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The lattice specific heat of Li and Na at high temperature and under pressure
PHYSICA
Physica B 179 (1992) 309-311 North-Holland
The lattice specific heat of Li and Na at high temperature and under pressure H . - M a t s u o Kagaya, K. Arai and T. Soma Department of Applied Physics and Mathematics, Mining College, Akita University, Akita 010, Japan Received 4 March 1992
Using our previous method with the anharmonic contributions from higher than cubic terms, the lattice specific heat of Li and Na is quantitatively calculated at high temperatures and under pressure. The obtained anharmonic coefficient for the specific heat A, defined by C~/3Nk = I + AT, for Li and Na decreases under pressure.
I. Introduction
The thermodynamical behaviour of materials under pressure is important in relation to the anharmonic effect. Alkali metals are an interesting subject to find an anharmonic effect due to a low melting point and high compression. Previously, the pressure dependence of the lattice specific heat for Li was studied theoretically, and a quadratic variation of the specific heat with pressure was predicted [1] from 90K to room temperature. Experimentally, for alkali metals, the lattice specific heat at constant volume Ca at high temperatures deviates largely from the constant value 3Nk [2]. The deviation of Cn is produced by the introduction of the anharmonic contributions from higher than cubic terms [3, 4]. Recently, we [5] have studied theoretically the anharmonic contributions to the specific heat of alkali metals such as K, Rb, Cs and alkali-alloys under pressure. In the present work, we report the obtained results of the pressure effect on the high-temperature lattice specific heat of alkali metals such as Li and Na.
Correspondence to: Dr. T. Soma, D e p a r t m e n t of Applied Physics and Math., Mining College, Akita University, Aki'ta 010, Japan.
2. A n h a r m o n i c coefficient for the specific heat
The deviation of C~ from the constant value
3Nk at high temperatures, namely the anharmonic coefficient for the specific heat A is given [3, 4] under pressure P by
C~,(P)/(3Nk) = 1 + A ( P ) T ,
(1)
A(P) = A2(P ) + A~(P) + A4(P ) ,
(2)
h2
A2(P)-
3M
V(2)(~
- - e f t ~,a~ 1 ,
p)
(kT) 3
k,
(3)
p)]2
[v(3)fR
A3(P ) = k[2C, f~(e) + ¼] t-elf, . (2)
~_,p)]3 ,
(4)
[Veff(R1,
and
kCJ4(p ) A,(P) -
2
p) ~2) . [VCff(n,, p)]2 l,ar (4) ( ~
--eff\*'l
(5)
The quantities A2(P), A3(P) and A4(P ) correspond to the effect of the thermal expansion, the cubic and quartic term, respectively. Then, V~z)CR v ~3)tp Yt/'(4)/'~ eff k'" 1 , P), --eff \ ' " 1 ' P) and --eff \ ' ' 1 P) represent the second, third and fourth derivative of the effective interatomic potential at the nearest-
0921-4526/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
310
H.-Matsuo Kagaya et al. / Lattice specific heat of Li and Na
neighbour distance R~ = V ~ a . (See refs. [3,4] and our previous work [5] with respect to the (# V- e( 3f)rf ~ R"l ) detailed contents of f~, f4, V(2) -~n~--L), and v(4)(# --elf,",)')
3. O b t a i n e d results and discussion
In performing numerical calculations, we introduce the conversion from the pressure P to the compressed volume (= 4a 3 = 4X/3R3,/9) using the equation of state [6] for alkali metals. When estimating f3(P) and f4(P), our lattice dynamical treatment [7, 8] is used based on the local H e i n e - A b a r e n k o v - t y p e pseudopotential I9]. In fig. l ( a ) - ( d ) , we give the P-dependence of A 2, A 3, A 4 and total A for Li and Na, respectively. In these figures, points are calculated data
with the H approximation [9] to the dielectric screening function and the full lines are leastsquares curves to the calculated data. In calculating A 2 ( P ) in eq. (3), we estimate A 2 ( P ) at the pressure-dependent melting temperature Tin(P) [101 of Li and Na. Then, in table 1, we show the obtained results of A2(P ), A3(P ), A4(P) and A ( P ) for Li and Na together with the variable interval due to the four approximations of the dielectric screening function [9]. From fig. l and table 1, we see that the pressure dependence of the anharmonic coefficient is monotonical for Li and Na, and the total anharmonic coefficient A ( P ) for the lattice specific heat decreases under pressure because of the tendency of A 3 ( P ) versus P. The numerical calculations were carried out with the ACOS-6 $2000 operating system in the Computer Center of T o h o k u University.
o
i •
i
; o,I
i
21
Na
OI I
I -0.2 l
0
o
ip(GPa)2,
z,
L~
0
.
