https:///astrobaki/api.php?action=feedcontributions&user=YunfanZhang&feedformat=atomAstroBaki - User contributions [en]2023-10-04T22:29:18ZUser contributionsMediaWiki 1.35.1Wouthuysen Field effect2015-04-30T23:36:16Z<p>YunfanZhang: /* Reference Materials */</p>
<hr />
<div>===Reference Materials===<br />
* [http://arxiv.org/PS_cache/astro-ph/pdf/0608/0608032v2.pdf Page 32 of Furlanetto et. al, 2009]<br />
* [http://bohr.physics.berkeley.edu/classes/221/1011/221a.html Detailed quantum mechanics lecture notes from UCB Physics 221A (bottom of page)]<br />
* [http://www.astro.sunysb.edu/fwalter/AST341/qn.html A Primer on Quantum Numbers and Spectroscopic Notation (Walter, Stony Brook)]<br />
* [http://www.columbia.edu/~crg2133/Files/CambridgeIA/Chemistry/Orbitals.pdf Orbitals (Guetta, Columbia)]<br />
<latex><br />
\documentclass{article}<br />
\usepackage{graphicx}<br />
\usepackage{amsmath}<br />
\usepackage{fullpage}<br />
\begin{document}<br />
<br />
\section*{The Wouthuysen-Field Effect}<br />
<br />
The Wouthuysen-Field effect is a coupling of the 21cm hyperfine<br />
transition to Ly-$\alpha$ radiation.<br />
<br />
This is important for possible<br />
high-redshift observations of the 21cm line, during the epoch of<br />
reionization. The 21cm hyperfine transition is forbidden by<br />
normal dipole selection rules, but transitions from one hyperfine<br />
state, up to the $n=2$ state, and back down to the other hyperfine<br />
state are not forbidden. So, if there is sufficient Ly-$\alpha$<br />
radiation to cause this intermediate transition, it will dominate over<br />
the direct (forbidden) hyperfine transition. Cosmologically speaking,<br />
this happens towards the end of<br />
the ``Dark Ages'' as reionization begins, and the WF effect remains<br />
the dominant effect until reionization is complete.<br />
<br />
\subsection*{Fine structure of hydrogen}<br />
<br />
Energy levels in hydrogen atoms are split due to spin orbit interaction<br />
(fine splitting), and the smaller effect of electron-proton spin interaction<br />
(hyperfine splitting). Anti-aligned spins lead to lower energy levels.<br />
The splittings of the lowest energy levels are:<br />
<br />
\begin{figure}<br />
\center{\includegraphics{WFfig1.jpg}}<br />
\caption{Energy level diagram with spectroscopic notation}<br />
\label{all levels}<br />
\end{figure}<br />
<br />
Here we have used spectroscopic notation $n\ _{F}L_{J}$, where $n$<br />
is the principal quantum number, $L=0(S),1(P)$ are the electron orbital<br />
angular momentum. $S_{e}$ ($S_{p}$) shall denote the electron (proton)<br />
spin (not to be confused with the $S$ orbital). Then $J=|L+S_{e}|$<br />
is the electron total angular momentum, and $F=|L+S_{e}+S_{p}|$ is<br />
the hydrogen total angular momentum. Note these are vector sums.<br />
}<br />
<br />
The splitting between the two $1\ S$ levels is of particular interest.<br />
It has energy difference $\nu=1.42GHz$, corresponding to wavelength<br />
$\lambda=21cm$. We also define $T_{*}=E_{21cm}/k_{B}=0.068K$. In<br />
the $21cm$ regime, the Rayleigh-Jeans limit holds and we can define<br />
the ``brightness temperature'' <br />
\[<br />
T_{b}\approx I_{\nu}c^{2}/2k_{B}\nu^{2}.<br />
\]<br />
$T_{b}$ only serves to redefine $I_{\nu}$, and should be distinguished<br />
from the ambient temperature of the CMB $T_{\gamma}=2.74(1+z)\mbox{ K}$. <br />
<br />
\subsection*{Cosmological context}<br />
<br />
After recombination $(z\sim1100)$, the photons in the cosmic fluid<br />
no longer has free charges to interact with and thus free stream through<br />
the ``dark ages'' until the Epoch of Reionization, believed to be<br />
around $z\sim 6-9$. Studying the 21cm absorption of neutral hydrogen<br />
can thus potentially probe the universe during the dark ages and epoch<br />
of reionization. The frequency $\nu$ of 21cm undergoes cosmological<br />
redshift and hence the actual observed brightness temperature is <br />
\[<br />
T_{b}=T'_{b}/(1+z).<br />
\]<br />
<br />
<br />
Denoting the lower $1S$ level by 0, the higher by 1, one can define<br />
the ``spin temperature $T_{S}$'' to characterize the relative abundance<br />
of these two states:<br />
\[<br />
\frac{n_{1}}{n_{0}}=\frac{g_{1}}{g_{0}}e^{-h\nu/k_{B}T_{S}}=3\exp(-T_{*}/T_{S})\approx3(1-T_{*}/T_{S}).<br />
\]<br />
During the dark ages till the end of reionization, several processes<br />
control the relative abundances. Among them are direct radiative transitions<br />
(Einstein coefficients) %<br />
\footnote{Direct transitions are disfavored by selection rules, as we shall<br />
explain later. %<br />
}, collisional excitation $C_{01}$, $C_{10}$ and the Wouthuysen-Field<br />
(WF) Effect $W_{01}$ and $W_{10}$. The WF effect will be explained<br />
in detail. For now let's just note that the overall statistical balance<br />
gives <br />
\begin{equation}<br />
n_{1}(A_{10}+B_{10}I_{CMB}+C_{10}+W_{10})=n_{0}(B_{01}I_{CMB}+C_{01}+W_{01}),\label{eq:balance}<br />
\end{equation}<br />
where $A_{10}=2.869\times10^{-15}\mbox{s}^{-1}$ and $B$ are the<br />
Einstein coefficients of the 21cm transition. Recall that in thermodynamic<br />
equilibrium, we had <br />
\[<br />
B_{01}=\frac{g_{1}}{g_{0}}B_{10}=\frac{3c^{2}}{2h\nu^{3}}A_{10}<br />
\]<br />
In the Rayleigh-Jeans limit, radiative coefficients satisfy<br />
\begin{equation}<br />
\begin{aligned}B_{01}I_{CMB} & =\frac{3kT_{\gamma}}{h\nu}A_{10}=\frac{3T_{\gamma}}{T_{*}}A_{10},\\<br />
B_{10}I_{CMB} & =\frac{T_{\gamma}}{T_{*}}A_{10},<br />
\end{aligned}<br />
\end{equation}<br />
Here $T_{\gamma}=2.74(1+z)\mbox{ K }$ is the temperature of the CMB. <br />
<br />
Remember that for collisional excitations of the hyperfine transition,<br />
in thermal equilibrium, the gas temperature is coupled directly to the<br />
spin temperature. In other words, collisions are the dominant effect<br />
that sets the hyperfine level population. This means the transition<br />
rates $C_{01}$ and $C_{10}$ are given by:<br />
<br />
\[<br />
\begin{aligned}\frac{C_{01}}{C_{10}} & =\frac{g_{1}}{g_{0}}e^{-T_{*}/T_{K}}\approx3\left(1-\frac{T_{*}}{T_{K}}\right).\end{aligned}<br />
\]<br />
Similarly, WF effect rates are given by the ``color temperature''<br />
$T_{W}$: <br />
\[<br />
\begin{aligned}\frac{W_{01}}{W_{10}} & =\frac{g_{1}}{g_{0}}e^{-T_{*}/T_{W}}\approx3\left(1-\frac{T_{*}}{T_{W}}\right).\end{aligned}<br />
\]<br />
Plugging the above results into the balance equation \eqref{eq:balance},<br />
we have the relation of the temperatures:<br />
\[<br />
\boxed{T_{S}^{-1}=\frac{T_{\gamma}^{-1}+x_{c}T_{K}^{-1}+x_{W}T_{W}^{-1}}{1+x_{c}+x_{W}},}<br />
\]<br />
where <br />
\begin{equation}<br />
\boxed{\begin{aligned}x_{c} & =\frac{C_{10}}{A_{10}}\frac{T_{*}}{T_{\gamma}},\\<br />
x_{W} & =\frac{W_{10}}{A_{10}}\frac{T_{*}}{T_{\gamma}},<br />
\end{aligned}<br />
}\label{eq:coeff}<br />
\end{equation}<br />
represents the relative rates. It remains to determine these coefficients. <br />
<br />
The rest of this article shall concentrate on the WF effect, and in<br />
particular the determination of $x_{W}.$ The WF effect involves absorption<br />
of an Lyman-$\alpha$ photon from the $1_{0}S_{1/2}$ state and subsequent<br />
decay into the $1_{1}S_{1/2}$ state. The CMB temperature during the<br />
dark ages have temperatures $3000K>T_{\gamma}>15K$, and is much smaller<br />
than the Lyman-$\alpha$ transition temperature of $T=13.6eV\sim1.6\times10^{5}\mbox{K}$,<br />
and thus we expect WF effect to be most important during the epoch<br />
of reionization, when the first stars provide the abundance of $\mbox{Ly-}\alpha$<br />
photons. To study the WF effect, we first look at the rules for allowed<br />
transitions. <br />
<br />
\subsection*{Parity and Selection Rules}<br />
Now we need to look at the selection rules for dipole<br />
transitions. First of all, there is the selection rule that $\Delta L<br />
= 0$ is forbidden by parity, a symmetry of electromagnetic interactions, so only transitions between $1S$ and $2P$<br />
will matter. (In other words, nevermind about the $2S$ states we wrote<br />
down above.) This is the selection rule that forbids the 21cm<br />
transition, which is why the WF effect can dominate over the direct<br />
21cm transition so long as enough Ly-$\alpha$ photons are around.<br />
<br />
Rotational symmetry is where the rest of our selection rules come<br />
from. There is a fancy thing in quantum called the Wigner-Eckart<br />
theorem which applies to ``irreducible tensor operators'' in general,<br />
and can generate selection rules for any such operator. The selection<br />
rule goes like this: a transition from $j$ to $j'$ is only allowed if<br />
$j'$ is in the range $|j-k|$ to $j+k$, where $k$ is the ``order'' of<br />
the tensor operator. For dipole transitions, $k = 1$: the dipole<br />
operator is an ``order 1 irreducible tensor operator'', so the rule<br />
becomes $j' = |j-1|, ..., j+1$. (Lowercase $j$ is a totally generic angular<br />
momentum quantum number, not the same as $J$.)<br />
<br />
To put illustrate all of these in simpler language, a photon is a vector $A^{\mu}$ and carries one unit<br />
of angular momentum. This means that a single photon must be circularly<br />
polarized (left or right) %<br />
\footnote{In general, a tensor of rank $k$ carries $k$ units of angular momentum,<br />
and has $2k+1$ possible spin states. Due to the lack of mass of a<br />
photon however, the electromagnetic wave is transverse in all reference<br />
frames (easily seen with Maxwell's equations), and thus only has two<br />
polarization states, with angular momentum (anti-)parallel to the<br />
direction of motion. %<br />
}. Since the vector does not has any spin dependence, it commutes with<br />
the electron spin operator:<br />
\[<br />
[A^{\mu},\ S]=0.<br />
\]<br />
Here $S=|\vec{S}_{e}+\vec{S}_{p}|$ is the vector sum of the electron<br />
and proton spins. <br />
<br />
This implies the first dipole selection rule:<br />
\[<br />
\Delta S=0.<br />
\]<br />
The spin along given directions ($m_{s}$), however, are not ``good''<br />
quantum numbers, and can change. <br />
<br />
To consider orbital angular momentum, recall the parity operator $P$.<br />
It reverses all physical space directions:<br />
\[<br />
P:x_{i}\rightarrow-x_{i}.<br />
\]<br />
If we fix the phase such that $P^{2}=1$, we have parity as a Hermitian<br />
and hence observable:<br />
\[<br />
P^{-1}=P^{\dagger}=P.<br />
\]<br />
Parity thus acts on other operators via conjugation. For example<br />
\[<br />
PA^{\mu}(t,x)P^{-1}=-A^{\mu}(t,-x).<br />
\]<br />
<br />
<br />
More generally, it is clear that quantities such as vector, pseudo-scalar<br />
change sign under parity while a pseudo-vector (such as angular momentum<br />
or magnetic field) or a true scalar do not change sign. For example,<br />
parity commutes with spin $[P,S]=0$, while it anti-commutes with<br />
momentum $\{P,p\}=0$. This means that while a spin state is a good<br />
quantum number when paired with parity, a momentum state is not. <br />
<br />
So how do states in a hydrogen atom behave under parity? To answer<br />
this question recall that such a state is proportional to the spherical<br />
harmonics:<br />
\[<br />
|nlm\rangle\propto R_{lm}(r)Y_{l}^{m}(\Omega)\propto R_{lm}(r)P_{l}^{m}(\cos\theta)e^{im\phi}.<br />
\]<br />
