Magnetic Susceptibility (χ): Dimensionless Response to Applied Fields

Magnetic susceptibility χ quantifies how a material polarises in response to an applied magnetic field H. Defined as χ = M/H, where M is magnetisation (A·m⁻¹) and H is field strength (A·m⁻¹), susceptibility is dimensionless. Positive χ indicates paramagnetic or ferromagnetic response, while negative χ signals diamagnetism. Engineers use χ to tailor sensors, shielding, medical devices, and geophysical surveys.

This article presents susceptibility definitions on volume, mass, and molar bases; recounts measurement advances from Gouy balances to superconducting quantum interference devices (SQUIDs); and demonstrates how χ feeds into design calculations for resonant circuits, MRI safety, and planetary exploration. Cross-links to the tesla article and magnetic permeability guide maintain consistent SI notation.

Definition, Units, and Material Classifications

Volume, mass, and molar susceptibilities

Volume susceptibility χv relates magnetisation per unit volume to field strength and is dimensionless. Mass susceptibility χm = χv/ρ carries units of m³·kg⁻¹, while molar susceptibility χmol = χv M/ρ uses m³·mol⁻¹, where ρ is density and M is molar mass. ISO 80000-12 recommends χ without subscript for volume susceptibility unless otherwise stated.

Magnetic material categories

Materials are classified by susceptibility magnitude: diamagnetics (χ ≈ −10⁻⁵), paramagnetics (χ ≈ 10⁻⁶ to 10⁻³), ferromagnetics (χ ≫ 1), and antiferro-/ferrimagnets with temperature-dependent χ. Relative permeability μᵣ relates to susceptibility via μᵣ = 1 + χ. Combining these relationships with the permeability explainer supports transformer, inductor, and shielding design.

Historical Development and Standardisation

Early magnetometry

Louis Néel and Pierre Curie pioneered susceptibility measurements in the late 19th and early 20th centuries. Gouy balances measured the force on a sample in a magnetic field gradient, while Quincke and Faraday methods examined liquids. The Curie law χ = C/T described paramagnetic behaviour, later extended by the Curie–Weiss law (χ = C/(T − Θ)) for ferromagnetics near transition temperatures.

Modern standards and instrumentation

Today, SQUID magnetometers and vibrating-sample magnetometers (VSMs) provide high-sensitivity χ measurements. ASTM A773 outlines magnetic permeability tests for soft magnetic materials, while IEC 60404 specifies methods for characterising electrical steels. Laboratories document calibration with reference materials such as palladium or gadolinium oxide to ensure traceability.

Conceptual Models and Tensor Behaviour

Anisotropy and tensor susceptibility

Crystalline materials often exhibit anisotropic susceptibility requiring tensor representation. The magnetisation vector M equals the susceptibility tensor χ̿ times H. Diagonal tensors arise in principal axes, while off-diagonal terms represent coupling. Single-crystal studies rely on neutron diffraction or torque magnetometry to resolve tensor components.

Nonlinear and frequency-dependent response

Ferromagnetic materials display nonlinear χ due to domain wall motion. AC susceptibility measurements reveal frequency-dependent behaviour, important for transformer cores and magnetic nanoparticles used in hyperthermia treatments. Complex susceptibility χ* = χ′ − jχ″ separates energy storage and loss, guiding design decisions for inductors and metamaterials.

Measurement Techniques

SQUID and VSM magnetometry

SQUID magnetometers detect magnetic flux changes down to 10⁻⁸ Φ₀, enabling ultra-sensitive susceptibility measurements for superconductors and biomaterials. VSM instruments oscillate samples within a uniform field; induced voltage in pickup coils reveals M and thus χ. Calibration routines include demagnetising factor corrections and background subtraction.

NMR and MRI-based susceptibility mapping

Magnetic resonance imaging (MRI) uses susceptibility to generate contrast and diagnose conditions such as microbleeds. Quantitative susceptibility mapping (QSM) reconstructs χ distributions from phase data. Nuclear magnetic resonance (NMR) spectroscopists correct chemical shift referencing using susceptibility data to avoid bulk magnetic susceptibility errors.

Applications Across Industries

Electronics and resonant circuits

Designers adjust susceptibility when selecting core materials for inductors and transformers. χ influences inductance, resonant frequency, and losses. Integrating material data with the LC resonant frequency calculator quantifies the impact on tuned circuits and wireless power systems.

Medical imaging and device safety

MRI compatibility standards limit χ differences between implants and body tissues to minimise artefacts and heating. Manufacturers characterise susceptibility to ensure devices meet ASTM F2052 and F2213. χ-driven analysis also informs targeted drug delivery using magnetic nanoparticles.

Geophysics and planetary exploration

Geologists map subsurface structures using magnetic susceptibility surveys. Portable kappameters log χ variations in drill cores, aiding mineral exploration. Planetary missions such as Mars rovers include magnet arrays to infer dust composition from magnetic adhesion. Geothermal developers combine susceptibility maps with the geothermal heat pump evaluator to optimise borehole placement in magnetically heterogeneous terrain.

Importance for Compliance, Safety, and Innovation

Regulatory and quality frameworks

Standards such as IEC 60601-2-33 specify susceptibility-related requirements for MRI systems. Aerospace and defence programmes mandate χ documentation for materials near magnetometers or navigation equipment to prevent bias. Quality systems record susceptibility drift due to ageing or thermal cycling to maintain calibration integrity.

Emerging technologies and sustainability

Novel materials—including high-entropy alloys and metamaterials—leverage engineered susceptibilities to achieve negative refractive indices or tunable magnetic responses. Recycling programmes test χ to sort ferrous and non-ferrous waste efficiently, reducing energy use. Energy storage systems monitor susceptibility changes in magnetocaloric materials to maintain performance.

Further Reading and Tools