5.2 Paramagnetism

Fundamentals of Paramagnetism

Paramagnetism describes how materials with unpaired electrons respond to external magnetic fields. Those unpaired electrons act as tiny magnetic dipoles that partially align with an applied field, producing a weak net attraction. Unlike ferromagnets, paramagnets lose their magnetization the moment you remove the field. Understanding paramagnetism gives you direct insight into the electronic structure of solids and connects quantum mechanics to measurable magnetic behavior.

Definition and Characteristics

A paramagnetic material has a positive magnetic susceptibility, meaning it's weakly attracted toward regions of stronger magnetic field. The mechanism is straightforward: unpaired electrons carry magnetic moments, and an external field tends to align those moments along the field direction. However, thermal energy constantly randomizes the orientations, so only a fraction of the moments align at any given instant.

Key features:

• Paramagnetism requires unpaired electrons (found in transition metals, rare earth elements, free radicals, and some molecules like $O_2$)
• The magnetization is proportional to the applied field at low to moderate field strengths
• The effect is inherently temperature-dependent: higher temperature means more thermal disorder and weaker net magnetization
• Once the external field is removed, the net magnetization drops to zero

Curie's Law

Curie's law captures the inverse relationship between paramagnetic susceptibility and temperature:

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

Here, $\chi$ is the magnetic susceptibility, $C$ is the Curie constant, and $T$ is the absolute temperature in Kelvin.

The Curie constant depends on the material's intrinsic properties, specifically the number density of magnetic ions and the magnitude of their magnetic moments. A larger $C$ means a stronger paramagnetic response at any given temperature.

The physical reasoning: as temperature rises, thermal agitation increasingly disrupts the alignment of magnetic moments, so susceptibility drops as $1/T$. This is one of the cleanest predictions in magnetism and holds well for dilute paramagnetic systems.

Langevin Theory

The Langevin theory provides a classical statistical-mechanics treatment of paramagnetism. It models each atom as a magnetic dipole free to point in any direction, then uses Boltzmann statistics to calculate the average alignment in a field.

The result is the Langevin function:

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

where $\alpha = \frac{\mu B}{k_B T}$ is the ratio of magnetic energy to thermal energy ($\mu$ is the magnetic moment, $B$ the applied field, and $k_B T$ the thermal energy).

Two important limits to know:

• Small $\alpha$ (weak field or high temperature): $L(\alpha) \approx \alpha/3$, which recovers Curie's law with magnetization linear in $B/T$
• Large $\alpha$ (strong field or low temperature): $L(\alpha) \rightarrow 1$, meaning all moments are fully aligned and magnetization saturates

Magnetic Susceptibility

Magnetic susceptibility ($\chi$) quantifies how strongly a material magnetizes in response to an applied field. For paramagnets, $\chi$ is positive but small (typically $10^{-3}$ to $10^{-5}$), and its behavior as a function of temperature and field reveals a lot about the underlying physics.

Temperature Dependence

In an ideal paramagnet, susceptibility follows $\chi \propto 1/T$. Plotting $1/\chi$ versus $T$ should give a straight line through the origin.

Deviations from this ideal behavior occur when:

• Magnetic moments interact with each other (exchange coupling), shifting the line away from the origin
• Crystal field effects modify the energy level spacing of the magnetic ions
• At very low temperatures, quantum effects dominate and the classical picture breaks down
• The system approaches a magnetic ordering transition

Field Dependence

For weak to moderate fields, the magnetization is linear in the applied field:

$M = \chi H$

This means $\chi$ is essentially constant, independent of field strength. Non-linear behavior shows up at high fields, where you start to saturate the magnetic moments. In the saturation regime, increasing the field further produces diminishing gains in magnetization because most moments are already aligned.

Curie-Weiss Law

Real paramagnetic materials often have interactions between magnetic moments. The Curie-Weiss law accounts for this:

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

The Weiss constant $\theta$ encodes the nature of the interactions:

• $\theta > 0$: ferromagnetic-type interactions (moments prefer parallel alignment)
• $\theta < 0$: antiferromagnetic-type interactions (moments prefer antiparallel alignment)
• $\theta = 0$: no interactions, reducing back to Curie's law

Experimentally, you extract $\theta$ from a plot of $1/\chi$ vs. $T$. The intercept on the temperature axis gives $\theta$, and the slope gives $1/C$.

Quantum Theory of Paramagnetism

Classical Langevin theory works reasonably well, but a proper quantum treatment is needed for quantitative accuracy. Quantum mechanics introduces discrete energy levels and quantum numbers that determine the allowed orientations of magnetic moments in a field.

Spin Paramagnetism

This contribution comes from the intrinsic spin angular momentum of unpaired electrons. Each electron has spin quantum number $S = 1/2$, giving two possible projections along the field direction (spin-up and spin-down).