Under parity, $\phi\rightarrow\pi+\phi$, $\theta\rightarrow\pi-\theta$,<br />
and since $P_{l}^{m}(-x)=(-1)^{l-m}P_{l}^{m}(x),$ we have <br />
\[<br />
P(|nlm\rangle)=(-1)^{l}|nlm\rangle.<br />
\]<br />
The parity of a state $|nlm\rangle$ does not depend on $m$, as expected<br />
since an intrinsic property such as parity should not depend on the<br />
orientation of the system. <br />
<br />
Thus the matrix element of the vector can be written as <br />
\[<br />
\begin{aligned}M & =\langle n'l'm'|A^{\mu}|nlm\rangle,\\<br />
& =\langle n'l'm'|P^{\dagger}PA^{\mu}P^{\dagger}P|nlm\rangle,\\<br />
& =(-1)^{l-l'+1}\langle n'l'm'|A^{\mu}|nlm\rangle.<br />
\end{aligned}<br />
\]<br />
Since a matrix element is a scalar and has even parity, we see that<br />
the end state must have different parity from the initial state. In<br />
other words, $l'-l$ must be odd!<br />
<br />
Furthermore, since the photon carries only one unit of angular momentum,<br />
any angular momentum change must be of order unity or less:<br />
\[<br />
|\Delta L|,|\Delta F|,|\Delta J|\le1.<br />
\]<br />
Thus we have the rule <br />
\[<br />
\Delta l=\pm1.<br />
\]<br />
<br />
<br />
To see this more explicitly, one can also write out the full expression<br />
\[<br />
|nlm\rangle=R_{lm}(r)P_{l}^{m}(\cos\theta)e^{im\phi}\frac{1}{\sqrt{2\pi}}.<br />
\]<br />
Since a dipole interaction, along a direction called $z$ has the<br />
form <br />
\[<br />
\vec{d}=E\vec{z}=Er\cos\theta\hat{z},<br />
\]<br />
The matrix element can be written as <br />
\[<br />
\begin{aligned}\langle n'l'm'|d|nlm\rangle & \propto\int_{0}^{\pi}P_{l'}^{m'}P_{l}^{m}\cos\theta\sin\theta d\theta\times\frac{1}{2\pi}\int_{0}^{2\pi}e^{i(m-m')\phi}d\phi.\end{aligned}<br />
\]<br />
From properties of associated Legendre polynomials, we can decompose<br />
\[<br />
P_{l}^{m}(\cos\theta)\cos\theta=bP_{l+1}^{m}+cP_{l-1}^{m},<br />
\]<br />
for some constants $b$ and $c$. And since they also satisfy the<br />
orthogonality rules<br />
\[<br />
\int_{-1}^{1}P_{l'}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{l'l},<br />
\]<br />
the above matrix element vanishes unless<br />
\[<br />
\begin{cases}<br />
m' & =m,\\<br />
l' & =l\pm1.<br />
\end{cases}<br />
\]<br />
In particular, the angular momentum along the dipole direction is<br />
unchanged. This is expected since the spin of the photon is pointed<br />
perpendicular to the field (a transverse wave). <br />
<br />
A difference between $l$ and $L$ is in place when there are more<br />
than one electron in the atom. We have seen that a single electron<br />
contributes a factor of $(-1)^{l}$ when acted on by parity. In the<br />
multi-electron case, this generalizes to<br />
\[<br />
P(|l_{1}l_{2}...\rangle)=(-1)^{\sum l_{i}}|l_{1}l_{2}...\rangle.<br />
\]<br />
Since in general the total orbital angular momentum $L\ne\sum l_{i}$,<br />
two states with the same $L$ can have different parity, and so one<br />
cannot conclude that $\Delta L=0$ is not allowed. One can conclude,<br />
however, that $(L=0)\rightarrow(L=0)$ transitions are not allowed.<br />
This is because the $S$ orbital has complete spherical symmetry,<br />
a symmetry broken by the vector photon, and thus an $S$ orbital cannot<br />
absorb a photon and stay an $S$ orbital. Similarly, considering L-S<br />
coupling, we have corresponding selection rules of forbidden $(J=0)\rightarrow(J=0)$<br />
or $(F=0)\rightarrow(F=0)$ transitions. Physically, these last two<br />
rules amounts to the combined effect of angular momentum conservation,<br />
and the first rule, that $\Delta S=0.$ For example, the transition<br />
from $1_{1}S_{1/2}$ to $2_{1}P_{1/2}$ is allowed, even though $\Delta F=0$,<br />
because in this transition the photon brought $\Delta L=1$, electron<br />
flips its spin to keep $\Delta J=0$, and proton also flips its spin,<br />
to preserve $\Delta S=0$ and $\Delta F=0$, in the process $m_{F}$<br />
is flipped. The transition $1_{0}S_{1/2}$ to $2_{0}P_{1/2}$ on the<br />
other hand, is forbidden, because as $\Delta L=1$, the two conditions<br />
$\Delta S=0$ and $\Delta F=0$ cannot be simultaneously satisfied. <br />
<br />
In summary, we have the following selection rules for dipole transitions<br />
in a neutral hydrogen atom<br />
\[<br />
\begin{cases}<br />
\Delta S & =0,\\<br />
\Delta L & =\pm1,\\<br />
\Delta J & =0,\pm1,\mbox{(except 0 to 0),}\\<br />
\Delta F & =0,\pm1,\mbox{(except 0 to 0). }<br />
\end{cases}<br />
\]<br />
<br />
<br />
\subsection*{The WF effect}<br />
<br />
<br />
\subsubsection*{Coupling coefficient}<br />
We would like to find the coupling coefficient $x_W$ defined at the end of the Cosmological Context subsection. <br />
The allowed transitions between the lowest levels are shown in Fig.<br />
2. The transitions relevant to the WF effect are traced with solid<br />
lines. Since we have denoted the $1S$ states by 0 and 1, we shall<br />
by convenience denote the 2$P$ states by 2,3,4,5, in order of increasing<br />
energy. <br />
\begin{figure}<br />
\center{\includegraphics{WFfig2.jpg}}<br />
\caption{Energy level diagram with allowed transitions}<br />
\label{all transitions}<br />
\end{figure}<br />
<br />
<br />
\begin{figure}<br />
\center{\includegraphics{WFfig3a.jpg}}<br />
\caption{Energy level diagram of WF effect}<br />
\label{wf transitions}<br />
\end{figure}<br />
<br />
<br />
Evidently, the (de-)excitation rate due to WF effect is given by:<br />
\begin{equation}<br />
\begin{aligned}W_{01} & =B_{03}J_{03}\frac{A_{31}}{A_{31}+A_{30}}+B_{04}J_{04}\frac{A_{41}}{A_{41}+A_{40}},\\<br />
& =\frac{3T_{\gamma}^{03}}{T_{*}^{03}}\frac{A_{30}A_{31}}{A_{31}+A_{30}}+\frac{3T_{\gamma}^{04}}{T_{*}^{04}}\frac{A_{40}A_{41}}{A_{41}+A_{40}},\\<br />
W_{10} & =B_{13}J_{13}\frac{A_{30}}{A_{31}+A_{30}}+B_{14}J_{14}\frac{A_{40}}{A_{41}+A_{40}},\\<br />
& =\frac{T_{\gamma}^{13}}{T_{*}^{13}}\frac{A_{30}A_{31}}{A_{31}+A_{30}}+\frac{T_{\gamma}^{14}}{T_{*}^{14}}\frac{A_{40}A_{41}}{A_{41}+A_{40}}.<br />
\end{aligned}<br />
\label{eq:W01}<br />
\end{equation}<br />
In the second and fourth lines I have used the relation from above:<br />
\begin{equation}<br />
\begin{aligned}B_{01}I_{CMB} & =\frac{3kT_{\gamma}}{h\nu}A_{10}=\frac{3T_{\gamma}}{T_{*}}A_{10},\\<br />
B_{10}I_{CMB} & =\frac{T_{\gamma}}{T_{*}}A_{10},<br />
\end{aligned}<br />
\end{equation} <br />
<br />
We would like to relate this rate to the total Ly$\alpha$ scattering<br />
rate<br />
\[<br />
P_{\alpha}=4\pi\chi_{\alpha}\int d\nu J_{\nu}\phi_{\alpha}(\nu),<br />
\]<br />
where $\sigma_{\nu}=\chi_{\nu}\phi_{\alpha}$ is the absorption cross<br />
section and $\chi_{\alpha}=f_{\alpha}\frac{\pi e^{2}}{m_{e}c^{2}}$,<br />
with $f_{\alpha}=0.4162$ the oscillator strength. The line $\phi_{\alpha}$<br />
can be assumed to be a Voigt profile. Thermal broadening leads to<br />
Doppler width<br />
\[<br />
\Delta\nu_{D}=\sqrt{\frac{2k_{B}T_{K}}{m_{H}c^{2}}}\nu_{\alpha},<br />
\]<br />
where $\nu_{\alpha}=2.47\times10^{15}\mbox{Hz}$ is the Lyman $\alpha$<br />
line center frequency. <br />
<br />
Since the dipole operator commutes with spin, the transition of $|nJFm_{F}\rangle$<br />
to $|n'J'F'm'_{F}\rangle$ is independent of $F$ and $m_{F}$. This<br />
means that the emission intensity of transition from $|nJF\rangle$,<br />
summed over all $m_{F}$, to $|n'J'\rangle$, summed over $m'_{F}$<br />
and $F'$, is the intensity of transition from a particular $|nJFm_{F}\rangle$,<br />
times $2F+1$, the degeneracy of the initial state. This is called<br />
the ``sum rule'' of the transition. <br />
<br />
The sum rules, together with the selected transitions in Fig. 2, immediately<br />
lead to<br />
<br />
\begin{equation}<br />
\begin{aligned}\frac{I_{51}}{I_{41}+I_{40}}=\frac{5}{3},\ & \frac{I_{41}+I_{40}}{I_{31}+I_{30}}=1,\ & \frac{I_{21}}{I_{31}+I_{30}}=\frac{1}{3},\\<br />
\frac{I_{40}}{I_{41}+I_{51}}=\frac{1}{3},\ & \frac{I_{30}}{I_{21}+I_{31}}=\frac{1}{3}.<br />
\end{aligned}<br />
\label{eq:int}<br />
\end{equation}<br />
The second line follow from the transition into excited states. <br />
<br />
Next we shall neglect radiative transfer effect and make the assumption<br />
that background intensities and temperatures are constant across all<br />
the hyperfine lines, and are given by the CMB values. Under this assumption<br />
\[<br />
P_{\alpha}=n_{(n=1)}B_{Ly\alpha}=\frac{3T_{\gamma}}{T_{*}}A_{Ly\alpha}.<br />
\]<br />
Let $I_{tot}=I_{51}+I_{40}+I_{41}+I_{31}+I_{30}+I_{21}$ be the total<br />
intensity of de-excitation. Then from the relations on $I$ above we get <br />
\[<br />
\begin{aligned}I_{30}=I_{41} & =\frac{1}{12}I_{tot},\\<br />
I_{31}=I_{40} & =\frac{1}{6}I_{tot}.<br />
\end{aligned}<br />
\]<br />
Furthermore, the intensities are related to the Einstein coefficients<br />
by <br />
\[<br />
\frac{I_{ki}}{I_{\alpha}}=\frac{g_{k}}{g_{tot}}\frac{A_{kj}}{A_{\alpha}},<br />
\]<br />
where $g_{k}=2F_{k}+1$ and $g_{tot}=1+3+3+5=12$ is the total degeneracy<br />
of $n=2$ level. <br />
<br />
Thus we have<br />
\[<br />
\begin{aligned}A_{30} & =A_{41}=A_{\alpha}/3,\\<br />
A_{31} & =A_{40}=2A_{\alpha}/3.<br />
\end{aligned}<br />
\]<br />
Thus finally $P_{10}=4P_{\alpha}/27$ and the coupling coefficient becomes <br />
\[<br />
\boxed{x_{W}=\frac{4P_{\alpha}}{27A_{10}}\frac{T_{*}}{T_{\gamma}}}.<br />
\]<br />
<br />
<br />
<br />
\subsubsection*{Color temperature and corrections}<br />
<br />
The effect of Ly$\alpha$ absorption on spin temperature also depends<br />
on the color temperature $T_{W}$. For an environment that's optically<br />
thick, such as the high-redshift IGM, we may take <br />
\[<br />
T_{W}=T_{K}.<br />
\]<br />
One commonly considered correction is due to energy loss in spin-exchange<br />
collisions. In these collisions, the electron spins of the two colliding<br />
hydrogen atoms exchange, while the total spin remains unchanged. This<br />
correction to the color temperature is worked out in for example \cite{key-1},<br />
section 2.3.3. We shall here just quote the result:<br />
\begin{equation}<br />
\boxed{T_{W}=T_{K}\left(\frac{1+T_{se}/T_{K}}{1+T_{se}/T_{S}}\right),}\label{eq:colort}<br />
\end{equation}<br />
where the spin exchange temperature <br />
\[<br />
T_{se}=\frac{2T_{K}\nu_{se}^{2}}{9\Delta\nu_{D}^{2}}\sim0.40\mbox{K}.<br />
\]<br />
To use Eq. \eqref{eq:colort}, the spin temperature must be determined<br />
iteratively. The iteration is shown to converge quickly. <br />
<br />
Another commonly considered correction are radiative transfer effects.<br />
Previously we assumed that the background intensity is constant and<br />
given by the CMB intensity. The Ly$\alpha$ absorption would decrease<br />
the background intensity and hence the estimated scattering rate.<br />
The effect is more important for lower kinetic temperature. This leads<br />
to a correction to the WF effect coupling coefficient $S_{\alpha}$<br />
such that <br />
\[<br />
x_{W}=S_{\alpha}\frac{J_{\alpha}}{J_{\nu}^{c}},<br />
\]<br />
where <br />
\[<br />
J_{\nu}^{c}=1.165\times10^{-10}\left(\frac{1+z}{20}\right)\mbox{cm}^{-2}\mbox{s}^{-1}\mbox{Hz}^{-1}\mbox{sr}^{-1}.<br />
\]<br />
<br />
<br />
Neglecting spin exchange, the suppression factor is given by <br />
\[<br />
S_{\alpha}\sim\exp\left[-0.803T_{K}^{-2/3}(10^{-6}\tau_{GP})^{1/3}\right],<br />
\]<br />
where the Gunn-Peterson optical depth can be written for overdensity<br />
$\delta$ as roughly: <br />
<br />
