Curies law

The curiesche law (also Curie law called) describes the dependence of the magnetic susceptibility of a substance on the absolute temperature , if ideal spin - paramagnetism is present. It was first set up in this form by Pierre Curie in 1896. In 1907, the French physicist Pierre-Ernest Weiss developed Curie's law into the Curie-Weiss law by including cooperative effects in the equation. ${\displaystyle \chi _{\mathrm {m} }}$ ${\displaystyle T}$

The law is obtained if one considers an ideal system of particles with spin ½ ( : particle number ). Ideal means that ${\displaystyle N}$ ${\displaystyle N}$

• the ground state of the particles is thermally isolated,
• there is no spin-orbit coupling ,
• there is no ligand field effect ,
• there is no magnetic anisotropy ,
• and there are no collective magnetic effects, d. H. there is no magnetic interaction between the particles.

description

The orientation of a spin ½ particle in an external magnetic field is taken as a model . The electron has a magnetic moment and behaves as a magnetic dipole . If you apply an external magnetic field, this exerts a directing force on the electron's spin. An orientation of the spin in the direction of the magnetic field is possible, which is energetically favorable, and an orientation opposite to the magnetic field, which is energetically unfavorable. First of all, one would expect that all spins in a substance are aligned parallel to the external magnetic field. In fact, however, there is a temperature dependency that can be attributed to:

• the Boltzmann statistic : with increasing temperature , the probability increases that spins adopt the unfavorable antiparallel orientation. and
• the thermal motion : with increasing temperature the proper motion of the particles counteracts a magnetic field orientation.

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

${\displaystyle \chi _{\mathrm {m} }={\frac {C}{T}}}$

with the Curie constant ${\displaystyle C=\mu _{0}\,n\,{\frac {\mu ^{2}}{3\,k_{\mathrm {B} }}}.}$

In it is

• ${\displaystyle \mu _{0}}$the magnetic field constant
• ${\displaystyle n={\frac {N}{V}}}$the particle density
• ${\displaystyle k_{\mathrm {B} }}$the Boltzmann constant
• ${\displaystyle \mu }$the magnitude of the permanent magnetic dipole moment ; Curie's law is assumed to be independent of temperature: ${\displaystyle \mu \neq \mu (T).}$

Often magnetic susceptibility and Curie constant are related to the amount of substance instead of volume : ${\displaystyle V}$ ${\displaystyle n}$

${\displaystyle \chi _{\mathrm {m,mol} }={\frac {C_{\mathrm {mol} }}{T}}}$

With ${\displaystyle C_{\mathrm {mol} }=\mu _{0}\,N_{\mathrm {A} }\,{\frac {\mu ^{2}}{3\,k_{\mathrm {B} }}},}$

wherein the Avogadro constant referred to. ${\displaystyle N_{\mathrm {A} }={\frac {N}{n}}}$

Derivation

The magnetic moment of an electron depends directly on its spin and thus on the spin quantum number ${\displaystyle {\vec {\mu }}}$ ${\displaystyle {\vec {s}}}$ ${\displaystyle s={\frac {1}{2}}:}$

${\displaystyle |{\vec {\mu }}|=g_{s}\,{\sqrt {s\,(s+1)}}\,\mu _{\mathrm {B} }.}$

Is in here

• ${\displaystyle g_{s}\approx 2}$the Landé factor for the spin of the electron
• ${\displaystyle \mu _{\mathrm {B} }}$the Bohr magneton .

In the external magnetic field (magnitude of the magnetic flux density ) there are only two possibilities for a particle to orientate itself (see Zeeman effect ): ${\displaystyle B}$ ${\displaystyle s={\tfrac {1}{2}}}$

• The magnetic spin quantum number belongs to the energetically favorable alignment in the field direction ${\displaystyle m_{S}=-{\tfrac {1}{2}}}$
• The magnetic spin quantum number belongs to the energetically unfavorable opposing alignment . ${\displaystyle m_{S}=+{\tfrac {1}{2}}}$

The associated energy is given by:

${\displaystyle E_{m_{s}}=g_{s}\,m_{s}\,\mu _{\mathrm {B} }\,B.}$

The energy difference between the two states is:

${\displaystyle \Delta E=E_{m_{s}=+{\frac {1}{2}}}-E_{m_{s}=-{\frac {1}{2}}}=g_{s}\,\mu _{\mathrm {B} }\,B.}$

In the canonical ensemble , d. H. At constant temperature and constant number of particles, the Boltzmann statistics give the occupancy probability of the respective state: ${\displaystyle p_{m_{s}}}$

${\displaystyle p_{m_{s}=\pm {\frac {1}{2}}}={\frac {\exp \left(-\beta \,E_{m_{s}=\pm {\frac {1}{2}}}\right)}{\exp \left(-\beta \,E_{m_{s}=+{\frac {1}{2}}}\right)+\exp \left(-\beta \,E_{m_{s}=-{\frac {1}{2}}}\right)}}}$

with the energy normalization , d. H. the reciprocal of the thermal energy . denotes the Boltzmann constant and the temperature . ${\displaystyle \beta ={\frac {1}{k_{\rm {B}}T}}}$ ${\displaystyle k_{\rm {B}}}$ ${\displaystyle T}$

