Paramagnetism

Simplified comparison of the permeabilities of ferromagnetic (μ _f ), paramagnetic (μ _p ) and diamagnetic materials (μ _d ) to vacuum (μ ₀ ). Here, μ is the slope of each curve B (H).
H: field strength of the external field
B: flux density of the induced field

Paramagnetism is one of the manifestations of magnetism in matter . Like diamagnetism , it describes the magnetic behavior of a material that is exposed to an external magnetic field . Paramagnets follow the external field in their magnetization, so that the magnetic field is stronger inside than outside. Paramagnetic materials therefore have a tendency to be drawn into a magnetic field. Without an external magnetic field, paramagnetic materials show no magnetic order (in contrast to the "spontaneous magnetization" that occurs even without a magnetic field, e.g. in ferromagnetism ).

The magnetic permeability of paramagnets is greater than 1 (or the magnetic susceptibility is positive). In the physical classification, all materials that meet this condition and have no persistent magnetic order are considered paramagnetic. ${\ displaystyle \ textstyle \ mu _ {r}}$ ${\ displaystyle \ chi = \ textstyle \ mu _ {r} -1}$

origin

Illustrations of a paramagnetic sample without an external magnetic field, ...

... in a weak magnetic field ...

... and in a strong magnetic field.

Paramagnetism only occurs in substances that have unpaired electrons (radicals, transition metal cations, lanthanoid cations) and whose atoms or molecules have a magnetic moment . Causes for this are quantum mechanical effects, u. a. the intrinsic angular momentum ( spin ) of the electrons.

As a model, one can imagine a paramagnetic sample made up of small bar magnets that rotate but cannot slip. If the sample is brought into a magnetic field, the bar magnets will preferentially align themselves in the direction of the magnetic field lines . An important feature here is that the bar magnets do not influence each other - they all align themselves independently of each other. The heat fluctuations cause a constant, random reorientation of the bar magnets. Disordered bar magnet orientations, i.e. configurations of the bar magnet orientations with a vanishing magnetic moment on average, are much more likely than an ordered distribution in which the magnetic moments are oriented in the same direction and lead to a non-vanishing total magnetic moment. Therefore, the stronger you want to align the magnets, the stronger the magnetic fields.

In physical terms: the cause of paramagnetic behavior lies in the alignment of the microscopic magnetic moments of a substance in a magnetic field. The individual magnetic moments are independent of one another. In contrast to ferromagnets , such an alignment is immediately destroyed by thermal fluctuations after the magnetic field is switched off. The magnetization of the material is proportional to the applied magnetic field ${\ displaystyle M}$ ${\ displaystyle H}$

{\ displaystyle \ left.M = \ chi H \ right.}

with .

{\ displaystyle \ left. \ chi> 0 \ right.}

The greater the magnetic susceptibility of the substance, the easier it is to magnetize it. The susceptibility is therefore a measure of the strength of paramagnetism. Because of the simple relationship between the susceptibility and the relative magnetic permeability , the latter is often used as a measure. ${\ displaystyle \ chi}$ ${\ displaystyle \ mu _ {r} = \ chi +1}$

One can often read that a very high susceptibility means that a sample is ferromagnetic. This statement is not entirely true. Although the susceptibility of ferromagnets is very high in many cases, the cause lies in said coupling. Ferromagnets show a magnetization even after switching off the magnetic field, the so-called remanence , while with paramagnets, as already mentioned, the magnetization disappears again after switching off the field.

species

A classical consideration does not provide an explanation for the presence of the magnetic moments discussed above. However, these can be understood quantum mechanically . The important for magnetism statement is that the total angular momentum of an atomic state always with a magnetic moment associated ${\ displaystyle {\ vec {J}}}$ ${\ displaystyle {\ vec {\ mu}}}$

{\ displaystyle {\ vec {\ mu}} = - g \ mu _ {\ mathrm {B}} {\ vec {J}} / {\ hbar}}

.

It is the Landé factor and the Bohr magneton . The total angular momentum is made up of three components: ${\ displaystyle g}$ ${\ displaystyle \ mu _ {\ mathrm {B}}}$

Spin and
Orbital angular momentum of the electrons as well
Nuclear spin of nucleons .

The magnetic moment associated with nuclear spin is - because of the significantly larger mass of the nucleons - too weak to be able to make a significant contribution to susceptibility. Therefore this is not considered further in the following. It should be noted, however, that the magnetic moment of the nucleus is definitely measurable, which is used in medicine for magnetic resonance imaging (MRT) (this is why the procedure is also called magnetic resonance imaging ).

The main contributions to susceptibility come from various sources, which are listed below. However, since there are always diamagnetic contributions to susceptibility, it is only an addition of all contributions that decides whether a substance is ultimately paramagnetic. However, if Langevin paramagnetism (see below) occurs, its contribution is usually dominant.

Magnetic moments of atoms in the ground state ( Langevin paramagnetism )

The total angular momentum of an atom in its ground state can be determined theoretically using Hund's rules . The most important essence from this is that the total angular momentum of a closed shell always adds up to zero. In all other cases, the atom has a magnetic moment. ${\ displaystyle {\ vec {J}}}$

The temperature dependence of this contribution is determined by Curies law

{\ displaystyle \ chi _ {\ mathrm {Langevin}} = {\ frac {C} {T}} \ sim 10 ^ {- 3 \ ldots -2}}

is described here, the Curie constant (a material constant ). ${\ displaystyle C}$

A more detailed analysis of the Langevin paramagnetism is done with the help of the Langevin and Brillouin functions .

