5.1 Diamagnetism

Fundamentals of Diamagnetism

Diamagnetism is the weakest form of magnetism, but it's also the most universal. Every material exhibits it. When you place a diamagnetic material in an external magnetic field, the material generates a small opposing field that tries to push the external field away. In most materials, this effect gets overshadowed by stronger magnetic responses like paramagnetism or ferromagnetism, but in materials with no unpaired electrons, diamagnetism is the dominant behavior.

Understanding diamagnetism gives you a window into how electrons respond to magnetic fields at the atomic level, which connects directly to broader topics in condensed matter physics like superconductivity and electronic band structure.

Definition and Basic Principles

Diamagnetism is the tendency of a material to create a magnetic field that opposes an externally applied field. This opposition produces a negative magnetic susceptibility, meaning the material's magnetization points opposite to the applied field direction.

• Present in all materials, but only observable as the dominant effect when stronger magnetic behaviors (paramagnetism, ferromagnetism) are absent
• The diamagnetic response originates from changes in the orbital motion of electrons when an external field is applied
• The induced magnetic moment always opposes the applied field, consistent with Lenz's law

Larmor Diamagnetism

When an external magnetic field is applied, electron orbits don't just sit there unchanged. They precess around the field direction at a characteristic rate called the Larmor frequency:

$\omega_L = \frac{eB}{2m}$

where $e$ is the electron charge, $B$ is the applied magnetic field, and $m$ is the electron mass.

This precession effectively adds a tiny circulating current to each electron orbit. By Lenz's law, that current generates a magnetic moment opposing the applied field. The strength of this diamagnetic response scales with both the number of electrons in the atom and the size of their orbits (larger orbits mean stronger response).

Langevin Theory of Diamagnetism

The Langevin model gives a classical derivation of diamagnetic susceptibility by treating electrons as charged particles in circular orbits around atomic nuclei. The result is:

$\chi = -\frac{Ne^2\langle r^2 \rangle}{6mc^2}$

where $N$ is the number of atoms per unit volume, $e$ is the electron charge, $\langle r^2 \rangle$ is the mean square orbital radius of the electrons, $m$ is the electron mass, and $c$ is the speed of light.

The negative sign confirms the opposing nature of diamagnetism. Notice that $\chi$ grows with $\langle r^2 \rangle$, which is why heavier atoms with more extended electron clouds tend to show stronger diamagnetism. This explains the general trend of increasing diamagnetic susceptibility with atomic number across the periodic table.

Microscopic Origin

Diamagnetism is rooted in how electrons respond quantum mechanically to an applied magnetic field. The external field modifies electron wave functions and orbital currents, producing the small opposing moments that define diamagnetic behavior.

Induced Magnetic Moments

An external magnetic field distorts the electron cloud around each atom, inducing a small magnetic moment. These induced moments point opposite to the applied field, creating a net repulsive interaction.

• The magnitude depends on the electronic structure: atoms with more electrons and larger orbitals produce stronger induced moments
• In a quantum mechanical treatment, the induced moments can be calculated using second-order perturbation theory, where the applied field acts as the perturbation to the unperturbed atomic Hamiltonian

Lenz's Law in Diamagnetic Materials

Lenz's law states that induced currents always oppose the change in magnetic flux that created them. At the atomic scale, this means:

1. An external magnetic field is applied to the material
2. The field modifies electron orbital motion, effectively inducing tiny current loops within each atom
3. These current loops generate magnetic moments that oppose the applied field
4. The material is therefore repelled by both poles of a magnet (since the opposition is to the field itself, not its direction)

One clarification: the diamagnetic susceptibility is a static property of the material in a field. It does not depend on the rate of change of the applied field. The opposition described by Lenz's law here refers to the fundamental tendency of the induced moments to oppose the field, not a time-dependent induction effect like in a solenoid.

Quantum Mechanical Description

A full quantum treatment modifies the Hamiltonian of the electron system to include the vector potential of the applied field. The diamagnetic susceptibility then takes the form:

$\chi = -\frac{\mu_0 e^2}{6m} \sum_n \langle n | r^2 | n \rangle$

where $|n\rangle$ represents the occupied electronic states and the sum runs over all electrons.

Landau diamagnetism is a distinct contribution that arises specifically from free (conduction) electrons in metals. When a magnetic field quantizes the orbital motion of free electrons into Landau levels, the resulting energy shift produces a diamagnetic contribution. This is separate from the core-electron diamagnetism described by the Langevin formula.

Superconductors represent the extreme case: below the critical temperature, Cooper pairs carry supercurrents that perfectly screen the interior from any applied field, giving $\chi = -1$ (perfect diamagnetism, or the Meissner effect).

Properties of Diamagnetic Materials

Magnetic Susceptibility

Magnetic susceptibility ($\chi$) quantifies how strongly a material magnetizes in response to an applied field. For diamagnetic materials:

• $\chi$ is small and negative, typically on the order of $-10^{-5}$ (in SI, dimensionless volume susceptibility)
• The negative sign means the magnetization opposes the applied field
• Unlike paramagnetic materials, diamagnetic susceptibility is essentially independent of temperature
• Common measurement tools include SQUID magnetometers and vibrating sample magnetometers (VSMs)

Temperature Dependence

Diamagnetic susceptibility shows very weak temperature dependence. This is because the effect comes from the structure of filled electron shells, which doesn't change much with temperature.

