Uniaxial Quadrupole Order in a Magnet with Strong Biquadratic Exchange

1. Biquadratic exchange interaction occurring in magnets with spins S ≥ 1 [1‐5] is of interest because it may change the character of paramagnet–ferromagnet temperature phase transitions from the second to first order. In addition, when the biquadratic exchange is sufficiently strong (in comparison with the bilinear one), peculiar quadrupole ordering of spin moments, which has neither spontaneous magnetization nor any other type of magnetic order with the presence of nonzero mean values of spin projection dipole operators \({\kern 1pt} \left\langle {{{S}_{{{\alpha }}}}\left( n \right)} \right\rangle \ne 0\) (\(\alpha = X,Y,Z\)) at magnetic lattice sites, may occur in these magnets. This circumstance was demonstrated by the example of a magnet with spin S = 1 [6, 7] (see also [8]); it was revealed that, when the uniaxial quadrupole state for S = 1 occurs from the paramagnetic state, it exists up to the temperature T = 0 (i.e., it may be the ground state for the system).

Note that the case of a system with spins S = 1 is distinguished; if one considers magnets with the biquadratic and weak bilinear ferromagnetic exchange with spins S > 1, quadrupole order cannot be the ground state of a magnet (as will be shown below), because even very weak bilinear ferromagnetic exchange makes the ferromagnetic state at T = 0 energetically more favorable. At the same time, at a sufficiently high ratio of the biquadratic and bilinear exchange parameters, the paramagnetic state of the system becomes unstable to a transition to the quadrupole state at a higher temperature (as compared to the temperature of transition to the ferromagnetic state). Thus, at S > 1, uniaxial quadrupole ordering in a magnet with strong biquadratic exchange may act as an intermediate magnetic state at the temperature transition from the paramagnetic to ferromagnetic state. Specific features of the behavior of thermodynamic characteristics of a magnet with the biquadratic exchange at the temperature transition from the paramagnetic to ferromagnetic state through an intermediate quadrupole state have not been studied; in this study, we solve this problem by the example of a magnet with spin S = 3/2.

2. The Hamiltonian of a magnet with the bilinear and biquadratic exchanges in magnetic field H parallel to the OZ axis of the cubic lattice is as follows:where n is a lattice site and the parameters of the bilinear (I > 0) and biquadratic (K > 0) exchange with z nearest magnetic neighbors are assumed positive.

To pass to the mean-field approximation in Hamiltonian (1), the biquadratic product of spins \({{\left( {{{S}_{n}}{{S}_{{n + {{\Delta }}}}}} \right)}^{2}}\) is expressed in terms of products of the quadrupole operators [7, 9]:

As a result, when passing to the mean-field approximation, the one-site Hamiltonian of the ferromagnetic state in a magnet with the biquadratic exchange takes the form ([7, 9], see also [8])Here, quantities \({{\sigma }_{Z}} = \left\langle {{{S}_{{Z,n}}}} \right\rangle \) (thermodynamic mean value of the Z spin projection), z is the number of nearest neighbors, and \({{q}_{0}} = \left\langle {{{Q}_{{0,n}}}} \right\rangle = 3\left\langle {S_{{Z,n}}^{2}} \right\rangle - S\left( {S + 1} \right)\) (mean value of the quadrupole operator \({{Q}_{{0,n}}}\)) act as the dipole and quadrupole order parameters of the magnetic state. Having set \({{\sigma }_{Z}} = 0\) and \(H = 0\) in Hamiltonian (3) (spontaneous magnetization \(m\sim {{\sigma }_{Z}}\) is absent in the quadrupole state) and retained \({{q}_{0}} \ne 0,\) we obtain the Hamiltonian of the uniaxial quadrupole state \(H_{{qu}}^{{{\text{MF}}}}\left( n \right)\) with the symmetry axis \(OZ.\) Note also that, in going from the biquadratic product of the spin operators \({{\left( {{{S}_{n}}{{S}_{{n + {{\Delta }}}}}} \right)}^{2}}\) to products of the quadrupole operators \({{Q}_{{\alpha ,n}}}\), the bilinear term \(\left( {{{S}_{n}}{{S}_{{n + {{\Delta }}}}}} \right)\) is additionally isolated, which leads to renormalization of the bilinear exchange parameter I and its replacement with \(J = I - \left( {1{\text{/}}2} \right){\kern 1pt} K\) [7, 9] (below, we will restrict ourselves to the case J > 0).

