7.4 Magnetization

Magnetization describes how materials respond to magnetic fields at the microscopic level. By understanding how individual magnetic dipole moments align (or don't) within a substance, you can predict the macroscopic magnetic behavior of everything from copper wiring to transformer cores to superconducting magnets.

Magnetization of Materials

When you place a material in an external magnetic field, the atomic-scale magnetic moments inside it respond. The nature of that response is what defines a material's magnetic classification:

• Diamagnetic: weakly opposes the applied field (e.g., copper, bismuth)
• Paramagnetic: weakly aligns with the applied field (e.g., aluminum, oxygen gas)
• Ferromagnetic: strongly aligns with the field and retains magnetization after the field is removed (e.g., iron, nickel, cobalt)
• Antiferromagnetic: neighboring moments align antiparallel, producing zero net magnetization (e.g., MnO, chromium)
• Ferrimagnetic: antiparallel alignment with unequal moments, yielding a net magnetization (e.g., magnetite, ferrites)

The magnetization you actually observe depends on the applied field strength, the temperature, and the intrinsic electronic structure of the material.

Magnetic Dipole Moments

Magnetic dipole moments are the microscopic sources of all magnetism in matter. They originate from two contributions:

• Orbital angular momentum of electrons orbiting the nucleus, which creates a current loop and thus a magnetic moment.
• Spin angular momentum, an intrinsic quantum property of the electron that also produces a magnetic moment.

The net magnetic dipole moment of a material is the vector sum of every individual atomic/molecular dipole moment. In most materials at zero field, these moments point in random directions and cancel out. An external field can partially or fully align them, producing macroscopic magnetization.

Magnetization Vector M

The magnetization vector $\mathbf{M}$ is defined as the net magnetic dipole moment per unit volume:

$\mathbf{M} = \frac{1}{V} \sum_i \mathbf{m}_i$

where the sum runs over all dipole moments $\mathbf{m}_i$ in volume $V$. For a uniformly magnetized sample with total moment $\boldsymbol{\mu}$, this simplifies to $\mathbf{M} = \boldsymbol{\mu}/V$.

• $\mathbf{M}$ is a vector field: it has both magnitude and direction at every point in the material.
• SI units: amperes per meter (A/m).
• CGS units: emu/cm³.

The direction of $\mathbf{M}$ points along the net alignment of the dipoles. In a linear, isotropic material, $\mathbf{M}$ is parallel to the applied field $\mathbf{H}$.

Magnetic Susceptibility χ

Magnetic susceptibility $\chi$ quantifies how readily a material magnetizes in response to an applied field. It's defined as:

$\chi = \frac{M}{H}$

This is a dimensionless ratio (in SI). A few key points:

• $\chi > 0$: paramagnetic (moments align with the field)
• $\chi < 0$: diamagnetic (moments oppose the field)
• $\chi \gg 1$: ferromagnetic (strong alignment, though the linear relation $M = \chi H$ only holds below saturation)

Susceptibility can depend on temperature (Curie law for paramagnets: $\chi = C/T$), field strength (in nonlinear materials), and crystallographic direction (anisotropic materials).

Diamagnetic vs Paramagnetic Susceptibility

• Diamagnetic susceptibility is small and negative, typically on the order of $\chi \sim -10^{-5}$. It arises from the induced change in orbital motion of electrons (Lenz's law at the atomic scale) and is present in all materials, though often masked by stronger effects. Examples: copper, silver, gold, most organic compounds.
• Paramagnetic susceptibility is small and positive, typically $\chi \sim 10^{-5}$ to $10^{-3}$. It arises from the partial alignment of pre-existing permanent dipole moments against thermal disorder. Examples: aluminum, platinum, liquid oxygen.

The magnitude of diamagnetic susceptibility is generally much smaller than paramagnetic susceptibility, which is why paramagnetism dominates whenever a material has unpaired electrons.

Magnetization Curves

A magnetization curve plots $M$ (or $B$) as a function of the applied field $H$. The shape tells you almost everything about a material's magnetic character:

• Diamagnets and paramagnets produce straight lines through the origin (linear response), with negative or positive slope respectively.
• Ferromagnets produce S-shaped curves that saturate at high fields, and if you cycle the field, you get a hysteresis loop.

Key quantities extracted from these curves:

• Saturation magnetization $M_s$: the maximum magnetization when all moments are aligned.
• Remanence $M_r$: the magnetization remaining after the applied field is reduced to zero.
• Coercivity $H_c$: the reverse field needed to bring the magnetization back to zero.

