EFFECTS OF THE SPIN AND MAGNETIC MOMENTS ON THE INTERACTION
ENERGY BETWEEN ELECTRONS. APPLICATION TO FERROMAGNETISM.
V. DOLOCAN1 AND V. O. DOLOCAN2
1
Faculty of Physics, University of Bucharest, Bucharest, Romania
IM2NP & Aix-Marseille University, Faculté des Sciences et Technique, Avenue Escadrille
Normandie Niemen, 13397 Marseille cedex 20, France
2
Abstract.
We study the influence of the spin and magnetic moments on the energy of interaction between electrons. First,
we separate the Heisenberg term and the second, we introduce the magnetic moment in the electron wave
function. We have found that in an isolated atom, with an incompletely filled shell, the interaction energy for the
triplet state is lower than that for the singlet state, so that the triplet state is favorable. Also, we have calculated
the interaction energy between the electrons from the adjacent atoms in fcc and bcc lattices and we have found
that the minimum interaction energy is attained for the triplet state.
Key words: ferromagnetism, electron-electron interaction, magnetic moment.
1. INTRODUCTION
An important problem is that of the origin of ferromagnetism. The physical effect that
produces magnetic ordering of adjacent magnetic moments is the same that leads to magnetic
ordering within an ion and produces Hund’s rules. Interactions between spins are actually just
a convenient way to record the end result of electrostatic repulsion[1]. Two electrons circling
a nucleus reduce their Coulomb interaction by adopting an anti symmetric wave function that
vanishes whenever they come near each other. The Pauli principle demands that the overall
wave function must be antisymmetric, so the spin wave function must be symmetric. That is
the electrons lower their energy by adopting the same spin state and developing a local
magnetic moment. The important concepts are introduced by Heisenberg[2] and Stoner[3]. It
was shown by Hubbard[4] that the correlation effects will lower the energy of non-magnetic
states more than that of the ferromagnetic states, and so make the condition for
ferromagnetism more stringent. The spin wave excitations give a way to explain the
temperature dependence of the magnetization in magnetic materials at low temperatures[5].
In this paper, using an effective field Hamiltonian (detailed in Ref.6), we investigate
the influence of the magnetic moments on the electron-electron interaction. We first discuss
the separation of the Heisenberg term in the energy of interaction. Next, we present the
modification of Coulomb’s law by the interaction between the magnetic moments of the
electrons. Further, we discuss the case of an isolated atom with applications to Hund’s rules
and the condition for ferromagnetism in fcc and bcc lattices.
2. MODEL AND FORMULATION
In this Section we present the electron-electron interaction by using an effective
Hamiltonian[6] and use an way to separate the Heisenberg term. In the interaction picture the
effective Hamiltonian of the electron-electron interaction is given by the expression (see
Appendix):
(1)
H eff
I1 =2 ℏ
,
∑
.
q , q ,k ,k , σ , σ
,
−1
∣g q ∣2 εk −εk −q−ω q a +q aq a +q a q c +k− q ,σ c+k ,σ c k ,σ c +k −q ,σ
o
0
o
,
,
,
,
(1)
gq =
o
q qo
ℏD
2 ω q ωq
8N mr (ρo+ Dr /c )
2
∑ ei∗q ∗r
oj
o
j
j
where D is a coupling constant, m is the mass of an electron, r is the distance between the two
electrons, ρo is the massive density of the interacting field, Dr/c2 is the “mass less density” of
the interacting field, ωq = cq is the classical oscillation frequency of the interacting field, ωqo is
the oscillation frequency of an electron, q is the wave vector of the interacting field, qo is the
wave vector of the boson associated with the electron, k is the wave vector of the electron and
ε k =ℏ k 2 /2m . Introducing creation and annihilation operators cjσ+ and cjσ of an electron of
spinσ in the orbital state φ ( r –rj ) one can also write
1
N
ck+−q ,σ =
ck ,σ =
1
N
∑e
−i ( k −q ) .r j
c +jσ ; ck+′,σ ′ =
j
∑e
j′
ik .r ′j
c j′σ ; ck ′−q ,σ ′ =
1
N
1
N
∑e
∑e
−ik ′.r ′j
j′
i ( k ′ −q ′ ) .r j
c +j ′σ ′
c jσ ′
j
and therefore,
ck+−q ,σ ck+′,σ ′ck ,σ ck ′−q ,σ ′ = ∑c +jσ c +j ′σ ′c j′σ c jσ ′
jj ′
(2)
where we have used that
1
N
∑e
i ( k −k ′ ) .r j
= δ (k′ − k )
j
We define the occupation number operator
n j = ∑c +jσ c jσ
σ
and the spin operators
 σ σασ ′ 
c jσ ′
s = ∑ c 
2
′
σσ