It follows from Eq. (3) that the energies of a magnet in the ground ferromagnetic and ground quadrupole states are

It is known for the case S = 1 [8] that the conditions \({{\sigma }_{Z}}\left( {T = 0} \right) = {{q}_{{0,f}}}\left( {T = 0} \right) = 1\) and \({{\sigma }_{Z}}\left( {T = 0} \right)\), \({{q}_{{0.qu}}}\left( {T = 0} \right) = - 2\) are satisfied in the ground ferromagnetic state and quadrupole state, respectively. Therefore, using (4) and imposing the requirement \(E_{{0,qu}}^{{{\text{MF}}}}\left( {S = 1} \right) < E_{{0,f}}^{{{\text{MF}}}}\left( {S = 1} \right),\) we find that the quadrupole state is energetically more favorable as the ground state of the system at K > 2J (or at K > I, in the notation of exchange parameters in initial Hamiltonian (1)).

For the case S = 3/2 in ground ferromagnetic state, spins at all sites are in the state with \({{S}_{{Z,n}}} = 3{\text{/}}2,\) so that \({{\sigma }_{Z}}\left( {T = 0} \right) = 3{\text{/}}2\) and \({{q}_{{0,f}}}\left( {T = 0} \right) = 3.\) The quadrupole state at T = 0 with \({{\sigma }_{Z}}\left( {T = 0} \right) = 0\) can be constructed in several ways; however, the version of quadrupole ordering with the maximum in magnitude \({{q}_{{0,qu}}}\left( {T = 0} \right)\) value should be considered as a contender for the ground state. For example, if at T = 0 S_Z = 3/2 at one half of the lattice sites distributed randomly, while \({{S}_{Z}} = - 3{\text{/}}2\) at the other half of the sites, then \({{\sigma }_{Z}}\left( {T = 0} \right) = 0\) and \({{q}_{{0,qu}}}\left( {T = 0} \right) = 3{{( \pm 3{\text{/}}2)}^{2}} - 15{\text{/}}4 = 3,\) i.e., \({{q}_{{0,qu}}}\left( {T = 0} \right)\) has the same value as the parameter \({{q}_{{0,f}}}\left( {T = 0} \right)\) of the ferromagnetic state. As a result, the contribution to the ground state energy from the biquadratic exchange is identical for both ferromagnetic and quadrupole orderings; however, due to the additional contribution from the bilinear exchange \( - 1{\text{/}}2Jz\sigma _{Z}^{2}\left( {T = 0} \right)\), the ferromagnetic state at T = 0 is always energetically more favorable than the quadrupole state.

A similar result that quadrupole ordering is energetically unfavorable as the ground state of the system at finite values of the bilinear exchange parameter J > 0 can be obtained at S = 2, 5/2, etc.; therefore, to clarify the question of possible existence of the quadrupole state at finite temperatures, we should carry out comparative study of the temperature behavior of thermodynamic potentials (TDPs) of the quadrupole and ferromagnetic states and possible temperatures of phase transitions to these states from the paramagnetic one. To this end, let us consider in more detail the case \(S = 3{\text{/}}2.\)

3. The thermodynamic potential of the system in the ferromagnetic state \(\left( {{{\sigma }_{Z}} \ne 0,{{q}_{0}} \ne 0} \right)\) for spins S = 3/2 in the presence of an external magnetic field H is

where \(\beta = 1{\text{/}}{{k}_{{\text{B}}}}T\) and the magnetic entropy S_M in the presence of the field (per atom) is

