# Hyperfine Structure of Spectral lines

The hyperfine structure of spectral lines refers to the splitting of atomic energy levels in atoms or ions due to interactions between the magnetic moments associated with the electron spin, nuclear spin, and the orbital angular momentum of electrons.

Here are some key points about hyperfine structure:

**Origin**: Hyperfine structure arises from the interaction between the magnetic dipole moment of the nucleus (due to nuclear spin), the magnetic dipole moment of the electron (due to electron spin), and the electric quadrupole moment of the nucleus.**Energy Levels Splitting**: This interaction causes the energy levels that would otherwise be degenerate (have the same energy) to split into multiple closely spaced levels. The splitting is typically very small compared to the overall energy levels but can be resolved with high-resolution spectroscopy.**Observation**: Hyperfine splitting is observed as multiple closely spaced lines in the spectrum of an atom or ion. For example, instead of a single spectral line, you might observe a doublet or a triplet of lines.**Measurement**: The hyperfine structure can be measured precisely using techniques such as Doppler-free spectroscopy, where the Doppler broadening of the lines due to thermal motion is eliminated.**Applications**: Understanding hyperfine structure is crucial in atomic physics and astrophysics. It provides information about nuclear properties (such as nuclear spin and magnetic moments) and can be used to test fundamental physical theories.**Notation**: In spectroscopy, hyperfine structure is often denoted by the letter F, which combines the total angular momentum of the electron and nucleus. The hyperfine levels are labelled by their total angular momentum quantum number F, and the individual levels within the F multiplet are labeled by their magnetic quantum numbers.

Overall, hyperfine structure is an important phenomenon in atomic spectroscopy that provides detailed insight into the internal structure of atoms and nuclei, contributing to both theoretical understanding and practical applications in various scientific fields. When a spectral line of heavy element is observed with an instrument of high resolving power, it is observed in a single spectral line consists of number of closely packed lines with the wavelength separation of 1A degree in spectral line is called hyperfine structure.

There are two types of hyperfine structure

1. The hyperfine structure due to nuclear magnetic moment and mechanical moment.

2. The hyperfine structure due to different isotopes of same chemical elements called isotopic structure.

**Term value in Hyperfine structure**

To explain the hyperfine structure of one valance electron system, we consider spin motion of nucleus. The nucleus spins around its axis. It has angular momentum I*ℏ where I* = square root of I(I+1); I is nuclear spin quantum.

In the figure, the nuclear spin vector I* interact with total angular momentum vector J* of an electron and give rise to a new vector F* so that F* = square root of F(F+1), F is hyperfine quantum number.

T find the effect of nuclear spin I* in one valance electron system, we have to consider the interaction between I* and l*, I* and S*

a. Interaction between I* and I*

According to classical electromagnetic theory, an electron of charge e produce an electric field E to nucleus at distance r from it is given by

Due to this electric field nucleus experiences a magnetic field H is given by

From angular momentum quantisation we have

From equation 2 and 3

In presence of magnetic field, nucleus precess around the magnetic field with an angular velocity given by Larmor's theorem.

where g_I = 𝜇/ I*ℏ is the ratio of nuclear magnetic moment and nuclear mechanical moment called g factor and M is nuclear mass.

Now the interaction energy is the product of w_L projection of nuclear spin angular momentum (I*h) on J*

Substituting H from 2, we get

Since l* precess around j*. Therefore angle between I* and l* is not constant. The cosine term in equation 6 must be averaged.

The averaged interaction energy will be

The averaged interaction energy will be

b. Interaction between I* and S*

According to classical electromagnetic theory, the mutual interaction energy between two magnetic dipole of moment 𝜇_I and 𝜇_s separated by a distance r is given by

Averaging the bracketted term using directional cosine we get

Bracketted term = -1/2 cos (I*J*) [ cos (j*s*) - 3 cos (j*l*)- cos(s*l*)]

Then the average interaction energy will be

Now total interaction energy will be

According to Goudsmith, the bracketted term can be replaced by l*²/j*²

Using 12 in 11 we get

In terms of term value

Thus hyperfine splitting is nearly 2000 times less than that observed in the case of spin orbit interaction.

This note is a part of the Physics Repository.