- Email: [email protected]
Radiative continua of noble gas plasmas
J. Quanr. .Specmsc.
Pergamon PII: s0022-4073(97)00136-2
RADIATIVE
CONTINUA
L. G. D’YACHKOVt§,
Radiar. Transfer Vol. 59, No. l/2, pp. 53-64. 1998 0 1998 Elsevier Science Ltd. All rights resewed Printed in Great Britain 0022-4073/98 $17.00 + 0.00
OF NOBLE
Y. K. KURILENKOVt
GAS PLASMAS and Y. VITELf
tlnstitute for High Temperatures, Russian Academy of Sciences, Izhorskaya 13/19, Moscow 127412. Russia and fLaboratoire des Plasmas Denses, Tour 12E5, Universite P et M Curie, boite 90,
4 Place Jussieu, 75252 Paris Cedex, France (Received
17 February
1997)
Abstract-A simple completely analytical method for the calculation of the plasma radiation continuum in the visible and neighbouring regions is derived. It is based on the semi-classical quantum defect theory. In spite of the simplicity the accuracy of the method is found to be comparable with those of Schliiter and Hofsaess calculations, but its main advantage is the reasonable description of the plasma density (weak coupling) effects in the near-threshold region. Thus, our calculation does not show the sharp threshold of the photoionization continuum and a correct comparison with experiment in the whole wavelength range under consideration can be made. A good agreement with experimental data on the continuous radiation of neon, argon, krypton and xenon plasmas is found. The calculation scheme presented may be useful for moderately dense plasma diagnostics and looks as reasonable basis for incorporation of strong coupling effects. 0 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
A considerable number of experimental studies on the emission and absorption continua of noble gas plasmas in the visible and neighbouring regions has been published in recent decades. In most of those papers experimental data are compared with calculations of Schliiter’ and Hofsaess’ in the quantum defect and scaled Thomas-Fermi approximations, respectively. However, a correct comparison can be made only for frequencies above the photoionization threshold (1 < 400-500 nm), since the calculations do not take into account the threshold shift due to the plasma effects; so they correspond to the case of ideal plasma, in fact, to the electron density N,+O. Spectra of real plasmas with finite values of N, will contain the manifestation of plasma density effects, i.e., the results of the influence of plasma surrounding on radiator. The level of this influence is different in dilute, moderately dense and very dense (non-ideal) plasmas. Correspondingly, different models should be applied for the description of plasma spectra: conceptions of isolated atom, weakly perturbed or strongly perturbed one would be used reasonably. The start of this problem for the upper members of spectral series concerns with the Inglis-Teller formula for apparent shift of photoionization thresholds: well-known overlapping of Stark components of split neighbour atomic levels in plasma microfields leads to merging of spectral lines before ideal photoionization thresholds. However, this formula was derived initially for low electron density astrophysical plasmas, and is suitable, strictly speaking, just for large principal quantum numbers. Atomic physics conceptions dominate in the calculations of opacities of dilute plasmas. The Inglis-Teller model extrapolated to plasmas of moderate density looks an insufficient one for proper description of near-threshold spectra. Under extreme densities the plasma coupling effects may even prevail in the reasonable physical picture for description of optical properties. Last case is the most complicated, strong coupling effects in plasma spectra are studied poorly until now and this subject is beyond the scope of this paper. A reasonable scheme for the calculations of near-threshold spectra for plasmas of moderate densities can be based on two physically clear key assumptions: first, statistical character of microfield action on atoms in a plasma and, second, undisturbed density of oscillator strengths for perturbed radiators (atoms or ions). A rather wide range of plasma parameters is available which §To whom all correspondence should be addressed. 53
54
L. G. D’yachkov
et al
correspond to these assumptions. Looking forward, such an approach may allow to clarify properly the density effects themselves beyond this scheme under extreme conditions. In this paper for the calculation of plasma radiation continuum, a new completely analytical method based on the semi-classical version of the quantum defect theory35 with a consequent description of the plasma effects in the near-threshold region is presented. Comparisons with the previous calculations’,2 and experimental data on radiative continua of neon, argon, krypton and xenon plasmas are performed and dependence of the radiation continua on the plasma parameters is inspected. 2. METHOD
OF CALCULATION
2.1. Photoionization cross-section The photoionization
cross-section of an atom is given by the expression 2n2 “@“@) = Jaa’
tie c cC(ly,l’y’) Ryd,,=,,, i’,
where n and 1 are the principal and angular momentum quantum numbers of y denotes the quantum numbers of angular momenta of the atom (y = jKJ y = SLJ for U-coupling), a is the fine structure constant, a,, is the Bohr dimensionless radial dipole matrix element and factor C(ly,I’y’) appears from the angles. The upper state is marked by a prime throughout the paper, E’ We calculate the radial matrix elements in the semi-classical approximation defect method and asymptotic expansion.3,4 Then we get
R,E,:!'Y'=
’
(
the optical electron, for j/-coupling and radius, Rf,? is the the integration over - E,,,, = lb. using the quantum
27
(2)
-
Z6crS
where 2 is the charge of the parent ion (Z = 1 for a neutral atom, Z = 2 for a singly charged ion, etc.), o = tio/Z2 Ryd, v = n - p = (Z’ Ryd/E,,;,)“’ is the effective principal quantum number of the bound state, p is its quantum defect, KP’ is the non-Coulomb part of the phase shift of the continuum state which is 71times quantum defect of the adjacent bound state series extrapolated to positive energy, fi = (~/o)~‘~(E,,;.-t E’)/2Z2 Ryd is the small parameter of the asymptotic expansion, x = c,(0/4)~‘~, f, = max(l,l’j = (I + 1’ + 1)/2 and S(x) = AZx”2Ai(x) - Ai’( T(x) = Alx”‘Gi(x) - Gi’(x) + x/2x, S,(x) = k (1 - 681~~‘~+ 4x3)Ai(x) + g x( 1 - Alx3j2)Ai’(x),
T(X) = k (1 - 6A1x312 + 4x3)Gi(x) + f x(1 - AIx3j2)Gi’(x) + &
AIx’12- &
x2,
s s n
Ai
= X-I
cos(xt + t3/3)dt,
0
a
Gi(x) = n-’
sinfxt + t3/3)drI
II
Here Al = 1’ = I+ 1, Ai is the Airy function, Gi(x) is the solution of a non-homogeneous equation similar to Airy’s equation6 and Ai’ and Gi’(x) are their derivatives.
55
Radiative continua of noble gas plasmas
and Ci(x) are expediently calculated In the case of interest x < 1 or at the least m 1. So, Ai by means of the power series. For Ai the corresponding expansion is given in Ref. [6]. It can be written in the simplest form by means of the recurrence relation between the coefficients:
Ai
= 2 ukxk, k=O
1 = 0.35503, a, = a’ = 32”1-(2/3)
3”3rl(l,3) = - 0.25882, az = 0, k 2 3.
ak-3 ak=
k(k-
A similar expansion for Gi(x) can be obtained too:
Gi(x) = 2 bkxk, k=O
bo=*,b,=
-+b,= fi
-&
d
bk-3 bk
=
k 2 3.
k(k - 1) ’
The factor C(ly,l’y’) is expressed in terms of 6j-symbols. In the case ofjl-coupling for excited states of rare gas atoms
applicable
CCjlKjl’K’)
and, taking into account the J splitting,
CCjlKJjl’K’J’)
= (2K + 1)(25’ + l){;
;,
“: J?C($KjlX’).
For all transitions allowed by the selection rules, the unified expression can be written C(jIKjl’K’)
[x: - (y, -j)‘][xt, - (& +j + I)‘] = ( - l)AK+’ 4(21+ 1)(21’ + l)K,,,(2K + 1 - Y,) ’
C(j/KJj,‘K’J’)
where
AK = K’ - K = 0,
6=4(J-K)(J’-K’)=
= 6 [2K + 1 + 6(25’ + l)]’ - 9 C(jlKjl’K’) 4(2K’ + 1)(2J + 1)
+ 1, Km = max{K,K’}, f 1.
