Anharmonic phonon shifts and widths for a centro symmetrical potential

.I. Phys. Chem. Solids

Pergamon Press 1968. Vol. 29, pp. 1805-l 82 1.

Printed in Great Britain.

ANHARMONIC PHONON SHIFTS FOR A CENTRO SYMMETRICAL

AND WIDTHS POTENTIAL

L. BOHLINt Institute of Physics, Theoretical Physics Dept., Umei University, Umei, Sweden and T. HiiGBERG AB Atomenergi, Studsvik, Nykijping, Sweden (Received 8 April 1968) Abstract-The anharmonic effects to lowest nonvanishing order, such as the shift and broadening of the one phonon neutron scattering peaks, in inert-gas solids have been calculated numerically. Approximations in the treatment have been avoided. The potential model employed is the LennardJones[6, 121 centro symmetrical potential, and interactions beyond third neighbour shells have been neglected. The calculation has been simplified by using the law of corresponding states, introducing the quantum parameter A* as the only formal difference between different elements. The results are presented in the reduced units of corresponding states. As there, to our knowledge, has been no measurements of phonon widths for inert-gas solids we have also calculated the wave number dependence of phonon widths for aluminum using the Lennard-Jones[6, 121 potential, adjusting the parameters to experimental thermal expansion data. The theoretical widths are compared with the experimental widths reported by Stedman and Nilsson. It must be emphasized that the employed potential of course is a very poor approximation for a metal potential but it might to some extent be excused as it, in this case, reasonably good reproduces the experimental phonon dispersion curves and frequency distribution.

1. INTRODUCTION

THE FORMALtheory of the anharmonic effects in solids on neutron scattering is by now well-known, being developed in different ways by many authors (see references [l-S]. There are, however, few numerical calculations for comparison with experiment. Kascheev and Krivoglaz[2] made some estimates of the order of magnitude of phonon shifts and widths but later Maradudin and Fein[4] and$4aradudin and Ambegaokar[S] made more explicit calculations for centrosymmetrical potential at the high temperature limit for nearest neighbour interactions. Cowley, R. A.[ l] and Cowley E. R. and R. A. [6] did corresponding calculations for alkali halides. In this paper we will present systematic ?Present address: AB Atomenergi, Studsvik, Nykoping, Sweden.

calculations for a centro-symmetrical potential using the law of corresponding states. In the calculations which are valid for arbitrary temperatures three nearest neighbour shells are taken into account and the derivatives of the potential are calculated without making any approximations. Although as yet there seem to be no measurements of phonon widths and shifts for inertgas solids (for dispersion curves of Kr, see reference [ 1S]), more or less systematic calculations have been made to demonstrate the temperature, wave number and frequency dependence of these quantities. Such data should afford a useful guide to measurements of temperature and wave number dependence and also assist with the comparison of different inert-gas solids in corresponding states. The equations for the widths (I(&, W)) and shifts (A(kj, o)), reformulated in accor1805

1806

L. BOHLIN

and T. H6GBERG

dance with the law of corresponding states are given (k is the wave number and w the frequency). These quantities are calculated and represented three-dimensionally with reference to a symmetry direction. This representation permits interpretation of the k-dependence of ‘half widths’ (r [kj, cu(&> ] ) , a quantity suitable for comparison with experiment, as the intersection between Ukj, w) and the dispersion curve. The k-dependence of A and I?, which is characteristic for a given direction, is also represented graphically. The strong o-dependence of T(kj, w) and A(&, o) for comparatively large w-values suggests that the temperature (T)-dependence of these quantities is sensitive to changes in k for the longitudinal branches. This is also found in the calculations and demonstrated graphically. We have studied how different kinds of phonon-phonon interaction contribute to the shift and width, for example, by separating the processes w = w1+m2 and o = w1-w,. The k- and T-dependences of the fourth order shift have been calculated. From the curves given for the k-dependence the validity of Maradudin and Fein’s approximate formulae can be ascertained. The only systematic measurements of phonon widths as a function of wave number that we are aware of are those for aluminium by Stedman and Nilsson[7]. We have done calculations for Al with a Lennard-Jones f6, 121 potential adjusting the parameters for thermal expansion. This choice of potential is justified only because it yields approximately the correct phonon spectrum and dispersion curves in the symmetry directions. It is, of course, a very poor approximation for a metal potential. The k-dependent widths as well as the dispersion curves in the symmetry directions are compared with Stedman and Nilsson’s experimental results. Although the calculated k-dependence of the widths is similar to the experimental value, there is a discrepancy especially in the magnitude of the transverse branches where the calculated values are smaller.

