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Simplified Theory of Line Broadening: Dependence of Spectral Line Parameters on Velocity and Temperature

Abstract

A simple version of the theory of line broadening is developed, which makes it possible to numerically and analytically calculate the parameters of collisional broadening, shifting, and narrowing of spectral lines based on the potentials of intermolecular interactions of the Lennard-Jones type. Namely, expressions are derived for the real and imaginary parts of the input and output frequencies of the collision integral and for the line width. The main simplifications consist in the model of nondegenerate states and one perturbing level. The eikonal approximation made it possible to express the constants under study in terms of the scattering S-matrices based on more general expressions in terms of the scattering amplitudes. The dependences of the parameters on the velocity of active molecules and gas temperature are determined.

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Translated by O. Ponomareva

APPENDIX

APPENDIX

Below, the subscripts qe (quasi-elastic) and qi (quasi-inelastic) at frequencies refer to the cases rie ⪡ 1 and rei ⪡ 1, respectively; Φ(x; y; z) is the degenerate hypergeometric function [83]. The resulting formulas for frequencies are:

$$\begin{gathered} \operatorname{Re} {{\nu }_{{qe}}} = \frac{{\Gamma \left( {\frac{{n - 2}}{{n - 1}}} \right)}}{{{{4}^{{\frac{1}{{n{\kern 1pt} - {\kern 1pt} 1}}}}}\Gamma \left( {\frac{1}{2} + \frac{1}{{n - 1}}} \right)\Gamma \left( {\frac{{n + 1}}{2}} \right)}} \\ \times \,\,{{\nu }_{0}}\left[ {2\left( {{{r}_{e}}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}}} + 1} \right)\Gamma \left( {2 - \frac{1}{{n - 1}}} \right)} \right. \\ \times \,\,\Gamma \left( {\frac{{n + 1}}{2}} \right)\Phi \left( {\frac{1}{{n - 1}} - \frac{1}{2};\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right) \\ + \,\,{{2}^{{\frac{5}{2}{\kern 1pt} - {\kern 1pt} \frac{n}{2}}}}{{c}_{n}}{{r}_{{ie}}}\left( {{{r}_{i}}{{r}_{e}}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}{\kern 1pt} - {\kern 1pt} 1}} + 1} \right)\Gamma \left( {\frac{{n + 7}}{4} - \frac{1}{{n - 1}}} \right) \\ \left. { \times \,\,\Phi \left( {\frac{1}{{n - 1}} - \frac{{n + 1}}{4};\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right)} \right], \\ \end{gathered} $$
(A1)
