273
Progress of Theoretical Physics, Vol. 15, No. 3, March 1956
Coulomb Interactions and the Diamagnetism of Free Electrons
Hideo KANAZAWA
Institute
of Physics,
College of Gt'Tleral Education, University of Tokyo, Tokyo
(Received December 2, 1955)
The effect of the long-range part of the Coulomb interactions on the Landau diamagnetism is
investigated, applying the collective description of the Coulomb interactions developed by Bohm and
Pines. It is shown that the magnitude of the diamagnetic susceptibility slightly increases due to the
Coulomb interactions between electrons.
§ 1. Introduction
The extension of Landau's1 > original work on the diamagnetism of a free electron gas
has been undertaken by several different methods. The magnetic moment of a free eleci::·on
gas consists of non-oscillatory (Landau diamagnetism) and oscillatory (de Haas-van AlFh-~n
effect) parts. When the use of a finite container to hold the electrons· is considered, the
surface states give rise to size dependent terms in both the oscillatory and non-oscillatory
parts of the magnetic moment. The effect brought about by using a finit~ c::mtainer has
been paid attention to and has been discussed by several workers. However the effect of the
Coulmb interactions between electrons on the diamagnetism has hardly been investigated.
A collective description of the Coulomb interactions in an electron gas has been developed by Bohm and Pines. 2 > The long-range part of the Coulmb interactions between
electrons is described in terms of collective fields, representing organized plasma oscillations of
the system as a whole. The· Hamiltonian describes these collective fields plus the system
of electrons interacting via screened Coulomb forces with a range of the order of the inter
electronic distance.
The aim of the present paper is to discuss the effect of the long-range part of the
Coulomb interactions on the Landau diamagnetism of a free electron gas, applying the
collective description of the Coulmb interactions. In § 2, the collective description of the
behavior of the electrons in a magnetic field is introduced. The Hamiltonian, subsidiary
conditions and the dispersion formula are derived. In § 3, the current density is calculated
to the first order in the magnetic field, applying the perturbation method d.~vebped by
Schafroth. 3> It is shown that the effect of the long-range part of the Coulomb interactions
on the diamagnetism is rather small and the increase in the magnitude of the diamagnetic
susceptibility for Na amounts to 6 percent.
§2.
Collective description of electron interaction in a magnetic field
We consider a system of electrons in a magnetic field embedded in a background of
H. Kanazawa
274
uniform positive charge whose density is equal to that of the electrons.
of our system may be written as
The Hamiltonian
where ~-t=p1 +_:__A(x1), A(xi) being the vector potential of the magnetic field.
The
c
second term corresponds to the Coulomb interactions expanded as a Fourier series in a box
of unit volume (since we are working in a box of· unit volume) . The prime in the summation over k denotes a sum in which k = 0 is excluded.*
Following Bohm and Pines, we shall introduce an equivalent Hamiltonian instead of
(2 · 1). Our equivalent Hamiltonian is given by
+Zrr l/m· ~(Ek·El)q,,qle•ln·<k+l>oo-t
i
kl<kc
-1/2 · ~PkP-k+2rrlfl ~ eil1i·k<oor"'J>jfC
k<kc
i"f'J
k>kc
-2rrne2 f} ~ lj!C,
(2·2)
k<kc
with the associated set of subsidiary conditions :
~k(j)=
{P-k-i(47re2 1ljlC) 112 ~ e-i/fi.k"'•} (j)=O
(k<k.),
(2·3)
i
where Ek=k/lkl·k, is the maximum momentum, beyond which the organiz:d oscillation is
not possible. According to the estimation by Bohm and Pines, k, for Na is"- 0.68 p0,
where Po is the Fermi momentum.
The equivalence of ( 2 · 2) with ( 2 · 1) may be seen by applying the unitary transformation f])=Scf;, where
We shall split up the third term in (2 · 2) into two parts.
That part for which
k+l=O is given by
where n is the total number of electrons. We shall neglect the remaining part for which
k l ~ 0 (random phase approximation) .
Let us introduce the creation and destruction operators for the collective field a" and
4, which is defin~d by
+
"' We shall drop this prime in the remamder of this paper.
275
Coulomb Interactions ond the Diamagnetism of Free Electrons
q1,= (hj2w) 112 (a~c-a"!.k),
P~c=i(bwj2) 1 '2 (a! +a_ ~c).
(2·6)
The commutation relations are given by
(2·7)
In terms of these variables, we write our Hamiltonian and subsidiary conditions as
(2. 8)
H=~+H1 +~+H. .....
~=~ n/j2m+ ~ bw(aJ: a~c+1/2),
i
(2 · 9a)
l~<kc
(2 · 9c)
(2 ·9d)
e=~c=a!+a-k- (8rre2bjw!l-) 1 ' 2 ~ il"R·km,.
(2 ·10)
i
In order to eliminate the field-particle interaction H.., we consider a canonical transformation from our operators ( x 1, p;, ak, a%) to a new set of operators (X;, P, A~c, At).
The relation between these two sets may be written as
(2 ·11)
The generating function of our canonical transformation is given by
S=- (eijm) ~ (27r"hjw)li2[
ik<kc
-e-ti"N·kXtAff
Akeilfl·kX"
E,.,· (H;-k/2)
bw-k·H;Jm+Pj2m
E~c· (H,-k/2) .
baJ- k · H;jm +k2/2m
J.
