- Email: [email protected]
The effects of non-LTE kinetics on Marshak wave propagation
J. Quant. Spectrosc.
Pergamon
Rodtot. Transfer Vol. 58. No. M. pp. 50%518, 1991 Q 1997 Elsevier Science Ltd. All rights reserved
Printed in Great Bntaln 0022-4073/97 $17.00 + 0.00
PII: SOO224073(97)00057-5
THE EFFECTS OF NON-LTE KINETICS ON MARSHAK WAVE PROPAGATION C. BOWEN CEA-Limeil,
and M. BUSQUET
94195, Villeneuve
St Georges,
France
Abstract-The absorption-emission mechanism involved in a non-LTE plasma immersed in a radiation field is responsible for the reduction of the coupling between the material and the radiation. We consider how this affects the propagation of a thermal wave. 0 1997 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION The Marshak wave problem’ refers to an initially cold halfspace of material with radiation incident upon the surface. It is of interest in indirect drive ICF studies where laser energy is converted into
x-rays by the wall of a confining cavity. The multiple absorption and re-emission of the radiation by the wall acts to thermalise the x-rays, which drive the implosion of a target. The fraction of the radiation which is absorbed, either by the wall or the ablator of the target, propagates in the form of a thermal (Marshak) wave, with a well-defined front and a quasi-equilibrium between emission and absorption of photons. It is the small difference between these two processes which heats the material. The propagation of the radiation field is determined by the opacity K,,and source function S,, which depend on the atomic populations. However, it is now well known that the hypothesis of local thermodynamic equilibrium (LTE) is not valid for the range of densities and temperatures encountered in laser studies (10” - 1O23cmm3, 100-1000 eV). This is the domain of collisional-radiative (C-R) models, whose complexity increases rapidly with the number of transitions and becomes untractable for mid-Z or high-i! materials unless some sort of approximation is made. Our interest is with atomic physics packages used in radiative hydrodynamics, in particular RADIOM.2 The moderate precision required of a multigroup radiative transfer solver allows the treatment of a reduced number of transitions between groups of levels3 In essence, one is effectively only concerned with global excitation and de-excitation rates between the ground state and the thermal band of a given ion species (which groups levels in LTE with each other) and ionisation and recombination from one species to the next. Assuming the populations to be steady-state, the model introduces a non-LTE (NLTE) source function and an ionisation temperature. This permits, at low computational cost, the use of existing LTE tables in the calculation of radiation-dependent NLTE emissivities. What we wish to address here is the effect of NLTE physics on the propagation of the Marshak wave. We therefore split up the problem into its essential components. First, we will examine line transitions for a given ionic species. From what we have seen above, this reduces in effect to a two-level atom problem, the transition taking place with a given global rate between the ground state and the thermal band. Second, we address the case of transitions between ion states. This will be modeled by an ion consisting of only a ground state and a continuum. We will see that NLTE can dramatically affect the rate at which absorption and emission take place, and how this in turn affects the profile of the radiation front. The results we get by using the exact source functions will be compared with those obtained from the steady-state, analytical forms characteristic of RADIOM. JQSRT 58/4-6-E
509
510
C. Bowen and M. Busquet 2. BOUND-BOUND
TRANSITIONS
2.1. Time-dependent rate equations We begin with a very simple C-R model of line absorption. We treat photon transport in slab geometry, and in the diffusion approximation-the field intensity is assumed to be nearly isotropic so that it can be replaced by its angular average J,. The plasma contains free electrons of density n, and two-level atoms of density n, = n, + n2, where n, and n2 are the densities of the lower and upper levels respectively. We choose 2 = 1, i.e., n, = n,, and consider the following processes: l l
Excitation and de-excitation by collision with the free electrons. This cools or heats the plasma. Radiative excitation and spontaneous emission.
The collisional excitation rate is given by the Mewe formula, C,, = Cn,gf,,X-‘T,-“2e-“,
C = 1.6 x 10e5 s-‘cm-3eV3’2,
(1)
where z is the electron temperature, g = 0.2 the Gaunt factor,4,5 f the oscillator strength, x the difference in energy levels, and u = x/I;. The collisional de-excitation rate C,, is given by detailed balance,
* (1
C 21=
!.?! n,
C,,
(2)
= ze”’
where * denotes a LTE value and o,, o2 are the statistical weights of the levels. The spontaneous de-excitation rate is6 R21
=
R = 4.3 x 10’ ss’eVd2.
