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William Francis Giauque's 1949 lecture, Some Consequences of Low Temperature Research in Chemical Thermodynamics, credits Peter Debye with an independent proposal. It describes gadolinium sulfate octahydrate in a jacket immersed in liquid helium, with an external solenoid. Exchange gas conducts away heat during magnetization; evacuation isolates the sample before field reduction. A liquid-nitrogen shield reduces radiative heat transfer. Its photograph identifies the 1933 apparatus operated with Duncan Peck MacDougall. An inductance bridge reads the sample's magnetic susceptibility; measuring coils remain outside the evacuated jacket in liquid helium. Increasing magnetic susceptibility indicates cooling, but converting it to temperature using Curie's law becomes approximate as that law fails.^[ref]

The measurement logic deserves a separate check. A changed inductance alone is not an absolute temperature calibration: the sample geometry, applied measuring magnetic field, and constitutive response enter the inference. A modern control would compare the susceptibility-based thermometer with an independently calibrated thermometer over their common range, check recovery after repeated sweeps, and measure subsequent warming to bound incoming heat. These are proposed controls derived from the readout principle, rather than additional procedures attributed to the 1933 experiment. The historical working salt has a richer magnetic level structure than the illustrative two-level paramagnetic cooling model.

A Gouy balance detects the force exerted by a nonuniform magnetic field on a sample of uniform cross-section extending between high- and low-field regions. TeachSpin's implementation reverses the usual arrangement: a rare-earth permanent-magnet assembly sits on a one-pan electronic balance, while a separate holder brings the sample into the field. The magnet's apparent mass change measures the opposite of the sample's vertical force, by Newton's third law. The described pole faces have diameter $25\mathrm{mm}$ and separation $15\mathrm{mm}$; samples occupy cuvettes of cross-section $(10\mathrm{mm}{)}^2$ and height about $32\mathrm{mm}$. ^[ref]

The same balance calibrates magnetic flux density by measuring the reaction force on the magnet when a wire of known horizontal length $L$ carries known electric current $i$ perpendicular to the field: $|F_z|=iL{B}_0$. The manufacturer describes water being repelled by the high-field region, identifying negative magnetic susceptibility. This observed sign supplies a qualitative test, not a measured Curie's law curve. ^[ref]

For weak reversible linear response, a uniform sample displacing a weakly magnetic surrounding medium has, to leading order, downward forceThis follows by integrating the induced-dipole force density $({\chi }_v-{\chi }_{\mathrm{medium}}){\partial }_z(B_0^2)/(2{\mu }_0)$ along a downward coordinate. $A$ is the cross-sectional area, and the two $B_0$ values are applied magnetic flux densities. The force depends on the endpoint fields; the entire field-gradient profile need not be measured under these assumptions. In the reversed apparatus, $F_{\downarrow }=-g\Delta m_{\mathrm{magnet}}$ after subtracting the holder background, where $g$ is gravitational acceleration. This sign must be reversed before inferring volume magnetic susceptibility.

Useful controls are an empty-holder reading at the same position, a distant-sample baseline, repeated wire-current reversals during field calibration, and stable temperature. They separate holder response, balance drift, force offsets, and thermal changes; these are proposed checks for the stated inference. Finite endpoint fields, nonuniform cross-section, uneven powder packing, stray forces and a demagnetizing field limit the simple formula. If magnetization is nonlinear, the weak-response force involves $A{\int }_{B_{0,\mathrm{top}}}^{B_{0,\mathrm{bottom}}}M(B_0)d{B}_0$ instead of a constant-${\chi }_v$ endpoint square. A balance reading then measures an integrated response, not directly the local differential magnetic susceptibility.

