ON THE EXPRESSIONS FOR LINE OPTICAL THICKNESS

J. Quant. Spectrosc. Radiat. ¹ransfer Vol. 60, No. 4, pp. 515—521, 1998( 1998 Elsevier Science Ltd. All rights reserved

Printed in Great BritainPII: S0022-4073(97)00222-7 0022—4073/98 $19.00#0.00

ON THE EXPRESSIONS FOR LINE OPTICAL THICKNESS

S. O. KASTNER1-A Ridge Road, Greenbelt, MD 20770, U.S.A.

(Received 8 July 1996)

Abstract—Expressions for emission line absorption coefficient/optical thickness, found in textsand in the literature, are examined and compared with the known, correct expressions whichare summarized at the beginning. Some ambiguities and discrepancies are noted, as well asunusual notations and conventions which may cause problems if not properly interpreted.( 1998 Elsevier Science Ltd. All rights reserved.

1 . I N T R O D U C T I O N

Analyses of optical thickness in spectral line emissions are becoming more feasible, and morerelevant, as computational resources and observational facilities improve. Workers who haveconsequently made increasing use of expressions given in the literature for calculating opticalthickness in spectral lines may have noticed differences in notation, disparities and even conflictsbetween such expressions in some cases. It will be appreciated that errors of a factor of two or similarfactors in computed line-center optical thickness can be especially significant in cases of low opacity,e.g. nebulae or stellar atmospheres in astronomy, or laboratory plasmas of limited extent. Thiscommunication is aimed at clarifying the situation by examining and comparing expressions, givenby various sources often referred to, with the correct expressions which are summarized at the outsetin Sec. 2 (in c.g.s. units). Most of the expressions are well known, but it is believed useful to gatherthem in one place.

2 . THE ABSORPTION COEFFICIENT AND OPTICAL THICKNESS

Optical thickness at frequency l is defined1,2 as q(l)"k (l)¸, where ¸ is the path length in cm andk(l) is the absorption coefficient in cm~1, so that optical thickness is dimensionless. The cross-section at frequency l (in cm2) is defined as k(l)/N

lwhere N

lis the population (in cm~3) of the lower

state (level or term) l.The absorption coefficient for a spectral line is defined in terms of the normalized frequency profile

P (l) of the line, as

k (l)"gu

gl

Aul

j20

8n1

*l C1!glN

uguN

lDN

lP(l), (1a)

"

ne2

mcflu

1

*l C1!glN

uguN

lDN

lP (l), (1b)

where giis the statistical weight of state (level or term) i; A

ulis the radiative rate (s~1) from upper

state u to lower state l; Nu, N

lare the respective population number densities in units of cm~3; j

0is

the line-center wavelength in cm; flu

is the line oscillator strength; and *l is the frequency intervalunit involved in the normalization of the profile P (l).

The three most important line profile functions are the Doppler (Gaussian) profilePD(x

D)"exp(!x2

D)/Jn, where x

Dis the dimensionless variable x

D"(l!l

0)/*l

Dmeasured in

HW(1/e)M Doppler halfwidths *lD; the Lorentz (damping) profile P

L(x

L)"[n (1#x2

L)]~1, where

xL

is the dimensionless variable xL"(l!l

0)/*l

Lmeasured in radiative-collisional damping line

(HWHM) halfwidths *lL; and the more general Voigt profile P

V(x

D, a), with parameter

a"*lL/*l

D, of which the other profiles are limiting cases (a"0 gives the Doppler profile; the

515

Lorentz profile is obtained in the limit aPR):

PV(x

D, a),

»(xD, a)

Jn"

a

n3@2 P=

~=

exp(!y2)

a2#(xD!y)2

dy. (2)

Other notations often used for the Voigt function »(xD, a) are H(a, v) (e.g. Refs. 3 and 4) in which the

variable v,xD; K (x,y) (e.g. Ref. 5) in which the variable y,a; and Jn º (a, x

D) (e.g. Ref. 6).