α
j
+
jσ
where σα, α = x,y,z are the Pauli matrices. Writing the components out explicitly, one has
1 +
1
(
c j c j′ + c +j′c j ); s y = i ( c +j ′c j − c +j c j )
2
2
1
s z = ( c +j c j − c +j′c j′ ); s + = s x + is y = c +j c j′ ; s − = s x − is y = c +j ′c j
2
sx =
With some algebra one obtains
1
c σc σ c σ c σ = n n
∑
2
σσ
′
+
j
j ′
+
j′ ′
j′
j
j′
+ 2 s zj s zj′ + s +j s −j ′ + s −j s +j ′ =
1
n j n j ′ + 2s j .s j ′
2
Therefore,
1

H Ieff1 = ∑ K jj′  n j n j′ + 2s j .s j′ 
2

jj ′
(3)
where
K j j=
'
ℏ 3 D2
DR
32 m R ρo + 2
c
2
2
(
)
2
∑
q , qo , k
( q . q o )2 1
i ( q . r −q
'
.r
'
)
e oj j o j j
( nq +1 ) ( nqo +1 )
ω2q ω2q 2 ( ε k −ε k−q )−ω q
o
(3a)
The first term enclosed between parentheses is a “pure” Coulomb term and the second term is
the Heisenberg term
H H = 2∑ K jj ′s j .s j′
(4)
jj ′
3. EFFECT OF MAGNETIC MOMENT ON THE INTERACTION ENERGY.
Let us consider two electrons separated at a distance r = |rj – rj’| . The expectation value of
the first term from Eq. (3) is
EC = ∑ K jj ′
jj ′
1
n j n j′
2
(5)
For nq = nqo = 0 and nj = nj’ = 1, we have
i q
1
(
e
∑
2 jj
(6)
'
oj
. r −q ' . r '
j
oj j
)
[
=1+ cos i ( q o1 .r 1−qo2 . r 2 )−
]
ie
∮ A ( r 1 ) dr 1−∮ A ( r 2 ) dr 2 ) =1+cos ( Γ )
ℏc(
where in the presence of the magnetic field we introduce the vector potential and thus we
substitute qo.r by qo.r – (e/ħc) ∮ A . dr .
The energy levels of atomic electrons are affected by the interaction between the electron
spin magnetic moment and the orbital magnetic moment of the electron. It can be visualized
as a magnetic field caused by the electron’s orbital motion interacting with the spin magnetic
moment. The effective magnetic field can be expressed in terms of the electron orbital angular
µ ×r
momentum. We consider the vector potential A = 3 where µ is the magnetic dipole
r
moment and r is a vector from the middle of the loop to an observation point. An electron in
a stationary state in an atom, having a definite angular momentum
(l )
projection Lz = ħml (ml the quantum magnetic number), has a magnetic moment µ z = µ B ml
where μB =e ℏ/2mc is the Bohr magneton. The theory and experiments demonstrate that the
free electron has a magnetic moment equal to the Bohr magneton µB , and a spin momentum
ms where ms = ±1/2 is the
s, the projection of which on a specified direction are s z = ± / 2 = 
(s)
spin quantum number. For µ z = µ B gms with g = 2 we obtain:
1
1
e e2
h mlj×r j
q j +q j )( r j−r j )+ ( q j −q j )( r j +r j ) −
. dr j +
(
ℏ 4π mc 2 e ∮ r 3j
2
2
e e2
h mlj ×r j
e e2
h 2m sj ×r jj
e e2
h 2m sj ×r jj
.
dr
−
dr
+
. dr jj
∮
∮
∮
2
2
j ℏ
jj ℏ
ℏ 4π mc2 e r 3
e r3
e r3
4π
m
c
4π
mc
j
jj
jj
Γ=
'
'
'
'
'
'
'
'
'
'
'
'
'
'
'
where ( h / e )m li and ( h / e ) m si are the flux vectors. For qoj = qoj’ =qo , one obtains
Γ=q o R cosθ−Γ o
Γ o=
[
2π×2m sj 2π×2m sj
e2
2π
2π
e2
m
−
m
+
−
+
lj
r j lj 2m c 2 r jj
rj j
2m c 2 r j
[
'
'
]
In these conditions q′o = q o −
'
'
'
]
e 2 ( ml + 2ms )

