dH) . Hm

=

+ l)nf'(H).

w2H1HmlylI(I

The first moment MI of the differentiated signal will be M1 =

lo" G(H)

*

(H

- Ho) d H

= w2H1HmlylZ(Z

+ 1)n

as is seen from a partial integration and the normalization of f ( H ) . Thus for the differentiated signals of two nuclei observed in turn by varying H while keeping w , HI, and Hm constant, we find (2.4.1.4.10)

The condition (yH1)2T1T2<< 1, necessary for the validity of Eq. (2.4.1.4.10), must be experimentally checked by making for each resonance a saturation curve, i.e., a plot of V , versus H I , from which a n upper limit for a nonsaturating value of H I can be determined. Also the modulation field H , should be reasonably small, compared to H l l z , to insure a true derivative in each case. In the event that the resonance curve is displayed as a function of frequency instead of field, then the area A' = V,(v)dv so determined may be used in Eq. (2.4.1.4.9), if the right side is multiplied by yI/y2; similarly the first moment M1' = G(v) * ( v - VO) dv of the differentiated curve may be used in Eq. (2.4.1.4.10), if the right side is multiplied by (y1/y2)2. From Eq. (2.4.1.4.9),it is seen that the area ratio is proportional to I(Z 1) a 3, 15, 35, 63, 99, . . . , for odd nuclei, for example, so that increasingly greater accuracy of measurement is necessary to distinguish

lom lom

+

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

119

between higher spin values. Particular care must then be taken to insure that there is no spurious signal height distortion by modulation effects, etc. It is evident that if I I and Iz are both known for two isotopes, then the first-moment ratio determines the relative isotopic abundance. This is the basis of a simple nondestructive technique for measuring isotopic abundance ratios to within a few per cent.43 2.4.1.4.4. QUADRUPOLE RESONANCES. 2.4.1.4.4.1. QuadrupoZe Splitting of Magnetic Resonance. In crystalline samples one must consider the total interaction X = -7hI * H Q AE. One of the simplest cases is that in which the magnetic interaction term is much larger than the quadrupole term. Using first-order perturbation theory, the energy levels are found to be, for electric fields of axial symmetry,4

+ -

E,(e)

=

e2qQ + 41(2I - 1) [3m2 - I ( I + 1)][+ cos2 8 - +]

(2.4.1.4.11)

where 0 is the angle between H and the axis ( 2 ) of field symmetry; eq = -2d2V/&2, where V is the electrostatic potential a t the nucleus due to the surrounding electronic charges; Q = scalar nuclear electric quadrupole moment.44 For crystalline fields of cubic symmetry, q vanishes; for fields of lower than axial symmetry, an additional parameter 7 = (azV/azz)- (aZV/dyz) is introduced and Eq. (2.4.1.4.11)is modified." For a single crystal, the energy levels of Eq. (2.4.1.4.11)have the discrete pattern as shown in Fig. 1 for a hypothetical spin I = 3. In general, there are 21 possible transitions of the magnetic dipole type (Am = + 1 ) between the levels. These can be induced by the applied rf field H I , and such resonances are readily observed by the magnetic resonance techniques discussed above. In contrast to liquids, the nuclear dipole-dipole interaction between neighboring nuclei is not averaged out, with the result that usually Tz<< TI in diamagnetic crystals; for this reason, the expected signal-to-noise ratio is less. A differentiating type of spectrometer is usually used, and the lines are observed by varying either the field or the frequency. A sample of -1 gram is usually required. The crystal is so mounted that 0 may be varied. At 0 = arccos (l/d3), the lines become nearly superposed, since by Eq. (2.4.1.4.11) the quadB. E. Holder and M. P. Klein, Bull. Am. Phys. SOC.[I]29, 17 (1954). Q is defined [see, e.g., H. B. G. Casimir, Interaction between atomic nuclei and electrons. Teyler's Tweede Genootschap Haarlem 11, 36 (1936)]as Q = e-'Jpr2(3 cos28 - 1) dr integrated over the nuclear volume for the state m = I , where p is the 43

44

nuclear charge density at a volume element dr at position T from the origin; 8 is the angle r makes with the nuclear spin axis.

120

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

rupole term vanishes to first order; higher order terms omitted from Eq. (2.4.1.4.11) may prevent an exact superposition. For polycrystalline samples, one observes a much less discrete spectrum representing the spatially averaged values of Em(@. In addition to those induced by the rf magnetic field HI, transitions corresponding to Am = f 2 , & 1 could be induced by an applied rf electric field gradient. Sufficiently intense gradients cannot be produced electrically in the laboratory; they may be produced, however, by strong lattice vibrations induced by supersonic t e c h n i q ~ e s . ~ ~ - 4 ~ 2.4.1.4.4.2. Pure Quadrupole Resonance. When single crystals are not available, and also when the quadrupole interaction is larger than the magnetic interaction, it is convenient to examine the so-called pure quadrupole spectrum. In zero magnetic field the energy becomes

e2qQ Em = 41(2I - 1) [3m2 - 1(1

+ l)]

(2.4.1.4.12)

for axially symmetric crystalline fields. Between these pure quadrupole levels an applied rf field HI may still induce magnetic dipole transitions (Am = & 1) by acting on the nuclear magnetic moment. For half-integral spin there are I - such allowed transitions; for integral spin there are I transitions. In a nonaxial field the levels are shifted but the Iml degeneracy is not removed for half-integral spin; for integral spin, there is a splitting of the m levels, and additional lines are observed. Since the energy, Eq. (2.4.1.4.12), is not orientation-dependent, powdered samples are usually used, and the resonances are observed with r e g e n e r a t i ~ eor~ super-regenerati~e~~ ~ detectors as the frequency is slowly varied. The signal is modulated by either Zeeman splitting, or by frequency-modulating the oscillator. Resonances over very wide ranges (42-700 Mc) have been observed. 2.4.1.4.4.3.Determinations of Nuclear Spin and Quadrupole Moments. In both cases 2.4.1.4.4.1and 2.4.1.4.4.2above, by measuring the number of lines and the resonance frequency for each, one can obtain the spin I and the coupling constant e2qQ in single crystals or powdered samples. For two isotopes of the same element, q will be the same, so the ratios of the nuclear quadrupole moments will be exactly equal to the ratio of the coupling constants. A number of such ratios have been measured.3 The sign of Q cannot be determined by these methods. The general problem is to evaluate q so that Q can be determined from the measured

+

45 W. G. Proctor and W. H. Tahttila, Phys. Rev. 101, 1757 (1956); W. Proctor and W. A. Robinson, Phys. Rev. 104, 1344 (1956). 4 6 M. Menes and D. I. Bolef, Phys. Rev. 109, 218 (1958).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

121

value of e2q&. Moderately accurate values of q have been estimated by the following procedure^.^' The value of q in the molecular solid is correlated with that for a free molecule;48 then, using the semiempirical considerations of Townes and D a i l e ~ ,the ~ ~ q for a free molecule is related to that for a free atom, which is then estimated by the established procedures of optical hyperfine structure,60including the corrections of Sternheimer61 and Koster.62 2.4.1.4.5. CORRECTIONS I N MAGNETIC RESONANCE. The actual magnetic field a t the nuclei may differ slightly from the applied magnetic field H O in which one observes magnetic resonance a t frequency wo. Several small corrections are necessary in calculating the gyromagnetic H 1 H 1 l . Here H1 is the ratio y and one usually writes w o / y = Ho magnetic shielding field a t the nuclei due to induced diamagnetic circulation of the electrons in the atom, ion, or molecule; and H” is the field at the nuclei due to the paramagnetism of the sample itself. 2.4.1.4.5.1. Diamagnetic Corrections. The field HI is proportional but opposite in sense to Ho and is written as H 1 = -uHo, where u is the magnetic shielding factor. Lamb63 calculated theoretically u for free atoms, and values have been tabulated by Lamb and by D i c k i n s ~ n . ~ ~ for hydrogen to u = for uranium. They range from u = 2 X Measured magnetic moments that have been multiplied by the correction factor (1 u ) are said to be diamagnetically corrected. If the resonances are observed in atoms or ions, the Lamb correction is probably adequate. However, many resonances are observed for nuclei contained in molecules, and Ramseys6 has calculated the additional corrections necessary. Unfortunately, they can be evaluated only for the protons in the H2 molecule; the result, including the Lamb term, is u = 2.66 X low6.By comparing the resonance frequency of protons in H2 to those in the HzO and in mineral oil, the shielding factors for the latter two are found and u = 2.60 X respectively. experimentally to be u = 2.82 x This shift in the resonance frequency for a given nucleus contained in different compounds is called the “chemical shift” and has been observed

+ +

+

47 See, for example, T. P. Das and E. L. Hahn, Nuclear quadrupole resonance spectroscopy. Solid State Phys. Suppl. 1, Chapters 7-9 (1958). 48 C. H. Townes and B. P. Dailey, J . Chem. Phys. 20, 35 (1952). 4 9 C. H. Townes and B. P. Dailey, J . Chem. Phys. 17, 782 (1949). 60 See, e.g., H. Kopfermann, “Kernmomente.” Akademische Verlagsges., Leipzig, 1940; reprinted by Edwards, Ann Arbor, Michigan. 6 1 R. M. Sternheimer, Phys. Rev. 84, 244 (1951); 86, 316 (19.52); B6, 736 (1954). 62 G. F. Koster, Phys. Rev. 86, 148 (1952). 53 W. E. Lamb, Phys. Rev. 60, 817 (1940). 54 W. C. Dickinson, Phys. Rev. 80, 563 (1950). 5 5 N. F. Ramsey, Phys. Rev. 77, 567 (1950); 78,699 (1950); 86,243 (1952).

122

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

for many nuclei.6s Although usually of the order of O.Ol%, a few cases have been found where the shifts are as large as l%.&' For this reason, it is important in measuring magnetic moments to observe the resonance in several different compounds, including free ions if possible; presumably the resonance which occurs at the lowest field (for fixed frequency) shows the least diamagnetic shielding. There are, of course, no diamagnetic corrections necessary in measuring the ratios of magnetic moments for two isotopes of the same element. Special high-resolution techniquesz6 have been developed to study these chemical shifts from the view point of structural chemistry.68 2.4.1.4.5.2. Paramagnetic Corrections. The field H" due to the magnetization M , of added paramagnetic ions has been experimentally measured by Dickinsons8for different ions and geometries of the sample. For a spherical sample, the depolarization field -(+r)M, cancels the Lorentz field (+r)M, and one might expect the field H" to be zero; experimentally, this is not found to be so, probably because of spinexchange interactions. Dickinson found that H" depends upon the sample geometry, the chemical compound containing the resonating nucleus, and the particular paramagnetic ion and its concentration. For example, a sample of HzO of roughly spherical proportions containing FeClz has a measured value H l l = 2cHo X oersted, when c is the molarity (moles/liter) of FeC12. For long cylindrical shapes, H l 1 may be negative. A general safety rule is, for ( H " / H ) < lo--*, it is necessary that c < 0.3 mole/liter. In powdered metal samples, Knightso has found that the paramagnetism of t,he conduction electrons contributes to H". This Knight shift in the resonance frequency may be as large as 1%; experimental and theoretical values have been tabulateds1 for many metals. 2.4.1.4.6. MEASUREMENT OF MAGNETIC FIELDS BY NUCLEAR MAGNETIC RESONANCE. Driscoll and Benders2 have measured the proton gyromagnetic ratio in absolute units by observing the resonance frequency w o in a magnetic induction field Bo,which was measured by a 68 A summary of data is given by H. E. Walchli, in "A Table of Nuclear Moment Data," ORNL-1469 and supplements. Oak Ridge National Laboratory, Oak Ridge, Tennessee. 67 W. G. Proctor and F. C. Yu, Phys. Rev. 81, 20 (1951). 68 W. A. Anderson, Phye. Rev. 102, 151 (1956); J. N. Schoolery, Discussions Faraday Sac. 18, 215 (1955). 60 W. C. Dickinson, Phys. Rat. 81, 717 (1951). So W. D. Knight, Phys. Rat. 78, 1269 (1949). 6 1 W. D. Knight, Solid State Phye. 2 , 93 (1956). (* R. L. Driscoll and P. L. Bender, Phys. Rev. Letters 1, 413 (1958); see also H. A. Thomas, J. A. Hipple, and R. L. DriscoU, Phys. Rev. 78, 787 (1950).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

123

current balance. They find y p = wo/Bo= (2.67513 & 0.00002) X lo4rad/ sec/gauss, without diamagnetic correction. Correspondingly, this result, can be used to measure a magnetic field in absolute units to high precision; one simply observes, in the field to be measured, the proton resonance frequency v P of protons in liquid Hz0 containing, say, 0.02 mole per liter of FeClz. The sample geometry may vary between a sphere and a long cylinder. The applied field is then given directly by the formula

B (in gauss)

=

234.873 X v p (in Mc/sec).

(2.4.1.4.13)

For high fields, it is sometimes more convenient to observe the deuteron resonance frequency Vd in DzO (containing, say, 0.5 mole/liter of FeClz), for which the following formula applies:

B (in gauss)

=

1530.06 X

Vd

(in Mc/sec)

(2.4.1.4.14)

For precision measurements of fields of a few gauss, the devices of Pack. ~ particularly ~ ~ suitable. ard and Varians3 and Abragam et ~ 2 1 are 2.4.1.4.7. SPECIAL METHODS. We discuss very briefly special methods, which, in addition to the usual steady-state methods described above, have some applications to nuclear physics. 2.4.1.4.7.1. Spin Echoes. By applying the rf field HI in the form of pulses instead of continuously, Hahn64 induced several transient responses of spin systems which he called spin echoes. This method is particularly useful for measurement of the relaxation times TI and Tz, and may be applied to spin systems either in magnetic interaction with a dc field, or in pure quadrupole interaction.O6I n the latter case, unusually small quadrupole coupling constants have been measured by the use of signals from spin system A transient double resonance a t a relatively high resonance frequency are observed as a function of a second relatively low frequency applied to a system B, which is in weak interaction with system A . At the resonance frequency of B , the signal strength of A is observed to change. In effect, one has a high frequency, and hence high sensitivity, probe for weak interactions. This method would be particularly appropriate for measuring quadrupole coupling constants smaller than a few Mc/sec. M. Packard and R. Varian, Phys. Rev. 93, 941 (1954). A. Abragam, J. Combrisson, and I. Solomon, Compt. rend. m a d . sci. 246, 157 (1957). 84 E. L. Hahn, Phys. Rev. 80, 580 (1950). 66 M. Bloom and R. E. Norberg, Phys. Rev. 93,638 (1954);E.1 1. Hahn and B. Heraog, ibid. 93, 639 (1954);W. G. Proctor and W. H. Tanttila, ibid. 98, 1854 (1955); M. Bloom, E. L. Hahn, and B. Heraog, ibid. 97, 1699 (1955). 66 B. Heraog and E. L. Hahn, Phys. Rev. 103, 148 (1956). 67 D. Kaplan and E. L. Hahn, Bull. Am. Phys. SOC.[2]2, 384 (1957). 6s

83s

124

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

2.4.1.4.7.2. Dynamic Nuclear Orientation. This method is closely related in technique to microwave paramagnetic resonance (Section 2.4.1.3), and is a consequence of some generalizationses of a suggestion by O v e r h a u ~ e r It .~~ has been used to determine the spins and magnetic moments of radioactive nuclei70~71by observing the change in angular distribution of y-rays accompanying certain resonance conditions. The principle is to enhance the population of certain nuclear magnetic states by saturating the microwave magnetic resonance of an electron-nuclear hyperfine-coupled system, in analogy t o the earlier work of Pound.72The resulting nuclear orientation, dynamically produced by the applied microwave field and associated relaxation processes, is to be contrasted to that produced by static hyperfine interactions a t very low temperatures, as discussed in Section 2.4.2.3. With reference to Section 2.4.1.3.2,we give a simplified illustration of the method by considering a hyperfine system with nuclear spin I = 1 and electron spin S = in an external field H along the x axis, the pertinent interactions being given by

+

X = gPHS,

+ AIgS,+ B(I,S, + I$$,).

(2.4.1.4.15)

The physical system may be, e.g., a single crystal of zinc sulfate in which about 0.1% of the Zn++ ions have been replaced by (paramagnetic) Co++ ions. The energy levels g@HM AmM are shown in Fig. 6, where the states are labeled by the high-field quantum numbers ( M = S,, m = I,); usually A m M << gPHM. The term in B may be rewritten as iB(I+S- I-S+), where I+ = I= iI,, etc.; and it is seen that this slightly admixes state b with state f, and also state c with state e. The consequence of this is that a microwave magnetic field applied parallel to H can induce the so-called forbidden transitions AM = k 1, A m = T 1, labeled 4 in Fig. 6. Suppose the transition b ~f is saturated, i.e., the populations are made equal by applying sufficient microwave power. Then the relative

+

+

+

F. Bloch, Phys. Rev. 93, 944 (1954); A. Overhauser, ibid. 94, 768 (1954);J. Kor ringa, ibid. 94, 1388 (1954); C. Kittel, ibid. 96, 589 (1954);P.Brovetto and G . CiniNuovo cimento [9]11, 616 (1954);P. Brovetto and S. Ferroni, ibid. [9]12, 70 (1954), A. Abragam, Phys. Rev. 98, 1729 (1955);A. Abragam, Compt. Tend. acad. sci. 242, 1720 (1956);A. Abragam and J. Combrisson, Compt. rend. acad. sci. 243, 576 (1956); G. Feher, Phys. Rev. 103, 500 (1956);G.Feher and E. A. Gere, ibid. 103, 501 (1956); C. D. Jeffries, ibid. 106, 164 (1957); D. Pines, J. Bardeen, and C. P. Slichter, ibid. 106, 489 (1957). 6p A. Overhauser, Phys. Rev. 89, 689 (1954);92, 411 (1953). M. ilbraham, R. W. Kedaie, and C. D. Jeffries, P h p . Rev. 106, 165 (1957);Bull. Am. Phys. SOC.[2]2, 382 (1957). 7 l F. M. Pipkin and J. W. Culvahouse,Phys. Rev. 106,1102 (1957);109,1423 (1955). 72 R. V. Pound, Phys. Rev. 79, 685 (1950).

2.4.

DETERMINATION OF SPIN, PARITY,

AND NUCLEAR MOMENTS

relative populations, saturating

(M 1 m)

b-f

I

e-A a ? ' I f

f

125

re1ative populations, saturating a-f I

I

I

FIG. 6. Energy levels, transitions, and populations for a hyperfine system with S = and Z = 1. Forbidden transition 4 and allowed transitions a are induced by microwave fields parallel and perpendicular, respectively, to the dc field H .

4

populations of the other states will be as shown, if we assume that the usual strong paramagnetic relaxation processes maintain thermal equilibrium between a and f, between b and el and between c and d ; it is also assumed that a time-dependent term B'(t) in the hyperfine interaction provides a relaxation process to keep c and e in thermal equilibrium. The resulting nuclear alignment

P

-

=

(N,

+Nf +N, +

N d

- 2Nb - 2 N , ) / Z N

-

x

-hv/3lcT

where v is the microwave frequency and T is the absolute temperature. For v 101O cps and T N 1"K, P 0.1, and this alignment will produce a detectable anisotropy in the angular distribution of the y-rays emitted by the nuclei. I n a typical experimental arrangement, Fig. 7, one observes a signal proportional to the difference in the y-ray intensities emitted parallel and perpendicular to the dc field H , as the field is slowly varied, keeping the oscillator frequency v fixed to the cavity resonance frequency. An anisotropy is recorded at the field corresponding to the transition b tf f, and also a t the field corresponding to c t)e ; in general, 21 anisotropy signals are observed, and the nuclear spin is thus directly determined. The field spacing between signals is proportional to the coupling constant A , from which the nuclear magnetic moment is determined,

126

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

either through hfs calculation, or by direct comparison to that of a known stable isotope. It is also possible to obtain nuclear orientation by saturating the usual allowed transitions AM = k 1, Am = 0, labeled a in Fig. 6. For example, by saturating the transition a ++ f, the populations shown are obtained if one assumes the same relaxation processes as before. The resulting

(;v3

microwove oscillator

radioactive porarnognetic crystal in microwove cavity

\

% I iq.

