Assuming that V (x) ≈ (1 − µ) G₁(x) + µL₁(x) is a very good approximation of the Voigt function, in this work we analytically find µ from mathematical properties of V (x). G₁(x) and L₁(x) represent a Gaussian and a Lorentzian function, respectively, with the same height and HWHM as V (x), the Voigt function, x being the distance from the function center. In this paper we extend the analysis that we have done in a previous paper, where µ is only a function of a; a being the ratio of the Lorentz width to the Gaussian width. Using one of the differential equation that V (x) satisfies, in the present paper we obtain µ as a function, not only of a, but also of x. Kielkopf first proposed µ(a, x) based on numerical arguments. We find that the Voigt function calculated with the expression µ(a, x) we have obtained in this paper, deviates from the exact value less than µ(a) does, specially for high |x| values.Facultad de Ciencias Astronómicas y GeofísicasInstituto de Astrofísica de La Plat

Cruzado, Alicia

Di Rocco, Héctor Oscar

Acta Physica Polonica A

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Vol. 122 (2012) ACTA PHYSICA POLONICA A No. 4The Voigt Prole as a Sum of a Gaussian and a LorentzianFunctions, when the Weight Coecient Dependson the Widths Ratio and the Independent VariableH.O. Di Roccoa,b,∗ and A. Cruzadoc,daInstituto de Física Arroyo Seco (IFAS), Universidad Nacional del Centro de la Pcia. de Buenos AiresPinto 399, 7000 Tandil, Buenos Aires, ArgentinabConsejo Nacional de Investigaciones Cientícas y Técnicas (CONICET)Rivadavia 1917, C1033AAJ Buenos Aires, ArgentinacFacultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo del Bosque s/n1900 La Plata, Buenos Aires, ArgentinadInstituto de Astrofísica de La Plata (IALP-CONICET), Paseo del Bosque s/n1900 La Plata, Buenos Aires, Argentina(Received December 20, 2011; revised version March 6, 2012; in nal form May 24, 2012)Assuming that V (x) ≈ (1 − µ) G1(x) + µL1(x) is a very good approximation of the Voigt function, in thiswork we analytically nd µ from mathematical properties of V (x). G1(x) and L1(x) represent a Gaussian anda Lorentzian function, respectively, with the same height and HWHM as V (x), the Voigt function, x being thedistance from the function center. In this paper we extend the analysis that we have done in a previous paper,where µ is only a function of a; a being the ratio of the Lorentz width to the Gaussian width. Using one of thedierential equation that V (x) satises, in the present paper we obtain µ as a function, not only of a, but alsoof x. Kielkopf rst proposed µ(a, x) based on numerical arguments. We nd that the Voigt function calculatedwith the expression µ(a, x) we have obtained in this paper, deviates from the exact value less than µ(a) does,specially for high |x| values.PACS: 32.70.−n, 32.70.Jz1. IntroductionLet Va(x) be the Voigt prole obtained by convolutionof G(x) with L(x).Approximating the Voigt prole asVa(x) ≈ (1− µ)G1(x) + µL1(x), (1)in a previous paper [1] (PI hereafter), we analyticallyfound µ as a function only on a, using the property ofthe normalized area.In PI we did some general considerations showing that,using the expression (1) to represent the Voigt prole, wewould rigorously calculate µ for each x value, to obtainµ as a function, not only on a, but also on x, as follows:µ(a, x) =Va(x)−G1(x)L1(x)−G1(x). (2)Even though the deviations of Va(x) calculated withµ(a) from its exact values are smaller than 0.5% relativeto the peak value, as it was found in PI, it is useful, theo-retically and practically, to calculate µ(a, x) from Eq. (2).It is important to emphasize