(2. 12)
Our Hamiltonian is expressed in terms of new variables as follows
H=e-is XeiS
=Xo+X1 +X2 +Xs.r. +i[X0, SJ+i[Xt> SJ
-1/2·[[X SJ, S]+···
0,
where
=Xo+He.p. +Xs.r. +X2+ij2 -[XI> SJ+ ···,*
H..p. = X 1 +i[X0, SJ. The elimination of H1 is incomplete
* We
have neglected i/2 [He.p., S].
(2 ·13)
owing to the non-
H. ·Kanazawa
276
commutability of D/ and E" · D,, and a ·weak interaction He,p. between electrons and plasrr_o.
oscillations remains. Making the sum of X 2 and terms in i/2 · [X 10 SJ, which are multiplied by (AfA,,+A"Af-A"A_"-AfA~"), zero, we get the follwing dispersion relation
2=mr>2+ 4 1re2/m · LJ
~21imk·H_tfm+k4j4m 2-(k·D./mr
•
[lilu-k·D;jmJ-k4 j4m 2
ltJ
(2·14)
.
For sufficiently small k, we may expand ( 2 · 14) in powers of k · R 1/ mli w, and k2/m li a.J,
and obtain
(2 ·15)
Our Hamiltonian may thus be expressed to the lowest order of our canonical transformacio:1
as follows
H= Helectron + Hron +
+
He,p,
+
[E"· (D,+k/2)] 2
liw-k·D;jm-k2j2m
(2 ·16)
Hr.p.>
J+l7reli2~e•t1i·k<X;-Xjljk2,
(2 ·17a)
i·'<'J
lc>k 0
J{ou= ~ liw(A% A"+1/2),
(2·17b)
k<k0
H..p. = (ej2m2) ~ (2rrlijw)
ik<lrc
'/2[{nl lim-k·D;jm+k2/2m
8~.,· (D.-k/2)
_
E~c· (D.-k/2)
D.2}A~ce•t1i·kX;_Ate-•11i·kX;
lilu-k·D./m+k2/2m
X
{nl
e,,.
E"· (D;-k/2)
_
(D.-k/2)
liw-k · D;jm +k2j2m
fjw-k · ll;jm +k2j2m
»l}].,
(l . 17c)
H .. P= _ (1re2li2/m2) ~ [ [E"· (0;-k/2)] [E"· (~j+k/2)] e•l1i·k(X;-Xjl
·
i~3
fjlu[fjw-k·D;jm-kj2m]
k<k 0
+e-ifi·k<X;-Xjl [E.· (D.-k/2)] [E"· (Dj+k/2)]
liw(fjw-k· D.jm-P/2m]
J.
(l . 17d)
In obtaining (2 ·16), we have neglected a number of terms which are quadratic in the
field variables and are multiplied by a phase · factor with a nonvanishing argument exp
[i(k+l)X.].
The second term in Heler.trum which may be interpreted as a sum of the self energies
of electrons, becoms approximatelr>
(2 ·18)
where n1 =47rk0 3j3h3•
Therefore we find
Coulomb Intnactions and the Diamagnetism of Free Electrons
Helectruu
= ::::8 fl/ /2m*+ 2'1C.lfl ::::8 eil1i•kCX,-Xj) j/C +const.,
277
(2 ·19)
i"v'J
''>kc
i
where m*=m X 3n/ (3n-n 1 ). Thus the "new" electrons behave as if they had an effective
mass m*, which is slightly larger than the "bar~" electron mass. Hr.p. describes very
weak electron-electron interaction. We may safely disregard Hr.p. in comparison with the
screened Coulomb interactions in considering the effects of electron-electron interactions.
Our new subsidiary conditions in lowest order of our canonical transformation are
given by
t:)
-~-'C'
( ~-k
newY'-.L.J
;;
1-
(
1
., )2(k · D/
11"w
;; m- k2j 2m )2
ei/1/i•kXitb=O (k<kc). (2·20)
1
It should be noted that in our new representation, the subsidiary condition (2 · 20) continue
to commute with the Hamiltonian (2 ·16) within the approximations we have made. This
follows since the commutation relations are unchanged by a canonical transformation.
Now let us consider the current density operator, which is given by
j(x)
= -ej2m · ::::8 [~,J(x-x;;) +iJ(x-x1 )~,:].
(2. 21)
i
The effect of our transformation on the current density op~rator may be obtained in similar
fashion. Our new current density operator is given by
j(x) =e-;;sJ(x) ets=J(x) +JriJ(x) +J'2l(x)
+ .. ·,
J(x) = -e/2m · ::::8 [D;;8(x-X1 ) +iJ(x-X;;)D,],
(2. 22)
(2 ·23a)
i
J'1>(x) =i[J(x), S]
= -/j2m2 •
::::8 (2nnjw) 12[ {DtiJ(x-X,)A.(k) -A,(k) (D,-k)iJ(x-X,)
ik<lfc
+iJ(x-X;;)D,A,(k) -A;;(k)iJ(x-X,) (D.-k)} Akeil1i·kX;
+A~e-•J;t;·kX;;
{A;;(k) o(x- X,)D.-iJ(x-X;) (ll;;-k)A;;(k)
+A;;(k)D;iJ(x-X,)- (D,-k)iJ(x-X;;)A;;(k)} ],
J' 2>(x) = -1/2 · [[J(x), S], SJ
= -e3/4m 3 • ::::8 (2d/w)[- ::::8 {D;;8(x-X;;)A.1 (k)
k<kc
ij
-A;;(k) (ll;;-k)iJ(x-X;;) +iJ(x-X;;)ll;;A;;(k) -A;;(k)iJ(x-X;;)
X (D,-k)} eilfi·kCX;;-X;> Aj(k)- ::::8 e-i/H-T"-'"t {A;;(k)iJ(x-X,)ll;;
ij
-iJ(x-X;;) (D.-k)A.(k) +A1 (k)H.iJ(x-X,)- (D;;-k)iJ(x-X.)