Consider a stipulated dilute collection of independent two-level systems, each with longitudinal magnetic moment $s\mu$, $s=\pm 1$, and energy $E_s=-s\mu B_0$. The prescribed vacuum/applied magnetic flux density is $B_0={\mu }_0{H}_{\mathrm{app}}$. Ignore interactions, orbital magnetic response, and the small sample-field correction. This is the magnetic version of the existing independent spin-one-half two-level system, with alignment probabilities rather than an interacting Curie–Weiss model.

For temperature $T>0$, the canonical ensemble gives the one-moment partition function $Z_1=2\cosh x$, with $x=\mu B_0/(k_{B}T)$. Taking the expected value over its two Boltzmann weights givesHere $n$ is the number density, $k_B$ the Boltzmann constant, and $\tanh$ the hyperbolic tangent. For $|x|\ll 1$, its small-argument expansion of the hyperbolic tangent gives $M\simeq {\mu }_{0}n{\mu }^2{H}_{\mathrm{app}}/(k_{B}T)$, a Curie's law response in the applied-$H$ convention. For large $|x|$, $M$ approaches the saturation magnetization $n\mu$ with sign set by the field, and its differential magnetic susceptibility falls towards zero.

Both plots use stipulated $n=10^{26}{\mathrm{m}}^{-3}$, $\mu =9.2740101\times 10^{-24}\mathrm{J}{\mathrm{T}}^{-1}$, $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, and ${\mu }_0=1.25663706\times 10^{-6}\mathrm{T}\mathrm{m}{\mathrm{A}}^{-1}$. The moment is approximately one Bohr magneton; the chosen vacuum permeability is a rounded model input. At $T=2\mathrm{K}$, $M_s=927.401\mathrm{A}{\mathrm{m}}^{-1}$ and ${\chi }_{\mathrm{model}}(0)=3.91410\times 10^{-4}$. The linear approximation overestimates $M$ by less than $3\%$ for $|x|\le 0.3$, corresponding to $|{\mu }_0{H}_{\mathrm{app}}|\le 0.89324\mathrm{T}$ or $|H_{\mathrm{app}}|\le 0.71082\mathrm{MA}{\mathrm{m}}^{-1}$. These parameters keep the modeled response weak; neglected magnetic feedback is of relative order ${\chi }_{\mathrm{model}}(0)$, rather than an explanation of a material's ordered phase.

At $B_0=0$, cooling increases the zero-field slope as $1/T$. At fixed $B_0\ne 0$, cooling instead gives ${\chi }_{\mathrm{model}}\sim 4{\mu }_{0}n{\mu }^2{e}^{-2\mu |B_0|/(k_{B}T)}/(k_{B}T)$: a saturated moment can hardly change further. The order of the zero-field and zero-temperature limits matters. This independent model has no finite-temperature collective phase transition; the diverging zero-field slope alone does not establish one.

With magnetic Helmholtz free energy convention $dF=-SdT-mdB$, a measured $(\partial S/\partial B{)}_T=-\alpha B/T^2$ fixes $(\partial m/\partial T{)}_B$ to the same value. Integrating determines the displayed inverse-temperature term but leaves a temperature-independent function $m_0(B)$. An independent constitutive or boundary condition is needed to fix it. Requiring zero moment at infinite temperature recovers Curie's law without that offset.

For two independent identical ions at sufficiently high temperature, Curie's law gives $\chi T=2(N_A{\mu }_B^2/3k_B)g^{2}S(S+1)$. In the cgs units used here, $N_A{\mu }_B^2/3k_B\simeq 0.125\mathrm{c}{\mathrm{m}}^3\mathrm{K}\mathrm{mo}{\mathrm{l}}^{-1}$.

Fe(II), Co(II) and Ni(II) have $S=2$, $3/2$ and 1, respectively. The corresponding high-spin occupancies, ordered as $(e^{\prime \prime },e^{\prime },a_1^{\prime })$, areHund filling puts one electron in each orbital before additional pairing. Spin-only dimer values at $g=2$ would be 6.0, 3.75 and 2.0; the observed larger values anticipate orbital contributions. The spin assignments require the chemically appropriate high-spin states, rather than rounding the measured $\chi T$ directly to a spin-only integer.

