715
Progress of Theoretical Physics, Vol. 20. No.5, November 1958
Equilibrium Properties of Classical Electron Gas
in Uniform Positive Ion Distribution
y oshi H. ICHIKAWA
Deparment of Physics, T ohoku University, Sendai
(Received July 21, 1958)
Thermal equilibrium properties of a classical electron gas are investigated by taking into account
the plasma oscillation mode. The free energy is calculated in terms of collective coordinate representation. It is shown explicitly that the Debye-Hiickel limiting term of the free energy is due to short
range correlation part of the Coulomb interaction, and that the long range correlation effect increases
the free energy by 22% of the Debye-Hiickel term.
§ 1.
Introduction
Recent developments in the researches on the fusion reaction draw our attention to
the investigation of the physical properties of a fully ionized gas, so called "plasma."J) 2)
Although the subject appears to belong to applied physics, physics of the plasma is a very
interesting fertile field even if one limits his interest to purely academic topics.
It is
essentially a many body problem of charged particles.
In the past years, many researches have been published concerning dynamical behaviour
of the plasma, 3)~!l) yet few works have been done about the thermal equilibrium properties
of the classical plasma. 10) Kihara has proposed to treat the plasma as a strong electrolyte
and to apply the Debye-Hiickel theory in deriving thermodynamical functions of the
plasma. H ) However, from the series of papersa)-!l) it is well known that the plasma shows
very characteristic dynamical behaviour, namely inside the plasma there occurs an oscillation of density fluctuations called the plasma oscillation. Experimental investigations have
·confirmed the existence of such a plasma oscillation both in arc discharge tubes]:?) and in
metals.13) Thus, in a thermal equilibrium state of the plasma, it is quite possible that
the plasma oscillation mode shares the partition of energy and plays an important role
to establish a thermal equilibrium state. Such a case certainly is out of the applicable
range of the Debye-Hiickel theory, since in the Debye-Hiickel theory the possible existence
of the plasma oscillation is not accounted for.
To derive the thermal properties of matter from the microscopic statistical mechanics,
it is sufficient to evaluate the partition function
(1)
716
Y. H. Ichikawa
with /9 =
1//CT,
Boltzman's constant, where N is the total number of particles, H ( ... Pi'
.. " ... Xi" .) is the Hamiltonian of the system.
In ordinary gases, since the inter-particle
interactions are of short range forces, the cluster expansion method certainly is a powerful
method to evaluate the partition function (1). But the method is completely unapplicable
to such a long range force as the Coulomb interaction between charged particles. Concerning this difficulty, Mayer l4 ) has presented a method to treat the long range Coulomb
interaction, and Zubarev16 ) has introduced auxiliary variables to treat the long range interaction.
In the following sections, we will discuss a possible way to deal with the thermal
equilibrium state of the plasma by taking into account the plasma oscillation mode. In
section 2, we will present an effective Hamiltonian for the non-degenerate electron gas in a
uniform positive ion background. In section 3, the partition function and the free energy
will be evaluated. The last section is devoted to discussions of the result, particularly of
our method in comparison with that of Mayer and Zubarev.
/C:
§ 2.
Effective Hamiltonian
According to the analyses of Bohm and Pines,4) we will present an effective Hamiltonian for a classical system of the plasma oscillation and electrons in the uniform
posltlve ion background.
For a quantum mechanical system, Bohm and Pines5) have
derived an effective Hamiltonian which is analogous to the effective Hamiltonian in
classical limit, yet they have made an over-simplification which has been sharply criticized
by Brueckner and others,'~) We will come back to this point in the last section.
The Hamiltonian for electrons in the uniform positive ion background is given as
p/
_ 2'"'"
1
2 -.;:-,
1
H=2: - . ~+2,.e 2..J-!,l' P 1~-27rne 2..J 2
•
i
2m
k k2
-,
k k
(2)
(3)
where V is the total volume of the system, N is the total number of electrons and n =
N,/V. The density fluctuation Ph can be separated into a component q,. , which is due
to the effect of the Coulomb interaction, and a component 'II,', which remains to be
present even if the interaction were absent. Then, one can show that under the random
phase approximation the q", executes a simple harmonic oscillation with an eigen frequency
(/)k,
which is determined through a dispersion relation
(4)
The dispersion relation (4) is determined to eliminate completely the coupling betweenthe q,., component and the 'Ik component of the density fluctuation. Explicit expressions,
of qk and 7) k are given as follows,
q,•. =V- 1/22:
i
w//[w2-
(kp/m) 2]e- ik ;Kii
,
(5)
Equilibrium Properties of Classical Electron Gas in Uniform Positi'Ye Ion Distribution 717
(6)
where (U p 2 47re2 n/m.
