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A phenomenological model to calculate electronic Grüneisen parameters of metals
Volume 55A, number 6
PHYSICS LETTERS
12 January 1976
A PHENOMENOLOGICAL MODEL TO CALCULATE ELECTRONIC GRUNEISEN PARAMETERS OF METALS J.B. BASTOS FILHO and M.M. SHUKLA Instituto de Filica “Gleb Wataghin”, Unirersidade Estadual de Campinas, Campinas, S.F., Brasil Received 15 October 1975 By describing the asphericity of the Fermi surface by means of electronic density fluctuation on phenomenological basis, an empirical expression is developed to calculate ‘Ye of metals. The theory has been found to explain satisfactorily well the experimental results.
In spite of some efforts done by theoreticians [e.g. I and refs. thereini to explain the experimental observation of the electronic GrUneisen parameter, ‘ye’ of the metals at low temperatures, no successful model has yet been proposed. In the present note an attempt is made to tackle such a problem. In the quasi harmonic approximation, ‘Ye’ may be written as ‘Ye
=
~iI~ YkC
C0(Wj~)(l’~ “
4~2 h2 =
/3,~~ 2/3 I J x 2m \87r/
.
2/3
2’3,
V~
(4)
where m is atomic mass and V is atomic volume. Noting the fact that the electronic heat capacity of metals at sufficiently low temperatures is given by the relation
C0
=
4
2EF T
(5)
and use of equations (4), (5), (2) and (1) yields
The vibrational frequencies of metals, wk, depend on the free energy of the system which comprises of the lattice part and electronic part. At extremely low temperatures the lattice part contributes a negligible amount in comparison to the electronic part so the dynamics of the system is entirely governed by the electron gas. It is impossible to calculate exactly the vibrational spectrum of electron gas [2]. We have, thus, adopted the simplified theory of Overhauser [31which replaces the complete exciton spectrum of the electron gas by a single plasmon branch hw(q). For small q, w(q) approaches2q2/2m. the plasma anddoubt, for large Thefrequency theory, no proq it approaches h intermediate q range where the mises reality in the thermal energy is found to be quite close to /Iw(q). Postulating that at sufficiently low temperature, the thermal energy of the electron gas, 1~w(q),is nothing but a fraction of the Fermi energy of the free electrons, we have x
=—~
/
(2)
‘Yk=—dlnwk/dlnV
=—EF
~ w(q)
~2
0(0.)k) / ~ / all the modes whereallCthe modes 0 is the specific heat function and
hw(q)
So,
(3)
‘Ye2/3 (6) for all metals. A similar kind of result was also derived by Collins and White [4] from an entirely different approach. The experimental observations of Ye of metals deviates from a common value of ‘~e= 2/3. A critical analysis of the free electron gas model and the spherical Fermi surface of metals shows that these conditions are never fulfilled in the reality. A consideration of exchange effects among conduction electrons as well as a proper account of the Coulomb interaction between the lattice and electron gas would deviate from a spherical surface and free electron gas model. Lots Fermi of theoretical and the experimental research on band structures of metals also support an aspherical Fermi surface. Let us assume that the effect of a non spherical Fermi surface is to raise the Fermi energy of the free electron gas by an amount which corrects the effect of the non sphericity over the spherical Fermi surface. To account for such an effect on phenomenological basis let us assume that the basic effect which 361
Volume 55A, number 6
l’IIYSICS LETTERS
12 January 1976
causes a non spherical Fermi surface is the electronic density fluctuations in metals which were absent for a spherical Fermi surface. Defining I~to be the new Fermi energy associated with a non spherical Fermi surface, we have =
x(~) x(~)is an empirical
EF +
(7) of
function taking account the electronic density fluctuations in metals. The above equation has got specific validity for extremely low temperatures when the electronic medium become so dense that the electronic density of states behaves like a degenerate electronic gas. We can, thus, easily obtain the modified frequency of vibrations of metals where
given by the following relation
hit (3N 2/3 213 + = mx -~ V— 8ir 1
x(~)
(8)
It is well known that the electronic density in metals is inversely proportional to the atomic volume. Thus one can postulate that the density fluctuation function should also be proportional to some power of the inverse to the atomic volume. Or writting x(~)as 2 ~ 2/3 x(~) 2h ~ N213 V~ (9)
(
We have his (3\2/3 —) 8ir
mx
{v—213
+ V—~}N2/3
(10)
Once the value of s is known the calculation of ‘Y~ becomes straightforward. It is easy to show that ‘Ye = for s = 1/3 ‘Ye
=
2/3
for
s= 0
for s1 The electronic density fluctuations in metals have got a particular character for each metal which, thus, gives rise to a different value of ‘Ye~By knowing the experimental value of’Y~we can calculate the value of s for any metal. In table 1 we have calculated the value of s for 19 metals.
