Magnetic Moments in Matter

A magnetic moment quoted for a material is not one context-free number. A Curie–Weiss fit, a high-field magnetization curve, magnetic diffraction, and an energy-integrated spectrum answer different operator, state, and resolution questions. Their answers can disagree even when every measurement and model is correct.

This page owns the material moment and projection ledger. It connects a physical magnetic-moment operator to a retained crystal-field, orbital, band, or pseudospin subspace; separates local, Curie-effective, saturation, ordered, and fluctuating moments; and decides whether a fixed-length low-energy moment is justified. It does not rederive atomic Landé algebra, fit raw magnetometry, construct crystal-field eigenstates, derive neutron cross sections, or select an ordered phase.

Required background. Magnetic Moments and g-Factors supplies magnetic-moment signs, magnetons, Zeeman conventions, and Landé algebra.

Helpful background. Hund’s Rules supplies the free-ion baseline. Spin–Orbit Coupling in Solids owns crystal-field, covalency, and spin–orbit projection. Magnetic Susceptibility owns measured Curie and high-field inference, while Itinerant Magnetism owns distributed and scale-dependent electronic moments.

Which Magnetic Moment Is Being Claimed?

Before comparing two quoted moments, write a ten-field material-moment record. Every field is required unless it is explicitly inapplicable.

1. Material, geometry, and state. Give composition, crystallographic and magnetic cell, field direction, temperature, pressure, filling or valence, doping, disorder, preparation, and history.
2. Physical operator and normalization. State spin, orbital, total dipole, magnetization density, or projected operator; charge and sign convention; moment, moment-squared, and per-ion, per-cell, per-mole, or per-volume units.
3. Retained subspace and matrix elements. Name the free-ion multiplet, crystal-field manifold, Kramers or non-Kramers doublet, band or Wannier subspace, projector rank, basis, and physical moment matrices.
4. Symmetry and g convention. Give time reversal, point or magnetic-space-group constraints, laboratory axes, pseudospin basis, principal values, and the combinations that are basis invariant.
5. Scale hierarchy. Compare charge, crystal-field, spin–orbit, exchange, anisotropy, Zeeman, thermal, longitudinal-fluctuation, damping, and probe-window scales.
6. Moment notion. Label the number as local equal-time, Curie-effective, saturation, ordered, fluctuating, sample moment, or magnetization and state its defining operation.
7. Spatial, temporal, and spectral window. Declare the local partition, form factor, momentum and energy coverage, elastic–inelastic split, integration range, and whether moment-squared weight is square rooted.
8. Probe and forward model. Record magnetometry, resonance, diffraction, inelastic scattering, local probe, or computation, including geometry, calibration, matrix elements, domains, backgrounds, and resolution.
9. Framework, provenance, and model readiness. State whether the model is ionic, projected-local, itinerant, mixed, or multipolar; how its parameters were derived or fitted; and whether a fixed-length reduction is controlled.
10. Claim, alternatives, and stopping rule. State the bounded comparison, uncertainty, competing explanations, falsifier, and the observation that would reject or escalate the moment model.

The record prevents three common category errors. A sample magnetic dipole moment $m$ in $\mathrm{A\,m^2}$ is not the magnetization $\mathbf M=\mathbf m/V$ in $\mathrm{A/m}$. A number in $\mu_B$ per formula unit is not a number per magnetic ion unless the occupancy is known. A fitted “local moment” is not an operator definition until its spatial and temporal windows are declared.

Free-Ion Baselines Are Starting Points

For an isolated, isotropic angular-momentum multiplet with quantum number $J$ and Landé factor $g_J$, two standard scales are

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

and

$$\mu_{\mathrm{sat}} = g_JJ\,\mu_B.$$

The first is the Curie fluctuation scale of a thermally randomized multiplet. The second is its maximum moment along a quantization axis. They differ even for an ideal ion because $\sqrt{J(J+1)}\ne J$. Magnetic Susceptibility owns the Curie–Weiss fit and its SI–cgs, background, and fit-window controls.

Hund’s Rules may supply an approximate free-ion term, but a crystal is not a free ion. Crystal fields split the multiplet, covalency moves magnetic density onto ligands, spin–orbit coupling changes the active basis, and exchange or hybridization can mix nominal configurations. A free-ion moment is therefore a baseline to test, not a value to impose on a solid.