Thus, under the random phase approximation, the exact Hamiltonian (2) is approximated by an effective Hamiltonian
p/....
H R.P.A. --~--,
Lr
i
2m
(7)
The terms qk 7J-k and q-T., 7J", are eliminated from the exact Hamiltonian according to
the random phase approximation and the dispersion relation (4). This reduction of the
exact Hamiltonian is a classical version of the procedure used by Sawada!) in studies of
the correlation energy of a high density degenerate electron gas.
Knowing that the
collective coordinate q,,- executes a simple harmonic oscillation, after introducing a Hamiltonian of the harmonic oscillator system H j given as eg. (9), we can express the H R P A.
as a sum of Hamiltonian of individual particle motion He and Hamiltonian of the collective motion H J as follows,
HR./"A.
=He+HJ'
(7')
2
He=~ Pi +27re2~
2m
k>k c
(8)
(9)
where
(10)
The factor I ((/) ) insures that the potential energy of the system of plasma oscillators
IS exactly the corresponding part of the Coulomb interaction 2r.:e'2
q,._ q_"./P, since
m(v'2/I(ev) =27re 2• The last term of He is the counterpart of the kinetic energy of the
plasma oscillators. The critical wave number kc is introduced as a parameter which separates
the effect of the Coulomb interaction into the long range part and the short range part.
In the later part of this paper, it will be shown that in the high density limit the kc
can be allowed to become infinite without affecting the result.
The kinetic energy term of H., (9) is added to give an explicit form of the Hamiltonian of a harmonic oscillator system, after finding that the ql.- executes a simple harmonic
oscillation with the freqency w. Hence, the counterpart expressed in terms of the indi-
718
Y. H. Ichikawa
vidual particle coordinates has to be 'subtracted from the Hamiltonian of the electron
system. In the discussions of the ground state properties of the system, these two terms
cancel each other; therefore they will not appear explicitly in the partition function (11).
However, one must realize that Bohm and Pines o) have disregarded the i ~ j terms of
He (8), while the kinetic energy term of H j (9) is included as a whole to calculate the
correlation energy of the degenerate electron gas.
Concerning the' potential energy term, after separating the density fluctuation PI.' into
the q". and 7Jk components, the qk 7)-lr and q-k 7)1" terms of the Coulomb interaction
have been eliminated by the dispersion relation (4), thus resulting in the simple harmonic
oscillation of the component ql,' Yet, it is very essential to realize that the dispersion
relation as well as the random phase approximation has no concern with the 7)lr y)-lr
term, which has been neglected by Bohm and Pines. Thus, it is clear now that the
neglect of "fjk 7J-Ti.' terms as well as of the i~ j terms of the last term of He (8), committed by Bohm and Pines5), is extra-simplification over the random phase approximation and
that these inconsistent points are responsible for those criticized by Brueckner and othersll).
Finally, we will add a few words to explain physical meaning of the transformation
of the Hamiltonian (2) into the effective Hamiltonian (7'), (8) and (9). The effect
of the transformation insures "diagonalization" of very complicated correlation effects
of the long range Coulomb interaction into the form of the collective oscillation.
In
other words, this transformation enables us to sum up all contributions of the mostdivergent clusters which cannot be handled in the scheme of Mayer's theory of ionic
solutionl ). In Mayer's theory only the parts of such most divergent clusters, namely the
ring clusters, have been summed up and it has been shown that the result gives the
Debye-Hiickel limiting term.
§ 3. Calculation of the partition function and the free energy
Using the Hamiltonian (7), (8) and (9), the partition function can be expressed
as follows,
~'!!k
c
where
Jdqk dq_k exp{
F(k)qk q-k} l(C, q)
a(k) =27Cne2/d/k 2
F(k)=a(k)/n,
(kl/2k2)
(11)
(12)
(13)
(14)
The transfomation function ] (C, q) is given as
Equilibrium Properties of Classical Electron Gas in Uniform Positive Ion Distribution 719
.na (Ck-V-1/2~ (l-fk (Pi) )exp( -ikxi » a(c_ k k
... ) •
i
(15)
where
(16)
for k>kc'
=0
The identity of eq. (11) with an ordinary phase space integral expression of the partition
function for the Hamiltonian (7) is self-evident if one carries out the integrations over
the collective coordinates (Ck, qk)'
Now, our problem is simply to calculate the transformation function] (C, q). Details
of the calculation will be described in Appendix. The result is
_( 2iTm )!lNJ'l N( i)2
J. ~.. ~~~
V
II.
