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Ab initio calculation of the pressure dependence of the lattice specific heat of aluminum
Volume 122, number 5
PHYSICS LETTERS A
15 June 1987
AB INITIO CALCULATION OF THE PRESSURE DEPENDENCE OF THE LATFICE SPECIFIC HEAT OF ALUMINUM G.J. VAZQUEZ and L.F. MAGARA Instituto de Fisica, UniversidadNacionalAutdnoma de Mexico, Apdo. Postal 20-364, 01000 Mexico DF, Mexico Received 17 September 1986; revised manuscript received 3 April 1987; accepted for publication 8 April 1987 Communicated by A.A. Maradudin
From a first-principles local pseudopotential we calculated the interionic potential, phonon dispersion curves, phonon spectra and lattice specific heat as functions ofpressure foraluminum. The pseudopotential was found from the electron density around an aluminum nucleus in an electron gas.
The knowledge ofthe behaviour of materials under pressure is useful for many applications. Making predictions of thermodynamic and transport properties of metals under pressure is not an easy task. A good starting point to make these predictions is to have a reliable, pressure-dependent pseudopotential. In particular in this work we are interested in the variation with pressure of the lattice specific heat of aluminum. Following the method of Manninen et al. [1], we have constructed a pressure-dependent first-principles local pseudopotential for aluminum. We obtained the interionic potential, the phonon dispersion curve, the phonon spectra and finally the lattice specific heat as functions of pressure. From pseudopotential theory and linear response theory [1,21 the interionic potential is given by ~(r) Z2
(
2
rk~+xZ2J
f dq sin(qr) ~(q) [6 n( q)12 \ o
q[le(q)l
), (1)
a nucleus embedded in a homogeneous electron gas at atmospheric pressure are far from experimental results [5]). We first calculated the electron density, n (r), in an electron gas perturbed by an aluminum nucleus embedded in a spherical vacancy made in the positive background and we also calculated the electron density, n~(r), around a pure jellium vacancy. Then, we calculated the difference i~n(r)=n(r)—n~(r)—Z~ IWb(r)I 2 b
where ~vb(r) refers to the bound electron wavefunctions. We made the calculation ofthe densities using the density functional formalism [6,71. The induced density ~n(r) shows wiggles for small values of r due to the orthogonalization of conduction states to core orbitals. In the pseudopotential formulation the pseudodensity must not have coi~e orbitals, thus we have smoothed our M( r) to remove the wiggles near ion following method of Manninen et al.the [11, using a thesecond-order polynomial, ön(r)=A—Br2,
where Z is the charge of the metal ion, ~(q) is the dielectric function of the electron gas and 6n(q) is the induced charge pseudodensity. In order to obtain 6n(q) we have considered the model of a nucleus embedded in a jellium vacancy [1,3,4] (the phonons generated using the model of
(2)
r~
(3)
for small values of r. The constants A, B and R0 are calculated under the conditions that 6n(r) and ô(6n(r))/ôr are continuous at R0 and that the electronic charge is conserved. This smoothed density is thepseudodensityweusedineqs. (1) and (2) tocal-
0375-960l/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
267
Volume 122, number 5
PHYSICS LETTERS A
culate the interionic potential for every value of pressure. The dielectric function we used satisfies the cornpressibility theorem, which is important in connection with the interionic potential [1,8] and involves the expression given by Gunnarson and Lundqvist
2)G(q), where for thefunction exchange correlation lectric e(q)=l+(4x/q is G given by energy [9J. This die__________________________________________________ 0( q) G(q)= l—(4x/k4F)G0(q)(l—L)’ where G0( q) is the usual Liadhard polarizabiity, kTF is the Fermi—Thomas screening constant and L— —
apJar~
(4)
ÔEF/ôrS
In eq. (4), ~ is the chemical potential, Fermi energy and
F
is the
~u(r~) =CF(rS) ~
where ~ r~)is the exchange—correlation contribution to the chemical potential. Using the expression ofGunnarson and Lundqvist [91for exchange—correlation (which we used in the calculation of the induced density), the corresponding value of L is 1/3
L= 1
_(~)rs(i +0.621
r~ r~+ 11.4
)
From the resulting interionic potential we calculated the phonon dispersion curves using the self consistent harmonic approximation [ 10~12 1. From the force constants obtained in the calculation of the phonon dispersion curve we calculated the phonon frequency distribution, F( v), following the method of Gilat and Raubenheimer [131. Finally, the lattice specific heat was obtained from 2 sinh( F(v) (5) ar — ~f3hv) di’, c a — kB 0 ~flh)
=
J
where is the average of the internal energy of the lattice, Tis the temperature, kB is the Boltzmann constant, ~1mis the maximum phonon frequency and /i is l/kBT. 268
15 June 1987
8
~qOO~q
q0 /
6-
U,
2-
i
/
£
~/ ~
/
‘1/7 /// K~~uuhhf~o
I! 1, / ,‘~ /‘
‘
‘
~
‘\ \
0
/ /
10
0
0,3
q(27r/a) Fig. 1. Phonon dispersion curves: experimental results [14] at atmospheric pressure 0), calculated at atmospheric pres-
(x,
sure (—)~ calculated for a value of r, which is 4% smaller than the corresponding value of r, at atmospheric pressure (__—).