~~

magnet pole

I ig h t f Pipe

L

A d iff e r e n- c

Fro. 7. Experimental arrangement for dynamic orientation of radioactive nuclei.

alignment is P = hv/6lcTl and in general one observes a y-ray anisotropy at 21 1 field values. In more general situations, the effects of relaxation processes between a and b, etc., and between a and el etc., must be taken into account. Feher6*has proposed and demonstrated that, independently of relaxation processes, if the populations of first a and f, and then a and b, are inverted by adiabatic fast passage (see Section 2.4.1.4.1.2),the resultant populations correspond to a nuclear orientation comparable in magnitude to those above.

+

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

127

Dynamic orientation experiments are usually done at 1- or 3-cm wavelengths a t the temperature of pumped liquid helium (-1.3"K). Conventional microwave resonance techniques are employed. Saturation powers between microwatts to milliwatts are required, and activities from microcuries to several millicuries have been used. Since the signal-to-noise ratio is determined by the number of decays rather than by the number of nuclei, this method is particularly appropriate for short-lived nuclei, in contrast to the method of direct paramagnetic resonance absorption, which is more suitable for long-lived and stable nuclei. The above discussion has been particularly appropriate for a, nucleus in relatively strong hyperfine coupling with its own paramagnetic ion, but similar ideas726 may be applied to the case of nuclei .inweak long-range dipole-dipole coupling with electron spins, for example, the protons in a piece of plastic containing free radicals. If the forbidden transitions involving a simultaneous flipping of an electron spin and a nuclear spin are saturated, an appreciable polarization of most of the nuclei in the sample may be achieved. For details we refer to a review article.72b

2.4.1.4.7.3. The Electron Nuclear Double Resonance (ENDOR) Technique.* I n this method,7aone deals with nuclear and electron paramagnetic resonances simultaneously (see Section 2.4.1.3). It is applicable to some paramagnetic substances in which the nucleus, in addition to the applied magnetic field, is subjected to the hyperfine field of the electron. The ENDOR method is illustrated in Fig. 8 for the simple case of a hyperfine system with I = +,S = +. The magnetic field is set to the proper value to observe the microwave transitions hv,, as indicated by the arrows in Fig. 8. The amplitude of the signal due to these transitions is proportional to the difference in population in levels in A and A'. If we partially saturate this resonance by increasing the microwave field, the population difference will be diminished and the signal reduced. By inducing now the transitions h v N or h v ~ (see , Fig. 8), the population difference in levels A and A' can be increased and thereby the electron spin resonance signal enhanced. Thus one monitors the electron spin resonance signal while nuclear transitions are being induced. Since the electronic moment is approximately three orders of magnitude A. Abragam and W. G. Proctor, Compt. rend. acad. sci. 246, 2253 (1958). D. Jeffries, Progr. i n Cryogenics 3, 129 (1961). 73 G. Feher, Phys. Rev. 103, 500 (1956).

72.8

nbC.

-

*Section 2.4.1.4.7.3 is by G. Feher.

128

2.

DETERMINATION O F FUNDAMENTAL Q U A N T I T I E S

larger than the nuclear moments, the sensitivity of the ENDOR technique greatly exceeds that of the ordinary nuclear resonance. It is also independent of the size of nuclear moments as long as they are large enough to give rise to a set of resolved ENDOR lines. For those two reasons, the method is particularly advantageous in determining moments of nuclei which have a small gyromagnetic ratio, or nuclei which are radioactive. The transition-inducing field at the microwave frequency v, is applied to the sample in a resonant microwave cavity, which may be thin-walled and slotted to allow the fields a t the radio frequency V N to be applied by coils wound around the cavity. m= ,

+ J-

?= I:-

m,- -I

ti-+

ms=+I

*

ml=

m5 = -I 2

mz= +'

A'

4.

FIG. 8. Energy levels of a system with Z = i,S = Arrows indicate the transitions involved in the ENDOR technique. [G. Feher, Phys. Rev. lOS, 500 (1956).]

By observing the two transitions hvrv and hvw, the nuclear are given in the high field region by the relation7 hvN

- hvN1

=

2grpH.

91

values

(2.4.1.4.16)

The method has been applied by Pipkin and Culvahouse71 to radioactive and by Feher et u Z . , ~ ~to P32. Pipkin and Culvahouse made use of the double resonance method to align the nuclei and used a y-counter to monitor the nuclear transitions. Halford et aZ.,76 have applied it to determine directly g1 for Nd143and Nd146;corrections similar to the chemical shift in nuclear resonance are required in this case.76a The ENDOR technique is also useful in the investigation of quadrupole interactions, as demonstrated in the work on F centers in KCl.V6 The quadrupole interactions of both the C136J7 and K39s41 were clearly reG. Feher, C. S. Fuller, and E. A. Gere, Phys. Rev. 107, 1462 (1957). D.Halford, C. A. Hutchison, Jr., and P. M. Llewellyn, Phys. Rev. 110,284(1958). 7ba €3. Bleaney, J . p h p . radium lS, 826 (1958). 76 G.Feher, Phys. Rev. 106, 1122 (1957). l4

76

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

129

solved. In this case the quadrupole interaction of the chlorines is only approximately 40 kc/sec, which indicates the relatively high resolution attainable by this method. It should be noted that the method is not applicable to all paramagnetic substances, but only to those that can be saturated and exhibit an inhomogeneously broadened77 line. If those two conditions are satisfied, transitions whose frequencies are much smaller than the electron spin resonance line width may be observed.73

2.4.2. Indirect Methods

2.4.2.1. Angular Correlation,* t 2.4.2.1.1. INTRODUCTION AND LITERAThe importance of angular correlation techniques for determining nuclear moments and parities stems mainly from two facts: a straightforward theoretical basis exists, and the experimental realization is relatively simple and inexpensive. These attractive features have instigated an enormous amount of theoretical and experimental work. The present discussion does not cover all of the major aspects, but is intended only as an outline of basic principles and a guide to the literature. For further studies and for all details, additional papers and reviews should be consulted. A first, naive introduction into the theory and a discussion of many experiments can be found in the reviews of Deutsch’ and F r a ~ e n f e l d e r . ~ . ~ A more complete and involved theory is given by Biedenharn and Rose.4 The influence of extranuclear fields has been discussed extensively by Steffen.s A recent survey, by Devons and Goldfarb,6 deals not only with cascade processes, but also with nuclear reactions and with the application to mesons and hyperons. An introduction into the modern theory TURE.

’7

A. M. Portis, Phys. Rev. 91, 1071 (1953).

t We

use the term angular correlation as comprising directional correZation and polarization correlation. In directional correlation only the directions of the radiations are observed; in polarization correlation one determines also the polarization of at least one radiation. M. Deutsch, Repts. Progr. i n Phys. 14, 196 (1950). 2 H. Frauenfelder, Ann. Revs. Nuclear En‘. 2 , 129 (1953). 3 H. Frauenfelder, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 531. Interscience, New York, 1955. L. C. Biedenharn and M. E. Rose, Revs. Modem Phys. 26,729 (1953). R. M. Steffen, Advances in Phys. 4, 293 (1955). (I S. Devons and L. J. B. Goldfarb, in “Handbuch der Physik-Encyclopedia of Physics” (S.Flugge, ed.), Vol. 42, p. 362. Springer, Berlin, 1957.

’

* Section 2.4.2.1 is by Hans Frauenfelder.

130

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

is provided by Biedenharn.’ All of these reviews contain extensive bibliographies. The information provided by angular correlation experiments alone very rarely is sufficient to determine all nuclear properties involved. I n most cases, angular correlation is only one tool out of many and other approaches (cf. Sections 2.4.1, 2.4.2.2, and 2.4.2.3) must be used in conjunction with it to get a complete picture.* 2.4.2.1.2. THEPHYSICAL PICTURE. The measurement of nuclear spins by angular correlation methods depends on the fact that the probability of emission of a particle or quantum by a nucleus is a function of the angle between the nuclear spin axis and the direction of emission. This function is determined by the spin of the decaying nucleus, the type of emitted radiation, and the angular momentum carried away by it. Observation of the radiation pattern with respect to the nuclear spin axis hence amounts to a measurement of the nuclear spin, provided the type of radiation and its angular momentum are known. The desired radiation pattern can be observed if the nuclei are partially or completely aligned, i.e., if their spins are parallel. Under ordinary circumstances, however, nuclei are randomly oriented in space and the resulting radiation pattern is isotropic. Three different methods can be used to produce an ensemble of nuclei which are not randomly oriented. (1) If the decaying nuclei are placed a t a very low temperature in an extremely strong magnetic field (electric field gradient) they are polarized (aligned). The directional distribution of the emitted radiation with respect to the direction of the applied field can then be measured and yields the desired information. (See Section 2.4.2.3.) (2) Nuclei can be produced in a preferential orientation by a spin dependent nuclear reaction. The directional distribution of the secondary radiation with respect to the incoming beam can then be used to determine the nuclear spin. (See Section 2.7.4.) (3) One can select only those nuclei whose spins happen to lie in a preferred direction. Such a selection can be executed if the nuclei decay through successive emission of two radiations R1 and Rz. The observation of R1in a fixed direction selects an ensemble of nuclei which has a nonisotropic distribution of spin orientations. The succeeding radiation Rz then shows a definite angular correlation with respect to R1. The observation of two directions, a8 discussed above, allows in general only the determination of the nuclear spins and the multipole

* For certain aspects of the problems discussed in this section see also Vol. 4, A, Chapters 3.5 and 4.5. ’I L. C. Biedenharn, in “Nuclear Spectroscopy” (F. Ajzenberg-Selove, ed.), Part B. Academic Press, New York, 1960.

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DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

131

orders of the emitted gamma rays, but not of their parities. In order to find the parities, one must, in addition to the directions, also measure the linear polarizations of the emitted radiations. (Experiments involving conversion electrons are an exception. There the directional correlation alone is already parity sensitive.) The idea underlying the determination of electric and magnetic nuclear moments can now be explained quite easily. Assume a certain nuclear orientation to be established by one of the methods discussed above. The nucleus remains in this excited state for a time T and then decays. The particle emitted in the decay is used to obtain information about the nuclear orientation. This information is correct only if the nucleus is unperturbed during its stay in the excited state. Any perturbation, magnetic or electric, will change the radiation pattern. This change can be used to extract information about the electromagnetic moments of excited nuclear states. If one, for instance, applies a magnetic field to the nucleus, it will precess during the time T.The corresponding rotation of the radiation pattern depends on the time T , the applied field H , and the nuclear g factor. It can thus be used to find the nuclear g factor. If, in addition, the nuclear spin is known, the nuclear magnetic dipole moment follows immediately. Similarly, the change of the correlation pattern in an applied electric field gradient yields the quadrupole coupling. The theoretical description of the correlation process can be patterned in such a way that it follows the physical picture given above rather closely :6-g Assume for instance a gamma-gamma cascade. The initial state is then described by a density matrix.'O Using first-order perturbation theory, one calculates the density matrix describing the intermediate nuclear state, which results after the emission of the first gamma ray. Again using first-order perturbation theory for the emission of the second gamma ray, one calcuIates the various interesting properties. The theory of angular correlation will not be discussed any further; all required formulas and results will be borrowed without derivation from the various theoretical papers cited above. 2.4.2.1.3. DETERMINATION OF SPINS.The schematic picture of a directional correlation measurement is shown in Fig. 1. A nuclear decay occurs between states a, b, c, . . . with spins I,, I*,I,, . . . through the successive emission of particles (or photons) R1, R z , . . . . (It is assumed here that the spins and parities of the states a, b, c, . . . are well defined.) The assignment of spins to the nuclear states and of angular momenta to the radiations then involves four steps. F. Coester and J. M. Jauch, Helu. Phys. Acta 26, 3 (1953). U. Fano, Phys. Rev. 90,577 (1953). l o U. Fano, Revs. Modern Phys. 29, 74 (1957). 8

132

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

I. The calculation of the various possible theoretical correlation functions W,,,,,(e). We denote here with W(e) dQ the relative probability that the radiation Rz is emitted into the solid angle dfl a t a n angle 0 with respect to R1 [Fig. l(b)]. For details see step I below. 11. The measurement of the coincidence rate C(6) as a function of the angle 6 included by the axes of two counters a t the source. See Fig. l(c) and step I1 below.

Source C

IC

a

b

C

FIQ.1. (a) Nuclear cascade. (b) W(0)dil is the relative probability that the radiation Rz is emitted into the solid angle dil a t an angle e with respect to R1. (c) The two counters 1 and 2 subtend an angle 29 at the source. From the coincidence rate as a function of angle C @ ) , one obtains after suitable correction the correIation function W(0).

111. Correction of C(8) in order to find W,,,(6). At each angle 19 between the counters, coincidences between radiations including a range of angles e are recorded. Corrections are hence needed to find w,,,(e) from the measured curve. These corrections are discussed in Section 2.4.2.1.7. IV. Comparison of W,,,(e) and Wthear(0).Once the experimental correlation function has been determined, i t has t o be compared with the various possible theoretical functions in order to determine the most likely assignment. I n most cases, a unique assignment is possible only if other methods (e.g. internal conversion, measurement of ground state spin, lifetime, etc.) contribute. Even so, the determination of a unique scheme more often resembles a jig-saw puzzle than a well-defined physics problem. I n the following, we discuss the first two steps in more detail. I. T h e theoretical directional correlation function. The most convenient form in which to express the correlation function W(6)is (2.4.2.1.1)

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

133

The functions P k are Legendre polynomials. (Sometimes, the much less convenient expansion in terms of cask 0 is used. Formulas for the conversion from one expansion into the other can be found in the papers by Steffen5 and Lloyd.") The coefficients A k contain all the physical information. Usually, one chooses Ao = 1, so that the correlation function, integrated over all angles, is unity: JV(0) dQ = 1. Three general rules for the coefficients Ak hold for all types of particles:

k integer, even. 0

I Ic, A k

I min(21b,2L1,2L2). =

Ak(a,b)Ak(b,c).

(2.4.2.1.2) (2.4.2.1.3) (2.4.2.1.4)

Rule (2.4.2.1.2), expressing the fact that only even Legendre polynomials appear, holds as long as one measures only the directions and linear polarizations of the radiations. The observation of the circular (longitudinal) polarization, however, introduces also odd integers (cf. Chapter 2.5). Rule (2.4.2.1.3) states that the highest term ,k in the expansion (2.4.2.1.1) is smaller than, or equal to, the smallest of the three numbers 21b, 2L1, 2L2, where LI and L2 are the angular momenta carried away by R1 and R2. This rule implies isotropic correlation if the angular momentum of either of the radiations or of the intermediate state equal 0 or +. Equation (2.4.2.1.4) expresses the fact that the coefficients Ak for a cascade can be broken up into two factors, each factor depending on only one transition of the cascade. The second factor Ak(b,c), for instance, is entirely determined by the properties of the levels b and c and of the radiation Rz. Numerical values for the factors Ak(a,b) for given properties of the levels a and b and of the radiation R have been calculated and tabulated for most cases of interest. Relevant references are collected in Table I. Consider now a simple example, the directional correlation between two successive gamma rays in the cascade shown in Fig. 2(a) ( I , = 2, I b = 2, I , = 0 ) . The second gamma ray in this cascade must be pure quadrupole, and from the references given in Table I, we find the corresponding directional correlation factors A2(b,c) = -0.5976, A4(b,c) = -1.069. If we assume the first transition to be pure also, for instance dipole radiation (L1 = l), we find immediately A2(a,b) = -0.4183, Ar(a,b) = 0 , and hence A2 = 0.2500 and A4 = 0. These values have to be compared with experiment. Generally, however, the first transition in a 2 -+ 2 3 0 cascade will be mixed, ie., carry away more than one single value of angular momentum. Angular momentum selection rules allow L1 = 1, 2, 3, or 4. As a rule, 11

S. P. Lloyd, Phys. Rev. 8S, 716 (1951).

134

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

TABLEI. References for Directional Correlation of Various Types of Radiations [This table is intended for quick reference only; no credit is given for original work. Compare Biedenharn” and Gibbs and Wayb also.] Radiation Pure ga.mma rays Mixed gamma rays More than two gamma rays Conversion electrons Alpha particles Heavy particles with spin Beta particles

Theory

Tables

a, c, 4 e a, c, d, e a, c, e

c, d, f c, d, f, u c, f, h c, i c, f j, k c, 1

c, e

a, c, e a, e c , el 1

0 L. C. Biedenharn, in “Nuclear Spectroscopy” (F. Ajzenberg-Selove, ed.), Part B. Academic Press, New York, 1960. * R . C. Gibbs and K. Way, “A Directory to Nuclear Data Tabulations,” U.S. Atomic Energy Commission, 1958. L. C. Biedenharn and M. E. Rose, Revs. Modern Phys. 26, 729 (1953). H. Frauenfelder, i n “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 531. Interscience, New York, 1955. S. Devons and L. J. B. Goldfarb, i n (‘Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 42, p. 362. Springer, Berlin, 1957. M. Ferenta and N. Rosenzweig, AEC Report ANL-5324 (1955). 0 R. G. Arns and M. L. Wiedenbeck, Phys. Rev. 111, 1631 (1958). J. M. Kennedy, B. J. Sears, and W. T. Sharp, AECL-106, CRT-569 (1954). ’ M. E. Rose, L. C. Biedenharn, and G. B. Arfken, Phys. Rev. 86, 5 (1952). j W. T. Sharp, J. M. Kennedy, B. J. Sears, and M. B. Hoyle, AECL-97, CRTO-556 (1953). A. J. Ferguson and A. R. Rutledge, AECL-420, CRP-615 (1957). I. Hauser, Nuovo cirnento Suppl. [lo] 6 , 182 (1957).

a

b

FIG.2. (a) 2 -+ 2 0 cascade, occurring frequently in even-even nuclei. (b) The first gamma ray 71 is usually mixed in this case; the two dominant angular momenta being L I = 1, LI‘ = 2.

2.4.

DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS

135

only the two lowest multipoles, L1 = 1 and L1'= 2, occur with measurable intensity [cf. Fig. 2(b)]. The coefficients Az(a,b) and Ad(a,b), and hence also A , and Ad, now become continuous functions of the mixing ratio 6, which in this case indicates the ratio of amplitude of the quadrupole to that of the dipole contribution. (6 is defined as the ratio of reduced matrix element for the emission of L' to that of L r a d i a t i ~ n , ~ ~ ~ - ~ * ~

FIG.3. Parametric plot1%of the directional correlation coefficients A2 and A P for the cascade 2 + 2 -+ 0. 6 ia the mixing ratio (quadrupole to dipole radiation) in the first transition. The curve is labeled with the values of the parameter 6' = [S/(l - ISI)]. Two experimental points are shown.

the ratio of the total intensity of the L' pole to that of the L pole is equal to az.) The most convenient representation of the dependence of A2 and A4 on 6 is a parametric pZot:12 one draws the curve A z = A z ( 6 ) , A4 = A4(8) in an A2 - A4 plane. [The functions Az(6) and A4(6) are easily calculated from the formulas and coefficients given by Frauenfelder,3 Biedenharn and Rose14Ferentz and Rosenzweig,13 and Arns and Wiedenbe~k.1~1 A parametric plot for the cascade 2 -+2 -+ 0 is shown in Fig. 3. The C. F. Coleman, Nuclear Phys. 6, 495 (1958). M. Ferenta and N . Rosenaweig, AEC Report ANL-5324 (1955). 14 R. G. Arns and M. L. Wiedenbeck, Phys. Rev. 111, 1631 (1958). 12 1s

136

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

experimental points with coordinates and must lie on this curve and thus yield values for sign and magnitude of the mixing ratio 6. 11. The experimental determination of C(8). The first problem is to decide when two radiations are genetically related, i.e., have been emitted by the same nucleus. It can be solved by observing the decays in a cloud chamber or a photographic emulsion (coincidence in space), but more often a coincidence in time is used. Two radiations from a source are assumed to originate from the same decay if they are observed within a very short time interval (resolving time). A correction for “accidental coincidences” is then applied. (See Section 2.4.2.1.7.)

FIQ.4. Slow-fast coincidence setup for directional correlation measurements.

