There are two complementary aspects to plasma behavior, the dynamical aspect and the atomic aspect. The plasma laser star theory is based mostly on the atomic properties and only depends on the dynamical aspects to the extent that the plasma is rapidly cooled by expansion or contact with colder gas.

  1. THE DYNAMICAL ASPECT: The collective behavior of the plasma under the influence of internal and external forces. This is usually the most talked about aspect of plasma. Historically the name 'plasma' was adopted from the field of biology because of the inherent emergent behavior such as self-organization into vortices compartmentalization and other instabilities that mimic living organisms. Lerner (1992) gives a good introduction to the subject, but the textbook by Perrat (1992) or Alfven (1965) cover the physics more comprehensively.
  2. THE ATOMIC ASPECT: The plasma is composed of dust, molecules, atoms, ions and electrons, an in-homogeneous mixture or 'protoplasm' if you like. The quantum mechanical behavior of the distribution among the various quantum states of these particles is computed using statistical thermodynamics. It is mostly these properties on which the laser star theory is based.
In the dynamical aspect, Lerner, Perrat and Alfven are all basically saying that many of the fundamental properties of a plasma are overlooked because the plasma particles are ignored in favour of a less general fluid dynamical approach which neglects many emergent plasma properties inherent in the full Maxwell's equations of electromagnetism. The full particle physics approach is a fundamentally more powerful framework for studying plasmas.

Since the main ingredient of the laser star theory is the rapid cooling of the stellar plasma, we focus on the quantum states of various component particles. We discuss in detail the cumulative statistical results of collision processes to re-distribute the internal quantum states of ions into various non-equilibrium configurations. As opposed to collective electromagnetic behavior these microscopic processes depend on the fundamental quantum behavior of each atom or ion.

The theoretical study of decaying plasma has been crucial in the development of short wavelength and X-Ray lasers (Matthews 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,

  1. Description of the plasma model.
  2. Various constraints the plasma parameters must obey.
  3. Important physical processes occurring in the plasma.
  4. Theoretical model used in our plasma simulation.
  5. How gain is computed.
  6. Results and astronomical evidence.
  7. Conclusion.


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), ... . Element A is characterized by a density nA cm^-3 and a temperature TA Kelvin, and the free electrons by a density ne and a temperature Te. All particles of the plasma are assumed to be at the same temperature T=Te=TA. Under the plasma conditions considered in this work, this is a reasonable approximation, especially since TA is used only to calculate the ionization equilibrium of element A (See appendix B)



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 ne of
                   ne << 1.2 X 10^15 T^(3/2) cm^-3                       (1)
In all cases studied here, we are well below this upper limit of validity. 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 ne for a lithium-like ion C IV is:

                   ne > 10^10 cm^-3                                        (2)
Since we consider plasmas with ne from 10^12 to 10^15 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.


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.


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.


Our description of quantum transitions uses of the following diagram notation:

 Collisional     Collisional    Radiative
 de-excitation   excitation     de-excitation

 ----+----       ----o----      ----+----     Upper quantum level
     |               |              $
     |               |              $
     |               |              $
 ----o----       ----+----      ----o----     Lower quantum level

Symbol  Meaning
------  ---------------------------------------------------------------------
  o     final state of the electron
  +     initial state of the electron
  |     collisional transition involving a collision with a free electron
  $     radiative transition involving a photon, hence the wigly line

  e     electron
 hv     photon
 Nix    X-times ionized atom
 N(x+1) X+1 times ionized atom
  i     quantum label of a state
  j     quantum label of a state
 n+     ion density
 ne     electron density
  T     electron temperature
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)

  + ==o===o==
      |   |        Nix + e  --->  N(x+1) + e + e                           (3)
  2 --|---+--
      |             The rate coefficient of this process is denoted by Si(T).
  1 --+------       The number of such processes occurring per unit volume
                    per unit time is given by :  ni ne Si(T)

     a1  a2      THREE-BODY RECOMBINATION (inverse of collisional ionization)
  + ==+===+==
      |   |        N(x+1) + e + e ---> Nix + e                             (4)
  2 --|---o--
      |              Rate coefficient: ai(T),
  1 --o------        number of processes: ne^2 n+ ai(T).

     B1  B2      RADIATIVE RECOMBINATION (dominates over three body at low ne)
  + ==+===+==
      $   $       N(x+1) + e ---> Nix + hv                                 (5)
  2 --$---o--
      $             Rate coefficient Bi(T),
  1 --o------       number of processes:  ne n+ Bi(T).

  + =========
                   Nix + e --->  Njx + e                                   (6)
  2 --o---+--
      |   |          Rate coefficient Cij(T),
  1 --+---o--        number of processes: ni ne Cij(T).