The foregoing procedure was repeated for five different values of the pressure. The effect of pressure was introduced by the electron gas density parameter, r5, where ~ 1/n0. We decreased the value of r~at atmospheric pressure by 1%, 2%, 3%, 4% and 5%, which correspond to changes in volume of 3%, 6%, 9%, 12% and 15%, respectively. In fig. 1 we show a comparison between the experimental [141 and calculated phonon dispersion curves at atmospheric pressure and the calculated phonon dispersion curve for a value of r~which is 4% smaller than the value at atmospheric pressure. There is a good agreement between theory and experiment at atmospheric pressure and there is the expected upwards shift of the phonon frequencies when the pressure is increased. Fig. 2 shows the phonon spectra for values of r~corresponding to atmospheric pressure and for values of r~which are 2% smaller and 4% smaller, respectively. Table 1 shows the results the calculationbetween for severalvalues of pressure. Theof correspondence the changes in volume, AVIV, and the applied pressure, p, was calculated by interpolating the experimental results of Bridgman [15] for aluminum. The variation of C~with pressure could be fitted by a quadratic form:
Volume 122, number 5
PHYSICS LETTERS A
15 June 1987
Table I Correspondence between the volume changes and pressure [15]. The variation of the specific heat with pressure for several values of the 2. The values ofA temperature is given. This variation could be fitted using a second degree polynomial: C 0=A0+A ,p+A2p 0, A, and A2 for every value ofthe temperature considered are also given. T(K)
P(atm) 1.000 (0)
2.418x10 (—3)
4
6.577x104 (—6)
l.024x103 (—9)
l.334x105 (—12)
1.609xl05 (—15)
0.0570 0.2180 0.5410 0.9920 1.5000 2.4730 3.2600 3.8510 4.2870 4.6100 5.0380 5.3440 5.5220
0.04600 0.1760 0.4480 0.8450 1.3090 2.2440 3.0360 3.6490 4.1100 4.4580 4.9260 5.2650 5.4640
0.0380 0.1430 0.3720 0.7200 1.1430 2.0320 2.8200 3.4480 3.9320 4.3030 4.8090 5.1820 5.4030
0.0310 0.1180 0.3100 0.6140 0.9960 1.8360 2.6120 3.2500 3.7530 4.1440 4.6880 5.0940 5.3380
0.0250 0.9600 0.2580 0.5220 0.8650 1.6520 2.4100 3.0530 3.5710 3.9810 4.5610 5.0010 5.2680
0.0210 0.0790 0.2130 0.4410 0.7480 1.4800 2.2140 2.8560 3.3870 3.8130 4.4270 4.9020 5.1930
a)
20 30 40 50 60 80 100 120 140 160 200 250 300
T(K) 30 20 40 50 60 80 100 120 140 160 200 250 300
A,,
A
A
0.2124 0.5554x10’ 0.5289 0.9731 0.l475x 10’ 0.2443x10’ 0.3231 10’ 0.3824x 10’ 04263x10’ 04589x10 0.5022xl0’ 0.5332x 10’ 0.5513xl0’
6 —0.3128xI0~ —0.l2l7xl0~’ —0.2704xl0~5 —0.4266x10~5 —0.5491 l0~~ —0.6492x10~5 —0.6220x l0~ —0.5510x l0~~ —0.4733xl0~5 —0.4003x105 —0.2887xl0~ —0.1999x i0~ —0.l438xl0~5
2 0.6258x10’2 0.2504x 10_li 0.4815xl0” 0.6284xl0” 0.6436x10” 0.3646x10” —0.7270x i0’~ —0.2621 x 10” —0.3935xl0~’ —0.4662x10~ —0.4695x 10_I —0.3959x 10—” —0.3245x10”
x
x
~ The value between parentheses is ~ V/V (%).
