(2008-03-20) Magnetization (M)
and Electric Polarization (P)
The densities of magnetic dipoles and electric dipoles, respectively.
Although Maxwell's equations
do describe electromagnetism both
in vacuum and in the midst of matter,
it's useful to make a distinction between
electromagnetic sources which are either
free or bound
to matter at the atomic level.
Ultimately, this allows a different presentation
of Maxwell's equations (where bound sources are suitably hidden)
which can be better suited to a description of electromagnetism
within the bulk of dense matter. First things first:
In one nice continuous model of matter, the microscopic electromagnetic sources
bound to matter are simply approximated by a distribution of
dipoles. (This dipolar approximation usually captures
directly the main aspects of things, but it may be awkward
in some cases, including antiferromagnetic materials.)
The fields created by those so-called
molecular sources is simply superimposed to the field
created by the free
charges and currents (henceforth subscripted with a nought).
The total current density j (which appears in
the ordinary Maxwell equations)
is the sum of three terms:
free current, magnetization current and polarization
current. Likewise, the total electric charge density is the sum of the
free charge density and the density
(-div P) implied by the conservation of
bound charges:
j = jo+
rot M +
¶P/¶t
r =
ro
- div P
Some authors (including
Tobias Brandes) argue that, from a purely mathematical perspective,
the traditional distinction between free and bound sources is arbitrary.
Thus, Brandes "simplifies" the above by discarding entirely the "free" parts of the
above (for which we use nought subscripts).
We do not adopt that viewpoint here.
Of the three components of the total current density,
the magnetization current or bound current
(rot M) is the most difficult to fathom. Let's explain...
(2008-03-23) Electric Polarization and Magnetization Gauge
Mathematically, the fields produced by any smooth distribution of
electromagnetic sources can be equated to what's produced by some
smooth distribution of electromagnetic dipoles.
However, this may involve "unphysical"
dipolar densities (P and M)
which grow without bounds over time, or over space.
Thus, there's a finiteness requirement which helps define
mathematically what
portion of the electromagnetic sources ought to be considered
free in the above sense.
The same distribution of charges and currents
is obtained if P and M are respectively
replaced by the following electric polarization and magnetization densities
(for arbitrary fields Z and k
of respective units C/m and A) :
P + rot ZM -
¶Z/¶t
- grad k
(2008-02-24) Electric Displacement D and
Magnetic Field Strength H Maxwell's equations "in matter" feature free
charges and currents only.
Defining magnetization
and polarization densities (M & P)
as above,
the two equations of Maxwell
involving electromagnetic sources become:
div (e0 E)
= ro
- div P rot (B / m0 )
- ¶
(e0 E)
/¶t =
jo +
rot M +
¶P/¶t
This strongly suggests bundling P with E
and M with B as follows...
The electric displacementD is defined as
a function of the electric fieldE and
electric polarization densityP
(in C/m2 ) namely:
D = e0E
+ P
Likewise, the magnetic field strengthH (also
called magnetizing field
or magnetizing force, in the magnet trade)
depends on the magnetic inductionB and
magnetizationM
(magnetic moment
per unit of volume, in A/m) :
H = B / m0
- M
Those definitions give Maxwell's equations the following simple form:
Maxwell's Equations in Matter (1864)
rot E +
¶B
= 0
div D =
ro
¶ t
rot H-
¶D
= jo
div B = 0
¶t
If and when there's no risk of confusion, the nought subscripts (denoting
free charges and currents) may be dropped.
The misleading term "displacement current" for
¶D/¶t
was coined by Maxwell himself, in 1861, when he had to introduce it to make
Ampère's Lawcome out right!
In a dense medium, some of it
can be interpreted as an actual current (the polarization current
¶P/¶t ).
In vacuum, however, none of this "current" is real;
it's simply a mathematical artefact which
makes Maxwell's equations consistent.
The above equations form a framework which must be supplemented by specific
relations giving D and H
in terms of E and B for a particular medium.
Such relations are known as electromagnetic constitutive relations.
This may be applied to the D and H fields
which result from non-dipolar expressions of bound sources,
although the constitutive relations for
multipolar expressions of
D and H are rarely considered.
(2008-02-24) Electric and magnetic susceptibilities
(ce
and cm )
A medium responds to a field with
polarization densities (P and M).