“Orbital quenching” also needs care. A nondegenerate real crystal-field state can have vanishing first-order $\langle\mathbf L\rangle$, while virtual excited states still generate anisotropic $g$ factors, Van Vleck response, and anisotropy. Magnetic Moments from Orbital Motion owns the bare current-loop moment. The crystalline material problem begins after local symmetry and hybridization have changed that atomic basis.

Project the Physical Moment into the Retained Subspace

Let $P$ project onto a selected low-energy manifold. If an orthonormal frame is stored as the columns of $U$, then

$$P = UU^\dagger, \qquad U^\dagger U = I.$$

Changing the frame by $U\mapsto UV$ with unitary $V$ does not change $P$. The projected Hamiltonian and every physical operator must be rotated in the same frame. Projecting only the Hamiltonian while continuing to use a bare-spin operator is not a consistent effective theory.

For a time-reversal-invariant Kramers doublet, choose a dimensionless pseudospin $\widetilde{\mathbf s}=\boldsymbol\sigma/2$. The physical moment in retained coordinates is

$$\hat\mu_\alpha^{\mathrm{eff}} = U^\dagger\hat\mu_\alpha U = -\mu_B \sum_\beta g_{\alpha\beta} \widetilde s_\beta.$$

Its lifted full-space restriction is $P\hat\mu_\alpha P=U\hat\mu_\alpha^{\mathrm{eff}}U^\dagger$. Here $\alpha$ labels a laboratory-space component and $\beta$ a pseudospin-frame component. This distinction keeps the full-space projector separate from its matrix in the retained coordinate frame.

The identity component vanishes because the moment is odd under time reversal while the two states form a Kramers pair. A generic projected subspace, or a subspace selected after time reversal is broken, may allow an identity offset. The Pauli matrices label the doublet; they are not automatically the matrices of bare electron spin, orbital angular momentum, or total $\mathbf J$.

A time-reversal-symmetric non-Kramers doublet has a different operator structure. In a basis where time reversal acts as complex conjugation, only one Pauli direction is time-reversal odd; point-group symmetry can restrict that magnetic dipole component further. The remaining pseudospin components are time-reversal even and may represent quadrupoles or higher multipoles. A three-component Kramers $g$-tensor description is therefore not universal. A nonmagnetic singlet can likewise have no permanent moment yet retain a finite Van Vleck response through virtual excited states.

The effective Zeeman Hamiltonian in retained coordinates is

$$H_Z^{\mathrm{eff}} = \mu_B \mathbf B^{\mathsf T} \mathbf g \widetilde{\mathbf s}.$$

In the original Hilbert space, $PH_ZP=UH_Z^{\mathrm{eff}}U^\dagger$.

For a field $\mathbf B=B\mathbf n$, the doublet splitting is

$$\Delta E(\mathbf n) = \mu_BB \sqrt{ \mathbf n^{\mathsf T} \mathbf G \mathbf n }, \qquad \mathbf G = \mathbf g\mathbf g^{\mathsf T}.$$

A pseudospin-basis rotation changes the displayed matrix $\mathbf g$ but leaves $\mathbf G$ and the splitting invariant. Relative signs and the full operator matrices can still matter when the moment is combined with exchange, neutron matrix elements, or another projected operator. Spin–Orbit Coupling in Solids owns how the crystal-field and spin–orbit states that define $P$ are obtained.

Five Moment Measures Are Not One Number

Curie-effective moment

For a degenerate projected spin $\widetilde S$ in a weak-field, high-temperature window above its interactions but below discarded levels, let $\nu$ be the number of identical active moments per formula unit. Per mole of formula units, the SI Curie tensor for susceptibility differentiated with respect to internal $\mathbf H$ is

$$C_{\alpha\beta}^{\mathrm{mol}} = \frac{ \mu_0N_A\nu\mu_B^2 }{3k_B} \widetilde S(\widetilde S+1) G_{\alpha\beta}.$$

For a mole of active moment centers, or one center per formula unit, set $\nu=1$. A diluted or partially occupied sublattice requires its audited concentration rather than silently using this value. The expression uses $\mathbf B\simeq\mu_0\mathbf H_{\mathrm{int}}$ in the weak-response regime; demagnetizing and local-field corrections belong in the experimental forward model. For the doublet defined above, $\widetilde S=1/2$; a larger retained multiplet requires its own dimensionless spin matrices.