Table 1 Metal
Experimental Grüneisen parameter
sin expressions of
-
-
~lg
1.4
Al K V Fe Co Ni Cu Nb Mo Pd Ag Cd
1.61 2/3 1.65 2.1 1.9 2.0 0.91 1.5 1.5 2.1 1.18 0.5
0.95 0 0.99 1.44 I 24 1 34 0.25 0.84 0.84 1.44 0.52 —0.16
Ta W Re
1.3 0.2 3.5
0.54 —0.46 2.84
Pt Au
2.4 1.6 1.7
1.74 0.94 1.04
-
() 74
-——
for any metal should never be less than 2/3. Our result is in confirmation with the recent findings of Couchman and Reynolds [5]. On going through the article of Collins and White [4] we have discovered that the experimental values of ‘Ye of cadmium and tungsten are given by the following values W : 0.2 ±0.2,
Cd : 0.5 ±1.0 It indicates that there are great uncertainties in the experimental values of ‘Ye of these two metals. Also if we take the positive sign of the experimental error, ‘Ye for W is 0.4 very close to 2/3 and for cadmium 1.5 much above 2/3. Thus, for a clearification of our ideas a fresh determination of experimental ‘Ye of tungsten and cadmium is strongly recommended.
‘Ye~/~
A study of table 1 shows that the value ofs is negative for tungsten and cadmium; On the other hand, from equation (9) the minimum value of s is found to be zero which indicates that the minimum value of; 362
References Eli J.B. Bastos Filho and M.M. Shukla, Phys. Letters 53A (1975) 225. 121 KS. Singwi, A. Sjolander, M.P. Tosi and Rh. Land, Phys. Rev. BI (1970) 1044. [31 A.W. Overhauser, Phys. Rev. B6 (1971) 1888. 141 J.G. Collins and G.K. White, Progr. Low. Temp. Phys. l’v (1964) 450. [5] P.R. Couchman and C.L. Reynolds Jr. J. Phys. Chcm. Solids 36 (1975) 834.
PHYSICS LETTERS
12 January 1976
A PHENOMENOLOGICAL MODEL TO CALCULATE ELECTRONIC GRUNEISEN PARAMETERS OF METALS J.B. BASTOS FILHO and M.M. SHUKLA Instituto de Filica “Gleb Wataghin”, Unirersidade Estadual de Campinas, Campinas, S.F., Brasil Received 15 October 1975 By describing the asphericity of the Fermi surface by means of electronic density fluctuation on phenomenological basis, an empirical expression is developed to calculate ‘Ye of metals. The theory has been found to explain satisfactorily well the experimental results.
In spite of some efforts done by theoreticians [e.g. I and refs. thereini to explain the experimental observation of the electronic GrUneisen parameter, ‘ye’ of the metals at low temperatures, no successful model has yet been proposed. In the present note an attempt is made to tackle such a problem. In the quasi harmonic approximation, ‘Ye’ may be written as ‘Ye
=
~iI~ YkC
C0(Wj~)(l’~ “
4~2 h2 =
/3,~~ 2/3 I J x 2m \87r/
.
