Engineering Physics 3(2+1)

LESSON 4. Theory of Magnetism

Where \[\theta\] is constant with dimension of temperature and have small values of the order of 10K or less for most paramagnetics.

It was Weiss who improved upon Langvien theory with considerable success by his new concept on an internal molecular field within the gas produced at any point by the neighboutring molecular magnets.

If this internal molecular field is represented by H_m , Weiss assume that H_{m =\[\beta I\]}

I is intensity  of magnetization and  a constant is called molecular field constant.

Effective magnetic field

\[{H_e}=H + {H_m}\]…………..(2)

\[{H_e}=H + \beta I\]……………(3)

\[I={{{\mu ^2}{\mu _o}HN} \over {3kT}}\]

Or

\[I=\mu NL\left( a \right)={I_s}L\left( a \right)\]

\[{I \over {{I_s}}}={{{\mu _o}\mu \left( {H + \beta I} \right){\rm{}}} \over {3kT}}\]…………….(4)

Taking one gram molecular weight of the material and expressing the gram molar magnetic moment  and its saturation value given by  σ_s

So,

\[{I\over{{I_s}}}\] =\[{{\sigma\over{{\sigma _s}{\rm{}}}}\]= \[{{{\mu _o}\mu \left( {H + \beta I} \right){\rm{}}} \over {3kT}}\]…………..(5)

 Let ρ is a density and  is molecular weight of the sample

And

\[{I_s} = {{{\sigma _s}\rho } \over M}\]………..(7)

\[\mu={{{\sigma _s}}\over N}\] ………..(8)     N  Avogadro’s number

Using equation (5) and (8)

We get,

\[{\sigma\over{{\sigma _c}}}={{{\mu _{o}}{\sigma _s}}\over {3NkT}}\left( {H + \beta I} \right)\]………….(9)

Where R is molecular gas constant = NΚ

Using equation (9) and (7)

\[\sigma={{{\mu _{o}}{\sigma _s}^2{_{}}}\over {3RT}}\left({H + \beta{{\sigma \rho }\over M}} \right)\]……………(10)

The molecular susceptibility

\[{\chi _m}={\sigma\over H}\]

Using equation (10)

 \[{\chi _m}\left( {T - {{{\mu _{o}}{\sigma _s}^2\beta {\rho _{}}} \over {3Rm}}} \right) = {C_m}\]

Or

\[{\chi _m}\left( {T - \theta } \right)={C_m}\]

\[\theta={{{\mu _{o}}{\sigma _s}^2\beta {\rho _{}}} \over {3Rm}}\] is called Curie point and C_m  = \[{{{\mu _{o}}{\sigma _s}^2{_{}}} \over {3R}}\] is called Curie molecular constant.

So

\[{\chi _m}={{{C_m}} \over {\left( {T - \theta } \right)}}\]………….(11)

This expression is called Curie-Weiss law.

 Ferromagnetism

The term ferromagnetism may be defined as a type of magnetism in which a material gets magnetized to a very large extent in the presence of an external field. The direction in which the material gets magnetized is the same as that of the magnetic field.

Properties

• Ferromagnetic materials acquire a high degree of magnetization in the same sense as the applied field some classical examples of ferromagnetic are iron, cobalt and nickel.
• These materials attract the line of forces strongly and hence permeability values are very high.
• The value of susceptibility of ferromagnetic is also very large and positive.
• When ferromagnetic material is placed in an external magnetic field, such that the length of the material is parallel to the field direction, a small change in the dimensions occurs. This effect is called magnetostriction.
• As the temperature, increases the value of susceptibility decreases. Above a certain temperature, ferromagnetic materials become ordinary paramagnetic and this temperature is called the curie temperature.

Weiss Molecular Field Theory and Ferromagnetism Or

Weiss theory of Ferromagnetism Or

Mean Field Theory of Ferromagnetism Or

Molecular Theory of Ferromagnetism

It has been observed that the ratio of the magnetic moment to the angular momentum    for the spinning electron is twice for an orbital electron. In case of ferromagnetic materials this ratio is almost same as that for a spinning electron. Hence, the spinning electrons make a major contribution to magnetic properties of ferromagnetic.

According to Weis, the effective field strength  may be regarded as the vector sum of external applied field strength  and the internal molecular field strength B_i

\[{B_e}=B + {B_i}=B + \lambda I\]…… (1)

Consider gram molecule of the substance whose density is  and molecular weight  and   and  being the gram molecular moment and its saturation value respectively then

And

\[{I_s} = {{{\sigma _s}\rho } \over M}\]……….. (3)

Since I  and I_s  refer to the unit volume

\[{I \over {{I_s}}}\] = \[{\sigma\over{{\sigma _s}{\rm{}}}}\]………… (4)

As the domains are assumed to obey the general theory of paramgnetism, then from Langevin’s theory of paramgnetism, we have

\[{I \over {{I_s}}}=L\left( a \right)=\left[{\cot a -{1\over a}}\right]\]…………. (5)

Equating (4) and (5)

\[{I \over {{I_s}}}\] = \[{\sigma\over{{\sigma _s}{\rm{}}}}\]  = \[L\left(a\right)=\left[{\cot a-{1\over a}}\right]\]………….. (6)

Where \[{{\mu {B_e}}\over {KT}}\] = a =  \[{{\mu \left( {B + \lambda I} \right){\rm{}}} \over {KT}}\]…….. (7)

When the external field is zero  B = 0

\[{B_e}=\lambda I={{\lambda \sigma \rho } \over M}\]…………. (8)

 Putting equation (8) in equation (7)

\[{\mu\over{KT}}{{\lambda\sigma\rho}\over M}\] = a = \[{{{\sigma _o}} \over N}\] \[{{\lambda \sigma \rho } \over {MKT}}={{{\sigma _o}} \over N}{{\left( {B + \lambda I} \right){\rm{}}} \over {KT}}\]

Hence

\[{\sigma\over{{\sigma _o}}}={{RMT} \over {\lambda \rho {\sigma _o}^2}}a\]……… (9)

Equation (9) and (7) simultaneously determine the condition for spontaneous magnetization.