Magnetic Materials

Magnetic moments and bound currents

Atoms and molecules can carry magnetic dipole moments through electronic orbital motion and intrinsic electron spin. A macroscopic specimen contains an enormous number of such moments. Their vector sum, rather than the magnitude of one atomic moment, determines the magnetic response observed outside the material.

A small permanent moment $\vec{m}$ in an applied magnetic field ${\vec{B}}_{\mathrm{app}}$ has interaction energy

Parallel alignment lowers this energy; antiparallel alignment raises it. Thermal motion competes with the alignment tendency. In an unmagnetized paramagnetic sample, moment directions are broadly distributed and the vector average is close to zero. A field produces a slight excess of moments pointing along the applied direction. The individual moments remain large on the atomic scale even when their macroscopic average is small.

Magnetization is the magnetic dipole moment per unit volume. A small volume $\Delta V$ containing moments ${\vec{m}}_i$ has

Its SI unit is ampere per metre. A moment has units $\mathrm{A}{\mathrm{m}}^2$, so division by volume leaves $\mathrm{A}{\mathrm{m}}^{-1}$. Magnetization is a vector field: both magnitude and direction can vary from point to point. A material with spatially varying moment alignment has spatially varying $\vec{M}$ even when its chemical composition is uniform.

The term saturation magnetization refers to the largest magnetization available when the relevant microscopic moments are nearly fully aligned. If a number density $n$ of atoms each contributes a moment of magnitude $m$, the limiting scale is

The expression is an upper scale, not a general room-temperature result. Thermal agitation, competing atomic configurations, and interactions among neighbouring moments commonly keep the measured magnetization far below saturation.

Bound-current interpretation

Magnetization can be represented macroscopically by effective currents. The representation does not require a literal tiny wire loop at every atom. It captures the magnetic field of the summed microscopic orbital and spin moments. The associated volume and surface current densities are

where $\hat{n}$ is the outward normal of a material surface. ${\vec{J}}_{\mathrm{b}}$ has units of $\mathrm{A}{\mathrm{m}}^{-2}$ and ${\vec{K}}_{\mathrm{b}}$ has units of $\mathrm{A}{\mathrm{m}}^{-1}$.

Uniform magnetization has $\nabla \times \vec{M}=0$ in the interior. Adjacent microscopic loops cancel there: a segment of current on one loop is opposed by a neighboring segment. At the exterior surface, there is no neighboring loop beyond the boundary to supply the cancellation. The remaining effective surface current accounts for the material’s external magnetic field in the bound-current description.

A long cylinder with uniform $\vec{M}=M\hat{z}$ has a cylindrical side with $\hat{n}=\hat{r}$ and therefore

The effective current circulates around the cylinder. Its magnitude per unit axial length is $M$. End faces carry no surface bound current in this ideal case because their normal is parallel or antiparallel to $\vec{M}$. A finite cylinder still has end-field structure; the simple long-cylinder picture describes the central region.

The bound-current picture separates internal magnetization from free current carried through an externally connected circuit. A copper wire feeding a coil has a transport current set by the circuit. A uniformly magnetized insulating sample can have a nonzero ${\vec{K}}_{\mathrm{b}}$ without any charge crossing its exterior surface from a power supply. Both forms of current contribute to magnetic fields, but their physical origins and measurement controls differ.

Susceptibility and field variables

In a weak, reversible response regime, magnetization is often proportional to the applied field. Tipler and Mosca express that relation as

The dimensionless susceptibility ${\chi }_m$ gives the sign and approximate size of the response. In the ideal central region of a long sample placed in a long solenoid,

The factor ${\kappa }_m=1+{\chi }_m$ is the relative permeability in this linear model. Geometry, field strength, temperature, and magnetic history determine whether a constant susceptibility is an adequate approximation.

Diamagnetic matter has a small negative susceptibility. Its response is induced by the applied field and opposes that field. Atoms with no permanent net magnetic moment can still develop a diamagnetic response because the applied field changes electronic orbital motion. Typical ordinary diamagnetic susceptibilities have magnitudes near $10^{-5}$, so the effect is usually weak.