\[<br />
\tau_{GP}=7\times10^{5}\left(\frac{\Omega_{b}h_{100}}{0.03}\right)\left(\frac{\Omega_{m}}{0.25}\right)^{-1/2}\left(\frac{1+z}{10}\right)(1+\delta).<br />
\]<br />
<br />
\end{document}<br />
</latex></div>YunfanZhangWouthuysen Field effect2015-04-30T23:33:46Z<p>YunfanZhang: /* Reference Materials */</p>
<hr />
<div>===Reference Materials===<br />
* [http://arxiv.org/PS_cache/astro-ph/pdf/0608/0608032v2.pdf Page 32 of Furlanetto et. al, 2009]<br />
* [http://bohr.physics.berkeley.edu/classes/221/1011/221a.html Detailed quantum mechanics lecture notes from UCB Physics 221A (bottom of page)]<br />
* [http://www.astro.sunysb.edu/fwalter/AST341/qn.html A Primer on Quantum Numbers and Spectroscopic Notation (Walter, Stony Brook)]<br />
* [http://www.columbia.edu/~crg2133/Files/CambridgeIA/Chemistry/Orbitals.pdf Orbitals (Guetta, Columbia)]<br />
<latex><br />
\documentclass{article}<br />
\usepackage{graphicx}<br />
\usepackage{amsmath}<br />
\usepackage{fullpage}<br />
\begin{document}<br />
<br />
\section*{The Wouthuysen-Field Effect}<br />
<br />
The Wouthuysen-Field effect is a coupling of the 21cm hyperfine<br />
transition to Ly-$\alpha$ radiation.<br />
<br />
This is important for possible<br />
high-redshift observations of the 21cm line, during the epoch of<br />
reionization. The 21cm hyperfine transition is forbidden by<br />
normal dipole selection rules, but transitions from one hyperfine<br />
state, up to the $n=2$ state, and back down to the other hyperfine<br />
state are not forbidden. So, if there is sufficient Ly-$\alpha$<br />
radiation to cause this intermediate transition, it will dominate over<br />
the direct (forbidden) hyperfine transition. Cosmologically speaking,<br />
this happens towards the end of<br />
the ``Dark Ages'' as reionization begins, and the WF effect remains<br />
the dominant effect until reionization is complete.<br />
<br />
\subsection*{Fine structure of hydrogen}<br />
<br />
Energy levels in hydrogen atoms are split due to spin orbit interaction<br />
(fine splitting), and the smaller effect of electron-proton spin interaction<br />
(hyperfine splitting). Anti-aligned spins lead to lower energy levels.<br />
The splittings of the lowest energy levels are:<br />
<br />
\begin{figure}<br />
\center{\includegraphics{WFfig1.jpg}}<br />
\caption{Energy level diagram with spectroscopic notation}<br />
\label{all levels}<br />
\end{figure}<br />
<br />
Here we have used spectroscopic notation $n\ _{F}L_{J}$, where $n$<br />
is the principal quantum number, $L=0(S),1(P)$ are the electron orbital<br />
angular momentum. $S_{e}$ ($S_{p}$) shall denote the electron (proton)<br />
spin (not to be confused with the $S$ orbital). Then $J=|L+S_{e}|$<br />
is the electron total angular momentum, and $F=|L+S_{e}+S_{p}|$ is<br />
the hydrogen total angular momentum. Note these are vector sums.<br />
}<br />
<br />
The splitting between the two $1\ S$ levels is of particular interest.<br />
It has energy difference $\nu=1.42GHz$, corresponding to wavelength<br />
$\lambda=21cm$. We also define $T_{*}=E_{21cm}/k_{B}=0.068K$. In<br />
the $21cm$ regime, the Rayleigh-Jeans limit holds and we can define<br />
the ``brightness temperature'' <br />
\[<br />
T_{b}\approx I_{\nu}c^{2}/2k_{B}\nu^{2}.<br />
\]<br />
$T_{b}$ only serves to redefine $I_{\nu}$, and should be distinguished<br />
from the ambient temperature of the CMB $T_{\gamma}=2.74(1+z)\mbox{ K}$. <br />
<br />
\subsection*{Cosmological context}<br />
<br />
After recombination $(z\sim1100)$, the photons in the cosmic fluid<br />
no longer has free charges to interact with and thus free stream through<br />
the ``dark ages'' until the Epoch of Reionization, believed to be<br />
around $z\sim 6-9$. Studying the 21cm absorption of neutral hydrogen<br />
can thus potentially probe the universe during the dark ages and epoch<br />
of reionization. The frequency $\nu$ of 21cm undergoes cosmological<br />
redshift and hence the actual observed brightness temperature is <br />
\[<br />
T_{b}=T'_{b}/(1+z).<br />
\]<br />
<br />
<br />
Denoting the lower $1S$ level by 0, the higher by 1, one can define<br />
the ``spin temperature $T_{S}$'' to characterize the relative abundance<br />
of these two states:<br />
\[<br />
\frac{n_{1}}{n_{0}}=\frac{g_{1}}{g_{0}}e^{-h\nu/k_{B}T_{S}}=3\exp(-T_{*}/T_{S})\approx3(1-T_{*}/T_{S}).<br />
\]<br />
During the dark ages till the end of reionization, several processes<br />
control the relative abundances. Among them are direct radiative transitions<br />
(Einstein coefficients) %<br />
\footnote{Direct transitions are disfavored by selection rules, as we shall<br />
explain later. %<br />
}, collisional excitation $C_{01}$, $C_{10}$ and the Wouthuysen-Field<br />
(WF) Effect $W_{01}$ and $W_{10}$. The WF effect will be explained<br />
in detail. For now let's just note that the overall statistical balance<br />
gives <br />
\begin{equation}<br />
n_{1}(A_{10}+B_{10}I_{CMB}+C_{10}+W_{10})=n_{0}(B_{01}I_{CMB}+C_{01}+W_{01}),\label{eq:balance}<br />
\end{equation}<br />
where $A_{10}=2.869\times10^{-15}\mbox{s}^{-1}$ and $B$ are the<br />
Einstein coefficients of the 21cm transition. Recall that in thermodynamic<br />
equilibrium, we had <br />
\[<br />
B_{01}=\frac{g_{1}}{g_{0}}B_{10}=\frac{3c^{2}}{2h\nu^{3}}A_{10}<br />
\]<br />
In the Rayleigh-Jeans limit, radiative coefficients satisfy<br />
\begin{equation}<br />
\begin{aligned}B_{01}I_{CMB} & =\frac{3kT_{\gamma}}{h\nu}A_{10}=\frac{3T_{\gamma}}{T_{*}}A_{10},\\<br />
B_{10}I_{CMB} & =\frac{T_{\gamma}}{T_{*}}A_{10},<br />
\end{aligned}<br />
\end{equation}<br />
Here $T_{\gamma}=2.74(1+z)\mbox{ K }$ is the temperature of the CMB. <br />
<br />
Remember that for collisional excitations of the hyperfine transition,<br />
in thermal equilibrium, the gas temperature is coupled directly to the<br />
spin temperature. In other words, collisions are the dominant effect<br />
that sets the hyperfine level population. This means the transition<br />
rates $C_{01}$ and $C_{10}$ are given by:<br />
<br />
\[<br />
\begin{aligned}\frac{C_{01}}{C_{10}} & =\frac{g_{1}}{g_{0}}e^{-T_{*}/T_{K}}\approx3\left(1-\frac{T_{*}}{T_{K}}\right).\end{aligned}<br />
\]<br />
Similarly, WF effect rates are given by the ``color temperature''<br />
$T_{W}$: <br />
\[<br />
\begin{aligned}\frac{W_{01}}{W_{10}} & =\frac{g_{1}}{g_{0}}e^{-T_{*}/T_{W}}\approx3\left(1-\frac{T_{*}}{T_{W}}\right).\end{aligned}<br />
\]<br />
Plugging the above results into the balance equation \eqref{eq:balance},<br />
we have the relation of the temperatures:<br />
\[<br />
\boxed{T_{S}^{-1}=\frac{T_{\gamma}^{-1}+x_{c}T_{K}^{-1}+x_{W}T_{W}^{-1}}{1+x_{c}+x_{W}},}<br />
\]<br />
where <br />
\begin{equation}<br />
\boxed{\begin{aligned}x_{c} & =\frac{C_{10}}{A_{10}}\frac{T_{*}}{T_{\gamma}},\\<br />
x_{W} & =\frac{W_{10}}{A_{10}}\frac{T_{*}}{T_{\gamma}},<br />
\end{aligned}<br />
}\label{eq:coeff}<br />
\end{equation}<br />
represents the relative rates. It remains to determine these coefficients. <br />
<br />
The rest of this article shall concentrate on the WF effect, and in<br />
particular the determination of $x_{W}.$ The WF effect involves absorption<br />
of an Lyman-$\alpha$ photon from the $1_{0}S_{1/2}$ state and subsequent<br />
decay into the $1_{1}S_{1/2}$ state. The CMB temperature during the<br />
dark ages have temperatures $3000K>T_{\gamma}>15K$, and is much smaller<br />
than the Lyman-$\alpha$ transition temperature of $T=13.6eV\sim1.6\times10^{5}\mbox{K}$,<br />
and thus we expect WF effect to be most important during the epoch<br />
of reionization, when the first stars provide the abundance of $\mbox{Ly-}\alpha$<br />
photons. To study the WF effect, we first look at the rules for allowed<br />
transitions. <br />
<br />
\subsection*{Parity and Selection Rules}<br />
Now we need to look at the selection rules for dipole<br />
transitions. First of all, there is the selection rule that $\Delta L<br />
= 0$ is forbidden by parity, a symmetry of electromagnetic interactions, so only transitions between $1S$ and $2P$<br />
will matter. (In other words, nevermind about the $2S$ states we wrote<br />
down above.) This is the selection rule that forbids the 21cm<br />
transition, which is why the WF effect can dominate over the direct<br />
21cm transition so long as enough Ly-$\alpha$ photons are around.<br />
<br />
Rotational symmetry is where the rest of our selection rules come<br />
from. There is a fancy thing in quantum called the Wigner-Eckart<br />
theorem which applies to ``irreducible tensor operators'' in general,<br />
and can generate selection rules for any such operator. The selection<br />
rule goes like this: a transition from $j$ to $j'$ is only allowed if<br />
$j'$ is in the range $|j-k|$ to $j+k$, where $k$ is the ``order'' of<br />
the tensor operator. For dipole transitions, $k = 1$: the dipole<br />
operator is an ``order 1 irreducible tensor operator'', so the rule<br />
becomes $j' = |j-1|, ..., j+1$. (Lowercase $j$ is a totally generic angular<br />
momentum quantum number, not the same as $J$.)<br />
<br />
To put illustrate all of these in simpler language, a photon is a vector $A^{\mu}$ and carries one unit<br />
of angular momentum. This means that a single photon must be circularly<br />
polarized (left or right) %<br />
\footnote{In general, a tensor of rank $k$ carries $k$ units of angular momentum,<br />
and has $2k+1$ possible spin states. Due to the lack of mass of a<br />
photon however, the electromagnetic wave is transverse in all reference<br />
frames (easily seen with Maxwell's equations), and thus only has two<br />
polarization states, with angular momentum (anti-)parallel to the<br />
direction of motion. %<br />
}. Since the vector does not has any spin dependence, it commutes with<br />
the electron spin operator:<br />
\[<br />
[A^{\mu},\ S]=0.<br />
\]<br />
Here $S=|\vec{S}_{e}+\vec{S}_{p}|$ is the vector sum of the electron<br />
and proton spins. <br />
<br />
This implies the first dipole selection rule:<br />
\[<br />
\Delta S=0.<br />
\]<br />
The spin along given directions ($m_{s}$), however, are not ``good''<br />
quantum numbers, and can change. <br />
<br />
To consider orbital angular momentum, recall the parity operator $P$.<br />
It reverses all physical space directions:<br />
\[<br />
P:x_{i}\rightarrow-x_{i}.<br />
\]<br />
If we fix the phase such that $P^{2}=1$, we have parity as a Hermitian<br />
and hence observable:<br />
\[<br />
P^{-1}=P^{\dagger}=P.<br />
\]<br />
Parity thus acts on other operators via conjugation. For example<br />
\[<br />
PA^{\mu}(t,x)P^{-1}=-A^{\mu}(t,-x).<br />
\]<br />
<br />
<br />
More generally, it is clear that quantities such as vector, pseudo-scalar<br />
change sign under parity while a pseudo-vector (such as angular momentum<br />
or magnetic field) or a true scalar do not change sign. For example,<br />
parity commutes with spin $[P,S]=0$, while it anti-commutes with<br />
momentum $\{P,p\}=0$. This means that while a spin state is a good<br />
quantum number when paired with parity, a momentum state is not. <br />
<br />
So how do states in a hydrogen atom behave under parity? To answer<br />
this question recall that such a state is proportional to the spherical<br />
harmonics:<br />
\[<br />
|nlm\rangle\propto R_{lm}(r)Y_{l}^{m}(\Omega)\propto R_{lm}(r)P_{l}^{m}(\cos\theta)e^{im\phi}.<br />
