physics: Diamagnetism

Diamagnetism is the property of an object or material that causes it to create a magnetic field in opposition to an externally applied magnetic field. It is a quantum mechanical effect that occurs in all materials; where it is the only contribution to the magnetism the material is called a diamagnet. Unlike a ferromagnet, a diamagnet is not a permanent magnet. Its magnetic permeability is less than μ₀ (the permeability of free space). In most materials diamagnetism is a weak effect, but a superconductor repels the magnetic field entirely, apart from a thin layer at the surface.
Diamagnets were first discovered when Sebald Justinus Brugmans observed in 1778 that bismuth and antimony were repelled by magnetic fields. The term diamagnetism was coined by Michael Faraday in September 1845, when he realized that every material responded (in either a diamagnetic or paramagnetic way) to an applied magnetic field

Materials

Notable diamagnetic materials^[1]
Material	χ_v (× 10⁻⁵)
Superconductor	−10⁵
Pyrolytic carbon	−40.0
Bismuth	−16.6
Mercury	−2.9
Silver	−2.6
Carbon (diamond)	−2.1
Lead	−1.8
Carbon (graphite)	−1.6
Copper	−1.0
Water	−0.91

Diamagnetism, to a greater or lesser degree, is a property of all materials and always makes a weak contribution to the material's response to a magnetic field. However, for materials that show some other form of magnetism (such as ferromagnetism or paramagnetism), the diamagnetic contribution becomes negligible. Substances that mostly display diamagnetic behaviour are termed diamagnetic materials, or diamagnets. Materials called diamagnetic are those that non-physicists generally think of as non-magnetic, and include water, wood, most organic compounds such as petroleum and some plastics, and many metals including copper, particularly the heavy ones with many core electrons, such as mercury, gold and bismuth. The magnetic susceptibility of various molecular fragments are called Pascal's constants.
Diamagnetic materials have a relative magnetic permeability that is less than or equal to 1, and therefore a magnetic susceptibility less than or equal to 0, since susceptibility is defined as χ_v = μ_v − 1. This means that diamagnetic materials are repelled by magnetic fields. However, since diamagnetism is such a weak property its effects are not observable in everyday life. For example, the magnetic susceptibility of diamagnets such as water is χ_v = −9.05×10⁻⁶. The most strongly diamagnetic material is bismuth, χ_v = −1.66×10⁻⁴, although pyrolytic carbon may have a susceptibility of χ_v = −4.00×10⁻⁴ in one plane. Nevertheless, these values are orders of magnitudes smaller than the magnetism exhibited by paramagnets and ferromagnets. Note that because χ_v is derived from the ratio of the internal magnetic field to the applied field, it is a dimensionless value.
All conductors exhibit an effective diamagnetism when they experience a changing magnetic field. The Lorentz force on electrons causes them to circulate around forming eddy currents. The eddy currents then produce an induced magnetic field opposite the applied field, resisting the conductor's motion.

Superconductors

Transition from ordinary conductivity (left) to superconductivity (right). At the transition, the superconductor expels the magnetic field and then acts as a perfect diamagnet.

Superconductors may be considered perfect diamagnets (χ_v = −1), since they expel all fields (except in a thin surface layer) due to the Meissner effect. However this effect is not due to eddy currents, as in ordinary diamagnetic materials (see the article on superconductivity).

Demonstrations

Curving water surfaces

If a powerful magnet (such as a supermagnet) is covered with a layer of water (that is thin compared to the diameter of the magnet) then the field of the magnet significantly repels the water. This causes a slight dimple in the water's surface that may be seen by its reflection.^[2]^[3]

Levitation

Main article: Magnetic levitation#Diamagnetism

A live frog levitates inside a 32 mm diameter vertical bore of a Bitter solenoid in a magnetic field of about 16 teslas at the Nijmegen High Field Magnet Laboratory.^[4]