xˆ where x
is a unit vector perpendicular to r and
2
2m
r
µ . When the interacting field is a photon field, then ρο = 0. For a quasi free electron εk - εk-q
2
<< ωq, ωq = qo / 2m ( m is an effective electron mass).
To calculate Kjj’ (3a) we consider the two sums separately. The first sum can be easily
calculated as:
o
∑
q
( q . qo )2
ω 3q ω 2q o
2m
=
ℏ
2
( )
π
qo
1
2m
Ω
cos2 α sin αdα ∫ qdq =
2 3
2∫
ℏ
qo c ( 2π ) 0
0
2
( )
r3
9πc 3
where we have considered Ω = 4πr3/3. Next
0.76π / r
π
Ω
2
(
1
+
cos
Γ
)
=
1
+
q
dq
cos( qo R cos θ − Γo ) sin θdθ =1 + 0.53 cos( Γo )
∑
( 2π ) 2 ∫0 o o ∫0
q
o
In these condition EC (5) becomes
EC =
ℏc
1+0 . 53 cos ( Γ o ) ]
144 π r [
(7)
The upper limit of q0 as 0.76π/r is imposed by the constraint condition
3
[( 4π r 3 / 3) / ( 2π ) ] × 4πqo3 / 3 = 1 . For msi = msj = ½ , Γo reduces to:
πe 2
m❑−mlj )
2 ( lj
mr o c
(8)
where ro = rj = rj’ is the radius of the electron orbit and r = rjj’. For mlj – mlj’ = 1 we get:
Γ o=
Γ0
triplet
'
=
πe 2
a
=
2
r
mr o c
o
(8a)
where a = 8.95.10-13 cm.
For msj = ½, msj’ = -1/2, we obtain:
2
2
Γ 0 = mrπe c ( mlj−mlj ) + 2πmrec
2
'
(9)
2
o
For mlj = mlj’ the above relation reduces to:
Γo
sin glet
=
2π e 2 2 a
=
mr c 2 r
(9a)
For an interacting pair of spins s1 = s2 = 1 / 2, the expectation value of the Heisenberg term
(4) is written
3 3