Along a unit vector $\mathbf n$, this corresponds to

$$\mu_{\mathrm{eff},\mathbf n}^2 = \mu_B^2 \widetilde S(\widetilde S+1) \mathbf n^{\mathsf T}\mathbf G\mathbf n.$$

This is a susceptibility coefficient, not a static expectation value. Excited crystal-field states, Van Vleck terms, exchange, Kondo physics, impurity tails, or a poor temperature window invalidate the simple identification.

Nor is a quantum static susceptibility generally just an ordinary equal-time variance. When $[\hat H,\hat\mu_\alpha]\ne0$, the equilibrium response uses the Kubo–Mori imaginary-time covariance. Equal-time fluctuations, thermodynamic response, and detector-bandwidth noise coincide only in stated commuting, classical, and limit conventions. Fluctuations and Susceptibilities owns that distinction.

Saturation moment

For an isolated doublet that remains valid to the applied field, the longitudinal saturated moment is

$$\mu_{\mathrm{sat},\mathbf n} = \frac{\mu_B}{2} \sqrt{ \mathbf n^{\mathsf T} \mathbf G \mathbf n }.$$

An anisotropic $g$ tensor can make the physical moment nonparallel to the field. Level mixing, metamagnetic transitions, unsaturated itinerant polarization, and sample heating can all prevent a measured high-field value from reaching this ideal limit.

Ordered moment

An ordered moment is the symmetry-selected static expectation of the physical moment density in a declared magnetic basis. It may be uniform or occur at a finite propagation vector. Quantum and thermal fluctuations, covalency, itinerancy, canting, and domain averaging can reduce its reported value. A finite-system symmetric eigenstate can have zero one-point expectation even when correlations diagnose incipient order.

Equal-time local moment

For one declared local operator, the instantaneous second moment is

$$m_{\mathrm{inst}}^2 = \left\langle \hat{\boldsymbol\mu}_i \cdot \hat{\boldsymbol\mu}_i \right\rangle.$$

This can remain large without long-range order. In an itinerant calculation, the word “local” additionally requires a region, orbital projector, or Wannier space; different defensible partitions need not give the same number.

Fluctuating moment

With $\delta\hat{\boldsymbol\mu}_i=\hat{\boldsymbol\mu}_i- \langle\hat{\boldsymbol\mu}_i\rangle$, define

$$m_{\mathrm{fluc}}^2 = \left\langle \delta\hat{\boldsymbol\mu}_i \cdot \delta\hat{\boldsymbol\mu}_i \right\rangle.$$

For the same operator, state, site, and normalization,

$$m_{\mathrm{inst}}^2 = m_{\mathrm{mean}}^2 + m_{\mathrm{fluc}}^2, \qquad m_{\mathrm{mean}}^2 = \left| \left\langle \hat{\boldsymbol\mu}_i \right\rangle \right|^2.$$

The mean contains field-induced polarization as well as spontaneous order. It may be called $m_{\mathrm{ord}}$ only in a source-selected zero-field ordered state with the same site, sublattice, and domain convention. For noncollinear, multi-sublattice, or multi-$\mathbf Q$ order, the corresponding statement uses the complete elastic Fourier weight rather than one scalar ordered moment. The identity is exact only when the static and connected pieces refer to the same complete operator record. Combining a bulk Curie fit, a site-projected calculation, and a resolution-limited neutron integral does not automatically satisfy it.

Elastic and Inelastic Weight Share a Sum Rule

Use the normalized Fourier operator

$$\hat\mu_{\mathbf q}^\alpha = \frac{1}{\sqrt N} \sum_j e^{-i\mathbf q\cdot\mathbf R_j} \hat\mu_j^\alpha$$

and the angular-frequency convention

$$S_\mu^{\alpha\beta}(\mathbf q,\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty}dt\, e^{i\omega t} \left\langle \hat\mu_{\mathbf q}^\alpha(t) \hat\mu_{-\mathbf q}^\beta(0) \right\rangle.$$

For equivalent sites and a complete Brillouin-zone and frequency integral,

$$\frac{1}{N} \sum_{\mathbf q,\alpha} \int_{-\infty}^{\infty}d\omega\, S_\mu^{\alpha\alpha}(\mathbf q,\omega) = m_{\mathrm{inst}}^2.$$

The disconnected elastic contribution carries the static mean, which may be field induced or ordered; the connected elastic and inelastic contributions carry fluctuations. Structure Factors owns the general correlation-function derivation, while Neutron Scattering owns polarization factors, magnetic form factors, absolute normalization, background, and resolution.

