Magnetic Susceptibility

Magnetic susceptibility describes how magnetization changes when a magnetic field is varied. A Curie-like increase on cooling can reveal thermally active moments; a weakly temperature-dependent response can come from itinerant spins or virtual transitions; a diamagnetic onset can signal superconducting screening. The same measured curve can contain several of these contributions.

A bulk magnetometer first measures a specimen’s total moment. Converting that signal into a material response requires the sample amount, a field convention, and a background model. Sample shape matters too: the specimen creates its own magnetic field, which can change both the amplitude and the apparent time scale of an ac response. We develop these steps before interpreting characteristic temperature dependences.

Required background. How Quantum Matter Is Measured supplies detector calibration, sample-environment and background checks. Susceptibilities supplies source–detector normalization, tensor channels and the distinction between static and dynamic response protocols.

Magnetization and its response to a field

Moment, magnetization, and field

The direct sample-scale quantity is the magnetic dipole moment $\mathbf m$, measured in $\mathrm{A\,m^2}$ in SI. Magnetization is moment per volume,

$$\mathbf M=\frac{\mathbf m}{V}, \qquad [M]=\mathrm{A\,m^{-1}}.$$

The macroscopic SI fields obey

$$\mathbf B=\mu_0(\mathbf H+\mathbf M).$$

For a quasistatic measurement at a specified bias and on a specified branch, the differential volume susceptibility is

$$\chi_{ij} = \left. \frac{\partial M_i}{\partial H_{\mathrm{int},j}} \right|_{T,\mathrm{branch},\ldots}.$$

It is dimensionless in SI. This definition already exposes several choices: the response can be a tensor, the relevant field is the internal $\mathbf H_{\mathrm{int}}$, and a dynamic or hysteretic response depends on protocol. The secant ratio $M/H$ equals this slope only for a response that is linear through the origin. A small ac perturbation instead measures a complex, frequency-dependent slope; it need not equal the equilibrated dc derivative.

Experimental papers also report:

• mass susceptibility $\chi_{\mathrm{mass}}$, in $\mathrm{m^3\,kg^{-1}}$;
• molar susceptibility $\chi_{\mathrm{mol}}$, in $\mathrm{m^3\,mol^{-1}}$;
• moment per formula unit, often in $\mu_B/\mathrm{f.u.}$;
• moment per magnetic ion, which additionally requires site occupancy and composition;
• apparent susceptibility formed with the applied rather than internal field.

If a specimen contains $n_{\mathrm{mol}}$ moles of formula units,

$$\frac{\text{moment per formula unit}}{\mu_B} = \frac{m} {n_{\mathrm{mol}}N_A\mu_B}.$$

The formula is simple; the chemical amount may not be. Hydration, flux inclusions, vacancies, mixed occupancy, oxidation, and substrate mass can dominate the uncertainty in a small specimen.

Unit conversions are part of the result

Legacy magnetic literature frequently uses emu, Oe, and cgs molar susceptibilities. Using the conventional magnetic-unit conversions tabulated by NIST,

$$\begin{aligned} 1\ \mathrm{emu\ of\ moment}&=10^{-3}\ \mathrm{A\,m^2},\\ 1\ \mathrm{emu\,cm^{-3}}&=10^3\ \mathrm{A\,m^{-1}},\\ \chi_{\rm mol}^{\rm SI}&=4\pi\times10^{-6}\chi_{\rm mol}^{\rm cgs}. \end{aligned}$$

The final line compares numerical molar susceptibilities expressed in $\mathrm{m^3\,mol^{-1}}$ (SI) and $\mathrm{cm^3\,mol^{-1}}$ (cgs). The factor $4\pi$ converts the susceptibility convention, in addition to the change of volume units. Here and in numerical examples, $\mu_0\simeq4\pi\times10^{-7}\ \mathrm{H\,m^{-1}}$ is sufficient; it is no longer an exact defining constant in the revised SI (Goldfarb, 2018). A table headed only “emu/mol” is incomplete because it may denote moment per field, moment at a stated field, or an older susceptibility convention. Preserve the original quantity and unit before converting it.

Linear response needs a field window

At each temperature, inspect $M(H)$ rather than assuming that a single low-field point lies in a linear equilibrium regime. A useful local model is

$$M(H_{\rm int})=M_0+\chi_1H_{\rm int} +\chi_2H_{\rm int}^2+\chi_3H_{\rm int}^3+\cdots.$$

In a time-reversal-symmetric paramagnet at zero bias, equilibrium oddness gives $M(-H)=-M(H)$ and removes the even terms. A remanent field, a chosen ferromagnetic branch, exchange bias, or an acquisition offset can spoil this pattern. Nonlinearity alone does not: an equilibrium $\chi_3H^3$ term is still odd. Field reversal separates odd and even contributions and is often more informative than collecting a denser one-direction sweep.

