Calculation of the Voigt Function in the Region of Very Small Values of the Parameter <i>a</i> Where the Calculation Is Notoriously Difficult

doi:10.4236/ajac.2013.412087

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American Journal of Analytical Chemistry, 2013, 4, 725-731

Published Online December 2013 (http://www.scirp.org/journal/ajac)

http://dx.doi.org/10.4236/ajac.2013.412087

Open Access AJAC

Calculation of the Voigt Function in the Region of Very

Small Values of the Parameter a Where the Calculation Is

Notoriously Difficult

Hssaïne Amamou1, Belkacem Ferhat2, André Bois1

1Laboratoire PROTEE-ISO, Université du Sud Toulon-Var, La Garde, France

2Laboratoire d’Electronique Quantique, Faculty of Physique, University of Sciences and Technology Houari Boumediene,

Alger, Algérie

Email: amam[email protected]

Received September 13, 2013; revised October 25, 2013; accepted November 15, 2013

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

The Voigt function is the convolution of a Lorentzian and a Guaussian density. The computation of these functions is re-

quired in several problems arising in a variety of physicochemical subjects; such as nuclear reactors, atmospheric transmit-

tance and spectroscopy. In this work we suggest using a new formula for the calculation of the Voigt function. Our for-

mula is a new integral representation for the Voigt function that gives the perfect results for the Voigt function calculation

and is easily calculable. We give also a comparison between our results of calculation of Voigt function for the very small

values of the parameter a, where the calculation is notoriously difficult, with those of the various algorithms of other authors.

Keywords: Convolution; Line Profile; Voigt Function; Lorentz Profile; Doppler Profile and Spectral Lines

1. Introduction

The shape of the spectral lines is a subject of great inter-

est in physics and chemistry. Indeed, several important

physical and chemical parameters are directly deducted

from the spectral lines whose shape is approximated by

the Gaussian profile or the Lorentzian profile. Therefore,

the parameters obtained do not correspond exactly to the

real physical conditions for which the spectral lines have

the shape of the distribution of Voigt. For this reason, the

study and the calculation of the Voigt function are very

interesting in many fields of physics and chemistry. In-

deed, for the signal emitted by a plasma, for example, the

phenomena that produce the enlargement of the spectral

lines are Doppler broadening caused by thermal agitation

of the particles, and the enlargement of pressure, which

is due to interactions between the transmitters and the

neighboring particles, the resulting profile of these phy-

sical phenomena is a Voigt profile.

The Voigt function results from the convolution prod-

uct between a Gaussian profile and a profile Lorentzian

and is expressed by the following formula:



exp

Vau x

aux













2ln2





 represents the relationship between

the distance from the center of the Lorentzian line and

the width of the Gaussian line.



ln 2





 determines the importance of Lor-

entzian in the profile, thus if this parameter tends to-

wards 0, the Lorentzian is negligible and if it tends to-

wards, the infinite the Lorentzian is dominant.



: Frequency.





: Lorentz half-width at half maximum in frequency.





: Doppler half-width at half maximum in frequency.



: Frequency in the center of the line.

This function has been studied recently by several

studies [1-8].

2. Calculation of the Voigt Function in the

Region near the u Axis

(1)

We give our new formula (That we have demonstrated in

the article of Amamou et al. [9] (Demonstration given in

Appendix 1 of this article)) of Voigt function in the fol-

lowing formula:

H. AMAMOU ET AL.

726



 



 



2π

,expcos2exp

sin 2expexpd

cos2cos2 expd

sin 2sin 2expd

Vauaau u

auux x

auuxx x























(2)

This analytical formula of the Voigt function gives a

solution to the mathematical problem which is due at the

infinite boundaries of the integral which defines the Voigt

function. This is a new integral representation for the

Voigt function that gives a perfect formula of Voigt func-

tion easily calculable and it’s different to the formula

given by Roston and Obaid [10] and gives a solution to

the problem of exponential growth described by Van

Synder [11].

This formula can be used for calculation of the spec-

tral lines whose profile is a convolution of a Lorentzian

profile and a Gaussian profile. This type of profile de-

scribes the actual physical conditions of several phys-

icochemical phenomena and its use is very interesting to

adjust the spectral lines by theoretical models.

In the Figures 1-3, the determination of the Lorentzian

profile and the Gaussian profile we have used the fol-

lowing parameters; 0

,, ,



 

. These figures shows

the three profiles; Voigt profile, Gaussian profile and

Lorentzian profile for different parameters a and u.

With:



: Wavelength.



: Lorentz half-width at half maximum in wave-

length.



: Doppler half-width at half maximum in wave-

length.



: Wavelength in the center of the line.

In the Figure 1 the Lorentzian profile, the Gaussian

profile and the Voigt profile are given for the following

parameters:

244:0.1:248 nm,246 nm,

0.4 nm,2.3 nm





 

thus the parameter .

4.79a

In the Figure 2 the Lorentzian profile, the Gaussian

profile and the Voigt profile are given for the following

parameters:

244:0.1:248 nm,246 nm,

3 nm,0.003 nm





 

thus the parameter .

