Multi-Species Saha Equation

On 24 Jun, 00:31, Agent Smith agent-sm...@two-blocks-on-your-
left.com wrote:

Does smog involve plasma chemistry, because NOx is two automotive
exhaust gases, with -1 ionization, which suggests a plasma, or does that
stuff instantly neutralize itself by going into solution to make acid
rain? Ozone also forms O3- at very low energies, but I may be
forgetting about the UV again.

Yes, apparently there is a reaction O3 + O2- O3- + O2, but the
problem would be to get the O2 ionized in the first place. A
temperature of 1000K simply won't be sufficient for this. So as
somebody indicated above already you need a spark or something else
sufficiently energetic to have some initial ionization in the first
place, and then you need chemical reactions that maintain this
ionization. In the latter respect there appears to exist a reaction
that could be of interest here, namely N + O NO+ + e- (see
http://www.evactron.com/sullivan.html ). This is an associative
ionization reaction which doesn't even need an initial ionization. The
spark (or whatever) would just need to be able to dissociate N2 and O2
into N and O and then the ionization could maintain itself at a rather
low temperature through chemical reactions (this is not really my
field though and I am just speculating here).
I wonder however whether the assumption of LTE can still be strictly
applied to cases where the equilibrium is maintained through chemical
reactions rather than elastic collisions, as the Maxwell-Boltzmann-
Saha distribution implies perfect symmetry of the collision process
(see my page http://www.plasmaphysics.org.uk/maxwell.htm ), but
chemical reactions are usually highly asymmetric.

Thomas

Thomas Smid wrote in
ps.com:

On 23 Jun, 19:33, John Schutkeker
wrote:
Thomas Smid wrote
roups.com:

On 23 Jun, 11:59, John Schutkeker
wrote:
Thomas Smid wrote
roups.com:

On 22 Jun, 12:46, Thomas Smid wrote:
On 22 Jun, 01:14, John Schutkeker
wrote:

If the Saha Equation gives the ionization of a plasma as a
function of temperature, does anybody know if it has been
extended to a multi-species plasma, and if so, what is the
name of the more general equation?

The Saha Equation is hardly ever appropriate as it assumes LTE
(Local Thermodynamic Equilibrium) i.e. the predominance of
collisions over all other processes. This does usually not
hold, and one has to calculate the ionization rate by a
detailed equlibrium assumption instead. For instance, the
plasma density n produced in the earth's ionosphere is given by
the photoionization-recombination equation

F*Q*N =alpha*n^2

where F is the ionizing solar UV photon flux, Q the
photoinization cross section, N the neutral density and alpha
the recombination coefficient (n the plasma density), i.e.

n=sqrt(F*Q*N/alpha)

and thus

n/N = sqrt(F*Q/alpha/N).

So the degree of ionization does here for instance not directly
depend on the local temperature T at all.

Thomas

I have actually a program which calculates the ion densities for
a multi constituent plasma on the basis as mentioned in my post
above i.e. assuming detailed photoionization-recombination
equilibrium and also taking charge exchange process between the
constituents into account.
Seehttp://www.plasmaphysics.org.uk/programs/iondens.htm.

Thanks, that was very helpful.

I said actually incorrectly above that the program assumes
photoionization as the ionizing source. It can in fact be any
source, as the corresponding input parameter (QP) is merely the
primary ionization rate, which could be due to anything. But it
must be a known external input parameter, which makes it probably
unsuitable for your problem.

I noticed that, but it's easy enough to throw away.

What struck me as most interesting is that you get a family of very
similar looking equations, plus one oddball in my case, or two
families in the general case of combustion of a complex mix of
hydrocarbons.

I haven't set up the matrix yet, but they lead to a general
polynomial of order n+1 (or 2n), where n is the number of chemically
reacting species. What's promising about this is that I just
recently learned from one of the sci.math gurus that pure
mathematicians have developed general techniques for solving such
polynomials, analytically.

When I was at MIT, in '82, it was common knowledge that order 5 was
the maximum, but now that important barrier has recently been
eliminated. Of course, are obviously still be serious practical
impediments to implementation, making it probably a dissertation
worthy topic.

