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# paramagnetism

[par-uh-mag-nit, par-uh-mag-]
paramagnetism: see magnetism.
Paramagnetism is a form of magnetism which occurs only in the presence of an externally applied magnetic field. Paramagnetic materials are attracted to magnetic fields, hence have a relative magnetic permeability greater than one (or, equivalently, a positive magnetic susceptibility). The force of attraction generated by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect. Unlike ferromagnets, paramagnets do not retain any magnetization in the absence of an externally applied magnetic field, because thermal motion causes the spins to become randomly oriented without it. Thus the total magnetization will drop 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 ferromagnets is non-linear and much stronger, so that it is easily observed, for instance, by opening and closing the door of one's refrigerator.

## Curie's law

For low levels of magnetisation, the magnetisation of paramagnets follows Curie's law to good approximation:

$boldsymbol\left\{M\right\} = chi cdotboldsymbol\left\{H\right\}=Ccdot frac\left\{boldsymbol\left\{H\right\}\right\}\left\{T\right\}$

where

M is the resulting magnetization
χ is the magnetic susceptibility
H is the auxiliary magnetic field, measured in amperes/meter
T is absolute temperature, measured in kelvins
C is a material-specific Curie constant

This law indicates that the susceptibility χ of paramagnetic materials is inversely proportional to their temperature. Curie's law is only valid under conditions of low magnetisation, since it does not consider the saturation of magnetisation that occurs when the atomic dipoles are all aligned in parallel. After everything is aligned, increasing the external field will not increase the total magnetisation since there can be no further alignment. However such saturation typically requires very strong magnetic fields.

## Examples of paramagnets

It is not easy to identify which materials should be called 'paramagnets', because the term is often used for rather different systems. In principle any system that contains atoms, ions or molecules with unpaired spins can be called a paramagnet, but the interactions between them do need consideration.

### 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. However, even the element hydrogen is usually not called a 'paramagnet' because at lower temperatures the monatomic gas is not stable. Two atoms will combine to form molecular H2 and in that interaction the magnetic moments are lost (quenched), because the spins will pair. As a substance hydrogen is therefore usually considered a diamagnet. This holds true for many elements. Although the electronic configuration of the individual atoms (and ions) of most elements contain unpaired spins, it is not correct to call these elements 'paramagnets' because at lower temperatures quenching is the rule rather than the exception. As pointed out above the quenching tendency is weakest for f-electrons.

Thus, condensed phase paramagnets are only possible if the interactions of the spins that lead either to quenching or to ordering are somehow kept at bay. There are two classes of materials for which this holds:

1. Molecular materials with a (isolated) paramagnetic center.
1. Good examples are organometallic compounds 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.
2. 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.
2. Dilute systems.
1. 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. $boldsymbol\left\{M\right\} = C frac\left\{boldsymbol\left\{H\right\}\right\}\left\{T- theta\right\}$. 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 (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 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

There are materials that 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 ferro- 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 microsopic level they are ordered. The materials do show an ordering temperature above which the behavior reverts to ordinary paramagnetism (with interaction). 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