The bracketted factor in expressions (1a) and (1b) accounts for the decrease in absorption andoptical thickness due to stimulated emission, important when the upper state population is notnegligible compared to the lower state population. For brevity it will be denoted below by F

SE. The

factor FSE

reduces to (1!exp(!hl0/k¹)) when Boltzmann equilibrium holds for the states u and l.

With *lD"(l

0/c)(2k¹/m)1@2, ¹ being the radiating atom’s (translational) temperature and m the

atomic mass in grams, one obtains numerically, for the Doppler or Voigt profiles:

k(l)"(1.1612]10~6) j0

fluS

M

¹

FSE

NlF (x

D) (3)

where F (xD) is the Doppler function exp(!x2

D) or the Voigt function »(x

D, a). Here M is the

atomic weight in atomic mass units (AMU).One has then for the Doppler profile absorption coefficient at line center (l"l

0, x

D"0):

kD(l

0)"(1.1612]10~6) j

0fluS

M

¹

FSE

Nl. (4a)

This is the case most commonly dealt with in astrophysics and plasma physics, holding reasonablywell for resonance lines which are most likely to be optically thick, so that expressions in theliterature are usually to be compared with Eq. (4a).

For the Voigt profile absorption coefficient at line center (l"l0, x

D"0)

kv(l

0)"(1.1612]10~6)j

0fluS

M

¹

FSE

Nlexp(a2) erfc(a) (4b)

because »(0, a)"exp(a2) erfc(a), where erfc(z) is the complementary error function. When pureDoppler broadening is assumed, a"0 and expression (4b) reduces to (4a).

If broadening by a (most probable) turbulent velocity » is to be taken into account, the Dopplerhalfwidth *l

Dis replaced by the Doppler/turbulent halfwidth *l

DV"(l

0/c)[(2k¹/m)#»2]1@2 in

expressions (4a) and (4b). This modifies (4a) [and (4b)] to

k (l0)"

Jne2

mcfluj0FSE

Nl[(2k¹/m)#»2]~1@2

"(1.4974]10~2) fluj0FSE

N1 C(1.6629]108)

¹

M#»2D

~1@2. (4c)

For the Lorentz profile absorption coefficient at line center (l"l0, x

L"0):

k (l0)"

j28n2

gu

gl

Aul

*lL

FSE

Nl"

e2

mc

flu

*lL

FSE

Nl

"(8.448]10~3) fluFSE

Nl/*l

L. (5a)

In terms of the radiative-collisional damping constant c"cu#cl,(tu)~1#(tl)~l,4n*lL

(thisrelation defines the Lorentz (HWHM) halfwidth *l

L,c/4n; see also Sec. 4 below), where ti is the

lifetime of state i, (5a) can be written also as

k(l0)"

4

cne2

mcfluFSE

Nl"(0.1062) f

luFSE

Nl/c. (5b)

516 S. O. Kastner

The numerical factors in expressions (5a) and (5b), for evaluation of the Lorentz profile line-centerabsorption coefficient in c.g.s. units, do not appear to have been previously given explicitly in theliterature.

A convenient expression for the HW(1/e)M Dopper halfwidth *lD, needed explicitly in evaluating

the Voigt parameter a in expression (4b) (see Sec. 4), is: *lD"(1.290]104) (¹/M)1@2/j, where ¹ is

the temperature, M is the atomic weight and the line wavelength j is in cm.Expressions (4a)— (4c), (5a) and (5b) are the appropriate expressions to use for the line-center

absorption coefficient factors k (l0) in the optical thickness q

0"k (l

0)¸, for a given profile (and when

complete frequency redistribution occurs at each scattering event), and will be referred to below asthe correct expressions.

The same correct expressions and numerical equivalents (4a) and (4b) are obtained by using the(more readily measured) FWHM Doppler width, which is a factor of 2 (ln 2)1@2 "1.66511 greaterthan the HW(1/e)HM Doppler halfwidth, and at the same time incorporating the factor 2(ln 2)1@2explicitly in the expression, i.e. the numerical coefficient in Eqs. (4a) and (4b) is then 1.9335]10~6.However consistency is thus required between the definition of ‘‘Doppler width’’ and the definitionof absorption coefficient; adopting the FWHM definition of Doppler width forces one to incorpor-ate the factor 2 (ln 2)1@2 also into the absorption coefficient.

