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Finite temperature model-potential theory of metals
Physica 105A (1981) 607-619 @ North-Holland
Publishing Co.
FINITE TEMPERATURE MODEL-POTENTIAL THEORY OF METALS
J.L. PELISSIER Commissariat BP
b I’Energie Atomique, Centre d’Etudes 27-94190 Villeneuue-St-George& France
de Limeil,
Received 3 June 1980 Received in final form 1 October 1980
The well-known model-potential theory of simple metals is extended to finite temperature range and applied to Na, K and Al equation of state calculation up to a few eV. An approximate expression of temperature dependent exchange-correlation effects has been derived. The results are quite encouraging.
1. Introduction Statistical models (i.e. Thomas-Fermi and its various refinements) are extensively used in electronic contribution to the equation of state calculation, even if they have proved to be quite unsuccessful in predicting zero temperature equilibrium volumes and compressibilities. More sophisticated theories - for instance band structure methods - are successfully applied to compressed metals zero isotherm determination. In the case of simple -that is free electron like-metals, model-potential theory has proved, since Ashcroft’) and Langreth’s pioneering work, to be quite adequate at 0 K over a fair range of densities*). In the present paper, we try to develop a finite temperature model-potential method and apply it to Na, K and Al equation of state calculation. On this occasion, the electron gas exchange-correlation contribution had to be approximately taken into account at non-zero temperature. The theoretical background is briefly sketched out in section 2. Zero temperature and finite temperature results are discussed in sections 3 and 4, respectively. 607
608
J.L. PELISSIER
2. Model-potential formalism 2.1. General survey Using a local potential V,, the simple metal Hamiltonian of ionic motion) is given in second quantized form as*:
k,k’,q#0
k,qfO
(without allowance
vqa:aisakf+qak_q.
The first term H,,+C.k(k2/2+ E,/Z + Wp)a;ak is the unperturbed Hamiltonian in which the Ewald energy Ew and the non-Coulombic part of the model potential
have been incorporated in the usual way3). The second term describes ion-electron interaction via V,. The last term is the Coulombic electron-electron interaction (v, .= 4r/1Roq2). If neglecting the “perturbation” H - Ho, the free electron like Hamiltonian HO gives rise to the well-known free energy: F,J = Ew + Z(~O + WY) - (l/p) 2 ln[ 1 + eP(ro-rk)] with lt = k*/2, k
the unperturbed dition: 2 =
&f =
chemical
x
(nk)o =
potential
p. being determined
by the usual con-
C [ 1 + eP(ek-@o)]-l. k
k
When the perturbation is “switched on”, instead of handling “bare” interactions vq and V,, it is more convenient -as in zero temperature theory- to use renormalized fiq and Vq in which electron gas screening is readily taken into account. Lowest order corrections to the thermodynamic potential 0 namely a band structure term aas (see 2.2.) and an exchange-correlation contribution &c (see 2.3.) -will give rise to extra terms in the free energy: F = Fo +
COBS
+
flx~)~=~~.
(1)
The chemical potential is not altered, as the free energy F = R + PN is stationary with respect to I_L.All thermodynamic quantities of interest may be derived from eq. (1). * All quantities of interest are referred to a unit cell of volume 00, which contains one atom of valence Z. The temperature is denoted by T(p = l/T). Calculations are in atomic units throughout the paper.
MODEL-POTENTIAL
THEORY OF METALS
609
2.2. The band structure term We consider the renormalized vq =
electron-ion
interaction
in the form:
“q
l(q, w = 0)’
where the static dielectric constant E(q, 0) contains an exchange correction, the spirit of Hubbard’s4) modification of the RPA c(q, w) = (1 +
fq)[ERPA(%
w>
-
in
11,
with
I
fq = - 1 q’4:kf cRPA(q,
w)
=
s
1 -
uqn(q,
k, is a screening parameter, k, = kTF= (4k&r)“*
w)
which is about the Thomas-Fermi
value:
atOK.
