Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, Magnetization in the direction of the applied magnetic field. In contrast with this behavior, diamagnetism materials are repelled by magnetic fields and form induced magnetic fields in the direction opposite to that of the applied magnetic field.[Miessler, G. L. and Tarr, D. A. (2010) Inorganic Chemistry 3rd ed., Pearson/Prentice Hall publisher, .] Paramagnetic materials include most and some compounds;[ they have a relative magnetic permeability slightly greater than 1 (i.e., a small positive magnetic susceptibility) 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.
]
Paramagnetism is due to the presence of unpaired electrons in the material, so all atoms with incompletely filled are paramagnetic. Due to their spin, unpaired electrons have a magnetic dipole moment and act like tiny magnets. An external magnetic field causes the electrons' spins to align parallel to the field, causing a net attraction. Paramagnetic materials include aluminium, oxygen, titanium, and iron oxide (FeO).
Unlike ferromagnetism, 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 even at absolute zero, meaning they are paramagnetic in the ground state, i.e. in the absence of thermal motion.) 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 nonlinear and much stronger, so that it is easily observed, for instance, in 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 orbital or molecular electron orbitals (see
Magnetic moment). 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 quantummechanical properties of spin and
angular momentum.
If there is sufficient energy exchange between neighbouring dipoles, they will interact, and may spontaneously align or antialign and form magnetic domains, resulting in ferromagnetism (permanent magnets) 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 .
Delocalization
+ Selected Pauliparamagnetic metals
!Material!!Magnetic susceptibility, $\backslash chi\_v$ 10^{−5}
(SI units) 
6.8 
5.1 
2.2 
1.4 
1.2 
0.72 
In conductive materials, the electrons are delocalized, that is, they travel through the solid more or less as Free particle. 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 spinup and spindown electrons. When a magnetic field is applied, the conduction band splits apart into a spinup and a spindown band due to the difference in magnetic potential energy for spinup and spindown 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 subbands 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 narrow (d)bands, 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 ptype metals are typically either Pauliparamagnetic or as in the case of gold even diamagnetic. In the latter case the diamagnetic contribution from the closed shell inner electrons simply wins over 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) (hence its use in
MRI). The high magnetic moments associated with lanthanides is one reason why superstrong magnets are typically based on elements like
neodymium or
samarium.
Molecular localization
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 (i.e. unpaired spins), some nonclosed shell moieties do occur in nature. Molecular oxygen is a good example. Even in the frozen solid it contains diradical 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 O
_{2} molecules. The distances to other oxygen atoms in the lattice remain too large to lead to delocalization and the magnetic moments remain unpaired.
Theory
The Bohr–van Leeuwen theorem proves that there cannot be any diamagnetism or paramagnetism in a purely classical system. The paramagnetic response has then two possible quantum origins, either coming from permanents magnetic moments of the ions or from the spatial motion of the conduction electrons inside the material. Both description are given below.
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,
$\backslash scriptstyle\; \backslash chi$, of paramagnetic materials is inversely proportional to their temperature, i.e. that materials become more magnetic at lower temperatures. The mathematical expression is:
 $\backslash boldsymbol\{M\}\; =\; \backslash chi\backslash boldsymbol\{H\}\; =\; \backslash frac\{C\}\{T\}\backslash boldsymbol\{H\}$
where:
 $M$ is the resulting magnetization, measured in /meter (A/m),
 $\backslash chi$ is the volume magnetic susceptibility (dimensionless),
 $H$ is the auxiliary magnetic field (A/m),
 $T$ is absolute temperature, measured in (K),
 $C$ is a materialspecific Curie constant (K).
