- Email: [email protected]
Electronic specific heat of simple metals
J. Phys. Chem. Solids
Pergamon Press 1968. Vol. 29, pp. 2 113-2 115.
ELECTRONIC
SPECIFIC
HEAT
Printed in Great Britain.
OF SIMPLE
METALS*
A. WASSl?RMANt and X3. E. DeWITT Lawrence Radiation Laboratory, University of California, Livermore, Calif. 94550, U.S.A. (Received 4 October 1967; in revisedform 20 May 1968) Abstract-The effect of lattice periodicity on the electronic specific heat of alkali metals is calculated by extending the work of Bellemans and deLeener to finite temperature. Psuedopotentials for these metals are simply taken from band splitting data. The caicuiation reproduces moderately well the general trend of experimentally measured electronic specific heats. Inclusion of electronphonon effects have no marked effect on these trends but in most cases improve numerical agreement. RESULTS of recent years suggest that many metals may be regarded as nearly free electron in their behavior, the idea being perhaps especially appropriate for the alkali metals. It is, therefore, a mild surprise to find that measured electronic specific heats for these metals differ from the free electron values. It is becoming recognized that electron correlations are relatively unimportant in removing discrepancies [l-3] and that electronphonon interactions alone [4,5,11] do not provide some of the observed trends among the alkalis. In contrast, Ham[6] finds from detailed energy-band studies that for some of the alkali metals electron interaction with the periodic potential is not at all negligible. Nevertheless, a nearly free electron model appears to be a fruitful and convenient way to express metallic properties. Within the spirit of this approximation, the effect of lattice periodicity on the electronic specific heat is calculated below by finding the contribution to the grand potential from the first nonzero electron-lattice diagram. Apart from formalism, this is just an extension of part of the work of Bellemans and deLeener [7] to finite temperat~es. The crystal potential is written as V(r) = E s eiG.r (the G’s are vectors of the reciprocal lattice) with all but
the V,,, being discarded. This coefficient is most simply estimated from the band gaps at the point IV in the Brillouin zone. We write the grand potential as f&- = C&+Q, where sl, is the contribution from the noninteracting electron gas,* 1 R,=-pp-
m
312 I
z3’2
o ez-Br+
1 ck =
(1)
and QL is the lowest-order electron-lattice inte~~tions
contribution [S],
from
where <= (21+1)+/@+~, with p the chemical potential for the interacting system, p = ( KbT)-l, Kb the Boltzmann constant, T the temperature. Summing over 5 and taking C + 21 (27r)3 J dK, the lattice contribution b&omes
where f-
(z) = l/eP(z-P’ + 1. Integrating once,
*work performed under the auspices of the U.S. Atomic Energy Commission. tPresent address: Oregon State University, Oregon 9733 1, U.S.A.
Corvallis,
*Units are used in which A = 2m = 1, energies in rydbergs and tengths in Bohr radii. The volume of the crystal is unity. 2113
2114
A. WASSERMAN
and H. E. DEWITT
=-_N
(4) The last integration can be performed in the case of very low temperature
x
-gf
where N is the number of particles in the noninteracting system,
to give
N = /_~0~‘~/3$
(w” the chemical potential for free electrons) &find to order IVi12and T2
12(R,(Z2) +&&t2~+R2~E2)}
1%
G#O
(5)
with (6)
Ro(t2) = l+(~)logl~~ ?? 12<(1+5)
R1= and
x %(A +R’(Po) ( _ Bd (P0)%(P0) - & (cLo)%(PlJ) [ MPO)
II
(9)
where the primes refer to derivatives with respect to the chemical potential. Returning this expression into fiRyl. and taking C, = - T (PF/-z-2) c’, x
{enPwL(1-z2,Ei
[-
@+(
1 -
52)]
-
e-nS/a-52)
xm[M41-t2)lI. The functions Ei(x) and Ei* (x) are exponential integrals [9] and t” = 1G 12/4p. R2(t2) an asymptotic form if has, fortunately, n/311(1 - 5”) % 1, a condition easily satisfied for electron densities and crystal lattices of interest at temperatures below 1°K. For Cs at 1°K &A( 1 - 5”) is approximately lo3 and this is the smallest value for the metals considered here. Thus, even for what seems dangerously close to G2 - 4 p d
1
R2 = -m(1
(7)
which can now be combined with R, to give 77-2
Rl --+R2
P2P2
=
R
=-
12p2p2(1_-2)’
(8)
Upon eliminating the chemical potential from equations (1) and (2) with the condition
with C,O = the specific heat of the free gas. Substituting appropriately from above, C&,0
= 1+ 0*00919?-,4c
I VGJ2
GfO 1
Xp 1
-bog
w-1)
25
It-5 I
l-5
II
.
