# Wouthuysen Field effect

## The Wouthuysen-Field Effect

The Wouthuysen-Field effect is a coupling of the 21cm hyperfine transition to Ly-${\displaystyle \alpha }$ radiation.

This is important for possible high-redshift observations of the 21cm line, during the epoch of reionization. The 21cm hyperfine transition is forbidden by normal dipole selection rules, but transitions from one hyperfine state, up to the ${\displaystyle n=2}$ state, and back down to the other hyperfine state are not forbidden. So, if there is sufficient Ly-${\displaystyle \alpha }$ radiation to cause this intermediate transition, it will dominate over the direct (forbidden) hyperfine transition. Cosmologically speaking, this happens towards the end of the “Dark Ages” as reionization begins, and the WF effect remains the dominant effect until reionization is complete.

### Spectroscopic Notation

To understand how the WF effect works, we need to have a reasonable notation for all of the angular momentum quantum numbers of the hydrogen atom. There are 3 sources of angular momentum:

• orbital angular momentum (${\displaystyle L=0,...,n-1}$)
• electron spin (${\displaystyle {\frac {1}{2}}}$)
• proton spin (${\displaystyle {\frac {1}{2}}}$)

Spectroscopic notation uses the following quantum numbers to describe the energy states of the hydrogen atom, with fine structure and hyperfine structure:

• orbital angular momentum (${\displaystyle L}$)
• orbital combined with electron spin (${\displaystyle J=L+{\frac {1}{2}}}$)
• total angular momentum (${\displaystyle F=J+{\frac {1}{2}}}$)

The orbital angular momentum is denoted by letters, as you probably remember vaguely from chemistry. For the purposes of this discussion, we’ll only be talking about ${\displaystyle L=0}$ and ${\displaystyle L=1}$, which are denoted by ${\displaystyle S}$ and ${\displaystyle P}$.

The format for spectroscopic notation is ${\displaystyle n~_{F}L_{J}}$. So, let’s list all of the ground state and first excited state energies in hydrogen using this notation.[1]

• ${\displaystyle n=1}$, ${\displaystyle L=0}$: ${\displaystyle 1~_{0}S_{\frac {1}{2}}}$, ${\displaystyle 1~_{1}S_{\frac {1}{2}}}$
• ${\displaystyle n=2}$, ${\displaystyle L=0}$: ${\displaystyle 2~_{0}S_{\frac {1}{2}}}$, ${\displaystyle 2~_{1}S_{\frac {1}{2}}}$
• ${\displaystyle n=2}$, ${\displaystyle L=1}$: ${\displaystyle 2~_{0}P_{\frac {1}{2}}}$, ${\displaystyle 2~_{1}P_{\frac {1}{2}}}$, ${\displaystyle 2~_{1}P_{\frac {3}{2}}}$, ${\displaystyle 2~_{2}P_{\frac {3}{2}}}$

For better visualization, figure 1 is an energy level diagram labeled with the same notation. It’s formatted with increasing energy going up (but not to scale), and with ${\displaystyle S}$ levels ${\displaystyle (L=0)}$ in the first column and ${\displaystyle P}$ levels ${\displaystyle (L=1)}$ in the second column.

Energy level diagram with spectroscopic notation

### Selection Rules

Now we need to look at the selection rules for dipole transitions. First of all, there is the selection rule that ${\displaystyle \Delta L=0}$ is forbidden by parity, so only transitions between ${\displaystyle 1S}$ and ${\displaystyle 2P}$ will matter. (In other words, nevermind about the ${\displaystyle 2S}$ states we wrote down above.) This is the selection rule that forbids the 21cm transition, which is why the WF effect can dominate over the direct 21cm transition so long as enough Ly-${\displaystyle \alpha }$ photons are around.

Rotational symmetry is where the rest of our selection rules come from. There is a fancy thing in quantum called the Wigner-Eckart theorem which applies to “irreducible tensor operators” in general, and can generate selection rules for any such operator. The selection rule goes like this: a transition from ${\displaystyle j}$ to $\displaystyle j’$ is only allowed if $\displaystyle j’$ is in the range ${\displaystyle |j-k|}$ to ${\displaystyle j+k}$, where ${\displaystyle k}$ is the “order” of the tensor operator. For dipole transitions, ${\displaystyle k=1}$: the dipole operator is an “order 1 irreducible tensor operator”, so the rule becomes $\displaystyle j’ = |j-1|, ..., j+1$ . (Lowercase ${\displaystyle j}$ is a totally generic angular momentum quantum number, not the same as ${\displaystyle J}$.)