Curie's law is valid under the commonly encountered conditions of low magnetization ( μ_{B} H ≲ k_{B} T), but does not apply in the highfield/lowtemperature regime where saturation of magnetization occurs ( μ_{B} H ≳ k_{B} T) 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,
 $C=\backslash frac\{n\}\{3k\_\backslash mathrm\{B\}\}\backslash mu\_\{\backslash mathrm\{eff\}\}^2\; \backslash text\{\; where\; \}\; \backslash mu\_\{\backslash mathrm\{eff\}\}\; =\; g\_J\; \backslash mu\_\backslash mathrm\{B\}\; \backslash sqrt\{J(J+1)\}.$
where n is the number of atoms per unit volume. The parameter μ_{eff} 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 μ_{eff}.
 >
!Click "show" to see a derivation of this law: 
Curie's Law can be derived by considering a substance with noninteracting magnetic moments with angular momentum J. If orbital contributions to the magnetic moment are negligible (a common case), then in what follows J = S. If we apply a magnetic field along what we choose to call the zaxis, the energy levels of each paramagnetic center will experience Zeeman splitting of its energy levels, each with a zcomponent labeled by M_{J} (or just M_{S} for the spinonly magnetic case). Applying semiclassical Boltzmann statistics, the magnetization of such a substance is
 $n\backslash bar\{m\}\; =\; \backslash frac\{n\backslash sum\backslash limits\_\{M\_\{J\}\; =\; J\}^\{J\}\{\backslash mu\_\{M\_\{J\}\}e^\}/\{k\_\{\backslash mathrm\{B\}\}T\}\backslash ;\}\}\}\{\backslash sum\backslash limits\_\{M\_\{J\}\; =\; J\}^\{J\}\{e^\}/\{k\_\{\backslash mathrm\{B\}\}T\}\backslash ;\}\}\}\; =\; \backslash frac\{n\backslash sum\backslash limits\_\{M\_\{J\}\; =\; J\}^\{J\}\{M\_\{J\}g\_\{J\}\backslash mu\_\{\backslash mathrm\{B\}\}e^H\}/\{k\_\{\backslash mathrm\{B\}\}T\}\backslash ;\}\}\}\{\backslash sum\backslash limits\_\{M\_\{J\}\; =\; J\}^\{J\}\{e^H\}/\{k\_\{\backslash mathrm\{B\}\}T\}\backslash ;\}\}\}.$
Where $\backslash mu\_\{M\_J\}$ is the zcomponent of the magnetic moment for each Zeeman level, so $\backslash mu\; \_\{M\_J\}\; =\; M\_J\; g\_J\backslash mu\_\backslash mathrm\{B\}\; \; \backslash mu\_\backslash mathrm\{B\}$ is called the Bohr magneton and g_{ J} is the Landé gfactor, which reduces to the freeelectron gfactor, g_{ S} when J = S. (in this treatment, we assume that the x and ycomponents of the magnetization, averaged over all molecules, cancel out because the field applied along the zaxis leave them randomly oriented.) The energy of each Zeeman level is $E\_\{M\_J\}\; =\; M\_J\; g\_J\; \backslash mu\_\backslash mathrm\{B\}\; H$. For temperatures over a few K, $M\_J\; g\_J\; \backslash mu\_\backslash mathrm\{B\}H/k\_\backslash mathrm\{B\}\; T\; \backslash ll\; 1$, and we can apply the approximation $e^\{M\_J\; g\_J\; \backslash mu\_\backslash mathrm\{B\}\; H\; /k\_\backslash mathrm\{B\}\; T\backslash ;\}\; \backslash simeq\; 1\; +\; M\_J\; g\_J\; \backslash mu\_\backslash mathrm\{B\}\; H/k\_\backslash mathrm\{B\}\; T\backslash ;$:
 $\backslash bar\{m\}=\backslash frac\{\backslash sum\backslash limits\_\{M\_J=J\}^J\; \{M\_J\; g\_J\; \backslash mu\_\backslash mathrm\{B\}\; e^\{M\_J\; g\_J\; \backslash mu\_\backslash mathrm\{B\}\; H/k\_\backslash mathrm\{B\}\; T\backslash ;\}\}\}\{\backslash sum\backslash limits\_\{M\_J=J\}^J\; e^\{M\_Jg\_J\backslash mu\_\backslash mathrm\{B\}\; H/k\_\backslash mathrm\{B\}\; T\backslash ;\}\}\backslash simeq\; g\_J\backslash mu\_\backslash mathrm\{B\}\; \backslash frac\{\backslash sum\backslash limits\_\{M\_J=J\}^J\; M\_J\; \backslash left(\; 1+M\_J\; g\_J\backslash mu\_\backslash mathrm\{B\}\; H/k\_\backslash mathrm\{B\}\; T\backslash ;\; \backslash right)\}\{\backslash sum\backslash limits\_\{M\_J=J\}^J\; \backslash left(\; 1+M\_J\; g\_J\; \backslash mu\_\backslash mathrm\{B\}\; H/k\_\backslash mathrm\{B\}\; T\; \backslash ;\; \backslash right)\}=\backslash frac\{g\_J^2\; \backslash mu\_\backslash mathrm\{B\}^2\; H\}\{k\_\backslash mathrm\{B\}\; T\}\; \backslash frac\{\backslash sum\backslash limits\_\{J\}^J\; M\_J^2\}\{\backslash sum\backslash limits\_\{M\_J=J\}^J\{(1)\}\},$
which yields:
 $\backslash bar\{m\}=\backslash frac\{g\_J^2\; \backslash mu\_\backslash mathrm\{B\}^2\; H\}\{3k\_\backslash mathrm\{B\}\; T\}\; J(J+1)$. The bulk magnetization is then $M=n\backslash bar\{m\}\; =\; \backslash frac\{n\}\{3k\_\backslash mathrm\{B\}T\}\; \backslash leftH,$
and the susceptibility is given by
 $\backslash chi=\backslash frac\{\backslash partial\; M\_\{\backslash rm\; m\}\}\{\backslash partial\; H\}\; =\; \backslash frac\{n\}\{3k\_\{\backslash rm\; B\}\; T\}\; \backslash mu\_\{\backslash mathrm\{eff\}\}^2\; \backslash text\{\; ;\; and\; \}\; \backslash mu\_\{\backslash mathrm\{eff\}\}\; =\; g\_J\; \backslash sqrt\{J(J+1)\}\; \backslash mu\_\{\backslash mathrm\; B\}.$