E H = 2 K12s1.s 2 = K12 [ S.S − s1.s 2 − s 2 .s1 ] = EC S ( S +1) − − 
4 4

(10)
with S = s1 +s2. In the singlet state, S = 0 and EH = - (3/2)EC . In the triplet state, S = 1, and EH
= (1/2)EC , where EC is given by Eq. (7). The spin dependent energy of interaction is
E S=
ℏ c 0 . 53cos ( Γ o ) ℏ c 1+0 . 53cos ( Γ o )
3
+
S ( S+ 1 )−
r 144 π
r 144 π
2
[
]
(11)
The term responsible for magnetic ordering is that containing the product of cos(Γo) and S( S
+ 1 ), that is
EM=
0 . 53 ℏ c cos ( Γ o )
144 πr
(12)
S ( S+ 1 )
For S = 0, singlet state, one obtains EMS = 0, while for S = 1, triplet state, one obtains
EtM =
1 . 06 ℏ c cos ( Γ to )
144 π r
(13)
It is observed that for cos(Γo) < 0, we have EMt < EMs and the interaction energy for the triplet
state is lower than that associated with the singlet state. But, in this case, as it appears from
relation (9a), the distance between the two electrons should be r < 5.7.10-13 cm. This situation
may occur in a shell of an atom or in an electron band, because the cosine term diminish the
repulsion Coulomb energy. Thus for an atom with two valence electrons in a shell with l > 1,
the interaction energy between the two electrons when they occupy different Lz states ( all
degenerate in energy) and adopt a triplet spin state, is smaller than the interaction energy
when they occupy the same Lz state, in which case they must have opposite spins. Therefore,
the triplet state is favorable. This may explain Hund’s rules related to the organization of
electrons in incompletely filled shells. Finally, we note that if we do not separate the
Heisenberg term, and do not take into account the magnetic moments, the expectation value
of HI (3) would be6
EI =
1 ℏc
ℏc
≈α
147 r
r
(14)
which is just the Coulomb’s law where the constant 1/147 differs some few on the fine
structure constant, α = 1/137; this difference appears due to constraint in the evaluation of the
integral over qo. For a better agreement with experimental data we must take in the above
equatuions the value of 0.65 than 0.53.
3. RESULTS FOR FCC AND BCC LATTICES
Suppose now that at some instant a given atom has its total spin in the up direction. The
intra-atomic interactions are of such a nature that this atom tends to attract electrons with spin
up and repel those with spin down. In the case of the face-centered cubic lattice every atom is
surrounded by 12 nearest neighbors, which have the coordinates (a/2)(±1,±1,0), (a/2)
(±1,0,±1), (a/2)(0,±1,±1). Suppose the magnetic moment µ(0,0,µ). Then, the vector potential
can be chosen in the following form A(Ax,Ay,0). For nj = nj’ = 1, 2s j .s j′ = S ( S +1) − 3 / 2 we
write:
( aq2 −Γ ) cos (aq2 −Γ )−
aq
aq
aq
aq
cos (
−Γ ) cos (
−cos (
−Γ ) cos (
]
)
2
2
2
2 )
1
N
i q j . r j −q j' . r j' )
∑e(
jj
ox
=4 [3−cos
oy
o
'
ox
oz
o
oy
oz
o
o
(15)
where Γo is given by Eq. (8a) for the triplet state and by Eq.(9a) for the singlet state. The sum
in the interaction energy becomes:
1
N3
( q . q'o)
2
i q oj . r −q ' . r '
j oj j
∑ ω3 ω2 ∑ e
q,q 'o
q 'o
q
(
jj'
)
2m
=
ℏ
2 3
( )
π /a
π /a
π /a
r
'
'
Ω 4 dq'
∫
ox ∫ dq oy ∫ dq oz
3
3
9πc ( 2π )
0
0
0
aq'ox
aq'oy
aq 'ox
aq 'oz
aq'oy
aq 'oz
[3−cos
−Γ o cos
−Γ o −cos
−Γ o cos
−cos
−Γ o cos
]
2
2
2
2
2
2
m2 64 r 3
3π 2−f fcc ( Γ o )
2
3 3
ℏ 9π c
(
) (
[
) (
) ( ) (
) ( )
]
where
f fcc ( Γo ) = ( cos Γo + sin Γo ) + 2( cos Γo + sin Γo )
2
Further
EC = ∑ K jj ′ =
jj ′