&Zfi,,
The radiative excitation rate is proportional detailed balance,
(3)
to the number of photons and the line profile &. By
1
J,,v - ‘&dv
R12 = RX2fi2e-”
J
9
(4)
B,,v- ‘&.dv s
where B,, = (2hv3/c2)exp( - hv/kT) is the Wien function. We will assume that B,. and J,, vary little over the line profile, and henceforth use v to denote the line center. One readily calculates the source function S, and the absorption coefficient k,,,
S,=
&
4,
(5)
hv RX2fi2e-” x)8= nI G 4%.. B,,
The evolution of the plasma is then described by the following system:
an,=4nh.g at
5 n,k 2
dJ
-$
(J,, -
W + nlC12- n2C2,
= X(n2C2, - n,C,,)
a
= crc,,(S,.- J,J + z
(7) (8)
(9)
The effects of non-LTE kinetics on Marshak wave propagation together
with
511
the Marshak condition’ for the external flux at the boundary, (10)
2.2. Steady-state approximation By assuming steady-state populations, we obtain an analytical expression for S, which gives us some insight into the physics. The NLTE population ratio is given by NLTE =
=
C,z
+
4,
C,,
+
R,,
(11)
* 1 + aJ,.lB,.
1+Gc
(12)
’
where o! = R2,/C2, =
1 34 x 10’3U3T7’2 e ’
~,WW
.
(13)
This parameter characterises the departure from LTE. When a 4 1, collisional de-excitation dominates radiative de-excitation and we recover the Boltzmann population ratio. We immediately deduce the NLTE steady-state source function
and the matter-radiation
exchange becomes V( = k,.(S,.- J,) = k,. +f$
.
(15)
Similar equations were obtained long ago by Mihalas.’ When J,.<
(16) This result is consistent with a diffusion process. For a S 1, the de-excitation which follows an absorption is much more likely to be radiative than collisional. The net result of this absorption-emission process is that the photons are in effect scattered rather than absorbed
512
C. Bowen and M. Busquet
F \. \
0.9 0.8 0.7 0.8 E t I=
0.5 0.4
20
80
100
140
180
220
280
300
340
380
x/lambda Fig. 1. Comparison of fronts in LTE (full lines) and NLTE (dotted lines), n, = 6 x 10” cm-‘. From left to right, r = 5, 40, 100, 200 and 300 ns. The mean free path I = 15 urn.
(destroyed by collisions), the effective absorption taking place over a time interval which is longer by a factor 0( 1 + a). In the diffusion approximation one therefore expects the photon destruction length to be longer than in LTE by a factor z J-- 1 + a. To understand the presence of the factor of l/2 in the result, we looked at a single mesh point. I; in fact relaxes roughly as 1 + 0.7x, the remaining discrepancy arising from the gradient of a over the front width. 3.
BOUND-FREE
TRANSITIONS
3.1. Time-dependent rate equations We now consider H-like ions of charge 2 consisting of only a ground state and a photoelectric edge at v,. The ions undergo radiative and collisional ionisation and recombination with the free
0.8 0.7 0.8 - .*-.-....._ . . -... *... -... *._. 0.5 -..* 0.4 0.3 a.-.-....-._._._. 0.2 -
\ \ I *..* I **.. , **.. , “*, I.
-.---._._._._*_
0.1 -
-0
1
2
3
4
5
: *. a.*a. . ;‘.&,___ .. L-
6
7
.-.-._
8
9
10
x/lambda Fig. 2. Comparison of LTE and NLTE fronts at 1 ns, grey approximation. The curves are as follows: LTE (full line), a = 64 (dashed), 640 (dotted) and 6400 (dot-dashed). The mean free path L = 40 urn.
The effects of non-LTE kinetics on Marshak wave propagation
513
0.5 i
0.4 -
-3
0.3 0.2 0.1 -
ol “““““““I 0
i
2
3
4
5
6
6
7
9 101112131415
x/lambda Fig. 3. Comparison of LTE and NLTE fronts at 1 ns, multigroup calculation. The curves are a~ follows: LTE (full line), a = 64 (dashed, 640 (dotted) and 6400 (dot-dashed). The mean free path L = 40 urn.
electrons. The charge state n, = Zn, + (Z Lotz’ formula
density of ions in the ground state will be denoted n, and the density of the next n +, with n, = n, + n, the total ion density. The free electron density + 1)n + is no longer constant but will be fixed by the state of ionisation. We use the for the collisional ionisation rate, c,,
= 42.5 x 10-6%-‘T;“*1.57EI(u),
where x = hv, and T is the free electron temperature,
(17)
u = x/z and E, is the exponential integral,
s 2 e-\
E,(u) =
dx .
