where ΔE_{I} is given by the above expression. For cobalt results T_{c} = 1411 K. A more elaborate method is the random phase approximation [12-14]. The Curie temperature can be also estimated numerically by employing the method of Monte Carlo simulations applied to effective Heisenberg Hamiltonian [15].

We take now into account the interaction with the atoms of the second group of coordination. If in a fcc structure we take into account the fact that every atom is surrounded by 6 neighbours of second order, which have the coordinates a(±1, 0, 0), a(0, ±1, 0), a(0, 0, ±1), then we must add to Equation (10) the term:

and the corresponding energy of interaction is:

Figure 2. Difference between the functions for the triplet and singlet state, Δf_{fcc}, as a function of Γ_{o}, in an fcc lattice filled shell.

Figure 3. Difference between the function of the triplet and the singlet state, Δf_{fcc}, as a function of Γ_{o} in a simple cubic lattice.

(15)

with

In Figure 3 Δf_{sc} = f_{sc}(Γ_{o}) – f_{sc}(2Γ_{o}) as a function of Γ_{o} is shown. It is observed that depending on the distance between the atoms the coupling may be ferromagnetic at Γ_{o} = 2.1 + 2sπ, as well as antiferromagnetic at This situation is also adequate for a simple cubic lattice. We note that in an fcc lattice of transition metals the ferromagnetic state is determined by the interaction between nearest neighbours.

4. Conclusion

Using an effective field Hamiltonian we have shown that the Coulomb’s law is modified by the spin-spin interaction. If we take into account the spin magnetic moment, when the spins are oriented antiparallel, 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 a fcc lattice of transition metals and the obtained results are in a good agreement with experimental data. By using the mean field approximation, we have estimated the Curie temperature. The coupling, due to the magnetic field generated by the two interacting electrons, oscillate with a “period” 1/k_{o}R where k_{o} = m/πe^{2}. This period of oscillation is dependent of R and for R = 3.3 Ǻ and m/m_{o} = 5, is equal to 0.4 Ǻ, a value which is smaller than the RKKY period of ~3 Ǻ. This means that the magnetic field generated by the magnetic moments of the electrons is modulated by an oscillation which has a period of π/k_{F}, where k_{F} is the Fermi wave vector.

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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.

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:

where

is the Hamiltonian of the electrons of mass m, are the electron creation and annihilation operators, k is the wave vector of an electron and σ is the spin quantum number,

where is the classical oscillation frequency, α is the restoring force constant, D is the coupling constant, ρ is the linear density of flux lines, , are the boson creation and annihilation operators and, ρ_{o} 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 H_{I} is given by the expression [16]:

where

are creation and annihilation operators associated with the electron oscillations, e_{q}_{l} denotes the polarization vectors and χ(σ) is the spin wave function. s_{n} 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:

q, q' are the wave vectors associated with the bosons of the connecting field, q_{o} 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:

(A1)

where Δ(x) = 1 for x = 0 and Δ(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:

where

Approximating

In Equation (A1) we choose q' =q_{o}, k' = k + q. In the interaction picture the effective Hamiltonian is given by:

The expectation value of the energy of H_{I}_{1}^{eff} is the energy of the electron-electron interaction given by Equation (1) in the text. The expectation value of the energy of H_{I}_{2}^{eff} is the self energy of the electron and is used to calculate, for example, the polaron energy [10].

NOTES

^{*}Corresponding author.