% ------------------------------------------------------------------------
%the following title page must be printed with PHYZZX flavor of LaTeX
%\date{April 8, 1994}
%\pubtype{PHY4005}
%\titlepage
%\title{LASER ACTION IN C IV PLASMAS}
%\author{John talbot}
%\footnote{\dag}{supervised by Y. P. Varshni}
%\address{University of Ottawa}
%\abstract{A theoretical investigation of population inversion in C IV
%recombining plasmas cooled by adiabatic expansion has been carried out.
%The non-equilibrium plasma is simulated on computer using the
%Collisional-Radiative model which requires rate coefficients for basic
%physical plasma processes. We show how gain per unit length is computed and
%why large gains occur only in limited regions of parameter space ($n_e, T_e$)
%we then compare these results with astronomical evidence. We also show how
%the rate coefficients and ionization equilibrium are computed.}
%
%\submit{F.Guillon, Student Project Coordinator}
%\endpage
%\end
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\def\title#1{\bigskip\\{\largetype #1}\bigskip}
\def\minus{{\scriptscriptstyle(-)}}
\def\plus{{\scriptscriptstyle(+)}}
\def\plusminus{{\scriptscriptstyle(\pm)}}
\def\zero{{\scriptscriptstyle(0)}}
\def\one{{\scriptscriptstyle(1)}}
\def\two{{\scriptscriptstyle(2)}}
\def\three{{\scriptscriptstyle(3)}}
\def\ion{{\scriptscriptstyle+}}
\def\CR{{\scriptscriptstyle\rm CR}}
\def\SS{{\scriptscriptstyle\rm SS}}
\def\teqzero{{\scriptstyle t=0}}
\def\m{{+m}}
\def\mplus{{+(m+1)}}
\def\R#1{\ \uppercase\expandafter{\romannumeral #1}}
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\\{\hugetype LASER ACTION IN C IV PLASMAS}
\title{A) INTRODUCTION}
The theoretical study of decaying plasma has been crucial in the development
of short wavelength and X-Ray lasers (Matthews {\sl et~al.}, 1988).
The purpose of this work is to study the adiabatic expansion of C IV plasmas
and to identify the conditions under which population inversions occur.
The results provide an explanation of the intensity anomalies of C IV lines in
Wolf-Rayet stars and Quasars.
This paper is divided into 9 sections,
B) Description of the plasma model.
C) Various constraints the plasma parameters must obey.
D) Important physical processes occurring in the plasma.
E) Detailed description of the theoretical model used in our
plasma simulation.
F) How gain is computed.
G) Results and astronomical evidence.
H) Conclusion.
Appendix A: Atomic data used in computing rate coefficients.
Appendix B: Ionization equilibrium computation.
Appendix C: Computer Program analysis
\title{B) THE PLASMA MODEL}
A plasma consists of one or more elements in various stages of ionization.
Its main constituents are positively charged ions, free electrons, neutral
atoms, and electromagnetic radiation from discrete and continuous spectra.
In this work, we consider a monatomic (element~A), stationary, and spatially
homogeneous plasma, free of magnetic fields. Label A refers to all ionization
stages of element A: neutral (A~I), singly ionized (A~II), doubly ionized
(A~III), $\ldots$ . Element A is characterized by a density $n_A$
${\rm cm}^{-3}$ and a temperature $T_A$ Kelvin, and the free electrons by a
density $n_e$ and a temperature $T_e$. Furthermore, each stage of ionization X
of element A is characterized by a density $n_A^{(x)}$; then
$n_A=\sum_x n_A^{(x)}$
All particles of the plasma are assumed to be at the same temperature
$T=T_e=T_A$. Under the plasma conditions considered in this work, this is a
reasonable approximation, especially since $T_A$ is used only to calculate
the ionization equilibrium of element A (See appendix B)
\title{C) APPLICABILITY OF THE PLASMA MODEL}
\medskip
\\ENERGY DISTRIBUTION OF THE FREE ELECTRONS
\medskip
The mass of the electron is much smaller than that of any ion. The free
electrons are thus much more mobile than the ions. Their energy distribution
in a plasma is described by Maxwell-Boltzmann statistics. This distribution
function holds if the mean de Broglie wavelength of the plasma electrons is
appreciably smaller than the mean distance between them (Vedenov, 1965).
This puts an approximate upper limit on $n_e$ of
$$n_e \ll 1.2\times 10^{15}\,T^{3/2} {\rm cm}^{-3},\eqno(1)$$
\\In all cases studied in this work, we are well below this upper limit of
validity.
\smallskip
An approximate lower limit of validity can be derived as follows. An electronic
Maxwellian distribution will hold if a sufficient number of elastic collisions
occur between the free electrons to allow enough energy exchanges between them
to establish a Maxwellian energy distribution. This will be the case if the
plasma is sufficiently dense. A crude estimate of the lower limiting value of
$n_e$ for a lithium-like ion C IV is:
$$ n_e>10^{10}{\rm cm}^{-3},\eqno(2)$$
Since we consider plasmas with $n_e=10^{12}$ to $10^{15}{\rm cm}^{-3}$,
we expect the Maxwellian distribution to be a reasonable approximation to
the actual distribution function of the free electrons in the plasma.
A sufficiently dense plasma also insures that the response time of the
distribution to a change in the plasma parameters is sufficiently rapid
that the distribution remains Maxwellian at all times.
\medskip
\\EQUILIBRIUM POPULATION OF THE LEVELS
\medskip
Plasmas in complete thermodynamic equilibrium (CTE) can only exist under very
special conditions and are thus very rarely observed. However, there exists a
large class of plasma in which each volume element fulfils all thermodynamic
equilibrium laws derived for plasmas in CTE except for Planck's radiation law.
Such plasmas are said to be in local thermodynamic equilibrium (LTE).
In these plasmas, the level population densities are determined solely by
collisional processes which are assumed to occur rapidly enough such that the
population densities respond instantaneously to any change in the plasma
conditions. The population densities thus depend entirely on the local values
of the plasma parameters. Furthermore, each physical process is accompanied
by its's inverse and, by the principle of detailed balance, these pairs of
processes occur at equal rates. The advantage of this model is that the atomic
cross-sections of the various physical processes are not needed to calculate
the population densities. These are determined solely by the laws of
statistical mechanics.
\medskip
\\NON-EQUILIBRIUM POPULATION OF THE LEVELS:
\medskip
Plasmas that don't obey the LTE model are called non-LTE plasmas.
The population densities of the energy levels of ions in non-LTE plasmas
must be obtained from the rate coefficients of the individual collisional and
radiative processes occurring within the plasma. Several models which take into
account some or most of these processes have been proposed; of these,
the most useful and general is the Collisional-Radiative (CR) model.
This model was first proposed and applied to hydrogenic ions by Bates
et~al.~(1962a,b). Since then, much work has been done using the CR model,
especially in modelling short wavelength lasers.
\title{D) PHYSICAL PROCESSES OCCURRING IN A PLASMA}
Let $e$ denote an electron, ${\rm h}\nu$ a photon, and $N_i^x$ an $X$-times
ionized atom (ion $X$) in state $i$, $n^\ion$ is the ion density ($N^{x+1}$)
and $n_e$ is the electron density. Then the physical processes occurring
within the plasma and included in the CR model can be described as follows
(rate coefficients are a function of $T$)
\medskip
\\i) {\sl Collisional ionization} by electron impact:
$$N_i^x+e\rightarrow N^{x+1}+e+e,\eqno(3)$$
The rate coefficient of this process is denoted by
$S_i(T)$. The number of such processes occurring per unit
volume per unit time is given by
$n_i n_e S_i(T) {\rm cm}^{-3}s^{-1}$
\medskip
\\ii) {\sl Three-body recombination}: inverse of process (i):
$$N^{x+1}+e+e\rightarrow N_i^x+e ,\eqno(4)$$
The rate coefficient: $\alpha_i(T)$, and the number of processes
$n_e^2n^\ion\alpha_i(T)$.
\medskip
\\iii) {\sl Radiative recombination}: It predominates over process (ii) in
low-density plasmas.
$$ N^{x+1}+e\rightarrow N_i^x+{\rm h}\nu,\eqno(5)$$
The rate coefficient $\beta_i(T)$, and the number of processes:
$n_en^\ion\beta_i(T)$.
\medskip
\\iv) {\sl Collisional excitation} by electron impact:
$$N_i^x+e\rightarrow N_j^x+e,\eqno(6)$$
Rate coefficient $C_{ij}(T)$, number of processes: $n_i n_e C_{ij}(T)$.
\medskip
\\v) {\sl Radiative transition}
$$N_j^x\rightarrow N_i^x+{\rm h}\nu,\eqno(7)$$
Einstein probability coefficient: $A_{ij}$; and number of processes:
$n_jA_{ji}$.
