Explaination of Normal Zeeman Effect & Zeeman Shift

The normal Zeeman effect can be explained by Classical electron theory as well as Quantum theory without taking electron spin. Now, let us explain Normal Zeeman effect with the help of mathematical expression.

Normal Zeeman Effect

Let us explain it by Classical Theory or Lorentz Classical electron Theory.
Consider an electron moving in a circular orbit of radius “r” with a velocity “v”, the centripetal force acting on an electron is given by :

When an external magnetic field B is applied perpendicular to the direction of motion of the electron, it will experience a Lorentz force i.e.

Now, we have:

Now, let us find dλ :

Also, we have,

Now,

Then,

The quantity eB/4πm is known as Normal Zeeman Separation.

Zeeman  Shift :

The quantity dυ = ±eB/4πm is called Zeeman Shift or Zeeman Separation. Thus,  when the light is seen perpendicular to the magnetic field, TV hen three components ” υ  – dυ”, ” υ”, ”  υ + d υ ” gives a normal triplet.
In terms of an atom consists of orbital angular momentum L and magnetic moment μl.
Then we have,
, where e and m are charge & mass of the electron.

The μl and L are oppositely directed when the atom is placed in magnetic field  B which is acting along the z-axis when L processes around the magnetic field ( Larmor Precession)  with quantized components.

When the atom is placed in the external magnetic field then quantized components are given by :

By Larmor’s theorem,  the angular velocity of Precession is given by :

The interaction energy of such precession is equal to the product of angular velocity and projection of L along that field, i.e.

Now,  ml takes (2l + 1 ) discrete values. It is each energy level for a given value of l,  is shifted into ( 2l + 1) energy levels called Zeeman level.

Let us consider, a line arising from l = 1 to l = 0 level in the magnetic field. The l = 1 level is splitted into (2l + 1 ) =  2 × 1 + 1 = 3 levels corresponding to ml = -1, 0, +1, while the level l = 0

=> 2(0) +1 = 1 level, i.e. this level is unstable.

The original line of frequency υ0 is shifted to :
where υ0  is the frequency of unshifted components & υ1 and υ2  are the frequency of shifted components.
Now,

The change in frequency in normal Zeeman effect is given by :

Thus,  measuring the change of wavelength λ we can find the value of e/m.