From Instrument Record to Sample Moment

SQUID magnetometry

A superconducting quantum interference device is an exceptionally sensitive flux detector. In a common moving-sample magnetometer, the specimen travels through a gradiometer pickup coil. The instrument records flux versus position and fits a calibrated response function to infer a dipole moment. The SQUID detects flux; susceptibility is inferred from the calibrated moment and field dependence (Clarke and Braginski, 2004).

Important controls include:

• center the specimen and inspect the entire scan residual, not only the fitted moment;
• use a holder whose signal is small, stable, and measured with the same mounting geometry;
• test whether sample length or off-axis position invalidates the point-dipole response;
• check the superconducting magnet’s remanent field near nominal zero;
• record approach direction, settling time, scan speed, and field history;
• repeat weak signals after remounting, because a tiny ferromagnetic contaminant can follow the holder rather than the crystal.

A fitted moment can look smooth even when the scan shape is wrong. Comparing full scans near an apparent transition helps distinguish a change of moment from a change in centering or sample motion.

Vibrating-sample magnetometry

In a vibrating-sample magnetometer, a periodic sample displacement modulates the flux through stationary pickup coils. For a small dipole and fixed coil geometry, the induced voltage amplitude scales schematically as

$$V_{\mathrm{ind}} \propto \omega a\,m\, \frac{dG}{dz},$$

where $a$ and $\omega$ are the vibration amplitude and angular frequency, and $G(z)$ is the pickup-flux coupling per unit moment. This is the small-displacement limit of Faraday induction; calibration supplies the actual coil response (Foner, 1959). Position, vibration amplitude, sample shape, vibration-induced heating, and mechanical background therefore belong in the uncertainty budget. VSMs are especially useful for rapid field loops, but speed does not remove sweep-rate dependence from a relaxing sample.

AC susceptibility

An ac measurement resolves the moment in phase and in quadrature with a small field. Let the drive below be the applied field along one principal axis:

$$\begin{aligned} H_a(t)&=H_{\rm dc}+h_a\cos\omega t, \\ M(t)-M_0&=h_a\chi_{\rm app}'\cos\omega t +h_a\chi_{\rm app}''\sin\omega t. \end{aligned}$$

Thus $\widetilde\chi_{\rm app}=\chi_{\rm app}'+i\chi_{\rm app}''$ for an $e^{-i\omega t}$ convention. A positive quadrature means that the magnetization lags the field. In the magnetoquasistatic, uniform-response model, the work dissipated per volume in a steady cycle is

$$W=\mu_0\oint H_a\,dM =\mu_0\pi h_a^2\chi_{\rm app}''.$$

One can verify the sign directly: multiply $H_a(t)$ by $dM/dt$ and integrate over one period. The term containing $\cos\omega t\sin\omega t$ vanishes, while the $\cos^2\omega t$ term is positive. For an ellipsoid, using $H_{\rm int}=H_a-NM$ gives the same closed-cycle work because $\oint NM\,dM=0$. However, the internal and applied drive amplitudes and susceptibilities differ; they must not be mixed in the loss formula.

The in-phase response is the reactive component, not necessarily the equilibrium or reversible dc susceptibility. Domain walls, spin freezing, particle blocking, eddy currents, and vortices can all produce loss. Vary frequency and amplitude separately, and check phase calibration. An amplitude-dependent peak can indicate nonlinear response or drive-induced heating; neither establishes a unique equilibrium transition.

Higher harmonics also contain information. In an instantaneous odd response $M=\chi_1H+\chi_3H^3$, a sinusoidal field produces a third harmonic of amplitude $\chi_3h_a^3/4$. Real dynamic nonlinear susceptibilities can have their own phases, and an instrument harmonic must be excluded before assigning the signal to the specimen.

Zero-field-cooled and field-cooled protocols

“Zero field” means the best characterized residual field, not the current setting printed as zero. A reproducible protocol states:

1. how the magnet was degaussed or reset;
2. the cooling field and cooling rate;
3. when the measurement field was applied;
4. whether data were taken on warming or cooling;
5. waiting and averaging times;
6. whether the same field sequence was used for holder and sample.

In a zero-field-cooled (ZFC) sequence, the specimen is cooled in nominal zero field, the measurement field is applied at low temperature, and data are usually taken on warming. “Field cooled” is incomplete unless it distinguishes cooling and warming branches. ZFC–FC splitting records irreversibility on the experimental time scale. It can arise from ferromagnetic domains, vortices, spin glasses, superparamagnetic particles, structural hysteresis, or instrument remanence.

Geometry and Demagnetizing Fields

A magnetized body generates a field that opposes its magnetization. For a uniformly magnetized ellipsoid along a principal axis,

$$H_{\mathrm{int}} = H_{\mathrm{appl}}-NM, \qquad 0\leq N\leq1,$$

with $N_x+N_y+N_z=1$ in SI. Combining this with $M=\chi H_{\mathrm{int}}$ gives

$$\chi_{\mathrm{app}} \equiv \frac{M}{H_{\mathrm{appl}}} = \frac{\chi}{1+N\chi},$$

and therefore

$$\frac{1}{\chi_{\mathrm{app}}} = \frac{1}{\chi}+N, \qquad \chi = \frac{\chi_{\mathrm{app}}} {1-N\chi_{\mathrm{app}}}.$$

The correction is negligible for many weak paramagnets because $|N\chi|\ll1$. It is essential near a ferromagnetic divergence and for strong diamagnetic screening. The uncertainty in $N$ should be propagated when the denominator approaches zero.