One of my biggest pet peeves about modern science is that
practitioners take recourse to the computer waay too soon. It's like
we're still stuck in the 19th century, when the only comprehensive
general techniques were to linearize and invert matrices, and
engineers beat feet the minute something nonlinear appears. It's
also like we foolishly idolize the whiz-bang, new digital calculating
toys, forgetting that, no matter how supremely powerful the computer
might be, algebra, calculus and ODE's are still mightier than any
other weapon of attack on a problem.

Lots of fully analytical, non-linear progress has been made since
then, as Poincare worked in the 1890's, and was a contemporary of
Einstein. Poincare was obviously nearly a century farther ahead of
his time that King Albert. Someday I dream of seeing those general
polynomial solving equations, to see if there are any practical uses
for them in plasma or dynamics.

My program under http://www.plasmaphysics.org.uk/programs/iondens.htm
uses an iterative method.

You said that already, which is what prompted my rant, because, since I
only deal in closed form solutions. it means I can't just put your
equations into my analysis,

It is based on the ionization-recombination
equilibrium equation I mentioned above already

q=alpha*n^2

Is it n^2 because ne=ni?

where q is the primary ionization rate and alpha the recombination
coefficient. For a multi-constituent plasma including charge exchange,
the equation becomes instead

q(i) + SSS[K(i,l,m,p)*n(l)*N(m)] =

alpha(i)*n(i)*S[n(l)] + n(i)*SS[K(i,m,p)*N(m)]

where the Ks are charge exchange coefficients and N the related
neutral densities (which must be given).

After "kinetic theory," "charge exchange" are the two most dreaded words in
my plasma vocabulary. Those reaction networks are so intricate that they
have always terrified me.

Is alpha just another equilibrium constant, perhaps called K2, if you like?
Likewise, aren't the two K's you have different matrices, so they should be
K1 and K3, respectively in that equation. I can throw q(i) away, if I have
no sources terms of anything except the main reactants, which in my
simplest case is CH4, methane, right?

I'm very nervous about using other people's equations if I can't derive
them myself, and given what I know about physical chemistry, I should be
able to rederive this. However, I'm not having any luck.

I wanted to simply start with A+B-C+D, which should give
ni*nj/(nl*nm)=k(i,j,l,m), where {i,j,l,m} correspond to {A,B,C,D}.
Ionization/recombination reactions are 3 species reactions, not four, so m
disappears. Same for dissociation/reattachment reactions.

My base equation become ni*nj = nl*nm * k(i,j,l,m), which looks like a dot
product, if there's an Einstein summation on the RHS. However, I can't
satisfy myself that that has to happen. As it stands now, I'm just looking
at four vectors and a gargantuan matrix (a tetrad), with scalar products of
nl,nm and k(i,j,l,m).

Oh no, wait, it's not the scalar product, but the lhs looks like the outer
product of nl and nm, while the rhs looks like the outer product of nl and
nm with the double dot product of k(i,j,l,m). But I still can't satisfy
myself, based on physical intuition alone that there must be a summation.

The SSS on the left-hand side is a triple sum that takes account of
the charge exchange reactions producing ions and the SS on the right
hand side is a double sum that takes account of the loss of ions due
to charge exchange (l and m designate here the indices of the source
or target constituents, p allows for more than 1 reaction channel).

p looks like a fudge factor to me. This matrix method should account for
an entire network of reactions, even if more than one pathway takes you to
the same destination.

In fact, it seems to me that your p is an attempt to account for the entire
chemistry of the non-ionized excitation states that lie between neutral
molecules and ions, each one of which will have its own activation
potential, it's own cross section, it's own sigma*v, and it's own
equilibrium constant.

It seems to me that that analysis will be identical to this one, which
suggests that maybe a recursive call into this very algorithm can eliminate
the fudge factor. Then, of course, you'll need a whole second "database"
of material parameters, for excitation states. Spectroscopists have that
information.

The single sum S[n(l)] is simply the total electron density. Because
of the term n(i)*S[n(l)], this is a quadratic equation for n(i), which
can be solved in a straightforward manner analytically. The program
just iterates this solution for n(i) starting from n(i)=sqrt[q(i)/
alpha(i)] as an initial guess. There is in principle no limit to the
number of constituents. I have successfully used it for 7, with the
densities varying over more than 6 orders of magnitude.

How many iterations does this code take to get to convergence, typically?

