- Email: [email protected]
New tables of the Voigt function
J. Quanr. Spcctrosc. Radiat. Transfer Vol. 42, No. 2, pp. Printed in Great Bntain. All rights reserved
NEW
TABLES VIVEK
1 I I-I 15, 1989
0022-4073/8Y
$3.00+0.00
Copyright Q 1989 Maxwell Pergamon Macmillan plc
OF THE BAKSHI
VOIGT
and R.
FUNCTION
J. KEARNEY
Department of Physics, University of Idaho, Moscow. ID 83843. U.S.A. (Received
21 November
1988)
Abstract-New tables are presented for fitting experimental data to the Voigt function. We used a 16,000-point line profile to generate new tables and have found up to 28% difference with existing tables.
INTRODUCTION
In the presence of Doppler and Lorentzian components, Voigt function,‘.’ which can be written as-’
the resultant line profile is given by the
x {exp(-t’)/[Y*+(X--r)*]} dt. (1) s mr Here, Y = (&lb,) m, 6, = Doppler width of the transition, b, = Lorentzian width of the transition, and X = [2(& - 2)/b*] @; i.e., X is the wavelength separation for the intensity peak at &,, scaled by the Doppler width of the transition. The Voigt function has been widely used to represent the line shape of the transitions emitted in arcs,4 d.c. plasmas’ and ICPS.~ There has also been some work on a general Voigt function, which takes into account the asymmetries and the wavelength shift present in the line shapes of the wavelengths emitted by these sources.6 During the last few decades, extensive calculations have been performed on the Voigt function for various values of X and Y. Armstrong’ published an excellent review of the work in this area prior to 1967. Most of the reviews of this work2’-24 are based on estimation of precision and computation times for the various algorithms used. There have been various approximations proposed to reduce the computation time in the calculation of the Voigt function.25-30In the present case, we are concerned with only one computation and hence the computation time is not important here. We have used algorithms from Armstrong’ for our calculations. Table 1 gives these algorithms, chosen for various values of X and Y. To determine the values of b, and b,, we need to fit the Voigt function to the experimental data. One way to do this is to use a grid-search method, which is usually very time-consuming. An easy way to perform this process is to read the values of the coefficients directly from tables by comparing ratios of full widths of the curves at various percentage intensities. Two such tables are available in the literature, one by van de Hulst et al” (VH) and another by Davies et al (JD).13 These tables have been reproduced by Alan3’ and Wiese.32 V(X, Y) = (Y/n)
TABLES
FOR VOIGT
FUNCTION
The use of Voigt-function tables is an easy procedure for fitting the experimental data to the function. The need for new tables arises for the following reasons: (1) In both of the tables of VH and JD, algorithms given by Born*’ and Hjterting14 have been used to calculate the values of the function. More accurate algorithms have since been proposed7.‘9 and we have found that the values calculated by the present algorithm’ differ significantly from the values of VH and JD. For given values of b,/b,, the percentage widths of VH differ up to 3%, and those of JD differ up to 4%. The differences in the values of b2/h and b,/h are still greater, up to 28% for VH and up to 18% for JD. Thus errors may result up to 28% error in the widths 6, and b,. The accuracy of VH is only 2 significant figures while that of JD is 5 significant figures. We have used 15 digits in our calculations, but report here only 5 digits in the tables for reasons given in the next section. (2) The tables of VH and JD do not cover the entire range of Y very well. There are only a few points 111
VIVEK BAKSHI and
0.2
R. J.
0.4
KEARNEY
0.6
0.8
1.0
Cb,/h)
Fig. I. Plots of the ratio of the full width of the Voigt function at 80 and 10% intensities, with respect to the full width at 50% intensity. as a function of h,/h. Also shown are piots of b,/b2 and h,/h as functions of h,/h. See the text for details on the use of this graph.