The spin magnetic moment is:

$\mu_s = -g_s \mu_B S$

where $g_s \approx 2.0023$ is the electron spin g-factor and $\mu_B$ is the Bohr magneton ($9.274 \times 10^{-24}$ J/T), the natural unit of electronic magnetic moment.

Spin paramagnetism is the dominant contribution in transition metal ions and free radicals, where orbital angular momentum is often quenched by the crystal field.

Orbital Paramagnetism

Electrons orbiting the nucleus also carry angular momentum, which generates an additional magnetic moment:

$\mu_l = -\mu_B L$

where $L$ is the orbital angular momentum quantum number. In free atoms, this contribution can be substantial. However, in most solids, the crystal field locks the orbital motion to the lattice, quenching the orbital contribution. This is why many transition metal compounds have magnetic moments close to the "spin-only" value.

The notable exception is the lanthanide (rare earth) ions, where the 4f electrons are buried deep inside the atom, shielded from the crystal field by the outer 5s and 5p shells. For these ions, orbital contributions remain significant.

Total Angular Momentum

The total angular momentum $J$ combines spin and orbital parts. For most atoms in condensed matter, Russell-Saunders (LS) coupling applies:

$\vec{J} = \vec{L} + \vec{S}$

The total magnetic moment is then:

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

where $g_J$ is the Landé g-factor:

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

This g-factor interpolates between $g = 1$ (pure orbital) and $g = 2$ (pure spin), depending on the relative contributions of $L$ and $S$. The value of $J$ is determined by Hund's rules, which you use to find the ground-state term of an ion.

Paramagnetic Materials

Rare Earth Elements

The lanthanide series (Ce through Lu) exhibits strong paramagnetism due to unpaired 4f electrons. Because the 4f shell is shielded by outer electron shells, crystal field effects are weak, and the free-ion approximation works well. These ions carry large magnetic moments and follow the Curie-Weiss law closely.

• Gadolinium ($Gd^{3+}$) has seven unpaired 4f electrons, giving $S = 7/2$ and one of the largest magnetic moments among the lanthanides
• The effective moments of rare earth ions generally agree well with the theoretical values predicted from Hund's rules and LS coupling

Transition Metals

Paramagnetism in transition metals arises from unpaired electrons in partially filled 3d orbitals. Unlike the lanthanides, the 3d electrons are exposed to the crystal field, which strongly affects their energy levels and magnetic behavior.

• The magnetic moment depends on the oxidation state and coordination geometry (octahedral, tetrahedral, square planar, etc.)
• Crystal field splitting can change the number of unpaired electrons. For example, $Fe^{2+}$ can be high-spin (4 unpaired electrons) or low-spin (0 unpaired electrons) depending on the ligand field strength
• Some transition metal compounds show temperature-independent paramagnetism (Van Vleck paramagnetism), arising from mixing of excited states into the ground state
• Copper(II) complexes, with a single unpaired electron, are classic examples of simple paramagnetic behavior

Alkali Metals

Alkali metals display a different kind of paramagnetism called Pauli paramagnetism, which comes from the conduction electrons rather than localized unpaired electrons.

In a free electron gas, most electrons are paired (one spin-up, one spin-down at each energy level). Only electrons near the Fermi surface can flip their spins in response to a field, so the susceptibility is small and nearly temperature-independent:

$\chi_P = \mu_B^2 \, g(E_F)$

where $g(E_F)$ is the density of states at the Fermi level. Pauli paramagnetic susceptibilities are typically orders of magnitude smaller than those of transition metal or rare earth compounds.

Experimental Techniques

Magnetic Resonance

Electron Paramagnetic Resonance (EPR), also called ESR, is the go-to technique for probing unpaired electrons. You place the sample in a magnetic field and irradiate it with microwave radiation. When the microwave photon energy matches the Zeeman splitting between spin states, resonance absorption occurs.

EPR provides:

• The g-factor of the paramagnetic species
• Hyperfine coupling constants (interactions between electron and nuclear spins)
• Information about the local symmetry and bonding environment
• Detection of free radicals in chemical and biological systems

Nuclear Magnetic Resonance (NMR) probes nuclear spins rather than electron spins. In paramagnetic materials, the large electron moments shift and broaden NMR lines, which can be used to map out the local magnetic environment around specific nuclei.

Susceptibility Measurements

• SQUID magnetometry (Superconducting Quantum Interference Device) is the most sensitive technique, capable of detecting magnetic moments as small as $\sim 10^{-8}$ emu. It measures the flux change as a sample moves through superconducting pickup coils
• Vibrating Sample Magnetometer (VSM) oscillates the sample near pickup coils and measures the induced voltage, giving magnetization as a function of field and temperature
• Faraday balance measures the force on a sample in a non-uniform magnetic field, from which susceptibility is extracted

These methods allow you to determine Curie constants, Weiss temperatures, and effective magnetic moments from $\chi(T)$ and $M(H)$ data.