Paramagnetic matter has a small positive susceptibility. Permanent atomic or molecular moments exist, but thermal motion randomizes their directions when the applied field is absent. An applied field biases the orientation distribution. The alignment remains partial when the dipole-field energy is small compared with thermal energy. In the weak-field limit, Curie’s law gives

The response increases with applied field and decreases with absolute temperature. Paramagnetism usually leaves no remanent magnetization after the applied field is removed.

Ferromagnetic matter has a strong positive response arising from cooperative interactions among neighboring moments. Microscopic regions called domains can have nearly aligned moments even when the macroscopic specimen has little net magnetization, because different domains point in different directions. A modest applied field can move domain boundaries or rotate domain alignment, producing a much larger net response than ordinary paramagnetism.

Ferromagnetic response is nonlinear and history dependent. Increasing and then decreasing the applied field traces a hysteresis loop. The loop can retain a nonzero magnetization or magnetic field at zero applied level, and a reversed applied field is required to bring the macroscopic response through zero. Thermal agitation above the Curie temperature disrupts the cooperative ordering, leaving a paramagnetic response.

The three classes describe response trends rather than a complete catalogue of magnetic behavior. Measured susceptibility should always be paired with field strength, temperature, orientation, and magnetic history. Those conditions determine whether a simple linear relation describes the sample or whether domain processes and hysteresis dominate the measurement.

Measuring a magnetic response

A response measurement separates the source field from the sample’s added magnetization. The sample magnetic moment can be found from a calibrated torque, force-gradient, or flux measurement, then divided by sample volume to obtain $M$. The applied source setting and the sample temperature must be recorded at the same time. A value of magnetization without a field scale cannot determine a susceptibility.

In a reversible weak-response regime,

Repeating the measurement at positive and negative source settings checks the sign. Diamagnetic response reverses with the source field and opposes it. Paramagnetic response reverses with the source field and follows it. A ferromagnetic sample can retain a response when the source setting returns to zero, so measurements must state the prior sweep direction and maximum applied level.

Finite sample shape changes the relation between an external source field and the internal field experienced by the material. The simple $\vec{B}={\vec{B}}_{\mathrm{app}}+{\mu }_0\vec{M}$ expression used above describes the central region of the long-cylinder idealization in the reference text. Ends, corners, gaps, and irregular sample shapes introduce spatial variation. A susceptibility reported from a finite specimen therefore includes the stated geometry and measurement location unless an appropriate shape correction has been made.

Temperature scans provide a second classification check. A weak paramagnetic response commonly decreases as temperature rises because random thermal motion weakens moment alignment. Ordinary diamagnetic susceptibility changes much less strongly with temperature. Ferromagnetic response can change abruptly near its Curie temperature, where domain-scale cooperative ordering is lost. A single room-temperature measurement cannot establish a complete magnetic class when the material composition or thermal history is unknown.

The response also depends on the timescale of the source sweep. Slow measurements that trace a ferromagnetic hysteresis loop give different values on increasing and decreasing branches. Reproducible reports identify the initial state, sweep range, sweep direction, and whether the specimen was demagnetized before the run. Those conditions turn susceptibility and magnetization from isolated numbers into reproducible material-response data.

B, H, and M

The magnetic flux density $\vec{B}$ is the field that appears in the magnetic force and flux laws. Its SI unit is the tesla. Magnetization $\vec{M}$ is the material dipole moment per unit volume and has units of amperes per metre. The magnetic field strength $\vec{H}$ separates the source-controlled part of a magnetostatic problem from the material response:

Both $\vec{H}$ and $\vec{M}$ have units of amperes per metre. Their sum has the same unit; multiplication by ${\mu }_0$ converts it to tesla. The definition separates a circuit's free current from the sample's bound-current response. In steady magnetostatics, the circulation of $\vec{H}$ counts free current:

The microscopic material currents are incorporated through $\vec{M}$ rather than appearing separately on the right side. This division makes coil currents, magnetization, and total magnetic flux distinguishable in one calculation.

A linear, isotropic material in a reversible range has

The relation uses a scalar ${\chi }_m$ only when the response has no preferred direction and remains proportional to $\vec{H}$. Crystals can be anisotropic, and ferromagnetic specimens can be nonlinear and history dependent. In those cases, a single scalar permeability cannot represent every direction and every point on a hysteresis loop. The slope of a local response curve may still characterize a stated operating point.