Initial vs Anhysteretic Magnetization

• The initial magnetization curve starts from a fully demagnetized state ($M = 0$, $H = 0$) and traces the magnetization as $H$ increases for the first time. Its shape reflects domain wall motion and rotation processes.
• The anhysteretic magnetization curve is obtained by superimposing a large AC field (which is slowly decreased to zero) on a DC bias field. This "shakes out" the domain walls from their pinning sites, giving an idealized, reversible curve free of hysteresis effects.

Comparing the two reveals how much of the magnetization process is governed by irreversible domain wall pinning versus reversible rotation.

Magnetic Permeability μ

Magnetic permeability $\mu$ measures how well a material supports the formation of magnetic flux within itself:

$\mu = \frac{B}{H}$

• SI units: henries per meter (H/m).
• The permeability of free space is $\mu_0 = 4\pi \times 10^{-7}$ H/m.
• High-permeability materials (like soft iron, $\mu_r \sim 10^3$ to $10^5$) concentrate magnetic flux and are used in transformer cores, inductors, and magnetic shielding.

Relative vs Differential Permeability

• Relative permeability $\mu_r = \mu / \mu_0$ is dimensionless and tells you how the material compares to vacuum. For paramagnets, $\mu_r$ is slightly greater than 1; for diamagnets, slightly less than 1; for ferromagnets, it can be enormous.
• Differential permeability $\mu_d = dB/dH$ is the local slope of the $B$-$H$ curve at a given operating point. This matters for nonlinear materials (ferromagnets) because $\mu$ is not constant. At saturation, $\mu_d$ drops toward $\mu_0$ since additional $H$ barely increases $B$.

For linear materials, $\mu_d = \mu$ everywhere. For ferromagnets, you need to specify which permeability you mean.

Magnetic Field Intensity H

The magnetic field intensity $\mathbf{H}$ (sometimes called the auxiliary field) represents the externally applied field contribution, separate from the material's own magnetization. It's related to the free current sources:

$\nabla \times \mathbf{H} = \mathbf{J}_f$

where $\mathbf{J}_f$ is the free current density. You can also write $\mathbf{H} = \frac{\mathbf{B}}{\mu_0} - \mathbf{M}$, which shows that $\mathbf{H}$ accounts for the applied field minus the material's magnetization contribution.

• SI units: A/m.
• CGS units: oersteds (Oe).

The field $\mathbf{H}$ is the natural quantity to use in boundary conditions and when working with magnetic circuits, because it depends only on free currents, not on the material response.

Magnetic Flux Density B

The magnetic flux density $\mathbf{B}$ is the total magnetic field inside a material, combining the contributions from both the external field and the material's magnetization:

$\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M})$

This is the constitutive relation that connects all three field quantities. You can also write it as $\mathbf{B} = \mu_0(1 + \chi)\mathbf{H} = \mu \mathbf{H}$ for linear materials.

• SI units: tesla (T).
• CGS units: gauss (G), where 1 T = 10⁴ G.

$\mathbf{B}$ is the field that enters the Lorentz force law ($\mathbf{F} = q\mathbf{v} \times \mathbf{B}$) and Faraday's law, so it's the physically measurable field that determines forces and induced EMFs.

B vs H Curves

• For linear materials (diamagnets, paramagnets), the $B$-$H$ curve is a straight line: $B = \mu H$. The slope gives you the permeability directly.
• For ferromagnets, the $B$-$H$ curve forms a hysteresis loop. As you cycle $H$ from positive to negative and back, $B$ traces a closed loop rather than retracing the same path. The loop encloses an area proportional to the energy dissipated per cycle.

The $B$-$H$ curve is distinct from the $M$-$H$ curve, though they contain equivalent information (since $B = \mu_0 H + \mu_0 M$). Engineers often prefer $B$-$H$ curves because $B$ directly gives the flux in a magnetic circuit.

Hysteresis in Ferromagnetic Materials

Hysteresis means the magnetization of a ferromagnet depends on its history, not just the current value of $H$. This happens because ferromagnets contain magnetic domains, regions where all dipoles are aligned in the same direction. The magnetization process involves:

1. Domain wall motion: walls between domains shift so that domains aligned with the field grow at the expense of others.
2. Domain rotation: at higher fields, the magnetization within domains rotates to align with the applied field.
3. Saturation: all moments are aligned; further increases in $H$ produce no additional $M$.

Domain walls get pinned by crystal defects, grain boundaries, and impurities. Unpinning them requires extra energy, which is why the magnetization path going up differs from the path going down.

Key hysteresis loop parameters:

• Saturation magnetization $M_s$: all moments aligned.
• Remanent magnetization $M_r$: magnetization at $H = 0$ after saturation.
• Coercive field $H_c$: the field needed to demagnetize the material.

Soft magnetic materials (low $H_c$, narrow loop) are used in transformers. Hard magnetic materials (high $H_c$, wide loop) are used as permanent magnets.

