7.5 Magnetic susceptibility

Types of Magnetic Materials

Magnetic materials respond differently to applied fields depending on their atomic structure and electron configuration. Three main categories cover most behavior you'll encounter, with two additional types that show up in more specialized contexts.

Diamagnetic vs paramagnetic vs ferromagnetic

Diamagnetic materials have a weak, negative susceptibility, meaning they're slightly repelled by magnetic fields. Examples include bismuth, copper, and water. Every material has a diamagnetic contribution, but it's usually masked if stronger magnetic behavior is present.

Paramagnetic materials have a small, positive susceptibility and are weakly attracted to magnetic fields. Aluminum, platinum, and molecular oxygen are common examples. Their magnetic moments exist but point in random directions until an external field partially aligns them.

Ferromagnetic materials have a large, positive susceptibility and can retain magnetization after the external field is removed. Iron, nickel, and cobalt are the classic examples. The key difference here is exchange interaction between neighboring atoms, which causes magnetic moments to align spontaneously within regions called domains. This alignment is what makes permanent magnets possible.

Antiferromagnetic and ferrimagnetic materials

Antiferromagnetic materials have magnetic moments on adjacent atoms that align antiparallel with equal magnitude, producing zero net magnetization. Manganese oxide (MnO) and chromium are typical examples. Above the Néel temperature ($T_N$), thermal energy destroys this antiparallel ordering and the material becomes paramagnetic.

Ferrimagnetic materials also have antiparallel alignment, but the opposing moments have different magnitudes, so there's a net magnetization in one direction. Magnetite ($\text{Fe}_3\text{O}_4$) and commercial ferrites fall into this category. Ferrimagnets are widely used in high-frequency applications, data storage, and magnetic sensors because they combine useful net magnetization with lower electrical conductivity than metals (reducing eddy current losses).

Magnetic Susceptibility

Susceptibility quantifies how strongly a material magnetizes in response to an applied field. It connects the microscopic behavior of electrons and atomic moments to macroscopic, measurable quantities.

Definition of magnetic susceptibility

The volume magnetic susceptibility $\chi$ is defined as the ratio of the induced magnetization $\mathbf{M}$ to the applied magnetic field intensity $\mathbf{H}$:

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

This is dimensionless. A few key points:

• $\chi > 0$: paramagnetic or ferromagnetic (material magnetizes along the field)
• $\chi < 0$: diamagnetic (material magnetizes opposite to the field)
• Typical diamagnetic values are on the order of $-10^{-5}$; paramagnetic values range from $10^{-5}$ to $10^{-3}$; ferromagnetic values can reach $10^3$ or higher

The magnitude of $\chi$ depends on composition, crystal structure, and temperature.

Magnetic susceptibility tensor

For isotropic materials, $\chi$ is a scalar and $\mathbf{M}$ points in the same direction as $\mathbf{H}$. In anisotropic materials (most single crystals), the magnetization induced along one axis can differ from that along another. Here, susceptibility becomes a second-rank tensor:

$M_i = \sum_j \chi_{ij} H_j$

This $3 \times 3$ matrix $\chi_{ij}$ captures the directional dependence. The tensor can be diagonalized along the crystal's principal axes, yielding three independent susceptibility values. Experimentally, techniques like torque magnetometry or single-crystal susceptibility measurements determine these components. Knowing the full tensor is essential for understanding magnetic anisotropy and for designing devices that exploit directional magnetic response.

Relation to permeability

Inside a linear magnetic material, the total magnetic field $\mathbf{B}$ is related to $\mathbf{H}$ through the permeability $\mu$:

$\mathbf{B} = \mu \mathbf{H} = \mu_0(1 + \chi)\mathbf{H}$

The relative permeability is:

$\mu_r = 1 + \chi$

For vacuum, $\chi = 0$ and $\mu_r = 1$. For diamagnets, $\mu_r$ is slightly less than 1. For paramagnets, slightly greater than 1. For ferromagnets, $\mu_r$ can be enormous (thousands or more), which is why iron cores concentrate magnetic flux so effectively in transformers and inductors.

Curie's Law

Curie's law describes how paramagnetic susceptibility depends on temperature. It applies when magnetic moments are non-interacting and thermal fluctuations compete with field-induced alignment.

Temperature dependence of susceptibility

For a paramagnetic material:

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

where $C$ is the Curie constant and $T$ is the absolute temperature in kelvin.

The physical picture: at higher temperatures, increased thermal energy randomizes the orientation of magnetic moments more effectively, reducing the net magnetization for a given applied field. So susceptibility drops as $1/T$.