Notice that this is the same rule as if you were combining angular momentum ${\displaystyle j}$ with another angular momentum ${\displaystyle k=1}$. It can be summarized as ${\displaystyle \Delta j=0,\pm 1}$, except that a transition from ${\displaystyle j=0}$ to $\displaystyle j’ = 0$ is not allowed. (Combining ${\displaystyle j=0}$ and ${\displaystyle k=1}$ can only give $\displaystyle j’=1$ .) This selection rule applies to every angular momentum quantum number, ${\displaystyle L}$ and ${\displaystyle J}$ and ${\displaystyle F}$. In contrast, the rule that ${\displaystyle \Delta L=0}$ is forbidden by parity only applies to ${\displaystyle L}$. So in total, our selection rules are:

• ${\displaystyle \Delta L=\pm 1}$ (see footnote)[2]
• ${\displaystyle \Delta J=0,\pm 1}$ (see footnote)[3]
• ${\displaystyle \Delta F=0,\pm 1}$, except ${\displaystyle F=0}$ to $\displaystyle F’ = 0$ is forbidden.

Figure 2 is another copy of the energy level diagram with the allowed transitions marked. The ${\displaystyle 2S}$ states are left off the diagram completely this time.

Energy level diagram with allowed transitions

But we care only about transitions that can cause an effective hyperfine transition, so the diagram really looks like figure 3.

250px
Energy level diagram of WF effect

Thus we can see there are 2 possible pathways for the WF effect to cause a hyperfine excitation or de-excitation. It’s possible to get into the details of transition rates for each intermediate transition, transition rates for each pathway, etc, but it’s not really worth the effort. Since Ly-${\displaystyle \alpha }$ radiation will usually be coupled to the kinetic temperature of the gas anyway, we’ll go straight into discussing temperatures.

### Spin Temperature

The Ly-${\displaystyle \alpha }$ line is so optically thick that we expect the radiation field to be coupled to the kinetic gas temperature. In other words, the Ly-${\displaystyle \alpha }$ radiation temperature will be the same as the gas temperature ${\displaystyle T_{K}}$. The other important temperature is ${\displaystyle T_{s}}$, the “spin temperature” or “excitation temperature”, which is defined by the following equation. ${\displaystyle T_{s}}$ only describes the hyperfine level populations, and may or may not be related to any physical temperatures in the problem.

${\displaystyle {\frac {n_{1}}{n_{0}}}={\frac {g_{1}}{g_{0}}}e^{-h\nu /kT_{s}}\,\!}$

Remember that for collisional excitations of the hyperfine transition, in thermal equilibrium, the gas temperature is coupled directly to the spin temperature. In other words, collisions are the dominant effect that sets the hyperfine level population. This means the transition rates ${\displaystyle q_{01}}$ and ${\displaystyle q_{10}}$ are given by:

${\displaystyle {\frac {q_{01}}{q_{10}}}={\frac {g_{1}}{g_{0}}}e^{-h\nu /kT_{K}}\,\!}$

So, the WF effect is very similar, since it couples ${\displaystyle T_{s}}$ and ${\displaystyle T_{K}}$ (indirectly through Ly-${\displaystyle \alpha }$). If you define the transition rates due to the WF effect as ${\displaystyle P_{01}}$ and ${\displaystyle P_{10}}$, you have:

${\displaystyle {\frac {P_{01}}{P_{10}}}={\frac {g_{1}}{g_{0}}}e^{-h\nu /kT_{K}}\,\!}$

In order to get an exact expression for ${\displaystyle P_{01}}$ or ${\displaystyle P_{10}}$, you would need to look at the Ly-${\displaystyle \alpha }$ radiation field, the einstein coefficients for each transition in the WF effect diagram, and all the other nasty details. All we need to know is that once the first stars and AGN turn on, the WF effect dominates the hyperfine level populations.

[1] This list comes from knowing all the possible ways to combine angular momenta. For example, ${\displaystyle J}$ is a combination of ${\displaystyle L}$ and ${\displaystyle {\frac {1}{2}}}$, so it can range from ${\displaystyle |L-{\frac {1}{2}}|}$ to ${\displaystyle L+{\frac {1}{2}}}$ in integer steps. This means for ${\displaystyle L=0}$ you can only have ${\displaystyle J={\frac {1}{2}}}$, and for ${\displaystyle L=1}$ you can have ${\displaystyle J={\frac {1}{2}}}$ or ${\displaystyle J={\frac {3}{2}}}$. The same rule applies when you add in proton spin; ${\displaystyle F}$ can range from ${\displaystyle |J-{\frac {1}{2}}|}$ to ${\displaystyle J+{\frac {1}{2}}}$. This means for ${\displaystyle J={\frac {1}{2}}}$ you can have ${\displaystyle F=0}$ or ${\displaystyle F=1}$, and for ${\displaystyle J={\frac {3}{2}}}$ you can have ${\displaystyle F=1}$ or ${\displaystyle F=2}$.

[2] Remember ${\displaystyle \Delta L=0}$ is forbidden by parity

[3] ${\displaystyle J=0}$ is impossible since ${\displaystyle J}$ is always half integers, so that exception for $\displaystyle J = J’ = 0$ doesn’t apply.