Three things inside an atom can produce a magnetic moment.
The orbital motion of the electrons. A circulating charge is a
current loop, and a current loop is a magnetic dipole.
The spin of the electrons. An intrinsic moment with no
classical analogue, and in most solids the dominant contribution.
The moment induced by the field itself. Applying a field
changes the orbital motion, and the change always produces a moment
opposing the field. This is diamagnetism, and unlike the first two it
is present in every atom.
The first two exist whether or not a field is applied and give moments that
align along the field: they are paramagnetic in character. The third
exists only because a field is applied and gives a moment against it.
Every solid has the third; only some have the first two.
The orbital moment.
An electron of charge \(-e\) circulating with
speed \(v\) in an orbit of radius \(r\) completes one circuit in \(T = 2\pi r/v\), so
the current is
\begin{equation}
I = \frac{e}{T} = \frac{ev}{2\pi r},
\end{equation}
and the moment of the equivalent loop is \(\mu = IA\):
The minus sign is not decoration: because the electron is negative, its moment
points opposite to its angular momentum, and every sign in the subject
depends on remembering it.
The spin moment.
Experiment (and Dirac theory) gives spin twice the
gyromagnetic ratio of orbital motion,
A useful number to carry: \(\mu_{B}B/k_{B} \approx 0.67\) K per tesla. Even in a
10 T laboratory field the magnetic energy is
\(\mu_{B}B \approx 5.8\times10^{-4}\) eV, some forty-five times smaller
than \(k_{B}T \approx 0.026\) eV at room temperature. Magnetic
energies are tiny compared with thermal energies. This single fact explains why
paramagnetic alignment is so feeble, why \(\chi \sim 10^{-3}\) at best, and why the
existence of iron magnets at 1000 K demands an explanation that has
nothing to do with magnetic forces between dipoles.
Dipole–dipole coupling cannot cause ferromagnetism
Two moments of one \(\mu_{B}\) a lattice spacing apart interact with energy
\(\sim \mu_{0}\mu_{B}^{2}/4\pi a^{3}\), which for \(a = 2\,\Ang\) is
\(1.1\times10^{-24}\) J — an equivalent temperature of only
0.08 K. Iron orders at 1043 K. Magnetic dipole coupling
is four orders of magnitude too weak, and the ordering must come from somewhere
else — the exchange
interaction of the section on ferromagnetism, which is electrostatic in origin and
magnetic only in its consequences.
\(\vb{B}\), \(\vb{H}\) and \(\vb{M}\)
Inside matter three vector fields are in play. \(\vb{M}\), the
magnetisation, is the moment per unit volume. \(\vb{H}\) is the field
intensity, fixed by the free currents in the coils. \(\vb{B}\) is the flux
density, and it is what a moving charge actually feels. In SI units
which matters because for ferromagnets \(M\) is not proportional to \(H\) and the
ratio \(M/H\) is not a material constant.
Reading the sign and size of \(\chi\)
Class
Sign of \(\chi\)
Typical \(|\chi|\)
\(T\) dependence
Diamagnet
negative
\(10^{-6}\)–\(10^{-5}\)
none
Paramagnet (ions)
positive
\(10^{-5}\)–\(10^{-3}\)
\(\chi = C/T\)
Pauli paramagnet (metal)
positive
\(10^{-6}\)
none
Ferromagnet
positive, huge
\(10^{2}\)–\(10^{5}\)
\(C/(T-T_{C})\), \(T>T_{C}\)
Antiferromagnet
positive, small
\(10^{-4}\)
\(C/(T+T_{N})\), \(T>T_{N}\)
Superconductor (type I)
\(\chi = -1\) exactly
\(1\)
none
Three questions identify any material: is \(\chi\) positive or negative, how big
is it, and how does it vary with \(T\)?