As the q = 0 term has been treated separately, the lowest order diagram to be retained is a second order one (fig. 1); its contribution may be written, __----__x
a
---_ 3 _-__-___* _ -
b
_____
-3
-b-b
k+q
c
__---__-___
X
t
Fig. 1. (a) Ion-electron diagram to be retained.
interaction;
(b) screened
ion-electron
interaction;
(c) lowest
order
610
3.L. PELISSIER
according to the diagram evaluation flBS
=
rule.?):
(l/%3)z, ~qv-qg(kiwl)g(k + 9, . .
iWl),
with WI= (21+ l)r/p (where I is an integer) and g(k, iwl) = (iwr - Ek+ p)-‘, the prime on summation indicating the absence of the q = 0 term. The I-summation is easily performed6):
kiwi- lk + CL)-‘(iW/- Ek+q+ p)-’ = p hkh
Ek -
(nk+,),J Ek+q
With a purely local model-potential, the k-summation introducing the RPA dielectric constant:
is immediate
when
According to (for a static lattice): V, = V-, = VCSq+, where G is a reciprocal lattice vector, we obtain the generalization of the well-known zero temperature “band structure energy”:
2.3. The exchange-correlation
term
We consider the w-dependent screening of the Coulombic that: Q(w) = uq/e(q, w). The corresponding mass operator C is given by’):
potential,
so
Z(k, iy) = - l/p 3 b&w, - iwl,)g(k + q, iwl,). If we multiply each interaction have’): * aRLc -=ah
vertex
v, by the coupling constant
1 Z(k, iwl) !@ z g-‘(k, iw,) - Z(k, iwr)’
At the lowest order, we may neglect the mass operator and write explicitly: h
A, we
aac_ _ ah
the I-summation
1
w
? ,I’ kq.
g(k + q, iwk(k
and the k-summation
iv)
in the denominator
AU,
1 - hv,(l + f,)II(q, iwl - iwjf)’
are easily performed
as in 2.2. and we
MODEL-POTENTIAL
THEORY
611
OF METALS
get: A aflic _ - ah - -$z
1_
rt$~~~~$‘~,
iwn)
with W, = y
(n integer).
But in doing so, we have introduced a spurious first order term, instead of the well-known exchange term Rx = - : Ct, ~,(n&(n~+,)~. So we have:
and 1
oxc = (aRxc/ah)dh= 0, + Rc, I0 with a, = +jz
(&ln[l
- (I+ f,)u,TVq, iwn)l + v,II(q, iwn) .
(2)
Without exchange correction (f, = 0), it is a well-known result?. The second term in the correlation contribution is easily evaluated: a$ = (1/2p) 2 v,lXq, iwn) q.n =(1/2p)
2
lt
~~CC~k+~X-C,U[g(,k+,-
+hJ']
(nk+q)o-hk>O =:pl
exp[P(ek+q-s)l-
1'
After some evident manipulations,
0: = i z
u,(bk+,)O(nk>O
where we have introduced w,(wi = 4?rZ/Ro). The first term
-
bk)O)
we get: =
-
flX
-
$12 c l/q*, 4
in the self-energy
term the plasma
pulsation
is much more difficult to evaluate. So that we shall invoke the so-called “plasmon-pole” approximation to calculate it. Lundqvist”) has introduced this approximation at 0 K and we may generalize it in the following form:
612 with
J.L. PELISSIER W: =
v&q,
wi(l 0)
+ aq* + bq4), the constants -
l/w*
and
¶-+O
QI(q,O)
a and b being determined
by:
- l/bq4. q-m
The corresponding approximate dielectric function retains the main features of the RPA: a plasmon pole for w = wq and a good static screening at both small and large wavenumbers. Now 0; could be written in a much more manageable form:
where we have used the new variable u* = (p2/4~*)wi(aq2 + bq4 + A + Af,), uI and u2 corresponding, respectively, to h = 0 and A = 1. The n-summation is immediately done using the identity: CiL l/(n*+ u*) = (r/u)coth(ru), and the subsequent u-integration is evident. We get:
1- exp[ - pw,(g, + A:)‘/*
1-
exp( - PwpAq)
+ ?[(A:
+ gq)“* - A,] ,
with the following definitions: g, = 1 + f, and Ai = aq2 + bq4. We shall return to the C!‘, calculation. From eq. (2), at large wavenumber 0: cancels the leading term of 0;. So that, for q > qc - where qc is a certain cut-off wavenumber -it is judicious to evaluate 0: in the plasmon pole approximation too. The cut-off qc is evidently determined from the condition ( vJI(q, iwn) ( = 1 for q = qc and all values of w,,. Using the plasmon pole approximation, we obtain: aqi -t bq’&= 1 and
flc(plasmon
pole) = - wp/4 x (l/A,)coth(:Pw,A,). cl
After having collected the various terms, we may write: fJxc=fix+*
1
Vqbk+,)O(nk)O
(3)
MODEL-POTENTIAL
THEORY OF METALS
613
3. Zero temperature results 3.1. Correlation energy In table I, the results obtained from eqs. (1) and (3) are compared in the range of metallic densities (r, being the electron gas parameter: (4/3)7& = C&/Z) and at zero temperature with two well-known expressions of the correlation energy - namely Pines-Nozieres”) and Wigner12) formulas. The exchange correction f, used in our calculation is that of Rice13). We see a quite fair agreement which subsists when we compare our calculations without exchange correction (f, = 0) with RPA Lam’s results’4) in table II. 3.2. Exchange-correlation
contribution
to the thermal efectiue
mass
We define it according to: am& =
a2nXClaT2
a2FolaT2
1
T=fj
It is characteristic for the interacting electron gas at low temperature. Table III shows a good agreement between our results with exchange correction, those of Silverstein (corrected by Rice13), who used the PinesNozieres approximation and those of Rice13) using Hubbard’s method. Our results with f, = 0 and the RPA Lam’s calculationsr4) are given in table IV.
