Purchase Solution

# Magnetization and susceptibility of non-interacting dipoles

Not what you're looking for?

By evaluating the canonical partition function, derive the magnetization and the susceptibility of a system of N non-interacting magnetic dipoles in the presense of an external magnetic field in the high temperature limit.

##### Solution Summary

We derive the partition function for a system of N non-interacting magnetic dipoles. From this we extract a formula for the magnetization. We then consider the high temperature expansion by expanding in the parameter mu H /kT <<1. The formula for the susceptibility in the high temperature limit is then derived.

##### Solution Preview

The partition function of the N noninteracting dipoles (Z_N)factorizes into N single dipole partition functions (Z_1):

Z_N = (Z_1)^N (1)

Evaluate Z_1 by integrating over mu:

Z_1 = Integral[d^2 mu Exp[beta (mu dot H)]] (2)

where beta = 1/(k T) and d^2 mu denotes integration over all directions of mu.

In Eq. (2) I've neglected the correct normalization. In classical statistical mechanics you can't get the normalization of the partition fuction right (it involves planck's constant). Usually (and in this case) the correct normalizaton isn't important. But if you are rigorous you should multiply the integral in (2) by 1/(2 pi g mu). To see this, consider Z_1 at infinite temperature. Then Z_1 is just the total number of states. The total number of states for a particle with spin total spin s is 2 s+1. The g factor ...

##### Classical Mechanics

This quiz is designed to test and improve your knowledge on Classical Mechanics.

##### Intro to the Physics Waves

Some short-answer questions involving the basic vocabulary of string, sound, and water waves.

##### The Moon

Test your knowledge of moon phases and movement.

##### Variables in Science Experiments

How well do you understand variables? Test your knowledge of independent (manipulated), dependent (responding), and controlled variables with this 10 question quiz.