X A. (k)} Aj(k) eilli·T.-X J +::::8 [e-i/li·kX;; {A;; (k) i3 (x- X;;) D.- iJ (x- X;;)
'
X (D;;-k)A;;(k) +A 1 (k)ll1o(x-X,)- (IJ,-k)o(x-X,.)
(2. 23b)
278
H. Kanazawa
(2 · 23c)
where
fj(J)-k·H./m+Pj2m
(2 ·24)
In obtaining ( 2 · 23c), we have neglected a number of terms, which are multiplied by
A% A1, AkAt* (k ~ 1), A kAt or A% At*. These terms make no contribution to the magnetic
moment as will be seen in the next section.
We have treated w as a pure number thus far, although we see from (2 ·14) that
(I) is, in fact, an operator since it contains
we have ignored this fact in carrying out
our canonical transformation. This approximation is justified, because the dependence of
tu on H, is of the order of ((k·H.jmfjw) 2)A•·• which is small. 2 >*
n..
§ 3. Landau diamagnetism
We shall define Hermitian operators ~% and
~%=
(1/2)
(~k+h.
fi by
c.),
fi= (1/2i) (~1,-h. c,),
( k<kc)
k.>O
(3 ·1)
where h. c. means Hermitian conjugate. Since H commute with ~~ within our approximation, a set of normalized orthogonal eigenfunctions of H forms an orthonormal set of
simultaneous eigenfunctions of H and ~~ 4 ) :
(3 ·2)
The true eigenfunctions of our system correspond to those with ~·',! = 0 :
Hljl,. 0 =Eno ~Jlno,
(3·3)
~f~Jlno=O.
The mean current density at temperature T is given by
i(x) =~(~Jlno,j(x) exp /3((n+5J-H) ·¢,.0 ) ,
n
where
5J =thermodynamic potential
(3=1/kT
(=chemical potential.
The mean current density ( 3 · 4) may be rewritten as
*For weak magnetic field, the value of ((k·Ili/mfi(J)) 2.A.v for Na is-1/16.
(3 ·4)
Coulomb Interactions and the Diamagnetism of Free Electrons
279
i(x) =~ ~(¢,.,,, j(x) exp f1(C:n+!2-H) · II a(~'t)a(f~-)¢..v)
n
~~
k<!.ln
A-,.>o
=TrU(x) exp f1(C:n+f2-H) ·
n a(ft)a(f;;)].
(3 ·5)
k<ltc
k,.<O
A perturbation tre<etment of the magnetic field leads to a linear relationship between
the current density i ( x) and the magnetic field :
(3 ·6)
Written in momentum space, this reads
(3. 7)
Gauge invariance and the equation of continuity require
(3 ·8)
If K (q
2)
is expanded as
K(q 2 ) =a0 +a1 if+ ... ,
(3 ·9)
then ( 3 · 6) can be written as
i(x) =a0 1l rotH+ ... ,
twhere H =rot A ( x) .
(3. 10)
Hence the susceptibility X may be obtained by the relation X= a 0 •
fljc.
As we are concerned with the effect of the long-range part of the Coulomb interactions
on the Landau diamagnetism, we shall ()mit the screened Coulomb interactions. Furthermore
we m<ey disregard the states in which plasma oscillations are excited, because the values of
he energies of plasma quanta are for alkali metals of the order of severa:l electron volts.
Therefore our Hamiltonian may be given by
H= ~ Hlj2m* + H•. p. + ~ ow/2
i
(3 ·11)
lc<kc
"""~
'
P, 2 j2m*+ (ej2m*c)~ {P, ·A(X,) +A(X,) ·Pi}
i
(3 ·12)
In obtaining (3 ·12) we have neglcct~d the terms which are quadratic in the magnetic
field~ and have approximated 1/ (ow-k ·Him +li/2m) by 1/ow in H •. p.· We shall
treat ltJ as a pure number, then we may dr:>p the zero point energy.
Using the formalism of seco.1d quantization, we get
280
H. Kanazawa
H= flo+ H' + H..p.•
(3 ·13)
H 0 = ~(p2j2m*)b:abpa
pa
(3 · 14a)
H'= (e/2m*c)~(p+p') ·A(p-p')b:,abvo~
(3·14b)
pplo
XA(p-p' +k)]] b:,abvaA~o+ (p+p' +k)
·[E,X [p-p' -k
(3·14-c)
XA(p-p' -k) ]] b:,abvaA%],
where
[bva• b:,a,]+ =iJpp/ aaal•
(3 ·15)
[hv'oo bplal]+ =[b:a, b;,a,]+ =0.
a- refers to the electron spin and takes on two values corresponding to the two orientations
of the electron spin. We have put
(3 ·16)
so that
A(x)
=~A(p)e-if~·poo.
(3 ·17)
p
The subsidiary conditions are given by
~k = (1/2) ~ g(k, p) (b:+kabva+b:abv+ka),
pa
(3 ·18)
where
g(k,p)
1
(3 ·19)
tr
Here we have neglected the dependence of
on the magnetic field. The justification
for this approximation is shown in Appendex II.
