Abstract. A technique based on the use of Fourier analysis is suggested for the computation of the frequency distribution function of a crystal. The method has.
ROC. PHYS. SOC., 1 9 6 5 , VOL. 8 6
The Fourier expansion method for computation of the frequency distribution function of crystals K. G. AGGARWAL, J. MAHANTY and V. K. TEWARY Department of Physics, Indian Insbtute of Technology, Kanpur, India
MS. received 29th April 1965, in revised form 17th August 1965 Abstract. A technique based on the use of Fourier analysis is suggested for the computation of the frequency distribution function of a crystal. The method has been applied to some two- and three-dimensionallattice models and it is shown that, compared with the usual methods, the present method saves time and labour with the added advantage of giving an analytic expression for the distribution function.
1. Introduction The object of this paper is methodological-that of developing a technique for obtaining analytic expressions which are good approximations to the frequency distribution function of a crystal lattice. With the advent of high speed electronic computers considerable progress has been made in computation of the frequency distribution functions of crystals. Mention may be made of the work of Gilat and Dolling (1964) whose method gives almost the exact spectrum, and that of Isenberg (1963) who has considerably extended the method of moments originally proposed by Montroll (1942, 1943). The latter method leads to an analytic expression for the frequency distribution function in the form of a series of Legendre polynomials, whereas the former does not admit of such a representation of the distribution function, so that the greatly increased accuracy of the former method is offset by the d8iculty in using the data for computation of the properties of the crystal, such as thermodynamic properties, which can only be computed numerically. The Legendre polynomial series for the frequency distribution function is poorly convergent and, though the convergence can be improved by special techniques such as that suggested by Lax and Lebowitz (1954), the computational problems associated with improving the convergence are formidable because of the need to know the location of the critical points. Perhaps the only reason Legendre polynomialshave been used traditionally is because the coefficientsof the expansion can be evaluated in terms of the moments. Use of other orthonormal sets, such as sine functions, does not offer this advantage. However, this difficulty is a minor one in this age of high speed computers. Our approach will be based on the use of sine functions as the basic functions for the expansion of the distribution function. There are natural reasons why sine functions should be used for the expansion of the frequencydistribution function of a three-dimensional crystal. The distribution function starts and ends with zero value and is usually continuous in between, its only peculiarities being some kinks leading to discontinuities in the derivative (Maradudin et al. 1963). The Fourier series of such a function is highly convergent (Lamb 1960) and, therefore, leads to a more accurate and convenient representation. As in the case of the Legendre Polynomial representation, the Fourier series representation of the distribution function 1225
K. G. Aggarwal, J. Mahanty and V. K. Tewary
1226
provides a means of obtaining a series expansion of any additive function of the normal mode frequencies of the lattice, such as the thermodynamic properties.
2. Fourier expansion of the distribution h c t i o n We-shall consider here the distribution function G(x) of the quantity x =
w2/w,,2
in one of the branches of the acoustic band. The results can be extended to other branches in the acoustic and optical bands in an obvious manner. The frequency distribution function g(o) is related to G(x) by the equation g(w) = ~oJ,,,-~x~’~G(x).
The dispersion relation for the branch under consideration can be written as
x
=
x(k)
where ~ ( k gives ) the functional dependence of x on the wave number k. The distribution function can be expanded in a Fourier series in the form m
G(x) =
2 A,sin(nrx) n=O
so that
A, = 2
r:
2J
G(x) sin(nxx) dx = -
v*
**
sin{nrx(k)) d3k
where V* is the volume of the first Brillouin zone. The evaluation of-the coefficients A , thus amounts to doing the integral in equation (4) using the explicit form of the dispersion relation in equation (2). In some cases the integral can be done analytically, and in all cases it can be evaluated numerically. Equation (4),when substituted in equation (3), gives the required representation for G(x) *
3. An example
As an example of application of the method to a case where the integral for A, can be evaluated analytically, let us consider an m-dimensional simple cubic lattice with nearest-neighbour central and non-central force constants being equal and each atom having only one degree of freedom (Montroll 1956). The dispersion relation in this case can be written as x(k)
=
1 -(m - cos K1 - cos k, 2m
... - cos K,)
(5)
where ki is the ith component of the m-dimensional vector k. Substituting this in equation (4),and noting that V* = (2r)”, we have
A,
=
“JJ;.
