Spin and orbital contributions to the magnetic moment

The value of A varies from a fraction of a cm for the very lightest atoms to a few thousand cm for the heaviest ones. 

For J-block metal ions, equation 20.15 gives results that correlate poorly with experimental data (Tables 20.7 and 

For many (but not all) first row metal ions, A is very small and the spin and orbital angular momenta of the electrons operate independently. For this case, the van Vleck formula (equation 20.16) has been derived strictly, equation 20.16 applies to free ions but, in a complex ion, the crystal field partly or fully quenches the orbital angular momentum. Data in Tables 20.7 and 20.8 reveal a poor fit between observed values of p.eff and those calculated from equation 20.16. 

If there is no contribution from orbital motion, then equation 20.16 reduces to equation 20.17 which is the spin-only formula we met earlier. Any ion for which L = 0 (e.g. high-spin Mn or Fe in which each orbital with mi = +2, +1, 0, —1, —2 is singly occupied, giving L = 0) should, therefore, obey equation 20.17. 

However, some other complex ions also obey the spin-only formula (Tables 20.7 and 20.8). In order for an electron to have orbital angular momentum, it must be possible to transform the orbital it occupies into an entirely equivalent and degenerate orbital by rotation. The electron is then effectively rotating about the axis used for the rotation of the orbital. In an octahedral complex, for example, the three t2g orbitals can be interconverted by rotations through 90° thus, an electron in a t2g orbital has orbital angular momentum. The Cg orbitals, having different shapes, cannot be interconverted and so electrons in Cg orbitals never have angular momentum. There is, however. 

The value of A varies from a fraction of a cm for the very lightest atoms to a few thousand cm for the heaviest ones. The extent to which states of different J values are populated at ambient temperature depends on how large their separation is compared with the thermal energy available, kT at 300 K, kT 200 cm or 2.6kJmoF. It can be shown theoretically that if the separation of energy 

However, some other complex ions also obey the spin-only formula (Tables 20.11 and 20.12). In order for an electron to have orbital angular momentum, it must be possible to 

Metal ion Ground Mefr Mb calculated Mefr Mb calculated Meff Mb calculated 

Metal ion Ground term Meff / Mb calculated from equation 21.20 Meff / Mb calculated from equation 21.21 Meff / Mb calculated from equation 21.22 

Some values of A are given in Table 21.13. Note that A is positive for less than half-filled shells and negative for shells that are more than half-filled. Thus, spin-orbit coupling leads to  

Worked example 21.8 Magnetic moments spin-orbit coupling 

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In the limit P —> 1, the a-polarized component of the magnetic scattering from a helix depends only on the spin magnetization density, I oc while the r-polarized component depends on the sum of the orbital and spin densities I oc (i + sY- Thus, it is in this sense that the spin and orbital contributions to the total moment may be separated. Semi-quantitative experiments to demonstrate these ideas in a helix have been performed on Ho (Gibbs et al. 1988, Gibbs et al. 1991). Both jt- and a-polarized scattered beams were observed. It was demonstrated that the nr-component is dominant, consistent with the large orbital moment in Ho, and that the Q dependence of the orbital and spin form... 

There may be a significant orbital contribution to the magnetic moment the spin-only formula, as the label implies, does not take this into account. The quantitative treatment of orbital contributions - which effectively alters the g-value from the free-electron value of 2.0023 - was one of the earliest and most important tasks of CF theory. The following generalisations can be made on the basis of the experimental results and the theoretical treatment for simplicity, we consider only octahedral and tetrahedral fields. 

Therefore, the orbital contribution to the magnetic moment is zero and we have only spin contribution. 

The atom is diamagnetic the orbital contribution to the magnetic moment is always zero because the charges are equal and opposite while the masses are identical the spin contribution is also zero in zero field since, if the spin momenta are parallel, the spin magnetic moments are opposed, while if the spin momenta are antiparallel there is no preferred direction and the time average of the magnetic moment in any particular direction is zero. 

Explain why in high-spin octahedral complexes, orbital contributions to the magnetic moment are only important for d, d, d and d configurations. 

Typically, values of //err for salts of [FeO/l lie in the range 2.8-3.0 //b Show that this is consistent with a //(spin-only) value for tetrahedral Fe(VI) and comment on why orbital contributions to the magnetic moment are not expected. 

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