(3
,2iT
n (1
k<kc
~ fkF
{Ql }
1~2~exp
-~~--2-
nfk
exp {
~(i_Pk~) z}
where Qk and Pk are defined by eqs. (A, 4c) and (A,
(A, 23). Thus, we obtain for the partition function,
( 2h:;
)WIV
N
4d).
fk
(17)
is defined by eq.
exp {:Ea (k) } .
·n{a(k)fl+l}-l fI{a(k) (1-f,J 2+1}-1.
The free energy
•
nfk
(18)
IS
Fo-KT:E{a(k) -log(a(k) (l-fk)2+1)k
-log (a (k) h2 + I)}
(19)
where Fo is the free energy for an ideal gas.
Now, to calculate the partition function (18) we need an explicit solution of the
dispersion relation (10). Though the question on existence of the root, which corresponds
to the plasma oscillation, of the dispersion relation is confronted with mathematical
difficulties as it has been discussed by Kampen,1 6) at least for the high density limit it
has an approximate solution
(20)
Hence, in the following, we will calculate the partition function at the high density limit.
However, one must remember that the result (18) is free from any restriction of the
magnitude of the density.
Then, corresponding to the approximate solution (20), we
have the following expression for the ftc,
(21)
Y. H. Ichikawa
720
Thus, the free energy due to the Coulomb interaction is obtained as
dF=-N/CT -!iN~
W(a) =
i:
3~2 [I+2~2 W(a)],
(22)
dx· 3X2{log(1 + (2x) -2)_
(1-f (x) /2X) 2)
-log (1
log (1
([(x) /2x) 2) }
(23)
x = k/kd
Fig. 1.
Variation of the integrand of eq. (23)
is the contribution arising from the upper limit of the k intergration, and it is the DebyeHuckel term.
The factor
is a characteristic factor of the present model of the
V.2
plasma, in which we treat the positive ions as uniform background of positive charge and
thus neglect the contribution of the inter-ion interactions. The second term in the bracket
of eq. (22) represents the contributions of the long range correlation effect.
It will be instructive to examine the structure of the long range correlation term
W (a). The term log (1
({(x) /2X) 2) is the contribution of the plasma oscillation mode
and the rest of the integrand of eq. (23) represents the contribution of long range effects
of the individual particle correlation. The integrand of IJI'(a) is shown in Fig. 1. The
figure shows that the main contribution comes from the region around k~O.7krl' Implications of Fig. 1 are just the same as those of Fig. 2 of reference 8. The integral of
W(a) is evaluated numerically and the result is shown in Fig. 2. From the figure, we
can see that the effect of the long range correlation is insensitive to the critical wave
number kc, if we take a reasonably large value of kc' This is in accordance with the
Equilibrium Properties of Classical Electron Gas in Uniform Positive Ion Distribution 721
__---0.244
0.20
0.10
ot-----0~5~--~--~~~--~--~--~--~--~--~--~--~oo
1.0
2.0
ex· kc/kd
Fig. 2.
Variation of the 1JI'(a) due to the long range correlation effects.
result obtained by Brueckner and others for the degenerate electron gas at the high density
limit.ll) The effect of the long range correlation amounts to be about 22% of the DebyeHuckel limiting term.
§ 4.
Discussions
The result obtained in the preceding section shows that the contribution of the long
range correlation to the free energy amounts to be about 22 % of that of the short range
correlation effect. At high temperature, the effect of the Coulomb interaction is so small
that the thermal properties of the plasma can be described as an ideal gas.
However,
To evaluate the
effects of the interaction are essential for the transport phenomena.
relaxation time, the effective Hamiltonian (7), with (8) and (9), must be used as a
basic Hamiltonian, which is transformed from the Hamiltonian (2) after the non-perturbational diagonalization of the strong long range correlation of the Coulomb interaction.
Here, one may realize that Zubarev's treatment is a perturbational calculation of the long
range correlation in terms of auxiliary variables.
Concerning the effective Hamiltonian (7), with (8) and (9), we will discuss effects
of the long range individual particle correlations, namely the effect of the 7)k 7)-1.' term
and of the last term of Hamiltonian (8). Bohm and Pines has disregarded the 7)k 7)-/r.
term by arguing that the term r,epresents part of a screened interaction beyond the screening radius k;;l. They have also neglected the last term by the same argument except the
term with i = j, which is involved as modification of the electron mass. However, since
the last term is as a whole the counterpart of the kinetic energy of the plasma oscillators,
it is hard to justify such inconsistent treatment of the last term of Hamiltonian (8) .