(11)
The result of applying equation (11) to specific metals is given in Table 1. In spite of deficiencies in the pseudopotential model used, this relatively simple calculation yields periodicity corrections to the electronic specific heats of alkali metals that are in fair agreement with those found by Ham[6] in direct band calculations. Taking account of additional Vc’s at the points P and H in the Brillouin zone would slightly improve agreement. Comparison with experiment is possible if electron-phonon and electron-
ELECTRONIC
SPECIFIC
HEAT OF SIMPLE
Table 1. Results ofequation Li r, V&a’
(C”/C,o)lb’ ( Cu/CuO)‘C’ (~C&O)~‘,,“,” (CiJICvO)erPeriment
3.25 0.105 1.30 166 0.14 _2.()@d’
Na
2115
(11)
K
3.93 0.009 1.0 1.0 0.13 1.25’d’
METALS
Rb
4.86 0.019 1.05 I.09 0.16
CS
5.20 0.032 1.18 1.21 0.18 1.26’“’
1.25(e)
5-63 0.044 1.47 1.76 0.22 1.43(e)
‘“‘Reference [6]. lb’Equation (1 1). “‘Reference[6]. ‘d’MARTIN D. L., Phys. Rev. 124,438 (1961). ‘e’LIEN W. H. and PHILLIPS N. E., Phys. Rev. 133,137OA (1964). (“Reference [5].
electron interactions are included. The former are taken from re,:ent work of Animalu and are shown in Ta.ple 1. If several current results [l-3] can be believed electronelectron effects are small and contribute negatively, but their actual magnitude is still open to question. It is clear that for sodium [ 111 and potassium deviations from free electron values are almost entirely due to phonon interactions. But for the other alkali metals periodicity effects cannot be ignored. The present result gives a compact expression for them. Apart from the nature of the model, the calculation suffers in some instances, from slow convergence and inability to completely describe band formation at the Brillouin zone boundaries. This defect should be particularly serious in the case of lithium. Here the large value of the pseudopotential likely requires a summation of higher order diagrams which can also produce mixing between states K and states K + G. No attempt has been made here to estimate the contribution of such terms. Other deviations from Ham’s results probably arise from the fact that the present calculation is based on minimal information about each metal, whereas Ham carried out detailed interpolation of the band structure from results at special points. Pseudopotentials of the simple analytic Hellman form r 101 were also used to comnute
the specific heat of alkali metals. Such forms could be a convenience if one were to pursue a theory of liquid metals since they are fitted from free atom spectral data independent of the solid, and are simple to handle arithmetically. The result for cesium, the heaviest alkali, is in exact agreement with the value in Table 1. However, the results grow progressively worse (too large) for the lighter metals. One is inclined to infer that a parallel exists between these Hellman forms and those from the Thomas-Fermi atom, i.e. the heavier the atom the better is the potential.
REFERENCES
1. HEDIN L., Phys. Rev. 139, A796 (1965). 2. WATABE M.,Prog. theor. Phys. 29,519 (1963). 3. GARRISON J., MORRISON H. L. and WONG J., Nuovo Cimento. B47.200 ( 1967). 4. QUINN J. J., The Fermi Surface (Edited by W. A. Harrison and M. B. Webb). Wiley, New York (1960). A. 0. E., BONSIGNORI F. and 5. ANIMALU BORTOLANI V., Nuovo Cimento B42,83, (1966). 6. HAM F., Phys. Rev. 128,82 (1962); Phys. Rev. 128, 2524 (1962).
7. BELLEMANS Chemical
A. and deLEENER M., Advances in Physics VI. Interscience, New York
(1964). 8. LUTTINGER
J. M. and WARD J. C., Phys. Rev. 118,1417 (1960). 9. ERDE’LYI A., et al., Higher Transcendental Functions, Vol. 2. McGraw-Hill, New York (1953). 10. SZASZ L. and McGINN G.,J. them. Phys. 42,2363 (1965). 11. ASHCROFT 14,285
(1965).
