Diamagnetic materials are repelled by a magnetic field; an applied magnetic field creates an
induced magnetic field in them in the opposite direction, causing a repulsive force. In contrast,
paramagnetic and ferromagnetic materials are attracted by a magnetic field. Diamagnetism
is a quantum mechanical effect that occurs in all materials; when it is the only contribution
to the magnetism, the material is called diamagnetic. In paramagnetic and ferromagnetic substances
the weak diamagnetic force is overcome by the attractive force of magnetic dipoles in
the material. The magnetic permeability of diamagnetic materials is less than μ0, the
permeability of vacuum. In most materials diamagnetism is a weak effect which can only
be detected by sensitive laboratory instruments, but a superconductor acts as a strong diamagnet
because it repels a magnetic field entirely from its interior.
Diamagnetism was first discovered when Sebald Justinus Brugmans observed in 1778 that bismuth
and antimony were repelled by magnetic fields. In 1845, Michael Faraday demonstrated that
it was a property of matter and concluded that every material responded (in either a
diamagnetic or paramagnetic way) to an applied magnetic field. On a suggestion by William
Whewell, Faraday first referred to the phenomenon as diamagnetic (the prefix dia- meaning through
or across), then later changed it to diamagnetism.
== Materials == Diamagnetism is a property of all materials,
and always makes a weak contribution to the material's response to a magnetic field. However,
other forms of magnetism (such as ferromagnetism or paramagnetism) are so much stronger that
when multiple different forms of magnetism are present in a material, the diamagnetic
contribution is usually negligible. Substances where the diamagnetic behaviour is the strongest
effect are termed diamagnetic materials, or diamagnets. Diamagnetic materials are those
that laypeople 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 values of various molecular fragments are called Pascal's constants.
Diamagnetic materials, like water, or water-based 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 χv = μv − 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 χv = −9.05×10−6. The most strongly diamagnetic material is
bismuth, χv = −1.66×10−4, although pyrolytic carbon may have a susceptibility of χv = −4.00×10−4
in one plane. Nevertheless, these values are orders of magnitude smaller than the magnetism
exhibited by paramagnets and ferromagnets. Note that because χv 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. In rare cases, the diamagnetic contribution
can be stronger than paramagnetic contribution. As is the case for gold, which has a magnetic
susceptibility less than 0, so is by definition a diamagnetic material, but when measured
carefully with X-ray magnetic circular dichroism, shows an extremely weak paramagnetic contribution
that is overcome by a stronger diamagnetic contribution.
=== Superconductors ===
Superconductors may be considered perfect diamagnets (χv = −1), because they expel
all magnetic fields (except in a thin surface layer) due to the Meissner effect.
== 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.
=== Levitation ===
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 applies only 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.In September 2009, NASA's Jet
Propulsion Laboratory (JPL) in Pasadena, California announced it had successfully levitated mice
using a superconducting magnet, an important step forward since mice are closer biologically
to humans than frogs. JPL said it hopes to perform experiments regarding the effects
of microgravity on bone and muscle mass. Recent experiments studying the growth of
protein crystals have led to a technique using powerful magnets to allow growth in ways that
counteract Earth's gravity.A simple homemade device for demonstration can be constructed
out of bismuth plates and a few permanent magnets that levitate a permanent magnet.
== Theory == The electrons in a material generally settle
in orbitals, with effectively zero resistance and act like current loops. Thus it might
be imagined that diamagnetism effects in general would be 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 the Pauli exclusion principle, many materials exhibit diamagnetism, but 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. However, the
classical theory of Langevin for diamagnetism gives the same prediction as the quantum theory.
The classical theory is given below.
=== Langevin diamagnetism === Paul Langevin's theory of diamagnetism (1905)
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)
I =
−
Z
e
2
B
4 π
m
.
{\displaystyle I=-{\frac {Ze^{2}B}{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
π
⟨
ρ
2
⟩
{\displaystyle \scriptstyle \pi \left\langle \rho ^{2}\right\rangle }
, where
⟨
ρ
2
⟩
{\displaystyle \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
μ =
−
Z
e
2
B
4 m
⟨
ρ
2
⟩ .
{\displaystyle \mu =-{\frac {Ze^{2}B}{4m}}\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
⟨
x
2
⟩
=
⟨
y
2
⟩
=
⟨
z
2
⟩
=
1 3
⟨
r
2
⟩
{\displaystyle \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
⟨
r
2
⟩
{\displaystyle \scriptstyle \left\langle r^{2}\right\rangle }
is the mean square distance of the electrons from the nucleus. Therefore,
⟨
ρ
2
⟩
=
⟨
x
2
⟩
+
⟨
y
2
⟩
=
2 3
⟨
r
2
⟩
{\displaystyle \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
volume diamagnetic susceptibility in SI units is
χ =
μ
0
n μ
B
= −
μ
0
e
2
Z n
6 m
⟨
r
2
⟩ .
{\displaystyle \chi ={\frac {\mu _{0}n\mu }{B}}=-{\frac {\mu _{0}e^{2}Zn}{6m}}\langle
r^{2}\rangle .}
=== In metals === The Langevin theory is not the full picture
for metals because they have non-localized electrons. The theory that describes diamagnetism
in a free electron gas is called Landau diamagnetism, named after Lev Landau, and instead considers
the weak counteracting field that forms when the electrons' 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. For the bulk
case of a 3D system and low magnetic fields, the (volume) diamagnetic susceptibility can
be calculated using Landau quantization, which in SI units is
χ =
−
μ
0
e
2
12
π
2
m ℏ
2 m
E
F
,
{\displaystyle \chi =-\mu _{0}{\frac {e^{2}}{12\pi ^{2}m\hbar }}{\sqrt {2mE_{\rm {F}}}},}
where
E
F
{\displaystyle E_{\rm {F}}} is the Fermi energy. This is equivalent to
−
μ
0
μ
B
2
g (
E
F
)
/
3
{\displaystyle -\mu _{0}\mu _{\rm {B}}^{2}g(E_{\rm {F}})/3}
, exactly
− 1
/
3
times Pauli paramagnetic susceptibility, where
μ
B
= e
ℏ
/
2 m
{\displaystyle \mu _{\rm {B}}=e\hbar /2m} is the Bohr magneton and
g (
E )
is the density of states (number of states
per energy per volume). This formula takes into account the spin degeneracy of the carriers
(spin ½ electrons). In doped semiconductors the ratio between
Landau and Pauli susceptibilities may change due to the effective mass of the charge carriers
differing from the electron mass in vacuum, increasing the diamagnetic contribution. The
formula presented here only applies for the bulk; in confined systems like quantum dots,
the description is altered due to quantum confinement. Additionally, for strong magnetic
fields, the susceptibility of delocalized electrons oscillates as a function of the
field strength, a phenomenon known as the de Haas–van Alphen effect, also first described
theoretically by Landau.
== See also == Antiferromagnetism
Magnetochemistry Stochastic electrodynamics