An external electromagnetic field can disturb the equilibrium of charges and spins in
ordinary matter. Some of the ensuing disturbances may be described classically in
such terms that the macroscopic electromagnetic fields appear to obey a modified
version of Maxwell's equations.
A simple way matter can react to a driving electromagnetic field is by creating
electric and magnetic dipoles in its midst with densities
P and M, respectively.
The simplest response of matter to a driving electromagnetic field at a given
frequency is the creation of varying dipoles
proportional to the fields.
The coefficients of proportionality are scalars in an isotropic medium,
but they are generally tensors.
The coordinates of those tensors are
complex numbers whose imaginary part vanishes at low frequency
(because the lag time in the response of matter to electromagnetic excitations
can then be neglected).
(2008-02-25) Electric Permittivity and Magnetic Permeability
Functions of electric susceptibility and
magnetic susceptibility.
In an isotropic nondispersive medium...
(2008-03-03) Paramagnetism (Pierre Curie, 1895)
A susceptibility inversely proportional to temperature.
Permanent magnetic dipoles in thermal equilibrium tend to align themselves with the
applied magnetic field.
Such a model of matter yields a magnetic susceptibility
which is inversely proportional to the temperature T:
cm = C / T
The constant of proportionality C is called the Curie constant.
Such a relation was first recorded (in the case of oxygen) by Pierre Curie in 1895.
To account for this, Langevin proposed (in 1905) that molecules have
permanent magnetic moments of magnitude
m, oriented according to Boltzmann statistics.
(2008-03-02) Diamagnetism
(Brugmans in 1778, Faraday in 1845)
Materials with negative susceptibilities repel both poles of a magnet.
In 1778, S.J. Brugmans (of Leyden University)
noted that bismuth
weakly repels both poles of a magnet.
In 1827, Le Baillif described the same effect for antimony
(see p. 144 of
Light on Electricity
by John Tyndall, 1871).
Also in 1827, [Antoine César]
Becquerel noticed the effect for wood.
In 1828, Seebeck
reported it for several other substances...
Antoine César
Becquerel (1788-1878; X1806) was the father of the physicist
[Alexandre] Edmond Becquerel
(1820-1891) who also investigated diamagnetism and paramagnetism.
The son of Edmond was
[Antoine] Henri Becquerel
(1852-1908; X1872) who shared the Nobel prize in physics
in 1903
for his discovery of natural radioactivity
Jean Becquerel
(1878-1953, X1897) was the son of Henri.
Like his great-grandfather, grandfather
and father before him, Jean held the chair of physics at the
Muséum national d'histoire naturelle (MNHN).
In 1845, Michael Faraday (1791-1867)
started to investigate
the phenomenon systematically and called it diamagnetism,
because a small rod of a diamagnetic substance
(like bismuth) tends to align itself across
the magnetic field lines (as each part of the rod tries to get as
far away from the nearest magnetic pole as possible).
It turns out that all substances have diamagnetic properties
but the diamagnetic repulsion
is usually masked by attractive paramagnetic or ferromagnetic
properties, which are much stronger if at all present
(especially the latter).
If measured for a given number of atoms or moles,
diamagnetism (unlike paramagnetism and ferromagnetism) does
not depend on temperature.
Thus, for a given volume of a certain substance, diamagnetism simply
varies with temperature as the density
of the substance (this amounts to very little dependence on temperature
for solids or liquids).
Here is how diamagnetism could be explained in semi-classical terms:
The Lorentz force
applied to an orbiting electron changes its centripetal acceleration
and modifies its orbital magnetic moment in a direction opposing
the applied external magnetic field. The size of the orbits
would have to be obtained from quantum considerations.
Classical Diamagnetism (Paul Langevin, 1905)
c =
-N
m0 q 2
åi
ri2
6 m
Paul Langevin
obtained that result in 1905 by a classical argument which takes into account
the Larmor precession of each electron about the applied magnetic field.
In the above formula,
N is the number of atoms per unit of volume,
q and m
are the charge and the mass of the electron.
The summation extends over all the electrons in each atom to yield the
sum of the mean squares of their orbital radii.
(2008-03-05) Magnetic Levitation
Levitation without active devices defies Earnshaw's Theorem (1842).