N. W. and WILKINS J. W., Phys. Lett.
Pergamon Press 1968. Vol. 29, pp. 2 113-2 115.
ELECTRONIC
SPECIFIC
HEAT
Printed in Great Britain.
OF SIMPLE
METALS*
A. WASSl?RMANt and X3. E. DeWITT Lawrence Radiation Laboratory, University of California, Livermore, Calif. 94550, U.S.A. (Received 4 October 1967; in revisedform 20 May 1968) Abstract-The effect of lattice periodicity on the electronic specific heat of alkali metals is calculated by extending the work of Bellemans and deLeener to finite temperature. Psuedopotentials for these metals are simply taken from band splitting data. The caicuiation reproduces moderately well the general trend of experimentally measured electronic specific heats. Inclusion of electronphonon effects have no marked effect on these trends but in most cases improve numerical agreement. RESULTS of recent years suggest that many metals may be regarded as nearly free electron in their behavior, the idea being perhaps especially appropriate for the alkali metals. It is, therefore, a mild surprise to find that measured electronic specific heats for these metals differ from the free electron values. It is becoming recognized that electron correlations are relatively unimportant in removing discrepancies [l-3] and that electronphonon interactions alone [4,5,11] do not provide some of the observed trends among the alkalis. In contrast, Ham[6] finds from detailed energy-band studies that for some of the alkali metals electron interaction with the periodic potential is not at all negligible. Nevertheless, a nearly free electron model appears to be a fruitful and convenient way to express metallic properties. Within the spirit of this approximation, the effect of lattice periodicity on the electronic specific heat is calculated below by finding the contribution to the grand potential from the first nonzero electron-lattice diagram. Apart from formalism, this is just an extension of part of the work of Bellemans and deLeener [7] to finite temperat~es. The crystal potential is written as V(r) = E s eiG.r (the G’s are vectors of the reciprocal lattice) with all but
the V,,, being discarded. This coefficient is most simply estimated from the band gaps at the point IV in the Brillouin zone. We write the grand potential as f&- = C&+Q, where sl, is the contribution from the noninteracting electron gas,* 1 R,=-pp-
m
312 I
z3’2
o ez-Br+
1 ck =
(1)
and QL is the lowest-order electron-lattice inte~~tions
contribution [S],
from
where <= (21+1)+/@+~, with p the chemical potential for the interacting system, p = ( KbT)-l, Kb the Boltzmann constant, T the temperature. Summing over 5 and taking C + 21 (27r)3 J dK, the lattice contribution b&omes
where f-
(z) = l/eP(z-P’ + 1. Integrating once,
*work performed under the auspices of the U.S. Atomic Energy Commission. tPresent address: Oregon State University, Oregon 9733 1, U.S.A.
Corvallis,
*Units are used in which A = 2m = 1, energies in rydbergs and tengths in Bohr radii. The volume of the crystal is unity. 2113
2114
A. WASSERMAN
and H. E. DEWITT
=-_N
(4) The last integration can be performed in the case of very low temperature
x
-gf
where N is the number of particles in the noninteracting system,
to give
N = /_~0~‘~/3$
(w” the chemical potential for free electrons) &find to order IVi12and T2
12(R,(Z2) +&&t2~+R2~E2)}
1%
G#O
(5)
with (6)
Ro(t2) = l+(~)logl~~ ?? 12<(1+5)
R1= and
x %(A +R’(Po) ( _ Bd (P0)%(P0) - & (cLo)%(PlJ) [ MPO)
II
(9)
where the primes refer to derivatives with respect to the chemical potential. Returning this expression into fiRyl. and taking C, = - T (PF/-z-2) c’, x
{enPwL(1-z2,Ei
[-
@+(
1 -
52)]
-
e-nS/a-52)
xm[M41-t2)lI. The functions Ei(x) and Ei* (x) are exponential integrals [9] and t” = 1G 12/4p. R2(t2) an asymptotic form if has, fortunately, n/311(1 - 5”) % 1, a condition easily satisfied for electron densities and crystal lattices of interest at temperatures below 1°K. For Cs at 1°K &A( 1 - 5”) is approximately lo3 and this is the smallest value for the metals considered here. Thus, even for what seems dangerously close to G2 - 4 p d
1
R2 = -m(1
(7)
which can now be combined with R, to give 77-2
Rl --+R2
P2P2
=
R
=-
12p2p2(1_-2)’
(8)
Upon eliminating the chemical potential from equations (1) and (2) with the condition
with C,O = the specific heat of the free gas. Substituting appropriately from above, C&,0
= 1+ 0*00919?-,4c
I VGJ2
GfO 1
Xp 1
-bog
w-1)
25
It-5 I
l-5
II
.