TABLE I Correlation energy (in lo-’ u.a.)
rs
1
2
3
4
5
Pines-Nozieres Wigner present work
- 57.5 - 50.
- 46.75 - 44.9
- 40.45 - 40.75
- 36.0 - 37.3
- 32.55 -31.9
- 55.7
-45.1
- 39.8
- 36.35
- 33.8
TABLE II Correlation energy, calculated without exchange correction.
rS
1
2
3
4
5
Lam present work
- 78.5 - 63.5
-61.5 - 50.7
- 52.5 - 44.2
- 46.6 - 40.0
- 42.3 - 36.9
614
J.L. PELISSIER TABLE III
Smffc rs
1
2
3
4
5
Silverstein Rice present work
- 0.04 - 0.029
+ 0.02 - 0.01 + 0.002
0.05 0.02 + 0.041
+ 0.10 0.06 + 0.080
+0.117
TABLE IV 6m kc calculated without exchange correction. r,
1
2
3
4
5
Lam present work
- 0.032 - 0.029
- 0.008 + 0.001
+ 0.026 + 0.039
+ 0.064 + 0.078
+0.105 +0.115
We notice that &I& is quite insensitive to the exchange correction and almost negligible compared to the free electron contribution. As our approximation of exchange-correlation effects seems reliable at 0 K and exhibits a good temperature derivative, we may expect sensible results when using it over a moderate temperature range. 3.3. Model-potential
determination
Since the present study is limited to local model-potentials, we have restricted our purpose to the most free electron like metals, in which non local effects are presumably negligible: Na, K and Al. We have used Vaks et a1.2)model-potential, which is (in real space) a square well of depth A and radius r,. Their values of the parameters for Na and K, and those of Senoo et a1.15)for Al were very slightly amended, as we used our correlation energy instead of Pines-Nozieres one (see table V). TABLE V Model-potential parameters A and r, (in a.u.) as given in ref. 2 or ref. 15, compared with the values used in the present work.
Na K Al
Ref. 2 or Ref. 15
This work
A
r,
A
r,
0.1872 0.1935 0.8815
2.095 3.042 1.350
0.1872 0.1965 0.8815
2.074 2.995 1.342
MODEL-POTENTIAL I
THEORY OF METALS
615
A’ .i-‘:/* .
_ 1.00
,_/+Y +7+L
010
./
/+ _
0.01
+
J
1
15
3
2
4
Fig. 2. Zero-temperature isotherm of Na (electronic pressure in Mbar versus compression). Thomas-Fermi; 0 Thomas-Fermi-Dirac; * APW (ref. 16); + model potential.
V
The question of the range of validity of such calculations immediately arises: as we have supposed that the valence, the well depth and the core radius remain unchanged when the atomic volume is reduced, compressioninduced ionization leads to a severe limitation in the high density region. As can be seen_ on figs. 2, 3 and 4, model-potential results are very close to the self-consistent APW equation of state calculations made by Perrot’6) up to compression (i.e. volume reduction) 2. In the case of aluminium, on the ground of McMahan and Ross conclusions”), model-potential method range 1.00 _
0.10
/
,://‘. A++~ / I J /
I
Fig. 3. Zero temperature
/’
I
isotherm of K (Same symbols as in fig. 2).
J.L. PELISSIER
616
10.0
1.00
0.10
Fig. 4. Zero femperature
isotherm of Al (same symbols as in fig. 2).
of application seems limited to compression 3. In the case of sodium and potassium, the appearance of imaginary frequencies’) in the phonon spectrum takes place respectively at compression 4 and 3.3. So that compression 3 was also retained as an upper bound in Na and K.
