The Spin Paramagnet

1 · What is a paramagnet?

Some materials — oxygen gas, aluminium, and salts of the transition and rare-earth metals — are weakly pulled into a magnetic field. Remove the field and the effect vanishes completely: there is no leftover magnetism. This is paramagnetism, and it is quite different from the permanent magnetism of iron (ferromagnetism).

The microscopic picture is simple. Each atom or ion carries a magnetic moment $\vec{\mu}$ — think of it as a tiny bar magnet, born from the spin and orbital motion of its electrons. In the absence of a field these moments point in random directions and cancel out, so the material is not magnetic. Switch on a field $\vec{B}$ and each moment feels a torque that favours alignment, because its energy

$$ E = -\,\vec{\mu}\cdot\vec{B} $$

is lowest when $\vec{\mu}$ points along $\vec{B}$. If the moments could settle into their lowest-energy state they would all line up perfectly. But they cannot: the material is at some temperature $T$, and thermal agitation constantly knocks them out of alignment.

The entire physics of the paramagnet is this contest between field (order) and temperature (disorder). To turn that sentence into equations we need a rule for how a system at temperature $T$ distributes itself among states of different energy. That rule is the heart of statistical mechanics, and we build it next — but first, some history.

2 · A little history

The paramagnet has been a proving ground for new physics for well over a century.

• 1895 — Pierre Curie measures the magnetic susceptibility of many substances across a huge range of temperature and distils an empirical law: for a paramagnet the susceptibility is inversely proportional to temperature, $\chi \propto 1/T$ [1]. This is Curie's law, and any theory of paramagnetism must reproduce it.

• 1905 — Paul Langevin gives the first statistical-mechanical explanation, treating each moment as a classical vector free to point in any direction and applying Boltzmann statistics [2]. He derives Curie's law and, along the way, the Langevin function $L(x)=\coth x - 1/x$.

• 1926–27 — Quantum theory. With the arrival of quantum mechanics, the magnetic moment is no longer a free classical vector: its projection on the field is quantised. Léon Brillouin [3] and Peter Debye [4] work out the resulting magnetisation, giving the Brillouin function $B_J(x)$ that we derive below. Langevin's classical result reappears as the $J\to\infty$ limit.

• 1927–1933 — W. F. Giauque turns the theory into a technology. Recognising that a paramagnet's entropy can be manipulated with a magnetic field, he proposes and then demonstrates adiabatic demagnetisation, reaching temperatures far below 1 K [5]. It earned him the 1949 Nobel Prize in Chemistry.

• 1952 — W. E. Henry measures the magnetisation of chromium, iron, and gadolinium salts at liquid-helium temperatures and in fields strong enough to drive them toward saturation, producing textbook-perfect agreement with the Brillouin function [6]. We reproduce his data at the end.

With the lineage in place, let us build the theory the way Boltzmann would.

3 · The simplest paramagnet: one spin, two states

Quantum mechanics hands us an enormous simplification. For a spin-$\tfrac12$ particle — a single unpaired electron is the canonical example — the component of the magnetic moment along the field can take only two values, $+\mu$ or $-\mu$. There is no "halfway". The moment is either

$$\text{**up} \;(\uparrow)\text{:}\quad \vec{\mu}\text{ along }\vec{B}, \qquad\text{or}\qquad \text{**down} \;(\downarrow)\text{:}\quad \vec{\mu}\text{ against }\vec{B}.$$

Using $E=-\vec\mu\cdot\vec B$ with a field $B$ pointing "up", the two states have energies

$$ E_\uparrow = -\mu B \qquad\text{(low energy — aligned)}, $$ $$ E_\downarrow = +\mu B \qquad\text{(high energy — anti-aligned)}. $$

So a single spin is a two-level system: a ground state and an excited state separated by an energy gap $\Delta E = E_\downarrow - E_\uparrow = 2\mu B$. That gap, set by the field, is the only energy scale in the problem. Everything that follows is about how a spin at temperature $T$ chooses between these two levels.

4 · Temperature, and the Boltzmann factor

Our spin is not isolated: it is in contact with a huge thermal reservoir — the rest of the material, the vibrating atoms of the crystal — at temperature $T$. Energy sloshes back and forth between spin and reservoir. We cannot say for certain which state the spin is in; we can only ask for the probability.