2/3
2’3,
V~
(4)
where m is atomic mass and V is atomic volume. Noting the fact that the electronic heat capacity of metals at sufficiently low temperatures is given by the relation
C0
=
4
2EF T
(5)
and use of equations (4), (5), (2) and (1) yields
The vibrational frequencies of metals, wk, depend on the free energy of the system which comprises of the lattice part and electronic part. At extremely low temperatures the lattice part contributes a negligible amount in comparison to the electronic part so the dynamics of the system is entirely governed by the electron gas. It is impossible to calculate exactly the vibrational spectrum of electron gas [2]. We have, thus, adopted the simplified theory of Overhauser [31which replaces the complete exciton spectrum of the electron gas by a single plasmon branch hw(q). For small q, w(q) approaches2q2/2m. the plasma anddoubt, for large Thefrequency theory, no proq it approaches h intermediate q range where the mises reality in the thermal energy is found to be quite close to /Iw(q). Postulating that at sufficiently low temperature, the thermal energy of the electron gas, 1~w(q),is nothing but a fraction of the Fermi energy of the free electrons, we have x
=—~
/
(2)
‘Yk=—dlnwk/dlnV
=—EF
~ w(q)
~2
0(0.)k) / ~ / all the modes whereallCthe modes 0 is the specific heat function and
hw(q)
So,
(3)
‘Ye2/3 (6) for all metals. A similar kind of result was also derived by Collins and White [4] from an entirely different approach. The experimental observations of Ye of metals deviates from a common value of ‘~e= 2/3. A critical analysis of the free electron gas model and the spherical Fermi surface of metals shows that these conditions are never fulfilled in the reality. A consideration of exchange effects among conduction electrons as well as a proper account of the Coulomb interaction between the lattice and electron gas would deviate from a spherical surface and free electron gas model. Lots Fermi of theoretical and the experimental research on band structures of metals also support an aspherical Fermi surface. Let us assume that the effect of a non spherical Fermi surface is to raise the Fermi energy of the free electron gas by an amount which corrects the effect of the non sphericity over the spherical Fermi surface. To account for such an effect on phenomenological basis let us assume that the basic effect which 361
Volume 55A, number 6
l’IIYSICS LETTERS
12 January 1976
causes a non spherical Fermi surface is the electronic density fluctuations in metals which were absent for a spherical Fermi surface. Defining I~to be the new Fermi energy associated with a non spherical Fermi surface, we have =
x(~) x(~)is an empirical
EF +
(7) of
function taking account the electronic density fluctuations in metals. The above equation has got specific validity for extremely low temperatures when the electronic medium become so dense that the electronic density of states behaves like a degenerate electronic gas. We can, thus, easily obtain the modified frequency of vibrations of metals where
given by the following relation
hit (3N 2/3 213 + = mx -~ V— 8ir 1
x(~)
(8)
It is well known that the electronic density in metals is inversely proportional to the atomic volume. Thus one can postulate that the density fluctuation function should also be proportional to some power of the inverse to the atomic volume. Or writting x(~)as 2 ~ 2/3 x(~) 2h ~ N213 V~ (9)
(
We have his (3\2/3 —) 8ir
mx
{v—213
+ V—~}N2/3
(10)
Once the value of s is known the calculation of ‘Y~ becomes straightforward. It is easy to show that ‘Ye = for s = 1/3 ‘Ye
=
2/3
for
s= 0
for s1 The electronic density fluctuations in metals have got a particular character for each metal which, thus, gives rise to a different value of ‘Ye~By knowing the experimental value of’Y~we can calculate the value of s for any metal. In table 1 we have calculated the value of s for 19 metals.
Table 1 Metal
Experimental Grüneisen parameter
sin expressions of
-
-
~lg
1.4
Al K V Fe Co Ni Cu Nb Mo Pd Ag Cd
1.61 2/3 1.65 2.1 1.9 2.0 0.91 1.5 1.5 2.1 1.18 0.5
0.95 0 0.99 1.44 I 24 1 34 0.25 0.84 0.84 1.44 0.52 —0.16
Ta W Re
1.3 0.2 3.5
0.54 —0.46 2.84
Pt Au
2.4 1.6 1.7
1.74 0.94 1.04
-
() 74
-——
for any metal should never be less than 2/3. Our result is in confirmation with the recent findings of Couchman and Reynolds [5]. On going through the article of Collins and White [4] we have discovered that the experimental values of ‘Ye of cadmium and tungsten are given by the following values W : 0.2 ±0.2,
Cd : 0.5 ±1.0 It indicates that there are great uncertainties in the experimental values of ‘Ye of these two metals. Also if we take the positive sign of the experimental error, ‘Ye for W is 0.4 very close to 2/3 and for cadmium 1.5 much above 2/3. Thus, for a clearification of our ideas a fresh determination of experimental ‘Ye of tungsten and cadmium is strongly recommended.
‘Ye~/~
A study of table 1 shows that the value ofs is negative for tungsten and cadmium; On the other hand, from equation (9) the minimum value of s is found to be zero which indicates that the minimum value of; 362
References Eli J.B. Bastos Filho and M.M. Shukla, Phys. Letters 53A (1975) 225. 121 KS. Singwi, A. Sjolander, M.P. Tosi and Rh. Land, Phys. Rev. BI (1970) 1044. [31 A.W. Overhauser, Phys. Rev. B6 (1971) 1888. 141 J.G. Collins and G.K. White, Progr. Low. Temp. Phys. l’v (1964) 450. [5] P.R. Couchman and C.L. Reynolds Jr. J. Phys. Chcm. Solids 36 (1975) 834.