--_
CHINESE
JOURNAL OF PHYSICS
VOL. 7, NO. I
.4PRIL
1968
Normal Vibration in Aluminum*
T U N- CH I E N L EE (*a/&) and JE N N- LIN HWANG (j$-@#)
Department of Physics, National Taiwan University, Taipei, Taiwan
(Received February 2, 1969)
In this paper, the atomic force constants of aluminum crystal determined
by Brockhouse and Stewart from their data of inelastic one-phonon neut’ron
scattering experiment are revised. The dispersion curves along 100 and 110
directions are calculated based on measurements of the intensity of the X-ray
diffuse scattering. Furthermore, the calcuated specific heat at low temperature
is in gocd agreement with the experimental data of Giauque and Meads after
a correction for the electronic contribution is made.
INTRODUCTION
there has been an increasing interest
R ECENTLY
distribution function of aluminum and its derived
in studying the frequency
properties, such as specific
heat. Brooks and Bingham”’ calculated specific heat of aluminum at constant
volume by using a simple Debye temperature. Their method of calculation is tco
simplified.. Gilat and Nicklow’*) used an axially symmetric method., including S
nearest neighbors to calculate g(w) as well as specific heat curve. Although their
results fit excellently well to the experimental data of Giauque and Meads@’ in
very low temperature, Phillips’4’ indicated that the quasiharmonic approximation
is invalid for many materials at temperatures (-300 ° K ) where most neutron
scattering experiments have been performed. F l i n n a n d Memanns”) u s e d a
first-nearest-neighbor model to derive the values of specific heat but its calculated
values above 200 °K are higher than the experimental values.
The information about the interatomic forces can be obtained. from inelastic
scattering experiments. Two different types of work are the X-ray diffuse
scattering (6) by Walker and the neutron inelastic scattering ”) by Brockhouse and
Stewart. The former analyzed experimenta. data with a 3-neighbor model and the
latter analyzed with many simple models.
In Brockhouse and Stewart ’s paper, they indicated that among various simple
* Work
supported by the National Council on Science Development.
(1) C. R. Brooks and R. E. Bingham, J Phys. Chem Solids, ‘29 1553 (1968)
( 2 ) G. Gilat and R. M. Micklow, Phys Rev. 142 487 (19663
( 3 j W. F. Giauque and P. F. Meads, J. Am. Chem. Sot 63 1897 (1941)
( 4 j N. E. Phillips, Phys. Rev. 114 676 (1959)
( 5 ) P. A. Finn and.G. M. McManus, Phys. Rev. I.32 2458 (1963)
( 6 ) C. B. Walker, Phys. Rev. 103 547 (1956)
( 7 j B. N. Brockhouse and A. T. Stewart, Revs. Modern Phys. 30 236 (1958)
33
T. C. LEE AND J. L. HWANG
34
models the model V (Z-neighbor model) seems to be able to fit to their data.
However, they did not calculate the frequency distribution function of aluminum
and also the specific heat heat curve. Furthermore, in their paper they did not
report the numerical value of rl and a2 respectively. The reported value of al.
81, and Pz did not satisfy the relation aC~~=2a1+281+2i?~. The valuse of a a n d
CJl we choose in this paper are shown in Table 2.
In this paper we calculate the frequency distribution function of aluminum
base on the Model V. The calculated dipersion curves in [loo] and [llO] directions
are compared with the result due to Walker and that due to Brockhouse and
Stewart. The calculated specific heat at low temperature are compared with
the experimental value of Giauque and Meads(‘) after a correction for the electronic contribution is made.
Although Squires (‘I has analyzed Walker’s data with 2-neighbor model but
our results and force constants used are different from him. Because the neutron
diffraction technique is more powerful, we believe that our result is better than
Squires’s .
FORCE CONSTANTS AND FREQUENCY SPECTRA OF NORMAL MODES
Aluminum is a face-centered cubic lattice. The space group is a% Each
atom has 12 first-nearest neighbors and 6 second nearest neighors. The Lattice
constant is a=4.041 x IO-‘cm. Because of this simple structure as well as its small
cross sections for a capture of neutrons and incoherent scattering, aluminum was
first chosen by Brockhouse and Stewart to perform neutron inelastic scattering
experiments. The experimental data are fitted with several simple models, They
indicated that the model V in which interatomic forces are restricted to interactions with the first and second neighbors gives the best fit to their data.