The relations between the variously defined ‘‘Doppler width’’ and ‘‘Doppler halfwidth’’ quantitiesare discussed in an Appendix of Kastner.7

3 . LITERATURE VARIATIONS

The writer’s survey finds the following points in texts and in the literature, in making comparisonswith the above expressions for absorption coefficient, usually with the Doppler expression (4a). Thesurvey also took note of varying notations, and sometimes misprints, which may have hinderedapplication of some authors’ expressions.

3.1. Text expressions

The Doppler profile expressions given by Refs. 3, 6, 8—13 are in agreement with expression (4a),because these authors define ‘‘Doppler width’’ as the HW(1/e)M value of the Doppler profile.Traving3 uses angular frequency u instead of frequency l. Unsold8 in particular distinguishes clearlybetween the HW(1/e)M ‘‘Doppler width’’ and the FWHM ‘‘Doppler width’’.

Mihalas4 gives expression (1b), using a notation for the absorption coefficient (p. 317) whichseparates out the profile normalization factor 1/(Jn *l). Shu14 gives the general expression (1b) inconnection with the Lorentz profile.

Two minor misprints were found in the relevant section of Jefferies’ book. In the expression for‘‘Doppler width’’ at the bottom of p. 86, frequency l should be replaced by line-center frequency l

0.

In the same expression, the lower-case symbol m—used for electron mass in his preceding expression(4.95)—should be replaced by atomic or molecular mass M.

Mitchell and Zemansky15 and more recently Lang16 on the other hand use the FWHM ‘‘Dopplerwidth’’, incorporating the factor 2 (ln 2)1@2 into the absorption coefficient to obtain the correctexpression.

Harwit17 gives the correct expression for absorption coefficient, using also the FWHM ‘‘Dopplerwidth’’; and using angular frequency.

Thorne18 uses the FWHM ‘‘Doppler width’’ as well, incorporating the factor 2 (ln 2)1@2 into theabsorption coefficient to obtain the correct expression. Her clear exposition of line radiation showsfurther that this numerical factor must for consistency then be explicitly included in relatedexpressions for intensity, etc.

The handbook of Allen,19 while very complete for its date, does not give the absorption coefficentexpression for the Doppler profile, but only for the Lorentz profile.

The notation for Boltzmann constant, used differently by different authors in the past, can causesome problems. For example Mitchell and Zemansky15 define R at the beginning of their book asthe Rydberg constant, yet use the same symbol R in their expression (33) for FWHM Doppler width.It is clear that in the latter expression the intended quantity is the molar or universal gas constantR"kM/m, where k is the Boltzmann constant, m the mass in grams, M the atomic weight in AMU

On the expressions for line optical thickness 517

and M/m the Avogadro constant NA. Thorne18 more clearly relates the symbols R and k, giving the

equivalence R/M"k/m. Jordan20 used R/M in her expression (13) for Doppler HWHM halfwidth.Liou21 uses a still differently defined symbol R, which he calls the ‘‘individual gas constant’’, for

the above defined ratio R/M, in his expression (2.2.9) for Doppler HW(1/e)M width aD,*l

D.

