Rare Earth Ions

ORLANDO: My sample contains rare earth - so we need to do simulations for rare earth materials.

EWALD: Yes, we can - a specific example of (1) is the following magnetic Hamiltonian $\mathcal H$ for rare earth ions, which may be treated with the program package:

  $${\mathcal H}= \sum_{n,lm} B_l^m O_{lm}({\mathbf J}^n) -\fra... ...hbf J}^n{\mathbf J}^{n'} - \sum_{n} g_{Jn} \mu_B {\mathbf J}^n {\mathbf H}$$ (2)

The first term describes the crystal field (Stevens Operators $O_l^m$, see table in appendix G), the second the magnetic exchange interaction, the third the Zeeman energy if an external magnetic field is applied. Note that by ”external magnetic field $\mathbf H$” we refer to the magnetic field in the sample, which is the applied field (e.g. generated by a coil) diminished by the demagnetizing tensor $\overline{n}_{\rm demag}$ times the magnetisation $\mathbf M$ of the sample, i.e. $\mathbf H= \mathbf H_{\rm applied} - \overline{n}_{\rm demag} \mathbf M$, e.g. for a spherical sample $\overline{n}_{\rm demag}=\overline{1}/3$. The unit of $\mathbf H$ is usually chosen to be Tesla, which actually refers to the quantity $\mu_0 \mathbf H$ (in SI units).

ORLANDO: I know about the demagnetisation factor, in our pulsed field experiments we sometimes have extremely large moments and for a small sample the corrections can be quite large.

SIMPLICIUS: I have done a lot of experiments on exotic matter, we need some other concepts than your theory. Can you do orbital excitations ?

EWALD: Instead (or rather in addition) to this it is also possible to treat the more general two ion exchange coupling⁹

  $${\mathcal H}_{JJ}= -\frac{1}{2} \sum_{nn'} \sum_ {ll'} \sum_... ...\mathcal K}_{ll'}^{mm'}(nn') O_{lm}({\mathbf J}^n) O_{l'm'}({\mathbf J}^{n'})$$ (3)

ORLANDO: Well - I have magnetostriction data, we need some theoretical explanation for the strain. Can you enter some magnetoelastic terms in the Hamiltonian ? I know a theoretician in Spain who has done so and has written two huge volumes about magnetostriction.

EWALD: Indeed, in addition to the above terms in the Hamiltonian the coupling of magnetic and lattice properties is possible by introducing magnetoelastic interactions: In order to calculate magnetoelastic effects the parameters $B_l^m$, ${\mathcal J}(ij)$ (or more general the ${\mathcal K}_{ll'}^{mm'}(ij)$) are expanded in a Taylor expansion in the strain tensor $\epsilon$ resulting in the magnetoelastic interaction (i.e. keeping only the terms linear in $\epsilon$). The equilibrium strain can be calculated by considering in addition the elastic energy and minimising the total free energy. ¹⁰

SIMPLICIUS: If I give you a student of mine and he starts doing McPhase, are there some exercises so he can learn how to use the program ?

Exercises:

• Write down the list of nonzero crystal field parameters in the crystal field Hamiltonian of a Nd$^{3+}$ ion in an orthorhombic crystal field (Section 6 describes how such a parameter set enters a McPhase calculation).
• Taking a primitive orthorhombic lattice of Nd$^{3+}$ ions write down the most general anisotropic bilinear interaction for the nearest neighbour exchange interaction (Section 7.1.4 describes how such a parameters set enters a McPhase calculation).