In the Bron-Von Krkman theory of lattice dynamics the frequency Y are
are determined by the following secular equation
I Tij-
M7E'Y26~j
/ =O,
where
Tii=al+281-alC1Ck- /3,Ci(Ci+Ck) +azSj+,3:(S,?+Sk2)
Tij = r,S;Sj (i+j)
and
Ci=cos0i, Si=sinBi, Bi=nqi/b
M=the mass of aluminum nucleus
dij =Kronecker delta
~______
i8
; G. L. Squires, Phys. Rev. 103 304 (1956)
(1)
NORMAL VIRRATIOS IN ALUkIINU4I
35
~1, 91, ~1 are force constants for the first-nearest neighbor and uz, Bz are for the
second-nearest neighbor. In order to solve the equation (I) the numerical value
of the five atomic force constants are required.
The numerical values of al, Bl, (rr-CL.) and 92 are given in ref. 7. In order
to obtain the numerical value of rl and a2 respectively, we use the Walker’s
ciatum T1=9.33 x lo3 cgs and from the numerical value of the rl-az reported in
reference 7. We get as=1.43XlO” cgs. The relation between elastic constants
and atomic force constants are(‘)
31+a2
=Lz
Cl1
4
(2a)
Although we can obtain the value of al, ,B1 and from reference 7, they do not
satisfy the equations (2) especially equation (Zb) ; PI is too small. In order to make
the five force constants satisfy equation (2) simultantously it is necessary to
change the value of P1 from--12.56~10~ cgs to -2.58~10~ cgs. The numerical
values of the five force constants determined by us, and those presented in reference 7 are given in Table 1 and 2. Inserting the numerical values of the five
force constants into Eqn (1)) we solve by an electronic computer the equations
for all points in the first Brillouin zone whose coordinates q; are integral multiples
Table 1. Numerical Values of Atomic Force Constants
Reported by Rrockhodse
and Stewart (lo3 cgs)
~. .___
Used in present
paper (10” cgs)
9.56
-2.58
~
9.34
yt--az=7.90
1.44
-0.62
9.56
-1.256
j
-0.62
I
Table 2. Numerical constants used in this paper
Elastic constants*
( x lOi dynes/cm2)
Cl1 = 1.092
Lattice Constant
C,z = 0.640
a34.041A
c,; =0.284
* cf. D. Lazarus, Phys. Rev. 76 545 (1949)
(9)
Atomic mass
M=44.78 x 1C-2’ gm
J. de Launay J. Chem. Phys. 21 1975 (1953)
T. C. LEE AND J. L. HWANG
36
of b/15. The frequency spectra of riormal modes g(v) obtained by us is shown
in Fig. 1.
8
0
2
4 ~10’2C;s
1
Ir
10
8 ’
The smoothed frequency spectrum of aluminum,
Fig. 1.
RESULTS AND DISCUSSION
The dispersion curves, along [lOOI and [llO] directions, are calculated and
shown in Table 3 and in Fig. 2. In the same figure the data from the experiments of neutron diffraction and of X-ray diffuse scattering as well as the
Table 3-A. Dispersion Data in
100 direction**
qiq max
11o’s cps)
( 10:s cps)
Table 3-B. Dispersion Data in
110 direction
ilO’ cpsl
.,
(lo’s CPS)
0.00
0.00
0.00
(
0.151
0.076
0.066
q / q m a x (lOI cpsj
-
,
~
0.00
0.00
0.00
0.07
0.104
0.053
0.09
0.13
0.206
0.106
0.18
0.295
0.152
0.132
0.20
0.306
0.159
0.27
0.429
0.229
0.195
0.27
0.401
0.210
0.36
0.540
0.308
0.255
0.00
0.33
0.490
0.261
0.45
0.642
0.389
0.310
0.40
0.4 ’7
0.573
0.310
0.55
0.716
0.469
0.361
0.648
0.357
0.64
0.763
0.549
0.405
0.400
0.73
0.784
0.626
0.444
0.440
0.82
0.810
0.629
0.523
0 53
0.714
0.60
0.771
0.67
0.820
0.73
0.80
0.87
0.913
0.547
0.93
0.926
0.558
1.00
0.930
0.562
/
0.476
0.91
0.827
0.606
0.596
0.960
I
!
