Paramagnetism is a form of magnetism whereby certain materials are attracted by an externally
applied magnetic field, and form internal, induced magnetic fields in the direction of
the applied magnetic field.
In contrast with this behavior, diamagnetic materials are repelled by magnetic fields
and form induced magnetic fields in the direction opposite to that of the applied magnetic field.
Paramagnetic materials include most chemical elements and some compounds; they have a relative
magnetic permeability greater than or equal to 1 and hence are attracted to magnetic fields.
The magnetic moment induced by the applied field is linear in the field strength and
rather weak.
It typically requires a sensitive analytical balance to detect the effect and modern measurements
on paramagnetic materials are often conducted with a SQUID magnetometer.
Paramagnetic materials have a small, positive susceptibility to magnetic fields.
These materials are slightly attracted by a magnetic field and the material does not
retain the magnetic properties when the external field is removed.
Paramagnetic properties are due to the presence of some unpaired electrons, and from the realignment
of the electron paths caused by the external magnetic field.
Paramagnetic materials include magnesium, molybdenum, lithium, and tantalum.
Unlike ferromagnets, paramagnets do not retain any magnetization in the absence of an externally
applied magnetic field because thermal motion randomizes the spin orientations.
Some paramagnetic materials retain spin disorder at absolute zero, meaning they are paramagnetic
in the ground state.
Thus the total magnetization drops to zero when the applied field is removed.
Even in the presence of the field there is only a small induced magnetization because
only a small fraction of the spins will be oriented by the field.
This fraction is proportional to the field strength and this explains the linear dependency.
The attraction experienced by ferromagnetic materials is non-linear and much stronger,
so that it is easily observed, for instance, by the attraction between a refrigerator magnet
and the iron of the refrigerator itself.
Relation to electron spins Constituent atoms or molecules of paramagnetic
materials have permanent magnetic moments, even in the absence of an applied field.
The permanent moment generally is due to the spin of unpaired electrons in atomic or molecular
electron orbitals.
In pure paramagnetism, the dipoles do not interact with one another and are randomly
oriented in the absence of an external field due to thermal agitation, resulting in zero
net magnetic moment.
When a magnetic field is applied, the dipoles will tend to align with the applied field,
resulting in a net magnetic moment in the direction of the applied field.
In the classical description, this alignment can be understood to occur due to a torque
being provided on the magnetic moments by an applied field, which tries to align the
dipoles parallel to the applied field.
However, the true origins of the alignment can only be understood via the quantum-mechanical
properties of spin and angular momentum.
If there is sufficient energy exchange between neighbouring dipoles they will interact, and
may spontaneously align or anti-align and form magnetic domains, resulting in ferromagnetism
or antiferromagnetism, respectively.
Paramagnetic behavior can also be observed in ferromagnetic materials that are above
their Curie temperature, and in antiferromagnets above their Néel temperature.
At these temperatures, the available thermal energy simply overcomes the interaction energy
between the spins.
In general, paramagnetic effects are quite small: the magnetic susceptibility is of the
order of 10−3 to 10−5 for most paramagnets, but may be as high as 10−1 for synthetic
paramagnets such as ferrofluids.
Delocalization In conductive materials the electrons are
delocalized, that is, they travel through the solid more or less as an electron gas.
Conductivity can be understood in a band structure picture as arising from the incomplete filling
of energy bands.
In an ordinary nonmagnetic conductor the conduction band is identical for both spin-up and spin-down
electrons.
When a magnetic field is applied, the conduction band splits apart into a spin-up and a spin-down
band due to the difference in magnetic potential energy for spin-up and spin-down electrons.
Since the Fermi level must be identical for both bands, this means that there will be
a small surplus of the type of spin in the band that moved downwards.
This effect is a weak form of paramagnetism known as Pauli paramagnetism.
The effect always competes with a diamagnetic response of opposite sign due to all the core
electrons of the atoms.
Stronger forms of magnetism usually require localized rather than itinerant electrons.
However in some cases a band structure can result in which there are two delocalized
sub-bands with states of opposite spins that have different energies.
If one subband is preferentially filled over the other, one can have itinerant ferromagnetic
order.
This situation usually only occurs in relatively narrowbands, which are poorly delocalized.
s and p electrons Generally, strong delocalization in a solid
due to large overlap with neighboring wave functions means that there will be a large
Fermi velocity; this means that the number of electrons in a band is less sensitive to
shifts in that band's energy, implying a weak magnetism.