The applied field in a long empty solenoid is often written ${\vec{H}}_{\mathrm{a}}=nI\hat{z}$. The associated empty-space flux density is ${\mu }_0{\vec{H}}_{\mathrm{a}}$. Once a sample is inserted, the material magnetization adds to the total $\vec{B}$. Referring only to “the magnetic field” obscures whether a numerical value denotes the source setting, the material moment density, or the total flux density.

Linear-media interfaces

Boundary conditions provide sharp tests of a linear-media model. Let $\hat{n}$ point from medium 1 to medium 2. Magnetic flux has no isolated sources, so the normal component of $\vec{B}$ is continuous:

The tangential jump in $\vec{H}$ is set by free surface current density ${\vec{K}}_{\mathrm{free}}$:

At an interface carrying no free sheet current, tangential $\vec{H}$ is continuous. If both media are linear and isotropic, the consequence is

The tangential component of $\vec{B}$ can therefore change at a permeability boundary even when no free surface current is present. The normal component does not jump. A map that crosses the boundary at different angles separates these two conditions and exposes a sensor alignment error.

The interface equations concern local components. They do not guarantee that the field is uniform within either medium. A finite specimen can have edge fringing and spatially varying magnetization even when its bulk susceptibility is well characterized. Boundary data are interpreted alongside the sample shape, probe position, and source geometry.

Demagnetization geometry and measurement protocol

Magnetization produces an internal field that often opposes the applied magnetizing direction in a finite sample. This demagnetizing field is a shape effect. An ellipsoid magnetized along one principal axis has it represented by

where $N$ is the dimensionless demagnetizing factor for that axis. The three principal factors of an ellipsoid sum to one in SI conventions. A long rod magnetized along its length has a small $N$. A thin disk magnetized perpendicular to its face has a factor approaching one. The same material can therefore show very different apparent response when cut into different shapes.

A linear sample has

Ignoring $N$ overestimates magnetization whenever ${\chi }_m$ is positive. The correction is modest for weak paramagnets and can dominate the apparent response of a high-susceptibility specimen. A long-rod geometry is widely used when a central region close to the applied field is required.

A careful measurement protocol begins with an empty-coil calibration of $H_{\mathrm{a}}$ at the intended current. Sample dimensions are measured before mounting, including the axis selected for magnetization. The moment measurement is converted to $M=m_{\mathrm{sample}}/V$. The demagnetizing factor is chosen for the measured shape and orientation, after which $H_{\mathrm{int}}=H_{\mathrm{a}}-NM$ is evaluated point by point.

Plotting $M$ against $H_{\mathrm{int}}$, rather than raw coil current, gives the material response in the linear model. A straight low-field branch gives ${\chi }_m$ from its slope. Reversing the source current removes a stationary background signal. Ferromagnetic specimens require a controlled initial state, full record of prior extrema, and separate increasing and decreasing sweeps. The report should include temperature, sample dimensions, coil calibration, probe location, field orientation, and the adopted demagnetizing factor.

Boundary measurement checks

A boundary experiment uses two probe orientations. A probe normal to a smooth interface tests continuity of the normal B component. A probe tangent to the same interface tests the permeability-dependent change in tangential B. The probe centre must be farther from the boundary than the active sensor radius; otherwise the reading averages material on both sides and cannot be compared with either limiting value.

Free surface current and bound surface current have different roles in the analysis. The tangential H jump condition uses free current supplied through a circuit. Magnetization contributes through M and can change B even when no free sheet current is present at the interface. Treating every observed B change as a free-current signal produces an incorrect boundary inference.

Record a reference scan with the specimen removed, then repeat the same scan with the specimen inserted and the same coil current. Current reversal removes stationary environmental offsets from both scans. Comparing the reversal differences isolates the material contribution while retaining a traceable source calibration. The resulting data can test the linear-media equations only over the stated field range and sample orientation.

Domains and hysteresis

Ferromagnetism arises from cooperative interactions among neighboring magnetic moments. The exchange interaction favors parallel alignment over microscopic regions. A large specimen usually divides into magnetic domains rather than maintaining one uniform direction everywhere. Within an individual domain the moments are strongly aligned; neighboring domains can point in different directions, leaving a small net moment for the whole specimen before an external field is applied.