2. CORRESPONDtNG

STATES

AND FORMALISM

Assuming central forces we can write the potential[9]

where we sum over m and n and define (o(O)= 0. Expansion of the potential Q, in a Taylor series (to the fourth order) yields

Q,,=ca(...R”.. of the atom

.) and R is the mean position

Here

rgty...

Q...=

i3("p(~Rm-Rn~)

axpax?... ’

With Rh = R and the operator 0 = (1lR) X (dldR) we obtain explicitly for the derivatives

ANHARMONIC

PHONON

The formulae for the potential derivatives are written in full to emphasize the absence of approximations in the calculations. It is convenient to use the method of corresponding states when interpreting data and making calculations. In the interpretation one can e.g. study the importance of quantum effects and in a natural way compare different solids. The calculations are shortened because asymptotically at high temperatures different elements give in reduced form the same shifts and widths, and because the only formal difference between different elements is a qu~tum parameter A *. Below are given the formulae needed to reduce variables. The potential and its derivatives can be obtained in reduced form if an energy E and a length CTare introduced. Thus

where (*) represents the ith derivative of the potential cp. Accordingly for the dynamical matrix:

SHIFTS

X

x

AND

1807

WIDTHS

A(-k”+k:+kf)

.

I@*

-k* “.?!) j

k:k:

’ .il’

.i2

x 12{(nf+n,*+ l)[s(w*-~w:-fiJo:)

x [s(w*+wt-02*)-~(0*-01*+~)]}

(4)

where A* = fift/(m~tr~) and the unreduced f is the same as in reference[4], The quantity T * = kT/e is the reduced temperature; of represents u(kiji) and n* = [exp(h*wi*/ T*) - I]-‘. The third order shift A(“*(k*j, w) is of the same form as the width (disregarding the factor l/rr) if &functions are replaced by principal values. The fourth order shift[3] is given by &4’*

(2) x 4>*

and therefore

for frequency and time. The Fourier transforms @“p?l..._ ti.... yield

of the derivatives

> (

,r/z,*

=-

XQ,

E

k.k,

. b-1 ’

(3)

above

the

j~l;“*,ji_l

From the expressions derived widths (I) in reduced form become

>

-k* j

k* kc -kP ’ j ‘jl’

jl

* (2nl*+ I).

(5)

A* is a parameter which multiplies the Laplacian in the reduced SchrMinger equation for the system treated; its magnitude is a measure of the importance of nuclear zero point motion. In the high temperature limit the shifts and widths are independent of A* and an asymptotic limit develops, which is the same for all substances with the form of potential treated here. However, calculation shows that this limit is seldom reached before the melting point for Ne. Specifying the reduced temperature (T *) and the quantum parameter A* the free energy is minimized

1808

L. BOHLIN

and T. HaGBERG

we need not specify e and a separatively this stage in the calculations.

at

3. PLAN OF CALCULATIONS AND PWKiRAM DBSCRWTION

with regard to the lattice constant in the quasiharmonic approximation[9]. The temperature dependence of the lattice constants calculated in this way are compared in Fig. 1 with experimental values. As the discrepancy due to higher anharmonic terms is rather large,

1.6lo

I

Ne

%oo

1.550 l.540 A 03

Q2

Q3 (Kc 0.5 0.6 Q7 T*

Fig. 1. Lattice constant as a function of temperature in reduced units. The solid Lines are experimental values (referenceI 121) reduced by the constants given in Table 1. The dashed Lines are theoretical values obtained from a ~ni~z~on of the qu~i-h~rno~c~ free energy using A* equal to O-0916 for Ne, O-0298 for Ar O-0165 for Kr and O-0106 for Xe. The conversion factors are defined by r* = r[cT and T* = k-T/e, where CJ-and l can be found in Table 1.

the experimental lattice constants are employed in the calculations of quasiharmonic frequencies and of shifts and widths. Using A* = h/~(mrrr2) (and T*) as parameters tThe lattice sums taken from [ I 41.

involved

in ~~(~~/~~)

were

It should be emphasized that the quasih~moni~ approximation employed is formally identieal to the harmonic theory and is different from that used by Cowley [ 11. The computer program used is an extension of an earlier programf8], described in detail in[ 111. Reduced units and a splitting in the final summation into terms are introduced, one for each value of the quantum parameter A*. The program consists of two independent parts: one part calculates the third order a~armo~i~ shift and width and the other computes the fourth order shift. The model is a f.c.c. lattice with centro symmetrical potential. No approximation have been made_ in the potential derivatives or in the polarization vectors and the program is valid for all temperatures. In its present form it takes three shells of nearest neighbours into account, but may easily be extended to include further shells. Given the reduced potential derivatives, lattice constant temperature and wave vectors as input data the program starts with a once and for all generation of at”, @Qh and Q$$th. The summation over the Brillouin zone is made by choosing k: vectors, uniformly distributed on a cubic net in the irreducible volume of the first Brillouin zone. The corresponding dynamical matrix is diagonalized using the method of Jacobi. The eigenfrequencies ~deigenvectors are thenobt~ned. In order to cover the entire zone, the chosen k? and calculated eigenvector is transformed by symmetry operations. After each transformation the vector k* - kj+ is formed, reduced if necessary and the corresponding are eigenvectors and eigenfrequencies calculated. The representation of the Dirac delta function and the p~ncipa1 value function have been chosen as

