Harmonics of phonons in Al

Physica 120B (1983) 286-290 North-Holland Publishing Company

§3.3. LATTICE DYNAMICS

H A R M O N I C S O F P H O N O N S IN Ai

Y. N A K A I and Y. T S U N O D A Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan

According to the Toda theory for a one dimensional exponential lattice, the lattice vibrations have higher harmonics without damping in time. This is contrasted to phonons with a finite life-time obtained by the perturbation theory in the pseudoharmonic approximation. A search for the higher harmonics in the phonons of AI was carried out by means of inelastic scattering of neutrons. Neutron groups observed at elevated temperatures are explained as the second harmonic wave expected for the Toda lattice, although other explanations cannot be excluded.

1. Introduction

The interatomic potential in a crystal can be expanded in a power series in x, the change in instantaneous pair separation of atoms with respect to the mean separation, as ¢~r = IJ)~)-}- ¢~X 2 -0- ~ X 3 -[- (~]X 4 -t-" • " .

(1)

In the harmonic approximation the real potential &r is replaced by a harmonic potential &h = ~/,0 h + &)x 2 '

(2)

where 4, h is assumed to be &~ or the averaged value of tfl~ in some sense. At relatively low temperatures, the harmonic approximation usually explains the lattice vibration of real crystals well. At elevated temperatures, however, the higher order terms of eq. (1), the anharmonic terms, cannot be ignored. In the quasiharmonic approximation to the anharmonic effects, only the change of 4,2h due to the lattice expansion has been considered [1]. There is also the pseudoharmonic approximation in which the frequency shift and the finite lifetime of phonons expressed by sinusoidal waves have been evaluated by considering the third and fourth order terms of eq. (1) on the basis of the perturbation theory [1]. In another way of describing the anharmonic phonons, the real potential &r is expressed as the

summation of an anharmonic potential which can be solved analytically, and small correction terms. The exact solutions for the appropriate potential would explain the real lattice vibration at elevated temperatures, even though the small correction terms are ignored. Here we propose the potential of an exponential lattice (Toda lattice) [2] as a solvable potential and we examined whether the characteristic feature of the Toda lattice can be found or not in the lattice vibrations of A1. Aluminum is a good candidate for the present study because of the large anharmonic effects even at moderately high temperatures, expected from the large thermal expansion and the simple dispersion relation for the phonons. Moreover, AI has convenient properties for neutron scattering experiments; the small incoherent and absorption cross sections and easy growth of the large single crystals.

2. H i g h e r h a r m o n i c s of lattice vibrations in a 1D Toda lattice

A periodic solution (cnoidal wave solution) of a one dimensional exponential lattice (Toda lattice) in which the interatomic potential is given as

cb e = ( a / b ) e bx,+ a x j ,

0378-4363/83/0000-0000/$03.00 © 1983 North-Holland and Yamada Science Foundation

(3)

Y. N a k a i and Y. Tsunoda / Harmonics of phonons in A l

can be obtained as follows [2], e -b*/- 1 = (2Kv)2/(ab/m) x [dn2{(K/Tr)(qR/- tot)} - E/K] n cos

= to2/(ab/m) n=l

(4)

n ( q R j - tot)

sinh(nTrK'/K)

'

(5)

where xj is the deviation of the interatomic distance between the jth and Q"+ 1)th atoms from the equilibrium value at zero temperature, (R/+~-Rj). H e r e a f t e r we neglect the quantum mechanical effect, because only the high temperature behavior is considered. Besides the usual notations of m, q, v and to, dn u, E and K ( K ' ) are Jacobi's elliptic dn-function, the complete elliptical integrals of the 2nd and 1st kind with modulus k(k'), respectively. If we expand the left-hand side of eq. (4) for small x/, the instantaneous displacement of the jth atom, u/, given by Y.{2] xt, can be expanded as u / = U~ sin(qRj - wt) + U2 sin 2 ( q R / - tot) + • • •,

(6) where 1 (2Try)2 1 f---., Ux = 2b sin qd/2 ab/m sinh(TrK'/K)

(7)

observe inelastic scattering peaks of neutrons at (q, toq), (2q, 2toq), (3q, 3toq) . . . . in q - t o space. This solution in the limit of small k closely resembles sinusoidal phonons with a small amplitude of vibration. On the other hand, the solution in the limit of k to l expresses a periodic array of solitons. The deformed sinusoidal lattice wave without damping in time can be expected for the T o d a lattice, as expressed by eq. (6), while the sinusoidal phonon with damping was expected on the pseudoharmonic approximation.

