Doing Single Ion Calculations

SIMPLICIUS: I always tell in my lectures that the crystal field is the electrostatic field, which is produced by the charges of the crystal environment of a rare earth atom. It acts on the 4f electrons of the rare earth and causes magnetic anisotropy. In addition to the electrostatic origin of the crystal field there is also a contribution due to the bonding.

EWALD: Look, Simplicius , in Fig. 3 and 4 I can show the you the effect of the crystal field on the charge distribution. Such plots can be made by using the programs pointc to calculate the crystal field parameters and display_density to plot the charge density ¹¹.

Figure 3: The crystal field model: the figure shows the influence of two positive charges on the charge density of 4f-electrons. The charge density is distorted by the electric field from the neighbouring atoms. Furthermore, this distortion leads to an anisotropy of the magnetic properties of the 4f shell. The resulting easy direction of the magnetic moment is shown by arrows for the different tri positive rare earth ions.

Figure 4: Calculated charge density of 4f-electrons in an orthorhombic crystal field. In this example the charge density of a Nd$^{3+}$ ion at a temperature of 10 K was calculated by taking the crystal field parameters determined by neutron spectroscopy and magnetic measurements in NdCu$_2$ [26]. [plot created by program display_density]

ORLANDO: Ewald , I appreciate the effort to do such beautiful plots of the charge density, but with your kind permission may I ask what relevance this has for experiments ? Normally in the literature I find the following expressions for the crystal field Hamiltonian ¹²

  $${\mathcal H}= \sum_{n,lm} B_{lm} O_{lm}({\mathbf J}^n) - \sum_{n} g_{Jn} \mu_B {\mathbf J}^n {\mathbf H}$$ (25)

EWALD: This is exactly what the so1ion module of McPhase uses to evaluate quantitatively the crystal field influence.

In the case of isolating materials the crystal field parameters $B_l^m$ can be obtained by the point charge model¹³, in the McPhase suite use programs makenn and pointc to evaluate the pointcharge model for a given crystal structure¹⁴. For metals the conduction electrons screen the point charges and the determination of the crystal field is usually only possible by fits to experimental data. The program package McPhase may be used to solve such crystal field problems.

SIMPLICIUS: In my lectures I am using a simpler way of dealing with crystal field anisotropy, it is to write instead of the first term in equ. 25

  $$\sum_s D_x^2 (J_x^s)^2 + D_y^2 (J_y^s)^2 +D_z^2 (J_z^s)^2$$ (26)

EWALD: Honorable Simplicius , indeed this is also possible - yet it is currently not possible to compute the parameters $D^2_{\alpha}$ from a point charge model.

ORLANDO: For a diagonalisation of the single ion crystal field Hamiltonian the module we have always used the program so1ion(cfield, written by P. Fabi né Hoffmann). It can calculate also transition elements and the inelastic neutron scattering cross section.

EWALD: Exactly this program has been included into McPhase. It can be used to calculate crystal field problems for rare earth ions. There is a program so1ion, which is self explaining, provided the user has a basic knowledge of crystal field theory (see e.g. the famous article by Hutchings [27]). However, I would recommend how to use the program singleion to do crystal field calculation for rare earth ions, because setting up the calculation in this way it can be easily extended to intermediate coupling schemes if necessary.