The Two-Level Paramagnet and the Brillouin Function

Cool a salt of magnetic ions in a field and its magnetization rises, saturates, and — crucially — its low-field susceptibility varies as $1/T$ : the colder the sample, the more strongly it responds. This is paramagnetism, and it is the single cleanest laboratory for the canonical ensemble. Each ion carries a fixed angular momentum $J$ ; in a field $B$ its $2J+1$ orientations become $2J+1$ energy levels split by $-g\mu_B B m$ . Because the ions are dilute and independent, the whole thermodynamics factorizes into one ion's partition function, and that partition function is a finite geometric series you can sum in closed form. The result is the Brillouin function $B_J(x)$ , which gives the magnetization as $M = Ng\mu_B J\,B_J(x)$ with $x = g\mu_B J B/k_BT$ . This one function contains everything: it reduces to $\tanh x$ for a spin- $\tfrac12$ ion, flows to the classical Langevin function $\coth x - 1/x$ as $J\to\infty$ , gives Curie's law $\chi = C/T$ in weak fields, and saturates to full alignment in strong fields. This article derives $B_J(x)$ from the sum over states, proves those four limits, extracts the Curie constant $C = Ng^2\mu_B^2 J(J+1)/3k_B$ , and works real numbers for a gadolinium-like ion. The magnetic heat capacity of the same level scheme (the Schottky bump) and its use as a refrigerant (adiabatic demagnetization) are named where they belong and treated in their own articles.

Key takeaways

$N$ independent moments of angular momentum $J$ in a field $B$ have $2J+1$ levels $E_m=-g\mu_B B m$ ; the single-ion partition function sums to $Z_1 = \sinh[(2J{+}1)x/2J]/\sinh[x/2J]$ with $x=g\mu_B J B/k_BT$ .
The magnetization is $M = Ng\mu_B J\,B_J(x)$ , where the Brillouin function is $B_J(x)=\tfrac{2J+1}{2J}\coth\tfrac{(2J+1)x}{2J}-\tfrac{1}{2J}\coth\tfrac{x}{2J}$ .
Spin- $\tfrac12$ : $B_{1/2}(x)=\tanh x$ . Classical limit $J\to\infty$ : the Langevin function $L(x)=\coth x-1/x$ .
Weak field: $B_J(x)\approx\tfrac{J+1}{3J}x$ , giving Curie's law $\chi=C/T$ with $C=Ng^2\mu_B^2J(J+1)/3k_B$ .
Strong field: $B_J\to1$ , so $M\to M_\text{sat}=Ng\mu_B J$ — every moment aligned.
Key idea: one geometric sum over $2J+1$ tilts of a spin produces a universal magnetization curve whose slope is the Curie law and whose plateau is saturation.

The core insight

A paramagnet is $N$ copies of a single quantum top in a field. Summing the Boltzmann factors over the top's $2J+1$ orientations gives a partition function whose field-derivative is the magnetization: $M = Ng\mu_B J\,B_J(x)$ with $x=g\mu_B J B/k_BT$ . The competition between the alignment energy $g\mu_B B$ and the thermal energy $k_BT$ is captured entirely by the ratio $x$ .

Where this lives today

The two-level and $2J+1$ -level paramagnet is the workhorse example in every statistical-mechanics course because it is exactly solvable and physically real. Its low-field $1/T$ susceptibility underpins the Curie-law thermometry used throughout low-temperature physics: a paramagnetic salt whose susceptibility is calibrated against a known $1/T$ line becomes a secondary thermometer valid down to the millikelvin range, and cerium magnesium nitrate (CMN) is a standard choice precisely because it follows Curie's law to very low $T$ . Its saturation curve is measured routinely in SQUID magnetometry of molecular magnets and rare-earth compounds, where fitting $B_J(x)$ extracts the ground-state $J$ and $g$ directly from the shape of the $M$ -versus- $B$ curve. The same level scheme carries entropy $S(B/T)$ that decreases as the field aligns the moments, and this field-tunable entropy is precisely what adiabatic demagnetization exploits to reach millikelvin and, with nuclear moments, microkelvin temperatures. When the moments are not independent — when exchange couples them — the same Brillouin function reappears inside mean-field (Weiss) theory as the self-consistency relation for ferromagnetism, with $B$ replaced by an effective molecular field $B+\lambda M$ ; the paramagnet is thus the dilute seed of the whole theory of ordered magnetism. The heat capacity of this level scheme is the Schottky anomaly, treated in its own article, and the itinerant-electron cousin is Pauli paramagnetism; both are named-and-deferred here so that this article owns just the magnetization function.