An ellipsoid has a uniform demagnetizing field under uniform magnetization (Osborn, 1945). A rectangular prism, platelet, cylinder, porous pellet, or irregular crystal has a spatially varying field. Tabulated factors for such shapes depend on the averaging and response model. Aharoni (1998) treats a uniformly magnetized prism; factors for nearly perfect diamagnetic screening require a different field solution (Prozorov and Kogan, 2018). Record dimensions and field orientation; quote the convention used; and do not report more precision in $N$ than the shape and alignment warrant.

For an ideal fully screening superconductor in SI,

$$B=0 \quad\Longrightarrow\quad \chi=-1$$

with respect to the internal $H$ field. The apparent susceptibility is instead

$$\chi_{\mathrm{app}} = -\frac{1}{1-N}.$$

A thin platelet measured perpendicular to its face can therefore have an apparent shielding signal whose magnitude is much larger than one. Calling that “more than 100% superconducting volume” confuses geometry with phase fraction.

Shape can shift an ac loss peak

For the same scalar ellipsoidal model, the correction acts on the complex susceptibility:

$$\widetilde\chi_{\rm app} =\frac{\widetilde\chi}{1+N\widetilde\chi}, \qquad \chi_{\rm app}'' =\frac{\chi''}{|1+N\widetilde\chi|^2}.$$

Correcting only the real part discards this coupling between the two quadratures. Its importance in measured spin-ice spectra is illustrated by Quilliam and collaborators (2011).

As a solvable example, suppose the internal response has one relaxation time, no instantaneous term, and positive static susceptibility $\chi_s$:

$$\tau\dot M+M=\chi_sH_{\rm int}, \qquad \widetilde\chi=\frac{\chi_s}{1-i\omega\tau}.$$

Substitution of $H_{\rm int}=H_a-NM$ gives

$$\tau\dot M+(1+N\chi_s)M=\chi_sH_a.$$

The measured response therefore has

$$\begin{aligned} \chi_{{\rm app},s}&=\frac{\chi_s}{1+N\chi_s},\\ \tau_{\rm app}&=\frac{\tau}{1+N\chi_s},\\ \omega_{\rm peak}\tau&=1+N\chi_s. \end{aligned}$$

Even at fixed temperature, equal intrinsic relaxation times can give unequal measured peak frequencies. This example assumes uniform scalar response, a frequency-independent geometric factor, and negligible skin effects. For an anisotropic specimen, tensor response and demagnetization must be solved together; the scalar fraction generally does not apply to an arbitrary direction.

One material model, three geometries: $\chi_s=3$ and the intrinsic $\tau$ are fixed. Dividing each measured loss by its own static apparent susceptibility makes the peak heights equal, but the peaks occur at $\omega\tau=1,2,3$ for $N=0,1/3,2/3$. The $N=0$ curve is the limiting geometry with no demagnetizing field along the drive. A frequency shift alone need not imply a changed microscopic relaxation rate.

Backgrounds, Normalization, and Uncertainty

The measured moment is a sum:

$$m_{\rm meas}=m_{\rm sample}+m_{\rm holder} +m_{\rm substrate}+m_{\rm grease} +m_{\rm contamination}+m_{\rm instrument}.$$

Background subtraction is reliable only when the subtracted object reproduces the geometry and history of the measured assembly. A linear diamagnetic substrate can be estimated from high field, but that procedure fails if the sample itself has a large linear term. A tiny ferromagnetic particle produces a saturating offset that can dominate a micrometre-scale film. Oxygen, steel tools, nickel coatings, magnetic dust, and Kapton or grease batches deserve controls.

Cullity and Graham (2008) discuss these magnetometry and normalization issues. For films and heterostructures, report both moment per area and the assumptions used to convert to a volume magnetization. A nominal thickness is not automatically the magnetic thickness. For crystals, record mass, dimensions, density source, composition, and orientation. Replicate measurements on an empty holder and, when feasible, on multiple sample masses: an intrinsic signal scales with sample amount, whereas many apparatus backgrounds do not.

A compact measurement model is

$$\chi_{\mathrm{mol}} = \frac{1}{n_{\mathrm{mol}}} \frac{\partial} {\partial H_{\mathrm{int}}} \left( m_{\mathrm{meas}}-m_{\mathrm{bg}} \right).$$

Its uncertainty combines moment calibration, fit residuals, field calibration, background model, amount, dimensions, orientation, temperature, and protocol reproducibility. These contributions are often correlated. For example, subtracting a fitted constant $\chi_0$ changes both the Curie constant and the Curie–Weiss temperature.

