AE-406
UDC 533.951
The Decay of Optically Thick Helium
Plasmas, Taking into Account Ionizing
Collisions between Metastable
Atoms or Molecules
J. Stevefelt
This report is intended for publication in a periodical. References may not be published prior to such publication without the
consent of the author.
AKTIEBOLAGET ATOMENERGI
STUDSVIK, NYKOPING, SWEDEN 1970
AE-406
THE DECAY OF OPTICALLY THICK HELIUM PLASMAS,
TAKING
INTO ACCOUNT IONIZING COLLISIONS BETWEEN METASTABLE
ATOMS OR MOLECULES
J . Stevefelt
ABSTRACT
The effective r e c o m b i n a t i o n r a t e of a helium afterglow p l a s m a ,
which is optically thick t o w a r d s the r e s o n a n c e l i n e s ,
is calculated
from the coupled r a t e equations for the number densities of free
e l e c t r o n s and of m e t a s t a b l e a t o m s or m o l e c u l e s .
is a n e u t r a l p l a s m a ,
metastables.
The model employed
consisting of one kind of ions and one kind of
The ions a r e lost by e l e c t r o n - i o n recombination only,
with subsequent formation of m e t a s t a b l e s ,
which a r e then deactivated
in collisions with free e l e c t r o n s or with other m e t a s t a b l e s : in the
l a t t e r c a s e one e l e c t r o n is r e g a i n e d to the free s t a t e .
When the r a t e
constants for t h e s e v a r i o u s p r o c e s s e s a r e time-independent,
it is
found that after a c e r t a i n t r a n s i t i o n t i m e a t r a n s i e n t equilibrium between
the number d e n s i t i e s of e l e c t r o n s and m e t a s t a b l e s is attained.
dense afterglow p l a s m a ,
large,
In a
where the r e c o m b i n a t i o n coefficient m a y be
the t r a n s i e n t equilibrium density of m e t a s t a b l e s m a y b e c o m e
significantly higher than the qua s i - e q u i l i b r i u m value obtained by equating
the t i m e derivative of the m e t a s t a b l e density to z e r o , and the effective
recombination coefficient m a y be reduced by much m o r e than a factor
of two.
P r i n t e d and distributed in November 1 97 0
- 2 -
LIST OF CONTENTS
Page
1.
Introduction
3
2.
Physical model
4
3.
Solution of differential equation for the ratio p
6
4.
The t r a n s i t i o n time
9
5.
The effective recombination coefficient
12
Acknowledgements
13
References
14
Captions of figures
15
Figures
- 3 -
1.
INTRODUCTION
A l a r g e amount of e x p e r i m e n t a l work on e l e c t r o n - i o n r e c o m b i n a t i o n
in helium d i s c h a r g e p l a s m a s has shown that,
conditions,
under c u r r e n t afterglow
the p l a s m a d i s i n t e g r a t e s in r e a s o n a b l e a g r e e m e n t with the
model of c o l l i s i o n a l - r a d i a t i v e recombination.
This model has b e e n
worked out in a r t i c l e s by B a t e s et a l , [ 1 ] , B a t e s and Khare [ 2 ] , Bates
et al. [ 3 ] , and Collins [ 4 ] .
In the t h e o r e t i c a l approach of B a t e s et a l . [ 1] a r a t e equation
is w r i t t e n for each excited level n of the n e u t r a l atom,
r a t e of i n c r e a s e of the population density N with t i m e .
finite set of coupled differential equations is obtained,
course of the recombination.
population densities N
free e l e c t r o n s ,
'
and N,
Thus an inwhich yields the
Except for e x t r e m e l y dense p l a s m a s , the
(n j£ 1) a r e much s m a l l e r than the density N of
and so the s a m e is valid for the t i m e derivatives of N
n
This m e a n s that a q u a s i - e q u i l i b r i u m number density is established
almost instantaneously for the excited l e v e l s ,
N
d e s c r i b i n g the
and the time d e r i v a t i v e s
(n / 1) o c c u r r i n g in the set of differential equations can be put equal
to z e r o , which g r e a t l y simplifies the solution.