available in the region of high Y( 32.5) where the Voigt function lies for most Stark-broadened lines in a d.c. plasma. Also, tables with smaller step sizes in Y are needed to maintain the accuracy of the interpolated values. CALCULATIONS
Our choice of aigorithm is the same as that of Armstr0ng.l No difference was found up to 15 digits between the values calculated by various algorithms near the boundaries of their range. We used I f&000-point profiles to calculate the widths. For this number of points, the widths converged at least up to 6 digits and hence we have shown only 5 digits in our final results. A straight line was fitted between points to obtain the widths at various percentage intensities. RESULTS
AND
TABLES
The results of our calculations are shown in Table 2. Column ho.sgives the ratio of the full width of the curve at 90 and 50% of maximum intensity. The results are plotted in Fig. 1 for the h,,, and h,, , values. To use the tables, one should measure the full width of the curve at 10, 50, 80% etc., of the maximum intensity. For example, the ratio of the 80% width to the 50% width will give the value of ho,#.Using this value, we can read the values of b,/h and b,/h from Fig. 1 or Table 2 (11is the value of the full width of the curve at half of maximum intensity). Since we know h, we can calculate the values of b, and bZ. The results may be further checked by measuring the widths itt other percentage intensities. Of course, the final accuracy of b, and b2 will depend on the accuracy of the measurements of the percentage widths. If the values of widths for a given experiment lie in a small range and a higher accuracy is desired, e.g., when the measurements of line shapes are performed with a Fabry-Perot interferometer, then it would be better to generate tables for a smaller step size.? iAll of the calculations were performed using Turbo Pascal (version 3.0) on an IBM/PC with an 8087 math co-processor. Software can be obtained from the authors and may be used to calculate desired tables in any given spectral region.
Table I. Algorithms are shown for calculation of the Voirrt function.7
HummerFadde
Table 2. Tabulations are given of Voigt profiles at various percentage intensities.
conrimed overleaf
1.8617E+OOO
8.02909-001
6_414OE+OOO
8.3718EtOOO
I.L762E+OOl
9.7509E-001
9.9509E-001
5.3570~000
9.850X-001
4.6724%+000
9.5508E-001
9.6506E-001
6.7410E-002
l.l766E-001
1.5202E-001
1.8WE-001
2.044x-001
3.3420~-001
3.3589E-001
3.3750E-001
3.3904E-001
3.4050E-001
3.4328E-001
2.26llE-001
3.4191E-001
3.4457E-001
3.4582E-001
2.6463E-001
2.8209f-001
3.4670E-001
3.4789E-001
3.4904E-001
3.5012E-001
3.5120E-001
3.5225E-001
3.5326E-001
3.5421E-001
3.552lE-001
3.5613E-001
3.57lOf-001
3.5796E-001
4.179'&000
9.2496E-001
5.3099E-001
5.3195E-001
5.3295~-001
5.3397E-001
5.3482E-001
5.0097E-001
5.0290f-001
5.0474E-001
5.065lE-001
5.0821E-001
S.U985E-001
5.11462-001
5.129?~-001
5.1446E-001
5.1549E-001
5.1690E-001
5.1827~-001
S.1956E-001
5.2084~-001
5.22lOE-001
5.2331~-001
5.2445E-001
5.2565E-001
5.2676E-001
5.2793~-001
5.2897~-001
3.588-5E-001 5.3005E-001
3.5964E-001
3.6044f-001
3.6127E-001
3.6212E-001
3.6282E-001
5.3568E-001
5.3657E-001
3&2bE-001 3.6353E-001
5.374X-001
5.3839E-001
5.3934E-001
5.4006E-001
5.4079E-001
b0.8