We next consider the current density operator which is given by ( 2 · 22). Since the
states in which plasma oscillations are excited are disregarded among the terms which
appear in ]<2J (x) as quadratic in the collective field variables only the terms which are
multiplied by AkA'f need to be retained. J<IJ (x) is linear in the collective field variables,
therefore it contributes to the magnetic moment, when combined with He.p.· As we are
going to calculate the mean current density to the first order in the magnetic field, we need
only the zeroth order terms in the magnetic field in J<IJ ( x). Thus we have
(3·20)
Covlomb Interactions and the Diamagnetism of Free Electrons
] 0
(x) = - (ej2m)~(p+p')eif1i•(p-pt)xb:, 0 bpo>
281
(3 · 21a)
pplrs
] 1
(x) = - (~jmc)~A(x)eif1i·<P-P'lx b:,obpa.
(3 ·21b)
pplo
-E~c·
(p-k/2) (p+p' -k)} t'i/1i•(P'-p+klxb:obptoAk
- {E~c· (p-k/2) (p+p' -k) -E~c· (p' -k/2) (p+p' +k)}
(3 · 21c)
] 0
< (x) = - (e/2m) ~ (p +p') [a -r(p'- p)1e•t1i·<p-p'l"'b:,obp0 ,
2>
(3·21d)
PP'·a
J/2>(x) = - (e2jmc) ~ A(x)[a-r(p' -p)J eil1i·<p-p'lxb:,obp0 ,
ppla
+r(ljmc) ~ [(n/i)p(A(x) · (p-p'))
ppla
+ (nji)
2
graddiv A(x)]
(3 · 21e)
where
(3·22)
(3 ·23)
In obtaining (3·21c), (3·21d) and (3·21e), we have approximated 1/(n(l)-k·Dim
+f2/2m) by 1/nw and we have replaced A~cA~c* by I because of (A~cA~*),•ac=(I+AfA~c)vac
=I. The terms which depend on two electron coordinates in ]<2 >( x) cancel with each
other to the first order in the magnetic field.
If we label the eigenstates of lfo by ).1; ).11, ).111, • • • and call the corresponding eigenvalues
of H 0 : E., E,,, E,,, ... and if we apply the perturbation method3>, we get the following
expression for the mean current density to the first order in the magnetic field :
i(x) =~( ll a(ft)a(tk') )n, exp (3((n+!10 -E,)
n' k<kc
kz>O
X [ (11 (x)
+1/ >(x) ),,, +~
2
1
e~<E,-E,nl
~.-E,,
X {(J0 (x) +Jo <2>(x) ),,,, (H' -!1'),,,+ (J/l (x) ),,,, (He.p. -!l•. p.),,,} ],
(3 ·24)
where !10 is the thermodynamic potential of the unperturbed system, and !1' ond !Je.p. are
the corrections to the thermodynamic potential due to H' and He.p. in first order respectively.
!10 , !1' and !le.p. are determined by
1 =Tr[exp ((n+!l-H) · IT ~(>"t)a(~:;;)]
k<...,k 0
kc>O
282
H. Kanazawa
1- e'(E~-E_,)
+------(!2' +!J•. p.-I-I' -H•. p.)~,~+ ···].
(3·25)
E~,-E~
We get
e-M.lo =~(1M
(ft) a(fi) ) ~~e~ r,,_ E~>,
(3. 26)
~
(3 ·27)
(3·28)
We can write
a(~~)= (1/2n) J:'!:t exp (ii.t~~).
(3. 29)
Therefore
(na(u)a(f;;) ),..
= (1j2n)"'J'IIdi.t d;,;;(exp i~().Z fZ +).;; fi) )~~'•
= (1j2n)"'Jrun d).;;~ (
n-o
{i~(..lt ft +).; fk")} n>~~,fn!.
(3. 30)
(IIo(;t)iJ(fk") >~~
= (1/27<)"'
J
Jld).t d..J;;~(
{i~().tf% + i.J,; fk)} n),,jnf
~(1/2rr)n' Jnd..Jtd..J;~,[1/2· ( {i~(AUZ+2;;~;)} 2 )w]n/nf
= (1j2n)"'J/Jd).t d..l;~[ -1/8 ·
n=O
~ [g2 (k,p)
{nva(1-np+ka)
TJk<k 0
kz>O
* The
process sl-,cwn in Fig. 1, which describes a two-step process leading back to the initial state :
contribute to the terms which are quadratic in Ak±· The contributions to the fourth order
terms in ,lk± come from the processes shown in Fig. 2. Fig. 2a describes a two-electron process in wh1ch each
electron suffers scatterings like those shown in Fig. 1. Fig. 2b corresponds to four-step processes of one electron :
p-7p+k~p~p+k'~p,etc. The ratio of the magnitude of the contributions from the processes shown m
Fig. 2b and Fig. 2a is of the order of 2J g2 (k,p)g2 (k 1,p)np11 (1-npua)(l-np+kta)/{ ~ g2 (k,p)n 1111
p~p+k~p,
a,p,k,kl <kc
a,p.k<k0
(1-np+ka)J2-0(1/n). ln general, the leading terms in 2n-th order terms in ).k± correspond to the processes
in which each of n electrons su'fers scatterings like those shown in Fig. 1. Terms cfo:!d powers in ).k± do
not contribute, because they are odd functiOns and so the integrals vanish.
Coulomb Interactions and the Diamagnetism of Free Electrons
283
(3·31)
where we have neglect~d. terms of the order of 1/n as compared with the terms of the
order of 1.
p
p
-----~·
p+k
-----~.
p
p'
-----~.
p+k
----- ~..
