Magnetic Susceptibility

Susceptibility is a working clinical quantity. Quantitative susceptibility mapping (QSM), now routine on research MRI scanners, inverts the tiny field distortions that tissue $\chi$ differences imprint on the phase of the MR signal, producing voxel-by-voxel maps of $\chi$ — iron-laden deep-gray nuclei light up positive (ferritin is paramagnetic), myelin and calcification run negative. Radiologists in 2026 track Parkinson's-related iron accumulation and grade microbleeds by reading, literally, this article's number as an image contrast; the same physics read as artifact — distortion near air/tissue interfaces where $\chi$ jumps — is what every echo-planar protocol fights with shimming. In the materials lab, the Gouy rod has largely retired in favor of SQUID and vibrating-sample magnetometers, but the deliverable is unchanged: $\chi(T)$ curves whose $1/T$ tails, flat floors, and hysteretic blowups are exactly the three fingerprints — the first characterization plot in essentially every new-quantum-material paper. And magnetic separation is quietly industrial: high-gradient separators pull $+\chi$ minerals and paramagnetically labeled cells out of slurries and blood using precisely the $(\chi V/\mu_0)B\,\mathrm{d}B/\mathrm{d}z$ force this article weighed.

Every material you can put in a magnet gap answers the field somehow. The chapter's opening articles built the two microscopic mechanisms: Diamagnetism showed that induced orbital currents oppose any applied field (Lenz's law at the atom — universal, feeble, negative); Paramagnetism showed that pre-existing moments partially align with it (positive, feeble at room temperature, growing as $1/T$). The susceptibility is where those stories become a single engineering number: the gain of the material. Dial a free current, establish an $\mathbf{H}$, and the material replies with $\mathbf{M} = \chi_m\mathbf{H}$ — a faithful, scaled echo whose scale factor is the material.

Two features of this landscape do most of the work in practice. First, the sign is the single most informative bit: diamagnetism is universal but weak, so any positive $\chi_m$ announces unpaired moments overpowering the induced response; any negative one announces their absence. A diamagnet is gently expelled from strong field, a paramagnet gently drawn in — a force measurable on a good balance, which is precisely how susceptibilities are weighed below. Second, the smallness. Electric susceptibilities of ordinary insulators are of order one; magnetic ones are $10^{-6}$–$10^{-3}$. Ordinary matter's magnetic response is a whisper — which is why $\mathbf{B}$ and $\mu_0\mathbf{H}$ are numerically indistinguishable inside water or aluminium, and why everything dramatic in magnetism runs on the one family where cooperation amplifies the whisper into a shout. That family, iron's, breaks the linear law entirely: its "susceptibility" is a slope on a history-dependent curve (Ferromagnetism owns that story). The response dial also arrives with fine print inherited from its twin. In Linear Dielectrics, Susceptibility and Permittivity, the polarization was written $\mathbf{P} = \varepsilon_0\chi_e\mathbf{E}$ — response proportional to the physical field. The magnetic convention instead uses the auxiliary field, and the reason is worth owning rather than memorizing.

Here is the constitutive law, the one every materials table assumes:

