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.
This paper is divided into 9 sections,
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)
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.
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.
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)
state
S1 S2 COLLISIONAL IONIZATION BY ELECTRON IMPACT
+ ==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).
C21 C12 COLLISIONAL EXCITATION BY ELECTRON IMPACT
+ =========
Nix + e ---> Njx + e (6)
2 --o---+--
| | Rate coefficient Cij(T),
1 --+---o-- number of processes: ni ne Cij(T).
A21 RADIATIVE TRANSITION
+ =========
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 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
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
underpopulated
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 *OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 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.