Orbital magnetic moments

Figure 7.2 Vectors I and s and magnetic moments and associated with orbital and spin angular momenta when the motions are (a) in the same direction and (b) in opposite directions...

The direction of the alignment of magnetic moments within a magnetic domain is related to the axes of the crystal lattice by crystalline electric fields and spin-orbit interaction of transition-metal t5 -ions (24). The dependency is given by the magnetocrystalline anisotropy energy expression for a cubic lattice (33) ... 

The magnetic moments of the heavy RE elements (Gd, Tb, Dy, etc) are coupled antiparallel to the magnetic moments of the TM elements (Fe, Co, etc). The REj TM alloys are therefore ferrimagnetic below their Curie temperature (T )- The heavy TM moments form one magnetic sublattice and the RE moments the other one. In contrast, the light RE moments (eg, Nd, Pr) couple parallel to the moments of TM. The RE spia is always antiparallel to the TM spia, but for the light RE elements, the orbital momentum is coupled antiparallel to the spia and larger than the spia. 

K4Mo(CN)7.2H20, has been precipitated from aqueous solution by the addition of ethanol. Its magnetic moment ( 1.75 BM) is consistent with 7-coordinate Mo in which the loss of degeneracy of the tjg orbitals has caused pairing of 2 of the three d electrons. 

The difference between the two extremes is essentially that, in the former, the Re retains its valence electrons in its d orbitals whereas in the latter it loses 6 of them to delocalized ligand orbitals. In either case paramagnetism is anticipated since rhenium has an odd number of valence electrons. The magnetic moment of 1.79 BM corresponding to 1 unpaired electron, and esr evidence showing that this electron is situated predominantly on the ligands, indicates that an intermediate oxidation state is involved... 

Figure A The variation with temperature and spin-orbit coupling constant, of the magnetic moments of octahedral, low-spin, d" ions, (The values of at 300 K are marked for individual ions).

Fe " complexes in general have magnetic moments at room temperature which are close to 5.92 BM if they are high-spin and somewhat in excess of 2BM (due to orbital contribution) if they are low-spin. A number of complexes, however, were prepared in 1931 by L. Cambi and found to have moments intermediate between these extremes. They are the iron(lll)-A,A-dialkyldithiocarbamates, [Fe(S2CNR2)3], in which the ligands are ... 

Eg term. A magnetic moment of around 5.5 BM (i.e. 4.90 BM- -orbital contribution) is expected for pure octahedral symmetry but, in practice, distortions produce values in the range 5.2-5.4BM. Similarly, in the electronic spectrum, the expected single band due to the Eg t ge g) T2g t ge ) transition is broadened... 

The T ground term of the tetrahedral ion is expected to lead to a temperature-dependent orbital contribution to the magnetic moment, whereas the A ground term of the octahedral ion is not, though mixing of the excited T2g(F) term into the AigiF) ground term is expected to raise its moment to ... 

The magnitude of the separation between the adjacent states of a term indicates the strength of the spin-orbit coupling, and in all but two cases (Sm and Eu ) it is sufficient to render the first excited state of the Ln ions thermally inaccessible, and so the magnetic properties are determined solely by the ground state. It can be shown that the magnetic moment expected for such a situation is given by ... 

The Stern-Gerlach experiment demonstrated that electrons have an intrinsic angular momentum in addition to their orbital angular momentum, and the unfortunate term electron spin was coined to describe this pure quantum-mechanical phenomenon. Many nuclei also possess an internal angular momentum, referred to as nuclear spin. As in classical mechanics, there is a relationship between the angular momentum and the magnetic moment. For electrons, we write... 

Figure 1 Orbital magnetic moments for bcc-Fe, fcc-Co and fcc-Ni. The columns denoted by E, K, L and K represent from left to right the experimental data [15] and the theoretical data obtained by the SPR-KKR-, the LMTO-SOC-OP [16] as well as the SOPR-KKR-methods.

The SOC induced orbital magnetic moments / oib as obtained by the SPR- and SOPR-KKR-CPA for the di.sordered alloy. sy.stem bcc-Fe Coi-a are given in Fig. 2. As for the pure elements one finds an enhancement of / oib by the OP-term by around 60 %. This enhancement brings the total theoretical orbital magnetic moment for the alloy in very satisfying agreement with experimental data derived from magneto mechanical as well as spectroscopic g-factor measurements [15]. 

Figure 2 Orbital magnetic moments in bcc-Fe Coi-a . The triangles pointing up-and downwards represent the theoretical moments of Fe and Co, respectively, while the concentration weighted sum is given by circles. Full and open symbols stand for results obtained with and without the OP-term included (SOPR- and SPR-KKR-CPA, resp.). Experimental data [15] for the average magnetic moment (bottom) stemming from magneto mechanical and spectroscopic g-factors are given by full squares and diamonds.

In summary, the OP-term introduced by Brooks and coworkers has been transferred to a corresponding potential term in the Dirac equation. As it is demonstrated this approach allows to account for the enhancement of the spin-orbit induced orbital magnetic moments and related phenomena for ordered alloys as well as disordered. systems by a corresponding extension of the SPR-KKR-CPA method. 

As seen in Fig. 8 the experimentally determined magnetic moments at room temperature are in general much lower than the free ion values. To extract the contribution of orbital reduction, the influence of intermediate coupling, crystal field effects and j-j mixing must be considered. 

The problem could be stated from another point of view. In an isostructural series the uranium and neptunium compounds tend to be itinerant electron magnets or band magnets (like iron) and their orbital contribution is at least partially quenched. For much heavier actinides we know that the compounds will make local moment magnets with orbital contributions. It is quite possible that in between these two clear cut forms of magnetism that the intermediate case could be dominated by fluctuations, and no recognizable form of magnetism would occur. To state that the... 

But similar calculations for the iron-group ions show marked disagreement with experiment, and many attempts were made to explain the discrepancies. The explanation is simple in many condensed systems the perturbing effect of the atoms or molecules surrounding a magnetic atom destroys the contribution of the orbital momentum to the magnetic moment, which is produced entirely by the spin moments of unpaired electrons.40... 

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