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 field is given by
U = −→µB.
−→
B (1)
U = −µBB cos θ (2)
According to Maxwell-Boltzmann statistics, at an absolute temperature T, the num-
ber of magnetic dipoles having energy U is proportional to exp −U
kBT
. Where, kB is the
Boltzmann’s constant.
In a bulk material magnetic dipoles are oriented in random directions, therefore, con-
tribution from all dipoles oriented between θ and θ + dθ with respect to the direction of
applied filed
−→
B per unit volume having energy U and is given by
dn = Cexp
−U
kBT
dΩ (3)
Here, dΩ is the solid angle between two hollow cones of semi-vertex angles θ and θ + dθ,
with c, a constant.
By definition, the solid angle Ω is given by
Ω = 2π(1 − cos θ)
dΩ = 2π sin θdθ (4)
from Eq.(3) and Eq.(4), we have
dn = Cexp
−U
kBT
2π sin θdθ (5)
Using Eq.(2) in (5), we have
dn = Cexp
µBB cos θ
kBT
2π sin θdθ (6)
substituting µBB
kBT
= x and 2πC = A in Eq.(6), we get
dn = Aexp(x cos θ) sin θdθ (7)
If we, integrate Eq.(7) between the limits (0, π), the total number of magnetic dipoles per
unit volume is given by
n =
π
0
dn =
π
0
Aexp(x cos θ) sin θdθ (8)
Using the substitution cos θ = u, i.e. sin θdθ = −du in Eq.(8), we have
n = −A
−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
sinh x (10)
Now, the magnetization due to contribution of dn magnetic dipoles parallel to the field
is given by the component µB cos θ. Whereas, the components perpendicular to the field
cancels one another the, by symmetry.
M =
π
0
µB cos θdn (11)
using Eq.(7) in Eq.(11), we have
M = µBA
π
0
cos θexp(x cos θ) sin θdθ
Now use the substitution cos θ = u, i.e. sin θdθ = −du in Eq. 8, we have
M = −µBA
−1
+1
uexu
du
M = −µBA u
ex
u
x
−
exu
x
du
−1
+1
M = µBA u
ex
u
x
−
exu
x2
+1
−1
M = µBA
ex
u
x
u −
1
x
+1
−1
M =
µBA
x
ex
1 −
1
x
− e−x
−1 −
1
x
M =
µBA
x
ex
+ e−x
−
1
x
ex
− e−x
M =
µBA
x
cos hx −
sinh x
x
M =
µBA
x
sinh x cothx −
1
x
(12)
Using Eq.(19) in Eq.(12), we obtain
M = nµB coth x −
1
x
M = nµBL(x)
where L(x) = coth x − 1
x
is known as Langevin’s function.
For small values of x, the series expansion of L(x) reduce to
L(x) = coth x −
1
x
(13)
L(x) ≈
x
3
Then the magnetization of the paramagnetic material is given by
M = nµB
x
3
µBB
kBT
= x (14)
2

3. M =
nµ2
BB
3kBT
(15)
Also, we know that M is very small for pramagnetic materials. Hence the magnetic
induction can be expressed as
B = µ0(M + H) ≈ µ0H
Then Eq.(15) takes the form
M =
nµ2
Bµ0H
3kBT
(16)
Hence, the magnetic susceptibility for a paramagnetic substances is given by
χ =
M
H
=
nµ2
Bµ0
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.
χ ∝
1
T
(18)
This is known as Curie’s law.
• Magnetic susceptibility has no explicit dependence on B.
n =
2A
x
sinh x (19)
References
[1] Solid state physics, Neil Ashcroft, Mermin, Brooks Cole, (1976).
[2] www.google.com
[3] www.arxiv.org
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