.
(a)
.
I
.
.
.
P(GPa)
2
.
3
(b) 0.20(
l
o
o.L 51~
Li
Li
l
i
Y
~'0 -0.5
0
-I.0l
0
O.lO l i
0.05 i
I
P(GPo)
2
3
o
oL
I
P(GPa)
2
5 3
(d) Fig. 1. The P-dependence of (a) A,, (b) A ~, (c) A~ and (d) A for Li and Na. The points & and • at P = 0 GPa in fig. l(d) arc the observed data [2] for Li and Na. (c)
H.-Matsuo Kagaya et al. / Lattice specific heat of Li and Na
311
Table 1 A2, A~, A 4 a n d A (in 10 3K ~) o f Li a n d N a at v a r i o u s p r e s s u r e s (in G P a ) . P
0
I
2
3
A2
Li Na
-0.129 ~ -0.121 -[).056 - -0.052
- 1.060 ~ - 1.[)55 -[).401 - -0.398
-I).922 ~ -0.918 -0.329 - -I).326
-0.834 ~ -0.831 - 0 . 2 9 2 -- - 0 . 2 9 0
A~
Li Na
0.255 ~ 0.296 0 . 2 2 9 -- 0 . 2 7 8
0.205 ~ 0.238 0.158 ~ 0.195
0.173 ~ 0.199 0.121 - (). 148
0.155 ~ 0.174 0 . 1 0 5 ~ (). 124
A4
Li Na
-0.026 ~ -0.024 - 0.073 ~ - 0.069
-0.017 ~ -0.[)15 - 0.061 ~ - 0 . 0 5 8
-0.012 ~ -0.011 - 0 . 0 5 3 - - 0.051
-0.010 ~ -0.009 - 0 . 0 5 0 - - 0.1)48
A
Li Na
0 . 1 0 7 ~ (). 146 0 . 1 0 7 -- 0 . 1 5 2
0.085 ~ 0.116 0.060 ~ 0.094
0.071 ~ 0 . 0 9 5 0.038 ~ 0.062
0.063 ~ 0.080 0.()29 ~ 0.()45
Ai',h~
Li Na
0 . 2 0 _+ 0 . 0 2 0.15 -+ 0.01
~' F r o m d a t a b y M a r t i n [2].
References
[1] L . F . M a g a n a a n d G . T . V a z q u e z , J. P h y s . F 17 ( 1 9 8 5 ) L137. [2] D . L . M a r t i n , P h y s . R e v . 139 (1965) A 1 5 0 . [3] P.C. T r i v e d i , J. P h y s . F 1 ( 1 9 7 1 ) 262. [4] P.C. T r i v e d i , H , O . S h a r m a a n d L . S . K o t h a r i , Phys. R e v . B 18 ( 1 9 7 8 ) 2668.
[5] T. S o m a , T. S h i n k e a n d H . - M a t s u o K a g a y a , P h y s i c a B 176 ( 1 9 9 2 ) 93. [6] T. S o m a , J, P h y s . F 1() (19811) 1401. [7] T. S o m a , P h y s . S t a t . Sol. B 99 ( 1 9 8 0 ) 195. [8] T. S o m a , Y. K i m u r a a n d H . - M a t s u o K a g a y a , Phys. Stat. Sol. B 112 ( 1 9 8 2 ) 151. [9] T. S o m a , P h y s i c a B 97 ( 1 9 7 9 ) 76. [10] T. S o m a , H . - M a t s u o K a g a y a a n d M. N i s h i g a k i , Phys. S t a t . Sol. B 116 ( 1 9 8 3 ) 673.
