Diamagnetism

Diamagnetism is a property of matter in which an applied magnetic field induces a magnetization opposing that field. A diamagnetic material has negative magnetic susceptibility and, in a nonuniform field, tends to move toward regions of lower field strength. The response is usually weak, so substances commonly described as “nonmagnetic” can nevertheless be diamagnetic. Diamagnetic contributions coexist with other magnetic responses but may be outweighed by paramagnetism or ferromagnetism. (feynmanlectures.caltech.edu)

Macroscopic description

For an isotropic material in the linear-response regime, magnetization is expressed as

where $\mathbf M$ is magnetic dipole moment per unit volume, $\mathbf H$ is magnetic field strength, and $\chi$ is volume susceptibility. In the International System of Units, $\chi$ is dimensionless, while both $M$ and $H$ are measured in amperes per metre. Diamagnetism corresponds to $\chi<0$, so the induced magnetization points opposite to $\mathbf H$. (physics.ucsb.edu)

The magnetic flux density satisfies

where $\mu_0$ is the vacuum permeability. Thus, for a given internal $H$, a linear diamagnet has lower $B$ than a vacuum. Ordinary diamagnetic susceptibilities are very small compared with unity: the material slightly modifies the field rather than completely excluding it. This distinction separates weak diamagnetism from the near-complete field exclusion possible in superconductors. (physics.ucsb.edu)

Microscopic origin

The principal microscopic origin is the response of electrons to an applied field. Electrons possess magnetic moments associated with orbital motion and spin. In an atom or molecule without a permanent magnetic moment, the field can still modify the electronic motion and induce an opposing moment. Cancellation of pre-existing moments therefore does not imply an absence of magnetic response. (feynmanlectures.caltech.edu)

A useful intuitive picture comes from electromagnetic induction: switching on a field produces an electric field that changes electronic motion, with the induced magnetic response opposing the change. This picture gives the direction of the effect, but a complete account requires quantum mechanics, not literal classical electron orbits. (feynmanlectures.caltech.edu)

For an isolated particle, paired electrons often cancel their spin contributions, making diamagnetism dominant. This is a useful connection with electron configuration, but it is not a universal rule for solids. Their magnetic response also depends on the electronic states of the entire material; elemental bismuth, for example, is diamagnetic despite the unpaired electrons of an isolated bismuth atom. (physics.ucsb.edu)

Quantum and classical descriptions

A classical orbital calculation can reproduce important features of atomic diamagnetism, including an induced moment proportional to the applied field and opposite in direction. However, classical mechanics cannot determine the appropriate electronic distribution within an atom. The electron’s spatial distribution must instead be described quantum mechanically. (feynmanlectures.caltech.edu)

The Bohr–van Leeuwen theorem establishes a deeper limitation: under its assumptions, a system of classical charged particles in thermal equilibrium has no equilibrium magnetization. Classical induction arguments alone therefore cannot provide a complete equilibrium theory of ordinary material magnetism. The theorem explains why apparently successful semiclassical pictures must ultimately be supported by quantum theory. (feynmanlectures.caltech.edu)

In conducting solids, delocalized electrons also have an orbital magnetic response. Landau diamagnetism describes this contribution for an electron gas in a magnetic field. In the ideal free-electron model, it is negative and has one-third the magnitude of the positive spin contribution, known as Pauli paramagnetism. The combined conduction-electron response is therefore still paramagnetic in that model. A periodic crystal potential changes the relationship, so the free-electron ratio cannot be applied indiscriminately to real materials. (is.muni.cz)

Materials and magnetic forces

Diamagnetic substances include water, diamond, many plastics, and numerous biological materials. Bismuth and graphite exhibit comparatively strong diamagnetism among ordinary materials, although their responses remain small. These examples show that diamagnetism is neither restricted to metals nor dependent on electrical conductivity. (mri-q.com)

A field gradient is needed to produce a net translational force on a small, homogeneous diamagnetic object. For weak susceptibility, negligible disturbance of the applied field, and an object small enough to treat the gradient as approximately uniform,

where $V$ is its volume. Because $\chi$ is negative, the force points toward decreasing $B^2$. A strong but spatially uniform field is therefore not sufficient for lifting such an object. (mri-q.com)

With suitable field strength and gradients, this force can balance gravity. Demonstrations include levitated water and a frog suspended in a 16-tesla magnet. Diamagnetic materials can also stabilize the suspension of permanent magnets without active feedback. Such arrangements do not contradict the restrictions on stable suspension of fixed magnetic dipoles, because the diamagnetic response changes with the applied field. (mri-q.com)

Relationship to superconductivity

Superconductivity provides a distinct, much stronger diamagnetic response. In the Meissner effect, a superconductor expels magnetic flux from its bulk when entering the superconducting state under suitable conditions. Screening currents confine the field to a thin surface region rather than merely reducing it slightly. (feynmanlectures.caltech.edu)

For an ideal bulk specimen in the Meissner state, $B$ approaches zero internally, corresponding to $\chi=-1$ when susceptibility is defined relative to the internal $H$. This is often called perfect diamagnetism. Ordinary diamagnets do not exhibit comparable bulk field exclusion, and magnetic repulsion or levitation alone does not establish that a material is superconducting. (physics.ucsb.edu)