P H Y S I C A L R E V I E W V O L U M E 1 0 9 , N U M B E R 4 F E B R U A R Y 1 5 , 1 9 5 8

Ultrasonically Induced Spin Transitions in Sodium Iodide* D. A. JENNINGS AND W. H. TANTTILA, Michigan State University, East Lansing, Michigan

AND

O. KRAUS, National Bureau of Standards, Washington, D. C. (Received September 30, 1957; revised manuscript received November 12, 1957)

We have examined the influence of ultrasonic vibrations on the nuclear magnetization of sodium and iodine nuclei in a single crystal of sodium iodide. Our results support Sternheimer's calculations of quad-rupole polarizability of ionic cores. In addition measurements made on the line width of the ultrasonic excitation effective in bringing about nuclear spin transitions show that the line width for iodine nuclei is narrower than the line width for the sodium nuclei. These results appear somewhat anomalous in the light of previous explanations of Pound, Watkins, and Reif.

I. INTRODUCTION

SEVERAL experiments1"3 have demonstrated that transitions between nuclear spin levels can be

produced in crystals by introducing ultrasonic energy into the crystal at the frequency of the Am= dz 1, or the Am=zi=2 transition. These transitions are caused by the interaction of the nuclear quadrupole moment with the time-varying electric field gradient at the site of the nucleus. When the lattice is distorted by the ultrasonic waves, the electric field gradient at the site of a nucleus varies periodically with the frequency of the ultrasonic waves. The variation of the electric field gradient produces the necessary time-dependent perturbation of the spin levels to bring about transitions. The informa­tion that one has hoped to obtain by these experiments is the strength and nature of the coupling between the lattice waves at ultrasonic frequencies and the nuclear quadrupole moments.

Sternheimer4 has shown that the potential arising from a quadrupolar nucleus can induce a quadrupole moment in the electron distribution of the atom. Depending on the state of the core electron there is an enhancement (antishielding) or a diminution (shielding) of quadrupole moment of the nucleus brought about by this polarization. In general, the atomic or ionic quad­rupole moment is larger than the nuclear quadrupole moment and often by a considerable amount. Lately Das and Bersohn5 have done variational calculations on the same effect. Van Kranendonk,6 who assumes that the quadrupolar coupling of a nuclear spin system to the crystalline lattice is responsible for the nuclear spin-lattice relaxation, replaces the nearest neighbors of a given nucleus with point charges of magnitude ye. Here e is the electronic charge; and in order to account for the experimental values of the nuclear spin-lattice

* Work supported, in part> by the National Science Foundation. 1 W. G. Proctor and W. H. Tanttila, Phys. Rev. 98,1854 (1955). 2 W. G. Proctor and W. H. Tanttila, Phys. Rev. 101, 1757

(1956). 3 W. G. Proctor and W. Robinson, Phys. Rev. 104, 1344 (1956). 4 R. M. Sternheimer, Phys. Rev. 105, 158 (1957). References to

other papers on this subject by Sternheimer and collaborators can be found in this paper.

5 T. P. Das and R. Bersohn, Phys. Rev. 102, 733 (1956). 6 J. Van Kranendonk, Physica 20, 781 (1954).

relaxation time, Van Kranendonk finds that the param­eter 7 has numerical values from 10 to 1000. On the basis of Van Kranendonk's model, one finds, therefore, that the quadrupolar coupling between a nucleus with quadrupole moment Q and neighboring monovalent ions is 10 to 1000 times greater than the value which one computes if one regards the ions as point charges with a single electronic charge.

Yosida and Moriya7 have attributed the fast relax­ation of quadrupolar nuclei in the alkali halide crystals to covalent bond formation. According to them the distortions produced by lattice waves are sufficient to bring about covalent bond formation between the halogen and alkali ions during the distortion. This covalent bond produces an electric field gradient at the site of the quadrupolar nucleus which is in excess of the field gradient arising from the relative motion of the point charges at the lattice sites by a factor which they indicate would be sufficient to account for the short relaxation times of quadrupolar nuclei in the alkali halide crystals.