$$\begin{gathered} \operatorname{Re}{{\nu }_{qi}}=\frac{c_{n}^{\frac{3{\kern 1pt} -{\kern 1pt} n}{n{\kern 1pt} -{\kern 1pt} 1}}\Gamma \left( \frac{n-2}{n-1} \right)}{\sqrt{2}(n-1)\Gamma {{\left( \frac{n-1}{2} \right)}^{\frac{2}{n{\kern 1pt} -{\kern 1pt} 1}}}\Gamma \left( \frac{1}{2}+\frac{1}{n-1} \right)} \\ \times \,\,{{\nu }_{0}}\left[ \sqrt{2}(n-1){{c}_{n}}\left( r_{i}^{\frac{2}{n{\kern 1pt} -{\kern 1pt} 1}}+1 \right)\Gamma \left( \frac{5}{2}-\frac{1}{n-1} \right) \right. \\ \times \,\,\Phi \left( \frac{1}{n-1}-1;\frac{3}{2};-\beta \frac{{{\mathsf{v}}^{2}}}{{{{\mathsf{\bar{v}}}}^{2}}} \right)+{{2}^{{n}/{2}\;}}{{r}_{ei}}\left( {{r}_{e}}r_{i}^{\frac{2}{n{\kern 1pt} -{\kern 1pt} 1}{\kern 1pt} -{\kern 1pt} 1}+1 \right) \\ \times \,\,\Gamma \left( \frac{11-n}{4}-\frac{1}{n-1} \right)\Gamma \left( \frac{n-1}{2} \right) \\ \left. \times \,\,\Phi \left( \frac{1}{n-1}+\frac{n-5}{4};\frac{3}{2};-\beta \frac{{{\mathsf{v}}^{2}}}{{{{\mathsf{\bar{v}}}}^{2}}} \right) \right], \\ \end{gathered}$$
(A2)
$$\begin{gathered} \operatorname{Im} {{\nu }_{{qe}}} = - \frac{{{{2}^{{2 - \frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}}}}\Gamma \left( {\frac{1}{2} - \frac{1}{{n - 1}}} \right)}}{{(n - 1)\Gamma \left( {\frac{1}{{n - 1}}} \right)\Gamma \left( {\frac{{n - 1}}{2}} \right)}} \\ \times \,\,{{\nu }_{0}}\left[ {\left( {{{r}_{e}}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}}} - 1} \right)\Gamma \left( {2 - \frac{1}{{n - 1}}} \right)\Gamma \left( {\frac{{n + 1}}{2}} \right)} \right. \\ \times \,\,\Phi \left( {\frac{1}{{n - 1}} - \frac{1}{2};\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right) + {{2}^{{\frac{{3{\kern 1pt} - {\kern 1pt} n}}{2}}}}{{c}_{n}}{{r}_{{ie}}}\left( {{{r}_{i}}{{r}_{e}}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}{\kern 1pt} - {\kern 1pt} 1}} - 1} \right) \\ \left. { \times \,\,\Gamma \left( {\frac{{n + 7}}{4} - \frac{1}{{n - 1}}} \right)\Phi \left( {\frac{1}{{n - 1}} - \frac{{n + 1}}{4};\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right)} \right], \\ n > 3, \\ \end{gathered} $$
(A3)
$$\begin{gathered} \operatorname{Im} {{\nu }_{{qe}}}({v} = 0) = - \frac{{{{\nu }_{0}}}}{{2\sqrt \pi }}\left[ {1.5635({{r}_{e}} - 1 + 2{{r}_{e}}\log {{r}_{e}})} \right. \\ \left. { - \,\,1.4632{{c}_{3}}{{r}_{{ie}}}({{r}_{i}} - 1 + 2{{r}_{i}}\log {{r}_{e}})} \right],\,\,\,\,n = 3, \\ \end{gathered} $$
(A4)
$$\begin{gathered} \operatorname{Im} {{\nu }_{{qi}}} = - \frac{{c_{n}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}{\kern 1pt} - {\kern 1pt} 1}}\Gamma \left( {\frac{1}{2} - \frac{1}{{n - 1}}} \right)}}{{\Gamma {{{\left( {\frac{{n - 1}}{2}} \right)}}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}}}}\Gamma \left( {\frac{1}{{n - 1}}} \right)}} \\ \times \,\,{{\nu }_{0}}\left[ {{{c}_{n}}\left( {r_{i}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}}} - 1} \right)\Gamma \left( {\frac{5}{2} - \frac{1}{{n - 1}}} \right)} \right. \\ \times \,\,\Phi \left( {\frac{1}{{n - 1}} - 1;\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right) + \frac{{{{2}^{{\frac{{n{\kern 1pt} - {\kern 1pt} 1}}{2}}}}}}{{n - 1}}{{r}_{{ei}}}\left( {{{r}_{e}}r_{i}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}{\kern 1pt} - {\kern 1pt} 1}} - 1} \right) \\ \times \,\,\Gamma \left( {\frac{{11 - n}}{4} - \frac{1}{{n - 1}}} \right)\Gamma \left( {\frac{{n - 1}}{2}} \right) \\ \left. { \times \,\,\Phi \left( {\frac{1}{{n - 1}} + \frac{{n - 5}}{4};\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right)} \right],\,\,\,\,n > 3, \\ \end{gathered} $$