The agreement at lower temperatures can be
2
C,,=A
0+A1p+A2p. The values of the constants A0 A1 and A2 for each value of the temperature we have considered are also given in table 1. Eq. (5) reproduced the calculated values of C,, within 0.1%. At atmospheric pressure, the theoretical values of the specific heat are less than 5% larger than experimental results for temperatures between 100 and 300 K [21,22]. For temperatures between 30 and 100 K the theoretical specific heat is about 10% larger than the experimental results [21,22]. For temperatures of 20 K and lower the theoretical values are about 20% larger than experimental results,
.
.
.
improved taking into account anharmonic contributions which make corrections to the temperature dependence [16]. There is, for aluminum, a more accurate calculation of the specific heat [17,18]. At atmospheric pressure the difference between the calculation in refs. [17,181 and experiment is smaller than the corresponding difference in our present work. This happens specially fortemperatures above 70 K for lower temperatures the corresponding differences are about the same. However, in the calculation ofrefs. [17,181 a two-parameter phenomenological pseudopotential is used. The parameters are chosen in order to fit, at 269
Volume 122, number 5
PHYSICS LETTERS A
15 June 1987
can be used to make predictions of the variation 1.2
08
04
other properties of aluminum (for which phonons play an important role) with pressure.
-
J
A
A
References [1] M. Manninen, P. Jena, R.M. Nieminen and J.K. Lee, Phys.
j
Rev.B24 (1981) 7057. [2] W.A. Harrison, Pseudopotentials in the theory of metals (Benjamin, New York, 1966).
•~\ I “
~
[3] G.W. Bryant and G.D. Mahan, Phys. Rev. B 17 (1978)1744.
I
0800________________ 2 4
6
8
10 I I
12
LI ~0’~Cps Fig. 2. Phonon spectra for values of r, corresponding to atmopheric pressure (—), for a value of r, 2% smaller ~_•~ for a value of r, 4% smaller (———).
atmospheric pressure, the phonon dispersion curve for aluminum. Because of its nature, it is not possible to use this pseudopotential to make predictions ofthe pressure dependence ofthe phonon spectra and specific heat ofaluminum. We do not have notice of measurements of the variation of C,, with pressure. However, Bastide et al.
[19] have reported a quadratic variation with pressure of the specific heat at constant pressure, Cj,,, and we can see that eq. (5) is consistent with the results of ref. [19] using the relationship between C,, and C,, [20].
Finally, it is clear that the calculation of the variation of the phonons under pressure for aluminum
270
of
[5] L.F. [4] E. Zaremba Magaflaand andD. G.J. Zobin, Vázquez, Phys.to Rev. be published. Lett. 44 (1980) 175. [6] H. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B964. [7]W.KohnandL.J.Sham,Phys.Rev. 140(1965)All33. [8] M.S. Duesbery and R. Taylor, Phys. Rev. B 7 (1970) 2870. [9] 0. Gunnarson and B.I. Lundqvist, Phys. Rev. 13 (1976) 4274. [10] N. Boccara andG. Sarma, Physics 1(1965)219. [11] N.S. Gillis, N.R. Werthamer and T.R. Koeler, Phys. Rev. 165 (1968) 951. [12] E.R. Gowley and R.C. Shulda, Phys. Rev. B 4 (1974) 1261. [13] G. Gilat and L.J. Raubenheimer, Phys. Rev. 144 (1966) 390. [14] and G. Nilsson, Phys. Arts Rev.Sci. 145 74 (1966) 492. [15] R. P.W.Stedman Bridgman, Proc. Am. Acad. (1942) 425;
76 (1945)l,9,55;76(1948) 71;77 (1949) 189. [161 C.B. Walker, Phys. Rev. 103 (1956) 547. [17] D.C. Wallace, Phys. Rev. 187 (1969) 991. [18] D.C. Wallace, Phys. Rev. B 1(1970) 3963.
[19] J.P. Bastide and153. C. Loriers-Susse, High Temp. High Pressures7 (1975) [20] N.V. Fomin, Soy. Phys. Solid State 23 (1981) 299. [211W.F. Giauque and P.F. Meads, J. Am. Chem. Soc. 63 (1941)
63. [22] D.H. Howling, E. Mendoza and J.E. Zimmerman, Proc. R. Soc. A 229 (1955) 86.