Pauli Susceptibility

Noninteracting benchmark

An itinerant Fermi system responds because a magnetic field shifts the energies of opposite spin projections. Let $D(E_F)$ be the total noninteracting density of states at the Fermi energy, including both spin species, per volume and energy. For an isotropic Zeeman coupling with effective $g$ factor,

$$\chi_{\mathrm{P}} = \mu_0 \frac{(g\mu_B)^2}{4} D(E_F).$$

For $g=2$, this becomes $\chi_{\mathrm P}=\mu_0\mu_B^2D(E_F)$. The formula is a benchmark, not a universal decomposition. Real materials can have anisotropic $g$ tensors, spin–orbit-entangled bands, multiple Fermi surfaces, exchange enhancement, and orbital response of comparable magnitude.

The corresponding low-temperature electronic heat-capacity coefficient is

$$\gamma_V = \frac{\pi^2k_B^2}{3} D(E_F).$$

This motivates a generalized Wilson ratio

$$R_W = \frac{\pi^2k_B^2} {3\mu_0(g\mu_B/2)^2} \frac{\chi_{\mathrm{spin}}}{\gamma_V}.$$

The noninteracting benchmark is $R_W=1$ when the same degrees of freedom, normalization, and $g$ factor enter both quantities. Before quoting $R_W$, subtract non-spin terms from $\chi$, use either molar quantities for both numerator and denominator or volume quantities for both, and state the assumed $g$. An enhanced ratio can reflect ferromagnetic correlations, but uncertainty in orbital response or crystal-field anisotropy can imitate the enhancement.

Interaction enhancement

For an isotropic, spin-rotation-invariant Landau Fermi liquid, the quasiparticle density of states $D^*(E_F)$ determines $\gamma_V$, while the uniform spin response also contains the interaction parameter $F_0^a$:

$$\begin{aligned} \chi_{\rm spin} &=\frac{\mu_0(g\mu_B/2)^2D^*(E_F)}{1+F_0^a},\\ R_W&=\frac{1}{1+F_0^a}. \end{aligned}$$

The stable paramagnetic regime requires $1+F_0^a>0$. Simply inserting an enhanced effective mass into the free-particle Pauli formula omits this spin-channel interaction. Strong spin–orbit coupling or multiple unequal bands requires more information than this single-parameter description; see Fermi-Liquid Theory.

What a flat susceptibility can mean

A weakly temperature-dependent positive susceptibility is compatible with Pauli paramagnetism, but it can also contain Van Vleck orbital mixing, core diamagnetism, and a broad exchange or coherence crossover. A useful fit model is

$$\chi(T)=\chi_{\rm core}+\chi_{\rm Landau}+\chi_{\rm VV} +\chi_{\rm P}(T)+\chi_{\rm imp}(T).$$

Calling the fitted constant $\chi_0$ “the Pauli susceptibility” silently assigns all temperature-independent terms to spin. Band calculations, anisotropy, NMR Knight shifts, and heat capacity can constrain that assignment. Stoner Criterion develops the exchange-enhanced spin response and its limits.

Curie and Curie–Weiss Regimes

Local moments

For one independent isotropic moment per formula unit, with all states of its multiplet thermally sampled and weak Zeeman splitting (Blundell, 2001),

$$\chi_{\mathrm{mol}}(T) = \frac{C}{T}, \qquad C = \frac{\mu_0N_A\mu_{\mathrm{eff}}^2} {3k_B}.$$

Thus an SI molar Curie constant gives

$$\frac{\mu_{\mathrm{eff}}}{\mu_B} = \left( \frac{3k_BC} {\mu_0N_A\mu_B^2} \right)^{1/2}.$$

For an isolated angular-momentum multiplet with quantum number $J$ and Landé factor $g_J$,

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

With $f$ identical moments per formula unit, $C$ acquires a factor $f$; the inferred moment per ion is then divided by $\sqrt f$. Several species contribute through a sum of squared effective moments. This effective moment is a thermal fluctuation scale, not the saturated moment $g_JJ\mu_B$. Confusing the two gives a systematic mismatch even for an ideal ion.

Interactions are often summarized by the Curie–Weiss form

$$\chi_{\mathrm{mol}}(T) = \chi_0 + \frac{C}{T-\Theta_{\mathrm{CW}}}.$$

Here $\Theta_{\mathrm{CW}}$ is a model-dependent intercept. In a simple local-moment mean-field model it reflects a weighted exchange scale, but its sign does not uniquely determine the eventual ordered state when interactions compete, sublattices are inequivalent, or anisotropic and long-range couplings matter.

Choose a physically consistent fit window

A straight-looking $1/\chi$ plot is not sufficient. If $\chi_0\neq0$, the linear quantity is

$$\frac{1}{\chi_{\mathrm{mol}}-\chi_0} = \frac{T-\Theta_{\mathrm{CW}}}{C},$$

and $\chi_0$, $C$, and $\Theta_{\mathrm{CW}}$ can be strongly covariant. Fit the untransformed $\chi(T)$ with uncertainties when possible, because inversion makes errors heteroscedastic and visually suppresses small systematic residuals.

