8.1 Diamagnetism and paramagnetism

Magnetic susceptibility

Magnetic susceptibility quantifies how strongly a material magnetizes in response to an applied magnetic field. Its sign and magnitude let you classify materials into three broad categories: diamagnetic (negative susceptibility), paramagnetic (positive susceptibility), or ferromagnetic (large positive susceptibility). In this section we focus on the first two.

Diamagnetic vs paramagnetic materials

• Diamagnetic materials have a negative susceptibility, meaning they are weakly repelled by an applied field. Common examples: bismuth, copper, water.
• Paramagnetic materials have a positive susceptibility, meaning they are weakly attracted to an applied field. Common examples: aluminum, platinum, molecular oxygen.
• Paramagnetic susceptibilities are typically several orders of magnitude larger than diamagnetic ones, though both are small compared to ferromagnets.

Magnetic susceptibility tensor

In anisotropic crystals the susceptibility depends on the direction of the applied field, so it becomes a second-rank tensor rather than a scalar. The susceptibility tensor is a 3×3 matrix relating the induced magnetization vector $\mathbf{M}$ to the applied field $\mathbf{H}$:

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

The principal axes of this tensor point along the directions of maximum and minimum susceptibility. For cubic crystals the tensor reduces to a scalar, but lower-symmetry structures (hexagonal, orthorhombic, etc.) can show measurably different susceptibilities along different crystallographic axes.

Measuring magnetic susceptibility

Several experimental techniques are used:

• Faraday balance — measures the force on a sample in a non-uniform magnetic field.
• Vibrating sample magnetometer (VSM) — oscillates the sample and detects the induced voltage in pickup coils.
• SQUID magnetometer — uses a superconducting quantum interference device for extremely sensitive measurements (down to $\sim 10^{-11}$ A·m²).

Temperature-dependent susceptibility measurements are especially useful: plotting $\chi$ vs. $T$ can reveal phase transitions and distinguish between Curie-law and Curie-Weiss behavior.

Diamagnetism

Diamagnetism is a weak, universal form of magnetism in which a material opposes an applied field. Every material is diamagnetic, but the effect is only observable when there are no unpaired electrons to produce a stronger paramagnetic signal.

Origin of diamagnetism

When an external magnetic field is applied, the orbital motion of electrons is slightly modified. By Lenz's law, the change in orbital current generates a magnetic moment that opposes the applied field. This is why the susceptibility is negative: the induced magnetization points opposite to the field.

Because this mechanism involves all electrons (paired or unpaired), diamagnetism is present in every material. It simply gets masked in paramagnetic or ferromagnetic materials by the much larger positive contribution from unpaired spins.

Diamagnetic susceptibility

• The diamagnetic susceptibility is negative and small, typically on the order of $10^{-5}$ to $10^{-6}$ in SI units.
• It is temperature-independent and does not vary with the strength of the applied field (the response is linear).
• The Langevin (or Larmor) expression for the diamagnetic susceptibility of an atom with $Z$ electrons is:

$\chi_{\text{dia}} = -\frac{\mu_0 n Z e^2}{6 m_e} \langle r^2 \rangle$

where $n$ is the number density of atoms, $e$ and $m_e$ are the electron charge and mass, and $\langle r^2 \rangle$ is the mean-square distance of the electrons from the nucleus. Larger, more diffuse electron clouds give a stronger diamagnetic response.

Diamagnetism in metals

Conduction electrons in metals contribute an additional diamagnetic term called Landau diamagnetism. It arises because the electron orbits are quantized into Landau levels in a magnetic field. The Landau susceptibility is:

$\chi_{\text{Landau}} = -\frac{1}{3}\chi_{\text{Pauli}}$

where $\chi_{\text{Pauli}}$ is the Pauli paramagnetic susceptibility of the conduction electrons. So in a free-electron metal, the Landau diamagnetic contribution is exactly one-third the magnitude of the Pauli paramagnetic contribution, and the net conduction-electron susceptibility remains paramagnetic. In real metals the ratio depends on the Fermi surface topology and band structure.

Diamagnetism in insulators

In insulators there are no conduction electrons, so the diamagnetic susceptibility comes entirely from the bound core electrons. You calculate it with the Langevin-Larmor formula above. Because there is no competing Pauli paramagnetism from free carriers, the net susceptibility of a closed-shell insulator is purely diamagnetic. Typical examples include noble gases (He, Ne, Ar) and ionic crystals like NaCl.

Diamagnetic levitation

Because diamagnetic materials are repelled by regions of strong field, a sufficiently powerful and non-uniform magnetic field can levitate them. Stable levitation occurs when the diamagnetic repulsive force exactly balances gravity. This has been demonstrated with water droplets, small frogs, and pieces of bismuth and graphite in fields of order 10–20 T. Practical applications include containerless processing of high-purity materials, where contact with a crucible would introduce contamination.