If a 7 item network is all you've ever used this for, I can't believe that
you haven't taken the time to write out the matrices longhand and look at
them to see what their structure suggests to you. Your sums just look like
2-D and 3-D dot products, making your matrix merely a symmetrical (?) list
of quadratics. That symmetry could be the foothold to begin reducing the
problem.

For 7 species, at worst, that would give a 14th rank polynomial with a very
specific structure, not at all the most general form of 14th rank
polynomial. Polynomial theory is a big research field these days, and
here's a chance to try to make a contribution to a fertile area, and get
some practical results. The mathematics should yield some interesting
insights.

I hope I can find a chance to look at it someday, but my wish list is
getting unmanageably long, and you're welcome to take the first crack at
the idea, if you'd like.

"N:dlzc D:aol T:com \(dlzc\)" wrote in news:gEkfi.390219
:

Dear John Schutkeker:

"John Schutkeker" wrote in
message
. 33.102...
"N:dlzc D:aol T:com \(dlzc\)" wrote in
news:S9ifi.218258
:

Dear Agent Smith:

"Agent Smith" wrote
in
message
. 17.102...
...
Does smog involve plasma chemistry,

In the combustion chamber, maybe.

Fascinating Mr. Spock,

;(

;)

a high pressure combustion plasma.
It must be initiated by the spark plug.

Probably only in the path of the spark.

Is there a situation where a mixture can get that
hot and that lean without knocking?

Not in an IC engine. At least one with valves on the exhaust
stream.

That's really not funny.

And why should a lean mixture burn hotter than
a stoichiometric one, anyhow? Shouldn't
stoichiometry maximized thermal energy output.

I don't know where that is coming from.

Common sense. Complete reaction of all species liberates all the
energy. Anything non-stiochiometric is pushing the reaction uphill
again. You did take high school chemistry, of course?

because NOx is two automotive exhaust
gases, with -1 ionization,

NO2, N2O, N2O5... no ionization (or it is lost quickly)

You forgot NO,

You don't find that coming out of a corona-based ozone generator
(which is what I know).

Maybe the presence of hycrocarbons lowers the energy barrier.

and I don't know about the pentox,

Unstable.

Of course. Is it a better oxidizer than N2O? How far do you have to
lower the temperature to get it to hold together?

but I doubt that N2O would survive,
since it's an even better oxidized than oxygen.

It survives.

I'll believe it when I see it.

It survives corona generation,

You yourself pointed out that an IC engine is not even vaguely similar
to a corona generator.

where the arc
"filaments" exceed 6000 K. But then it has plenty of
"ammunition" to divert the filament away. And bulk gas
temperatures in a corona generator are usually less than 50 degC.

which suggests a plasma, or does that
stuff instantly neutralize itself by going
into solution to make acid rain?

Of course water is present, because it's a combustion
product. As to when it liquefies is another matter,
because it's a very hot exhaust gas. But water from
combustion often drips from tailpipes, in some
special circumtances, although I forget the technical details.

Condensation on the inside of a cooler exhaust system. I had not
been following the whole thread. Seems somehow out of place on
sci.astro. But NOx is observed to exit the tailpipe too...

I cross-posted to sci.astro, because a tiny point came up that mattered
to them.

Not unless water is present. NOx with visible light
and a VOC can form ozone.

Are you sure that's visible light and not UVA or UVB?
They're all a part of the ground level sunlight of a hot,
sunny summer day.

http://www.atmosci.ceh.ac.uk/docs/MC...hesis06_C1.pdf
... says 280 - 430 nm, which is UV-B, UV-A, to violet. So
"visible" isn't quite right, no.

Thanks for making me find that!

Thanks, I thought so.

"Timo A. Nieminen" wrote in
:

On Sat, 23 Jun 2007, Agent Smith wrote:

Timo Nieminen wrote:
On Fri, 22 Jun 2007, John Schutkeker wrote:

If the Saha Equation gives the ionization of a plasma as a function
of temperature, does anybody know if it has been extended to a
multi-species plasma, and if so, what is the name of the more
general equation?

It's still Saha's equation. What happens is that the electron
density Ne is not a function of the ionisation of the single
element, but depends on the ionisation of all of the species. This
is pretty much the case for any astrophysical plasma.

So, in the electron density (ne) term, on the left side, I want to
put the total ne, not just the contribution from just one reaction of
a number, right? Then I have several Saha equations to solve
simulatneously, for several ionization reactions.