For $B=5$ T, the same spin-only two-state probabilities apply, but the low-temperature argument is no longer small. At 300 K, $x=0.0111952$; at 2 K, $x=1.67928$. HenceThe 2 K sample is about $93.3\%$ of its saturation magnetization, with about $96.6\%$ of the moments in the aligned state. The two percentages describe different observables. A linear extrapolation would predict $M/M_s\simeq x=1.679$, exceeding saturation and therefore contradicting the bounded two-state population.

For completeness, at this 2 K, 5 T bias the dimensionless secant susceptibility is approximately ${\chi }_0\tanh x/x=0.01199$, while the differential susceptibility is ${\chi }_0{\mathrm{sech}}^{2}x=0.002805$ under the same prescribed-field approximation. Neither equals the zero-field Curie slope $0.02159$. Thus saturation reduces the incremental response even while the magnetization itself is large.

For $B=0.05$ T, use the two spin-only moment states. Their probabilities and normalized magnetization areAt 300 K, $x=1.11952\times 10^{-4}$; at 2 K, $x=0.0167928$. ThereforeThe phrase “fraction aligned” is ambiguous if it is used to mean net alignment. Half the moments are already in the parallel state at zero field, but the net magnetization is then zero because the antiparallel half cancels them. The table reports both the literal parallel-state population and the imbalance that produces magnetization. At this weak field even the 2 K sample is far from saturation, so the Curie approximation is accurate within this independent-spin model.

A $3d^9$ shell has one hole in the otherwise filled $d$ shell. Hund's rules give $S=1/2$, $L=2$, and, for a more-than-half-filled shell, $J=L+S=5/2$: the free-ion term is ${}^2{D}_{5/2}$. Its Landé factor isDistinguish the Curie effective moment from a saturated projection:The question then explicitly asks us to use orbital quenching in the crystal. Ligand fields split and mix orbital states, suppressing their freely orientable angular momentum. The spin-only approximation has $g\simeq 2$, $S=1/2$, so ${\mu }_{\mathrm{eff}}=\sqrt{3}{\mu }_B$ but the two field-axis projections are $\pm {\mu }_B$. The latter, not $\sqrt{3}{\mu }_B$, belong in the tanh argument and saturation magnetization.

The sulfate and water environment supplies mainly closed-shell, weakly magnetic surroundings, stabilizes localized Cu2+ sites and separates their moments so that exchange can be weak on the temperature scale considered. Paramagnetism here is an approximation of weakly interacting local spins, not a claim that dilution guarantees no magnetic ordering at any temperature. Elemental copper has a filled $3d$ band and itinerant conduction electrons rather than one localized Cu2+ spin per atom. Pauli spin response and orbital/core contributions then replace a local-spin Curie law; ordinary pure copper has a weak net diamagnetic response.^[ref]

There is one Cu2+ per formula unit, hence $n_s=N_A/V_m=5.5148\times 10^{27}$ m${}^{-3}$. The spin-only weak-field SI volume susceptibility isThese neglect exchange, anisotropic $g$ factors and background diamagnetism. At high field and low temperature, saturation requires the nonlinear response in a; a field-independent Curie value then describes only the zero-field slope.

Let ${\mu }_a=2.13{\mu }_N$ be the magnitude of the saturated nuclear moment of helium-3, with ${\mu }_N=e\hslash /(2m_p)$. A spin-one-half nucleus has the two Zeeman energies $\mp {\mu }_{a}B$. For independent nuclei the one-site partition function and magnetization density areIn the weak-field limit ${\mu }_{a}B\ll k_{B}T$, expand the hyperbolic tangent:The nuclear moment's negative sign means that its moment points oppositely to its spin; it does not make the thermal magnetic susceptibility negative. The response is paramagnetism governed by Curie's law.