Physica B 179 (1992) 309-311 North-Holland
The lattice specific heat of Li and Na at high temperature and under pressure H . - M a t s u o Kagaya, K. Arai and T. Soma Department of Applied Physics and Mathematics, Mining College, Akita University, Akita 010, Japan Received 4 March 1992
Using our previous method with the anharmonic contributions from higher than cubic terms, the lattice specific heat of Li and Na is quantitatively calculated at high temperatures and under pressure. The obtained anharmonic coefficient for the specific heat A, defined by C~/3Nk = I + AT, for Li and Na decreases under pressure.
I. Introduction
The thermodynamical behaviour of materials under pressure is important in relation to the anharmonic effect. Alkali metals are an interesting subject to find an anharmonic effect due to a low melting point and high compression. Previously, the pressure dependence of the lattice specific heat for Li was studied theoretically, and a quadratic variation of the specific heat with pressure was predicted [1] from 90K to room temperature. Experimentally, for alkali metals, the lattice specific heat at constant volume Ca at high temperatures deviates largely from the constant value 3Nk [2]. The deviation of Cn is produced by the introduction of the anharmonic contributions from higher than cubic terms [3, 4]. Recently, we [5] have studied theoretically the anharmonic contributions to the specific heat of alkali metals such as K, Rb, Cs and alkali-alloys under pressure. In the present work, we report the obtained results of the pressure effect on the high-temperature lattice specific heat of alkali metals such as Li and Na.
Correspondence to: Dr. T. Soma, D e p a r t m e n t of Applied Physics and Math., Mining College, Akita University, Aki'ta 010, Japan.
2. A n h a r m o n i c coefficient for the specific heat
The deviation of C~ from the constant value
3Nk at high temperatures, namely the anharmonic coefficient for the specific heat A is given [3, 4] under pressure P by
C~,(P)/(3Nk) = 1 + A ( P ) T ,
(1)
A(P) = A2(P ) + A~(P) + A4(P ) ,
(2)
h2
A2(P)-
3M
V(2)(~
- - e f t ~,a~ 1 ,
p)
(kT) 3
k,
(3)
p)]2
[v(3)fR
A3(P ) = k[2C, f~(e) + ¼] t-elf, . (2)
~_,p)]3 ,
(4)
[Veff(R1,
and
kCJ4(p ) A,(P) -
2
p) ~2) . [VCff(n,, p)]2 l,ar (4) ( ~
--eff\*'l
(5)
The quantities A2(P), A3(P) and A4(P ) correspond to the effect of the thermal expansion, the cubic and quartic term, respectively. Then, V~z)CR v ~3)tp Yt/'(4)/'~ eff k'" 1 , P), --eff \ ' " 1 ' P) and --eff \ ' ' 1 P) represent the second, third and fourth derivative of the effective interatomic potential at the nearest-
0921-4526/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
310
H.-Matsuo Kagaya et al. / Lattice specific heat of Li and Na
neighbour distance R~ = V ~ a . (See refs. [3,4] and our previous work [5] with respect to the (# V- e( 3f)rf ~ R"l ) detailed contents of f~, f4, V(2) -~n~--L), and v(4)(# --elf,",)')
3. O b t a i n e d results and discussion
In performing numerical calculations, we introduce the conversion from the pressure P to the compressed volume (= 4a 3 = 4X/3R3,/9) using the equation of state [6] for alkali metals. When estimating f3(P) and f4(P), our lattice dynamical treatment [7, 8] is used based on the local H e i n e - A b a r e n k o v - t y p e pseudopotential I9]. In fig. l ( a ) - ( d ) , we give the P-dependence of A 2, A 3, A 4 and total A for Li and Na, respectively. In these figures, points are calculated data
with the H approximation [9] to the dielectric screening function and the full lines are leastsquares curves to the calculated data. In calculating A 2 ( P ) in eq. (3), we estimate A 2 ( P ) at the pressure-dependent melting temperature Tin(P) [101 of Li and Na. Then, in table 1, we show the obtained results of A2(P ), A3(P ), A4(P) and A ( P ) for Li and Na together with the variable interval due to the four approximations of the dielectric screening function [9]. From fig. l and table 1, we see that the pressure dependence of the anharmonic coefficient is monotonical for Li and Na, and the total anharmonic coefficient A ( P ) for the lattice specific heat decreases under pressure because of the tendency of A 3 ( P ) versus P. The numerical calculations were carried out with the ACOS-6 $2000 operating system in the Computer Center of T o h o k u University.
o
i •
i
; o,I
i
21
Na
OI I
I -0.2 l
0
o
ip(GPa)2,
z,
L~
0
.
