How Does Magnetic Moment Change with Temperature and Field Strength?

[patrickmoloney]

Homework Statement

Find the magnetic moment of a crystal when placed

(i) in a weak field at high temperature

(ii) in a strong field at low temperature

Homework Equations

This is the last part of a question which I feel I have solved correctly up until this point.

The mean magnetic moment I found is

$$\langle M \rangle = N_{\mu} \frac{2 \sinh \Big{(}\dfrac{\mu \beta}{kT}\Big{)}}{1+ \cosh \Big{(}\dfrac{\mu \beta}{kT}\Big{)}}$$

The Attempt at a Solution

Well at high temperature and low magnetic field strength

$$\dfrac{\mu \beta}{kT} \ll 1$$

And at low temperature high magnetic field strength

$$\dfrac{\mu \beta}{kT} \gg 1$$

What happens to the hyperbolic functions as one goes to $0$ and one goes to $\infty$?

EDIT: I know what happens as $\lim_{x \rightarrow \infty}\sinh x = e^x$ and $\lim_{x \rightarrow \infty}\cosh x = e^x$

 

[kuruman]

I assume your crystal is paramagnetic. You might see what's going more easily if you cast the hyperbolic functions in terms of exponentials, then take the limits.

 

[patrickmoloney]

(i)Weak field at high temperature

$$\dfrac{\mu B}{kT} \ll 1 \implies e^{\frac{\mu B}{kT}} \approx 1 \pm \dfrac{\mu B}{kT}$$

From my mean magnetic moment$$M= N\mu \dfrac{e^{\frac{\mu B}{kT}}-e^{\frac{\mu B}{kT}}}{1 + e^{-\frac{\mu B}{kT}} + e^{\frac{\mu B}{kT}}}$$

substituting $e^{\frac{\mu B}{kT}} = 1 \pm \dfrac{\mu B}{kT}$

$$M = \dfrac{1+\frac{\mu B}{kT} -1 + \frac{\mu B}{kT}}{1+1-\frac{\mu B}{kT} + 1 + \frac{\mu B}{KT}} = \dfrac{2N\mu^2 B}{3kT}$$

(ii) Strong field at low temperature

$$\dfrac{\mu B}{kT} \gg 1 \implies e^{\frac{\mu B}{kT}} \gg 1 \pm e^{-\frac{\mu B}{kT}} \gg 1$$

Hence

$$M= N\mu \dfrac{e^{\frac{\mu B}{kT}}}{e^{\frac{\mu B}{kT}}}= N\mu$$

 

[kuruman]

Looks right. At the high temperature end you get the 1/T Curie law dependence. At low temperature, only (mostly) the ground state is occupied and the magnetization is the number of magnetic moments times the value of one moment.

 

[patrickmoloney]

Looks right. At the high temperature end you get the 1/T Curie law dependence. At low temperature, only (mostly) the ground state is occupied and the magnetization is the number of magnetic moments times the value of one moment.

Much appreciated. Insight into these types of problems helps a lot. Thanks