\medskip
\\These are the main physical processes which occur in a plasma under most
conditions. (see appendix A for computation of rate coefficients)
The plasma is also assumed to be optically thin such that all
radiation emitted within the plasma escapes without being absorbed. (Thus
neglecting photoexcitation and photoionization)
By including all the above collisional and radiative processes into a rate
equation for each quantum level of a specific stage of ionization of our
atom we obtain the following differential equation for the time development
of the population density $n_i$ of a level $i$.
$$\eqalignno{\dot n_i\equiv {dn_i\over {dt}}&=-n_en_iS_i-n_en_i
\sum_{j\ne i}^\infty C_{ij}-n_i\sum_{j=1}^{i-1} A_{ij}\cr
&\qquad+n_e\sum_{j\ne i}^\infty n_jC_{ji}
+\sum_{j=i+1}^\infty n_jA_{ji}+\{ \beta_i+\alpha_i
n_e\}n_en^\ion\quad,&(8)\cr}$$
There is such an equation for each and every discrete level $i$ = 1, 2, 3,
$ \cdots,\infty$. Thus, we obtain an infinite number of coupled differential
equations describing the population densities of all the discrete levels.
\title{E) THE THEORETICAL MODEL}
We consider what happens when the plasma in the outer layers of a star
rapidly expands. To focus our discussion we consider Wolf-Rayet stars.
The atmospheres of these stars depart seriously from LTE. We consider the
state of plasma at the base of the extended atmosphere of a Wolf-Rayet star,
roughly where its `photosphere' would lie. We make the reasonable assumption
that the conditions there will also correspond to non-LTE. Hence to obtain
the relative concentrations of differing ionization stages of an element at
a particular electron density ($n_e$) and electron temperature ($T_e$) we
use the non-LTE method of House (1964) ( see Appendix B ).
It is well known that Wolf--Rayet stars are undergoing a high--speed
mass loss. We assume that this plasma expands adiabatically, for which the
plasma density $N$ and $T_e$ are related by $T_eN^{1-\gamma}$=const.
We assume $\gamma = 5/3$; for the actual plasma the value will be slightly
smaller. The flow is supersonic and to a first approximation, the plasma is
assumed to be `frozen' during the rapid fall of temperature, for which a
factor of 5 is assumed. (The same factor for the fall in the temperature
has been used by Gudzenko {\sl et~al.\/}, 1966,
and Varshni and Lam, 1976.)
\bigskip
\\To examine deviations from LTE it is useful to define the normalized
population density of level $i$ as
$$\rho_i={n_i\over n_i^E}\quad,\eqno(9)$$
$$n_i^E\equiv Z_in_en^\ion\quad,\eqno(10)$$
$$Z_i = \{\omega_i/2\omega^\ion\}(h^2/2\pi m_ekT_e)^{3/2}
\exp(\chi_i/kT_e)\quad,\eqno(11)$$
\\where $n_i^E$ is the Saha equilibrium population density of level $i$,
$\omega_i$ is the statistical weight of level $i$, $\omega^\ion$ is the
partition function of the ion and $\chi_i$ is the ionization potential of
the level $i$.
\medskip
\\Dividing eq.(8) by $n_i^E$ and using eq.(9), the set of
equations (8) becomes
$$\eqalignno{\dot\rho_i \equiv{\dot n_i\over n_i^E}&= -\left( n_eS_i
+n_e\sum_{j\ne i}^\infty C_{ij}+
\sum_{j=1}^{i-1} A_{ij}\right) \rho_i\cr
&+n_e\sum_{j\ne i}^\infty C_{ij}\rho_j +
\sum_{j=i+1}^\infty {Z_j\over Z_i} A_{ji}\rho_j+{1\over Z_i}
\{ \alpha_in_e +\beta_i\},&(12)\cr}$$
\\with $i$ = 1,2,3,$ \cdots, \infty.$
(We used the fact that $Z_iC_{ij}=Z_jC_{ji}$ from the principle of detailed
balance)
\bigskip
\\To solve this set of equations, the following assumptions are used:
\bigskip
\item{(i)} For all levels $i$ located above a sufficiently high lying level,
$r$, the population density is assumed to be given by the Saha-Boltzmann
equilibrium equation instead of by Eq. (8). Thus, in this case, we have
$\rho_{i>r}=1$. The infinite set of equations (12) thus become a
finite set of $r$ coupled equations which can be solved for $\rho_i,
\ i=1,2,\cdots ,r$.
\medskip
\item{(ii)} The infinite sums appearing in Eq. (12) can be cut off at a
sufficiently high--lying level $s>r$ above which the rate coefficients
involving these states contribute little.
\medskip
\item{(iii)} The time derivative, Eq. (12), can be put equal to zero for all the
levels $ i\le r $ except for the lowest levels without causing significant
error.
\vfill\eject %----------------------------------------------------------------
{\bigtype
\\TABLE 1. Population coefficients for various Collisional-Radiative models.
\bigskip}
\halign to \hsize{\strut#&\vrule#\tabskip=0em plus2em&
\quad#\hfil\quad&\vrule#&
\quad$r_#$\hfil\quad&\vrule#&
\quad$r_#$\hfil\quad&\vrule#&
\quad$r_#$\hfil\quad&\vrule#&
\quad$\dot\rho_#$\hfil\quad&\vrule#\cr
\noalign{\hrule}
&&&&\multispan5 &&\omit&\cr
&&&&\multispan5 $n_i=Z_ir_i^\zero n^\ion n_e$\hfil&&\omit&\cr
&&&&\multispan5 &&\omit&\cr
&&\hfil 0&&\multispan5 $\rho_i=r_i^\zero$\hfil&&{i}=0&\cr
&&&&\multispan7\hrulefill&\cr
&&Steady&&{i>r}^\zero =1&&\NA&&\NA&&\omit&\cr
&&State&&{i>s}^\zero=0&&\NA&&\NA&&\omit&\cr
&&&&\omit&&\omit&&\omit&&\omit&\cr
\noalign{\hrule}
&&&&\multispan5 &&\omit&\cr
&&&&\multispan5 $n_i=Z_i[r_i^\zero n^\ion n_e+r_i^\one n_1/Z_1]$\hfil&&\omit&\cr
&&&&\multispan5 &&\omit&\cr
&&\hfil 1&&\multispan5 $\rho_i=r_i^\zero+r_i^\one\rho_1$\hfil&&{i>1}=0&\cr
&&&&\multispan7\hrulefill&\cr
&&&&1^\zero=0&&1^\one=1&&\NA&&1\ne 0&\cr
&&H I&&{i>r}^\zero =1&&{i>r}^\one=0&&\NA&&\omit&\cr
&&&&{i>s}^\zero=0&&{i>s}^\one=0&&\NA&&\omit&\cr
&&&&\omit&&\omit&&\omit&&\omit&\cr
\noalign{\hrule}
&&&&\multispan5 &&\omit&\cr
&&&&\multispan5 $n_i=Z_i[r_i^\zero n^\ion n_e+r_i^\one n_1/Z_1+
r_i^\two n_2/Z_2]$\hfil&&\omit&\cr
&&&&\multispan5 &&\omit&\cr
&&\hfil 2&&\multispan5 $\rho_i=r_i^\zero+r_i^\one\rho_1+
r_i^\two\rho_2$\hfil&&{i>2}=0&\cr
&&&&\multispan7\hrulefill&\cr
&&&&1^\zero=0&&1^\one=1&&1^\two=0&&1\ne 0&\cr
&&C IV&&2^\zero=0&&2^\one=0&&2^\two=1&&2\ne 0&\cr
&&&&{i>r}^\zero=1&&{i>r}^\one=0&&{i>r}^\two=0&&\omit&\cr
&&&&{i>s}^\zero=0&&{i>s}^\one=0&&{i>s}^\two=0&&\omit&\cr
&&&&\omit&&\omit&&\omit&&\omit&\cr
\noalign{\hrule}}
\bigskip
\\ $r_i^\zero$ is the relative contribution to level $i$ from
the ion level ($n^\ion$),
\\ $r_i^\one$ is the contribution to level $i$ from level 1, and
\\ $r_i^\two$ is the contribution to level $i$ from level 2.
\bigskip
\item{0)} In the steady state model all the time derivatives are zero and the
population density $n_i$ depends only on the ion level density $n^\ion$.
\medskip
\item{1)} In the quasi-steady state model some time derivatives are non-zero.
For Hydrogen-like ions, the time derivative of the lowest level is non-zero;
the excited level density $n_i$ depends on the ground level density $n_1$,
and the ion level density $n^\ion$.
\medskip
\item{2)} For Lithium-like ions, such as C IV ions, the unusually small energy
gap between the first and second quantum states leads to a relatively large
population in the second level. Thus the time derivative of state 1 and 2 is
non-zero and all excited states depend on the first and second quantum levels
as well as the ion level.
