# Spin system in an external magnetic field

I need to know how to start the problem, just how to do part a), and then I'll probably be able to do the rest on my own.

Problem :

Consider two spin systems A and A' placed in an external field H. System A consist of N weakly interacting localized particles of spin 1/2 and magntic moment u. Similarly, system A' consists of N' weakly interacting localized particles of spin 1/2 and magnetic moment u'. The two systems are initially isolated with respective total energies bNuH and b'N'u'H. They are then placed in thermal contact with each other. Suppose that |b|<<1 and |b'|<<1 so that the simple expressions of Problem 2.4c can be used for the densities of stats of the two systems.

In the most probable situation corresponding to the final thermal equilibrium, how is the energy Et of systemA related to the energy Et' of system A' ?

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#### Solution Preview

If you do the derivation a bit more precisely, then you find the result:

Omega(E)* (delta E) = sqrt[2/(pi N)] 2^N Exp[-E^2/(2N mu^2 H^2)] (delta E)/(2 mu H)

for the density of states, see attached file. The prefactor is just the normalization which you automatically get right if you include the sqrt(2 pi N) term in Stirling approximation for N! In the attached file I work out the so-called multiplicity which gives the number of ways you can arrange the spins so the number of spins pointing in the direction of the field is

N/2 + eta. This then means that (N/2 - eta) spins point downward and therefore the energy is

E = 2 eta mu H ----->

eta = E/(2 mu H)

Reif defines Omega(E)*( delta E) to be the number of states between energy E and E + Delta E. If you look at the energy on a macroscopic scale you don't see that it ...

#### Solution Summary

A detailed solution is given.

The Energy of Spin-Orbit Coupling

Can you show that the energy of interaction is proportional to the scalar product s∙l.? See attachment for symbols.

The energy of a magnetic moment mu in a magnetic field B is equal to their scalar product (see attachment). If the magnetic field arises from the orbital angular momentum of the electron, it is proportional to the l: if the magnetic moment mu is that of the electron spin, then it is proportional to s. It then follows that the energy of the interaction is proportion to the scalar product s-l.

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