3.650lE-001
3.6578E-001
3&55E-001
3.6715E-001
3_6775E-001
b0.9
9.3492E-001 3.79X%+000 2.4632E-001
3.2430E+OOO
3.4953E+OOO
9.1483E-001
2.9400E-001
3.0999E-001
3.2533~001
3.3961E-001
3.537lE-001
3.6747f-001
3.8067f-001
3.9312E-001
4.0630f-001
4.1846E-001
4.3124E-001
4.4277E-001
4.548lE-001
4.6526E-001
4.76tlE-001
4.8739E-001
4.WllE-001
5.0882E-001
5.1883E-001
5.2917~-001
5_3983E-001
5.5084E-001
5.6221E-001
5.7097E-001
5.7995E-001
b2/h
9.45lOf-001
2.8947E+OOO
3.0869~+000
8.9733f-001
9.07SS~-001
2.5824E+OOO
2.726SE+OOO
8.8702E-001
8.6670E-001
8_77OOf-001
2.3302EcOOO
2.4503f+OOO
8.5626E-001
2.1260E+OOO
2.222lf+OOO
8.35788-001
8.4588E-001
1.9458EtOOO
1.7897f+OOO
7_9242E-001
2.029%+000
1.7176WOOO
7.8118E-001
8.142SE-001
1.657SE+OOO
7.7118E-001
8.2475E-001
1.5374~+000
1.5975E+OOO
1_4774E+OOO
7.3737E-001
7.4933~-001
1.4293~+000
7.2727~-001
7.6058f-001
1.33326+000
1.3813~+000
7_055tE-001
1_2852E+OOO
6.9379f-001
7.16&E-001
l.l891E+OOO
1.2372f+OOO
6.6853f-001
6.8148f-001
l.l170f+OOO
l.l531E+OOO
6,4783f-001
b/b2
6.58386-001
bl/h
6.7472E-001
6.5550f-001
6.572OE-001
6.5883E-001
6.6042~-001
6.619SE-001
6.6343f-001
6.6489E-001
6.6627f-001
6.6764E-001
6.6859E-001
6.6989E-001
6.7116f-001
6.7236E-001
&7355E-001
l.OOOOE+OOO
1.0oooE+000
1.0000E+000
l~OOOOE+OOO
1.0000E+000
1.0000E+000
1.0000E+000
1.0000E+000
l.OOOOf+OOO
l.OOOOE+OOO
1.OOOOf+OOO
1.0000E+000
1.0000E+000
1.0000E+000
b0.5
8.3101f-001
8.3177E-001
8.3247E-001
8.2953E-001
8.1703E-001
8.1809E-001
8.1913E-OOl
8.2014E-001
8.2112E-001
8.2208E-001
8.2303E-001
8.23938-001
8.2483E-001
8.254%-001
8.263lE-001
8.2715~-001
8.2795E-001
8.28758-001
1.2116E+000
1.2lOSE+OOO
1.209lE+OOO
1.2076E+OOO
1.2062E+OOO
1.2049~+000
1.2035~+000
1.2022E+OOO
1.2009~+000
l.l995E+OOO
l.l982E+OOO
l.l969E+OOO
l.l957E+OOO
1.1944E+OOO
l.l933E+OOO
l.l922f+OOO
1.1910E+000
1.1897E+OOO
l.l887f+OOO
l.l877E+OOO
I.l866E+OOO
1.1855E+OOO
1.1844E+OOO
l.l832E+OOO
1.1824~+000
1.1815E+OOO
b0.4
1.4939E+OOO
1.4911E+OOO
1.4873E+OOO
1.4834E+OOO
1.4798E+oOO
1.4761E+OOO
1.4723~+000
1.4687~+000
1.4652E+OOO
1.4614f+OOO
1.4579E+OOO
1.4542E+OOO
1.4508E+OOO
1.4472~+000
1.4441E+OOO
1.4409E+OOO
1.4375E+OOO
1.4340E+OOO
1.43lIE+000
1.428lE+OOO
1.425Of+OOO
1.4218E+OOO
1.4185E+OOO
1.4152E+OOO
1.4126E+OOO
1.4100E+OOO
b0.3
2.76921+000
2.7526EtOOO
2.73648+000
2.7188f+OOO
2_7021f+000
2.6841EtOOO
2,6675f+OOO
2.6497E+OOO
2.6339E+OOO
2.6173E+OOO
2.599&+000
2.5810EsOOO
2.565X+000
2.5489ftOOO
2.5317E+OOO
2.5137E+OOO
2.4949E+OOO
2.4753E+OOO
2.46008+000
2.4442f+OOO
bO.l
1.9326E+OOO
1.9269E+OOO
1.9190E+OOO
1.91llE+OOO
1.9034E+OOO
2.8632E+OOO
2.8SlSE+OOO
2.835lf+OOO
2.8185f+ooo
2.8024E+OOO
l.f3956E+OOO 2.7859E+OOO
1.8877E+OOO
1.8799E+OOO
1.8724~+000
1.8643E+OOO
1.8566E+OOO
I.&LBbE+OOO
1.8409E+OOO
1.8330E+OOO
1.8260~+000
1.8186E+OOO
1.8110E+000
1.8029E+OOO
1.7962E+OOO
1.7892E+OOO
1.782OE+OOO
1.7746E+OOO
1.7669E+OOO
1.7590E+OOO
1.7529E+OOO
1.7466E+OOO
b0.2
1.0000E+000
l.OOOOE+000
1.0000E+000
1.0000E+000
1.0000E+000
l.O00OE+000
1.0000E+000
1.2239~+000
1.2223E+OOO
1.2207E+OOO
1.2192~+000
1.2176E+OOO
1.216lE+OOO
1.2145EtOOO
l.S255~+000
1.5215~+000
t.S17SE+OOO
1.5134E+OOO
1.5095E+OOO
1.5056~+000
l~S016E+OOO
1.996lE+OOC
1.9881EtOOO
1.9801E+000
l.9721~+000
1.964X+000
1.9563E+OOO
1.9483E+OOO
2.9921EtoOO
2.976lE+OOO
2.9601E+OOO
2.9440E+OOO
2.9279ftOOO
2.9119E+OOO