\vfill\eject %----------------------------------------------------------------
\\For hydrogen-like ions if we substitute from the above table back into
Eq.(12)
\\we obtain the following set of coupled linear equations for $\quad
i=2,3,\ldots r$,
$$\eqalignno{
-\biggl(n_eS_i+n_e\sum_{j\ne i}^sC_{ij}+\sum_{j=1}^{i-1}A_{ij}\biggr)
(r_i^\zero+&r_i^\one\rho_1)\cr
+n_eC_{i1}\rho_1&+
n_e\sum^r_{\scriptstyle j\ne i\atop\scriptstyle j=2}C_{ij}
(r_j^\zero+r_j^\one\rho_1)+\sum_{j=i+1}^r{Z_j\over Z_i}A_{ji}
(r_j^\zero+r_j^\one\rho_1)\cr
&+\sum_{j>r}^s\{n_eC_{ij}+{Z_j\over Z_i}A_{ji}\}+
{1\over Z_i}\{\beta_i+\alpha_in_e\}=0, &(13)\cr}$$
\\We can group the $\rho_1$ terms and obtain,
$$\eqalignno{
\biggl[-\biggl(n_eS_i+n_e\sum_{j\ne i}^sC_{ij}+\sum_{j=1}^{i-1}A_{ij}\biggr)
r_i^\zero&+ n_e\sum^r_{\scriptstyle j\ne i\atop\scriptstyle j=2}
C_{ij}r_j^\zero+\sum_{j=i+1}^r{Z_j\over Z_i}A_{ji}r_j^\zero\cr
&+\sum_{j>r}^s\{n_eC_{ij}+{Z_j\over Z_i}A_{ji}\}+
{1\over Z_i}\{\beta_i+\alpha_in_e\}\biggr]+&(14)\cr
\biggl[-\biggl(n_eS_i+n_e\sum_{j\ne i}^sC_{ij}+\sum_{j=1}^{i-1}A_{ij}\biggr)
r_i^\one&+ n_e\sum^r_{\scriptstyle j\ne i\atop\scriptstyle j=2}
C_{ij}r_j^\one+\sum_{j=i+1}^r{Z_j\over Z_i}A_{ji}r_j^\one+
n_eC_{i1}\biggr]\rho_1=0;\cr}$$
\\This set of equations is of the form $a_i+b_i\rho_1=0$ for $i=2,3,\ldots r$.
The general solution for an arbitrary value of $\rho_1$ is $a_i=0$, $b_i=0$
thus,
$$\eqalignno{
-\biggl(n_eS_i+n_e\sum_{j\ne i}^sC_{ij}+\sum_{j=1}^{i-1}A_{ij}
\biggr)r_i^\zero&+n_e\sum^r_{\scriptstyle j\ne i\atop\scriptstyle j=2}
C_{ij}r_j^\zero+\sum_{j=i+1}^r{Z_j\over Z_i}A_{ji}r_j^\zero=\cr
&-\sum_{j>r}^s\{n_eC_{ij}+{Z_j\over Z_i}A_{ji}\}-
{1\over Z_i}\{\beta_i+\alpha_in_e\}&(15)\cr
-\biggl(n_eS_i+n_e\sum_{j\ne i}^sC_{ij}+\sum_{j=1}^{i-1}A_{ij}\biggr)
r_i^\one&+ n_e\sum^r_{\scriptstyle j\ne i\atop\scriptstyle j=2}
C_{ij}r_j^\one+\sum_{j=i+1}^r{Z_j\over Z_i}A_{ji}r_j^\one
=- n_eC_{i1}&(16)\cr}$$
\vfill\eject %----------------------------------------------------------------
\\this is equivalent to 2 matrix equations for Hydrogen-like ions
$(\rho_i=r_i^\zero+r_i^\one\rho_1)$
$$ \eqalignno{
\sum_{j=2}^rB_{ij}r_j^{\scriptscriptstyle(k)}&=
R_i^{\scriptscriptstyle(k)}\quad,\qquad i=2,3,\ldots,r\qquad
\hbox{and}\qquad k=0\ \hbox{or}\ 1&(17)\cr
\cr
R_i^\zero&=-\sum_{j>r}^s\{ n_eC_{ij}+{Z_j\over Z_i}A_{ji}\}
-{1\over Z_i}\{ \beta_i+\alpha_in_e\}\quad,&(18)\cr
R_i^\one&=-n_eC_{i1}\quad,&(19)\cr
B_{ij}&=\cases{n_eC_{ij}+{\displaystyle{Z_j\over Z_i}}A_{ji}\quad,&
if $ij$\qquad (lower);\cr}&(20)\cr}$$
\\Similarly for C IV ions, $(\rho_i=r_i^\zero+r_i^\one\rho_1+r_i^\two\rho_2)$
we obtain 3 matrix equations:
$$\eqalignno{
\sum_{j=3}^rB_{ij}r_j^{\scriptscriptstyle(k)}&=
R_i^{\scriptscriptstyle(k)}\quad,\qquad i=3,4,\ldots,r\qquad
\hbox{and}\qquad k=0,1,2\qquad\qquad&(21)\cr
\cr
R_i^\two&=-n_eC_{i2}\quad,&(22)\cr}$$
\\To compute the steady state population density we must solve only one matrix
equation:
$$\sum_{j=1}^rB_{ij}r_j^{\scriptscriptstyle(0)}=
R_i^{\scriptscriptstyle(0)}\quad,\qquad i=1,2,3,\ldots,r,\qquad\qquad
(\rho_i=r_i^\zero)\eqno(23)$$
The iterative method of solving linear equations, Pan and Reif (1985) is
ideally suited for scanning the $n_e$,$T_e$ parameter space, since we can use as
initial matrix inverse the previous solution at $n_e+\delta n_e$,$T_e$. The rate
coefficients only need be re-computed when $T_e$ changes.
Except for some sharp transitions near higher $n_e$, small changes in these
parameters will in general produce smoothly varying solutions, thus the matrix
inverse at a nearby point is a very good initial guess and the iteration
converges quadratically:
We start with some matrix ${\bf A}_0$ that is assumed to be an
{\sl approximate\/}
inverse of the matrix ${\bf B}$, so that ${\bf A}_0\cdot{\bf B}$ is
approximately the identity matrix {\bf 1}. Then the recurrence,
$${\bf A}_{2n+1}=2{\bf A}_n-{\bf A}_n\cdot{\bf B}\cdot{\bf A}_n\quad,\qquad
n=0,1,3,7,\ldots\eqno(24)$$
\\will converge {\sl quadratically\/} to the unknown inverse matrix
${\bf B}^{-1}$.
This equation is known as {\sl Hotelling's Method\/}.
For the very first initial inverse a suitable approximation can be obtained from
$${\bf A}_0=\epsilon {\bf B}^{\rm T},\eqno(25)$$
\\$\epsilon$ is a small constant which can be set to the reciprocal of the
maximum row sum times the maximum column sum of ${\bf B}$.
\\For Hydrogen-like ions, after solving the 2 matrix equations we are left with
a differential equation for $n_1$. Since the excited state densities linearly
depend on $n_1$, population inversion will be time dependent and a search must
be made for a critical time when the laser transition has maximum gain.
We can solve the differential equation for $n_1$ by substituting the
$(r_i^\zero,r_i^\one)$ solutions of Eq.(17) back into Eq.(8) and obtain:
$$\eqalignno{\dot n_1&=\quad-n_en_1S_1\quad-\quad n_en_1\sum_{j=2}^sC_{1j}\cr
&+\quad\sum_{j=2}^r\{n_eC_{j1}+A_{j1}\}Z_j[r_j^\zero n^\ion n_e+r_j^\one n_1/Z_1
]\cr
&+\quad\sum_{j>r}^s\{n_eC_{j1}+A_{j1}\}Z_jn^\ion n_e
\quad+\quad\{\beta_1+\alpha_1n_e\}n_en^\ion\quad,&(26)}$$
$$\eqalignno{
\dot n_1&=\alpha_\CR n^\ion n_e - S_\CR n_1 n_e\quad,&(27)\cr
\cr
S_\CR&=S_1+\sum_{j=2}^s C_{1j} - {1\over Z_1n_e}
\sum_{j=2}^r\{n_eC_{j1}+A_{j1}\}Z_jr_j^\one\quad,&(28)\cr
\alpha_\CR&=\beta_1+\alpha_1n_e+\sum_{j=2}^r\{n_eC_{j1}
+A_{j1}\}Z_jr_j^\zero+\sum_{j>r}^s\{n_eC_{j1}+A_{j1}\}Z_j\quad,
&(29)\cr}$$
\\Solving the differential equation:
$$\eqalignno{
n_1(t)&={\alpha_\CR \over S_\CR}n^\ion+\bigg[n_1(\teqzero)-
{\alpha_\CR \over S_\CR}n^\ion\bigg]e^{-S_\CR n_et}\quad,&(30)\cr
n_1(t\rightarrow\infty)&={\alpha_\CR \over S_\CR}n^\ion\quad,&(31)\cr}$$
\\Thus the final solution for H I excited state densities ($i>1$) is,
\medskip
$$ n_i=Z_i[r_i^\zero n^\ion n_e+r_i^\one n_1(t)/Z_1],\eqno(32)$$
\medskip
\\$n_1(t)$ is taken from Eq.(30), $n_1(\teqzero)$ and
$n^\ion$ are obtained from the ionization equilibrium calculations of
Appendix B, computed for the $n_e, T_e$ values before the expansion. Then after
the expansion we assume that these densities are 'frozen', so we divide them
by the expansion factor. Eventually $n_1$ and $n^\ion$ relax to the cooler
ionization equilibrium with an exponential time factor $S_\CR n_e$, we note
that $n_1(t\rightarrow\infty)$ is merely a re-arrangement of Eq.(B-1) for the
lower temperature and density.