The founding result of statistical mechanics, due to Boltzmann and Gibbs, answers exactly this. A system in equilibrium with a reservoir at temperature $T$ occupies a state of energy $E$ with a probability proportional to the Boltzmann factor

$$ \boxed{\; P(E) \;\propto\; e^{-E/k_B T} \;=\; e^{-\beta E}\;} $$

Two pieces of notation earn their keep here:

• $k_B = 1.381\times10^{-23}\,\mathrm{J/K}$ is Boltzmann's constant. It is just a unit-conversion factor between temperature and energy: $k_B T$ is the characteristic thermal energy available at temperature $T$. (From here on we choose units with $k_B = 1$, so temperature is measured directly in energy.)

• $\displaystyle \beta \equiv \frac{1}{k_B T}$ is the inverse temperature. It appears so often that it gets its own symbol. Large $\beta$ means cold; small $\beta$ means hot.

How to read the Boltzmann factor. Low-energy states are more likely than high-energy ones, and the ratio of likelihoods for two states depends only on their energy gap measured in units of the thermal energy:

$$ \frac{P(E_2)}{P(E_1)} = e^{-\beta (E_2 - E_1)} = e^{-(E_2-E_1)/k_B T}. $$

When $k_B T$ is much larger than the gap, this ratio is near 1 — heat wins, all states nearly equally likely. When $k_B T$ is much smaller than the gap, the ratio collapses to 0 — cold wins, the system freezes into its ground state. That single exponential contains the entire field-versus-temperature contest.

5 · The partition function $Z$

"Proportional to" is not good enough — probabilities must sum to 1. We fix the constant by adding up the Boltzmann factors of every accessible state. That sum is the partition function:

$$ Z \;\equiv\; \sum_{\text{states } s} e^{-\beta E_s}. $$

(The letter is $Z$, for the German Zustandssumme — literally "sum over states".) With $Z$ in hand the probability of any single state is properly normalised:

$$ P_s = \frac{e^{-\beta E_s}}{Z}. $$

$Z$ looks like a mere bookkeeping device, but it is the most powerful object in the subject: once you know $Z$ as a function of temperature and field, every thermodynamic quantity follows by differentiation. We will use three such identities:

$$ \langle E\rangle = -\frac{\partial \ln Z}{\partial \beta}, \qquad F = -k_B T \ln Z, \qquad S = -\frac{\partial F}{\partial T}, $$

where $F$ is the Helmholtz free energy and $S$ the entropy. For our two-level spin the sum has just two terms:

$$ Z = e^{-\beta E_\uparrow} + e^{-\beta E_\downarrow} = e^{+\beta \mu B} + e^{-\beta \mu B} = 2\cosh(\beta \mu B). $$

That compact $2\cosh(\beta\mu B)$ is the seed from which the whole paramagnet grows. Let's plant it in code.

6 · Magnetization: the signature curve

The quantity a laboratory actually measures is the magnetization — the net magnetic moment. For one spin it is the average of $\mu_z$ over the two states:

$$ \langle \mu_z \rangle = (+\mu)\,P(\uparrow) + (-\mu)\,P(\downarrow) = \mu\,\frac{e^{\beta\mu B} - e^{-\beta\mu B}}{e^{\beta\mu B} + e^{-\beta\mu B}}. $$

That ratio is the definition of the hyperbolic tangent, so

$$ \boxed{\;\langle \mu_z \rangle = \mu\,\tanh\!\left(\frac{\mu B}{k_B T}\right) = \mu\tanh(\beta\mu B)\;} $$

For $N$ independent spins the total magnetization is just $M = N\langle\mu_z\rangle$. This one formula is the paramagnet's fingerprint. Read off its two limits:

• Strong field / low temperature ($\beta\mu B \gg 1$): $\tanh \to 1$, so $M \to N\mu$. Every spin is aligned — the material is saturated and cannot be magnetised any further.
• Weak field / high temperature ($\beta\mu B \ll 1$): $\tanh(x)\approx x$, so $M \approx N\mu \cdot \dfrac{\mu B}{k_B T} = \dfrac{N\mu^2}{k_B T}B$. Magnetization is linear in field and falls off as $1/T$ — this is Curie's law, which we return to in §9.

The plot below is the master curve $\tanh(x)$ against $x=\mu B/k_BT$. Slide the temperature and watch the same physical spin move between the linear (hot) and saturated (cold) regimes.

7 · Energy, entropy, and heat capacity

Now we cash in on the promise that everything flows from $Z = 2\cosh(\beta\mu B)$. Write $x \equiv \beta\mu B$.

Average energy. Using $\langle E\rangle = -\partial_\beta \ln Z$,

$$ \langle E\rangle = -\mu B\,\tanh(\beta\mu B). $$

It is just $-B$ times the magnetization, as it must be since $E=-\mu_z B$. Cold spins sit in the ground state $\langle E\rangle\to-\mu B$; hot spins average to $\langle E\rangle\to 0$.

