Spectral Distributions in Nuclei and Statistical Spectroscopy

Cover......Page 1
Title......Page 4
Dedication......Page 6
Contents......Page 12
Foreword......Page 8
Preface......Page 10
1.1 Introduction.......Page 20
1.1.1 Spectral distributions and fluctuations......Page 23
1.1.2 Average-Fluctuation separation......Page 25
1.1.3 Random matrix ensembles......Page 28
1.1.4 Fluctuations and quantum chaos......Page 31
1.1.5 Laws of statistical spectroscopy......Page 34
1.2.1 Moments and distributions......Page 35
1.2.2 Central Limit Theorems for NIP and IP systems......Page 43
1.2.3 Partitioning and partial densities......Page 50
1.3.1 Five methods for moment expansions for expectation values......Page 56
1.3.2 Gram-Charlier and Edgeworth expansions......Page 60
1.3.3 Partitioning, convergence and multi-variate expectation values......Page 62
1.4 Unitary Decompositions, Geometry and Configuration Traces: State and Spin Cutoff Densities......Page 66
1.4.1 Unitary decompositions and renormalization of the NIP Hamiltonian......Page 67
1.4.2 Geometry, norms and correlation coefficients......Page 73
1.4.3 State density and variances of IP spreading Gaussians......Page 76
1.4.4 Moments for spin cutoff densities......Page 78
1.5.1 Level densities using spherical configurations......Page 84
1.5.2 Level densities using convolution forms......Page 87
1.6 Group Symmetries and Trace Propagation......Page 95
1.6.1 Introduction......Page 96
1.6.2 Classification of group symmetries......Page 97
1.6.3 Propagation of fixed symmetry trace matrix......Page 98
1.6.4 Casimir propagation......Page 100
1.6.5 Propagation with I.B. operators......Page 103
1.6.6 Trace of the trace matrix......Page 105
1.6.7 Fixed-J traces......Page 106
1.6.8 Reduced traces......Page 112
1.6.9 Regular structures with random interactions......Page 115
1. 7.1 Preliminaries......Page 116
1. 7.2 Bivariate distributions......Page 119
1. 7.3 NIP strength densities: Convolution form......Page 123
1.7.4 Bivariate Gaussian form for NIP strength densities......Page 127
1.7.5.2 (k\' = 1, k = 1) OT\' one-body transition operators......Page 129
1.7.5.3 (k\' = 2, k = 2) or two-body transition operators......Page 130
1.7.7 IP strength densities: Convolution form......Page 132
1.8 Applications to Atoms, Mesoscopic Systems and Interacting Bosons......Page 139
Bibliography......Page 142
Original Papers......Page 164
2. Combinatorial Methods for Spectroscopic Averaging and Spectral Moments J. B. French Phys. Lett. 23, 248-250 (1966)......Page 165
References......Page 167
3. Measures for Group-Symmetry Breaking in Nuclei J. B. French Phys. Lett. B 26, 75-77 (1967)......Page 168
References......Page 170
4. Validity of Random Matrix Theories for Many-Particle Systems J. B. French and S. S. M. Wong Phys. Lett. B 33, 449-452 (1970)......Page 171
References......Page 174
1. INTRODUCTION......Page 175
2. SPECTRAL DISTRIBUTIONS......Page 176
3. SCALAR DISTRIBUTION AS A SIMPLE EXAMPLE......Page 180
4. OPERATOR AVERAGING......Page 181
S. CONFIGURATION DISTRIBUTIONS......Page 187
6. SCALAR AND CONFIGURA nON MOMENTS IN THE SPHERICAL SHELL MODEL......Page 188
7. MOMENT RENORMALIZATIONS......Page 194
8. SUMMARY AND FINAL REMARKS......Page 196
I. INTRODUcnON......Page 199
A. Eigenvalue Distributions and Spectra......Page 200
B. Spectral Moments and Structure of Eigenvectors......Page 205
III. SCALAR DISTRIBUTIONS AND GROUND-STATE ENERGIES......Page 207
A. Configuration Moments and Vector-Space Truncation......Page 212
B. Partial Variances and Symmetry Breaking......Page 215
V. GROUND STATE OF 0 16......Page 216
A. Core Centroid and Partial Dispersions......Page 217
B. Model for Core Distribution......Page 221
C. Role of Complex Configurations......Page 223
VI. CONCLUSIONS......Page 224