\vfill\eject %---------------------------------------------------------------
\\For Lithium-like ions, after solving the 3 matrix equations we are left with
a differential equations for $n_1$ and for $n_2$. These must be solved since
as mentioned before, gain is time dependent. We can solve for $n_1$ and $n_2$
by substituting the solutions of Eq.(21) $(r_i^\zero,r_i^\one,r_i^\two)$ back
into Eq.(8) and using the constraints of table (1) we find:
$$\eqalignno{\dot n_1&=\quad-n_en_1S_1\quad-\quad n_en_1\sum_{j=2}^sC_{1j}\quad
+\quad n_en_2C_{21}\quad+\quad n_2A_{21}\cr
&\qquad\qquad+\quad
\sum_{j=3}^r\{n_eC_{j1}+A_{j1}\}Z_j[r_j^\zero n^\ion n_e+r_j^\one n_1/Z_1
+r_j^\two n_2/Z_2]\cr
&\qquad\qquad+\quad
\sum_{j>r}^s\{n_eC_{j1}+A_{j1}\}Z_jn^\ion n_e
\quad+\quad\{\beta_1+\alpha_1n_e\}n_en^\ion\quad,&(33)}$$
$$\eqalignno{\dot n_2&=\quad-n_en_2S_2\quad-\quad n_en_2\sum_{j\ne2}^sC_{2j}
\quad-\quad n_2A_{21}\quad+\quad n_en_1C_{12}\quad\cr
&\qquad\qquad+\quad
\sum_{j=3}^r\{n_eC_{j2}+A_{j2}\}Z_j[r_j^\zero n^\ion n_e+r_j^\one n_1/Z_1
+r_j^\two n_2/Z_2]\cr
&\qquad\qquad+\quad
\sum_{j>r}^s\{n_eC_{j2}+A_{j2}\}Z_jn^\ion n_e
\quad+\quad\{\beta_2+\alpha_2n_e\}n_en^\ion\quad,&(34)}$$
$$\eqalignno{
\dot n_1&=\alpha_1^\CR n^\ion n_e+M_{21}^\CR n_2n_e-S_1^\CR n_1n_e\quad,
&(35)\cr
\dot n_2&=\alpha_2^\CR n^\ion n_e+M_{12}^\CR n_1n_e-S_2^\CR n_2n_e\quad,
&(36)\cr
\cr
S_1^\CR&=S_1+\sum_{j=2}^sC_{1j}-{1\over Z_1n_e}
\sum_{j=3}^r\{n_eC_{j1}+A_{j1}\}Z_jr_j^\one\quad,&(37)\cr
S_2^\CR&=S_2+\sum_{j\ne 2}^sC_{2j}+{1\over n_e}A_{21}-{1\over Z_2n_e}
\sum_{j=3}^r\{n_eC_{j2}+A_{j2}\}Z_jr_j^\two\quad,&(38)\cr
M_{21}^\CR&=C_{21}+{1\over n_e}A_{21}+{1\over Z_2n_e}
\sum_{j=3}^r\{n_eC_{j1}+A_{j1}\}Z_jr_j^\two\quad,&(39)\cr
M_{12}^\CR&=C_{12}+{1\over Z_1n_e}
\sum_{j=3}^r\{n_eC_{j2}+A_{j2}\}Z_jr_j^\one\quad,&(40)\cr
\alpha_1^\CR&=\beta_1+\alpha_1n_e+\sum_{j=3}^r\{n_eC_{j1}+A_{j1}\}Z_jr_j^\zero
+\sum_{j>r}^s\{n_eC_{j1}+A_{j1}\}Z_j\quad,&(41)\cr
\alpha_2^\CR&=\beta_2+\alpha_2n_e+\sum_{j=3}^r\{n_eC_{j2}+A_{j2}\}Z_jr_j^\zero
+\sum_{j>r}^s\{n_eC_{j2}+A_{j2}\}Z_j&(42)\cr}$$
\\The general solution of the coupled system of equations (35),(36) can be
written as
$$\eqalignno{
n_j(t)&=n_j^\SS+n_j^\plus e^{-\lambda^\plus t}-
n_j^\minus e^{-\lambda^\minus t},\qquad j=1,2&(43)\cr
\cr
\lambda^\plusminus&={n_e\over2}\bigg[{S_1^\CR+S_2^\CR\pm
\sqrt{(S_1^\CR-S_2^\CR)^2+4M_{12}^\CR M_{21}^\CR}}\bigg]\quad,&(44)\cr
\cr
n_j^\SS&={K_j^\SS\over\lambda^\plus\lambda^\minus}\quad,&(45)\cr
\cr
n_j^\plusminus&={n_j(\teqzero)\lambda^{\plusminus2}
-K_j\lambda^\plusminus+K_j^\SS\over\lambda^\plusminus
(\lambda^\plus-\lambda^\minus)}\quad,&(46)\cr
\cr
K_1^\SS&=n_e^2n^\ion[\alpha_1^\CR S_2^\CR+\alpha_2^\CR M_{21}^\CR]\quad,
&(47)\cr
\cr
K_2^\SS&=n_e^2n^\ion[\alpha_2^\CR S_1^\CR+\alpha_1^\CR M_{12}^\CR]\quad,
&(48)\cr
\cr
K_1&=n_e[\alpha_1^\CR n^\ion+S_2^\CR n_1(\teqzero)
+M_{21}^\CR n_2(\teqzero)]\quad,&(49)\cr
\cr
K_2&=n_e[\alpha_2^\CR n^\ion+S_1^\CR n_2(\teqzero)
+M_{12}^\CR n_1(\teqzero)]\quad,&(50)\cr}$$
\\The steady state population densities as $t\rightarrow\infty$
$$\eqalignno{
n_1(t\rightarrow\infty)\equiv n_1^\SS&=
{\alpha_1^\CR S_2^\CR+\alpha_2^\CR M_{21}^\CR \over
S_1^\CR S_2^\CR -M_{12}^\CR M_{21}^\CR }n^\ion\quad,&(51)\cr
\cr
n_2(t\rightarrow\infty)\equiv n_2^\SS&=
{\alpha_2^\CR S_1^\CR+\alpha_1^\CR M_{12}^\CR \over
S_1^\CR S_2^\CR -M_{12}^\CR M_{21}^\CR }n^\ion\quad,&(52)\cr}$$
\bigskip
\\The final solution for excited states ($i>2$) of C IV :
\bigskip
$$n_i=Z_i[r_i^\zero n^\ion n_e+r_i^\one n_1(t)/Z_1+r_i^\two n_2(t)/Z_2],
\eqno(53)$$
\bigskip
\\Where $n_1(t)$ and $n_2(t)$ are taken from Eq.(43), to find $n_1(\teqzero)$
and $n_2(\teqzero)$ we first need to find the $n^\ion$ density by using the
ionization equilibrium calculations of Appendix B on the plasma before it
expands. Then by solving the steady state model Eq.(23) we can compute $n_1$
and $n_2$ for this initial state, then after the expansion we assume that these
densities are 'frozen', so we divide them by the expansion factor.
\vfill\eject %----------------------------------------------------------------
Once we have obtained this solution for C IV we may want to compare it to the
solution we would have obtained if we had used the Hydrogen-like model instead.
Rather than solving two more matrix equations we can use a shortcut
by setting Eq.(36) to zero ($\dot n_2=0 $) we find,
$$\eqalignno{
n_2&={\alpha_2^\CR\over S_2^\CR}n^\ion + {M_{12}^\CR\over S_2^\CR}n_1\quad,
&(54)\cr
\cr
\rho_2&=r_2^{*\zero}+r_2^{*\one}\rho_1\quad,&(55)\cr
\cr
r_2^{*\zero}&={\alpha_2^\CR\over S_2^\CR}{1\over n_eZ_2} ,
\qquad r_2^{*\one}={M_{12}^\CR\over S_2^\CR}{Z_1\over Z_2}\quad,&(56)\cr
\cr
\rho_i&=r_i^\zero + r_i^\one\rho_1
+ r_i^\two[ r_2^{*\zero} + r_2^{*\one}\rho_1]\quad,&(57)\cr
\cr
\rho_i&=r_i^{*\zero} + r_i^{*\one}\rho_1\quad,&(58)\cr
\cr
r_i^{*\zero}&=r_i^\zero + r_i^\two r_2^{*\zero}\quad,&(59)\cr
\cr
r_i^{*\one}&=r_i^\one + r_i^\two r_2^{*\one}\quad,&(60)\cr}$$
\\inserting Eq.(54) into Eq.(35) we find we can express $\dot n_1$
as a function of $\alpha^\CR$ and $S^\CR$:
$$\eqalignno{
\dot n_1&=\alpha_1^\CR n^\ion n_e + {M_{21}^\CR \alpha_2^\CR\over S_2^\CR}
n^\ion n_e + {M_{21}^\CR M_{12}^\CR\over S_2^\CR}n_1 n_e - S_1^\CR n_1 n_e
\quad,&(61)\cr
\dot n_1&=\alpha^\CR n^\ion n_e - S^\CR n_1 n_e\quad,&(62)\cr
\alpha^\CR&=\alpha_1^\CR + {M_{21}^\CR \alpha_2^\CR\over S_2^\CR}\quad,
&(63)\cr
S^\CR&=S_1^\CR-{M_{21}^\CR M_{12}^\CR\over S_2^\CR}\quad,&(64)\cr}$$
\\More simply put, the C IV model in which the excited states depended on
$n_1$ and $n_2$ reduces to a Hydrogen-like model in which excited states now
only depend on $n_1$. This can be expressed as: (for $i>1$)
$$n_i=Z_i\bigg(\bigg[r_i^\zero+{r_i^\two\alpha_2^\CR\over S_2^\CR n_e Z_2}
\bigg]n^\ion n_e +\bigg[r_i^\one+r_i^\two{M_{12}^\CR Z_1\over S_2^\CR Z_2}
\bigg]n_1(t)/Z_1\bigg),\eqno(65)$$
\bigskip
\\we can substitute Eq.(63) and (64) into Eq.(30) to find $n_1(t)$, and to
find $n_1(\teqzero)$ and $n^\ion$ we use the same method used for the
Hydrogen-like model.
