# Fine Structure

## Introduction to Fine Structure

In atomic physics and quantum effects on atoms, the study of the hydrogen atom and their spectrum plays an important role. When the hydrogen spectrum was studied, physics noticed that the familiar red spectral line of the hydrogen atom consists of two closely spaced lines. That means the spectral line was split into two closely spaced lines or closely spaced doublet. The splitting of spectral lines is known as the fine structure or fine structure of spectral lines and it is considered one of the first pieces of experimental evidence for the electron spin.

### Fine Structure of Hydrogen Atom

The fine structure of the hydrogen atom is also known as the hydrogen fine spectrum. We know that the hydrogen atom is one of the simplest forms of atom available, which consists of a single electron in its valence shell. Before we start with the fine structure of the hydrogen atom let us have a look at the spectrum of the hydrogen atom. The spectrum of a hydrogen atom consists of different series of spectral lines and these sets of spectral lines fall into a different region of the electromagnetic spectrum, for example, the Balmer series lies in the visible region of the electromagnetic spectrum.

Now, what is the fine structure of a hydrogen atom? When we examine the Balmer series of spectral lines we know that it consists of four different spectral lines corresponding to violet, blue, green and red wavelengths. When spectral lines of the hydrogen spectrum examined under a high-resolution spectrometer it was found that a single spectral line appears to be resolved into two pairs of closely spaced single lines such that these split lines will be having slightly different wavelengths. This splitting spectral line is known as the fine structure of a hydrogen atom.

When the red spectral line or which is also known as the Hthe line is closely examined with high-resolution spectrometers, physicists found that it consists of two closely spaced doublet lines due to spin-orbit coupling. We know that the electrons are revolving around the nucleus in definite orbitals and due to the orbital motion of electrons a magnetic field is generated. When the spin electron magnetic moment interacts with the magnetic field, this interaction is familiarly known as spin-orbit coupling.

In atomic spectroscopy, the energy levels of electrons of an atom are given by the formula:

n2s+1 lj .....(1)

Where,

n - The principal quantum number

s -The spin angular momentum quantum number

l -The orbital angular momentum quantum number

j -The total angular momentum quantum number (i.e., the sum of both spin and orbital angular momentum i.e., $(l\pm s)$

Depending upon the value of l different orbits or energy levels are designated, for example, for l =0 we have S-orbit, for l =1 we have P-orbit, and so on.

### Fine Spectrum

The fine structure of the spectral line describes the splitting of spectral lines due to the electron spin and the relativistic correction to the total energy of the hydrogen atom electron. When electrons transit from lower energy level to higher energy level by absorbing the energy, it will be unstable and hence loses its energy in the form of photons of different wavelengths that further results in a spectrum.

The interaction between the magnetic field generated due to the relative motion of the nucleus and the electron spin angular momentum will result in the splitting of the energy of electrons into two energy levels.

The electron with +½ will have a magnetic spin momentum and experiences a torque due to the presence of a magnetic field and hence it will rotate it, at the same time, the electron with -½ will also have some magnetic spin momentum and experiences a torque due to the presence of magnetic field and hence it will rotate it in opposite direction. As the electron rotates, there will be a change in its internal energy and it is given by:

⇒ U = - μB

(Note: since they rotate by a different amount, hence they will also have a different amount of energy)

Suppose that the electron in hydrogen atom transit from 1s level to 2P level, we know that the motion of the electron is associated with the orbital quantum number and the spin quantum number. When the electron is in the 1S state it is in its own orbit and hence a single energy level is obtained, whereas the 2p state due to spin-orbit interaction splits into two levels. Mathematically, we write:

j = $(l\pm s)$ ...(1)

For P-orbit the value of l is 1and we know that the spin quantum number of the electron is 12. substituting, these values in equation (1) we get,

⇒ j =  $\frac{3}{2}$ , $\frac{1}{2}$

Thus 2P level is split into two energy levels. Thus, when the transition of an electron from 1S to 2P is observed we notice only a single spectral line, when it is observed through high resolution spectrometer, we notice that there are two closely spaced spectral lines with slightly different wavelengths and this splitting of spectral lines is known as the fine structure of hydrogen atom or the fine spectrum.

### Fine Structure of H Alpha Line

The H-alpha(H)line is a specific deep red visible spectral line found in the Balmer series and the wavelength of the H-alpha is around 656 nm. The H-alpha line originates when the electron transit from its third to second lowest energy level. The H-alpha line is one of the brightest spectral lines in the Balmer series.

### Did You Know?

• Spectral lines give information on the nucleus. The main effects are isotope shift and hyperfine structure.

• The study of the hyperfine structure of the H alpha line is of importance in many fields of science. The emission of the H alpha line determines many features of the solar atmosphere including prominences and the chromosphere.