A real experiment covers a finite region of $\mathbf q$ and $\omega$. Missing weight may be elastic, above the incident-energy window, hidden by a phonon or background, transferred to ligand magnetization, or carried by degrees of freedom outside a spin-only model. A partial integral is evidence about that window, not a proof that the remaining moment has vanished.

Local and Itinerant Are Scale-Dependent Limits

An ion-centered moment can be useful at short times while the same material has coherent itinerant quasiparticles at low energy. Conversely, a metal can contain localized rare-earth or impurity moments carried by electrons distinct from its Fermi surface. The useful question is not whether the material is permanently “local” or “itinerant,” but which variables retain their amplitude over the declared energy, temperature, and length window.

Itinerant Magnetism owns band reconstruction, longitudinal fluctuations, the Stoner continuum, and the Rhodes–Wohlfarth diagnostic. A large Curie-effective moment together with a small spontaneous or saturation moment can support an itinerant or mixed interpretation, but it is not a binary theorem. Crystal-field population, orbital terms, Kondo screening, incomplete saturation, and fit backgrounds can produce similar ratios.

Spatial decomposition is also model dependent. A moment integrated inside an atomic sphere, assigned to a Wannier orbital, or reported on a ligand cluster depends on that partition. The total sample dipole and bulk magnetization are physical, and a spin-density observable can be defined with a specified operator and resolution. A unique atom-by-atom division, especially of orbital magnetization, generally is not. From Quantum Mechanics to Materials explains how such projectors and parameter choices must retain their provenance.

Bulk crystalline orbital magnetization is likewise not generally the sum of site-local $\langle\mathbf L\rangle$. It can include itinerant circulation and Berry-geometric contributions from occupied Bloch states. An atomic orbital moment remains a useful local diagnostic only when its projection and scope are declared.

Decide Whether a Fixed-Length Model Is Licensed

A spin or pseudospin Hamiltonian is a reduction, not a synonym for magnetism. Accept a fixed-length description only when all of the following tests pass.

1. Stable subspace. The retained multiplet or doublet has fixed rank across the relevant material, field, pressure, and structural regime.
2. Scale separation. Charge transfer, higher crystal-field levels, and longitudinal amplitude modes lie well above exchange, $k_BT$, Zeeman, drive, probe, linewidth, and resolution scales.
3. Projected operators. The magnetic moment, exchange, anisotropy, and probe operators are all projected into the same frame with declared axes.
4. Static consistency. Susceptibility, saturation, and ordered moments are compatible after temperature, domain, covalency, and background corrections.
5. Dynamic consistency. Elastic plus inelastic weight, mode polarization, continua, and longitudinal response agree with the retained Hilbert space.
6. Transferability. Parameters inferred from one observable predict at least one independent state, field direction, or probe without refitting.

Escalate to a multilevel spin–orbital, cluster, or itinerant-electron theory if discarded states approach the working window, the moment amplitude changes strongly with state, ligand weight is essential, longitudinal spectral weight is low energy, or the projected model fails an independent observable. Only after this audit should Exchange Interactions or Common Spin Hamiltonians be used to assign couplings to the retained variables.

Worked Audit: An Anisotropic Kramers Doublet

Consider a hypothetical tetragonal Ce$^{3+}$ insulator. The free-ion baseline has $J=5/2$ and $g_J=6/7$. For this idealized audit, a crystal field isolates a pure $|m_J=\pm1/2\rangle$ ground doublet by $\Delta_{\mathrm{CF}}=18\,\mathrm{meV}$. In its principal axes the projected tensor is

$$\mathbf g = \operatorname{diag} \left( \frac{18}{7}, \frac{18}{7}, \frac{6}{7} \right).$$

Write all ten fields before using the doublet.

