4.4 The paramagnet at fixed temperature

Take-home message: Understand the paramagnet and you are close to mastering the subject!

First, recap previous sections on the isolated spin- ${\textstyle \frac 1 2}$ paramagnet at zero and non-zero magnetic field.

The ideal paramagnet is a lattice of $N$ sites at each of which the spin points either up or down. Each of these has a magnetic moment $\pm \mu$. In an external field, these two states will have different energy; spin-up has energy $-\mu B$, and spin-down, $\mu B$. As we saw previously the partition function for a single atom is therefore

$$Z_1=e^{ \mu B/ k_{\scriptscriptstyle B}T}+e^{-\mu B/ k_{\sc... ...er k_{\scriptscriptstyle B}T}\right)=2{\rm cosh }(\mu B\beta)$$

(Remember $\beta=1/k_{\scriptscriptstyle B}T$.)

Since the atoms are non-interacting, the total energy and magnetisation of the system are just $N$ times the average energy and magnetisation of a single spin. The energy is

$$\left\langle E \right\rangle = -N{\partial \ln Z_1\over\partial\beta}=-N\mu B \tanh (\mu B\beta)$$

(For a refresher on hyperbolic trig functions, see here.)

At low $T$, all the spins are aligned with the field and the energy per spin is close to $-\mu B$. However as $T$ increases, thermal fluctuations start to flip some of the spins; this is noticeable when $k_{\scriptscriptstyle B}T$ is of the order of $\mu B$. As $T$ gets very large, the energy tends to zero as the number of up and down spins become more nearly equal. Remember, $\left\langle n_\downarrow/n_\uparrow \right\rangle =\exp(-2\mu B/k_{\scriptscriptstyle B}T)$, so it never exceeds one.

We can also calculate the heat capacity :

$$C_V = {\partial E\over\partial T}=Nk_{\scriptscriptstyle B} (\mu B\beta)^2 {\rm sech }^2(\mu B\beta)$$

We see that the heat capacity tends to zero both at high and low $T$. At low $T$ the heat capacity is small because $k_{\scriptscriptstyle B}T$ is much smaller than the energy gap $2\mu B$, so thermal fluctuations which flip spins are rare and it is hard for the system to absorb heat. This behaviour is universal; quantisation means that there is always a minimum excitation energy of a system and if the temperature is low enough, the system can no longer absorb heat.

The high-$T$ behaviour arises because the number of down-spins never exceeds the number of up-spins, and the energy has a maximum of zero. As the temperature gets very high, that limit is close to being reached, and raising the temperature still further makes very little difference. This behaviour is not universal, but only occurs where there is a finite number of energy levels (here, there are only two). Most systems have an infinite tower of energy levels, there is no maximum energy and the heat capacity does not fall off.

Up to now we've cheated a bit, (though the results are correct,) in that we didn't calculate the partition function for the whole system, only for a single spin. It is easy to show however that the partition function for $N$ non-interacting spins on a lattice is

$\mbox{\large\colorbox{yellow}{\rule[-3mm]{0mm}{10mm} \ $\displaystyle Z_N=(Z_1)^N$ }}$

(Details and a caveat here.) Since $\left\langle E_N \right\rangle$ is derived from $\ln Z_N$, we see immediately that the results for $N$ particles will just be $N$ times the single particle values. We can also calculate the Helmholtz free energy, $F=-k_{\scriptscriptstyle B}T\ln Z_N$, and the entropy, from $S=-(\partial F/\partial T)_{\scriptscriptstyle B,N}$ or from $S=(E-F)/T$. We can also calculate the magnetisation, $m=-(\partial F/\partial B)_{\scriptscriptstyle T,N}$ and find that $m=-E/B$ as expected.

Below we plot $S$ and $m$ against temperature for several different external fields. (For details of the algebra behind the plots see here.)

At zero temperature, the magnetisation goes to $N\mu$: all the spins are up. There is no disorder, and so the entropy is zero.

The stronger the field, the higher the temperature has to be before the spins start to be appreciably disordered.

At high temperatures the spins are nearly as likely to be up as down; the magnetisation falls to zero and the entropy reaches a maximum. The entropy of this state is $Nk_{\scriptscriptstyle B}\ln2$, as we have already seen.

References

• Mandl 3
• (Bowley and Sánchez 5.7)
• (Kittel and Kroemer 3)