ANHARMONIC

PHONON

The numerical error source involved in the calculations is due to the difficulty in establishing a one to one correspondence between the division of the zone and the parameter E in the delta- and principal-value functions. We have examined the influence of E vs. division, both in the case of shift and width as a function of energy, and as a function of wave vector. In Fig. 2 we have given the o-dependence of the width for a number of subdivisions of the first Brillouin zone (the shift behaves in a similar way). The heavy oscillations, due to the delta function representation combined with poor statistics in the frequency spectrum, are smoothed out as the number of wave vectors in the zone increases. With the exception, perhaps of the small kink between the last two maxima, the curve obtained by combining 17,280 wave vectors is thought to be close to the final form realised when the num-

1.41.2

Line

_______ _ -_.___.

Number of points in the zone 17280 4632 2362

SHIFTS

AND

WIDTHS

1809

ber of points approaches infinity and l (in the representation of the delta functions) approaches zero. In view of the computer time requirements the calculations were limited, employing 2368 points in the zone. The resulting curves were then smoothed using as a guide a number of curves with good statistics (17,280 points). The nfluence of c on the wave vector dependence can be seen in Fig. 3, the curves are based on a summation over 2400 points in the zone. The value E = 0.1 is then too large. One cheque on E is the sharp cut off of r(kj, w). If E is too large this is smoothed out as is the fact for l = O-1 according to our calculations. The value E = 0.01, however, gives the sharp cut off at the maximum frequency according to formulae (4). In the calculation of the fourth order shift the numerical error is very small, the lattice sum converging extremely rapidly. The relative difference between calculations with 134 and 17,280 points in the Brillouin zone is only 0*002.

Fj

ii ii i jI

Fig. 2. Third order anharmonic shift as a function of energy shown for different subdivisions of the Brillouin zone. The calculations correspond to Kr at T* = 0.4 in [OXMJ. The value of l used in the representation of the delta- and principal value functions is 0.01. (The heavy oscillations are due to the representation at the delta function combined with poor statistics in the frequency distribution.)

1810

L. BOHLIN

and T. H&BERG 4. WIDTHS

WAVE

To

3-

z B L

2-

Z= 0

1

transverse in [lfIO]

branch

.

-

.

v

v -I

.2

.L

.6

.6

1.0

k/k max Fig. 3. The curves in Fig. 3 afford examples of how the wave vector dependence of the width changes with l . (The calculation is for Kr at 79°K. (I = 5.725 A). The solid lines are obtained with c = 0.01 which is a good value for the subdivision used in this case. The number of points in the zone is approximately 2.500. The circles and triangles are from a similar calculation with l = 0.1.

AND SHIFTS

NUMBER

AS A FUNCTION

The (kj, W) dependence of the widths and shifts for different temperatures is calculated according to formulae (4) and (5) above. These widths and shifts do not contain more than one polarization direction. In reference [8] the calculations include the ‘polarization mixing’. We have used a Lennard-Jones [ 6, 12]-potential (P(r) = - 4 E x [(~/lr)~- ((r/r)‘*] with the constants given in Table 1 and frequencies in the quasiharmonic approximation defined by Leibfried (reference [9]) have been calculated. The temperature dependence of the lattice constants is derived from experimental values: Xe, Kr and Ar which have approximately the same temperature dependence can be treated simultaneously, while separate calculations were made for Neon. Dispersion curves are given in the symmetry directions at a single temperature in Table 2 as an example and for comparison with obtained shifts and widths. To demonstrate general trends we have given three dimensional figures of T* (k*j, w*)A*(k*j, w*) in the [ 1001 direction in Fig. 4(a) and 4(b) (j = 1 longitudinal branch). Calculations for the other branches show that the (R*j, w*) dependence is very nearly identical. If it is possible to define a ‘half-

Table 1. Parameters are listed which have been used in the LennardJones potential together with the reduction parameters Solid Neon

Argon

(T in A e in 1O-I4erg k/cin K-’ m in erg+/cm* .10-m A* (e/m+) I’* in

2.815[12] 0~501[12] 0.0275

340[13] 1.65[12] 0.0084

3.35 0.0916

6.63 0.0298

I O”.n-’

4.36

4.67

Type

OF

AND FREQUENCY

Krypton

Xenon

Aluminum

3.95t 2.91t 0.0047

2.577 18.6 0.0074

13.89 0.0166

21.8 0.0106

4.476 0.0142

3.45

2,92

3.63[13] 2.19[12] 0+063

25.2

tThese values on c and (T have been adjusted from (reference1 121) keeping A* and phonon frequencies constant.