---- ~.
P+ll
p
--- _e!
-----~~'
Fig. 1
p
p+k+k'
~;'---
~;---
p+k
p
. P(""p)
P+k'
-----e.
p+k+k'
----e.
p
p
e-~,---
e.'-----
~:-----
p'+k'
-----~.
p
~:-----
p+k
-----~.
p
Fig. 2a
p+k
-----e•
p
Fig. zb
Substituting (3 · 31) int:> (3 · 26), we obtain
+f9 H(l- [ 9 )} (lt +i.2k) J·//7,(1 +e-~(<(P)-tlJ
= (1/2rr)n' n
41!'
2:,; g2 (k, q) {fq(1-{9 +k) +h+· (1-{q)}
k<kc
k.>o g
X flv[1
+e-~(eCvH:>J,
(3 · 32)
where J;,= 1/ {1 +exp {9(c(p) -()} is the Fermi function.
In (3 · 27) the part for which II=IJ1 is proportional to iA(x)dx, which can be made
to vanish by a gauge transformation. Accordingly we find
12' ""4-c- 1 e~uo 2:.; (1/2rr)"'' f ll di.t di.;; ~ [1/2 · 2:.;( {i 2j
'"""'
Jk<kr.
n=O
p
Ot ~t +i.; fk')} 2 )pp]"'/n!
k<kc
(3·33)
(H!),,, corresponds to an inelastic scattering of one electron
change must be compensated by ~;.
(p~p+k).
This momentum
Then the integral is seen to be of the form
H. Kanazawa
284
X [A(k, p)At +B(k, p)Ak"
+ C(k, k', p) Ot-kl ).j';,-).;_,.,).f,
+i).t-kl ).,:;; +i).jj Xi-k') ]=o,
{3 ·34)
because
J?:
e-<h·").
= 0,
{3·35)
H?~+J;.-e-a•v.•++>."-> (l+ -).2-) =0.
Therefore
{3·36)
He.p. has no diagonal matrix elements and f"l; do not depend on the collective field
variables, hence (IJ{}(ft)(J(f;) ).,.(He.p.).,,=O. Therefore
(3. 37)
!Je.p.=O.
We now proceed to calculate the mean current density.
current density (3 · 24) into two parts: the i 1 (x) for which
i 2 (x) for which 1.1~1.1 1 •
(a) i 1 (x)
We shall split up the mean
and the remaining part
1.1=111
~~(1/2rr)"' f fl d).tJ).; '£ [1/2 · ( {i~ (Atft +).;f;;) }2) ..]" jn!
J
n-=0
i 1 (x)
'lol
= ..~}_1/2rr)"'1 ffdltd).;;~[ -1/B·r'JJ_,.f(k, r) {nr
0
(1-nr+ko)
k 0 >o
+nr+ko(l-nro)} ().2%+A2k") ]"jn! · exp .B[(~nro+fl0 - ~c(r)nro]
ro
ra
X[- (l/mc) (1 +a)A(x)~n,o
po
+r(ljmc) (o/i) 2 grad div A(x) ~n,0
po
-
(lj4mm*c)~(1+a-rif)
(2p-q)e-•11i·qoo
pqo
XA(q)·(2p-q) 1-exp.B{c(p)-c(p-a)} n,"(l-n,_qo)
c(p) -c(p-q)
-
~(e4 j2m 4c)
pqa
k<ko
(ljow)2(27l"OjaJ)[2kE,.· (p-k/2) {(p · q) (8 1,,- A(q))
Coulomb Interactions and the Diamagnetism of Free Electrcns
- (p·A(q))
(8~c·
q)}
+k8~c·
(p-k/2) {(8,,· q) (q·A(q))
28:>
-q 2 (8~c·A(q))}
+ {q8~c· (p-k/2) -2p(8,,· q)} {(p· q) (8~c·A(q))- (p·A(q)) (8,,· q)}]
X e-im•qoo_ _1_-_e_x_,_p-'-~-:'-{c_(=p-'-)-,--_c_(-"'-p_-_k--,-------'q'-'-)--_fj_w--'-}-- nva ( 1 - nv-Jc-qa) ].
c(p) -c(p-k-q) -fiw
(3. 38)
Carrying out the summations over n11 a, using (3 · 32) and transforming to the momentum
space, we get for the kernel, which is defined by (3 · 7),
(3. 39)
where KIL/(q) is the contribution to the kernel from J(x) and J(2l(x), and K.,_/'(q)
is that from J(lJ ( x) . They are given by
+ (p3j6m2 *)2JqJ.qpp!Lp~p;.pp
{6,/;, (1-fvY- fv(l-fv)}
J.p
2
- (q~'-q~/4)~fv(1- {;,)] +O (q4) ,
(3. 40)
KIL/'(q) = - (e4jm~c) 2J (1jfiw) 2 (2rrfijw)[kiL8"· (p-k/2)
pk<kc
X {(8~c· q)q~-q2 k~Jk}
(1-e~'ll) /r; 'fv(l-fp-k)
+ {q~'-8"· (p-k/2) -2p!L (8~:· q)}
{(p · q)k~fk- (8~c· q) P~} (1-er.'ll) /~
Xfp(l-{11 - 1,) +2k~'-8,,· (p-k/2) {(p· q)k~fk- (8~c· q)p~}
X { -q · (p-k) /m· (~~eB'II-e~'fl+ 1) /rl 'fv(l-{11 -A.)
+q· (p-k)jm· (1-e~'ll)j~·~fvfv-k(1-fv-k)]+O(q4 ),
(3·41)
where
~===-fiw+k·p/m-P/2m.