In addition, using the minimization condition for TDP f in the order parameters \(\partial {{f}_{f}}{\text{/}}\partial {{\sigma }_{Z}} = 0\) and \(\partial {{f}_{f}}{\text{/}}\partial {{q}_{0}} = 0\) or directly calculating the thermodynamic means \({{\sigma }_{Z}} \equiv \left\langle {{{S}_{{Zn}}}} \right\rangle \) and \({{q}_{0}} \equiv \left\langle {{{Q}_{{0n}}}} \right\rangle \) for the statistical ensemble with Hamiltonian \(H_{f}^{{{\text{MF}}}}\left( n \right)\) (3), we obtain the self-consistent equations for the order parameters \({{\sigma }_{Z}}\) and \({{q}_{0}}\):

We investigate the spontaneous magnetic behavior of the system in the limit \(H \to 0.\) Then, the system of equations (7) for the dipole (\({{\sigma }_{Z}}\)) and quadrupole (\({{q}_{0}}\)) order parameters takes the form

The obtained system of equations (8) has solutions of three types: (i) solution \({{\sigma }_{Z}} = 0,\,\,{{q}_{0}} = 0\) (no order parameters), which corresponds to the paramagnetic state; (ii) solution \({{\sigma }_{Z}} = 0,\,\,{{q}_{0}} \ne 0,\) when only the quadrupole order parameter exists but the dipole order parameter (magnetization) is absent; and (iii) solution \({{\sigma }_{Z}} \ne 0,\,\,{{q}_{0}} \ne 0,\) describing the ferromagnetic state. Let us analyze the signs of the second derivatives of the TDPs with respect to the order parameters for specific solution types to determine the conditions of their thermodynamic stability (i.e., the conditions of correspondence to local minima of the TDPs).

It follows for the paramagnetic state \({{\sigma }_{Z}} = 0,\,\,{{q}_{0}} = 0\) that the following conditions should be satisfied:

Therefore, for spin S = 3/2, the paramagnetic state should be stable at high temperatures, both at \(T > {{T}_{{c0}}} = \left( {5{\text{/}}4} \right)Jz{\text{/}}{{k}_{{\text{B}}}}\) and at \(T > {{T}_{{qu,0}}} = \left( {3{\text{/}}2} \right)Kz{\text{/}}{{k}_{{\text{B}}}}.\) Then, the higher temperature among the obtained values \({{T}_{{qu,0}}}\) and \({{T}_{{c0}}}\) (which depends on the ratio between the bilinear (J) and biquadratic (K) exchange parameters) is considered as the lower temperature stability boundary of the paramagnetic state. For \(J{\text{/}}K < 6{\text{/}}5\), the paramagnetic state becomes unstable to the transition to the quadrupole state at temperature \({{T}_{{qu,0}}},\) whereas at \(J{\text{/}}K > 6{\text{/}}5\) it is unstable to the transition to the ferromagnetic state at \({{T}_{{c0}}}.\)

Note also that, if the transition from the paramagnetic to magnetically ordered state is of a second-order, these temperatures \({{T}_{{qu,0}}}\) or \({{T}_{{c0}}}\) become critical temperatures of the second-order phase transition. If the transition from the paramagnetic to magnetically ordered state is of a first-order, the corresponding critical temperatures of the first-order transition (\({{T}_{{qu,1}}}\) or \({{T}_{{c1}}}\)) lie somewhat above the thermodynamic stability boundaries (\({{T}_{{qu,0}}}\) or \({{T}_{{c0}}}\)).

Similarly, for the stability of the quadrupole state \({{\sigma }_{Z}} = 0,\) \({{q}_{0}} \ne 0\), we obtain the conditionnote that the quadrupole order parameter q₀ depends on temperature. The q₀ value in the quadrupole phase is determined by the self-consistent equation

which is obtained from Eq. (8) at \({{\sigma }_{Z}} = 0\) and can also be presented in the useful alternative formwhich is used to simplify expression (10).