There is no extra factor $1/3$ here: the stated $2.13{\mu }_N$ is the moment of a fully polarized spin-one-half state. Equivalently, writing the moment operator as $g_N{\mu }_N\mathbf{I}$ requires $g_N=-4.26$, and $g_N^2{\mu }_N^{2}I(I+1)/3=(2.13{\mu }_N{)}^2$. The formula presumes independent localized nuclear moments; exchange correlations invalidate it close to the much lower ordering scale discussed in part (e).

For weak fields in a linear isotropic material, magnetization is $\mathbf{M}={\chi }_m\mathbf{H}$, with magnetic susceptibility ${\chi }_m$. Paramagnetism has ${\chi }_m>0$, while diamagnetism has ${\chi }_m<0$. In either case, $\mathbf{B}={\mu }_0(\mathbf{H}+\mathbf{M})={\mu }_0(1+{\chi }_m)\mathbf{H}$. These weak responses generally disappear when the applied field is removed, unlike hysteretic ferromagnetism.

A paramagnet can contain permanent magnetic moments from unpaired electrons or atomic angular momentum. Alignment lowers the energy $-\mathbf{\mu }\cdot \mathbf{B}$, while thermal disorder opposes it. For independent classical moments of magnitude $\mu$, their weak-field average projection is ${\mu }^{2}B/(3k_{B}T)$, soThis is Curie's law; it assumes weak magnetization and no important interactions between moments. With quantum angular momentum $J$, replace ${\mu }^2$ by $(g{\mu }_B{)}^{2}J(J+1)$. Strong-field magnetization saturates instead of growing linearly without limit. Conduction electrons also exhibit Pauli paramagnetism: only occupations near the Fermi surface readily redistribute between spin states, giving a nearly temperature-independent susceptibility when $T\ll T_F$.

Diamagnetism is an induced orbital response opposing the applied magnetic field. For bound electrons, a weak field changes orbital motion and induces an opposing magnetic moment. For an isotropic collection of atoms, the elementary orbital estimate iswhere the sum runs over the electrons of one atom. It is usually small and weakly temperature dependent. Delocalized electrons also have orbital Landau diamagnetism; for an ideal three-dimensional free electron gas with a quadratic isotropic dispersion and electron spin with $g=2$, ${\chi }_L=-{\chi }_P/3$. Both spin and orbital contributions must be included to determine the total response. Diamagnetism is not a consequence of permanent dipoles choosing to align against the field; that orientation would increase their energy.

At weak field or high temperature, ${\mu }_{B}B\ll k_{B}T$ and $\tanh x\simeq x$, soUsing $B\simeq {\mu }_{0}H$ for the applied-field response and neglecting demagnetizing corrections, the SI volume magnetic susceptibility isThis is Curie's law. If susceptibility is instead defined as $\partial M/\partial B$, its Curie constant is $n{\mu }_B^2/k_B$ without ${\mu }_0$; stating which field is used prevents a units error.

A magnetic field favors alignment of the magnetic moments of a paramagnet, reducing the number of substantially occupied magnetic configurations at fixed temperature. Its magnetic entropy therefore decreases as the field magnitude increases, explaining the negative sign of $(\partial S/\partial B{)}_T=-\alpha B/T^2$.

Choose the magnetic Helmholtz free energy convention $dF=-SdT-mdB$. Equality of its mixed partial derivatives gives the magnetic Maxwell relationThe magnetic moment inferred from an entropy derivative, integrated at fixed $B$, isThe given entropy derivative alone cannot determine the temperature-independent function $m_0(B)$. With the usual independent-paramagnet condition $m\to 0$ as $T\to \infty$ at fixed $B$, $m_0(B)=0$, giving Curie's law, $m=\alpha B/T$. The sketch uses that additional condition. The inverse-$T$ expression applies in the weak-field regime; its divergence cannot describe the actual low-temperature saturation of a finite set of magnetic moments.