\vfill\eject %-----------------------------------------------------------
\title{F) PLASMA LASER GAIN}
\bigskip
%\def\xb{\lambda}
%\def\t{\times}
The initial density of carbon atoms before expansion is taken
to be $1\times 10^{14}\ {\rm cm}^{-3}$ . Calculations will be
carried out for the population
densities of 32 levels of C IV for a grid of $n_e$ and $T_e$ values.
The population inversion is often measured in terms of $P$, where
$P$ is given by
\smallskip
$$P=n_k/\omega_k-n_i/\omega_i, \eqno(66)$$
\smallskip
\noindent where $n_i$ and $n_k$ are the atomic populations of the lower
and upper levels, and $\omega_i$ and $\omega_k$ are the
respective statistical weights. $P$ is sometime called the
`overpopulation density' (Oda and Furukane, 1983). $P$
is related to the fractional gain per unit distance, $\alpha$ , at the
centre of a Doppler-broadened line by the following expression
(cf. Willett, 1974)
\smallskip
$$\alpha =\left( {\ln 2\over \pi }
\right)^{1/2}\left[ {\omega_kA_{ki}\over
4\pi}\right] {P\lambda_o^2 \over \Delta \nu}, \eqno(67)$$
\smallskip
\\where $\lambda_o$ is the centre wavelength of the
transition, and $\Delta \nu$ is the linewidth. $\alpha$
describes the intensity of a plane wave at $\lambda_o$ according to
$I=I_oe^{\alpha\ell}$ where $\ell$ is the length over which gain occurs.
In astrophysical sources, $\ell$ is very large, thus even a very small positive
$\alpha$ can lead to an intense line. Unlike laboratory lasers which contain
mirrors, gain is achieved only by amplified spontaneous emission (ASE) i.e.
single pass lasers.
Recombination dominates in this decaying plasma, rapid cooling is
necessary to favour the over-population of upper levels compared to lower
levels. The upper levels are dominated by electron collisions, however below
a certain level radiative decay begins to take over. Hence a population
inversion can build-up between an upper level above this threshold and a
lower level below this threshold.
Upper level recombination occurs mainly via three-body collisions,
since radiative recombination occurs preferentially into the lower-lying
quantum states. In the rate equation, the three-body recombination term depends
quadratically on $n_e$ while the radiative recombination term depends only
linearly on $n_e$, this extra sensitivity to $n_e$ by the three-body term
requires higher electron densities to establish an over-population of the upper
levels. Yet if the electron density is too high, the collisional de-excitation
of the laser transition dominates.
The $T_e^{-2}$ dependence of the three-body recombination rate along with
the contradictory requirements of high temperature to initially strip the atom
of the required number of electrons leads to the conclusion that significant
gain occurs in a limited region of plasma parameter space.
\vfill\eject %--------------------------------------------------------------------------
\title{G) RESULTS AND COMPARISON WITH ASTRONOMICAL SPECTRA}
Population inversions in C IV were found to occur mostly in $n=6\rightarrow 5$
transitions: $6s\rightarrow 5p, 6p\rightarrow 5d, 6d\rightarrow 5p,
6f\rightarrow 5d, 6g\rightarrow 5f$ and $6h\rightarrow 5g$, figure 1 shows the
$6g\rightarrow 5f$ (4658\AA) gain contour plot ($\alpha\Delta\nu$), the gain
decreases very rapidly at high values of $n_e$, decreases less rapidly at high
values of $T_e$, and decreases slowly at high values of $T_e$ and low values
of $n_e$.
Wolf-Rayet stars are known to have an expanding envelope of hot ionized gases.
When the speed of expansion is low, the expansion will be closer to being
isothermal than adiabatic. However as the speed of expansion increases, the
expansion will become more and more adiabatic, and certain spectral lines can
be expected to display laser action. WC8 stars have relatively sharp lines;
since the widths of the lines in an expanding shell arise from the Doppler
effect, the speed of expansion of the plasma must be low and the degree of
laser action is also expected to be low. The lines become wider in WC7 stars,
indicating that the speed of ejection is greater than in WC8 stars;
correspondingly, the degree of laser action is also expected to be greater.
The lines become still wider in WC6 stars; we thus expect the degree of laser
action to further increase. The three Wolf-Rayet spectra in Figure 2
demonstrate the correlation between expansion velocity and laser gain in
the $6g\rightarrow 5f$ (4658\AA) transition.
Laser action has also been shown to occur in He I (Nasser,1987) and many other
ions, which leads us to wonder if the so-called red-shifted lines in quasars
are due to laser action instead ?
The low spectral resolution survey by Craig~{\sl et~al.}~(1989) shows
that many quasars seem to have one or two dominant lines, at various
locations in the spectrum. Many of these lines are broad possibly indicating
the blending of adjacent lines or high velocities. Some of the lines have
P-Cygni profiles typical of high velocity outflows. All of the spectra have
absorption lines not belonging to cold interstellar gases which indicates that
the objects are surrounded by a thick envelope of colder gases.
These last two observations indicate that laser action could be responsible
for the unusual line intensities, which requires rapid cooling by
adiabatic expansion and/or gas contact. However, quasar surveys suffer from the
selection effect: Spectra which show similar spectral emission features as
quasars but come from identifiable galactic objects are ignored or not
published, such as Wolf-Rayets and central star of planetary nebula.
High spatial resolution 6cm radio surveys by the VLA (A\&A) indicate that many
quasars occur between pairs of highly collimated jets. Morphologically the jets
look similar to the bipolar high-velocity-molecular-gas (HVMG) outflows observed
by Lada~{\sl et~al.}~(1990) in proto-stars veiled by molecular dark clouds.
Radio VLBI studies of quasar jets indicate possible super-luminal velocities
in portions of the jet, however if we drop the assumption that these objects are
at cosmological distances, the expansion velocity becomes reasonable for example
if they were objects within our galaxy.
\vfill\eject
\title{H) CONCLUSION}
\bigskip
In this thesis we have explained the difference between LTE and non-LTE
plasmas and why we must use the rate coefficients for basic physical plasma
processes, instead of the simpler statistical model. We used these rates
in a collisional radiative model of a recombining C IV plasma expanding
adiabatically. From the solution of the level population we show how gain per
unit length is computed, and why large gains occur only in limited regions of
parameter space ($n_e, T_e$). We then compared our results to stellar spectra
of Wolf-Rayet stars, and Quasars. Closer to earth, many experimental research
labs have succeeded in producing laser action in decaying plasmas, and the
future looks 'bright' for high power short wavelength UV or X-Ray lasers.
(Matthews {\sl et~al.}, 1988)
\vfill
\centerline{Figure.1 : $6g\rightarrow 5f$ (4658\AA) gain contour plot of C IV.}
\eject %----------------------------------------------------------------
\vbox{ }
\vfill
\centerline{Figure.2 : Laser action in Wolf-Rayet stars at 4658\AA}
\eject %----------------------------------------------------------------
\\{\largetype APPENDIX A : ATOMIC DATA}
\bigskip
We describe in more detail how we obtain the rate coefficients.
Using cross sections derived from quantum mechanical models
we integrate these transition rates over the appropriate electron distribution
obtained from statistical mechanics.
\medskip
{\it (a) Energy levels.}
\smallskip
All of the levels having a principal quantum number
$n\le 6$ are treated as individual levels. Levels 7,8 and 9 remain discrete
only if $\ell=0$, otherwise they are grouped together to form a single level.
For the levels with $n\ge 10$, all of the S,P,D,$\cdots$ levels are grouped
together in one level. The upper limit of the levels, $r$ and $s$ in CR model
calculation are taken as $n_r=$Variable and $n_s=32$, respectively. Therefore
we have 32 distinct levels in the main system, and the total number of
levels whose population densities is calculated is 30.
\smallskip
{\it (b) Radiative Transitions.}
The rates of spontaneous radiative transitions are determined by Einstein
coefficients $A_{n\ell,n'\ell'}$. Einstein coefficients were calculated
using the formula (Menzel and Pekeris, 1935)
$$A_{n\ell,n'\ell'} = {8\pi ^2e^2\omega_{n'\ell'}\over mc^3
\omega_{n\ell}}\nu_{n\ell,n'\ell'}^2f_{n'\ell',n\ell},
\eqno\hbox{(A-1)}$$
\\where $\omega_{n'\ell'}$ and $\omega_{n\ell}$
are the statistical weights of the level $n'\ell'$ and $n\ell$,
respectively, $\nu_{n\ell,n'\ell'}$ the frequency of the
photon emitted as result of the transition and $f_{n'\ell',n\ell}$
is the absorption oscillator strength for the transition.