Ultrasonic experiments also indicate that there is a stronger coupling between the ions and the nuclei than one would get from a simple model having charges e and quadrupole moments Q at the lattice sites. How­ever, it is difficult to measure the ultrasonic energy density in a crystal. In previous experiments, the ultra­sonic energy density has been calculated from the power transmitted to the crystal and the phonon relaxation time of the ultrasonic phonons. However, phonon re­laxation times derived from measurements appearing in the literature of the attenuation of pulsed ultrasonic waves are unreliable since the attenuation measurements themselves are somewhat uncertain and the meaning of a phonon relaxation time derived from such a measure­ment is ambiguous. We have circumvented the problem of finding the ultrasonic energy density in the crystal at the cost of obtaining information about the absolute value of quadrupole coupling between a sodium or iodine nucleus and its neighbors in a sodium iodide crystal.

T K. Yosida and T. Moriya, J. Phys. Soc. Japan 11, 33 (1956).

1059

1060 J E N N I N G S , T A N T T I L A , A N D K R A U S

Using a model8 of a cubic lattice having charges ye at the lattice points with quadrupole moments Q, we derive an expression for the net magnetization in the direction of a constant external magnetic field, Ho, of a set of sodium nuclei in equilibrium with thermal and ultrasonic lattice vibrations and find an analogous ex­pression for the iodine nuclei. We determine experi­mentally the equilibrium magnetization in the z direc­tion, the direction of Ho, of the sodium nuclei as a function of ultrasonic power transmitted to the sodium iodide crystal and repeat the experiment for iodine in the same crystal. We relate the experimental results to the theoretical expression and find a value of 7i/7Na which is not dependent on knowledge of the ultrasonic energy density, since the ultrasonic energy density is the same for measurements on both nuclei.

Part I I is devoted to the experimental method, and part I I I is devoted to results and discussion.

II. EXPERIMENTAL METHOD

The experimental method used was essentially that of Proctor and Robinson.3 A quartz crystal was glued to one end of a sodium chloride rod \ inch in diameter and one inch long. The sodium iodide crystal, \ inch in diameter and one inch long, was glued to the other end of the sodium chloride rod. The extreme end of the sodium iodide crystal was ground so that it was about 3° out of parallel with the quartz face. This was to eliminate unidirectional standing waves which would make the acoustic impedance a critical function of the frequency.

The purpose of the sodium chloride was merely to provide an extension of the ultrasonic path for con­venience. The pulsed nuclear induction was at 5 Mc and the longitudinal ultrasonic waves were introduced continuously in the [100] direction from an X-cut quartz while the attenuation measurements were being made. The Ho magnetic field was along the [001] direction of the sodium iodide crystal.

The thermal relaxation times of the iodine nuclei and the sodium nuclei were measured at 5 Mc/sec by the method given by Proctor and Robinson.3

Voltage for the quartz was derived from a Heathkit Model DX-35 transmitter driven by a General Radio Model 1001-A signal generator. The voltage across the quartz crystal was measured by a peak-reading diode voltmeter.

The procedure for making the ultrasonic attenuation measurements was as follows. With no voltage on the quartz the sodium or iodine resonance was located at 5 Mc/sec by adjusting the magnetic field until the nuclear induction signal appeared on the oscilloscope. The 10 Mc/sec voltage was then applied to the quartz and the ultrasonic frequency adjusted for maximum attenuation of the induction signal. The attenuation

was then measured as a function of the voltage on the quartz.

The ultrasonic line width was measured by monitoring the ultrasonic frequency for various values of the ultra­sonic attenuation at constant voltage on the quartz.