Neutron Scattering

Neutrons carry a magnetic moment, making them uniquely suited to probe magnetism at the atomic scale.

• Elastic neutron scattering (diffraction) reveals the spatial arrangement of magnetic moments, distinguishing between ferromagnetic, antiferromagnetic, and more complex orderings
• Inelastic neutron scattering measures magnetic excitations such as spin waves and crystal field transitions
• Particularly valuable for studying paramagnetic correlations above ordering temperatures, where short-range magnetic order may persist even without long-range order

Applications of Paramagnetism

Magnetic Cooling

Adiabatic demagnetization exploits the magnetocaloric effect to reach ultra-low temperatures (below 1 K). The process works in three steps:

1. Isothermal magnetization: Apply a strong magnetic field to a paramagnetic salt while it's in thermal contact with a heat bath. The moments align, reducing magnetic entropy, and the released heat flows into the bath.
2. Thermal isolation: Disconnect the sample from the heat bath.
3. Adiabatic demagnetization: Slowly remove the magnetic field. The moments randomize, increasing magnetic entropy. Since total entropy is conserved (adiabatic process), the lattice entropy must decrease, cooling the sample.

Materials with large magnetic moments work best. Gadolinium gallium garnet (GGG) and cerium magnesium nitrate (CMN) are commonly used paramagnetic refrigerants.

Contrast Agents in MRI

Paramagnetic ions, especially $Gd^{3+}$, are used as contrast agents in medical MRI. The large magnetic moment of gadolinium shortens the $T_1$ (spin-lattice) relaxation time of nearby water protons, brightening the signal in $T_1$-weighted images.

• Gadolinium is administered in chelated form (e.g., Gd-DTPA) to prevent toxicity from free $Gd^{3+}$ ions
• These agents improve visualization of blood vessels, tumors, and inflammation
• Manganese-based agents offer an alternative, particularly for liver imaging

Sensors and Detectors

• Paramagnetic oxygen analyzers exploit the fact that $O_2$ is one of the few common gases that is paramagnetic. By measuring the magnetic susceptibility of a gas mixture, you can determine the oxygen concentration. These analyzers are widely used in hospitals, industrial processes, and environmental monitoring
• Magnetoresistive sensors incorporating paramagnetic materials detect changes in magnetic field strength for various applications

Paramagnetism vs Other Magnetic States

Diamagnetism vs Paramagnetism

Diamagnetism is always present, but in materials with unpaired electrons, the paramagnetic contribution is typically much larger and dominates. Superconductors are a special case, exhibiting perfect diamagnetism (the Meissner effect) with $\chi = -1$.

Paramagnetism vs Ferromagnetism

Ferromagnets have strong exchange interactions that cause neighboring moments to align spontaneously below the Curie temperature ($T_C$). Above $T_C$, thermal energy overcomes these interactions and the material becomes paramagnetic.

Key differences:

• Ferromagnets retain magnetization without an applied field; paramagnets do not
• Ferromagnets show hysteresis and magnetic domains; paramagnets show neither
• Ferromagnetic susceptibility is enormous ($10^1$ to $10^5$) compared to paramagnetic values
• Common ferromagnets at room temperature: iron ($T_C = 1043$ K), nickel ($T_C = 627$ K), cobalt ($T_C = 1394$ K)

Paramagnetism vs Antiferromagnetism

In antiferromagnets, neighboring moments align antiparallel due to exchange interactions, producing zero net magnetization below the Néel temperature ($T_N$). Above $T_N$, the material becomes paramagnetic.

• Both paramagnets and antiferromagnets have zero net magnetization without an applied field, but for different reasons: paramagnets have randomly oriented moments, while antiferromagnets have ordered but canceling moments
• Antiferromagnets show a characteristic peak in $\chi(T)$ at $T_N$, whereas paramagnetic susceptibility decreases monotonically with temperature
• Manganese oxide (MnO), with $T_N = 118$ K, is a classic antiferromagnet

Temperature Effects

Curie Temperature

The Curie temperature marks the boundary between ordered (ferromagnetic) and disordered (paramagnetic) phases. Above $T_C$, thermal fluctuations destroy the long-range magnetic order.

• Near $T_C$, the susceptibility diverges and follows the Curie-Weiss law: $\chi = C/(T - T_C)$
• Critical phenomena near the transition involve power-law behavior with universal critical exponents
• Iron becomes paramagnetic above 1043 K; nickel above 627 K

Low Temperature Behavior

At low temperatures, the classical Curie law breaks down and quantum mechanics takes over.

