Question #39a95

Consider a single electron in an atom revolving around the nucleus of atomic number #Z#.

Then the Coulombic attraction between electron and nucleus supplies the necessary centripetal force of revolution and hence,

#(Ze^2)/(4piepsilon_0r^2) = momega_0^2r# where #omega_0# is the circular frequency.

Thus, #omega_0 = sqrt ((Ze^2)/(4piepsilon_0mr^3)#

Now suppose a magnetic field #B# is applied perpendicular to the electronic motion, the electron now revolves with a different (modified) frequency as we shall calculate below.

Then Lorentz force on the electron shall be such that it will tend to act opposite to coulomb interaction and hence, net force which is again the centripetal force is,

#momega^2r = (Ze^2)/(4piepsilon_0r^2) - evB#

But #v = omegar#

#implies momega^2r = (Ze^2)/(4piepsilon_0r^2) - eBomegar#

#implies omega^2 = (Ze^2)/(4piepsilon_0mr^3) - (eB)/momega#

#implies omega^2 + (eB)/momega - (Ze^2)/(4piepsilon_0mr^3) = 0#

#implies omega^2 + (eB)/momega - omega_0^2 = 0#

Where we have substituted #(Ze^2)/(4piepsilon_0mr^3) = omega_0^2#

Now solving for #omega#,

#omega = -(eB)/(2m) + sqrt (((eB)/(2m))^2 + omega_0^2)#

and

#omega = -(eB)/(2m) - sqrt (((eB)/(2m))^2 + omega_0^2)#

For #omega_0# sufficiently greater than #(eB)/(2m)# which is called the Larmor frequency, we get,

#omega = -(eB)/(2m) + omega_0# and #omega = -(eB)/(2m) - omega_0#

This is a remarkable result, the electrons with orbital moments along the direction of the field slow down while the electrons with orbital moments opposing the field speed up, by an amount given by the Larmor's frequency,

#Deltaomega = -(eB)/(2m)#

Thus, corresponding change in electronic current in orbital motion,

#I = eDeltanu = (eDeltaomega)/(2pi)#

#implies I = -(e^2B)/(4pim)#

For #Z# electrons per atom and #N# atoms in unit volume, the net induced diamagnetic moment (equal to the magnetization) is,

#M = -(ZNe^2B)/(4m)< r^2>#

Where #< r^2> = < x^2> + < y^2># is taken to be a quantum mechanical average distance.

But, for spherically symmetric electron cloud,

#< x^2> = < y^2> = < z^2>#

And #< rho^2> = < x^2> + < y^2> + < z^2> implies < r^2> = 2/3< rho^2>#

Thus the magnetization,

#M = -(ZNe^2B)/(6m)< rho^2>#

Therefore, diamagnetic suseptibility,

#chi_m = (mu_0M)/B#

#implies chi_m = -(mu_0ZNe^2)/(6m)< rho^2>#

As consistent with experiment, the suseptibility is independent of temperature.