2.8955f+OOO
1.00OOf+000 1.213OE+OOO1.4977E+OOO 1.9405f+OOO 2.8795E+OOO
1.0000E+000
1.0000E+000
1.OOOOE+OOO
l.O00OE+000
l.OOOOE+OOO
1.0000E+000
1.0000f+000
1.0000f+000
1.0000f+000
l.OOOOE+OOO
i.OOOOE+OOO
8.3321f-001l.0000E+OOO
8.3387E-001
8.3457E-001
8.3516E-001
8.3578E-001
8.3642E-001
8.3708E-001
8.3763E-001
8.3818E-001
8.387%-001
8.39346-001
8.3993f-001
8.4054E-001
8.410lE-001
B-4148&-001
bO.6
6.7!!8Sf-001 8.3029E-001
6.7691f-001
6.7804E-001
6.7908E-001
6.8018f-001
6.81161-001
6.8218E-001
6.8306E-001
6.8397f-001
6.8491~-001
6.85688-001
6.6668E-001
6.8749E-001
6.8833E-001
6.89186-001
6.9006E-001
6.9095E-001
6.9163E-001
6.9232E-001
b0.7
New tables of the Voigt function
115
AcknoH’l~d~ement-The work described in this paper was supported by the U.S. Department of the Interior Bureau of Mines under Contract No. 50134035 through the Department of Energy, under Contract Number DE-AC07-761D01570. REFERENCES 1. (a) A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and E.xcited Atoms, Macmillan, New York, NY (1934); (b) S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities, Addison-Wesley, Reading, MA (1959). 2. Deconvolution with Applications in Spectroscopy,, P. A. Jannson ed., Academic Press, New York, NY (1987). 3. T. Hasegawa and H. Haraguchi, Spectrochim. Acta 40B, 123 (1985). 4. D. W. Jones, K. Musiol, and W. L. Wiese, Spectral Line Shapes 2, K. Burnet ed., de Gruyter, New York, NY (1983). 5. V. Bakshi and R. J. Kearney. JQSRT 41, 369 (1989). 6. D. Cope, R. Khoury, and R. J. Lovett, JQSRT 39, 163 (1988). 7. B. H. Armstrong, JQSRT 7, 61 (1967). 8. B. D. Fried and S. D. Conte, The Plasma Dispersion Function, Academic Press, New York, NY (1961). Table and Generating Procedure,” 9. D. G. Hummer, “The Voigt Function: An Eight-Significant-Figure Univ. of Colorado, NBS JILA Report 24, unpublished (1964). IO. D. W. Posner, Aust. J. Phys. 12, 184 (1959). 11. V. N. Faddeyeva and N. M. Terentev, Tables of Values of the Function W(Zkfi)r Complex Argument, Pergamon Press, New York, NY (1961). 12. H. C. Van de Hulst and J. J. M. Reesink, Astrophys. J. 106, 121 (1947). 13. J. T. Davies and J. M. Vaughan, Astrophys. J. 137, 1302 (1963). 14. F. Hjerting, Astrophys. J. 88, 508 (1938). 15. K. Hunger, Z. Astrophys. 39, 36 (1956). 16. B. W. Fowler and C. C. Sung, JOSA 65, 949 (1975). 17. J. H. Pierluissi and P. C. Vanderwood, JQSRT 18, 555 (1977). 18. J. R. Drummond and M. Steckner, JQSRT 34, 517 (1985). 19. A. K. Hui. B. H. Armstrong. and A. A. Wray, JQSRT 19, 509 (1978). 20. M. Born, Optik, Springer, Berlin (1933). 21. J. J. Oliver0 and R. J. Longbothum, JQSRT 17, 233 (1977). 22. A. Klim, JQSRT 26, 537 (1981). 23. J. T. Twitty, P. L. Rarig, and R. E. Thompson, JQSRT 24, 529 (1980). 24. N. Allard and J. Kielkopf, Rev. Mod. Phljs. 54, 1103 (1982). 25. X. Zhu, JQSRT 39, 421 (1988). 26. K. G. P. Sulzmann, JQSRT 29, 89 (1983). 27. L. M. Faires, B. A. Palmer, and J. W. Bra&, Spectrochim. Acta 40B, 135 (1985). 28. J. Humlicek, JQSRT 27, 437 (1982). 29. S. R. Drayson, JQSRT 16, 611 (1976). 30. A. H. Karp. JQSRT 20, 379 (1978). 31. C. W. Allen, Astrophysical Quantities 3rd edn. Athlone Press, London (1976). 32. W. L. Wiese, Plasma Diagnostic Techniques, R. M. Huddelston and S. L. Leonard eds., Academic Press, New York, NY (1964).