About the effect of the 'lk 7)-1.,.. term, the present analysis shows that the presence of the
term is essential to get the kc-insensitive result. The kc- insensitive result in the high
Y. H. Ichikawa
'722
density limit has been obtained by Brueckner and othetsR) by correctly including the term
~(VPl (e2~O) . Hence it is clear now that if one includes the long range individual particle
correlation term to the Bohm-Pines effective Hamiltonian, namely if one uses the effective
Hamiltonian (7), with (8) and (9) for the degenerate high density electron gas the
resulting correlation energy should agree completely with the result obtained by Brueckner
and others ..~)
To close the present discussions, a few remarks will be given on the random phase
approximation. The random phase approximation picks up the most divergent correlations
of particles in which the same amount of momentum is transferred among the particles.
On the other hand, Mayer's sum of the ring clusters, which results in the Debye-Hiickel
term, con.sists only of the part of the most divergent correlations, or the random-phasecorrelations. Therefore, if one gathers the random-phase-correlation parts from other various
complicated clusters, they give as a whole the contribution which is also proportional to
the square root of the density, v/n. At the high density limit, according to the present
investigation, it has been shown that the contribution is identified with that of the plasma
oscillation mode. At the low density limit, the contribution mayor may not be identified
with that of the plasma oscillation mode, yet so far no theory has carefully checked if
the contribution, which is also proportional to V/ n, is negligible compared with that of
the ring clusters. We will discuss these points in some details in a separate paper.
It is. very essential to extend the present theory to a realistic case, in which the
motion of the positive ions is also taken into consideration. It has been shown that in
the binary system of electrons and ions there may occur two kinds of collective motion,
-electronic plasma oscillation and the acoustic oscillation. 7) Detailed studies of these problems will be discussed in another separate paper.
It is a great pleasure to thank Professors T. Kihara, S. Hayakawa and N. Fukuda
for their stimulating discussions at the Symposium on Plasma Physics held in May 1958
at Research Institute for Fundamental Physics, Kyoto University. The author is obliged
to Professor Hayakawa for his suggestive discussions in the course of preparing this paper.
Thanks are also due to Dr. Sumi for his critical discussions.
Appendix
We will present here detail accounts of the derivation of the transformation function
.l(C, q), eq. (17) and of the partition function, eq. (18). We have an expression (15)
for the transformation fun.ction ] (C, q).
First, let us introduce the Fourier representation of the a-function as follows,
() (qk- V-II'.! 'Sfk (Pi) exp (-ikx i »
i
(A· 1)
(J (C Io - V- 1 /2 ~
i
(jk
(Pi.) exp (- ikxi »
Equilibriun Properties of Classical Electron Gas in Uniform Positi'Ye Ion Distribution 723
(A·2)
where
gk (p.~)
1-
fie (pJ .
Thus, we have
(A·3)
Next, we will change the variables
by the following relation,17) respectively:
(Uk
(I)_k=Uk sin
(tJk -
(I)
-1~ =
+iU
k
(vI;., v_I;.), (qh, q-k) and (C,.., C_J..)
(U_h) ,
(O)k,
Cfk
}
(A· 4a)
cos ~k
C,..+C_ k =2Pk cos
rk
}
(A· 4b)
}
(A· 4c)
L
(A· 4d)
The integrations over the original variables are transformed as follows, corresponding tothe change of variables (A· 4a) - (A . 4d) ,
CA· 5a)
fdVk fdv_k= (i12) fdVk V k fd~k'
CA· 5b)
fdqk Jdq-k= -2if dQk Qk fdo-k'
CA· 5c)
Y. H. Ichikawa
724
CA· 5d)
Then, we get
q_k(O_l., = QkUk sin (CPk-O"k),
(A·6)
Ck )..I/,,+C_ k )..I_k=PkVk sin (¢k- r k),
(A· 7)
ql.' (Ok
+Yk (Pi) (e
-ikxi
)..Ik+e
=fic (p.i,) Uk sin (kXi+ CPk)
+ gk (Pi) V k
+ikxi
)..1_,..)
+
sin (kXi
¢,) .
(A·8)
By using relations (A. 5) - (A. 8), the expression (A. 3 ) becomes
(2i'!) -4 (iI2) 2~dUk Uk ~dCPk JdV!c V'c ~d¢k
. exp[iQ!c Uk sin (CPk- O"k) ]exp[iP!c Vic sin (¢Ic- rk) J.