If you know the concentration of each species, and you know the
temperature, and they're all in LTE, in principle, yes. In practice,
uncertainties in the concentrations of some easily ionisable species
might be problematic.

OTOH, you might already know the temperature and the electron
concentration beforehand, in which case you just use Saha's equation
to find the ionisation of whatever element is of interest.

I have no worries about LTE, but one problem that we have in fusion
plasmas is that TeTi in a low pressure glow discharge. My biggest
point of curiousity would be about what conditions in a plasma chemical
reactor vessle could cause such a divergence. Or rather, not what
conditions in the vessel, but what conditions in a similar vessel with
the most convenient changes of parameters to force that splitting to
take place.

Here is a gedanken experiment, to look at it the other way. Supposing I
were to start with a low pressure glow that were combusting a minuscule
trickle of CH4 & O2 into CO2 & H2O. Then I slowly started raising the
pressure, remaining at stoichiometry, could I get to a Ti=Te condition,
without a plasma quench, before I left the range of parameters that
define a glow discharge?

Or is TiTe one possible way to define the parameter range for a glow
discharge?

Thomas Smid wrote in news:1182678081.469505.158990
@g4g2000hsf.googlegroups.com:

On 24 Jun, 00:31, Agent Smith agent-sm...@two-blocks-on-your-
left.com wrote:

Does smog involve plasma chemistry, because NOx is two automotive
exhaust gases, with -1 ionization, which suggests a plasma, or does that
stuff instantly neutralize itself by going into solution to make acid
rain? Ozone also forms O3- at very low energies, but I may be
forgetting about the UV again.

Yes, apparently there is a reaction O3 + O2- O3- + O2, but the
problem would be to get the O2 ionized in the first place. A
temperature of 1000K simply won't be sufficient for this. So as
somebody indicated above already you need a spark or something else
sufficiently energetic to have some initial ionization in the first
place, and then you need chemical reactions that maintain this
ionization. In the latter respect there appears to exist a reaction
that could be of interest here, namely N + O NO+ + e- (see
http://www.evactron.com/sullivan.html ).

There's no such thing as NO+ . The nitric oxide ion we're looking at is
NO-, which is one of the NOx gasses that make the nitric acid in smog. If
that stuff existed, it would be a revolutionary concept in chemistry, a
radical that has both + and - states in the same plasma.

That guy's scam ranks right up there with the dietary supplement scams,
where they sell you L-carnitine, when only the D- isomers work in nature.
That guy's brain is inverted, and all he wants is the $$.

This is an associative
ionization reaction which doesn't even need an initial ionization. The
spark (or whatever) would just need to be able to dissociate N2 and O2
into N and O and then the ionization could maintain itself at a rather
low temperature through chemical reactions (this is not really my
field though and I am just speculating here).

Yeah, I think you're writing the crook's product literature for him.
There's no such thing as "an associative ionization reaction that doesn't
need an initial ionization" to trigger it.

All ionization has to be triggered somehow. That's what the avalanche and
breakdown are all about, and it's the central concept in plasma, the
difference between a plasma and a neutral gas. What you just said was that
there are some kinds of neutral gasses that can spontaneously become
plasmas. I'm afraid that dog won't hunt, Billy Bob.

I wonder however whether the assumption of LTE can still be strictly
applied to cases where the equilibrium is maintained through chemical
reactions rather than elastic collisions, as the Maxwell-Boltzmann-
Saha distribution implies perfect symmetry of the collision process

Yeah, I think that's the classical assumption that makes chemistry work,
and it may be a different way to state LaVoisier's Law. Krall &
Trivelpiece does discuss how to make higher order corrections to the
Maxwellian, that are kind of like the second and higher terms in a Taylor
Series expansion, but that's not the same thing you're saying.

I believe that this is one of the fundamental assumptions of physics, and
probably one of the most rigid, an absolute, unequivocal invariance under
time reversal.

If you look at a dynamical system of particles, the simplest case of which
is two colliding billiard balls, the laws of physics operate exactly the
same if you run the film forward or backward. From looking at the film,
you can't tell which one is time run forward, and which one is time run
backward.

