# 21cm Transition

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## Hyperfine transition of hydrogen atoms

The ground state of atomic hydrogen split into two hyperfine levels by the interactions between the spins of electron and proton. Parallel spin has higher energy than antiparallel spin. The energy difference between these two levels is which corresponds with photons with frequency 1.4204 GHz or wavelength 21.105 cm.

## Einstein coefficient

The Einstein coefficient is the probability for a system in the excited level to return spontaneously to the lower level . Therefore, if is the electron density in level then is the number of such spontaneous transition per second per unit volume.

The probability that incoming photon is absorbed is where is the average energy density of the radiation field. So, the number of photons absorbed by electron in level to jump to is . There is another emission process proportional to that need to include: which equal to the number of photons emitted by ”stimulated emission”.

For system in stationary state, the number of absorbed and emitted photons must be equal, so

For hyperfine transition,

.

This transition probability is about smaller than that of an allowed optical transition.

Characteristic time for hyperfine transition is

.

## Spin temperature

Spin temperature describes the ratio of atoms in the excited states () to the ground state (). According to Boltzmann distribution:

There are three processes which determine the population of the hyperfine levels in ground state of hydrogen: collisions, 21 cm radiation, and Lyman- radiation. Their relationship with spin temperature is

where , , and is the brightness temperature of 21 cm radiation, kinetic temperature, and the temperature of Lyman-, respectively, while and are coefficients which determine the relative efficiencies of the processes.

Spin temperature becomes equal to thermal temperature after many interactions between atoms via collisions or radiative transfer.

## HI Column Density

HI column density is the number of neutral hydrogen atoms per unit area of line of sight. If the spin temperature is constant along the line of sight then

If the line is gaussian then this approximation is useful:

where is the Gaussian peak and is the Full Width Half Maximum (FWHM) of the Gaussian.