The physical picture

Imagine a crystal in which magnetic ions sit far enough apart that they do not talk to each other — a paramagnetic salt like gadolinium sulfate, where each Gd $^{3+}$ ion carries a well-defined total angular momentum $J$ and magnetic moment $\boldsymbol\mu = -g\mu_B\mathbf{J}/\hbar$ . Switch on a field $B$ along $z$ . Space quantization allows only $2J+1$ projections $m=-J,-J+1,\dots,+J$ , and each has energy $E_m=-g\mu_B B m$ : the moment most aligned with the field ( $m=+J$ ) sits lowest, the most anti-aligned ( $m=-J$ ) highest. Nothing continuous — a ladder of $2J+1$ rungs. Temperature blurs the ladder. At high $T$ the ion visits all $2J+1$ orientations almost equally, so the average moment is small and grows only linearly with $B$ — this is the Curie regime, $\chi\propto1/T$ . At low $T$ (or high $B$ ) the ion is pinned in its ground orientation $m=+J$ , the average moment reaches its maximum $g\mu_B J$ , and further field does nothing — saturation. The whole story is the balance of two energies: the Zeeman splitting $g\mu_B B$ that wants to align the moment, and $k_BT$ that wants to randomize it. Their ratio is the dimensionless field $x = g\mu_B J B/k_BT$ .

Figure

Brillouin magnetization curves B_J(x): from tanh (J=½) down to Langevin (J→∞), all saturating to 1

There is a decisive prediction hiding in that ratio. Because $B_J$ depends on $B$ and $T$ only through $x=g\mu_B J B/k_BT$ , magnetization data taken at many different fields and temperatures must collapse onto a single master curve when plotted against $B/T$ . W. E. Henry's 1952 measurements on chromium, iron, and gadolinium salts did exactly this — points spanning fields up to several tesla and temperatures down to $1.3$ K fell on the three theoretical $B_J$ curves for $J=\tfrac32,\tfrac52,\tfrac72$ with no adjustable parameters. That scaling collapse, and the straight $1/\chi$ –versus– $T$ line it implies at low field, is the fingerprint of independent moments.

Figure

Curie's law: inverse susceptibility is a straight line through the origin

The core equations

The whole thermodynamics of $N$ independent moments follows from one ion's partition function, so we build that first and differentiate it for the magnetization.

E_m = -g\mu_B B\,m, \qquad m = -J,\,-J+1,\,\dots,\,+J \qquad (1)

where: $E_m$ is the Zeeman energy of the orientation labelled by the magnetic quantum number $m$ , $g$ is the Landé $g$ -factor, $\mu_B=e\hbar/2m_e=9.274\times10^{-24}\,\mathrm{J/T}$ is the Bohr magneton, $B$ the applied field, and $J$ the total angular momentum. There are exactly $2J+1$ levels, equally spaced by $g\mu_B B$ .

Z_1 = \sum_{m=-J}^{J} e^{-\beta E_m} = \sum_{m=-J}^{J} e^{a m} = \frac{\sinh\!\big[(2J+1)\tfrac{x}{2J}\big]}{\sinh\!\big[\tfrac{x}{2J}\big]} \qquad (2)

where: $\beta=1/k_BT$ , $a\equiv\beta g\mu_B B$ is the level spacing in units of $k_BT$ , and $x\equiv aJ = g\mu_B J B/k_BT$ is the dimensionless field. The single-ion partition function is a finite geometric series (derived below). For $N$ independent, distinguishable ions on lattice sites the total factorizes, $Z=Z_1^N$ — the factorization theorem recalled from Factorization of the Partition Function.

M = -\frac{\partial F}{\partial B} = N g\mu_B J\, B_J(x), \qquad B_J(x)=\frac{2J+1}{2J}\coth\!\frac{(2J+1)x}{2J}-\frac{1}{2J}\coth\!\frac{x}{2J} \qquad (3)

where: $F=-Nk_BT\ln Z_1$ is the Helmholtz free energy, $M$ the total magnetization, and $B_J(x)$ the Brillouin function — the central object of this article. $M_\text{sat}=Ng\mu_B J$ is the fully aligned magnetization, reached as $B_J\to1$ .

