## LaTeX Version of Equations

eq 4
$$\Psi(\alpha J)=\sum_{\gamma\,S\,L} |c(\gamma\,S\,L\,J)\,\Phi(\gamma\,S\,L\,J)\quad.$$

eq 5
$$\sum_{\gamma\,S\,L} |c(\gamma\,S\,L\,J)|^2 = 1\quad.$$

eq 8
$$g_{\beta S L J}= 1+(g_{\rm e}-1) ~ \frac{J(J+1)-L(L+1)+S(S+1)}{2J(J+1)}\quad.$$

eq 9
$$g_{\beta J}= \sum_{\gamma S L}\,g_{S L J} \, |c(\gamma S L J |^2\quad,$$

eq 10
$$E_n = -\frac{Z^2}{n^2}\quad,$$

eq 11
$$E_{nl} = - ~ \frac{Z^2_c}{(n-\delta_1)^2} = - ~ \frac{Z^2_c}{(n*)^2} ,$$

eq 12
\begin{eqnarray*}
\delta&=& n - n*\\         &~& ~\\
&=& \delta_0 + \frac{a}{(n-\delta_0)^2} + \frac{b}{(n-\delta_0)^4} +

eq 13
$\epsilon_{\rm line} = (4\pi)^{-1} h^\nu A_{ki}\, N_k ~ ,$

eq 14
\begin{eqnarray*}
I_{\rm line}=\epsilon_{\rm line}l&=& \int_0^{+\infty} \, I(\lambda){\rm d}\lambda\\
&~& \\
&~& (4\pi)^{-1} \, (hc/\lambda_0) \, A_{ki}\, N_kl ~ ,\end{eqnarray*}

eq 15
$W(\lambda)=[I(\lambda)-I^\prime(\lambda)]/I(\lambda)\quad ,$

eq 16
$$W_{ik} = \int_0^{+\infty} \, W(\lambda){\rm d}\lambda = \frac{e^2}{4\epsilon_0\,m_{\rm e}\,c^2} ~ \lambda_0^2 \, N_i \, f_{ik}\,l\quad ,$$

eq 17
$$S=S(i,k) = S(k,i)= |R_{ik}|^2\quad,$$

eq 18
$$R_{ik}=\langle\psi_k| \, P ~ |\psi_i\rangle\quad,$$

eq 19
$$A_{ki}=\frac{2\pi e^2}{m_{\rm e}c\epsilon_0\lambda^2} ~ \frac{g_i}{g_k} f_{ik}= \frac{16\pi^3}{3h\epsilon_0\lambda^3 g_k} ~S\quad,$$'| t2g 130

eq 20
$$A_{ki}=\frac{6.6702\times10^{15}}{\lambda^2} \frac{g_i}{g_k} f_{ik} = \frac{2.0261\times10^{18}}{\lambda^3 g_k} S\quad ,$$

eq 21
$$f_{ik} = \frac{2}{3} (\Delta E/g_1) S\quad .$$

eq 22
\begin{eqnarray*}\bar g_{i(k)} &=& \sum_{i(k)} (2J_{i(k)} + 1)\\
&=& (2L_{i(k)} + 1) (2S_{i(k)} + 1) \quad . \end{eqnarray*}

eq 23
$S_M=\sum S_{\rm line} ~ ,$

eq 24
$$f_M=(\bar\lambda \bar{g}_i)^{-1} \sum_{J_k J_i} g_i\lambda(J_i, J_k) f(J_i, J_k) \quad .$$

eq 25
$$\bar\lambda=n\bar\lambda_{\rm air} = hc/\overline{\Delta E} \quad ,$$

eq 26
$$\overline{\Delta E}=\overline{E_k}-\overline{E_i}= (\bar g_k)^{-1} \sum_{J_k} g_k E_k-(\bar g_i)^{-1} \sum_{J_i} g_i E_i\quad ,$$

eq 27
$$\tau_k=\left(\sum_i \, A_{ki}\right)^{-1}\quad.$$

eq 28
$$A_{ki\prime} \Big/ \sum_i \, A_{ki} = A_{ki\prime} \, \tau_k\quad.$$

eq 29
$$\tau_k=1/A_{ki\prime} \tau_k\quad.$$

eq 30
$$(\Delta E)_Z=(E_k-E_i)_Z=R_M\, hc\,Z^2(1/n_i^2-1/n_k^2)\quad.$$

eq 31
$$(\Delta E)_Z=Z^2(\Delta E)_{\rm H}\quad,$$

eq 32
$$(\lambda_{\rm vac})_Z=Z^2(\lambda_{\rm vac})_{\rm H}\quad,$$

eq 33
$$S_Z=Z^{-2}\, S_{\rm H}\quad,$$

eq 34
$$f_Z=f_{\rm H}\quad,$$

eq 35
$$A_Z=Z^4 A_{\rm H}\quad,$$

eq 36
$$f(n,l\rightarrow n^\prime, l\pm1)\, \alpha(n^\prime)^{-3}\quad.$$

eq 40
$$\Delta\lambda_{1/2}^D = (7.16\times10^{-7})\,\lambda(T/M)^{1/2}\quad.$$

eq 41
$$\Delta\lambda_{1/2}^R \simeq 8.6\times10^{-30}(g_i/g_k)^{1/2} \, \lambda^2 \, \lambda_r \, f_r\, N_i\quad.$$

eq 42
$$\Delta\lambda_{1/2}^W \simeq 30\times10^{16} \lambda^2 C_6^{2/5} (T/\mu)^{3/10} N \quad,$$

eq 43
$$\Delta\lambda_{1/2}^{S,H}=(2.50\times10^{-9}) \, \alpha_{1/2} N_{\rm e}^{2/3}\quad,$$

eq 44
\begin{eqnarray*} \epsilon_{\rm cont}&=&
\frac{\epsilon^6}{2\pi\, \epsilon_0^3 (6\pi\, m_{\rm e})^{3/2}}~
N_{\rm e}\, N_Z\, Z^2 \times \frac{1}{(kT)^{1/2}} ~
\exp \left(- \frac{hc}{\lambda kT}\right)~ \frac{\Delta\lambda}{\lambda^2}\\
&~&\times \left\{\frac{2Z^2\, I_{\rm H}}{kT} ~
\sum_{n\geq(Z^2 \,I_{\rm H}\, \lambda/hc)^{1/2}}^{n^\prime}~
\frac{\gamma_{\rm fb}}{n^3}~ \exp\left(\frac{Z^2 \, I_{\rm H}}{n^2kT}\right)
+ \bar\gamma_{\rm fb}
\left[\exp\left(\frac{Z^2 \,I_{\rm H}}{(n^\prime+1)^2\, kt}\right)
-1\right] + \gamma_{\rm ff} \right\} \quad, \end{eqnarray*}

eq
\begin{eqnarray*}
(nl-n^\prime l^\prime)_{\rm Li}&\rightarrow&
\left[ (n+1)l-(n^\prime+1) l^\prime \right]_{\rm Na}\\
&\rightarrow& \left[ (n+2)l-(n^\prime+2) l^\prime \right]_{\rm Cu}~\rightarrow~\ldots \quad .
\end{eqnarray*}