\]<br />
Under parity, $\phi\rightarrow\pi+\phi$, $\theta\rightarrow\pi-\theta$,<br />
and since $P_{l}^{m}(-x)=(-1)^{l-m}P_{l}^{m}(x),$ we have <br />
\[<br />
P(|nlm\rangle)=(-1)^{l}|nlm\rangle.<br />
\]<br />
The parity of a state $|nlm\rangle$ does not depend on $m$, as expected<br />
since an intrinsic property such as parity should not depend on the<br />
orientation of the system. <br />
<br />
Thus the matrix element of the vector can be written as <br />
\[<br />
\begin{aligned}M & =\langle n'l'm'|A^{\mu}|nlm\rangle,\\<br />
& =\langle n'l'm'|P^{\dagger}PA^{\mu}P^{\dagger}P|nlm\rangle,\\<br />
& =(-1)^{l-l'+1}\langle n'l'm'|A^{\mu}|nlm\rangle.<br />
\end{aligned}<br />
\]<br />
Since a matrix element is a scalar and has even parity, we see that<br />
the end state must have different parity from the initial state. In<br />
other words, $l'-l$ must be odd!<br />
<br />
Furthermore, since the photon carries only one unit of angular momentum,<br />
any angular momentum change must be of order unity or less:<br />
\[<br />
|\Delta L|,|\Delta F|,|\Delta J|\le1.<br />
\]<br />
Thus we have the rule <br />
\[<br />
\Delta l=\pm1.<br />
\]<br />
<br />
<br />
To see this more explicitly, one can also write out the full expression<br />
\[<br />
|nlm\rangle=R_{lm}(r)P_{l}^{m}(\cos\theta)e^{im\phi}\frac{1}{\sqrt{2\pi}}.<br />
\]<br />
Since a dipole interaction, along a direction called $z$ has the<br />
form <br />
\[<br />
\vec{d}=E\vec{z}=Er\cos\theta\hat{z},<br />
\]<br />
The matrix element can be written as <br />
\[<br />
\begin{aligned}\langle n'l'm'|d|nlm\rangle & \propto\int_{0}^{\pi}P_{l'}^{m'}P_{l}^{m}\cos\theta\sin\theta d\theta\times\frac{1}{2\pi}\int_{0}^{2\pi}e^{i(m-m')\phi}d\phi.\end{aligned}<br />
\]<br />
From properties of associated Legendre polynomials, we can decompose<br />
\[<br />
P_{l}^{m}(\cos\theta)\cos\theta=bP_{l+1}^{m}+cP_{l-1}^{m},<br />
\]<br />
for some constants $b$ and $c$. And since they also satisfy the<br />
orthogonality rules<br />
\[<br />
\int_{-1}^{1}P_{l'}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{l'l},<br />
\]<br />
the above matrix element vanishes unless<br />
\[<br />
\begin{cases}<br />
m' & =m,\\<br />
l' & =l\pm1.<br />
\end{cases}<br />
\]<br />
In particular, the angular momentum along the dipole direction is<br />
unchanged. This is expected since the spin of the photon is pointed<br />
perpendicular to the field (a transverse wave). <br />
<br />
A difference between $l$ and $L$ is in place when there are more<br />
than one electron in the atom. We have seen that a single electron<br />
contributes a factor of $(-1)^{l}$ when acted on by parity. In the<br />
multi-electron case, this generalizes to<br />
\[<br />
P(|l_{1}l_{2}...\rangle)=(-1)^{\sum l_{i}}|l_{1}l_{2}...\rangle.<br />
\]<br />
Since in general the total orbital angular momentum $L\ne\sum l_{i}$,<br />
two states with the same $L$ can have different parity, and so one<br />
cannot conclude that $\Delta L=0$ is not allowed. One can conclude,<br />
however, that $(L=0)\rightarrow(L=0)$ transitions are not allowed.<br />
This is because the $S$ orbital has complete spherical symmetry,<br />
a symmetry broken by the vector photon, and thus an $S$ orbital cannot<br />
absorb a photon and stay an $S$ orbital. Similarly, considering L-S<br />
coupling, we have corresponding selection rules of forbidden $(J=0)\rightarrow(J=0)$<br />
or $(F=0)\rightarrow(F=0)$ transitions. Physically, these last two<br />
rules amounts to the combined effect of angular momentum conservation,<br />
and the first rule, that $\Delta S=0.$ For example, the transition<br />
from $1_{1}S_{1/2}$ to $2_{1}P_{1/2}$ is allowed, even though $\Delta F=0$,<br />
because in this transition the photon brought $\Delta L=1$, electron<br />
flips its spin to keep $\Delta J=0$, and proton also flips its spin,<br />
to preserve $\Delta S=0$ and $\Delta F=0$, in the process $m_{F}$<br />
is flipped. The transition $1_{0}S_{1/2}$ to $2_{0}P_{1/2}$ on the<br />
other hand, is forbidden, because as $\Delta L=1$, the two conditions<br />
$\Delta S=0$ and $\Delta F=0$ cannot be simultaneously satisfied. <br />
<br />
In summary, we have the following selection rules for dipole transitions<br />
in a neutral hydrogen atom<br />
\[<br />
\begin{cases}<br />
\Delta S & =0,\\<br />
\Delta L & =\pm1,\\<br />
\Delta J & =0,\pm1,\mbox{(except 0 to 0),}\\<br />
\Delta F & =0,\pm1,\mbox{(except 0 to 0). }<br />
\end{cases}<br />
\]<br />
<br />
<br />
\subsection*{The WF effect}<br />
<br />
<br />
\subsubsection*{Coupling coefficient}<br />
We would like to find the coupling coefficient $x_W$ defined at the end of the Cosmological Context subsection. <br />
The allowed transitions between the lowest levels are shown in Fig.<br />
2. The transitions relevant to the WF effect are traced with solid<br />
lines. Since we have denoted the $1S$ states by 0 and 1, we shall<br />
by convenience denote the 2$P$ states by 2,3,4,5, in order of increasing<br />
energy. <br />
\begin{figure}<br />
\center{\includegraphics{WFfig2.jpg}}<br />
\caption{Energy level diagram with allowed transitions}<br />
\label{all transitions}<br />
\end{figure}<br />
<br />
<br />
\begin{figure}<br />
\center{\includegraphics{WFfig3a.jpg}}<br />
\caption{Energy level diagram of WF effect}<br />
\label{wf transitions}<br />
\end{figure}<br />
<br />
<br />
Evidently, the (de-)excitation rate due to WF effect is given by:<br />
\begin{equation}<br />
\begin{aligned}W_{01} & =B_{03}J_{03}\frac{A_{31}}{A_{31}+A_{30}}+B_{04}J_{04}\frac{A_{41}}{A_{41}+A_{40}},\\<br />
& =\frac{3T_{\gamma}^{03}}{T_{*}^{03}}\frac{A_{30}A_{31}}{A_{31}+A_{30}}+\frac{3T_{\gamma}^{04}}{T_{*}^{04}}\frac{A_{40}A_{41}}{A_{41}+A_{40}},\\<br />
W_{10} & =B_{13}J_{13}\frac{A_{30}}{A_{31}+A_{30}}+B_{14}J_{14}\frac{A_{40}}{A_{41}+A_{40}},\\<br />
& =\frac{T_{\gamma}^{13}}{T_{*}^{13}}\frac{A_{30}A_{31}}{A_{31}+A_{30}}+\frac{T_{\gamma}^{14}}{T_{*}^{14}}\frac{A_{40}A_{41}}{A_{41}+A_{40}}.<br />
\end{aligned}<br />
\label{eq:W01}<br />
\end{equation}<br />
In the second and fourth lines I have used the relation from above:<br />
\begin{equation}<br />
\begin{aligned}B_{01}I_{CMB} & =\frac{3kT_{\gamma}}{h\nu}A_{10}=\frac{3T_{\gamma}}{T_{*}}A_{10},\\<br />
B_{10}I_{CMB} & =\frac{T_{\gamma}}{T_{*}}A_{10},<br />
\end{aligned}<br />
\end{equation} <br />
<br />
We would like to relate this rate to the total Ly$\alpha$ scattering<br />
rate<br />
\[<br />
P_{\alpha}=4\pi\chi_{\alpha}\int d\nu J_{\nu}\phi_{\alpha}(\nu),<br />
\]<br />
where $\sigma_{\nu}=\chi_{\nu}\phi_{\alpha}$ is the absorption cross<br />
section and $\chi_{\alpha}=f_{\alpha}\frac{\pi e^{2}}{m_{e}c^{2}}$,<br />
with $f_{\alpha}=0.4162$ the oscillator strength. The line $\phi_{\alpha}$<br />
can be assumed to be a Voigt profile. Thermal broadening leads to<br />
Doppler width<br />
\[<br />
\Delta\nu_{D}=\sqrt{\frac{2k_{B}T_{K}}{m_{H}c^{2}}}\nu_{\alpha},<br />
\]<br />
where $\nu_{\alpha}=2.47\times10^{15}\mbox{Hz}$ is the Lyman $\alpha$<br />
line center frequency. <br />
<br />
Since the dipole operator commutes with spin, the transition of $|nJFm_{F}\rangle$<br />
to $|n'J'F'm'_{F}\rangle$ is independent of $F$ and $m_{F}$. This<br />
means that the emission intensity of transition from $|nJF\rangle$,<br />
summed over all $m_{F}$, to $|n'J'\rangle$, summed over $m'_{F}$<br />
and $F'$, is the intensity of transition from a particular $|nJFm_{F}\rangle$,<br />
times $2F+1$, the degeneracy of the initial state. This is called<br />
the ``sum rule'' of the transition. <br />
<br />
The sum rules, together with the selected transitions in Fig. 2, immediately<br />
lead to<br />
<br />
\begin{equation}<br />
\begin{aligned}\frac{I_{51}}{I_{41}+I_{40}}=\frac{5}{3},\ & \frac{I_{41}+I_{40}}{I_{31}+I_{30}}=1,\ & \frac{I_{21}}{I_{31}+I_{30}}=\frac{1}{3},\\<br />
\frac{I_{40}}{I_{41}+I_{51}}=\frac{1}{3},\ & \frac{I_{30}}{I_{21}+I_{31}}=\frac{1}{3}.<br />
\end{aligned}<br />
\label{eq:int}<br />
\end{equation}<br />
The second line follow from the transition into excited states. <br />
<br />
Next we shall neglect radiative transfer effect and make the assumption<br />
that background intensities and temperatures are constant across all<br />
the hyperfine lines, and are given by the CMB values. Under this assumption<br />
\[<br />
P_{\alpha}=n_{(n=1)}B_{Ly\alpha}=\frac{3T_{\gamma}}{T_{*}}A_{Ly\alpha}.<br />
\]<br />
Let $I_{tot}=I_{51}+I_{40}+I_{41}+I_{31}+I_{30}+I_{21}$ be the total<br />
intensity of de-excitation. Then from Eq. \eqref{eq:int} we get <br />
\[<br />
\begin{aligned}I_{30}=I_{41} & =\frac{1}{12}I_{tot},\\<br />
I_{31}=I_{40} & =\frac{1}{6}I_{tot}.<br />
\end{aligned}<br />
\]<br />
Furthermore, the intensities are related to the Einstein coefficients<br />
by <br />
\[<br />
\frac{I_{ki}}{I_{\alpha}}=\frac{g_{k}}{g_{tot}}\frac{A_{kj}}{A_{\alpha}},<br />
\]<br />
where $g_{k}=2F_{k}+1$ and $g_{tot}=1+3+3+5=12$ is the total degeneracy<br />
of $n=2$ level. <br />
<br />
Thus we have<br />
\[<br />
\begin{aligned}A_{30} & =A_{41}=A_{\alpha}/3,\\<br />
A_{31} & =A_{40}=2A_{\alpha}/3.<br />
\end{aligned}<br />
\]<br />
Thus finally $P_{10}=4P_{\alpha}/27$ and the coupling coefficient becomes <br />
\[<br />
\boxed{x_{W}=\frac{4P_{\alpha}}{27A_{10}}\frac{T_{*}}{T_{\gamma}}}.<br />
\]<br />
<br />
<br />
<br />
\subsubsection*{Color temperature and corrections}<br />
<br />
The effect of Ly$\alpha$ absorption on spin temperature also depends<br />
on the color temperature $T_{W}$. For an environment that's optically<br />
thick, such as the high-redshift IGM, we may take <br />
\[<br />
T_{W}=T_{K}.<br />
\]<br />
One commonly considered correction is due to energy loss in spin-exchange<br />
collisions. In these collisions, the electron spins of the two colliding<br />
hydrogen atoms exchange, while the total spin remains unchanged. This<br />
correction to the color temperature is worked out in for example \cite{key-1},<br />
section 2.3.3. We shall here just quote the result:<br />
\begin{equation}<br />
\boxed{T_{W}=T_{K}\left(\frac{1+T_{se}/T_{K}}{1+T_{se}/T_{S}}\right),}\label{eq:colort}<br />
\end{equation}<br />
where the spin exchange temperature <br />
\[<br />
T_{se}=\frac{2T_{K}\nu_{se}^{2}}{9\Delta\nu_{D}^{2}}\sim0.40\mbox{K}.<br />
\]<br />
To use Eq. \eqref{eq:colort}, the spin temperature must be determined<br />
iteratively. The iteration is shown to converge quickly. <br />
<br />
Another commonly considered correction are radiative transfer effects.<br />
Previously we assumed that the background intensity is constant and<br />
given by the CMB intensity. The Ly$\alpha$ absorption would decrease<br />
the background intensity and hence the estimated scattering rate.<br />
The effect is more important for lower kinetic temperature. This leads<br />
to a correction to the WF effect coupling coefficient $S_{\alpha}$<br />
such that <br />
\[<br />
x_{W}=S_{\alpha}\frac{J_{\alpha}}{J_{\nu}^{c}},<br />
\]<br />
where <br />
\[<br />
J_{\nu}^{c}=1.165\times10^{-10}\left(\frac{1+z}{20}\right)\mbox{cm}^{-2}\mbox{s}^{-1}\mbox{Hz}^{-1}\mbox{sr}^{-1}.<br />
\]<br />
<br />
<br />
Neglecting spin exchange, the suppression factor is given by <br />
\[<br />
S_{\alpha}\sim\exp\left[-0.803T_{K}^{-2/3}(10^{-6}\tau_{GP})^{1/3}\right],<br />
\]<br />
where the Gunn-Peterson optical depth can be written for overdensity<br />