that, although Eq. (2)can be calculated for each x value for every prole Va(x),there is not a simple analytical expression for the right∗ corresponding author; e-mail: hdirocco@exa.unicen.edu.arhand side of this expression. Indeed, if we expand in se-ries, we see that the coecients of Va(x) depend compli-catedly on a, involving Kummer functions (or conuenthypergeometric functions), as was shown in [2].The dependence on x was already introduced byKielkopf [3] based on numerical arguments. In thepresent paper we nd an analytical expression for µ(a, x)from one of the dierential equation that V (x) satises,which allow a better t of Va(x) than µ(a) does, speciallyfor high |x| values.We emphasize that, beyond the usefulness, thissemitheoretical approach produces, in our opinion, a cer-tain aesthetic satisfaction.2. Analytical deduction of µ(a, x) using one ofthe dierential equations satised by Va(x)Va(x) satises several dierential equations, that canbe found in [4] and [5]. Of direct usefulness in our caseis the following:V ′′(x) +4xV ′(x)w2G+(4a2 + 2w2G+4x2w4G)V (x) =4aπw3G.(3)If µ = µ(a, x) is assumed in Eq. (1), Eq. (3) re-sults very complicated to our purposes, with terms like∂µ/∂x and ∂2µ/∂x2, and other such as G′1(x), G′′1(x),(670)The Voigt Prole as a Sum . . .Part II 671L′1(x) and L′′1(x). But, if the derivatives like ∂µ/∂x and∂2µ/∂x2 are negligible with respect to the derivativesG′1(x), G′′1(x), L′1(x) and L′′1(x), Eq. (3) becomes simpleenough to obtain a function µ(a, x) that permits a bet-ter t as compared with which has been obtained in PI,specially for high |x| values.Taking into account that the derivatives G′1(x) andG′′1(x) are proportional to G1(x) and that L′1(x) andL′′1(x) are proportional to L1(x), from Eq. (3) we ob-tain−2 ln 2(1− µ)G1(x)γ2V+4(ln 2)2(1− µ)x2G1(x)γ4V+8µx2L1(x)(x2 + γ2V)2 −2µL1(x)x2 + γ2V+4x[−2 ln 2(1− µ)xG1(x)w2Gγ2V− 2µxL1(x)w2G(x2 + γ2V)]+P (x)[(1− µ)G1(x) + µL1(x)] = Q (4)with P (x) = (4a2 + 2)/w2G + 4x2/w4G, Q = 4a/(πw3G)and µ = µ(a, x). Developing the former expression andcollecting terms, we obtainµ(a, x) =fG(x)G1(x) + fQ(x)QfL(x)L1(x) + fG(x)G1(x), (5)wherefQ(x) = γ8Vw2G + 2γ6Vw2Gx2 + γ4Vw2Gx4, (6)fG(x) = −8(ln 2)2γ2Vw2Gx4 − 4(ln 2)2γ4Vw2Gx2+2(ln 2)γ6Vw2G − 2P (x)γ6Vw2Gx2 − P (x)γ4Vw2Gx4− 4(ln 2)2w2Gx6 + 8(ln 2)γ6Vx2 + 16(ln 2)γ4Vx4+8(ln 2)γ2Vx6 − P (x)γ8Vw2G + 4(ln 2)γ4Vw2Gx2+2(ln 2)γ2Vw2Gx4 (7)andfL(x) = 6γ4Vw2Gx2 − 2γ6Vw2G + 2P (x)γ6Vw2Gx6+P (x)γ4Vw2Gx4 + P (x)γ8Vw2G − 8γ6Vx2 − 8γ4Vx4. (8)In spite of neglecting the derivatives ∂µ/∂x and∂2µ/∂x2, the µ(a, x) values we obtain from Eq. (5) de-part very little from those we obtain from Eq. (2).2.1. Limit casesIn order to achieve the nal analytic expression we aresearching, we need to know the behavior of µ(a, x) nearx = 0 and for x → ∞.Taking into account that µ(a, x) is an even functionof x, we analyse those limit cases in the following sections.2.1.1. µ(a, x) value at x = 0Evaluating fG(x), fL(x) and fQ(x) at x = 0, Eq. (5)can be written asµ(a, 0) = −2.258891354 + 6.517782707a2γ2Vw2G+3.258891354γ2Vw2G− 2.888117265× 4a× γ3Vπw3G. (9)Taking into account that γV/wG = b1/2(a), as we seein PI, we nally obtainµ(a, 0) = −2.258891354 + 6.517782707a2b21/2(a)+ 3.258891354b21/2(a)−2.888117265× 4a× b31/2(a)π. (10)Comparing µ(a, 0) given by the former expression withµ(a) obtained in PI and given byµ(a) =b1/2(a)ea2Φc(a)−√ln(2)b1/2(a)ea2Φc(a)(1−√π ln(2)) , (11)we conclude that µ(a, 0) < µ(a) for every a value.2.1.2. µ(a, x) value near x = 0It is known that a even function can be expanded as aneven power series about zero as f(x) ≃ A0 + A1x2 + . . .But, since µ(a, x) is expressed as a quotient (Eq. (5)), weexpand numerator and denominator to obtain µ(a, x) asa quotient of power series about zero.Then, at x ≃ 0 it is veriedµ(a, x → 0) =µapp(a, 0) +Ax2(+Bx4 + . . .)1 + Cx2(+Dx4 + . . .). (12)2.1.3. µ(a, x) value at x → ∞Taking into account that G1(x) tends to zero muchfaster than Q and L1(x), from Eq. (5) it is obtainedlimx→∞µ(a, x) ∼ limx→∞fQ(x)QfL(x)L1(x),wherelimx→∞fQ(x) ∼ γ4Vw2Gx4,limx→∞fL(x) ∼4γ4Vx6w2G,andlimx→∞L1(x) ∼b1/2(a) exp(a2)Φc(a)γV√πx2.It would be worthwhile to point out that, both,limx→∞(fQ(x)Q) and limx→∞(fL(x)L1(x)) are of fourthdegree on x.Taking into account that γV = wGb1/2(a), as we seein PI, we obtainµ(a,∞) = limx→∞µ(a, x) ∼ a√π[b1/2(a)]2exp(a2)Φc(a),(13)depending on a, as expected.Comparing the µ(a) values given by the last expres-sion with that obtained from Eq. (11), µ(a,∞) < µ(a) isveried.3. On the values of the tting formulaat x = 0 and x = ±γVSince G1(x), L1(x), and Va(x) take, all of them, bydenition, the same value at x = 0 and x = ±γV, it isclear that numerator and denominator in Eq. (2) are nullat those x values, making it impossible to evaluate µ.672 H.O. Di Rocco, A. CruzadoThe denominator of expression (5), fL(x)L1(x) +fG(x)G1(x), instead, is null at x ≈ ±γL; please note thesymbol ≈. For all those x values for which fL(x)L1(x)+fG(x)G1(x) is null, µ(a, x) must not be calculated fromEq. (5).All the x values for which µ(a, x) cannot be calculatedfrom Eq. (5), divided by a × γG/√ln 2, depend only ona = wL/wG and are given by the following universal algo-rithm: we call ρ(a) the function that results after ndingthe roots offL(x)L1(x) + fG(x)G1(xa× γG/√ln 2. (14)ρ(a) can be numerically tted, with a correlation coe-cient of R2 = 1, by the following universal function on a:ρ(a) =a0 + a2a2 + a4a4 + a6a61 + β2a2 + β4a4 + β6a6, (15)where a0 = 5.9966, a2 = 72.0531, a4 = −37.0930,a6 = 156.6913, β2 = 102.8163, β4 = −66.6862, andβ6 = 156.4950.Though a× γG/√ln 2 is exactly wL, it is clear that wLis not an independent parameter entering Eq. (14). wL =a× wG = a× γG/√ln 2 enter Eq. (14) only through a.Fig. 1. Roots of [fL(x)L1(x)+fG(x)G1(x)] divided bywL ≡ aγG/√ln 2. Points of this plot multiplied by wLshould not be used to calculate µ(a, x) from Eq. (5) (seetext).ρ(a) is represented in Fig. 1 for 0.25 ≤ a ≤ 4.5. Forlower or higher a values Va(x) depart very little from aGaussian and/or a Lorenztian prole, if the noise is ≥ 1%(easily reached in laboratory experiments).4. Indicators of the t qualityAs it was set in PI, we can test the quality of our tby normalizing the deviation of V (µa, x) from the exactvaluea) to the peak value, Va(0)∆1 =Va(x)− V (µ(a), x)Va(0)oror b) to the value at x, Va(x)∆2 =Va(x)− V (µ(a), x)Va(x).Criterion (b) is much more sensitive, since, both, thenumerator and the denominator of ∆2 are increasinglysmall numbers.We are going to see in Sect. 6 that the t obtainedusing V [µ(a, x), x] is better than the one obtained in PIusing V [µ(a), x], specially for high |x| values.5. The recipe to construct µ(a, x)In order to calculate µ(a, x), Eqs. (5)-(8) should beused, with γV obtained from the experimental data. Fora selected value of a, the needed wG value is determinedby considering the relation between γV and wG given byγV = wGb1/2(a), as it is explained in PI.The calculation must be done taking into account thealgorithm (14), that considers the problem of annulmentof both, numerator and denominator cited in Sect. 3.Then, in rst place, ρ(a) must be obtained from Eq. (15)for the selected value of a, in order to excluded the valuesx ≈ ±ρ(a) × a × γG/√ln 2 (≡ ±wL × ρ(a)) from thecalculations.Although at this point we have all that