Magnetic moments of conduction electrons ( Pauli paramagnetism )

Electrons can move practically freely in metals . Every electron has a magnetic moment as a result of its spin - one expects a Curie-like contribution to susceptibility. However, due to the Pauli principle , only the excited conduction electrons have the freedom to align their spin in the magnetic field. Their number is proportional to ( is the Fermi temperature , another material constant ): ${\ displaystyle T / T _ {\ mathrm {F}}}$ ${\ displaystyle T _ {\ mathrm {F}}}$

{\ displaystyle \ chi _ {\ mathrm {Pauli}} \ sim {\ frac {C} {T}} \ cdot {\ frac {T} {T _ {\ mathrm {F}}}} = {\ frac {C } {T _ {\ mathrm {F}}}} \ sim 10 ^ {- 6 ..- 5}}

.

However, a closer look shows that there is a dependency on the strength of the external magnetic field.

Magnetic moments of atoms in excited states ( Van Vleck paramagnetism )

Even if the total angular momentum of an atom is zero in its ground state, this does not have to apply to excited states . At a finite temperature, some atoms are always in an excited state, which is why this contribution occurs in all substances. However, it is only of significant size in molecular crystals; there it can even surpass Langevin's paramagnetism in strength. Calculating the size of this contribution is, however, quite time-consuming, especially for molecules .

Comparison of the orders of magnitude

${\ displaystyle \ chi ^ {\ text {van Vleck}} \ approx \ chi ^ {\ text {Pauli}} \ approx | \ chi ^ {\ text {Diamag.}} | \ approx 10 ^ {- 6 \ ldots -5} \ ll \ chi ^ {\ text {Langevin}} \ approx 10 ^ {- 3 \ ldots -2}}$

Superparamagnetism

The magnetic properties of granular ferromagnetic solids depend on the grain size . When the grain size is reduced, the number of magnetic regions ( Weiss region ) per grain decreases. Below a critical size, it is energetically unfavorable to form several of these areas. So there is only one Weiss area per grain, ie all atomic magnetic moments of a grain are arranged parallel to one another. Below a further critical value, a stable alignment of the total magnetic moment is no longer possible at finite temperatures, since the energy required for remagnetization becomes smaller than the thermal energy . The solid as a whole now behaves paramagnetically with the peculiarity that the magnetic moments do not react individually, but in blocks to external magnetic fields. This particular form of paramagnetism is known as superparamagnetism.

application

The paramagnetism of oxygen is used in physical gas analysis.

Examples

Alkali metals

The electron shell of the alkali metals consists of a noble gas configuration and an additional s-electron. According to Hund's rules , the atoms in their ground state have a magnetic moment. This is the first case (see above) that makes a strong contribution to susceptibility. The alkali metals are therefore paramagnetic.

Alkaline earth metals

In contrast to the alkali metals, the alkaline earth metals have two s-electrons and thus a closed lower shell. However, they belong to the group of metals and thus concern the second case. With the exception of beryllium , this contribution outweighs the diamagnetic contribution, which means that the alkaline earth metals are weakly paramagnetic.

Rare earth

The rare earth metals are among the technically most important materials for alloys in permanent magnets . The reason is that the crucial shell that is not fully occupied is inside the electron shell (f electrons) and therefore has practically no influence on the chemical properties of the atoms. Almost all of these metals are therefore paramagnetic (after the first case), but the strength of the paramagnetism varies greatly; Gadolinium is even ferromagnetic. This makes them ideal candidates in alloys with ferromagnetic metals, which can make very strong permanent magnets.

Molecules

Since molecules often have a closed electronic configuration and are not metals, they only show a contribution after the third case. Some examples of paramagnetic substances are:

Magnetite

Magnetite (Fe ₃ O ₄ ) normally shows ferrimagnetic behavior ( ferrimagnetism ).
With particle sizes which are smaller than 20 to 30 nm, superparamagnetic behavior is shown at room temperature. In the presence of an external magnetic field, all particles align themselves in the direction of this field. After removing the external field, the thermal energy is large enough that the mutual alignment of the particles relaxes and the magnetization approaches zero again.

literature

Dieter Meschede : Gerthsen Physics. 18th edition. Springer-Verlag, Berlin 1995, pp. 390f., ISBN 3-540-59278-4 - Brief overview of paramagnetism.
Neil W. Ashcroft , N. David Mermin : Solid State Physics. International Edition. Harcourt, Orlando 1976, pp. 643-670, ISBN 0-03-049346-3 (English) - detailed theoretical treatment of para- and diamagnetism.

Web links

Classification of Magnetic Elements (English, PDF) - from the Applied Alloy Chemistry Group at the University of Birmingham. (238 kB)
Quantum Mechanics - The Key To Understanding Magnetism (English, PDF) - Lecture by JH Van Vleck at the Nobel Prize award 1977. (137 kB)
Video: Paramagnetic Matter . Institute for Scientific Film (IWF) 2004, made available by the Technical Information Library (TIB), doi : 10.3203 / IWF / C-14890 .

Individual evidence

↑ Wolfgang Nolting: 2 parts, part 1, basics 1st edition. Teubner Verlag, 1986, pp. 214f., ISBN 3-519-03084-5 - Paramagnetism of Localized Moments.

[1] Wolfgang Nolting: 2 parts, part 1, basics 1st edition. Teubner Verlag, 1986, pp. 214f., ISBN 3-519-03084-5 - Paramagnetism of Localized Moments.