The formula for the magnetization for pure spin 1/2 paramagnetism results from the occupation probabilities: ${\displaystyle M}$

${\displaystyle M=n\;\sum _{m_{s}=-{\frac {1}{2}}}^{m_{s}=+{\frac {1}{2}}}\;\mu _{z,m_{s}}\,p_{m_{s}}.}$

In this case, referred to the component of the electronic (spin) the magnetic moment in the field direction: ${\displaystyle \mu _{z,m_{s}}}$

${\displaystyle \mu _{z,m_{s}}=g_{s}\,\mu _{\mathrm {B} }\,m_{s}.}$

The magnetic susceptibility is related to the magnetization as follows:

${\displaystyle \chi _{\mathrm {m} }={\frac {\partial M}{\partial B}}.}$

The curiesche law gives an approximation assuming that the magnetic influence is small compared to the effect of temperature, so at relatively low magnetic fields and relatively high temperatures:

${\displaystyle {\frac {\Delta E}{k_{\mathrm {B} }\,T}}\ll 1}$

${\displaystyle \Rightarrow \chi _{\mathrm {m} }={\frac {C}{T}}.}$

Herein is the substance-specific Curie constant . ${\displaystyle C}$

Multi-electron systems

The Curie law can only be applied to a limited extent for multi-electron systems, since interelectronic interaction and spin-orbit coupling lead to complications. In the case of a pure LS coupling in which the electronic ground state is thermally isolated, the Curie constant can be formulated as follows:

${\displaystyle C={\frac {\mu _{0}\,n}{3\,k_{\mathrm {B} }}}\,\mu ^{2}={\frac {\mu _{0}\,n}{3\,k_{\mathrm {B} }}}\,g_{J}^{2}\,J\,(J+1)\,\mu _{\mathrm {B} }^{2}}$

With

• the total angular momentum that results from the LS coupling for the ground state ${\displaystyle J}$
• the Landé factor with LS coupling: ${\displaystyle g_{J}}$

${\displaystyle g_{J}=1+{\frac {J\,(J+1)+S\,(S+1)-L\,(L+1)}{2\,J\,(J+1)}}}$

• the total spin quantum number ${\displaystyle S}$
• the total track angular momentum quantum number . ${\displaystyle L}$

The quantum numbers and belong to the basic state of the LS coupling. ${\displaystyle L}$ ${\displaystyle S}$

The quantum numbers , and can be determined with the help of Hund's rules . ${\displaystyle S}$ ${\displaystyle L}$ ${\displaystyle J}$

Spin-only systems

In the case of multi-electron systems which, in addition to LS coupling and thermal insulation of the ground state, also have a half-occupancy of a lower shell, one speaks of spin-only systems. The name comes from the fact that with half occupation the total orbital angular momentum is quantum number . As a result, the magnetic behavior of the atom is determined solely by its total spin. ${\displaystyle L=0}$

The Landé factor is then : ${\displaystyle J=S}$

${\displaystyle g_{S}=2.}$

The Curie constant results from:

${\displaystyle C={\frac {\mu _{0}\,n}{3\,k_{\mathrm {B} }}}\;2^{2}\,S\,(S+1)\,\mu _{\mathrm {B} }^{2}}$

Substances with Curie paramagnetism

The ideal Curie paramagnetic behavior occurs relatively seldom, since numerous factors (interelectronic interaction, spin-orbit coupling, anisotropy, ligand field effects, collective effects) strongly influence the magnetic behavior of a substance. For the main group elements , radicals show spin-paramagnetic behavior, e.g. B. the oxygen molecule with two unpaired electrons. In the case of the subgroup elements , Curie paramagnetism is only found in atoms with LS coupling and thermally isolated ground state.

Spin-only paramagnetism can be found in some compounds with a weak ligand field of Mn ${\displaystyle ^{2+}}$ or Fe ${\displaystyle ^{3+}}$ (both: 3d - electron configuration ) or Gd (4f -electronic configuration ). The ligand field effect must be weak enough that a high-spin configuration is present. ${\displaystyle ^{5}}$ ${\displaystyle ^{3+}}$ ${\displaystyle ^{7}}$

Curie-Weiss law

If collective magnetic effects occur, i.e. ferromagnetism , antiferromagnetism or ferrimagnetism , the Curie-Weiss law applies instead of Curie's law :

${\displaystyle \chi _{\mathrm {m} }={\frac {C}{T-\Theta }}.}$

Is in here

• C is the Curie-Weiss constant
• ${\displaystyle \Theta }$the Curie temperature . If it is positive, the ferromagnetic effects predominate; if it is negative, antiferromagnetic or ferrimagnetic effects predominate (see Néel temperature ).

literature

• Heiko Lueken : Magnetochemistry . BG Teubner, Stuttgart / Leipzig 1999, ISBN 3-519-03530-8 .