\chi = \lim_{B\to0}\frac{\partial M}{\partial B}\Big/\ \cdots \;=\; \frac{C}{T}, \qquad C = \frac{N g^2\mu_B^2\,J(J+1)}{3k_B} \qquad (4)

where: $\chi$ is the low-field magnetic susceptibility, $C$ the Curie constant, and $\mu_\text{eff}=g\sqrt{J(J+1)}\,\mu_B$ the effective moment (so $C=N\mu_\text{eff}^2/3k_B$ ). In SI a factor $\mu_0$ accompanies $\chi$ depending on whether one differentiates with respect to $B$ or $H$ ; the physics — $\chi\propto1/T$ — is convention-independent.

Derivation ASumming the ladder: the single-ion partition function

The goal is to turn the sum over $2J+1$ Boltzmann factors in equation (2) into the compact $\sinh$ -ratio, because every thermodynamic quantity is a derivative of $\ln Z_1$ . The only tool needed is the finite geometric series, and the one subtlety is bookkeeping the offset exponent so that the series starts cleanly at zero; the payoff is a closed form valid for every integer and half-integer $J$ . We work with $a=\beta g\mu_B B$ and restore $x=aJ$ at the end.

Step 1. Write the sum with the substitution $a=\beta g\mu_B B$ : $Z_1=\sum_{m=-J}^{J}e^{am}$ . Factor out the lowest term to make it an ordinary geometric series: $Z_1=e^{-aJ}\sum_{k=0}^{2J}e^{ak}$ , where $k=m+J$ runs over $0,1,\dots,2J$ .

Step 2. Sum the geometric series: $\sum_{k=0}^{2J}e^{ak}=\dfrac{e^{a(2J+1)}-1}{e^{a}-1}$ . Hence $Z_1=e^{-aJ}\dfrac{e^{a(2J+1)}-1}{e^{a}-1}$ .

Step 3. Symmetrize by multiplying top and bottom by $e^{-a/2}$ : the denominator becomes $e^{a/2}-e^{-a/2}=2\sinh(a/2)$ , and the numerator becomes $e^{-aJ}\!\left(e^{a(2J+1)}-1\right)e^{-a/2}=e^{a(J+1/2)}-e^{-a(J+1/2)}=2\sinh[(J+\tfrac12)a]$ .

Step 4. Collect: $Z_1=\dfrac{\sinh[(J+\tfrac12)a]}{\sinh(a/2)}$ . Since $(J+\tfrac12)a=(2J+1)\tfrac{a}{2}$ and $a=x/J$ , this is $Z_1=\dfrac{\sinh[(2J+1)x/2J]}{\sinh[x/2J]}$ , equation (2). Verified: the companion script sums $\sum_{m=-J}^{J}e^{am}$ term by term for $J=\tfrac12,1,\tfrac32,\tfrac52$ and confirms it equals $\sinh[(2J+1)a/2]/\sinh(a/2)$ at four values of $a$ each.

Go deeper · Derivation B · 1 minFrom the partition function to the Brillouin function and Curie's law

Derivation BFrom the partition function to the Brillouin function and Curie's law

Here the aim is to obtain the magnetization (equation 3) and its weak-field limit (equation 4) directly from $\ln Z_1$ , and to establish the article's owned result: the single universal function $B_J(x)$ that interpolates between Curie linearity and saturation. The strategy is that the average moment is a logarithmic derivative of $Z_1$ ; the subtlety is a chain-rule factor between the variables $a$ and $x=aJ$ , which is exactly what produces the prefactor $J$ in $M=Ng\mu_B J\,B_J$ . The small- $x$ expansion then uses $\coth y\approx 1/y+y/3$ .

Step 1. The average projection is $\langle m\rangle=\dfrac{1}{Z_1}\sum_m m\,e^{am}=\dfrac{\partial\ln Z_1}{\partial a}$ . Differentiating the $\sinh$ -ratio, $\dfrac{\partial\ln Z_1}{\partial a}=\dfrac{2J+1}{2}\coth\!\big[\tfrac{(2J+1)a}{2}\big]-\tfrac12\coth\!\big[\tfrac{a}{2}\big]$ .