A simple Curie–Weiss window should suppress the relevant exchange, Kondo, ordering, and blocking corrections while sampling a well-defined set of local levels. It may lie above the splittings of a full multiplet, or below the first excited crystal-field level but above interactions within a ground-state doublet. Those two windows need not give the same effective moment or anisotropy. A low-lying excited multiplet can curve $1/\chi$ without any change in exchange. Temperature-independent terms matter most at the high-temperature end, exactly where a broad fit may seem most convincing.

Useful robustness tests are:

• vary both lower and upper fit bounds and plot parameter drift;
• fit crystallographic directions separately before averaging;
• compare $C$ with the known magnetic-ion fraction and valence;
• inspect $M(H,T)$ for saturation, impurity tails, and nonlinearity;
• compare $\Theta_{\mathrm{CW}}$ with ordering, heat-capacity, and spectroscopic scales;
• report the full parameter covariance, not only marginal standard errors.

A large ratio $|\Theta_{\mathrm{CW}}|/T_{\mathrm{order}}$ is often called a frustration parameter. It is a useful screening metric only when the Curie–Weiss model is valid and $T_{\mathrm{order}}$ is independently established. Low dimensionality, anisotropy, disorder, Kondo screening, and a poor fit can also suppress or obscure ordering.

Diamagnetism and Orbital Response

Core, Landau, and Van Vleck terms

Closed electron shells generate core diamagnetism. Tabulated ionic increments can provide a rough subtraction, but covalency and solid-state charge distribution limit their accuracy. Conduction electrons also have orbital diamagnetism. For an ideal three-dimensional free-electron gas with orbital mass $m^*=m_e$ and $g=2$,

$$\chi_{\mathrm{Landau}} = -\frac{1}{3}\chi_{\mathrm{P}}.$$

The mass assumption matters: for an isotropic noninteracting parabolic band with orbital mass $m^*$ and spin moment $g\mu_B/2$, the ratio is instead

$$\frac{\chi_{\mathrm{Landau}}}{\chi_{\mathrm P}} = -\frac{1}{3} \left(\frac{2m_e}{g m^*}\right)^2.$$

Neither ratio is a general rule for interacting, multiband, low-dimensional, Dirac, flat-band, or strong spin–orbit systems. Near band degeneracies, orbital susceptibility can be large, anisotropic, and nonanalytic.

Virtual transitions between crystal-field or spin–orbit-coupled states generate Van Vleck susceptibility. For one nondegenerate ion per formula unit, with excited levels thermally unoccupied, the molar response along direction $\alpha$ contains (Van Vleck, 1932)

$$\chi_{\mathrm{VV},\alpha}^{\mathrm{mol}} = 2\mu_0N_A \sum_{n\neq0} \frac{ \left| \langle n|\hat{\mu}_{\alpha}|0\rangle \right|^2 } {E_n-E_0}.$$

It is positive in this elementary form and can remain nearly temperature independent when the excited states are well above $k_BT$. Degeneracies, thermal populations, exchange, and band formation require a fuller treatment. Most importantly, a constant positive term need not come from a Fermi surface.

Superconducting screening

Diamagnetic onset is a powerful superconductivity diagnostic when its geometry and history are controlled. ZFC screening tests whether magnetic flux is excluded from much of the specimen after the field is applied at low temperature. Field-cooled magnetization probes flux expulsion under a different history and is often reduced by pinning. These are not interchangeable measurements.

To interpret a superconducting onset, compare (Tinkham, 1996):

• field magnitude, direction, remanent-field procedure, and ZFC/FC sequence;
• raw moment, sample amount, density or dimensions, and demagnetizing correction;
• onset criterion and transition width;
• field dependence relative to lower critical and irreversibility scales;
• whether the signal scales with sample amount and survives holder controls;
• complementary zero resistance, heat capacity, local-field, microwave, or diffraction evidence as appropriate.

The ZFC shielding magnitude is not a direct superconducting volume fraction in a platelet, porous pellet, powder, granular composite, or specimen whose dimensions are comparable to the penetration depth. Conversely, a small field-cooled signal does not imply a small superconducting volume when vortex pinning is strong.

Ordered Phases

Ferromagnets

A ferromagnet is not identified merely by an upturn in $\chi(T)$. Bulk evidence includes a spontaneous equilibrium magnetization below the transition, anisotropy-consistent behavior, and reproducible field dependence after geometry correction. Real low-field measurements are dominated by domain nucleation, wall motion, pinning, and history. The measured susceptibility can therefore be very large, nonlinear, frequency dependent, and dissipative.

Record complete $M(H)$ loops with sweep rate and maximum-field history. Subtract holder and linear backgrounds without erasing the high-field approach to saturation. A small hysteretic component can come from a dilute ferromagnetic impurity even when the host is paramagnetic. Scaling of saturation moment with sample amount, composition, and impurity concentration is often decisive. Ferromagnetism explains spontaneous symmetry breaking, domains, and Arrott analysis.

