Langevin theory of Paramagnetism

Paramagnetism has been explained using the classical approach. Derivation of Magnetization and Susceptibility in case of paramagnetism using Langevin Theory of Paramagnetism.

1. Langevin's Theory of Paramagnetism
The potential energy of the magnetic dipole in the external eld is given by
U=
!
B:
!
B (1)
U=BBcos (2)
According to Maxwell-Boltzmann statistics, at an absolute temperatureT, the num-
ber of magnetic dipoles having energyUis proportional toexp

U
kBT

. Where,kBis the
Boltzmann's constant.
In a bulk material magnetic dipoles are oriented in random directions, therefore, con-
tribution from all dipoles oriented betweenand+dwith respect to the direction of
applied led
!
Bper unit volume having energyUand is given by
dn=Cexp

U
kBT

d (3)
Here,dis the solid angle between two hollow cones of semi-vertex anglesand+d,
with c, a constant.
By denition, the solid angleis given by
= 2(1cos)
d = 2sind (4)
from Eq.(3) and Eq.(4), we have
dn=Cexp

U
kBT

2sind (5)
Using Eq.(2) in (5), we have
dn=Cexp

BBcos
kBT

2sind (6)
substituting
BB
kBT
=xand2C=Ain Eq.(6), we get
dn=Aexp(xcos) sind (7)
If we, integrate Eq.(7) between the limits(0; ), the total number of magnetic dipoles per
unit volume is given by
n=
Z

0
dn=
Z

0
Aexp(xcos) sind (8)
Using the substitutioncos=u, i.e.sind=duin Eq.(8), we have
n=A
Z
1
+1
exp(ux)du
n=A

exp(ux)
x

1
+1
n=A

e
x
e
+x
x

n=
A
x
(e
+x
e
x
)
(9)
1

2. n=
2A
x
sinhx
(10)
Now, the magnetization due to contribution ofdnmagnetic dipoles parallel to the eld
is given by the componentBcos. Whereas, the components perpendicular to the eld
cancels one another the, by symmetry.
M=
Z

0
Bcosdn (11)
using Eq.(7) in Eq.(11), we have
M=BA
Z

0
cosexp(xcos) sind
Now use the substitutioncos=u, i.e.sind=duin Eq. 8, we have
M=BA
Z
1
+1
ue
xu
du
M=BA

u
e
x
u
x

Z
e
xu
x
du

1
+1
M=BA

u
e
x
u
x

e
xu
x
2

+1
1
M=BA

e
x
u
x

u
1
x

+1
1
M=
BA
x

e
x

1
1
x

e
x

1
1
x

M=
BA
x


e
x
+e
x


1
x

e
x
e
x


M=
BA
x

coshx
sinhx
x

M=
BA
x
sinhx

cothx
1
x

(12)
Using Eq.(19) in Eq.(12), we obtain
M=nB

cothx
1
x

M=nBL(x)
whereL(x) =

cothx
1
x

is known as Langevin's function.
For small values ofx, the series expansion ofL(x)reduce to
L(x) =

cothx
1
x

(13)
L(x)
x
3
Then the magnetization of the paramagnetic material is given by
M=nB
x
3

*
BB
kBT
=x

(14)
2

3. M=
n
2
B
B
3kBT
(15)
Also, we know thatMis very small for pramagnetic materials. Hence the magnetic
induction can be expressed as
B=0(M+H)0H
Then Eq.(15) takes the form
M=
n
2
B
0H
3kBT
(16)
Hence, the magnetic susceptibility for a paramagnetic substances is given by
=
M
H
=
n
2
B
0
3kBT
(17)
In above Eq.(17), few important results can be pointed out from expression for the
magnetic susceptibility
Magnetic susceptibility of the paramagnetic materials are positive.
Magnetic susceptibility varies inversely with temperature,i.e.
_
1
T
(18)
This is known as Curie's law.
Magnetic susceptibility has no explicit dependence onB.
n=
2A
x
sinhx
(19)
References
[1]
[2]
[3]
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