In 1842,
Samuel Earnshaw
(1805-1888) proved that permanent magnets are unable to produce stable levitation.
This theorem can be extended to include ferromagnetic or paramagnetic materials.
In 1845, Faraday rediscovered diamagnetism.
In 1847, Lord Kelvin recognized
that Earnshaw's theorem would not apply to diamagnetic materials.
Static magnetic levitation is indeed possible if diamagnets are
involved.
Because of their negative susceptibility,
diamagnetic bodies seek equilibrium at a minimum of the magnetic
field... Although diamagnetic effects are small, they can be large enough
to oppose Earth gravity (for thin shim of pyrographite) or, at least,
combine with stronger magnetic fields to obtain stable
levitation in midair, at room temperature.
Martin D. Simon designed a diamagnetic levitation stand
(see video)
at UCLA which included two disks of pyrolytic graphite (PG).
According to Meredith Lamb,
it was on January 17, 2000 that those PG disks were first used to make thin floaters
that could levitate over a pattern of
alternating poles formed by four neodymium block magnets.
We are not discussing here the use of electromagnets (which consume
some power) to achieve the illusion of stability
by a dynamic control of the magnetic field, using sensors which monitor the position of
a permanent magnet floating over another one.
This does have great entertainment value,
though.
(2008-06-23) Pyrolytic Carbon (Pyrolytic Graphite, or
PyroCarbon)
At room temperature, this is the most diamagnetic substance known.
Pyrolytic graphite
(PG or PyC) is a layered form of pure carbon with a density between 1.7 and 2.0.
It's obtained from short hydrocarbon gases (mostly methane or propane)
by chemical vapor deposition
(CVD)
at high temperature (up to 2000°C)
under a low partial pressure (10 mmHg or less)
which prevents the formation of carbon black
(this can be achieved by dilution in an inert gas,
like helium, argon, nitrogen or hydrogen).
The process is fairly slow: A thickness of just 1 mm
requires typically 48 hours
(but as little as 1 hour for low-grade stuff).
In medical applications (replacement joints and heart valves)
material coated with pyrolytic carbon is marketed under the name of
PyroCarbon by companies like
Ascension
Orthopedics (Austin, Texas)
and Nexa
(the Tornier group acquired the relevant implant technology from the French
firm BioProfile).
PyroCarbon was first used to manufacture heart valves in 1968.
For experimental purposes, pyrolytic graphite is available from
SciToys.
(2008-03-18) The theorem of Bohr and Van Leeuwen (1911, 1919)
Classical diamagnetism and paramagnetism cancel each other...
As part of his doctoral dissertation (Copenhagen 1911)
Niels Bohr (1885-1962) introduced a
classical
argument which would later be developed by
Hendrika Johanna
Van Leeuwen (1887-1974) in her own doctoral dissertation (Leiden 1919,
Journal de Physique 1921) under the guidance of
H.A. Lorentz and Paul Ehrenfest.
The remark, known as the Bohr-Van Leeuwen Theorem,
is that the ordinary laws of classical and statistical physics
(outside of quantum theory) imply that an external magnetic field will not
induce any net magnetization in a set of moving electric charges
at thermal equilibrium.
Thus, classically, the diamagnetic and paramagnetic effects cancel each other
exactly !
Of course, this flies in the face of experimental results and merely goes to show
that classical physics by itself cannot
produce an adequate theory of magnetism.
Some form of quantization is needed to resolve this and other issues and reconcile
theory with experiment (the magnetic dipoles postulated by
Langevin in his theory of paramagnetism can be construed as
a good substitute for such a quantization).
John
Hasbrouck Van Vleck (1899-1980)
discusses the theorem in
Theory of Electric and Magnetic Susceptibilities (1934). In his
Nobel
lecture (1970) he argues that this particular point
may have been one of the main motivations which led Niels Bohr
himself to propose quantum conditions for the structure of the atom, in 1913
(thereby founding the so-called Old Quantum Theory).
Proof :
(The following argument is based on what
Richard Feynman says
in section 34-6 (vol. 2 and vol. 3) of
The Feynman Lectures on Physics.)
A system of moving charges has a probability proportional to
e-U/kT to have a state of motion of energy
U at thermal equilibrium (temperature T).