Antiferromagnets and ferrimagnets

The uniform susceptibility of an antiferromagnet commonly shows a cusp or slope change near the Néel temperature, but the shape depends strongly on anisotropy, dimensionality, domain population, and field direction. Short-range correlations can produce a broad maximum above the ordering temperature. A uniform field couples primarily to the net moment, not directly to the staggered order parameter. Antiferromagnetic order may enlarge the crystallographic unit cell, or occur between sites within it with propagation vector zero; a finite propagation vector is not required.

For some simple antiferromagnets, Fisher’s (1962) relation motivates comparing the magnetic heat capacity with

$$\frac{d(\chi T)}{dT}.$$

This can sharpen a transition estimate, but it is not a universal identity for arbitrary anisotropic, itinerant, frustrated, or multi-sublattice magnets. Diffraction or a local magnetic probe is needed to establish the magnetic structure. Antiferromagnetism develops the staggered order parameter and its direct signatures.

Ferrimagnets contain oppositely aligned sublattices with unequal moments. Their inverse susceptibility can curve substantially, and a compensation temperature can drive the net moment through zero without destroying sublattice order. A one-component Curie–Weiss fit can therefore return parameters with little microscopic meaning. Ferrimagnetism explains the two-sublattice physics.

Glassiness, Blocking, and Slow Dynamics

A cusp in ac susceptibility that shifts with frequency, together with ZFC–FC splitting, signals slow magnetic dynamics. It does not by itself distinguish a collective spin glass from superparamagnetic blocking, domain-wall freezing, cluster glassiness, or vortex dynamics.

A compact empirical frequency-shift metric is

$$K_f = \frac{\Delta T_f} {T_f\,\Delta\log_{10}f},$$

where $T_f$ is a consistently defined peak or inflection temperature. Numerical ranges quoted for “canonical” spin glasses and superparamagnets overlap across materials and frequency windows, so $K_f$ is a comparator rather than a phase criterion.

For a noninteracting single-domain particle with anisotropy barrier $E_b=K_uV$, the simplest blocking model gives

$$\tau(T) = \tau_0 \exp\left( \frac{K_uV}{k_BT} \right).$$

Here $K_u$ is an anisotropy energy density, unrelated to $K_f$. The observed blocking temperature solves $\tau(T_B)\sim t_{\mathrm{obs}}$ and therefore depends on measurement time. A distribution of sizes produces broad peaks and relaxation. Interactions can make even this description inadequate.

A collective-glass claim should combine frequency and amplitude sweeps with aging, memory, nonlinear susceptibility, relaxation scaling, and a structural account of disorder. Local probes or scattering can test whether static moments develop throughout the sample. Glasses and Spin Glasses develops collective freezing and its relation to these experiments; Mydosh (1993) discusses the measurement signatures.

Spin-Liquid and Heavy-Fermion Signatures

Spin-liquid-compatible behavior

A frustrated magnet may show a Curie–Weiss regime at high temperature, broad susceptibility features as short-range correlations develop, and no sharp ordering anomaly down to the lowest measured temperature. Those observations are compatible with a quantum spin liquid, but they are also compatible with weak ordering below the base temperature, disorder, random singlets, glassiness, or a valence-bond state.

Low-temperature Curie tails are especially dangerous:

$$\chi(T) = \chi_{\mathrm{intrinsic}}(T) + \frac{C_{\mathrm{imp}}}{T-\Theta_{\mathrm{imp}}}.$$

A small fraction of orphan spins can dominate the bulk response while contributing little to the intrinsic correlations. Constrain that fraction with field dependence, saturation scale, composition, NMR line shapes, muon response, and diffuse scattering. Susceptibility can establish active moments, anisotropy, interaction scales, and absence of a bulk uniform anomaly within a stated window. It cannot by itself establish long-range entanglement or fractionalization. Quantum Spin Liquids and Balents (2010) explain what further observations can distinguish these states.

Heavy-fermion-compatible behavior

Many $f$-electron materials cross from high-temperature local-moment Curie–Weiss behavior to a lower-temperature Kondo or lattice-coherence regime. The low-temperature susceptibility can become large and Pauli-like while the heat-capacity coefficient is also strongly enhanced. A broad maximum or anisotropic crossover may mark crystal-field depopulation, Kondo screening, intersite correlations, or coherence; susceptibility alone rarely separates them.

Compare susceptibility with the electronic $C/T$, transport, and spectroscopy on the same specimen. Enhanced $\chi$ and $\gamma$ together are more informative than either alone, but crystal-field and impurity contributions still need separation (Stewart, 1984). Heavy Fermions and Kondo Lattices develop the microscopic interpretation.

Which measurement resolves the ambiguity?