A typical experimental setup for gamma rays is shown in Fig. 4 (slowfast circuit). The gamma rays y1 and yz are counted in the scintillation counters 1 and 2 (cf. Section 1.4.1). The outputs of these counters are divided into a OW^' and a “fast” channel. The slow channels, using linear amplifiers, preserve the pulse heights and determine the energies of y1 and y z . I n the fast channels, only the very first part of each pulse is used. Wide band amplifiers “blow up” the pulse; the very first part of the leading edge is then shaped into a very short pulse, usually with delay lines. The widths of these pulses essentially determine the resolving time of the circuit. The fast coincidence circuit responds only if the shaped pulses from both counters arrive within the “resolving time.” The output of the two slow channels and of the fast coincidence circuit are fed into a slow coincidence circuit. An output signal results only if the two gamma rays possess proper energies and if the two pulses are simultaneous (i.e.,

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

137

arrive within the resolving time). The simple arrangement shown in Fig. 4 is adequate for many y-y directional correlation experiments. Improvements and changes, however, can be made in various directions. For work requiring extremely short resolving times, the ordinary photomultipliers can be replaced by types not requiring additional amplification. The NaI counters are replaced by other detectors if radiations other than y rays are observed (cf. Chapters 1.2-1.5, 1.8, 2.2, and papers by Frauenfelder3 and Devons and GoldfarbO). If one radiation is in coincidence with many others or with a continuous spectrum, as for instance in a beta-gamma coincidence experiment, then the use of multichannel pulse-height analyzers permit,s the simultaneous recording of the directional correlation function a t various energies.en15 Data gathering, quite often the limiting factor in correlation work, can be speeded up by the use of more than two counters.16 Detailed discussions of the electronics involved in directional correlation experiments can be found in papers by Bell,16v17 Kandiah and Chaplin, l 8 de Benedetti and Findley,lg and in the present series.20 2.4.2.1.4. DETERMINATION OF PARITIES; LINEARPOLARIZATION OF GAMMARAYS. The parity of a nuclear state with respect to that of another state (usually the ground state) can be found by determining the parity of a transition linking the two. As in the case of the spin determination, the decaying nucleus is first partially or completely oriented by one of the three methods discussed under Section 2.4.2.1.1.The parity of a t,ransition can then be inferred from the observation with respect to the direction of orientation of (a) the directional distribution of conversion electrons, or (b) the directional distribution and linear polarization of gamma rays. The first approach is better suited to low energy transitions (say, E < 0.3 Mev), where the conversion coefficient is appreciable; the second is more favorable for high-energy gamma rays (say, E > 0.2 Mev). 2.4.2.1.4.1.Conversion Electrons. The directional correlation involving conversion electrons is similar to that of gamma rays treated in Section 2.4.2.1.3.The coefficients Ah(a,b),however, depend now also on the parity of the conversion electrons; i.e., for a given multipole order L, they differ ‘6 T. R. Gerholm, in “Proceedings of the Rehovoth Conference on Nuclear Structure’’ (H. J. Lipkin, ed.), p. 483. Interscience, New York, 1958. 18 R. E. Bell, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), Chapter 18. Interscience, New York, 1955. l 7 R. E. Bell, Ann. Revs. Nuclear Sci.4, 93 (1954). K. Kandiah and G. B. B. Chaplin, Progr. in NucEear Phys. 6, 89 (1956). 19 S. de Benedetti and R . W. Findley, in “Handbuch der Physik-Encyclopedia of Physics” (S. FlCgge, ed.), Vol. 45, p. 222. Springer, Berlin, 1958. 2 0 S P Vol. ~ 2 of this series.

138

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

for electric and magnetic radiation. Numerical values for the coefficients A,(a,b) can be found in the references given in Table I. The experimental realization requires special care: the source must be very thin and uniform in order to minimize scattering of the conversion electrons, scattering between source and counter has to be avoided, K electrons must be selected and recorded in order to permit comparison with the theoretical calculations, and extranuclear effects must be avoided or taken into account. Details regarding all these points can be found in the reviews,ass in the references given there, and in the papers by Gimmi, Heer, and Scherrer,21Coburn, Kane, and FrankeljZ2and Snyder and FrankeLZs 2.4.2.1.4.2. Linear Polarization of Gamma Rays. Electric and magnetic radiation of the same multipole order L have opposite parity; their electric and magnetic vectors are related through the transformation E 4 H, H 4 -E. This transformation leaves the Poynting vector and hence the directional distribution of the radiation unaltered. A measurement of the directional correlation thus yields L, but gives no information about the parity. With L known, the observation of the direction of E (i.e., of the linear polarization of the gamma ray) furnishes the parity, however. I n angular distribution experiments, one can directly detect the linear polarization of emerging y rays, without a coincidence setup. I n cascade decays, one must observe the linear polarization of y rays in coincidence with a preceding or following radiation. Relevant formulas for such polarization-direction correlations can be found in papers by Biedenharn and RoseI4Devons and Goldfarb16and Biedenharn.' Basically, such a measurement is similar to that treated in Section 2.4.2.1.3 and hence the discussion here is restricted to the new aspect, namely the detection of the linear polarization of y rays. Before treating methods for measuring the linear polarization, a few remarks concerning the description of the state of polarization of photons are in order. The complete description in terms of Stokes parameters can be found in various r e f e r e n ~ e s ~ J ~(cf. , ~ 4Section 2.5.2). For the following discussion, an outline of selected pure cases is sufficient. Circularly polarized y rays have their spin ( l h ) parallel (right c.P.) or antiparallel (left c.P.) to their direction of motion, Linear polarization cannot be described so simply in terms of the spin of photons. Classically, a beam of y rays with complete linear polarization corresponds to electromagnetic radiation in which the electric vector E lies always in the direction of polarization. Polarimeter, i.e., detectors sensitive to the state of polarization of 2'

F. Gimmi, E. Heer, and P. Schemer, Helv. Phys. Acta 2B, 147 (1956).

** H. C. Coburn, J. V. Kane, and S. Frankel, Phys. Rev. 106, 1293 (1957). 2a 24

E. S. Snyder and S. Frankel, Phys. Rev. 106, 755 (1957). w. H. McMaster, Am. J . Phy8. 22, 351 (1954).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

139

gamma rays, are always based on some kind of scattering. Methods to determine the degree of circular polarization are generally based on the Compton effect2S*26 for low-energy y rays ( < 6 Mev), and on the nuclear p h o t ~ e f f e c t ~ ’ for - ~ ~higher energies ( E > 3 Mev). In addition to these two major methods, some additional polarization sensitive effects can probably be used-the atomic photoeffect30 and pair p r o d ~ c t i o n . ~ ~ As yet, no detectors based on these effects have been built. The Compton p ~ l a r i m e t e r , ~treated ~ - ~ ~ here in some detail as an example, is based on the Klein-Nishina f o r m ~ l afor ~ the ~ * differential ~~ cross section du/dQ for Compton scattering

@

dn

-

[

k !f 2 ko2 ko

+ ko-k - 2 sin2 6

0082

1.

3

(2.4.2.1.5)

Here, ko is the initial energy of the y ray and k the energy of the scattered quantum k0 (2.4.2.1.6) k = 1 (ko/mc2)(1 - cos 6)‘

+

6 is the scattering angle, 3 the angle between the direction of polarization of the incident photon and the plane of scattering, ro = e2/mc2 the classical radius of the electron. Equation (2.4.2.1.5)shows that du/dQ depends on 4 and that the observation of Compton scattering hence can be used for the determination of the direction and degree of linear polarization. Figure 5 shows the geometry used in a n ideal experiment; Fig. 6 displays t8heintensity of scattered radiation, for fixed values of ICO and 6, as a function of the angle 4. The idealized polarimeter shown in Fig. 5 operates by recording the number of Compton scattered photons as a function of the angle 3. The minimum of the corresponding curve (Fig. 6) indicates the direction A. Wightman, Phys. Rev. 74, 1813 (1948). N. R. Steenberg, Can. J . Phys. 31, 204 (1953). 2’ D. H. Wilkinson, Phil. Mag. [7] 43, 659 (1952). z8 L. W. Fagg and S . S. Hanna, Phys. Rev. 92, 372 (1953). 2’) I. S. Hughes and D. Sinclair, Proc. Phys. SOC.(London) A89, 125 (1956). F. L. Hereford and J. P. Keuper, Phys. Rev. 90, 1043 (1953). s1 M. M. May, Phys. Rev. 84, 265 (1951). F. Metzger and M. Deutsch, Phys. Rev. 78, 551 (1950). s3 J. J. Kraushaar and M. Goldhaber, Phys. Rev. 89, 1081 (1953). * 4 D. R. Hamilton, A. Lemonick, and F. M. Pipkin, Phys. Rev. 92, 1191 (1953). s6 G. R. Bishop and J. P. Perez y Jorba, Phys. Rev. 98,89 (1955). 9 B C. F. Coleman, Phil. Mag. [S] 1, 166 (1956). 3’ P. H. Stelson and F. K. McGowan, Phys. Rev. 106, 1346 (1957); 109,901 (1958). 3* W. Heitler, “Quantum Theory of Radiation,” Oxford Univ. Press, London and New York, 1954. 2s

z8

140

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Polarization Vector scatterer

FIQ.5. Geometry for a polarimeter based on the Compton scattering. Polarization Vector Counter

FIQ.6. Polardiagram showing the cross section o!u/ds2 for Compton scattering as a function of the angle ip between the direction of polarization of the incident gamma ray and the plane of scattering. The incident photons with energy ko are completely polarized and enter the scatterer perpendicular to the plane of the figure. The scattering angle 6 is fked at 80". The intensity distribution is shown for two incident energies, 0.5 Mev and 3 MeV.

of polarization. To determine the degree of polarization, one assumes that the incident y-ray beam of arbitrary linear polarization consists of two completely polarized and mutually orthogonal beams, with intensities I,,and II. The polarization vector of 111lies in the direction of the scattering minimum (as shown in Fig. s), that of II a t right angle to it. The

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

141

degree of polarization of the incident beam can then be described by (2.4.2.1.7) The measured numbers of scattered y rays in the two extreme points are given by N,i, = I , ,&(a = 0") 11 dU(* = 90") (2.4.2.1.8) N,,, = I , ,dU(* = 90") 11 &(a = 0").

+ +

From (2.4.2.1.7)and (2.4.2.1.8),the degree of polarization P follows as (2.4.2.1.9) where R, called the asymmetry ratio of the polarimeter, is defined by du(@ =

R = &(a

=

90") 0")

(2.4.2.1.10)

For the idealized polarimeter of Fig. 5, R can be calculated from the Klein-Nishina formula, Eq. (2.4.2.1.5). An actual polarimeter can be characterized by five quantities: asymmetry ratio, efficiency, solid angle, energy resolution, and signal-to-noise ratio. The asymmetry ratio R is defined by a n obvious generalization of Eq. (2.4.2.1.10).For intensity reasons, scatterer and counter subtend large solid angles. The angles 6 and a hence possess considerable variation (e.g., A 6 = A0 = 50"), and for each energy ko, R must be either calculated by integration, or it must be determined experimentally. (See Section 2.4.2.1.7 and the papers by Metzger and D e u t s ~ h Kraushaar ,~~ and G ~ l d h a b e r Hamilton ,~~ et ~ l . Bishop , ~ ~ and Perez y J ~ r b aColeman,36 ,~~ and Stelson and M~Gowan.~') A good polarimeter should of course combine a small energy resolution with the largest possible values of the other characteristic quantities. The simplest Compton polarimeter consists of a y-ray collimator (lead or tungsten), a scatterer (e.g., lead or copper), and a counter.3g The collimator shields the counter from direct radiation. The mean scattering angle 6 is so chosen that the asymmetry ratio R becomes a maximum. The major disadvantage of such a polarimeter is a small signal-to-noise ratio and a bad energy resolution. Considerable improvement is achieved by using a scintillation counter as scatterer and measuring coincidences between signals in the' scatterer and the counter (now called analyzer) .32-40 The signal-to-noise ratio for S. de Benedetti, in "Beta- and Gamma-Ray Spectroscopy" (K. Siegbahn, ed.),

3%

p. 674. Interscience, New York, 1955. 40 J. I. Hoover, W. R. Faust, and C. F. Dohne, Phys. Rev. 86, 58 (1952).

142

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

coincidences is much larger than that for single counts. The energy resolution is still bad, since the spread A6 in the scattering angle 6 causes a corresponding unavoidable spread of pulse height in the scatterer and the analyzer. The spread of the energy can be seen by calculating (klko) from Eq. (2.4.2.1.6) for the various angles 6. Energy resolution and signal-to-noise ratio can, however, be further improved.36The scintillation crystal of the analyzer is made sufficiently large so that the scattered y rays are nearly always completely absorbed when being counted. The output pulses from the scatterer and the analyzer are again fed into a coincidence circuit, but they are also added. A true event is recorded only if a coincidence occurs and if simultaneously the sum pulse possesses a pulse height corresponding to the initial energy ko. 2.4.2.1.5. DETERMINATION OF GYROMAGNETIC RATIOS.The basic idea underlying the determination of gyromagnetic ratios has been sketched in Section 2.4.2.1.2.Detailed treatment of the theoretical and experimental aspects can be found in the papers by Frauenfelder,* Biedenharn and Rose,4Steffeq6 Devons and Goldfarb,* Gerholm,I6Alder et u Z . , ~ ~Abragam and Aeppli et Krohn and R a b ~ ySteffen , ~ ~ and Z0be1,~~ and Bodenstedt et al.46aThe present discussion is restricted to an outline of the experimental approach. The gyromagnetic ratio of an excited nuclear state can be measured if the following conditions are fulfilled. sec and sec. (a) The lifetime T of the state lies between about (b) The state is either the intermediate state in a nuclear cascade or it can be produced through a nuclear reaction (e.g., Coulomb excitation), (c) The state decays by emission of a radiation (usually a y ray) which displays a nonisotropic directional distribution with respect to either the direction of the incoming beam or the direction of emission of the first radiation. (d) The directional distribution or correlation is not appreciably disturbed by quadrupole interaction. [Quadrupole interaction can always be present, since condition (c) requires that the intermediate state possess a spin of at least 1.1 Assume now that a cascade, yI-yz, has been found which fulfills the first three conditions. The next step consists in preparing a source in 41 K. Alder, H. Albers-Schonberg, E. Heer, and T. B. Novey, Helv. Phys. Actu 26, 761 (1953). 48 A. Abragam and R. V. Pound, Phys. Rev. 93, 943 (1953). 4) H.Aeppli, H. Albers-Schonberg, H. Frauenfelder, and P. Schemer, Helv. Phys. Acta 26, 339 (1952). 4 4 V. E. E o h n and S. Raboy, Phys. Rev. 97, 1017 (1955). 45 R. M.Steffen and W. Zobel, Phys. Rev. 103, 126 (1956). 46s E. Bodenstedt, H. J. Kroner, G. Strube, C. Giinther, J. Radeloff, and E. Gerdau, 2.Physik 163, 1 (1961).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

143

which the nuclei experience little or no quadrupole interaction. This problem can often be solved by preparing a nonviscous liquid ~ o u r c e , ~ , ~ . ~ , 21,42,44,46-49 by using the substance in the melted p h a ~ e ,by ~~ embedding ~,~ the radioactive atoms in a cubic crystal3 or in a n oriented single crystal.60 Next, the directional correlation function W ( 0 ) , Eq. (2.4.2.1.1) is determined (Section 2.4.2.1.3).Assume that the measured coefficients Az, Ah, together with other data yield an unambiguous value for the spin I of the intermediate nuclear state. Now, the gyromagnetic ratio can be found by applying to the source a magnetic field H which is perpendicular to the plane of the two counters and the source [Fig. 7(a)J.Each of the two counters shall be sensitive to only one y ray of the cascade. During the time t , which elapses between the emission of y 1 and that of yz, the nucleus precesses in the field H by an angle W H t , where W H is the Larmor frequency (2.4.2.1.11)

the nuclear magneton. Instead of the correlation W ( 0 ) ,one observes W(O - W H t ) , averaged over the range of time t which the coincidence circuit accepts. Suppose that it registers only if yz is emitted within the time interval T I to Tz after the emission of y 1 [Fig. 7(b)]. The measured correlation function is then given by p o is

w(e,H)

-

JTT’exp(-t/T)W(B - W H t , H

=

0) dt.

(2.4.2.1.12)

In practice, the time T Iand Tz are determined by the resolving time T , of the coincidence circuit and a delay Ta inserted into the channel which counts y l (Fig. 7). The choice of Ta and T,depends on the lifetime of the nuclear state under investigation. The discussion of the following three cases shows all the important aspects of the method. Assume for simplicity that the highest term in Eq. (2.4.2.1.1) is given by lc, = 2. Moreover, instead of using Eq. (2.4.2.1.1),it is more convenients1to employ the equivalent expansion in terms of cos n0: w(e) = 1 B~ cos 20 (2.4.2.1.13) B , = 3A2/(4 Ad.

+

+

Three important special cases can now be treated easily. H. Albers-Schonberg, E. Heer, and P . Schemer, Helv. Phys. Acta 27, 637 (1954). R. M. Steffen, Phys. Rev. 103, 116 (1956). 48 V. E. Krohn, T. B. Novey, and S. Raboy, Phys. Rev. 106, 234 (1957). 4s G. Goldring and R. P. Scharenberg, Phys. Rev. 110,701 (1958). SOH. Albers-Schonberg, E. Heer, T. B. Novey, and P. Schemer, Helv. Phys. Acta 27, 547 (1954). 6 1 K. Alder, Helv.Phys. Acta 26, 235 (1952). 4fi

47

144

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Magnetic Field H

b 1

Td

_I Coincidences

Resolving TimeTr

Decoys in this intervol ore accepted by the coincidence circuit

Emission ot

'1

3;

I_ Delay Td in

Channel I

Td T2

_I

(b) FIG.7. (a) Setup for the determination of the gyromagnetic ratio of the intermediate state in a y-7 cascade. The magnetic field H is applied perpendicular t o the plane containing the source and the two counters. (b) N ( t ) dt denotes the probability of emission of ya in the time interval t to t dl after emission of 7 , . T indicates the lifetime of the intermediate nuclear state. The coincidence circuit accepts only decays which occur in the interval T Ito Tz.T Iand TPare fixed by the delay Td in channel 1 and the resolving time T,: TI= Td - T,,T2 = T d T,.