4. Finite temperature results 4.1. Interacting
electron gas behavior
Among the various results that may be obtained from eq. (3), the temperature variation of the Helmoltz free energy was retained. The ratio [R&T) - &(O)]/[Fo(T) - F,-,(O)], which is characteristic of the relative importance of exchange-correlation correction, is displayed versus temperature on fig. 5 for a typical metallic density, namely that of sodium (Y, = 3.93). As could be seen, exchange-correlation effects represent only a few per cent of the free electron gas Helmoltz energy and become completely negligible about 15 eV. The correlations cancel the major part of the exchange contribution: the well-known logarithmic singularity of the specific heat at T = 0 K disappears and, for instance at 5 eV, the exchange correction is reduced from - 20% to - 5% when the correlations are taken into account. 4.2. Complete results Two isotherms obtained by the model potential method are compared with the corresponding Thomas-Fermi-Dirac results”) on figs. 6, 7 and 8 respec-
MODEL-POTENTIAL
THEORY OF METALS
Fig. 5. Aflxc/AF~ (in percentage) versus temperature
1
1.5
2
617
(in eV) for r, = 3.93.
3
Fig. 6. Finite temperature isotherms of Na (electronic pressure in Mbar versus compression). Thomas-Fermi-Dirac; + model potential.
0
tively for Na, K and Al. A quite similar behaviour and an increasing agreement between the two models when the temperature is raised may be noticed. Temperature induced ionization limits the temperature range of validity in the case of model potential calculations. We have tried to estimate roughly this effect in the following way: if, at 0 K, the highest core level is occupied by Zc electrons and distant of AE from the bottom of the free electron like
618
J.L. PELISSIER
Fig. 7. Finite temperature
isotherms of K (same symbols as in fig. 6).
band (as given, for instance, by a compressed atom self-consistent culation’*), the corresponding ionization is taken as: &z=zc
{
l-
1 1 + exp[ - p(I AE I+ k)]
’
where the free electron chemical potential is supposed unaffected.
r 4
IO /
./
+
IOeV
/
/_-;-
+ ./
/ 5eV 1
:/ t 1
Fig. 8. Finite temperature
1.5
2
3
isotherms of Al (same symbols as in fig. 6).
cal-
MODEL-POTENTIAL
THEORY OF METALS
619
At T = 3 eV for Na and K, and T = 10 eV for Al, the valence modification is respectively of l%, 5% and 1%. In the case of potassium, it may be considered as the very extreme limit of validity.
5. Conclusion In the case of simple metals, model-potential theory seems to be a quite valuable tool at finite temperature too, in a limited range of temperatures and densities.
Acknowledgment The author is greatly indebted to F. Perrot (Centre d’Etudes de Limeil) for having supplied him with many unpublished results.
References 1) N.W. Ashcroft and D.C. Langreth, Phys. Rev. 155 (1967) 682. 2) V.G. Vaks, S.P. Kravchuk and A.V. Trefilov, Fiz. Tverd. Tela (Leningrad) 19 (1977) 1271 [Sov. Phys. Solid State 19 (1977) 7401. 3) J. Hammerberg and N.W. Ashcroft, Phys. Rev. B9 (1974) 409. 4) J. Hubbard, Proc. Roy. Sot. A240 (1957) 539. 5) W.E. Parry, The Many-Body Problem (Clarendon, Oxford, 1973). 6) Ref. 5, appendix D. 7) Ref. 5, p. 114. 8) Ref. 5, p. 67. 9) D.J. Thouless, The Quantum Mechanics of Many-Body Systems (AC. Press, New York and London, 1961) p. 139. 10) B.I. Lundqvist, Physik Kond. Mat. 6 (1967) 206. 11) P. Nozieres and D. Pines, Phys. Rev. 111 (1958) 442. 12) E.P. Wigner, Trans. Faraday Sot. 34 (1938) 678. 13) T.M. Rice, Ann. Phys. (N.Y.) 31 (1%5) 100. 14) J. Lam, Phys. Rev. B5 (1972) 1254. 15) M. Senoo, H. Mii and I. Fujishiro, J. Phys. Sot. Japan 41 (1976) 1562. 16) F. Perrot, private communication, 17) A.K. McMahan and M. Ross, High-Pressure Science and Technology vol. 2, K.D. Timmerhaus and M.S. Barber, eds. (Plenum, New York, 1979) p. 920. 18) F. Perrot, private communication.