Magnetic observation	Main ambiguity	Useful comparison
Curie–Weiss-looking response	Background or changing crystal-field population	Fit-window stability, composition, spectroscopy
Small hysteretic component	Bulk order or dilute contamination	Sample-amount scaling, diffraction, local magnetic probe
Cusp or broad maximum	Ordering, short-range correlations, or slow response	Heat capacity, direction dependence, frequency sweeps
Diamagnetic onset	Screening geometry versus superconducting fraction	Penetration-depth scale, heat capacity, local fields
Frequency-shifting loss peak	Shape, blocking, or collective freezing	Complex demagnetization correction, aging, amplitude sweeps
Large low-temperature response	Heavy quasiparticles, orbital terms, or dilute moments	Electronic heat capacity, field dependence, spectroscopy
No resolved ordering anomaly	Disordered phase or an unresolved transition	Local fields, magnetic scattering, lower temperatures

Choose the comparison that separates the competing explanations in the particular sample. None of these rows supplies a phase diagnosis by itself.

Exercises

Exercise 1: Convert a molar susceptibility

A paper reports $\chi_{\mathrm{mol}}^{\mathrm{cgs}}=3.20\times10^{-3}\ \mathrm{cm^3\,mol^{-1}}$. Convert it to SI molar susceptibility.

Solution

Use the molar conversion including the $4\pi$ factor:

$$\begin{aligned} \chi_{\rm mol}^{\rm SI}&=4\pi(3.20)\times10^{-9}\\ &=4.02\times10^{-8}\ \mathrm{m^3\,mol^{-1}}. \end{aligned}$$

The numerical value is not $3.20\times10^{-9}$ because susceptibility conventions, not only cubic centimetres to cubic metres, contribute to the conversion.

Exercise 2: Correct a platelet’s apparent susceptibility

A platelet measured perpendicular to its face has $N=0.70$ and $\chi_{\mathrm{app}}=-2.50$. Find the internal-field susceptibility in the ellipsoidal approximation. Is the result compatible with nearly complete superconducting screening?

Solution

Apply

$$\chi = \frac{\chi_{\mathrm{app}}} {1-N\chi_{\mathrm{app}}}.$$

Therefore

$$\begin{aligned} \chi&=\frac{-2.50}{1-(0.70)(-2.50)}\\ &=\frac{-2.50}{2.75}=-0.909. \end{aligned}$$

The corrected value is close to the ideal SI value $-1$. The apparent magnitude larger than one is a geometry effect, not evidence for a superconducting fraction above $100\%$. A real platelet is not an ellipsoid, so uncertainty in the effective $N$ and finite penetration depth still matter.

Exercise 3: Extract an effective moment

An SI Curie–Weiss fit gives

$$C=1.26\times10^{-5}\ \mathrm{m^3\,K\,mol^{-1}}.$$

Find $\mu_{\mathrm{eff}}/\mu_B$. Compare it with a spin-only $S=1$ ion with $g=2$.

Solution

The Curie constant produced by one Bohr magneton squared is

$$\frac{\mu_0N_A\mu_B^2}{3k_B} \simeq 1.57\times10^{-6} \ \mathrm{m^3\,K\,mol^{-1}}.$$

Hence

$$\frac{\mu_{\mathrm{eff}}}{\mu_B} = \sqrt{ \frac{1.26\times10^{-5}} {1.57\times10^{-6}} } \simeq 2.83.$$

For $S=1$ and $g=2$,

$$g\sqrt{S(S+1)} = 2\sqrt{2} \simeq 2.83.$$

The Curie constant is therefore consistent with one such moment per formula unit, subject to occupancy, orbital, and fit-window checks. The corresponding saturation moment would be $gS\mu_B=2\mu_B$, not $2.83\mu_B$.

Exercise 4: Diagnose a neglected constant background

Suppose the true susceptibility is

$$\chi(T)=3.0\times10^{-9}+\frac{1.26\times10^{-5}}{T+20} \mathrm{m^3\,mol^{-1}}.$$

Using only the points at $100\ \mathrm K$ and $300\ \mathrm K$, estimate the apparent Curie–Weiss temperature obtained by forcing $1/\chi$ to be linear with no $\chi_0$ term.

Solution

At the two temperatures,

$$\begin{aligned} \chi(100\ \mathrm K) &= 1.08\times10^{-7} \ \mathrm{m^3\,mol^{-1}},\\ \chi(300\ \mathrm K) &= 4.2375\times10^{-8} \ \mathrm{m^3\,mol^{-1}}. \end{aligned}$$

Their inverses are approximately $9.26\times10^6$ and $2.36\times10^7\ \mathrm{mol\,m^{-3}}$. A line through these points has slope

$$a \simeq 7.17\times10^4 \ \mathrm{mol\,m^{-3}\,K^{-1}}$$

and intercept $b\simeq2.09\times10^6\ \mathrm{mol\,m^{-3}}$. Written as $a(T-\Theta_{\mathrm{app}})$, it gives

$$\Theta_{\mathrm{app}} = -\frac{b}{a} \simeq -29\ \mathrm K.$$

The true denominator contains $T+20$, so the true Curie–Weiss temperature is $-20\ \mathrm K$. Neglecting a small positive constant biases the inferred intercept by almost $50\%$ in this two-point example.