Small variations can arise from thermal expansion altering interatomic distances and thus electron orbital radii. In materials that also have paramagnetic contributions, the paramagnetic part follows Curie's law ($\chi \propto 1/T$) and can mask the underlying diamagnetic signal at low temperatures. Separating these contributions often requires careful measurements across a wide temperature range.

Field Strength Effects

For typical laboratory fields, the diamagnetic response is linear: doubling the applied field doubles the induced magnetization (in the opposing direction). This linearity holds because the perturbation to electron orbits is tiny compared to the internal atomic fields.

Non-linear effects can appear in extremely strong fields (tens of tesla or more), where the magnetic energy becomes comparable to other energy scales in the material. Pulsed magnets and high-field superconducting magnets are used to probe these regimes.

Diamagnetism in Different States of Matter

Diamagnetism in Solids

• Crystalline solids can show anisotropic diamagnetism, meaning the susceptibility depends on the direction of the applied field relative to the crystal axes. Graphite is a classic example: its diamagnetic response is much stronger perpendicular to the carbon layers than parallel to them.
• In metals, conduction electrons contribute Landau diamagnetism on top of the core-electron (Langevin) diamagnetism. The total diamagnetic response in metals is the sum of both contributions.
• Insulators and semiconductors show diamagnetism primarily from tightly bound core electrons.

Diamagnetism in Liquids and Gases

Liquids generally exhibit weaker diamagnetism than their solid forms because molecular tumbling averages out anisotropic contributions. Noble gases are the cleanest examples of pure atomic diamagnetism since they have completely filled electron shells and no molecular interactions.

A note on molecular gases: $O_2$ is actually paramagnetic (it has two unpaired electrons), so its paramagnetic response dominates over its diamagnetic background. $N_2$, with all electrons paired, is purely diamagnetic.

Superconductors as Perfect Diamagnets

Superconductors below their critical temperature $T_c$ exhibit the Meissner effect: complete expulsion of magnetic flux from the bulk of the material. This corresponds to $\chi = -1$, or perfect diamagnetism.

• Type I superconductors maintain perfect flux expulsion up to a single critical field $H_c$, above which superconductivity is destroyed
• Type II superconductors allow partial flux penetration through quantized vortices between a lower critical field $H_{c1}$ and an upper critical field $H_{c2}$
• The microscopic origin is the formation of Cooper pairs, whose macroscopic quantum coherence sustains persistent screening currents that cancel the interior field

Measurement Techniques

SQUID Magnetometry

The Superconducting Quantum Interference Device (SQUID) is the most sensitive magnetometer available, capable of detecting fields as small as $\sim 10^{-14}$ T.

1. The sample is placed in a uniform applied field inside the SQUID system
2. The sample's magnetic moment induces a flux change in a superconducting pickup coil
3. This flux change is transferred to a SQUID sensor, which converts it to a measurable voltage via quantum interference effects in a superconducting loop containing Josephson junctions
4. The output voltage is proportional to the sample's magnetic moment

SQUID magnetometry is ideal for measuring weak diamagnetic signals, especially in small samples or thin films where the total moment is tiny.

Vibrating Sample Magnetometer

A vibrating sample magnetometer (VSM) works by oscillating the sample mechanically in a uniform magnetic field:

1. The sample vibrates at a known frequency within a set of detection coils
2. The time-varying magnetic flux from the moving sample induces a voltage in the coils (Faraday's law)
3. This voltage is proportional to the sample's magnetic moment

VSMs offer good sensitivity and fast measurement times. They're well-suited for samples with somewhat stronger diamagnetic signals or when you need to sweep the field quickly.

Faraday Balance Method

The Faraday balance measures the force on a sample placed in a region of known field gradient:

1. The sample is suspended from a sensitive balance in a non-uniform magnetic field
2. A diamagnetic sample experiences a force pushing it toward weaker field regions
3. The change in apparent weight is measured and related to the susceptibility through the field gradient

This method is particularly useful for studying how susceptibility varies with temperature, since the sample can be heated or cooled while suspended.

Applications of Diamagnetism

Magnetic Levitation

Because diamagnetic materials are repelled by magnetic fields, they can be stably levitated in a sufficiently strong field gradient. This is one of the few cases where stable passive levitation is possible (Earnshaw's theorem normally forbids it for ferromagnets, but diamagnets are the exception).