..fsin
(24
2m
.
(m - cos k1 - cos K , .. - cos k,)
dk, dk2 ... dkm. (6)
Obviously A , will be non-vanishing only for odd values of n. The non-vanishing efficients are
n” [-2n1 f
Azn+l = 2( - 1)”
,=1
2n 0
\(2n+1)7T cos,2m cos k,
(2n+ 1)r
CO-
Computation of the frequency distributionfunction of crystals
1227
The normalized form of G(x) is given by
For the one- and two-dimensional cases, i.e. for m = 1 and 2, the convergence of the series in equation (8) is expected to be poor owing to the occurrence of infinities in G(x). In the three-dimensional case, taking 30 terms in the series gives a distribution function which agrees with the correct value to within 0.6% (table 1). In the two-dimensional case the agreement is surprisingly good when the same number of terms are taken, except at x = 0-5 and at x = 0.0 and 1.0, considering the fact that no attempt has been made to handle the logarithmic singularity at x = 0.5 and non-zero values of G(x) at x = 0.0 and 1.0 in the manner suggested by Lax and Lebowitz (1954). It may be noted in this case that, at x = 0-0 and 1.0, equation (3) would lead to zero values of G(x). While this is unrealistic for the two-dimensional case, it is possible to extrapolate to x = 0.0 and 1.0 from the neighbouring points to get values close to the true ones. Table 1 gives the true and computed values of G(x) for the two- and three-dhensiond cases. Table 1 X
0.00
0.05 0 10
0.15 0.20 0.25 0 30 0.35 0.40 0.45 0.50
G(x)-two-dimensional True Computed 0 6366 0 6706 0.7096 0 7549 0 8086 0 8740 0 9560 10650 1 2224 14977 *tY
0~0000 0.7117 0.6988 0.7608 0 8160 0.8685 0 9760 10537 1.2445 1:4887 2.6689
G(x)-three-dimensional True Computed
0~0000 0.2525 0.3917 0.5302 0.6871 0.8847 1.1808 1.7306 1.7184 1.7121 1,7100
0 0000 0.2569 0.3934 0.5296 0.6912 0.8807 1.1878 1.7452 1.7201 1.7135 1.7137
It will be shown later that one can get a very satisfactory representation for the G(x) of a laminar crystal by modifying this method in the manner suggested by Lax and Lebowitz ( 1954).
4. The distribution function for face-centred cubic and body-centred cubic lattices The example of the previous section illustrates the usefulness of the method. We now proceed to apply the method to compute the distribution functions of some monatomic face- and body-centred cubic lattice models. For the face-centred cubic lattice we assume a central force model (de Launay 1956). For the body-centred cubic lattice we assume a model (Henclricks et al. 1963) in which each atom interacts with its nearest neighbours through central and non-central forces. /3 and D are the nearest-neighbour non-central and next-neighbour central force constants expressed in units of the nearestneighbour central force constant a. In terms of the functions C, = COS A, and S, = sin k, where k, is the jth component of k,the secular equations can be written as follows:
K. G. Aggarwal, J. Mahanty and V. K. Tewary
1228
face-centred cubic lattice
body-centred &c
lattice
-fl)s1c2s3
I
where h2
= [Mw2r;E
3Mw
/
for face-centred cubic lattice for body-centred cubic lattice
(11)
M being the mass of the atom in the lattice. Solving the cubic equation for h2, we get the dispersion relation for each frequency branch. For each branch G(x) can be evaluated by using equations ( 3 ) and (4),taking care to ensure that the integration in equation (4)is done through the Brillouin zone corresponding to the lattice model. I n these cubic lattices the symmetry of the constant frequency surfaces can be exploited by doing the integration through only 1/48 of the Brillouin zone, which in the face-centred cubic case is the part of the zone enclosed in the trihedral angle defined by (OOl), (101) and (111) directions, and in the body-centred cubic case is the part enclosed within the tetrahedron bounded by the planes k, = 0, k3 = T,k, = k, and k, = k3, with their ranges defined by