In a l a t e r paper by Bates et a l , [ 3 ] the influence of ionizing
collisions bet-ween two m e t a s t a b l e a t o m s on the overall recombination
r a t e was taken into account.
This calculation was also b a s e d on an
assumption of qua s i - e q u i l i b r i u m number density of m e t a s t a b l e s , and
it was shown that the effective recombination coefficient m a y then be
reduced by at m o s t a factor of two. However,
as was pointed out
already in the f o r m e r paper [ 1 ] , in a p l a s m a which is optically
thick to the r e s o n a n c e lines the density of a t o m s in the quantum level
n = 2 may b e c o m e comparable to the free e l e c t r o n density,
and the
a s s u m p t i o n of q u a s i - e q u i l i b r i u m m e t a s t a b l e density is then no longer
justified.
This is a common situation in an afterglow p l a s m a ,
and the
- 4 -
calculation of the m e t a s t a b l e density and effective recombination
coefficient in such a p l a s m a will be the subject of the p r e s e n t p a p e r ,
2.
PHYSICAL MODEL
We a s s u m e a two-level model of the helium a t o m or m o l e c u l e :
a ground level with quantum number n = 1,
molecule is r e p u l s i v e ,
number n = 2,
and an excited,
In addition,
p l a s m a is n e u t r a l ,
that i s ,
m e t a s t a b l e level with quantum
we consider the unbound level n = <». The
the free e l e c t r o n density is equal to the
number density of ions ; further,
one kind of m e t a s t a b l e s
which for the h e l i u m
t h e r e is just one kind of ions and
(atomic or m o l e c u l a r ) .
The free e l e c t r o n s
recombine with the ions in an unspecified way to f o r m a m e t a s t a b l e
state,
tion
and the r a t e constant for this p r o c e s s is called the r e c o m b i n a coefficient a .
The m e t a s t a b l e s a r e deexcited to the ground level
in collisions with free e l e c t r o n s with a r a t e constant p .
m e t a s t a b l e s may collide with each other,
and the other b e c o m e s ionized.
t h r e e r a t e constants a, p,
two
such that one is deexcited
The r a t e coefficient for this p r o c e s s
is denoted T, -A-ll other p r o c e s s e s ,
a s s u m e d to be negligibly slow.
Finally,
such as ambipolar diffusion,
Further,
are
in a first consideration,
the
and T a r e a s s u m e d to be t i m e - i n d e p e n d e n t .
This simplified model might be applied to s e v e r a l limiting c a s e s .
First,
when the dominant ion is the atomic He ,
the collisional-
radiative recombination r e s u l t s in the formation of atomic m e t a s t a b l e s
He(2), which undergo mutual ionizing collisions to again produce
atomic ions:
a
He
+ e + X —> He(2) + X
(1)
T
He(2) + He(2) — » H e ( l ) + H e + + e
Here,
the stabilizing p a r t i c l e X m a y be an e l e c t r o n or a neutral
(2)
- 5 -
atom.
If,
on the other hand,
in the afterglow,
only m o l e c u l a r ions H e ?
are present
c o l l i s i o n a l - r a d i a t i v e r e c o m b i n a t i o n r e s u l t s in
the formation of m o l e c u l a r m e t a s t a b l e s He ? (2),
which,
colliding
with each other, produce atomic ions, as was d e m o n s t r a t e d byCollins and Hurt [5] . If the p r e s s u r e is high enough, the atomic
ions a r e immediately converted into m o l e c u l a r ions:
,
He2
a
+ e + X —> He 2 (2) + X
(3)
He 2 (2) + He 2 (2) —> 3He(l) + H e + + e
(4)
H e + + 2 He(l) - > H e 2 + + He(l)
(5)
Finally,
the dominant ion m a y be H e ? ', undergoing dissociative
recombination to f o r m atomic m e t a s t a b l e s He(2):
,
a
He
+ e —-> He(l) + He(2)
These m2 e t a s t a b l e s m a y obey the p r o c e s s (2),
which again,
if the p r e s s u r e is high,
(6)
forming atomic ions,
a r e rapidly converted according
to (5). In this c a s e we have m o l e c u l a r ions but atomic m e t a s t a b l e s ,
and the recombination r a d i a t i o n observed will be atomic line
emission.