\(\chi\) is dimensionless in SI, but not in CGS
In SI, \(\vb{M}\) and \(\vb{H}\) share units of A m\(^{-1}\) and \(\chi\) is a pure
number. Older literature and many examination questions use CGS, where
\(\vb{B} = \vb{H} + 4\pi\vb{M}\) and susceptibilities are quoted per gram or per
mole; the two conventions differ by \(4\pi\), so
\(\chi_{\text{SI}} = 4\pi\,\chi_{\text{CGS}}\). If a problem quotes the Curie
constant in “CGS units” it is not being decorative — the numerical answer
depends on it. Check which convention is in force before substituting.
The perfect diamagnet
A type-I superconductor expels magnetic flux entirely (the Meissner effect), so
\(\vb{B} = 0\) inside. Setting \(\vb{B} = 0\) in the relation above gives
\(\vb{M} = -\vb{H}\), hence
This is the strongest diamagnetism possible: no material can have
\(\chi < -1\), since that would require \(\vb{B}\) and \(\vb{H}\) to point in opposite
directions. On an \(M\)–\(H\) plot drawn to the same scale in both axes, a type-I
superconductor is the unique line of slope exactly \(-1\), an ordinary diamagnet
has a small negative slope, and a paramagnet a small positive slope. That
diagram is a recurring examination question, and \(\chi = -1\) is the only entry
on it that is exact.
The five classes
The five types of magnetic order. The arrows show the arrangement of
moments on neighbouring sites in the absence of an applied field.
{1.3}
Type
Moments without field
Net \(M\) at \(H=0\)
Example
Diamagnetic
none
0
Cu, Bi, H\(_{2}\)O, Ar
Paramagnetic
present, randomly oriented
0
Al, Pt, O\(_{2}\), Gd\(^{3+}\) salts
Ferromagnetic
present, parallel
\(\neq 0\)
Fe, Co, Ni, Gd
Antiferromagnetic
present, antiparallel, equal
0
MnO, Cr, NiO
Ferrimagnetic
antiparallel, unequal
\(\neq 0\)
Fe\(_{3}\)O\(_{4}\), ferrites
Arrangements of moments at
\(H=0\). The paramagnet has moments but no correlation between them; the
ferromagnet aligns them; the antiferromagnet alternates equal moments and
cancels exactly; the ferrimagnet alternates unequal moments and so
retains a net magnetisation. A diamagnet (not shown) has no moments at all
until a field is applied.
Paired electrons contribute no net spin moment
A frequently garbled statement. In a filled shell or a filled orbital the
electrons occur in pairs of opposite spin, and their spin moments cancel
exactly; the orbital moments cancel too, since for every occupied
\(m_{\ell}\) there is an occupied \(-m_{\ell}\). A closed-shell atom therefore has
\(L = S = J = 0\) and no permanent moment at all, which is why the inert
gases, and ions like Na\(^{+}\) and Cl\(^{-}\), are purely diamagnetic. Permanent
moments require unpaired electrons, and that is why paramagnetism is
found in transition-metal and rare-earth ions with partly filled \(d\) or \(f\)
shells.
The molecular oxygen case is often quoted as an “exception”, which invites
the wrong conclusion. O\(_{2}\) is paramagnetic because molecular orbital
theory puts two electrons singly into degenerate \(\pi^{*}\) orbitals with
parallel spins. It is an exception to the expectation that a stable molecule has
all electrons paired — not an exception to the rule that paramagnetism needs
unpaired spins. The rule has no exceptions.
Diamagnetism
Diamagnetism is the only magnetic effect that every material has. It arises
because switching on a field changes the orbital motion of the electrons, and by
Lenz's law the change opposes the flux that caused it. The induced moment is
therefore antiparallel to \(\vb{B}\), \(\chi < 0\), and the material is weakly
repelled from a field gradient.
In an atom with unpaired electrons the paramagnetic response is one to three
orders of magnitude larger and completely masks the diamagnetic one. So although
diamagnetism is universal, it is observed only in materials with no
permanent moments: closed-shell atoms and ions, most covalent molecules, and
superconductors.