[It may be noted that the atmospheric workers use aD

for the astrophysical *lD. The latter

notation appears to be a more logical notation because a is often used for other quantities, e.g.polarizability, or even the absorption coefficient itself, and because the dimension of frequency ismore clearly exhibited. The ambiguous or even conflicting use of spectroscopic notation, betweenastrophysical and atmospheric workers, is pointed up most clearly perhaps in the use of the symbolS which for most spectroscopists is the line strength defined as (the square of ) a multipole matrixelement, as introduced by Condon and Shortley,22 and is therefore a purely atomic propertyindependent of the atomic environment (temperature, density, etc.). The atmosphericworkers21,23~25 on the other hand use the same symbol S, referred to variously as integratedabsorption, line strength or even line intensity, to refer to the integrated absorption coefficient, i.e.they define the line absorption coefficient, k(l) as k (l)"Sg (l) where g(l),P (l)/*l is the normalizedprofile (‘‘shape factor or line shape’’), and S": k (l) dl. This S is thus defined quite differently,involving level populations (defined by ambient density and temperature, etc.) in addition to theintrinsic atomic multipole transition matrix element. (One astrophysical text which similarly definesline emission as ‘‘line strength’’ is that of Shu.14)]

3.2. Literature expressions

A limited selection of such expressions is given here, to illustrate the practical problems which canbe encountered in comparing/validating them against expressions (4) or the more general expres-sions (1a), (1b) and (3). It is stressed that in each case ambiguities or conflicts, when noted, do notaffect the main conclusions of the articles cited.

Doyle and McWhirter,26 following Mitchell and Zemansky,15 use the correct expression foroptical thickness in treating C III emission lines at the solar limb. Clegg and Walsh27 also give thecorrect expression.

Furukane et al28 give the correct expression and numerical coefficient, including the stimulatedemission factor. Davis and McFarlane29 give the correct expression (1a), using the FWHM Dopplerwidth in differently defined profile functions g (l).

Acton,30 in treating X-ray emission in the solar corona, gives an expression (3) for absorptioncoefficient which is specialized in terms of relative abundance of the elements and contains a factor0.80 representing an assumed value of the ratio N

H/N

e, and which therefore must be coupled with

electron column density instead of element column density to give the correct optical thickness. Inhis Table 1, the heading ‘‘column density’’ thus refers to electron column density. The correspondingentries must be multiplied by the factor 0.80 to give (conventional) elemental column densities.

Jaegle et al,31 in discussing optically thick line formation in laboratory plasmas, give an expres-sion (1) for absorption coefficient coupled with an expression (4) for ‘‘line width’’. A factor of J2 ismissing in expression (4), and a factor of 2 is missing in expression (1), so that their absorptioncoefficient is too small by a factor of 1/J2"0.707. (There is also a suffix ‘‘g’’ implying gramsfollowing ‘‘atomic mass’’ M, in their expression (4), which can be misleading. Jaegle et al howevergive a following concrete example which shows that their M is to be interpreted as atomic weightrather than atomic mass in grams.)

Jordan,20 in her important paper on the effect of optical depth on lines from a common upperlevel, gives an expression (12) for absorption coefficient and an expression (13) for Doppler‘‘half-width’’. The latter expression includes a factor (ln 2)1@2, being therefore the HWHM value,while the former expression includes a factor 2 (ln 2)1@2, so that one would expect the resultingnumerical equivalent—Jordan’s expression (14)—to be twice the value of the correct expression (1)above. However, the numerical factor in her expression (14) is half the correct value of 1.16]10~14.In a later article,32 Jordan gives the correct numerical factor (note a typographical omission of theelemental abundance factor N(E)/N(H) in the integrand of the expression in question, on p. 1468).

Park,33 in presenting a useful treatment of curves of growth for Van der Waals-broadened spectrallines, gives however an expression for optical depth—not numbered—which appears to have

518 S. O. Kastner

a number of misprints. The coefficient is given as ne/mc rather than ne2/mc, and the given numericalequivalent appears to be incorrect. It was noted also that Park uses the frequency symbol l forwavenumber, instead of the more usual p.

Colket34 similarly uses the frequency symbol l for wavenumber, and therefore includes an extrafactor of c in the denominator of his (unnumbered) expression for absorption coefficient, which alsocontains the FWHM factor 2 (ln 2)1@2. Though his expression appears to be correct, it is not easilycomparable with expression (1b) above because of the non-standard notation.