I think you should get Chen and Krall & Trivelpiece and start reading.
This copy (http://tinyurl.com/2fp2jb) is only $22, after postage. You're
missing some key fundamentals background classes, and if your city doesn't
have colleges that teach them, as most don't, you're going to have to teach
yourself. It's a long haul, but worth every second, as those are the two
best plasma books ever written. Life is a journey, not a destination.

(see my page http://www.plasmaphysics.org.uk/maxwell.htm ), but
chemical reactions are usually highly asymmetric.

Only in energy. It's the difference between an energy barrier of
exp(-e*phi/kT) in one direction, and exp(+e*phi/kT) in the other. Your
asymmetry is different. Your asymmetry is in time.

John Schutkeker ) writes:
"N:dlzc D:aol T:com \(dlzc\)" wrote in news:gEkfi.390219
:

Dear John Schutkeker:

"John Schutkeker" wrote in
message
. 33.102...
"N:dlzc D:aol T:com \(dlzc\)" wrote in
news:S9ifi.218258
:

Dear Agent Smith:

"Agent Smith" wrote
in
message
. 17.102...
...
[...]

And why should a lean mixture burn hotter than
a stoichiometric one, anyhow? Shouldn't
stoichiometry maximized thermal energy output.

I don't know where that is coming from.

Common sense. Complete reaction of all species liberates all the
energy. Anything non-stiochiometric is pushing the reaction uphill
again.

Except that what happens in an IC engine's cylinder isn't one reaction.
It's a horribly messy series of parallel reactions, some competing with others
and all going at different rates. Just for starters, both the fuel (a
mixture of hydrocarbons) and the nitrogen in the air can react with oxygen.
You have a mixture of nitrogen oxides, some excited oxygen species, both the
major oxides of carbon, hydroxyl radicals and a whole raft of radicals
drived from the hydrocarbons, all interacting. And the individual reaction
rates and equilibria are also functions of the temperature, which depends
on the amount of energy liberated at any instant . . . .

There's no guarantee that every reaction will go to completion in the time
available. (Traces of benzene can be formed in hydrocarbon combustion and
survive to be emitted in the exhaust; the oxidation of CO is relatively slow
and quite temperature dependent, &c &c.)

(2) Since both nitrogen and hydrocarbons are competing for the oxygen, one
*might* (subject to the uncertainties outlined above) expect that a rich
mixture will produce energy most quickly and so lead to the highest
temperatures.

Empirically, the stoichiometric mixture is close to the composition that
produces the most oxides of nitrogen. And the maximum engine power does
come from a rich mixture, but maximum fuel economy comes from a lean.
It's complicated.

--John Park

On 24 Jun, 21:02, John Schutkeker
wrote:
Thomas Smid wrote
It is based on the ionization-recombination
equilibrium equation I mentioned above already

q=alpha*n^2

Is it n^2 because ne=ni?

Yes indeed, it implies overall charge neutrality.

Is alpha just another equilibrium constant, perhaps called K2, if you like?

alpha(i) is the recombination coefficient for ion constituent i.
I have reproduced and explained the equation under
http://www.plasmaphysics.org.uk/imgs/iondensequa.gif . Hopefully this
makes it somewhat clearer.

Likewise, aren't the two K's you have different matrices, so they should be
K1 and K3, respectively in that equation.

I use only one matrix to store the charge exchange coefficients k .
Basically, the matrix elements contain the coefficients for all
reactions of all ions i with a neutrals m. For instance, if N2(+) has
the index 1 and O(+) the index 2, and I have the 2 reaction channels
N2+ +O N2 +O+ and N2+ +O N +NO+, then k(1,2,1) would be given the
value 6*10^-12 cm^3/sec and k(1,2,2) the value 5*10^-10 cm^3/sec. This
is what has to be passed to the the array KR in my program (see
http://www.plasmaphysics.org.uk/programs/iondens.htm ). This on its
own works however obviously only for the loss rates as it doesn't
contain any information about what products each reaction has. So for
the production rates through charge exchange I have set up a separate
matrix KRL which provides the key for identifying the corresponding
values in the matrix KR. If for instance NO(+) has the index 3 in the
scheme and I want to take into account that NO+ is produced by the
second reaction N2+ +O mentioned above, I set KRL(3,1,2,2)=1 , which
then would correctly assign the value stored in KR(1,2,2) (5*10^-10
cm^3/sec) to this reaction.

