Two-state paramagnet: numerical solution

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problems 3.17 – 3.18.

We can apply the formulas for entropy, temperature and heat capacity to a real-life system by looking at a two-state paramagnet. This is a system of {N} magnetic dipoles which, when placed in a magnetic field {B}, align themselves so that their magnetic moment {\mu} points either parallel or antiparallel to the field. The energy of a dipole that is aligned with the field is lower, and we’ll call it {-\mu B}, so that the antiparallel dipole has energy {+\mu B}, and the total energy of the system is

\displaystyle U=\mu B\left(N_{\downarrow}-N_{\uparrow}\right)=\mu B\left(N-2N_{\uparrow}\right) \ \ \ \ \ (1)

where an up arrow indicates parallel alignment and a down arrow antiparallel.

The net magnetization is then

\displaystyle M=\mu\left(N_{\uparrow}-N_{\downarrow}\right)=-\frac{U}{B} \ \ \ \ \ (2)

The multiplicity of states is the same as a set of {N} coins with {N_{\uparrow}} heads, so

\displaystyle \Omega=\binom{N}{N_{\uparrow}}=\frac{N!}{N_{\uparrow}!\left(N-N_{\uparrow}\right)!} \ \ \ \ \ (3)

For small systems, we can find the entropy directly as

\displaystyle \frac{S}{k}=\ln\Omega \ \ \ \ \ (4)

For {N_{\uparrow}=98}, we get {U/\mu B=-96}, {M/N\mu=0.96}, {\Omega=4950} and {S/k=8.507}.

For each value of {N_{\uparrow}} from 0 up to {N}, we can evaluate {U} and {S} from the formulas above and then plot {S} versus {U} (I used Maple for the plot):

For {-100\le U/\mu B<0}, the curve is a ‘normal’ entropy curve in that the entropy increases with increasing energy, and the curve is concave down.

From this we can get the temperature

\displaystyle \frac{1}{T}=\frac{\partial S}{\partial U} \ \ \ \ \ (5)


Thus the temperature increases with energy in the region {-100\le U/\mu B<0}. At {U=0}, however, the derivative is zero implying an infinite temperature, and for {U>0}, the slope is negative, indicating a negative temperature. Since, in this case, negative temperatures occur at higher energies than positive temperatures, we have to interpret a negative temperature as actually being higher than a positive one, in fact, higher than an ‘infinite’ positive temperature.

In this case, it’s probably better to use the entropy of the system as a physical measure of what’s going on, since the second law implies that the system will tend to the energy with the greatest entropy, which is {U=0}. At this energy, there are equal numbers of up and down dipoles, so the system is maximally randomized.

For a system with {N=100}, we can approximate 5 by taking finite differences. Thus for a given value of {N_{\uparrow}} we can estimate the temperature as

\displaystyle T\left(N_{\uparrow}\right) \displaystyle = \displaystyle \frac{\Delta U}{\Delta S}\ \ \ \ \ (6)
\displaystyle \displaystyle = \displaystyle \frac{U\left(N_{\uparrow}+1\right)-U\left(N_{\uparrow}-1\right)}{S\left(N_{\uparrow}+1\right)-S\left(N_{\uparrow}-1\right)} \ \ \ \ \ (7)

For {N_{\uparrow}=98}, we get

\displaystyle T\left(98\right)=\frac{U\left(99\right)-U\left(97\right)}{S\left(99\right)-S\left(97\right)}=0.541\frac{\mu B}{k} \ \ \ \ \ (8)

This allows us to plot temperature versus energy:

We can see the flip over from {+\infty} to {-\infty} as the energy increases through zero.

The heat capacity can be obtained similarly as

\displaystyle C \displaystyle = \displaystyle \frac{\Delta U}{\Delta T}\ \ \ \ \ (9)
\displaystyle \displaystyle = \displaystyle \frac{U\left(N_{\uparrow}+1\right)-U\left(N_{\uparrow}-1\right)}{T\left(N_{\uparrow}+1\right)-T\left(N_{\uparrow}-1\right)} \ \ \ \ \ (10)

where we use 7 to calculate the temperatures. For {N_{\uparrow}=98} we have

\displaystyle C\left(98\right)=0.310Nk \ \ \ \ \ (11)

The plot of {C} versus temperature is

[The plot does actually extend down to 0 at {T=0} if we use the analytic solution, but because we’re dealing with discrete values of {N_{\uparrow}}, it cuts out early.] This curve is similar in shape to that of an Einstein solid at low temperatures.

Finally, we can plot the magnetization as a function of temperature:

If we start off with {T>0} and lower the temperature towards zero, the magnetization gradually increases until at {T=0}, the system is frozen into the state where all dipoles are parallel to the field. As we increase {T} to {+\infty}, we approach maximum randomness with equal numbers of dipoles pointing up and down so {M\rightarrow0}. Increasing the temperature beyond {+\infty} so it becomes negative (starting at {-\infty}) the magnetization again increases, but now the dipoles are aligned antiparallel to the field, eventually saturating as {T\rightarrow0} from the negative side.

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