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Kinetics of ionization-recombination processes in nonideal hydrogen plasmas
Physic:] A 170 (1YYl)
682Z68X
North-Holland
KINETICS OF IONIZATION-RECOMBINATION NONIDEAL HYDROGEN PLASMAS
PROCESSES
IN
W. EBELING and I. LEIKE .SekriorzPhwik. Humboldt- Uniwrsitiit Berlin, O- IO4~~ Berlin. (~rrmon~
Received
The
11 May 1YYO
time evolution
H-plasma
is studied
of the relative
account via energy level shifts. essentially
smaller
the thermal
populations
of cxcitcd atomic levels in a reacting dense
on the basis of rate equations. It is found
Many-particle
that the occupation
effects
arc taken
into
of the bound state levels is
than for the ideal cast. It is shown that the mechanism of the relaxation
equilibrium
depends on the density.
Inversions
of the level populations
to
can he
observed.
1. Introduction Using modern experimental techniques such as e.g. high power lasers, ion beams, shock fronts and high-current discharges, very dense completely ionized plasmas may be produced [ 11. The problem we want to attack in this paper is how a dense will approach
Fig.
completely the thermal
ionized equilibrium.
plasma,
1. Scheme of the atomic levels and the transitions
arc left.
produced
For simplicity
by a high-energy we shall assume
for two densities
pulse, that the
where two or three levels
W. Ebeling and I. Leike
I Ionization-recombination
processes in plasmas
683
whole process is isothermal and isochoric. Then the dominant processes are the transitions from the continuum to the high-lying bound states, a ladder of transitions to the lower bound states up to the time where the thermal equilibrium is reached. One of the basic effects in a dense plasma is the shift of the levels which leads to a finite number of bound states [2-71. This number will be called nmax in the following. The principle scheme of the levels and the transitions is shown in fig. 1 for two densities where two or three levels are left. Our approach is based on the method of kinetic equations for the population of the levels [g-16]. Many-particle effects are taken into account via energy level shifts [5,6].
2. Kinetics of nonideal plasmas Let us consider a nondegenerate H-plasma which consists of free electrons with number density n, and H+-ions with density ni. Further we assume the existence of bound states k = 1,2, . . . with the densities nk. Then the rate equations for the temporal variation of the population density of an excited atom in a hydrogen plasma can be written as [15,16]
dn,_dt
k-l
“IllaX
-(Y$,&,
+ &njflz -
c
akmnkne
-
2,
Pkmnkne
m=k+l k-l +
2 ItI=1
“lll=X %k%%
+
2 m=k+l
&k%%
7
(1)
where (Yeand Pk are the ionization rate coefficient and three-body recombination rate coefficient of level k, respectively, and akrn and Pkrn are the excitation and de-excitation rates from level k to m, respectively. Radiation can be neglected as is justified in dense plasmas. Because of electroneutrality we have
(2)
n, = ni .
Particle conservation
n, +
c
yields for an isochor plasma
nk = q) ,
(3)
k=l
where n,, the total number density of the heavy particles, is constant. Ebeling and Kilimann [5,6] assume that in a dense thermal plasma the
upward
transition
the ionization
coefficients
energy,
dependence
on the density,
[S, 61. These
assumptions
of Klimontovitch can then ak
Here
[ll]
be written =
show an Eyring-like
whereas
which
may
are in agreement
and Kremp in the form
a;” exp(pd,)
the downward
exponential transitions
be neglected
only
on
a weak
in first approximation
with the quantum
et al. [12-141.
dependence
show
The nonideal
statistical
results
rate coefficients
[S, 6)
.
(4)
p,=pI;‘.
(6)
Px,,,= P;I,,.
(7)
A, is the lowering
of the ionization
energy
of level k due to
(i) (ii) (iii) (iv)
dynamic self-energy. exchange self-energy, dynamically screened Coulomb interaction. phase space occupation [2,3.6.7].
The [IO]. In some The
ideal ionization and excitation coefficients are due to Drawin and Emard The shifts of the levels have been calculated by Ebeling and Kilimann [7]. order to avoid that the effective ionization energy becomes negative for KU~ < ,yx, we take into account an additional term of the Taylor series. shifts then take the form
fh, = (1 -
;xJp)h, + :,x&‘)r,
- iixn,b’)r/ - !2k‘x,/%x:,
(9)
as long as ~a~ is smaller than or equal to xk,. Otherwise J,, = (E,,I. so that at energy I,, = lE,I - A,, disappears due to the Kan = xA, the effective ionization Mott transition. The values of xn, have been calculated by Rogers and Graboske [3]. are the average values of the atomic radii r. r’, r’ [7] W,l? (a,. w>,, and p = r/a,. K is the inverse Debye radius K = qm and uR the Bohr radius.