Hysteresis Loss vs Eddy Current Loss

Both are sources of energy dissipation in ferromagnetic materials under AC fields, but they have different origins:

• Hysteresis loss comes from the irreversible work of moving pinned domain walls. The energy lost per cycle equals the area enclosed by the $B$-$H$ loop. It scales linearly with frequency (one loop per cycle).
• Eddy current loss comes from currents induced in the conducting material by the time-varying $\mathbf{B}$ field (Faraday's law). These currents dissipate energy via Joule heating. Eddy current loss scales as the square of frequency and the square of the peak flux density.

To minimize these losses in devices like transformers:

• Use soft magnetic materials with narrow hysteresis loops (reduces hysteresis loss).
• Use laminated cores or high-resistivity materials like ferrites (reduces eddy current loss by limiting the path length for induced currents).

Curie Temperature of Ferromagnets

The Curie temperature $T_C$ is the critical temperature above which a ferromagnet loses its spontaneous magnetization and becomes paramagnetic. Named after Pierre Curie (1895).

Below $T_C$, the quantum-mechanical exchange interaction between neighboring spins is strong enough to maintain long-range magnetic order, even without an applied field. Above $T_C$, thermal energy $k_B T$ overwhelms the exchange energy, randomizing the spin orientations.

Some representative Curie temperatures:

• Iron: $T_C \approx 1043$ K
• Nickel: $T_C \approx 631$ K
• Cobalt: $T_C \approx 1394$ K

Above $T_C$, the susceptibility follows the Curie-Weiss law:

$\chi = \frac{C}{T - T_C}$

where $C$ is the Curie constant. This diverges as $T \to T_C^+$, reflecting the phase transition.

Magnetization of Superconductors

Superconductors exhibit perfect diamagnetism below their critical temperature $T_c$ and critical field $H_c$: they completely expel magnetic flux from their interior. This gives them an effective susceptibility of $\chi = -1$ (in SI, for a long cylinder parallel to the field), the most extreme diamagnetic response possible.

The microscopic origin lies in the formation of Cooper pairs, bound states of two electrons with opposite spin and momentum. These pairs condense into a macroscopic quantum state that supports persistent supercurrents on the surface, which generate a field that exactly cancels the applied field inside the bulk.

This perfect diamagnetism is distinct from perfect conductivity. A perfect conductor would trap whatever flux was present when it became conducting; a superconductor actively expels flux, regardless of whether the field was applied before or after cooling.

Meissner Effect in Superconductors

The Meissner effect (discovered by Meissner and Ochsenfeld, 1933) is the complete expulsion of magnetic flux from the interior of a superconductor as it transitions below $T_c$.

How it works:

1. A superconductor sits in a weak external magnetic field above $T_c$ (normal state, flux penetrates).
2. The sample is cooled below $T_c$.
3. Surface screening currents spontaneously arise, generating a field that exactly cancels the applied field inside the material.
4. The interior reaches $\mathbf{B} = 0$.

The Meissner effect proves that superconductivity is a true thermodynamic phase, not just zero resistance. A hypothetical "perfect conductor" cooled in a field would maintain the internal flux forever (by Faraday's law, no changing flux means no EMF to drive currents that would expel it). The superconductor, by contrast, expels the flux upon entering the superconducting state.

The Meissner effect is the principle behind magnetic levitation of superconductors and is used in magnetic shielding applications.

Magnetization Measurement Techniques

Measuring magnetization accurately requires sensitive instruments, since many materials have weak magnetic signals. The three most common techniques are:

• Vibrating Sample Magnetometry (VSM): the sample vibrates mechanically in a uniform field, inducing a voltage in nearby pickup coils proportional to the sample's magnetic moment. Sensitivity: ~$10^{-6}$ emu. Relatively fast and inexpensive.
• SQUID Magnetometry: uses superconducting quantum interference devices based on the Josephson effect to detect extremely small changes in magnetic flux. Sensitivity: up to ~$10^{-12}$ emu. The gold standard for weak signals, thin films, and small samples.
• Alternating Gradient Magnetometry (AGM): applies an alternating field gradient to the sample and measures the resulting oscillating force. Sensitivity is comparable to or slightly better than VSM.

All three techniques work by placing the sample in a known applied field and detecting the response. From the raw data, you extract $M$ vs. $H$ curves and derive quantities like $\chi$, $M_s$, $M_r$, and $H_c$.

Vibrating Sample vs SQUID Magnetometers

SQUID magnetometers exploit the quantum interference of superconducting currents through Josephson junctions. The sample moves through a superconducting pickup loop coupled to the SQUID sensor. Their extreme sensitivity makes them indispensable for research on nanostructures, dilute magnetic systems, and any sample where the signal is too weak for a VSM.