PHYSICS LETTERS A
15 June 1987
AB INITIO CALCULATION OF THE PRESSURE DEPENDENCE OF THE LATFICE SPECIFIC HEAT OF ALUMINUM G.J. VAZQUEZ and L.F. MAGARA Instituto de Fisica, UniversidadNacionalAutdnoma de Mexico, Apdo. Postal 20-364, 01000 Mexico DF, Mexico Received 17 September 1986; revised manuscript received 3 April 1987; accepted for publication 8 April 1987 Communicated by A.A. Maradudin
From a first-principles local pseudopotential we calculated the interionic potential, phonon dispersion curves, phonon spectra and lattice specific heat as functions ofpressure foraluminum. The pseudopotential was found from the electron density around an aluminum nucleus in an electron gas.
The knowledge ofthe behaviour of materials under pressure is useful for many applications. Making predictions of thermodynamic and transport properties of metals under pressure is not an easy task. A good starting point to make these predictions is to have a reliable, pressure-dependent pseudopotential. In particular in this work we are interested in the variation with pressure of the lattice specific heat of aluminum. Following the method of Manninen et al. [1], we have constructed a pressure-dependent first-principles local pseudopotential for aluminum. We obtained the interionic potential, the phonon dispersion curve, the phonon spectra and finally the lattice specific heat as functions of pressure. From pseudopotential theory and linear response theory [1,21 the interionic potential is given by ~(r) Z2
(
2
rk~+xZ2J
f dq sin(qr) ~(q) [6 n( q)12 \ o
q[le(q)l
), (1)
a nucleus embedded in a homogeneous electron gas at atmospheric pressure are far from experimental results [5]). We first calculated the electron density, n (r), in an electron gas perturbed by an aluminum nucleus embedded in a spherical vacancy made in the positive background and we also calculated the electron density, n~(r), around a pure jellium vacancy. Then, we calculated the difference i~n(r)=n(r)—n~(r)—Z~ IWb(r)I 2 b
where ~vb(r) refers to the bound electron wavefunctions. We made the calculation ofthe densities using the density functional formalism [6,71. The induced density ~n(r) shows wiggles for small values of r due to the orthogonalization of conduction states to core orbitals. In the pseudopotential formulation the pseudodensity must not have coi~e orbitals, thus we have smoothed our M( r) to remove the wiggles near ion following method of Manninen et al.the [11, using a thesecond-order polynomial, ön(r)=A—Br2,
where Z is the charge of the metal ion, ~(q) is the dielectric function of the electron gas and 6n(q) is the induced charge pseudodensity. In order to obtain 6n(q) we have considered the model of a nucleus embedded in a jellium vacancy [1,3,4] (the phonons generated using the model of
(2)
r~
(3)
for small values of r. The constants A, B and R0 are calculated under the conditions that 6n(r) and ô(6n(r))/ôr are continuous at R0 and that the electronic charge is conserved. This smoothed density is thepseudodensityweusedineqs. (1) and (2) tocal-
0375-960l/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
267
Volume 122, number 5
PHYSICS LETTERS A
culate the interionic potential for every value of pressure. The dielectric function we used satisfies the cornpressibility theorem, which is important in connection with the interionic potential [1,8] and involves the expression given by Gunnarson and Lundqvist
2)G(q), where for thefunction exchange correlation lectric e(q)=l+(4x/q is G given by energy [9J. This die__________________________________________________ 0( q) G(q)= l—(4x/k4F)G0(q)(l—L)’ where G0( q) is the usual Liadhard polarizabiity, kTF is the Fermi—Thomas screening constant and L— —
apJar~
(4)
ÔEF/ôrS
In eq. (4), ~ is the chemical potential, Fermi energy and
F
is the
~u(r~) =CF(rS) ~
where ~ r~)is the exchange—correlation contribution to the chemical potential. Using the expression ofGunnarson and Lundqvist [91for exchange—correlation (which we used in the calculation of the induced density), the corresponding value of L is 1/3
L= 1
_(~)rs(i +0.621
r~ r~+ 11.4
)
From the resulting interionic potential we calculated the phonon dispersion curves using the self consistent harmonic approximation [ 10~12 1. From the force constants obtained in the calculation of the phonon dispersion curve we calculated the phonon frequency distribution, F( v), following the method of Gilat and Raubenheimer [131. Finally, the lattice specific heat was obtained from 2 sinh( F(v) (5) ar — ~f3hv) di’, c a
=
J
where
15 June 1987
8
~qOO~q
q0 /
6-
U,
2-
i
/
£
~/ ~
/
‘1/7 /// K~~uuhhf~o
I! 1, / ,‘~ /‘
‘
‘
~
‘\ \
0
/ /
10
0
0,3
q(27r/a) Fig. 1. Phonon dispersion curves: experimental results [14] at atmospheric pressure 0), calculated at atmospheric pres-
(x,
sure (—)~ calculated for a value of r, which is 4% smaller than the corresponding value of r, at atmospheric pressure (__—).