W. Ebeling and I. Leike
I Ionization-recombination
processes in plasmas
685
3. Results
Starting from eqs. (4), (8) and (9) we calculated the ionization coefficient of the ground state level CC,as a function of the free electron density for different temperatures. The results are shown in fig. 2. Our curves end up at the Mott density since the ionization coefficient makes no sense if the level already disappeared. For comparison we have drawn the results of Schlanges and Bornath [14] which have been obtained using the Debye approximation. This approximation, however, leads to an overestimation of the ionization coefficient at high densities. Furthermore the disappearance of the level due to the Mott transition is not included there. The time evolution of the population densities of excited atomic levels has been calculated solving the system of differential equations (1) numerically under the constraints (2) and (3). Due to Rogers and Graboske [3] at a temperature T = 30 000 K for plasma densities n,, = 1018 (lo”, lo*‘) cmP3 five (three, two) atomic levels are left, respectively. The other levels already disappeared in the continuum. We started the relaxation process with a completely ionized plasma. The
Fig. 2. Ionization coefficient of the ground state level as a function of the free electron density for different temperatures: (1) T = 16 000 K, (2) T = 30 000 K, (3) T = 64 000 K for the present study (solid curves) compared to the ones of Schlanges and Bornath [14] (dashed curves).
~~~~~._ L.-L-15
-IL
~13
-12
-11
tgt is!
evolution of the relative population densities of the atomic levels for T = 30 000 K and plasma densities ,I,, of (a) 1O’x cm ‘, (b) lO”‘cm~’ and (c) 10” cm ‘. The dashed curves represent an idcal. the solid curves a nonideal plasma. The curves are labeled with the main
Fig. 3. Time
quantum
numbers
of the levels. At time I = 0 a completely
ionized
plasma k assumed.
W. Ebeling and I. Leike
I Ionization-recombination
processes in plasmas
687
resulting curves are presented in figs. 3a-3c. The dashed curves show the behavior of an ideal plasma, where A,, = 0. The difference between nonideal population densities and ideal population densities increases with the density. At n, = 10” crnm3 the nonideal curves are up to four times smaller than the ideal ones. Bates et al. [15], who studied ideal low-density plasmas, found that at low densities the relaxation times for the excited levels are much shorter than for the ground level. They computed as a limit for this behavior a free electron density of 1018 (102”) cme3 at T = 16 000 (64 000) K. We agree with their results and found furthermore, that nonideal plasmas behave in a similar way. At higher plasma densities (figs. 3b, 3c) the population densities show a different temporal dependence. The levels reach the thermal equilibrium simultaneously for both ideal and nonideal plasmas. In addition at high plasma densities the population densities of the excited levels exceed the equilibrium population densities up to a factor of 10 for ideal plasmas and 6 for nonideal plasmas. This comes from the fact that the variation of the population density due to recombination is favored at higher densities according to the factor n: in the recombination term. In nonideal plasmas this effect is suppressed because of the strong enhancement of the ionization coefficient. Inversions of the level populations can be observed. The inversions break down faster with higher density. Fig. 4 shows for comparison the relaxation if one starts with a low temperature Boltzmann distribution. Here the population densities of nonideal and ideal plasmas differ from one another by a factor of 1.5.
lg% I
-----__
“0
1
I
-12
-11
-10
k!trs7
Fig. 4. Time evolution of the relative population densities of the atomic levels for T = 30 000 K and n,, = 10”’ cm-‘. The time evolution is shown in dashed curve for ideal plasmas and in solid curve for nonideal plasmas. At time t = 0 a Boltzmann distribution for T = 1000 K is assumed.
To
summarize,
temporal
we
behavior
H-plasmas.
have
discussed
of the population
The effects discussed
higher
temperatures,
higher
collision
however.
the
influence
densities
above
of nonideality
of excited
atomic
do not depend
the equilibrium
on
on the temperature.
is reached
the
levels in dense
faster
because
At of the
frequency.
Acknowledgements The authors discussions.
would
like to thank
Th. Bornath
and D. Kremp
for valuable
References 1l] V.E.
Fortov.
in: W. Eheling
et al.. l‘ransport
Properties
ol Dense Plasmas (Birkhauser,
Bascl.
IYX‘I). [2] W.D.
Kraett,
D. Kremp.
W. Ebcling
Systems (Akademie-Vcrlag.
Berlin.
and Ci. Rijpkc,
IYX5) (Plenum.
Quantum
New York.