When the angular momentum states $\ell$ or $\ell'$
are grouped together then,
\smallskip
$$A_{n{\ell_i}{\ldots}{\ell_k},n'{\ell_i}'{\ldots}{\ell_k}'} =
{8\pi ^2e^2\nu^2\sum_{\ell_i}^{\ell_k} \sum_{\ell_i'}^{\ell_k'}
\omega_{n'\ell'}f_{n'\ell',n\ell}\over mc^3\sum_{\ell_i}^{\ell_k}
\omega_{n\ell}}, \eqno\hbox{(A-2)}$$
\smallskip
For transitions between unresolved levels and between resolved and unresolved
levels,hydrogenic formula was used to calculate the oscillator strength
( Kramers, 1923).
$$f_{n',n}={2^2\over 3\sqrt{3} \pi \omega_n'}\left({1\over n^{\prime 2}}
-{1\over n^2}\right)^{-3}\left({1\over n^3n^{\prime 3}}\right)g_I(n',n)
\eqno\hbox{(A-3)}$$
\\where $\omega_n'$ is the statistical weight of the level
$n'$, and $g_I(n',n)$ is the bound--bound Kramers--Gaunt factor.
Analytic form for $g_I(n',n)$ has been given by Menzel and Pekeris
(1935),
$$g_I(n,n)={\pi\sqrt{3}\lbrack (n-n')/(n+n')\rbrack^{2n+2n'}nn'\over
(n-n')}\left|\Delta (n,n')\right| \eqno\hbox{(A-4)}$$
where
$$\Delta (n,n')=\left\lbrack _2F_1\left(-n+1,-n;1;{-4nn'\over (n-n')^2}
\right)\right\rbrack^2-\left\lbrack\,_2F_1\left(-n'+1,-n;1;{-4nn'\over
(n-n')^2}\right)\right\rbrack^2 \eqno\hbox{(A-5)}$$
\\and the $_2F_1$ are hypergeometric functions.
\smallskip
{\it (c) Collisional excitation and de--excitation rate coefficients.}
The rate coefficient for collisional excitation of the $n'\ell'
\rightarrow n\ell$ transition by electron impact is given by
$$C_{n'\ell',n\ell}=\int_v\sigma_{n'\ell',n\ell}(v)\,v\,f(v)\, dv
\eqno\hbox{(A-6)}$$
\\where $v$ is the velocity and $f(v)$ the velocity
distribution of the incident electrons. Using the Maxwellian velocity
distribution
$${dn_e\over n_e}=f(v)\,dv={4v^2\over \sqrt{\pi}(2kT/m)^{3/2}}e^{-{mv^2
\over 2kT}}\,dv \eqno\hbox{(A-7)}$$
\\and expressing Equations (2-14) and (2-15) in terms of the
kinetic energy $E$ of the incident electron, we obtain
$$C_{n'\ell',n\ell}={1\over \sqrt{\pi m}}\left(2\over kT\right)^{3/2}
\int_E \sigma_{n'\ell',n\ell}(E)\,E\,e^{-{E\over kT}}\,dE
\eqno\hbox{(A-8)}$$
\\with threshold energy units $U=E/E_{n'\ell',n\ell}$,
Equation (98) becomes
\smallskip
$$C_{n'\ell',n\ell}=\sqrt{{8\over \pi mk^3}}{E^2_{n'\ell',n\ell}\over
T^{3/2}}\int_1^{\infty}\sigma_{n'\ell',n\ell}(U)\,e^{-{E_{n'\ell',n\ell}
\over kT}U}\,U\,dU \eqno\hbox{(A-9)}$$
\\where $E_{n'\ell',n\ell}$ is the threshold
energy for the excitation of the $n'\ell'\rightarrow n\ell$
transition in Rydberg.
\smallskip
A very large number of excitation cross sections is required in this
work. Unfortunately, the available data are but a very small fraction
of the required cross sections. Most of the available data from
theoretical calculations and work
experimental are on the excitation from
the ground state and low lying excited states. For all of the cross
sections for which data exist the best experimental and theoretical data
were fitted to semi--empirical formulas.
\smallskip
For allowed transitions between low-lying levels, the modified form
of the Drawin cross section formula (1963, 1964, 1967) proposed by Millette and
Varshni (1980) was used:
$$\sigma_{n'\ell',n\ell}(U)=4\pi a_o^2\alpha
{f_{n'\ell',n\ell}\over E_{n'\ell',n\ell}^2}{U-\phi
\over U^2}\ln{(1.25\beta U)}, \eqno\hbox{(A-10)}$$
\\ $f_{n'\ell',n\ell}$ is the absorption
oscillator strength, $\alpha$,
$\beta$ and $\phi$ are the fitting parameters which depend on the
transition.
$E$ is the kinetic energy of the incident electron,
$U=E/E_{n'\ell',n\ell}$ is energy in threshold units,
$E_{n'\ell',n\ell}$ is the threshold
energy for the excitation of the $n'\ell'\rightarrow n\ell$
transition in Rydberg.
For transitions with $\Delta n>3$, we used the effective Gaunt factor
or $\bar g$ approximation developed by Van Regemorter (1962).
For optically forbidden transitions without change in spin, we
used the modified form of Drawin, proposed by Millette and
Varshni (1980):
$$\sigma_{n'\ell',n\ell}(U)=4\pi a_o^2({n'/n})^3
{\alpha \over E_{n'\ell',n\ell}^2}{U-\phi
\over U^2}, \eqno\hbox{(A-11)}$$
\\where the symbols have their usual meaning.
For other optically forbidden transitions without change in spin
between levels with $n\le 6$ cross sections were
calculated using the symmetrized binary
encounter theory (Burgess, 1964a,b). The basic idea of the binary--encounter
collision theory is that excitation of atoms by electrons is described
approximately as a collision between two free electrons. In the
derivation of cross section formulas it is further assumed that the
atomic electrons initially have an isotropic velocity direction
distribution with one constant magnitude of the velocity. This theory
has been developed by Thomson (1912) and Gryzinski (1959).
Mathematical corrections and discussions of the theory are given by
Stabler (1964) and Vriens (1964a,b).
Burgess (1964a,b) introduced a new model with a symmetrical treatment of
the two interacting electrons; the binary encounter may be treated as
a quantum mechanical collision between identical particles with
proper symmetrization and treatment of interference between direct
and exchange scattering. The initial and final atomic states are
treated classically in terms of an orbiting electron with definite
initial and final kinetic energy, the change in kinetic energy being
related to the angle of scattering. Burgess also allowed for the
acceleration of the incident electron in the same way as Thomas
(1927), the kinetic energy of the incident electron before the binary
encounter being set equal to $E_1+E_2+U$, where $E_1$ is the initial
kinetic energy of the incident electron, $E_2$ is the initial kinetic
energy of the atomic electron, and $U$ is the ionization energy. In
this way, Burgess showed that cross sections for direct and exchange
excitation could be calculated, together with the effect of interference
between direct and exchange scattering. Vriens (1966) obtained simpler
formulas for the cross section than those Burgess has given, using
the same transformations in Burgess's collision model,
\def\var{\varepsilon}
$$\sigma(\var_1,\var_2)= (\sigma_d+\sigma_{ex}+\sigma_{int})\,F
\eqno\hbox{(A-12)}$$
\\where $\sigma_d$ is the cross section for direct excitation,
$$\sigma_d={\pi e^4\over E_1+E_2+U}\left\lbrack {1\over \var_1}-{1\over
\var_2}+{2E_2\over 3}\,\left({1\over \var^2_1}-{1\over \var^2_2}\right)
\right\rbrack \eqno\hbox{(A-13)}$$
\\ $\sigma_{ex}$ is the cross section for exchange excitation,
$$\eqalignno{\sigma_{ex}&={\pi e^4\over E_1+E_2+U}\,\left\lbrack
{1\over E_1+U-\var_2}-{1\over E_1+U-\var_1}\right.\cr
&\left.+{2E_2\over 3}\,\left( {1\over (E_1+U-\var_2)^2}-{1\over (E_1
+U-\var_1)^2}\right)\right\rbrack &\hbox{(A-14)}\cr}$$
\\ $\sigma_{int}$ gives the influence of interference
between direct and exchange scattering:
$$\sigma_{int}=-{\pi e^4\over E_1+E_2-U}\,{1\over E_1+U}\,\ln{
\left\lbrack{(E_1+U-\var_1)\var_2\over \var_1(E_1+U-\var_2)}
\right\rbrack} \eqno\hbox{(A-15)}$$
\\ $F$ is the focussing factor, which was introduced by Burgess
(1964a,b) to account for electron--positive ion collision, and is given
by
$$F=1+(ze^4/E_1{\bar r}) \eqno\hbox{(A-16)}$$
\\ where $\bar r$ is the mean radius and $z$ is the charge
number of the ion.
\smallskip
Equations (A-13)-(A-15) are valid for $E_1\geq\var_2$, while $\var_2$
should be replaced by $E_1$ for $\var_1\leq E_1\leq \var_2$.