Performing the summations, we get the following gauge invariant expressions for the kernels
(see Appendix I)
KIL/ (q) = - (q~'-q~-a~'-~if)[- (2rre2j3mc) (2m*() 1 ' 2 (1 +a
+Bm*(r)/h3]+0(q4) ,
(3·42)
(3·43)
where
286
H. Kanazawa
+ ((jow) (lj!C) {-2/9-f-5Kjl2-38K /225+2K /9
2
4
(3. 44)
-7K5j36-2K6/27+K 7/45} ],
and K=kc/p0 , Po being the Fermi momentum.
terms which vanish as T ~0.
The susceptibility is given by
In ( 3 · 43) and ( 3 · 44) we have neglected
X=- ( 4mp.2 j3h2 ) (3rr2 ii) 118 (1 +a+8m*(r -E)
= Xo + X0 (a + 8m*(r -E)
=xo+LIXo•
(3 ·45)
where p.=efij2mc is the Bohr magneton and Xo is the diamagnetic susceptibility for the
perfect electron gas.
From (3·22) and (3·23) we get
a=(n'jn)(k/jlOmfi(l)),
(3·46)
r= (n'/n) (l/l2mfi(l)).
(3 ·47)
According to the estimation by Bohm and Pines, kc for Na is "-0.68 p0, so that n1 ,..._
0.16 n and k/j2m-....0.46 (. Inserting these values and (/hor..... l/1.56, we find that the
correction to the susceptibility for Na amounts to about 6 percent :
Xo (a+ 8m*(r) "'- 0.077 Xo
-x E--0.02lz
0
0
(contribution from ]< 2>( x) ) ,
(contribution from ]<1>( x) ) ,
Llxo!Xo"'-0.056.
In carrying out the summations over npa in (3 · 38), we have neglected the cases
where rrr or r+krr in (fia(~t)a(~J;) >~~coincides with prr, p-qrr or p-k-qrr. These
cases represent the effect of the subsidiary conditions. However this effect is very small.
It can be shown that the contribution to the susceptibility is of the order of LIX0/;;:
(b)
i 2 (x)
=~,exp ~((n+!20 -E~) · (lj2n)n'J lld..tk+d-l,;;~[l/2 · ~
X ( {i ~(,lU% +Ak' ~k')} 2 )aa]njn! · [(i ~(,lH'.t +..t:;f;) )~~'
+ 1/2( {i ~ O.tf.t +..tk' fk')}
2 ) . .,] [ ( ]
1
(x) +N2>(x) \'~
(3. 48)
(x) +J, <2 >),,, in (3 · 48) corresponds to an inelastic scattering of one electron. The
2
momentum defect arising in ] 1 ( x)
must be compensated by the scattering ·by
1 < >( x)
,;=;!;. The same argument as in the evaluation of 52' leads to that the contribution from
( ]1
+]
Coulomb Interactions and the Diamagnetism of Free Electrons
287
this one electron process to the mean current density vanishes. Since ~r do not depend
on th: collective field variables, J0 <I> ( x) makes no contribution. Hence the sole contribution
comes from the terms which contain (J0 (x) +Jo <2>(x) )(H'). As the momentum change
of th ~ electron by ~t is ± k ( k < kc) , the dependence of the mean current density on the
vector potential of the magnetic field is of the form
(3. 49)
i. e., there are no Fourier components with k > k.. Therefore this current density has nothing
to do with the Landau diamagnestism. Thus we find that our final result is given by
( 3 · 4 5) . (Actually i 2 (X) vanishes. See Appendix III) .
§ 4. Discussion
We have developed the collective description of the behavior of electrons in the magnetic field and have investigated the effect of the long-range part of the Coulomb interactions
on the Landau diamagnetism. It has been shown that the long-range part of the Coulomb
interactions slightly increases the magnitude of the diamagnetic susceptibility. Our model
may be applied to alkali metals because the effect of the periodicity of the crystal lattice
may be taken into account by the replacement of the electron mass m by m.tr5> Our
result is qualitatively in agreement with that obtained by Pines.* Pines6 > has calculated the
energy of one electron in the absence of the magnetic field and inserted it into the formula
for the diamagnetic susceptibility. He has estimated the increase in the magnitude of the
diamagnetic susceptibility for alkali metal to be about 10 percent.
The experimental value of the diamagnetic susceptibility for Na seems to be considerably
smaller than the theoretical one, therefore other effects would have to be considered. The
screened Coulomb interactions, of which effect we have not considered in the present paper,
would not have a much effect on the diamagnetism. However this effect remains to be
examined.
The author wishes to express his sincere thanks to Professors S. Tomonaga, M.
Nogami and R. Kubo for helpful discussions.
Appendix I
K~> (q) is given by
K~.(q) = - (2ljmc) (1
+ a)a~'-.::Sfp+ (21/mc)rq~'-q" ::8 fp
p
p
- (2ejmm*c)::8(1 +a-rq2 ) (p~'- -q~'-/2) (p.-q 11/2)
p
X
p {E(p) -c(p-q)}
c(p) -c(p-q)
1-exp
fp(1-fp-q).
(A1)
* While preparing· our manuscript, we were informed of Prof. Pines recent work through his preprint
sent to Mr. H. Watanabe. We thank Mr. Watanabe for his kindness.
H. Kanazawa
288
Expanding (AI) in powers of q, we get
K~~(q) = - (2e2 jmm*c) (I +a)~ [ (/12/4m*) YP~tP~+ ~(q~tq>-P~P>P
).