x
(18)
u
Using detailed balance, the 3-body recombination
rate is
i >*c,,
c,, = 2
(19)
= ze”C, +
= An:
+ e
(20)
2.5 x lo-” X-‘T;‘i21.57eUE,(u),
(21)
Table 1, Comparison of multigroup relaxation times r and profile widths L, as a function of a, normalised to theu LTE values. The superscript ‘c’ labels the values calculated using Eq. (52) with yrnUr = 23 a 6.4 6.4 6.4 6.4 6.4 6.4
5 x x x x x x
IO-’ IO-’ IO” IO’ IO’ IO’
1.0 I.1 6.1 I8 22 23
TC
Lf
1.0 1.1 5.8 17 22 23
1.0 1.0 2.0 2.6 6.3 7.1
J-G 1.0 1.3 2.7 8.1 25 80
J; 1.0 1.0 2.5 4.2 4.7 4.8
514
C. Bowen and M. Busquet
where o, and w, are the statistical weights of the Z and Z + 1 ion ground states respectively, q= o+o,n;’ is the weight of the ionised state, and w, = AT:‘* is the free electron weight, A = 6.04 x lo*’ crnm3 eV-3’2.‘o We can use the same type of collisional rates as the for the line by defining bound-free oscillator strengths fi +, f+, and Gaunt factor g, + ,3 C
&+.h+ =
zfi+.
.f+, =
The radiative recombination
(22)
Cn,X-‘;e-“2e-”
(23)
rate is the one given by Seaton,”
R,
&*f+
I =
(24)
I
(25)
f+, = z 1.96
Unlike the line, the absorption profile 4” = (v,/v)~ must be taken into account. Recalling that the ionisation rate is proportional to the number of photons, we find
s cc
J,,v- 4dv
R,, = R&+e-”
h+ = Introducing
” ot B,,v- 4dv s p5
(26)
$f,I.
(27)
as before the source function and the absorption coefficient, s
K =
=
n+f+&
(28)
nd +e-’
n, RX*f, +e-"v-3 G
Jj
(29)
3
I,
B,.v- 4dv
we obtain for the evolution of the plasma the time-dependent
an,= 471 at
C-R system
(J,, - S,,)dv + n,C, + - n +C, ,
(30)
(31)
aJ
$
a
= CK($ - J,,) + 5
(32)
The effects of non-LTE
kinetlcs
on Marshak
wave propagation
515
By discretising the frequency variable into G groups, we obtain the multigroup equations dn, at =4+yS,)+n,C,+
--n+C+,
(33)
(34)
( >
!?L_!?_ LaJ, at - ax 31c, ax
+f@g-Jg),
g=
1,. .., G,
(35)
where, integrating over the group width,
Jg =
J,,dv, sg
K=p E
S, =
S,.dv sk-
(36)
s
rcB,,dv Lz (37)
B,,dv sR
s
v - “B,.dv
(v-“)s
=
g
(38)
B,.dv sR
3.2. Steady-state approximation As for the line, we derive an analytical form for the source function by assuming steady-state populations, =
I + d/B l+c?
=J,.v - 4dv,
B=
SyE
j =
6, c(=R+I= C +1
Wf+ 1 Cn,g,+f+,~-‘T;“*
Bt
(39)
=B,,~-~dv s,P
(401
1.34 x lo”$T:‘* =
n&5 + IO.3
’
(41)
It is then possible to write the exchange term in the formI
w = ic,.(S, - J,.) = K,. 1 - &
:
>
(B,, - J,.) + *
=[cr,,,_,.(JV,- B,.,) - a&J,, s “S
- B,.)]dv’ ,
(42)
516
C. Bowen and M. Busquet
where
s I
B,.v ‘dv “I vg = > K B,v - 4dv s “I -
(43)
and the frequency scattering coefficient (J,,,_v
-4 y
(f)
=
.
For each group, the frequency scattering contributes rc,B,,dvx(J,, - Bg,) R’
(vi),
_
(J,. - B,)v4dv B sg = rcgBg~(Jg,- Bg,)
rcB(Jg- Bg) v
,
(45)
6 where K is the Planck mean opacity, Cc rc,.B,.dv K=
d
s
B
B=
’
Cc B,dv . 1.
(46)
The scattering terms are conservative and therefore do not contribute
s
JC~= xc, 1 - &
kKdv = c~;(Bg - Jg), g “s
to the energy exchange, ;
.
(47)
g>
We see that the reduction in radiation-matter coupling is only significant in the vicinity of uO, typically close to o,, and vanishes for frequencies well above threshold. We can then exhibit the multigroup equations for steady-state source function,
an,= +~L+~,$(J~- B,)+nG+ at
--n+C+,
(48)
s
= 471i K~'(.& - Bg) - x 2 g=l
+
clc;(B, -
Jg) + &
(49)
cK,Bg~(Jg, - Bg,) g’
CKB(J, -
B,) q
g= 1, . . . . G.
}, (50)
The coupling term in Eq. (49) may be rewritten as xKg(
-
g
1 - $Jg
- 4) C%(Jg - B,) .