+

+

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

145

I. Td = 0, T , >> r , each counter sensitive to one y ray only. This case applies particularly to nuclei with very short lifetimes (T = 10-lo sec). Present technique then does not permit using coincidence circuits with shorter resolving times and the assumptions made represent actual experiments quite well. The selection of y 1 in channel 1 and yz in channel 2 is performed with a slow-fast circuit (Fig. 4). The effect of the magnetic field on the correlation pattern under the assumptions I can be calculated by inserting Eq. (2.4.2.1.13) into Eq. (2.4.2.1.12) and setting T1 = 0, T 2 = 0 0 . The result

+

WI(@,H)= 1 [B,/d1 f (2WHT)2]cos2(8 - AO) (2.4.2.1.14) where A0 = & arc tan 2 ~ ~ clearly 7 , shows an attenuation [by a factor l/dl ( ~ O R Tand ) ~ ] a rotation (by an angle A@) of the correlation pattern. For small angles, AO is approximately equal to the classical angle of precession W H T . The magnitude of the g factor can be determined from the attenuation or from the rotation. The sign of g follows from the direction of rotation. 11. Td = 0, T , >> r , each counter accepts both y rays. Such a situation occurs if both y rays possess very nearly equal energy. The corresponding effect on the correlation pattern can be seen by calculating the function WII(O,H)= +(Wl(O,H) Wl(O, - H ) j , since exchange of y1 and y z corresponds to a transformation H -+ -H. Hence

+

+

As expected, only an attenuation occurs, but no rotation. Only the magnitude, but not the sign of g can be found. 111. Td > T,, T , << T . In this case, coincidences occur only if y 1 enters channel 1 and yz channel 2 [cf. Fig. 7(b)]. From Eq. (2.4.2.1.12),one finds directly WIII(@,H) = - WHTd, H = 0). (2.4.2.1.16) The original correlation function is hence only rotated, but not attenuated. The g factor must be found from the rotation. Case I11 can be applied to relatively long-lived states ( ~ : 1 0 -sec ~ to lop7see). The preceding outline displays the most important features of the directional correlation method for determining nuclear g factors. A number of additional remarks are in order, however. (1) The assumption kmaX= 2, used in I and I1 above, can easily be removed. Relevant formulas can be found in papers by Biedenharn and Rose,4 Steffen,6 Devons and Goldfarb,s Alder et uZ.,~' Abragam and Pound,42 Alder,61and Nierenberg.62

w(@

62

W. A. Nierenberg, Ann. Revs. Nuclear Sci. 7, 349 (1957).

146

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

(2) Cases intermediate between the ones treated above can also be handled without too much complication. Calculations for correlation functions with arbitrary values of T d , T,, and r can be found in papers by Alder et aZ.41 and Steffen and Z ~ b e l . ~ ~ (3) The limits for the applicability of the angular correlation method, sec are given by the following considerarain= sec and rmX = tions. The presently available maximum field strengths, H,, 5 lo6 gauss, determine rain.For shorter lifetimes, the resulting rotation of the correlation pattern is too small to be observed. The upper limit, rmX,is fixed by the circumstance that an optimum measurement of the attenuation or rotation requires a resolving time T, which is comparable to r. The number of accidental coincidences, however, is proportional to T , and outweighs the true coincidences for r > sec. (4) The experiments discussed above all yield the nuclear gyromagnetic ratio. If the spin I of the intermediate state is also known, then the magnetic moment follows CC = glro. (2.4.2.1.17) (5) Assumption (d) above, demanding the absence of quadrupole interactions, is often not fulfilled. Nevertheless, the true g factor can be found, either by applying corrections to the g factor inferred from attenuation, or by using the g factor given by the r o t a t i ~ n . ~ ~ , ~ ~ . ~ ~ (6) Experimental details can be found in papers by Gerholm,16Aeppli et al.,43Krohn and R a b ~ ySteffen , ~ ~ and Z0be1,~~ Krohn et Goldring and S ~ h a r e n b e r g and , ~ ~ Albers-Schonberg et U Z . ~ O It is obvious that the strong magnetic field present a t the source necessitates, in addition to the usual requirements for anguIar correlation experiments, light pipes between scintillation crystals and photomultipliers and very good magnetic shielding of the multipliers. (7) I n the discussion of cases I and 111, it is assumed that the complete correlation function is measured as a function of the applied field. Actually, it is advantageous to use three (or more) counters in fixed positions, e.g., one at 0', one a t 135', and one a t 225'. Coincidences 0'-135' and 0'-225' are measured. The ratio R = C(0'-135')/C(0'-225') as a function of the applied field H then is a measure for the rotation of the pattern and allows a very accurate determination of W H . (See Krohn and Raboy44 and Steffen and Z0be1.~~) (8) Further improvement results if more than three counters are used and if coincidences between all pairs of counters are recorded. Such an arrangement greatly improves statistical precision.16B6a (9) Even in strong magnetic fields, and with states of very long life6)

T. R. Gerholm, T. Lindqvist, and H. de Waard, Nuclear Instr. 1, 107 (1957).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

147

time, the nuclei perform only a small number of revolutions. Magnetic resonance experiment^^^ hence seem not feasible but it may be possible to use the method of “stroboscopic coincidence^."^^ (10) Gyromagnetic ratios can also be determined if the excited state is reached by a nuclear.reaction. The magnetic field is applied to the target and the corresponding rotation of the directional distribution of the de-excitation radiation is observed. No coincidences are involved and all nuclei, regardless of the actual time 2 they spend in the excited state, are detected. The approach thus corresponds to case I above. Usually, two counters are used in fixed positions and the ratio R [see remark (7)]as a function of the applied field yields the g factor. (See the papers by Goldring and S ~ h a r e n b e r gLehmann ,~~ et u Z . , ~ ~TreacylK6Sugimoto and Mizob u ~ h iPhilips , ~ ~ and Jones,68and S ~ g i m o t o . ~ ~ ) 2.4.2.1.6. DETERMINATION OF ELECTRIC QUADRUPOLE MOMENTS. The measurement of gyromagnetic ratios, as discussed in Section 2.4.2.1.5, is quite straightforward. The application of similar ideas to the determination of electric quadrupole moments is much more intricate and has yielded far fewer results. The discussion here is hence very brief. Two facts render experiments on quadrupole moments difficult. (a) The interaction between an electric quadrupole moment and an inhomogeneous electric field depends quadratically on the magnetic quantum number m, which describes the orientation of the nucleus with respect to the extranuclear field. Classically, this m degeneracy means that the nuclei precess with equal probability in both directions. No rotation of the correlation pattern occurs and all the information must be gathered from the attenuation. This approach is less precise and also does not yield the sign of the quadrupole moment. (b) No artificial field gradients can be produced which are large enough to perturb an angular correlation, hence the natural field gradients in solids and liquids must be used for the study of the interaction. Since the radioactive atoms are usually foreign to the lattice in which they are embedded, and since they also recoil when undergoing a decay, it is extremely difficult to estimate reliably the electric field gradient present a t the nucleus. Results of most measurements are expressed in terms of the quadrupole coupling, i.e., the product Q * VE, where Q is the nuclear quadrupole moment and V E the electric field gradient. The separation into the two factors is almost always extremely difficult. V. L. Telegdi, unpublished. P. Lehmann, A. LBvBque, and R. Pick, Phys. Rev. 104, 411 (1956). 68 P. B. Treacy, NuckaT Phys. 2, 239 (1956). 67 K. Sugimoto and A, Mizobuchi, Phys. Rev. 103, 739 (1956). 68 W. R. Philips and G. A. Jones, Phit. Mag. [S] 1, 576 (1956). 69 K. Sugimoto, J . Phys. SOC. Japan 15, 240 (1958). 64

66

148

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

Information on the determination of the quadrupole coupling has been described in the following papers: F r a ~ e n f e l d e r ,Steffen16 ~ Devons and Albers-Schonberg et aZ.,46 Goldfarb,6 Alder et aZ.,41Abragam and Steffen,47 Krohn et al.,48 Albers-Schonberg et a l l s o Lehmann and Miller,6oand Harris.6Oa 2.4.2.1.7. EXPERIMENTAL CORRECTIONS AND TESTS.Three experimental phases will be outlined briefly. Tests must establish that the equipment works properly; a procedure for getting reliable data has to be worked out in each experiment; the raw data must be corrected for all deviations of the actual setup from the ideal one. 2.4.2.1.7.1. Tests. The invention of tests forms a n essential part of the preparation for any experiment. During the setting-up of the equipment, ideas for tests occur naturally. The following list is hence not a complete guide, but it is intended only as a n example. In a directional correlation experiment, the following tests are valuable. (1) For each channel, one determines stability, energy resolution, energy calibration, linearity of response, and efficiency with radioactive isotopes of well-known radiation characteristics. (2) The performance of the counting systems as a function of source strength is observed. (3) The solid angle Q of the counters is determined, either by calculation61or by experiment.62 (4) The resolving time T r of the coincidence circuit is investigated as a function of source strength and energy of incident radiation. T , can be found either by measuring the coincidences from a “prompt” cascade (i.e., a cascade with a very short lifetime of the intermediate state) with a delay Td > Tr in one channel, or by using two independent sources and separating the counters. ( 5 ) Scattering from one counter into the other, or from material close to source or counters, is checked. (6) The over-all performance of the directional correlation system is tested by measuring V(6)for well-known cascades (Nieo,Pd106). Most of these tests apply also to polarization-direction correlations. I n addition, however, the asymmetry ratio R of the polarimeter must be determined by one or more of the following experiments. (7) R can be found by measuring the linear polarization of gamma rays which were scattered by a n angle of about 90°.32--34,40 (8) The polarization-direction correlation of a well-known cascade (NiSo,Pdlo6)can serve to fix R.33 P. Lehmann and J. Miller, J . phys. radium 17, 526 (1956). S. M.Harris, Nuclear Phys. 11, 387 (1959). M.E.Rose, Phys. Rev. 91, 610 (1953). 82 J. S. Lawson, Jr., and H. Frauenfelder, Phys. Rev. 91, 649 (1953).

2.4.

DETERMINATION OF SPIN, PARITY, .4ND NUCLEAR MOMENTS

149

(9) The linear polarization of y rays following Coulomb excitation in even-even nuclei is well known and hence can also be used to find R.37 (10) The linear polarization of annihilation quanta yields a value for R a t the energy 0.51 MeV. This experiment, however, requires two polarimeter. 39 2.4.2.1.7.2.Procedure. It is simplest to discuss a y-y directional correlation. Other measurements are obvious generalizations of this basic experiment. Before a run, and at intervals during runs, resolving time and back, are determined. N a n d C denote single counts ground counts [ N i 0 ( 8 ) Co(9)] and coincidences, respectively, per unit time. 8 denotes the angle subtended a t the source by the counters (Fig. 1). A source of proper strength is prepared. The maximum source strength is determined by the ratio of accidental to true coincidences. For accurate measurements, this ratio should be smaller than about 0.2;under unfavorable conditions, one may be forced to make it unity or even higher. The source is then centered to about 1 %; i.e., the counting rate N ( 9 ) of the movable counter as a function of the angle 6 should be constant to about 1 %. After the source is centered, the coincidences and the single counts are recorded for various angles. The angle should be changed frequently, e.g., about every 20 min. For short-lived radioisotopes, more frequent changes are advisable. The choice of angles depends on the form of the correlation function. I n general, it is best, to use a few angles only (e.g., go", 120", . . . 270") and to obtain a t each angle a large number of coincidences. For extended experiments, automatic equipment for changing the angles and recording the data is very convenient. From the measured data, one obtains the "true" values Ct(S) and N t ( 8 ) by subtracting background, accidental coincidences, and contributions due to disturbing radiation^;^^ e.g. :

Cf(fi)= C m e a s (9) - Co(8) - Carc(9) @(9) = 2TrNr;1""(9)Ny""(8).

- Cd(9) (2.4.2.1.18) (2.4.2.1.19)

The contribution Cd(8),due to other coincident y rays present in the source must be found in a separate measurement, with the energy selection adjusted in such a way th at the various contributions can be singled out. The true number of single counts and coincidences can now be written as : Nit(&) = Mpifli~i (2.4.2.1.20) Ct(8) = ~ P , P , Q l E l Q 2 ~ 2 E , ~ ( i ) . (2.4.2.1.21) 63

N. Levine, H. Frauenfelder, and A. Rossi, Z. Physik 161, 241 (1958).

150

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

M is the number of nuclear disintegrations per unit time, pi the probability per disintegration that the radiation selected in counter i is emitted, Qi the solid angle in units of 4n, ei the efficiency of channel i, and ec the efficiency of the coincidence circuit. K (8) denotes the directional correlation function, as measured with the equipment under discussion. Comparison of Eq. (2.4.2.1.21) with Eq. (2.4.2.1.19) shows that the accidental coincidences increase with the square of the source strength, the true coincidences only linearly. As mentioned above, an optimum source strength exists which can be calculated from Eqs. (2.4.2.1.19)(2.4.2.1.21). Decreasing the resolving time T , permits the use of correspondingly stronger sources. Equations (2.4.2.1.19)-(2.4.2.1.21) also show that the ratio of true to accidental coincidences is unaffected by the solid angle. Very large solid angles hence are often advantageous. (Optimum solid angles are discussed in Devons and Goldfarb.e) For the further evaluation, the coincidence ratio C f ( 8 ) / N 1 ' ( 8is ) formed. From Eqs. (2.4.2.1.20) and (2.4.2.1.21), one then finds for the desired uncorrected experimental correlation function K ( 8 ): (2.4.2.1.22)

If one uses movable counters, it is seen from this equation that one best divides through by the counting rate in the niovable counter, i.e., that one identifies counter 1 with the movable one. The ratio K ( 8 ) then becomes independent of the solid angle of the movable counter. Furthermore, it is independent of the decay of the source, of the branching probability pZ and of the efficiency of the movable counter. Small errors in the centering of the source are corrected in first order. The numbers K ( 8 ) are now fitted by

+ A;~'~(COS8) +

K ( 8 ) = Ka(1

*

*

.

+ A~maPkmar(COS 8))

(2.4.2.1.23)

using the method of least square^.^^^^^-^' It usually suffices to take = 4. Details concerning the determination of the coefficients A k ' and their errors are given in RoselB1 Breitenbergerl64 and Price;6s applications of the method are treated in Klema and McGowanBsand Breitenberger.67 2.4.2.1.7.3. Corrections. K ( 8 ) corresponds to the correlation function W(0)only under the assumption of centered point sources, point detectors,

,k

E. Breitenberger, Proc. Phys. Soc. (London) A69, 489 (1956). 86

P.C.Price, PhiE. Mug.[7]46,237 (1954);Proc. Cumbridge Phil. Soc. 60,491 (1954).

s8E. D.Klema and F. K. McGowan, Phys. Rev. 91, 616 (19531. E. Breitenberger, Proc. Phys. SOC.(London) A69, 453 (1956).

2.4.

DETERMINATION

OF SPIN, PARITY,

AND NUCLEAR MOMENTS

151

TABLE11. Corrections in Angular Correlation Experiments [The references are not complete; in some cases, only a few out of a large number of papers are given as examples.] Correction for

References

Perturbing influence of extra nuclear fields Finite size of counters Finite sire of source Decentering of source Scattering in the source Count er-to-counter scattering Disturbing radiations

a, b, c, d b, eel, r b, c, e, f, i, k

m

f,

c, n c, e, 0 j , P, q

A. Abragam and R. V . Pound, Phys. Rev. 92, 943 (1953). Devons and L. J. B. Goldfarb, in “Handbuch der Physik-Encyclopedia of Physics” (5.Flugge, ed.), Vol. 42, p. 362. Springer, Berlin, 1957. H. Frauenfelder, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 531. Interscience, New York, 1955. R. M. Steffen, Advances in Phys. 4, 293 (1955). M. Walter, 0. Huber, and W. Ziinti, Helv. Phys. Acta 23, 697 (1950). f S. Frankel, Phys. Rev. 83, 673 (1951). 0 M. E. Rose, Phys. Rev. 91, 610 (1953). E. Breitenberger, Proc. Phys. Soc. (London) A66, 846 (1953). i A. M. Feingold and S. Frankel, Phys. Rev. 97, 1025 (1955). i E. L. Church and J. J. Kraushaar, Phys. Rev. 88,419 (1952). J. S. Lawson, Jr., and H. Frauenfelder, Phys. Rev. 91, 649 (1953). E. D. Klema and F. K. McGowan, Phys. Rev. 92, 1469 (1953). E. Breitenberger, Phil. Mug.171 46, 497 (1954). E. Breitenberger, Proc. Phys. SOC.(London) A67, 1108 (1954). H. Frauenfelder, Ann. Revs. Nuclear Sci. 2, 129 (1953). p E. D. Klema, Phys. Rev. 100, 66 (1955). q N. Levine, H. Frauenfelder, and A. Rossi, 2.Physik 161, 241 (1958). H. I. West, Jr., Univ. of California Report UCRL 5451 (1959). a

* S.

no scattering, no disturbing radiations, and no extranuclear effects. In order to compare experimental and theoretical results, K(6) must be corrected for all deviations from such an ideal arrangement. All of the necessary corrections have been discussed in the literature ; relevant references are collected in Table 11. After applying the corrections, the final form of the experimental correlation function becomes P P ( 6 )=

K O (1 + AYP~(cos6) + . . .

+ A ~ s x P k , , ( ~6)). ~~ (2.4.2.1.24)

One now makes the assumption 6 = 0 and compares theoretical coefficients A k with AYp in order to extract the desired physical information.

152

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

2.4.2.2. Conversion Coefficients.* 2.4.2.2.1. INTRODUCTION; THEOCONSIDERATIONS. The ratio of the probabilities for emitting Y-shell conversion electrons (see Section 2.2.3.1; Y = K , LI * * Mv * Ltotal *) to that of nuclear gamma rays is called the Y-shell conversion coefficient ay. This quantity, as well as the ratio of conversion coefficients in different shells depend on the character of the transition under consideration, i.e., on whether the parities of the initial and the final state are the same or not, and on the distribution of the angular momenta of the emitted gamma rays. As discussed in Section 2.4.2.1, the last datum is connected with the difference between the spins of the initial and the final levels. Information about spins and parities of nuclear levels can, therefore, be obtained by comparing measured conversion coefficients, or ratios of conversion coefficients for the same transition in different shells, with the theoretically computed values. We will therefore first discuss some results of conversion theory and then some experimental procedures for measuring conversion coefficients. Conversion coefficients also depend on the transition energy and on the nuclear charge. Since these quantities are mostly known with sufficient precision, their influence will not be discussed here (except for remarking that conversion coefficients mostly increase strongly with increasing Z and with decreasing E ) . Moreover, only theoretical results for pure multipole transitions will be considered; the probabilities for emitting gamma rays and those for emitting conversion electrons of different multipole orders have to be added incoherently. Conversion coefficients depend on the nuclear structure, both in a static as well as in a dynamic way.' The static effect arises through the action of the average nuclear charge distribution on the atomic electrons. Practically, it is dependent on the nuclear radius only. The dynamic effect originates in the circumstance that nuclear matrix elements due t o penetration of the electron wave function inside the nucleus allow ejection of conversion electrons but no emission of nuclear gamma rays. These matrix elements are normally much smaller than the gamma ray matrix elements which then determine the conversion coefficients (almost independent of nuclear structure). But in cases where the gamma matrix elements become very small, either by I-forbiddenness* or through violation of asymptotic

RETICAL

- -

--

1 L. A. Sliv and M. Listengarten, Zhur. Ekuptl. i Teoret. F i z . 29, 29 (1952); T. A. Green and M. E. Rose, Phys. Rev. 110, 105 (1958); E. L. Church and J. Weneser, Ann. Rev. Nuclear Sci. 10, 193 (1960). 2 E. L. Church and J. Weneser, Phys. Rev. 104, 1382 (1956).

-

* Section 2.4.2.2 is by A. H. Wapstra.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

153

selection rules13the penetration matrix elements become prominent, thus giving rise t o relatively large conversion coefficients. Evidently, large deviations can occur only for delayed transitions. K selection rules apply to both kinds of matrix elements, so that transitions delayed by K forbiddenness alone should have normal conversion coefficients.3 I n only slightly delayed transitions, interference between the two kinds of matrix elements may cause conversion coefficients t o be even somewhat smaller than those for undelayed transition^.^ The influence of the nuclear size effect on K, LI, and ,511 conversion coefficients is approximately the same; that on LIIIconversion coefficients is very much less, even in a case6 where LI and LII conversion coefficients are large by a factor 20. Theoretical conversion coefficients for the K , LI, LII, and LIII shells have been computed by RoselBtaking into account the influence of the static correction for the finite nuclear size and of the screening of the electric field of the nucleus by the electron shells around it. The same book gives M-conversion coefficients uncorrected for the last two effects. As a result, these conversion Coefficients are somewhat large (e.g., about 40% for E2 and 2 = 80); the ratios of the conversion in the M subshells are more nearly correct. Sliv and Band’ published K and L conversion coefficients computed from a model in which the nuclear currents are located on the surface of the nucleus, thus including the static correction and part of the dynamic correction. Their results differ from those of Rose by less than 5% in almost all cases of physical importance. Data permitting the calculation of conversion coefficients for any nuclear model have been published by Green and R0se.l The precision in measuring conversion coefficients is mostly limited due t o the difficulties in measuring gamma ray intensities. Ratios of Conversion coefficients for one gamma transition, which are identical with ratios of conversion line intensities, can be determined much more accurately. Since these ratios too depend on multipolarity and character, they can also be used to get information about these data. The K to L S. G. Nilsson and J. 0. Rasmussen, Nuclear Phys. 6, 617 (1958). L. S. Kisslinger, Bull. Am. Phys. SOC.2,358 (1957); A. H. Wapstra and C. J. Nijgh, Nuclear Phys. 1, 245 (1955); K. 0. Nielsen, 0. B. Nielsen, and M. A. Waggoner, ibid. 3

4

2, 476 (1956). 5 F. Asaro, F. S. Stephens, J. M. Hollander, and I. Perlman, Phys. Reu. 117,492 (1960). 6 M. E. Rose, “Internal Conversion Coefficients.’’ Interscience, New York, 1958. ‘L. A. Sliv and I. M. Band, “Coefficients of Internal Conversion.’’ Acad. 8ci. U.S.S.R., Moscow, 1956 and 1958; translated as Reports 57-ICC-K1 and 59-ICC-L, Department of Physics, University of Illinois, Urbana.

2.

164

DETERMINATION OF FUNDAMENTAL QUANTITIES

conversion line ratios can be measured most easily, but are often not very sensitive to the multipolarity (see Fig. 1). The L subshell ratios give much more information (see Fig. 2), but they can only be measured with instruments of high resolving power. IOC

2-

K / L ratios

Gamma Energy

FIQ.1. Ratio of K and L conversion lines for different pure multipole transitions, for Z = 79. At other nuclear charges, the same K / L ratios appear at approximately the same values for E / Z z .