NEW
TABLES VIVEK
1 I I-I 15, 1989
0022-4073/8Y
$3.00+0.00
Copyright Q 1989 Maxwell Pergamon Macmillan plc
OF THE BAKSHI
VOIGT
and R.
FUNCTION
J. KEARNEY
Department of Physics, University of Idaho, Moscow. ID 83843. U.S.A. (Received
21 November
1988)
Abstract-New tables are presented for fitting experimental data to the Voigt function. We used a 16,000-point line profile to generate new tables and have found up to 28% difference with existing tables.
INTRODUCTION
In the presence of Doppler and Lorentzian components, Voigt function,‘.’ which can be written as-’
the resultant line profile is given by the
x {exp(-t’)/[Y*+(X--r)*]} dt. (1) s mr Here, Y = (&lb,) m, 6, = Doppler width of the transition, b, = Lorentzian width of the transition, and X = [2(& - 2)/b*] @; i.e., X is the wavelength separation for the intensity peak at &,, scaled by the Doppler width of the transition. The Voigt function has been widely used to represent the line shape of the transitions emitted in arcs,4 d.c. plasmas’ and ICPS.~ There has also been some work on a general Voigt function, which takes into account the asymmetries and the wavelength shift present in the line shapes of the wavelengths emitted by these sources.6 During the last few decades, extensive calculations have been performed on the Voigt function for various values of X and Y. Armstrong’ published an excellent review of the work in this area prior to 1967. Most of the reviews of this work2’-24 are based on estimation of precision and computation times for the various algorithms used. There have been various approximations proposed to reduce the computation time in the calculation of the Voigt function.25-30In the present case, we are concerned with only one computation and hence the computation time is not important here. We have used algorithms from Armstrong’ for our calculations. Table 1 gives these algorithms, chosen for various values of X and Y. To determine the values of b, and b,, we need to fit the Voigt function to the experimental data. One way to do this is to use a grid-search method, which is usually very time-consuming. An easy way to perform this process is to read the values of the coefficients directly from tables by comparing ratios of full widths of the curves at various percentage intensities. Two such tables are available in the literature, one by van de Hulst et al” (VH) and another by Davies et al (JD).13 These tables have been reproduced by Alan3’ and Wiese.32 V(X, Y) = (Y/n)
TABLES
FOR VOIGT
FUNCTION
The use of Voigt-function tables is an easy procedure for fitting the experimental data to the function. The need for new tables arises for the following reasons: (1) In both of the tables of VH and JD, algorithms given by Born*’ and Hjterting14 have been used to calculate the values of the function. More accurate algorithms have since been proposed7.‘9 and we have found that the values calculated by the present algorithm’ differ significantly from the values of VH and JD. For given values of b,/b,, the percentage widths of VH differ up to 3%, and those of JD differ up to 4%. The differences in the values of b2/h and b,/h are still greater, up to 28% for VH and up to 18% for JD. Thus errors may result up to 28% error in the widths 6, and b,. The accuracy of VH is only 2 significant figures while that of JD is 5 significant figures. We have used 15 digits in our calculations, but report here only 5 digits in the tables for reasons given in the next section. (2) The tables of VH and JD do not cover the entire range of Y very well. There are only a few points 111
VIVEK BAKSHI and
0.2
R. J.
0.4
KEARNEY
0.6
0.8
1.0
Cb,/h)
Fig. I. Plots of the ratio of the full width of the Voigt function at 80 and 10% intensities, with respect to the full width at 50% intensity. as a function of h,/h. Also shown are piots of b,/b2 and h,/h as functions of h,/h. See the text for details on the use of this graph.