Thus, it follows from condition (10) that the quadrupole state (as thermodynamically stable) exists only at temperatures \(T > {{T}_{{c0,qu - f}}}.\) The characteristic temperature \({{T}_{{c0,qu - f}}}\) (temperature of the second-order phase transition from the quadrupole to ferromagnetic phase) and the quadrupole order parameter at this temperature \({{q}_{0}}\left( {{{T}_{{c0,qu - f}}}} \right)\) are determined from the joint solution of the two equations

4. Let us consider the range of ratios of the bilinear and biquadratic exchange parameters J/K, in which the paramagnetic state passes to the quadrupole (rather than ferromagnetic) state with a decrease in temperature. Assuming that this transition is of a second-order, we expand the TDP of the quadrupole state in series of small values of the order parameter \({{q}_{0}} \ll 1\) near the transition point \({{T}_{{qu,0}}}\):

where \(\tau = 1 - T{\text{/}}{{T}_{{qu,0}}} \ll 1\) and \({{T}_{{qu,0}}} = 3{\text{/}}2Kz{\text{/}}{{k}_{B}}\) is the critical temperature of the second-order phase transition. We find from (14) that near \({{T}_{{qu,0}}}\) the order parameter \({{q}_{0}}\left( T \right)\) depends on temperature as

the internal energy of the quadrupole state \({{U}_{{0,qu}}}\) is

Hence, we can obtain the magnetic heat capacity of the quadrupole state \({{C}_{{M,qu}}}\) (per atom):note that \(d{{q}_{0}}{\text{/}}dT\) in (18) is found by differentiation of Eq. (11) for q₀. In particular, using expression (15) for q₀, we find the jump in the magnetic heat capacity at the temperature T_qu,0 of transition to the quadrupole phase:

For comparison, the jump in the magnetic heat capacity at the Curie point \({{T}_{{\text{C}}}}\) in a conventional ferromagnet with only bilinear exchange, calculated for the case S = 3/2 in the mean-field approximation [10], is \({{C}_{M}}\left( {{{T}_{{\text{C}}}} - {{0}^{ + }}} \right) = \left( {75/34} \right){{k}_{{\text{B}}}} \approx 2.206{{k}_{{\text{B}}}}.\) Thus, the jump in the heat capacity at the transition from the paramagnetic to uniaxial quadrupole state is somewhat smaller than that at the transition between the paramagnetic and ferromagnetic states but is quite comparable in order of magnitude.

To consider the initial magnetic susceptibility \({{\chi }_{{0,qu}}}\left( T \right)\) of the quadrupole state, we should consider that switching on a finite magnetic field H induces magnetization m, proportional to the mean value of the Z spin projection \({{\sigma }_{Z}}\left( H \right) = \left\langle {{{S}_{Z}}\left( H \right)} \right\rangle \sim H\), both in the quadrupole and paramagnetic states. Then, the bilinear exchange makes an additional contribution \( - Jz{{\sigma }_{Z}}\left( H \right){{S}_{Z}}\) to the Hamiltonian for the quadrupole state in the field. Hence, the Hamiltonian of the quadrupole state in a field becomes formally identical to the Hamiltonian of the ferromagnetic state in a field (3), with the only difference: the dipole order parameter \({{\sigma }_{Z}}\left( H \right)\) disappears in the temperature range of the quadrupole state in the limit \(H \to 0\) (\({{\sigma }_{Z}}\left( {H \to 0} \right) \to 0\)).

Using Hamiltonian (3) and, accordingly, Eq. (7) to determine the derivative \(d{{\sigma }_{Z}}\left( H \right){\text{/}}dH,\) we obtain the initial magnetic susceptibility \({{\chi }_{{0,qu}}}\left( T \right)\) of the quadrupole state for temperatures \(T < {{T}_{{qu,0}}}{\kern 1pt} :\)and the susceptibility \({{\chi }_{{0,p}}}\left( T \right)\) of the paramagnetic state at \(T > {{T}_{{qu,0}}}\):

It can be seen that the second-order phase transition between the paramagnetic and uniaxial quadrupole states does not exhibit any sharp features in the initial susceptibility \({{\chi }_{0}}\left( T \right)\) at the transition point \({{T}_{{qu,0}}},\) but only induces a kink in the temperature dependence. At the same time, a comparison of expression for \({{\chi }_{{0,qu}}}\left( T \right)\) (20) and equations for the lower temperature boundary \({{T}_{{c0,qu - f}}}\) of thermodynamic stability of the quadrupole phase (13) shows that the initial magnetic susceptibility of the quadrupole state diverges at the temperature \({{T}_{{c0,qu - f}}}.\) As will be shown below, \({{T}_{{c0,qu - f}}}\) is the temperature of the second-order phase transition between the ferromagnetic and uniaxial quadrupole states.