+q.qAP~LP>J} {fv(l-fv) -2fv2(1-fv)}
+ (/13/6m2*) ~q>- qPP~'-P~P>-PP {6 [/ (I-fv) 2-fr,(I-fv)}
>.p
- (q~'-q.j4)f1fv(l-fv)]+ (2ijmc)r ~ [q~tq~fv
p
(A2)
Replacing the summations over p by the integration, we obtain
~fv=
(8rr/3hs) (2ms*)lf2[(:lf2+ (rr2/8f)Cl/2+ ···],
p
~fv(l-j~)
= (4-:rjhs) (2m3*) 1/2~-1[(1/2_ ('lf/24f)C3/2+ ···1,
p
~ P~LP>fv(l-fv) =aiL.(8rrj3ha)m* (2m3*)lf2J3-1 [(3/2+ ('If/8f)Cl/2+
.. ·],
p
~ P11-P•fv2 (1-fv) =a~t.(8rrj3h3 )m* (2m 3*) 112 {1- 1[(g12j2- (3/4/1)(112
p
+ (n2jl6J92)Cl/2+ ···],
~P~'-P•P~-Ppfr,(l-j~)
p
= (a~t~a}.p+a~tJ.a•P+a~'-Pa•J.) (16rrji5h3)
Xm2• (zma .. )l'2;3-1[e'2+ (Sn2j8f92)(1'2+ ···],
~ P~tP~P}. pp[/(1-fvY= caiL.a}.p+alJ.Aa~p+aiLPa~}.) (16rr/I5h3)
p
X m2* (2m3*) 1/2 /1-1[(5/2/6+ (5/8f) (n2j6-1)(1/Z +. ·J
(A3)
Inserting ( A3) into ( A2) , we get
K~~(q) = - (q~tq~-a~'-~q2 )[- (2ne2 /3mc) (2m*()'' 2 (I+a+8m*(r)/h3]
(A4)
+0(q4).
K~.(q) is given by
K~.(q) = - (e4 jm4c) ~ (ljow) 2 (2rro/w) [zk~'-(E~c· (p-k/2))
pk<kc
X {(p·q)k.jk-(E,,·q)p~} +kt£(E~c· (p-k/2)) {(E~c·q)q~
-tk~fk}
+ {q~'-(E~c· (p-k/2)) -2p~t(E~c· q)}
-(E~c·q)p. }]
{ (p· q)k~fk
1-exp/1(r;+p)
-fv(l-fv-Tc-q),
r;+p
where
1)- -ooJ+ k
· pjm -k2/2m,
p=q· (p-k)jm-q 2/2m.
(A5)
Coulomb Interactions and the Di-amagnetism of Free Electrons
Expanding
{1-e~c·~+Pl}
289
/(7J+P) in powers of p, we get
1-e'C"IJ+Pl
(A6)
Also we find
Inserting (A6) and (A7) into (AS), we obtain (3·41), which yields in the limit of
T=O
K~, ( q) = (e4 jm4 c)
2:i
(1/baJ ) 3 (2nbjaJ) [kJL (Ek · (p-k/2)) {q, (Ek · q)
pk<kc
-q2 k,jk} [fp(1-fp-k)]T=o+ {qJL(E,,· (p-k/2)) -2pJL(8,,- q)}
X {(p. q)k,jk- (E~c· q)p,}[fp(1-fp-k) ]T=O
+2kf.L (E 1,· (p-k/2)) {(p · q)k,jk- (E~c· q) p,} {q · (p-k) jmbaJ
X [{P(1-fp-k) ]T=o+ q · (p-k) /m ·[ftfPh-k(1-fp-k) ]T-o}]
(As)
+0(q4),
where we have put 7}~ -bw. The integrations over p and k are elementary, but tedious.
For example the first term becomes
= (1/h6 ) )dkkJL {q,(E 1,· q) -ifk,jk} .\' l dp.r (p cos 8-k/2)sin fJdfJdr.p
P<Po, lp-kl >Po
Po
p+k
= (2njh6 ) ) d kkJL {q,(Ek· q) -q2 k,jk} /kf pdp)
Po-k
k<kc
{(l +k -r) /2k-k/2} sds
2
Po
where r=l+P-2pk cos 8. Other terms may be evaluated similarly.
by (3·43).
The mean total number of electrons may be given by
T,.[n exp (1((n-H) ·ITa(~t)a(f;:-)]
T,.[exp (1((n-H) ·llB(~t)a(f,;)]
The result is given
(A9)
which, with the help of ( 3 · 26), can be written as
n= ~''
(.}-1
a,
___!__ log ("-~oo.
The insertion of (3 · 32) into (A10) gives
(A10)
H. Kanazawa
290
(All)
The second term tends to zero as T ~0.
Therefore
(A12)
Appendix IT
The subsidiary conditions are given by ( 3 · 1):
~%=!~{
•
1
1- (1/hw) 2 (k· llJm-k2j2m) 2
eif1iol•X•+h.c.}.
(Al3)
Expanding ( A13), we get to a good appoximation
+ (k·A(X.)) (k·P.) -P(k ·A(X;))}}
e•t1i·~<X•+h.
c.].
(A14)
Using the formalism of second quantization, we obtain
(A15)
where
O(k, p, p')
= (1/hoJ)
O*(k, p, p')
2
(e/m 2c) {k· (p+p' +k) -k2} k·A(p-p' +k),
= (1/hw) 2 (e/m2c) {k· (p+p' +k) -P} k ·A(p'-p-k).