c  3π 2
2
− ( cos Γo + sin Γo ) − 2( cos Γo + sin Γo ) 
3 
9π r  4

and the interaction energy
EI=
[
]{
ℏ c 3π2
1
3
− ( cos Γ o +sin Γ o ) 2−2 ( cos Γ o +sin Γ o )
+S ( S+1 ) −
3
2
2
9π r 4
The term responsible for magnetic ordering is
}
(16)
E M =−
[
ℏ c ( cos Γ o + sin Γ o ) 2 +2 ( cos Γ o +sin Γ o )
] S (S+ 1)=−ΔE S ( S+ 1 )
I
9π3 r
(16a)
which has a minimum for Γo = 2nπ - π/4.
In a bcc structure, every atom is surrounded by 8 nearest neighbors which have the
coordinates (a/2)( ± 1, ± 1, ± 1 ). One obtains
1
N
∑e
(
i q oj .r j −q oj′ .r j′
jj ′
)

 aq

 aq

 aq
= 81 − cos ox − Γo  cos oy − Γo  cos oz
 2

 2
 2





Further,
i q
( q .q o )2
1
(
e
∑
∑
3
3 2
N q,q ω q ω q jj
oj
.r −q ' . r '
j oj j
)= 4m2 r 3
ℏ2
'
o
o
π /a
π /a
π /a
[
8 ∫ dq ox ∫ dq oy ∫ dq oz 1−cos
0
0
2
3
0
[
(
Ω
9πc 3 π 3
]
aqox
aq oy
aq
− Γ o cos
−Γ o cos oz =
2
2
2
) (
32 m r
8
1− 3 ( cos Γ o +sin Γ o ) 2
2
3
ℏ
9πc
π
)
]
The energy of interaction is given by the expression
EI=
[
][
ℏc
8
1
3
1− 3 ( cos Γ o +sin Γ o )2
+S ( S +1 )−
18 π r
2
2
π
]
and the term responsible for magnetic ordering is
4ℏ c
f bcc S ( S +1 ) =−ΔE I S ( S+ 1 )
9π 4 r
f bcc= ( cos Γ o +sin Γ o )2
E M =−
(17)
which is some few different from the energy (16a) obtained for the fcc structure.
From Eqs. (16) and (17) it appears that there is an oscillating interaction between the two
electrons belonging to different atoms. The nature of this interaction may be not different than
that which is encountered in the case of interaction between two ions by indirect exchange
[7]. In the later case, roughly speaking, the electrons belonging to an ion flip a conduction
electron which then travels to another site and interacts with the spin of the ion of the second
site. Perhaps the most significant application of the RKKY theory has been to the theory of
giant magneto resistance (GMR)[8,9]. GMR was discovered when the coupling between thin
layers of magnetic materials separated by a non-magnetic spacer material was found to
oscillate between ferromagnetic and anti ferromagnetic as a function of the distance between
the layers. The period of an oscillation is determined by the Fermi wave vector, in the case of
free electron gas, via λ = 2π/2kF = π/kF. The phase of the cosine in RKKY theory is kFr while
in our theory this phase is Γo = πe2/mrc2= 1/kor where ko = mc2/πe2 = 1.296.1012 cm-1 . In this
case the “period” of an oscillation is dependent of r. From the condition Γo(n) - Γo(n+1) = 2π ,
one obtains λn = rn –rn+1=2kornrn+1. For rn ~ 10-5 Ǻ and m=mo , results λn ~ 10-5 Å a value
which is much shorter than the Compton wavelength, ħ/mc = 8×10-3Ǻ, and than the period of
the RKKY oscillation, which is of the order of 3Ǻ. The oscillatory behavior is the result of
the interference of two oscillating fields generated by the magnetic moments of the two
electrons. Each fringe arises from a definite difference in phase. In the RKKY model the
effect is an oscillatory polarization of the conduction electrons. Due to the small values of
