LANGEVIN THEORY ON PARA MAGNETISM

Langevin theory on para magnetism

Para magnetic substances have similar properties as Ferro magnetic substances  but with less intensity. The example  are aluminium, platinum, sodium, manganese , calcium, etc.

Para magnetic substances have permanent magnetic dipole  moment . The dipoles are randomly arranged in the absence of external magnetic field. When an external field is applied the dipoles of para substances align in the direction of external field.

The alignment of dipoles in the direction of external field depends on two factors.

1.  The magnitude of external field

2.   The temperature ( thermal agitation) of the atoms or molecules of the para substance

Let the number of atoms or molecules per unit volume of para substance is equal to  N.

The dipole moment of each atomic or molecular magnet = M

And the angle between the direction of field and the dipole of pare substance = ϴ

Then the potential energy of the dipole in the direction of external field   U =  MB cosϴ

Now imagine a sphere of radius r  within the para substance .

The atoms or molecules are aligned at all possible angles with the direction of the external field.

Let the number of atoms or molecules aligned between the angle  θ and ( θ + dθ ) is equal to dN

According to Langevin theory ,  dN is directly proportional to the potential energy of the atoms and also the solid angle between ϴ and ( ϴ + dϴ )

dN  ∝ e^(- U /KT)   ꭥ

  dN = A e^(- U /KT)   ꭥ

Where ꭥ = solid angle.

ꭥ = area  between  ϴ and ( ϴ + dϴ ) / r^2

ꭥ =  2π sinϴ dϴ

Substituting the values of U  = MB sinΘ  and ꭥ =  2π sinϴ dϴ

For dN

dN = A e^(-MB  sin ϴ/KT) *   2π sinϴ dϴ

A *2𝛑 =  constant , say C

dN = C

dN = C e^((-MB  sin ϴ/KT))  sinϴ dϴ

Also  -MB/KT = a  and cosϴ = 𝞪, then sinϴ dϴ = d𝞪

Then dN = C e^aα d𝞪

Integrating on both sides within the limits -1 to +1 for 𝞪 

and 0 to N for N

∫ dN = ∫ C e^aα d𝞪
 N  =  C[e^a -  e^(-a)]  / a
Intensity of magnetization I 

 I = ∫ M cosϴ dN
Integrating and substituting the value of N
we get

 I =  MN [ coth a - (1/a)  ]
 coth a - (1/a)  is known as Langevin function .it is written as

  coth a - (1/a)  = L(a)
Hence the intensity of magnetization I =  MN * L(a)
When all the atoms or molecules align in the direction of external field the para substance has maximum intensity of magnetization ( saturation )  I0  =  MN
 I  / I0   =  L(a)

Graph  

A graph is drawn between  I / I0  and  Langevin function  L(a)