4. THE VOIGT PARAMETER a

It will be noticed that while most texts and articles have given an expression for line-centerabsorption coefficient equivalent to (4a) above, which is appropriate for the case of pure Dopplerbroadening of the line profile and which will hold reasonably well for most strong resonance lines,few have incorporated the decrease at line center encountered when appreciable line damping bycollisions is present [expression (4b)], represented by a significantly large Voigt parameter a. This isthe case for many subordinate lines, and even for some resonance lines under conditions of highdensity. The variation from almost pure Doppler line profiles to almost pure Lorentz line profiles asone moves from the rarefied upper terrestrial atmosphere to the denser lower atmosphere isa concrete example discussed by the atmospheric workers. As noted above the decrease in line-centerintensity, included in expression (4b), is given by the Voigt line-center factor »(0, a)"exp(a2) erfc(a)which can be readily computed using one of the algorithms given in the Handbook of Abramowitzand Stegun,35 or by Hui et al.36 What is first needed, however, is an estimate of the Voigt parametera for the particular line considered. A brief summary of the definitions and evaluation of thisparameter and associated quantities is therefore given here.

The definition of a is

a,(HWHM)*l

L(HW(1/e)M)*l

D

,

c4n*l

D

, (6)

where the total damping constant is given in general by c,4n*lL"c

N#c

C#c

R, the sum of the

natural broadening constant cN"&

lA

ul, the collisional broadening constant c

C"c

CE#c

CI(c

CEbeing the elastic contribution and c

CIthe inelastic contribution), and the broadening constant c

Rdue

to stimulated emission and absorption when a radiation field is present. A general expression for thelast quantity is given for example by Allen19 as

cR" +

j:u

Bujº(l

uj)# +

k;u

Bukº(l

uk)#+

j:l

Bljº(l

lj)#+

k;l

Blkº (l

lk), (7)

where º(l) is the radiation field density, given in terms of the mean (direction-averaged) radiationfield intensity J (l) by º(l)"4nJ/c, and the B

ijare the Einstein stimulated emission coefficients.

The inelastic contribution cCI

to the collisional damping constant is essentially the sum of thecollisional excitation and de-excitation rates +

jOuC

ujand +

kOlC

lkleaving the upper and lower

levels of the transition. In terms of the effective collision strengths ¶ij

involved, it may be expressedas

cCIK(8.63]10~6)N

e¹~1@2

e G1

guC +r:u

¶ru# +

s;u

¶us

exp(!Eus/k¹

e)D

#

1

glC +r:l

¶rl#+

s;l

¶lsexp(!E

ls/k¹

e)DH .

An example of its calculation for the particular case of C III lines is given by Bhatia and Kastner.37In the same reference, examples of calculation of the elastic contribution c

CEare given, using the

hyperbolic-path approximation of Sahal-Brechot and Van Regemorter.38 Values of the resultingVoigt parameter a are illustrated in Fig. 2 of that reference for some resonance, subordinate andintercombination lines of C III, as functions of electron density.


5. APPLICABILITY OF THE VOIGT PROFILE

For completeness, a brief description is given here of conditions for which the Voigt profile maynot be applicable.

The Voigt profile is obtained as a convolution of the Doppler and radiative/collisional broadeningprofiles, which assumes independence of the two contributions. More detailed analyses show thatthis assumption does not hold completely. A more general Voigt profile, which takes into accountthe interdependence of the two contributions due to coupling by emitter—perturber relative vel-ocities, has been discussed and evaluated by Cope et al39,40 (and see references therein). The moregeneral Voigt profile involves line shift as well as width modification. At higher gas densitiesnarrowing of the profile occurs also, due to an effectively decreased Doppler contribution, as shownby Dicke41 and Gersten and Foley.42

Other line broadening mechanisms besides Doppler broadening and (Lorentz) radiative broaden-ing and electron collisional broadening may be present, e.g. in ionized plasmas where Starkbroadening produces a different and asymmetric (Holtsmark) line profile. Such mechanisms cannotbe described by the Lorentzian parameter a. The profiles resulting from the contributions of severalbroadening mechanisms have been discussed by some authors, e.g. Mijatovic et al43 who combineDoppler and Stark contributions; Rosznyai44 who combined Doppler, Lorentz and Stark contribu-tions; and Humlic\ ek45 who combines Doppler, Lorentz, Stark and instrumental broadening.

