12 CHAPTER 3. PARAMAGNETISM OF FREE IONSMfinding an atom in a state with energy is given by
The magnetization of the system can then be found from the statistical average
of the magnetic moment This statistical average is obtained by weighing
the magnetic moment of each state by the probability that this state is occupied and
summing over all states:
The calculation of the magnetization by means of this formula is a cumbersome procedure
and eventually leads to Eq. (3.1.10). For the readers who are interested in how this result
has been reached and in the approximations made, a simple derivation is given below. Since
there is no magnetism but merely algebra involved in this derivation, the average reader will
not lose much when jumping directly to Eq. (3.1.10), keeping in mind that the magnetization
given by Eq. (3.1.10) is a result of the thermal averaging in Eq. (3.1.4), involving 2J +1
equidistant energy levels.
By substituting into Eq. (3.1.4), and using the relations in
and one may write
Since there cannot be any confusion with here, we have dropped the subscript J of
and simply write g from now on.
From the standard expression for the sum of a geometric series, one finds