X, = All,,,+ AKK,,
Y, = (1 - AK’)K
and
2.2. Absorption coeficient
The bound-free absorption coefficient kbC is defined as a sum of the photoionization cross-sections multiplied by the populations of the corresponding levels. Summation is performed over all states with the ionization energy less than 1To. For highly excited states the sum is replaced by an integral and can be written as an unified integral expression’ together with the free-free absorption contribution having the same structure. Let us denote by E. the lower limit of the
56
L. G. D’yachkov et al
integral. If the radial matrix element is approximated approximated by a linear function of energy P
by Eq. (2) and the quantum defects are
E
=a+bZ2
Ryd
(3)
3
this integral can be taken analytically. The corresponding theory is given in Ref. j. Here we present the final expression in a some more general form (and with some other denotes) than that in Ref. 5, allowing for the excited states of the parent ion and atomic level series converging to them. This is important for rare gas atoms, ions of which have low-lying excited levels 2P,i2. As a result, for the absorption coefficient we have
k(o) = kbf(0)+ kR(o)=
-
1
N,a(p,o) + k,,,(o)
)
(4)
hw < Ep < E.
(5) where NP is the concentration the quantum numbers),
of atoms (ions with charge Z - 1) in a state p (here p denotes all
16,/%xe4Z2
G!,(w)=
(6)
3w3$2G
is the integral expression for a hydrogenic atom (Kramers-Unsold formula), G,(o) is the Biberman-Norman function’ allowing for the specific features of a non-hydrogenic atom, N, and Ni are the electron and ion densities, o0 = min{o,]E,]/12) and G is the average Gaunt factor (over - o0 < E < co). The function tin,(o) is calculated from the following equation (pi is the state of the parent ion):
tint
=
1 + 1 D(piN’yY’)
(7)
3
p,&’
D(PiaYY’) = &
W,W)exp
( -
$){@I
+
+ SQ,)(COS $, COS V, - COS IjGoc0S 6)
7Q,[cos’h @VI + $1)- cos211/, cos(vO+ b)l} , (8)
where g(y) is the statistical weight of the y state of the atom, &(pi)and Ci are the excitation energy and the partition function of the parent ion, s = (a/2 - a,)(0/4)-~‘~, crO= fim$Z’ Ryd, z = (kT/Z2 Ryd)(o/4) - 213and tiO= arctan(2n7X,), v, = +!I + 27r(xo+ SX,), Qo = 167r2x1,-‘[S’(x) +
$I = arctan {2rc[rx, + (b’ - b)kT/Z* Ryd]}, K = $1 + 27r[x0+ s& + a’ - a + b’a - (6’ - b)a,J,
T2(x)1, Q, = 327c2xI,-‘[S(x)S,(x) + T(x)ZT(x)],
1 T(x) x0 = ; arctan S(X) ,
xl =
W) I;(x) - T(xMx) W’(x)
+ T2(x)]
Simple formula for the average Gaunt factor, including both bound-free and free-free transitions, is derived in Ref. *. In the linear approximation [by analogy with Eq. (2) and Eq. (3)], it is written as G = 1 + 0.1372[s +
7 -
&~~/2)“~].
It is easily seen that Eq. (8) reduces to zero if the quantum defects are negligible. So, the sum in Eq. (7) is restricted to the non-hydrogenic transitions. Accounting of the contributions from l,l’ I 3 is usually enough.
51
Radiative continua of noble gas plasmas
To calculate the free-free absorption coefficient kR (and corresponding rR function), one should put & = 0 in the expression for kin,. The difference between kin, and kR gives k:,. Then one can get kbf, adding kkl to the sum in Eq. (4).
2.3. Plasma efects in the near-threshold region It is well known that the interaction between the radiating atom and surrounding plasma particles gives rise to a ‘red’ shift and spread of the photoionization thresholds. For the description of the near-threshold region, we use the microfield model approach which was developed by many authors9-” For the electron on an atomic level a potential barrier is formed in the direction of the plasma microfield. If the microfield is strong enough, the level may be turned out above the barrier top and disappear. The realization probability of the level E, is then defined as the integral of the microfield distribution function
s FP
W(4) =
P(FNF,
(9)
0
where F, is the critical value of the field, for which the level is found at the top of the barrier and ceases to be bound. This leads to the weakening of the upper members of the spectral line series. However, unbound states appear above the barrier instead of the destroyed levels and a continuum replaces the lines with a probability 1 - W. So, we continue the photoionization continuum multiplied by a factor 1 - W over the zero-field threshold towards the long wavelengths. Near the threshold W+O, while W+l far off it. Therefore, the line series passes gradually to a continuum. Each of the terms in the sum (1) corresponds to a line series; consequently, it is continued with the own factor 1 - W. (Strictly speaking, the photoionization continuum would be continued with the factor 1 - W/W,, where W, is the realization probability of the lower state of the continuum and adjacent spectral series. However, in the spectral region under consideration W, 1 1 for all the spectral series of noble gas atoms if the electron density is not very high.) This way of the continuation of the photoionization continuum is based on the assumption of the conservation of the density of oscillator strengths when the lines transform to a continuum.‘.‘9.20 Using a similar way of the continuation, detailed description of the near-threshold region has early been obtained for hydrogen.‘3,2’,22To estimate the critical values of the microfield for the hydrogenic states, the uniform field model has been applied. In the present calculation we use Hooper’s microfield distributions23 and the simple analytical approximation derived in Ref. 24for the cumulative of these distributions. Thus, our calculation method is a completely analytical one. The choice of the critical values of the microfield is not a simple problem (a review can be found in Ref. I”). In this paper we use the simplest expressions from the uniform field (UF), Fp = Ei/4e3, and nearest neighbour (NN), F, = E,/ 16e’, approximations.‘6, ‘*We shall show that in contrast to hydrogen, NN is more appropriate for noble gas atoms (see also Ref. “). 3. RESULTS
AND COMPARISON WITH OTHER AND EXPERIMENT
CALCULATIONS
Results of calculations and measurements of the absorption (emission) coefficient are frequently presented, by analogy with Eq. (5) in terms of r function
k(w) = k”(W)<(W),
(10)
where k”(W) is defined by k”(w) =
16,/2 7rSue4Z2 NJ, exp 303JmT
fiw kT
(11)
or any similar way. Eq. (11) is the same as Eq. (6) with G = 1 and o. = o. In such a traditional form we present the results of the derived method. In our calculations the photoionization cross-sections of all states belonging to ns, (n + l)s, (n + 2)s, np, (n + l)p, nd, (n + l)d, ns’, (n + l)s’, (n + 2)s’, np’, (n + I)$, nd’ and (n + 1)d’ configurations (56 states for each atom
58
L. G. D’yachkov et al
allowing for the K and J splitting; n = 3,4, 5, 6 for neon, argon, krypton and xenon, respectively) are calculated and included into the sum in Eq. (4), while the photoionization of other states is taken into account by the integral expression (5) (E,, = - 0.065 Ryd for argon, krypton and xenon, while E,, = - 0.0627 Ryd for neon). Energy levels are taken from Refs. 25and 26.The quantum defects are extrapolated to the continuum states by means of Eq. (3). 3.1. Comparison with previous calculations Schliiter’ and Hofsaess’ have calculated the gb’ function defined by the equation kbf = 2a4Z2kTA3 3&c2e2
Na$exp(
-
&)[exp(
-$$-
l]ib’,
(12)
where N,, C, and Zare the total concentration, partition function and ionization energy of the atom, respectively, and y0 = 6 is the statistical weight of the ground state of the parent ion without allowing for the multiplet splitting. The function rbf in Eq. (12) is related to the function 5 in Eq. (10) [with k” according to Eq. (ll)] by the equation
5=
$[l-
exp(
&)]tbi+exp(
-
- &)Cfl
if the relation between N,, Ni and N, is established by the Saha equation for ideal plasma. Our method is compared with those calculations in Figs 1 - 4 for neon, argon, krypton and xenon. Simple analytical calculation (solid curve) is in a good agreement with calculations of Schltiter’ (dashed-dotted curve) and Hofsaess2 (dashed curve). However, these calculations cannot correctly be compared with experiment data for dense plasmas in the near-threshold region (A 2 400-500 nm), since no plasma effects leading to the ‘red’ shift and spread of the threshold are taken into account. In the same figures we also show our results taking into account the plasma effects in the near-threshold regions for electron density values lOI and 10” cm” according to Sec. 2.3 (dotted curves) with the critical values of the microfield from the NN approximation.