• Energy level spacings become comparable to $k_B T$, so you can't treat the moment orientations as continuous
• Crystal field splitting and spin-orbit coupling determine which energy levels are thermally populated
• Some paramagnetic materials undergo magnetic ordering transitions at very low temperatures
• Susceptibility tends toward a finite value as $T \rightarrow 0$ rather than diverging, consistent with the third law of thermodynamics
• Cerium magnesium nitrate (CMN) remains paramagnetic down to millikelvin temperatures, making it useful as a magnetic thermometer in that regime

High Temperature Limit

When thermal energy greatly exceeds all other energy scales (crystal field splittings, spin-orbit coupling), the system approaches classical behavior and Curie's law holds well.

• Plotting $1/\chi$ vs. $T$ at high temperatures gives a reliable straight line for extracting $C$ and $\theta$
• Orbital contributions that were quenched at lower temperatures may become partially unquenched
• At sufficiently high temperatures, structural phase transitions or chemical decomposition can occur before the true high-temperature paramagnetic limit is reached

Microscopic Origins

Unpaired Electrons

Unpaired electrons are the primary source of paramagnetism. They occur whenever an atom, ion, or molecule has partially filled orbitals.

• Transition metal ions: unpaired 3d electrons (e.g., $Cu^{2+}$ has one, $Mn^{2+}$ has five)
• Rare earth ions: unpaired 4f electrons (e.g., $Gd^{3+}$ has seven)
• Free radicals: molecules with an odd number of electrons
• Molecular oxygen ($O_2$): has two unpaired electrons in antibonding $\pi^*$ orbitals

The number and arrangement of unpaired electrons directly determine the magnitude of the paramagnetic response.

Magnetic Moments

The magnetic moment of an ion is determined by its total angular momentum quantum number $J$:

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

For transition metal ions where orbital momentum is quenched, the spin-only formula applies:

$\mu_{eff} = 2\mu_B\sqrt{S(S+1)} = \mu_B\sqrt{n(n+2)}$

where $n$ is the number of unpaired electrons. This formula predicts, for example, $\mu_{eff} \approx 1.73 \, \mu_B$ for one unpaired electron and $\mu_{eff} \approx 5.92 \, \mu_B$ for five.

For rare earth ions, you need the full expression with $J$ and $g_J$ because orbital momentum is not quenched.

Spin-Orbit Coupling

Spin-orbit coupling is the interaction between an electron's spin magnetic moment and the magnetic field generated by its orbital motion around the nucleus. Its strength scales roughly as $Z^4$ (where $Z$ is the atomic number), so it's much more important for heavy elements.

Effects of spin-orbit coupling:

• Mixes spin and orbital angular momenta, making $J$ (not $L$ and $S$ separately) the good quantum number
• Produces fine structure in atomic spectra
• Leads to magnetocrystalline anisotropy in solids, where the magnetization prefers certain crystallographic directions
• In rare earth ions, strong spin-orbit coupling is responsible for their large and anisotropic magnetic moments

Paramagnetic Ions

Free Ion Approximation

The free ion approximation treats each paramagnetic ion as if it were isolated, ignoring the surrounding crystal environment. You apply Hund's rules to determine the ground-state term ($L$, $S$, and $J$), then calculate the expected magnetic moment.

This works well for:

• Rare earth ions, where the 4f electrons are shielded from the crystal field
• Dilute paramagnetic systems where ion-ion interactions are negligible

For trivalent rare earth ions ($Gd^{3+}$, $Dy^{3+}$, $Er^{3+}$, etc.), the measured effective moments typically agree closely with free-ion predictions.

Crystal Field Effects

When a paramagnetic ion sits inside a crystal, the surrounding ions and ligands create an electrostatic potential (the crystal field) that breaks the degeneracy of the d or f orbitals.

• In an octahedral crystal field, the five d orbitals split into a lower $t_{2g}$ set (three orbitals) and an upper $e_g$ set (two orbitals)
• In a tetrahedral field, the splitting is reversed and smaller in magnitude
• The competition between crystal field splitting energy and electron pairing energy determines whether an ion is high-spin or low-spin
• Crystal field symmetry also determines the magnetic anisotropy of the ion

Quenching of Orbital Momentum

In many transition metal compounds, the crystal field locks the d-electron orbitals into fixed spatial orientations, preventing them from carrying orbital angular momentum. This is called orbital quenching.

When orbital momentum is fully quenched, the magnetic moment reduces to the spin-only value. This is why the spin-only formula works so well for first-row transition metals in octahedral or tetrahedral environments.

Quenching is incomplete when:

• The ground state is orbitally degenerate (e.g., $T$ ground terms in octahedral symmetry)
• Spin-orbit coupling mixes in orbital contributions from excited states
• The crystal field is weak (as in rare earth compounds)

For example, $Ni^{2+}$ in octahedral coordination has a quenched orbital moment, giving $\mu_{eff} \approx 2.83 \, \mu_B$, close to the spin-only value for two unpaired electrons.