Thompson - Computing for Scientists and Engineers (523188), страница 74
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nThe Voigt function is often also called the Voigt profile. Note that H (a,v ) is aneven function of v, since it is the convolution of two functions that are even functions of their arguments. This result is useful in numerical work, both in programming and in debugging code.It is interesting and relevant that the results about Gaussians together convolutingto Gaussians and Lorentzians with Lorentzians producing Lorentzians implies thatVoigt functions have the same property, so that these convolutions all stay in thesame families.Exercise 10.33(a) Prove that the convolution of two Voigt functions is also a Voigt function.In order to do this use the results from Exercise 10.26 that the convolution oftwo Gaussians produces a Gaussian and from Exercise 10.30 that two Lorentzians when convolved produce a Lorentzian.
In addition, you need the resultsthat the process of convolution is associative and commutative.(b) Demonstrate this analytical result, at least approximately, by using Program 10.1, Convolute Arrays, to convolute two Voigt-profile arrays that arethe outputs from Program 10.2, Voigt Profile. Compare the convolution408FOURIER INTEGRAL TRANSFORMSthat results with that obtained by this program, using as input an appropriateGaussian of total width given by (10.49) and a Lorentzian of total width from(10.56), in terms of the widths input to form the two Voigt profiles.(c) Generalize the result in (a) to prove that convolution of any number of Voigtprofiles produces a Voigt profile, similarly to (and dependent on) the results derived in Exercises 10.26 and 10.30 for Gaussians and Lorentzians.
nThese results have a very practical consequence, as follows. Suppose that there areseveral sources of Gaussian-shape line broadening, and possibly several contributions to Lorentzian-type line widths, all modifying the same spectral line, with otherconditions being constant.
The resulting spectral profile will be a Voigt profile thatcan readily be obtained by first combining all the Gaussian widths quadratically by(10.51) and all the Lorentzian widths additively by the generalization of (10.55),then computing a single Voigt profile.Computationally, the result in Exercise 10.33 is a convenience, but scientificallyit is a hindrance because it prevents one from recognizing the individual line-broadening contributions, and thereby having some possibility of determining their physical origin. Sources of such line broadening in stellar environments and models forestimating the broadening are discussed in Chapter 9 of the book by Mihalas.The integral expression for the Voigt function H (a,v ) in (10.65) generally cannot be simplified. For any given a and v it may be estimated by numerical integration, which is one option considered in Project 10 below.
A very common situationis that the resonance, which has FWHM is very much broadened by convolutionwith the Gaussian, that is, one has a << 1. It is then practical to expand H (a,v ) ina Taylor series in powers of a by(10.66)If this series converges rapidly. then it is a practical series, especially since the auxiliary functions Hn (v) may be computed independently of the choice of a.Exercise 10.34(a) In the integrand of the Voigt function (10.65) make a Maclaurin expansionof the exponential, then interchange integration and summation in order to showthat(10.67)(b) Differentiate this expression twice with respect to variable v, which may betaken under the integral sign because the only dependence of the integral on v isin the cosine function. Identify the result as proportional to the function with then value two larger.
Thus derive the recurrence relation10.3CONVOLUTIONS AND FOURIER TRANSFORMS409(10.68)which enables all the Hn (v) to be calculated by recurrence from H0(v ) andH 1 (v). nOur computation is now reduced to finding appropriate formulas for the two lowestH functions, then being able to differentiate these with respect to v.The function H0 (v) may be derived either by some laborious mathematics or bynoting that it is proportional to the value of the convolution when a = 0, which isjust the Gaussian function with altered normalization. Thus we can show that(10.69)Exercise 10.35For those who are skeptical of the above reasoning, derive H0(v) from the definition (10.64).