I -
Neon T= 15000 K
400
600
500
Wavelength
700
800
(nm)
Fig. 1. The lb’ function of neon. Comparison of present semi-classical calculation (solid curves) with previous calculations of Schliiter’ (dasheddotted curve) and Hofsaes$ (dashed curve). Dotted curves show the results of the present calculation taking into account the plasma effects in the near-threshold region with the NN approximation for the estimation of the critical values of the microfield.
59
Radiative continua of noble gas plasmas
L
,
N,=lO"~m'~ -
Argon T= 12000
200
300
400
500
600
K
700
800
Wavelength (nm) Fig. 2. As in Fig. 1 for argon.
3.2. Comparison with experiment Our calculation results are compared with experimental data on the radiative continua of neon, argon, krypton and xenon plasmas in Figs 5 - 8 respectively. Presented 5 functions correspond to the absorption (emission) coefficient taking into account both bound-free and free-free transitions and are defined by Eq. (10) and Eq. (11). Solid curves in Figs 5 - 8 show the calculation results with the critical values of the microfield from the NN approximation, while dashed curves correspond to the UF approximation. 3.2. I. Neon. Continuum radiation of neon plasma was measured by Schnapauff 27in a cascade arc and Gavrilov2* in pulsed discharges through fused quartz tubes. In Fig. 5 our calculations are compared with their results. Though the calculations underestimate the radiative continuum, they give the correct behaviour of the spectrum as the plasma density increases: the minimum in the near-threshold region (12= 450-600 nm) disappears gradually. Furthermore we can conclude that the better agreement between the calculation and experiment is observed with using the NN model. 3.2.2. Argon. The most of the experimental studies of the noble gas plasma spectra deals with argon. Plenty of experimental data on argon spectra in a wide range of the plasma conditions has
: ,’ 200
1
I
I
I
I
I
300
400
500
600
700
800
Wavelength
(nm)
Fig. 3. As in Fig. 1 for krypton.
60
L. G.
D’yachkovet al
Xenon T= I 1:1
200
I
I
I
300
10000 K
I
I
400 500 600 Wavelength (nm)
I
I
700
800
J
Fig. 4. As in Fig. 1 for xenon. been obtained in recent decades. Some of them are shown in Fig. 6 as well as the results of our calculations. We have selected the experimental data29-39obtained after 1975 from plasmas with close temperatures and electron densities (but produced in differ plasma sources) and being in an agreement with each other within the measurement error. (More detailed analysis of available experimental data is given by D’yachkov et a1.40)The data of Refs. [29, 31-36,38, 391 correspond to a temperature range 10 000-14 500 K, while the data of Refs. [30] and [37]correspond to higher temperatures 16 900 and 20 000 K respectively; the electron densities are near 10” cm - ’ (the pressure is one or a few atmospheres). Argon plasmas were generated in shock tubes29,38and wall-stabilized 30-34,36.39, gas-flow-stabilized” and free-burning35 arcs. A good agreement between the shown experimental data allows to believe that they are sufficiently reliable. In the short wavelength region (1 < 400 nm) free from the significant spectral lines, the experimental data except Ref. [30] have a small scatter, while some excess of the data of Behringer and Thoma30 over those of Refs. [31-34,36,39] is due to the temperature dependence of the
0.1 200
300
400
500
600
700
800
900
1000
1100
Wavelength (nm) Fig. 5. The r function of neon. Comparisonof experimentdata (symbols)and present semi-classical calculationsusingboth NN (solid curves) and UF (dashed curves) approximations for the estimation of the critical values of the microfield: + (Schnapauff’) and 1, N, = 3.79 x 10tbcme3, T = 14680 K; q (GavriloP) and 2, NC= 1.4 x 10” cm- ‘, T= 21 000 K; l (Gavrilov 28)and 3, N. = 1.1 x lOI8cm-‘, T= 21 000 K.
Radiative continua of
200
300
400
61
noble gas plasmas
500
600
700
800
Wavelength (nm) Fig. 6. The C function of argon. Comparison of experiment data (symbols) and present semi-classical calculations using both NN (solid curves) and UF (dashed curves) approximations for the estimation of the critical values of the microfield. Experiment: Meiners and Weiss”, +, N. = (0.4-1.7) x 10” cm-‘, T = 10 5O(rl2 500 K; Behringer and ThornaM, +,N.=1.99x10”cm~‘,T=169OOK;Goldbachet al”, 0 N = 1 2 x IO” cm-‘, T = 13 200 K; PrestotQ2-“, x , N = 2 07 x IO” cm - ‘, T = 13 780 K; Akamaisu gnd Takata’5, A, N, = 1.09 x 10” cm-‘, T= 13 000 K; Schnehage et alJ6, 0, N, = (0.711.67)x IO” cm-‘, T = 12 1It&l4 430 K; Belov et al”, n , N, = 1.8x 10” cm-‘, T = 20 000 K; Zangers T = IO 000-12 500 K; Wilbers et al”‘, 0, and MeinersJ8, ., N = (0.42-I 5) x 10” cm-’ N, = 3.12 x IO” cm-‘, Ti 13 500 K: Present calculations: 1. N, = 2.07 x IO” cm-‘, T= 13 780 K; 2, N,= 1.99 x 10”cm-‘, T= 16900 K.
t function. Approximately the same temperature dependence is demonstrated by our calculations in this spectral region. Curves 1 and 2 in Fig. 6 show the calculation results for the experimental conditions of Preston3z-34 (N, = 2.07 x 10” cm-‘, T = 13 780 K) and Behringer and Thoma” (IV, = 1.99 x 10” cm - 3, T = 16 900 K), respectively. For longer wavelengths (,I > 400 nm) the
3
g
2
.S B z 0 0.; 0.8 0.7 0.6 0.5 300
400
500
600
700
800
Wavelength (nm) Fig. 7. The l function of krypton. Comparison of experiment data (symbols) and present semi-classical calculations using both NN (solid curves) and UF (dashed curves) approximations for the estimation of the critical values of the microfield. Experiment: Meiners and Weiss29, 0. N, = 4 x lOI cm-‘, T= 9500 K and +, N, = 2 x 10” cm-‘, T= 11 400 K; Baessler et a/.4’, 0, N, = 4.3 x lOI cm-‘, T=10150K and 0, N,=1.4xlO”cm-‘, T=12200K; Manola et al@, +, N,=(4.548.25) x lOI cm-‘, T= 9520-10 750 K; Vitel et al”, 0, N, = 9.6 x IO” cm-‘, T= 13 600 K and n , N. = 1.72 x IO’*cm-‘, T= 14900 K. Present calculations: 1, N. = 4.3 x lOI cm-‘, T= 10 150 K; 2, N,= 1.72 x IO’*cm-‘, T= 14900 K.