(A5)
$$\begin{gathered} \operatorname{Im} {{\nu }_{{qi}}}({v} = 0) \\ = {{\nu }_{0}}\left\{ {\frac{2}{{\sqrt \pi }}{{c}_{3}}\left[ {(\log {{c}_{3}} - 0.4228)({{r}_{i}} - 1) + {{r}_{i}}\log {{r}_{i}}} \right]} \right. \\ \left. {^{{^{{^{{^{{}}}}}}}} + \,\,{{r}_{{ei}}}\left[ {(\log {{c}_{3}} + 0.5772)({{r}_{e}} - 1) + {{r}_{e}}\log {{r}_{i}}} \right]} \right\},\,\,\,\,n = 3, \\ \end{gathered} $$
(A6)
$$\begin{gathered} \operatorname{Re} {{\gamma }_{{qe}}} = \frac{{{{2}^{{\frac{{n{\kern 1pt} - {\kern 1pt} 3}}{{n{\kern 1pt} - {\kern 1pt} 1}}}}}\Gamma \left( {\frac{{n - 2}}{{n - 1}}} \right){{{({{r}_{e}} - 1)}}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}}}}}}{{\Gamma \left( {\frac{1}{2} + \frac{1}{{n - 1}}} \right)\Gamma \left( {\frac{{n + 1}}{2}} \right)}} \\ \times \,\,{{\nu }_{0}}\left[ {2\Gamma \left( {2 - \frac{1}{{n - 1}}} \right)\Gamma \left( {\frac{{n + 1}}{2}} \right)} \right. \\ \times \,\,\Phi \left( {\frac{1}{{n - 1}} - \frac{1}{2};\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right) \\ + \,\,{{2}^{{\frac{{3{\kern 1pt} - {\kern 1pt} n}}{2}}}}{{c}_{n}}{{r}_{{ie}}}\frac{{{{r}_{i}} - 1}}{{{{r}_{e}} - 1}}\Gamma \left( {\frac{{n + 7}}{4} - \frac{1}{{n - 1}}} \right) \\ \left. { \times \,\,\Phi \left( {\frac{1}{{n - 1}} - \frac{{n + 1}}{4};\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right)} \right], \\ \end{gathered} $$
(A7)
$$\begin{gathered} \operatorname{Re} {{\gamma }_{{qi}}} = \frac{{c_{n}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}}}{{{({{r}_{i}} - 1)}}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}}}}\Gamma \left( {\frac{{n - 2}}{{n - 1}}} \right)}}{{\Gamma {{{\left( {\frac{{n - 1}}{2}} \right)}}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}}}}\Gamma \left( {\frac{1}{2} + \frac{1}{{n - 1}}} \right)}} \\ \times \,\,{{\nu }_{0}}\left[ {\Gamma \left( {\frac{5}{2} - \frac{1}{{n - 1}}} \right)\Phi \left( {\frac{1}{{n - 1}} - 1;\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right)} \right. \\ + \,\,\frac{{{{2}^{{\frac{{n{\kern 1pt} - {\kern 1pt} 1}}{2}}}}}}{{{{c}_{n}}(n - 1)}}{{r}_{{ei}}}\frac{{{{r}_{e}} - 1}}{{{{r}_{i}} - 1}}\Gamma \left( {\frac{{11 - n}}{4} - \frac{1}{{n - 1}}} \right) \\ \left. { \times \,\,\Gamma \left( {\frac{{n - 1}}{2}} \right)\Phi \left( {\frac{1}{{n - 1}} + \frac{{n - 5}}{4};\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right)} \right], \\ \end{gathered} $$
(A8)