III. RESULTS AND DISCUSSION

Figure 1 shows the plot of the ratio (Ao/A)2 as a function of the peak voltage squared applied to the quartz for both the sodium and iodine nuclei. A 0 is the amplitude of the free-induction decay envelope appear­ing on the oscilloscope with no ultrasonic energy intro­duced and the time interval between successive radio-frequency pulses applied to the sodium iodide long compared to T\. A is the amplitude of the free-induction decay envelope with the ultrasonic excitation present. Ao and A are proportional to the magnetization in the z-direction. The straight-line function for iodine ap­pearing in Fig. 1 can be written

(Ao/Ay=l+hV*, (1)

where V is the peak voltage. In order to associate the measured slope with the

derived expression, we merely replace Mz and Mzo in Eq. (57) of the accompanying paper8 by A and Ao and find that, for dMz/dt=0,

kiV^fc. (2)

To evaluate ft we use the definition of /3 given in Eq. (56) and Eq. (31) of the accompanying paper.8 From Eq. (31) we find

Wu(i^) = Wu(-^-h) = 81 e*Qfyi2B2k2

160 fi2a65n sin25,

WU(A 729 e*QiWB2k2

(3)

-i) = ̂ (-f<=4H : sin25, 800 fiWvi

6

5

(fl4

3

2

1

V

-

*T/jO-$x -

!

/

7 ;

x ^ ^

i

/

X • •

•

i

y ^

-

-= S O D I U M

= I O D I N E

•

8 See O. Kraus and W. H. Tanttila, Phys. Rev. 109,1052 (1958), preceding paper.

0 500 1000 1500 2000

QUARTZ VOLTAGE SQUARED

FIG. 1. (Ao/A)2 is shown as a function of the square of the peak voltage applied to the X-cut quartz transducer. A 0 is the nuclear induction amplitude appearing on the oscilloscope with the ultra­sonic excitation absent. A is the attenuated amplitude in the presence of the Am = ± 2 ultrasonic excitation. The upper curve is for iodine nuclei and the lower curve for sodium nuclei.

U L T R A S O N I C A L L Y I N D U C E D S P I N T R A N S I T I O N S I N N a l 1061

where Qi is the quadrupole moment of iodine, B is the amplitude of the ultrasonic vibrations, k is 2w/\, X is the ultrasonic wavelength, a is the lattice constant, and Svi is the ultrasonic line width. The quantity 71 is the 7 of our model applicable to the iodine nuclei and should not be confused with the magnetogyric ratio 7. Similarly, for the sodium nuclei,

27 e4QNa27Na2£2£2

Wu(§*±-±) = Wu(-i^)= sin2S. (4) 16 tiW8vxa

Using the fact that B=CV, where C is a constant and V is the voltage applied to the quartz, we have, from Eq. (57) of the accompanying paper8:

ki Tn Q r W <5*>Na — = (0.240). (5) &Na ^iNa 8vi ( W 7 N a 2

We measure Tn=0.0065 second and jTiNa=5 seconds. The ratio of the slopes ki/k^&~2.2. Finally we have

(7l/7Na)2=124(^i/^Na).

From the measured ultrasonic line widths, we have

5*>Na= 4.65 kc/sec, 5z>i = 4.49 kc/sec.

This gives a final value

7i/YNa= 10.9.

The results indicate agreement with the theoretical calculations of Sternheimer4 and Das and Bersohn.6

Our 7 is to be compared with 1+7*,, where y^ is the polarization calculated by Sternheimer. He gives 7oo = 4.2 for the sodium ion, and although he does not calculate y^ for the iodide ion, one can expect to obtain a value of at least 143 which is the value of yw that he calculates for the cesium ion. Sternheimer's results would give 7i/7Na=28.

There is only qualitative agreement between our results and the results of Sternheimer. Part of the dis­crepancy between these results can be attributed to the oversimplified theoretical treatment we have used.8

I t is certainly true that the iodine nuclei have their satellite lines broadened considerably by imperfections in the crystal. This has been ignored in the theory. The effect of the broadening would be to reduce the effect of the ultrasonic waves, and if introduced properly into the theory the value of 7i/7Na would be larger than the value we arrive at.

