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The radiation escape factor
J. Quant. Spectrosc. Radiat. Transfer Vol. 39, No. 6, pp. 489-491, 1988
0022-4073/88 $3.00 + 0.00
Printed in Great Britain. All rights reserved
Copyright 0 1988Pergamon Press plc
NOTE THE RADIATION
ESCAPE
FACTOR
ERNST E. FILL Max-Planck-Institut
fiir Quantenoptik,
D-8046 Garching, F.R.G.
(Received 14 October 1987)
Abstract-An expression is derived for the effective radiative decay rate of the population of a bound level in gases or plasmas. The basic idea of the analysis is that the radiation generated throughout the volume of a radiator must equal the radiation emitted from its surface.
1. INTRODUCTION Radiation trapping is an important phenomenon affecting the level populations of gases or plasmas. On reabsorption of a spontaneously-emitted photon, a particle initially in the lower state of a transition is put into the upper state, thus reducing the apparent decay rate of this level. An
exact treatment of the problem shows that the decay of the upper level is no longer exponential. However, it is common usage to define a radiation escape factor G < 1 (Biberman-Holstein coefficient) as the ratio of the effective radiative decay rate to the decay rate of an isolated atom; thus,’ 4, = GA,, ,
(1)
where A,, is the spontaneous emission rate from the upper to the lower level. The escape factor defined by Eq. (1) is conveniently used in collisional-radiative models applied to problems such as line radiation,’ ionization equilibrium,* and gain calculations.3,4 Various expressions for G have been derived. In early papers, the problem was treated as a diffusion process. ‘-’ Holstein recognized that the description by the diffusion equation was not adequate.* He showed that the exact treatment requires solution of an integro-differential equation and derived simple expressions that are valid at high optical density.’ A general expression for the escape factor that is independent of the shape of the radiating body was derived by Drawin and Emard,” who characterized the geometry, somewhat arbitrarily, by the shortest distance of the actual position to the boundary. Extensive calculations of the escape factor and a systematic review of various definitions have been given by Irons.” In this Note, we show that a simple expression for the escape factor, as defined in Eq. (1) can be derived from consideration of the radiative power emerging from a gas or plasma. The expression may be readily used to yield approximate values of G under various conditions.
2 ANALYSIS
The basic idea is that for any radiator, the number of particles in the upper state multiplied by the effective decay rate and the photon energy must be equal to the power emitted from the surface. By expressing the radiation from the surface into a solid angle of 2~ in terms of Planck’s radiation law, we obtain iV, VA,&
= 2~WAv,,S/c’[exp(hv/kT)
- 11,
(2)
where N, = number density of particles in the upper state, V = volume of the radiator, S = surface area of the radiator, kT = Boltzmann temperature of the radiating line (not the electron temperature). Also, Av, is the equivalent line width, which is well known in astrophysics’2-14 and Qs R.T39,&E
489
490
ERNST E. FILL
is defined as the width of a line with rectangular shape that has the same integrated absorption (or emission) as the line in question. It is given by the expression’3,‘4 Av, =
[l - exp( -k,L)]
dv,
(3)
s line
where k, is the absorption coefficient at frequency v and L is the path length through the medium. Neglecting stimulated emission (hv >>kT) and noting that the number density in the lower level is fi = (gl/gU)%exp(hvlkO
(4)
even under non-LTE conditions, one can rearrange Eq. (2) to obtain the main result of this paper: Ae,r= 2@,Av,,P2
V&N,,
(5)
where g, and g, are the degeneracies of the upper and lower level, respectively. Equation (5) is a very simple expression for the effective radiative decay rate. The geometry of the radiator is represented by the surface-area-to-volume ratio and the opacity is expressed by the equivalent line width, a quantity extensively used in astrophysical calculations related to the curve The only approximations made are that the plasma is assumed to be homogeneous of growth. 12-14 while its conditions vary slowly.
3.