Statistic\
01 Charged
Particle
19X5) (Mir.
Moscow.
IYXX) [in
Russian]. [-iI F.J.
Rogers and H.C.
141 W. Eheling.
on Phenomena Swansea.
1.51 W. Ehelinp.
Contrih.
H.W.
(1 I ] Yu.1..
Inwted
(lY70)
1577.
and M. Schlangcs. WT.
Paper\.
Phys. 2Y (IYXY)
165.
Z. Naturforsch.
34~
Z. Naturford.
45s
and W. Eheling. V.S.
Vol-ohjov
Plnsmy (Nauka.
Plasma Spcctroacopy
Drawin
and F. Emard.
Klimontovitch,
Williams.
Proc. XVlllth
1111.C‘ont.
cd. (l!nivcr
of Wale\.
Kremp.
Kinetic
Schlangea
[IS]
D.R.
[lh]
T. Fujimoto.
Bates.
Th.
and Th. A.E.
Yakuhov.
51’). 613.
Kinctika
Ner;tvmrvesnoi
Ni\kotcm-
(McGraw-Hill.
Thcorq
New
( lY77)
York.
IY6X).
333.
of Electromagnetic
Processw
(Nauka.
A 8X (IYXX)
7171.
Mo~~o~~.
IYXO)
lYH2) [in Russian].
Bornath
M. Schlangcs,
LT.
(IVW) ( IYYO)
lYX2).
Physica C 85
Berlin-Heidelberg,
1121 M. Schlanges,
and
Moaco~.
Griem.
(Springer.
[ 1.31D. [13] M.
Gases.
I
A
Plasma
Bihcrmann.
pcraturnoi [Y] H.R.
in Ionized
Rev.
‘1. Rother
and K. Kilimann.
171 K. Kilimann
[IO]
Phy\.
R. Redmcr.
lY87).
(61 W. Ehcling [8] L..M.
C~ruboske.
A. Fiirstcr.
and I>. Kremp, Th.
Bornath,
Kingston
Bornath Contrih.
and R.W.P.
Phys. Rev.
and M. Bonitz, Plasma 373.
Plasma I’hys. 3
Phys. 3Y (IYXY)
McWhirtcr.
J. Phys. Sot. Jpn. 47 (lY7Y)
(‘ontrih.
Proc. R. Sot.
( IYXY) .i I I.
527. A X7
( IO62 )
37.
682Z68X
North-Holland
KINETICS OF IONIZATION-RECOMBINATION NONIDEAL HYDROGEN PLASMAS
PROCESSES
IN
W. EBELING and I. LEIKE .SekriorzPhwik. Humboldt- Uniwrsitiit Berlin, O- IO4~~ Berlin. (~rrmon~
Received
The
11 May 1YYO
time evolution
H-plasma
is studied
of the relative
account via energy level shifts. essentially
smaller
the thermal
populations
of cxcitcd atomic levels in a reacting dense
on the basis of rate equations. It is found
Many-particle
that the occupation
effects
arc taken
into
of the bound state levels is
than for the ideal cast. It is shown that the mechanism of the relaxation
equilibrium
depends on the density.
Inversions
of the level populations
to
can he
observed.
1. Introduction Using modern experimental techniques such as e.g. high power lasers, ion beams, shock fronts and high-current discharges, very dense completely ionized plasmas may be produced [ 11. The problem we want to attack in this paper is how a dense will approach
Fig.
completely the thermal
ionized equilibrium.
plasma,
1. Scheme of the atomic levels and the transitions
arc left.
produced
For simplicity
by a high-energy we shall assume
for two densities
pulse, that the
where two or three levels
W. Ebeling and I. Leike
I Ionization-recombination
processes in plasmas
683
whole process is isothermal and isochoric. Then the dominant processes are the transitions from the continuum to the high-lying bound states, a ladder of transitions to the lower bound states up to the time where the thermal equilibrium is reached. One of the basic effects in a dense plasma is the shift of the levels which leads to a finite number of bound states [2-71. This number will be called nmax in the following. The principle scheme of the levels and the transitions is shown in fig. 1 for two densities where two or three levels are left. Our approach is based on the method of kinetic equations for the population of the levels [g-16]. Many-particle effects are taken into account via energy level shifts [5,6].
