Материал: Маслов ИНТРОДУЦТИОН ТО ПХЫСИЦС ОФ СЕЦОНД-ОРДЕР МАГНЕТИЦ 2015

(magnetic moment) to all other atoms (magnetic moments), the prime implies that Rl ≠ 0. This sum is the same for every atom, since we con-

sider that all atoms (magnetic moments) located periodically, i.e., they form the crystal. Thus, the interaction term takes the following form

ergy (sum of the energies of exchange interactions of randomly selected atom with all other atoms). Taking into account that J (Rl ) falls off ex-

ponentially as the Rl increases, denote the J (Rl )= J for the nearest neighbors and considering that J (Rl )= 0 for other atoms, we obtain ∑/ J (Rl )≈ zJ, where z is the number of nearest neighbors of each

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where θ is the some characteristic temperature (next, we shall omit the Boltzmann constant).

Thus, taking into account the notation introduced, we can write down the Hamiltonian of the Ising model in the following form

i=1

The main differences between this Hamiltonian and the Hamiltonian

i=1

of the constant E0′ (note that E0′ does not affect on the σz calculation)

and the field h substituted by the “effective field” heff = h + θσz .

Finally, in the mean-field approximation system of interacting magnetic moments can be approximately described as the system of nonin-

teracting magnetic moments placed into the “effective field” heff ≠ h.

12. Curie-Weiss equation and Curie-Weiss law

We now apply the mean-field approximation to the system of magnetic moments with Jij ≥ 0 (i, j) . In particular, we assume Jij = J

for the nearest neighbors and Jij = 0 for other atomic pairs. Magnetization of such system (projection on the z-axis) equals to

−1≤ x ≤1, and x = 0 in the case of µz = 0. Let us find x = x(T, H ).

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Recall that the Hamiltonian of the Ising model in the mean-field approximation is

i=1

N

Constant E0′ does not affect on the σz calculation, term −heff ∑σiz is

i=1

the operators sum of the interaction between each magnetic moment and “effective field”. In other words, in our approximation magnetic moments do not depend on each other, but they located in the “effective

field” heff . Therefore, the value σz for arbitrary magnetic moment can be obtained in terms of single magnetic moment located in the field

Recall that the heff = h + θσz ≡ h + θx . Therefore the Eq. (12.3) takes the form

Such dependence of x is referred to as Curie-Weiss equation.

Let us find the magnetization and magnetic susceptibility at high

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The latter temperature dependence of χ is referred to as Curie-Weiss law. Note that if the exchange interaction is absent ( Jij = 0 ), then

θ = 0 , so, we obtain the Curie law. The value θ is called the Curie temperature.

13. Ferromagnetic transition in the Ising model. Curie temperature. Order parameter

In the previous paragraph we have shown that at high temperatures

which something happens. Maybe it is a transition to ferromagnetic state? Let us study the Curie-Weiss equation in more detail.

Consider the h = 0 in the Curie-Weiss equation, thus the latter takes the following form

This equation can not be solved analytically. We shall try to solve it graphically. From the Fig. 13.1 and 13.2 one can see that at T > θ the Curie-Weiss equation has only one solution ( x = 0 ), but at T < θ it has three solutions ( x = 0; x ≠ 0 ).

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