3. Experimental results and discussions The m e a s u r e m e n t s of inelastic scattering of neutrons for Al single crystals were p e r f o r m e d at 300 to 770 K by using a triple axis spectrometer T U N S installed at JRR-2, J A E R I . The longitudinal phonon modes were observed because these would be m o r e reasonably approximated by a 1D T o d a lattice than the transverse modes. The [110]L branch is convenient to separate the weak higher harmonic peak of (2q, 2toq) from the strong fundamental phonon peak of (2q, to2q), as the dispersion curve has a m a x i m u m at about 0 . 6 q m a x [3], as seen in the inset of fig. I. Hereafter, for convenience, the c o m p o n e n t s of q/qmax along the [ll0] direction, ~ and 2sr, are assigned for the fundamental and 2nd harmonic peaks,

1 (2Try)2 U2 = 2b sin qd ab/rn

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It is noticeable that the cnoidal wave solution has higher harmonics with the wavevector and angular frequency, nq and ntoq (n = 1, 2, 3 . . . . ). As the ratio U2/U~ was roughly estimated to be a few tenths, as described below, we can directly

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60

Fig. I. N e u t r o n g r o u p s o f t h e f u n d a m e n t a l p h o n o n s at sr = 0.35 a n d ~ ' = 0.7 a n d t h e 2 n d h a r m o n i c p e a k at 2sr = 0.7. S p u r i o u s p e a k is s h a d e d . T h e b r o k e n c u r v e in t h e i n s e t s h o w s the positions where the 2nd harmonic peaks are observed. The positions of the constant Q scan are shown by the d o t t e d - b r o k e n lines.

Y. N a k a i and Y. Tsunoda I Harmonics of phonons in A I

288

respectively. T h e n e u t r o n groups of the fund a m e n t a l p h o n o n peaks at (" = 0.35 and 0.7 and of the 2rid h a r m o n i c p e a k at 2(" : 0.7 are shown in fig. I. T h e shaded peak is spurious, as discussed below. A s the 2nd h a r m o n i c peak at 26 : 0.7 is relatively strong in the intensity and is well separated from the tail of the f u n d a m e n t a l p h o n o n peak, m a n y m e a s u r e m e n t s except a few at 2(' = 0.6 and 0.8 were d o n e at 2(' = 0.7. As the p e a k positions at 2(' = 0.6 and 0.7, as seen in fig. 2, are located on the expected positions from 900~

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~6(,0 ( meV ) Fig. 2. Phonon groups at 2 ( ' = 0.7 observed at different ex perimental conditions, together with the ones at 2 ( = 0.6. The energy of incident neutrons, Ei, of 40 or 35 m e V and the specimen with large or small size (~1 or #2, respectively) were used on the scattering plane of (110) or (001).

(2q, 2wq), the 2nd harmonics would also have the dispersion curve displayed as the b r o k e n curve of the inset of fig. 1. In o r d e r to distinguish spurious peaks from the higher harmonic peaks, measurements at different experimental conditions were performed, as seen in fig. 2. The spurious peaks at 2 ( = 0.7, shaded in fig. 2, d e p e n d on the energy of incident neutrons, /7i, and the crystal setting. These peaks m a y be due to the Bragg reflections of the A/2 or A/3 c o m p o n e n t of the incident and scattered neutrons. For an example, the spurious peak in the case of/Tg = 35 meV, would be due to the successive scattering process of A/3(006) reflection in the m o n o c h r o m a t o r crystal, 480 in the specimen and A/2(004) in the analyzer crystal. As A1 has no incoherent scattering, the higher o r d e r c o n t a m i n a t i o n takes place only when the Bragg condition is accidentally satisfied. In o r d e r to subtract the double scattering contributions by p h o n o n s , we estimated it as follows. T h e intensity as a function of energy gain tim at the (400) reciprocal lattice point was compared with the numerical calculations of the double scattering cross sections, as seen in fig. 3a, in o r d e r to obtain the instrumental constant. T h e calculation at (2.7, 2.7, 0) showed a m o n o t o n o u s decrease of the double scattering contribution with increasing h~o as well as the one at (400). T h e double scattering intensity at (2.7, 2.7, 0) estimated with the instrumental constant is c o m p a r e d with the observed intensity at 2(" = 0.7 at 640 and 300 K, as seen in fig. 3b. A f t e r the subtraction of the double scattering contributions, the higher h a r m o n i c peak is f o u n d only at elevated t e m p e r a t u r e s just b e y o n d the statistical uncertainty, as seen in fig. 4, although the tail of the f u n d a m e n t a l p h o n o n peak at (" = 0.7 and the t e m p e r a t u r e i n d e p e n d e n t spurious peaks are also seen. T h e integrated intensity of the 2rid h a r m o n i c peak at 2(' = 0.7 can be roughly evaluated by the s m o o t h - e x t r a p o l a t i o n of the tail of the fundamental p h o n o n at ( ' = 0.7. T h e ratio of the amplitude of the 2rid h a r m o n i c vibration at 2(' = 0.7 and t e m p e r a t u r e T, Uz(T), to that of the f u n d a m e n t a l p h o n o n at (' = 0.35 and T = 650 K, U~(650), was o b t a i n e d as a function of T in fig. 5,