  + =========
                   Njx  --->  Nix + hv                                     (7)
  j ----+----
        $            Einstein probability coefficient: Aij;
  i ----o----        number of processes: nj Aji.

The plasma is also assumed to be optically thin such that all radiation emitted within the plasma escapes without being absorbed. (Thus neglecting photo- excitation and photoionization) see appendix A for computation of rate coefficients


The collisional radiative model is a differential equation of the relative contributions to the quantum levels from various microscopic quantum processes. It takes into account the rate for population and depopulation of each quantum level by radiation or collision from the others levels.

The main thermodynamic process leading to population inversion is the rapid cooling of the free-electron gas, either by rapid adiabatic expansion and/or by collisions with a cold neutral gas such as Hydrogen (atomic or molecular) and Helium.

The three body recombination rate varies with the square of the electron density, it is thus more important at higher plasma densities; while on the other hand the radiative recombination rate varies only linearly with electron density. This and the balance between other microscopic plasma processes produce strong population inversions in very narrow ranges of electron density and electron temperature.

The following two diagrams represents the various microscopic processes contributing to the population or de-poppulation of level 4, the first diagram contains contributions into level 4 from the other levels 1,2,3,5 and the ion level (+):


              Collisional              Radiative

 Level   S4  C64 C54 C34 C24 C14      B4  A64 A54
    + ====+============================+============  Ion level (continuum)
    6 ----|---+------------------------$---+--------
          |   |                        $   $
    5 ----|---|---+--------------------$---$---+----
          |   |   |                    $   $   $
    4 ----o---o---o---o---o---o--------o---o---o----  upper laser level
                      |   |   |
                      |   |   |
    3 ----------------+---|---|---------------------  lower laser level
                          |   |
                          |   |
    2 --------------------+---|---------------------
    1 ------------------------+---------------------  ground level
The upper quantum levels are more closely spaced and as the excitation energy increases the levels merge into the continuum, where the quantum states are described by a continuous positive energy parameter corresponding to the velocity of free electrons, which are distributed according to Maxellian statistics. The next diagram represents level 4 de-population mechansisms, losses out of level 4 into the other levels 1,2,3,5 and the ion level (+) :


              Collisional             Radiative

 Level   a4  C46 C45 C43 C42 C41      A43 A42 A41
    + ====o=========================================  Ion level (continuum)
    6 ----|---o-------------------------------------
          |   |
    5 ----|---|---o---------------------------------
          |   |   |
    4 ----+---+---+---+---+---+--------+---+---+----  upper laser level
                      |   |   |        $   $   $
                      |   |   |        $   $   $
    3 ----------------o---|---|--------o---$---$----  lower laser level
                          |   |            $   $
                          |   |            $   $
    2 --------------------o---|------------o---$----
                              |                $
                              |                $
                              |                $
    1 ------------------------o----------------o----  ground level
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 a differential equation for the population density ni of a level i as a function of time. (see the Appendix for the equations)


   |<---------RADIATIVE------------>| |<-COLLISIONAL->|         LEVEL

 3 =======  =======  ==+=+==  ===+===  =+=+=+=  ===+===  ======= Ion
                       $ $       $      | | |      |             (Continuum)
                       $ $       $      | | |      |
 2 -------  ---+---  --$-$--  ---o---  -o-o-o-  ---|---  o-o-o-o Upper
               $       $ $                         |
               $       $ $                         |
 1 -+-+-+-  ---$---  --o-o--  -------  -------  ---o---  ---o--- Lower
    $ $ $      $
    $ $ $      $
 0 -o-o-o-  ---o---  -------  -------  -------  -------  ------- Ground

    A10      A21      A31      A32      C32      C31     INVERSION



INVERSION occurs between level 2 and 1 due to A10+A21+A31+A32+C32+C31+...

The first diagram (A10) has more transitions than the second diagram (A21) to represent the fact that the radiative decay rate of the lower levels is larger than the the rate of the upper levels ( A10> A21 ).

The electron density plays an important role in determining which process dominates for any given set of plasma parameters. The three body recombination rate (C32, C31) depends on the square of the electron density, however the radiative recombination rate (A31, A32) depends only linearly on the electron density. Therefore the three body recombination rate dominates at the higher electron densities while the radiative recombination rate dominates for intermediate densities, while the spontaneous decay rate is independent of electron density and dominates at the extremely low densities; (i.e. Forbidden radiation in planetary nebula.) The three body collisional recombination rate is stronger into upper level than into lowel levels ( C32 is larger than C31 ); which contributes to the over-population of the upper laser level relative to the lower laser level.