C’%(Jg - 4) 8
g
(51)
The effects of non-LTE
kinetics
on Marshak
wave propagation
517
The second term in the { } vanishes in the grey approximation, and the reduction of the radiation-matter coupling by a factor 1 + a is evident. However, for a multigroup calculation, this second term contributes a correction to the exchange rate. Eq. (51) indicates that, compared to the grey result, the NLTE to LTE ratio of relaxation times T(a) no longer increases indefinitely with a, but saturates at some value TV”. We therefore expect z(a) to satisfy approximately -I
z(a) = i
j&+f&
I
.
(52)
3.3. Results We consider a photoelectric edge at 2.1 keV, n, = 6 x 1022,2 = 12, wI = w + = 1 (this system is close to H-like Al). The drive temperature T’“’ = 700 eV is chosen so that the peak of the Wien function coincides with the absorption edge. The Planck mean free path A= K- ’ = 40 urn is calculated at the drive temperature. 3.3.1. Grey approximation. Fig. 2 shows some LTE and NLTE profiles at 1 ns. The ratio of profile widths is given to within 20% by fi for a < lo-‘, which is consistent with the corresponding relaxation times for a single mesh point. For larger a, the profile becomes very flat and the front difficult to define. 3.3.2. Multigroup calculations. Fig. 3 shows the multigroup fronts at 1 ns, in LTE and for a = 64,640 and 6400. As for the line, the same fronts are obtained by using the steady-state source function. However, it was found that it is not possible to ignore the frequency scattering terms in the source function, even though they do not contribute to the total energy, since the resulting profile is strongly deformed. As we can see in Table 1, the difference in width between a = 640 and 6400 is only about lo%, where it should be a factor of 3 if the widths scaled as fi as in the grey approximation. This shows clearly the impact of the saturation. The table also gives the single mesh point x relaxation times, which are seen to be well reproduced by Eq. (52). The last column of the table gives the square root of the relaxation time. Comparison with the profile widths shows that the two are not related through a simple diffusion-like relation, as here frequency scattering adds to the effective scattering process described in Sec. Sec. 2.3. 4. CONCLUSION
The two models presented here indicate that NLTE kinetics have little incidence on the velocity of the Marshak wave, but that the front is stretched compared to LTE. In the case of bound-bound transitions, the width of the front is consistent with a diffusion process, being related to the increased lifetime of the photons. As regards bound-free transitions, frequency scattering makes it difficult to establish a simple scaling law. Nevertheless, both types of transitions exhibit a precursor to the front which is absent in LTE. This could have some incidence on the predicted implosion of a target, and cause some discrepancy in the interpretation of measurements of the position of a wave front. It remains now to look carefully at realistic cases where the full opacity is taken into account, as well as the whole range of transport and hydrodynamic processes. In a real plasma, we will have a competition for these effects between lines and each photoionisation continuum. In the thermal band approximation, which is generally valid, the whole array of lines in a given ion acts more or less as a single transition with a broad spectral width Av which is of the order of the group width. The effect of frequency scattering is small, mainly because the frequency spread Av/l is only a fraction of unity (typically 0.2-0.6). Multiple excited states and ions will not change significantly the size of the effects of bound-bound transitions, which will average over the range of frequencies dominating radiation transport (3 to 5 kYJ. On the other hand, we have seen that bound-free transitions show a saturation of NLTE effects, which could under certain conditions dominate the larger effect of the bound-bound transitions. Preliminary investigations indicate that multiple photoionisation continua do not wash out the NLTE effects seen with a single threshold. We therefore expect the results presented here to be representative of NLTE in a real plasma. The importance of these effects in a Megajoule/NIF target is currently under investigation.
518
C. Bowen and M. Busquet
REFERENCES 1. Marshak, R. E., Phys. Fluids, 1958, 1, 24. 2. Busquet, M., Phys. Fluids B, 1993, 5, 4591. 3. Busquet, M., Phys. Rev. A, 1982, 25, 2302. 4. Mewe, R., Astron. Astrophys., 1972, 20, 215. 5. Van Regemorter, H., Astrophys. J., 1962, 136, 906. 6. Griem, H.R., Plasma Spectroscopy. McGraw-Hill, Maidenhead, 1964. 7. Pomraning, G. C., JQSRT, 1978, 21, 249. 8. Mihalas, D., Stellar Atmospheres. Freeman, London, 1970. 9. Lotz, W., Astrophys. J. Suppl., 1967, 14, 207. 10. Zel’dovich, Y. B. and Raizer, Y. P., Physics of Shock Waves and High-Temperature Phenomena. Academic Press, New York, 1966. 11. Seaton, M. J., Mon. Not. R. Astron. Sot., 1959, 119, 81. 12. Busquet, M., Unpublished, 1993.