2.4.2.2.2. DETERMINATION OF BETA AND GAMMA RAY INTENSITIES, The most natural way for obtaining conversion coefficients is to measure electron intensities in a beta ray spectrometer and gamma rays in a gamma ray spectrometer. Measurements of relative electron intensities in a magnetic or electric beta ray spectrometer with moderate precision offers no difficulties, since in a well designed instrument of this type the efficiency is nearly independent of the energy (see Section 2.2.1.1). It is, however, necessary t o take several precautionss if high precision is desired. 8 See, e.g., A. H. Wapstra, “Report on the Israel Conference on Nuclear Physics.” North-Holland Publ., Amsterdam, 1957.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

155

2-

I

0.2

1.0E

,

I

0.4

0.3

.

I

I

I

I

l

0.5 0.7 Gamma Energy

l

I

I MeV

2

3-

#/-

0.2

I

I

0.3

0.4

t

1

I

I

I

l

0.7 Gamma Energy 0.5

l

I MeV

I

1

2

2-

=--_-_ ---------------

0.01

I

I

I

I

I

I

-------_

MI I

I

l

l

I

156

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

For measuring absolute intensities, beta ray spectrometers can be calibrated using sources with simple spectra, the strength of which has been determined using 4~ counters or coincidence techniques. In cases with few, well separated electron lines, electron intensities may be measured in proportional or scintillation counter spectrometers, or with the new solid state detectors (see Sections 1.3.1, 1.4.1, and 1.8.1). Efficiencies of gamma spectrometers vary with energy in a way which, in many cases, cannot be computed with the necessary accuracy. These instruments have, therefore, t o be calibrated even for measurements of ratios of intensities of gamma ray lines with different energies. The most accurate measurements can be obtained, in principle, with crystal diffraction spectrometers.9 Such instrhments demand very strong sources. The more popular scintillation spectrometer needs much weaker sources, but has a much lower resolving power. Even for sources with few, well separated gamma lines, intensity measurements in a scintillation spectrometer with a precision of 5 % or better are mostly hard to make due to the difficulty of separating the lines from the Compton continua, caused by lines of higher energies in an unambiguous way. Secondary electrons emitted by a thin “converter” placed at the source position of a beta ray spectrometer and irradiated by the gamma ray source can be measured as much sharper lines than those found in scintillation spectrometer (see Section 2.2.3.1.2). By choosing a converter of material with high atomic number, the ratio of the intensities of the photo lines and the Compton continua can, moreover, in principle be made larger than is possible in scintillation spectra. Uranium converters are most suitable, but for many purposes lead converters, which can be obtained more easily, are sufficient. Sources for this type of measurement should be about a hundred times stronger than those used in scintillation spectrometry. In the computation of gamma ray intensities from the measured photopeak areas, one needs to know the photo cross sections as a function of the direction at which the photo electron is emitted. Habitually, the influence of this function is expressed as the product of the total photo cross section T Y and an angular distribution factor fY, so that the photopeak area A Y (if plotted as number of electrons per momentum interval versus momentum) becomes A y = 1 7 T y f y db Q the quantity d being the thickness of the converter and b a dimensional factor, and Q being the solid angle of acceptance of the spectrometer used. Values for the total cross sections can be derived from the tables of White 9

See Vol. 5, A, Section 2.2.3.2.

2.4.

DETERMINATION O F S P I N , PARITY, .4ND NUCLEAR MOMENTS

157

Grodstein or from older ones by Davisson.’O The recent work of Hultberg and his collaborators on the computation of the angular distribution factor fK is extremely important for the use of this method in intensity measurements. Yet stronger sources are necessary if one wants t o measure Compton lines,12 since then the gamma rays have to be collimated before they strike the converter, which in this case is preferably of low 2 material. The last method has the advantage that it produces only one line for every gamma ray. 2.4.2.2.3. DETERMINATION O F CONVERSION COEFFICIENTS; SPECIAL METHODS. Accurate measurements of conversion coefficients can be made in special cases, e.g., a single y transition preceded by a single /3- transition. The conversion coefficients can then be found by measurement of the intensities of the conversion lines and the beta continuum in a beta spectrometer: 1/aK

=

/3/K

- (K +L + M

+

. . .)/K

(2.4.2.2.1)

The symbols K , L , M , . . . stand for the intensities of the K,L,M, . . . conversion electrons. The intensity /3 of the beta continuum is mostly obtained by extrapolating the high energy part of the spectrum t o lower energies with help of the theoretical spectrum shape, since the measured spectrum is always distorted at low energies by counter window absorption and by the occurrence of scattered electrons. This extrapolation can, in allowed spectra, be made most easily with help of a Fermi plot. Forbidden spectra in the so-called “unique” class have also well-known shapes.I3 The other forbidden beta spectra have shapes depending on the accidental values of some nuclear matrix elements which are unknown. This fact introduces an additional uncertainty in the measurement of the lo G. White Grodstein, Natl. Bur. Standards (U.S.) Circ. No. 683 (1957); C. M. Davisson, in “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 857. Interscience, New York, 1955. 1 1 S. Hultberg, Nuclear Instr. & Methods 10, 24 (1960); S. Hultberg, D. Horen, and J. Hollander, Nuclear P h p . 28, 471 (1961); W. F. Frey, J. H. Hamilton, and S. Hultberg, Arkiv Fysik 21, 383 (1962). 12 M. Mladjenovic and A. Hedgran, Arkiv Fysik 8, 49 (1954); B. S. Dzhelepov et al., Nuclear Phys. 2, 408 (1956). 13 N. Dismuke, M. E. Rose, C. L. Perry, and P. R. Bell, U.S. Atomic Energy Comm. Rept. 0.R.N.L.-1222 (1952); M. E. Rose, C. L. Perry, and M. M. Dismuke, U.S. Atomic Energy Comm. Rept. 0.R.N.L.-1459 (1953); K. Siegbahn, ed., “Beta- and Gamma-Ray Spectroscopy,” cf. p. 875. Interscience, New York, 1955; A. H. Wapstra, G. J. Nijgh, and R. van Lieshout, “Nuclear Spectroscopy Tables.” Interscience, New York, 1959.

158

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

intensity of the continuous beta spectrum, though in the most frequent case of first forbidden transitions, the deviations are mostly small.14 The influence of this uncertainty on the conversion coefficients can in principle be avoided by measuring the electron spectrum in a beta ray spectrometer in coincidence with K X-rays due to internal conversion; the X-rays may be detected by a scintillation counter. In the special case discussed above we now find 1 / a ~=

TL,-K/TLSK

- (K

+L + M +

*

*

.)/K

(2.4.2.2.2)

in which npg indicates the number of coincidences per nuclear electron selected by the beta spectrometer (of course corrected for backgrounds and accidental coincidences), and n,-K the same per K conversion electron. Equations (2.4.2.2.1) and (2.4.2.2.2) are rather similar; the difference is that, using Eq. (2.4.2.2.2),n g K can be measured at places in the beta spectrum where the influence of disturbing effects and backgrounds is lowest. In the same simple cases, the K conversion coefficient may be obtained by comparing the gamma ray intensity with that of the K X-rays caused by the conversion process: a~ =

(E,/EK)~K/TWK

where X K and y represent the photopeak arcas in the scintillation spectrum, W K is the fluorescent yield and E, and EK are the respective efficiencies for detecting the gamma and the X-rays in the photopeak of the scintillation spectra. Sources used in this method should be thin enough to avoid absorption or fluorescent excitation of X-rays. The ratio E J E K can be computed accurately if the gamma energy is so low that the mean free path for absorption is smali compared with the dimensions of the counting crystal. I n this case, this method is capable of high precision. A somewhat more involved coincidence method was applied by Petterson et aZ.16on Aulg8.They measured coincidences between the 412 kev gamma rays and the nuclear electrons in Aul9s, and between 134 kev gamma rays and the L conversion line of the 165 kev transition in Hglg7 (decay schemes see Fig, 3). In this case,

in which n are again the number of coincidences per detected gamma ray; N ~ ~and Z KNp are the count rates of nuclear and conversion electrons, at the top position of the conversion line and at the point where the Py coinl4

l6

A. H. Wapstra, Nuclear Phya. 9, 519 (1958). B. G.Petterson, J. E, Thun, and T,R. Gerholm, NucZeur Phys. 24,243 (1961).

2.4.

DETERMINATION

OF SPIN, PARITY, A N D NUCLEAR MOMENTS

159

cidences were taken respectively. The factor c is the L conversion probability of the 165 kev line except for a 4 % correction for the difference between the line shapes of the 412K line and the complex 134L line. Another met,hod was used by Lewin et a1.16 in a study of TIzoo.They compared coincidences between the group of 1200 kev lines (see Fig. 3) preceding the 368 kev transition and the K conversion line of the last TLOO

ffl

Au 198

7

E @ I97

h

' .

O+

O+ ~ 1 9 8

YZ+

HqKX>

p" ~1203

FIQ. 3. Decay schemes of Hg197, Aulgg, Tlzo'J, and HgZoa,the middle two partly simplified. The values give transition energies in kev.

transition with coincidences between the 412 kev gamma rays and the nuclear electrons in Aulg8.Then, l / a ~= c(nsr41z/n,-,izoo)(n,/ns>

-

(K

+L + M + .

*

.)/K.

Here, ne is the ratio of top counting rate and area of a monoenergetic electron line, and ng is the ratio of the counting rate of nuclear electrons in AulS8a t the point where nar is measured and the area of the continuous beta spectrum. The quantities ng, and ne-,are the numbers of coincidences per recorded gamma ray; the factor c, nearly unity, corrects for the dependence of the coincidence and counting efficiencies on th e electron energy, and for the fact that the actual decay schemes are somewhat more involved than depicted in Fig. 3. I n complicated cases, some of the above methods cannot be applied t o yield accurate results. Then, the method of comparison of external and internal conversion lines may become very useful. If these lines are measured in the same spectrometer, both are proportional t o the same solid angle, which therefore does not occur in the final result allowing higher precision. l6

W. H. G. Lewin, B. van Nooijen, and A. H. Wapstra, Nuclear Phys. 27, 681

(1961).

160

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

If no strong enough sources are available to allow external conversion measurements, one still has not necessarily to know the absolute efficiencies of electron and gamma spectrometers. Often, a conversion coefficient of one of the transitions in the nuclide studied may be known; it can then be used to calibrate the electron and gamma counters relative to one another. Or, they may be relatively calibrated by measuring a standard nuclide in both. In nuclides decaying by negaton decay alone, the total amount of K X-radiation should be equal to the number of K conversion electrons multiplied by the fluorescent yield.17 Then, the ratio of K Auger electrons to that of K X-rays is known with considerable accuracy. And in nuclides emitting positrons, two annihilation quanta of 511 kev are emitted for every positon. Thus, one of the last three possibilities may be used in calibrating beta and gamma spectrometers relative to one another in order to determine conversion coefficients. In cases where the above methods fail, one may adopt a theoretical conversion coefficient for one of the transitions studied. Thus, the transition between the first excited 2+ state and the ground state in even-even nuclei is necessarily a pure E2 transition; and its conversion coefficient is often found to agree well with theory (though a few real exceptions may exist la). 2.4.2.2.4. EXAMPLE OF USEOF CONVERSION DATA.As a rather typical example for the use of conversion coefficient measurements we consider the case of HgZo3. This nuclide decays as depicted in Fig. 3 to a 279.12 kev excited level in TlZo3 decaying to the ground state by a single transition; the conversion coefficients can therefore be measured by the special methods outlined above. Accurate meas~rernents’~ yielded the results given in Table I; all experimental data have errors of only a few per cent. Theoretical values for the conversion coefficients of pure multipole transitions of a 279.12 kev gamma transition in TI are also given in Table I. Since parity is a good quantum number for electromagnetic transitions, only those multipole transitions can occur mixed in which the change in parity is the same (the upper and lower parts in Table I, respectively). The conversion coefficient for a mixture of two multipolarities is a! = aa1 (1 - a)a!,

+

1’ A. H. Wapstra, G. J. Nijgh, and R. van Lieshout, “Nuele&rSpectroscopy TabIes.” Interscience, New York, 1959. 18 J. F. W. Jansen, S. Hultberg, P. F. A. Goudsmit, and A. H. Wapstra, Nucleaf Phys. (to be published). 1 9 G. J. Nijgh, A. H. Wapstra, L. Th. M. Omstein, N. Salomons-Grobben, J. R. Huizenga, and 0. Almh, Nuclear Phys. 9,528 (1958), and other work discussed there.

2.4.

DETERMINATION O F SPIN, PARITY, AND NUCLEAR MOMENTS

161

and LYZ being the theoretical conversion coefficients for these multipolarities and a the percentage gamma rays of the first type (a is therefore the same for conversion in different shells). Comparison of the measured K / L ratio (the most easily obtained datum) with the theoretical values shows that the transition is a combiLYI

TABLEI. Conversion Coefficients ( x 104) and Ratios for the 279 kev Transitions in T I 2 0 3 (Theoretical data for transitions in which the parity does not change are printed above the experimental data printed in bold-face type, the other below. All theoretical data are obtained from Rose's tables6 except those printed in italics which are due to Sliv.') Conversion ratio

fif 1

41 10 3920

E2 M3

Exp. El M2 E3

764 770 41200 1620 278 14650 2060

646 684 479

586

55

616

64

108

618

110

19700 480 46 3890 4530

12700 242 34 3170 499

243 271 2400 166 7 434 2890

5 5 127 137 4550 82 6 291 1135

6.37 6.76 1.60 1.49

2.10 3.37 6.08 3.76 0.45

0.907

0.086

0.900

0,093

0.226 0.212 0.647

0.508 0.623 0.122

0.604 0.732 0.814 0.110

0.326 0.145 0.111 0.639

0.007 0.007 0.266 0.265 0.231 0.171 0.123 0.075 0.251

TABLE 11. K Conversion Coefficients and L Subshell Ratios Giving the Correct K / L Ratio for the 279 kev Transition in T l * ' J 3

+ + M2 + E3, Rose

E2, Rose MI E2, Sliv Experimenta1 MI

OIK

Lr / L

LllIL -

LIIIIL

0.179

0.479 0.517 0.604 0.732

0.351 0.332

0.170 0.151

0.326 0.173

0.171 0.095

0.196

0.162 1.33

+

+

nation of either M 1 E2, or M 2 M3. The theoretical K conversion coefficients and the LI :LII :L I I Iratios for the mixtures yielding the correct K / L ratio are given in Table 11. Comparison with the experimental values E 2 mixture. The ground state has a spin points very strong t,o a M 1 and positive parity; our result therefore proves that the 279 kev level in TI2O3has a spin and positive parity. The theoretical values in Table I1 do not agree completely with the experimental result, probably due to the dynamic finite size effect discussed in the beginning of this section. According t o an estimate by Kiss-

+

+

162

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

linger,20the influence of this correction would decrease the theoretical value of LYK by about 10%. Unfortunately, this arbitrariness in the theoretical conversion coefficients for pure multipoles hampers an accurate determination of the multipole mixing ratios a from conversion coefficients alone: the values for a as computed from Rose’s theoretical values and the experimental K , LI, LII, and LIII conversion coefficients are 0.256 0.012, 0.280 k 0.013, 0.46 f 0.03, and 0.37 k 0.03, respectively. One would be inclined to reason that the LIII conversion coefficients are least affected by the dynamic finite size effects since the LrII electron wave function has a minimum at the nucleus. The value for a as obtained above from the LIII result agrees indeed fairly well with the result a = 0.31 5 0.025 found by McGowan and StelsonZ1from angular distribution measurements. The theoretical results of Sliv lead to a value a = 0.42 f 0.03, in somewhat worse agreement with McGowan and Stelson’s result.

2.4.2.3. Nuclear Orientation.* 2.4.2.3.1. INTRODUCTION. A nucleus of total angular momentum I is oriented when the ( 2 1 1) magnetic substates have unequal occupational probabilities. Such a nucleus defines a certain spatial anisotropy and, when subsequent emission or absorption processes occur, this anisotropy is carried over into the final state as a consequence of the conservation laws involved. The detailed theories of these effects are closely related to those for angular correlations, and both can be treated by the same very general and refined

+

L. S. Kisslinger, Bull. Am. Phys. SOC.2, 358 (1958). F. K. McGowan and P. H. Stelson, Phys. Rev. 107, 1647 (1957);and private communication. 1 L. C. Biedenharn and M. E. Rose, Revs. Modern Phys. 26, 729 (1953). F. Coester and J. M. Jauch, Helv. Phys. Acta 40, 3 (1953). a S. Devons and L. J. B. Goldfarb, i n “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 43, p. 362. Springer, Berlin, 1957. M. E. Rose, “Elementary Theory of Angular Momentum.” Wiley, New York, 1957. 5 U. Fano and G. Raoah, “Irreducible Tensorial Sets.” Academic Press, New York, 1959. * M. Jacob and G. C.Wick, Ann. Phys. (N.Y.) 7,404 (1959). 7 L. J. B. Goldfarb, i n “Nuclear Reactions” (P. M. Endt and M. Demeur, eds.). North-Holland Publ., Amsterdam, 1959. 8 L. C. Biedenharn, in “Nuclear Spectroscopy” (F. Ajzenberg-Selove, ed.), Part B, Chapter V.C. Academic Press, New York, 1960. 9 S. R. de Groot and H. A. Tolhoek, in “Beta and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 613. Interscience, New York, 1955. $0

$1

-

* Section 2.4.2.3is by E. Ambler.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

163

Indeed, it is appropriate to regard nuclear orientation and angular correlation as special cases of the most general observation, where a complete set of observations of the angular distribution and polarization of particles and radiation is made. Of course, for any particular purpose, e.g., for extracting given quantities of physical interest, many of these observations would be irrelevant or redundant, and one would have to select those which are appropriate. In particular, one would need to determine whether it is a n advantage to know the orientation of the initial nucleus. At this point we should note that there are two types of nuclear orientation: nuclear polarization, where there is a net nuclear moment; and nuclear alignment, where orientations parallel and antiparallel to a given axis are preferred over other orientations although no net moment exists. The experiments that have been carried out to date involve mainly the orientation of radioactive nuclei and the observation of the angular distribution of the decay products, although work in other fields, e.g., that of nuclear reactions, has begun. It should be remembered, however, that the method is an indirect one, and quantities such as spins and parities are seldom determined in a straightforward and unambiguous way starting from a completely unknown situation. The experimental method separates naturally into the following parts: (I) the production of oriented nuclei; (2) the determination of the degree of nuclear orientation produced; (3) the measurement of the relevant angular distributions, including in some cases polarization or correlation measurements; (4) the computation of theoretical curves and their subsequent comparison with experiments.

As a general rule, the measurement of spins and parities is possible simply because, for a given degree of nuclear orientation, angular' distributions for certain spin sequences can be fit to the data and others cannot. The measurement of nuclear moments, on the other hand, is possible when a decay scheme is known so that the degree of nuclear orientation, which is directly related to a nuclear moment, can be determined. The method is entirely reliable for the measurement of spins provided sufficient initial information concerning the decay scheme is available. The same can hardly be claimed for the measurement of nuclear moments, where the sensitivity is seldom such as to give accuracies better than lo%, and where agreement with more direct determinations is often not very good. The method has the advantage that it can be carried out with radioactive nuclei of which only a small quantity is available.

2.

164

DETERMINATION O F FUNDAMENTAL QUANTITIES

TABLE I. Experiments Using Oriental Nucleia Nucleus

Ref.

So46 Cr61 Mn 62 Mn MnKZ MnKZ Mn 6zm MnK4 Mn6* MnK4 MnK6 Mn66 Mn 68 Mn 68

1 2 3 4

co

C066

co

co CO67 co co co Cob8 C068

co

c o 80 coeo

co

ED

COSO CoEO

co co 80 go

Co60 As16 In114m

In116 SblZ2 Sblz2 I131

Cela7 ce1a7m

Cela9 Cela9 CeI4l Cel4I

Method S6

5 6

53 53 53 s3 53

7

53

8 9 7

53 53 s3 53

10 11

12 13

14 13 15,16 17 18 19-21 22 23 24 6 17 25

20,26 22 27 28 29-31 1 24,32,33 34 35 36,37 35 38,39 40 41 41 42 43 43 44

52

53 53 s3

s3 82 53 52 s2 s2 s2

53

s3 53 s3 s2 52 53 53 s5 S6 53 D5 S6

s1 S6

D 4 & D5 54 52 s2 52 52 s2 s2

Substance

Observation

Iron alloy “Brine holes” in CMN CMN CMN CMN CMN CMN CMN CMN CMN Mn (ND4)2(S04)2.6D~0 NFS CMN CMN CMN CMN CU(NHdz(S04)~.6HzO CMN CU(NHJ z (SO,) 2.6H20 CU(NHdz(S04)~6HzO CU(NHAz(S04)2.6H20 NFS CMN CMN CMN CMN CU(NH~Z(SO~)Z.~HZ~ CU(NH,)z(SOdz.6HzO CMN CMN Co metal Iron alloy CMN As in silicon Iron alloy Metal Iron alloy Sb in silicon Co (Zn)p-t,p-i NES NES CES CMN CMN CES

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

165

TABLE I (Continued) Nucleus

Ref.

