The Magnetic Moment

A 3D cubic lattice of spin sites may adopt several different AFM states, for example, the G-, C- and A-types, as depicted in 24.5 where the gray and white spheres indicate up- and down-spin sites. The G-type structure has AFM interactions in three different directions, the C-type in two different directions, and the A-type in 

A magnetic system is isotropic if its magnetic moment is nonzero in all directions, but is uniaxial if its magnetic moment is nonzero in only one direction. Magnetic ions are 

As a specific racample, consider that one electron is present in one of the three p orbitals p, (i=x, y, z), for which the effective one-electron Hamiltonian satisfies 

The diagonalization of this matrix leads to three energies 

Suppose that the ground state is given by . Then the orbital angular momentum L of the single electron is given by 

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In addition, there could be a mechanical or electromagnetic interaction of a system with an external entity which may do work on an otherwise isolated system. Such a contact with a work source can be represented by the Hamiltonian U p, q, x) where x is the coordinate (for example, the position of a piston in a box containing a gas, or the magnetic moment if an external magnetic field is present, or the electric dipole moment in the presence of an external electric field) describing the interaction between the system and the external work source. Then the force, canonically conjugate to x, which the system exerts on the outside world is... 

If the angular momentum of a free electron is represented by a spin vector S=(S, S, S ), the magnetic moment... 

Here Ti is the spin-lattice relaxation time due to the paramagnetic ion d is the ion-nucleus distance Z) is a constant related to the magnetic moments, i is the Larmor frequency of the observed nucleus and sis the Larmor frequency of the paramagnetic elechon and s its spin relaxation time. Paramagnetic relaxation techniques have been employed in investigations of the hydrocarbon chain... 

It is possible to systematically alter the net magnetic moment of ferrites by chemical substitutions. A very important industrial appHcation is the increase of the magnetic moment in mixed MnZn-ferrites and NiZn-ferrites. When Zn ions are introduced in Mn-ferrite or Ni-ferrite, these ions prefer to occupy A-sites. Because is nonmagnetic, the A-sublattice magnetization is reduced and consequendy the total net magnetic moment is increased. 

Titanium trifluoride [13470-08-17, TiF, is a blue crystalline solid that undergoes oxidation to Ti02 upon heating in air at 100°C (see Titanium compounds). In the absence of air, disproportionation occurs above 950°C to give TiF and titanium metal. TiF decomposes at 1200°C, has a density of 2.98 g/cm, and is insoluble in water but soluble in acids and alkafles. The magnetic moment is 16.2 x 10 J/T (1.75 -lB). 

Fig. 8. Principle of the magnetooptical read-out of domain patterns by the polar Kerr effect. The polarisation plane of the incoming laser beam is rotated clock- or counterclockwise according to the orientation (up or down) of the magnetic moments.

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. 

Other ferrate salts include calcium ferrate [35764-67-1], and sodium ferrate [13773-03-0]. The magnetic moments of these materials are 2.63.0 X 10 J/T (2.8-3.2 ), which is consistent with the expectation of two unpaired electrons. The [FeOJ ion is an extremely strong oxidizing... 

The electronic stmcture of cobalt is [Ar] 3i/4A. At room temperature the crystalline stmcture of the a (or s) form, is close-packed hexagonal (cph) and lattice parameters are a = 0.2501 nm and c = 0.4066 nm. Above approximately 417°C, a face-centered cubic (fee) aHotrope, the y (or P) form, having a lattice parameter a = 0.3544 nm, becomes the stable crystalline form. The mechanism of the aHotropic transformation has been well described (5,10—12). Cobalt is magnetic up to 1123°C and at room temperature the magnetic moment is parallel to the ( -direction. Physical properties are Hsted in Table 2. 

Figure 1.1. Nuclear precession nuclear charge and nuclear spin give rise to a magnetic moment of nuclei such as protons and carbon-13. The vector n of the magnetic moment precesses in a static magnetic field with the Larmor frequency vo about the direction of the magnetic flux density vector Bo...

In neutron reflectivity, neutrons strike the surface of a specimen at small angles and the percentage of neutrons reflected at the corresponding angle are measured. The an jular dependence of the reflectivity is related to the variation in concentration of a labeled component as a function of distance from the surface. Typically the component of interest is labeled with deuterium to provide mass contrast against hydrogen. Use of polarized neutrons permits the determination of the variation in the magnetic moment as a function of depth. In all cases the optical transform of the concentration profiles is obtained experimentally. 

One further important difference between neutron and X-ray difliaction is the former s sensitivity to magnetic structure. The magnetic moments of neutrons... 

For specimens where gradients in the ms etic moment are of interest, similar arguments apply. Here, however, two separate reflectivity experiments are performed in which the incident neutrons are polarized parallel and perpendicular to the surfiice of the specimen. Combining reflectivity measurements under these two polarization conditions in a manner similar to that for the unpolarized case permits the determination of the variation in the magnetic moments of components parallel and perpendicular to the film surface. This is discussed in detail by Felcher et al. and the interested reader is referred to the literature. 

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