Step 2. Convert to $x=aJ$ (so $\partial/\partial a=J\,\partial/\partial x$ is not needed here; just substitute $a=x/J$ ): $\langle m\rangle=J\!\left[\dfrac{2J+1}{2J}\coth\tfrac{(2J+1)x}{2J}-\dfrac{1}{2J}\coth\tfrac{x}{2J}\right]=J\,B_J(x)$ . The bracket defines $B_J(x)$ . Each ion contributes moment $g\mu_B\langle m\rangle$ , so $M=Ng\mu_B\langle m\rangle=Ng\mu_B J\,B_J(x)$ .

Step 3. Weak field. Expand each $\coth y\approx 1/y+y/3$ with $y=\tfrac{(2J+1)x}{2J}$ and $y=\tfrac{x}{2J}$ . The $1/y$ pieces cancel, leaving $B_J(x)=\dfrac{x}{12J^2}\big[(2J+1)^2-1\big]+O(x^3)=\dfrac{x(4J^2+4J)}{12J^2}=\dfrac{J+1}{3J}\,x$ .

Step 4. Curie constant. Insert $x=g\mu_B J B/k_BT$ : $M\approx Ng\mu_B J\cdot\dfrac{J+1}{3J}\cdot\dfrac{g\mu_B J B}{k_BT}=\dfrac{Ng^2\mu_B^2 J(J+1)}{3k_B}\dfrac{B}{T}$ . Thus $\chi\propto1/T$ with $C=Ng^2\mu_B^2J(J+1)/3k_B$ , equation (4). Verified: the script confirms $\langle m\rangle=J\,B_J(x)$ numerically for $J=\tfrac12,\tfrac52$ , the slope $B_J'(0)=(J+1)/3J$ symbolically (giving $1,\tfrac23,\tfrac{7}{15}$ for $J=\tfrac12,1,\tfrac52$ ), and that $M_\text{small}=C\,B/T$ with $C$ as quoted.

Limiting cases

Each limit of $B_J(x)$ is a physical check on the construction.

Case	Result	Meaning
Spin- $\tfrac12$ , $J=\tfrac12$	$B_{1/2}(x)=\tanh x$	Two levels only; $M=Ng\mu_B\tfrac12\tanh(\mu B/k_BT)$ — the textbook two-state paramagnet.
Classical, $J\to\infty$	$B_J(x)\to\coth x-\dfrac1x=L(x)$	A freely rotating classical moment; Langevin's 1905 result recovered as the continuum limit.
Weak field, $x\ll1$	$B_J\approx\dfrac{J+1}{3J}x$	Linear response; Curie's law $\chi=C/T$ , susceptibility diverging as $T\to0$ .
Strong field, $x\gg1$	$B_J\to1$ , $M\to Ng\mu_B J$	Every moment pinned to $m=+J$ ; saturation, field-independent.

Common trapConfusing localized-moment paramagnetism with Pauli paramagnetism

The $1/T$ Curie law derived here belongs to localized moments — ions with a fixed $J$ sitting on lattice sites, each independently sampling its $2J+1$ orientations. It does not describe the paramagnetism of conduction electrons in a metal. There the electrons are itinerant and degenerate, only those within $\sim k_BT$ of the Fermi energy can flip, and the susceptibility is essentially temperature-independent — Pauli paramagnetism, treated in its own article. A quick tell: if $\chi\propto1/T$ you are looking at local moments (insulating salts, rare-earth ions); if $\chi$ is small and flat in $T$ you are looking at a Fermi gas. Mixing the two — for instance, applying the Curie constant to the conduction electrons of a metal — overestimates the electronic susceptibility by orders of magnitude.

Worked examples

Example 1 — A gadolinium-like ion in a modest field. Take $\mathrm{Gd}^{3+}$ : half-filled $4f$ shell, $L=0$ , so $g=2$ and $J=S=\tfrac72$ . Put it in $B=1$ T at $T=300$ K. How aligned is it, and what is its effective moment?

Step 1. Dimensionless field: $x=\dfrac{g\mu_B J B}{k_BT}=\dfrac{2\,(9.274\times10^{-24})\,(3.5)(1)}{(1.381\times10^{-23})(300)}=\mathbf{0.0157}$ . This is tiny — we are deep in the Curie regime.

Step 2. Fractional alignment: $M/M_\text{sat}=B_{7/2}(0.0157)\approx\dfrac{J+1}{3J}x=\dfrac{4.5}{10.5}(0.0157)=\mathbf{0.0067}$ . Only about $0.7\%$ of the moments' worth of alignment at room temperature and $1$ T — which is why paramagnets look magnetically inert unless you go cold or strong.