Radiative recombination into the upper levels is usually weaker than into the lower levels ( A32 < A31 ), this selectivity works againsts the strong over-population mechanism.

However, the lower laser level has a stronger radiative decay rate (A21) than that of the upper laser level (A10 > A21),this mechanism plays a significant role in the de-population of the lower lasing level, contributing to the population inversion between levels 2 and 1.

Therefore strong population inversions can build up between level 2 and 1 only if the electron density is large enough to reduce the influence of radiative recombination which 'poisons' the lower laser level. But electron density cannot be too large otherwise the collisional de-exciation (C21) rate between the upper and lower laser levels drains the inversion that is building up between them.

Although is general terms there are many other processes which contribute the level population of each level in principle depends on the population of all the other levels, and is beyond our qualitative approximation.

The dominant processes leading to population inversions are the collisional mixing among the crowded upper levels and strong three body recombination preferentially into the upper levels, also rapid radiative de-population of the lower quantum levels.


 Level   C65 C54 C56 C46 C54 C64    a6  a5  a4
    + ===============================+===+===+======  Ion level (continuum)
    6 ----o-------+---o-------+------o---|---|------
          |       |   |       |          |   |
    5 ----+---o---o---|---+---|----------o---|------
              |       |   |   |              |        overpopulated
    4 --------+-------+---o---o--------------o------  upper laser level

    3 ------------------+---+-----------------------  lower laser level
                        $   $
                        $   $
    2 ------------------o---$---+-------------------
                            $   $
                            $   $
                            $   $
    1 ----------------------o---o-------------------  ground level
                       A32 A31 A21
When these processes dominate, it is clear that a population inversion will build up between level 4 and level 3, leading to amplified spontaneous emission of radiation corresponding to the 4-3 transition. i.e. laser action, the laser gain will be proportional to the radiative transition probability (A43) and the degree of overpopulation density (P43) :


Random spontaneous emission photons
causes a linear increase of intensity  Incoherent sum of waves with
 |       |       |       |       |     no fixed phase relationship
 |       |       |       |       V                                    Weak
 |       |       |       V       *################################### emission
 |       |       V       *@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ line
 V       *+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++


                                  Coherent sum of in-phase waves

Single           Zone of         *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
spontaneous      maximum         *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
emission         plasma          *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
photon           laser           *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
 |               gain    *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
 |                       *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\ Strong
 V               *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\ laser
 *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\ emission
         *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\ line
                 *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\ arrives
         ^               *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\ at earth
         |       ^       *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
         |       |               *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
         |       |       ^       *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
         |       |       |       *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
         |       |       |       *\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\
         |       |       |       ^
       Exponential amplification by           Interstellar space
       stimulated emission photons
Every photon can 'stimulate' the emission of another photon of identical wavelength, phase and polarization. Quantum mechanics tells us that the emission of stimulated photons increases when the ambient radiations field contains a large number of previously stimulated photons, as demonstrated in the above diagram. The exponential growth of the amplified spontaneous emission due to the 'chain reaction' nature of stimulated emission and the very large distances involved in stellar plasmas, lead us to conclude that even for very small gains the laser emission line will dominate the spectrum.

Even though G may sometimes be small x is always very large, therefore the factor G x in the exponent for the laser intensity is usually much greater than one, leading to extremely large magnifications of initial intensity ( I > > Io ). Stimulated emission creates many other unusual effects such as significant polarization in the laser line and sensitivity to initial plasma conditions.


Intensity^                     Exponential Growth of the emission line.
         |       I(x)         /
10000 Io +-                  /
         |   Growth curve   /
         |   for G=2.30    /   ni = Upper level population density
         |                /    nj = Lower level population density
         |               /     gi = Upper level quantum statistical weight
 1000 Io +-             /      gj = Lower level quantum statistical weight
         |             /
         |            /        Pij = ni/gi - nj/gj    Overpopulation density
         |           /
         |          /
  100 Io +-        /
         |        /  lambda= transition wavelength
         |       /      dnu= linewidth
         |      /        G = (ln2/pi)^(1/2) [gi Aij/4pi] Pij lambda^2/dnu
         |     /
   10 Io +-   /         Io = Initial intensity
         |   /           x = length over which gain occurs
         |  /
         | /           I(x)= Io exp [ G x ]   Intensity as a function of x
      Io +----+----+----+----+---> x  (length over which gain occurs)
         0    1    2    3    4
We had restricted ourselves to a simplified 6 level model, for the sake of clarity during the previous discussions; however rigorous collisional- radiative computer simulations using from 32 X 32 up to 62 X 62 matrices resulted in strong population inversions between many levels of C III, C IV, N V, O VI, He II, He I, N III, C III, H I, and many other ions.

Next Section : Carbon Plasmas