APPLICATION
To apply Eq. (5) to any real situation, the equivalent line width must be evaluated. If the line shape is known, the problem reduces to determination of an effective path length to be used in Eq. (3). For an exact calculation of Av,s at a given geometry, the detailed radiation transfer problem must be solved. Equation (5) then requires a tedious evaluation. Fortunately, however, an approximate determination of Av, is possible by choosing the particular pathlength which makes Aeff E A,, in the optically-thin case, at all optical densities. We find for k,L a 1,
Av,~ = L
(6)
k, dv = (J2/8~)4(gu/g’)L s line
and obtain the following representative
results:
For an infinite slab of thickness d,
S/V=2/dand
For an infinitely long cylinder of diameter D,
S/V =4/D
For a sphere of diameter D,
S/V = 6/D and L = (2/3)D.
L=d;
and L = D;
(7)
As an example, the escape coefficient for an infinite cylinder may be calculated and compared with the result derived by Holstein for large optical depth at the line center. For a Lorentzian line with AvL equal to the full width at half maximum, the equivalent line width at large optical depth may be approximated byI Av,s = (~/2Jn)[A,,Av,N,(g,/g,)Ll”2.
(8)
Taking L = D from Eq. (7), A,R= (4&/A)
t-%,g,Av,lN,g,D)“2.
To compare this result with Holstein’s formula, we write for the absorption line centre ko = (~2/4x2)N,(g,/g,)A,llAv,
(9) coefficient at the (10)
and find G = A,,/A,,
= 1. 13/(k,,D)“2,
(11)
Note
491
which differs only by a numerical factor close to 1 from Holstein’s formula G = l.l15/(nk,R)“’
= 0.89/(k,,D)“‘,
where R is the radius of the cylinder. For a Gaussian line with a width Av, a useful approximation a large range of k,,L >>1 is
(12) for the equivalent line width at
Av,~ = 1.3Av,[ln(k,,L)]“2,
(13)
which yields &= The absorption
(87~1.3Av,g,/~2N,Dg,)[ln(k,,D)]‘~2.
(14)
coefficient at line center is k,, = (122A,,g,/8~Av,g,)ZV,2(1n 2)“2/&
(15)
G = 1.22[ln(k,D)]“2/k,D.
(16)
and the escape factor is
At large optical depth, Eq. (16) yields somewhat higher values for G than Holstein’s relation G = 1.6/k,R[n
ln(kOR)]“*.
(17)
Acknowledgement-This work was supported, in part, by the commission of the European Communities in the framework of the Association Euratom/IPP.
REFERENCES 1. R. W. P. McWhirter, in Plasma Diagnostic Techniques, p. 233, R. H. Huddlestone and S. Leonard eds., Academic Press, New York, NY (1965). 2. C. Chenais-Popovics, P. Alaterre, P. Audebert, J. P. Geindere, and J. C. Gauthier, JQSRT 36, 355 (1986). 3. G. J. Pert, J. Phys. B: Atom. Molec. Phys. 9, 3301 (1976). 4. E. E. Fill and D. G. Goodwin, Proc. SPIE 31st Annual Znt. Tech. Symp., San Diego, CA (August 1987). 5. K. T. Compton, Phys. Rev. 20, 283 (1922). 6. E. A. Milne, J. Lond. Math. Sot. 1, 1 (1926). 7. C. Kenty, Phys. Rev. 42, 823 (1932). 8. T. Holstein, Phys. Rev. 72, 1212 (1947). 9. T. Holstein, Phys. Rev. 83, 1159 (1951). 10. H. W. Drawin and F. Emard, Beitr. Plasma Phys. 13, 143 (1973). 11. F. E. Irons, JQSRT 22, 1 (1979). 12. A. Undid, Physik der Sternatmosphtiren, p. 288, Springer, Berlin (1955). 13. D. Mihalas, Stellar Atmospheres, 2nd ed., p. 269, W. H. Freeman, San Francisco, CA (1978). 14. A. P. Thome, Spectrophysics, p. 306, Chapman & Hall, London (1974).