I can throw q(i) away, if I have
no sources terms of anything except the main reactants, which in my
simplest case is CH4, methane, right?

No, you can't throw q(i) away as it provides the primary ionization
for the system. If you would set q(i)=0 all the ion densities would be
zero. So q(i) must be given and independent of the ion densities n(i)
(it could be due to photoionization, certain chemical reactions or
whatever).


How many iterations does this code take to get to convergence, typically?

It is a long time ago since I wrote, tested and used the program, and
I can't remember it exactly anymore, but I am vaguely certain that
the iteration converged rather quickly (somewhere in the single
figures in most cases). It all depends of course on the required
accuracy you specify. I think I always used an accuracy of 10^-2 or
10^-3 (a higher accuracy is pretty much pointless anyway given that
all the parameters entering into the problem are usually much less
accurate in the first place).
It is also likely that the number of iterations increases with the
number of constituents, but that should be only linearly. More
critically in this respect is that the number of matrix operations
would go up quadratically or so.
And also, as mentioned in the program documentation already, one might
in certain cases need better than double precision for the program to
work correctly (you'll know if suddenly you obtain negative ion
densities).

Just a correction here by the way: I said earlier that the first
iteration would assume n(i)=sqrt(q(i)/alpha(i)). As you can see from
the program however, the initial ion density is given as DEN(I)=-L(I)
+SQRT(QP(I)+L(I)**2) so the losses due to charge exchange are already
taken into account (I think I changed this as the latter is closer to
the true solution and thus requires fewer iterations).


For 7 species, at worst, that would give a 14th rank polynomial with a very
specific structure, not at all the most general form of 14th rank
polynomial.

I don't know how you would obtain a 14th order polynomial here. Pair
reactions shouldn't result in anything higher than quadratic (at least
not with the methods I know), and that shouldn''t depend on the number
of constituents in the first place. Or am I getting something wrong
here?

Thomas

On 25 Jun, 01:53, John Schutkeker
wrote:
Thomas Smid wrote in
Yes, apparently there is a reaction O3 + O2- O3- + O2, but the
problem would be to get the O2 ionized in the first place. A
temperature of 1000K simply won't be sufficient for this. So as
somebody indicated above already you need a spark or something else
sufficiently energetic to have some initial ionization in the first
place, and then you need chemical reactions that maintain this
ionization. In the latter respect there appears to exist a reaction
that could be of interest here, namely N + O NO+ + e- (see
http://www.evactron.com/sullivan.html).

There's no such thing as NO+ .

I don't know what leads you to this conclusion. NO+ is an important
ion in the upper atmosphere (see for instance
http://pubs.acs.org/ncw/2003/articles/ar50025a002.pdf ). The neutral
NO density is actually several orders of magnitudes smaller than the NO
+ density, with the latter being largely produced through charge
exchange reactions like N2+ +O N +NO+ , O+ +N2 N +NO+ , N+ +O2
O +NO+ , O2+ +N O + NO+. I found NO+ also mentioned in publications
dealing with flame physics (http://links.jstor.org/sici?
sici=0080-4630%2819600510%29255%3A1283%3C520%3AMSOIIF%3E2.0. CO%3B2-
X&size=LARGE&origin=JSTOR-enlargePage ).

There's no such thing as "an associative ionization reaction that doesn't
need an initial ionization" to trigger it.

Just google for 'associative ionization' and you get a lot of
references. See for instance http://www.iop.org/EJ/abstract/1402-4896/71/6/008
.. This one even explicity deals with the production of NO+ through
associative ionization. There is in principle nothing that
distinguishes ionizing reactions from other chemical reactions that
for instance produce molecules in excited states. It is only that
usually the ionization energy required is too high for this to happen.
But NO has a rather low ionization energy (9.3 eV) so it should lend
itself to this process.

Thomas

Thomas Smid wrote in
oups.com:

On 24 Jun, 21:02, John Schutkeker
wrote:
Thomas Smid wrote
It is based on the ionization-recombination
equilibrium equation I mentioned above already

q=alpha*n^2

Is it n^2 because ne=ni?

Yes indeed, it implies overall charge neutrality.

Is alpha just another equilibrium constant, perhaps called K2, if you
like?

alpha(i) is the recombination coefficient for ion constituent i.
I have reproduced and explained the equation under
http://www.plasmaphysics.org.uk/imgs/iondensequa.gif . Hopefully this
makes it somewhat clearer.

