Consider a system of N atoms each of which has magnetic moment of magnitude μ. The Hamiltonian in presence of external magnetic field (B) is

H (p,q) - μB Σ cos αi

where H(p,q) is the hamiltonian of the system, the absence of an external magnetic field and αi is the angle between B and magnetic moment of ith atom.

Therefore partition function of a dipole is

where the sum is overall the allowed orientation of spin magnetic moment μ.

The partition function of the system for N particle system is

Now, using the element of solid angle representing a small range of orientation of the dipole,

Theory of paramagnetism

We know the partition function of the magnetized atomic magnet is

i. Induced magnetic moment

The mean magnetic moment M is oriented along the direction of B. If we choose B = Bƶ i.e B along ƶ axis. Then

M = <Mz>

or, M = N <μcos α>

ii. Magnetic susceptibility

Magnetic susceptibility is defined as the magnetic moment per unit volume and can be expressed as

iii. At high temperature susceptibility satisfies the Curie law.

We know,

χ = <Mz >/N

= μ L(x)

where L(x) = cot hx - 1/x where x = β μB. L(x) is called Langevin function. if temperature is very high then,

x = β μB = μB/KBT << 1 and Langevin function becomes

L(x) = cothx - 1/x = 1/x + x/3 + ...............-1/x = x/3

L(x) = μB/3KBT

So χ ∝ 1/T

This is Curie Law.

This note is taken from Statistical Mechanics, MSC physics, Nepal.

This note is a part of the Physics Repository.

Next topic : Polarisability per molecule in terms of Langevin's function

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