·exp[ _iV-l/2 ~
/'kepi) Ulc
sin (kxi+cplc) ]
i
(A'9)
Since the factors exp {- F (k) Ck C -h} and exp {- F (k) q,.. q-Ii'} of the expresslon
of the partition function eq. (11) are indepenent of O"k and r,C) as one can see from
the changes of variables (A.4c) and (A.4d), which transform these factors into the
following,
exp {-F(k) Ck C_ k } =exp {-F (k) Ql}'
(A· lOa)
exp{-F(k) qkq_l.,}=exp{
(A· lob)
we can average (A.9) over the angles
O"b
and ric'
F(k)Pl}'
Then, using relations
(2i'!) -1 ~ dO"Ic exp[iQIc Ulc sin (CPk- O"k) ] = Jo (Qk Uk),
(A·lla)
(2i'!)-ljdZ'1c exp [iPk Vlc sin (¢k-r'c)]=JO(Pk Vk),
(A· lIb)
we can write (A.9) as
. exp[ - iV- 1 /2::S g,c (Pi) Vic sin (kXi+¢k)] .
(A· 12)
Equilibrium Properties of Classical Electron Gas in Uniform Positive Ion Distribution 725
Here, we may carry out the integration over the angles lfk and ¢k approximately as
follows,
fdSOk exp [ _iV-liZ ~fk(pJ Uk sin (kXt+¢k)
::::::"27r[1-
(Uk2/4V)~fk(Pi)/"Jc(Pj)
J
cos (k.Xi-X j ) ]
~,J
::::::"27r exp[ - (Uk 2/4V) tifk (Pi)fk (P:) cos (k· Xt-X j )
fd¢k exp [ -
iV-lIZ
'tJ 9k (PI,) V k sin (kXi
~27rexp[ -
¢k)
J'
(A· 13a)
J.
(A· 13b)
J
(Vk2 /4V)'ti gk(Pi)gk(Pj)cos(k·xi-x j)
Thus, (A.12) becomes
(i/47r)ZJdUk Uk lo(Qk Uk) JdVk Vklo(Pk V k )·
. exp [ - (4V)
-1
ti (Uk2fk (PI,) fk (Pj) + Vlgk(Pi) gk (P.1»
cos (k· xt-X:J)
J
(A· 14)
Now, let us turn back to the expression of l(C, p) and carry out the integrations
,over the spatial and momentum coordinates. We have
l(C, q)
(i/47r)2{IJdUk Ut:lC(Qk Uk) JdVk Vklo(Pk V k) .
.Jd
. exp {
N
J
Pi exp {- p::8 pl/2m} d N Xi'
( 4V)
-1
ti (Ul fk (Pi) fk (Pd)
V k2 g,c (p~) glc (Pj) ) cos (k· Xi - Xj )
}.
(A· IS)
Here the contribution of terms with i ~ j is smaller by a factor 1/V N
terms with i = j, hence we get
than that of
for the integrand of (A.15). The approximations employed in (A.13) and in (A.16)
are exact in calculating the free energy up to the order of
A factor of (A.16),
vn.
fdp exp {-
pp2/2m} exp{ - (4V) -1 (Ul fk2 (p)
+ Vl gl (p»
},
(A·17)
Y. H .. Ichikawa
726
can be replaced by
(A·I8)
where lfl) and (gl) are averages over the momentum space with the weight of exp
{-f3p2/2m}. Thus, putting (A.I8) into (A.IS), we get for l(C, q) the following,
J (C,
q)
= (2'TCm/(3) 3N/2VN(i/4'TC)
.If·)dUk UkJo (Qk Uk) exp {
2
(n/4) (/'k 2)Ul}
(A· 19)
''IJdVkVklo(PkVk)exp{- (n/4) (gl)Vl} .
The integrations over
J (C,
Uk
and V k give the final result for
1 (C,
q) as
(2'TCm/f3r N/2 V N(i/2'TC) 2
q)
. fl(n(fk 2
k
» -lexp {-Qk
2
/
n(fl)}
(A· 20)
For the partition function, we get
Z= (2'TCm/h 2f3)3NI2 V N exp{~a(k)}
.If JdQk (2Qk/ n(j;/) ) exp {
Ql/ n(fl)}
»
. -'lJdPk (2Pk/n(gk 2 exp {-Pl/n(gk2)}.
(A·21)
For the mean values (fl) and (gl), we will approximate them by expressions
(fk(P) 2)
(gk (p) 2) = (I_fie)2 .
(A·22)
where fk is defined by
(A·23)
Equilibrium Properties of Classical Electron Gas in Uniform Positive Ion Distribution 727
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A similar method of evaluation has been introduced independently by; J. K. Percus and G. j~
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