Entropy. Entropy measures the number of ways the system can arrange itself — its disorder. From $F=-k_BT\ln Z$ and $S=-\partial F/\partial T$,

$$ \frac{S}{k_B} = \ln\!\big(2\cosh x\big) - x\tanh x. $$

Its limits tell a clean story. Hot ($x\to0$): $S\to k_B\ln 2$ — the spin is equally likely up or down, the maximum disorder of a two-state system. Cold ($x\to\infty$): $S\to 0$ — the spin is frozen into its single ground state, zero disorder. This vanishing of entropy at $T\to0$ is a concrete instance of the third law of thermodynamics.

Heat capacity. How much heat must we add to warm the spins by $\mathrm{d}T$? That is the heat capacity $C = \partial\langle E\rangle/\partial T$:

$$ \frac{C}{k_B} = (\beta\mu B)^2\,\operatorname{sech}^2(\beta\mu B) = x^2\operatorname{sech}^2 x. $$

Unlike an ideal gas, whose heat capacity is flat, this one is zero at both extremes and shows a hump in between — the celebrated Schottky anomaly. Physically: a spin can only absorb energy when $k_BT$ is comparable to the gap $2\mu B$. Much colder and there isn't enough thermal energy to promote it; much hotter and both states are already near-equally full, so extra heat changes little. Let's see all three at once.

8 · Beyond spin-½: the Brillouin function

Real magnetic ions are rarely simple spin-$\tfrac12$ objects. An ion with total angular momentum quantum number $J$ has $2J+1$ equally spaced levels, with the moment's projection along the field taking the values $m_J = -J, -J+1, \dots, +J$ (in units of $g\mu_B$, where $g$ is the Landé factor and $\mu_B$ the Bohr magneton). The energies are $E_m = -m_J\, g\mu_B B$.

The recipe is exactly the same, only the sum over states is longer. The partition function is a finite geometric series that collapses to

$$ Z_J = \frac{\sinh\!\big[(2J+1)y/2\big]}{\sinh(y/2)}, \qquad y \equiv \frac{g\mu_B B}{k_B T}. $$

Differentiating $\ln Z_J$ gives the magnetization as $M = N g\mu_B J\, B_J(x)$, where $x \equiv gJ\mu_B B/k_BT$ and $B_J$ is the Brillouin function:

$$ \boxed{\; B_J(x) = \frac{2J+1}{2J}\coth\!\left(\frac{2J+1}{2J}x\right) - \frac{1}{2J}\coth\!\left(\frac{x}{2J}\right)\;} $$

It looks fearsome but it is just "$\tanh$ for general $J$". Two checks confirm it:

• $J = \tfrac12$: the formula reduces to $B_{1/2}(x) = \tanh(x)$ — our two-state result. ✓
• $J \to \infty$ (moment free to point anywhere, the classical case): it becomes the Langevin function $B_\infty(x) = L(x) = \coth x - 1/x$, exactly what Langevin derived in 1905. ✓

So a single function interpolates between the quantum two-level spin and Langevin's classical spinning compass. Let's implement it and watch $J$ sweep between the two.

9 · Curie's law and the effective moment

Expand any Brillouin function for small argument, $B_J(x) \approx \dfrac{J+1}{3J}\,x$ as $x\to0$. Feeding this into $M = Ng\mu_B J\,B_J(x)$ with $x = gJ\mu_B B/k_BT$ gives the weak-field, high-temperature susceptibility $\chi = M/B$:

$$ \boxed{\;\chi = \frac{C}{T}, \qquad C = \frac{N g^2 \mu_B^2\, J(J+1)}{3 k_B} = \frac{N\mu_{\text{eff}}^2}{3k_B}\;} $$

This is Curie's law — exactly the $\chi\propto 1/T$ that Pierre Curie measured in 1895, now derived from first principles. The proportionality constant $C$ is the Curie constant, and it encodes the size of the moment through the effective moment

$$ \mu_{\text{eff}} = g\,\mu_B\sqrt{J(J+1)}. $$

Two things worth savouring:

1. It is a thermometer of the microscopic world. Measuring $\chi(T)$ in the lab and reading off $C$ tells you $J(J+1)$ — the quantum numbers of individual ions — from a bulk magnetic measurement.
2. The classic test is a straight line. If Curie's law holds, a plot of $1/\chi$ against $T$ is a straight line through the origin whose slope is $1/C$. Deviations from that line (a nonzero intercept) reveal interactions between spins — the story that leads on to ferromagnetism and the Curie–Weiss law. The figure below shows the ideal paramagnet passing this test.