(A16)
Analogously
(A17)
The contribution from ft to the mean current density in first order in the the magnetic
field is given by
~ (1/2tr)n'[ JldJ.tdX;(i~(i.tft+X;;f;;,_))n'
llll'p
J
Coulomb Interactions and the Diamagnetzsm of Free Electrons
291
X (exp i ~Ot ~~~ +i;; ~k.,) )PP • (J0 (x) +lo '2l (x) )~,~
(AlB)
Xexp ~(t;n+!20 -E~).
We see that the integral is of the form
(A19)
which vanishes.
Appendix III
i 2 (x)
:i; [
~ ~ exp ~(t;n+f20 -E~) · (1/21l')n'\.., J/dit d;.-;; n=O
11=\:ll'
(A20)
We need to consider only the processes in which one electron suffers an inelastic scattering
(the elastic scattering should be excluded) by H' and another electron is scattered inelastically by ] 0 (x) +10 ' 2J (x).
If the inelastic scattering of two electrons (p1, p2 ~p 1 ±k, p 2 ±k) is done by H'
and J +J'2l, (JJ(}(~t)(J(~-;;) )w, must correspond to the scattering (p1 ± k, p 2 ± k~pi> p 2 ) .
Then the expression for the mean current density contains the factor
Jrn d;.t d;.;;
~ g( =Fk, PJ)g(± k,p2) or-;.;; 2 ± zi;.t ;.;;)
Pl,.PO
k<k0
kz>O
Xexp
[-1/4·
~ g(k, q) {fq(l-fq+k) +fq+k(l-fg)} Vt+J.l-)]
qk<k0
kz>O
(A21)
XF(k, Pl• P2),
which vanishes, because
J"'JJ;.+J..r(l+-;. -)exp [ -a
2
2 ().2
++i. 2 -)]=0,
-co
11
(A22)
00
-«>
J;.+J;.-;.+ ;.- exp [ -a2(l++J.2-)]=0.
H. Kanazawa
292
Thus we are left wirh the process in which two electrons (pH p 2) are scattered into the
states (p 1 ±k, p2 ~k) by H' and J+J<2l. We have
i2(x)
"""e~Uo[fl (1 +e-~(e(r\-t)J. (1/21I")n' r II dJ.t dJ:;;
Jk<k0
k0 >o
Xexp [-1/4· ~ g2 (k, q) {fq(1-{q+lc) +fq+k(1-{q)} ().~+
+J.t-)]
qk<k0
k.>o
X[ -1/8 · ~ g( -k, p 1 )g(k, P2) ().~+ +J.~-)
"1"""•
k<kc
k.>o
X {- (2ljmm*c) (1 +a-rP) [ (p 2 +k/2)e-•l1i·k"'(p1 ·A(k))
X 1-exp ~ {c(p1) -c(p1 -k)}
c(p1 ) -c(p1 -k)
+ (p -kjZ)e'i1i·'k.KJ(p ·A( -k))
1
2
X 1-exp~{c(p2 )-c(p2 +k)} .Jt" P ( 1 _t" _ )( 1 _t" )}
c(p2 ) -c(p2 +k)
. JP1JP2
JP1 k
JP2+k
-1/8·
~
"1"""•
k<k
g(k, p,)g( -k, p2 )().~+ +J.~-) {- (2ljmm*c)
0
kz>O
X(1 +a-rP)[ (p
2
-kj2)e.if1i·k"'(p1 ·A( -k))
X 1-exp ~ {c(p1) -c(p1 +k)} + (p1 +k/2 )e-•tM<x(p2 ·A(k))
c(p1 ) -c(p1 -k)
X 1-exp~{c(p2 )-c(p2-k)} ]"
c(p2) -c(p2 -k)
t" (
JP1JP2
Here we have fixed rhe gauge by divA (x) =0.
sum as a factor
1 _t"
JP1+k
)( 1 _t" _ )}].
Each term in (A23) has rhe following
~(p·A( ± k) )g( ~k, p) 1-exp ~ {c(p) -c(p~k)} {,( 1 -j;,'f").
,
c(p)
(AZJ)
}Po k
-c(p~k)
(A24)
Since rhe summand except (p ·A ( k)) is a function of p and (p · k), ( A24) is easily
seen to be proportional to (k·A(k))jk, which vanishes on account of divA(x)=O.
Therefore we find
(A25)
Coulomb Interactions and the Diamagnetism of Free Electrons
293
References
1) L. Landau, Z. Physik. 64 (1930), 629.
2) D. fohm and D. Pines, Phys. Rev. 92 (1953), 609.
3) M. F.. Schafroth, Helv. Phys. Acta 24 (1951), 645; Nuov. Cim. 9 (1952), 291.
H. Ichimura, Prog. Theor. Phys. 11 (1954), 274.
4) E. C. Kemble, Fundamental Principles of Quantum Mechanics (McGraw.Hill, New York, 1937),
p. 284.
5) H. Kanazawa, Prog. Theor. Phys. 13 (1955), 227.
J. Hubbard, Proc. Phys. Soc. A 67 (1954), 1058.
6) F. Se1tz and D. Turnbull, Advances in Solid State Physics Vol. 1 (Academic Press, New York,
1955).
Note added in proof: It is expected that the following term wh1ch should be added to the current
density operator (2·21) will make some contributiOn to the susceptibility:
The effect of this term on the diamagnetism together with the effect of the screened Coulomb interactions
will be considered in a subsequent paper.