Domain formation balances several energy contributions. Exchange energy favors alignment within a domain. Domain walls cost energy because the moment direction changes across a finite region. Magnetostatic energy favors patterns that reduce stray magnetic flux outside the specimen. The observed domain structure minimizes the combined energy subject to sample shape, defects, temperature, and prior magnetic history.

An applied magnetizing field changes that balance. Domains already oriented near the applied direction can grow by domain-wall motion. Moments within a domain can also rotate toward the applied direction. Both mechanisms increase the macroscopic magnetization. Pinning by defects and grain boundaries makes the wall motion partly irreversible, which produces hysteresis in a bulk measurement.

Thermal agitation weakens the domain-scale order. Above the Curie temperature, the cooperative ferromagnetic ordering is lost and the material responds paramagnetically. The transition temperature and domain behavior depend on composition and microstructure. A magnetic classification therefore applies to a stated temperature range, processing history, and field scale.

Hysteresis, remanence, and coercivity

A hysteresis loop records the response during a complete forward-and-reverse magnetizing cycle. Start from a demagnetized state and increase $H$ in the positive direction. Magnetization rises rapidly while favorable domains grow, then approaches saturation magnetization $M_{\mathrm{s}}$ as most moments align. Further increase of the source mainly raises the flux density through the source-field term once $M$ changes slowly.

Reducing $H$ from saturation follows a different branch. At $H=0$, a ferromagnetic sample can retain a remanent flux density $B_{\mathrm{r}}$ or remanent magnetization $M_{\mathrm{r}}$. A reversed field is required to reduce the measured response to zero. Its magnitude is the coercive field $H_{\mathrm{c}}$. Continuing the reverse sweep reaches negative saturation; returning to positive field closes the loop.

The energy dissipated per unit volume in a quasistatic cycle is proportional to the enclosed loop area. With $B$ plotted against $H$, the loss density is

The expression has units $\mathrm{J}{\mathrm{m}}^{-3}$. It measures irreversible domain processes during the cycle. A narrow loop has lower hysteresis loss than a wide loop at comparable operating amplitude.

Soft and hard material tradeoffs

Magnetically soft materials have low coercivity and a narrow hysteresis loop. Their magnetization changes readily when the source field changes, and their cycle loss is relatively small. Such behavior suits transformer and inductor cores, where the intended operating state repeatedly reverses. High permeability can be valuable in a restricted low-field range, yet saturation and frequency-dependent losses still set an operating limit.

Magnetically hard materials have larger coercivity and substantial remanence. They retain a magnetized state after the source field is removed, making them appropriate for permanent magnets and recorded magnetic states. Their larger loop area represents a larger energy cost for repeated reversal. A material chosen to preserve a stored magnetic state is therefore poorly suited to a low-loss alternating cycle.

Permeability and susceptibility require an operating-point label in a ferromagnet. A secant estimate, such as $B/H$, depends on the initial state and the endpoint of a sweep. A differential permeability,

describes a local slope on one specified branch. Near saturation, $\mathbf{d}M/\mathbf{d}H$ becomes small and $\mathbf{d}B/\mathbf{d}H$ approaches ${\mu }_0$. A numerical permeability quoted without field amplitude, branch direction, temperature, and sample shape has limited predictive value.

A coercivity measurement uses a calibrated magnetizing coil and a signal proportional to either $B$ or $M$. The sample is first driven to a stated positive saturation condition. The current is then swept through zero at a controlled rate while the response is sampled. Interpolation between the two samples that bracket zero response gives $H_{\mathrm{c}}$. The intercept at $H=0$ on the return branch gives remanence. Repeating the full loop after reversing the initial saturation tests symmetry and can indicate drift in the measurement chain.

Saturation and data reduction

Saturation is a statement about magnetization, not a statement that flux density becomes constant. As the available microscopic moments align, $M$ approaches a limiting value $M_{\mathrm{s}}$. The total flux density still contains the source contribution:

Above magnetic saturation, an increase in $H$ produces little additional magnetization, while $B$ continues to increase with a slope approaching ${\mu }_0$. A graph of B alone can therefore appear to keep rising after the material response has saturated. Separating M from B prevents the source-field slope from being misidentified as additional moment alignment.