In addition, neglecting spin conduction is not war­ranted8 in the experiment we have done. The ultrasonic vibrations generate hot spots in the spin system whose distance is certainly less than the wavelength of the ultrasonic radiation. This is due to our having the opposite faces of the sodium iodide rod 3° out of parallel. In the crystal there are not only the compressional waves introduced but also standing shear waves which, like the compressional waves, are not restricted to the x direction. The distance between the hot spots can,

under these circumstances, be small enough so that spin conduction plays an important role. However, the experimental results are virtually unchanged if the spin temperature is assumed uniform. I t is very possible that the spin conduction of the iodine nuclei is sensibly diminished by quadrupole broadening. This would partially account for our low value for the ratio 7i/7Na.

Perhaps as significant as the qualitative agreement we get with Sternheimer's results are the ultrasonic line widths that we measured. First of all, the usual inter­pretation of the small signal obtained from quadrupolar nuclei in the alkali halide crystals has been that the satellite lines are broadened beyond detection9-10 and that the only transitions detected are the J<=>—- J tran­sitions which are not perturbed in the first order by the interaction of the quadrupole moment and the strains in the crystal. Apparently this is true for a large number of iodine nuclei since we observed induction signals that were considerably less than what one might expect if all the iodine nuclei participated in the resonance. On the other hand, the iodine resonance was 1.58 gauss wide while the sodium resonance was 2 gauss wide as measured by a modified Pound-Watkins marginal oscil­lator. The usual explanation for this has been that the iodine line widths are narrower since the transitions involving satellite lines are broadened beyond detection. However, when one examines the ultrasonic line widths for the A w = ± 2 transitions one sees that the iodine line width is narrower than the sodium line width. Since the Am=.±2 transitions involve satellite lines, one would expect from the previous explanations that the ultrasonic line width of iodine would be greater than that of sodium.

From the above, one is almost forced into accepting the following explanation for the situation with regard to strains in the sodium iodide crystal. There are regions of little, if any, strain where none of the iodine levels (or sodium levels) are significantly broadened. There are apparently very small regions of intermediate strain. I t appears that, at least in the crystal we worked with (Harshaw optical quality), more nuclei were in the highly strained regions than in the slightly strained regions. This behavior has been observed in pure quadrupole transitions of chlorine in some organic chlorides.11,12

Another interesting observation was that for the sodium nuclei the ultrasonic line width increased only slightly (less than 20%) with a change from 60 v to 80 v of rf applied to the quartz, whereas, the iodine ultrasonic line width increased by a factor of 2. Ap­parently the increased temperature in the sodium

9 G. D. Watkins and R. V. Pound, Phys. Rev. 89, 658 (1953). 10 F. Reif, Phys. Rev. 100, 1597 (1955). 11 Kraus, Michel, and Tanttila, Bull. Am. Phys. Soc. Ser. II, 1,

215 (1956). In this abstract there is an error in the second to the last sentence. I t should read: "Contrary to results reported elsewhere on other compounds, the integrated intensity falls off with an increase in impurity although the resonance shows no appreciable broadening.''

12 R. E. Michel and R. D. Spence, J. Chem. Phys. 26,954 (1957).

1062 J E N N I N G S , T A N T T I L A , A N D K R A U S

iodide crystal arising from the increased energy supplied to the crystal did bring about thermal strains that broadened the iodine resonance in the region that was otherwise almost completely free of strains. The broad­ening was not because of the increased ultrasonic power as such since we permitted the temperature to come to equilibrium, whereas, measurements made at 80 v before the crystal had time to increase its tem­perature gave ultrasonic line widths for iodine that were essentially the same as those we obtained at 50 v applied to the quartz.