1. Material, geometry, and state: one Ce$^{3+}$ ion per formula unit in a tetragonal insulator; weak-field susceptibility is fitted over $5$–$15\,\mathrm K$, while directional splitting is tested at $5\,\mathrm K$ up to $5\,\mathrm T$.
2. Physical operator and normalization: the electronic dipole is projected from $-g_J\mu_B\hat{\mathbf J}/\hbar$ and reported in $\mu_B$ per Ce. Below, $\hat{\mathbf J}/\hbar$ is represented by dimensionless matrices with eigenvalues $m_J$; nuclear moments are outside scope.
3. Retained subspace and matrix elements: the rank-two ground Kramers doublet is retained and the higher crystal-field states are eliminated. The physical moment matrices give the principal $\mathbf g$ displayed above.
4. Symmetry and g convention: $z$ is the tetragonal axis, $x,y$ are basal axes, the pseudospin frame follows those principal axes, and zero-field time reversal forbids an identity offset. The measurable splitting depends on $\mathbf g\mathbf g^{\mathsf T}$.
5. Scale hierarchy: at $5\,\mathrm K$, $k_BT=0.43\,\mathrm{meV}$; the exchange upper bound is $0.06\,\mathrm{meV}$, the largest Zeeman splitting is $0.74\,\mathrm{meV}$, and probe transfer is below $2\,\mathrm{meV}$, all well below $18\,\mathrm{meV}$.
6. Moment notion: compare the free-ion baseline, isolated-doublet Curie-effective coefficient, and asymptotic projected saturation moment. The finite-field measurement is not assumed saturated and no ordered moment is claimed.
7. Spatial, temporal, and spectral window: the local partition is one Ce ion, the susceptibility and resonance windows are those above, and neutron spectroscopy checks the $18\,\mathrm{meV}$ gap and moment matrix elements. Ligand covalency is neglected here and would require an enlarged cluster.
8. Probe and forward model: resonance measures directional splitting; susceptibility uses an internal-field Curie model plus a fitted Van Vleck term; neutron spectra test the level energies and transition intensities.
9. Framework, provenance, and model readiness: this is an ideal ionic-to-projected-local construction whose $g$ tensor follows analytically from the pure doublet. Its fixed-length rank-two use is provisionally controlled only over the stated scale window.
10. Claim, alternatives, and stopping rule: the same projected moment must fit all field directions. Substantial covalent redistribution, additional low-energy spectral weight, or field-induced higher-level admixture rejects the rank-two claim and escalates to a cluster or multilevel model.

The free-ion numbers are

$$\mu_{\mathrm{eff}}^{\mathrm{ion}} = \frac67\sqrt{\frac{35}{4}}\,\mu_B \simeq 2.54\,\mu_B$$

and

$$\mu_{\mathrm{sat}}^{\mathrm{ion}} = \frac67\frac52\,\mu_B \simeq 2.14\,\mu_B.$$

Inside the doublet, the zero-temperature asymptotic basal and axial saturation moments are instead

$$\mu_{\mathrm{sat},\perp} = \frac97\,\mu_B \simeq 1.29\,\mu_B, \qquad \mu_{\mathrm{sat},\parallel} = \frac37\,\mu_B \simeq 0.43\,\mu_B.$$

The corresponding isolated-doublet directional Curie-effective coefficients are about $2.23\,\mu_B$ and $0.74\,\mu_B$, and the ideal powder root-mean-square value is $1.87\,\mu_B$. They apply only in the weak-field window above the interaction scale and below the crystal-field gap. Likewise, $5\,\mathrm T$ at $5\,\mathrm K$ need not realize the asymptotic saturation moments quoted above. At temperatures or fields that populate or admix the next doublet, the rank-two projection must be reopened.

Worked Audit: A Cross-Probe Moment Budget

Consider a hypothetical correlated metal with one transition-metal site per formula unit. The reported data are a Curie-effective moment $3.10\,\mu_B$, an extrapolated high-field moment $1.05\,\mu_B$, a diffraction ordered moment $0.65\,\mu_B$, an absolutely normalized equal-time $m_{\mathrm{inst}}=2.60\,\mu_B$ reported from a converged neutron integral, and a spin-density calculation reporting $0.80\,\mu_B$ inside one atomic sphere.