+
and 0 < k3 6 T. 0 < k,, k2 < &r The integration required in equation (4)was carried out by a CDC-3600 electronic computer. For the face-centred cubic lattice we have used the procedure given by Overton and Dent (1959, US. Naval Research Lab., Rep. 5252)to evaluate the frequencies in a particular branch in the region of integration. For the body-centred cubic lattice the volume of the region of integration was divided into cubes with edges along the cubic axes and of length 0.01~. Table 2 contains the first fifteen coefficientsA , of the function sin(") in equation (3) for three values of G for face-centred cubic lattices. Tables 3(a) and 3(b) contab the Same number of coefficients for three values of fl with s = 0 and three values of 5 With fl = 0, respectively, for the body-centred cubic lattices. Here L, T1and T2refer to longitudinal and two transverse branches respectively. The dimensionless variable the ratio A2/Ams2 so that the range of x is from 0 to 1 for each branch. The coefficients have been so normalized that the area enclosed by G(x) for each branch is 1 when G($)
Computation of the frequency distribution function of crystals
1229
is expressed in a Fourier series with the number of terms in the series limited to the maximum value of n given in the tables. Extensive tables of these coefficientsfor eleven values of 5 for the face-centred cubic case and six values of 5 with three values of fl corresponding to each value of a for the body-centred cubic case are also available and cm be obtained from the Editorial Office of the Institute of Physics and the Physical Society on request. These tables may be used for approximate interpolation of the Table 2. Face-centred cubic lattice U
= -0-25
u = o
U
= 0.25
n
L
Ti
Tz
L
Ti
Tz
L
TI
Tz
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1.273 -1.167 0.984 -0.605 0.233 0.132 -0.368 0.489 -0.455 0.330 -0.133 -0.054 0.211 -0.286 0.291
1 468 0.479 0.216 0.416 0.022 0.035 0.012 0 046 0.111 --0 036 0.035 0.091 0.060 0.032 0.059
1.581 -0 206 0 097 0 067 -0.183 0 132 -0 040 0.061 0.047 0.003 0.003 0.038 -0.073 0.069 0.006
1,051 -1.135 1.147 -0.982 0.751 -0446 0.156 0.104 -0.276 0.364 -0.350 0.266 -0.126 -0.021 0.157
1.620 0.356 -0'133 -0.237 0.001 0.145 0.015 0.161 -0.180 0.029 0.069 -0.110 0.137 0'014 -0.056
1.596 -0'363 -0.138 0.209 0.050 -0.112 0.097 -0.026 0.012 0048 -0.059 0.034 0027 -0.013 -0.027
1.214 -1.378 1.229 -0'753 0 185 0 324 -0'593 0.598 -0 373 0057 0230 -0.382 0.383 -0 252 0068
1 694 -0.310 -0.533 0.353 0.253 -0.365 0.148 0.026 -0.018 0.054 -0.102 0,085 -0.008 0.025 -0.091
1.407 -0.816 0.487 -0.188 0.105 -0.025 -0.020 0.106 -0.126 0.112 -0.048 0.015 0.004 0.001 0 011
L refers to the longitudinal branch, and Tiand Tzto the transverse branches.
Table 3(a). Body-centred cubic lattice, E = 0
p n
L
1 2
1.081 -0.994 0.925 -0.718 0.546 -0.361 0.242 -0.141 0.100 -0.079 0 095 -0.107 0 127 -0 124 0 115
3 4
5 6 7 8 9 10 11 12 13 14
15
=
p=0
-0.05
Ti 1.072 0.940 0839 0619 0 502 0.401 0 368 0.306 0.265 0.207 0.183 0.155 0151 0 136 0 133
Tz
L
1.703 0.153 -0'614 -0070 0.519 0 079 -0347 -0070 0.317 0 070 -0234 -0.058 0224 0053 -0.178
1.066 -0 990 0 921 -0.717 0.544 -0.365 0.253 -0'164 0.134 -0.124 0.146 -0.161 0.177 -0.168 0.152
Ti 1.075 0.945 0.840 0616 0.495 0.393 0.358 0.296 0.253 0.197 0475 0.150 0 148 0 134 0.134
Tz
L
1.703 0.150 -0'616 -0070
1.056 -0 988 0 919 -0.714 0 541 -0 365 0.260 -0.181 0 161 -0.160 0 186 -0 201 0 212 -0 197 0 173
0515 0 076 -0.349 -0070 0.318 0.068 -0236 -0059 0.224 0.052 -0.180
p = 0.05 Ti 1 082 0.952 0.841 0611 0.486 0.382 0.346 0.283 0.240 0.184 0.164 0.142 0.141 0.129 0.129
L refers to the longitudinalbranch, and Tiand Tzto the transverse branches.