The ultrasonic line width is larger than the line width measured by the Pound-Watkins marginal oscillator by a factor of about three. Part of this broadening may be attributed to the quadrupole broadening of the satellite

1. INTRODUCTION

IN the preceding papers of this series1 we have con­sidered in some detail the influence on electronic

motion of the long-range correlations introduced by the Coulomb interaction between the electrons. We have seen that these correlations, in most solids, give rise to a collective excitation of the electron system as a whole, the plasmon. We have developed a technique for iso­lating the plasmon excitations by introducing a set of extra variables and carrying out a series of canonical transformations on the system Hamiltonian. We have postponed until now any inquiry into the nature of the remaining elementary excitations in solids.

Most of the problems of solid-state physics are treated within the framework of a one-electron approxi­mation. Considering how crude such an approximation is, its impressive success is very puzzling, as has been emphasized anew recently by Mott.2 Detailed effects, such as the de Haas-van Alphen effect in metals or cyclotron resonance in semiconductors, appear under-

i p . Nozieres and D. Pines, Phys. Rev. 109, 741, 762 (1958); hereafter referred to as NP I and NP II.

2 N. F. Mott, Nature 178, 1205 (1956).

lines,3 but it is also possible that part of the broadening at low levels of ultrasonic radiation is due to the band width of the ultrasonic radiation which, though intro­duced at the monochromatic frequency of 10 Mc/sec, decays to other neighboring modes by phonon-phonon collision. Measurements made on iodine nuclei in potassium iodide at low ultrasonic intensities give an ultrasonic line width of 3 kc/sec. This is somewhat less than the ultrasonic line width of iodine nuclei in sodium iodide.

ACKNOWLEDGMENTS

We wish to acknowledge the assistance of Mr. Charles Kingston, Mr. Ernest Brandt, and Mr. Richard Hoskins who assisted in the construction of apparatus.

standable only within the framework of the concept of a Fermi surface and of independent electron excitations. There is, consequently, little question that certain elementary excitations in solids bear a close formal resemblance to those postulated in a one-electron model. In this paper, we consider the present theoretical basis for the one-electron approximation. Our argu­ments are somewhat qualitative in character; we propose lines along which one may hope to make them quantitative.

The justification of the one-electron model in the free electron gas has been considered recently by Landau3

and by Gell-Mann.4 Both assume that as one switches on the charge of the electrons, the energy levels vary continuously from their free-electron values. With this assumption, one may establish a one-to-one corre­spondence between the energy levels of the systems of interacting and noninteracting electrons. I t is then possible to justify a one-particle approximation in the limit of low-energy excitations.

3 L. D. Landau, J. Exptl. Theoret. Phys. (U.S.S.R.) 30, 1058 (1956) [translation: Soviet Phys. JETP 3, 920 (1957)].

4 M. Gell-Mann, Phys. Rev. 106, 369 (1957).

P H Y S I C A L R E V I E W V O L U M E 1 0 9 , N U M B E R 4 F E B R U A R Y 1 5 , 1 9 5 8

Electron Interaction in Solids. The Nature of the Elementary Excitations

PHILIPPE NOZIERES AND DAVID PINES Palmer Physical Laboratory, Princeton University, Princeton, New Jersey

(Received May 27, 1957)

Possible elementary excitations in solids are studied with the aid of the general theoretical approach developed in the preceding papers of this series. Particular attention is paid to the basic theoretical justi­fication for the individual-particle-like elementary excitations ("effective" electrons). It is concluded that good qualitative arguments may now be given for the existence of effective electrons in solids, but that a detailed quantitative deduction has yet to be made. The presence of an energy gap is shown to be a neces­sary condition for the existence of strong spatial correlations between minority carriers in solids (excitons, conduction electron plasmons in semiconductors, etc.) and the nature of such correlated minority electron excitations is discussed. The plasmon spectrum of various solids is discussed and compared with experiment.