(11)
The result of applying equation (11) to specific metals is given in Table 1. In spite of deficiencies in the pseudopotential model used, this relatively simple calculation yields periodicity corrections to the electronic specific heats of alkali metals that are in fair agreement with those found by Ham[6] in direct band calculations. Taking account of additional Vc’s at the points P and H in the Brillouin zone would slightly improve agreement. Comparison with experiment is possible if electron-phonon and electron-
ELECTRONIC
SPECIFIC
HEAT OF SIMPLE
Table 1. Results ofequation Li r, V&a’
(C”/C,o)lb’ ( Cu/CuO)‘C’ (~C&O)~‘,,“,” (CiJICvO)erPeriment
3.25 0.105 1.30 166 0.14 _2.()@d’
Na
2115
(11)
K
3.93 0.009 1.0 1.0 0.13 1.25’d’
METALS
Rb
4.86 0.019 1.05 I.09 0.16
CS
5.20 0.032 1.18 1.21 0.18 1.26’“’
1.25(e)
5-63 0.044 1.47 1.76 0.22 1.43(e)
‘“‘Reference [6]. lb’Equation (1 1). “‘Reference[6]. ‘d’MARTIN D. L., Phys. Rev. 124,438 (1961). ‘e’LIEN W. H. and PHILLIPS N. E., Phys. Rev. 133,137OA (1964). (“Reference [5].
electron interactions are included. The former are taken from re,:ent work of Animalu and are shown in Ta.ple 1. If several current results [l-3] can be believed electronelectron effects are small and contribute negatively, but their actual magnitude is still open to question. It is clear that for sodium [ 111 and potassium deviations from free electron values are almost entirely due to phonon interactions. But for the other alkali metals periodicity effects cannot be ignored. The present result gives a compact expression for them. Apart from the nature of the model, the calculation suffers in some instances, from slow convergence and inability to completely describe band formation at the Brillouin zone boundaries. This defect should be particularly serious in the case of lithium. Here the large value of the pseudopotential likely requires a summation of higher order diagrams which can also produce mixing between states K and states K + G. No attempt has been made here to estimate the contribution of such terms. Other deviations from Ham’s results probably arise from the fact that the present calculation is based on minimal information about each metal, whereas Ham carried out detailed interpolation of the band structure from results at special points. Pseudopotentials of the simple analytic Hellman form r 101 were also used to comnute
the specific heat of alkali metals. Such forms could be a convenience if one were to pursue a theory of liquid metals since they are fitted from free atom spectral data independent of the solid, and are simple to handle arithmetically. The result for cesium, the heaviest alkali, is in exact agreement with the value in Table 1. However, the results grow progressively worse (too large) for the lighter metals. One is inclined to infer that a parallel exists between these Hellman forms and those from the Thomas-Fermi atom, i.e. the heavier the atom the better is the potential.
REFERENCES
1. HEDIN L., Phys. Rev. 139, A796 (1965). 2. WATABE M.,Prog. theor. Phys. 29,519 (1963). 3. GARRISON J., MORRISON H. L. and WONG J., Nuovo Cimento. B47.200 ( 1967). 4. QUINN J. J., The Fermi Surface (Edited by W. A. Harrison and M. B. Webb). Wiley, New York (1960). A. 0. E., BONSIGNORI F. and 5. ANIMALU BORTOLANI V., Nuovo Cimento B42,83, (1966). 6. HAM F., Phys. Rev. 128,82 (1962); Phys. Rev. 128, 2524 (1962).
7. BELLEMANS Chemical
A. and deLEENER M., Advances in Physics VI. Interscience, New York
(1964). 8. LUTTINGER
J. M. and WARD J. C., Phys. Rev. 118,1417 (1960). 9. ERDE’LYI A., et al., Higher Transcendental Functions, Vol. 2. McGraw-Hill, New York (1953). 10. SZASZ L. and McGINN G.,J. them. Phys. 42,2363 (1965). 11. ASHCROFT 14,285
(1965).
N. W. and WILKINS J. W., Phys. Lett.