The foregoing procedure was repeated for five different values of the pressure. The effect of pressure was introduced by the electron gas density parameter, r5, where ~ 1/n0. We decreased the value of r~at atmospheric pressure by 1%, 2%, 3%, 4% and 5%, which correspond to changes in volume of 3%, 6%, 9%, 12% and 15%, respectively. In fig. 1 we show a comparison between the experimental [141 and calculated phonon dispersion curves at atmospheric pressure and the calculated phonon dispersion curve for a value of r~which is 4% smaller than the value at atmospheric pressure. There is a good agreement between theory and experiment at atmospheric pressure and there is the expected upwards shift of the phonon frequencies when the pressure is increased. Fig. 2 shows the phonon spectra for values of r~corresponding to atmospheric pressure and for values of r~which are 2% smaller and 4% smaller, respectively. Table 1 shows the results the calculationbetween for severalvalues of pressure. Theof correspondence the changes in volume, AVIV, and the applied pressure, p, was calculated by interpolating the experimental results of Bridgman [15] for aluminum. The variation of C~with pressure could be fitted by a quadratic form:
Volume 122, number 5
PHYSICS LETTERS A
15 June 1987
Table I Correspondence between the volume changes and pressure [15]. The variation of the specific heat with pressure for several values of the 2. The values ofA temperature is given. This variation could be fitted using a second degree polynomial: C 0=A0+A ,p+A2p 0, A, and A2 for every value ofthe temperature considered are also given. T(K)
P(atm) 1.000 (0)
2.418x10 (—3)
4
6.577x104 (—6)
l.024x103 (—9)
l.334x105 (—12)
1.609xl05 (—15)
0.0570 0.2180 0.5410 0.9920 1.5000 2.4730 3.2600 3.8510 4.2870 4.6100 5.0380 5.3440 5.5220
0.04600 0.1760 0.4480 0.8450 1.3090 2.2440 3.0360 3.6490 4.1100 4.4580 4.9260 5.2650 5.4640
0.0380 0.1430 0.3720 0.7200 1.1430 2.0320 2.8200 3.4480 3.9320 4.3030 4.8090 5.1820 5.4030
0.0310 0.1180 0.3100 0.6140 0.9960 1.8360 2.6120 3.2500 3.7530 4.1440 4.6880 5.0940 5.3380
0.0250 0.9600 0.2580 0.5220 0.8650 1.6520 2.4100 3.0530 3.5710 3.9810 4.5610 5.0010 5.2680
0.0210 0.0790 0.2130 0.4410 0.7480 1.4800 2.2140 2.8560 3.3870 3.8130 4.4270 4.9020 5.1930
a)
20 30 40 50 60 80 100 120 140 160 200 250 300
T(K) 30 20 40 50 60 80 100 120 140 160 200 250 300
A,,
A
A
0.2124 0.5554x10’ 0.5289 0.9731 0.l475x 10’ 0.2443x10’ 0.3231 10’ 0.3824x 10’ 04263x10’ 04589x10 0.5022xl0’ 0.5332x 10’ 0.5513xl0’
6 —0.3128xI0~ —0.l2l7xl0~’ —0.2704xl0~5 —0.4266x10~5 —0.5491 l0~~ —0.6492x10~5 —0.6220x l0~ —0.5510x l0~~ —0.4733xl0~5 —0.4003x105 —0.2887xl0~ —0.1999x i0~ —0.l438xl0~5
2 0.6258x10’2 0.2504x 10_li 0.4815xl0” 0.6284xl0” 0.6436x10” 0.3646x10” —0.7270x i0’~ —0.2621 x 10” —0.3935xl0~’ —0.4662x10~ —0.4695x 10_I —0.3959x 10—” —0.3245x10”
x
x
~ The value between parentheses is ~ V/V (%).
