Magnetic Susceptibility

Magnetic susceptibility, usually denoted by $\chi$, measures how strongly a material becomes magnetized in response to a magnetic field. In its simplest form, it is the proportionality coefficient between magnetization and magnetic field strength. Its sign, magnitude, and dependence on temperature, field, and frequency distinguish different magnetic responses and provide information about microscopic magnetic moments and their interactions. Static susceptibility describes the response to a steady field, whereas dynamic susceptibility describes the response to a time-varying field. (arxiv.org)

Definition and relation to permeability

For a linear, isotropic material,

where $\mathbf M$ is magnetic dipole moment per unit volume, $\mathbf H$ is the internal magnetic field strength, and $\chi_v$ is volume susceptibility. In the International System of Units (SI), both $M$ and $H$ are measured in amperes per metre, so $\chi_v$ is dimensionless. The magnetic flux density satisfies

giving

Here $\mu_0$ is vacuum permeability and $\mu_r$ is relative magnetic permeability. This scalar relation assumes a linear, isotropic response. (iupac.org)

When magnetization is nonlinear, the ratio $M/H$ differs from the local slope of the magnetization curve. Differential susceptibility is defined by

It describes the response to a small field change around a specified operating point. Materials with magnetic hysteresis can have different responses depending on their field history, so susceptibility need not be a single constant. (qdusa.com)

Volume, mass, and molar conventions

Susceptibility may be normalized by volume, mass, or amount of substance. If $\rho$ is density and $M_{\mathrm{mol}}$ is molar mass, then

Their SI units are respectively $\mathrm{m^3\,kg^{-1}}$ and $\mathrm{m^3\,mol^{-1}}$. Equivalently, molar susceptibility equals volume susceptibility multiplied by molar volume. Normalization must therefore be specified when comparing measurements. (old.iupac.org)

Magnetic unit conventions introduce additional factors. For corresponding dimensionless volume susceptibilities in SI and conventional Gaussian cgs notation,

Converting conventional cgs molar susceptibility in $\mathrm{cm^3\,mol^{-1}}$ to SI requires multiplication by $4\pi\times10^{-6}$, not merely conversion from cubic centimetres to cubic metres. (media.iupac.org)

Magnetic behavior and temperature dependence

Diamagnetism has negative susceptibility: the induced magnetization opposes the applied field. Paramagnetism has positive susceptibility, corresponding to a net induced moment along the field. A positive susceptibility alone does not establish ferromagnetism, which involves collective magnetic ordering and can exhibit spontaneous magnetization and hysteresis. (goldbook.iupac.org)

Microscopic moments arise principally from electron orbital motion and spin. The observed susceptibility combines contributions from different mechanisms; a material containing local moments may also have a temperature-independent background. Consequently, interpreting susceptibility requires more than identifying its sign. (arxiv.org)

For approximately independent localized moments in a weak field, Curie’s law gives

where $C$ is the Curie constant and $T$ is absolute temperature. Thermal agitation competes with field-induced alignment. The constant is related to the number and effective magnitude of the moments, allowing susceptibility measurements to constrain molecular electronic structure. (goldbook.iupac.org)

Interactions are often represented approximately by the Curie–Weiss law,

where $\chi_0$ is a background contribution. Positive and negative Weiss temperatures commonly indicate predominantly ferromagnetic and antiferromagnetic interactions, respectively, but are not conclusive descriptions of the ordered state. The Weiss temperature need not equal the actual ordering temperature, and fitting too close to a phase transition can invalidate the approximation. (arxiv.org)

Anisotropic and dynamic susceptibility

In an anisotropic material, susceptibility is a second-rank tensor:

Magnetization can therefore point in a different direction from the applied field. In specified coordinates, the coefficients form a matrix, expressing directional differences in magnetic response. (iupac.org)

For an oscillating field of angular frequency $\omega$, dynamic susceptibility is represented by a complex number. With time dependence $e^{i\omega t}$, a common convention is

The real component describes the in-phase response; the imaginary component describes the out-of-phase response and magnetic dissipation. Its sign convention depends on the chosen time dependence. Frequency-dependent measurements probe relaxation and irreversible processes that steady-field measurements cannot resolve. (qdusa.com)

Measurement and demagnetizing effects

Magnetometers measure magnetic moment as field or temperature varies. Sensitive instruments use a superconducting quantum interference device (SQUID); other techniques detect forces or voltages induced by moving a sample relative to pickup coils. AC susceptometers instead detect the response to an oscillating field. (qdusa.com)

The internal field generally differs from the externally applied field because the sample creates a demagnetizing field. For a uniformly magnetized ellipsoid along a principal axis,

where $N$ is the SI demagnetizing factor. Shape effects are especially important for large susceptibilities. In ideal bulk superconductivity, complete field exclusion corresponds to intrinsic $\chi_v=-1$, but apparent susceptibility depends on geometry; the same correction cannot generally be applied to arbitrary shapes using a universal constant. (tsapps.nist.gov)