Deviations from Curie's law occur when interactions between moments are non-negligible. For ferromagnetic materials above their ordering temperature, the Curie-Weiss law provides a better description:

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

Here $T_C$ is the Curie temperature. The susceptibility diverges as $T \to T_C^+$, signaling the phase transition into the ferromagnetic state.

Curie constant and Curie temperature

The Curie constant $C$ is material-specific and depends on the effective magnetic moment $p_{\text{eff}}$ of the magnetic ions:

$C = \frac{n \mu_0 p_{\text{eff}}^2 \mu_B^2}{3 k_B}$

where $n$ is the number density of magnetic ions, $\mu_B$ is the Bohr magneton, and $k_B$ is Boltzmann's constant. The effective moment itself depends on the number of unpaired electrons and the total angular momentum quantum numbers.

The Curie temperature $T_C$ is the critical temperature above which a ferromagnet loses its spontaneous magnetization and becomes paramagnetic. For iron, $T_C \approx 1043 \text{ K}$; for nickel, $T_C \approx 627 \text{ K}$. At $T_C$, thermal energy overcomes the exchange interaction responsible for long-range magnetic order.

Measuring Magnetic Susceptibility

Several experimental techniques exist for measuring susceptibility, each suited to different sample types and sensitivity requirements.

Faraday balance method

The Faraday balance measures the force on a sample placed in a region of known magnetic field gradient.

1. Suspend the sample from a sensitive balance between the poles of an electromagnet shaped to produce a controlled field gradient $\frac{dB}{dz}$.
2. The force on the sample is $F = \chi V H \frac{dH}{dz}$, where $V$ is the sample volume.
3. Measure the apparent weight change at several field strengths.
4. Extract $\chi$ from the slope of force vs. $H \frac{dH}{dz}$.

This method works well for small samples and detects both paramagnetic and diamagnetic responses.

Vibrating sample magnetometer

A vibrating sample magnetometer (VSM) measures magnetization by vibrating the sample at a fixed frequency in a uniform field and detecting the induced voltage in nearby pickup coils.

• The induced voltage is proportional to the sample's magnetic moment and the vibration amplitude and frequency.
• VSMs can sweep applied field, temperature, and time, making them versatile for measuring hysteresis loops, saturation magnetization, and remanence.
• Sensitivity is typically on the order of $10^{-6}$ emu, sufficient for most bulk and thin-film samples.

SQUID magnetometer

The superconducting quantum interference device (SQUID) magnetometer is the most sensitive instrument for measuring magnetic moments, with sensitivity reaching $\sim 10^{-8}$ emu or better.

• A SQUID consists of a superconducting loop interrupted by one or two Josephson junctions. Changes in magnetic flux through the loop produce measurable voltage oscillations.
• The sample is moved through superconducting pickup coils coupled to the SQUID, and the resulting flux change is recorded.
• This extreme sensitivity makes SQUIDs ideal for weakly magnetic samples, very small specimens, and research requiring precise susceptibility data across wide temperature ranges.

Applications of Magnetic Susceptibility

Identifying unknown materials

Every material has a characteristic susceptibility value (or set of values). Comparing a measured $\chi$ against tabulated data helps identify composition and purity. This is used in archaeology for artifact analysis, in geology for mineral identification, and in manufacturing for quality control.

Studying phase transitions

Susceptibility measurements are a direct probe of magnetic phase transitions. A sharp peak or divergence in $\chi(T)$ signals a transition: the Curie temperature for ferromagnets, the Néel temperature for antiferromagnets. Tracking how $\chi$ changes with temperature reveals the nature of the transition (first-order vs. second-order) and helps characterize new magnetic materials.

Magnetic resonance imaging (MRI)

MRI exploits susceptibility differences between tissues to generate image contrast. The local magnetic field experienced by hydrogen nuclei depends on the susceptibility of surrounding tissue, which shifts their resonance frequency and alters relaxation times. Susceptibility-weighted imaging (SWI) is a specific MRI technique that enhances contrast between regions of different $\chi$, useful for visualizing blood vessels, microbleeds, and iron deposits in the brain.

Magnetic separation techniques

Differences in susceptibility allow physical separation of materials in a field gradient. More magnetic components experience a stronger force and migrate toward the high-field region.

• Mining: separating magnetic ores from non-magnetic gangue
• Environmental remediation: removing magnetic contaminants from soil or water
• Biotechnology: magnetic nanoparticles functionalized with antibodies label and separate specific cells or biomolecules from complex mixtures

Quantum Mechanical Description

Classical models can't fully account for magnetic susceptibility. A quantum mechanical treatment is needed to explain why some materials are diamagnetic, others paramagnetic, and why susceptibility takes the specific values it does.