parameter a, for the distance between electrons of ~ 3 Å in a lattice, it results Γo << 1 and the
effect of cosine terms is negligible.
4. CONCLUSIONS
Using an effective field Hamiltonian (as in Ref.6) we have shown that the Coulomb’s law is
modified by the spin-spin interaction. First, we separate the Heisenberg term in the energy of
interaction. Next, we take into account the effect of the spin magnetic moment. The Coulomb
interaction is modulated by a cosine term whose argument depends on the spin magnetic
moment and on the distance between the two electrons. Evidently, this situation occurs in the
absence of the magnetic field, for example in superconductors. Further, we have studied the
influence of the electron magnetic moments, both orbital and spin, on the electron-electron
interaction in an isolated atom. We have found that in an incomplete shell the interaction
energy of the triplet state is smaller than the energy of the singlet state. On this basis may be
explained Hund’s rules. The condition of ferromagnetism is studied in both fcc and bcc
lattices.. The coupling energy, due to the magnetic field generated by the two interacting
electrons, oscillate with a “period” 1/kor where ko = mc2/πe2 . This period of oscillation is
dependent of r and is of the order of 10 -5 Ǻ, a value which is much smaller than the RKKY
period which is of ~ 3Ǻ.
APPENDIX
The electron will always carry with it a lattice polarization field. The composite particle,
electron plus phonon field, is called a polaron; it has a larger effective mass than the electron
in the unperturbed lattice. By analogy, in our model, we consider a coupling between an
electron and a boson.
Let us consider a linear chain of N bodies, separated at a distance R. The Hamiltonian
operator of the interacting bodies ( electrons ) and the boson connecting field takes the general
form
H = H o ,el + H o , ph + H I
where
H o,el =∑
k,σ
ℏ 2 k2 +
c c
2m k,σ k,σ
+
is the Hamiltonian of the electrons of mass m, ck ,σ , ck ,σ are the electron creation and
annihilation operators, k is the wave vector of an electron and σ ls the spin quantum number,
(
H o,ph=∑ ℏ ωq a+q aq +
q
(
ωq=
α+DRq 2
ρ
1 /2
)
1
2
)
where ωq is the classical oscillation frequency, α is the restoring force constant, D is the
coupling constant, ρ is the linear density of flux lines, aq+, aq are the boson creation and
2
annihilation operators and ρ = ρ o + DR / c , ρo is the density of the interacting field, if this is a
massive field, c is the velocity of the boson waves. The interaction Hamiltonian operator HI is
given by the expression[10]
2
HI =
DR
 ∂u 
+
sn 
 Ψ ( z ) Ψ( z )dz
∑
∫
2
 ∂z 
n
where
1
ℏ 1 /2
e
a ql e iq . z +a+ql e−iq . z )
∑
(
ql
2
ρω
NR
√ q .l
ql
1
Ψ ( z )=
c kσ eik . z χ ( σ )
∑
√ NR k,σ
1
Ψ + ( z )=
∑ c + e−ik . z χ ( σ )
√ NR k,σ kσ
iq ( z− z )
1
1 ℏ
n
s n= ∑ S q e
; Sq =
b +b+−q )
N q,q
R 2mωq ( q
(
u ( z )=
)
o
o
o
o
bq+o , bq o are creation and annihilation operators associated with the electron oscillations, eql