Method

Substance

45 46 47 13 48 49 43 50 51 52 53 54 55,59 56 56 57 58 59 60 35 61 61 61,62

s2 & s 3 s2 & 53 s2 s3 s2 S2,S3 s2 52 s2 & S3 s3 Sl S2,S3 S2,S3 s2 s2 s3 52 s3 52 S6 s4

NES NES CES CMN N ES NES CMN NES NES SES Metal NES NES NES NES HES NES NES YES Iron alloy URN URN URN

54 S2$4

Observation

a Column one refers to the nucleus oriented, column two to the original publication, column three to the method used (cf. Table 11), column four to the substance used, and column five to the observations made. In the fourth column we have used the following abbreviations: CMN = cerous magnesium nitrate, NFS = nickel fluosilicate, Co (Zn)p-t,p-i = cobalt (zinc) p-toluenesulfonate, p-iodobenzenesulfonate, NES, CES, SES, HES, YES = neodymium, cerium, samarium, holmium, ytterbium ethyl sulfates respectively, and URN = uranyl rubidium nitrate. I n the fifth column the angular distribution of alpha, beta, and gamma rays is denoted by w(a,q), w(p,q), and w(k,v) respectively; w(k,v)& and w(k,q).5 refer to plane polarization and circular polarization measurements of gamma rays respectively; an refers to a measurement of slow neutron capture cross section; w(p,k,v) refers to beta-gamma angular correlation; w(kl,k2,q)to gamma-gamma angular correlation, and w(f,q) to the angular distribution of fission fragments.

REFERENCES TO TABLE I 1. A. V. Kogan, V. D. Kul'kov, L. P. Nikitin, N. M. Reinov, I. A. Sokolov, and

M. F. Stel'makh, Zhur. Eksptl. i Teoret. Fiz. 99, 47 (1960); Soviet Phys. JETP 12, 34 (1961). 2. M. Kaplan and D. A. Shirley, Phys. Rev. Letters 6, 361 (1961). 3. W. J. Huiskamp, M. J. Steenland, A. R. Miedema, H. A. Tolhoek, and C. J. Gorter, Physica 22, 587 (1956).

166

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

REFERENCES TO TABLE I (Continued) 4. W. J. Huiskamp, A. N. Diddens, J. C. Severiens, A. R. Miedema, and M. J. Steenland, Physica 23, 605 (1957). 5. E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 110,787 (1958). . . 6. H. Postma, W. J. Huiskamp, A. R. Miedema, H. A. Tolhoek, and C. J. Gorter, Physim 24, 157 (1958); H. Postma, Thesis, University of Leiden, Netherlands,

unpublished. 7. R. W. Bauer, M. Deutsch, G. 8. Mutchler, and D. G. Simons, Phys. Rev. 120, 946 (1960). 8. M. A. Grace, C. E. Johnson, N. Kurti, H. R. Lemmer, and F. N. H. Robinson, Phil. Mag.[71 46,1192 (1954). 9. G. R. Bishop, J. M. Daniels, H. Durand, C. E. Johnson, and J. Perez y Jorba, Phil. Mag.L71 46,1197 (1954). 10. S. Bernstein, L. D. Roberts, C. P. Stanford, J. W. T. Dabbs, and T. E. Stephenson, Phys. Rev. 94, 1243 (1954); J. W. T. Dabbs and L. D. Roberts, ibid. 96, 970 (1954). 11. P. Dagley, M. A. Grace, M. Gregory, J. S. Hill, and C. V. Sowter, reported by N. Kurti, Physica Suppl. S164 (September, 1958). 12. R. W. Bauer and M. Deutsch, Phys. Rev. 117, 519 (1960). 13. H. Postma and W. J. Huiskamp, PTOC.7th Intern. Conf. Low Temp. Phys. p . 183 (1961). 14. R. W. Bauer and M . Deutsch, Nudear Phys. 16, 264 (1960). 15. L. J. Gallaher, C. Whittle, J. A. Beun, A. N. Diddens, C. J. Gorter, and M. J. Steenland, Physica 21, 117 (1955). 16. 0. J. Poppema, J. E. Siekman, and R. van Wageningen, Physica 21, 223 (1955). 17. E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 108,503 (1957). 18. G. R. Bishop, M. A. Grace, C. E. Johnson, A. C. Knipper, H. R. Lemmer, J. Perez y Jorba, and R. G. Scurlock, Phil. Mag. [7] 46, 951 (1951). 19. J. M. Daniels, M. A. Grace, H. Halban, N. Kurti, and F. N. H. Robinson, Phil. Mag.[7] 43, 1297 (1952). 20. J. C. Wheatley, D. F. Griffing, and R. D. Hill, Phys. Rev. 99,334 (1955). 21. D. F. Griffig and J. C. Wheatley, Phys. Rev. 104, 389 (1956). 22. G. R. Bishop, J. M. Daniels, G. Goldschmidt, H. Halban, N. Kurti, and F. N. H. Robinson, Phys. Rev. 88, 1432 (1952). 23. P. Dagley, M. A. Grace, J. S. Hill, and C. V. Sowter, Phil. Mag. [S] 5, 489 (1958). 24. E. Ambler, R. W. Hayward, D. D. Hoppes, R. P. Hudson, and C. 6. Wu, Phys. Rev. 106, 1361 (1957). 25. E. Ambler, M. A. Grace, H. Halban, N. Kurti, H. Durand, C. E. Johnson, and H. R. Lemmer, Phil. Mag. [7] 44, 216 (1953). 26. B. Bleaney, J. M. Daniek, M. A. Grace, H. Halbaa, N. Kurti, F. N. H. Robinson, and F. E. Simon, Proc. Roy. Soc. A221, 170 (1954). 27. P. S. Jastram, R. C. Sapp, and J. G. Daunt, Phye. Rev. 101, 1381 (1956). 28. J. C. Wheatley, W. J. Huiskamp, A. N. Diddens, M. J. Steenland, and H. A. Tolhoek, Physica 21, 841 (1955). 29. M. A. Grace, C. E. Johnson, N. Kurti, R. G. Scurlock, and R. T. Taylor, in “ConfBrence de physique des basses temphratures,” p. 263. Centre National de la Recherche Scientifique and UNESCO, Paris, 1956; BuZl. Am. Phys. Soc. 2, 136 (1957).

2.4.

DETERMINATION OF SPIN, PARITY, A N D NUCLEAR MOMENTS

167

REFERENCES TO TABLEI (Continued) 30. G. R. Khutsishvili, Zhur. Eksptl. i Teoret. Fiz. 29, 894 (1955); Soviet Phys. J E T P 2, 744 (1956). 31. J. M . Daniels and M. A. R. LeBlanc, Can. J . Phys. 37, 1321 (1959). 32. C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson, Phys. Rev. 106, 1413 (1957). 33. C. E . Johnson, M. A. Grace, R. G. Scurlock, and C. V . Sowter, Phil. Mag. [8] 2, 1050 (1957). 34. F. M. Pipkin and J. W. Culvahouse, Phys. Rev. 109, 1423 (1958). 35. B. N. Samoilov, V. V. Sklyarevskii, and E. P. Stepanov, Zhur. Eksptl. i Teoret. Fiz. 38, 359 (1960); Soviet Phys. J E T P 11, 261 (1960). 36. J. W. T. Dabbs, L. D. Roberts, and S. Bernstein, Phys. Rev. 98, 1512 (1955). 37. A. Stolovy, Phys. Rev. 118, 211 (1960). 38. F. M. Pipkin, Phys. Rev. 112, 935 (1958). 39. G. E. Bradley, F. M. Pipkin, and R. E. Simpson, Phys. Rev. 123, 1824 (1961). 40. C. E. Johnson, J. F. Schooley, and D. A . Shirley, Phys. Rev. 120, 1777 (1960). 41. J. N. Haag, C . E. Johnson, D, A. Shirley, and D. H. Templeton, Phys. Rev. 1% 591 (1961). 42. M. A. Grace, C. E. Johnson, R. G. Scurlock, and R. T. Taylor, Phil. Mag.,to be published. 43. E. Ambler, R. P. Hudson, and G . M . Temmer, Phys. Rev. 101, 196 (1956). 44. C. F. M. Cacho, M. A. Grace, C. E. Johnson, A. C. Knipper, R. G. Scurlock, and R. T. Taylor, Phil. Mag. 171 46, 1287 (1955). 45. D. D. Hoppes, E. Ambler, R. W, Hayward, and R. S. Kaeser, Phys. Rev. Letters 6, 115 (1961). 46. D. D. Hoppes, Natl. Bur. Standards (US.)Rept. 93 (unpublished). 47. M. A. Grace, C. E. Johnson, R. G. Scurlock, and R. T. Taylor, Phil. Mag. [81 3, 456 (1958). 48. C.A. Lovejoy, J. 0. Rasmussen, and D. A . Shirley, Phys. Rev. 123, 954 (1961). 49. D. A. Shirley, J. F. Schooley, and J. 0. Rasmussen, Phys. Rev. 121,558 (1961). 50. G. R. Bishop, M. A. Grace, C. E. Johnson, H. R. Lemmer, and J. Perez y Jorba, Phil. Mag. [Sl 2, 534 (1957). 51. G. A. Westenbarger and D. A. Shirley, Phys. Rev. 123, 1812 (1961). 52. L. D. Roberts, S. Bernstein, J. W. T. Dabbs, and C. P. Stanford, Phys. Rev. 95, 105 (1954). 53. A. Stolovy, Bull. Am. Phys. SOC.[II] 6 (3), 275 (1961). 54. C. A. Lovejoy:andLD. A . Shirley, Proc. 7th Intern. Cmf. Low Temp. Phys. p. 164 (1961). 55. C. E. Johnson, J. F. Schooley, and D, A. Shirley, Phys. Rev. 120, 2108 (1960). 56. Q. 0. Navarro and D. A . Shirley, Phys. Rev. 123, 186 (1961). 57. H. Postma, H. Marshak, V. L. Sailor, F. J. Shore, and C. A . Reynolds, Bull. Ant. Phys. SOC.[I11 6 (3), 275 (1961). 58. H. Postma, A. R. Miedema, M. C. Eversdijk Smulders, Physiea 26,671 (1959). 59. H. Postma, M. C. Eversdijk Smulders, and W . J. Huiskamp, Physica 27, 245 (1961). 60. M. A. Grace, C . E. Johnson, G. R. Scurlock, and R.:T. Taylor, Phil. Mag. [81 2, 1079 (1957). 61. J. W. T. Dabbs, L. D. Roberts, and G. W. Parker, Physica Suppl. 569 (September 1958); Proe. 7th Intern. Cmf. Low Temp. Phys. p. 174 (1961). 62. S. H. Hanauer, J. W. T. Dabbs, L. D. Roberts, and G. W. Parker, ORNL Report 2919 (unpublished).

168

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

It is clear that the analysis of the data of each experiment will be very different in detail, and that the results of such analysis may be complicated. It would be neither possible nor appropriate, therefore, to redescribe all the experiments that have been done to date. We have tried to list them concisely in Table I, however, so that the reader may refer to the original papers for important and special details. The aim of the greater part of the article is to give an account of the general physical principles, some of the special techniques, and some of the mathematical methods of the theory that are fast becoming developed to the point where the experimentalist might be expected to use them. The reader will find useful a number of review articles.1°-17 2.4.2.3.2. THEPRODUCTION O F O R I E N T E D NUCLEI-STATICMETHODS. We may divide into three groups the methods proposed for orienting nuclei, and call them the static, dynamic, and transient methods. We have listed these methods together with certain information about them in Table I1 (a), (b), and (c) and have numbered them S1, S2, etc., for easy reference. The basic ideas of the static methods are as follows. The degeneracy of the ( 2 1 1) spatial orientations of the nucleus can be removed either by a coupling of the nuclear magnetic moment to a magnetic field or by a coupling of the nuclear quadrupole moment t o an electric field gradient. If the nuclei are then cooled to a temperature, T , such that kT ( k is Boltzmann’s constant) is of the order of the hyperfine splitting, the lower levels, which correspond to preferred orientations of the nuclear spin, become preferentially populated. The use of external electric fields is not possible since the magnitudes required exceed the dielectric strength of the substances to which they must be applied. The use of a n external

+

10 R. J. Blin-Stoyle and M. A. Grace, i n “Handbuch der Physik-Encyclopedia of Physics” (S. Fliigge, ed.), Vol. 42, p. 555. Springer, Berlin, 1957. 1lR. J. Blin-Stoyle, M. A. Grace, and H. Halban, i n “Beta- and Gamma-Ray Spectroscopy” (K. Siegbahn, ed.), p. 600. Interscience, New York, 1955. 12 M. J. Steenland and H. A. Tolhoek, i n “Progress in Low Temperature Physics” (C. J. Gorter, ed.), Vol. 11, p. 293. North-Holland Publ., Amsterdam, 1957. 13E. Ambler, in “Progress in Cryogenics” (K. Mendelssohn, ed.), Vol. 2, p. 235. Heywood, London, 1960. 14 C. D. Jeffries, in “Progress in Cryogenics” (K. Mendelssohn, ed.), Vol. 3, p. 129. Heywood, London, 1961. 14& G. R. Khutsishvili, U s p e k h i Piz. Nauk 71, 9 (1960); Soviet P h y s . - U s p e k h i 3, 285 (1960). R. P. Hudson, in “Progress in Cryogenics” (K. Mendelssohn, ed.), Vol. 3, p. 99. Heywood, London, 1961. 16 N. Kurti, Nuouo cimenfo Suppl. [lo] 6, 1101 (1957). 17 W. J. Huiskamp and H. A. Tolhoek, in “Progress in Low Temperature Physics” (C. J. Gorter, ed.), Vol. 111, p. 333. North-Holland Publ., Amsterdam, 1961.

2.4.

DETERMINATION

OF SPIN,

PARITY,

AND NUCLEAR MOMENTS

169

magnetic field, on the other hand, provides a most direct and general way of orientating nuclei. Although it has been used successfully to a certain extent, rapid development of this so-called “brute-force” method has been hindered largely by the slow technical development of reliable steady high magnetic fields. With the continued development of high power solenoids, and the further possibility of high field superconducting solenoids, however, this situation is likely to change, and since the method has such a wide applicability we shall discuss it in some detail. We shall then discuss the other methods that have been proposed, which rely upon the presence in certain compounds of high internal fields, i.e., large hyperfine splitting. Some of the methods in this last group have met with considerable success. 2.4.2.3.2.1. External Field Polarization-El. The feasibility of the method in principle,’* and the appreciation of the practical problems in detai1,lg have long been known. More recent and extensive studies have allowed the method to be used in a significant although limited way.20-22 If an assembly of nuclei of magnetic moment p is coupled to an external field H a t temperature T , the nuclear polarization is given by: fl

= &(X)

=

21

+

1 ___ coth 21

(T + 21

1

1

x) - 2f coth

($x )

(2.4.2.3.1)

where &(x) is the Brillouin function and x = p H / k T . The value of f l is the same as the ratio of the nuclear magnetization to that a t saturation, and this quantity has been tabulated already in connection with electronspin m a g n e t i ~ m . 2 ~If~we 2 ~ adopt the figure of f l = 0.2 as being a reasonably large nuclear polarization for carrying out experiments, we find that for nuclear magnetic moments normally encountered, we require H I T = 106-107 gauss/”K. Now, as we shall discuss later, a value of T = 0.05”K can be obtained fairly easily, whereas a value of T = 0.01”K should be possible; therefore, magnetic fields in the range 104-106 gauss are suitable. As an illustration, we take what we believe to be a reasonable figure, viz., H / T = 4 X lo6 gauss/”K (e.g., 100 kilogauss a t 0.025”K) F. E. Simon, Conrpt. rend. ccungr. sur le magnt%isme, Strasbourg, 1939,p. 1 (1939). N. Kurti, “Les phknombnes cryomagn&iques,” p. 29. Coll&gede France. Paris, 1948. 20 J. W. T. Dabbs, L. D. Roberts, and S. Bernstein, ORNL Report, central files number 55-5-126(un-published). 2 1 J. W. T. Dabbs, L. D. Roberts, and S. Bernstein, Phys. Rev. 98, 1512 (1955). 22 A. Stolovy, Phys. Rev. 118, 211 (1960). 2s J. R. Hull and R. A. Hull, J. Chem. Phys. 9, 465 (1941). 24 L.P. Schmid and J. S. Smart, LLTables of Some Thermodynamic Functions which Occur in the Theory of Magnetism.” Navard Report 3640, U.S. Naval Ordnance Laboratory, White Oak, Maryland. 1s 19

TABLE 11. List of Methods Proposed for Orienting Nuclei (a). Static Methods of Orienting Nuclei Method

No.

Ref.

External field polarization

s1

1-3

Magnetic hfs alignment

52

4

w

l

0

Magnetic hfs polarization Electric hfs alignment

S3

5,6

s4

7

Ferromagnetics and anti-ferromagnetics

55

8-10

Dilute solid solutions in above

S6

10-13

Criteria

-

&upling p"H.1 HIT lohi07 gauss per degree K Coupling AS,Z,or

+

B(SJ, S J d S = 4 and A >> B or B >> A S>&andA,B#O Z > $ As above, but S = $ and A = B included P#O

Large enough hfs; coupling for alignment as in S2 and for polarization as in 53 As above

Applicability

All nuclei with pn. Long nuclear relaxation times may limit method to metals and alloys Paramagnetic compounds, e.g.9 Ce2Mg,(N03)1~24H20 Nd(C,H,SO,)a.SH,O Rare earth metals (See

Special Requirements Adiabatic demagnetization and contact cooling. Poor thermal conductivity is a severe problem Adiabatic demagnetization, usually

55) As above

Covalently bonded complexes, e.g., UOZ in (U02)Rb(N03)r High density of relevant nuclei is a very important advantage for work on nuclear reactions Whenever mechanism (e.g. diamagnetic core polarization in iron alloys) producing large hfs is operative

As above, but external field

102-103 gauss needed Temperatures < 1"K, usually

As above

As above

(b). Dynamic Methods of Orienting Nuclei Method Optical pumping

No.

Ref.

Criteria

D1

1

Resonance with low-field Zeeman splitting; F = I S, is oriented by transitions induced by a polarized light beam As D1 but orientation is passed o n t o other atoms in vapor by collisions. Large cross section for spin-flip exchange with atoms of added vapor. Depolarization at walls must be small Z relaxes via A S . I coupling; saturate electron spin resonance

Applicability

+

Optical pumping with ‘spin transfer’

D2

2

rf and microwave pumping in metals

D3

3

Extension t o nonmetals and semiconductors

D4

4-6

Use of ‘forbidden’ transitions

D5

Atoms in vapor

Metals with unpaired s-electrons

Nuclear relaxation via Paramagnetic centers with resolved hfs AS . I; or A S J , B(SzZz &ZV) with H along z-axis hfa with some anisotropy _ _ in order to give reasonable transition probabilities with rf and external fields parallel. Relaxation transitions, which produce orientation in D3, are ‘driven’ by rf field. Differential effect with inhomogeneously broadened line

+

7

Atoms in vapor, e.g., alkali metals, mercury. Possible extension t o solids

+

Special Requirements Vapor at ordinary temperaturea. When radioactive species used, large background due t o wall absorption is troublesome A s above

T

‘V 1°K preferably; relatively high rf powers needed, which can be a disadvantage T ‘V 1°K preferably

As above

TABLE I1 (Continued) (c). Transient Methods of Orienting Nuclei Method

No.

Ref.