The initial magnetization curve is measured from a reproducibly demagnetized state. It differs from a major hysteresis loop because domain populations and wall positions begin in a different configuration. Minor loops trace only a restricted range between previous extrema. They are relevant to low-amplitude devices, but their slope and loss area cannot be substituted for a full saturation-loop specification.

Data reduction begins with the coil calibration that converts current to applied $H$. The sample geometry then sets the demagnetizing correction needed for $H_{\mathrm{int}}$. A calibrated magnetic-flux or moment signal is converted to B or M with the sensor scale and sample volume. Each point is stored with the sweep direction and the preceding maximum field. A ferromagnet has memory; two readings at the same applied current can represent different states after different histories.

Noise treatment must preserve the loop shape. Averaging repeated points at a fixed state reduces random sensor noise. Averaging values from opposite sweep branches would erase the hysteresis that the experiment intends to measure. Baseline offsets are measured with the sample absent or with a reversal protocol appropriate to the sensor. Drift is checked by returning to a previous reference current after a sweep.

The final report identifies whether a quoted permeability is an initial slope, a differential slope at an operating point, a secant ratio over a stated interval, or an effective value inferred from a particular geometry. These quantities can differ by orders of magnitude in a ferromagnet. The field range, temperature, frequency or sweep rate, sample shape, and magnetic prehistory are part of the numerical value rather than optional experimental detail.

Traceable hysteresis data also retain the specimen dimensions and the selected magnetization axis. Those records allow later comparison of coercivity, remanence, and loss measurements made with different sample geometries.

Magnetic energy and material selection

A reversible magnetization process stores magnetic energy in the coupled source and material system. A quasistatic reversible branch has incremental energy density

Integrating from an unmagnetized reference state gives

The integrand uses the actual constitutive branch. A linear medium with constant permeability has

The units are joules per cubic metre. This result applies only while the response is single-valued and reversible. A ferromagnetic path contains history-dependent domain changes, so the energy supplied during an increase and the energy returned during a decrease differ.

The B–H loop area gives this difference,

The energy per unit volume becomes heat during one complete quasistatic cycle. A small loop area reduces core heating in a repeated magnetic drive. The loop area is independent of the graphical scale only when the axes are calibrated in their physical units; an unscaled chart cannot determine energy loss.

The partition between stored and dissipated energy changes with the drive amplitude. A minor loop far below saturation can have a much smaller loss than a major loop driven close to saturation. Frequency also matters in a real component because magnetic processes may lag the applied drive and circulating currents can be induced within conductive material. A quasistatic loop provides the baseline hysteresis loss; additional dynamic loss requires a measurement at the intended waveform and frequency.

Transformer-core choices

A transformer core guides magnetic flux through a closed low-reluctance path so that the windings couple efficiently. The core material is usually magnetically soft: low coercivity, low remanence, and a narrow loop reduce the energy needed to reverse magnetization each cycle. A high low-field permeability reduces the magnetizing current required for a specified operating flux density.

Core selection also requires a saturation margin. If the operating B approaches the material’s saturation range, incremental permeability falls and the magnetizing current rises sharply. The waveform can distort and losses increase. Designs therefore specify a maximum B below saturation at the highest intended temperature and the lowest expected supply frequency.

Conductive cores also support induced circulating currents. Laminating the core into insulated thin sheets interrupts large current loops and reduces this heating mechanism. Ferrite materials use high electrical resistivity for the same reason at higher frequencies. These loss controls supplement, rather than replace, the choice of a narrow hysteresis loop.

Core loss is measured in watts per mass or volume at stated frequency, peak flux density, waveform, and temperature. A quoted loss number without those conditions cannot be transferred directly to another drive. Mechanical stress, grain orientation, air gaps, and manufacturing heat treatment can alter both permeability and loss. The finished component must be characterized in its assembled geometry, especially when a deliberate air gap changes the magnetic path.

Temperature dependence and characterization errors

Weak paramagnets follow Curie-like behavior over an appropriate temperature and field range:

where $C$ is a material-dependent Curie constant. Heating weakens the orientation bias by increasing thermal agitation. Diamagnetic susceptibility usually changes more weakly with temperature. Ferromagnetic order weakens substantially as temperature rises toward the Curie temperature and the material becomes paramagnetic above that transition.