For reversible quasistatic thermal isolation, $dS=0$. With positive field-dependent heat capacity $C_B=T(\partial S/\partial T{)}_B$,Reducing a positive $B$ therefore lowers $T$: this is adiabatic demagnetization. Equivalently, reducing $B^2$ cools for either field sign. Reduced alignment would raise magnetic entropy at fixed $T$; cooling compensates so total entropy remains constant. Insulation without reversibility would allow additional entropy production and need not follow this adiabat exactly.

Use the independent spin-one-half two-level system with fixed $N$ distinguishable sites and Hamiltonian levels $E_{\pm }=\pm \mu B$, where $\mu >0$ is the magnitude of the field-axis magnetic moment and $B>0$ is a uniform externally controlled magnetic field. Assume positive temperature, no interactions between sites, and no appreciable lattice or load entropy. The two levels differ by $2\mu B$. With inverse temperature $\beta$, the canonical ensemble gives the one-site and total partition functionsHere the minus subscript denotes the lower energy $-\mu B$. Factorization of the partition function requires additive energy and independent sites; there is no particle-exchange factor $N!$ for these fixed distinguishable sites.

Differentiate the partition function, holding $B$ fixed when differentiating with respect to $\beta$. The internal energy, Helmholtz free energy, total field-axis magnetic moment $m$, and magnetic entropy areThe extensive moment $m$ is distinct from magnetization, the moment per unit volume. Direct substitution of $p_{\pm }$ into the Gibbs entropy gives the same $s(x)$ independently of the free-energy derivative.

At fixed $B$, the magnetic heat capacity isStrict decrease makes $s$ invertible on $x>0$. Constant magnetic entropy therefore fixes $x$, not $B$ or $T$ separately:The partition function may be used to label these equilibrium states even though the isolated cooling leg has no external heat bath. Actual thermalization of the lattice and load additionally requires a sufficiently fast spin-lattice relaxation; perfectly uncoupled spins by themselves cannot refrigerate an attached object.

The limiting cases are instructive. For $x\ll 1$, $s(x)=\ln 2-x^2/2+O(x^4)$, $m\simeq N{\mu }^{2}B/(k_{B}T)$, and $C_{B,\mathrm{mag}}\simeq N{k}_B{x}^2$. The moment response reproduces Curie's law. For $x\gg 1$, $s(x)\simeq (1+2x)e^{-2x}$ and the upper-state occupation is exponentially small. Taking $B\to 0$ at fixed $T>0$ gives $s\to \ln 2$; taking $T\to 0$ at fixed $B>0$ gives $s\to 0$. The origin has a path-dependent limit. Extrapolating a line of constant $x$ to $B=T=0$ does not construct a usable zero-temperature state or prove that a real material retains independent degenerate spins there.

Volume magnetic susceptibility is the dimensionless ratio of longitudinal magnetization to internal magnetic field intensity in a linear regime through the origin. Positive response gives paramagnetism. Always distinguish magnetization $M=\mathcal{M}/V$ from total magnetic moment $\mathcal{M}$: dividing by volume changes the normalization. Mass-normalized and mole-normalized values likewise require the mass density or the volume per amount of substance before comparison with ${\chi }_v$.

In the existing Pauli paramagnetism formula, $M$ denotes total magnetic moment, and $g(E_F)$ is the total spin-summed density of states. Relabel that response asHere ${\mu }_B$ is the Bohr magneton. If its $B$ is a prescribed vacuum/applied magnetic flux density $B_0={\mu }_0{H}_{\mathrm{app}}$, the dimensionless apparent slope is ${\mu }_0{\chi }_B^{\mathrm{vol}}$. It agrees approximately with intrinsic ${\chi }_v$ only when the demagnetizing field correction and magnetic feedback are negligible. The existing Curie's law formula also uses a total-magnetic moment, $B$-derivative convention; its $N$ counts moments rather than giving a number density.