denotes the polarization vectors and χ(σ) is the spin wave function. sn is the displacement of
a body near its equilibrium position in the direction of R and, in the approximation of nearest
neighbours, it is assumed that D does not depend on n. The Hamiltonian of interaction
between electrons via bosons becomes
H I =−
1 D
∫∑ ∑
N 3 2R2
n
'
k,k ' ,q,q ,q
o
e
iq o . z
n
,σ,σ
ℏ
ℏ q . q'
+
b
+b
( qo −qo ) 2 ρω
' 2mω q o
q
−i( q+q' +k −k' −q ). z
o
+
( a q +a−q
) ( aq' +a+−q' ) c+' ' c kσ e
kσ
dz
q, q’ are the wave vectors associated with the bosons of the connecting field, qo is the wave
vector associated with the oscillations of the electron, and k, k’ are the wave vectors of the
electrons. Consider the integral over z
∫e
− i ( q + q′ + k − k ′ + q o ).z
dz = NR∆ ( q + q′ + k − k ′ − q o )
(A1)
where ∆(x ) = 1 for x = 0 and ∆(x) = 0, otherwise. In the bulk crystal NR is replaced by V =
NΩ where Ω is the volume of a unit cell and N is the number of unit cells. We write
∑
H I=−ℏ
'
'
'
g q (a q +a+−q ) ( a−q +a+q ) c + ' c kσ bq Δ ( q+q ' +k+k ' −q o )+
k,k ,q,q ,q ,σ,σ
o
+
gq o a−q +a q b q ' +b+−q '
o
o
(
where
)(
'
o
'
k' σ
o
) c k σ' c +kσ bq Δ ( q+q ' +k −k ' +q o)
+
'
o
gq =
o
ℏD
qq'
8N2 mR ( ρo +DR /c 2 ) ω q ω q
'
∑e
iq o . z
n
n
If we omit the terms with two creators and two annihilators, it may be written
(a
q
)(
)
+ a−+q a−q′ + aq+′ ≈ aq aq+′ + a−+q a−q′
In equation (A1) we choose q’ =qo, k’ = k + q. In the interaction picture the effective
Hamiltonian is given by
eff
eff
H eff
I =H I1 +H I2
H eff
I1 =ℏ
∑ ∣g q ∣2
q,q o ,k
+
o
ωq
( ε k −εk −q )2 −ω 2q
+
+
+
( a q a q aq a q +a+q aq a q aq ) c +k−q,σ c k σ' c kσ c k −q,σ'
o
'
o
'
2
H eff
I2 =2 ℏ ∑ ∣gq∣
q,k
o
'
o
'
1
( ε k −ε k −q )−ωq
'
'
a+q a q c+k−q,σ c k −q,σ
The expectation value of the energy of HI1eff is the energy of the electron-electron interaction
given in the text. The expectation value of the energy of HI2eff is the self energy of the electron
and is used to calculate, for example, the polaron energy[6]. Finally, we specify that we have
considered the interaction, via bosons, of two oscillators which move with the momentum ħk .
The result is also valid for k = 0.
REFERENCES
1. W. Heitler and F. London,Z. Phys. 44,455(1927).
2. W. Heisenberg, On the theory of ferromagnetism, Z. Phys. 49,619(1926).
3. E. C. Stoner, Magnetism and Matter, (Methuen, Lodon, 1934).
4. J. Hubbard, Proc. Roy. Soc. of London A266 ,278(1963).
5. K. Hüller, J. Mag. Magn. Mater. 61, 347(1986)
6. V. Dolocan, A. Dolocan, and V. O. Dolocan, Int. J. Mod. Phys. B 24,479(2010); ibid
Rom. J. Phys. 55, 153 (2010).
7. C. Kittel, Indirect exchange interaction in metals, Solid State Physics: Advances in
Research and Applications, 22, 1(1968).
8. P. Grünberg, R. Schreiber, Y. Pang, M. B. Brodsky, and H. Sowers, Phys. Rev.
Lett.57, 2442(1986).
9. M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen van Dan, F. Petroff, P. Etienne, G.
Creuzet, A. Friederich, and J. Chazelos, Phys. Rev. Lett. 61, 2472(1988)
10. A. Dolocan, V. O. Dolocan and V. Dolocan, Mod. Phys. Lett. 19, 669(2005).
Fig.1