Step 3. Effective moment: $\mu_\text{eff}=g\sqrt{J(J+1)}\,\mu_B=2\sqrt{3.5\times4.5}\,\mu_B=\sqrt{63}\,\mu_B=\mathbf{7.94\,\mu_B}$ , the standard tabulated value for $\mathrm{Gd}^{3+}$ . The Curie constant per mole then follows from $C=N_A\mu_\text{eff}^2/3k_B$ . The takeaway: at ordinary temperatures a paramagnet responds linearly and weakly; the $1/T$ growth of $\chi$ is what makes it a useful thermometer and refrigerant at low $T$ .

Example 2 — The same ion, cold and strong: approaching saturation. Now cool to $T=1$ K and raise the field to $B=5$ T.

Step 1. Dimensionless field: $x=\dfrac{2(9.274\times10^{-24})(3.5)(5)}{(1.381\times10^{-23})(1)}=\mathbf{23.5}$ . We are now far into the strong-field regime, $x\gg1$ .

Step 2. Alignment: $B_{7/2}(23.5)=\dfrac{8}{7}\coth\!\big(\tfrac{8}{7}\cdot23.5\big)-\dfrac{1}{7}\coth\!\big(\tfrac{23.5}{7}\big)=\mathbf{0.9997}$ . The moments are essentially fully aligned; further field buys almost nothing.

Step 3. Contrast: raising the temperature back to $300$ K at fixed $5$ T would drop $x$ by a factor $300$ to $\approx0.078$ , and $M/M_\text{sat}$ back to a few percent. Saturation is a low-temperature, high-field corner. The takeaways: the magnetization is controlled entirely by $x=g\mu_B J B/k_BT$ — $0.7\%$ aligned at $1$ T/ $300$ K, but $99.97\%$ aligned at $5$ T/ $1$ K.

Origins · 1 minléon brillouin and how this was discovered

léon brillouin

1889–1969 · French physicist who quantized the theory of paramagnetism, giving the magnetization function that bears his name

Léon Brillouin, working in Paris in the late 1920s, replaced Langevin's classical freely-rotating moment with a quantum top restricted to $2J+1$ orientations, and in 1927 obtained the quantized magnetization curve $B_J(x)$ now universally called the Brillouin function. It was an early, concrete triumph of space quantization: the same discreteness that Stern and Gerlach had seen in a beam now dictated the bulk magnetization of a salt. Brillouin's name recurs across physics — Brillouin zones in solids, Brillouin scattering of light by sound, and the "B" in the WKB(J)/Brillouin semiclassical method — and late in life he wrote influentially on the link between entropy and information (his "negentropy" principle). His paramagnet remains the standard first worked example of the canonical ensemble applied to a finite-level system.

History

1895

— Pierre Curie measures the paramagnetic susceptibility and finds χ ∝ 1/T

1905

— Paul Langevin builds the classical statistical theory of paramagnetism and the Langevin function

1907

— Pierre Weiss adds the molecular field, extending the picture to ferromagnetism

1922

— Stern and Gerlach demonstrate space quantization of magnetic moments

1927

— Léon Brillouin derives the quantized magnetization curve, the Brillouin function B_J(x)

1952

— W. E. Henry's paramagnetic-salt data collapse onto the B_J curves confirms the theory across J and B/T

Going further

References. Reif, Fundamentals of Statistical and Thermal Physics, §7.8 (ideal paramagnet) · Kittel, Introduction to Solid State Physics, ch. 11 (paramagnetism, the Henry data collapse) · Kardar, Statistical Physics of Particles, §4 · Ashcroft & Mermin, Solid State Physics, ch. 31 · Blundell, Magnetism in Condensed Matter, ch. 2.

