The "fine structure" in the hydrogen spectrum refers to the splitting of spectral lines that occurs due to the interaction between the magnetic moment of the electron and the magnetic field arising from its own motion. This phenomenon was first observed by Pieter Zeeman in 1896 and later explained by Arnold Sommerfeld using Niels Bohr's atomic model in 1916.

In the context of the hydrogen spectrum, the fine structure arises from the interaction between the electron's intrinsic spin and its orbital motion around the nucleus. This interaction leads to a slight splitting of the spectral lines into multiple components, which can be observed in high-resolution spectroscopy.

The fine structure of the hydrogen spectrum is significant because it provides additional information about the energy levels and behaviour of electrons in the hydrogen atom. It also serves as experimental evidence supporting the validity of quantum mechanics and its ability to describe atomic phenomena accurately.

The fine structure splitting in the hydrogen spectrum is relatively small compared to other atomic interactions, but it can be measured precisely with advanced spectroscopic techniques. It contributes to our understanding of atomic structure and helps refine theoretical models of the hydrogen atom.

When the spectral line is observed that a single spectral line consists of a number of closely packed line separated by a certain wavelength. Thus observed arrangement of closely packed lines in the spectral line called fine structure line. The fine structure of spectral line can be explained by the combined effect of spin orbit interaction and the relativistic variation of mass with velocity.

Expression for term value in fine structure line

According to somerfield relativistic theory the change in term value due to relativistic variation of mass with velocity is

R = Rydberg constant

z = atomic number

𝛼 = fine structure

k = somerfield azimuthal quantum number

Rewriting equation 1 we get

If we compare classical and quantum mechanical result for same phenomena, then it shows classical value k can be replaced by ( l + 1/2).

Equation 2 can be written as

The term value in spin orbit interaction is given by

where j can have two values

j = l + 1/2 for upper level

j = l - 1/2 for lower level

Also,

Now the change in term value due to combined effect of spin orbit interaction and the relativistic variation of mass with velocity can be obtained by adding either equation 3 and 5 or 3 or 6. So adding 3 and 5 we get

The total term value for spectral line will be

T = To + ΔT

We have selection rule

We have selection rule

Δj = 0 , ±1. Using selection rule in equation 9 we can explain fine structure of spectral line.

Here Rz²: This term represents the energy associated with the electron's motion around the nucleus, where Rz²/n² is the expression for the energy level of the electron in a hydrogen-like atom according to the Bohr model.

The second term seems to represent corrections to the energy levels due to fine structure. It involves the fine-structure constant 𝛼 and introduces corrections to the energy levels beyond what is predicted by the simple Bohr model. This correction accounts for relativistic and quantum mechanical effects.

This formula used to calculate the energy levels of an electron in a hydrogen-like atom, incorporating both the basic Bohr model and corrections due to fine structure. It's essential for understanding the behaviour of electrons in atoms and for accurately predicting spectral lines in atomic spectra.

This note is a part of the Physics Repository.

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