Publishing Co.
FINITE TEMPERATURE MODEL-POTENTIAL THEORY OF METALS
J.L. PELISSIER Commissariat BP
b I’Energie Atomique, Centre d’Etudes 27-94190 Villeneuue-St-George& France
de Limeil,
Received 3 June 1980 Received in final form 1 October 1980
The well-known model-potential theory of simple metals is extended to finite temperature range and applied to Na, K and Al equation of state calculation up to a few eV. An approximate expression of temperature dependent exchange-correlation effects has been derived. The results are quite encouraging.
1. Introduction Statistical models (i.e. Thomas-Fermi and its various refinements) are extensively used in electronic contribution to the equation of state calculation, even if they have proved to be quite unsuccessful in predicting zero temperature equilibrium volumes and compressibilities. More sophisticated theories - for instance band structure methods - are successfully applied to compressed metals zero isotherm determination. In the case of simple -that is free electron like-metals, model-potential theory has proved, since Ashcroft’) and Langreth’s pioneering work, to be quite adequate at 0 K over a fair range of densities*). In the present paper, we try to develop a finite temperature model-potential method and apply it to Na, K and Al equation of state calculation. On this occasion, the electron gas exchange-correlation contribution had to be approximately taken into account at non-zero temperature. The theoretical background is briefly sketched out in section 2. Zero temperature and finite temperature results are discussed in sections 3 and 4, respectively. 607
608
J.L. PELISSIER
2. Model-potential formalism 2.1. General survey Using a local potential V,, the simple metal Hamiltonian of ionic motion) is given in second quantized form as*:
k,k’,q#0
k,qfO
(without allowance
vqa:aisakf+qak_q.
The first term H,,+C.k(k2/2+ E,/Z + Wp)a;ak is the unperturbed Hamiltonian in which the Ewald energy Ew and the non-Coulombic part of the model potential
have been incorporated in the usual way3). The second term describes ion-electron interaction via V,. The last term is the Coulombic electron-electron interaction (v, .= 4r/1Roq2). If neglecting the “perturbation” H - Ho, the free electron like Hamiltonian HO gives rise to the well-known free energy: F,J = Ew + Z(~O + WY) - (l/p) 2 ln[ 1 + eP(ro-rk)] with lt = k*/2, k
the unperturbed dition: 2 =
&f =
chemical
x
(nk)o =
potential
p. being determined
by the usual con-
C [ 1 + eP(ek-@o)]-l. k
k
When the perturbation is “switched on”, instead of handling “bare” interactions vq and V,, it is more convenient -as in zero temperature theory- to use renormalized fiq and Vq in which electron gas screening is readily taken into account. Lowest order corrections to the thermodynamic potential 0 namely a band structure term aas (see 2.2.) and an exchange-correlation contribution &c (see 2.3.) -will give rise to extra terms in the free energy: F = Fo +
COBS
+
flx~)~=~~.
(1)
The chemical potential is not altered, as the free energy F = R + PN is stationary with respect to I_L.All thermodynamic quantities of interest may be derived from eq. (1). * All quantities of interest are referred to a unit cell of volume 00, which contains one atom of valence Z. The temperature is denoted by T(p = l/T). Calculations are in atomic units throughout the paper.
MODEL-POTENTIAL
THEORY OF METALS
609
2.2. The band structure term We consider the renormalized vq =
electron-ion
interaction
in the form:
“q
l(q, w = 0)’
where the static dielectric constant E(q, 0) contains an exchange correction, the spirit of Hubbard’s4) modification of the RPA c(q, w) = (1 +
fq)[ERPA(%
w>
-
in
11,
with
I
fq = - 1 q’4:kf cRPA(q,
w)
=
s
1 -
uqn(q,
k, is a screening parameter, k, = kTF= (4k&r)“*
w)
which is about the Thomas-Fermi
value:
atOK.