$\delta$ as roughly: <br />
<br />
\[<br />
\tau_{GP}=7\times10^{5}\left(\frac{\Omega_{b}h_{100}}{0.03}\right)\left(\frac{\Omega_{m}}{0.25}\right)^{-1/2}\left(\frac{1+z}{10}\right)(1+\delta).<br />
\]<br />
<br />
\end{document}<br />
</latex></div>YunfanZhangWouthuysen Field effect2015-04-30T23:29:54Z<p>YunfanZhang: /* Reference Materials */</p>
<hr />
<div>===Reference Materials===<br />
* [http://arxiv.org/PS_cache/astro-ph/pdf/0608/0608032v2.pdf Page 32 of Furlanetto et. al, 2009]<br />
* [http://bohr.physics.berkeley.edu/classes/221/1011/221a.html Detailed quantum mechanics lecture notes from UCB Physics 221A (bottom of page)]<br />
* [http://www.astro.sunysb.edu/fwalter/AST341/qn.html A Primer on Quantum Numbers and Spectroscopic Notation (Walter, Stony Brook)]<br />
* [http://www.columbia.edu/~crg2133/Files/CambridgeIA/Chemistry/Orbitals.pdf Orbitals (Guetta, Columbia)]<br />
<latex><br />
\documentclass{article}<br />
\usepackage{graphicx}<br />
\usepackage{amsmath}<br />
\usepackage{fullpage}<br />
\begin{document}<br />
<br />
\section*{The Wouthuysen-Field Effect}<br />
<br />
The Wouthuysen-Field effect is a coupling of the 21cm hyperfine<br />
transition to Ly-$\alpha$ radiation.<br />
<br />
This is important for possible<br />
high-redshift observations of the 21cm line, during the epoch of<br />
reionization. The 21cm hyperfine transition is forbidden by<br />
normal dipole selection rules, but transitions from one hyperfine<br />
state, up to the $n=2$ state, and back down to the other hyperfine<br />
state are not forbidden. So, if there is sufficient Ly-$\alpha$<br />
radiation to cause this intermediate transition, it will dominate over<br />
the direct (forbidden) hyperfine transition. Cosmologically speaking,<br />
this happens towards the end of<br />
the ``Dark Ages'' as reionization begins, and the WF effect remains<br />
the dominant effect until reionization is complete.<br />
<br />
\subsection*{Fine structure of hydrogen}<br />
<br />
Energy levels in hydrogen atoms are split due to spin orbit interaction<br />
(fine splitting), and the smaller effect of electron-proton spin interaction<br />
(hyperfine splitting). Anti-aligned spins lead to lower energy levels.<br />
The splittings of the lowest energy levels are:<br />
<br />
\begin{figure}<br />
\center{\includegraphics{WFfig1.jpg}}<br />
\caption{Energy level diagram with spectroscopic notation}<br />
\label{all levels}<br />
\end{figure}<br />
<br />
Here we have used spectroscopic notation $n\ _{F}L_{J}$, where $n$<br />
is the principal quantum number, $L=0(S),1(P)$ are the electron orbital<br />
angular momentum. $S_{e}$ ($S_{p}$) shall denote the electron (proton)<br />
spin (not to be confused with the $S$ orbital). Then $J=|L+S_{e}|$<br />
is the electron total angular momentum, and $F=|L+S_{e}+S_{p}|$ is<br />
the hydrogen total angular momentum. Note these are vector sums.<br />
}<br />
<br />
The splitting between the two $1\ S$ levels is of particular interest.<br />
It has energy difference $\nu=1.42GHz$, corresponding to wavelength<br />
$\lambda=21cm$. We also define $T_{*}=E_{21cm}/k_{B}=0.068K$. In<br />
the $21cm$ regime, the Rayleigh-Jeans limit holds and we can define<br />
the ``brightness temperature'' <br />
\[<br />
T_{b}\approx I_{\nu}c^{2}/2k_{B}\nu^{2}.<br />
\]<br />
$T_{b}$ only serves to redefine $I_{\nu}$, and should be distinguished<br />
from the ambient temperature of the CMB $T_{\gamma}=2.74(1+z)\mbox{ K}$. <br />
<br />
\subsection*{Cosmological context}<br />
<br />
After recombination $(z\sim1100)$, the photons in the cosmic fluid<br />
no longer has free charges to interact with and thus free stream through<br />
the ``dark ages'' until the Epoch of Reionization, believed to be<br />
around $z\sim 6-9$. Studying the 21cm absorption of neutral hydrogen<br />
can thus potentially probe the universe during the dark ages and epoch<br />
of reionization. The frequency $\nu$ of 21cm undergoes cosmological<br />
redshift and hence the actual observed brightness temperature is <br />
\[<br />
T_{b}=T'_{b}/(1+z).<br />
\]<br />
<br />
<br />
Denoting the lower $1S$ level by 0, the higher by 1, one can define<br />
the ``spin temperature $T_{S}$'' to characterize the relative abundance<br />
of these two states:<br />
\[<br />
\frac{n_{1}}{n_{0}}=\frac{g_{1}}{g_{0}}e^{-h\nu/k_{B}T_{S}}=3\exp(-T_{*}/T_{S})\approx3(1-T_{*}/T_{S}).<br />
\]<br />
During the dark ages till the end of reionization, several processes<br />
control the relative abundances. Among them are direct radiative transitions<br />
(Einstein coefficients) %<br />
\footnote{Direct transitions are disfavored by selection rules, as we shall<br />
explain later. %<br />
}, collisional excitation $C_{01}$, $C_{10}$ and the Wouthuysen-Field<br />
(WF) Effect $W_{01}$ and $W_{10}$. The WF effect will be explained<br />
in detail. For now let's just note that the overall statistical balance<br />
gives <br />
\begin{equation}<br />
n_{1}(A_{10}+B_{10}I_{CMB}+C_{10}+W_{10})=n_{0}(B_{01}I_{CMB}+C_{01}+W_{01}),\label{eq:balance}<br />
\end{equation}<br />
where $A_{10}=2.869\times10^{-15}\mbox{s}^{-1}$ and $B$ are the<br />
Einstein coefficients of the 21cm transition. Recall that in thermodynamic<br />
equilibrium, we had <br />
\[<br />
B_{01}=\frac{g_{1}}{g_{0}}B_{10}=\frac{3c^{2}}{2h\nu^{3}}A_{10}<br />
\]<br />
In the Rayleigh-Jeans limit, radiative coefficients satisfy<br />
\begin{equation}<br />
\begin{aligned}B_{01}I_{CMB} & =\frac{3kT_{\gamma}}{h\nu}A_{10}=\frac{3T_{\gamma}}{T_{*}}A_{10},\\<br />
B_{10}I_{CMB} & =\frac{T_{\gamma}}{T_{*}}A_{10},<br />
\end{aligned}<br />
\end{equation}<br />
Here $T_{\gamma}=2.74(1+z)\mbox{ K }$ is the temperature of the CMB. <br />
<br />
Remember that for collisional excitations of the hyperfine transition,<br />
in thermal equilibrium, the gas temperature is coupled directly to the<br />
spin temperature. In other words, collisions are the dominant effect<br />
that sets the hyperfine level population. This means the transition<br />
rates $C_{01}$ and $C_{10}$ are given by:<br />
<br />
\[<br />
\begin{aligned}\frac{C_{01}}{C_{10}} & =\frac{g_{1}}{g_{0}}e^{-T_{*}/T_{K}}\approx3\left(1-\frac{T_{*}}{T_{K}}\right).\end{aligned}<br />
\]<br />
Similarly, WF effect rates are given by the ``color temperature''<br />
$T_{W}$: <br />
\[<br />
\begin{aligned}\frac{W_{01}}{W_{10}} & =\frac{g_{1}}{g_{0}}e^{-T_{*}/T_{W}}\approx3\left(1-\frac{T_{*}}{T_{W}}\right).\end{aligned}<br />
\]<br />
Plugging the above results into the balance equation \eqref{eq:balance},<br />
we have the relation of the temperatures:<br />
\[<br />
\boxed{T_{S}^{-1}=\frac{T_{\gamma}^{-1}+x_{c}T_{K}^{-1}+x_{W}T_{W}^{-1}}{1+x_{c}+x_{W}},}<br />
\]<br />
where <br />
\begin{equation}<br />
\boxed{\begin{aligned}x_{c} & =\frac{C_{10}}{A_{10}}\frac{T_{*}}{T_{\gamma}},\\<br />
x_{W} & =\frac{W_{10}}{A_{10}}\frac{T_{*}}{T_{\gamma}},<br />
\end{aligned}<br />
}\label{eq:coeff}<br />
\end{equation}<br />
represents the relative rates. It remains to determine these coefficients. <br />
<br />
The rest of this article shall concentrate on the WF effect, and in<br />
particular the determination of $x_{W}.$ The WF effect involves absorption<br />
of an Lyman-$\alpha$ photon from the $1_{0}S_{1/2}$ state and subsequent<br />
decay into the $1_{1}S_{1/2}$ state. The CMB temperature during the<br />
dark ages have temperatures $3000K>T_{\gamma}>15K$, and is much smaller<br />
than the Lyman-$\alpha$ transition temperature of $T=13.6eV\sim1.6\times10^{5}\mbox{K}$,<br />
and thus we expect WF effect to be most important during the epoch<br />
of reionization, when the first stars provide the abundance of $\mbox{Ly-}\alpha$<br />
photons. To study the WF effect, we first look at the rules for allowed<br />
transitions. <br />
<br />
\subsection*{Parity and Selection Rules}<br />
Now we need to look at the selection rules for dipole<br />
transitions. First of all, there is the selection rule that $\Delta L<br />
= 0$ is forbidden by parity, a symmetry of electromagnetic interactions, so only transitions between $1S$ and $2P$<br />
will matter. (In other words, nevermind about the $2S$ states we wrote<br />
down above.) This is the selection rule that forbids the 21cm<br />
transition, which is why the WF effect can dominate over the direct<br />
21cm transition so long as enough Ly-$\alpha$ photons are around.<br />
<br />
Rotational symmetry is where the rest of our selection rules come<br />
from. There is a fancy thing in quantum called the Wigner-Eckart<br />
theorem which applies to ``irreducible tensor operators'' in general,<br />
and can generate selection rules for any such operator. The selection<br />
rule goes like this: a transition from $j$ to $j'$ is only allowed if<br />
$j'$ is in the range $|j-k|$ to $j+k$, where $k$ is the ``order'' of<br />
the tensor operator. For dipole transitions, $k = 1$: the dipole<br />
operator is an ``order 1 irreducible tensor operator'', so the rule<br />
becomes $j' = |j-1|, ..., j+1$. (Lowercase $j$ is a totally generic angular<br />
momentum quantum number, not the same as $J$.)<br />
<br />
To put illustrate all of these in simpler language, a photon is a vector $A^{\mu}$ and carries one unit<br />
of angular momentum. This means that a single photon must be circularly<br />
polarized (left or right) %<br />
\footnote{In general, a tensor of rank $k$ carries $k$ units of angular momentum,<br />
and has $2k+1$ possible spin states. Due to the lack of mass of a<br />
photon however, the electromagnetic wave is transverse in all reference<br />
frames (easily seen with Maxwell's equations), and thus only has two<br />
polarization states, with angular momentum (anti-)parallel to the<br />
direction of motion. %<br />
}. Since the vector does not has any spin dependence, it commutes with<br />
the electron spin operator:<br />
\[<br />
[A^{\mu},\ S]=0.<br />
\]<br />
Here $S=|\vec{S}_{e}+\vec{S}_{p}|$ is the vector sum of the electron<br />
and proton spins. <br />
<br />
This implies the first dipole selection rule:<br />
\[<br />
\Delta S=0.<br />
\]<br />
The spin along given directions ($m_{s}$), however, are not ``good''<br />
quantum numbers, and can change. <br />
<br />
To consider orbital angular momentum, recall the parity operator $P$.<br />
It reverses all physical space directions:<br />
\[<br />
P:x_{i}\rightarrow-x_{i}.<br />
\]<br />
If we fix the phase such that $P^{2}=1$, we have parity as a Hermitian<br />
and hence observable:<br />
\[<br />
P^{-1}=P^{\dagger}=P.<br />
\]<br />
Parity thus acts on other operators via conjugation. For example<br />
\[<br />
PA^{\mu}(t,x)P^{-1}=-A^{\mu}(t,-x).<br />
\]<br />
<br />
<br />
More generally, it is clear that quantities such as vector, pseudo-scalar<br />
change sign under parity while a pseudo-vector (such as angular momentum<br />
or magnetic field) or a true scalar do not change sign. For example,<br />
parity commutes with spin $[P,S]=0$, while it anti-commutes with<br />
momentum $\{P,p\}=0$. This means that while a spin state is a good<br />
quantum number when paired with parity, a momentum state is not. <br />
<br />
So how do states in a hydrogen atom behave under parity? To answer<br />
this question recall that such a state is proportional to the spherical<br />
harmonics:<br />
\[<br />
