Physics

I. A. Kvasnikov

On the Application of the Variational Principle in Problems on a Binary Alloy and the Ising System

(Presented by Academician N. N. Bogolyubov, 14 XI 1956)

As was indicated in the preceding note \((^{2})\), in order to determine the statistical sum of a binary alloy it is necessary to solve the auxiliary problem of an Ising antiferromagnet. In the present note, when applying N. N. Bogolyubov’s variational principle \((^{1})\), a different division into parts of the Hamiltonian of the antiferromagnet \(((^{2}), (1))\) will be made, consisting in the fact that some of the terms from \(\mathcal H_1\) are transferred to \(\mathcal H_0\), with the sole condition that \(Z_0 = \operatorname{Sp}\{\exp(-\mathcal H_0)\}\) be computable. Namely, \(\mathcal H_0\) includes the interaction terms of groups of neighboring sites; the interaction between these groups and the interaction with distant neighbors remains in \(\mathcal H_1\). The simplest choice of these groups is in the form of pairs of sites. For simplicity we neglect the interaction with distant neighbors, which can be taken into account analogously to \((^{2})\). Apparently, increasing the number of sites in each of the groups included in \(\mathcal H_0\) (as the next approximation, it is natural to choose groups each of which consists of a central site and a “shell” of all its nearest neighbors) will, at the cost of increasing the computational part, lead to an increasingly good approximation.

Let us agree to denote by the letter \(\mathcal H\) the Hamiltonian divided by the temperature \(\theta\); \(B\) is the magnetic field in units of the Bohr magneton, divided by \(\theta\); \(N\) is the number of lattice sites; \(E = e/\theta\) is the exchange integral divided by the temperature. We shall also assume that all lattice sites are equivalent and that opposite boundaries of the crystal are joined to one another. In accordance with what has been said, for the antiferromagnet we have

\[ \mathcal H = \mathcal H_0 + \mathcal H_1 = \frac{1}{2} E \sum_{\langle ij\rangle} \sigma_i \sigma_j - B \sum_{i=1}^{N} \sigma_i, \tag{1} \]

\[ \mathcal H_0 = \sum_{l=1}^{N/2} \mathcal H_{2l}, \qquad \mathcal H_k = -(B-\beta_0)\sigma_k - (B-\beta_1)\sigma_{k+1} + \varepsilon \sigma_k \sigma_{k+1}. \]

Here \(\beta_0\) and \(\beta_1\) are variational parameters referring to two sublattices; \(\varepsilon\) is a parameter characterizing the effective interaction of a pair of nearest sites. It is easy to see that for \(\varepsilon = 0\) the above choice of \(\mathcal H_0\) coincides with formula (6) of note \((^{2})\). Proceeding in accordance with the method described earlier \(((^{1}), (4))\), after simple calculations we obtain:

\[ \begin{aligned} -\frac{2}{N}\ln Z_{inf} ={}& 2\ln 2 + \ln[\operatorname{ch}\gamma_0\, \operatorname{ch}\gamma_1\, \operatorname{ch}\varepsilon - \operatorname{sh}\gamma_0\, \operatorname{sh}\gamma_1\, \operatorname{sh}\varepsilon] \\ &+ \beta_0 \frac{\operatorname{th}\gamma_0 - \operatorname{th}\gamma_1 \operatorname{th}\varepsilon} {1 - \operatorname{th}\gamma_0 \operatorname{th}\gamma_1 \operatorname{th}\varepsilon} + \beta_1 \frac{\operatorname{th}\gamma_1 - \operatorname{th}\gamma_0 \operatorname{th}\varepsilon} {1 - \operatorname{th}\gamma_0 \operatorname{th}\gamma_1 \operatorname{th}\varepsilon} \\ &+ (E-\varepsilon) \frac{\operatorname{th}\varepsilon - \operatorname{th}\gamma_0 \operatorname{th}\gamma_1} {1 - \operatorname{th}\gamma_0 \operatorname{th}\gamma_1 \operatorname{th}\varepsilon} - E(c-1) \frac{(\operatorname{th}\gamma_0 - \operatorname{th}\gamma_1 \operatorname{th}\varepsilon) (\operatorname{th}\gamma_1 - \operatorname{th}\gamma_0 \operatorname{th}\varepsilon)} {(1-\operatorname{th}\gamma_0 \operatorname{th}\gamma_1 \operatorname{th}\varepsilon)^2}, \end{aligned} \tag{2} \]

where, for convenience, the notation \(\gamma_0 = B-\beta_0,\ \gamma_1 = B-\beta_1\) has been introduced.

The variational parameters \(\beta_0\), \(\beta_1\), and \(\varepsilon\) are then determined from the system of transcendental equations:

\[ \frac{\partial \ln Z_{inf}}{\partial \beta_0} = \frac{\partial \ln Z_{inf}}{\partial \beta_1} = \frac{\partial \ln Z_{inf}}{\partial \varepsilon} =0. \tag{3} \]

As in notes \((^{1,2})\), we shall regard \(Z_{inf}\) as an approximate expression for the statistical sum of an antiferromagnet.

