Three and four generalized Lorentzian approximations for the Voigt line shape

Julio Puerta; Pablo Martin

doi:10.1364/AO.20.003923

Applied Optics
Vol. 20,
Issue 22,
pp. 3923-3928
(1981)
•https://doi.org/10.1364/AO.20.003923

Three and four generalized Lorentzian approximations for the Voigt line shape

Julio Puerta and Pablo Martin

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Julio Puerta and Pablo Martin, "Three and four generalized Lorentzian approximations for the Voigt line shape," Appl. Opt. 20, 3923-3928 (1981)

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Abstract

Three and four generalized Lorentzians in two variables have been obtained to approximate the Voigt function by using the asymptotical Padé method. The accuracy has been greatly improved with respect to the one and two generalized Lorentzian approximations reported in a previous paper. Furthermore, the four Lorentzian function is always positive for all values of the normalized collision width and line separation. This approximation gives an accuracy for most of the values of better than 0.0001, and in the worst region the absolute error is ~0.001. In the limits of low and high pressure adequate limit functions are obtained. A four generalized Lorentzian gives a reliable and easily calculable approximation to the Voigt function for most of the experimental needs.

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Corrections

Julio Puerta and Pablo Martin, "Three and four generalized Lorentzian approximations for the Voigt line shape: errata," Appl. Opt. 22, 19-19 (1983)
https://opg.optica.org/ao/abstract.cfm?uri=ao-22-1-19

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Equations (25)

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d	0.0	1.0	2.0	3.0	4.0	5.0	6.0	7.0	8.0	9.0	10.0
</hr>
p
0.0	0.0	+25.4	62.7	−11.0	−14.3	−8.3	−4.8	−2.8	−1.7	−1.1	−0.7
0.1	0.0	−13.4	51.4	−7.6	−13.0	−8.1	−4.7	−2.8	−1.8	−1.1	−0.8
0.5	0.7	2.8	25.2	−0.4	−8.5	−6.7	−4.3	−2.7	−1.8	−1.2	−0.8
1.0	2.7	6.0	12.7	2.3	−4.6	−4.9	−3.7	−2.6	−1.7	−1.2	−0.9
2.0	4.1	4.9	5.2	2.2	−1.0	−2.3	−2.4	−1.9	−1.5	−1.1	−0.8
3.0	3.4	3.5	3.0	1.6	−0.0	−0.1	−1.4	−1.3	−1.1	−0.9	−0.8
4.0	2.6	2.5	2.0	1.2	0.4	−0.3	−0.7	−0.9	−0.8	−0.7	−0.6
5.0	1.9	1.8	1.5	1.0	0.4	−0.0	−0.3	−0.5	−0.6	−0.6	−0.5
6.0	1.4	1.3	1.1	0.8	0.4	0.1	−0.1	−0.3	−0.4	−0.4	−0.4
7.0	1.0	1.0	0.9	0.6	0.4	0.2	−0.0	−0.1	−0.2	−0.3	−0.3
8.0	0.8	0.8	0.7	0.5	0.4	0.2	0.1	−0.1	−0.1	−0.2	−0.2
9.0	0.6	0.6	0.5	0.4	0.3	0.2	0.1	0.0	−0.1	−0.1	−0.1
10	0.5	0.5	0.4	0.4	0.3	0.2	0.1	0.0	−0.0	−0.1	−0.1
20	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.0	0.0	0.0
30	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
40	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
50	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
60	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
70	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
80	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
90	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
100	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0

d	0.0	1.0	2.0	3.0	4.0	5.0	6.0	7.0	8.0	9.0	10.0

p
0.0	0.00	3.9	14.9	−4.6	−2.9	−0.9	−0.2	−0.3	0.03	0.00	0.00
0.1	0.00	−2.8	11.1	−3.4	−2.7	−0.9	−0.2	−0.04	0.02	0.03	0.03
0.5	0.04	−0.3	4.4	−0.1	−1.9	−0.9	−0.3	−0.1	−0.02	0.01	0.02
1.0	0.20	0.5	1.8	−0.1	−1.1	−0.7	−0.3	−0.1	−0.05	−0.1	0.00
2.0	0.40	0.5	0.6	0.1	−0.4	−0.4	−0.3	−0.2	−0.1	−0.04	−0.02
3.0	0.32	0.3	0.3	0.02	−0.1	−0.2	−0.2	−0.1	−0.1	−0.05	−0.03
4.0	0.23	0.2	0.15	0.06	−0.04	−0.1	−0.1	−0.1	−0.1	−0.05	−0.03
5.0	0.15	0.1	0.1	0.05	−0.01	−0.05	−0.07	−0.07	−0.05	−0.04	−0.03
6.0	0.10	0.01	0.01	0.07	0.04	0.00	−0.02	−0.04	−0.04	−0.03	−0.03
7.0	0.07	0.07	0.05	0.03	0.00	−0.01	−0.02	−0.03	−0.03	−0.03	−0.02
8.0	0.05	0.05	0.04	0.03	0.00	−0.00	−0.01	−0.01	−0.02	−0.02	−0.02
9.0	0.04	0.03	0.03	0.02	0.00	0.00	−0.00	−0.01	−0.01	−0.01	−0.01
10	0.03	0.03	0.02	0.02	0.00	0.00	−0.00	−0.01	−0.01	−0.01	−0.01
20	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
30	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
40	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
50	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
60	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
70	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
80	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
90	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