Double resonance

T1

1

hfs must be resolved, but Paramagnetic centers with detailed form not so resolved hfs important. Necessary to make adiabatic rapid passage through electron spin resonance and nuclear spin resonance successively. Nuclear spin relaxation must not be too short

Three other transient effects

T2

2

Z and S relax via A S . I coupling, i.e., main relaxation process is that where S and Z flip together

c. -3 t 9

T3

T4

3

Criteria

Hyper6ne coupling is mainly of the type AS * I

Applicability

Special Requirements

T

‘v

1°K preferably

Only necessary to turn on external field Saturate two electron resonances in turn where Z does not flip After thermal equilibrium, decrease external field from Paschen-Back to Zeeman region (a) so that change is at all times adiabatic, and (b) so that the time taken is << electron spin-lattice relaxation time

REFERENCES TO TABLEII(a) C. J. Gorter, Physik. Z. 36, 923 (1934). N. Kurti and F. E. Simon, Proc. Roy. SOC.A149, 152 (1935). F. E. Simon, Compt. rend. sur le magndtisme, Strasbourg p, 1 (1935). B. Bleaney, Phil. Mag. (71 42, 441 (1951). C. J. Gorter, Physica 14,504 (1948). M. E. Rose, Phys. Rev. 76, 213 (1949). R. V. Pound, Phys. Rev. 76, 1410 (1949). 8. J. G. Daunt, Proc. Intern. Conf. Low Temp. Phys., Ozfordp. 157 (1951). 9. C. J. Gorter, Proc. Zntern. Conf.Low Temp. Phys., O z f o r d p . 158 (1951). 10. W. Marshall, Phys. Rev. 110, 1280 (1958). 11. B. N. Samoilov, V. V. Sklyarevskii, and E. P. Stepanov, Zhur. Eksptl. i Teoret. Fiz. 38, 359 (1960); Soviet Phys. J E T P 11, 261 1. 2. 3. 4. 5. 6. 7.

2

(1960). 12. D. A. Coodings and V. Heine, Phys. Rev. Letters 6, 370 (1960). 13. R. E. Watson and A. J. Freeman, Phys. Rev. 123, 2027 (1961).

REFERENCES TO TABLEII(b) A. Kastler, Proc. Phys. SOC.(London)A67, 853 (1954); J. Opt. Soc. Am. 47, 460 (1957). H. G. Dehmelt, Phys. Rev. 106, 1487 (1957); 109, 381 (1958). A. W. Overhauser, Phys. Rev. 92, 411 (1953). J. Korringa, Phys. Rev. 94, 1388 (1954). F. Bloch, Phys. Rev. 93, 944 (1954). A. Abragam, Phys. Rev. 98, 1729 (1955). 7. C. D. Jeffries, Phys. Rev. 106, 164 (1957).

1. 2. 3. 4. 5. 6.

REFERENCES TO TABLEII(c) 1. G. Feher, Phys. Rev.

103, 500 (1956). 2. D. Pines, J. Bardeen, and C. P. Slichter, Phys. Rev. 106, 489 (1957). 3. A. Abragam, Compt. rend. 242, 1720 (1956).

2.

174

DETERMINATION OF FUKDAMENTAL QUANTITIES

and plot f l against nuclear moment in Fig. 1. We also indicate the actual polarization that could be obtained in a number of favorable but useful cases. Since the polarization goes practically linearly with x over a significant range, the following high temperature approximations are useful : (2.4.2.3.2) f2

+ 3)

1 (21 - 1)(21 * I ( I 1)

+

= fl2

1

(2-4.2.3.3)

where f l is the polarization and fz the alignment (cf. Section 2.4.2.3.5). The quantity in the square bracket varies from 0.25 for I = 1 to 0.4 for 1 = m, so that the alignment, being proportional to flz, would be very small in this range.

fI

50%

40% 30 % 2 0 Yo I0

0

0

I

2

3

4

5

6

Pn FIG.1. Nuclear polarization, f l , in a magnetic field of 100 kilogauss and at a temperature of 0.025"K, for various values of nuclear spin, plotted as a function of nuclear moment, p a , in nuclear magnetons. Values corresponding to a few specific isotopes are indicated. The dotted line on hydrogen isotopes applies to the room temperature ortho-para ratios for moIecular hydrogen.

The main obstacles in the way of the full utilization of this method have been practical ones associated with the simultaneous production of very high steady magnetic fields and very low temperatures. I n order to

2.4.

DETERMINATION O F SPIN, PARITY, AND NUCLEAR MOMENTS

175

cool substances to very low temperatures, they must be brought into contact with a paramagnetic salt which has itself been cooled by the process of adiabatic demagnetization. This may be done by taking many silver or copper wires in parallel, embedding one set of ends in the paramagnetic salt, and attaching the other ends to the sample. With such an arrangement the main resistance to heat flow is usually found to be surface contact resistance, e.g., between the wires and the salt. In addition to this practical difficulty, there is a more fundamental one associated with a very long nuclear spin-lattice relaxation time found in insulating substances at low temperatures. This should not present any serious difficulty in the case of metals, however, and in the case of insulators it is often possible to devise ways of shortening the relaxation time, e.g., by including a trace of paramagnetic impurity. When we are interested in cooling relatively small numbers of nuclei, e.g., in experiments on radioactivity, the amount of heat to be transported is quite small, so that cooling can be effected fairly quickly. If we are interested in cooling a large number of nuclei, on the other hand, e.g., for experiments on nuclear reactions, the cooling may be rather slow. The rate of cooling may be computed quite simply since the algebraic expressions for all the relevant quantities, such as the entropy, S , and polarization, fl, of the nuclei in an external magnetic field are known and have been tabulated for different values of the parameter 5 . 2 3 , 2 4 Thus, for example, in the “high” temperature range (cf. Section 2.4.2.3.8) considered above, the reduction in entropy per mole, over the value of R ln(21 1) for random orientation ( R is the gas constant), is given for a change in temperature dT, by:

+

dS where

p

= =

p d T / T 3 erg deg-I 0.037pn2H2(1 l)/Ierg deg, and

+

(2.4.2.3.4) (2.4.2.3.5)

is given in nuclear magnetons, H in kilogauss. The heat flow Q is given by Q = T dS/dt. (2.4.2.3.6)

pn

In a given experimental arrangement the heat flow between the nuclear stage and the electronic stage (a demagnetized paramagnetic salt at temperature, I’E,say) will be governed by a number of thermal resistances R1, Rz, . . . etc., in series. Thermal resistance will arise through finite nuclear-spin relaxation times, the thermal resistance of the metallic link between the stages, the contact between the link and the salt and the finite thermal diffusivity of the paramagnetic salt. In addition to the heat of magnetization to be extracted from the nuclei, there will be spurious heat leaks such as are due to vibrations, residual gas conduction,

2.

176

DETERMINATION O F FUNDAMENTAL QUANTITIES

eddy current heating, and so on. These factors have been discussed in some detai1,20*26 and ways of coping with the difficulties that arise have been discussed. Perhaps the most significant factor is the thermal resistance of the boundary26 between the paramagnetic salt and the wires connected to the nuclear stage. Good bonds are necessary a t the interface, therefore, and a number of ways for making them have been discussed.26-28A good working rule is t o take the heat flow across an area A em2 to be given by Q = lo3* A(T3 - T E ~ergs/sec ) (2.4.2.3.7) where T is the temperature of the nuclear stage, and to assume that a very large mass of paramagnetic salt is used so that TE remains sensibly constant. If a more detailed analysis is required, the necessary data may be found in the references already quoted. For the present, we prefer to follow a simple procedure in order to obtain an idea of the practical limits of the method. Thus, for example, suppose we wish to polarize the nuclear spins in one mole of Li7, having at our disposal H = 100 kilogauss, and assuming T E = 0.01”K. Take A = 2 X lo4 em2. Using Eqs. (2.4.2.3.4-7), we can compute the temperature and nuclear polarization of the sample as a function of t, the time after the cooling starts, i.e., after demagnetization of the upper stage. The result is shown in Fig. 2; it will be seen that a significant nuclear polarization is built up after about an hour. The heat content of the nuclear spins in an external field is quite large a t these low temperatures, so that it is possible to do experiments on the nuclei in which a significant amount of heat is developed. By way of illustration we have shown the projected behavior of the system under “cyclical” operation, and we see, for example, that with a heat dissipation of 30 pw an average nuclear polarization of over 20% can be held for 10 minutes. Thus experiments with incident particles that produce small heat dissipation in the source, e.g., photons and neutrons, could be done very easily, whereas experiments with incident particles such as protons, though very possible, may prove to be laborious. 2.4.2.3.2.2.Magnetic hfs Alignment2g--S.2, and Magnetic hfs Polarization30J--XS. These methods rely upon the fact that in certain paramag26

N. Kurti, F. N. H. Robinson, Sir Francis Simon, and D. A. Spohr, Nature 178

450 (1956). 28 21

See, e.g., W. A. Little, Can. J . Phys. 57,334 (1959), and references quoted therein. A. R. Miedema, H. Postma, N. J. Van der Vlugt, and M. J. Steenland, Physica

26, 509 (1959). 28 A. C. Anderson, G . L. Salinger, and J. C. Wheattley, Rev. Sci. Instr. 52,1110 (1961). 28 B. Bleaney, Phil. Mag. [7] 42, 441 (1951). 30 C. J. Gorter, Physica 14, 504 (1948). 31 M. E. Rose, Phys. Rev. 76, 213 (1949).

2.4.

DETERMINATION

OF SPIN,

PARITY, AND NUCLEAR MOMENTS

177

netic salts there are large magnetic fields (-lo6 gauss) a t the nuclei of the paramagnetic ions due to the unpaired electrons, and at temperatures of the order of 0.01"K the nuclear magnetic moments become oriented with respect to these fields. When the electron moments are oriented a t these very low temperatures, therefore, nuclear orientation will automatically follow. In the case of magnetic hfs polarization the

T "K

fI

.I 2

.6 0

.I 0

.50

.O 0

.40

.O 6

.3 0

.O 4

.2 0

.o 2

.I0 0

0 -

0

60

I20 TIME - mins

180

240

FIG.2. Temperature, T,and nuclear polarization, fl, plotted as a function of time for a specimen of 1 mole of Li7 cooled by contact with a paramagnetic salt at a temperature, TE,of 0.01"K. The heat flow, 4, is supposed to be governed by contact resistance .

electron moments are polarized by the application of an external field, and since the electron magnetism is easily saturated a t low temperatures, a field of a few hundred gauss suffices. In the case of the magnetic hfs alignment method, use is made of the interaction between the paramagnetic ion and the electric field arising from its neighbors in the crystalline lattice, and in a rough way one may say that the electric field aligns the electron moment. This is an oversimplified picture, however, as is the way of describing the interaction between the electron and nuclear spins in terms of a magnetic field. The behavior of paramagnetic ions in a crystal

2.

178

DETERMINATION OF FUNDAMENTAL QUANTITIES

lattice is complex, although the spin Hamiltoniana2provides a very concise way of describing this behavior a t low temperatures. The use of the spin Hamiltonian for describing the behavior of paramagnetic ions in the solid state is based upon the fact that the energy levels of the free ion are perturbed by the electric fields in the crystal in such a way that there is a low-lying group of levels more or less wellseparated in energy from the rest. At very low temperatures only these levels are populated, and their behavior, for most purposes, can be considered independently from the rest. They are described collectively by a fictitious spin operator S , the value of which is obtained by setting (25 1) equal to the number of electronic levels in the group. These levels, which are not necessarily all degenerate, often consist of a complicated mixture of orbital and spin eigenstates of the free ion. Their magnetic properties may be quite anisotropic, a fact that is taken into account by combining S with an anisotropic g-factor. Usually the environment of the ion is such that there is symmetry around a given axis, and this is chosen to be the z-axis in the spin Hamiltonian. It happens frequently, although it is not invariably the case, that all ions of a given kind are equivalent in the crystal, i.e., they have the same axis of symmetry and are described by the same spin Hamiltonian. The z-axis then usually coincides with some crystal axis that may be identified easily from the natural shape of the single crystals. For a single axis of symmetry the spin Hamiltonian 52 is

+

where /3 is the Bohr magneton, fin the nuclear magneton, and gn the nuclear g-factor. D represents an electronic splitting originating solely from the action of the crystal electric field. A and B represent the magnetic hyperfine splitting and are proportional to the nuclear moment of a given isotope. P represents the electric hyperfine splitting and is proportional to the product of the nuclear spectroscopic quadrupole moment and the electric field gradient a t the nucleus, The terms Az and A, are present in the cases of ions with an even number of electrons, e.g., Pr, Ho, Tb. I n these cases the energy levels are not necessarily degenerate in the sense of Kramers’ Theorem,89 although in many substances, owing to the particular crystal symmetry, this may happen. I n particular for some cases A. Abragam and M. H. L. Pryce, PTOC.Roy. SOC.A206, 135 (1951). H. A. Kramers, Koninkl. Ned. Akad. Wetenschap., PTOC. 93,953 (1930); Physica 1, 182 (1934). See also M. J. Klein, Ant. J . Phys. 90, 65 (1952). 52

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

179

the ground state may be an “accidental” doublet. This doublet may then be split by very slight distortions of the crystal field from its assumed symmetry, and the terms A, and Au are included to describe this effect. The effects they describe are not usually sharp, so that mean values about a statistical distribution are assigned to A%and Au. For the cases mentioned above, a Gaussian distribution has been found to describe adequately experimental data on paramagnetic resonance.34 Note that for electric interactions the magnetic substates + M referring to the electron spin, and the magnetic substates f m referring to the nuclear spin, are degenerate, i.e., electric interaction alone can never produce polarization, only alignment. To produce a nuclear polarization an external magnetic field must be applied which couples either directly to the nuclear spin (method Sl) as represented by the last term in Eq. (2.4.2.3.8), or indirectly through the electron spin as represented by the terms in A and B (method 53). It should be noted that for a given paramagnetic compound, different isotopes of the same element have spin Hamiltonians that differ only in the hyperfine coupling constants, and that A and B are proportional to p / I , and P to &/1(21 - 1) [cf. Eq. (2.4.2.3.11)l. This is an essential point in the method usually adopted for the determination of nuclear moments. For a further discussion of the spin Hamiltonian, as well as for numerical values of the constants in particular cases, the reader is referred to a number of review articles.36-38 As classified in Table 11, the methods of nuclear orientation can then be discussed by assuming that certain terms in the spin Hamiltonian are dominant.29-39Thus, for example, for method S1 we need consider only the last term in Eq. (2.4.2.3.8). For method 52, two cases are important, the case of S = 4 where the D term automatically vanishes, and the case of S > +. For S = 4 there are two extreme cases. When A >> B the energy levels are as shown in Fig. 3a, with the nuclear magnetic substates I , = + I lowest. I n this case the nuclear spin aligns parallel and antiparallel to the z-axis. When B >> A the energy levels are as shown in Fig. 3b, the nuclear magnetic (for I half-integral) or I , = 0 (for I integral) lie substates I , = lowest, and nuclear spin “aligns,” or rather precesses, in a plane perpendicular to the z-axis. When S > p there are again two cases, viz., D negative and D positive. We have shown the energy levels in the former

*+

s4 J.

M. Baker and B. Bleaney, Proc. Roy. Soc. A245, 156 (1958). B. Bleaney and K. W. H. Stevens, Repfs. Progr. in Phys. 16, 108 (1953). K. D. Bowers and J. Owen, Repts. Progr. in Phys. 18, 304 (1955). J . w. Orton, Repts. Progr. in Phys. 22, 204 (1959). M. H. L. Pryce, Nuovo cimento Suppl. [lo] 6, 817 (1957). 39 A. Simon, M. E. Rose, and J. M. Jauch, Phys. Rev. 84, 1155 (1951).

36

180

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES rn

m f 1/2

=T

( i 3 / 2 ,i1/2) f f 3/2

if51/2 /2

f 5/2

i 5/2

f 3/2 f 1/2

f 1/2 f3/2

f 5/2

f 3/2

L2P

+1/2

L-

(C)

FIG. 3. Energy levels for various values of the parameters A , B, D and P in Eq. 2.4.2.3.8;in (a) D = 0, P = 0, and A >>B, in (b) D = 0, P = 0, and B >> A , in (c) P = 0, D >> A , B and positive, in (d) D = 0, A = 0, B = 0 and P positive. The energy levels are given assuming I = $, with S = 6in (a) and (b),and S = Q in ( c ) . The quantity m labels the eigenstates of I..

case in Fig. 3c, and have assumed for simplicity that A 2 B and D >> A . The case of D negative corresponds to the case where the electron spin levels 8,= + S are lowest, and is more favorable for producing nuclear orientation than the case of D positive. We have listed in Table I11 the paramagnetic salts most frequently used

2.4.

181

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

TABLE 111. List of Paramagnetic Salts Used for Nuclear Orientation by Methods 52 and 538

~~

8

Mn55 (X)(5) (Y) C O (X)(5) ~ ~ (Y) M+++

1.997 1.997 4.108 7.29

f g4 -_I

S

911

4+ 3i

CeI4l(6) Pr141 NdI4'(6)

4

1.997 - 80 1.997 -215 4.385 2.338 -1.84 2.72

0.25 1.55 0.45

+

D

91 ___-

90 90 85 283

- 90 - 90 103 5 1

A

B

P

A

126 237

-

-

-

-

-

-

(-20) 770 (-40)

-

-

-

-

-

-

Ethylsulfates: M + + + ( C ~ H S S O ~ ) ~ . ~ H ~ O Cooling substances: Ce(C2HSS04)3.9H20; gll(7) = 3.80, 91 = 0.22, T-T*correl.(8) Nd(C2HoS04)3.9HzO;911 = 3.535, g 1 = 2.072, T-T* correL(9)

I Ce(l0) ~r141(11) Nd14a ~m147(12) Sm147 ~b169(11 Dy167 Ho l E 616) ( Yb17s(17)

IS/

-

911

i i

D 91 50.4

3.80 $ 1.525 3 3.535 2.072 4 0.432 0.604 4 0.596 8 i 17.72 < 0 . 3 (#)i 10.76(14: 0 4 15.36 0 C4)+ 3 . 4

-

+

+

-

-

A

I

B l

755 380 165 60 2090 691(15 3340 02

A

-

(288) ( 3) -

3870 -

650 -

P

-

.