Temperature measurement requires a sensor thermally coupled to the specimen. A sensor mounted on the coil former or chamber wall can miss the temperature gradient between an internally heated core and the thermometer during alternating drive. Stabilization time, drive amplitude, and temperature ramp direction belong in the characterization record.

Several systematic errors recur in magnetic measurements. An incorrect sample volume scales every reported magnetization. An uncertain demagnetizing factor distorts the inferred internal H. A pickup coil with unknown effective area miscalibrates B. Mechanical vibration, sensor offset, and trapped remanence shift low-field readings. Electrical resistance in the drive winding changes with temperature, so a nominal voltage source does not guarantee a constant magnetizing current.

Characterization therefore uses a sequence of checks: empty-fixture baseline, coil-current calibration, sample geometry measurement, controlled initial state, temperature stabilization, and repeated forward/reverse sweeps. Results are reported with the drive waveform, frequency, peak level, field direction, sample orientation, and uncertainty method. This information separates a material property measured under defined conditions from a response produced by one particular fixture.

Separating magnetic losses

Core loss is obtained from an electrical power balance with stated fixture terms. One reduction is

where $R_w$ is the winding resistance at operating temperature and $P_{\mathrm{empty}}$ is the no-sample or empty-fixture loss defined by the same drive and measurement chain. The subtraction is valid only when those terms are measured under the same frequency, waveform, and thermal conditions as the specimen.

• Frequency series. At fixed peak $B$, measure a sequence of frequencies rather than one power reading. Hysteresis loss per cycle follows loop area, so its power contribution rises approximately with cycle rate when loop shape is fixed. Induced-current loss rises more rapidly and depends on lamination thickness, resistivity, and magnetic-path geometry. The series identifies whether the core remains in its intended low-loss range.
• Magnetic amplitude. Determine peak $B$ from a calibrated sense winding or flux probe. Drive voltage alone is insufficient: it includes winding drops, and saturation changes the current-to-$B$ relation. Inspect the magnetic waveform before assigning a sinusoidal-loss specification to a distorted drive.
• State and uncertainty. Propagate core mass or volume, input power, winding resistance, temperature, frequency, and peak $B$ into the reported loss density. Repeat after a controlled demagnetization or saturation sequence as well as during repeated runs. Retracing the same minor loop can understate variation caused by magnetic history.
• Matched specimens. Compare samples with matched processing and geometry. Cutting, pressing, machining, and mounting stress alter domain-wall motion. An air gap changes the effective magnetic path and lowers apparent permeability even when bulk material is unchanged. Report intrinsic sample data separately from the finished magnetic-circuit response.

The record distinguishes a material property measured under defined conditions from a loss value set by one fixture, waveform, and magnetic history.

Measurement and model limits

A measured flux density becomes a material response only after the source field and sample shape are accounted for.

Uncertainty propagation follows the same sequence. The B calibration, $H_{\mathrm{a}}$ calibration, and N estimate contribute separately to M. Near $N=1$, the denominator $1-N$ amplifies every uncertainty, which makes a thin-disk geometry a poor choice for extracting a high-susceptibility response. A long axial specimen with small N reduces that amplification. Select and document the specimen geometry during measurement design, then include its contribution in the final uncertainty budget.

Minor loops and magnetic history

The full saturation loop is one possible magnetic history. Components often operate on minor loops after a limited change in drive rather than traversing the major loop. Begin from a point on a major branch, reverse the applied field before saturation, then return. The resulting minor loop encloses its own loss area and has its own local slope. Its remanence and effective permeability depend on the reversal point.

The same applied H can therefore give several possible B or M values. A statement such as “the permeability at 200 amperes per metre” is incomplete for a hysteretic material unless it includes the initial state, sweep direction, previous extrema, and waveform amplitude. This history dependence is a measurable property of domain-wall motion and pinning, not random instrument scatter.

Minor-loop measurements use a fixed reset protocol. One sequence drives the sample to a documented reference state, reaches the chosen upper drive level, and cycles between the specified lower and upper values. The cycle is repeated until consecutive loops agree within the measurement uncertainty. A different reset or a larger previous excursion can change the loop even when the final drive amplitude is identical.

