ErNi$_2$B$_2$C - single ion module

The crystal field ground state of the $J=15/2$ Er$^{3+}$ ion in ErNi$_2$B$_2$C can be described by a quasi-quartett consisting of two doublets separated by a energy interval $\Delta$.

In order to calculate efficiently the single ion property (for small effective magnetic fields in comparison to the total CF splitting) the Hamiltonian $H=H_{cf}- g_J \mu_b {\mathbf H}{\mathbf J}$ is projected to the quasi quartett and may be written as

$$H=\left ( \begin{array}{cccc} -\Delta/2 & 0 & 0 & 0 \\ 0 & ... ... \end{array}\right) -g_J \mu_B (H_a J_a + H_b J_b + H_c J_b)$$ (166)

with the angular momentum operators given by the 4x4 matrices

$$J_a= \left ( \begin{array}{cccc} 0 & b & 0 & c \\ b & 0 &-c & 0 \\ 0 &-c & 0 & e \\ c & 0 & e & 0 \end{array}\right )$$ (167)

$$J_b= \left ( \begin{array}{cccc} 0 & -ib & 0 & ic \\ +ib & ... ... 0 & -ic & 0 & -ie \\ -ic & 0 & +ie & 0 \end{array}\right )$$ (168)

$$J_c= \left ( \begin{array}{cccc} +a& 0 & 0 & 0 \\ 0 &-a & 0 & 0 \\ 0 & 0 &-d & 0 \\ 0 & 0 & 0 &+d \end{array}\right )$$ (169)

The constants $a$-$e$ can be computed from the crystal field parameters, if these are known. On the other hand, they are connected to the saturation magnetic moments $\mathbf M$ by the following equations ($\max(...)$ denotes the maximum of the argument values)

$$M_{001}=g_J \mu_B \max(\vert a\vert,\vert d\vert)$$ (170)

$$M_{100} = g_J \mu_B \lambda$$ (171)

$$2\lambda^2 = e^2+2c^2+b^2 + \sqrt{(e^2+2c^2+b^2)^2-4(be{\mathbf +}c^2)}$$ (172)

$$M_{110} = g_J \mu_B \lambda$$ (173)

$$2\lambda^2 = e^2+2c^2+b^2 + \sqrt{(e^2+2c^2+b^2)^2-4(be{\mathbf -}c^2)}$$ (174)

The module given in /examples/erni2b2c/1ion_mod/quartett.c diagonalises the Hamiltonian (175) and calculates the thermal expectation value $<>_T$ of the vector $\mathbf J$, which is returned to the McPhas program.