5. The temperature phase transition between the ferromagnetic and quadrupole states can be either a first-order one (with jumps in the order parameters at the transition point \({{T}_{{c1,qu - f}}}\)) or a second-order transition. In the latter case, when approaching the transition point \({{T}_{{c1,qu - f}}}\) from below, the dipole order parameter (magnetization) \({{\sigma }_{Z}}\left( T \right)\) gradually disappears, while the quadrupole order parameter \({{q}_{0}}\left( T \right)\) of the ferromagnetic state gradually (without jumps) passes to \({{q}_{0}}\left( T \right)\) of the quadrupole state. This suggests that deviations of the order parameter \({{q}_{0}}\left( T \right)\) from its value at the transition point \({{q}_{0}}\left( {{{T}_{{c0,qu - f}}}} \right) \equiv {{q}_{{00}}}\) are small in a narrow temperature range \({{T}_{{c0,qu - f}}} - T \ll {{T}_{{c0,qu - f}}}\) below the transition point \({{T}_{{c0,qu - f}}}\). Therefore, on the assumption that the deviations \(\gamma = {{q}_{0}}\left( T \right) - {{q}_{{00}}} \ll 1\) and the order parameter \({{\sigma }_{Z}} \ll 1\) are small quantities and considering the smallness of the relative temperature deviation \({{\tau }_{f}} = 1 - T{\text{/}}{{T}_{{c0,qu - f}}} \ll 1,\) one can expand the TDP of the ferromagnetic state f_f (5) in series of these small parameters:

Let us clarify that, to simplify the form of the coefficients, the expansion of the TDP \({{f}_{f}}\) (5) is performed using relation (12), which makes it possible to eliminate exponential factors of \({\text{exp}}\left( { - \beta Kz{{q}_{{00}}}} \right)\) and the equality \(1 - \left( {\left( {15 + 4{{q}_{{00}}}} \right){\kern 1pt} {\text{/}}12} \right)Jz/\left( {{{k}_{{\text{B}}}}{{T}_{{c0,qu - f}}}} \right) = 0,\) obtained from the coefficient at \(\sigma _{Z}^{2}.\)

We find from the expansion of the TDP (22) that

where

It is of importance that the sign of the coefficient a in \(\sigma _{Z}^{2} = a{{\tau }_{f}}\) depends on the ratio between the exchange parameters \(J{\text{/}}K,\) and it changes from positive to negative at the point, which is determined by the equation

Since the order parameters \({{q}_{{00}}} \equiv {{q}_{0}}\left( {{{T}_{{c0,qu - f}}}} \right)\) and the temperatures \({{T}_{{c0,qu - f}}}\) determined from the system of equations (13) also depend on the ratio of the exchange parameters \(J{\text{/}}K\), we have to find the critical value \({{(J{\text{/}}K)}_{{tr}}}\) from the joint solution of three interrelated Eqs. (25) and (13) for three quantities \({{(J{\text{/}}K)}_{{tr}}}\), \({{q}_{{00}}}\), and \({{T}_{{c0,qu - f}}}\) determining the position of the triple point in the curve of the second-order phase transitions \({{T}_{{c0,qu - f}}}\left( {J{\text{/}}K} \right).\)

Note that the negative sign of the coefficient a < 0 in \(\sigma _{Z}^{2} = a{{\tau }_{f}}\) indicates a change in the order of the phase transition and the occurrence of the first-order transition with jumps in the order parameters.