This is why s- and p-type metals are typically either Pauli-paramagnetic or as in the case
of gold even diamagnetic.
In the latter case the diamagnetic contribution from the closed shell inner electrons simply
wins from the weak paramagnetic term of the almost free electrons.
d and f electrons Stronger magnetic effects are typically only
observed when d- or f-electrons are involved.
Particularly the latter are usually strongly localized.
Moreover the size of the magnetic moment on a lanthanide atom can be quite large as it
can carry up to 7 unpaired electrons in the case of gadolinium(III).
The high magnetic moments associated with lanthanides is one reason why superstrong
magnets are typically based on elements like neodymium or samarium.
Molecular localization Of course the above picture is a generalization
as it pertains to materials with an extended lattice rather than a molecular structure.
Molecular structure can also lead to localization of electrons.
Although there are usually energetic reasons why a molecular structure results such that
it does not exhibit partly filled orbitals, some non-closed shell moieties do occur in
nature.
Molecular oxygen is a good example.
Even in the frozen solid it contains di-radical molecules resulting in paramagnetic behavior.
The unpaired spins reside in orbitals derived from oxygen p wave functions, but the overlap
is limited to the one neighbor in the O2 molecules.
The distances to other oxygen atoms in the lattice remain too large to lead to delocalization
and the magnetic moments remain unpaired.
Curie's law
For low levels of magnetization, the magnetization of paramagnets follows what is known as Curie's
law, at least approximately.
This law indicates that the susceptibility of paramagnetic materials is inversely proportional
to their temperature, i.e. that materials become more magnetic at lower temperatures.
The mathematical expression is:
where: is the resulting magnetization
is the magnetic susceptibility is the auxiliary magnetic field, measured
in amperes/meter is absolute temperature, measured in kelvins
is a material-specific Curie constant Curie's law is valid under the commonly encountered
conditions of low magnetization, but does not apply in the high-field/low-temperature
regime where saturation of magnetization occurs and magnetic dipoles are all aligned with
the applied field.
When the dipoles are aligned, increasing the external field will not increase the total
magnetization since there can be no further alignment.
For a paramagnetic ion with noninteracting magnetic moments with angular momentum J,
the Curie constant is related the individual ions' magnetic moments,
. The parameter μeff is interpreted as the
effective magnetic moment per paramagnetic ion.
If one uses a classical treatment with molecular magnetic moments represented as discrete magnetic
dipoles, μ, a Curie Law expression of the same form will emerge with μ appearing in
place of μeff.
When orbital angular momentum contributions to the magnetic moment are small, as occurs
for most organic radicals or for octahedral transition metal complexes with d3 or high-spin
d5 configurations, the effective magnetic moment takes the form,
, where n is the number of unpaired electrons.
In other transition metal complexes this yields a useful, if somewhat cruder, estimate.
Examples of paramagnets Materials that are called 'paramagnets' are
most often those that exhibit, at least over an appreciable temperature range, magnetic
susceptibilities that adhere to the Curie or Curie–Weiss laws.
In principle any system that contains atoms, ions, or molecules with unpaired spins can
be called a paramagnet, but the interactions between them need to be carefully considered.
Systems with minimal interactions The narrowest definition would be: a system
with unpaired spins that do not interact with each other.
In this narrowest sense, the only pure paramagnet is a dilute gas of monatomic hydrogen atoms.
Each atom has one non-interacting unpaired electron.
Of course, the latter could be said about a gas of lithium atoms but these already possess
two paired core electrons that produce a diamagnetic response of opposite sign.
Strictly speaking Li is a mixed system therefore, although admittedly the diamagnetic component
is weak and often neglected.
In the case of heavier elements the diamagnetic contribution becomes more important and in
the case of metallic gold it dominates the properties.
Of course, the element hydrogen is virtually never called 'paramagnetic' because the monatomic
gas is stable only at extremely high temperature; H atoms combine to form molecular H2 and in
so doing, the magnetic moments are lost, because the spins pair.
Hydrogen is therefore diamagnetic and the same holds true for many other elements.
Although the electronic configuration of the individual atoms of most elements contain
unpaired spins, they are not necessarily paramagnetic, because at ambient temperature quenching is
very much the rule rather than the exception.
The quenching tendency is weakest for f-electrons because f orbitals are radially contracted
and they overlap only weakly with orbitals on adjacent atoms.