The appropriate classical energy band specified by $\var_1$ and $\var_2$
is not well defined; in this calculations $\var_1$ has been taken as
the excitation energy of the transition $n'\ell'\to n\ell$ and
$\var_2$ as the excitation energy to the next higher
level with the same $L$ or $S$ (Cacciatore and Capitelli, 1976).
\medskip
The rate coefficients for de--excitation of a bound electron from
state $n\ell$ to $n'\ell'$ by electron impact are calculated from
the principle of detailed balance (Drawin, 1963)
$$Z_{n\ell}C_{n\ell,n'\ell'}=Z_{n'\ell'}C_{n'\ell',n\ell} \eqno\hbox{(A-17)}$$
\\where $Z_{n'\ell'}$ and $Z_{n\ell}$ are the partition functions
of the states $n'\ell'$ and $n\ell$, respectively. (Eq.11)
\medskip
{\it (d) Ionization and three-body recombination rate coefficients.}
The rate coefficient of the ionization by electron impact is obtained
by integrating the cross section over the free electron velocity
distribution, $f(v)$:
$$S_{n\ell}=\int_v\,\sigma_{n\ell}(v)\,v\,f(v)\,dv.\eqno\hbox{(A-18)}$$
Using Equation (97), we obtain
$$S_{n\ell}=2\sqrt{{2\over \pi m}}\,{I_{n\ell}^2\over (kT_e)^{3/2}}
\int_1^\infty\,\sigma_{n\ell}(U)Ue^{-{I_{n\ell}\over
kT_e}U}\,dU, \eqno\hbox{(A-19)}$$
\noindent where $I_{n\ell}$ is the ionization potential.
\\Experimental cross section data was fitted to the
semi-empirical formula (Drawin, 1967):
\smallskip
$$\sigma_{n\ell}(U)=2.66\pi a_o^2\alpha {\xi\over E_{n\ell}^2}{U-1\over
U^2}\ln{(1.25\beta U)}, \eqno\hbox{(A-20)}$$
The numerical value of $\xi$ for
ionization from the ground state is given by Drawin (1967), for $n>1$,
$\xi =1$, $\alpha$ and $\beta$ are fitting parameters,
and the other symbols have their usual meaning.
The cross sections for other levels were calculated using this formula
with $\alpha=1$, and $\beta$ determined from the following expression
(Drawin, 1967)
$$\beta=1+{z_{eff}-1\over z_{eff}+2} \eqno\hbox{(A-21)}$$
\\ $z_{eff}$ denotes the effective charge number of the nucleus
acting on the electron.
The rate for three--body recombination of a free electron into state
$n\ell$ of an atom or ion is obtained from the principle of detailed
balance.
$$\alpha_{n\ell}=Z_{n\ell}S_{n\ell} \eqno\hbox{(A-22)}$$
\\where $Z_{n\ell}$ is the partition function of the state $n\ell$. (Eq.11)
\medskip
{\it (e) Radiative recombination.}
\smallskip
The radiative recombination rate coefficient is obtained from the
photoionization cross section by applying the principle of detailed
balancing. The rate coefficient is then given by
$$\beta_{n\ell}={1\over c^2}\sqrt{{2\over \pi}}(mkT_e)^{-3/2}{\omega_{
n\ell}\over \omega_i}\,e^{I_{n\ell}/kT_e}\int_{I_{n\ell}}^\infty\,
(h\nu)^2\,\sigma_{n\ell}(\nu)\,e^{-{h\nu\over kT_e}}\,d(h\nu)
\eqno\hbox{(A-23)}$$
\noindent where $\omega_i$ is the statistical weight of the parent ion,
and the other symbols have their usual meaning. Using threshold energy
units, Equation (125) becomes
$$\beta_{n\ell}={1\over c^2}\sqrt{{2\over \pi}}(mkT_e)^{-3/2}{\omega_{
n\ell}\over \omega_i}\,e^{I_{n\ell}/kT_e}I^3_{n\ell}\int_1^\infty\,
U^2\,\sigma_{n\ell}(U)\,e^{-{I_{n\ell}\over kT_e}U}\,d(U)
\eqno\hbox{(A-24)}$$
\medskip
Cross sections were fitted to a semi-empirical formula suggested by Millette
and Varshni (1980),
\smallskip
$$\sigma_{n\ell}(U)={C\over U^p}\left[ 1+{a_1\over U}+{a_2\over U^2}+\ldots +
{a_m\over U^m}\right] , \eqno\hbox{(A-25)}$$
\noindent where $C$, $p$ and $a_k,\ k=1,\ldots,m$ are fitting parameters,
and $U$ is the energy of the incident photon in threshold units.
The hydrogenic approximation was adopted to calculate the photoionization
cross sections for other levels; the cross section being given by
$$\sigma_{n\ell}(\nu)=2.815\times 10^{29}\,{g_{II}(n,\nu)\over n^5\nu^3}
\eqno\hbox{(A-26)}$$
\noindent where $g_{II}(n,\nu)$ is the Gaunt-factor, it can be
approximated by the expression (Seaton, 1959)
$$g_{II}(n,\nu)=1+{0.1728(U_n-1)\over n^{2/3}(U_n+1)^{2/3}}-{0.0496
(U^2_n+{4\over 3}U_n+1)\over (U_n+1)^{4/3}}+\ldots\eqno\hbox{(A-27)}$$
\noindent where $U_n$ is the threshold energy of the state $n\ell$.
\vfill\eject %----------------------------------------------------------------
\title{APPENDIX B : IONIZATION EQUILIBRIUM}
\medskip
The relative concentrations of various stages of ionization of an element
in a monatomic non-LTE plasma under statistical equilibrium are calculated with
the model of House (1964). Since the plasma is not in LTE, the individual
physical processes contributing to the ionization equilibrium must be
considered. House (1964) calculated the ionization equilibrium of elements
between hydrogen and iron, over a wide range of temperatures, including the
processes of collisional ionization, radiative and collisional recombination.
The model is based on the following assumptions: (1) Each stage of ionization
of element $X$ consists only of a ground state and a continuum. (2) The
monatomic plasma is optically thin. Furthermore, collisional excitation
followed by auto-ionization and di-electronic recombination processes are
neglected, which under certain conditions, can restrict the applicability of
the model.
Let $S^\m$ represent the total rate coefficient for ionization from
$m$-times ionized atom of element $X$ to $m+1$ , $\beta^\m$ and $\alpha^\m$
the radiative and three body recombination rate coefficients from ion
$m+1$ to $m$. Then under statistical equilibrium, we have
$$ -n_eN^\m S^\m + \{\beta^\m + \alpha^\m n_e\}n_eN^\mplus=0\quad,
\eqno\hbox{(B-1)}$$
\\$N^\m$ and $N^\mplus$ are the density of ions $m$ and $m+1$,
respectively.Eq.(B-1) is a simplified version of Eq.(8) without the terms
$C_{ij}$ and $A_{ij}$, only the $S_1,\beta_1$ and $\alpha_1$ terms remain.
Equilibrium is equivalent to the steady state solution $dn_1/dt=0$,
$N^\m\equiv n_1$ and $N^\mplus\equiv n^\ion$.
The collisional ionization rate coefficient is calculated with the following
simple approximate formula (Allen, 1961):
$$S^\m=1.15\times 10^{-8}f_m{\zeta_m\over I_m^2}\sqrt{T_e}e^{-I_m/kT_e}
\quad,\eqno\hbox{(B-2)}$$
\\$\zeta_m$ is the number of electrons in the outer shell of ion $m$,
$I_m$ is the ionization potential of ion $m$ in eV,
$$f_m=3.1-{1.2\over z_m}-{0.9\over z_m^2}\quad,\eqno\hbox{(B-3)}$$
\\$z_m$ is the core charge of ion $m$, and all other symbols have their usual
meaning. The three-body recombination rate coefficient is obtained from the
condition of detailed balancing
$$\alpha^\m={h^3\over(2\pi m_ekT_e)^{3/2}}{U_m\over 2U_{m+1}}e^{I_m/kT_e}
S^\m\quad,\eqno\hbox{(B-4)}$$
\\$U_m$ is the partition function of ion $m$, given approximately as
$$U_m=\sum_{i=1}^{\infty}\omega_i^\m e^{-E_{1i}^\m/kT_e}
\approx\omega_1^\m\quad,\eqno\hbox{(B-5)}$$
\\The radiative recombination rate coefficient has been calculated using the
well known formula of Elwert (1952) given by
$$\beta^\m=5.16\times 10^{-14}f_1\bigg({I_H\over kT_e}\bigg)^{1/2}{I_m\over I_H}
nG_1(I_m/kT_e)g\quad,\eqno\hbox{(B-6)}$$
\\where
$$ G_1(x)=xe^{x}E_1(x)\quad,\eqno\hbox{(B-7)}$$
\\$n$ is the principle quantum number of the ground state of the lower stage of
ionization, $I_m$ is the ionization potential of ion of charge $+m$ in eV,
$I_H$ is the ionization potential of hydrogen, $E_1(x)$ is the first
exponential integral, $f_1=0.8,g=3$. The value of $f_1$ and $g$ come from
Allen (1961). The fraction of atoms of element $X$ that have been ionized
$m$ times is given by
$$ N^\m\bigg/\sum_j N^{+j},\eqno\hbox{(B-8)} $$
{\bigtype
\\TABLE 2. Parameters used to calculate the ionization equilibrium }
\medskip
\halign to \hsize{\strut#&\vrule#&
\quad#\ \ \hfil&\vrule#&
\quad#\quad\hfil&\vrule#&
\quad#\quad\hfil&\vrule#&
\quad#\quad\hfil&\vrule#&
\quad#\quad\hfil&\vrule#&
\quad#\quad\hfil&\vrule#&
\quad#\quad\hfil&\vrule#&
\quad#\quad\hfil&\vrule#\cr
\noalign{\hrule}
&&\multispan{15}&\cr
&&\multispan{15} \hfil\hbox{\largetype Ionization stages of carbon}\hfil&\cr
&&\multispan{15}&\cr
\noalign{\hrule}
&& &&C\ I&&C\ II&&C\ III&&C\ IV&&C\ V&&C\ VI&&C\ VII&\cr
\noalign{\hrule}
&&$I_m$(eV)&&11.260&&24.383&&47.887&&64.492&&392.077&&489.981&&---&\cr
&&$z_m$&&1&&2&&3&&4&&5&&6&&---&\cr
&&$\zeta_m$&&4&&3&&2&&1&&2&&1&&---&\cr
&&$n$&&2&&2&&2&&2&&1&&1&&---&\cr
&&$U_m$&&9&&6&&1&&2&&1&&2&&1&\cr
\noalign{\hrule} }
\vfill\eject %----------------------------------------------------------------
\title{APPENDIX C : COLLISIONAL-RADIATIVE PROGRAM}
\medskip
We modified the FORTRAN4 program originally written by Nasser (1987) so that
it takes advantage of the cleaner syntax of FORTRAN77 by using indentation
and replacing confusing {\tt GOTO}s with {\tt IF..THEN..ELSE} statements.