This energy U includes only the kinetic energy of the particles
and their electric potential energy.
It's unaffected
by the existence of any additional magnetic field.
Thus, the exact same statistical distribution of charge velocities is achieved
at thermal equilibrium whether an external magnetic field is applied or not.
If we assume, as we do within a strict classical framework, that magnetic
moments are entirely due to the circulating currents formed by moving
charges, then we come to the conclusion that no magnetic moments at all are induced.
In other words, the net magnetic susceptibility is zero!
This conclusion cannot be reached if intrinsic
magnetic moments exists which are not explained by the classical motions
of point charges.
Such things are allowed in
Quantum Theory and they can be postulated
ad hoc in semiclassical models, like the Langevin
theory of paramagnetism. For example, a pointlike electron has a quantum
spin which endows it with angular momentum and
magnetic moment in spite of its lack of structure and the
subsequent lack of a dubious explanation in terms of the rotation
of some other stuff...
(2008-03-18) Thermodynamics of Electromagnetism
Electromagnetic interactions of moving charges and magnetic dipoles.
To avoid the blatant contradiction of experimental evidence embodied
by the above Theorem of Bohr and Van Leeuwen,
a semiclassical discussion of magnetism should at least allow
the existence of fundamental magnetic dipoles
(elementary particles endowed with a magnetic moment not
due to a rotation of electric charges).
The energy of such a beast does depend on the magnetic field it is subjected to.
Pierre-Ernest Weiss (1865-1940)
was born in Mulhouse, France. He attended the Zürich Polytechnikum
(ETH)
graduating at the head of his class in 1887 with a degree in mechanical engineering
In 1888, Weiss was admitted to the Ecole normale supérieure (ENS)
where his classmates included Elie Cartan (1869-1951)
whose son Henri Cartan (1904-2008)
would marry his daughter Nicole in 1935 (5 children).
Pierre Weiss became agrégé in 1893.
He was a student of
Jules Violle (1841-1923) and
Marcel Brillouin
(1854-1948). His doctoral dissertation (1896) on magnetite and iron-antimony alloys
established a relation between magnetization and crystal symmetry.
In 1902, he returned to the ETH of Zürich as professor and director of the
Physics Laboratory.
In 1918, Weiss went on to Strasbourg, where he created his own laboratory
dedicated to magnetism
(in 1928, he hired Louis Néel, who would get a
Nobel prize in 1970 for discovering antiferromagnetism).
Weiss was one of the key founders of the modern study of magnetism.
The Weiss magneton (empirical molecular magneton) is roughly
equal to 1.853 10-24 J/T
(or about 20% of a Bohr magneton).
In ferromagnetic materials,
the magnetization of the medium itself can create a magnetic
field which greatly exceeds a typical external field.
Furthermore, a remanent magnetization may exist in the
absence of any external field.
In 1906-1907, Pierre Weiss
discovered that such materials are always subdivided
into variously oriented domains
where the magnetization has its full saturation value.
Those domains are now known as Weiss domains.
To explain this, Weiss proposed the so-called
molecular field hypothesis whereby molecules could be endowed
with tiny magnetic dipole moments which
tend to align with their neighbors within each Weiss domain.
The boundaries between Weiss domains are called
Bloch walls, in honor of the Swiss physicist
Felix Bloch (1905-1983;
Nobel 1952)
who investigated them.
Saturation Magnetization :
Ferromagnetism is such that the magnetic moments created at the atomic level
tend to be aligned in each Weiss domain. It's useful to estimate what the
maximum magnetization can be under such conditions.
The contribution of each atom in the material is mostly due to its electrons,
either from their orbital motion or their intrinsic spin
which are respectively quantized (nonrelativistically) to a whole
or half-integer multiple of the
Bohr magneton.
In the main, we neglect the
interesting
magnetic effects due to the nucleons, which are 3 orders of magnitude smaller.
The magnetic moment of an atom can be attributed to its unpaired electrons.
It's typically equal to 1 or 2 Bohr magneton,
but can be as high as 10.6 Bohr magnetons
in the case of Holmium (67).
Thus, holmium pole pieces can
concentrate magnetic flux
by up to 3.96 T (this theoretical iimit is obtained by multiplying the
permeability of the vacuum into the volume density of the magnetic moments).