Denoting the m e t a s t a b l e density by M,
by N,
the free e l e c t r o n density
and the t i m e d e r i v a t i v e s by M and N,
this physical model
leads to the coupled r a t e equations
M = a - N 2 - p MN - 2 T M 2
2
,,2
N = -arN +vM
If M is substituted against the r a t i o p = M/N,
p =
-N[TP3
2
+ 2 r p 2 - (a - p ) p - Of]
2
N = -N (a-rP )
(7)
(8)
the r a t e equations b e c o m e
(9)
(io)
- 6 -
F r o m eq.' (10) it is seen that the effective r e c o m b i n a t i o n coefficient
is given by the e x p r e s s i o n
a
eff
3.
= a
"rP
2
(n)
SOLUTION OF DIFFERENTIAL EQUATION FOR THE RATIO p
Next we shall d e t e r m i n e the time development of the different
particle concentrations.
It is then simplest to study the ratio p
between the m e t a s t a b l e density M and the free e l e c t r o n density N.
The r a t e equations (9) and (10) can be combined into one s e c o n d - o r d e r
differential equation for the density ratio p :
r -
2rP
+ 4rP + p
p
.2
,
3
p
(1£)
2
yp + 2yp - (a - p )p - a
After one integration, equation (12) yields
p =c]p-Pll
^P-P-Zj
2
3
JP-P31
(13)
p . , p~, and p - a r e the solutions of the equation
3
yp
2
+ 2Tp
-(a-p)p-a = 0
(14)
and A , , A ? , and A^ a r e d e t e r m i n e d by the identity
2rp2 + 4rp+ P
A
AA
l
A
22
+
3
rP
+ 2rP
2
- (a - P ) P - a
The integration constant C,
;P " Pi
3
+
p
p
" 2
(15)
p
p
" :
appearing in equation (13),
by equation (9) together with the initial values N and p
is d e t e r m i n e d
at t i m e z e r o .
It is s e e n that the values of p . and A. (i = 1, 2, 3) a r e functions
of the two r a t i o s
a = a/r
,
b = 3/y
- 7-
only.
Further,
if b - 1,
a condition which is always fulfilled
according to the data published by Bates et al. [3] and Collins [ 6 ] ,
all' the zeros p. are real: one is positive and two are negative. Equation (9) can be written
p = - T N(p -
Pl)(p
- p 2 )(p - p 3 )
(16)
where p. is the positive solution.
are then positive,
since the density ratio p, as well as y and N,
are positive quantities.
p> 0 i f p < p , ,
Hence,
and
p < 0
and during the whole afterglow,
solution p . .
if p >
p, ,
p tends to approach the positive
Computed values of p. and A. are given in tables 1-3
for different values of a and b.