As the q = 0 term has been treated separately, the lowest order diagram to be retained is a second order one (fig. 1); its contribution may be written, __----__x
a
---_ 3 _-__-___* _ -
b
_____
-3
-b-b
k+q
c
__---__-___
X
t
Fig. 1. (a) Ion-electron diagram to be retained.
interaction;
(b) screened
ion-electron
interaction;
(c) lowest
order
610
3.L. PELISSIER
according to the diagram evaluation flBS
=
rule.?):
(l/%3)z, ~qv-qg(kiwl)g(k + 9, . .
iWl),
with WI= (21+ l)r/p (where I is an integer) and g(k, iwl) = (iwr - Ek+ p)-‘, the prime on summation indicating the absence of the q = 0 term. The I-summation is easily performed6):
kiwi- lk + CL)-‘(iW/- Ek+q+ p)-’ = p hkh
Ek -
(nk+,),J Ek+q
With a purely local model-potential, the k-summation introducing the RPA dielectric constant:
is immediate
when
According to (for a static lattice): V, = V-, = VCSq+, where G is a reciprocal lattice vector, we obtain the generalization of the well-known zero temperature “band structure energy”:
2.3. The exchange-correlation
term
We consider the w-dependent screening of the Coulombic that: Q(w) = uq/e(q, w). The corresponding mass operator C is given by’):
potential,
so
Z(k, iy) = - l/p 3 b&w, - iwl,)g(k + q, iwl,). If we multiply each interaction have’): * aRLc -=ah
vertex
v, by the coupling constant
1 Z(k, iwl) !@ z g-‘(k, iw,) - Z(k, iwr)’
At the lowest order, we may neglect the mass operator and write explicitly: h
A, we
aac_ _ ah
the I-summation
1
w
? ,I’ kq.
g(k + q, iwk(k
and the k-summation
iv)
in the denominator
AU,
1 - hv,(l + f,)II(q, iwl - iwjf)’
are easily performed
as in 2.2. and we
MODEL-POTENTIAL
THEORY
611
OF METALS
get: A aflic _ - ah - -$z
1_
rt$~~~~$‘~,
iwn)
with W, = y
(n integer).
But in doing so, we have introduced a spurious first order term, instead of the well-known exchange term Rx = - : Ct, ~,(n&(n~+,)~. So we have:
and 1
oxc = (aRxc/ah)dh= 0, + Rc, I0 with a, = +jz
(&ln[l
- (I+ f,)u,TVq, iwn)l + v,II(q, iwn) .
(2)
Without exchange correction (f, = 0), it is a well-known result?. The second term in the correlation contribution is easily evaluated: a$ = (1/2p) 2 v,lXq, iwn) q.n =(1/2p)
2
lt
~~CC~k+~X-C,U[g(,k+,-
+hJ']
(nk+q)o-hk>O =:pl
exp[P(ek+q-s)l-
1'
After some evident manipulations,
0: = i z
u,(bk+,)O(nk>O
where we have introduced w,(wi = 4?rZ/Ro). The first term
-
bk)O)
we get: =
-
flX
-
$12 c l/q*, 4
in the self-energy
term the plasma
pulsation
is much more difficult to evaluate. So that we shall invoke the so-called “plasmon-pole” approximation to calculate it. Lundqvist”) has introduced this approximation at 0 K and we may generalize it in the following form:
612 with
J.L. PELISSIER W: =
v&q,
wi(l 0)
+ aq* + bq4), the constants -
l/w*
and
¶-+O
QI(q,O)
a and b being determined
by:
- l/bq4. q-m
The corresponding approximate dielectric function retains the main features of the RPA: a plasmon pole for w = wq and a good static screening at both small and large wavenumbers. Now 0; could be written in a much more manageable form:
where we have used the new variable u* = (p2/4~*)wi(aq2 + bq4 + A + Af,), uI and u2 corresponding, respectively, to h = 0 and A = 1. The n-summation is immediately done using the identity: CiL l/(n*+ u*) = (r/u)coth(ru), and the subsequent u-integration is evident. We get:
1- exp[ - pw,(g, + A:)‘/*
1-
exp( - PwpAq)
+ ?[(A:
+ gq)“* - A,] ,
with the following definitions: g, = 1 + f, and Ai = aq2 + bq4. We shall return to the C!‘, calculation. From eq. (2), at large wavenumber 0: cancels the leading term of 0;. So that, for q > qc - where qc is a certain cut-off wavenumber -it is judicious to evaluate 0: in the plasmon pole approximation too. The cut-off qc is evidently determined from the condition ( vJI(q, iwn) ( = 1 for q = qc and all values of w,,. Using the plasmon pole approximation, we obtain: aqi -t bq’&= 1 and
flc(plasmon
pole) = - wp/4 x (l/A,)coth(:Pw,A,). cl
After having collected the various terms, we may write: fJxc=fix+*
1
Vqbk+,)O(nk)O
(3)
MODEL-POTENTIAL
THEORY OF METALS
613
3. Zero temperature results 3.1. Correlation energy In table I, the results obtained from eqs. (1) and (3) are compared in the range of metallic densities (r, being the electron gas parameter: (4/3)7& = C&/Z) and at zero temperature with two well-known expressions of the correlation energy - namely Pines-Nozieres”) and Wigner12) formulas. The exchange correction f, used in our calculation is that of Rice13). We see a quite fair agreement which subsists when we compare our calculations without exchange correction (f, = 0) with RPA Lam’s results’4) in table II. 3.2. Exchange-correlation
contribution
to the thermal efectiue
mass
We define it according to: am& =
a2nXClaT2
a2FolaT2
1
T=fj
It is characteristic for the interacting electron gas at low temperature. Table III shows a good agreement between our results with exchange correction, those of Silverstein (corrected by Rice13), who used the PinesNozieres approximation and those of Rice13) using Hubbard’s method. Our results with f, = 0 and the RPA Lam’s calculationsr4) are given in table IV.