Diamagnets may be levitated in stable equilibrium in a magnetic field, with no power consumption. Earnshaw's theorem seems to preclude the possibility of static magnetic levitation. However, Earnshaw's theorem only applies to objects with positive susceptibilities, such as ferromagnets (which have a permanent positive moment) and paramagnets (which induce a positive moment). These are attracted to field maxima, which do not exist in free space. Diamagnets (which induce a negative moment) are attracted to field minima, and there can be a field minimum in free space.
A thin slice of pyrolytic graphite, which is an unusually strong diamagnetic material, can be stably floated in a magnetic field, such as that from rare earth permanent magnets. This can be done with all components at room temperature, making a visually effective demonstration of diamagnetism.
The Radboud University Nijmegen, the Netherlands, has conducted experiments where water and other substances were successfully levitated. Most spectacularly, a live frog (see figure) was levitated.^[5]
In September 2009, NASA's Jet Propulsion Laboratory in Pasadena, California announced they had successfully levitated mice using a superconducting magnet,^[6] an important step forward since mice are closer biologically to humans than frogs.^[7] They hope to perform experiments regarding the effects of microgravity on bone and muscle mass.
Recent experiments studying the growth of protein crystals has led to a technique using powerful magnets to allow growth in ways that counteract Earth's gravity.^[8]
A simple homemade device for demonstration can be constructed out of bismuth plates and a few permanent magnets that levitate a permanent magnet.^[9]

Theory

The electrons in a material generally circulate in orbitals, with effectively zero resistance and act like current loops. Thus it might be imagined that diamagnetism effects in general would be very, very common, since any applied magnetic field would generate currents in these loops that would oppose the change, in a similar way to superconductors, which are essentially perfect diamagnets. However since the electrons are rigidly held in orbitals by the charge of the protons, and are further constrained by quantum mechanics, although many materials exhibit diamagnetism, they typically respond very little to the applied field.
The Bohr–van Leeuwen theorem proves that there cannot be any diamagnetism or paramagnetism in a purely classical system. Yet the classical theory for Langevin diamagnetism gives the same prediction as the quantum theory.^[10] The classical theory is given below.

Langevin diamagnetism

The Langevin theory of diamagnetism applies to materials containing atoms with closed shells (see dielectrics). A field with intensity

B

, applied to an electron with charge

e

and mass

m

, gives rise to Larmor precession with frequency

ω = eB / 2m

. The number of revolutions per unit time is

ω / 2π

, so the current for an atom with

Z

electrons is (in SI units)^[10]

$I = -\frac{Ze^2B}{4 \pi m}.$

The magnetic moment of a current loop is equal to the current times the area of the loop. Suppose the field is aligned with the

z

axis. The average loop area can be given as $\scriptstyle \pi\left\langle\rho^2\right\rangle$ , where $\scriptstyle \left\langle\rho^2\right\rangle$ is the mean square distance of the electrons perpendicular to the

z

axis. The magnetic moment is therefore

$\mu = -\frac{Ze^2B}{4 m}\langle\rho^2\rangle.$

If the distribution of charge is spherically symmetric, we can suppose that the distribution of

x,y,z

coordinates are independent and identically distributed. Then $\scriptstyle \left\langle x^2 \right\rangle \;=\; \left\langle y^2 \right\rangle \;=\; \left\langle z^2 \right\rangle \;=\; \frac{1}{3}\left\langle r^2 \right\rangle$ , where $\scriptstyle \left\langle r^2 \right\rangle$ is the mean square distance of the electrons from the nucleus. Therefore $\scriptstyle \left\langle \rho^2 \right\rangle \;=\; \left\langle x^2\right\rangle \;+\; \left\langle y^2 \right\rangle \;=\; \frac{2}{3}\left\langle r^2 \right\rangle$ . If $N$ is the number of atoms per unit volume, the diamagnetic susceptibility in SI units is

$\chi = \frac{\mu_0 N \mu}{B} = -\frac{\mu_0 N Z e^2}{6 m}\langle r^2\rangle.$

In metals

The Langevin theory does not apply to metals because they have non-localized electrons. The theory for the diamagnetism of a free electron gas is called Landau diamagnetism, and instead considers the weak counter-acting field that forms when their trajectories are curved due to the Lorentz force. Landau diamagnetism, however, should be contrasted with Pauli paramagnetism, an effect associated with the polarization of delocalized electrons' spins.^[11]^[12]

physics

Friday, 18 October 2013

Diamagnetism

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