0.506
1.00
0.834
0.681
0.54;
0.890
~
0.529
** Notation in this table cf. reference 8
-.__
37
NORMAT. VIBRATION IN AI.IJMINIJM
0
.2
k
.6
q/%lx
a
1.0
Fig 2. Dispersion curves for phonons travelling along 100 and 110 directions. The circles 0
represent neutron diffraction data from Brockhouse and Stewart; the crosses, i,
neutron diffraction data for Carter et al; the dotts X-ray diffuse scattering data of
Walker. The solid curves are calculated by this paper and the dashed curves were
obtained by Walker.
calculated dispersion curves by Walker are shown also. The comparison shows
.that the dispersion curves calculated by Walker and those by us can fit equally
well to the experimental data. However the Walker’s model was based on the
assumption that the interatomic forces are restricted to the first, second and third
neighbors. The model is more complicated than ours. Since the dispersion curves
based on the two models are equally good, we believe that the simple model is
more realistic than the complicated one.
In Fig 3 our results are compared with those of Squires(“). At the position
1.0
t
(100)
Fig. 3. This figure is the same as Fig. 2 except that the dashed curves were obtained by Squires.
(10) See the dispersion curve shown in Reference 11.
(11) R. S. Carter et al Phys. Rev. 106 1168 (1957)
__.-
..,I
.,,
.,.
.
T. C. LEE AND J. 1~. HWANG
3s
A and B shown in Fig 3, our results fit to the experimental value of neutron
diffraction. At the position C and D our results can fit not only to the expermental values of neutron diffraction but also to the data of X-ray scattering.
The agreement with the X-ray scattering experiment is more satisfactory than
the Squire’s result.
The specific heat at constant volume due to lattice vibrations can be calculated
from
C,=3R
s0
Urn
x2ez
(ex-1)’
g(x)dz
where x=hv/KT. Using the results of g(b) shown in Fig 1, we calculate some
values of C, and. show them in Table 4. In Fig. 4 our calculated specific heats
are plotted in terms of Debye temperature. Since Phillips’*) has indicated that
the low calorimeteric values of Giauque and Meads at moderate temperature
have some ambiguities, it is difficult to assess the degree of agreement between
theoretical and. experimental values. If Giauque and Mead’s results at higher
temperature are correct, then the agreement between our results and the experimental values is quite good. Furthermore it is easy to find from Fig. 4 that
our results (solid curve) is better than one of the Walker’s results (curve A).
If the anharmonic term is considered our results will be much better than the
other one of the Walker’s (curve B).
Table 4. Specific heat of aluminum at constant voltme
calculated in the present paper
Temperature (OK:
Specific Hezt f j /mole-deg)
40
2.14
60
6.24
80
10.28
100
13.55
120
16.01
140
17.84
160
17.20
180
20.23
200
21.01
220
21.63
240
22.11
260
22.50
280
2281
300
23.07
NORMAL VIBRATION IN Al.IJMINLX
39
Fig. 4. The variation of Debye t e m p e r a t u r e w i t h t e m p e r a t u r e . T h e
dots represent the experimental vaIues of Giauque and Meads ’,
after a correction for the electronic contribution. The solid
curve represents the Debye temperature determined from our
calculated specific heats; the dashed curve a and b are Walker’s
results. C u r v e a i s o b t a i n e d f r o m t h e d i r e c t l y c a l c u l a t e d
specific heats and curve b is obtained from similar calculations
including an approximation for anharmonic terms.
ACKNOWLEDGEMENT
We wouId Iike to thank Prof. G. C. Shu for granting permission to use the
electronic computer IBM 1620 at the Computer Center of National Taiwan University and Miss H. Ho for punching the cards and running the programs.