Fig. 4(a).

Figs. 4(a-b). The third order shift and width as a function of energy and wave vector is given as three dimensional figures using equidistant profiles. The figures show in a simple way how the wave vector dependence is formed. The intersecting curves of lighter material are the dispersion curves. The curves are based on a calculation for Kr at T* = 0.4 in the [lOO] direction and corresponds to the longitudinal branch. Fig. 4(a) is the width and 4(b) is the shift. Note that in (b) the wave vector scale is reversed with respect to the same in (a) and that the zero level is represented by the horizontal line in the background. The conversion factors are defined by the relationships [w*] = z/(m . u”/.s) . [w] for frequency, shift and width and T* = kT/c for temperature. The wave vector is always given in units of the reciprocal (reduced) lattice constant, the stars being omitted. Numerical values of the conversion factors are given in Table 1.

[Facing page 1810]

Fie. 4(h)

ANHARMONIC

PHONON

SHIFTS

AND

1811

WIDTHS

Table 2. Dispersion curves for the inert gas crystals Ne-Xe in the symmetry directions for T* = 0.4. The phonon frequencies have been calculated with the proper lattice constant (see Fig. 1). The conversion factors from w* to w,~elnx$, are given in Table 1 Direction

w*(l ,O,O) transv. 10llg

klkmax

o*(l,l,l) long

transv.

long

o*(l,l,o) h.transv.

l.transv.