Larmor precession
Apply a field \(\vb{B}\) to an electron in a closed shell. The extra force is
\(-e\,\vb{v}\times\vb{B}\), and its effect on a bound orbit is not to distort the
orbit but to make the whole orbit precess about \(\vb{B}\) at the Larmor
frequency
This is Larmor's theorem: for a spherically symmetric charge distribution, and
to first order in \(B\), the motion in a frame rotating at \(\omega_{L}\) is the same
as the motion without the field.
The precession is itself a circulation of charge and so a current. For \(Z\)
electrons circulating with angular frequency \(\omega_{L}\), the induced current is
the charge \(-Ze\) passing a point once per period,
\begin{equation}
I = -Ze\,\frac{\omega_{L}}{2\pi} = -\frac{Ze^{2}B}{4\pi m_{e}} .
\end{equation}
\(Z\) is a count of electrons, not a mass
In the diamagnetic susceptibility, \(Z\) is the number of electrons per atom
contributing to the response — the atomic number, or in practice the
number of core electrons. It is not the atomic mass, and confusing the two
changes the answer by a factor of roughly two for light elements and more for
heavy ones. The distinction also has physical content: diamagnetism grows with
the number of electrons and with the size of their orbits, which is why bismuth
and graphite are the strongest ordinary diamagnets and helium among the
weakest.
The Langevin diamagnetic susceptibility
The moment of a current loop of area \(\pi\rho^{2}\), where \(\rho\) is the distance
from the field axis, is \(\mu = I\pi\langle\rho^{2}\rangle\):
Now \(\langle\rho^{2}\rangle = \langle x^{2}\rangle + \langle y^{2}\rangle\) is a
projection onto the plane perpendicular to \(\vb{B}\), whereas the quantity a
quantum calculation naturally supplies is the full
\(\langle r^{2}\rangle = \langle x^{2}\rangle + \langle y^{2}\rangle
+ \langle z^{2}\rangle\). For a spherically symmetric charge cloud the three are
equal, so
The sign is negative and cannot be otherwise. There is no
material parameter in the relation above that can change sign; \(N\), \(Z\),
\(e^{2}\) and \(\langle r^{2}\rangle\) are all positive. Diamagnetism always
opposes.
There is no temperature in the formula. Nothing is being aligned
against thermal disorder — the moment is induced, not oriented — so
\(\chi_{\text{dia}}\) is essentially independent of \(T\). This is the sharpest
experimental distinction from paramagnetism.
It scales as \(\langle r^{2}\rangle\). The outermost electrons
dominate, since \(r^{2}\) weights them heavily. Large, polarisable, many-electron
atoms are the strongest diamagnets.
Worked example 8.1.
Estimate the diamagnetic susceptibility of solid argon, with
\(N = 2.66\times10^{28}\) m\(^{-3}\), \(Z = 18\) and
\(\langle r^{2}\rangle^{1/2} = 0.62\,\Ang\).
Solution. Substituting into the relation above with
\(\langle r^{2}\rangle = (0.62\times10^{-10})^{2} = 3.84\times10^{-21}\) m\(^{2}\):
A number of order \(10^{-5}\), negative, and dimensionless, exactly as the
classification table of the section on magnetic Moments, Fields and Susceptibility promises. Measured values for
the solid inert gases lie in the range \(-1\) to \(-2 \times 10^{-5}\).
The two signatures of a diamagnet: \(M\) is
linear in \(H\) with a small negative slope, and \(\chi\) does not vary with
temperature. Compare the paramagnetic \(\chi = C/T\) hyperbola of
the figure above.
Landau diamagnetism
The Langevin result applies to electrons bound in atoms. Conduction electrons in
a metal are not bound, and one might expect them to give no diamagnetic
contribution at all — classically they give none, by the Bohr–van Leeuwen
theorem. Quantum mechanically the field quantises their orbital motion into
Landau levels, and the resulting susceptibility is
with \(\chi_{\text{Pauli}}\) the spin contribution derived in
the section on paramagnetism. The free-electron gas is therefore net
paramagnetic, with two thirds of the bare Pauli value. In a real metal the
factor is modified by the band mass, since the orbital term depends on \(m^{*}\)
while the spin term does not, and in bismuth — where \(m^{*}\) is very small —
the Landau term wins outright and the metal is strongly diamagnetic.
