Question

Question #38003

Answer 1

When a magnetic field is applied to some atom, the line spectra due to transitions between atomic energies split into closely lying components. This is called Zeeman Effect.

This is due to magnetic interaction between the field and the electrons. Depending on whether or not there is a net spin contribution in the atom this effect is classified into :

1) Normal Zeeman effect - No net spin in the atom
2) Anamolous Zeeman effect - Net spin is present

In the following section, I'll explain Normal Zeeman Effect.

Explanation:

The magnetic moment of an electron is,

`vecmu_l = -gammavecL` where `gamma = e/(2m)` is the gyromagnetic ratio and `vec L` is the orbital angular momentum.

Now, suppose that an external magnetic field has been applied, then energy of interaction would be, ''

`E_B = -vecmu_l*vecB`

If `theta` be angle between `vecL` and `vecB`, then

`E_B = gamma|vecL|BCos theta`

But from the quantum theory, `|vecL| = sqrt[l(l+1)]barh` where `l` is azimuthal quantum number.

Therefore, the electron-field interaction energy,

`E_B = gammasqrt[l(l+1)]barhBcos theta`
`implies E_B = mu_Bsqrt[l(l+1)]Bcos theta` where `mu_B = (ebarh)/(2m) = gammabarh` is the Bohr magneton.

Now, from the rule of space quantization,

`sqrt[l(l+1)]barhcos theta = m_lbarh` where `m_l` is the magnetic quantum number.

Substituting this,

`E_B = mu_Bm_lB`

If this is the interaction energy, then total energy of an electron with principal and azimuthal quantum numbers `n` and `l` is,

`E_(nlm) = E_(nl) + E_B`

This shows that the atomic energies are split into closely lying components depending on the value of `m_l` for a fixed `n` and `l`.

But, it can be shown that transitions can occur only for `Deltam_l = +1,0,-1`. This is called the selection rule for the transition.

Thus when a transition happens from an initial level to a final level, '

`DeltaE_(nlm) = DeltaE_(nl) + mu_BBDeltam_l`
The frequency of corresponding photon,

`nu = (DeltaE_(nl))/h + (mu_BBDeltam_l)/h`
`nu = nu_0 + (mu_BBDeltam_l)/h` where `mu_0` is the frequency of photon that would have been emitted if magnetic field was absent (`B = 0`).

Thus according to the selection rule for `Deltam_l` taking 3 possible values, three possible frequencies of photons are emitted.

Thus, each spectral line splits into three closely lying lines.

This is Normal Zeeman effect.