0.1

Ne Ar Kr Xe

2.979 3.257 3.337 3.384

2.320 2.400 2.540 2.569

3.131 3-377 3.490 3.538

1.567 1607 1.707 1.726

3.663 3.960 4.087 4.143

2.466 2.551 2,700 2.731

1.585 1.611 1.717 1.735

o-2

Ne Ar Kr Xe

5.932 6.478 6.635 6.728

4.578 4.750 5.013 5.070

6.202 6687 6.911 7.005

3.092 3.171 3.369 3.406

7.178 7.758 8.004 8.113

4900 5.079 5.368 5.429

3.156 3.217 3.422 3.457

0.3

Ne Ar Kr Xe

8.819 9.618 9.849 9.984

6.715 6.999 7.355 7.439

9.152 9.864 10.192 10.329

4.535 4.650 4,942 4.9%

10400 11.235 11.587 11.743

7.270 7-559 7.968 8,060

4.700 4.811 5.100 5.153

0.4

Ne Ar Kr Xe

11.587 12.617 12.913 13.088

8.677 9.093 9.505 9.615

11.913 12.834 13.258 13.435

5-859 6.008 6-386 6.456

13.195 14.246 14.685 14.881

9.542 9.962 10.466 10.588

6.202 6.385 6.738 6.809

0.5

Ne Ar Kr Xe

14.161 15.393 15.748 15.957

10.416 10.981 11.413 11.544

14.419 15.524 16.033 16.245

7.032 7.210 7.666 7.750

15.447 16.667 17.168 17.394

11.684 12.265 12.828 12.978

7646 7.924 8-321 8.410

0.6

Ne Ar Kr Xe

16.451 17.855 18.258 18.495

11.891 12.612 13.032 13.183

16.591 17.858 18.438 18.680

8.026 8.229 8.752 8.849

17.067 18.401 18.941 19.187

13665 14.407 15.017 15.195

9.017 9.409 9.827 9.934

o-7

Ne Ar Kr Xe

18.363 19.903 20.345 20.605

13.071 13.938 14.328 14.493

18.369 19.763 20.400 20666

8.820 9.043 9.620 9.726

18.006 19.400 19.951 20.206

15.454 16.380 17.002 17.205

10.294 10.814 11.236 11.360

0.8

Ne Ar Kr Xe

19.805 2 1.445 21.914 22.190

13.930 14,918 15.272 15449

19.687 21.175 21.853 22.137

9.398 9.636 10.252 10.365

18.268 19.669 20.206 20.456

17.025 18.138 18.752 18.978

1 l-455 12.109 12.519 12.660

0.9

Ne Ar Kr Xe

20.703 22.403 22.889 23.175

14.452 15.520 15.846 16.030

20.499 22.044 22.747 23.041

9-749 9.9% 10.636 10.753

18.356 19.650 20.238 20.485

17.925 19.286 19.788 20.032

12.473 13.259 13648 13.802

1.0

Ne Ar Kr Xe

21-009 22.728 23.220 23.509

14.626 15.723 16.038 16.224

20.774 22.338 23.049 23.347

9-867 10-l 17 1O-765 10.884

19.429 20.880 21441 21.705

17.121 18.414 18.865 19.094

13.322 14.225 14.520 14.757

1812

UN

dd

UC-4

dd

cv

d

*-IJi

NOCDWUN

,G-dddd

wo~wuJcu

44dddd

z v-

wuc*I

ddd

ANHARMONIC

PHONON

width’ (shift) in neutron measurements this can be obtained from the figure as the interception of the dispersion curve and r *(k*j, w *) (A*(k*j, w *)) (see Fig. 4) the interception giving the k-dependence of the widths (shifts) demonstrated in Figs. 5-7 opposite. The temperature dependence of T*(k*j, o*) is mainly determined by the temperature dependence of the frequency spectrum and the factors (n: +n$ + 1) and (n: -n;). Since the contribution to the first ‘hill’ in Fig. 4(a) is mainly from the term with the factor (nr - nz) (see insert in Fig. 8(a)) which approaches zero for small temperatures, the widths are small, especially for the transverse branches and for small k*-values when T * approaches zero. At higher temperatures the frequency spectrum is shifted towards lower frequencies and (nf-n$) becomes larger. Accordingly, changes in r*(k*j, o*), shown in Figs. 10-l 1 are exhibited. Figures 8(a) and 8(b) demonstrate how Uk*, w*) is built up. The partial sums in r* originating in different polarization directions (ij) are given separately as are the sums containing (n:+n$+ 1) and (nr -nz*) (see formula (4)). Naturally, the maximum of w in the above sums is determined by the sum of the maximum or minimum frequencies in the corresponding parts of the frequency spectra; These simple facts explains the maximum w* in T*(k*, a*) and the position of the ‘valley’ in Fig. 4(a). The minima of the ‘valley’ occurs where the rate of decrease in the width due to phonons interacting in processes such as w = w1- wz equals the rate of growth in the width due to phonons involved in processes such as o = o1 - 02. The different k-dependence of Ne is, as mentioned above, due to its larger quantum parameter. The Ne widths for the low transverse branches are lower than the other (Figs. 5-7) and in the longitudinal and high transverse branches (Figs. 6 and 7) the Ne widths are lower for low k-values and higher for high k-values than are the Ar-Kr-Xe widths. This is due to zero point motion which

1813

SHIFTS AND WIDTHS

7

2

4

6 w

6

10

12

14

16

(in1012radrcc' 1

I

2

4

6

812 -1

10

12

14

16

inX) tadset?)

Fig. 8. In (a) and (b) it is shown how different kinds of interactions contribute to the total width (¬ed with SUM in Fig. 8(a)). The curves have been evaluated for Kr at 79°K with a lattice constant of 5.725 A near the intersection point in the [i lo] direction, and belongs to the longitudinal branch. The curves denoted by jl,j, are the contributions from all processes involving phonons of polarizations j, and j,. At the insert in the upper left of Fig. 8(a) we have separated the two terms of formula (4). Curve A is the (nf -II;) term and B is the (n: + n$ + 1) term.

is still dominant in Ne at this temperature (T* = O-4). (n(6) = O-02 for Ne,n(r3) = 2 for Xe). This leads to the depression of the first hill (from the (n, - n&term, see Fig. 8(a)) and the raising of the remainder (the (n, + n2 + 1) term) by comparison with Ar-Xe. (To accord with this it is essential to observe that the quantum parameter multiplies the total sums and that A& s A&X 10). The explanation then follows from the model (Fig. 4(a)) for the formation of the k-dependent width.

L. BOHLIN

1814

and T. HGGBERG

The k*-dependence of the shifts and widths in reduced units in the symmetry directions is given in Figs. 5-7 and 9 at T” = O-4 for the inert gas solids. The shift from the fourth order term in the potential behaves almost like the frequency (Fig. 9) (see also reference [8]). The third order shifts and widths are obtained as intersections of the o*-dependent shifts and widths with the dispersion

0.08 0.07

z’b. 1

-I

k,.a Fig. 9. An example is given of the fourth order shift as a function of wave vector. The curves have been plotted as shift divided by corresponding frequency and are calculated for Ar at T* = O-4 in the [ 1001 direction. The conversion factors are defined by the relationships [o*] = v’(m-~~/r).[w] for frequency, shift and width and T* = kT/e for temperature. The wave vector is always given in units of the reciprocal (reduced) lattice constant, the stars being omitted. Numerical values of the conversion factors are given in Table I.

curve, as examplified in Fig. 4(a) and (b) for the [loo] direction. For large k*-values and the longitudinal branches it is doubtful if phonon data can be interpreted in terms of ‘widths’ and ‘shifts’ because of the distortion of the symmetry of the phonon peaks produced by large and strongly @*dependent shifts and widths (see reference@] Fig. 13, where also the influence of the polarization mixing is demonstrated. See also e.g. Fig. 5). On the other hand the transverse widths are very small and in the [lOO] direction even decrease for large k*-values (also demonstrated in Fig. 5).