Diamagnetism in one line each
Universal: present in every atom, since every atom has orbiting
electrons.
Weak: \(\chi \sim -10^{-5}\), so it is visible only when nothing
paramagnetic is present.
Temperature-independent: nothing is competing with thermal
disorder.
Strongest in large-\(Z\), large-\(r\) systems: Bi, graphite, and
(as a special case with \(\chi = -1\)) type-I superconductors.
Paramagnetism
A paramagnet has permanent moments that are randomly oriented at \(H=0\) and
partially aligned by a field. The alignment is opposed by thermal agitation, and
the balance between the two produces the characteristic \(1/T\) susceptibility
known as Curie's law. Because the mechanism is orientation rather than
induction, temperature enters in an essential way — the single sharpest
contrast with diamagnetism.
The moment of a free ion: Hund's rules
Before any statistical mechanics we need the size of the moment. For an ion with
a partly filled shell the electrons arrange themselves to minimise energy subject
to the Pauli principle, and the resulting ground state is fixed by three rules,
applied in order.
Hund's rules
Maximise \(S\). Fill orbitals with parallel spins as far as the
exclusion principle allows. Parallel spins keep electrons apart and lower the
Coulomb repulsion.
Then maximise \(L\), consistent with rule 1. Electrons orbiting in
the same sense meet less often.
Then fix \(J\) by spin–orbit coupling:
\begin{equation*}
J = |L - S| \quad\text{if the shell is }\textit{less}\text{ than half full},
\qquad
J = L + S \quad\text{if }\textit{more}\text{ than half full}.
\end{equation*}
For an exactly half-filled shell \(L = 0\) and \(J = S\), so the two prescriptions
agree.
The two limits are worth memorising: pure spin (\(L=0\), \(J=S\)) gives \(g_{J}=2\);
pure orbital (\(S=0\), \(J=L\)) gives \(g_{J}=1\).
\(\sqrt{J(J+1)}\), not \(J\)
The moment that enters the susceptibility is the root-mean-square value
\(g_{J}\mu_{B}\sqrt{J(J+1)}\), because it comes from \(\langle\mu^{2}\rangle\) in
the high-temperature expansion. The saturation moment — the largest
value \(\mu_{z}\) can take, and the one that matters at low temperature or high
field — is \(g_{J}\mu_{B}J\). These are different numbers, and using the wrong
one is the commonest error in this section. For Gd\(^{3+}\) (\(J = 7/2\), \(g=2\))
they are \(7.94\mu_{B}\) and \(7\mu_{B}\) respectively.
Worked example 8.2.
Find the ground state, \(g_{J}\) and \(p\) for a free ion with an \(f^{7}\)
configuration (e.g. Gd\(^{3+}\), Eu\(^{2+}\)).
Solution. The \(f\) shell has \(\ell = 3\), so \(m_{\ell}\) runs over
\(-3,-2,-1,0,+1,+2,+3\): seven orbitals, holding fourteen electrons when full.
With seven electrons the shell is exactly half full.
Rule 1: all seven spins parallel, so
\begin{equation*}
S = 7 \times \tfrac{1}{2} = \tfrac{7}{2}.
\end{equation*}
Rule 2: with one electron in each orbital there is no freedom left, and
Gd\(^{3+}\) is the largest free-ion moment among the rare earths, which is why
gadolinium compounds dominate magnetic refrigeration and MRI contrast agents.
The Brillouin function and Curie's law
Place \(N\) such ions per unit volume in a field \(B\). The component of \(\vb{J}\)
along the field is quantised, \(J_{z} = M_{J}\hbar\) with
\(M_{J} = -J,\dots,+J\), so the energy levels are