|nlm\rangle\propto R_{lm}(r)Y_{l}^{m}(\Omega)\propto R_{lm}(r)P_{l}^{m}(\cos\theta)e^{im\phi}.<br />
\]<br />
Under parity, $\phi\rightarrow\pi+\phi$, $\theta\rightarrow\pi-\theta$,<br />
and since $P_{l}^{m}(-x)=(-1)^{l-m}P_{l}^{m}(x),$ we have <br />
\[<br />
P(|nlm\rangle)=(-1)^{l}|nlm\rangle.<br />
\]<br />
The parity of a state $|nlm\rangle$ does not depend on $m$, as expected<br />
since an intrinsic property such as parity should not depend on the<br />
orientation of the system. <br />
<br />
Thus the matrix element of the vector can be written as <br />
\[<br />
\begin{aligned}M & =\langle n'l'm'|A^{\mu}|nlm\rangle,\\<br />
& =\langle n'l'm'|P^{\dagger}PA^{\mu}P^{\dagger}P|nlm\rangle,\\<br />
& =(-1)^{l-l'+1}\langle n'l'm'|A^{\mu}|nlm\rangle.<br />
\end{aligned}<br />
\]<br />
Since a matrix element is a scalar and has even parity, we see that<br />
the end state must have different parity from the initial state. In<br />
other words, $l'-l$ must be odd!<br />
<br />
Furthermore, since the photon carries only one unit of angular momentum,<br />
any angular momentum change must be of order unity or less:<br />
\[<br />
|\Delta L|,|\Delta F|,|\Delta J|\le1.<br />
\]<br />
Thus we have the rule <br />
\[<br />
\Delta l=\pm1.<br />
\]<br />
<br />
<br />
To see this more explicitly, one can also write out the full expression<br />
\[<br />
|nlm\rangle=R_{lm}(r)P_{l}^{m}(\cos\theta)e^{im\phi}\frac{1}{\sqrt{2\pi}}.<br />
\]<br />
Since a dipole interaction, along a direction called $z$ has the<br />
form <br />
\[<br />
\vec{d}=E\vec{z}=Er\cos\theta\hat{z},<br />
\]<br />
The matrix element can be written as <br />
\[<br />
\begin{aligned}\langle n'l'm'|d|nlm\rangle & \propto\int_{0}^{\pi}P_{l'}^{m'}P_{l}^{m}\cos\theta\sin\theta d\theta\times\frac{1}{2\pi}\int_{0}^{2\pi}e^{i(m-m')\phi}d\phi.\end{aligned}<br />
\]<br />
From properties of associated Legendre polynomials, we can decompose<br />
\[<br />
P_{l}^{m}(\cos\theta)\cos\theta=bP_{l+1}^{m}+cP_{l-1}^{m},<br />
\]<br />
for some constants $b$ and $c$. And since they also satisfy the<br />
orthogonality rules<br />
\[<br />
\int_{-1}^{1}P_{l'}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{l'l},<br />
\]<br />
the above matrix element vanishes unless<br />
\[<br />
\begin{cases}<br />
m' & =m,\\<br />
l' & =l\pm1.<br />
\end{cases}<br />
\]<br />
In particular, the angular momentum along the dipole direction is<br />
unchanged. This is expected since the spin of the photon is pointed<br />
perpendicular to the field (a transverse wave). <br />
<br />
A difference between $l$ and $L$ is in place when there are more<br />
than one electron in the atom. We have seen that a single electron<br />
contributes a factor of $(-1)^{l}$ when acted on by parity. In the<br />
multi-electron case, this generalizes to<br />
\[<br />
P(|l_{1}l_{2}...\rangle)=(-1)^{\sum l_{i}}|l_{1}l_{2}...\rangle.<br />
\]<br />
Since in general the total orbital angular momentum $L\ne\sum l_{i}$,<br />
two states with the same $L$ can have different parity, and so one<br />
cannot conclude that $\Delta L=0$ is not allowed. One can conclude,<br />
however, that $(L=0)\rightarrow(L=0)$ transitions are not allowed.<br />
This is because the $S$ orbital has complete spherical symmetry,<br />
a symmetry broken by the vector photon, and thus an $S$ orbital cannot<br />
absorb a photon and stay an $S$ orbital. Similarly, considering L-S<br />
coupling, we have corresponding selection rules of forbidden $(J=0)\rightarrow(J=0)$<br />
or $(F=0)\rightarrow(F=0)$ transitions. Physically, these last two<br />
rules amounts to the combined effect of angular momentum conservation,<br />
and the first rule, that $\Delta S=0.$ For example, the transition<br />
from $1_{1}S_{1/2}$ to $2_{1}P_{1/2}$ is allowed, even though $\Delta F=0$,<br />
because in this transition the photon brought $\Delta L=1$, electron<br />
flips its spin to keep $\Delta J=0$, and proton also flips its spin,<br />
to preserve $\Delta S=0$ and $\Delta F=0$, in the process $m_{F}$<br />
is flipped. The transition $1_{0}S_{1/2}$ to $2_{0}P_{1/2}$ on the<br />
other hand, is forbidden, because as $\Delta L=1$, the two conditions<br />
$\Delta S=0$ and $\Delta F=0$ cannot be simultaneously satisfied. <br />
<br />
In summary, we have the following selection rules for dipole transitions<br />
in a neutral hydrogen atom<br />
\[<br />
\begin{cases}<br />
\Delta S & =0,\\<br />
\Delta L & =\pm1,\\<br />
\Delta J & =0,\pm1,\mbox{(except 0 to 0),}\\<br />
\Delta F & =0,\pm1,\mbox{(except 0 to 0). }<br />
\end{cases}<br />
\]<br />
<br />
<br />
\subsection*{The WF effect}<br />
<br />
<br />
\subsubsection*{Coupling coefficient}<br />
<br />
The allowed transitions between the lowest levels are shown in Fig.<br />
2. The transitions relevant to the WF effect are traced with solid<br />
lines. Since we have denoted the $1S$ states by 0 and 1, we shall<br />
by convenience denote the 2$P$ states by 2,3,4,5, in order of increasing<br />
energy. <br />
\begin{figure}<br />
\center{\includegraphics{WFfig2.jpg}}<br />
\caption{Energy level diagram with allowed transitions}<br />
\label{all transitions}<br />
\end{figure}<br />
<br />
<br />
\begin{figure}<br />
\center{\includegraphics{WFfig3a.jpg}}<br />
\caption{Energy level diagram of WF effect}<br />
\label{wf transitions}<br />
\end{figure}<br />
<br />
<br />
Evidently, the (de-)excitation rate due to WF effect is given by:<br />
\begin{equation}<br />
\begin{aligned}W_{01} & =B_{03}J_{03}\frac{A_{31}}{A_{31}+A_{30}}+B_{04}J_{04}\frac{A_{41}}{A_{41}+A_{40}},\\<br />
& =\frac{3T_{\gamma}^{03}}{T_{*}^{03}}\frac{A_{30}A_{31}}{A_{31}+A_{30}}+\frac{3T_{\gamma}^{04}}{T_{*}^{04}}\frac{A_{40}A_{41}}{A_{41}+A_{40}},\\<br />
W_{10} & =B_{13}J_{13}\frac{A_{30}}{A_{31}+A_{30}}+B_{14}J_{14}\frac{A_{40}}{A_{41}+A_{40}},\\<br />
& =\frac{T_{\gamma}^{13}}{T_{*}^{13}}\frac{A_{30}A_{31}}{A_{31}+A_{30}}+\frac{T_{\gamma}^{14}}{T_{*}^{14}}\frac{A_{40}A_{41}}{A_{41}+A_{40}}.<br />
\end{aligned}<br />
\label{eq:W01}<br />
\end{equation}<br />
In the second and fourth lines I have used the relation from above:<br />
\begin{equation}<br />
\begin{aligned}B_{01}I_{CMB} & =\frac{3kT_{\gamma}}{h\nu}A_{10}=\frac{3T_{\gamma}}{T_{*}}A_{10},\\<br />
B_{10}I_{CMB} & =\frac{T_{\gamma}}{T_{*}}A_{10},<br />
\end{aligned}<br />
\end{equation} <br />
<br />
We would like to relate this rate to the total Ly$\alpha$ scattering<br />
rate<br />
\[<br />
P_{\alpha}=4\pi\chi_{\alpha}\int d\nu J_{\nu}\phi_{\alpha}(\nu),<br />
\]<br />
where $\sigma_{\nu}=\chi_{\nu}\phi_{\alpha}$ is the absorption cross<br />
section and $\chi_{\alpha}=f_{\alpha}\frac{\pi e^{2}}{m_{e}c^{2}}$,<br />
with $f_{\alpha}=0.4162$ the oscillator strength. The line $\phi_{\alpha}$<br />
can be assumed to be a Voigt profile. Thermal broadening leads to<br />
Doppler width<br />
\[<br />
\Delta\nu_{D}=\sqrt{\frac{2k_{B}T_{K}}{m_{H}c^{2}}}\nu_{\alpha},<br />
\]<br />
where $\nu_{\alpha}=2.47\times10^{15}\mbox{Hz}$ is the Lyman $\alpha$<br />
line center frequency. <br />
<br />
Since the dipole operator commutes with spin, the transition of $|nJFm_{F}\rangle$<br />
to $|n'J'F'm'_{F}\rangle$ is independent of $F$ and $m_{F}$. This<br />
means that the emission intensity of transition from $|nJF\rangle$,<br />
summed over all $m_{F}$, to $|n'J'\rangle$, summed over $m'_{F}$<br />
and $F'$, is the intensity of transition from a particular $|nJFm_{F}\rangle$,<br />
times $2F+1$, the degeneracy of the initial state. This is called<br />
the ``sum rule'' of the transition. <br />
<br />
The sum rules, together with the selected transitions in Fig. 2, immediately<br />
lead to<br />
<br />
\begin{equation}<br />
\begin{aligned}\frac{I_{51}}{I_{41}+I_{40}}=\frac{5}{3},\ & \frac{I_{41}+I_{40}}{I_{31}+I_{30}}=1,\ & \frac{I_{21}}{I_{31}+I_{30}}=\frac{1}{3},\\<br />
\frac{I_{40}}{I_{41}+I_{51}}=\frac{1}{3},\ & \frac{I_{30}}{I_{21}+I_{31}}=\frac{1}{3}.<br />
\end{aligned}<br />
\label{eq:int}<br />
\end{equation}<br />
The second line follow from the transition into excited states. <br />
<br />
Next we shall neglect radiative transfer effect and make the assumption<br />
that background intensities and temperatures are constant across all<br />
the hyperfine lines, and are given by the CMB values. Under this assumption<br />
\[<br />
P_{\alpha}=n_{(n=1)}B_{Ly\alpha}=\frac{3T_{\gamma}}{T_{*}}A_{Ly\alpha}.<br />
\]<br />
Let $I_{tot}=I_{51}+I_{40}+I_{41}+I_{31}+I_{30}+I_{21}$ be the total<br />
intensity of de-excitation. Then from Eq. \eqref{eq:int} we get <br />
\[<br />
\begin{aligned}I_{30}=I_{41} & =\frac{1}{12}I_{tot},\\<br />
I_{31}=I_{40} & =\frac{1}{6}I_{tot}.<br />
\end{aligned}<br />
\]<br />
Furthermore, the intensities are related to the Einstein coefficients<br />
by <br />
\[<br />
\frac{I_{ki}}{I_{\alpha}}=\frac{g_{k}}{g_{tot}}\frac{A_{kj}}{A_{\alpha}},<br />
\]<br />
where $g_{k}=2F_{k}+1$ and $g_{tot}=1+3+3+5=12$ is the total degeneracy<br />
of $n=2$ level. <br />
<br />
Thus we have<br />
\[<br />
\begin{aligned}A_{30} & =A_{41}=A_{\alpha}/3,\\<br />
A_{31} & =A_{40}=2A_{\alpha}/3.<br />
\end{aligned}<br />
\]<br />
Thus finally $P_{10}=4P_{\alpha}/27$ and the coupling coefficient becomes <br />
\[<br />
\boxed{x_{W}=\frac{4P_{\alpha}}{27A_{10}}\frac{T_{*}}{T_{\gamma}}}.<br />
\]<br />
<br />
<br />
<br />
\subsubsection*{Color temperature and corrections}<br />
<br />
The effect of Ly$\alpha$ absorption on spin temperature also depends<br />
on the color temperature $T_{W}$. For an environment that's optically<br />
thick, such as the high-redshift IGM, we may take <br />
\[<br />
T_{W}=T_{K}.<br />
\]<br />
One commonly considered correction is due to energy loss in spin-exchange<br />
collisions. In these collisions, the electron spins of the two colliding<br />
hydrogen atoms exchange, while the total spin remains unchanged. This<br />
correction to the color temperature is worked out in for example \cite{key-1},<br />
section 2.3.3. We shall here just quote the result:<br />
\begin{equation}<br />
\boxed{T_{W}=T_{K}\left(\frac{1+T_{se}/T_{K}}{1+T_{se}/T_{S}}\right),}\label{eq:colort}<br />
\end{equation}<br />
where the spin exchange temperature <br />
\[<br />
T_{se}=\frac{2T_{K}\nu_{se}^{2}}{9\Delta\nu_{D}^{2}}\sim0.40\mbox{K}.<br />
\]<br />
To use Eq. \eqref{eq:colort}, the spin temperature must be determined<br />
iteratively. The iteration is shown to converge quickly. <br />
<br />
Another commonly considered correction are radiative transfer effects.<br />
Previously we assumed that the background intensity is constant and<br />
given by the CMB intensity. The Ly$\alpha$ absorption would decrease<br />
the background intensity and hence the estimated scattering rate.<br />
The effect is more important for lower kinetic temperature. This leads<br />
to a correction to the WF effect coupling coefficient $S_{\alpha}$<br />
such that <br />
\[<br />
x_{W}=S_{\alpha}\frac{J_{\alpha}}{J_{\nu}^{c}},<br />
\]<br />
where <br />
\[<br />
J_{\nu}^{c}=1.165\times10^{-10}\left(\frac{1+z}{20}\right)\mbox{cm}^{-2}\mbox{s}^{-1}\mbox{Hz}^{-1}\mbox{sr}^{-1}.<br />
\]<br />
<br />
<br />
Neglecting spin exchange, the suppression factor is given by <br />
\[<br />
S_{\alpha}\sim\exp\left[-0.803T_{K}^{-2/3}(10^{-6}\tau_{GP})^{1/3}\right],<br />
\]<br />
where the Gunn-Peterson optical depth can be written for overdensity<br />
$\delta$ as roughly: <br />
<br />
\[<br />