Since the widths have a characteristic k*dependence (Fig. 5-7) which depends only on the third derivative of the potential to the lowest order in the anharmonicity, their becomes highly desirable. measurement The anharmonic shifts are more difficult to analyse because they consist of three parts which are ‘added’ to the dispersion curve. The shifts and widths of Xe, Kr and Ar are indistinguishable in terms of the units used and the actual reduced temperature (Fig. 5-7). On the other hand, neon which has a larger quantum parameter than the other rare gas solids, behaves differently in k-dependence, a fact which should be experimentally demonstrable. It would also be interesting to measure and compare the phonon shifts and widths for the same reduced units for different rare gas solids. Because of the rapid change in I-*(k*j, o*) and A*(k*j, o*) with w* for some w*values (Fig. 4(a) and 4(b)) higher order approximations may be necessary for large k-values. 5. TEMPERATUREDEPENDENCE In Figs. 10 and 11 an example of the temperature dependence of A*(k*j, w *) and I’*(k*j, w*) is given, showing that the two main effects are that these quantities increase with temperature and are ‘shifted’ in w*. If half widths and shifts are defined from r(k*j, o*) and A(k*j, w*) prediction of the temperature dependence is clearly impossible without detailed calculation. In Figs. 12-18 results from such calculations are presented for different k*-values and polarization directions. When T* -+ ~0formula (4) gives the same limit for all substances with the potential treated here. The temperature when the asymptotic form is reached in practice is clearly different for different substances and k*-values as is demonstrated in Figs. 12-18. For Ne this limit seldom occurs before the melting point (for an exception see top of Fig. 18, transverse branch). It might be pos-

ANHARMONIC

PHONON

SHIFTS

AND

WIDTHS

1815

1816

L. BOHLIN

and T. HijGBERG

sible to draw conclusions about the quantum parameter A* or the potential by measuring widths (shifts) at low and asymptotic temperatures. The sensitiveness of the third order longitudinal widths and shifts for large k*values to changes in temperature is also demonstrated in Figs. 12-l 8. In this connection compare for example the ‘temperature coeffi-

_ k,=1.0 1.2 1.0 0.8 0.6 ; 0.4 *LX' 0.2

0.4

1

k,-0.1875

0.2

0.2

:e

0.7 0.2 0.1

*t 0.2

0.1

0.2

0:3

0.4

0.5

0.6

0.7

*Xi I-

T"

0.6

Fig. 12.

0.4 0.2

0.6 0.4

Fig. 14.

0.2

1.0 k,=0.5

0.8

/

J Ne

0.6

*

0.6 c

0.2

*a Q ' 1.0 0.8 0.6 0.4 0.2

0.1

0.2

o.3T*0.4

Fig. 13.

05

0.6

0.7

Fig. 15.

ANHARMONIC

PHONON

SHIFTS

1817

AND WIDTHS

2.0 1.8

J

k, =O.E

1.6 1.4 0.6

*- 1.2 t *$1.0 , 0.8 0.6

1 k, =0.5

.

.

/

0.4 0.2

0.8 0.l

0.2

0.3

0.4 TX

65

0.6

0.7

0.6

Fig. 16.

Fig. 18.

0.2 *t *L-’ 0.6

Pig, 17.

cients’ of the longitudinal and transverse branches in Fig. 12 for k, = 1.0. It would be interesting to demonstrate ex~~menta~y this striking behavio~r in temperature dependence. The explanation is that the transverse frequency lies in the ‘valley’ in the whole temperature interval, while the longitudinal frequency climbs the ‘mountain side’ (Fig. 4(a) and IO). For tbe longitudinal branch and large k*-values the dispersion curve lies on the steep slope of r*(k*j, CO*) (Fig. 4(a)),

Figs. 12-18. The temperature dependence of the shift and width for three wave vectors in each of the symmetry directions. Figures 12 and 13 are in the [ 1001 direction, 14, 15 and 16 in [IlO] and 17 and 18 in [ill]. The solid lines correspond to the longitudinal branch, the dashed lines to the high transverse branch and the dotted tines to the low transverse branch. The notations Ne-Xe and Ar-Xe are ~breviations for respectively Ne, Ar, Kr Xe and Ar, Kr, Xe. The conversion factors are defined, by the relationships [~a*]= t/(m . a*/~) * [a] for frequency shift and width and T* = kT/e for temperature. The wave vector is always given in units of the reciprocal (reduced) lattice constant, the stars being omitted. Numerical values of the conversion factors are given in Table 1.