TABLE I Correlation energy (in lo-’ u.a.)
rs
1
2
3
4
5
Pines-Nozieres Wigner present work
- 57.5 - 50.
- 46.75 - 44.9
- 40.45 - 40.75
- 36.0 - 37.3
- 32.55 -31.9
- 55.7
-45.1
- 39.8
- 36.35
- 33.8
TABLE II Correlation energy, calculated without exchange correction.
rS
1
2
3
4
5
Lam present work
- 78.5 - 63.5
-61.5 - 50.7
- 52.5 - 44.2
- 46.6 - 40.0
- 42.3 - 36.9
614
J.L. PELISSIER TABLE III
Smffc rs
1
2
3
4
5
Silverstein Rice present work
- 0.04 - 0.029
+ 0.02 - 0.01 + 0.002
0.05 0.02 + 0.041
+ 0.10 0.06 + 0.080
+0.117
TABLE IV 6m kc calculated without exchange correction. r,
1
2
3
4
5
Lam present work
- 0.032 - 0.029
- 0.008 + 0.001
+ 0.026 + 0.039
+ 0.064 + 0.078
+0.105 +0.115
We notice that &I& is quite insensitive to the exchange correction and almost negligible compared to the free electron contribution. As our approximation of exchange-correlation effects seems reliable at 0 K and exhibits a good temperature derivative, we may expect sensible results when using it over a moderate temperature range. 3.3. Model-potential
determination
Since the present study is limited to local model-potentials, we have restricted our purpose to the most free electron like metals, in which non local effects are presumably negligible: Na, K and Al. We have used Vaks et a1.2)model-potential, which is (in real space) a square well of depth A and radius r,. Their values of the parameters for Na and K, and those of Senoo et a1.15)for Al were very slightly amended, as we used our correlation energy instead of Pines-Nozieres one (see table V). TABLE V Model-potential parameters A and r, (in a.u.) as given in ref. 2 or ref. 15, compared with the values used in the present work.
Na K Al
Ref. 2 or Ref. 15
This work
A
r,
A
r,
0.1872 0.1935 0.8815
2.095 3.042 1.350
0.1872 0.1965 0.8815
2.074 2.995 1.342
MODEL-POTENTIAL I
THEORY OF METALS
615
A’ .i-‘:/* .
_ 1.00
,_/+Y +7+L
010
./
/+ _
0.01
+
J
1
15
3
2
4
Fig. 2. Zero-temperature isotherm of Na (electronic pressure in Mbar versus compression). Thomas-Fermi; 0 Thomas-Fermi-Dirac; * APW (ref. 16); + model potential.
V
The question of the range of validity of such calculations immediately arises: as we have supposed that the valence, the well depth and the core radius remain unchanged when the atomic volume is reduced, compressioninduced ionization leads to a severe limitation in the high density region. As can be seen_ on figs. 2, 3 and 4, model-potential results are very close to the self-consistent APW equation of state calculations made by Perrot’6) up to compression (i.e. volume reduction) 2. In the case of aluminium, on the ground of McMahan and Ross conclusions”), model-potential method range 1.00 _
0.10
/
,://‘. A++~ / I J /
I
Fig. 3. Zero temperature
/’
I
isotherm of K (Same symbols as in fig. 2).
J.L. PELISSIER
616
10.0
1.00
0.10
Fig. 4. Zero femperature
isotherm of Al (same symbols as in fig. 2).
of application seems limited to compression 3. In the case of sodium and potassium, the appearance of imaginary frequencies’) in the phonon spectrum takes place respectively at compression 4 and 3.3. So that compression 3 was also retained as an upper bound in Na and K.