.
(a)
.
I
.
.
.
P(GPa)
2
.
3
(b) 0.20(
l
o
o.L 51~
Li
Li
l
i
Y
~'0 -0.5
0
-I.0l
0
O.lO l i
0.05 i
I
P(GPo)
2
3
o
oL
I
P(GPa)
2
5 3
(d) Fig. 1. The P-dependence of (a) A,, (b) A ~, (c) A~ and (d) A for Li and Na. The points & and • at P = 0 GPa in fig. l(d) arc the observed data [2] for Li and Na. (c)
H.-Matsuo Kagaya et al. / Lattice specific heat of Li and Na
311
Table 1 A2, A~, A 4 a n d A (in 10 3K ~) o f Li a n d N a at v a r i o u s p r e s s u r e s (in G P a ) . P
0
I
2
3
A2
Li Na
-0.129 ~ -0.121 -[).056 - -0.052
- 1.060 ~ - 1.[)55 -[).401 - -0.398
-I).922 ~ -0.918 -0.329 - -I).326
-0.834 ~ -0.831 - 0 . 2 9 2 -- - 0 . 2 9 0
A~
Li Na
0.255 ~ 0.296 0 . 2 2 9 -- 0 . 2 7 8
0.205 ~ 0.238 0.158 ~ 0.195
0.173 ~ 0.199 0.121 - (). 148
0.155 ~ 0.174 0 . 1 0 5 ~ (). 124
A4
Li Na
-0.026 ~ -0.024 - 0.073 ~ - 0.069
-0.017 ~ -0.[)15 - 0.061 ~ - 0 . 0 5 8
-0.012 ~ -0.011 - 0 . 0 5 3 - - 0.051
-0.010 ~ -0.009 - 0 . 0 5 0 - - 0.1)48
A
Li Na
0 . 1 0 7 ~ (). 146 0 . 1 0 7 -- 0 . 1 5 2
0.085 ~ 0.116 0.060 ~ 0.094
0.071 ~ 0 . 0 9 5 0.038 ~ 0.062
0.063 ~ 0.080 0.()29 ~ 0.()45
Ai',h~
Li Na
0 . 2 0 _+ 0 . 0 2 0.15 -+ 0.01
~' F r o m d a t a b y M a r t i n [2].
References
[1] L . F . M a g a n a a n d G . T . V a z q u e z , J. P h y s . F 17 ( 1 9 8 5 ) L137. [2] D . L . M a r t i n , P h y s . R e v . 139 (1965) A 1 5 0 . [3] P.C. T r i v e d i , J. P h y s . F 1 ( 1 9 7 1 ) 262. [4] P.C. T r i v e d i , H , O . S h a r m a a n d L . S . K o t h a r i , Phys. R e v . B 18 ( 1 9 7 8 ) 2668.
[5] T. S o m a , T. S h i n k e a n d H . - M a t s u o K a g a y a , P h y s i c a B 176 ( 1 9 9 2 ) 93. [6] T. S o m a , J, P h y s . F 1() (19811) 1401. [7] T. S o m a , P h y s . S t a t . Sol. B 99 ( 1 9 8 0 ) 195. [8] T. S o m a , Y. K i m u r a a n d H . - M a t s u o K a g a y a , Phys. Stat. Sol. B 112 ( 1 9 8 2 ) 151. [9] T. S o m a , P h y s i c a B 97 ( 1 9 7 9 ) 76. [10] T. S o m a , H . - M a t s u o K a g a y a a n d M. N i s h i g a k i , Phys. S t a t . Sol. B 116 ( 1 9 8 3 ) 673.