where: $\mathbf{M}$ — magnetization, dipole moment per volume, A/m (Magnetization owns it); $\mathbf{H}$ — the auxiliary field, A/m (The Auxiliary Field H built it); $\chi_m$ — the magnetic susceptibility, dimensionless because $\mathbf{M}$ and $\mathbf{H}$ share units. The law claims linearity and isotropy; both hold superbly for dia- and paramagnets at laboratory fields, and fail by design for ferromagnets. Feed (1) into $\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M})$ and the response packages into $\mathbf{B} = \mu_0(1 + \chi_m)\mathbf{H}$ — the permeability ladder, built properly in Magnetic Permeability. Now the confession. The electric twin defines $\chi_e$ through $\mathbf{P} = \varepsilon_0\chi_e\mathbf{E}$, with $\mathbf{E}$ the total macroscopic field — the physical one. By strict analogy $\chi_m$ "should" be defined through $\mathbf{B}$: it is $\mathbf{B}$ that is physical, $\mathbf{B}$ that exerts forces. Instead, convention defines it through $\mathbf{H}$ — the bookkeeping field. Griffiths admits the inconsistency in a footnote to §6.4 and moves on; since this chapter has spent two articles teaching that $\mathbf{H}$ is accounting and $\mathbf{B}$ is physics, we owe the choice a defense. The defense is operational. In a dielectric experiment you set a voltage, which maps most directly onto $\mathbf{E}$. In a magnetic experiment you set a current, which maps onto $\mathbf{H}$: wind a toroid, drive $I$ through $N$ turns, and $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = NI$ pins $H$ from the ammeter alone, whatever fills the core — the material-blind reading Ampere's Law for H made the whole point of its toroid flagship. $\mathbf{B}$ inside the sample, by contrast, already contains the answer ($\mathbf{M}$) you are trying to characterize. A response coefficient is most useful when its "cause" axis is the experimentalist's knob; $\chi_m$ against $\mathbf{H}$ is exactly that. There is also a bookkeeping bonus: with (1), the constitutive chain closes in one line, $\mathbf{B} = \mu_0(1+\chi_m)\mathbf{H}$, with no self-consistency loop to solve — as the derivation now shows, together with the exact exchange rate between the two dictionaries.

One more line of honesty completes the mirror. The E3 convention is physically pure but operationally awkward — the total $\mathbf{E}$ inside a dielectric already contains the screening you are computing, which is why Dielectric Sphere in a Uniform Field had to solve a boundary-value problem rather than multiply two numbers. The magnetic convention is operationally clean but physically impure. Neither community chose wrong; they optimized different virtues, and the $(1+\chi)$ exchange rate is the peace treaty between them.

Real numbers, SI volume susceptibilities, room temperature unless flagged:

Where do the sizes come from? The microscopic theory was earned earlier in the chapter, and this article only quotes it, with credit. Diamagnetism derived the induced-orbit (Langevin) susceptibility, $\chi_m^{\rm dia} = -\mu_0 n e^2\langle r^2\rangle/6m_e$ — negative, temperature-flat, a few parts in $10^6$ for tightly bound electrons. Paramagnetism derived the two-level Boltzmann response and its high-temperature (Curie) tail, $\chi_m^{\rm para} \approx \mu_0 n\mu^2/k_B T$ — positive, growing as the thermal disorder that fights alignment weakens. Together they explain the landscape's left-right structure: all-paired atoms show only the dia floor; one unpaired moment per atom typically buys a $+10^{-5}$-scale para response that buries it. The quantitative Curie constant and the Curie–Weiss upgrade near ordering transitions are owned by Curie Law and Curie Temperature — here the $1/T$ shape is used only as a fingerprint.

The tree earns its keep on real unknowns. Sign comes from the direction of the force in a field gradient (next section) — one weighing. Temperature separates the two positive mechanisms: a Curie paramagnet's response doubles when you halve $T$; a diamagnet's does not move; a metal whose positive $\chi$ is temperature-flat is showing Pauli paramagnetism of its conduction electrons, named here and deferred to the solid-state track by Paramagnetism. And a sample whose "susceptibility" depends on what you did to it last week is not linear at all: hysteresis is the third fingerprint, and it removes the sample from this article's jurisdiction into Ferromagnetism's. One refinement flag, for the honest record. In precision work the $\mathbf{H}$ inside a finite sample is not the applied $\mathbf{H}_0$: the sample's own poles feed back a demagnetizing contribution (the $-N\mathbf{M}$ of The Auxiliary Field H's shape trio), so measured ratios need a shape correction before they equal the material's $\chi_m$. For $|\chi_m| \sim 10^{-5}$ the correction is parts in $10^5$ — utterly negligible; for large-$\chi$ materials it dominates. The systematic machinery is owned by Magnetic Boundary Conditions.