which implies that the number of interacting phonons is changing rapidly. The shape of the curves in Figs. 12- 18 indicates the possibility for extrapolating zero point widths and shifts for the longitudinal branch (j = 1) using measurements for not too small k*-values, at least for Neon. It might also be possible to compare reduced zero point shifts for Neon and Argon ex~~ment~ in this way (see

1818

L. BOHLIN

and T. HGGBERG

e.g. Fig. 16 in this respect). The temperature dependence of the fourth order shift is examplified in Fig. 19. 5.

,

I

T* Fig. 19. The temperature dependence of the fourth order shift at the zone boundary in the [ 1 IO] direction. The curves have been drawn for Ar. The conversion factors are defined by the relationships [o*] = q(m * rP’/e)+[o] for frequency, shift and width and T* = kTfe for temperature. The wave vector is always given in units of the reciprocal (reduced) Iattice constant, the stars being omitted. Numerical values of the conversion factors are given in Table 1. k,.a Ul,Ol -high transv. 0.375 0.75 7'

-

The experimentally measured widths given up to now are not accurate enough for a detailed comparison with theory. To our knowledge there are no measurements of widths on rare gas solids, but Stedman and Nilsson [71 have published systematic measurements on the k-dependence of widths in the symmetry directions for Aluminum. The Lennard-Jones [6,12] potential used in our present calculations is clearly a very poor approximation for a metal potential, but since it yields approximately the right phonon spectrum and dispersion curves (Figs. 20-21) it seemed worthwhile to make a comparison with Stedman and Nilsson’s measurements. (Calculations with a more realistic potential are in progress). The parameters in the [6,12] potential (Table 1) are adjusted to experimental values for thermal expansion. In Figs. 22-27 our numericai results are 0 0.5

k,ra ~1.1.110.25

0

0.75 0.375 k,,a tl.l,Ol long.andlmvtransv.-

Fig. 20. Phonon frequencies in the symmetry directions for Al at 300°K. For experimental points given by Stedman, circles representing longitudinal phonons, triangles, high transverse phonons and squares, low transverse phonons, (reference[7]). The solid lines are theoretical quasi-harmonic phonon frequencies obtained with a Lennard-Jones[6,12] potential adjusting the parameters to the zero temperature lattice constant and thermal expansion.

ANHA~MONIC

\

1y-y I

PHONON

2

3

C

5

6

J 7

(in &adscE’,

SHIFTS AND WIDTHS

1819

The agreement might be somewhat better for a more long-range potential. It is inconceivable, however, that widths as large as the experimental values for the transverse branches can be obtained without radically changing the phonon-spectrum. This, in turn, is reasonably reproduced with the [6,12] potential (Fig. 2 1). In this connection a more detailed experimental investigation of the

Fig. 21. Phonon frequency dis~bution for Al at 800K obtained with the ~nn~d-Jones potential from a sample of 5740 wave vectors in the irreducibie volume of the Briliouin zone. No interpolation has been made, the dynamical matrix has been diagonalized for each wave

l~i~inal

bFanch 1

0.3 *

vector. (The variable on the horizontal axis is frequency.) 0.2 .

Longitudinalbranch

i

'0 t

0.1 .

P 0Z

0.3

- high transverse branch T

0.2

0.4

06 k. ,a

0.8

l&_&..j

1.0

O.l

Fig. 22.

given together with the experimental values. Although it may seem overambitious to give results in all symmetry directions for this poor potential model we have done so because it is the tendencies in the change in k-dependence with direction which are significant. For comparison, we also give the widths from the electron-phonon interaction calculated by Bjorkman et aL[lO] (and R. Johnson[ 151) assuming them to be same at 80” and at 300°K. For convenience we have summed the widths although they are calculated with different potentials and the result is therefore difficult to interpret. Although the calculated kdependence of the widths agrees in shape roughly with the ex~~rnent~ values our values are, in general, smaller particularly for the transverse branches and for small k-values.

_

035 030 O.L5 0.60 0.75 k,.a

Fig. 23,

to~itudina~ branch

*

;p

0.3

P 9

0.2 -

E %

0.1 .

‘3

transverse branch

I _

.

1

0.2

=

0.1 . 0.1 0.2

0.3 k, .a

Fig. 24.