To do this, you will need to change variables to a complex variable z, then to use the normalization integral for the Gaussian function. Aftersome tedious and error-prone steps you should obtain (10.69). nNow that we have H0(v), it is straightforward to generate by differentiation thesuccessive H n (v) with n even. For example, the second derivative of H0(v)leads immediately to(10.70)Exercise 10.36Carry out the indicated differentiation and normalization by using (10.69) in(10.68) with n = 0 in order to derive (10.70). nFIGURE 10.10 The Hn functions in (10.66) for n = 0.1.2.3.
and the Dawson integral F in(10.72), each as a function of v.410FOURIER INTEGRAL TRANSFORMSFor usual values of a in the expansion (10.66), Hn (v) with n > 3 is not required, but you can produce H4, etc., if you are willing to keep on differentiating. InFigure 10.10 you can see the exponentially damped behavior of H0 and H2.The values of Hn(v) for n odd require more mathematical analysis, and eventually more computation, than for n even. Although one may calculate H1 directlyfrom the definition (10.67), some manipulation of the integral puts it into a form thatis more suitable for differentiating to produce the Hn(v) for larger odd-n values.As usual, you can exercise your analysis skills by completing the intermediate steps.Exercise 10.37(a) Use integration by parts in (10.67) for n = 1, with cos (vx) being the quantity that is differentiated. Transform variables to z = x2 on the part that is integrated. Thus show that(10.71)where F (v) is called Dawson's integral, defined by(10.72)(b) From the definition, show that Dawson’s integral is an odd function of v,and therefore that H1 (v) is an even function of v, as required.(c) By expressing the sine function in (10.72) in terms of the complex exponential of vx by Euler’s theorem, then introducing the variable u = -ix/2 - v andmanipulating the limits of integration before using the normalization integral forthe Gaussian, show that Dawson’s integral can be written as(10.73)This form is more suitable than (10.72) for analysis and numerical work.
nIt appears that we are sinking deeper and deeper into the quicksands of mathematical complexity; fear not, we will drag ourselves out by our bootstraps. We nowhave H1 in terms of a reasonable-looking integral, (10.73). For H3 we will need derivatives of F (v), which are straightforward to obtain from this second form ofDawson’s integral.Exercise 10.38(a) Differentiate F (v) given by (10.73) with respect to v, noting that the derivative of an integral with respect to its upper limit is just the integrand evaluated atthis limit.
in order to show that(10.74)10.4COMPUTING AND APPLYING THE VOIGT PROFILE411(b) By reusing this result, show that the second derivative of Dawson’s integralis given by(10.75)Note that the first derivative is an even function of v and that the second derivative is odd in v, as required from the evenness of F under reflection of v. nThis completes our analysis of Dawson’s integral.
In Project 10 we address thepractical concerns of computing this function, making use of the results that we havejust derived.We now return to the job of obtaining formulas for the Hn. The only one thatwe will require but do not yet have is H3. This can be obtained from H1 by usingthe derivative relation (10.68).Exercise 10.39Differentiate H1 from (10.71) twice with respect to v, substitute the derivativesof F according to (10.74) and (10.75), then use the recurrence relation (10.68)in order to show that(10.76)which is again explicitly even in v.
nThe functions H1 and H3, given by (10.71) and (10.76), are shown in Figure 10.10. They slowly damp out as v increases because they are basically components of a Gaussian. The computation of F (v) that is needed for these functions isdescribed in Section 10.4.We have finally produced an expression for the Voigt profile through terms inthe ratio of Lorentzian to Gaussian widths, a, up to cubic. By continuing to higherderivatives the function H (a,v) may be expressed in terms of exponential functionsand Dawson’s integral function F (v), (10.73).10.4 PROJECT 10: COMPUTING AND APPLYINGTHE VOIGT PROFILEIn this section we develop the numerics of computing the Voigt profile, starting fromthe analysis in the preceding subsection.
Then we apply this profile to relate temperatures to the widths of atomic emission lines from stellar sources. This project alsoillustrates how to make the step-by-step development and programming of algorithms for numerical computation that results in effective and tested programs that canbe adapted to various uses in science and engineering.412FOURIER INTEGRAL TRANSFORMSThe numerics of Dawson’s integralDawson’s integral, the function F (v) defined as (10.72) or (10.73), arose as an intermediate step in our calculation of the functions Hn (v) for the series expansion inpowers of a for the Voigt profile, formula (10.66). Although we display F (v) inFigure 10.10, we have not yet shown how to approximate it numerically.Series expansion is the most obvious way to express F (v), since inspection ofFigure 10.10 or of tables (such as Abramowitz and Stegun, Table 7.5) shows thatit is a smooth function of small range that varies approximately proportional to vwhen v is small.
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