6. Using the results obtained to calculate the lines of the temperature phase transitions in the case of spins S = 3/2, we present the phase diagram of the magnetic states for the range of ratios of the bilinear and biquadratic exchange parameters \(0 < J{\text{/}}K < 1.5\) (Fig. 1). It can be seen that, at \(J{\text{/}}K > 0.936\), the paramagnetic state directly passes to the ferromagnetic one with a decrease in temperature through the first-order phase transition, whereas at \(J{\text{/}}K < 0.936\), the transition from the paramagnetic state to the ferromagnetic one is through an intermediate uniaxial quadrupole state. In this case, the second-order transition from the paramagnetic to uniaxial quadrupole state first occurs, and then, with a further decrease in temperature, the phase transition to the ferromagnetic state takes place, which is a second-order one for the range of exchange parameter ratios \(0 < J{\text{/}}K < 0.714\) and a first-order transition for the range of ratios \(0.714 < J{\text{/}}K < 0.936\). Thus, it can be seen that there are two versions of temperature changes in the magnet properties related to the passage through an intermediate quadrupole state: one version at \(J{\text{/}}K < 0.714\) with a sequence of two second-order phase transitions and the other at \(0.714 < J{\text{/}}K < 0.936\) with a sequence of second- and first-order phase transitions. As an example of these two behaviors, we present the calculations of temperature changes in the magnetic heat capacity and the initial magnetic susceptibility for \(J{\text{/}}K = 0.5\) (two second-order transitions) and \(J{\text{/}}K = 0.85\) (second- and first-order transitions).

Figure 2 shows the temperature behavior of the magnetic heat capacity \({{C}_{M}}\) for \(J{\text{/}}K = 0.5\) (curve 1) and \(J{\text{/}}K = 0.85\) (curve 2). In both cases, the magnetic contribution to the heat capacity arises stepwise at the second-order phase transition point to the quadrupole state \({{t}_{{qu,0}}} = 1\) and equals to \({{C}_{{M,qu}}}\left( {{{t}_{{qu,0}}} = 1} \right) = \left( {3{\text{/}}2} \right){{k}_{B}},\) and then begins to decrease with temperature to the point of the next phase transition. A new jump in the heat capacity occurs at the second-order transition point at \({{t}_{{c0,qu - f}}} = 0.695\) in the case of \(J{\text{/}}K = 0.5\) and at the first-order transition point at \({{t}_{{c1,qu - f}}} = 0.953\) in the case of \(J{\text{/}}K = 0.85\). Note that the jump in the heat capacity at the first-order transition point is much larger in magnitude than that at the second-order transition point.