When orbital angular momentum contributions to the magnetic moment are small, as occurs for most organic radicals or for octahedral transition metal complexes with d^{3} or highspin d^{5} configurations, the effective magnetic moment takes the form ( with gfactor g_{e} = 2.0023... ≈ 2),
 $\backslash mu\_\{\backslash mathrm\{eff\}\}\backslash simeq\; 2\backslash sqrt\{S(S+1)\}\; \backslash mu\_\backslash mathrm\{B\}\; =\backslash sqrt\{N\_\{\backslash rm\; u\}(N\_\{\backslash rm\; u\}+2)\}\; \backslash mu\_\backslash mathrm\{B\},$
where N_{u} is the number of unpaired electrons. In other transition metal complexes this yields a useful, if somewhat cruder, estimate.
Pauli paramagnetism
For some alkali metals and noble metals, conductions electrons are weakly interacting and delocalized in space forming a Fermi gas. For these materials one contribution to the magnetic response comes from the interaction between the electron spins and the magnetic field known as Pauli paramagnetism. For a small magnetic field $\backslash mathbf\{H\}$, the additional energy per electron from the interaction between an electron spin and the magnetic field is given by:
 $\backslash Delta\; E=\; \backslash mu\_0\backslash mathbf\{H\}\backslash cdot\backslash boldsymbol\{\backslash mu\}\_e=\backslash mp\; \backslash mu\_0\backslash mathbf\{H\}\backslash cdot\backslash left(g\_e\backslash frac\{\backslash mu\_\backslash mathrm\{B\}\}\{\backslash hbar\}\backslash mathbf\{S\}\backslash right)=\backslash pm\; \backslash mu\_0\; \backslash mu\_\backslash mathrm\{B\}\; H,$
where $\backslash mu\_0$ is the vacuum permeability, $\backslash boldsymbol\{\backslash mu\}\_e$ is the electron magnetic moment, $\backslash mu\_B$ is the Bohr magneton, $\backslash hbar$ is the reduced Planck constant, and the gfactor cancels with the spin $\backslash mathbf\{S\}=\backslash pm\backslash hbar/2$. The $\backslash pm$ indicates that the sign is positive (negative) when the electron spin component in the direction of $\backslash mathbf\{H\}$ is parallel (antiparallel) to the magnetic field.
For low temperatures with respect to the Fermi energy $T\_F$ (around 10^{4} Kelvin for metals), the number density of electrons $n\_\{\backslash uparrow\}$ ($n\_\{\backslash downarrow\}$) pointing parallel (antiparallel) to the magnetic field can be written as:
 $n\_\{\backslash uparrow\}\backslash approx\backslash frac\{n\_e\}\{2\}\backslash frac\{\backslash mu\_0\backslash mu\_\backslash mathrm\{B\}\}\{2\}g(E\_\backslash mathrm\{F\})H\backslash quad;\backslash quad\; \backslash left(n\_\{\backslash downarrow\}\backslash approx\backslash frac\{n\_e\}\{2\}+\backslash frac\{\backslash mu\_0\backslash mu\_\backslash mathrm\{B\}\}\{2\}g(E\_\backslash mathrm\{F\})H\backslash right),$
with $n\_e$ the total freeelectrons density and $g(E\_\backslash mathrm\{F\})$ the electronic density of states (number of states per energy per volume) at the Fermi energy $E\_\backslash mathrm\{F\}$.
In this approximation the magnetization is given as the magnetic moment of one electron times the difference in densities:
 $M=\backslash mu\_\backslash mathrm\{B\}(n\_\{\backslash downarrow\}n\_\{\backslash uparrow\})=\backslash mu\_0\backslash mu\_\backslash mathrm\{B\}^2g(E\_\backslash mathrm\{F\})H,$
which yields a positive paramagnetic susceptibility independent of temperature:
 $\backslash chi\_\backslash mathrm\{P\}=\backslash mu\_0\backslash mu\_\backslash mathrm\{B\}^2g(E\_\backslash mathrm\{F\}).$
The Pauli paramagnetic susceptibility is a macroscopic effect and has to be contrasted with Landau diamagnetic susceptibility which is equal to minus one third of Pauli's and also comes from delocalized electrons. The Pauli susceptibility comes from the spin interaction with the magnetic field while the Landau susceptibility comes from the spatial motion of the electrons and it is independent of the spin. In doped semiconductors the ratio between Landau's and Pauli's susceptibilities changes as the effective mass of the charge carriers $m^*$ can differ from the electron mass $m\_e$.
The magnetic response calculated for a gas of electrons is not the full picture as the magnetic susceptibility coming from the ions has to be included. Additionally, this formulas may break down for confined systems that differ from the bulk, like , or for high fields, as demonstrated in the de Haasvan Alphen effect.
Pauli paramagnetism is named after the physicist Wolfgang Pauli. Before Pauli's theory, the lack of a strong Curie paramagnetism in metals was an open problem as the Drude model could not account for this contribution without the use of quantum statistics.
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 noninteracting unpaired electron. 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. 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 H_{2} and in so doing, the magnetic moments are lost ( quenched), because of the spins pair. Hydrogen is therefore diamagnetic and the same holds true for many other elements. Although the electronic configuration of the individual atoms (and ions) 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 felectrons because f (especially 4 f) orbitals are radially contracted and they overlap only weakly with orbitals on adjacent atoms. Consequently, the lanthanide elements with incompletely filled 4forbitals are paramagnetic or magnetically ordered.
+μ_{eff} values for typical d^{3} and d^{5} transition metal complexes.[Orchard, A. F. (2003) Magnetochemistry. Oxford University Press.]
!Material!!μ_{eff}/μ_{B} 
3.77 
3.87 
3.79 
3.78 
5.92 
5.92 
5.89 

