Question #38003

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.