for
The factors (p - p_) and (p - p_)
a >> 1,
The following may be noted:
NTa
b< 1
(17)
for a «
1,
b< a
Pj—> ^ a / 2
(18)
for
1,
b >> a
Pj—> a/b = a/p
(19)
Ax£ 1
(20)
a «
and,
in addition
A
l
p
+ A
+p
2
+ A
3 =
+p
(21)
2
= -2
(22)
Table 1. b = 0
a
P
l
P
p
2
3
A
l
A
2
A
3
0. 0191
0.1
-0. 10
- 2.00
1.022
0.973
0.005
0.0733
0.2
-0.18
- 2. 02
1.039
0.942
0.018
0.5
-0.40
- 2. 10
1.071
0.830
0.098
1.50
1.0
-0.63'
-
1.091
0.612
0.297
5.33
2.0
-0.85
-3.15
1.091
0.297
0. 612
5.0
-0.97
-
6.03
1.063
0. 066
0.871
10.0
-0.99
-11.01
1.039
0.018
0.942"
. 0.417 "
29.2
109
2.37
- 8 -
Table 2. b = 0. 5
a
0.0645
0.157
0.583
1.75
5.67
29.6
110
0.1
0.2
0.5
1.0
2. 0
5.0
10.0
-0.37
-0.45
-0.62
-0.79
-0.92
-0.98
-1.00
A
P3
?2
Pi
-
1.73
-
1.75
1.88
- 2.21
- 3.08
- 6.02
-11.00
A
l
1.063
1.092
1.125
1.130
1.112
1.070
1.041
2
A3
1. 117 - 0. 181
1.051 - 0.143
0.860
0.015
0.558
0.312
0.236
0.652
0.050
0.881
0.014
0.945
Table 3. b = 1. 0
a
0.110
0.240
0.750
2.00
6.00
30.0
110
P
A
l
P2
P3
0.1
0.2
0.5
1.0
2.0
5.0
10.0
-1.00
-1.00
-1.00
-1.00
-1.00
-1.00
-1.00
- 1. 10
- 1.20
- 1.50
- 2.00
- 3.00
- 6.00
-11.00
A
l
1.076
1.119
1. 167
1. 167
1.133
1.076
1.043
2
9.091
4.167
1.333
0.500
0.167
0.033
0.009
When the density r a t i o p is closely equal to p . ,
A
3
-8.167
-3.286
-0.500
0.333
0.700
0.891
0.948
we have what
is generally called a " t r a n s i e n t equilibrium" between the concent r a t i o n s of free electrons and m e t a s t a b l e s .
The r a t e of p l a s m a
decay under this condition will be d i s c u s s e d below.
ever,
First,
how-
we shall calculate the t i m e of t r a n s i t i o n from the initial
value p to the asymptotic value p , .
It will be shown that this
t r a n s i t i o n t i m e is much s h o r t e r then the c h a r a c t e r i s t i c decay t i m e
for the e l e c t r o n density.
- 9-
4.
THE TRANSITION TIME
Equation (13) for p can be solved exactly,
exponents A. are integers.
a,
One such case is in the limit of large
for all values of b < 1,
A.
= 1,
provided that the
where
A 2 = 0,
A3 = 1
Comparing equations (13) and (16) one finds
T N 0 (P 0
-
Pl)
C =.
(23)
A
A
l
lPo " Pl I
(Po '
P
2 "
1
2}
A
(Po " P3>
where N and p are taken at the time t .
o
^o
o
C < 0
if
and C > 0
if
For large a,
rp
o
>
3 "
1
Hence,
p.
^1
p < p,
o
-^1
r
equation (13) may then be written
P = " | C | ( p - Px)(p - P 3 )
(24)
with the solution
P " Pl
P " p3
Po " P l
P
"(t " V ^
(25)
o " p3
and where the characteristic time T is
=
1
lPl-P3i|Cf
(26)
Inserting asymptotic values for p.,
this expressionbecom.es
(27)
2 T N 0 (p 0 + !)*£"
2 N ^ ( M Q + NQ)
Also when a and b are both small,
A = 1,
A2 = 1,
A
3
the exponents A. are integers:
=
°
- 10 -
The solution for p is then again given by the e x p r e s s i o n (25), with
p_ r e p l a c e d by p ? ,
and with
(28)
T =
C
(Pj " P 2 ) | |
T
N
0(P0
+ 2)
^
a
^r(
M
+
0
2N
0)
It is seen that the c h a r a c t e r i s t i c t r a n s i t i o n t i m e s T for t h e s e two
considered c a s e s ,
respectively,
given by the e x p r e s s i o n s (27) and (28),
differ by at m o s t a factor \f2: this m a x i m u m difference
o c c u r s for ^o
p = 0.
In the g e n e r a l c a s e ,
when A. a r e not i n t e g e r s ,
to define a c h a r a c t e r i s t i c t r a n s i t i o n t i m e for the
it is convenient
quasi-exponential
approach in t i m e of p to the asymptotic value p , :
T
P - Pi
= -. - _ 1
• P
(29)
We want to evaluate this t r a n s i t i o n t i m e for the situation where the
density r a t i o p is only "slightly" deviating from the " t r a n s i e n t equil i b r i u m " value p , ,
which could occur for instance when the r a t e
coefficients &t fj , and y v a r y slowly in t i m e .