ACKNOWLEDGMENTS......Page 225
I. INTRODUCTION......Page 226
2. MOMENT POLYNOMIALS......Page 230
3. UNITARY GROUP DECOMPOSITIONS......Page 233
4. THE DECOMPOSITION AND STRUCTURE OF THE HAMILTONIAN......Page 241
5. GROUND-STATE AND Low-LYING ENERGIES AND STRUCTURES......Page 249
6. LEVEL DENSITIES......Page 257
7. STRENGTH DISTRIBUTIONS......Page 270
8. CONCLUSION......Page 274
REFERENCES......Page 276
8. Two-Body Random Hamiltonian and Level Density O. Bohigas and J. Flores Phys. Lett. B 34, 261-263 (1971)......Page 278
References......Page 280
9. Spin-Isospin SU(4) Symmetry in the (ds) Shell J. B. French and J. C. Parikh Phys. Lett. B 35, 1-4 (1971)......Page 281
References......Page 284
10. Angular-Momentum Dependence of the Density of States J. N. Ginnochio Phys. Rev. Lett. 31, 1260- 1263 (1973)......Page 285
11. Occupation Probabilities and Spin Cut-Off Factors F. S. Chang and J. B. French Phys. Lett. B 44, 135-138 (1973)......Page 289
References......Page 292
12. Recent Results on Energy-Level Fluctuations O. Bohigas, J. Flores, J. B. French, M. J. Giannoni, P. A. Mello, and S. S. M. Wong Phys. Rev. C 10, 1551-1553 (1974)......Page 293
1. INTRODUCTION......Page 296
2. THE GAUSSIAN AND SEMICIRCULAR LIMITS......Page 299
3. NONINTERACTING PARTICLES......Page 300
4. DENSITIES FOR INTERACTING PARTICLES......Page 303
5. VARIANCE, ERGODICITY, AND FLUCTUATIONS......Page 307
6. CONCLUSION......Page 316
REFERENCES......Page 317
2. NONINVARIANCE GROUPS AND COMPLEMENTARY SUBGROUPS......Page 318
3. PROPAGATION OF OPERATOR AVERAGES......Page 320
4. DISCUSSION OF SOME PARTICULAR CASES......Page 321
15. Strength Distributions and Statistical Spectroscopy. I. General Theory J . P. Draayer, J. B. French, and S. S. M. Wong Ann. Phys. (N. Y.) 106, 472-502 (1977)......Page 323
1. INTRODUCTION......Page 324
2. POLYNOMIAL EXPANSIONS......Page 328
3. CONVERGENCE AND THE CENTRAL LIMIT THEOREM; CORRELATION COEFFICIENTS AND SIMPLE GEOMETRY......Page 331
4. LINEAR RESPONSE THEORY......Page 336
5. PARTITIONING FOR STRENGTHS, SUM RULES, AND EXPECTATION VALUES......Page 339
6. MULTIPOLE STRENGTHS AND SUM RULES......Page 343
7. UNITARY GEOMETRY......Page 347
8. FLUCTUATIONS AND CORRELATIONS......Page 350
9. CONCLUSION......Page 352
REFERENCES......Page 353
1. Introduction......Page 354
2. Elementary Principles......Page 355
3. A Simple Example: Single-state Occupancies......Page 357
4. Subgroup Structures, Spectra and Characteristic Fonns......Page 358
5. Propagation of Information......Page 360
6. Infonnation Propagation in Dilute Spaces......Page 365
References......Page 367
I. FLUCTUATIONS......Page 368
II. MODEL SPACES......Page 375
III. INFORMATION PROPAGATION......Page 377
IV. FURTHER PROBLEMS......Page 383
REFERENCES......Page 384
18. Shape of Eigenvalue Distribution for Bosons in Scalar Space V. K. B. Kota and V. Potbhare Phys. Rev. C 21, 2637-2642 (1980)......Page 386
APPENDIX......Page 390
19. Fluctuation Properties of Nuclear Energy Levels: Do Theory and Experiment Agree? R. U. Haq, A. Pandey, and O. Bohigas Phys. Rev. Lett. 48, 1086- 1089 (1982)......Page 392
20. Shell Model and Spectral Distribution Study of SU(4) Symmetry Mixing in the S-D Shell R. U. Haq and C. Quesne Phys. Lett. B 114, 91 - 94 (1982)......Page 396
References......Page 399
II. REVIEW OF SPECTRAL DISTRmUTION METHODS......Page 400
A. Polynomial expansions......Page 401
B. Central limit theorem......Page 402
D. Unitary geometry and trace propagation......Page 403
E. Spectra and occupancies......Page 404
A. Sum rule analysis......Page 405
B. Eigenvalue bounds......Page 406
D. Examples......Page 407
1. Electric quadrupole......Page 409
2. Magnetic dipole......Page 411
IV. LINEAR-ENERGY-WEIGHTED SUM RULES......Page 413
A. The LBWSR operator......Page 414