The ten-field audit is:

1. Material, geometry, and state: use one transition-metal site per formula unit at fixed composition and structure. Curie data cover $180$–$350\,\mathrm K$ and high-field magnetization is at $2\,\mathrm K$. Diffraction and neutron quantities in the exact budget use the same $2\,\mathrm K$, zero-measurement-field ordered state and declared domain population after one field-cooling protocol.
2. Physical operator and normalization: experiment uses the total electronic magnetic density and reports $\mu_B$ per transition-metal formula unit; the calculation reports only projected spin density. Sample mass, occupancy, and magnetic-ion fraction have been audited.
3. Retained subspace and matrix elements: correlated multiorbital bands are retained with no assumed rigid spin or fixed-rank local doublet. Each probe uses the physical moment matrix appropriate to that band space.
4. Symmetry and g convention: the magnetization field follows the easy axis; diffraction declares one refined domain and its ordered representation. No single-ion $g$ tensor is assumed applicable to all five quantities.
5. Scale hierarchy: the Curie window lies above the ordering scale, while magnetization, elastic order, and equal-time weight are compared at $2\,\mathrm K$. Low-energy longitudinal response and band reconstruction remain active candidate scales rather than discarded corrections.
6. Moment notion: the five values are respectively Curie-effective, high-field, ordered, instantaneous, and partitioned computational moments.
7. Spatial, temporal, and spectral window: experimental values use a full formula-unit partition. Neutrons span the Brillouin zone and $0\le\hbar\omega\le150\,\mathrm{meV}$, include elastic weight and every magnetic component, use detailed balance for negative frequency, and show energy-tail convergence. The atomic sphere omits interstitial and ligand density.
8. Probe and forward model: susceptibility subtracts $\chi_0$ and impurity tails; magnetization corrects demagnetizing field and background; neutron analysis uses magnetic form factors, polarization, absolute units, domains, and the declared resolution boundary.
9. Framework, provenance, and model readiness: the working description is multiorbital and itinerant or mixed; its moments come from separate fitted, integrated, and projected records. A fixed-length local spin is not yet licensed because amplitude, ligand, and longitudinal sectors remain active.
10. Claim, alternatives, and stopping rule: the data license a hierarchy of moment notions, not one hidden spin length. Compare field- and temperature-dependent longitudinal weight, ligand-sensitive probes, and band reconstruction; only a model that predicts them without changing its moment length may justify the fixed-spin alternative.

For the same complete local operator, the connected fluctuation scale would be

$$m_{\mathrm{fluc}} = \sqrt{ (2.60\,\mu_B)^2 - (0.65\,\mu_B)^2 } \simeq 2.52\,\mu_B.$$

Here the static mean is the ordered moment because the state and domain conditions were declared. This exact subtraction is licensed only by the stated common temperature, field, domain population, operator, partition, normalization, component and elastic coverage, detailed-balance completion, and momentum- and energy-window convergence. Without them, the neutron integral is a partial lower bound and the displayed quadratic subtraction is not an exact cross-probe identity.

The hierarchy does not describe one hidden spin quantum number measured five ways. It says that substantial instantaneous magnetic weight survives while a much smaller component is static, and that the high-field and atomic-sphere partitions do not exhaust the same operator. The low-energy model should remain multiorbital or itinerant until a restricted spin model passes both the static and dynamic transfer tests.

Common Claim Failures

• Calling every Pauli matrix a spin. A pseudospin labels a subspace; only projected physical operators determine what it measures.
• Using $g=2$ by habit. Crystal-field and spin–orbit projection can produce anisotropic, off-diagonal, or nearly vanishing components.
• Equating effective and saturation moments. Their free-ion factors are $\sqrt{J(J+1)}$ and $J$, and their experimental windows differ.
• Equating a local moment with order. Equal-time weight can survive in a paramagnet, spin liquid, Kondo regime, or above an ordering transition.
• Treating an ordered moment as the full sum rule. Elastic weight is only one part of the equal-time moment budget.
• Calling a partial spectral integral missing physics. First audit energy, momentum, polarization, elastic, form-factor, and background coverage.
• Assigning a unique moment to an atom in a covalent solid. State the spatial projector or integration region and test another partition.
• Summing site-local $\langle\mathbf L\rangle$ for a crystal. Modern orbital magnetization can contain itinerant and Berry-geometric contributions.
• Inferring a rigid spin from one good dispersion. A fit to mode energies can fail intensities, linewidths, longitudinal response, or another state.
• Using one moment ratio as a phase theorem. Rhodes–Wohlfarth and related ratios are diagnostics whose backgrounds and scale windows must be checked.

Exercises

1. Compare free-ion effective and saturation moments

An isolated multiplet has $J=3/2$ and $g_J=4/5$. Find $\mu_{\mathrm{eff}}$, $\mu_{\mathrm{sat}}$, and their ratio.