I"< Nakai and Y. Tsunoda / Harmonics of phonons in A I (o)

(400) T= 640 K

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Fig. 3. Neutron groups at (a) (400) and (b) (2.7, 2.7, 0). (a) Multiple scattering contribution compared with the numerical calculations of double scattering contributions by phonons (solid curve) within the constant factor. (b) Shaded parts are the double scattering contributions estimated from the numerical calculation and the measurements at (400), as seen in (a). Solid curves are as a guide to the eye. The temperature independent spurious peaks are also seen.

and can be c o m p a r e d with the rough estimation based on the T o d a lattice model, as follows. The p a r a m e t e r s a and b of the potential of eq. (3),

289

were determined by using the experimental values of the thermal expansion coefficient and of the m a x i m u m frequency of phonons. As the amplitude at each T, U~(T), can be roughly evaluated on the harmonic approximation with B o s e - E i n s t e i n distribution, the values of U2(T) were calculated by using eqs. (7) and (8). The estimated ratio, U2(T)/Ul(650), as a function of T is shown by the solid curve in fig. 5, and is c o m p a r e d with the observed ones after the corrections for the D e b y e - W a l l e r factors and of the reflectivity of the analyzer crystal. In spite of the crude approximations, the agreement is fair, as seen in fig. 5. Recently, the scattering law including the higher harmonic peaks was calculated numerically for the 1D T o d a lattice by Diederich [4]. However, as his calculation is for extremely high temperatures, the quantitative comparison with the present experiments is difficult. A n o t h e r explanation of the fine structure of the scattering law, especially of the tail parts of the phonon peaks, can be obtained by considering the interference effect of one-phonon scattering and multi-phonon scattering, which is also an anharmonic effect. The interference effect may explain the present results, as in the case of Ne [5]. Nevertheless, the semi-quantitative agreement in fig. 5 and the question whether the present peak is the 2nd or 3rd harmonics,

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Fig. 4. Second harmonic peaks after the subtraction of the double scattering contributions at 2~" = 0.7, wi'th E~ = 35 meV and the #1 specimen.

Fig. 5. Amplitude of the function of temperature, fundamental phonon at experimental conditions. tion (see text).

2nd harmonic lattice vibrations as a normalized by the amplitude of the 650 K, observed on the different Solid curve shows the rough estima-

290

Y. N a k a i and Y. Tsunoda / Harmonics of phonons in A I

e n c o u r a g e us t o c o n t i n u e t h e p r e s e n t s t u d y in spite of the weak scattering intensity.

Acknowledgements T h e a u t h o r s e x p r e s s t h e i r s i n c e r e t h a n k s to Professor Kunitomi f o r his e n c o u r a g e m e n t throughout the present study.

References [1] P. Brfiesch, Phonons: Theory and Experiments I. (Springer Verlag, Berlin, 1982) p. 152. [2] M. Toda, J. Phys. Soc. Jpn 22 (1%7) 431; 23 (1967) 501. [3] R. Stedman and G. Nilsson, Phys. Rev. 145 (1966) 492. [4] S. Diederich, Phys. Rev. B24 (1981) 3186, 3193. [5] W.M. Collins and H.R. Glyde, Phys. Rev. Bl7 (1978) 2766.