4. Finite temperature results 4.1. Interacting
electron gas behavior
Among the various results that may be obtained from eq. (3), the temperature variation of the Helmoltz free energy was retained. The ratio [R&T) - &(O)]/[Fo(T) - F,-,(O)], which is characteristic of the relative importance of exchange-correlation correction, is displayed versus temperature on fig. 5 for a typical metallic density, namely that of sodium (Y, = 3.93). As could be seen, exchange-correlation effects represent only a few per cent of the free electron gas Helmoltz energy and become completely negligible about 15 eV. The correlations cancel the major part of the exchange contribution: the well-known logarithmic singularity of the specific heat at T = 0 K disappears and, for instance at 5 eV, the exchange correction is reduced from - 20% to - 5% when the correlations are taken into account. 4.2. Complete results Two isotherms obtained by the model potential method are compared with the corresponding Thomas-Fermi-Dirac results”) on figs. 6, 7 and 8 respec-
MODEL-POTENTIAL
THEORY OF METALS
Fig. 5. Aflxc/AF~ (in percentage) versus temperature
1
1.5
2
617
(in eV) for r, = 3.93.
3
Fig. 6. Finite temperature isotherms of Na (electronic pressure in Mbar versus compression). Thomas-Fermi-Dirac; + model potential.
0
tively for Na, K and Al. A quite similar behaviour and an increasing agreement between the two models when the temperature is raised may be noticed. Temperature induced ionization limits the temperature range of validity in the case of model potential calculations. We have tried to estimate roughly this effect in the following way: if, at 0 K, the highest core level is occupied by Zc electrons and distant of AE from the bottom of the free electron like
618
J.L. PELISSIER
Fig. 7. Finite temperature
isotherms of K (same symbols as in fig. 6).
band (as given, for instance, by a compressed atom self-consistent culation’*), the corresponding ionization is taken as: &z=zc
{
l-
1 1 + exp[ - p(I AE I+ k)]
’
where the free electron chemical potential is supposed unaffected.
r 4
IO /
./
+
IOeV
/
/_-;-
+ ./
/ 5eV 1
:/ t 1
Fig. 8. Finite temperature
1.5
2
3
isotherms of Al (same symbols as in fig. 6).
cal-
MODEL-POTENTIAL
THEORY OF METALS
619
At T = 3 eV for Na and K, and T = 10 eV for Al, the valence modification is respectively of l%, 5% and 1%. In the case of potassium, it may be considered as the very extreme limit of validity.
5. Conclusion In the case of simple metals, model-potential theory seems to be a quite valuable tool at finite temperature too, in a limited range of temperatures and densities.
Acknowledgment The author is greatly indebted to F. Perrot (Centre d’Etudes de Limeil) for having supplied him with many unpublished results.
References 1) N.W. Ashcroft and D.C. Langreth, Phys. Rev. 155 (1967) 682. 2) V.G. Vaks, S.P. Kravchuk and A.V. Trefilov, Fiz. Tverd. Tela (Leningrad) 19 (1977) 1271 [Sov. Phys. Solid State 19 (1977) 7401. 3) J. Hammerberg and N.W. Ashcroft, Phys. Rev. B9 (1974) 409. 4) J. Hubbard, Proc. Roy. Sot. A240 (1957) 539. 5) W.E. Parry, The Many-Body Problem (Clarendon, Oxford, 1973). 6) Ref. 5, appendix D. 7) Ref. 5, p. 114. 8) Ref. 5, p. 67. 9) D.J. Thouless, The Quantum Mechanics of Many-Body Systems (AC. Press, New York and London, 1961) p. 139. 10) B.I. Lundqvist, Physik Kond. Mat. 6 (1967) 206. 11) P. Nozieres and D. Pines, Phys. Rev. 111 (1958) 442. 12) E.P. Wigner, Trans. Faraday Sot. 34 (1938) 678. 13) T.M. Rice, Ann. Phys. (N.Y.) 31 (1%5) 100. 14) J. Lam, Phys. Rev. B5 (1972) 1254. 15) M. Senoo, H. Mii and I. Fujishiro, J. Phys. Sot. Japan 41 (1976) 1562. 16) F. Perrot, private communication, 17) A.K. McMahan and M. Ross, High-Pressure Science and Technology vol. 2, K.D. Timmerhaus and M.S. Barber, eds. (Plenum, New York, 1979) p. 920. 18) F. Perrot, private communication.