Paramagnetism

Paramagnetic materials contain atoms or ions with unpaired electrons, giving each atom a permanent magnetic dipole moment. Without an external field these moments point in random directions due to thermal agitation, so the net magnetization is zero. An applied field partially aligns them, producing a positive (attractive) magnetization.

Origin of paramagnetism

Each unpaired electron carries a magnetic moment from its spin angular momentum (and, in many ions, orbital angular momentum as well). In zero field, thermal energy $k_B T$ randomizes the orientation of these moments. When a field $\mathbf{B}$ is applied, the energy of a moment depends on its orientation: moments aligned with the field have lower energy. The Boltzmann distribution then favors partial alignment, giving a net magnetization in the direction of the field.

Paramagnetic susceptibility

• The paramagnetic susceptibility is positive and typically in the range $10^{-3}$ to $10^{-5}$ (SI).
• It is temperature-dependent: higher temperature means more thermal disorder, so the susceptibility decreases.
• The functional form follows Curie's law or the Curie-Weiss law, depending on whether interactions between moments are significant.

Curie's law

For non-interacting magnetic moments, the susceptibility obeys:

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

where $C$ is the Curie constant. The Curie constant is related to the effective moment by:

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

where $n$ is the number density of magnetic ions. Curie's law works well at high temperatures where $k_B T$ greatly exceeds the magnetic interaction energy between neighboring moments.

Curie-Weiss law

When interactions between moments are non-negligible, the susceptibility is modified to:

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

where $\theta$ is the Weiss constant (also called the Weiss temperature).

• $\theta > 0$: ferromagnetic interactions (moments tend to align parallel). The material undergoes a ferromagnetic transition near $T = \theta$.
• $\theta < 0$: antiferromagnetic interactions (moments tend to align antiparallel).
• $\theta = 0$: recovers Curie's law (non-interacting moments).

Plotting $1/\chi$ vs. $T$ gives a straight line whose slope is $1/C$ and whose $T$-intercept is $\theta$. This is one of the most common ways to extract $C$ and $\theta$ from experimental data.

Paramagnetic materials

• Transition metal ions such as $\text{Fe}^{3+}$ and $\text{Mn}^{2+}$ are paramagnetic because of partially filled 3d shells.
• Rare earth ions such as $\text{Gd}^{3+}$ and $\text{Er}^{3+}$ are paramagnetic due to partially filled 4f shells. Their moments are often well described by Hund's rules because the 4f electrons are shielded from crystal field effects.
• Molecular oxygen ($\text{O}_2$) is paramagnetic with two unpaired electrons, which is why liquid oxygen visibly clings to the poles of a strong magnet.
• Paramagnetic salts like Mohr's salt (ammonium iron(II) sulfate) are standard calibration materials for susceptibility measurements.

Langevin theory of paramagnetism

The Langevin theory is a classical treatment that models each magnetic moment as a vector free to point in any direction. It captures the essential competition between the aligning effect of the field and the randomizing effect of thermal energy.

Classical approach

1. Treat each magnetic moment $\boldsymbol{\mu}$ as a classical vector of fixed magnitude $\mu$.
2. In a field $\mathbf{B}$, the energy of a moment at angle $\theta$ to the field is $U = -\mu B \cos\theta$.
3. The probability of finding a moment at angle $\theta$ is given by the Boltzmann factor: $P(\theta) \propto \exp(\mu B \cos\theta / k_B T)$.
4. Average $\cos\theta$ over all solid angles using this probability distribution.

Langevin function

The result of step 4 is the Langevin function:

$L(\alpha) = \coth(\alpha) - \frac{1}{\alpha}$

where $\alpha = \mu B / k_B T$ is the ratio of magnetic energy to thermal energy. The average magnetization per moment is $\langle \mu_z \rangle = \mu \, L(\alpha)$.

Key limits:

• Small $\alpha$ (high $T$ or weak field): $L(\alpha) \approx \alpha/3$, which gives $\chi = n\mu_0\mu^2 / 3k_BT$. This recovers Curie's law.
• Large $\alpha$ (low $T$ or strong field): $L(\alpha) \to 1$, meaning all moments are fully aligned and the magnetization saturates at $M_{\text{sat}} = n\mu$.

Limitations of Langevin theory

• It treats moments as classical vectors with continuous orientations. Real atomic moments are quantized ($m_J$ takes discrete values).
• It ignores interactions between moments, so it cannot describe ordering transitions.
• It does not account for crystal field effects, which can quench orbital angular momentum in transition metal ions and significantly alter the effective moment.

For these reasons, the quantum theory of paramagnetism is needed for quantitative accuracy, especially at low temperatures.

Quantum theory of paramagnetism

The quantum treatment replaces the continuous classical moment with a quantized total angular momentum $J$. This gives the correct low-temperature behavior and accurately predicts effective moments for specific ions.