62
L. G. D’yachkov et al
200
300
400
500
600
700
800
900
Wavelength (nm) Fig. 8. The i function of xenon. Comparison of experiment data (symbols) and present semi-classical calculations using both NN (solid curves) and UF (dashed curves) approximations for the estimation of the critical values of the microfield. Experiment: Meiners and Wei@, 0, N, = 2.2 x lOu cm-‘, T= 7900 K, n , N, = 3.2 x 1Or6cm-‘, T= 8200 K, 0, N. = 7.8 x lOr6cm-‘, T= 9000 K and +, N = 1.5 x 10” cm-3 , T= 9800 K; Goldbach et al”, IJ, N. = 10” cm-‘, T= 10 050 K and A, N: = 2.2 x IO” cm-‘, T = 12 300 K; Gavrilov and GavrilovaQ, +, N, = 3.35 x 10” cm-’ and x , N. = 8.03 x IO” cm-‘, T = 14 000 K; Vitel et al”, 0, N, = 7.4 x 10” crnm3, T= 11 400 K. Present calculations: 1, N, = 10” cme3, T= 10050 K; 2, N,=2.2 x 10”cm-‘, T= 12300 K; 3, N,=7.4x lO”cm-‘, T= 11400 K.
temperature dependence become weak, but the scatter of the experimental data increases. The last can apparently be explained by difficulty in the problem of the separation of the line-wing contributions.38 From the comparison of the calculation results with experimental data one can conclude that the agreement between them is quite good, if the critical values of the microfield are defined from the NN model, while it is some worse for the UF model. 3.2.3. Krypton. Radiation continuum of krypton plasma has been measured in shock tubes by Berge et a14’,Meiners and Weiss2g and Manola et a1,42wall-stabilized arc by Baessler et a1,43and flashlamps by Vitel et al.* Electron density range of about two orders (3 x lOI - 1.7 x lo’* cme3) is covered; temperature values are between 9500 and 15 700 K. A part of the measurement data giving a clear view of the whole collection of the data is displayed in Fig. 7. Results of our calculations for low and high values of the plasma parameters corresponding to the experimental conditions of Baessler et a143 (NC= 4.3 x lOI cmm3, T = 10 150 K) and Vitel et al” (N, = 1.72 x lo’* cme3, T = 14 900 K) are represented by curves 1 and 2, respectively. The calculation agrees well with the measurements of Baessler et a143and in the mean with data of Manola et a14*obtained for close plasma parameters and having a considerable scatter. Small scatter in the data of Baessler et a143gives the possibility of seeing a trend in the decrease of the ( function as the electron density increases up to 1.4 x 10” cm-3 at T = 12 200 K. A similar result has been obtained by Meiners and Weissz9studying emission from krypton plasma at I = 456.1 nm as a function of N$T”*. Two points from Ref. [29] corresponding to the minimum and maximum values of the electron density and temperature are presented in Fig. 7. Subsequent decrease of the 5 function can be seen from the comparison of the measurement results of Vitel et al” for the higher electron densities with Refs. [29,42,43]. It should however be noted that this effect is small and comparable with the accuracy of the measurements (lO-25%). On the whole we can conclude that our calculation results agree with the experimental data within the experimental error, although they are slightly higher than the data of Vitel et al.” But in this case it is difficult to choose decidedly the model (NN or UF) for the estimation of the critical values of the microfield. However, taking into account the corresponding choice for neon and argon, we think that the NN model is more appropriate for krypton too. 3.2.4. Xenon. In Fig. 8 are shown the experimental data on the radiation from xenon plasmas produced in shock tubes,29 wall-stabilized arcs4s and flashlamps. 44.46Here we restrict ourself to the
Radiative continua of noble gas plasmas
63
consideration of electron density values up to 10” cm - 3, although measurements with xenon, as well as argon, were performed also for higher densities. The shown experimental data are in a satisfactory agreement (except the one point at I = 320 nm from Ref. [47]), but, as in the case of krypton, a decrease of the 5 function with increasing electron density can be noted.29,46 Our calculation results are presented for the plasma parameters corresponding to the experimental conditions of Goldbach et a14’(curve 1, T = 10 050 K and N, = 10” cm - ‘; curve 2, T = 12 300 K and N, = 2.2 x 10” cmm3) and Vitel et al4 (curve 3, T = 11 400 K and N, = 7.4 x 10” cm-‘). In this case, in contrast to krypton, we can do the evident choice of the NN model (solid curves), since it gives rise to a good agreement with experiment, while the UF model (dashed curves) gives the r function that is considerably lower than the experimental data for I > 600 nm. Moreover, comparing solid curves l-3 with corresponding experimental data, one can see that the theory (using NN) correctly reflect the variations of the 5 function with varying temperature and electron density. 4. CONCLUSIONS We present a simple completely analytical method for the calculations of plasma radiation continua generated from bound-free and free-free transitions of electrons in the field of a positive ion. It is based on the semi-classical version of the quantum defect theory, uses the microfield model approximation and assumes the conservation of the density of oscillator strengths for the extrapolation of the photoionization continuum over its long-wavelength edge. A comparison of the calculation results with experimental data for neon, argon, krypton and xenon plasmas having a temperature in the range 8000-21 000 K and electron density between lOI and 10” cm - 3, has been performed and it is shown that this method gives a rather good approximation of the radiation continuum in the spectral range 200-1000 nm. In contrast to the well-known previous calculations,‘.* our method does not show the sharp edge of the photoionization continuum, since it takes into account the plasma effects in the near-threshold region and gives a reasonable description of this spectral region important for experimental studies and applications. In the framework of the microfield model approximation, for the estimation of the critical values of the microfield we use the simplest expressions from two of the rather crude models, uniform field and nearest neighbour (an accurate physical model would apparently lay between them). Comparison with experimental data shows that the NN model provides a considerably better agreement between theory and experiment in the near-threshold region for noble gas plasmas. We think that this model would apparently be more appropriate in the general case of complex atom plasmas. Our simple method may be used, looking forward, for comparison with experimental data on very dense plasma radiation to clarify the density (or coupling) effects in plasma optical properties.47 In particular, it would be useful as the basis for the construction of the method for the calculations of the opacities of dense plasmas: if the density effects are known independently, they can easily be incorporated into the simple calculation scheme presented above. Note, that further comparison with experimental data on dense plasma radiation is a tool to test or check any predictions on coupling effects in radiation too. Last but not least, some complex modern problems of moderately dense plasmas diagnostics based on radiation continuum may be solved easily and accurately on the basis of the method described in this paper. Acknowledgements-We would like to thank G. A. Kobzev and G. Maynard for stimulating discussions. We acknowledge with thanks also the partial support of this work by the NATO International Scientific Exchange Program under Linkage Grant No. HTECH.LG 960803.
REFERENCES I. Schliiter, D., Z. Phys., 1968, 210, 80. 2. Hofsaess, D., JQSRT, 1978, 19, 339. 3. D’yachkov, L. G. and Pankratov, P. M., J. Phys. B, 1991, 24, 2267. 4. D’yachkov, L. G. and Pankratov, P. M., J. Phys. B, 1994, 27, 461. 5. D’yachkov, L. G., High Temp., 1992, 30, 709. 6. Abramowiz, M. and Stegun, I. A., eds., Handbook of Mathematical Functions. Dover, New York, 1965.