\tau_{GP}=7\times10^{5}\left(\frac{\Omega_{b}h_{100}}{0.03}\right)\left(\frac{\Omega_{m}}{0.25}\right)^{-1/2}\left(\frac{1+z}{10}\right)(1+\delta).<br />
\]<br />
<br />
\end{document}<br />
</latex></div>YunfanZhangWouthuysen Field effect2015-04-30T23:18:32Z<p>YunfanZhang: /* Reference Materials */</p>
<hr />
<div>===Reference Materials===<br />
* [http://arxiv.org/PS_cache/astro-ph/pdf/0608/0608032v2.pdf Page 32 of Furlanetto et. al, 2009]<br />
* [http://bohr.physics.berkeley.edu/classes/221/1011/221a.html Detailed quantum mechanics lecture notes from UCB Physics 221A (bottom of page)]<br />
* [http://www.astro.sunysb.edu/fwalter/AST341/qn.html A Primer on Quantum Numbers and Spectroscopic Notation (Walter, Stony Brook)]<br />
* [http://www.columbia.edu/~crg2133/Files/CambridgeIA/Chemistry/Orbitals.pdf Orbitals (Guetta, Columbia)]<br />
<latex><br />
\documentclass{article}<br />
\usepackage{graphicx}<br />
\usepackage{amsmath}<br />
\usepackage{fullpage}<br />
\begin{document}<br />
<br />
\section*{The Wouthuysen-Field Effect}<br />
<br />
The Wouthuysen-Field effect is a coupling of the 21cm hyperfine<br />
transition to Ly-$\alpha$ radiation.<br />
<br />
This is important for possible<br />
high-redshift observations of the 21cm line, during the epoch of<br />
reionization. The 21cm hyperfine transition is forbidden by<br />
normal dipole selection rules, but transitions from one hyperfine<br />
state, up to the $n=2$ state, and back down to the other hyperfine<br />
state are not forbidden. So, if there is sufficient Ly-$\alpha$<br />
radiation to cause this intermediate transition, it will dominate over<br />
the direct (forbidden) hyperfine transition. Cosmologically speaking,<br />
this happens towards the end of<br />
the ``Dark Ages'' as reionization begins, and the WF effect remains<br />
the dominant effect until reionization is complete.<br />
<br />
\subsection*{Fine structure of hydrogen}<br />
<br />
Energy levels in hydrogen atoms are split due to spin orbit interaction<br />
(fine splitting), and the smaller effect of electron-proton spin interaction<br />
(hyperfine splitting). Anti-aligned spins lead to lower energy levels.<br />
The splittings of the lowest energy levels are shown in Fig. 1:<br />
<br />
\begin{figure}<br />
\center{\includegraphics{WFfig1.jpg}}<br />
\caption{Energy level diagram with spectroscopic notation}<br />
\label{all levels}<br />
\end{figure}<br />
<br />
Here we have used spectroscopic notation $n\ _{F}L_{J}$, where $n$<br />
is the principal quantum number, $L=0(S),1(P)$ are the electron orbital<br />
angular momentum. $S_{e}$ ($S_{p}$) shall denote the electron (proton)<br />
spin (not to be confused with the $S$ orbital). Then $J=|L+S_{e}|$<br />
is the electron total angular momentum, and $F=|L+S_{e}+S_{p}|$ is<br />
the hydrogen total angular momentum %<br />
\footnote{Note these are vector sums. %<br />
}. <br />
<br />
The splitting between the two $1\ S$ levels is of particular interest.<br />
It has energy difference $\nu=1.42GHz$, corresponding to wavelength<br />
$\lambda=21cm$. We also define $T_{*}=E_{21cm}/k_{B}=0.068K$. In<br />
the $21cm$ regime, the Rayleigh-Jeans limit holds and we can define<br />
the ``brightness temperature'' <br />
\[<br />
T_{b}\approx I_{\nu}c^{2}/2k_{B}\nu^{2}.<br />
\]<br />
$T_{b}$ only serves to redefine $I_{\nu}$, and should be distinguished<br />
from the ambient temperature of the CMB $T_{\gamma}=2.74(1+z)\mbox{ K}$. <br />
<br />
\subsection*{Cosmological context}<br />
<br />
After recombination $(z\sim1100)$, the photons in the cosmic fluid<br />
no longer has free charges to interact with and thus free stream through<br />
the ``dark ages'' until the Epoch of Reionization, believed to be<br />
around $z\sim 6-9$. Studying the 21cm absorption of neutral hydrogen<br />
can thus potentially probe the universe during the dark ages and epoch<br />
of reionization. The frequency $\nu$ of 21cm undergoes cosmological<br />
redshift and hence the actual observed brightness temperature is <br />
\[<br />
T_{b}=T'_{b}/(1+z).<br />
\]<br />
<br />
<br />
Denoting the lower $1S$ level by 0, the higher by 1, one can define<br />
the ``spin temperature $T_{S}$'' to characterize the relative abundance<br />
of these two states:<br />
\[<br />
\frac{n_{1}}{n_{0}}=\frac{g_{1}}{g_{0}}e^{-h\nu/k_{B}T_{S}}=3\exp(-T_{*}/T_{S})\approx3(1-T_{*}/T_{S}).<br />
\]<br />
During the dark ages till the end of reionization, several processes<br />
control the relative abundances. Among them are direct radiative transitions<br />
(Einstein coefficients) %<br />
\footnote{Direct transitions are disfavored by selection rules, as we shall<br />
explain later. %<br />
}, collisional excitation $C_{01}$, $C_{10}$ and the Wouthuysen-Field<br />
(WF) Effect $W_{01}$ and $W_{10}$. The WF effect will be explained<br />
in detail. For now let's just note that the overall statistical balance<br />
gives <br />
\begin{equation}<br />
n_{1}(A_{10}+B_{10}I_{CMB}+C_{10}+W_{10})=n_{0}(B_{01}I_{CMB}+C_{01}+W_{01}),\label{eq:balance}<br />
\end{equation}<br />
where $A_{10}=2.869\times10^{-15}\mbox{s}^{-1}$ and $B$ are the<br />
Einstein coefficients of the 21cm transition. Recall that in thermodynamic<br />
equilibrium, we had <br />
\[<br />
B_{01}=\frac{g_{1}}{g_{0}}B_{10}=\frac{3c^{2}}{2h\nu^{3}}A_{10}<br />
\]<br />
In the Rayleigh-Jeans limit, radiative coefficients satisfy<br />
\begin{equation}<br />
\begin{aligned}B_{01}I_{CMB} & =\frac{3kT_{\gamma}}{h\nu}A_{10}=\frac{3T_{\gamma}}{T_{*}}A_{10},\\<br />
B_{10}I_{CMB} & =\frac{T_{\gamma}}{T_{*}}A_{10},<br />
\end{aligned}<br />
\label{eq:BI}<br />
\end{equation}<br />
Here $T_{\gamma}=2.74(1+z)\mbox{ K }$ is the temperature of the CMB. <br />
<br />
Remember that for collisional excitations of the hyperfine transition,<br />
in thermal equilibrium, the gas temperature is coupled directly to the<br />
spin temperature. In other words, collisions are the dominant effect<br />
that sets the hyperfine level population. This means the transition<br />
rates $C_{01}$ and $C_{10}$ are given by:<br />
<br />
\[<br />
\begin{aligned}\frac{C_{01}}{C_{10}} & =\frac{g_{1}}{g_{0}}e^{-T_{*}/T_{K}}\approx3\left(1-\frac{T_{*}}{T_{K}}\right).\end{aligned}<br />
\]<br />
Similarly, WF effect rates are given by the ``color temperature''<br />
$T_{W}$: <br />
\[<br />
\begin{aligned}\frac{W_{01}}{W_{10}} & =\frac{g_{1}}{g_{0}}e^{-T_{*}/T_{W}}\approx3\left(1-\frac{T_{*}}{T_{W}}\right).\end{aligned}<br />
\]<br />
Plugging the above results into the balance equation \eqref{eq:balance},<br />
we have the relation of the temperatures:<br />
\[<br />
\boxed{T_{S}^{-1}=\frac{T_{\gamma}^{-1}+x_{c}T_{K}^{-1}+x_{W}T_{W}^{-1}}{1+x_{c}+x_{W}},}<br />
\]<br />
where <br />
\begin{equation}<br />
\boxed{\begin{aligned}x_{c} & =\frac{C_{10}}{A_{10}}\frac{T_{*}}{T_{\gamma}},\\<br />
x_{W} & =\frac{W_{10}}{A_{10}}\frac{T_{*}}{T_{\gamma}},<br />
\end{aligned}<br />
}\label{eq:coeff}<br />
\end{equation}<br />
represents the relative rates. It remains to determine these coefficients. <br />
<br />
The rest of this article shall concentrate on the WF effect, and in<br />
particular the determination of $x_{W}.$ The WF effect involves absorption<br />
of an Lyman-$\alpha$ photon from the $1_{0}S_{1/2}$ state and subsequent<br />
decay into the $1_{1}S_{1/2}$ state. The CMB temperature during the<br />
dark ages have temperatures $3000K>T_{\gamma}>15K$, and is much smaller<br />
than the Lyman-$\alpha$ transition temperature of $T=13.6eV\sim1.6\times10^{5}\mbox{K}$,<br />
and thus we expect WF effect to be most important during the epoch<br />
of reionization, when the first stars provide the abundance of $\mbox{Ly-}\alpha$<br />
photons. To study the WF effect, we first look at the rules for allowed<br />
transitions. <br />
<br />
\subsection*{Parity and Selection Rules}<br />
Now we need to look at the selection rules for dipole<br />
transitions. First of all, there is the selection rule that $\Delta L<br />
= 0$ is forbidden by parity, a symmetry of electromagnetic interactions, so only transitions between $1S$ and $2P$<br />
will matter. (In other words, nevermind about the $2S$ states we wrote<br />
down above.) This is the selection rule that forbids the 21cm<br />
transition, which is why the WF effect can dominate over the direct<br />
21cm transition so long as enough Ly-$\alpha$ photons are around.<br />
<br />
Rotational symmetry is where the rest of our selection rules come<br />
from. There is a fancy thing in quantum called the Wigner-Eckart<br />
theorem which applies to ``irreducible tensor operators'' in general,<br />
and can generate selection rules for any such operator. The selection<br />
rule goes like this: a transition from $j$ to $j'$ is only allowed if<br />
$j'$ is in the range $|j-k|$ to $j+k$, where $k$ is the ``order'' of<br />
the tensor operator. For dipole transitions, $k = 1$: the dipole<br />
operator is an ``order 1 irreducible tensor operator'', so the rule<br />
becomes $j' = |j-1|, ..., j+1$. (Lowercase $j$ is a totally generic angular<br />
momentum quantum number, not the same as $J$.)<br />
<br />
To put illustrate all of these in simpler language, a photon is a vector $A^{\mu}$ and carries one unit<br />
of angular momentum. This means that a single photon must be circularly<br />
polarized (left or right) %<br />
\footnote{In general, a tensor of rank $k$ carries $k$ units of angular momentum,<br />
and has $2k+1$ possible spin states. Due to the lack of mass of a<br />
photon however, the electromagnetic wave is transverse in all reference<br />
frames (easily seen with Maxwell's equations), and thus only has two<br />
polarization states, with angular momentum (anti-)parallel to the<br />
direction of motion. %<br />
}. Since the vector does not has any spin dependence, it commutes with<br />
the electron spin operator:<br />
\[<br />
[A^{\mu},\ S]=0.<br />
\]<br />
Here $S=|\vec{S}_{e}+\vec{S}_{p}|$ is the vector sum of the electron<br />
and proton spins. <br />
<br />
This implies the first dipole selection rule:<br />
\[<br />
\Delta S=0.<br />
\]<br />
The spin along given directions ($m_{s}$), however, are not ``good''<br />
quantum numbers, and can change. <br />
<br />
To consider orbital angular momentum, recall the parity operator $P$.<br />
It reverses all physical space directions:<br />
\[<br />
P:x_{i}\rightarrow-x_{i}.<br />
\]<br />
If we fix the phase such that $P^{2}=1$, we have parity as a Hermitian<br />
and hence observable:<br />
\[<br />
P^{-1}=P^{\dagger}=P.<br />
\]<br />
Parity thus acts on other operators via conjugation. For example<br />
\[<br />
PA^{\mu}(t,x)P^{-1}=-A^{\mu}(t,-x).<br />
\]<br />
<br />
<br />
More generally, it is clear that quantities such as vector, pseudo-scalar<br />
change sign under parity while a pseudo-vector (such as angular momentum<br />
or magnetic field) or a true scalar do not change sign. For example,<br />
parity commutes with spin $[P,S]=0$, while it anti-commutes with<br />
momentum $\{P,p\}=0$. This means that while a spin state is a good<br />
quantum number when paired with parity, a momentum state is not. <br />
<br />
So how do states in a hydrogen atom behave under parity? To answer<br />
this question recall that such a state is proportional to the spherical<br />
harmonics:<br />
\[<br />