How do you measure a number like $-9\times10^{-6}$? Not with a compass. The classical answer, essentially unchanged since Faraday's 1840s survey, is to convert susceptibility into weight. A magnetized sample in a nonuniform field feels a net force — the $\mathbf{F} = \nabla(\mathbf{m}\cdot\mathbf{B})$ law that Forces and Torques on Magnetic Dipoles derived — proportional to $\chi_m$, sign included. Hang the sample from an analytical balance, energize the magnet, and read the apparent mass change: paramagnets get heavier, diamagnets lighter. Milligram resolution on a gram sample reads susceptibility to parts in $10^6$.

Every working scientist eventually opens a handbook and finds a susceptibility that seems wrong by orders of magnitude. Two traps guard the tables. The first is the $4\pi$: in Gaussian (CGS) units the constitutive relation reads $\mathbf{M} = \chi^{\rm CGS}\mathbf{H}$ with $\mathbf{B} = \mathbf{H} + 4\pi\mathbf{M}$, so the same physical response carries a susceptibility smaller by $4\pi$:

where: both numbers are dimensionless — which is exactly why nothing warns you. Worked once, remembered forever: water is $\chi^{\rm SI} = -9.05\times10^{-6}$, hence $\chi^{\rm CGS} = -9.05\times10^{-6}/4\pi = -7.20\times10^{-7}$, the "$-0.72\times10^{-6}$" of older tables. A CGS-vs-SI misread is a silent factor-12.57 error in any force or field estimate built on it. The second trap is which susceptibility the column reports. Volume susceptibility $\chi$ (dimensionless) is what field equations want; chemists tabulate per gram and per mole, because samples are weighed, not volumed:

where: $\rho$ — mass density; $M_{\rm mol}$ — molar mass. These are dimensional, so unit checking saves you. Water again, the full dictionary: $\chi = -9.05\times10^{-6}$ (SI volume) $\Rightarrow \chi_\rho = -9.05\times10^{-9}\,\mathrm{m^3/kg} \Rightarrow \chi_{\rm mol} = -1.63\times10^{-10}\,\mathrm{m^3/mol}$; and in the CGS-molar currency handbooks favor, $-13.0\times10^{-6}\,\mathrm{cm^3/mol}$ ("$-13.0$ emu/mol") — the standard literature value, recovered exactly, which is the round-trip check that certifies the dictionary.

1 · The Faraday balance reads water (confidence builder)

A $1.0\,\mathrm{cm^3}$ sealed water sample hangs from a microbalance at a calibrated spot where $B\,\mathrm{d}B/\mathrm{d}z = 10.0\,\mathrm{T^2/m}$, strong field below the sample. Predict the apparent mass change when the magnet energizes.

Step 1. Identify the response. Water: $\chi_m = -9.05\times10^{-6}$, diamagnetic — the induced moment opposes the field, so the force points away from strong field, i.e. upward here. Expect an apparent lightening.

Step 2. Apply the balance formula (2): $F_z = (\chi_m V/\mu_0)\,B\,\mathrm{d}B/\mathrm{d}z = (9.05\times10^{-6} \times 1.0\times10^{-6}/1.2566\times10^{-6}) \times 10.0 = 7.2\times10^{-5}\,\mathrm{N}$, directed up.

Step 3. Convert to balance language: $\Delta m_{\rm app} = F/g = 7.2\times10^{-5}/9.81 = 7.3\,\mathrm{mg}$ lighter — a $0.73\%$ effect on the sample's $1.00\,\mathrm{g}$ weight, enormous by microbalance standards. $\Delta m = -7.3\,\mathrm{mg}$.

Step 4. Sanity-check the scaling. The force is quadratic in field, so halving the magnet current cuts the reading to a quarter — a two-point check that distinguishes a genuine susceptibility signal from a ferromagnetic contamination speck, whose saturated moment gives a force merely linear in the gradient. Practitioners run exactly this test.

2 · The Gouy balance interrogates an unknown (exam level)

A cylindrical pellet column of cross-section $A = 0.50\,\mathrm{cm^2}$ hangs in a Gouy magnet with $B = 0.80\,\mathrm{T}$ at the bottom face and $\approx 0$ at the top. At $300\,\mathrm{K}$ the sample reads $1.50\,\mathrm{g}$ heavier with the field on; at $150\,\mathrm{K}$ it reads $3.0\,\mathrm{g}$ heavier. Classify the material and extract $\chi_m$ at both temperatures; give the $300\,\mathrm{K}$ value in CGS too.