The program was adapted to Carbon (C IV or ${\rm C}^{+3}$) by replacing the
Helium oscillator strengths by carbon data from Millette (1980) as well as
semi-empirical fit coefficients for collisional excitation and photo-
recombination. This Ion-specific data was moved to a separate text file to
reduce clutter in the main program and make it more general, so it can read
other ions such as N~V, O~VI and He~I.
The subprograms were re-written to take most of their parameters via the
parameter list rather than the more anonymous {\tt COMMON BLOCK}s, this
resulted in more self-contained subroutines reducing the possibility of
error from uninitialized or re-used variables.
\medskip
\\We replaced many functions in the program by their equivalent subroutines
found in Press {\sl et al.}
(1992), which are widely used and well verified numerical recipes in FORTRAN:
\medskip
\item{-} {\tt DE1X(X)}, was replaced by {\tt EXPINT(1,X)}.
\item{-} {\tt DGAMMA(X)} was replaced by {\tt DEXP(GAMMLN(X))}
\item{-} {\tt DHANKL()} and {\tt BKL()} were both replaced by
{\tt BESSIK()} which uses {\tt BESCHB} and
{\tt CHEBEV}.
\item{-} {\tt LEQT2F}($\ldots$) was replaced by the iterative matrix inversion
scheme eq. (2.5.11) in Press~{\sl et~al.} (1992).
\\The matrix inversion subroutine now uses a high speed iterative scheme whose
performance improves for each subsequent electron density scan. Separate
testing of the inversion subroutine on a matrix filled with random numbers
generated by {\tt RAN2} of Press {\sl et al.}(1992) confirms that it always
converges, even for the most ill-conditioned matrix. General improvements to
optimize the speed of the program were performed such as computation of
oscillator strengths and Einstein radiative rate coefficient are now only
performed once at the beginning of the program. The benefit of the increased
speed was an 10-fold increase in sampling resolution of temperature and
density scans of the parameter space (now computed on a regularly spaced
logarithmic grid). The density sweep has been moved into the {\tt R0COF()}
subprogram.
The {\tt R0COF()} subprogram which initializes the matrices and computes the
population coefficients was verified by performing from basic principles the
complete derivation of the quasi-steady state collisional-radiative model for
one, two and three independent ground levels. {\tt R0COF()} can now switch
between collisional-radiative models based on 1,2 or 3 independent lower
levels by setting {\tt NG} to 1,2 or 3 Number of Ground levels.
The House (1964) ionization equilibrium was improved by including all
ionization stages with a relative contribution of greater than $10^{-40}$.
To prevent overflow and increase floating point precision, higher ionization
stages are included as a function of increased temperature. The improved
subroutine was tested separately and a trial plot for Carbon as a function of
$T_e$ was found to be stable for all $T_e$ and the shape of the plots were
found to be very close to Figure 1.5 in Millette (1980). The computation of
various rate coefficients specific to the House (1964) ionization equilibrium
subroutine were made electron density independent. They are now only computed
once at the beginning of the temperature sweep. The {\tt RATIO} subroutine
was also optimized for execution speed.
Many of the subroutines were shortened by removing intermediate variables,
and {\tt RETURN} statements were all but eliminated so that conditional
clauses would dominate for a clearer self-evident case-by-case analysis.
Program execution control now returns to the main program only at the end
of every subroutine. The layout of the program itself was shuffled in order
to group the subroutines nearer to their execution points and subprograms
required by other subprograms are declared first to simplify preliminary
read-through of the source code by others.
Program diagnostic controls were added to the ion data file, each printout
option can be easily be changed with a text-editor without the need to
re-compile the main program, allowing a variable level of numerical detail
at run-time.
The main fortran77 program was compiled by {\tt f2c} a fortran-to-C converter
in the public domain which is easily obtainable for most other platforms.
Then we used SAS C V6.5, an optimizing C compiler for amiga computers.
On an A4000 Amiga with a 68040 CPU at 25MHz, the inversion of a 32 X 32 matrix
takes less than one second. Some of the plots were produced by {\tt GnuPlot}
a multi-platform public domain plotting program, with terminal type set
to LaTEX in order to include equations in the plot borders.
\vfill\eject
\title{REFERENCES}
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\def\AA{{\sl Astron.\ Astrophys. }}
\def\ARAA{{\sl Ann.\ Rev.\ Astron.\ Astrophys. }}
\def\AO{{\sl Appl.\ Optics }}
\def\AJ{{\sl Astron.\ J. }}
\def\ApJ{{\sl Astrophys.\ J. }}
\def\ApJS{{\sl Astrophys.\ J.\ Suppl. }}
\def\ApJL{{\sl Astrophys.\ J.\ (Letters) }}
\def\APL{{\sl Appl.\ Phys.\ Letters }}
\def\ASS{{\sl Astrophys.\ Space Sci. }}
\def\BAAS{{\sl Bull.\ Am.\ Astron.\ Soc. }}
\def\BAPS{{\sl Bull.\ Am.\ Phys.\ Soc. }}
\def\CJC{{\sl Can.\ J.\ Chem. }}
\def\CJP{{\sl Can.\ J.\ Phys. }}
\def\CPC{{\sl Computer Phys.\ Comm. }}
\def\CRC{{\sl CRC }}
\def\CRASP{{\sl C.\ R.\ Acad.\ Sci.\ Paris }}
\def\JAP{{\sl J.\ Appl.\ Phys. }}
\def\JCP{{\sl J.\ Chem.\ Phys. }}
\def\JP{{\sl J.\ Phys. }} % J. Phys. B : Atomic and Molecular Physics
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\def\JQS{{\sl J.\ Quant.\ Spectrosc.\ Radiat.\ Transfer. }}
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\def\JRASC{{\sl J.\ Roy.\ Astron.\ Soc.\ Canada }}
\def\LPB{{\sl Laser and Particle Beams }}
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\def\MSRS{{\sl M\'em.\ Soc.\ Roy.\ Sci.\ Li\`ege }}
\def\NP{{\sl Nucl.\ Phys. }}
\def\MRAS{{\sl Mem.\ Roy.\ Astron.\ Soc }}
\def\NAT{{\sl Nature }}
\def\NIM{{\sl Nucl.\ Instr.\ and Methods }}
\def\OC{{\sl Opt.\ Comm. }}
\def\OS{{\sl Optics and Spectrosc. }}
\def\OL{{\sl Opt.\ Letters }}
\def\PA{{\sl Physica }}
\def\PC{{\sl Physics Canada }}
\def\PF{{\sl Phys. Fluids }}
\def\PL{{\sl Phys.\ Letters }}
\def\PLA{{\sl Phys.\ Letters\ A }}
\def\PPS{{\sl Proc.\ Phys.\ Soc. }}
\def\PR{{\sl Phys.\ Rev. }}
\def\PRA{{\sl Phys.\ Rev.\ A }}
\def\PRE{{\sl Phys.\ Rev.\ E }}
\def\PRS{{\sl Proc.\ Roy.\ Soc. }}
\def\PRL{{\sl Phys.\ Rev.\ Letters }}
\def\PSS{{\sl Planetary Space Sci. }}
\def\PZ{{\sl Phys.\ Z. }}
\def\PubASP{{\sl Publ.\ Astron.\ Soc.\ Pacific }}
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\def\ZT{{\sl Zh.\ Tekh.\ Fiz. }} % Zhurnal Tekhnicheskoi Fiziki QC 1. A
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\vfill\eject
\end