0022-4073/88 $3.00 + 0.00
Printed in Great Britain. All rights reserved
Copyright 0 1988Pergamon Press plc
NOTE THE RADIATION
ESCAPE
FACTOR
ERNST E. FILL Max-Planck-Institut
fiir Quantenoptik,
D-8046 Garching, F.R.G.
(Received 14 October 1987)
Abstract-An expression is derived for the effective radiative decay rate of the population of a bound level in gases or plasmas. The basic idea of the analysis is that the radiation generated throughout the volume of a radiator must equal the radiation emitted from its surface.
1. INTRODUCTION Radiation trapping is an important phenomenon affecting the level populations of gases or plasmas. On reabsorption of a spontaneously-emitted photon, a particle initially in the lower state of a transition is put into the upper state, thus reducing the apparent decay rate of this level. An
exact treatment of the problem shows that the decay of the upper level is no longer exponential. However, it is common usage to define a radiation escape factor G < 1 (Biberman-Holstein coefficient) as the ratio of the effective radiative decay rate to the decay rate of an isolated atom; thus,’ 4, = GA,, ,
(1)
where A,, is the spontaneous emission rate from the upper to the lower level. The escape factor defined by Eq. (1) is conveniently used in collisional-radiative models applied to problems such as line radiation,’ ionization equilibrium,* and gain calculations.3,4 Various expressions for G have been derived. In early papers, the problem was treated as a diffusion process. ‘-’ Holstein recognized that the description by the diffusion equation was not adequate.* He showed that the exact treatment requires solution of an integro-differential equation and derived simple expressions that are valid at high optical density.’ A general expression for the escape factor that is independent of the shape of the radiating body was derived by Drawin and Emard,” who characterized the geometry, somewhat arbitrarily, by the shortest distance of the actual position to the boundary. Extensive calculations of the escape factor and a systematic review of various definitions have been given by Irons.” In this Note, we show that a simple expression for the escape factor, as defined in Eq. (1) can be derived from consideration of the radiative power emerging from a gas or plasma. The expression may be readily used to yield approximate values of G under various conditions.
2 ANALYSIS
The basic idea is that for any radiator, the number of particles in the upper state multiplied by the effective decay rate and the photon energy must be equal to the power emitted from the surface. By expressing the radiation from the surface into a solid angle of 2~ in terms of Planck’s radiation law, we obtain iV, VA,&
= 2~WAv,,S/c’[exp(hv/kT)
- 11,
(2)
where N, = number density of particles in the upper state, V = volume of the radiator, S = surface area of the radiator, kT = Boltzmann temperature of the radiating line (not the electron temperature). Also, Av, is the equivalent line width, which is well known in astrophysics’2-14 and Qs R.T39,&E
489
490
ERNST E. FILL
is defined as the width of a line with rectangular shape that has the same integrated absorption (or emission) as the line in question. It is given by the expression’3,‘4 Av, =
[l - exp( -k,L)]
dv,
(3)
s line
where k, is the absorption coefficient at frequency v and L is the path length through the medium. Neglecting stimulated emission (hv >>kT) and noting that the number density in the lower level is fi = (gl/gU)%exp(hvlkO
(4)
even under non-LTE conditions, one can rearrange Eq. (2) to obtain the main result of this paper: Ae,r= 2@,Av,,P2
V&N,,
(5)
where g, and g, are the degeneracies of the upper and lower level, respectively. Equation (5) is a very simple expression for the effective radiative decay rate. The geometry of the radiator is represented by the surface-area-to-volume ratio and the opacity is expressed by the equivalent line width, a quantity extensively used in astrophysical calculations related to the curve The only approximations made are that the plasma is assumed to be homogeneous of growth. 12-14 while its conditions vary slowly.
3.