64
L. G. D’yachkov et al
7. Biberman, L. M. and Norman, G. E., Sov. Phys.-Usp., 1967, 10, 51. 8. D’yachkov, L. G., J. Phys. B, 1990, 23, L429. 9. Unsold, A., Z. Astrophys., 1948, 24, 355. 10. Avilova, I. V. and Norman, G. E., Teplojiz. Vys. Temp., 1964, 2, 517. 11. Giindel, H., Be&r. Plasmaphys., 1971, 11, 1. 12. Kobzev, G. A., Sou. Phys. JETP, 1972, 34, 310. 13. Gurovich, V. Ts. and Engelsht, V. S., Sov. Phys. JETP, 1977, 45, 232. 14. Kobzev, G. A. and Kurilenkov, Yu. K., High Temp., 1978, 16, 385. 15. Ruzdjak, V. and Vujnovic, V., Astron. Astrophys., 1977, 54, 751. 16. Sevastyanenko, V., Contrib. Plasma Phys., 1985, 25, 151. 17. Gavrilov, V. E. and Gavrilova, T. V., Opt. Spectrosc., 1987, 63, 429. 18. Hummer, D. G. and Mihalas, D., Astrophys. J., 1988, 331, 794. 19. D’yachkov, L. G., Kobzev, G. A. and Pankratov, P. M., J. Phys. B, 1988, 21, 1939. 20. D’yachkov, L. G., Kobzev, G. A. and Pankratov, P. M., JQSRT, 1990, 44, 123. 21. DIppen, W., Anderson, L. and Mihalas, D., Astrophys. J., 1987, 319, 195. 22. Seaton, M. J., J. Phys. B., 1990, 23, 3255. 23. Hooper, C. F., Jr., Phys. Rev., 1968, 165, 215. 24. D’yachkov, L. G., JQSRT (submitted). 25. Moore, C. E., Atomic Energy Levels: NBS Circular No 467. US Government Printing Office, Washington, DC., 194991958. 26. Baskin, S. and Stoner, J., Jr., Atomic Energy Levels and Grotrians Diagrams, Vol. 1, 1975; Vol. 2, 1978, Amsterdam. 27. Schnapauff, R., Z. Astrophlls., 1968, 68, 431. 28. Gavrilov, V. E., Opt. Spectrosc., 1992, 72, 8. 29. Meiners, D. and Weiss, C. O., JQSRT, 1976, 16, 273. 30. Behringer, K. and Thoma, P., JQSRT, 1976, 16, 671. 31. Goldbach, C., Nollez, G. and Plomdeur, P., J. Phys. B, 1977, 10, 1181. 32. Preston, R. C., J. Phys. B, 1977, 10; 1377. 33. Preston, R. C., J. Phys. B, 1977, 10, 1573. 34. Preston, R. C., JQSRT, 1977, 18, 337. 35. Akamatsu, I. and Takata, S., JQSRT, 1977, 18, 459. 36. Schnehage, S. E., Kock, M. and Schulz-Gulde, E., J. Phys. B, 1982, 15, 1131. 37. Belov, S. N., Nikolaevsky, L. S. and Podmoshensky, I. V., TeploJiz. Vys. Temp., 1988, 26, 169. 38. Zangers, J. and Meiners, D., JQSRT, 1989, 42, 25. 39. Wilbers, A. T. M., Kroesen, G. M. W., Timmermans, G. J. and Schram, D. C., JQSRT, 1991, 45, 1. 40. D’yachkov, L. G., Kobzev, G. A. and Pankratov, P. M., High Temp., 1996, 34, 855. 41. Berge, 0. E., Bohm, A. and Rehder, L., Z. Naturforsch., 1964, 20a, 120. 42. Manola, S., Devic, S. and Lesage, A., JQSRT, 1991, 46, 577. 43. Baessler, P., Obbarius, H.-U. and Schulz-Guide, E., Physica C, 1979, 96, 147. 44. Vitel, Y., Siyacoun, A., Giry, L. and Louvet, G., J. Phys. B, 1993, 26, 4333. 45. Goldbach, C., Nollez, G. and Stack, D., J. Phys. B, 1976, 9, 1191. 46. Gavrilov, V. E. and Gavrilova, T. V., Opt. Spectrosc., 1990, 69, 26. 47. Kobzev, G. A., Iakubov I. T. and Popovich, M. M., eds., Transport and Optical Properties of Nonideal Plasma. Plenum Press, New York, 1995.
Pergamon PII: s0022-4073(97)00136-2
RADIATIVE
CONTINUA
L. G. D’YACHKOVt§,
Radiar. Transfer Vol. 59, No. l/2, pp. 53-64. 1998 0 1998 Elsevier Science Ltd. All rights resewed Printed in Great Britain 0022-4073/98 $17.00 + 0.00
OF NOBLE
Y. K. KURILENKOVt
GAS PLASMAS and Y. VITELf
tlnstitute for High Temperatures, Russian Academy of Sciences, Izhorskaya 13/19, Moscow 127412. Russia and fLaboratoire des Plasmas Denses, Tour 12E5, Universite P et M Curie, boite 90,
4 Place Jussieu, 75252 Paris Cedex, France (Received
17 February
1997)
Abstract-A simple completely analytical method for the calculation of the plasma radiation continuum in the visible and neighbouring regions is derived. It is based on the semi-classical quantum defect theory. In spite of the simplicity the accuracy of the method is found to be comparable with those of Schliiter and Hofsaess calculations, but its main advantage is the reasonable description of the plasma density (weak coupling) effects in the near-threshold region. Thus, our calculation does not show the sharp threshold of the photoionization continuum and a correct comparison with experiment in the whole wavelength range under consideration can be made. A good agreement with experimental data on the continuous radiation of neon, argon, krypton and xenon plasmas is found. The calculation scheme presented may be useful for moderately dense plasma diagnostics and looks as reasonable basis for incorporation of strong coupling effects. 0 1998 Elsevier Science Ltd. All rights reserved
1. INTRODUCTION
A considerable number of experimental studies on the emission and absorption continua of noble gas plasmas in the visible and neighbouring regions has been published in recent decades. In most of those papers experimental data are compared with calculations of Schliiter’ and Hofsaess’ in the quantum defect and scaled Thomas-Fermi approximations, respectively. However, a correct comparison can be made only for frequencies above the photoionization threshold (1 < 400-500 nm), since the calculations do not take into account the threshold shift due to the plasma effects; so they correspond to the case of ideal plasma, in fact, to the electron density N,+O. Spectra of real plasmas with finite values of N, will contain the manifestation of plasma density effects, i.e., the results of the influence of plasma surrounding on radiator. The level of this influence is different in dilute, moderately dense and very dense (non-ideal) plasmas. Correspondingly, different models should be applied for the description of plasma spectra: conceptions of isolated atom, weakly perturbed or strongly perturbed one would be used reasonably. The start of this problem for the upper members of spectral series concerns with the Inglis-Teller formula for apparent shift of photoionization thresholds: well-known overlapping of Stark components of split neighbour atomic levels in plasma microfields leads to merging of spectral lines before ideal photoionization thresholds. However, this formula was derived initially for low electron density astrophysical plasmas, and is suitable, strictly speaking, just for large principal quantum numbers. Atomic physics conceptions dominate in the calculations of opacities of dilute plasmas. The Inglis-Teller model extrapolated to plasmas of moderate density looks an insufficient one for proper description of near-threshold spectra. Under extreme densities the plasma coupling effects may even prevail in the reasonable physical picture for description of optical properties. Last case is the most complicated, strong coupling effects in plasma spectra are studied poorly until now and this subject is beyond the scope of this paper. A reasonable scheme for the calculations of near-threshold spectra for plasmas of moderate densities can be based on two physically clear key assumptions: first, statistical character of microfield action on atoms in a plasma and, second, undisturbed density of oscillator strengths for perturbed radiators (atoms or ions). A rather wide range of plasma parameters is available which §To whom all correspondence should be addressed. 53
54
L. G. D’yachkov
et al
correspond to these assumptions. Looking forward, such an approach may allow to clarify properly the density effects themselves beyond this scheme under extreme conditions. In this paper for the calculation of plasma radiation continuum, a new completely analytical method based on the semi-classical version of the quantum defect theory35 with a consequent description of the plasma effects in the near-threshold region is presented. Comparisons with the previous calculations’,2 and experimental data on radiative continua of neon, argon, krypton and xenon plasmas are performed and dependence of the radiation continua on the plasma parameters is inspected. 2. METHOD
OF CALCULATION
2.1. Photoionization cross-section The photoionization
cross-section of an atom is given by the expression 2n2 “@“@) = Jaa’
tie c cC(ly,l’y’) Ryd,,=,,, i’,
where n and 1 are the principal and angular momentum quantum numbers of y denotes the quantum numbers of angular momenta of the atom (y = jKJ y = SLJ for U-coupling), a is the fine structure constant, a,, is the Bohr dimensionless radial dipole matrix element and factor C(ly,I’y’) appears from the angles. The upper state is marked by a prime throughout the paper, E’ We calculate the radial matrix elements in the semi-classical approximation defect method and asymptotic expansion.3,4 Then we get
R,E,:!'Y'=
’
(
the optical electron, for j/-coupling and radius, Rf,? is the the integration over - E,,,, = lb. using the quantum
27
(2)
-
Z6crS
where 2 is the charge of the parent ion (Z = 1 for a neutral atom, Z = 2 for a singly charged ion, etc.), o = tio/Z2 Ryd, v = n - p = (Z’ Ryd/E,,;,)“’ is the effective principal quantum number of the bound state, p is its quantum defect, KP’ is the non-Coulomb part of the phase shift of the continuum state which is 71times quantum defect of the adjacent bound state series extrapolated to positive energy, fi = (~/o)~‘~(E,,;.-t E’)/2Z2 Ryd is the small parameter of the asymptotic expansion, x = c,(0/4)~‘~, f, = max(l,l’j = (I + 1’ + 1)/2 and S(x) = AZx”2Ai(x) - Ai’( T(x) = Alx”‘Gi(x) - Gi’(x) + x/2x, S,(x) = k (1 - 681~~‘~+ 4x3)Ai(x) + g x( 1 - Alx3j2)Ai’(x),
T(X) = k (1 - 6A1x312 + 4x3)Gi(x) + f x(1 - AIx3j2)Gi’(x) + &
AIx’12- &
x2,
s s n
Ai
= X-I
cos(xt + t3/3)dt,
0
a
Gi(x) = n-’
sinfxt + t3/3)drI
II
Here Al = 1’ = I+ 1, Ai is the Airy function, Gi(x) is the solution of a non-homogeneous equation similar to Airy’s equation6 and Ai’ and Gi’(x) are their derivatives.