Atomic and molecular magnetism

The magnetic moment of an atom comes from two sources: the spin angular momentum of electrons and their orbital angular momentum around the nucleus. The total magnetic moment is the vector sum of all spin and orbital contributions.

In a filled subshell, spin-up and spin-down electrons cancel, and orbital contributions cancel as well. Only atoms or ions with unpaired electrons carry a net magnetic moment. In molecules, the situation is modified by molecular orbital formation, which can pair previously unpaired electrons or create new unpaired ones.

Hund's rules and electron configuration

Hund's rules determine the ground-state quantum numbers ($S$, $L$, $J$) of an atom or ion, which in turn fix its magnetic moment:

1. Maximize $S$ (total spin): electrons fill orbitals singly with parallel spins before pairing. This minimizes electron-electron repulsion via the exchange interaction.
2. Maximize $L$ (total orbital angular momentum): consistent with rule 1, electrons occupy orbitals that maximize $L$.
3. Determine $J$ (total angular momentum): for a subshell less than half-filled, $J = |L - S|$; for more than half-filled, $J = L + S$. This reflects spin-orbit coupling.

The ground-state $J$ value directly determines the effective magnetic moment:

$p_{\text{eff}} = g_J \sqrt{J(J+1)}$

where $g_J$ is the Landé g-factor. Getting the electron configuration right through Hund's rules is the starting point for predicting any material's susceptibility.

Spin and orbital angular momentum contributions

The spin magnetic moment of an electron is:

$\boldsymbol{\mu}_s = -g_s \mu_B \mathbf{S}/\hbar$

where $g_s \approx 2$. The orbital magnetic moment is:

$\boldsymbol{\mu}_L = -g_L \mu_B \mathbf{L}/\hbar$

where $g_L = 1$. The factor-of-two difference between $g_s$ and $g_L$ means spin contributes disproportionately to the total moment per unit angular momentum.

In many 3d transition metal ions, the crystal field quenches the orbital angular momentum, so the "spin-only" approximation ($p_{\text{eff}} \approx 2\sqrt{S(S+1)}$) works well. For 4f rare-earth ions, spin-orbit coupling is strong and both $L$ and $S$ contribute significantly, requiring the full $J$-based expression.

Magnetic Anisotropy

Magnetic anisotropy is the directional dependence of a material's magnetic properties. It determines the easy axis (preferred magnetization direction) and the energy cost of rotating magnetization away from it.

Magnetocrystalline anisotropy

This type originates from spin-orbit coupling interacting with the crystal lattice. The non-spherical charge distribution of electron orbitals, locked to the lattice by the crystal field, creates energetically preferred orientations for the magnetization.

For a uniaxial crystal (e.g., hexagonal cobalt), the anisotropy energy density to lowest order is:

$E_a = K_1 \sin^2\theta$

where $K_1$ is the first anisotropy constant and $\theta$ is the angle between $\mathbf{M}$ and the easy axis. Materials with large $K_1$ (like rare-earth magnets such as Nd$_2$Fe$_{14}$B) resist demagnetization and are used as permanent magnets.

Shape anisotropy

Non-spherical geometries create anisotropy through the demagnetizing field. The demagnetizing field is stronger along shorter dimensions, so magnetization prefers to lie along the longest dimension where the demagnetizing factor is smallest.

For a prolate ellipsoid, the shape anisotropy energy density is:

$E_{\text{shape}} = \frac{1}{2}\mu_0 (N_a - N_c) M^2 \sin^2\theta$

where $N_a$ and $N_c$ are demagnetizing factors along the short and long axes. In elongated nanoparticles and thin films, shape anisotropy often dominates over magnetocrystalline anisotropy, dictating the preferred magnetization direction.

Anisotropy in nanostructures and thin films

At the nanoscale, surface and interface effects become significant because a large fraction of atoms sit at boundaries with broken symmetry and reduced coordination.

• Nanoparticles: Surface anisotropy can compete with or dominate bulk contributions. Below a critical size, nanoparticles become single-domain and can exhibit superparamagnetism, where thermal fluctuations flip the magnetization direction on experimentally accessible timescales.
• Thin films: Interfacial anisotropy from the interaction between a magnetic layer and its substrate (or adjacent layers) can induce perpendicular magnetic anisotropy (PMA), where the easy axis points out of the film plane. PMA is critical for technologies like spin valves, magnetic tunnel junctions, and magnetoresistive random-access memory (MRAM).

Controlling anisotropy in nanostructures is one of the central challenges in modern magnetic device engineering.