REVIEWS OF MODERN PHY SIC S VOLUME 43, NUMBER 4 OCTOBER i97i :xcitations in Quantum Crysta. s (A Survey ot:;yI'V, :;4 "-xperiments in So. . ic. ", . -'. .e. . turn) R. A. GUYER Department of Physics and Astronomy, University of 3Iassachusetts, Amherst, 3fassachusetts 0100Z R. C. RICHARDSON* Department of Physics, Cornell University, Ithaca, 1Vem York 146'50 I. I. ZANE Departmertt of Physics, Duke Urtioersity, Durham, North Carolina 27706 This article gives a systematic review of the many NMR experiments in solid helium to date, emphasizing the view- point that the results may be interpreted primarily in terms of the effects of three fundamental excitations in the solid: the vacancy waves; the 'He tunneling interaction or exchange; and the 'He — 'He tunneling interaction or mass fluctuation waves. CONTENTS 1. INTRODUCTION 1. Introduction. . ... . , . . .. ~. .. . .. , . . . , .... . . .. . ~ . , .. . 2. Background (NMR). .. . . .. . . .. .. . . ... . , . .. . . . , .. . . 3. Excitations in Pure 'He. . .. . . . . .. . .. . .. . . . . .. .. . . . . 3.1 Kxcitations. . . . . , . . ... . . . . . . . . . . . . .... .. , ... . .. 3. 2 Interactions. . . . . .. . ... . . .. .... . .. .. . . .... . . . . . 4. NMR in Pure 'He (Experimental}. .. . . .. , .. ... . . . .. 4. 1 Ti Relaxation, Theory. . . .. .. . . . . ... . . . .. . ... .. . 4.2 T1 Experiments, Results. . . . . .. . .. , .... . . , . . . . , . 4. 3 T2 Relaxation, Theory. ... .. , .. . .. . . . . . . ... . . .. . 4. 4 T2 Experiments, Results. .. .. . . . . . . . , . . . .. .. . .. . 4.5 Diffusion, Theory. . . .. , .. . . . . . ~. .. .. . . . . . . . . . .. 4. 6 Diffusion Experiments, Results. . . .. . .. , . . . .. . . , . 4.7 Properties of the Excitations in Pure 'He. . . . , . . . . 5. Excitations in Dilute 'He — 4He Mixtures. . .... . .. . , ... . 5.1 Mass Fluctuation%aves . 5. 2 Interactions. ... . . . . . .. . . .. . . .. . . . . .. . .. . . . .. . . 6. NMR in Dilute 'He — 4He Mixtures. . . . . . . . ... . . . . . . . . 6. 1 Dilute 'He in 'He Mixtures, Theory. .... . . ... . . . , 6. 2 Experiments on Dilute 4He in 'He Mixtures, Results. . . . ... .. . ...... .. .. ... .. . .. . .... . . . .. . 6. 3 Dilute 'He in 4He Mixtures, Theory. .... . . .... .. . 7. Nondilute 'He — 4He Mixtures .. 7.1 Introduction. . ... . ... . .. . . . .. . .. . . . ... . . .. . . . . . 7.2 T1 Relaxation, Theory. .. . .. . . . . .. . . .. . . ... . . .. . 7. 3 T1 Experiments, Results. .. ..... . . . .. . . . , .. . ~. . . 7. 4 T~ and. Diffusion Experiments. . . . . . .. .. . . ... . . . . 7.5 Properties of Excitations in 'He — 'He Mixtures. . . . . 8. Concluding Remarks. .. . . . . . , . . .. . . . . . . .......... . . 9. Acknowledgment. . .. .. . . ~ . . , ...... .. . .. , .... .. ... . Appendices. . .. . . . .. . .. .. .. . . . . , .. . . . .. .. , . . .. . . . . A. Relaxation Times. . . . . . . .. ... , .. . .. .. . . . . .. . . . . . .. . A. O Introduction to Ti. .. , .. . . . .. . . ... .. . . . .. . ... .. A. 1 Zeeman — Vacancy Wave Relaxation. ..... . .. .. ... A. 2 Zeeman — Tunneling Relaxation. .. , . . ... . . . . , . .. . A. 3 Tunneling — Vacancy Wave Relaxation. . . . . . ...... A. 4 Vacancy Wave — Phonon Relaxation. . . .. , , . . ... . . A. 5 Tunneling — Mass Fluctuation Wave Relaxation. . .. A. 6 Mass Fluctuation Wave — Phonon Relaxation. .. . .. A. 7 Relaxation Topologies. . . . ~. .. . .. . . .. .. , . . .. .... B. Equilibrium Times, T2, 10/3 Effect, etc. . . . .. . .. . . . . . . 8 B. 2 10(3 Effect and Nonadiabatic Frequency Shift. . . . C. D. Diffusion. ~. .... .. . . . . . , , , ..... . ... ... , . . . C. 1 Introduction. . .. ... . ... . , . .. . . .. . . , . . . C.2 Diffusion Constants. . ~. . . , .. . . .. .. .. . . . Specific Heats, Etc. . .. . ... . . . ... . . . . .. . . .. . ~ Present Address, Physics Department Colorado State Univ. , Ft. Collins, Colorado. 532 A quantum solid is one in which the zero-point 534 motion of the atoms about the equilibrium lattice sites 537 is a large fraction of the near-neighbor distance. This 541 large zero-point motion has three important con- 545 sequences (e. g. , Guyer, 1969): 543 (a) Neighboring atoms in the lattice encounter one another away from their respective lattice sites at distances comparable with the hard core radius. (b) An atom visits a large region of space in the vicinity of its lattice site. The small parameter of con- ventional lattice dynamics (rms displacement/near- neighbor distance) is not small so that there is large anharmonicity. (c) Neighboring atoms tunnel around one another and exchange lattice sites. 573 573 The difficulties caused by an atom's visit to a rela- 574 tively large region of space near its lattice site or its encounters with its near neighbors at the hard-core radius have a significant effect on how one does a theory of quantum crystals. But, the aggregate of conventional thermostatic and thermodynamic experiments on the quantum crystals exhibit few remarkable or unusual 584 features that are a consequence of large anharmonicity 586 or close approach (Guyer, 1969) . 588 However, the third consequence of the large zero- 589 point motion of the atoms in a quantum crystal has 589 important experimental implications. There is a finite overlap between the wavefunction of an atom localized near lattice site 1 and the wavefunction for an atom p96 localized near lattice site 2, a near neighbor site of 1. 596 Because of this overlap, the atoms can tunnel about one another and change place. In solid 'He, the atoms are 598 fermions (there is one unpaired nuclear spin) so there is a nuclear exchange process due to the finite overlap. The energies associated with the exchange process are on the order of 1 mK. Thus this process is unimportant 532