Spin and orbital angular momentum

• The spin quantum number $S$ comes from unpaired electrons. Each electron contributes $s = 1/2$.
• The orbital quantum number $L$ comes from the orbital motion of electrons around the nucleus and takes integer values.
• The total angular momentum $J$ is the vector coupling of $L$ and $S$. Its allowed values range from $|L - S|$ to $L + S$ in integer steps.

The magnetic moment associated with $J$ is:

$\mu_J = g_J \sqrt{J(J+1)} \, \mu_B$

where $g_J$ is the Landé g-factor and $\mu_B$ is the Bohr magneton.

Hund's rules

Hund's rules determine the ground-state values of $S$, $L$, and $J$ for a free ion:

1. Maximize $S$: fill orbitals with parallel spins first (minimizes electron-electron repulsion via exchange).
2. Maximize $L$: consistent with rule 1, place electrons to give the largest total orbital angular momentum.
3. Determine $J$:
   • If the shell is less than half-filled: $J = |L - S|$
   • If the shell is more than half-filled: $J = L + S$
   • If the shell is exactly half-filled: $L = 0$, so $J = S$

These rules work very well for rare earth ions (4f electrons are well shielded). For transition metal ions (3d), crystal field effects often quench the orbital contribution, and the effective moment is closer to the spin-only value $\mu_{\text{eff}} \approx 2\sqrt{S(S+1)} \, \mu_B$.

Effective magnetic moment

The effective magnetic moment $\mu_{\text{eff}}$ is defined through the Curie constant:

$\mu_{\text{eff}} = g_J \sqrt{J(J+1)} \, \mu_B$

The Landé g-factor is:

$g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$

You can extract $\mu_{\text{eff}}$ experimentally from a $1/\chi$ vs. $T$ plot: the slope gives $1/C$, and from $C$ you calculate $\mu_{\text{eff}}$. Comparing the measured value with the Hund's rule prediction tells you whether orbital angular momentum is quenched.

Van Vleck paramagnetism

Van Vleck paramagnetism is a temperature-independent paramagnetic contribution. It appears in ions whose ground state has $J = 0$ (no first-order paramagnetic response), but where the applied field mixes in low-lying excited states through second-order perturbation theory.

The Van Vleck susceptibility involves matrix elements of the magnetic moment operator between the ground state and excited states:

$\chi_{\text{VV}} = 2n \sum_{n \neq 0} \frac{|\langle 0 | \hat{\mu}_z | n \rangle|^2}{E_n - E_0}$

Because the energy denominators are fixed (not thermal), this contribution does not depend on temperature. It is significant in ions like $\text{Eu}^{3+}$ (ground state $^7F_0$, $J = 0$) and $\text{Sm}^{3+}$, where the $J$-multiplet spacing is comparable to $k_BT$.

Applications

The distinct magnetic responses of diamagnetic and paramagnetic materials underpin a range of technologies across medicine, engineering, and materials science.

Magnetic resonance imaging (MRI)

MRI exploits the paramagnetism of hydrogen nuclei (proton spin) in the body. A strong static field partially aligns the nuclear spins, and radiofrequency pulses tip them away from equilibrium. The relaxation signal encodes spatial information to build detailed images of soft tissue. Paramagnetic contrast agents, most commonly gadolinium-based complexes ($\text{Gd}^{3+}$, with seven unpaired 4f electrons), shorten relaxation times of nearby water protons, enhancing contrast between different tissues.

Magnetic levitation

Diamagnetic levitation uses the repulsive force on a diamagnetic object in a strongly non-uniform field. While maglev trains primarily rely on superconducting or electromagnetic levitation (not simple diamagnetism), true diamagnetic levitation is used in laboratory settings for containerless processing. Removing contact with a crucible prevents contamination, which is valuable for growing ultra-pure crystals or studying molten materials.

Magnetic refrigeration

Magnetic refrigeration exploits the magnetocaloric effect in paramagnetic (or ferromagnetic) materials. The cycle works in two steps:

1. Magnetize the material isothermally: the moments align, reducing magnetic entropy, and the material releases heat to a sink.
2. Demagnetize adiabatically: the moments randomize, absorbing thermal energy from the lattice, and the material cools.

Paramagnetic salts like cerium magnesium nitrate are used for sub-kelvin cooling (adiabatic demagnetization refrigeration). Near room temperature, materials with ferromagnetic transitions (e.g., gadolinium near its Curie point of 294 K) are more practical. Magnetic refrigeration avoids greenhouse-gas refrigerants and can achieve high thermodynamic efficiency.

Magnetic separation

Magnetic separation uses field gradients to sort materials by their susceptibility. Paramagnetic minerals like ilmenite ($\text{FeTiO}_3$) and wolframite are pulled toward regions of high field, while diamagnetic gangue minerals like quartz and calcite are not. In biotechnology, paramagnetic beads coated with antibodies or other ligands selectively bind target biomolecules (proteins, nucleic acids), which are then pulled out of solution with a magnet for purification.