|nlm\rangle\propto R_{lm}(r)Y_{l}^{m}(\Omega)\propto R_{lm}(r)P_{l}^{m}(\cos\theta)e^{im\phi}.<br />
\]<br />
Under parity, $\phi\rightarrow\pi+\phi$, $\theta\rightarrow\pi-\theta$,<br />
and since $P_{l}^{m}(-x)=(-1)^{l-m}P_{l}^{m}(x),$ we have <br />
\[<br />
P(|nlm\rangle)=(-1)^{l}|nlm\rangle.<br />
\]<br />
The parity of a state $|nlm\rangle$ does not depend on $m$, as expected<br />
since an intrinsic property such as parity should not depend on the<br />
orientation of the system. <br />
<br />
Thus the matrix element of the vector can be written as <br />
\[<br />
\begin{aligned}M & =\langle n'l'm'|A^{\mu}|nlm\rangle,\\<br />
& =\langle n'l'm'|P^{\dagger}PA^{\mu}P^{\dagger}P|nlm\rangle,\\<br />
& =(-1)^{l-l'+1}\langle n'l'm'|A^{\mu}|nlm\rangle.<br />
\end{aligned}<br />
\]<br />
Since a matrix element is a scalar and has even parity, we see that<br />
the end state must have different parity from the initial state. In<br />
other words, $l'-l$ must be odd!<br />
<br />
Furthermore, since the photon carries only one unit of angular momentum,<br />
any angular momentum change must be of order unity or less:<br />
\[<br />
|\Delta L|,|\Delta F|,|\Delta J|\le1.<br />
\]<br />
Thus we have the rule <br />
\[<br />
\Delta l=\pm1.<br />
\]<br />
<br />
<br />
To see this more explicitly, one can also write out the full expression<br />
\[<br />
|nlm\rangle=R_{lm}(r)P_{l}^{m}(\cos\theta)e^{im\phi}\frac{1}{\sqrt{2\pi}}.<br />
\]<br />
Since a dipole interaction, along a direction called $z$ has the<br />
form <br />
\[<br />
\vec{d}=E\vec{z}=Er\cos\theta\hat{z},<br />
\]<br />
The matrix element can be written as <br />
\[<br />
\begin{aligned}\langle n'l'm'|d|nlm\rangle & \propto\int_{0}^{\pi}P_{l'}^{m'}P_{l}^{m}\cos\theta\sin\theta d\theta\times\frac{1}{2\pi}\int_{0}^{2\pi}e^{i(m-m')\phi}d\phi.\end{aligned}<br />
\]<br />
From properties of associated Legendre polynomials, we can decompose<br />
\[<br />
P_{l}^{m}(\cos\theta)\cos\theta=bP_{l+1}^{m}+cP_{l-1}^{m},<br />
\]<br />
for some constants $b$ and $c$. And since they also satisfy the<br />
orthogonality rules<br />
\[<br />
\int_{-1}^{1}P_{l'}^{m}(x)P_{l}^{m}(x)dx=\frac{2(l+m)!}{(2l+1)(l-m)!}\delta_{l'l},<br />
\]<br />
the above matrix element vanishes unless<br />
\[<br />
\begin{cases}<br />
m' & =m,\\<br />
l' & =l\pm1.<br />
\end{cases}<br />
\]<br />
In particular, the angular momentum along the dipole direction is<br />
unchanged. This is expected since the spin of the photon is pointed<br />
perpendicular to the field (a transverse wave). <br />
<br />
A difference between $l$ and $L$ is in place when there are more<br />
than one electron in the atom. We have seen that a single electron<br />
contributes a factor of $(-1)^{l}$ when acted on by parity. In the<br />
multi-electron case, this generalizes to<br />
\[<br />
P(|l_{1}l_{2}...\rangle)=(-1)^{\sum l_{i}}|l_{1}l_{2}...\rangle.<br />
\]<br />
Since in general the total orbital angular momentum $L\ne\sum l_{i}$,<br />
two states with the same $L$ can have different parity, and so one<br />
cannot conclude that $\Delta L=0$ is not allowed. One can conclude,<br />
however, that $(L=0)\rightarrow(L=0)$ transitions are not allowed.<br />
This is because the $S$ orbital has complete spherical symmetry,<br />
a symmetry broken by the vector photon, and thus an $S$ orbital cannot<br />
absorb a photon and stay an $S$ orbital. Similarly, considering L-S<br />
coupling, we have corresponding selection rules of forbidden $(J=0)\rightarrow(J=0)$<br />
or $(F=0)\rightarrow(F=0)$ transitions. Physically, these last two<br />
rules amounts to the combined effect of angular momentum conservation,<br />
and the first rule, that $\Delta S=0.$ For example, the transition<br />
from $1_{1}S_{1/2}$ to $2_{1}P_{1/2}$ is allowed, even though $\Delta F=0$,<br />
because in this transition the photon brought $\Delta L=1$, electron<br />
flips its spin to keep $\Delta J=0$, and proton also flips its spin,<br />
to preserve $\Delta S=0$ and $\Delta F=0$, in the process $m_{F}$<br />
is flipped. The transition $1_{0}S_{1/2}$ to $2_{0}P_{1/2}$ on the<br />
other hand, is forbidden, because as $\Delta L=1$, the two conditions<br />
$\Delta S=0$ and $\Delta F=0$ cannot be simultaneously satisfied. <br />
<br />
In summary, we have the following selection rules for dipole transitions<br />
in a neutral hydrogen atom<br />
\[<br />
\begin{cases}<br />
\Delta S & =0,\\<br />
\Delta L & =\pm1,\\<br />
\Delta J & =0,\pm1,\mbox{(except 0 to 0),}\\<br />
\Delta F & =0,\pm1,\mbox{(except 0 to 0). }<br />
\end{cases}<br />
\]<br />
<br />
<br />
\subsection*{The WF effect}<br />
<br />
<br />
\subsubsection*{Coupling coefficient}<br />
<br />
The allowed transitions between the lowest levels are shown in Fig.<br />
2. The transitions relevant to the WF effect are traced with solid<br />
lines. Since we have denoted the $1S$ states by 0 and 1, we shall<br />
by convenience denote the 2$P$ states by 2,3,4,5, in order of increasing<br />
energy. <br />
\begin{figure}<br />
\center{\includegraphics{WFfig2.jpg}}<br />
\caption{Energy level diagram with allowed transitions}<br />
\label{all transitions}<br />
\end{figure}<br />
<br />
<br />
\begin{figure}<br />
\center{\includegraphics{WFfig3a.jpg}}<br />
\caption{Energy level diagram of WF effect}<br />
\label{wf transitions}<br />
\end{figure}<br />
<br />
<br />
Evidently, the (de-)excitation rate due to WF effect is given by:<br />
\begin{equation}<br />
\begin{aligned}W_{01} & =B_{03}J_{03}\frac{A_{31}}{A_{31}+A_{30}}+B_{04}J_{04}\frac{A_{41}}{A_{41}+A_{40}},\\<br />
& =\frac{3T_{\gamma}^{03}}{T_{*}^{03}}\frac{A_{30}A_{31}}{A_{31}+A_{30}}+\frac{3T_{\gamma}^{04}}{T_{*}^{04}}\frac{A_{40}A_{41}}{A_{41}+A_{40}},\\<br />
W_{10} & =B_{13}J_{13}\frac{A_{30}}{A_{31}+A_{30}}+B_{14}J_{14}\frac{A_{40}}{A_{41}+A_{40}},\\<br />
& =\frac{T_{\gamma}^{13}}{T_{*}^{13}}\frac{A_{30}A_{31}}{A_{31}+A_{30}}+\frac{T_{\gamma}^{14}}{T_{*}^{14}}\frac{A_{40}A_{41}}{A_{41}+A_{40}}.<br />
\end{aligned}<br />
\label{eq:W01}<br />
\end{equation}<br />
In the second and fourth lines I have used Eq. \eqref{eq:BI}. <br />
<br />
We would like to relate this rate to the total Ly$\alpha$ scattering<br />
rate<br />
\[<br />
P_{\alpha}=4\pi\chi_{\alpha}\int d\nu J_{\nu}\phi_{\alpha}(\nu),<br />
\]<br />
where $\sigma_{\nu}=\chi_{\nu}\phi_{\alpha}$ is the absorption cross<br />
section and $\chi_{\alpha}=f_{\alpha}\frac{\pi e^{2}}{m_{e}c^{2}}$,<br />
with $f_{\alpha}=0.4162$ the oscillator strength. The line $\phi_{\alpha}$<br />
can be assumed to be a Voigt profile. Thermal broadening leads to<br />
Doppler width<br />
\[<br />
\Delta\nu_{D}=\sqrt{\frac{2k_{B}T_{K}}{m_{H}c^{2}}}\nu_{\alpha},<br />
\]<br />
where $\nu_{\alpha}=2.47\times10^{15}\mbox{Hz}$ is the Lyman $\alpha$<br />
line center frequency. <br />
<br />
Since the dipole operator commutes with spin, the transition of $|nJFm_{F}\rangle$<br />
to $|n'J'F'm'_{F}\rangle$ is independent of $F$ and $m_{F}$. This<br />
means that the emission intensity of transition from $|nJF\rangle$,<br />
summed over all $m_{F}$, to $|n'J'\rangle$, summed over $m'_{F}$<br />
and $F'$, is the intensity of transition from a particular $|nJFm_{F}\rangle$,<br />
times $2F+1$, the degeneracy of the initial state. This is called<br />
the ``sum rule'' of the transition. <br />
<br />
The sum rules, together with the selected transitions in Fig. 2, immediately<br />
lead to<br />
<br />
\begin{equation}<br />
\begin{aligned}\frac{I_{51}}{I_{41}+I_{40}}=\frac{5}{3},\ & \frac{I_{41}+I_{40}}{I_{31}+I_{30}}=1,\ & \frac{I_{21}}{I_{31}+I_{30}}=\frac{1}{3},\\<br />
\frac{I_{40}}{I_{41}+I_{51}}=\frac{1}{3},\ & \frac{I_{30}}{I_{21}+I_{31}}=\frac{1}{3}.<br />
\end{aligned}<br />
\label{eq:int}<br />
\end{equation}<br />
The second line follow from the transition into excited states. <br />
<br />
Next we shall neglect radiative transfer effect and make the assumption<br />
that background intensities and temperatures are constant across all<br />
the hyperfine lines, and are given by the CMB values. Under this assumption<br />
\[<br />
P_{\alpha}=n_{(n=1)}B_{Ly\alpha}=\frac{3T_{\gamma}}{T_{*}}A_{Ly\alpha}.<br />
\]<br />
Let $I_{tot}=I_{51}+I_{40}+I_{41}+I_{31}+I_{30}+I_{21}$ be the total<br />
intensity of de-excitation. Then from Eq. \eqref{eq:int} we get <br />
\[<br />
\begin{aligned}I_{30}=I_{41} & =\frac{1}{12}I_{tot},\\<br />
I_{31}=I_{40} & =\frac{1}{6}I_{tot}.<br />
\end{aligned}<br />
\]<br />
Furthermore, the intensities are related to the Einstein coefficients<br />
by <br />
\[<br />
\frac{I_{ki}}{I_{\alpha}}=\frac{g_{k}}{g_{tot}}\frac{A_{kj}}{A_{\alpha}},<br />
\]<br />
where $g_{k}=2F_{k}+1$ and $g_{tot}=1+3+3+5=12$ is the total degeneracy<br />
of $n=2$ level. <br />
<br />
Thus we have<br />
\[<br />
\begin{aligned}A_{30} & =A_{41}=A_{\alpha}/3,\\<br />
A_{31} & =A_{40}=2A_{\alpha}/3.<br />
\end{aligned}<br />
\]<br />
Thus finally $P_{10}=4P_{\alpha}/27$ and the coupling coefficient<br />
in Eq. \eqref{eq:coeff} becomes <br />
\[<br />
\boxed{x_{W}=\frac{4P_{\alpha}}{27A_{10}}\frac{T_{*}}{T_{\gamma}}}.<br />
\]<br />
<br />
<br />
<br />
\subsubsection*{Color temperature and corrections}<br />
<br />
The effect of Ly$\alpha$ absorption on spin temperature also depends<br />
on the color temperature $T_{W}$. For an environment that's optically<br />
thick, such as the high-redshift IGM, we may take <br />
\[<br />
T_{W}=T_{K}.<br />
\]<br />
One commonly considered correction is due to energy loss in spin-exchange<br />
collisions. In these collisions, the electron spins of the two colliding<br />
hydrogen atoms exchange, while the total spin remains unchanged. This<br />
correction to the color temperature is worked out in for example \cite{key-1},<br />
section 2.3.3. We shall here just quote the result:<br />
\begin{equation}<br />
\boxed{T_{W}=T_{K}\left(\frac{1+T_{se}/T_{K}}{1+T_{se}/T_{S}}\right),}\label{eq:colort}<br />
\end{equation}<br />
where the spin exchange temperature <br />
\[<br />
T_{se}=\frac{2T_{K}\nu_{se}^{2}}{9\Delta\nu_{D}^{2}}\sim0.40\mbox{K}.<br />
\]<br />
To use Eq. \eqref{eq:colort}, the spin temperature must be determined<br />
iteratively. The iteration is shown to converge quickly. <br />
<br />
Another commonly considered correction are radiative transfer effects.<br />
Previously we assumed that the background intensity is constant and<br />
given by the CMB intensity. The Ly$\alpha$ absorption would decrease<br />
the background intensity and hence the estimated scattering rate.<br />
The effect is more important for lower kinetic temperature. This leads<br />
to a correction to the WF effect coupling coefficient $S_{\alpha}$<br />
such that <br />
\[<br />
x_{W}=S_{\alpha}\frac{J_{\alpha}}{J_{\nu}^{c}},<br />
\]<br />
where <br />
\[<br />
J_{\nu}^{c}=1.165\times10^{-10}\left(\frac{1+z}{20}\right)\mbox{cm}^{-2}\mbox{s}^{-1}\mbox{Hz}^{-1}\mbox{sr}^{-1}.<br />
\]<br />
<br />
<br />
Neglecting spin exchange, the suppression factor is given by <br />
\[<br />
S_{\alpha}\sim\exp\left[-0.803T_{K}^{-2/3}(10^{-6}\tau_{GP})^{1/3}\right],<br />
\]<br />
where the Gunn-Peterson optical depth can be written for overdensity<br />
$\delta$ as roughly: <br />
<br />
\[<br />
\tau_{GP}=7\times10^{5}\left(\frac{\Omega_{b}h_{100}}{0.03}\right)\left(\frac{\Omega_{m}}{0.25}\right)^{-1/2}\left(\frac{1+z}{10}\right)(1+\delta).<br />
\]<br />
<br />
\end{document}<br />
</latex></div>YunfanZhang