Solution

The two definitions give

$$\mu_{\mathrm{eff}} = \frac45\sqrt{\frac{15}{4}}\,\mu_B \simeq 1.55\,\mu_B,$$ $$\mu_{\mathrm{sat}} = \frac45\frac32\,\mu_B = 1.20\,\mu_B.$$

Their ratio is

$$\frac{\mu_{\mathrm{eff}}}{\mu_{\mathrm{sat}}} = \sqrt{\frac{J+1}{J}} = \sqrt{\frac53} \simeq 1.29.$$

The mismatch exists before crystal fields, covalency, or itinerancy enter.

2. Read an anisotropic g tensor

A Kramers doublet has $\mathbf g=\operatorname{diag}(2.4,1.2,0.6)$. A field points along $\mathbf n=(1,1,0)/\sqrt2$. Find the splitting per unit field, the longitudinal saturated moment, and the powder Curie-effective coefficient. Show that a pseudospin-basis rotation $\mathbf g\mapsto\mathbf g\mathbf R$ with orthogonal $\mathbf R$ changes none of these quantities.

Solution

Here

$$\mathbf n^{\mathsf T}\mathbf G\mathbf n = \frac{2.4^2+1.2^2}{2} = 3.60.$$

Therefore

$$\frac{\Delta E}{B} = \sqrt{3.60}\,\mu_B \simeq 1.90\,\mu_B,$$

where the right-hand side has the units of magnetic moment. The saturated longitudinal moment is half that value, about $0.95\,\mu_B$. The full moment vector need not be parallel to the field. For $\widetilde S=1/2$, the powder coefficient satisfies

$$\mu_{\mathrm{eff,powder}}^2 = \mu_B^2 \frac{\widetilde S(\widetilde S+1)}{3} \operatorname{Tr}\mathbf G = 1.89\,\mu_B^2,$$

so $\mu_{\mathrm{eff,powder}}\simeq1.37\,\mu_B$. Under the basis rotation,

$$\mathbf G' = (\mathbf g\mathbf R)(\mathbf g\mathbf R)^{\mathsf T} = \mathbf g\mathbf R\mathbf R^{\mathsf T}\mathbf g^{\mathsf T} = \mathbf G.$$

Directional splittings, saturation moments, and the powder trace are therefore invariant even though the displayed $\mathbf g$ matrix changes.

3. Repair a normalization mismatch

A $4.0\,\mathrm{mg}$ crystal of molar mass $400\,\mathrm{g\,mol^{-1}}$ contains two equivalent magnetic ions per formula unit and four formula units per crystallographic cell. Its background-corrected saturated sample moment is $1.79\times10^{-4}\,\mathrm{A\,m^2}$. Find the moment per formula unit and per magnetic ion in $\mu_B$, per cell in $\mu_B$, and per mole of formula units in $\mathrm{A\,m^2\,mol^{-1}}$. State what additional information is needed to quote magnetization in $\mathrm{A/m}$.

Solution

The sample contains

$$n = \frac{4.0\times10^{-3}\,\mathrm g} {400\,\mathrm{g\,mol^{-1}}} = 1.0\times10^{-5}\,\mathrm{mol}$$

of formula units. The number of formula units is $nN_A$. Thus

$$\mu_{\mathrm{f.u.}} = \frac{1.79\times10^{-4}\,\mathrm{A\,m^2}} {(1.0\times10^{-5})N_A} \simeq 3.21\,\mu_B.$$

With two fully occupied equivalent magnetic sites, the value is $1.61\,\mu_B$ per ion. Four formula units give $12.84\,\mu_B$ per cell. The molar dipole moment is

$$m_{\mathrm{mol}} = (3.21\,\mu_B)N_A \simeq 17.9\,\mathrm{A\,m^2\,mol^{-1}}.$$

The sample volume, including the convention for porosity or packing if relevant, is needed for $M=m/V$ in $\mathrm{A/m}$.

4. Balance elastic and inelastic weight

For one consistently normalized local operator in a source-selected zero-field ordered state with a single declared sublattice and domain, $m_{\mathrm{inst}}=2.40\,\mu_B$ and $m_{\mathrm{ord}}=0.80\,\mu_B$. Find the connected fluctuating moment. If an experiment captures $70\%$ of its connected spectral weight, what moment scale would that partial integral report?