Step 1. Fingerprint 1 — sign. Heavier means pulled into strong field: $\chi_m > 0$. Not a diamagnet.

Step 2. Extract the number. Invert the Gouy formula (3): $\chi_m = 2\mu_0 F/(A B^2)$ with $F = \Delta m\,g = 1.50\times10^{-3} \times 9.81 = 1.47\times10^{-2}\,\mathrm{N}$:

A strong paramagnet — transition-metal-salt territory, three orders above aluminium.

Step 3. Fingerprint 2 — temperature. Halving $T$ doubled the reading, so $\chi_m(150\,\mathrm{K}) = 2.3\times10^{-3} \propto 1/T$: Curie behavior, the signature of Paramagnetism's free-moment mechanism. (Flat reading: Pauli paramagnetism of a metal. History-dependent reading: ferromagnetic ordering — fingerprint 3 — and this whole linear analysis would be void.) The Curie constant this slope encodes belongs to Curie Law and Curie Temperature.

Step 4. Currency conversion. $\chi^{\rm CGS} = \chi^{\rm SI}/4\pi = 1.16\times10^{-3}/12.566 = 9.2\times10^{-5}$ — the number a 1960s handbook would print. $\chi_m(300\,\mathrm{K}) = 1.2\times10^{-3}$ SI $= 9.2\times10^{-5}$ CGS, Curie paramagnet. Note what made the whole measurement honest: $H$ inside the sample is the applied one to a part in $10^{3}$ (demagnetizing correction $\sim N\chi_m$, with the shape machinery of Magnetic Boundary Conditions) — for weak susceptibilities, geometry forgives.

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.4 (incl. the convention footnote) · Jackson, Classical Electrodynamics, 3rd ed., §5.8 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11 · Zangwill, Modern Electrodynamics, Ch. 13.

Provenance. All constitutive relations and conventions follow Griffiths §6.4 and Jackson §5.8. A verifier should recompute: (1) $\mu_r=1+\chi_m$ from the definition of $\mathbf{H}$; (2) the Curie-law coefficient $C=n\mu_0 m^2/3k_B$ via the Langevin small-$x$ expansion; (3) $\mathbf{J}_b=\chi_m\mathbf{J}_f$ from $\nabla\times\mathbf{M}$. Quoted susceptibility values (water $-9\times10^{-6}$, aluminium $+2.2\times10^{-5}$, bismuth $-1.7\times10^{-4}$, mu-metal $\mu_r\sim10^4$–$10^5$) are standard CRC/textbook figures and should be cross-checked against CRC Handbook tables. $\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}$. Addendum for this edition: all worked numbers recomputed in SI with $\mu_0 = 1.25663706\times10^{-6}$ N/A² — dictionary $\chi_B(\text{water}) = -9.050082\times10^{-6}$; CGS conversion $-9.05\times10^{-6}/4\pi = -7.20\times10^{-7}$; mass/molar chain $\chi_\rho = -9.05\times10^{-9}$ m³/kg, $\chi_{\rm mol} = -1.63\times10^{-10}$ m³/mol $= -13.0\times10^{-6}$ cm³/mol CGS; Faraday balance $7.2\times10^{-5}$ N ⇒ 7.3 mg on 1 cm³ water at 10 T²/m; Gouy bismuth 0.67 g at 1.0 T on 1 cm²; exam pellet $\chi = 1.16\times10^{-3}$ from 1.50 g at 0.80 T on 0.50 cm²; the microscopic $\chi$ formulas are quoted from the Diamagnetism and Paramagnetism articles, not re-derived here, and the Curie-law coefficient item in the verifier list above is discharged there.

Builds on: Magnetization · The Auxiliary Field H · Magnetic Dipole and Magnetic Moment · Ampere's Law for H

Leads to: Magnetic Permeability · Diamagnetism · Paramagnetism · Ferromagnetism · Linear and Nonlinear Magnetic Media