The agreement at lower temperatures can be
2
C,,=A
0+A1p+A2p. The values of the constants A0 A1 and A2 for each value of the temperature we have considered are also given in table 1. Eq. (5) reproduced the calculated values of C,, within 0.1%. At atmospheric pressure, the theoretical values of the specific heat are less than 5% larger than experimental results for temperatures between 100 and 300 K [21,22]. For temperatures between 30 and 100 K the theoretical specific heat is about 10% larger than the experimental results [21,22]. For temperatures of 20 K and lower the theoretical values are about 20% larger than experimental results,
.
.
.
improved taking into account anharmonic contributions which make corrections to the temperature dependence [16]. There is, for aluminum, a more accurate calculation of the specific heat [17,18]. At atmospheric pressure the difference between the calculation in refs. [17,181 and experiment is smaller than the corresponding difference in our present work. This happens specially fortemperatures above 70 K for lower temperatures the corresponding differences are about the same. However, in the calculation ofrefs. [17,181 a two-parameter phenomenological pseudopotential is used. The parameters are chosen in order to fit, at 269
Volume 122, number 5
PHYSICS LETTERS A
15 June 1987
can be used to make predictions of the variation 1.2
08
04
other properties of aluminum (for which phonons play an important role) with pressure.
-
J
A
A
References [1] M. Manninen, P. Jena, R.M. Nieminen and J.K. Lee, Phys.
j
Rev.B24 (1981) 7057. [2] W.A. Harrison, Pseudopotentials in the theory of metals (Benjamin, New York, 1966).
•~\ I “
~
[3] G.W. Bryant and G.D. Mahan, Phys. Rev. B 17 (1978)1744.
I
0800________________ 2 4
6
8
10 I I
12
LI ~0’~Cps Fig. 2. Phonon spectra for values of r, corresponding to atmopheric pressure (—), for a value of r, 2% smaller ~_•~ for a value of r, 4% smaller (———).
atmospheric pressure, the phonon dispersion curve for aluminum. Because of its nature, it is not possible to use this pseudopotential to make predictions ofthe pressure dependence ofthe phonon spectra and specific heat ofaluminum. We do not have notice of measurements of the variation of C,, with pressure. However, Bastide et al.
[19] have reported a quadratic variation with pressure of the specific heat at constant pressure, Cj,,, and we can see that eq. (5) is consistent with the results of ref. [19] using the relationship between C,, and C,, [20].
Finally, it is clear that the calculation of the variation of the phonons under pressure for aluminum
270
of
[5] L.F. [4] E. Zaremba Magaflaand andD. G.J. Zobin, Vázquez, Phys.to Rev. be published. Lett. 44 (1980) 175. [6] H. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B964. [7]W.KohnandL.J.Sham,Phys.Rev. 140(1965)All33. [8] M.S. Duesbery and R. Taylor, Phys. Rev. B 7 (1970) 2870. [9] 0. Gunnarson and B.I. Lundqvist, Phys. Rev. 13 (1976) 4274. [10] N. Boccara andG. Sarma, Physics 1(1965)219. [11] N.S. Gillis, N.R. Werthamer and T.R. Koeler, Phys. Rev. 165 (1968) 951. [12] E.R. Gowley and R.C. Shulda, Phys. Rev. B 4 (1974) 1261. [13] G. Gilat and L.J. Raubenheimer, Phys. Rev. 144 (1966) 390. [14] and G. Nilsson, Phys. Arts Rev.Sci. 145 74 (1966) 492. [15] R. P.W.Stedman Bridgman, Proc. Am. Acad. (1942) 425;
76 (1945)l,9,55;76(1948) 71;77 (1949) 189. [161 C.B. Walker, Phys. Rev. 103 (1956) 547. [17] D.C. Wallace, Phys. Rev. 187 (1969) 991. [18] D.C. Wallace, Phys. Rev. B 1(1970) 3963.
[19] J.P. Bastide and153. C. Loriers-Susse, High Temp. High Pressures7 (1975) [20] N.V. Fomin, Soy. Phys. Solid State 23 (1981) 299. [211W.F. Giauque and P.F. Meads, J. Am. Chem. Soc. 63 (1941)
63. [22] D.H. Howling, E. Mendoza and J.E. Zimmerman, Proc. R. Soc. A 229 (1955) 86.