(13) and (23),
and approximating A, = 1, p = p , ,
1
a s s - f(a, b , p )
T«
Inserting the e x p r e s s i o n s
we obtain
(30)
o
where the function f(a, b , p ) is given by
f(a, b , p ) =
1-A 2
(PQ - P 2 )
dl
1-A 3
A2
A3
(p o " P 3 )
(Pj - P 2 )
(Pj " P 3 )
(31)
- 11 -
It is e a s i l y verified,
by inspection of t a b l e s 1-3,
that the
e x p r e s s i o n s (30) and (31) develop into the e x p r e s s i o n s (27) and (28)
for the two r e s p e c t i v e s p e c i a l c a s e s c o n s i d e r e d above.
values a and b , T has its m a x i m u m value for p
F o r fixed
= 0: this value
then r e p r e s e n t s an upper bound for the t r a n s i t i o n t i m e T at those
p a r t i c u l a r values a, b , yf and N Q .
The function f(a, b , p Q = 0) is
shown graphically in F i g . 1: its v a r i a t i o n is quite s m a l l ,
and
f(a, b , 0 ) ^ 1/2
Hence
T
<
.
~ 2 N o ^
(32)
F r o m equation (16) t m a y be e x p r e s s e d in t e r m s of the instantaneous
e l e c t r o n density N,
T =
rN(Pi
if again p is n e a r l y equal to p , :
(33)
- PzKP! - P 3 )
Rewriting t h i s e x p r e s s i o n as
T=
- r g(a> *>)
yN(a + \Ji)
(34)
it is found that the function
/
g(a
, \
'
b) =
a + \!a.
( P l " P 2 ) ( P l " P3)
is v e r y s i m i l a r to the function f(a, b , p
slowly with a and b .
f, g —> 1/2
ft g ^
(35)
= 0),
in p a r t i c u l a r it v a r i e s
Common features of the two functions a r e
for a —> oo
1/2
(36)
(37)
f, g —-> \fr/4
for a - » 0, b = 0
(38)
f» g °=
for s m a l l a, b = 1
(39)
^Ja.
F r o m the upper bound of g it follows that
- 12 -
T
<
i
=
i
2yN(a + \}&) 2N(a+Vc?Y)
In a helium afterglow experiment,
and y
(40)
the r e a c t i o n r a t e coefficients <x, £ »
cannot be expected to be time-independent c o n s t a n t s .
their t i m e v a r i a t i o n is quite slow,
v a r i a t i o n in e l e c t r o n density,
However,
and in any case not faster than the
which might be e x p r e s s e d as a c h a r a c t e r -
istic t i m e for quasi-exponential decay (during a short t i m e i n t e r v a l ) :
'N~
it is seen,
In g e n e r a l ,
N
^eff1^"^
aTN
by c o m p a r i s o n with the e x p r e s s i o n (40),
that
between five and ten c h a r a c t e r i s t i c t i m e s for approaching
" t r a n s i e n t equilibrium" between the densities of m e t a s t a b l e s and free
e l e c t r o n s have elapsed before the e l e c t r o n density has decayed to l / e
of its initial value N .
o
This also m e a n s that,
even in an afterglow,
where the recombination coefficient v a r i e s in t i m e ,
the density ratio
p between m e t a s t a b l e s and free e l e c t r o n s r e l a x e s to a value close to
the t r a n s i e n t equilibrium value p , ,
coefficients en, p » and y only.
which is a function of the r a t e
The r a t e of p l a s m a decay under such
a t r a n s i e n t equilibrium will next be d i s c u s s e d .
5.
THE E F F E C T I V E RECOMBINATION COEFFICIENT
The free e l e c t r o n density decays at a r a t e given by the
(8) or (10).
equations
Under " t r a n s i e n t equilibrium" the positive solution p
of
equation (14) is to be i n s e r t e d for the density r a t i o p in the e x p r e s s i o n
(10).
The resulting effective recombination coefficients Q!e££,
as d e -
- 13 -
fined by the e x p r e s s i o n (11),
a
eif/<X v e r s u s a = <x/f,
and with b = p / y a s p a r a m e t e r .