2. Kinetics of nonideal plasmas Let us consider a nondegenerate H-plasma which consists of free electrons with number density n, and H+-ions with density ni. Further we assume the existence of bound states k = 1,2, . . . with the densities nk. Then the rate equations for the temporal variation of the population density of an excited atom in a hydrogen plasma can be written as [15,16]
dn,_dt
k-l
“IllaX
-(Y$,&,
+ &njflz -
c
akmnkne
-
2,
Pkmnkne
m=k+l k-l +
2 ItI=1
“lll=X %k%%
+
2 m=k+l
&k%%
7
(1)
where (Yeand Pk are the ionization rate coefficient and three-body recombination rate coefficient of level k, respectively, and akrn and Pkrn are the excitation and de-excitation rates from level k to m, respectively. Radiation can be neglected as is justified in dense plasmas. Because of electroneutrality we have
(2)
n, = ni .
Particle conservation
n, +
c
yields for an isochor plasma
nk = q) ,
(3)
k=l
where n,, the total number density of the heavy particles, is constant. Ebeling and Kilimann [5,6] assume that in a dense thermal plasma the
upward
transition
the ionization
coefficients
energy,
dependence
on the density,
[S, 61. These
assumptions
of Klimontovitch can then ak
Here
[ll]
be written =
show an Eyring-like
whereas
which
may
are in agreement
and Kremp in the form
a;” exp(pd,)
the downward
exponential transitions
be neglected
only
on
a weak
in first approximation
with the quantum
et al. [12-141.
dependence
show
The nonideal
statistical
results
rate coefficients
[S, 6)
.
(4)
p,=pI;‘.
(6)
Px,,,= P;I,,.
(7)
A, is the lowering
of the ionization
energy
of level k due to
(i) (ii) (iii) (iv)
dynamic self-energy. exchange self-energy, dynamically screened Coulomb interaction. phase space occupation [2,3.6.7].
The [IO]. In some The
ideal ionization and excitation coefficients are due to Drawin and Emard The shifts of the levels have been calculated by Ebeling and Kilimann [7]. order to avoid that the effective ionization energy becomes negative for KU~ < ,yx, we take into account an additional term of the Taylor series. shifts then take the form
fh, = (1 -
;xJp)h, + :,x&‘)r,
- iixn,b’)r/ - !2k‘x,/%x:,
(9)
as long as ~a~ is smaller than or equal to xk,. Otherwise J,, = (E,,I. so that at energy I,, = lE,I - A,, disappears due to the Kan = xA, the effective ionization Mott transition. The values of xn, have been calculated by Rogers and Graboske [3]. are the average values of the atomic radii r. r’, r’ [7] W,l? (a,. w>,, and p = r/a,. K is the inverse Debye radius K = qm and uR the Bohr radius.
W. Ebeling and I. Leike
I Ionization-recombination
processes in plasmas
685
3. Results
Starting from eqs. (4), (8) and (9) we calculated the ionization coefficient of the ground state level CC,as a function of the free electron density for different temperatures. The results are shown in fig. 2. Our curves end up at the Mott density since the ionization coefficient makes no sense if the level already disappeared. For comparison we have drawn the results of Schlanges and Bornath [14] which have been obtained using the Debye approximation. This approximation, however, leads to an overestimation of the ionization coefficient at high densities. Furthermore the disappearance of the level due to the Mott transition is not included there. The time evolution of the population densities of excited atomic levels has been calculated solving the system of differential equations (1) numerically under the constraints (2) and (3). Due to Rogers and Graboske [3] at a temperature T = 30 000 K for plasma densities n,, = 1018 (lo”, lo*‘) cmP3 five (three, two) atomic levels are left, respectively. The other levels already disappeared in the continuum. We started the relaxation process with a completely ionized plasma. The
Fig. 2. Ionization coefficient of the ground state level as a function of the free electron density for different temperatures: (1) T = 16 000 K, (2) T = 30 000 K, (3) T = 64 000 K for the present study (solid curves) compared to the ones of Schlanges and Bornath [14] (dashed curves).
~~~~~._ L.-L-15
-IL
~13
-12
-11
tgt is!
evolution of the relative population densities of the atomic levels for T = 30 000 K and plasma densities ,I,, of (a) 1O’x cm ‘, (b) lO”‘cm~’ and (c) 10” cm ‘. The dashed curves represent an idcal. the solid curves a nonideal plasma. The curves are labeled with the main
Fig. 3. Time
quantum
numbers
of the levels. At time I = 0 a completely
ionized
plasma k assumed.