Tz 1.703 0.147 -0.617 -0068 0.519 0.074 -0.350 -0.068 0.319 0.067 -0.238 -0.058 0.224 0.052 -0.180
K.G. Aggamal, J. Mahunty and V. K. Tewary
1230
Table 3(b). Body-centred cubic lattice, p = 0 U
=
0=04
0.3
0 1 0 5
n
L
Ti
Tz
L
TI
Tz
L
TI
1 2 3 4 5 6 7 8 9 10 11 12 13 14 i5
1.065 -1.145 1.133 -0.934 0.688 -0.398 0.159 0.031 -0.135 0.180 -0.161 0.117 -0.056 0012 OGi4
1.530 0.948 0.223 -0.244 -0.256 -0.154 0.026 0.121 0.129 0.036 -0.020 -0.034 0.002 0.012 0.012
1.630 -0415 -0.228 0.329 -0 023 -0088 0 110 -0080 0064 0052 -0107 0.048 0.002 -0096 C.029
1.103 -1 202 1.167 -0912 0593 -0243 -0024 0201 -0254 0.230 -0147 0.061 0011 -0.045 0.053
1.629 0.712 -0.175 -0.426 -0.133 0.120 0'224 0.083 -0.067 -0.110 -0.005 0.061 0.053 -0.007 -0.020
1.591 -0.524 -0.036 0206 -0 074 0.089 -0059 -0038 0.169 -0.122 0.013 0.058 -0.038 0.038 -0028
1.131 -1.242 1.176 -0 864 0489 -0 109 -0 144 0.271 -0 261 0 189 -0 090 0.017 0.031 -0.052 0.070
1686 0397 -0476 -0330 0 178 0 271 0054 -0 178 -0097 0 091 0127 -0021 -0 081 -0 018 0 063
T, 1.531 -0.589
0129 0025 0034 0 088 -0166 0132 0012 -0054 0056 -0056 0088 -0053 0000
L refers to the longitudinal branch, and Tiand TZto the transverse branches.
coefficientsfor the intermediate values of the force constants. It may be mentioned that the total machine time needed for all these computations on the computer CDC-3600 was about 45 minutes. The frequency distribution function G(A) for Q = - 0.25 in a face-centred cubic lattice is shown in figure 1. The frequency distribution function of vanadium obtained by
h
Figure 1. Frequency distribution function G(A) of a facecentred cubic lattice model with U = -0.25. Here A2 = Mw2/2a.
Figure 2. Frequency distnbution function G O of vanadium taking the noncentral force model. f is the normhed frequency U/ umaX.
the values of the elastic constants given by Alers (1960) is shown in figure 2. our results in the latter case are in qualitative agreement with those obtained by Hendicks et al. (1963) using the root sampling method.
U s h
5. Conclusion
The method for calculating the frequency distribution function of a crystal lattice presented in the preceding sections has two distinct advantages over the conventiona1
Computation of the frequency distribution function of mystak 1231 root sampling method-a very considerable economy of time and numerical work, and yielding simple analytic expressions for the distribution function which can conveniently be used in computation of quantities that require the distribution function. The accuracy of the method can be enhanced by increasing the number of terms in the series in equation (3) and, at the same time, using a finer mesh for the integration involved in equation (4). This method does not require a close examination of the topology of the constant frequency surfaces, particularly the points of degeneracy where two or more branches cross. The method of Gilat and Dolling (1964) requires prior knowledge of these points of degeneracy i n k space. The convergence of the Fourier series is much better than the corresponding series in Legendre polynomials. However, the number of terms one must retain to get a satisfactory representation of G(x) in a given case is best determined by trial. The extreme complexity of the dispersion relations makes it d a c u l t to establish a priori rules on this point. We have found that 15 to 20 terms for the transverse branches and 20 to 25 terms for the longitudinal branches give quite satisfactory results. The convergence of the Fourier series for the longitudinal branch is poorer than that for a transverse branch because of the peaking that occurs in the former in the high frequency end of the spectrum. Since a finite Fourier series can never reproduce the critical points exactly, the question of how closely the location of the critical points can be determined by this method is of some interest. The critical points in G(x)are usually such that they lead to S-function singularities in G(x). If one first evaluates G(x) with sufficient accuracy, i.e. by keeping an adequate number of terms in the Fourier series such that the convergence has become obvious in the form of the smallness of the last few coefficients, G ( x ) can then be evaluated in the form of the series G(x) = - x(nn)2Ansin(nnx). n