Provenance. A verifier reproduces: (1) the geometric-sum identity $Z_1=\sinh[(2J{+}1)a/2]/\sinh(a/2)$ for $J=\tfrac12,1,\tfrac32,\tfrac52$ ; (2) $B_{1/2}(x)=\tanh x$ and $B_{1/2}(1)=0.76159$ ; (3) the $J\to\infty$ Langevin limit $\coth x-1/x$ with $L(2)=0.53731$ ; (4) the small- $x$ slope $(J+1)/3J$ giving $1,\tfrac23,\tfrac{7}{15}$ ; (5) $\langle m\rangle=J\,B_J(x)$ ; (6) the Curie constant $C=Ng^2\mu_B^2J(J+1)/3k_B$ and $\mu_\text{eff}=7.94\,\mu_B$ for $\mathrm{Gd}^{3+}$ ; (7) the worked values $x=0.0157$ , $M/M_\text{sat}=0.0067$ at $1$ T/ $300$ K and $x=23.5$ , $M/M_\text{sat}=0.9997$ at $5$ T/ $1$ K.

Builds on: Canonical Partition Function · Boltzmann Factor · Factorization of the Partition Function. Leads to: The Schottky Anomaly · Adiabatic Demagnetization and Magnetic Cooling · Pauli Paramagnetism.

Why are there exactly 2J+1 energy levels?

Because a quantum angular momentum $J$ has $2J+1$ allowed projections along any chosen axis, $m=-J,-J+1,\dots,+J$ — this is space quantization. In a field the energy depends only on that projection, $E_m=-g\mu_B B m$ , so each projection is one energy level. For a spin- $\tfrac12$ ion that is two levels; for $\mathrm{Gd}^{3+}$ with $J=\tfrac72$ it is eight. The count $2J+1$ is exactly what makes the partition function a finite geometric series rather than an integral.

Where does the Landé g-factor come from and why is it often about 2?

The $g$ -factor relates the magnetic moment to the angular momentum, $\boldsymbol\mu=-g\mu_B\mathbf{J}/\hbar$ . For pure spin $g\approx2$ (the Dirac value, $2.0023$ with QED corrections); for pure orbital motion $g=1$ ; for a general ion with both, the Landé formula interpolates using $J$ , $L$ , and $S$ . $\mathrm{Gd}^{3+}$ has $L=0$ (half-filled shell), so it is pure spin and $g=2$ exactly to good approximation, which is why it is the cleanest textbook paramagnet.

Why does the magnetization depend only on B/T and not on B and T separately?

Because the only place $B$ and $T$ enter the single-ion problem is through the Boltzmann factor $e^{-E_m/k_BT}=e^{g\mu_B B m/k_BT}$ , and there $B$ and $T$ appear solely as the ratio $B/T$ (inside $x=g\mu_B J B/k_BT$ ). Everything thermodynamic is built from that partition function, so it too can depend only on $B/T$ . This is why data at many fields and temperatures collapse onto one curve when plotted against $B/T$ — a direct, parameter-free test that the moments are independent.

How is this different from Curie–Weiss behavior in a ferromagnet?

Curie's law $\chi=C/T$ is for independent moments and its $1/\chi$ line passes through the origin. When moments interact (exchange), mean-field theory replaces $B$ by $B+\lambda M$ , and the susceptibility becomes $\chi=C/(T-\theta)$ — the Curie–Weiss law, whose $1/\chi$ line intercepts the temperature axis at the Weiss temperature $\theta$ (positive for ferromagnets). Above $\theta$ the material is still paramagnetic but "assisted"; at $\theta$ it orders. The Brillouin function reappears there as the self-consistency equation, so this article's machinery is the seed of the theory of magnetism, not just its dilute limit.

What sets the saturation magnetization, and can you exceed it?

Saturation is reached when every moment sits in its lowest Zeeman level $m=+J$ , giving $M_\text{sat}=Ng\mu_B J$ — the maximum the localized moments can provide. You cannot exceed it by raising the field further; $B_J(x)$ approaches $1$ but never surpasses it. (You can get additional magnetization from other sources — conduction electrons, orbital contributions, or induced moments — but those are separate mechanisms, not more of the same localized alignment.)

Is the classical Langevin theory just wrong, then?

Not wrong — incomplete. Langevin's 1905 model of a freely rotating classical moment gives $M=Ng\mu_B J\,L(x)$ with $L(x)=\coth x-1/x$ , which is exactly the $J\to\infty$ limit of the quantum result. For ions with large $J$ it is an excellent approximation. What it misses is the discreteness: for small $J$ (especially spin- $\tfrac12$ , where $B_J=\tanh$ ) the finite number of levels changes both the initial slope $(J+1)/3J$ and the detailed shape of the approach to saturation. The quantum Brillouin function contains Langevin as one limit and the two-state $\tanh$ as the other.