B. Isovector magnetic dipole......Page 415
C. Isoscalar magnetic dipole......Page 419
D. Electric quadruPQle......Page 420
E. Other multipoles......Page 421
F. Strength centroids......Page 422
V. CONCLUSION......Page 423
APPENDIX A: ELECTRIC AND MAGNETIC MULTIPOLE OPERATORS......Page 424
APPENDIX B: SCALAR PRODUCT 0 .0  OF TENSORIAL OPERATOR 0 .......Page 426
APPENDIX C: EVALUATION OF [O ,[H,O ] -]-......Page 427
22. Nuclear Level Densities and Partition Functions with Interactions J. B. French and V. K. B. Kota Phys. Rev. Lett. 51, 2183-2186 (1983)......Page 430
2.GOE Predictions for the Energy Levels......Page 434
3.Comparison with Experiment for Energy Levels......Page 435
4. Width Fluctuations......Page 436
References......Page 438
24. Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws O. Bohigas, M. J. Giannoni, and C. Schmit Phys. Rev. Lett. 52, 1- 4 (1984)......Page 439
1. Introduction......Page 443
2. Bivariate Cumunts for the Operators H and Jz......Page 444
3. Bivariate Edgeworth Expansion for  (E, M)......Page 446
4. Fixed-l Averages of Powers of H......Page 447
5. Spin Cut-Off Factors......Page 448
References......Page 449
26. Higher-Order Correlations in Spectra of Complex Systems O. Bohigas, R. U. Haq, and A. Pandey Phys. Rev. Lett. 54, 1645-1648 (1985)......Page 451
I. INTRODUCTION......Page 455
II. A SURVEY ON PROPAGATION COEFFICIENTS......Page 456
III. THE SUM OF FIXED-ISOSPIN SPECTROSCOPIC FACTORS......Page 457
IV. THE PROPAGATION COEFFICIENT Z......Page 458
V. THE PROPAGATION COEFFICIENT R......Page 459
VII. USE OF Z AND S AS SPECTROSCOPIC FACTORS FOR NUCLEAR FRAGMENTATION......Page 461
28. Symmetries and Quantum Chaos: Time-Reversal Invariance in the Nucleon-Nucleon Interaction J. B. French, V. K. B. Kota, A. Pandey, and S. Tomsovic Phys. Rev. Lett. 58, 2400-2403 (1987)......Page 463
1. Introduction......Page 467
2. Wigner Universality and the Statistical Model......Page 469
3. Spectral and Width Fluctuations......Page 471
4. Ericson Fluctuations and Detailed Balance......Page 474
5. Fluctuation-Free Spectroscopy and the     a Reduction......Page 477
6 . Summary and Outlook......Page 483
References......Page 484
1. INTRODUCTION......Page 487
2. SPECTRAL AVERAGING AND LEVEL DENSITIES......Page 491
3.1. Preliminaries......Page 495
3.2. Strength Distributions in Noninteracting Particle Systems......Page 496
3.3. Strength Distributions in Interacting-Particle Systems......Page 498
3.4. Shell-Model Tests......Page 500
3.5. Ensemble Derivations......Page 503
4. THE SPECTROSCOPY OF HEAVY NUCLEI;  -BOUNDS ON TRNI......Page 506
5. CONCLUSION......Page 510
REFERENCES......Page 511
1. Introduction......Page 513
3. Level density decomposition......Page 514
4. Nature and magnitude of fluctuations......Page 517
5. Conclusions......Page 520
Acknowledgments......Page 522
Appendix......Page 523
32. Influence of Saturation Properties on Shell-Model Calculations A. Abzouzi, E. Caurier, and A. P. Zuker Phys. Rev. Lett. 66, 1134-1137 (1991)......Page 525
1. lntroduction......Page 529
3. Formalism for -decay rates at finite temperature......Page 530
4.1. State densities......Page 531
S. GT NEWSR strength......Page 532
6. Results for  -decay rates at finite temperature......Page 533
7. Conclusions......Page 534
References......Page 535
1. Introduction......Page 536
2.1. Preliminaries......Page 538
2.2. Convolution form for IP densities......Page 540
3.1. Preview......Page 543
3.2.2. Reference energy......Page 544
3.2.3. Choice of the single-particle orbits, single-particle energies and the two-body interaction......Page 546
3.2.4. Unitary orbits for spreading variances and NIP calculations......Page 547
3.2.5. Procedure for calculations......Page 548
3.3.1. 55Mn......Page 549
3.3.2. 56Fe......Page 551
3.3.4. 60Co......Page 553
