Exact Calculation of the Paramagnetic Susceptibility of a Single Crystal with Arbitrary Crystal Field and Exchange Interactions

Explicit expressions for the first few terms of the $\frac{1}{T}$ expansion of the paramagnetic susceptibility are exactly calculated as a function of the parameters of the magnetic Hamiltonian. The calculation is done without approximations such as molecular field. It is done for a single crystal with one kind of magnetic ion in equivalent crystalline sites and can be used when $J$ or $S$ is a good quantum number, or when the lowest energy states can be described using an effective spin $S^{\prime }$. The $\frac{1}{T}$ and $\frac{1}{T^2}$ terms of the susceptibility are calculated as a function of the direction of the applied magnetic field for a crystal with an arbitrary symmetry, dipolar interactions, and arbitrary exchange interactions between the magnetic ions. These interactions, which can give rise to ferromagnetic or antiferromagnetic ordering, may be bilinear, biquadratic or of higher order. The first term of the susceptibility is the well-known $\frac{C}{T}$ law, where $C$ is the Curie constant. Two selection rules allow the determination of the second term which depends only on the bilinear exchange and on the second-order crystal field. The third term and the fourth-order saturation term are calculated in the case of a bilinear exchange for a single crystal having at least three axes of symmetry, where each has at least a twofold symmetry.