# Brillouin function

Brillouin function
for different values ​​of  J

The Brillouin function (after the French-American physicist Léon Brillouin (1889–1969)) is a special function that emerges from the quantum mechanical description of a paramagnet : ${\ displaystyle B (x)}$

{\ displaystyle {\ begin {alignedat} {2} B_ {J} (x) & = {\ frac {2J + 1} {2J}} \ cdot \ coth \ left ({\ frac {2J + 1} {2J }} \, x \ right) && - {\ frac {1} {2J}} \ cdot \ coth \ left ({\ frac {1} {2J}} \, x \ right) \\ & = \ left ( 1 + {\ frac {1} {2J}} \ right) \ cdot \ coth \ left [\ left (1 + {\ frac {1} {2J}} \ right) x \ right] && - {\ frac { 1} {2J}} \ cdot \ coth \ left ({\ frac {1} {2J}} \, x \ right) \ end {alignedat}}}

The symbols represent the following quantities :

## use

With the Brillouin function, the magnetization of a paramagnet of the amount of substance in an external magnetic field can be formulated: ${\ displaystyle M}$ ${\ displaystyle N}$

{\ displaystyle {\ begin {aligned} M & = NmB_ {J} (\ xi) \\\ Leftrightarrow B_ {J} (\ xi) & = {\ frac {M} {Nm}}. \ end {aligned}} }

With

• the magnetic moment of a particle${\ displaystyle m}$
• the parameter ${\ displaystyle \ xi = {\ frac {mB} {k _ {\ mathrm {B}} \, T}} = {\ frac {g \ mu _ {\ mathrm {B}} \, JB} {k _ {\ mathrm {B}} \, T}}}$

Another, semi-classical description of a paramagnet occurs with the help of the Langevin function , which results in the Limes and at the same time from the Brillouin function (whereby the total magnetic moment remains constant): ${\ displaystyle L}$ ${\ displaystyle J \ to \ infty}$${\ displaystyle g \ mu _ {\ mathrm {B}} \ to 0}$

{\ displaystyle {\ begin {aligned} M & = NmL (\ xi) \\\ Leftrightarrow L (\ xi) & = {\ frac {M} {Nm}}. \ end {aligned}}}

## literature

• Torsten Fließbach: Statistical Physics - Textbook on Theoretical Physics IV . Elsevier Spectrum Academic Publishing House, Heidelberg 2006.