3.3.6. 63Cu......Page 556
3.4. Summary of 8-orbit calculations......Page 557
4.1. Choice of s values, single-particle energies, centroid separations and other calculational details......Page 558
4.2.1. 62Ni......Page 559
4.2.3. 65Cu......Page 560
4.3. Summary......Page 561
Acknowledgements......Page 562
Appendix A. Unitary group decompositions......Page 563
B.1. Fermi gas forms......Page 564
B.2. Experimental data......Page 566
References......Page 567
Abstract......Page 570
References......Page 575
I. INTRODUCTION......Page 576
A. Definitions......Page 577
C. General description and strong coupling limit......Page 578
B. Shape of the strength function......Page 580
A. Two-step diagonalization and the spectral function......Page 582
B. Dependence on the interaction strength......Page 583
V. CONCLUSION......Page 584
1. Introduction......Page 586
3. Partial state densities in a 2 x 2 p-EGOE......Page 588
4. Shell-model results......Page 591
5. Conclusions and future outlook......Page 593
References......Page 594
I. INTRODUCTION......Page 595
II. COMPILATION OF LEVEL DENSlTY PARAMETERS......Page 596
Ill. COMPARISON WITH THEORY......Page 598
IV. COMPARISON WITH OTHER COMPILATIONS......Page 602
V. SUMMARY......Page 605
I. INTRODUCTION......Page 607
II. STATISTICAL AND NONSTATISTICAL LIMITS......Page 608
III. FORMULATION OF STATISTICAL THEORY FOR PARITY VIOLATION......Page 609
A. Statistical spectroscopy of French and collaborators......Page 610
1. Model space and strong interaction......Page 612
3. Higher-order contributions arising from vPNC(1) SUC......Page 613
IV. CALCULATIONS......Page 614
A. Size of the model space......Page 616
1. Permanent deformation......Page 617
D. Choice of weak-coupling parameters......Page 618
VI. SUMMARY AND CONCLUSION......Page 619
APPENDIX: MA TRJX ELEMENTS OF THE EFFECTIVE PARJTY-VIOLATING INTERACTION......Page 620
PACS number(s):......Page 624
Introduction.......Page 628
Second moment.......Page 629
1. Introduction......Page 632
2. Moments, state densities, random matrix ensembles, and quantum chaos......Page 634
3. General methods for expectation values......Page 637
3.2. Parametric derivative method......Page 638
4. Convergence of moment expansions with the action of EGOE(l + 2)......Page 639
5. Expectation values with partitioning......Page 641
5.1. Convergence with partitioning according to K subspaces......Page 642
5.2. Convolution factoring......Page 644
6. Conclusions and future outlook......Page 645
Appendix A......Page 647
Appendix B......Page 648
References......Page 649
PACS number(s):......Page 652
PACS number(s):......Page 659
I. INTRODUCTION......Page 664
II. BASIC DEFINITIONS AND RESULTS......Page 665
III. MATRIX OF SECOND MOMENTS......Page 667
IV. EIGENVALUES AS SU(N) RACAH COEFFICIENTS AND THEIR APPLICATIONS......Page 668
ACKNOWLEDGMENT......Page 671
I. INTRODUCTION......Page 673
II. SYMPLECTIC sp(4) PAIRING MODEL INTERACTION......Page 674
III. THEORY OF SPECTRAL DISTRIBUTIONS......Page 675
IV. UNDERLYING SYMMETRIES OF REALISTIC INTERACTIONS......Page 676
A. The sp(4) model and pairing character......Page 677
B. Pairing+quadrupole model interaction......Page 678
C. Energy spectrum......Page 680
D. Correlations between interactions for nuclear systems with more than two nucleons......Page 681
V. CONCLUSIONS......Page 682
ACKNOWLEDGMENTS......Page 683
PACS Nos.:......Page 685
1. Introduction......Page 686
2. State densities: unitary group decompositions and convolutions......Page 689
3. Spin-cutoff densities and spin-cutoff factors......Page 693
4. An example: the nucleus 234U......Page 696
5. Results for heavy nuclei......Page 699
6. Strength distributions and two-point functions......Page 710
Acknowledgments......Page 711
References......Page 712
Appendix A.......Page 713