Hydrodynamic
Pergamon
Rodtot. Transfer Vol. 58. No. M. pp. 50%518, 1991 Q 1997 Elsevier Science Ltd. All rights reserved
Printed in Great Bntaln 0022-4073/97 $17.00 + 0.00
PII: SOO224073(97)00057-5
THE EFFECTS OF NON-LTE KINETICS ON MARSHAK WAVE PROPAGATION C. BOWEN CEA-Limeil,
and M. BUSQUET
94195, Villeneuve
St Georges,
France
Abstract-The absorption-emission mechanism involved in a non-LTE plasma immersed in a radiation field is responsible for the reduction of the coupling between the material and the radiation. We consider how this affects the propagation of a thermal wave. 0 1997 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION The Marshak wave problem’ refers to an initially cold halfspace of material with radiation incident upon the surface. It is of interest in indirect drive ICF studies where laser energy is converted into
x-rays by the wall of a confining cavity. The multiple absorption and re-emission of the radiation by the wall acts to thermalise the x-rays, which drive the implosion of a target. The fraction of the radiation which is absorbed, either by the wall or the ablator of the target, propagates in the form of a thermal (Marshak) wave, with a well-defined front and a quasi-equilibrium between emission and absorption of photons. It is the small difference between these two processes which heats the material. The propagation of the radiation field is determined by the opacity K,,and source function S,, which depend on the atomic populations. However, it is now well known that the hypothesis of local thermodynamic equilibrium (LTE) is not valid for the range of densities and temperatures encountered in laser studies (10” - 1O23cmm3, 100-1000 eV). This is the domain of collisional-radiative (C-R) models, whose complexity increases rapidly with the number of transitions and becomes untractable for mid-Z or high-i! materials unless some sort of approximation is made. Our interest is with atomic physics packages used in radiative hydrodynamics, in particular RADIOM.2 The moderate precision required of a multigroup radiative transfer solver allows the treatment of a reduced number of transitions between groups of levels3 In essence, one is effectively only concerned with global excitation and de-excitation rates between the ground state and the thermal band of a given ion species (which groups levels in LTE with each other) and ionisation and recombination from one species to the next. Assuming the populations to be steady-state, the model introduces a non-LTE (NLTE) source function and an ionisation temperature. This permits, at low computational cost, the use of existing LTE tables in the calculation of radiation-dependent NLTE emissivities. What we wish to address here is the effect of NLTE physics on the propagation of the Marshak wave. We therefore split up the problem into its essential components. First, we will examine line transitions for a given ionic species. From what we have seen above, this reduces in effect to a two-level atom problem, the transition taking place with a given global rate between the ground state and the thermal band. Second, we address the case of transitions between ion states. This will be modeled by an ion consisting of only a ground state and a continuum. We will see that NLTE can dramatically affect the rate at which absorption and emission take place, and how this in turn affects the profile of the radiation front. The results we get by using the exact source functions will be compared with those obtained from the steady-state, analytical forms characteristic of RADIOM. JQSRT 58/4-6-E
509
510
C. Bowen and M. Busquet 2. BOUND-BOUND
TRANSITIONS
2.1. Time-dependent rate equations We begin with a very simple C-R model of line absorption. We treat photon transport in slab geometry, and in the diffusion approximation-the field intensity is assumed to be nearly isotropic so that it can be replaced by its angular average J,. The plasma contains free electrons of density n, and two-level atoms of density n, = n, + n2, where n, and n2 are the densities of the lower and upper levels respectively. We choose 2 = 1, i.e., n, = n,, and consider the following processes: l l
Excitation and de-excitation by collision with the free electrons. This cools or heats the plasma. Radiative excitation and spontaneous emission.
The collisional excitation rate is given by the Mewe formula, C,, = Cn,gf,,X-‘T,-“2e-“,
C = 1.6 x 10e5 s-‘cm-3eV3’2,
(1)
where z is the electron temperature, g = 0.2 the Gaunt factor,4,5 f the oscillator strength, x the difference in energy levels, and u = x/I;. The collisional de-excitation rate C,, is given by detailed balance,
* (1
C 21=
!.?! n,
C,,
(2)
= ze”’
where * denotes a LTE value and o,, o2 are the statistical weights of the levels. The spontaneous de-excitation rate is6 R21
=
R = 4.3 x 10’ ss’eVd2.
