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 :
B.
(
x
)
{\ displaystyle B (x)}
B.
J
(
x
)
=
2
J
+
1
2
J
⋅
coth
(
2
J
+
1
2
J
x
)
-
1
2
J
⋅
coth
(
1
2
J
x
)
=
(
1
+
1
2
J
)
⋅
coth
[
(
1
+
1
2
J
)
x
]
-
1
2
J
⋅
coth
(
1
2
J
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:
M.
{\ displaystyle M}
N
{\ displaystyle N}
M.
=
N
m
B.
J
(
ξ
)
⇔
B.
J
(
ξ
)
=
M.
N
m
.
{\ displaystyle {\ begin {aligned} M & = NmB_ {J} (\ xi) \\\ Leftrightarrow B_ {J} (\ xi) & = {\ frac {M} {Nm}}. \ end {aligned}} }
With
the magnetic moment of a particle
m
{\ displaystyle m}
the parameter
ξ
=
m
B.
k
B.
T
=
G
μ
B.
J
B.
k
B.
T
{\ 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):
L.
{\ displaystyle L}
J
→
∞
{\ displaystyle J \ to \ infty}
G
μ
B.
→
0
{\ displaystyle g \ mu _ {\ mathrm {B}} \ to 0}
M.
=
N
m
L.
(
ξ
)
⇔
L.
(
ξ
)
=
M.
N
m
.
{\ 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. Web links
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
![{\ 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27c2820358e7683e738656e51d548303e1f5b549)