The magnetic energy density of a ferromagnet
(in joules per cubic meter or, equivalently, in pascals)
is the product of the remanent flux
density (i.e., the magnetic induction B,
in teslas) by the density of magnetization M.
A non-SI unit commonly used in the trade for this is the megagauss-oersteds (MG.Oe) :
1 T = 104 G
1 A/m = 4 p 10-3 Oe
1 T.A/m = 1 J/m3 = 1 Pa =
40 p G.Oe = 125.6637... G.Oe
Conversely, 1 MG.Oe =
106/40p Pa = 7957.747... J/m3
For example, the theoretical maximum for a neodymium-iron-boron magnet
("NIB" or "neo") is quoted
to be 64 MGOe while the best available grade is currently 54 MGOe
(that's what the designation "N54" means). In more readable SI units,
those numbers correspond respectively to 0.51 MPa and 0.43 MPa.
In other words, an N54 neodymium magnet
packs ideally a magnetic energy of 0.43 J per cubic centimeter.
(2008-03-09) Antiferromagnetism (Néel 1932, Landau 1933)
Magnetic multipoles dominate when adjacent dipoles cancel.
Louis Néel
(1904-2000)
was a towering figure in French physics.
He helped transform Grenoble into
a major research center and earned a belated
Nobel prize in 1970.
Louis Néel started a scientific career
dedicated to magnetism in Strasbourg in 1928, as an assistant to
Pierre Weiss (1865-1940).
He is the eponym for the Néel temperature (above which antiferromagnetism
disappears) and Néel walls (planes separating domains
whose magnetizations differ only by components parallel to the walls).
Louis Néel
Antiferromagnetism occurs below a certain transition temperature, called the
Néel temperature TN , which varies from
one antiferromagnetic material to the next:
(2008-03-09) Ferrimagnetism (Louis Néel, 1947)
Several types of dipoles may partially cancel each other in a crystal.
The most famous example of a ferrimagnetic substance is
lodestone (which
Gilbert
spelled loadstone ) which is the traditional name for
magnetite,
the most magnetic substance among
naturally occurring minerals. In fact, magnetism derives its name
from magnetite, not the other way around...
Magnetite, magnesium and manganese, are actually named after
the Greek region of Magnesia (central Greece)
because they were first discovered in minerals from that area.
Magnetite (Fe3O4 )
is also called ferrous-ferric oxide.
An expanded chemical formula
(FeO, Fe2O3 )
better reflects the structure of its crystal...
In the lattice, ferrous ions (Fe++ ) and ferric ions (Fe+++ )
tend to have antiparallel dipole moments.
However, the ferrous and the ferric magnetic moments are
not equal in magnitude, so there's a net local magnetization.
(2008-04-04) Magneto-Optical Effect (Michael Faraday, 1845)
In an active crystal, light polarization is rotated by a magnetic field.
Faraday effect (transmitted beam) and Kerr effect (reflected beam).
The magnetization may be polar (perpendicular to the diopter) longitudinal
(parallel to both the diopter and the plane of incidence) or transverse
(parallel to the diopter, perpendicular to the plane of incidence).
The first nonlinear-optical effect was the quadratic
Kerr effect
(quadratic electro-optic effect, QEO effect)
described in 1875 by the Reverend John C.
Kerr (1824-1907).
In 1893, Pockels
(1865-1913) discovered that a birefringence proportional to the applied field exists in some crystals
(Pockels Effect).
(2008-02-24) Classical Conductor of Conductivity
s. Ohm's law.
Current density (j) is proportional to the electric field
(E) : j = s E
In an ideal conductor (a superconductor)
the conductivity is infinite and, therefore, E = 0.
There's no electric field and the magnetic field doesn't change.
Ordinary substances
have a finite conductivity s
which varies with temperature.
(2015-04-18) Aimé Auguste Cotton (1869-1951)
Interaction of light with chiral molecules. The Cotton effect.
In 1953, a prestigious yearly prize for promising French researchers was
created to honor the memory of Aimé Cotton.
That prize was awarded in 1971 to Serge Haroche
(b. 1944) for his doctoral work. Haroche went on to earn the Nobel
Prize in Physics, in 2012.
