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

The crystal field ground state of the $J=15/2$ Dy$^{3+}$ ion in DyNi$_2$B$_2$C can be described by a quasi-quartet 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 quartet and may be written as


\begin{displaymath}
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)
\end{displaymath} (124)

with the angular momentum operators given by the 4x4 matrices

\begin{displaymath}
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 )
\end{displaymath} (125)


\begin{displaymath}
J_b=
\left (
\begin{array}{cccc}
0 & -ib & 0 & ic \\
+ib & ...
...
0 & +ic & 0 & -ie \\
-ic & 0 & +ie & 0
\end{array}\right )
\end{displaymath} (126)


\begin{displaymath}
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 )
\end{displaymath} (127)

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)


\begin{displaymath}
M_{001}=g_J \mu_B \max(\vert a\vert,\vert d\vert)
\end{displaymath} (128)


$\displaystyle M_{100}$ $\textstyle =$ $\displaystyle g_J \mu_B \lambda$ (129)
$\displaystyle 2\lambda^2$ $\textstyle =$ $\displaystyle e^2+2c^2+b^2 + \sqrt{(e^2+2c^2+b^2)^2-4(be{\mathbf -}c^2)}$ (130)


$\displaystyle M_{110}$ $\textstyle =$ $\displaystyle g_J \mu_B \lambda$ (131)
$\displaystyle 2\lambda^2$ $\textstyle =$ $\displaystyle e^2+2c^2+b^2 + \sqrt{(e^2+2c^2+b^2)^2-4(be{\mathbf +}c^2)}$ (132)

The module given in /examples/dyni2b2c/1ion_mod/quartett.c diagonalises the Hamiltonian (131) and calculates the thermal expectation value $<>_T$ of the vector $\mathbf J$,which is returned to the McPhas program.

Martin Rotter 2017-01-10