The loss associated with a periodic minor loop is found from its own $\left|\oint H\mathbf{d}B\right|$ area, multiplied by the operating frequency and magnetic volume for a power estimate. Substituting the major-loop area overstates loss for a small-signal component. Using a minor loop measured at one temperature and frequency to predict another requires a separate temperature and rate validation.

Shape and calibration reconciliation

The B/H/M inversion can be checked by measuring the same material in two shapes. A long rod and a thin disk have different raw slopes when M is plotted against applied $H_{\mathrm{a}}$, because their demagnetizing factors differ. After converting each point to

the low-field data should collapse onto one material-response curve within uncertainty. Persistent separation after the correction indicates a problem with the assumed N, sample volume, sensor calibration, anisotropy, or history control.

Calibration reconciliation starts with an empty-fixture current sweep that relates the actual coil current to applied H. A second reference checks the B sensor or pickup coil against a known field or a traceable reference sample. Sample dimensions are measured along the magnetization axis, and volume is calculated from those dimensions rather than from a nominal stock size. The same temperature sensor, drive waveform, and reset protocol are used for both shapes.

Agreement is evaluated with residuals rather than visual overlap alone. At a common corrected H, form

Values of $|z|$ near unity support agreement within the stated uncertainty. Large systematic residuals that grow with H often indicate nonlinear response or a demagnetizing correction outside its valid range. Residuals that remain constant can indicate a sensor zero offset or sample-volume bias.

Model boundary statement

A concise material-data statement names the response branch, temperature, frequency or sweep rate, field interval, sample geometry, magnetization axis, demagnetizing correction, calibration references, and uncertainty convention. It also says whether the reported quantity is B, H, M, secant permeability, differential permeability, remanence, coercivity, or loop-loss density.

The linear B/H/M model applies only to the reversible interval used for the inversion. Saturation, hysteresis, anisotropy, thermal gradients, large air gaps, and dynamic loss require an expanded model or direct measurement. This boundary statement makes the quoted data usable without claiming a wider range than the experiment supports.

Inversion uncertainty audit

The worked inversion contains three independent input classes: the B scale, the applied-H scale, and the shape factor. Let their standard uncertainties be $u_B$, $u_H$, and $u_N$. Differentiating

gives the first-order uncertainty estimate

For the numerical example, use $u_B=0.020\mathrm{mT}$, $u_H=8\mathrm{A}{\mathrm{m}}^{-1}$, and $u_N=0.010$. The three contributions to $u_M$ are approximately $19.9\mathrm{A}{\mathrm{m}}^{-1}$, $10.0\mathrm{A}{\mathrm{m}}^{-1}$, and $7.4\mathrm{A}{\mathrm{m}}^{-1}$. Their quadrature sum is

The result is reported as $M=(5.92\pm 0.23)\times 10^2\mathrm{A}{\mathrm{m}}^{-1}$ for the stated field, temperature, ellipsoidal shape, and reversible branch. The B calibration dominates this particular uncertainty budget. Improving the demagnetizing-factor estimate alone would not substantially improve the final magnetization uncertainty.

The audit also catches inconsistent measurements. A B reading below the empty coil value in a material expected to have positive low-field response may indicate an incorrect sensor sign, a reversed coil lead, a different sample orientation, or a diamagnetic contribution. A recovered M larger than an independently measured saturation scale indicates that the linear central-field model has been applied outside its domain or that volume and calibration data need review.

Raw readings, calibration constants, geometry measurements, and all reduction equations should remain with the published material curve. A future measurement can then update one calibration factor or one shape estimate without recreating the full experiment. This traceability supports later measurement of the same material in another core, at another temperature, or on a different hysteresis branch.

The uncertainty expression assumes independent input errors. A shared coil calibration or common temperature drift introduces correlation between points on a curve. Such effects are recorded as systematic terms rather than reduced by averaging repeated readings. Separating random and systematic uncertainty keeps an apparently precise fit from overstating the confidence of a material-data inversion.

Independent replication with a second calibrated fixture gives the strongest check on a reported response curve. Agreement across fixture geometries, after the stated shape corrections, tests both the material model and the calibration chain.