0.00083255a

In the Figure 3 the Lorentzian profile, the Gaussian

profile and the Voigt profile are given for the following

parameters:

Figure 1. Voigt profile: “black” Voigt, “blue” Gauss and

“red” Lorentz.

Figure 2. Voigt profile: “black” Voigt, “blue” Gauss and

“red” Lorentz.

Figure 3. Voigt profile: “black” Voigt, “blue” Gauss and

“red” Lorentz.

Open Access AJAC

H. AMAMOU ET AL. 727

244:0.1:248 nm,246 nm,

0.3 nm,0.3 nm





 

thus the parameter .

0.8326a

Our formula is also a very interesting method for easy

calculation of the Voigt function. For the calculation of

the integrals of Equation (2) the trapezoidal rule method

and the adaptive Simpson’s method give very good re-

sults.

Table A1 (Appendix 2) gives the values of the Voigt

function calculated with the Formula (2) for the very

small values of the parameter a where the calculation is

notoriously difficult [12]. This table gives also the com-

putation time(s) for the values of each column of the

table. This calculation time depends obviously on the

performances of the computer. The computer that we

have used has a processor Intel pentium 2.3 GHz and a

memory (RAM) 4 GHz. This table gives the reference

values of the Voigt function calculated from Equation

(2).

Table 1 gives a comparison between our results of

calculation of Voigt function in the in the region of very

small values of the parameter a with those of the various

algorithms of other authors.

3. Conclusion

The new representation integral for Voigt function that

we have demonstrated and used to adjustment “fitting”

of lines spectral in a precedent article is used in this work

for calculation of Voigt function. Thus, this function is

easily calculable. We also made a comparison between

the results obtained by our formula and those obtained by

the various algorithms of other authors in the region of

Table 1. Comparison between our results and the results of

various algorithms of other authors (D is here for 10).

Calculation of Voigt in the region

of very small values of the parameter a

Author u = 5.4 a = 10−10 u = 5.5a = 10−14

Amstrong et al. [13] 12

2.260842D 14

7.307387Da



Humliek [14] 12

2.260842D 16

1.966215Da



Humliek [15] 12

2.260845D 14

7.307387Da



Hui [16] 8

2.667847

 9

9.238980Da





Lether and Wenston [17] 12

2.260845D 14

7.307386Da



Mclean et al. [18] 5

4.89872D

 5

4.24886Da





Poppe and Wiyers [19] 12

2.260850D 14

7.307805Da



Shippony and Read [1] 12

2.260845D 14

7.287724Da



Zaghloul [8] 12

2.260844D 7.287724Da



This Works 12

2.260844D 14

7.307387Da



very small values of the parameter a where the calcula-

tion is notoriously difficult.

REFERENCES

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Accurate Voigt Function Algorithm,” Journal of Quanti-

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[2] M. R. Zaghloul and A. N. Ali, “Algorithm 916: Comput-

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[3] J. He and Q. G. Zhang, “An Exact Calculation of the

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[5] F. Schreier, “Optimized Implementations of Rational

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http://dx.doi.org/10.1016/j.jqsrt.2010.12.010

[6] S. P. Limandri, R. D. Bonetto, H. O. Di Rocco and J. C.

Trincavelli, “Fast and Accurate Expression for the Voigt

Function. Application to the Determination of Uranium

M Linewidths,” Spectrochimica Acta Part B: Atomic

Spectroscopy, Vol. 63, No. 9, 2008, pp. 962-967.

http://dx.doi.org/10.1016/j.sab.2008.06.001

[7] S. M. Abrarov, B. M. Quine and R. K. Jagpal, “A Simple

Interpolating Algorithm for the Rapid and Accurate Cal-

culation of the Voigt Function,” Journal of Quantitative

Spectroscopy and Radiative Transfer, Vol. 110, No. 6-7,

2009, pp. 376-383.

http://dx.doi.org/10.1016/j.jqsrt.2009.01.003

[8] M. R. Zaghloul, “On the Calculation of the Voigt Line

Profile: A Single Proper Integral with a Damped Sine In-

tegrand,” Monthly Notices of the Royal Astronomical So-

ciety, Vol. 375, No. 3, 2007, pp. 1043-1048.

http://dx.doi.org/10.1111/j.1365-2966.2006.11377.x

[9] H. Amamou, A. Bois, M. Grimaldi and R. Redon, “Exact

Analytical Formula for Voigt Function which Results

from the Convolution of a Gaussian Profile and a Lor-

entzian Profile,” Physical Chemical News PCN, Vol. 43,

2008, pp. 1-6.

[10] G. D. Roston and F. S. Obaid, “Exact Analytical Formula

for Voigt Spectral Line Profile,” Journal of Quantitative

Spectroscopy & Radiative Transfer, Vol. 94, No. 2, 2005,

pp. 255-263. http://dx.doi.org/10.1016/j.jqsrt.2004.09.007

[11] S. Van, “Comment on ‘Exact Analytical Formula for

Voigt Spectral Line Profile’,” Journal of Quantitative

Spectroscopy & Radiative Transfer, Vol. 95, No. 4, 2005,

pp. 557-558. http://dx.doi.org/10.1016/j.jqsrt.2005.03.001

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H. AMAMOU ET AL.