available in the region of high Y( 32.5) where the Voigt function lies for most Stark-broadened lines in a d.c. plasma. Also, tables with smaller step sizes in Y are needed to maintain the accuracy of the interpolated values. CALCULATIONS
Our choice of aigorithm is the same as that of Armstr0ng.l No difference was found up to 15 digits between the values calculated by various algorithms near the boundaries of their range. We used I f&000-point profiles to calculate the widths. For this number of points, the widths converged at least up to 6 digits and hence we have shown only 5 digits in our final results. A straight line was fitted between points to obtain the widths at various percentage intensities. RESULTS
AND
TABLES
The results of our calculations are shown in Table 2. Column ho.sgives the ratio of the full width of the curve at 90 and 50% of maximum intensity. The results are plotted in Fig. 1 for the h,,, and h,, , values. To use the tables, one should measure the full width of the curve at 10, 50, 80% etc., of the maximum intensity. For example, the ratio of the 80% width to the 50% width will give the value of ho,#.Using this value, we can read the values of b,/h and b,/h from Fig. 1 or Table 2 (11is the value of the full width of the curve at half of maximum intensity). Since we know h, we can calculate the values of b, and bZ. The results may be further checked by measuring the widths itt other percentage intensities. Of course, the final accuracy of b, and b2 will depend on the accuracy of the measurements of the percentage widths. If the values of widths for a given experiment lie in a small range and a higher accuracy is desired, e.g., when the measurements of line shapes are performed with a Fabry-Perot interferometer, then it would be better to generate tables for a smaller step size.? iAll of the calculations were performed using Turbo Pascal (version 3.0) on an IBM/PC with an 8087 math co-processor. Software can be obtained from the authors and may be used to calculate desired tables in any given spectral region.
Table I. Algorithms are shown for calculation of the Voirrt function.7
HummerFadde
Table 2. Tabulations are given of Voigt profiles at various percentage intensities.
conrimed overleaf
1.8617E+OOO
8.02909-001
6_414OE+OOO
8.3718EtOOO
I.L762E+OOl
9.7509E-001
9.9509E-001
5.3570~000
9.850X-001
4.6724%+000
9.5508E-001
9.6506E-001
6.7410E-002
l.l766E-001
1.5202E-001
1.8WE-001
2.044x-001
3.3420~-001
3.3589E-001
3.3750E-001
3.3904E-001
3.4050E-001
3.4328E-001
2.26llE-001
3.4191E-001
3.4457E-001
3.4582E-001
2.6463E-001
2.8209f-001
3.4670E-001
3.4789E-001
3.4904E-001
3.5012E-001
3.5120E-001
3.5225E-001
3.5326E-001
3.5421E-001
3.552lE-001
3.5613E-001
3.57lOf-001
3.5796E-001
4.179'&000
9.2496E-001
5.3099E-001
5.3195E-001
5.3295~-001
5.3397E-001
5.3482E-001
5.0097E-001
5.0290f-001
5.0474E-001
5.065lE-001
5.0821E-001
S.U985E-001
5.11462-001
5.129?~-001
5.1446E-001
5.1549E-001
5.1690E-001
5.1827~-001
S.1956E-001
5.2084~-001
5.22lOE-001
5.2331~-001
5.2445E-001
5.2565E-001
5.2676E-001
5.2793~-001
5.2897~-001
3.588-5E-001 5.3005E-001
3.5964E-001
3.6044f-001
3.6127E-001
3.6212E-001
3.6282E-001
5.3568E-001
5.3657E-001
3&2bE-001 3.6353E-001
5.374X-001
5.3839E-001
5.3934E-001
5.4006E-001
5.4079E-001
b0.8