Consequently, the lanthanide elements with incompletely filled 4f-orbitals are paramagnetic
or magnetically ordered.
Thus, condensed phase paramagnets are only possible if the interactions of the spins
that lead either to quenching or to ordering are kept at bay by structural isolation of
the magnetic centers.
There are two classes of materials for which this holds:
Molecular materials with a paramagnetic center.
Good examples are coordination complexes of d- or f-metals or proteins with such centers,
e.g. myoglobin.
In such materials the organic part of the molecule acts as an envelope shielding the
spins from their neighbors.
Small molecules can be stable in radical form, oxygen O2 is a good example.
Such systems are quite rare because they tend to be rather reactive.
Dilute systems.
Dissolving a paramagnetic species in a diamagnetic lattice at small concentrations, e.g. Nd3+
in CaCl2 will separate the neodymium ions at large enough distances that they do not
interact.
Such systems are of prime importance for what can be considered the most sensitive method
to study paramagnetic systems: EPR.
Systems with interactions
As stated above, many materials that contain d- or f-elements do retain unquenched spins.
Salts of such elements often show paramagnetic behavior but at low enough temperatures the
magnetic moments may order.
It is not uncommon to call such materials 'paramagnets', when referring to their paramagnetic
behavior above their Curie or Néel-points, particularly if such temperatures are very
low or have never been properly measured.
Even for iron it is not uncommon to say that iron becomes a paramagnet above its relatively
high Curie-point.
In that case the Curie-point is seen as a phase transition between a ferromagnet and
a 'paramagnet'.
The word paramagnet now merely refers to the linear response of the system to an applied
field, the temperature dependence of which requires an amended version of Curie's law,
known as the Curie–Weiss law:
This amended law includes a term θ that describes the exchange interaction that is present albeit
overcome by thermal motion.
The sign of θ depends on whether ferro- or antiferromagnetic interactions dominate and
it is seldom exactly zero, except in the dilute, isolated cases mentioned above.
Obviously, the paramagnetic Curie–Weiss description above TN or TC is a rather different
interpretation of the word 'paramagnet' as it does not imply the absence of interactions,
but rather that the magnetic structure is random in the absence of an external field
at these sufficiently high temperatures.
Even if θ is close to zero this does not mean that there are no interactions, just
that the aligning ferro- and the anti-aligning antiferromagnetic ones cancel.
An additional complication is that the interactions are often different in different directions
of the crystalline lattice, leading to complicated magnetic structures once ordered.
Randomness of the structure also applies to the many metals that show a net paramagnetic
response over a broad temperature range.
They do not follow a Curie type law as function of temperature however, often they are more
or less temperature independent.
This type of behavior is of an itinerant nature and better called Pauli-paramagnetism, but
it is not unusual to see e.g. the metal aluminium called a 'paramagnet', even though interactions
are strong enough to give this element very good electrical conductivity.
Superparamagnets Some materials show induced magnetic behavior
that follows a Curie type law but with exceptionally large values for the Curie constants.
These materials are known as superparamagnets.
They are characterized by a strong ferromagnetic or ferrimagnetic type of coupling into domains
of a limited size that behave independently from one another.
The bulk properties of such a system resembles that of a paramagnet, but on a microscopic
level they are ordered.
The materials do show an ordering temperature above which the behavior reverts to ordinary
paramagnetism.
Ferrofluids are a good example, but the phenomenon can also occur inside solids, e.g., when dilute
paramagnetic centers are introduced in a strong itinerant medium of ferromagnetic coupling
such as when Fe is substituted in TlCu2Se2 or the alloy AuFe.
Such systems contain ferromagnetically coupled clusters that freeze out at lower temperatures.
They are also called mictomagnets.
See also Bohr magneton
Curie temperature Diamagnetism
Ferromagnetism Magnetochemistry
References
General reference texts Charles Kittel, Introduction to Solid State
Physics.
Neil W. Ashcroft and N. David Mermin, Solid State Physics.
John David Jackson, Classical Electrodynamics.
External links http:www.ndt-ed.orgCommunityCollegePhysics/MagneticMatls.htm
Magnetism: Models and Mechanisms in E. Pavarini, E. Koch, and U. Schollwöck: Emergent Phenomena
in Correlated Matter, Jülich 2013, ISBN 978-3-89336-884-6