[Wien] Physical significance of magnetization direction with -so

[pieper]

I hope my five cent might be usefull:

If you do have magnetic moments, be they ferro-, ferri-, or 
antiferromagnetic, or induced by an external field, the results can 
depend on the orientation of the moments. In addition, keep in mind that 
the various magnetic moments one likes to think of may not be constants 
of motion, or good quantum numbers, so they cannot be used to specify 
the eigenstates.

Inasmuch as S and L are good quantum numbers Hund's first and second 
rules for single ions state that the ground state of an electron shell 
in the Coulomb potential of the nuclear charge will have maximum total 
spin moment S=\sum s_i and maximum total angular momentum L=\sum l_i 
compatibel with the Pauli principle. L and S are constants of motion, 
they commute with the Hamiltonian of the nuclear Coulomb potential. L 
and S can, therefore, be used to enumerate the eigenstates. The energy 
is given by the size of L and S. The direction of L and S is unimportant 
(for a single ion without magnetic field applied!) the nuclear Coulomb 
potential is radial symmetric. Each state of the shell is 
(2L+1)(2S+1)-fold degenerate. This degeneracy is partially split by 
spin-orbit coupling and by the electrostatic crystal field.

You find the Hamiltionian for spin-orbit coupling H_so=\lambda (S*L) 
with S and L the vectors of the total spin and angular momentum of the 
electrons in one shell (say, d-, or f-shell) from considering the energy 
of an electron spin s in the magnetic field due to the relative motion 
of the nuclear charge with respect to the electron on its path at 
angular momentum l around the nucleus. With this H_so only J=L+S stays a 
constant of motion, neither S nor L. You can find the magnitude of J by 
Hund's third rule if H_so stays the most important player down to that 
energy. However, this is still just the radial symmetric potential of a 
single nuclear charge, the energy of the eigenstates is given by 
specifying J and you can point J in any direction.

In a crystal the charge distribution of the surrounding ions also acts 
on the electrons in the shell. This crystal field Hamiltonian obviously 
is not spherical symmertric, it will reduce the rotational symmetry to 
the point symmetry of the site. If there is a magnetic moment on an ion 
the energy will depend on its direction in the lattice. This is the 
source of the magnetic single-ion anisotropy.

What the eigenstates of the combined Hamiltonian look like depends on 
the relative size and symmetry of the contributions. For outer d-shells 
crystal fields usually dominate over H_so leading e.g. to what is called 
the quenching of the orbital angular momentum (L=0) which is sensitive 
to magnetic fields. For 4f-shells which are shielded by outer d- and 
s-shells H_so is frequently dominant and Hund's third rule often 
survives, allowing to calculate J and consider the dependance of the 
energy on its direction in the lattice.

Martin

---
Dr. Martin Pieper
Karl-Franzens University
Institute of Physics
Universitätsplatz 5
A-8010 Graz
Austria
Tel.: +43-(0)316-380-8564


Am 03.05.2015 19:37, schrieb Laurence Marks:
> An elementary question: do the results of -so depend upon the
> magnetization direction used in initso, or should they in principle be
> independent of it?
> 
> --
> 
> Professor Laurence Marks
> Department of Materials Science and Engineering
> Northwestern University
> www.numis.northwestern.edu [1]
> Corrosion in 4D: MURI4D.numis.northwestern.edu [2]
> Co-Editor, Acta Cryst A
> "Research is to see what everybody else has seen, and to think what
> nobody else has thought"
> Albert Szent-Gyorgi
> 
> Links:
> ------
> [1] http://www.numis.northwestern.edu
> [2] http://MURI4D.numis.northwestern.edu
> 
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