Magnetic Ordering in Materials with Singlet Crystal-Field Ground State. II. Behavior in the Ordered State or in an Applied Field

The theory of magnetic ordering and of collective excitation behavior in materials with a single crystal-field ground state has been extended to finite temperature and to the regime where the system has a magnetic moment due to ordering or an applied field. For simplicity, we still consider a system where the only excited state in the absence of exchange is also a singlet at an energy $\Delta$ (in units of °K) above the ground state. We show that in the random-phase approximation (RPA), the self-consistent molecular-field eigenstates serve as a good basis for determining the collective excitation energies; while in the two-site correlation approximation (TSCA), a modification of the molecular-field states is required in the ordered phase. The energies of the collective excitations are calculated with and without an external magnetic field in both the paramagnetic and magnetically ordered phases. At finite temperatures, a Green's-function formalism is employed to facilitate the calculation of thermodynamic quantities. For $\frac{T_C}{\Delta }\ge 0.1$, we find in the RPA a discontinuity in the magnetization at the critical point. The discontinuity is most prominent when $T_C$ is comparable to the crystal-field splitting $\Delta$, and vanishes both as $\frac{T_C}{\Delta }$ decreases toward 0.1 and at $\Delta =0\mathrm{}$. We show that in the TSCA the magnetic transition is also first order, and, in fact, for the TSCA the transition is first order even at $T=0\mathrm{}$. The specific heat and susceptibility are calculated in both the RPA and the TSCA and compared with the molecular-field-theory results.