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728

[12] R. J. Wells, “Rapid Approximation to the Voigt/Faddeeva

Function and Its Derivatives,” Journal of Quantitative

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1999, pp. 29-48.

http://dx.doi.org/10.1016/S0022-4073(97)00231-8

[13] B. H. Armstrong, “Spectrum Line Profiles: The Voigt

Unction,” Journal of Quantitative Spectroscopy and Ra-

diative Transfer, Vol. 7, No. 1, 1967, pp. 61-88.

http://dx.doi.org/10.1016/0022-4073(67)90057-X

[14] J. Humlicek, “Optimized Computation of the Voigt and

Complex Probability Functions,” Journal of Quantitative

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http://dx.doi.org/10.1016/0022-4073(82)90078-4

[15] J. Humlicek, “An Efficient Method for Evaluation of the

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Its Derivatives,” Journal of Quantitative Spectroscopy &

Radiative Transfer, Vol. 21, No. 4, 1978, pp. 309-313.

http://dx.doi.org/10.1016/0022-4073(79)90062-1

[16] A. K. Hui, B. H. Armstrong and A. A. Wray, “Rapid

Computation of the Voigt and Complex Error Functions,”

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[17] F. G. Lether and P. R. Wenston, “The Numerical Com-

putation of the Voigt Function by a Corrected Midpoint

Quadrature Rule for (−∞, ∞),” Journal of Computational

and Applied Mathematics, Vol. 34, No. 1, 1991, pp. 75-

92. http://dx.doi.org/10.1016/0377-0427(91)90149-E

[18] A. B. McLean, C. E. J. Mitchell and D. M. Swanston,

“Implementation of an Efficient Analytical Approxima-

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[19] G. P. M. Poppe and C. M. J. Wijers, “More Efficient

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No. 1, 1990, pp. 38-46.

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H. AMAMOU ET AL. 729

Appendix 1

The spectral radiant intensity of Voigt profile is given

by:

 

GLGtL tt

 





 

 (4)

With:

  

2ln2 4ln2

exp

πD

















(5)

is the Gaussian profile whose



 the Doppler half-

width at half maximum and



is the frequency.

And



2π

















(6)

is the Lorentzian profile whose



 the Lorentz half-

width at half maximum and 0



is the frequency in the

center of the line.

By an adequate change of variables, the convolution

Equation (4) can be put in the following form:

  

ln 22,



Vau



 (7)

With



exp

Vau x

aux









 (8)





,Vau is the Voigt function whose parameters are:

We can also put the expression (4) in the following

form:







 

exp2 πexp πd

4ln 2

it t



 





















t

(9)

The two integrals







,II



are given by the following relations:



exp2 πexp πd

4ln 2

exp2 πexp πd

4ln 2

it t

itt











































t





(10)

which can be also written in the following form:

 



 



 

ππ

expln2exp2 πexpln2d

2ln2

expln2exp2 πexpln 2d

2ln2

D D

























 

















 















 



 











 







t





(11)

By making the change of variable according to:



ln 2

2ln





















2

(12)

Thus, the preceding relation could be in the following form:



expexp2 πexp d

2ln2

expexp2 πexp d

2ln2

ait a

aita































































t







(13)

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H. AMAMOU ET AL.

730

By using a suitable change of variable, the expression (13) can be formulated as follows:

 



 



 



 



2ln2expexp2exp 2expd

aiauiux xx

ai auiuxxx





























(14)

Thereafter, we can write the two precedent integrals like:

 



 



 



 



200

2ln2expexp2exp 2expdexp 2expd

aiauiuxx xiuxx x



 













 









(15)

Thereafter:

 



 



 



4ln2expcos 2cos 2expdsin 2sin 2expd

Iaauuxx xauuxx xI



 





 (16)

with:

 



 



 



300

4ln2expcos2cos2expdsin 2sin 2expd

aauuxxxau uxx







x (17)

By a relatively simple mathematical analysis we can give the following solutions:





 



 

00 0

22 π

cos 2expd1expdexp

1.2 1.2.3.42

sin2expd2expdexpexpd

1.2.3

ux ux

uxxxxxu

uxx xuxx xux x

 



  







 







 



(18)

Thus, the Voigt profile can be written in the following form:

 



 



2222

4ln2 π

expcos2 expsin2 expexpd

π2

IaauuauuxxI













(19)

From this manner and according to the Equation (7) we express the Voigt function in the following way:



 



 



222

,expcos2 expsin2 expexpd

cos 2cos 2expdsin 2sin2expd

Vauaau uau uxx

auuxx xauuxx x

















(20)

Open Access AJAC

H. AMAMOU ET AL.

Open Access AJAC

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Appendix 2

Table A1. Calculation of Voigt function very small values of the parameter a. We give also the calculation times of each 12 values of each column (D is here for 10).