d	0.0	1.0	2.0	3.0	4.0	5.0	6.0	7.0	8.0	9.0	10.0
</hr>
p
0.0	0.0	+25.4	62.7	−11.0	−14.3	−8.3	−4.8	−2.8	−1.7	−1.1	−0.7
0.1	0.0	−13.4	51.4	−7.6	−13.0	−8.1	−4.7	−2.8	−1.8	−1.1	−0.8
0.5	0.7	2.8	25.2	−0.4	−8.5	−6.7	−4.3	−2.7	−1.8	−1.2	−0.8
1.0	2.7	6.0	12.7	2.3	−4.6	−4.9	−3.7	−2.6	−1.7	−1.2	−0.9
2.0	4.1	4.9	5.2	2.2	−1.0	−2.3	−2.4	−1.9	−1.5	−1.1	−0.8
3.0	3.4	3.5	3.0	1.6	−0.0	−0.1	−1.4	−1.3	−1.1	−0.9	−0.8
4.0	2.6	2.5	2.0	1.2	0.4	−0.3	−0.7	−0.9	−0.8	−0.7	−0.6
5.0	1.9	1.8	1.5	1.0	0.4	−0.0	−0.3	−0.5	−0.6	−0.6	−0.5
6.0	1.4	1.3	1.1	0.8	0.4	0.1	−0.1	−0.3	−0.4	−0.4	−0.4
7.0	1.0	1.0	0.9	0.6	0.4	0.2	−0.0	−0.1	−0.2	−0.3	−0.3
8.0	0.8	0.8	0.7	0.5	0.4	0.2	0.1	−0.1	−0.1	−0.2	−0.2
9.0	0.6	0.6	0.5	0.4	0.3	0.2	0.1	0.0	−0.1	−0.1	−0.1
10	0.5	0.5	0.4	0.4	0.3	0.2	0.1	0.0	−0.0	−0.1	−0.1
20	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.0	0.0	0.0
30	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
40	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
50	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
60	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
70	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
80	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
90	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
100	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0

d	0.0	1.0	2.0	3.0	4.0	5.0	6.0	7.0	8.0	9.0	10.0

p
0.0	0.00	3.9	14.9	−4.6	−2.9	−0.9	−0.2	−0.3	0.03	0.00	0.00
0.1	0.00	−2.8	11.1	−3.4	−2.7	−0.9	−0.2	−0.04	0.02	0.03	0.03
0.5	0.04	−0.3	4.4	−0.1	−1.9	−0.9	−0.3	−0.1	−0.02	0.01	0.02
1.0	0.20	0.5	1.8	−0.1	−1.1	−0.7	−0.3	−0.1	−0.05	−0.1	0.00
2.0	0.40	0.5	0.6	0.1	−0.4	−0.4	−0.3	−0.2	−0.1	−0.04	−0.02
3.0	0.32	0.3	0.3	0.02	−0.1	−0.2	−0.2	−0.1	−0.1	−0.05	−0.03
4.0	0.23	0.2	0.15	0.06	−0.04	−0.1	−0.1	−0.1	−0.1	−0.05	−0.03
5.0	0.15	0.1	0.1	0.05	−0.01	−0.05	−0.07	−0.07	−0.05	−0.04	−0.03
6.0	0.10	0.01	0.01	0.07	0.04	0.00	−0.02	−0.04	−0.04	−0.03	−0.03
7.0	0.07	0.07	0.05	0.03	0.00	−0.01	−0.02	−0.03	−0.03	−0.03	−0.02
8.0	0.05	0.05	0.04	0.03	0.00	−0.00	−0.01	−0.01	−0.02	−0.02	−0.02
9.0	0.04	0.03	0.03	0.02	0.00	0.00	−0.00	−0.01	−0.01	−0.01	−0.01
10	0.03	0.03	0.02	0.02	0.00	0.00	−0.00	−0.01	−0.01	−0.01	−0.01
20	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
30	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
40	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
50	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
60	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
70	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
80	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
90	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

Abstract

Corrections

Cited By

Figures (3)

Tables (2)

Equations (25)

Applied Optics