Standard texts on line profiles (some referenced in the references just given) can be consulted forfurther information on these departures from the Voigt profile. For the present purpose it suffices tonote that such modified line profiles need only to be suitably normalized to be incorporated into thegeneral line absorption coefficient expression (1a) and (1b) of Sec. 2 above, giving then the associatedoptical thickness s(l)"k(l)¸. On the other hand, analyses of line scattering require more thanknowledge of s(l) alone, because they involve escape probability expressions which are integrals overthe line profiles; or in the case of full radiative transfer treatments, scattering kernels which areintegrals over the line profiles. Even for the Doppler and Voigt profile functions the expressions arenot integrable but have been represented by series or integrated numerically. For the modified lineprofile functions just mentioned, the corresponding scattering kernels and escape probabilities donot appear yet to have been evaluated.

Finally, a profile-independent definition of optical thickness itself (Ref. 6, p. 62) is more appropri-ate than line-center optical thickness e.g. when comparing scattering by different line profiles, orwhen complete frequency redistribution does not hold during the scattering process.

REFERENCES

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spheres. Pergamon, Oxford, 1972.6. Ivanov, V. V., ¹ransfer of Radiation in Spectral ¸ines. NBS Special Publication 385, Washington, D. C.,

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10. Swihart, T. L., Astrophysics and Stellar Astronomy. Wiley, New York, 1968.11. Jefferies, J. T., Spectral ¸ine Formation. Blaisdell, Waltham, MA, 1968.12. Rybicki, G. B. and Lightman, A. P., Radiative Processes in Astrophysics. Wiley, New York, 1979.13. Osterbrock, D. E., Astrophysics of Gaseous Nebulae and Active Galactic Nuclei. University Science Books,

Mill Valley, CA, U.S.A., 1989.14. Shu, F. H., ¹he Physics of Astrophysics, »ol. 1: Radiation. University Science Books, Mill Valley, CA,

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520 S. O. Kastner

20. Jordan, C., Solar Phys., 1967, 2, 441.21. Liou, K. N., Radiation and Cloud Processes in the Atmosphere. Oxford University Press, Oxford, 1992.22. Condon, E. U. and Shortley, G. H., ¹heory of Atomic Spectra. Cambridge University Press, New York,

1951.23. Penner, S. S., Quantitative Molecular Spectroscopy and Gas Emissivities. Addison-Wesley, Reading, MA,

U.S.A., 1959.24. Lenoble, J., Atmospheric Radiative ¹ransfer. Deepak Publishing, Hampton VA, U.S.A., 1993.25. Goody, R., Principles of Atmospheric Physics and Chemistry. Oxford University Press, New York, 1995.26. Doyle, J. G. and McWhirter, R. W. P., Mon. Not. R. Astron. Soc., 193, 947.27. Clegg, R. E. S. and Walsh, J. R., Mon. Not. R. Astron. Soc., 1985, 215, 323.28. Furukane, U., Yokota, T., Kawasaki, K. and Oda, T., J. Quant. Spectrosc. Radiat. ¹ransfer, 1983, 29, 75.29. Davis, C. C. and McFarlane, R. A., J. Quant. Spectrosc. Radiat. ¹ransfer, 1977, 18, 151.30. Acton, L. W., Astrophys. J., 1978, 225, 1069.31. Jaegle, P., Jamelot, G. and Carillon, A., Progress in Stellar Spectral ¸ine Formation ¹heory. D. Reidel,

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¹ransfer, 50, 1993.44. Rozsnyai, B. F., J. Quant. Spectrosc. Radiat. ¹ransfer, 1978. 19, 641.45. Humlic\ ek, J., J. Quant. Spectrosc. Radiat. ¹ransfer, 1983, 29, 125.


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ON THE EXPRESSIONS FOR LINE OPTICAL THICKNESS