W. Ebeling and I. Leike
I Ionization-recombination
processes in plasmas
687
resulting curves are presented in figs. 3a-3c. The dashed curves show the behavior of an ideal plasma, where A,, = 0. The difference between nonideal population densities and ideal population densities increases with the density. At n, = 10” crnm3 the nonideal curves are up to four times smaller than the ideal ones. Bates et al. [15], who studied ideal low-density plasmas, found that at low densities the relaxation times for the excited levels are much shorter than for the ground level. They computed as a limit for this behavior a free electron density of 1018 (102”) cme3 at T = 16 000 (64 000) K. We agree with their results and found furthermore, that nonideal plasmas behave in a similar way. At higher plasma densities (figs. 3b, 3c) the population densities show a different temporal dependence. The levels reach the thermal equilibrium simultaneously for both ideal and nonideal plasmas. In addition at high plasma densities the population densities of the excited levels exceed the equilibrium population densities up to a factor of 10 for ideal plasmas and 6 for nonideal plasmas. This comes from the fact that the variation of the population density due to recombination is favored at higher densities according to the factor n: in the recombination term. In nonideal plasmas this effect is suppressed because of the strong enhancement of the ionization coefficient. Inversions of the level populations can be observed. The inversions break down faster with higher density. Fig. 4 shows for comparison the relaxation if one starts with a low temperature Boltzmann distribution. Here the population densities of nonideal and ideal plasmas differ from one another by a factor of 1.5.
lg% I
-----__
“0
1
I
-12
-11
-10
k!trs7
Fig. 4. Time evolution of the relative population densities of the atomic levels for T = 30 000 K and n,, = 10”’ cm-‘. The time evolution is shown in dashed curve for ideal plasmas and in solid curve for nonideal plasmas. At time t = 0 a Boltzmann distribution for T = 1000 K is assumed.
To
summarize,
temporal
we
behavior
H-plasmas.
have
discussed
of the population
The effects discussed
higher
temperatures,
higher
collision
however.
the
influence
densities
above
of nonideality
of excited
atomic
do not depend
the equilibrium
on
on the temperature.
is reached
the
levels in dense
faster
because
At of the
frequency.
Acknowledgements The authors discussions.
would
like to thank
Th. Bornath
and D. Kremp
for valuable
References 1l] V.E.
Fortov.
in: W. Eheling
et al.. l‘ransport
Properties
ol Dense Plasmas (Birkhauser,
Bascl.
IYX‘I). [2] W.D.
Kraett,
D. Kremp.
W. Ebcling
Systems (Akademie-Vcrlag.
Berlin.
and Ci. Rijpkc,
IYX5) (Plenum.
Quantum
New York.
Statistic\
01 Charged
Particle
19X5) (Mir.
Moscow.
IYXX) [in
Russian]. [-iI F.J.
Rogers and H.C.
141 W. Eheling.
on Phenomena Swansea.
1.51 W. Ehelinp.
Contrih.
H.W.
(1 I ] Yu.1..
Inwted
(lY70)
1577.
and M. Schlangcs. WT.
Paper\.
Phys. 2Y (IYXY)
165.
Z. Naturforsch.
34~
Z. Naturford.
45s
and W. Eheling. V.S.
Vol-ohjov
Plnsmy (Nauka.
Plasma Spcctroacopy
Drawin
and F. Emard.
Klimontovitch,
Williams.
Proc. XVlllth
1111.C‘ont.
cd. (l!nivcr
of Wale\.
Kremp.
Kinetic
Schlangea
[IS]
D.R.
[lh]
T. Fujimoto.
Bates.
Th.
and Th. A.E.
Yakuhov.
51’). 613.
Kinctika
Ner;tvmrvesnoi
Ni\kotcm-
(McGraw-Hill.
Thcorq
New
( lY77)
York.
IY6X).
333.
of Electromagnetic
Processw
(Nauka.
A 8X (IYXX)
7171.
Mo~~o~~.
IYXO)
lYH2) [in Russian].
Bornath
M. Schlangcs,
LT.
(IVW) ( IYYO)
lYX2).
Physica C 85
Berlin-Heidelberg,
1121 M. Schlanges,
and
Moaco~.
Griem.
(Springer.
[ 1.31D. [13] M.
Gases.
I
A
Plasma
Bihcrmann.
pcraturnoi [Y] H.R.
in Ionized
Rev.
‘1. Rother
and K. Kilimann.
171 K. Kilimann
[IO]
Phy\.
R. Redmcr.
lY87).
(61 W. Ehcling [8] L..M.
C~ruboske.
A. Fiirstcr.
and I>. Kremp, Th.
Bornath,
Kingston
Bornath Contrih.
and R.W.P.
Phys. Rev.
and M. Bonitz, Plasma 373.
Plasma I’hys. 3
Phys. 3Y (IYXY)
McWhirtcr.
J. Phys. Sot. Jpn. 47 (lY7Y)
(‘ontrih.
Proc. R. Sot.
( IYXY) .i I I.
527. A X7
( IO62 )
37.