Curie's law

The curiesche law (also called Curie law ) describes the dependence of the magnetic susceptibility of a substance on the absolute temperature, provided that an ideal Paramagnetism spin present. It was set up by Pierre Curie in 1896 for the first time in this form. Developed in 1907, the French physicist Pierre -Ernest Weiss Curie's law continues to Curie -Weiss law, he included by cooperative effects in the equation.

This gives the law when considering an ideal system of particles with spin ½ (: particle number ). Ideal means that

• The ground state of the particles is thermally insulated,
• No spin -orbit coupling is present,
• No ligand-field effect is present,
• No magnetic anisotropy is present,
• And no collective magnetic effects are present, i.e., no magnetic interaction between the particles.

Description

As a model taking the orientation of a spin ½ particle in an external magnetic field. The electron as a rotating carrier has a magnetic moment and behaves as a magnetic dipole. If you create a external magnetic field, so this exercise from a directing force to the spin of the electron. It is an alignment of the spins in the direction of the magnetic field is possible, which is energetically favorable, and a magnetic field to the opposite orientation, which is energetically unfavorable. First, one would expect that align parallel to the external magnetic field in a substance all spins. However, actually, there is a temperature dependence is due to:

• The Boltzmann statistics: with increasing temperature increases the probability that spins occupy the unfavorable antiparallel alignment. and
• The thermal motion: with increasing temperature counteracts the proper motion of the particles of a magnetic field orientation.

The magnetic susceptibility is a physical quantity which depends on the number of spins in the magnetic field aligned in the field direction, and how many are opposite. To calculate the susceptibility, therefore, the directing effect of the external magnetic field and the counteracting thermal effects must be taken into account. The quantum- mechanically correct function for this task is the Brillouin function. The Curie 's law is a special case of this function for weak magnetic fields, and not too low temperatures:

With the Curie constants

It is

• The magnetic field constant
• The particle density
• Is Boltzmann's constant
• The amount of the permanent magnetic dipole moment; the Curie's law is assumed, that it is independent of temperature:

Often magnetic susceptibility and the Curie constant are based instead on the volume to the amount of substance:

With

The Avogadro constant.

Derivation

The magnetic moment of an electron is directly related to its spin and thus on the spin quantum number

Herein is

• The Landé factor for the spin of the electron
• The Bohr magneton.

The external magnetic field (magnitude of magnetic flux density ), there is for a particle with only two alignment possibilities (see Zeeman effect ):

• To the energetically favorable alignment in the field direction include the magnetic spin quantum number
• Energetically unfavorable to the opposing alignment includes the magnetic spin quantum number.

Each associated energy is given by:

The difference in energy between the two states is:

In the canonical ensemble, that is, at constant temperature and constant number of particles, is derived from the Boltzmann statistics, the occupation probability of each state:

With the energy normalization, that is, the reciprocal of the thermal energy. denotes the Boltzmann's constant and the temperature.

From the occupation probabilities gives the formula for the magnetization in pure Spin-1/2-Paramagnetismus:

The electronic component of the (spin - ) denotes the magnetic moment in the field direction:

The magnetic susceptibility is related to the magnetization as follows:

The curiesche law is obtained as an approximation under the assumption that the magnetic influence compared to the influence of temperature is small, ie, at relatively weak magnetic fields and relatively high temperatures:

Herein, the material-specific Curie constant.

Many-electron systems

For many-electron systems, the Curie law can be applied only limited because interelectronic interaction and spin -orbit coupling lead to complications. In the case of a pure LS- coupling wherein the electronic ground state is thermally insulated, the Curie constant can be formulated as follows:

With

• The total angular momentum quantum number, which is given by the LS- coupling for the ground state
• The Landé factor in LS coupling:

• The total spin quantum number
• The total orbital angular momentum quantum number.

The quantum numbers and belong to the ground state of the LS- coupling.

The quantum numbers, and can be determined using the Hund's rules.

Spin -only systems

For multi- electron systems, which in addition to the LS- coupling and thermal insulation of the ground state and half fill a subshell, it is called spin -only systems. The name comes from the fact that in the half- occupation is the total orbital angular momentum quantum number. Thus, the magnetic behavior of the atom is determined solely by its total spin.

The Landé factor is then at:

The Curie constant is given by:

Materials with Curie paramagnetism

The ideal Curie paramagnetic behavior occurs relatively rarely, because many factors (Inter electronic interaction, spin -orbit coupling, anisotropy, ligand field effects, collective effects ) strongly influence the magnetic behavior of a material. In the main group elements radicals show spin paramagnetic behavior, for example, the oxygen molecule with two unpaired electrons. In the subgroup elements are found Curie paramagnetism only for atoms with LS- coupling and thermally insulated ground state.

Spin -only paramagnetism is found in some compounds with weak ligand field of Mn or Fe (both: 3d electron configuration ) or Gd (4f -electron configuration). The ligand field effect must be weak enough that a high-spin configuration.

In case of collective magnetic effects, ie with ferromagnetism, antiferromagnetism or ferrimagnetism, the Curie- Weiss law applies instead of the curie 's law:

Herein is

• The Curie temperature. If it is positive, outweigh the ferromagnetic effects; if it is negative, outweigh antiferromagnetic or ferrimagnetic effects (see Neel temperature).