3.650lE-001
3.6578E-001
3&55E-001
3.6715E-001
3_6775E-001
b0.9
9.3492E-001 3.79X%+000 2.4632E-001
3.2430E+OOO
3.4953E+OOO
9.1483E-001
2.9400E-001
3.0999E-001
3.2533~001
3.3961E-001
3.537lE-001
3.6747f-001
3.8067f-001
3.9312E-001
4.0630f-001
4.1846E-001
4.3124E-001
4.4277E-001
4.548lE-001
4.6526E-001
4.76tlE-001
4.8739E-001
4.WllE-001
5.0882E-001
5.1883E-001
5.2917~-001
5_3983E-001
5.5084E-001
5.6221E-001
5.7097E-001
5.7995E-001
b2/h
9.45lOf-001
2.8947E+OOO
3.0869~+000
8.9733f-001
9.07SS~-001
2.5824E+OOO
2.726SE+OOO
8.8702E-001
8.6670E-001
8_77OOf-001
2.3302EcOOO
2.4503f+OOO
8.5626E-001
2.1260E+OOO
2.222lf+OOO
8.35788-001
8.4588E-001
1.9458EtOOO
1.7897f+OOO
7_9242E-001
2.029%+000
1.7176WOOO
7.8118E-001
8.142SE-001
1.657SE+OOO
7.7118E-001
8.2475E-001
1.5374~+000
1.5975E+OOO
1_4774E+OOO
7.3737E-001
7.4933~-001
1.4293~+000
7.2727~-001
7.6058f-001
1.33326+000
1.3813~+000
7_055tE-001
1_2852E+OOO
6.9379f-001
7.16&E-001
l.l891E+OOO
1.2372f+OOO
6.6853f-001
6.8148f-001
l.l170f+OOO
l.l531E+OOO
6,4783f-001
b/b2
6.58386-001
bl/h
6.7472E-001
6.5550f-001
6.572OE-001
6.5883E-001
6.6042~-001
6.619SE-001
6.6343f-001
6.6489E-001
6.6627f-001
6.6764E-001
6.6859E-001
6.6989E-001
6.7116f-001
6.7236E-001
&7355E-001
l.OOOOE+OOO
1.0oooE+000
1.0000E+000
l~OOOOE+OOO
1.0000E+000
1.0000E+000
1.0000E+000
1.0000E+000
l.OOOOf+OOO
l.OOOOE+OOO
1.OOOOf+OOO
1.0000E+000
1.0000E+000
1.0000E+000
b0.5
8.3101f-001
8.3177E-001
8.3247E-001
8.2953E-001
8.1703E-001
8.1809E-001
8.1913E-OOl
8.2014E-001
8.2112E-001
8.2208E-001
8.2303E-001
8.23938-001
8.2483E-001
8.254%-001
8.263lE-001
8.2715~-001
8.2795E-001
8.28758-001
1.2116E+000
1.2lOSE+OOO
1.209lE+OOO
1.2076E+OOO
1.2062E+OOO
1.2049~+000
1.2035~+000
1.2022E+OOO
1.2009~+000
l.l995E+OOO
l.l982E+OOO
l.l969E+OOO
l.l957E+OOO
1.1944E+OOO
l.l933E+OOO
l.l922f+OOO
1.1910E+000
1.1897E+OOO
l.l887f+OOO
l.l877E+OOO
I.l866E+OOO
1.1855E+OOO
1.1844E+OOO
l.l832E+OOO
1.1824~+000
1.1815E+OOO
b0.4
1.4939E+OOO
1.4911E+OOO
1.4873E+OOO
1.4834E+OOO
1.4798E+oOO
1.4761E+OOO
1.4723~+000
1.4687~+000
1.4652E+OOO
1.4614f+OOO
1.4579E+OOO
1.4542E+OOO
1.4508E+OOO
1.4472~+000
1.4441E+OOO
1.4409E+OOO
1.4375E+OOO
1.4340E+OOO
1.43lIE+000
1.428lE+OOO
1.425Of+OOO
1.4218E+OOO
1.4185E+OOO
1.4152E+OOO
1.4126E+OOO
1.4100E+OOO
b0.3
2.76921+000
2.7526EtOOO
2.73648+000
2.7188f+OOO
2_7021f+000
2.6841EtOOO
2,6675f+OOO
2.6497E+OOO
2.6339E+OOO
2.6173E+OOO
2.599&+000
2.5810EsOOO
2.565X+000
2.5489ftOOO
2.5317E+OOO
2.5137E+OOO
2.4949E+OOO
2.4753E+OOO
2.46008+000
2.4442f+OOO
bO.l
1.9326E+OOO
1.9269E+OOO
1.9190E+OOO
1.91llE+OOO
1.9034E+OOO
2.8632E+OOO
2.8SlSE+OOO
2.835lf+OOO
2.8185f+ooo
2.8024E+OOO
l.f3956E+OOO 2.7859E+OOO
1.8877E+OOO
1.8799E+OOO
1.8724~+000
1.8643E+OOO
1.8566E+OOO
I.&LBbE+OOO
1.8409E+OOO
1.8330E+OOO
1.8260~+000
1.8186E+OOO
1.8110E+000
1.8029E+OOO
1.7962E+OOO
1.7892E+OOO
1.782OE+OOO
1.7746E+OOO
1.7669E+OOO
1.7590E+OOO
1.7529E+OOO
1.7466E+OOO
b0.2
1.0000E+000
l.OOOOE+000
1.0000E+000
1.0000E+000
1.0000E+000
l.O00OE+000
1.0000E+000
1.2239~+000
1.2223E+OOO
1.2207E+OOO
1.2192~+000
1.2176E+OOO
1.216lE+OOO
1.2145EtOOO
l.S255~+000
1.5215~+000
t.S17SE+OOO
1.5134E+OOO
1.5095E+OOO
1.5056~+000
l~S016E+OOO
1.996lE+OOC
1.9881EtOOO
1.9801E+000
l.9721~+000
1.964X+000
1.9563E+OOO
1.9483E+OOO
2.9921EtoOO
2.976lE+OOO
2.9601E+OOO
2.9440E+OOO
2.9279ftOOO
2.9119E+OOO