10 · Meeting the experiment: Henry, 1952

Theory is only worth as much as its confrontation with the laboratory. The definitive test came from W. E. Henry at the U.S. Naval Research Laboratory [6]. Working at liquid-helium temperatures (1.30–4.21 K) and in fields up to 50 000 gauss, he measured the magnetic moment per ion for three salts and drove all three past 99.5% saturation — something Curie-era experiments, stuck in the weak-field linear regime, had never seen.

His three ions are ideal "spin-only" paramagnets (orbital angular momentum quenched by the crystal, so $g\approx2$ and $J=S$):

Henry's masterstroke was to plot the moment not against field alone but against the ratio $H/T$ — the very combination the Brillouin function depends on. Data taken at different temperatures then collapse onto a single universal curve per ion. The solid lines below are the Brillouin functions $B_S$ with no adjustable parameters — only the known $S$ and $g=2$; the markers are the measurements.

On the data. The markers were digitized by eye from the classic reproduction of Henry's figure in Kittel's Introduction to Solid State Physics [7, Fig. 3.3]; the values are approximate readings, not archival numbers. The whole point of Henry's paper — and of the figure — is that they fall on the parameter-free Brillouin curves to within the plotted line width.

11 · Putting it to work: cooling with a magnet

The paramagnet is not just a pedagogical toy — it is the working substance of the coldest refrigerators ever built. The key is the entropy curve from §7. Because entropy depends on temperature and field only through the ratio $B/T$, we can move entropy around by changing the field.

Adiabatic demagnetisation (Giauque, 1927 [5]; Nobel Prize 1949) runs in two strokes:

1. Magnetise isothermally. Apply a strong field while the sample is held at temperature $T_i$ by a bath. The field aligns the spins, lowering their entropy — the disorder is dumped into the bath as heat.
2. Demagnetise adiabatically. Thermally isolate the sample and slowly reduce the field. With no heat allowed in or out, the entropy stays constant. But at low field the same entropy corresponds to a much lower temperature — so the sample cools, dramatically.

The spins carry their disorder budget down in temperature with them. Since $S$ is a function of $B/T$, holding $S$ fixed while dropping $B$ by a factor forces $T$ down by the same factor: $T_f/T_i \approx B_f/B_i$. Starting near 1 K, Giauque's team reached a few thousandths of a kelvin; the nuclear-spin version of the same trick reaches microkelvin. The figure shows the two strokes as a path on the entropy diagram.

12 · Where to go next

We assumed the spins are independent — each feels only the external field. That single assumption is what makes the paramagnet exactly solvable, and it is also the doorway to almost everything else in the physics of magnetism. Relax it and a new world opens up:

• Interacting spins → ferromagnetism. Let each spin also feel its neighbours. The simplest way to do this, mean-field theory [8], replaces the neighbours by an average internal field $B_{\text{eff}} = B + \lambda M$ proportional to the magnetization itself. The magnetization equation becomes self-consistent, $M = \tanh\!\big(\beta\mu(B+\lambda M)\big)$, and below a critical Curie temperature $T_c = \lambda\mu^2/k_B$ it develops a nonzero solution even at $B=0$: spontaneous magnetization — a permanent magnet, and a genuine phase transition. The cell below solves exactly this. Try turning the coupling on.

• Negative temperature. Because the two-level spin system has a bounded energy spectrum, one can invert the populations ($P_\downarrow > P_\uparrow$) — a state formally hotter than infinity, with $T<0$. Purcell and Pound demonstrated it in a crystal of nuclear spins in 1951 [9].

• Lower and lower. Nuclear adiabatic demagnetisation reaches microkelvin temperatures; the same statistical ideas underpin laser cooling and the pursuit of quantum degeneracy.

• Quantum magnetism. Add quantum-mechanical exchange and geometry and you reach the Ising and Heisenberg models, frustration, and spin liquids — frontier research today.

Suggested exercises. (1) Derive the heat capacity $C(T)$ by differentiating $\langle E\rangle$ and confirm the Schottky peak sits at $k_BT\approx0.83\,\mu B$. (2) Show $B_{1/2}(x)=\tanh x$ analytically. (3) In the mean-field cell below, locate $T_c$ numerically and check it scales linearly with the coupling $\lambda$.

Further reading. For the statistical mechanics from scratch, Schroeder [10] is superb; for magnetism specifically, Blundell [11]; and Kittel [7] is the standard solid-state reference (and the source of the reproduced Henry figure).