APPLICATION
To apply Eq. (5) to any real situation, the equivalent line width must be evaluated. If the line shape is known, the problem reduces to determination of an effective path length to be used in Eq. (3). For an exact calculation of Av,s at a given geometry, the detailed radiation transfer problem must be solved. Equation (5) then requires a tedious evaluation. Fortunately, however, an approximate determination of Av, is possible by choosing the particular pathlength which makes Aeff E A,, in the optically-thin case, at all optical densities. We find for k,L a 1,
Av,~ = L
(6)
k, dv = (J2/8~)4(gu/g’)L s line
and obtain the following representative
results:
For an infinite slab of thickness d,
S/V=2/dand
For an infinitely long cylinder of diameter D,
S/V =4/D
For a sphere of diameter D,
S/V = 6/D and L = (2/3)D.
L=d;
and L = D;
(7)
As an example, the escape coefficient for an infinite cylinder may be calculated and compared with the result derived by Holstein for large optical depth at the line center. For a Lorentzian line with AvL equal to the full width at half maximum, the equivalent line width at large optical depth may be approximated byI Av,s = (~/2Jn)[A,,Av,N,(g,/g,)Ll”2.
(8)
Taking L = D from Eq. (7), A,R= (4&/A)
t-%,g,Av,lN,g,D)“2.
To compare this result with Holstein’s formula, we write for the absorption line centre ko = (~2/4x2)N,(g,/g,)A,llAv,
(9) coefficient at the (10)
and find G = A,,/A,,
= 1. 13/(k,,D)“2,
(11)
Note
491
which differs only by a numerical factor close to 1 from Holstein’s formula G = l.l15/(nk,R)“’
= 0.89/(k,,D)“‘,
where R is the radius of the cylinder. For a Gaussian line with a width Av, a useful approximation a large range of k,,L >>1 is
(12) for the equivalent line width at
Av,~ = 1.3Av,[ln(k,,L)]“2,
(13)
which yields &= The absorption
(87~1.3Av,g,/~2N,Dg,)[ln(k,,D)]‘~2.
(14)
coefficient at line center is k,, = (122A,,g,/8~Av,g,)ZV,2(1n 2)“2/&
(15)
G = 1.22[ln(k,D)]“2/k,D.
(16)
and the escape factor is
At large optical depth, Eq. (16) yields somewhat higher values for G than Holstein’s relation G = 1.6/k,R[n
ln(kOR)]“*.
(17)
Acknowledgement-This work was supported, in part, by the commission of the European Communities in the framework of the Association Euratom/IPP.
REFERENCES 1. R. W. P. McWhirter, in Plasma Diagnostic Techniques, p. 233, R. H. Huddlestone and S. Leonard eds., Academic Press, New York, NY (1965). 2. C. Chenais-Popovics, P. Alaterre, P. Audebert, J. P. Geindere, and J. C. Gauthier, JQSRT 36, 355 (1986). 3. G. J. Pert, J. Phys. B: Atom. Molec. Phys. 9, 3301 (1976). 4. E. E. Fill and D. G. Goodwin, Proc. SPIE 31st Annual Znt. Tech. Symp., San Diego, CA (August 1987). 5. K. T. Compton, Phys. Rev. 20, 283 (1922). 6. E. A. Milne, J. Lond. Math. Sot. 1, 1 (1926). 7. C. Kenty, Phys. Rev. 42, 823 (1932). 8. T. Holstein, Phys. Rev. 72, 1212 (1947). 9. T. Holstein, Phys. Rev. 83, 1159 (1951). 10. H. W. Drawin and F. Emard, Beitr. Plasma Phys. 13, 143 (1973). 11. F. E. Irons, JQSRT 22, 1 (1979). 12. A. Undid, Physik der Sternatmosphtiren, p. 288, Springer, Berlin (1955). 13. D. Mihalas, Stellar Atmospheres, 2nd ed., p. 269, W. H. Freeman, San Francisco, CA (1978). 14. A. P. Thome, Spectrophysics, p. 306, Chapman & Hall, London (1974).