2.8955f+OOO
1.00OOf+000 1.213OE+OOO1.4977E+OOO 1.9405f+OOO 2.8795E+OOO
1.0000E+000
1.0000E+000
1.OOOOE+OOO
l.O00OE+000
l.OOOOE+OOO
1.0000E+000
1.0000f+000
1.0000f+000
1.0000f+000
l.OOOOE+OOO
i.OOOOE+OOO
8.3321f-001l.0000E+OOO
8.3387E-001
8.3457E-001
8.3516E-001
8.3578E-001
8.3642E-001
8.3708E-001
8.3763E-001
8.3818E-001
8.387%-001
8.39346-001
8.3993f-001
8.4054E-001
8.410lE-001
B-4148&-001
bO.6
6.7!!8Sf-001 8.3029E-001
6.7691f-001
6.7804E-001
6.7908E-001
6.8018f-001
6.81161-001
6.8218E-001
6.8306E-001
6.8397f-001
6.8491~-001
6.85688-001
6.6668E-001
6.8749E-001
6.8833E-001
6.89186-001
6.9006E-001
6.9095E-001
6.9163E-001
6.9232E-001
b0.7
New tables of the Voigt function
115
AcknoH’l~d~ement-The work described in this paper was supported by the U.S. Department of the Interior Bureau of Mines under Contract No. 50134035 through the Department of Energy, under Contract Number DE-AC07-761D01570. REFERENCES 1. (a) A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and E.xcited Atoms, Macmillan, New York, NY (1934); (b) S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities, Addison-Wesley, Reading, MA (1959). 2. Deconvolution with Applications in Spectroscopy,, P. A. Jannson ed., Academic Press, New York, NY (1987). 3. T. Hasegawa and H. Haraguchi, Spectrochim. Acta 40B, 123 (1985). 4. D. W. Jones, K. Musiol, and W. L. Wiese, Spectral Line Shapes 2, K. Burnet ed., de Gruyter, New York, NY (1983). 5. V. Bakshi and R. J. Kearney. JQSRT 41, 369 (1989). 6. D. Cope, R. Khoury, and R. J. Lovett, JQSRT 39, 163 (1988). 7. B. H. Armstrong, JQSRT 7, 61 (1967). 8. B. D. Fried and S. D. Conte, The Plasma Dispersion Function, Academic Press, New York, NY (1961). Table and Generating Procedure,” 9. D. G. Hummer, “The Voigt Function: An Eight-Significant-Figure Univ. of Colorado, NBS JILA Report 24, unpublished (1964). IO. D. W. Posner, Aust. J. Phys. 12, 184 (1959). 11. V. N. Faddeyeva and N. M. Terentev, Tables of Values of the Function W(Zkfi)r Complex Argument, Pergamon Press, New York, NY (1961). 12. H. C. Van de Hulst and J. J. M. Reesink, Astrophys. J. 106, 121 (1947). 13. J. T. Davies and J. M. Vaughan, Astrophys. J. 137, 1302 (1963). 14. F. Hjerting, Astrophys. J. 88, 508 (1938). 15. K. Hunger, Z. Astrophys. 39, 36 (1956). 16. B. W. Fowler and C. C. Sung, JOSA 65, 949 (1975). 17. J. H. Pierluissi and P. C. Vanderwood, JQSRT 18, 555 (1977). 18. J. R. Drummond and M. Steckner, JQSRT 34, 517 (1985). 19. A. K. Hui. B. H. Armstrong. and A. A. Wray, JQSRT 19, 509 (1978). 20. M. Born, Optik, Springer, Berlin (1933). 21. J. J. Oliver0 and R. J. Longbothum, JQSRT 17, 233 (1977). 22. A. Klim, JQSRT 26, 537 (1981). 23. J. T. Twitty, P. L. Rarig, and R. E. Thompson, JQSRT 24, 529 (1980). 24. N. Allard and J. Kielkopf, Rev. Mod. Phljs. 54, 1103 (1982). 25. X. Zhu, JQSRT 39, 421 (1988). 26. K. G. P. Sulzmann, JQSRT 29, 89 (1983). 27. L. M. Faires, B. A. Palmer, and J. W. Bra&, Spectrochim. Acta 40B, 135 (1985). 28. J. Humlicek, JQSRT 27, 437 (1982). 29. S. R. Drayson, JQSRT 16, 611 (1976). 30. A. H. Karp. JQSRT 20, 379 (1978). 31. C. W. Allen, Astrophysical Quantities 3rd edn. Athlone Press, London (1976). 32. W. L. Wiese, Plasma Diagnostic Techniques, R. M. Huddelston and S. L. Leonard eds., Academic Press, New York, NY (1964).