55
Radiative continua of noble gas plasmas
and Ci(x) are expediently calculated In the case of interest x < 1 or at the least m 1. So, Ai by means of the power series. For Ai the corresponding expansion is given in Ref. [6]. It can be written in the simplest form by means of the recurrence relation between the coefficients:
Ai
= 2 ukxk, k=O
1 = 0.35503, a, = a’ = 32”1-(2/3)
3”3rl(l,3) = - 0.25882, az = 0, k 2 3.
ak-3 ak=
k(k-
A similar expansion for Gi(x) can be obtained too:
Gi(x) = 2 bkxk, k=O
bo=*,b,=
-+b,= fi
-&
d
bk-3 bk
=
k 2 3.
k(k - 1) ’
The factor C(ly,l’y’) is expressed in terms of 6j-symbols. In the case ofjl-coupling for excited states of rare gas atoms
applicable
CCjlKjl’K’)
and, taking into account the J splitting,
CCjlKJjl’K’J’)
= (2K + 1)(25’ + l){;
;,
“: J?C($KjlX’).
For all transitions allowed by the selection rules, the unified expression can be written C(jIKjl’K’)
[x: - (y, -j)‘][xt, - (& +j + I)‘] = ( - l)AK+’ 4(21+ 1)(21’ + l)K,,,(2K + 1 - Y,) ’
C(j/KJj,‘K’J’)
where
AK = K’ - K = 0,
6=4(J-K)(J’-K’)=
= 6 [2K + 1 + 6(25’ + l)]’ - 9 C(jlKjl’K’) 4(2K’ + 1)(2J + 1)
+ 1, Km = max{K,K’}, f 1.
X, = All,,,+ AKK,,
Y, = (1 - AK’)K
and
2.2. Absorption coeficient
The bound-free absorption coefficient kbC is defined as a sum of the photoionization cross-sections multiplied by the populations of the corresponding levels. Summation is performed over all states with the ionization energy less than 1To. For highly excited states the sum is replaced by an integral and can be written as an unified integral expression’ together with the free-free absorption contribution having the same structure. Let us denote by E. the lower limit of the
56
L. G. D’yachkov et al
integral. If the radial matrix element is approximated approximated by a linear function of energy P
by Eq. (2) and the quantum defects are
E
=a+bZ2
Ryd
(3)
3
this integral can be taken analytically. The corresponding theory is given in Ref. j. Here we present the final expression in a some more general form (and with some other denotes) than that in Ref. 5, allowing for the excited states of the parent ion and atomic level series converging to them. This is important for rare gas atoms, ions of which have low-lying excited levels 2P,i2. As a result, for the absorption coefficient we have
k(o) = kbf(0)+ kR(o)=
-
1
N,a(p,o) + k,,,(o)
)
(4)
hw < Ep < E.
(5) where NP is the concentration the quantum numbers),
of atoms (ions with charge Z - 1) in a state p (here p denotes all
16,/%xe4Z2
G!,(w)=
(6)
3w3$2G
is the integral expression for a hydrogenic atom (Kramers-Unsold formula), G,(o) is the Biberman-Norman function’ allowing for the specific features of a non-hydrogenic atom, N, and Ni are the electron and ion densities, o0 = min{o,]E,]/12) and G is the average Gaunt factor (over - o0 < E < co). The function tin,(o) is calculated from the following equation (pi is the state of the parent ion):
tint
=
1 + 1 D(piN’yY’)
(7)
3
p,&’
D(PiaYY’) = &
W,W)exp
( -
$){@I
+
+ SQ,)(COS $, COS V, - COS IjGoc0S 6)
7Q,[cos’h @VI + $1)- cos211/, cos(vO+ b)l} , (8)
where g(y) is the statistical weight of the y state of the atom, &(pi)and Ci are the excitation energy and the partition function of the parent ion, s = (a/2 - a,)(0/4)-~‘~, crO= fim$Z’ Ryd, z = (kT/Z2 Ryd)(o/4) - 213and tiO= arctan(2n7X,), v, = +!I + 27r(xo+ SX,), Qo = 167r2x1,-‘[S’(x) +
$I = arctan {2rc[rx, + (b’ - b)kT/Z* Ryd]}, K = $1 + 27r[x0+ s& + a’ - a + b’a - (6’ - b)a,J,
T2(x)1, Q, = 327c2xI,-‘[S(x)S,(x) + T(x)ZT(x)],
1 T(x) x0 = ; arctan S(X) ,
xl =
W) I;(x) - T(xMx) W’(x)
+ T2(x)]
Simple formula for the average Gaunt factor, including both bound-free and free-free transitions, is derived in Ref. *. In the linear approximation [by analogy with Eq. (2) and Eq. (3)], it is written as G = 1 + 0.1372[s +
7 -
&~~/2)“~].
It is easily seen that Eq. (8) reduces to zero if the quantum defects are negligible. So, the sum in Eq. (7) is restricted to the non-hydrogenic transitions. Accounting of the contributions from l,l’ I 3 is usually enough.
51
Radiative continua of noble gas plasmas
To calculate the free-free absorption coefficient kR (and corresponding rR function), one should put & = 0 in the expression for kin,. The difference between kin, and kR gives k:,. Then one can get kbf, adding kkl to the sum in Eq. (4).
