Magnetism of lanthanide and actinide ions

It will be recalled that in Section 9.1 it was shown that the spin-only model for molecular paramagnetism gave rise to the equation 

It will therefore not be surprising that, essentially by replacing 5 by J in the derivation, the J-only model gives rise to the equation 

The equality sign in the spin-only equation has been replaced by a proportionality because, whereas there can be no ambiguity about how a given value of S arises—one simply has to count the number of unpaired electrons and divide by two—as has been seen, a given J value can arise from a variety of different spin and orbital components. Spin magnets and orbital magnets are not immediately interchangeable and so the constant of proportionality in the J-only equation, denoted g, has to reflect the particular mix involved. The equation for g is 

The quantity corresponding to g in the spin-only formula is the factor 2 (which appears as 4 when placed inside the square root). In Table 11.11 is given a comparison of observed and calculated values for the lanthanides. As has already been commented, the general agreement is very good. Two additional points should be made. First, a small contribution from temperature independent paramagnetism (TIP, the mixing of excited states into 

Low and high here are relative to adjacent elements only. So, lanthanum is four or five times more abundant than either gadolinium or dysprosium. 

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