Exercise 5: AC loss with an applied-field convention

An ac experiment uses an applied amplitude $h_{\mathrm{ac}}=80\ \mathrm{A\,m^{-1}}$ at $f=1.0\ \mathrm{kHz}$ and measures the apparent quadrature $\chi''_{\rm app}=0.030$ under the convention defined above. Find the energy dissipated per unit volume per cycle and the average dissipated power density.

Solution

The loss per cycle is

$$\begin{aligned} W_{\mathrm{cycle}} &= \mu_0\pi h_{\mathrm{ac}}^2\chi''_{\rm app}\\ &= \left( 4\pi\times10^{-7} \right) \pi(80)^2(0.030)\\ &\simeq 7.58\times10^{-4} \ \mathrm{J\,m^{-3}}. \end{aligned}$$

Multiplying by cycles per second,

$$\frac{P}{V} = fW_{\mathrm{cycle}} \simeq 0.758\ \mathrm{W\,m^{-3}}.$$

This is the sample’s magnetic loss density under the linear-response convention. Coil and eddy-current backgrounds still require independent subtraction.

Exercise 6: Compare spin response and heat capacity

Explain why it is invalid to combine a molar susceptibility that still contains core diamagnetism with a volumetric heat-capacity coefficient in a Wilson ratio. Explain how to make the comparison physically meaningful.

Solution

The ratio is dimensionless only when susceptibility and $\gamma$ refer to the same amount convention. Molar $\chi$ divided by volumetric $\gamma_V$ leaves an unphysical molar-volume factor. Core diamagnetism also does not arise from the low-energy quasiparticles counted by $\gamma$, so retaining it contaminates the spin response.

A consistent comparison uses the following choices:

1. convert both observables to molar quantities or both to volume quantities using a documented molar volume;
2. estimate and subtract core, Landau, Van Vleck, impurity, and ordered contributions to isolate $\chi_{\mathrm{spin}}$ as far as the data permit;
3. use the normal-state electronic $\gamma$ for the same specimen and regime;
4. state the effective $g$ factor and its direction;
5. propagate the subtraction and normalization uncertainties.

These steps remove avoidable biases. Any remaining deviation must still be interpreted within a model appropriate to the spin and band structure.

Exercise 7: What does the missing anomaly establish?

A frustrated insulator has $\Theta_{\mathrm{CW}}=-120\ \mathrm K$, no sharp anomaly in $\chi(T)$ above $2\ \mathrm K$, and a low-temperature Curie tail. The authors call it a quantum spin liquid. What can the susceptibility establish, and what measurements are needed before the phase claim is credible?

Solution

The data can establish a high-temperature antiferromagnetic interaction scale if the Curie–Weiss window and crystal-field assumptions are valid. They can also establish the absence of a resolved bulk uniform-susceptibility anomaly above $2\ \mathrm K$. They do not distinguish a quantum spin liquid from ordering below $2\ \mathrm K$, weak or broadened order, a spin glass, random singlets, a valence-bond state, or disorder.

The Curie tail should first be tested against a dilute-moment model using field dependence and saturation, chemical analysis, and local line shapes. Heat capacity should test for bulk anomalies and entropy release. Muon or nuclear-resonance measurements should test for static internal fields and heterogeneous freezing. Neutron scattering should search for magnetic Bragg peaks and characterize diffuse or continuum response. AC susceptibility, aging, and memory should test glassiness. Measurements should extend below the lowest inferred intrinsic scale and be repeated across specimens.

A defensible intermediate statement is “a frustrated magnet with no detected long-range order above $2\ \mathrm K$ and a low-temperature response requiring impurity or intrinsic modeling.” Quantum spin liquid remains a hypothesis until complementary measurements distinguish the principal alternatives.

Exercise 8: Extract the intrinsic relaxation time

At one temperature, an ellipsoidal specimen with $N=1/3$ has static apparent susceptibility $\chi_{{\rm app},s}=1.5$ and an ac-loss peak at $f_p=200\ \mathrm{Hz}$. Assume the single-relaxation model above. Find $\chi_s$ and $\tau$. Why does $1/(2\pi f_p)$ underestimate the intrinsic time?

Solution

The static correction gives

$$\chi_s=\frac{1.5}{1-(1/3)(1.5)}=3.$$

The measured peak determines $\tau_{\rm app}=1/(2\pi f_p)=0.796\ \mathrm{ms}$, so

$$\tau=(1+N\chi_s)\tau_{\rm app} =1.59\ \mathrm{ms}.$$

The demagnetizing field supplies an additional restoring response in the equation for $M(t)$. Ignoring it assigns this geometric effect to a faster intrinsic relaxation. For a loss spectrum with several relaxation times, this one-peak inversion need not apply.

References

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