Likewise, aren't the two K's you have different matrices, so they
should be K1 and K3, respectively in that equation.

I use only one matrix to store the charge exchange coefficients k .
Basically, the matrix elements contain the coefficients for all
reactions of all ions i with a neutrals m. For instance, if N2(+) has
the index 1 and O(+) the index 2, and I have the 2 reaction channels
N2+ +O N2 +O+ and N2+ +O N +NO+, then k(1,2,1) would be given the
value 6*10^-12 cm^3/sec and k(1,2,2) the value 5*10^-10 cm^3/sec. This
is what has to be passed to the the array KR in my program (see
http://www.plasmaphysics.org.uk/programs/iondens.htm ). This on its
own works however obviously only for the loss rates as it doesn't
contain any information about what products each reaction has. So for
the production rates through charge exchange I have set up a separate
matrix KRL which provides the key for identifying the corresponding
values in the matrix KR. If for instance NO(+) has the index 3 in the
scheme and I want to take into account that NO+ is produced by the
second reaction N2+ +O mentioned above, I set KRL(3,1,2,2)=1 , which
then would correctly assign the value stored in KR(1,2,2) (5*10^-10
cm^3/sec) to this reaction.

I can throw q(i) away, if I have
no sources terms of anything except the main reactants, which in my
simplest case is CH4, methane, right?

No, you can't throw q(i) away as it provides the primary ionization
for the system. If you would set q(i)=0 all the ion densities would be
zero. So q(i) must be given and independent of the ion densities n(i)
(it could be due to photoionization, certain chemical reactions or
whatever).


How many iterations does this code take to get to convergence,
typically?

It is a long time ago since I wrote, tested and used the program, and
I can't remember it exactly anymore, but I am vaguely certain that
the iteration converged rather quickly (somewhere in the single
figures in most cases). It all depends of course on the required
accuracy you specify. I think I always used an accuracy of 10^-2 or
10^-3 (a higher accuracy is pretty much pointless anyway given that
all the parameters entering into the problem are usually much less
accurate in the first place).
It is also likely that the number of iterations increases with the
number of constituents, but that should be only linearly. More
critically in this respect is that the number of matrix operations
would go up quadratically or so.
And also, as mentioned in the program documentation already, one might
in certain cases need better than double precision for the program to
work correctly (you'll know if suddenly you obtain negative ion
densities).

Just a correction here by the way: I said earlier that the first
iteration would assume n(i)=sqrt(q(i)/alpha(i)). As you can see from
the program however, the initial ion density is given as DEN(I)=-L(I)
+SQRT(QP(I)+L(I)**2) so the losses due to charge exchange are already
taken into account (I think I changed this as the latter is closer to
the true solution and thus requires fewer iterations).

For 7 species, at worst, that would give a 14th rank polynomial with
a very specific structure, not at all the most general form of 14th
rank polynomial.

I don't know how you would obtain a 14th order polynomial here. Pair
reactions shouldn't result in anything higher than quadratic (at least
not with the methods I know), and that shouldn''t depend on the number
of constituents in the first place. Or am I getting something wrong
here?

It's because you've got 7 coupled sets of pair reactions, each generating a
quadratic, and they all have to be solve simultaneously for the components.
I finally see how to set up the equation, although I haven't written it
down yet. Basically, I have to take your dot/outer product expressions
above and feed them into the matrix method for solving n equations in n
unknowns.

That will yield the master equation for the polynomial, which could
hopefully then be fed into the new equations for solving polynomials of
arbitrary order. I really do need to start writing this stuff down, before
these insights slip my mind. Aren't there several different matrix methods
for solving coupled sets of equations of identical form?

Bruce Scott TOK ] wrote in
:

John Schutkeker wrote:

I have no worries about LTE, but one problem that we have in fusion
plasmas is that TeTi in a low pressure glow discharge. My biggest

For tokamak plasmas ionisation is by electron collision, so the T you
have to know is T_e

point of curiousity would be about what conditions in a plasma chemical
reactor vessle could cause such a divergence. Or rather, not what [...]

In a tokamak, the usual reason is transport (transport time scale for
the local layer shorter than tau_e times M_i/m_e)

What's the "local layer"?