&Zfi,,
The radiative excitation rate is proportional detailed balance,
(3)
to the number of photons and the line profile &. By
1
J,,v - ‘&dv
R12 = RX2fi2e-”
J
9
(4)
B,,v- ‘&.dv s
where B,, = (2hv3/c2)exp( - hv/kT) is the Wien function. We will assume that B,. and J,, vary little over the line profile, and henceforth use v to denote the line center. One readily calculates the source function S, and the absorption coefficient k,,,
S,=
&
4,
(5)
hv RX2fi2e-” x)8= nI G 4%.. B,,
The evolution of the plasma is then described by the following system:
an,=4nh.g at
5 n,k 2
dJ
-$
(J,, -
W + nlC12- n2C2,
= X(n2C2, - n,C,,)
a
= crc,,(S,.- J,J + z
(7) (8)
(9)
The effects of non-LTE kinetics on Marshak wave propagation together
with
511
the Marshak condition’ for the external flux at the boundary, (10)
2.2. Steady-state approximation By assuming steady-state populations, we obtain an analytical expression for S, which gives us some insight into the physics. The NLTE population ratio is given by NLTE =
=
C,z
+
4,
C,,
+
R,,
(11)
* 1 + aJ,.lB,.
1+Gc
(12)
’
where o! = R2,/C2, =
1 34 x 10’3U3T7’2 e ’
~,WW
.
(13)
This parameter characterises the departure from LTE. When a 4 1, collisional de-excitation dominates radiative de-excitation and we recover the Boltzmann population ratio. We immediately deduce the NLTE steady-state source function
and the matter-radiation
exchange becomes V( = k,.(S,.- J,) = k,. +f$
.
(15)
Similar equations were obtained long ago by Mihalas.’ When J,.<
(16) This result is consistent with a diffusion process. For a S 1, the de-excitation which follows an absorption is much more likely to be radiative than collisional. The net result of this absorption-emission process is that the photons are in effect scattered rather than absorbed
512
C. Bowen and M. Busquet
F \. \
0.9 0.8 0.7 0.8 E t I=
0.5 0.4
20
80
100
140
180
220
280
300
340
380
x/lambda Fig. 1. Comparison of fronts in LTE (full lines) and NLTE (dotted lines), n, = 6 x 10” cm-‘. From left to right, r = 5, 40, 100, 200 and 300 ns. The mean free path I = 15 urn.
(destroyed by collisions), the effective absorption taking place over a time interval which is longer by a factor 0( 1 + a). In the diffusion approximation one therefore expects the photon destruction length to be longer than in LTE by a factor z J-- 1 + a. To understand the presence of the factor of l/2 in the result, we looked at a single mesh point. I; in fact relaxes roughly as 1 + 0.7x, the remaining discrepancy arising from the gradient of a over the front width. 3.
BOUND-FREE
TRANSITIONS
3.1. Time-dependent rate equations We now consider H-like ions of charge 2 consisting of only a ground state and a photoelectric edge at v,. The ions undergo radiative and collisional ionisation and recombination with the free
0.8 0.7 0.8 - .*-.-....._ . . -... *... -... *._. 0.5 -..* 0.4 0.3 a.-.-....-._._._. 0.2 -
\ \ I *..* I **.. , **.. , “*, I.
-.---._._._._*_
0.1 -
-0
1
2
3
4
5
: *. a.*a. . ;‘.&,___ .. L-
6
7
.-.-._
8
9
10
x/lambda Fig. 2. Comparison of LTE and NLTE fronts at 1 ns, grey approximation. The curves are as follows: LTE (full line), a = 64 (dashed), 640 (dotted) and 6400 (dot-dashed). The mean free path L = 40 urn.
The effects of non-LTE kinetics on Marshak wave propagation
513
0.5 i
0.4 -
-3
0.3 0.2 0.1 -
ol “““““““I 0
i
2
3
4
5
6
6
7
9 101112131415
x/lambda Fig. 3. Comparison of LTE and NLTE fronts at 1 ns, multigroup calculation. The curves are a~ follows: LTE (full line), a = 64 (dashed, 640 (dotted) and 6400 (dot-dashed). The mean free path L = 40 urn.
electrons. The charge state n, = Zn, + (Z Lotz’ formula
density of ions in the ground state will be denoted n, and the density of the next n +, with n, = n, + n, the total ion density. The free electron density + 1)n + is no longer constant but will be fixed by the state of ionisation. We use the for the collisional ionisation rate, c,,
= 42.5 x 10-6%-‘T;“*1.57EI(u),
where x = hv, and T is the free electron temperature,
(17)
u = x/z and E, is the exponential integral,
s 2 e-\
E,(u) =
dx .
