$$\begin{aligned} g (\omega ^\prime _0, \omega ) = \frac{\varDelta \omega _0}{2 \pi }\, \frac{1}{(\omega ^\prime _0 - \omega )^2 + \varDelta \omega _0^2/4}. \end{aligned}$$

(14.21)

where

\(\varDelta \omega _0\) is the halfwidth of a transition. The probability of a transition in the frequency interval

\(\omega ^\prime _0, \omega ^\prime _0+\text {d}\omega ^\prime _0\) is equal to

$$\begin{aligned} P(\omega ^\prime _0){\text {d}}{\omega ^{\prime }_0}=\frac{2\sqrt{\text {In}}2}{\sqrt{\pi \varDelta \omega _c}} \text {exp}(-\frac{\text {In}2{(\omega ^{\prime }_0-\omega _0)}^2}{\varDelta \omega _\text {c}^2/4})\text {d}\omega ^{\prime }_0. \end{aligned}$$

(14.22)

where

\(\varDelta \omega _c\) is the halfwidth and

\(\omega _0\) the center frequency of the Gaussian profile. We obtain the spectral profile of a line by averaging,

$$\begin{aligned} S(\omega )=\int ^\infty _{0}{g}(\omega ^{\prime }_0,\omega )P({\omega ^{\prime }_0})\text {d}{\omega ^{\prime }_0}. \end{aligned}$$

(14.23)

It follows that

$$\begin{aligned} S(\omega )=\frac{\varDelta \omega _0}{\pi ^{3/2}\varDelta \omega _c}\int ^\infty _{0} \frac{1}{(\omega ^\prime _0 - \omega _0)^2 + \varDelta \omega _0^2/4} \text {exp}(-\frac{\text {In}2{(\omega ^{\prime }_0-\omega _0)}^2}{\varDelta \omega _\text {c}^2/4})\text {d}\omega ^{\prime }_0. \end{aligned}$$

(14.24)

The equation has to be solved numerically.

- (a)
Show that the limits of the Voigt profile are the Lorentzian or the Gaussian profile, depending on the ratio of the two halfwidth \(\varDelta \omega _0\) and \(\varDelta \omega _c\).

- (b)
Show that (14.24) is consistent with ( 7.52).