• Pyrolytic graphite and bismuth are commonly used for levitation demonstrations because of their relatively strong diamagnetic susceptibilities
• Practical applications include frictionless bearings and containerless materials processing, where the sample floats without touching any surface
• Maglev trains primarily use electromagnetic or electrodynamic levitation rather than pure diamagnetic levitation, though superconducting (Meissner effect) levitation is used in some designs

Diamagnetic Materials in Technology

• Magnetic shielding: diamagnetic enclosures can partially attenuate external fields around sensitive electronics, though high-permeability ferromagnetic shields (mu-metal) are far more effective for most applications
• Non-magnetic tools made from diamagnetic materials (certain copper alloys, plastics) are used in environments where stray magnetism would interfere with measurements
• Diamagnetic properties factor into the design of certain magnetic sensors and precision instruments

Biomedical Applications

• Diamagnetic levitation of cells and small organisms enables studies in simulated microgravity without going to space
• The diamagnetic susceptibility difference between tissues contributes to contrast in MRI: variations in local susceptibility create field inhomogeneities that affect signal intensity
• Magnetic manipulation of diamagnetic microparticles is being explored for targeted drug delivery and lab-on-a-chip biosensing platforms

Diamagnetism vs. Paramagnetism

Key Differences

Relative Strengths

Paramagnetic effects are typically 10 to 1000 times stronger than diamagnetic effects in materials with unpaired electrons. That's why diamagnetism is only the dominant response in materials where all electrons are paired.

Diamagnetism is always present as a background contribution, even in paramagnetic and ferromagnetic materials. Its strength increases with atomic number (more electrons, larger orbitals), while paramagnetic strength depends on the number and arrangement of unpaired electrons.

Coexistence in Materials

Most real materials exhibit both diamagnetic and paramagnetic contributions simultaneously. The measured susceptibility is the algebraic sum:

$\chi_{\text{total}} = \chi_{\text{dia}} + \chi_{\text{para}}$

Since $\chi_{\text{dia}} < 0$ and $\chi_{\text{para}} > 0$, the sign of the total susceptibility tells you which effect wins. Temperature is a useful tool for separating the two: as you cool a sample, the paramagnetic part grows (Curie law), while the diamagnetic part stays constant. Plotting $\chi$ vs. $1/T$ and extrapolating to $1/T = 0$ gives the diamagnetic contribution.

Notable Diamagnetic Materials

Bismuth and Graphite

Bismuth has the strongest diamagnetic susceptibility of any element, with $\chi \approx -1.66 \times 10^{-4}$. This makes it the go-to material for demonstrating diamagnetic effects.

Pyrolytic graphite is even more interesting because its diamagnetism is highly anisotropic. The susceptibility perpendicular to the carbon layers is much larger in magnitude than parallel to them, due to the delocalized $\pi$-electron system in the graphene planes. A small piece of pyrolytic graphite can visibly float above an array of permanent magnets.

Graphene and other 2D carbon materials show distinctive diamagnetic behavior tied to their unusual electronic band structure (Dirac cones), which is an active area of condensed matter research.

Water and Organic Compounds

Water is weakly diamagnetic ($\chi \approx -9.0 \times 10^{-6}$), which matters for MRI and for understanding how biomolecules behave in aqueous environments.

Aromatic organic compounds like benzene and naphthalene show enhanced diamagnetism due to ring currents: the delocalized $\pi$-electrons circulate around the aromatic ring in response to an applied field, generating a significant opposing moment. This ring-current effect is actually used in NMR spectroscopy to identify aromatic structures, since it shifts the resonance frequencies of nearby protons.

Liquid crystals exhibit anisotropic diamagnetism that can be exploited to align them with magnetic fields during manufacturing of display panels.

Noble Gases

All noble gases are diamagnetic because their electron shells are completely filled, leaving no unpaired electrons. Among them, xenon has the strongest diamagnetic response due to its large number of electrons and extended orbitals.

Noble gases serve as clean model systems for testing theories of atomic diamagnetism, since there are no complications from bonding, crystal fields, or unpaired spins.

Diamagnetism in Everyday Life

Earth's Atmosphere

The atmosphere is composed primarily of diamagnetic gases ($N_2$, $Ar$, $CO_2$), with the notable exception of paramagnetic $O_2$. The net magnetic character of air is very weak and doesn't meaningfully contribute to Earth's magnetic field, which is generated by convection currents in the liquid iron outer core. However, the magnetic susceptibility of air does need to be accounted for as a background correction in precision magnetometry experiments.

Biological Systems

Most biological molecules, including DNA, proteins, and lipids, are diamagnetic because their electrons are predominantly paired in covalent bonds. Water, the dominant component of living tissue, is also diamagnetic.

These diamagnetic properties are directly relevant to MRI, where susceptibility differences between tissues (e.g., between oxygenated and deoxygenated blood) create the contrast that makes imaging possible. Research into using magnetic fields to manipulate cell behavior and tissue growth exploits the small but real diamagnetic forces on biological matter.

Consumer Products

Most everyday materials you encounter, including plastics, glass, wood, and ceramics, are diamagnetic. Their magnetic response is so weak that you'd never notice it without specialized equipment.

A fun demonstration: place a thin piece of pyrolytic graphite on top of four small neodymium magnets arranged with alternating poles, and the graphite will hover stably. This is one of the simplest ways to see diamagnetism with your own eyes.