The most prominent peaks in G(x) can usually be associated with the critical points of G(x). This test, applied to the three-dimensional lattice whose G(x) is given in table 1, gives the location of the critical points very closely. In general, this test can be used to supplement the information one can get by visual inspection of the G(x) curve obtained by the present method. For a two-dimensional lattice this method leads to incorrect results, such as giving zero values to G(x) at the end points and finite peaks at the logarithmic singularities. However, as discussed in 5 3, the distribution function obtained by this method agrees closely with the actual one over most of the region, so that in those cases where the finer details of the distribution function are unimportant, such as in the calculation of the thermodynamic properties, the distribution function computed by the present method can be used. To obtain a more exact form of the distribution function one can include cosine functions in the series for G(x)to give non-vanishing values at the end points and a suitable number of singular functions to reproduce the desired singularities in the manner suggested by Lax and Lebowitz (1954). For the purpose of illustration, we have chosen a graphite-type laminar hexagonal lattice with nearest-neighbour interactions which had been studied by Rosenstock (1953). The distribution function G(x) for the acoustic branch has been obtained by this modified method and is given by the expression G ( ~=) 0.302 cos ,,~-0-275 loglx-$l+8.275 x 10-2sinm - 1.909 x sin 2 ~ x -1.680 x sin 3nx - 1 . 9 2 8 ~10-2sin 4nx+1.962x 57i-x - 8 . 7 7 2 ~ 10-3sin 6~x-6.53 x 10-3sin 7 7 ~ ~
K. G. Aggarwal, J. Mahanty and V . K. Tewary
1232
where x = w2/umax2 for this particular branch. The logarithmic term reproduces &e singularity at x = 8. The distribution function for the other branch can be obtained by
X
Figure 3. G(x) for a graphite-type two-dimensional lattice with nearest-neighbour central force interaction. The fullcurve represents the exact value due to Rosenstock (1953), and the open circles show the results obtained by the present method.
reflection of the above function about the x = 1.0 axis for this model. In figure 3 the full curve is the exact result obtained by Rosenstock and the points represent the values obtained from the above formula. The agreement is satisfactory considering that only nine functions have been taken. For a two-dimensional lattice the numerical integration in equation (4)can be performed by the standard two-dimensional Simpson's rule. For three-dimensionallattices the integration procedure is somewhat complicated because of the irregular shape of the region of integration, and one has to be careful in writing the machine programme SO as to carry on the integration in the entire region. Programmes for face- and body-centred cubic lattices in FORTRAN 11 are available from the authors and can be obtained on request.
Acknowledgments This research is supported by the U.S. National Bureau of Standards through a research contract. The authors are indebted to Dr. J. Mathews for useful commenR to Mr. R. N. Basu and Mr. P. Vashishta for their help in the computational work at fie 1.I.T. Computer Centre, and to the staff of the Computer Division of Tata Institute of Fundamental Research, Bombay, for providing the facilities for use of the CDC-3600 computer.
References
ALERS, G.A., 1960,Phys. Rev., 119,1532-35. GILAT,G., and DOLLING, G., 1964,Phys. Letters, 8,304-6.
Computation of the frequency distributionfunction of crystals -\TRICKS,
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J. B., RISER,H. N., and CLARK,C. B., 1963, Phys. Rev., 130,1377-80.
ISENBERG, C., 1963, Phys. Rev., 132, 2427-33. LAMB,H., 1960, The Dynamical Theory of Sound (New York: Dover Publications), chap. DE LAUNAY, J., 1956, Solid State Phys., 2, 220-303 (New York: Academc Press).
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LAX,M., and LEBOWITZ, J. L., 1954, Phys. Rev.,96,594-8. "N I, A. A., MONTROLL, E. W., and WEISS,G. H., 1963, Solid State Phys. (Suppl. 3) (New York: Academic Press), chap. 3. MONTROLL, E. W., 1942, J . Chem. Phys., 10,218-25. 1943, J. Chem. Phys., 11,481-95. 1956, Proc. Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. 3 (Berkeley, Cahf: University of California Press), pp. 209-46. RosENSTOCK,H. B., 1953, J. Chem. Phys., 21, 2064-9.
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