2.3. Plasma efects in the near-threshold region It is well known that the interaction between the radiating atom and surrounding plasma particles gives rise to a ‘red’ shift and spread of the photoionization thresholds. For the description of the near-threshold region, we use the microfield model approach which was developed by many authors9-” For the electron on an atomic level a potential barrier is formed in the direction of the plasma microfield. If the microfield is strong enough, the level may be turned out above the barrier top and disappear. The realization probability of the level E, is then defined as the integral of the microfield distribution function
s FP
W(4) =
P(FNF,
(9)
0
where F, is the critical value of the field, for which the level is found at the top of the barrier and ceases to be bound. This leads to the weakening of the upper members of the spectral line series. However, unbound states appear above the barrier instead of the destroyed levels and a continuum replaces the lines with a probability 1 - W. So, we continue the photoionization continuum multiplied by a factor 1 - W over the zero-field threshold towards the long wavelengths. Near the threshold W+O, while W+l far off it. Therefore, the line series passes gradually to a continuum. Each of the terms in the sum (1) corresponds to a line series; consequently, it is continued with the own factor 1 - W. (Strictly speaking, the photoionization continuum would be continued with the factor 1 - W/W,, where W, is the realization probability of the lower state of the continuum and adjacent spectral series. However, in the spectral region under consideration W, 1 1 for all the spectral series of noble gas atoms if the electron density is not very high.) This way of the continuation of the photoionization continuum is based on the assumption of the conservation of the density of oscillator strengths when the lines transform to a continuum.‘.‘9.20 Using a similar way of the continuation, detailed description of the near-threshold region has early been obtained for hydrogen.‘3,2’,22To estimate the critical values of the microfield for the hydrogenic states, the uniform field model has been applied. In the present calculation we use Hooper’s microfield distributions23 and the simple analytical approximation derived in Ref. 24for the cumulative of these distributions. Thus, our calculation method is a completely analytical one. The choice of the critical values of the microfield is not a simple problem (a review can be found in Ref. I”). In this paper we use the simplest expressions from the uniform field (UF), Fp = Ei/4e3, and nearest neighbour (NN), F, = E,/ 16e’, approximations.‘6, ‘*We shall show that in contrast to hydrogen, NN is more appropriate for noble gas atoms (see also Ref. “). 3. RESULTS
AND COMPARISON WITH OTHER AND EXPERIMENT
CALCULATIONS
Results of calculations and measurements of the absorption (emission) coefficient are frequently presented, by analogy with Eq. (5) in terms of r function
k(w) = k”(W)<(W),
(10)
where k”(W) is defined by k”(w) =
16,/2 7rSue4Z2 NJ, exp 303JmT
fiw kT
(11)
or any similar way. Eq. (11) is the same as Eq. (6) with G = 1 and o. = o. In such a traditional form we present the results of the derived method. In our calculations the photoionization cross-sections of all states belonging to ns, (n + l)s, (n + 2)s, np, (n + l)p, nd, (n + l)d, ns’, (n + l)s’, (n + 2)s’, np’, (n + I)$, nd’ and (n + 1)d’ configurations (56 states for each atom
58
L. G. D’yachkov et al
allowing for the K and J splitting; n = 3,4, 5, 6 for neon, argon, krypton and xenon, respectively) are calculated and included into the sum in Eq. (4), while the photoionization of other states is taken into account by the integral expression (5) (E,, = - 0.065 Ryd for argon, krypton and xenon, while E,, = - 0.0627 Ryd for neon). Energy levels are taken from Refs. 25and 26.The quantum defects are extrapolated to the continuum states by means of Eq. (3). 3.1. Comparison with previous calculations Schliiter’ and Hofsaess’ have calculated the gb’ function defined by the equation kbf = 2a4Z2kTA3 3&c2e2
Na$exp(
-
&)[exp(
-$$-
l]ib’,
(12)
where N,, C, and Zare the total concentration, partition function and ionization energy of the atom, respectively, and y0 = 6 is the statistical weight of the ground state of the parent ion without allowing for the multiplet splitting. The function rbf in Eq. (12) is related to the function 5 in Eq. (10) [with k” according to Eq. (ll)] by the equation
5=
$[l-
exp(
&)]tbi+exp(
-
- &)Cfl
if the relation between N,, Ni and N, is established by the Saha equation for ideal plasma. Our method is compared with those calculations in Figs 1 - 4 for neon, argon, krypton and xenon. Simple analytical calculation (solid curve) is in a good agreement with calculations of Schltiter’ (dashed-dotted curve) and Hofsaess2 (dashed curve). However, these calculations cannot correctly be compared with experiment data for dense plasmas in the near-threshold region (A 2 400-500 nm), since no plasma effects leading to the ‘red’ shift and spread of the threshold are taken into account. In the same figures we also show our results taking into account the plasma effects in the near-threshold regions for electron density values lOI and 10” cm” according to Sec. 2.3 (dotted curves) with the critical values of the microfield from the NN approximation.
I -
Neon T= 15000 K
400
600
500
Wavelength
700
800
(nm)
Fig. 1. The lb’ function of neon. Comparison of present semi-classical calculation (solid curves) with previous calculations of Schliiter’ (dasheddotted curve) and Hofsaes$ (dashed curve). Dotted curves show the results of the present calculation taking into account the plasma effects in the near-threshold region with the NN approximation for the estimation of the critical values of the microfield.
59
Radiative continua of noble gas plasmas
L
,
N,=lO"~m'~ -
Argon T= 12000
200
300
400
500
600
K
700
800
Wavelength (nm) Fig. 2. As in Fig. 1 for argon.
3.2. Comparison with experiment Our calculation results are compared with experimental data on the radiative continua of neon, argon, krypton and xenon plasmas in Figs 5 - 8 respectively. Presented 5 functions correspond to the absorption (emission) coefficient taking into account both bound-free and free-free transitions and are defined by Eq. (10) and Eq. (11). Solid curves in Figs 5 - 8 show the calculation results with the critical values of the microfield from the NN approximation, while dashed curves correspond to the UF approximation. 3.2. I. Neon. Continuum radiation of neon plasma was measured by Schnapauff 27in a cascade arc and Gavrilov2* in pulsed discharges through fused quartz tubes. In Fig. 5 our calculations are compared with their results. Though the calculations underestimate the radiative continuum, they give the correct behaviour of the spectrum as the plasma density increases: the minimum in the near-threshold region (12= 450-600 nm) disappears gradually. Furthermore we can conclude that the better agreement between the calculation and experiment is observed with using the NN model. 3.2.2. Argon. The most of the experimental studies of the noble gas plasma spectra deals with argon. Plenty of experimental data on argon spectra in a wide range of the plasma conditions has
: ,’ 200
1
I
I
I
I
I
300
400
500
600
700
800
Wavelength
(nm)
Fig. 3. As in Fig. 1 for krypton.
60
L. G.
D’yachkovet al
Xenon T= I 1:1
200
I
I
I
300
10000 K
I
I
400 500 600 Wavelength (nm)
I
I
700
800
J
Fig. 4. As in Fig. 1 for xenon. been obtained in recent decades. Some of them are shown in Fig. 6 as well as the results of our calculations. We have selected the experimental data29-39obtained after 1975 from plasmas with close temperatures and electron densities (but produced in differ plasma sources) and being in an agreement with each other within the measurement error. (More detailed analysis of available experimental data is given by D’yachkov et a1.40)The data of Refs. [29, 31-36,38, 391 correspond to a temperature range 10 000-14 500 K, while the data of Refs. [30] and [37]correspond to higher temperatures 16 900 and 20 000 K respectively; the electron densities are near 10” cm - ’ (the pressure is one or a few atmospheres). Argon plasmas were generated in shock tubes29,38and wall-stabilized 30-34,36.39, gas-flow-stabilized” and free-burning35 arcs. A good agreement between the shown experimental data allows to believe that they are sufficiently reliable. In the short wavelength region (1 < 400 nm) free from the significant spectral lines, the experimental data except Ref. [30] have a small scatter, while some excess of the data of Behringer and Thoma30 over those of Refs. [31-34,36,39] is due to the temperature dependence of the
0.1 200
300
400
500
600
700
800
900
1000
1100
Wavelength (nm) Fig. 5. The r function of neon. Comparisonof experimentdata (symbols)and present semi-classical calculationsusingboth NN (solid curves) and UF (dashed curves) approximations for the estimation of the critical values of the microfield: + (Schnapauff’) and 1, N, = 3.79 x 10tbcme3, T = 14680 K; q (GavriloP) and 2, NC= 1.4 x 10” cm- ‘, T= 21 000 K; l (Gavrilov 28)and 3, N. = 1.1 x lOI8cm-‘, T= 21 000 K.
Radiative continua of
200
300
400
61
noble gas plasmas
500
600
700
800
Wavelength (nm) Fig. 6. The C function of argon. Comparison of experiment data (symbols) and present semi-classical calculations using both NN (solid curves) and UF (dashed curves) approximations for the estimation of the critical values of the microfield. Experiment: Meiners and Weiss”, +, N. = (0.4-1.7) x 10” cm-‘, T = 10 5O(rl2 500 K; Behringer and ThornaM, +,N.=1.99x10”cm~‘,T=169OOK;Goldbachet al”, 0 N = 1 2 x IO” cm-‘, T = 13 200 K; PrestotQ2-“, x , N = 2 07 x IO” cm - ‘, T = 13 780 K; Akamaisu gnd Takata’5, A, N, = 1.09 x 10” cm-‘, T= 13 000 K; Schnehage et alJ6, 0, N, = (0.711.67)x IO” cm-‘, T = 12 1It&l4 430 K; Belov et al”, n , N, = 1.8x 10” cm-‘, T = 20 000 K; Zangers T = IO 000-12 500 K; Wilbers et al”‘, 0, and MeinersJ8, ., N = (0.42-I 5) x 10” cm-’ N, = 3.12 x IO” cm-‘, Ti 13 500 K: Present calculations: 1. N, = 2.07 x IO” cm-‘, T= 13 780 K; 2, N,= 1.99 x 10”cm-‘, T= 16900 K.