0.4

0.5

1820

L. BOHLIN longitudinal

Tf$ P 0 e z ‘5 5? s $ a

and T. HGGBERG

branch

longitudinal

T

0.3

branch

0.2 0.1

transverse

0.3

branch

transverse

0.2 0.1

l&_+_&&J 0.2

0.4

0.6

0.8

1.0

k, .a

1 longitudinal

0.3

1

70 m

0.2

-z L 00

01

branch

i

i&$f$HJg :if

:

___‘----;-c2_~ . . l .

high transverse

l

branch

:

low

transverse

1

;;f:r;LJ 0.1

0.2

0.3 k, ,a

04

0.5

Fig. 27,

Fig. 25. 0.L

branch

branch

015 030 O.L5 0.60

075

k,.a

Fig. 26.

temperature dependence should prove valuable. The width as a function of temperature at some k-values would give valuable information about the anharmonicity although the absolute value of the widths is poorly determined at present. Although we appear to have reproduced roughly the same temperature dependence, a significant comparison of the two investigations requires more detailed experimental results.

Figs. 22-27. The phonon line-width in Al as a function of wave vector in the symmetry directions at 80” and 300°K. The open circles are experimental phonon widths reported by Stedman[7]. The filled circles are our calculated phonon widths originating from the phononphonon interaction. The contribution from electronphonon interaction to the phonon width as given by Bjiirkman et al. [lo] (and R. Johnson[lS]) is drawn with dashed lines. The solid lines are the sum of the contributions from phonon-phonon and electron-phonon interactions to the line widths. The electron-phonon interaction width was taken to be the same at 80” and 300°K. Figures 22, 23 and 24 are in order respectively in the [ 1001, [ 1101 and [ 1111 directions at 300°K. Figures 2.5, 26 and 27 the same at 80°K.

Singularities in r(kj - o) can be found by methods similar to those used by Maradudin and Peretti [ 161 in studying frequency spectra. This gives, for a three-dimensional lattice, singularities in the dependence of P on o of the form (w--w(k&) -~(k~---kj,))~‘~. In our calculations we have also obtained indications of wave number dependent singularities in the derivatives as can be seen from Figs. 4(a) and 22-27 although they are smeared out because of our representation of the Dirac &function. Stedman and Nilsson’s measurements also give sudden changes in the width at large k-values as can be expected on the basis of theory. However, lattice imperfections will also give singularities in the width of this type (reference[ 17]), but then they occur where the the singularities of the frequency-spectrum appears and are wavenumber and temperature independent.

ANHARMONIC

PHONON

Acknowledgements-The

authors are indebted to Professors A. Claesson. A. Sjiilander and I. Wallerforvaluable discussions and constructive criticism and to Dr R. Johnson for allowing us to use his results prior to its publication. One of us (L.B.) wishes to thank AB Atomenergi, Studsvik, Sweden for a fellowship.

REFERENCES COWLEY R. A..Adu. Phvs. 12.421 (1963). V. N. and KRIVOGLAi

:: KASCHEEV

Soviet Phvs. solid St. 3. 1107 ( 196 3. K~KKEDEE J. J. J. Phys&28,374

M. A.,

1).

(1962). A. A. and FEIN A. E., Phys. Rev. 4. MARADUDIN 128.2589 (1962). A. A. and AMBEGAOKAR V., 5. MARADUDIN Phys. Rev. 135, A IO7 1 ( 1964). 6. COWLEY E. R. and COWLEY R. A., Proc. R. Sot. A287,259 (1965). 7. STEDMAN R. and NILSSON G., Phys. Rev. 145. 492

( 1966).

SHIFTS

AND WIDTHS

1821

8. HOGBERG T., BOHLIN L. and EBBSJO I., AB Atomenergi, Stockholm, Sweden 1967. (AE274). 9. LEIBFRIED G. and LUDWIG W., Solid State Physics (Edited by F. Seitz and D. Tumbull) Vol. 12, p. 276. Academic Press, New York (1961). IO. BJORKMAN G., LUNDQVIST B. I. P. and SJOLANDER A., Phys. Rev. 159.55 I (1967). 11. EBBSJO I., BOHLIN L. and HOGBERG T., AE-report. To be published. 12. POLLACK G. L., Rev. Mpd. Phys. 36. 748 (1964). 13. GUGGENHEIM E. A. and McGLASHAN M. L., Mol. Phys. 3,563 ( 1960). 14. KIHARA P. and KOBA S. J. phys. Sot. Japan 7, 348 (1952).

15. JOHNSON R., To be published. 16. MARADUDIN A. and PERETTI

J.. C. r. hebd. SPanc. Acad. Sci. Paris 247.23 IO (I 958). 17. KRIVOGLAZ M. A.. Societ Phys. JETP 13, 397(1961). 18. DANIELS W. B. et al., Phys. Rev. Lett. 18, 548 (1967).