The magnetization per particle from (2) and (3) is determined by the expression:

\[ M=\frac{1}{2}\, \frac{(\operatorname{th}\gamma_0+\operatorname{th}\gamma_1)(1-\operatorname{th}\varepsilon)} {1-\operatorname{th}\gamma_0\operatorname{th}\gamma_1\operatorname{th}\varepsilon}. \tag{4} \]

In order to avoid cumbersome calculations, let us consider the application of the results obtained to a \(50\%\) binary alloy, using relations (2) and (4) of note \((^2)\). Setting \(M=0\), we obtain from equations 3) and (4):

\[ B=0;\qquad \gamma_0=-\gamma_1=\gamma;\qquad \varepsilon=E=\frac{e}{\theta}. \tag{5} \]

The mean thermodynamic energy per particle of the alloy, determined by relation (20) of work \((^2)\), will have the form:

\[ \overline{E}= \frac{\varepsilon_0}{8} -\frac{e}{2}\, \frac{c\,\operatorname{th}^2\gamma(1+\operatorname{th}E)^2+\operatorname{th}E(1-\operatorname{th}^2\gamma)^2} {(1+\operatorname{th}^2\gamma\,\operatorname{th}E)^2}, \tag{6} \]

where \(c\) is the number of nearest neighbors for the given lattice; \(\varepsilon_0\) is a constant. The quantity \(\gamma\) is determined from the equation

\[ \gamma = E(c-1)\, \frac{\operatorname{th}\gamma(1+\operatorname{th}E)} {1+\operatorname{th}^2\gamma\,\operatorname{th}E}. \tag{7} \]

From this equation it follows, in particular, that at temperature \(\theta>\theta_0\) the quantity \(\gamma=0\), while the critical temperature \(\theta_0\) is determined by the equation:

\[ 1=E_0(c-1)(1+\operatorname{th}E_0), \tag{8} \]

where \(E_0=e/\theta_0\).

Thus, in contrast to the results of \((^2)\), we have that for \(\theta>\theta_0\) the mean energy and the heat capacity of the alloy, calculated in units of Boltzmann’s constant, are functions of temperature:

\[ \overline{E} = \frac{\varepsilon_0}{8} -\frac{e}{2}\operatorname{th}\frac{e}{\theta}, \tag{9} \]

\[ C = \frac{\partial \overline{E}}{\partial \theta} = \frac{e^2}{2\theta^2} \left( 1-\operatorname{th}^2\frac{e}{\theta} \right). \tag{10} \]

The value of the heat capacity at the point \(\theta_0\) from the left can be obtained from (6)—(3):

\[ C= \frac{[2(c-1)E_0-1](6+4E_0)+3(c-1)} {2(c-1)[3-2(c-1)E_0]}. \tag{11} \]

In the case of a simple cubic lattice, the numerical solution of equation (8) gives the value \(E_0\simeq 0.171\); then the value of the heat capacity to the left of the Curie point is \(C\simeq 1.535\), and to the right, according to formula (10), \(C\simeq 0.0142\), so that the jump in heat capacity only slightly exceeds the magnitude of the jump that follows from the Bragg—Williams theory (see also \((^2)\)). It should also be noted that the values of \(E_0\) obtained from (8) and formula (11) of work \((^2)\) are quantities of the same order; for example, for \(c=6\), according to formula (11) of work \((^2)\), we have \(E_0=1.166\).

Applying the scheme set forth above to the Ising ferromagnetic system, we obtain for the average magnetic moment of a particle a formula which, from the point of view of the variational principle (1), is an improvement of the Weiss formula:

\[ M=\frac{1}{N}\frac{\partial \ln Z}{\partial B} =\operatorname{th}(B+J(c-1)M)\, \frac{1+\operatorname{th}J} {1+\operatorname{th}^{2}(B+J(c-1)M)\cdot \operatorname{th}J}, \tag{12} \]

where \(J\) is the exchange integral divided by the temperature \(\theta\). The equation for the critical temperature \(\theta_0\) is analogous to (8):

\[ 1=J_0(c-1)(1+\operatorname{th}J_0). \tag{13} \]

In the practically realizable case \(B\ll 1\), for \(\theta>\theta_0\) we obtain for the magnetic susceptibility

\[ \chi \simeq \frac{1+\operatorname{th}J} {\theta-\theta_0 J_0(c-1)(1+\operatorname{th}J)}, \tag{14} \]

in contrast to the Curie–Weiss law (see also (1)):

\[ \chi \simeq \frac{1}{\theta-\theta_0}. \]

Putting \(B=0\) in (12), we obtain the equation for the spontaneous magnetization as a function of temperature. Direct calculation for temperatures \(\theta/\theta_0=1;\ 0.9;\ 0.8;\ 0.7;\ 0.6\) leads, respectively, to the values \(M=0.0;\ 0.533;\ 0.718;\ 0.838;\ 0.912\), which are somewhat larger than the quantities obtained from Weiss theory: \(M=0.0;\ 0.525;\ 0.710;\ 0.828;\ 0.907\).

The relations for the energy and heat capacity of the Ising ferromagnet are analogous to (9)—(11).

Under an analogous consideration of the Heisenberg model, instead of (12), for the susceptibility one obtains the more complicated expression

\[ M=\operatorname{th}\gamma\, \frac{1+\operatorname{th}J} {1+\operatorname{th}J\cdot \operatorname{th}^{2}\gamma +e^{-J}\operatorname{th}J\cdot \operatorname{sh}J\cdot (1-\operatorname{th}^{2}\gamma)}, \]

where \(\gamma=B+J(c-1)M\).

I express my deep gratitude to Academician N. N. Bogolyubov, under whose supervision this work was carried out.

Moscow State University
named after M. V. Lomonosov

Received
12 XI 1956

REFERENCES

\(^{1}\) I. A. Kvasnikov, DAN, 110, No. 5 (1956).
\(^{2}\) I. A. Kvasnikov, DAN, 113, No. 3 (1957).
\(^{3}\) F. Nix, W. Shockley, Advances in Physical Sciences, 20, 344 (1938); G. F. Newell, E. W. Montroll, Rev. Mod. Phys., 25, 353 (1953).