x
(18)
u
Using detailed balance, the 3-body recombination
rate is
i >*c,,
c,, = 2
(19)
= ze”C, +
= An:
+ e
(20)
2.5 x lo-” X-‘T;‘i21.57eUE,(u),
(21)
Table 1, Comparison of multigroup relaxation times r and profile widths L, as a function of a, normalised to theu LTE values. The superscript ‘c’ labels the values calculated using Eq. (52) with yrnUr = 23 a 6.4 6.4 6.4 6.4 6.4 6.4
5 x x x x x x
IO-’ IO-’ IO” IO’ IO’ IO’
1.0 I.1 6.1 I8 22 23
TC
Lf
1.0 1.1 5.8 17 22 23
1.0 1.0 2.0 2.6 6.3 7.1
J-G 1.0 1.3 2.7 8.1 25 80
J; 1.0 1.0 2.5 4.2 4.7 4.8
514
C. Bowen and M. Busquet
where o, and w, are the statistical weights of the Z and Z + 1 ion ground states respectively, q= o+o,n;’ is the weight of the ionised state, and w, = AT:‘* is the free electron weight, A = 6.04 x lo*’ crnm3 eV-3’2.‘o We can use the same type of collisional rates as the for the line by defining bound-free oscillator strengths fi +, f+, and Gaunt factor g, + ,3 C
&+.h+ =
zfi+.
.f+, =
The radiative recombination
(22)
Cn,X-‘;e-“2e-”
(23)
rate is the one given by Seaton,”
R,
&*f+
I =
(24)
I
(25)
f+, = z 1.96
Unlike the line, the absorption profile 4” = (v,/v)~ must be taken into account. Recalling that the ionisation rate is proportional to the number of photons, we find
s cc
J,,v- 4dv
R,, = R&+e-”
h+ = Introducing
” ot B,,v- 4dv s p5
(26)
$f,I.
(27)
as before the source function and the absorption coefficient, s
K =
=
n+f+&
(28)
nd +e-’
n, RX*f, +e-"v-3 G
Jj
(29)
3
I,
B,.v- 4dv
we obtain for the evolution of the plasma the time-dependent
an,= 471 at
C-R system
(J,, - S,,)dv + n,C, + - n +C, ,
(30)
(31)
aJ
$
a
= CK($ - J,,) + 5
(32)
The effects of non-LTE
kinetlcs
on Marshak
wave propagation
515
By discretising the frequency variable into G groups, we obtain the multigroup equations dn, at =4+yS,)+n,C,+
--n+C+,
(33)
(34)
( >
!?L_!?_ LaJ, at - ax 31c, ax
+f@g-Jg),
g=
1,. .., G,
(35)
where, integrating over the group width,
Jg =
J,,dv, sg
K=p E
S, =
S,.dv sk-
(36)
s
rcB,,dv Lz (37)
B,,dv sR
s
v - “B,.dv
(v-“)s
=
g
(38)
B,.dv sR
3.2. Steady-state approximation As for the line, we derive an analytical form for the source function by assuming steady-state populations, =
I + d/B l+c?
=J,.v - 4dv,
B=
SyE
j =
6, c(=R+I= C +1
Wf+ 1 Cn,g,+f+,~-‘T;“*
Bt
(39)
=B,,~-~dv s,P
(401
1.34 x lo”$T:‘* =
n&5 + IO.3
’
(41)
It is then possible to write the exchange term in the formI
w = ic,.(S, - J,.) = K,. 1 - &
:
>
(B,, - J,.) + *
=[cr,,,_,.(JV,- B,.,) - a&J,, s “S
- B,.)]dv’ ,
(42)
516
C. Bowen and M. Busquet
where
s I
B,.v ‘dv “I vg = > K B,v - 4dv s “I -
(43)
and the frequency scattering coefficient (J,,,_v
-4 y
(f)
=
.
For each group, the frequency scattering contributes rc,B,,dvx(J,, - Bg,) R’
(vi),
_
(J,. - B,)v4dv B sg = rcgBg~(Jg,- Bg,)
rcB(Jg- Bg) v
,
(45)
6 where K is the Planck mean opacity, Cc rc,.B,.dv K=
d
s
B
B=
’
Cc B,dv . 1.
(46)
The scattering terms are conservative and therefore do not contribute
s
JC~= xc, 1 - &
kKdv = c~;(Bg - Jg), g “s
to the energy exchange, ;
.
(47)
g>
We see that the reduction in radiation-matter coupling is only significant in the vicinity of uO, typically close to o,, and vanishes for frequencies well above threshold. We can then exhibit the multigroup equations for steady-state source function,
an,= +~L+~,$(J~- B,)+nG+ at
--n+C+,
(48)
s
= 471i K~'(.& - Bg) - x 2 g=l
+
clc;(B, -
Jg) + &
(49)
cK,Bg~(Jg, - Bg,) g’
CKB(J, -
B,) q
g= 1, . . . . G.
}, (50)
The coupling term in Eq. (49) may be rewritten as xKg(
-
g
1 - $Jg
- 4) C%(Jg - B,) .
