21cm Transition

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\section*{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 ${6 \times 10^{-6} eV}$ which corresponds with photons with frequency ${\nu} =$ 1.4204 GHz or wavelength 21.105 cm.

\section*{Einstein coefficient}

The Einstein coefficient ${A_{21}}$ is the probability for a system in the excited level ${E_{2}}$ to return spontaneously to the lower level ${E_{1}}$. Therefore, if ${N_{2}}$ is the electron density in level ${E_{2}}$ then ${N_{2}}{A_{21}}$ is the number of such spontaneous transition per second per unit volume.

The probability that incoming photon is absorbed is ${B_{12}{U}$ where ${U} = 4\pi{I}/{c}$ is the average energy density of the radiation field. So, the number of photons absorbed by electron in level ${E_{1}}$ to jump to ${E_{2}}$ is ${N_{1}}{B_{12}}{U}$. There is another emission process proportional to ${U}$ that need to include: ${N_{2}{B_{21}{U}$ 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

$$ {N_{2}}{A_{21}} + {N_{2}}{B_{21}}{U} = {N_{1}}{B_{12}}{U} $$

For hyperfine transition,

$${A_{10}} = 2.86888(7) \times 10^{-15} s^{-1}$$.

This transition probability is about $10^{23}$ smaller than that of an allowed optical transition.

Characteristic time for hyperfine transition is

$${t_{1/2}} \approx 1/{A_{10}} = 3.49 \times 10^{14} s \approx 1.11 \times 10^7 yr$$.

\section*{Spin temperature}

Spin temperature ${T_{s}}$ describes the ratio of atoms in the excited states (${N_1}$) to the ground state (${N_0}$). According to Boltzmann distribution:

$$ \frac{N_1}{N_0} = \frac{g_1}{g_0} \exp (-\frac{h\nu}{kT_s})$$

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

$${T_s} = \frac{T_R + y_c T_K + y_L T_L}{1 + y_c + y_L}$$

where ${T_R}$, ${T_K}$, and ${T_L}$ is the brightness temperature of 21 cm radiation, kinetic temperature, and the temperature of Lyman-$\alpha$, respectively, while ${y_c}$ and ${y_L}$ 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.

\section*{HI Column Density}

HI column density ${N_{H}}$ is the number of neutral hydrogen atoms per unit area of line of sight. If the spin temperature ${T_{s}$ is constant along the line of sight then

$${N_{H}} = 1.8224(3) \times 10^{18} (\frac{T_{s}}K) \int{\tau(\upsilon)d(\frac{\upsilon}{km s^{-1}}) cm^{-2}$$

If the line is gaussian then this approximation is useful:

$${N_{H}} \approx 1.94 \times 10^{18} (\frac{T_{s}}K) {\tau_0} (\frac{\Delta V}{km s^{-1}}) cm^{-2}$$

where ${\tau_0}$ is the Gaussian peak and ${\Delta V}$ is the Full Width Half Maximum (FWHM) of the Gaussian.