Figures 3a and 3b show the results of calculating the temperature behavior of the initial magnetic susceptibility \({{\chi }_{0}}\left( T \right)\) for the exchange parameter ratio \(J{\text{/}}K = 0.5\) (the transition from the quadrupole to ferromagnetic state occurs according to the second-order scheme) and for \(J{\text{/}}K = 0.85\) (the transition to the ferromagnetic state is a first-order one). In both cases, the behavior of \({{\chi }_{0}}\left( T \right)\) at \(T > {{T}_{{qu,0}}}\) in the paramagnetic region is described by the Curie−Weiss law with the paramagnetic Curie point \({{{{\Theta }}}_{p}} = \frac{{S\left( {S + 1} \right)}}{3}{{|}_{{S = 3/2}}}Jz{\text{/}}{{k}_{B}} = \left( {5{\text{/}}4} \right)Jz{\text{/}}{{k}_{B}}\) (in dimensionless temperatures t, the paramagnetic Curie point is \(\widetilde {{{{{\Theta }}}_{p}}} = {{{{\Theta }}}_{p}}{\text{/}}{{T}_{{qu,0}}} = 5/6\left( {J{\text{/}}K} \right)\)). When the temperature decreases and the paramagnetic state passes to the quadrupole one at the point \({{T}_{{qu,0}}}\) of the second-order transition, the temperature dependence \({{\chi }_{0}}\left( T \right)\) does not diverge but only undergoes a kink, because \({{\chi }_{0}}\left( T \right)\) begins to increase with a decrease in temperature with a higher rate than it should be according to the Curie–Weiss law. In the case \(J{\text{/}}K = 0.5\), the susceptibility \({{\chi }_{0}}\left( T \right)\) at the interface with the ferromagnetic phase diverges with a further decrease in temperature to the critical temperature of the second-order phase transition \({{T}_{{c0,qu - f}}}\), which in turn indicates that the inverse susceptibility becomes zero: \(\chi _{0}^{{ - 1}}\left( {{{T}_{{c0,qu - f}}}} \right) = 0\) (Fig. 3a, inset). Thus, the behavior of the second-order transition between the ferromagnetic and uniaxial quadrupole phases does not differ radically from that of the second-order transition between the ferromagnetic and paramagnetic phases. If the transition from the quadrupole to ferromagnetic state is a first-order one, a sharp but finite peak is formed from the side of the quadrupole phase when approaching the critical temperature \({{T}_{{c1,qu - f}}}\) from above; afterwards, there is a sharp jump at the point \({{T}_{{c1.qu - f}}}\) with a significant decrease in the \({{\chi }_{0}}\left( T \right)\) value at a wing of the peak from the side of the ferromagnetic phase (the behaviors of the normalized “direct” susceptibility \({{\tilde {\chi }}_{0}}\left( t \right) = {{\chi }_{0}}\left( t \right){\text{/}}\left( {\mu _{0}^{2}{\text{/}}{{k}_{B}}} \right)\) and inverse susceptibility \(\tilde {\chi }_{0}^{{ - 1}}\left( t \right)\) are presented in Fig. 3b). It can be seen that there are no qualitative differences in the behavior of susceptibility for the cases of first-order transition between the quadrupole and ferromagnetic states and first-order transition between the paramagnetic and ferromagnetic states.

For a system of localized magnetic moments with the biquadratic exchange and spin S = 3/2, the “paramagnetism–uniaxial quadrupole order” phase transition is a second-order one and can be distinguished by a significant jump in the heat capacity C_M and occurrence of a kink in the temperature dependence of the initial magnetic susceptibility at the critical transition point \({{T}_{{qu,0}}}\). The lower-temperature “quadrupole order–ferromagnetism” phase transition may be of either the second or first order; it is also accompanied by a jump in the magnetic heat capacity. It is more important that the critical temperature of this transition is characterized by a sharp anomaly in the initial magnetic susceptibility: at the second-order transition, the susceptibility \({{\chi }_{0}}\left( T \right)\) diverges at the critical point \({{T}_{{c0,qu - f}}}\) (correspondingly, the inverse susceptibility \(\chi _{0}^{{ - 1}}\left( T \right)\) turns to zero), while at the first-order transition, a finite peak arises at the critical point \({{T}_{{c1,qu - f}}}\). Thus, the physical manifestations of the phase transition between the uniaxial quadrupole and ferromagnetic states are qualitatively the same as for the phase transition between the paramagnetic and ferromagnetic states. This is not surprising because, roughly speaking, the uniaxial quadrupole state (regarding its structure) is an anisotropic modification of the isotropic paramagnetic state.

Finally, we should note that our calculations, performed for a three-dimensional cubic lattice in the mean-field approximation, have a model character and that the implementation of the quadrupole state in a system implies the presence of strong biquadratic exchange (in comparison with the bilinear one). The question of the value of the non-Heisenberg biquadratic exchange K in three-dimensional magnets has been poorly investigated, although some attempts of such calculations have been made [3]. At the same time, much attention is paid to this problem in layered, quasi-two-dimensional, and single-layer magnetic systems (superconductors based on iron, etc.) [4]. Most of these compounds are characterized by fairly weak biquadratic exchange; however, there are some peculiar examples, such as CuBr₃ and 2H–FeS₂, where the biquadratic exchange parameter K is several times larger than the bilinear exchange parameter J (see Table 1 in [4]), so that the biquadratic exchange (rather than bilinear) induces the inherent magnetic order.