C’%(Jg - 4) 8
g
(51)
The effects of non-LTE
kinetics
on Marshak
wave propagation
517
The second term in the { } vanishes in the grey approximation, and the reduction of the radiation-matter coupling by a factor 1 + a is evident. However, for a multigroup calculation, this second term contributes a correction to the exchange rate. Eq. (51) indicates that, compared to the grey result, the NLTE to LTE ratio of relaxation times T(a) no longer increases indefinitely with a, but saturates at some value TV”. We therefore expect z(a) to satisfy approximately -I
z(a) = i
j&+f&
I
.
(52)
3.3. Results We consider a photoelectric edge at 2.1 keV, n, = 6 x 1022,2 = 12, wI = w + = 1 (this system is close to H-like Al). The drive temperature T’“’ = 700 eV is chosen so that the peak of the Wien function coincides with the absorption edge. The Planck mean free path A= K- ’ = 40 urn is calculated at the drive temperature. 3.3.1. Grey approximation. Fig. 2 shows some LTE and NLTE profiles at 1 ns. The ratio of profile widths is given to within 20% by fi for a < lo-‘, which is consistent with the corresponding relaxation times for a single mesh point. For larger a, the profile becomes very flat and the front difficult to define. 3.3.2. Multigroup calculations. Fig. 3 shows the multigroup fronts at 1 ns, in LTE and for a = 64,640 and 6400. As for the line, the same fronts are obtained by using the steady-state source function. However, it was found that it is not possible to ignore the frequency scattering terms in the source function, even though they do not contribute to the total energy, since the resulting profile is strongly deformed. As we can see in Table 1, the difference in width between a = 640 and 6400 is only about lo%, where it should be a factor of 3 if the widths scaled as fi as in the grey approximation. This shows clearly the impact of the saturation. The table also gives the single mesh point x relaxation times, which are seen to be well reproduced by Eq. (52). The last column of the table gives the square root of the relaxation time. Comparison with the profile widths shows that the two are not related through a simple diffusion-like relation, as here frequency scattering adds to the effective scattering process described in Sec. Sec. 2.3. 4. CONCLUSION
The two models presented here indicate that NLTE kinetics have little incidence on the velocity of the Marshak wave, but that the front is stretched compared to LTE. In the case of bound-bound transitions, the width of the front is consistent with a diffusion process, being related to the increased lifetime of the photons. As regards bound-free transitions, frequency scattering makes it difficult to establish a simple scaling law. Nevertheless, both types of transitions exhibit a precursor to the front which is absent in LTE. This could have some incidence on the predicted implosion of a target, and cause some discrepancy in the interpretation of measurements of the position of a wave front. It remains now to look carefully at realistic cases where the full opacity is taken into account, as well as the whole range of transport and hydrodynamic processes. In a real plasma, we will have a competition for these effects between lines and each photoionisation continuum. In the thermal band approximation, which is generally valid, the whole array of lines in a given ion acts more or less as a single transition with a broad spectral width Av which is of the order of the group width. The effect of frequency scattering is small, mainly because the frequency spread Av/l is only a fraction of unity (typically 0.2-0.6). Multiple excited states and ions will not change significantly the size of the effects of bound-bound transitions, which will average over the range of frequencies dominating radiation transport (3 to 5 kYJ. On the other hand, we have seen that bound-free transitions show a saturation of NLTE effects, which could under certain conditions dominate the larger effect of the bound-bound transitions. Preliminary investigations indicate that multiple photoionisation continua do not wash out the NLTE effects seen with a single threshold. We therefore expect the results presented here to be representative of NLTE in a real plasma. The importance of these effects in a Megajoule/NIF target is currently under investigation.
518
C. Bowen and M. Busquet
REFERENCES 1. Marshak, R. E., Phys. Fluids, 1958, 1, 24. 2. Busquet, M., Phys. Fluids B, 1993, 5, 4591. 3. Busquet, M., Phys. Rev. A, 1982, 25, 2302. 4. Mewe, R., Astron. Astrophys., 1972, 20, 215. 5. Van Regemorter, H., Astrophys. J., 1962, 136, 906. 6. Griem, H.R., Plasma Spectroscopy. McGraw-Hill, Maidenhead, 1964. 7. Pomraning, G. C., JQSRT, 1978, 21, 249. 8. Mihalas, D., Stellar Atmospheres. Freeman, London, 1970. 9. Lotz, W., Astrophys. J. Suppl., 1967, 14, 207. 10. Zel’dovich, Y. B. and Raizer, Y. P., Physics of Shock Waves and High-Temperature Phenomena. Academic Press, New York, 1966. 11. Seaton, M. J., Mon. Not. R. Astron. Sot., 1959, 119, 81. 12. Busquet, M., Unpublished, 1993.
Hydrodynamic