Cooling substance: (Ni 15%, Zn 85%) SiFr6HzO; S = 1, g = 2.29, D = -1400, T-T* correL(l8)

_ -s

4$

co59

4

911

2.000 5.82

_

91 _

_

D

_

2.000 -134 3.44 -

~

-

A

91 184

A

P

A

47

-

-

A

P

A

B

- +150(19) -1654(21) +178 +301 862

-

-

-

a For the cooling substances references are given t o g-factors and t o T-T* correlation data. For other ions, constants in spin Hamiltonian are given for one isotope;

182

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

those for other isotopes differ only in the values of A , B[ a r / Z ] and P [ a Q / I ( 2 I - l)]. Except where explicit references are given (in parentheses), the data are taken from references 35-37 of text. The units for A , B, P, and A are lo-' cm-'. It should be noted that the constants have not usually been measured directly in the cooling substances, but rather in an komorphous diamagnetic salt (see references 35-37), and in some cases differences have been noted. REFERENCES A N D FOOTNOTES TO TABLE I11 1. M+++, M++, and Mf refer to trivalent, divalent and monovalent ions respectively. 2. A. H. Cooke, H. J. Duffus, and W. P. Wolf, Phil. Mag. [7] 44, 623 (1953). 3. R. P. Hudson, R. 5. Kaeser, and H. E. Radford, Proc. 7th Intern. Conf. Low Temp. Phys. p. 100 (1961). 4. J. M. Daniels and F. N. H. Robinson, Phil. Mag. [71 44, 630 (1953). 5. There are two sets of nonequivalent ions in unit cell; the ratio of X: Y appear to be about 1 6 : l [B. Bleaney, K. D. Bowers, and R. S. Trenam, Proc. Roy. Soc. A228, 157 (1955)l. 6. R. W. Kedzie, M. Abraham, and C. D. Jeffries,Phys. Rev. 108,54 (1957). 7. A. H. Cooke, 5. Whitley, and W. P. Wolf, Proc. Phys. Soc. (London) B68, 415 (1955). (See also reference 14 of text for discussion.) 8. C. E. Johnson and H. Meyer, Proc. Roy. Soc. A263, 199 (1959). 9. H. Meyer, Phil. Mag. [8] 2, 521 (1957). 10. No direct measurement of hyperfine splitting; see reference 41 of Table I. 11. J. M. Baker and B. Bleaney, Proc. Roy. Soc. A246, 156 (1958). 12. H. J. Stapleton, C. D. Jeffries, and D. A. Shirley, Phys. Rev. 124, 1455 (1961.) 13. Not directly determined, but see references 48 and 49 of Table I. 14. A. H. Cooke, D. T. Edmonds, F. R. McKim, and W. P. Wolf, Proc. Roy. SOC. A262, 246 (1959). 15. See reference 56 of Table I. 16. J. M. Baker and B. Bleaney, Proc. Phys. Soc. (London) A68, 1090 (1955). (See also reference 11.) 17. See reference 60 of Table I. 18. See reference 14 of text. 19. See reference 61 of Table I. 20. M. H. L. Pryce, Phys. Rev. Letters 3, 375 (1959). 21. See reference 62 of Table I. 9

for nuclear orientation by methods S2 and 53, together with the appropriate spin Hamiltonian constants. It will be noticed that almost all the substances used are those met with in work on magnetic cooling, and the nuclei to be oriented (usually radioactive isotopes) are incorporated substitutionally. This method, using a crystal of mixed constitution, works particularly well since the nuclei are in excellent thermal contact with the coolant. It is important to note, however, that spin Hamiltonian constants are usually obtained from paramagnetic resonance measurements made on a salt considerably diluted with isomorphous diamagnetic ions, and it is not invariably the case that they are the same as for the magnetically concentrated salt.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

183

To produce nuclear polarization we must, of course, apply an external magnetic field large enough to cause appreciable polarization of the electron moment. At the temperatures reached by magnetic cooling a few hundred gauss will usually suffice, provided the direction is suitably chosen with respect to the crystal field axes. Thus if we have a large D term in Eq. (2.4.2.3.8),a very anisotropic g-factor with 911 >> gl, or a very anisotropic hfs with A >> B , a magnetic field applied perpendicular to the z-axis will produce only a second order Zeeman effect. I n this case the spins remain oriented with respect to the z-axis, which is determined by the crystal field symmetry and not the external magnetic field. It is clear that the desired effect can be achieved by applying the magnetic field along the z-axis. There is an obvious advantage, therefore, in using single crystals with one kind of magnetic ion in unit cell instead of either polycrystalline specimens or single crystals with ions with different magnetic axes, e.g., the alums. The methods described in Section 2.4.2.3.8may be used to obtain the following “high” temperature approximations for the polarization and alignment : fl

= g,PHzAS(S

fz = -P(I

+ 1)(1 + 1)/9k2T2

+ 1)(21 + 3)(21 - 1)/451kT

(2.4.2.3.9)

+ (A2 - B2)S(S+ 1 ) ( 1 + 1)(21 - 1)(21 + 3)/2701k2T2 + P2(1 + 1)(21 + 5)(21+ 3)(21 - 1)(21 - 3)/18901k2T2. (2.4.2.3.10) 2.4.2.3.3. Electric hfs Alignrnent4O-S4. The coefficient P in Eq. (2.4.2.3.8)is given by41 3eQ d2V P= (2.4.2.3.11) 41(21 - 1)

where e is the absolute value of the electronic charge and Q the nuclear electric quadrupole moment. V is the electrostatic potential a t the nucleus, and therefore d2V/dz2 is the electric field gradient along the symmetry axis. Now we require P to be a t least 10V ev to obtain an appreciable nuclear alignment, and since values of Q may be of the order of one barn in favorable cases, we can see that electric field gradients of the order of 1Ol6 esu are required. Such intense field gradients are found locally in certain compounds, where the electron cloud surrounding the nucleus is very asymmetrical, for example, of the type encountered in certain covalent bonds. A notable series of compounds where this R. V. Pound, Phys. Rau. 76, 1410 (1949). See, e.g., A. R. Edmonds, “Angular Momentum in Quantum Mechanics,” p. 115. Princeton Univ. Press, Princeton, New Jersey, 1957. 40

41

184

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

situation obtains is the actinyl double nitrate^,^^^^^ where the hyperfine splittings are very large and where alignment can be achieved without having recourse to magnetic cooling. I n order to give a n idea of the pattern of the energy levels when the hfs is due to quadrupole splitting only, we have drawn the case of P negative in Fig. 3d. It will be noted from Eq. (2.4.2.3.10) that whereas with magnetic hfs alignment f 2 a 1/T2 a t high temperatures, in quadrupole alignment fz cc 1/T. 2.4.2.3.2.4. The Use of Ferromagnetics and Anti-ferromagnetics. Several of the transition and rare earth metals and alloys, as well as numerous intermetallic compounds, are ferromagnetic, and some are known to have a sufficiently large hfs so th at nuclear orientation becomes practicable (see Table I). The hfs is produced through magnetic coupling, although with transition metals, unlike the situation that obtains in paramagnetic salts, the electrons that give rise to the magnetic moment are not tightly bound. It is also a remarkable fact44that normally diamagnetic ions, when dissolved in small quantities in a ferromagnetic material such as iron, show sufficiently large hfs to make nuclear orientation possible. The mechanism that brings this about seems to be due to a polarization of the inner electron shells, which, in the case of S-shells is particularly potent in producing hfs. The method promises to have very wide application, especially for the investigation of radioactive nuclei that are normally nonmagnetic in the soIid state. Before going on to discuss the dynamic and transient methods for nuclear orientation, we shall now discuss some points of technique which pertain especially to the static methods. TECHNIQUES.2.4.2.3.3.1. Specimen 2.4.2.3.3. SOMEEXPERIMENTAL Preparation.* Most of the salts shown in Table I11 may be obtained as single crystals b y growing them from an aqueous solution. Small seed crystals are obtained, which are placed one in each of a number of small beakers of a few cm3 capacity. The beakers are filled with saturated solution which is allowed to evaporate very slowly. As the seed crystal grows, a layer of faults often occurs a t the place where the original surface of the seed was located. This can be avoided very often by warming the solution slightly (about +"C) before pouring it over the seed. The seed tends to dissolve slightly and if the warming is overdone may dissolve

* See also Vol. VI, A, Part 41

2.

J. C. Eisenstein and M. H. L. Pryce, Proc. Roy. SOC.A229, 20 (1955); A238, 31

(1956). 43 B. Bleaney, Discussions Faraday Sac. No. 19 (1955). 44 B. N. Samoilov, V. V. Sklyarevskii, and E. P. Stepanov, Zhur. Eksptl. i Teoret. Fit. 38, 359 (1960); So& Phys. J E T P 11, 261 (1960).

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

185

completely! A slight dissolving, however, seems to be of value in avoiding faults at the surface of the seed. It is preferable to start with the highest grade of materials available, and sometimes a number of further purifications by recrystallization are beneficial. The rate of growth must be kept very slow and regular (one or two cm3 per week) in order to obtain clear single crystals, and even then all the crystals obtained are unlikely to be entirely free from occlusions of solution. To achieve the best results the ambient temperature should be controlled to better than i-1”C,and it is preferable to control the humidity also. The beaker should be covered to avoid dust and other contaminants entering the solution, which might cause fresh seeds to form. In practice it is not always possible to prevent the formation of new seeds, although these can be kept from interfering with the growth of the main crystal by a suitable arrangement, e.g., by suspending the main crystal in the middle of the solution away from the newly formed seeds which generally fall to the bottom of the beaker. It is always possible to stop t,he growth temporarily and to filter the solution, but there is a chance of new faults appearing when growth is restarted. Special care must be exercised in heating some solutions, particularly the ethyl sulfates since there is a tendency for the ethyl sulfate to break down into acid and alcohol. This is particularly serious because further deterioration is rapid. In growing ethyl sulfates the acidity of the solution should be checked a t regular intervals. It is also possible to grow the crystals from alcohol solutions, and often an advantage to grow them a t temperatures below normal room temperatures. The anhydrous uranyl rubidium nitrate crystals may be grown satisfactorily from solution in concentrated nitric acid. Controlled evaporation of the solvent may be effected by slowly passing dry nitrogen gas over the solution. If it is necessary to include radioactive nuclei in the crystal for gammaray measurements, the activity may be placed in the solution. Allowance should be made for the fact that in this case, as in the general case of growing crystals of mixed composition, the relative concentration of ions in the solution may not be the same as that in the crystal. When beta radioactivity is to be investigated, however, or when only a small amount of activity is available or when the half-life is very short, the activity can be mixed with a drop of solution which is then placed upon the surface of an inert crystal and allowed to crystallize. Similar remarks apply to specimens which are alpha radioactive although, because of the greatly differing scattering properties of alpha particles and electrons in matter, the need for a thin source is not so critical. The obtaining of an “ideal” beta-particle source under the circumstances is of course not possible. The preparation of a “good” beta-particle source is a t best somewhat of an

186

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

art, although it is possible in most cases to check the quality by comparing beta and gamma counting rates. In the case of the preparation of metal specimens, a number of alternative methods of preparation have been employed. It is possible to irradiate metal single crystals in a pile, for example, and subsequently anneal them.44s46 It is also possible to plate the activity on the surface and then heat treat so as to allow a slight diffusion to take place.44 2.4.2.3.3.2. Magnetic Cooling. Although there are a few useful experiments that can be done at about 1°K (the practical low temperature limit which can be reached by causing liquid He4 to boil under reduced pressure), and a t about 0.25”K (the practical low temperature limit reached by similarly using liquid He9, in most cases temperatures of about 0.01”K are required and the method of magnetic cooling is used. The principle of the method of magnetic cooling is to apply a magnetic field H O(typically in the range 10-30 kilogauss) to a suitable paramagnetic salt a t a low temperature T O (usually about 1°K). After thermal equilibrium has been established the salt is isolated thermally, usually by pumping away exchange gas, and the magnetic field is removed. The temperature then falls to a value T, as the field falls to a value H such that T, = -To *H. (2.4.2.3.12) Ho When the external field becomes very small, Eq. (2.4.2.3.12) ceases to hold and the temperature becomes dependent upon internal rather than external constraining forces, such as crystal fields and hfs. When we approach a cooperative region, as is often the case in practice, the relation is more complicated, and in fact at temperatures below a Nee1 point, the application of an external field may actually cause a slight cooling of the sample. For a detailed discussion of these and other points we refer the reader to more detailed treatments of the subject of magnetic co0ling,4~-~~ and also to Section 2.4.2.3.8. We shall refer here only to a few points of technique that are somewhat special to nuclear orientation. 41 M. A. Grace, C. E. Johnson, N. Kurti, R. G. 8curlock, and R. T. Taylor, in “Conference de physique des basses temptktures,” p 263. Centre National de la Recherche Scientifique and UNESCO, Paris, 1956. 48 C. G. B. Garrett, “Magnetic Cooling.” Harvard Univ. Press, Cambridge, Massachusetts, 1954. 47 E. Ambler and R. P. Hudson, Repts. P r o p . in Phys. 18, 251 (1955). 48 D. de Klerk and M. J. Steenland, in “Progress in Low Temperature Physics” (C. J. Gorter, ed.), Vol. 1, p. 273. North-Holland Publ., Amsterdam, 1955. 49 D. de Klerk, in “Handbuch der Physik-Encyclopedia of Physics” (S. Flugge, ed.), Vol. 15, p. 38. Springer, Berlin, 1956.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

187

In experiments with radioactive nuclei the activity is included in a host crystal essentially as an impurity. The type of paramagnetic ion in the host crystal can then be chosen to produce efficient cooling. Among the considerations that affect this choice is the condition that the coolant ion have little or no hfs. The reason for this can best be illustrated by considering the entropies involved. For a given type of ion the maximum entropy of the electron and nuclear spin systems is R ln[(2S 1)(2Z l)]. In order to produce considerable orientation, it is necessary to remove as much of this entropy as possible from the nuclear spin system. By magnetizing at 1”K, however, the maximum amount of entropy that can be l), i.e., the electron removed from the system as a whole is R ln(2S spins are completely ordered, whereas the nuclear spins are completely disordered. Since the demagnetization process is isentropic this is also the final entropy, although it is now shared between the two systems. For I appreciably greater than S, it is clear that the final state will not be a highly ordered one, i.e., the nuclear orientation will be small. When a second type of ion, of the kind mentioned above, is incorporated in small concentration, however, the entropy balance is very favorable for producing a large nuclear orientation. The best example of this is the case of cerous magnesium nitrate,60*61 which may be cooled by demagnetization to a temperature of 0.003”K1partly because it is a very dilute salt magnetically and partly because there is no hfs. This salt has the further advantage that it is a double salt into which both trivalent rare earth ions and divalent transition metal ions may be incorporated. Another point to be noted about magnetic cooling is that it is a “single shot” process, i.e., after demagnetization the specimen begins to warm up again, and measurements are taken during this warming period. Since the heat capacities of the salts are often very small, the question of thermal insulation becomes most important. Particular attention must be paid to technique in this connection, and sources of heat influx such as arise from radiation, and more particularly from vibration of the sample and conduction by residual exchange gas desorbing from the walls of the cryostat, must be minimized. For a carefully designed apparatus, a heat influx of 10 ergs/min into a sample weighing a few grams would be considered very satisfactory. For details of specimen support inside the cryostat, the use of exchange gas, thermal “guard-rings,” etc., the reader is referred to references 10-22, 25, and 4 6 4 9 . When a radioactive substance is being investigated, the heating due to absorption of some of the energy released during decay may become noticeable. For the more commonly used

+

+

+

J. M. Daniels and F. N. H. Robinson, Phil. Mag. [7] 44, 630 (1953). R. P. Hudson, R. S. Kaeser, and H. E. Radford, Proc. 7th Intern. Conf. Low Temp. Phys. p. 100 (1961). Kt

188

2.

DETERMINATION OF FUNDAMENTAL QUANTITIES

isotopes with beta-gamma cascades, a useful average working figure is 1 erg/min per microcurie of activity. The amount of activity required is usually in the range 1-100 microcurie. Thus even with cooling substances such as cerous magnesium nitrate, warm up times of one hour can be achieved, although since much larger effects are observed at the lowest temperature, not all of this time is equally useful. I n fact, the more usual procedure is to take data only at the lowest temperatures, and then deliberately to warm up the sample and take the “warm” data needed for normalization purposes. Besides requiring less stringent conditions for counter stability, this method overcomes inaccuracies due to inhomogenous heating in the sample, which arises on account of the rather low thermal diffusivity of the salts a t low temperatures. In fact, for the most accurate work, the taking of “cold” data is often confined to the first few minutes after demagnetization. There is a further point to note when experiments are being performed with polarized nuclei, I n this case, there is an external field present which affects both the specific heat, C , and the temperature of the sample. Now we have c = [b CH2J/T2 (2.4.2.3.13)

+

where c is the Curie constant and b is the “specific heat constant.” For work on radioactivity, or on nuclear reactions where the incident particles are uncharged, no serious trouble due to small heat capacities should be encountered. For charged particle reactions, however, the situation is likely to be very different, and a change of technique would be needed. One could consider producing continuous refrigeration by the use of He3 as ref1igerant~2-~4or by the development of ways of producing cyclical magnetic as well as resorting to experiments with a rather poor duty cycle, as illustrated for a somewhat extreme case in Fig. 2. In the case considered above for producing nuclear polarization, the residual external magnetic field limits the extent of the cooling and hence the polarization. An alternative procedure is to use two different specimens separated sufficiently in space but connected by a thermal link, the first specimen being cooled directly by demagnetization and the second by contact with the first, in the manner outlined in the previous section. This method has the further advantage that the substances in the V. P. Peshkov and K. N. Zinov’era, Repts. Progress in Phys. 22, 504 (1959). W. Taconis, in “Progress in LOW Temperature Physics” (C. J. Gorter, ed.), Vol. 111, p. 153. North-Holland Publ., Amsterdam, 1961. 54 E. Ambler and R. B. Dove, Rev. Sci. ZnstT. 32, 737 (1961). 56 C. V. Heer, C. B. Barnes, and J. G . Daunt, Rev. Sci. Instr. 26, 1088 (1954). 52

6s K.

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

189

second stage need not necessarily be a paramagnetic salt. There are a number of points of technique involved, as we have already indicated, the most important one being that of obtaining good thermal contact between the thermal link and the salt. I n certain cases, demagnetization to a nonzero field can be achieved most advantageously without sacrificing the degree of cooling by using single crystals which are magnetically very anisotropic and by essentially rotating the sample with respect to the magnetic field.66Equation (2.4.2.3.12)is then modified to read: (2.4.2.3.14) where go and g are the g-factors in the directions of H o and H respectively. The advantage of using this method when g o >> g , as is the case with cerous magnesium nitrate, is evident. When the g-factor along H is not small, i t is necessary to allow for saturation effects. This is the case, for example, with neodymium ethyl sulfate where very often a polarizing field is applied along the z-axis while susceptibility measurements are made perpendicular. I n this case a useful relation is i

T*

=

( T H *tanh y)/y

(2.4.2.3.15)

where y = gllpH/2kT, TH* is the measured magnetic temperature, and T* is the one to be substituted in the T-T* correlation (see below). Finally, we should point out th at when we are in the range of cooperative effects in paramagnetic salts, changes in temperature may be complicated but, can be found by a procedure based upon the thermodynamic relati~n~’*~~*~~ (2.4.2.3.16) where M is the magnetic moment of the sample. I n order t o determine the degree of nuclear orientation (see Section 2.4.2.3.5),it is usually necessary to know the temperature of the specimen. When working with liquid He4, 68 or liquid He3, 69 temperatures can be obtained simply from the vapor pressure, although corrections for thermomolecular pressure differences may be significant.6oI n the magnetic cooling 56A.

H. Cooke, S. Whitley, and W. P. Wolf, Proc. Phys. SOC.(London) B68, 415

(1955).

W. F. Giauque, Phys. Rev. 92, 1339 (1953). F. G. Brickwedde, H. van Dijk, M. Durieux, J. R. Clement, and J. K. Logan, J. Research Natl. Bur. Standards A64 1 (1960). 59 S. G. Sydoriak and T. R. Roberts, Phys. Rev. 106, 175 (1957). 60 T. R. Roberts and S. G. Sydoriak, Phys. Rev. 102, 304 (1956). 57 68

190

2.

DETERMINATION O F FUNDAMENTAL QUANTITIES

region the absolute temperature is usually correlated with the susceptibility of the paramagnetic salt and published as a T-To correlation. The procedure is to measure T*, compute T o , and then use the published data to find T46-61[cf. refs. in Table 1111. The following relation defines T*:

c/T* = M / H ,

(2.4.2.3.17)

where c is the Curie constant and M the magnetic moment caused by a small external measuring field, He. T* is thus related only to measurable quantities, although it will be different for differently shaped specimens. The symbol To refers to a spherically shaped sample, and it is customary to relate all T* values to To through the relationship

+ /3

(2.4.2.3.18)

(n,- n)c

(2.4.2.3.19)

T o = T* where

/3

=

where n, is the demagnetizing coefficient for a sphere (= 47r/3), and n that for the specimen used. The following points are worth noting: at higher temperatures, T o 3 T; a t lower temperatures the determination of the T-Tb correlation is a lengthy, and also somewhat inaccurate, procedure; for magnetically anisotropic substances, the T-To correlation will depend upon the direction of measurement with respect to the crystallographic axes; and finally the value of T o measured in the presence of a magnetic field, is not necessarily equal to that which would be measured in zero field at the same temperature, since saturation effects can occur at low fields in the magnetic cooling range. Once the T-TD correlation is known, secondary thermometers, such as carbon thermometers-or, more particularly, y-ray anisotropies of radioactive nuclei, can be calibrated and used. This latter method, the “nuclear thermometer” as it has come to be called, has many advantages. It can be used, for example, to give the temperature of a very specific region (e.g., for work on beta decay of the surface layer of a paramagnetic salt). It is also convenient to use in experiments on radioactivity, since the “thermometric” isotope can be simply included in the crystal and the gamma ray intensity, and hence the temperature, recorded simultaneously with the other data. 2.4.2.3.4. THE PRODUCTION OF ORIENTED NUCLEI-DYNAMICAND TRANSIENT METHODS. The relevant properties of substances used in this class of experiment can usually be described by a spin Hamiltonian. Preferential nuclear population distributions are not caused primarily by lowering temperature; however, they are caused by driving transitions between different energy levels by means of oscillating fields a t a fre-

2.4.

DETERMINATION OF SPIN, PARITY, AND NUCLEAR MOMENTS

191

quency chosen to satisfy some resonance condition. The oscillating field is usually an electromagnetic field (although the use of acoustic vibrations for the purpose has been proposed) and the frequency ranges from the optical region down through the microwave to the rf region. The basic idea is to induce transitions a t rates comparable to the natural relaxation rates of the system. Saturation effects then occur, and the system is driven away from thermal equilibrium into a state of dynamic equilibrium where cer