Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment . Suppose the direction of can be only parallel or antiparallel to an externally applied magnetic field (this will be the case if is due to the spin of a single electron). According to statistical mechanics, it can be shown that the probability of an atom being in a state with energy U is proportional to where T is the temperature and k is the Boltzmann’s constant. Thus since , the fraction of atom whose dipole moment is parallel to is proportional to and the fraction of atoms whose dipole moment is antiparallel to is proportional to . (a) We have to show that the magnetization of this solid is . (b) We have to show that (a) reduces to for . (c) We have to show that (a) reduces to for .

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Consider a solid containing N atoms per unit volume, each atom having a magnetic dipole moment . We suppose the direction of can be only parallel or antiparallel to an externally applied magnetic field (this will be the case if is due to the spin of a single electron). According to statistical mechanics, it can be shown that the probability of an atom being in a state with energy U is proportional to where T is the temperature and k is the Boltzmann’s constant.

As

,

the fraction of the atoms whose dipole moment is parallel to , ,

where is a constant, and N is the total number of atoms per unit volume. The fraction of the atoms whose dipole moment is antiparallel to, , will be

.

As

we note that

.

The magnetization of the material, which is defined as the magnetic moment per unit volume, will therefore be given by the expression

Let us call

,

we thus find that

.

(b)

We will consider two limiting cases, one when

. In this limit as

The expression for magnetization simplifies to

(c)

When

,

and

.