Solution

The complete connected variance is

$$m_{\mathrm{fluc}} = \sqrt{2.40^2-0.80^2}\,\mu_B = \sqrt{5.12}\,\mu_B \simeq 2.26\,\mu_B.$$

Spectral weights add as squared moments, so the partial integral reports

$$m_{\mathrm{partial}} = \sqrt{0.70}\,m_{\mathrm{fluc}} \simeq 1.89\,\mu_B.$$

The remaining weight is not automatically absent; it may lie outside the measured momentum, energy, polarization, or elastic window.

5. Classify five quoted moments

A paper quotes: (a) $3.0\,\mu_B$ from a Curie constant; (b) $1.2\,\mu_B$ at the highest measured field; (c) $0.7\,\mu_B$ from magnetic Bragg intensity; (d) $2.5\,\mu_B$ from a complete equal-time sum rule; and (e) $2.4\,\mu_B$ from its connected part. Quantities (c)–(e) refer to the same source-selected zero-field ordered state, local operator, sublattice, domain, and normalization. Classify all five and state which three must obey an exact quadratic identity.

Solution

(a) is Curie-effective, (b) is a high-field value that is a saturation moment only after saturation and level-mixing checks, (c) is ordered, (d) is instantaneous local, and (e) is fluctuating. The exact relation is

$$m_{\mathrm{inst}}^2 = m_{\mathrm{ord}}^2 + m_{\mathrm{fluc}}^2.$$

Here $0.7^2+2.4^2=2.5^2$. The Curie and high-field values are not terms in that identity.

6. Decide whether a fixed-length model is ready

A candidate doublet is separated by $12\,\mathrm{meV}$ from the next level. The working window has exchange $4\,\mathrm{meV}$, maximum Zeeman energy $3.5\,\mathrm{meV}$, probe transfer $5\,\mathrm{meV}$, and $k_BT=2.6\,\mathrm{meV}$. Is a rank-two fixed-length model controlled? What would improve the case?

Solution

The largest retained scale is already about $42\%$ of the discarded-level gap, and several effects can combine. The hierarchy is not parametrically strong; virtual or real excited-level admixture may affect the moment and interactions. One should include the next level or quantify its corrections.

At lower temperature, field, and probe energy, a hierarchy such as all retained scales below $0.5\,\mathrm{meV}$ would be much better. Even then, the projected $g$ tensor, static moments, dynamic sum rule, and transfer to an independent observable must be checked.

7. Diagnose local and itinerant windows

A metal shows a large femtosecond-scale local spin correlation above $T_C$, a small low-temperature ordered moment, a coherent spin-split Fermi surface, and low-energy longitudinal spectral weight. Does this establish a purely local or purely itinerant magnet? Choose an adequate starting description.

Solution

Neither exclusive label follows. The short-time correlation supports moment formation on that window, while the spin-split Fermi surface and low-energy amplitude response show that mobile electrons and longitudinal fluctuations remain active. A multiorbital itinerant or mixed spin–fermion description is the safe starting point. A fixed-spin reduction may be derived only for a narrower low-energy transverse sector and must inherit state-dependent parameters and a documented cutoff.

8. Audit a cross-probe claim

A study claims a rigid spin-$1/2$ material because the Curie fit gives $1.73\,\mu_B$ for $g=2$. Neutron diffraction finds $0.25\,\mu_B$, inelastic data cover only $0$–$8\,\mathrm{meV}$, and a $6\,\mathrm{meV}$ crystal-field excitation disperses into the magnetic continuum. Accept, stop, or escalate the claim, and name decisive tests.

Solution

Escalate. The Curie value is compatible with an ideal spin-$1/2$ coefficient, but it does not prove an isolated doublet. The low ordered moment can reflect fluctuations, covalency, itinerancy, or incomplete static order. More importantly, the crystal-field excitation lies inside the measured dynamical window and hybridizes with it, so the assumed rank-two space is not isolated.

Fit the full crystal-field and moment matrices, extend the absolute spectral integral in momentum and energy, separate elastic and connected weight, measure directional Zeeman splittings, and test whether a multilevel model predicts field and temperature evolution. A rigid spin-$1/2$ claim can resume only if a controlled lower-energy window emerges.

The canonical next owners are Spin–Orbit Coupling in Solids for the multilevel projection and Neutron Scattering with Structure Factors for the absolute moment budget. If a low-energy longitudinal continuum persists, the claim instead escalates to Itinerant Magnetism.

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