is necessary tocarry out the calculation of Va(x) ≈ (1−µ(a, x))G1(x)+µ(a, x)L1(x), it can be interesting to see the behavior ofµ(a, x), which will be presented in the following section.6. Results and discussionsIn Fig. 2 µ(a, x) as a function of a, for two dierent xvalues, is shown. Note the divergences for a ≈ x/wG.Fig. 2. µ(a, x) as a function on a for two dierent xvalues. µ(a, 1) and µ(a, 2) are displayed as functions ona for wG = 1. µ(a), as obtained in PI, is also shown forcomparison.In Fig. 3 µ(a, x) as a function of x is shown, for twodierent a values. In this case, divergences are observedfor x ≈ awG ≡ wL.All the curves in Fig. 2 and Fig. 3 have been obtainedby adopting wG = γG/√ln 2 = 1 in the calculations, buta similar behavior is observed for all wG values. In bothgures µ(a, x) is displayed including the divergences, inorder to see its real behaviour. In both gures µ(a), asobtained in PI, is also shown for comparison.The Voigt Prole as a Sum . . .Part II 673Fig. 3. µ(a, x) as a function on x for two dierent avalues. µ(1, x) and µ(2, x) are displayed as functions onx for wG = 1. µ(1) and µ(2), as obtained in PI, are alsoshown for comparison.In Fig. 1 all x values for which fL(x)L1(x) +fG(x)G1(x) is null, and, therefore, for which µ(a, x) can-not be calculated from Eq. (5), are represented (dividedby a× γG/√ln 2) for 0.25 ≤ a ≤ 4.5.In Figs. 4 and 5 we show ∆1 = [Va(x)− V (µ(a, x), x)]/Va(0) and ∆2 = [Va(x)−V (µ(a, x), x)]/Va(x) for µ as afunction only on a, as we have obtained in PI, and for µ asa function on a and x, as we have obtained in the presentpaper. It is clear from these gures that an improved tis obtained when µ as a function, not only on a, but alsoon x, is considered. A good t is also obtained for high|x| values, which is not obtained by us in PI, nor by otherauthors in [3] and Liu [6], when µ as a function only ona is considered.Fig. 4. ∆1 as a function on |x| using µ(a), as obtainedin PI, and µ(a, x), as obtained at the present paper. Thenumerical divergences can be observed.Fig. 5. As in Fig. 4, but for ∆2.7. ConclusionsAt the present paper we let µ to be a function, notonly on a, but also on x. Using one of the dierentialequations that V (x) satises, and neglecting the deriva-tives ∂µ/∂x and ∂2µ/∂x2 with respect to the derivativesG′1(x), G′′1(x), L′1(x), and L′′1(x), we have been able toobtain: (i) an analytic expression for µ(a, x), (ii) rela-tive ts, ∆1 and ∆2, better than those we have obtainedin PI, (iii) a universal table of values, so that, if we mul-tiply them by wL ≡ a×γG/√ln 2, we obtain the x valuesfor which µ(a, x) cannot be calculated with the expres-sion we have found.AcknowledgmentsWe wish to thank the Referees for their valuable sug-gestions that allowed us to signicantly improve ourarticle.References[1] H.O. Di Rocco, A. Cruzado, Acta Phys. Pol. A 122,666 (2012).[2] H.O. Di Rocco, D.I. Iriarte, J.A. Pomarico, Appl.Spectrosc. 55, 822 (2001).[3] J.L. Kielkopf, J. Opt. Soc. Am. 63, 987 (1973).[4] R. Ciurylo, J. Domyslawska, R.S. Trawinski, ActaPhys. Pol. A 83, 425 (1993).[5] T. Andersen, J. Quant. Spectrosc. Radiat. Transfer19, 169 (1978).[6] Y. Liu, J. Lin, G. Huang, Y. Guo, C. Duan, J. Opt.Soc. Am. B 18, 666 (2001).

The Voigt Profile as a Sum of a Gaussian and a Lorentzian Functions, when the Weight Coefficient Depends on the Widths Ratio and the Independent Variable

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The Voigt Profile as a Sum of a Gaussian and a Lorentzian Functions, when the Weight Coefficient Depends on the Widths Ratio and the Independent Variable

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