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 (isolated) paramagnetic center.

Good examples are coordination complexes of d or fmetals 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 O_{2} 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. Nd^{3+} in CaCl_{2} 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 felements 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 Néelpoints, 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 Curiepoint. In that case the Curiepoint 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:
 $\backslash boldsymbol\{M\}\; =\; \backslash frac\{C\}\{T\; \backslash theta\}\backslash boldsymbol\{H\}$
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 T_{ N} or T_{ C} 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 antialigning antiferromagnetic ones cancel. An additional complication is that the interactions are often different in different directions of the crystalline lattice (anisotropy), 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 Pauliparamagnetism, but it is not unusual to see, for example, 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 (with interaction). 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 TlCu_{2}Se_{2} or the alloy AuFe. Such systems contain ferromagnetically coupled clusters that freeze out at lower temperatures. They are also called mictomagnetism.
See also
Further reading

Charles Kittel, Introduction to Solid State Physics (Wiley: New York, 1996).

Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).

John David Jackson, Classical Electrodynamics (Wiley: New York, 1999).
External links

http://www.ndted.org/EducationResources/CommunityCollege/MagParticle/Physics/MagneticMatls.htm

Magnetism: Models and Mechanisms in E. Pavarini, E. Koch, and U. Schollwöck: Emergent Phenomena in Correlated Matter, Jülich 2013,