It is especially noted that,
tends to unity,
a r e p r e s e n t e d in F i g . 2 as the r a t i o
for s m a l l a, b / 0,
the r a t i o 0Ceff/a
exactly like the r e s u l t obtained under the q u a s i -
equilibrium approximation for the m e t a s t a b l e density: for the e x p r e s s i o n (19) is valid in both c a s e s .
a = a/ft
however,
F o r larger values of the r a t i o
it is e a s i l y seen f r o m the r a t e equation (7) that,
since the t i m e derivative M in r e a l i t y is negative and different f r o m
zero,
the m e t a s t a b l e density M must be l a r g e r than the q u a s i -
equilibrium value obtained f r o m the approximation M = 0.
for l a r g e a the r a t i o a rr/oc d e c r e a s e s like l/\/a,
Thus,
while in the
qua s i - e q u i l i b r i u m approximation it approaches the value l / 2 .
Inserting e x p e r i m e n t a l values of £ and y for helium,
reported
by Bates et al. J"3] and Collins [ 6 ] :
P = 6.4 • 10
m /sec,
y - 1.85 • 10
m /sec
it is s e e n that the r a t i o 0Le^/a, is reduced to about one half even for
recombination coefficients CH as small as 10-15 m3 // s e c . This
implies that ionizing collisions between m e t a s t a b l e a t o m s or^rnolecules a r e of importance in p r a c t i c a l l y all l a b o r a t o r y helium a f t e r glow p l a s m a s ,
a fact which must be taken into account in any
e x p e r i m e n t a l d e t e r m i n a t i o n of the recombination coefficient for
helium ions,
based upon afterglow m e a s u r e m e n t s of the decay of
e l e c t r o n density.
ACKNOWLEDGEMENTS
The author wishes to thank Dr. K. Nygaard and D r . S. P a l m g r e n for
many stimulating and valuable d i s c u s s i o n s during the c o u r s e of this work
- 14 -
REFERENCES
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Phenomena in ionized g a s e s . 9th Int. Conf. B u c h a r e s t
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- 15 -
• CAPTIONS OF FIGURES
Fig. 1
A graph of the function f(a, b , p
= 0 ) v e r s u s a with b as
parameter.
Fig. 2
A graph of the r e c o m b i n a t i o n coefficient ratio ®e£f/ci
v e r s u s a with b as p a r a m e t e r .
c
.3
_
.H/
_
^
w^T*-
P/Y=0
O
|3/Y=05^
o
II
d
9_
y
(3/V=>^
1
o.()i
0.1
10
O/Y
FIGURE 1
1i
d
LIST O F PUBLISHED AE-REPORTS
380. An expansion method to unfold proton recoil spectra. By J. Kockum. 1970.
20 p. Sw. cr. 10:-.
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348. Computer program for inelastic neutron scattering by an anharmonic crystal.
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350. Formation of negative metal ions in a field-free plasma. By E. Larsson.
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351. A determination of the 2 200 m/s absorption cross section and resonance
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353. Diffusion from a ground level point source experiment with thermoluminescence dosimeters and Kr 85 as tracer substance. By Ch. Gyllander, S.
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356. On the theory of compensation in lithium drifted semiconductor detectors.
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the biological and medical research laboratory. By K. Samsahl. 1970. 18 p.
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List of published AES-reports (In Swedish)
1. Analysis by means of gamma spectrometry. By D. Brune. 1961. 10 p. Sw.
cr. 6 : - .
2. Irradiation changes and neutron atmosphere in reactor pressure vesselssome points of view. By M. Grounes. 1962. 33 p. Sw. cr. 6 : - .
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1966. 321 p. Sw. cr. 15:-.
7. Building materials containing radium considered from the radiation protection point of view. By Stig O. W . Bergstrom and Tor Wahlberg. 1967.
26 p. Sw. cr. 1 0 : Additional copies available from the Library of AB Atomenergi, Fack, S-611 Q1
Nykoping 1 , Sweden.
EOS-tryckerierna, Stockholm 1970