t function. Approximately the same temperature dependence is demonstrated by our calculations in this spectral region. Curves 1 and 2 in Fig. 6 show the calculation results for the experimental conditions of Preston3z-34 (N, = 2.07 x 10” cm-‘, T = 13 780 K) and Behringer and Thoma” (IV, = 1.99 x 10” cm - 3, T = 16 900 K), respectively. For longer wavelengths (,I > 400 nm) the
3
g
2
.S B z 0 0.; 0.8 0.7 0.6 0.5 300
400
500
600
700
800
Wavelength (nm) Fig. 7. The l function of krypton. Comparison of experiment data (symbols) and present semi-classical calculations using both NN (solid curves) and UF (dashed curves) approximations for the estimation of the critical values of the microfield. Experiment: Meiners and Weiss29, 0. N, = 4 x lOI cm-‘, T= 9500 K and +, N, = 2 x 10” cm-‘, T= 11 400 K; Baessler et a/.4’, 0, N, = 4.3 x lOI cm-‘, T=10150K and 0, N,=1.4xlO”cm-‘, T=12200K; Manola et al@, +, N,=(4.548.25) x lOI cm-‘, T= 9520-10 750 K; Vitel et al”, 0, N, = 9.6 x IO” cm-‘, T= 13 600 K and n , N. = 1.72 x IO’*cm-‘, T= 14900 K. Present calculations: 1, N. = 4.3 x lOI cm-‘, T= 10 150 K; 2, N,= 1.72 x IO’*cm-‘, T= 14900 K.
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200
300
400
500
600
700
800
900
Wavelength (nm) Fig. 8. The i function of xenon. Comparison of experiment data (symbols) and present semi-classical calculations using both NN (solid curves) and UF (dashed curves) approximations for the estimation of the critical values of the microfield. Experiment: Meiners and Wei@, 0, N, = 2.2 x lOu cm-‘, T= 7900 K, n , N, = 3.2 x 1Or6cm-‘, T= 8200 K, 0, N. = 7.8 x lOr6cm-‘, T= 9000 K and +, N = 1.5 x 10” cm-3 , T= 9800 K; Goldbach et al”, IJ, N. = 10” cm-‘, T= 10 050 K and A, N: = 2.2 x IO” cm-‘, T = 12 300 K; Gavrilov and GavrilovaQ, +, N, = 3.35 x 10” cm-’ and x , N. = 8.03 x IO” cm-‘, T = 14 000 K; Vitel et al”, 0, N, = 7.4 x 10” crnm3, T= 11 400 K. Present calculations: 1, N, = 10” cme3, T= 10050 K; 2, N,=2.2 x 10”cm-‘, T= 12300 K; 3, N,=7.4x lO”cm-‘, T= 11400 K.
temperature dependence become weak, but the scatter of the experimental data increases. The last can apparently be explained by difficulty in the problem of the separation of the line-wing contributions.38 From the comparison of the calculation results with experimental data one can conclude that the agreement between them is quite good, if the critical values of the microfield are defined from the NN model, while it is some worse for the UF model. 3.2.3. Krypton. Radiation continuum of krypton plasma has been measured in shock tubes by Berge et a14’,Meiners and Weiss2g and Manola et a1,42wall-stabilized arc by Baessler et a1,43and flashlamps by Vitel et al.* Electron density range of about two orders (3 x lOI - 1.7 x lo’* cme3) is covered; temperature values are between 9500 and 15 700 K. A part of the measurement data giving a clear view of the whole collection of the data is displayed in Fig. 7. Results of our calculations for low and high values of the plasma parameters corresponding to the experimental conditions of Baessler et a143 (NC= 4.3 x lOI cmm3, T = 10 150 K) and Vitel et al” (N, = 1.72 x lo’* cme3, T = 14 900 K) are represented by curves 1 and 2, respectively. The calculation agrees well with the measurements of Baessler et a143and in the mean with data of Manola et a14*obtained for close plasma parameters and having a considerable scatter. Small scatter in the data of Baessler et a143gives the possibility of seeing a trend in the decrease of the ( function as the electron density increases up to 1.4 x 10” cm-3 at T = 12 200 K. A similar result has been obtained by Meiners and Weissz9studying emission from krypton plasma at I = 456.1 nm as a function of N$T”*. Two points from Ref. [29] corresponding to the minimum and maximum values of the electron density and temperature are presented in Fig. 7. Subsequent decrease of the 5 function can be seen from the comparison of the measurement results of Vitel et al” for the higher electron densities with Refs. [29,42,43]. It should however be noted that this effect is small and comparable with the accuracy of the measurements (lO-25%). On the whole we can conclude that our calculation results agree with the experimental data within the experimental error, although they are slightly higher than the data of Vitel et al.” But in this case it is difficult to choose decidedly the model (NN or UF) for the estimation of the critical values of the microfield. However, taking into account the corresponding choice for neon and argon, we think that the NN model is more appropriate for krypton too. 3.2.4. Xenon. In Fig. 8 are shown the experimental data on the radiation from xenon plasmas produced in shock tubes,29 wall-stabilized arcs4s and flashlamps. 44.46Here we restrict ourself to the
Radiative continua of noble gas plasmas
63
consideration of electron density values up to 10” cm - 3, although measurements with xenon, as well as argon, were performed also for higher densities. The shown experimental data are in a satisfactory agreement (except the one point at I = 320 nm from Ref. [47]), but, as in the case of krypton, a decrease of the 5 function with increasing electron density can be noted.29,46 Our calculation results are presented for the plasma parameters corresponding to the experimental conditions of Goldbach et a14’(curve 1, T = 10 050 K and N, = 10” cm - ‘; curve 2, T = 12 300 K and N, = 2.2 x 10” cmm3) and Vitel et al4 (curve 3, T = 11 400 K and N, = 7.4 x 10” cm-‘). In this case, in contrast to krypton, we can do the evident choice of the NN model (solid curves), since it gives rise to a good agreement with experiment, while the UF model (dashed curves) gives the r function that is considerably lower than the experimental data for I > 600 nm. Moreover, comparing solid curves l-3 with corresponding experimental data, one can see that the theory (using NN) correctly reflect the variations of the 5 function with varying temperature and electron density. 4. CONCLUSIONS We present a simple completely analytical method for the calculations of plasma radiation continua generated from bound-free and free-free transitions of electrons in the field of a positive ion. It is based on the semi-classical version of the quantum defect theory, uses the microfield model approximation and assumes the conservation of the density of oscillator strengths for the extrapolation of the photoionization continuum over its long-wavelength edge. A comparison of the calculation results with experimental data for neon, argon, krypton and xenon plasmas having a temperature in the range 8000-21 000 K and electron density between lOI and 10” cm - 3, has been performed and it is shown that this method gives a rather good approximation of the radiation continuum in the spectral range 200-1000 nm. In contrast to the well-known previous calculations,‘.* our method does not show the sharp edge of the photoionization continuum, since it takes into account the plasma effects in the near-threshold region and gives a reasonable description of this spectral region important for experimental studies and applications. In the framework of the microfield model approximation, for the estimation of the critical values of the microfield we use the simplest expressions from two of the rather crude models, uniform field and nearest neighbour (an accurate physical model would apparently lay between them). Comparison with experimental data shows that the NN model provides a considerably better agreement between theory and experiment in the near-threshold region for noble gas plasmas. We think that this model would apparently be more appropriate in the general case of complex atom plasmas. Our simple method may be used, looking forward, for comparison with experimental data on very dense plasma radiation to clarify the density (or coupling) effects in plasma optical properties.47 In particular, it would be useful as the basis for the construction of the method for the calculations of the opacities of dense plasmas: if the density effects are known independently, they can easily be incorporated into the simple calculation scheme presented above. Note, that further comparison with experimental data on dense plasma radiation is a tool to test or check any predictions on coupling effects in radiation too. Last but not least, some complex modern problems of moderately dense plasmas diagnostics based on radiation continuum may be solved easily and accurately on the basis of the method described in this paper. Acknowledgements-We would like to thank G. A. Kobzev and G. Maynard for stimulating discussions. We acknowledge with thanks also the partial support of this work by the NATO International Scientific Exchange Program under Linkage Grant No. HTECH.LG 960803.
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