$$\begin{gathered} \operatorname{Im}{{\gamma }_{qe}}=-\frac{{{2}^{1{\kern 1pt} -{\kern 1pt} \frac{2}{n{\kern 1pt} -{\kern 1pt} 1}}}\Gamma \left( \frac{1}{2}-\frac{1}{n-1} \right){{({{r}_{e}}-1)}^{\frac{2}{n{\kern 1pt} -{\kern 1pt} 1}}}}{\Gamma \left( \frac{1}{n-1} \right)\Gamma \left( \frac{n+1}{2} \right)} \\ \times \,\,{{\nu }_{0}}\left[ \Gamma \left( 2-\frac{1}{n-1} \right)\Gamma \left( \frac{n+1}{2} \right)\Phi \left( \frac{1}{n-1}-\frac{1}{2};\frac{3}{2};-\beta \frac{{{\mathsf{v}}^{2}}}{{{{\mathsf{\bar{v}}}}^{2}}} \right) \right. \\ +\,\,{{2}^{\frac{3{\kern 1pt} -{\kern 1pt} n}{2}}}{{c}_{n}}{{r}_{ie}}\frac{{{r}_{i}}-1}{{{r}_{e}}-1}\Gamma \left( \frac{n+7}{4}-\frac{4}{n-1} \right) \\ \left. \times \,\,\Phi \left( \frac{1}{n-1}-\frac{n+1}{4};\frac{3}{2};-\beta \frac{{{\mathsf{v}}^{2}}}{{{{\mathsf{\bar{v}}}}^{2}}} \right) \right],\,\,\,\,n>3, \\ \end{gathered}$$
(A9)
$$\begin{gathered} \operatorname{Im} {{\gamma }_{{qe}}}({v} = 0) = {{\nu }_{0}}\left\{ {({{r}_{e}} - 1){{{\left[ {\log ({{r}_{e}} - 1) - 0.4410} \right]}}^{{^{{^{{^{{}}}}}}}}}} \right. \\ \left. { + \,\,\frac{{{{c}_{3}}}}{{\sqrt \pi }}{{r}_{{ie}}}({{r}_{i}} - 1)\left[ {2\log ({{r}_{e}} - 1) + 0.7316} \right]} \right\},\,\,\,\,n = 3, \\ \end{gathered} $$
(A10)
$$\begin{gathered} \operatorname{Im} {{\gamma }_{{qi}}} = - \frac{{c_{n}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}}}{{{({{r}_{i}} - 1)}}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}}}}\Gamma \left( {\frac{1}{2} - \frac{1}{{n - 1}}} \right)}}{{\Gamma {{{\left( {\frac{{n - 1}}{2}} \right)}}^{{\frac{2}{{n{\kern 1pt} - {\kern 1pt} 1}}}}}\Gamma \left( {\frac{1}{{n - 1}}} \right)}} \\ \times \,\,{{\nu }_{0}}\left[ {\Gamma \left( {\frac{5}{2} - \frac{1}{{n - 1}}} \right)\Phi \left( {\frac{1}{{n - 1}} - 1;\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right)} \right. \\ + \,\,\frac{{{{2}^{{\frac{{n{\kern 1pt} - {\kern 1pt} 1}}{2}}}}}}{{{{c}_{n}}(n - 1)}}{{r}_{{ei}}}\frac{{{{r}_{e}} - 1}}{{{{r}_{i}} - 1}}\Gamma \left( {\frac{{11 - n}}{4} - \frac{1}{{n - 1}}} \right) \\ \left. { \times \,\,\Gamma \left( {\frac{{n - 1}}{2}} \right)\Phi \left( {\frac{1}{{n - 1}} + \frac{{n - 5}}{4};\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right)} \right],\,\,\,\,n > 3, \\ \end{gathered} $$
(A11)
$$\begin{gathered} \operatorname{Im} {{\gamma }_{{qi}}}({v} = 0) \\ = \frac{2}{{\sqrt \pi }}{{\nu }_{0}}\left\{ {{{c}_{3}}({{r}_{i}} - 1)\left[ {\log {{c}_{3}} + \log ({{r}_{i}} - 1) - 0.4228} \right]} \right. \\ \left. { + \,\,\frac{{\sqrt \pi }}{2}{{r}_{{ei}}}({{r}_{e}} - 1)\left[ {\log {{c}_{3}} + \log ({{r}_{i}} - 1) + 0.5772} \right]} \right\}, \\ n = 3. \\ \end{gathered} $$
(A12)

Averaging the above frequencies over the velocities is equivalent to replacing the functions \(\Phi \left( {\frac{1}{{n - 1}} + f;\frac{3}{2}; - \beta \frac{{{{{v}}^{2}}}}{{{{{{\bar {v}}}}^{2}}}}} \right),\) \(f = - \frac{1}{2},\) \(\frac{{n - 5}}{4},\) −1, \( - \frac{{n + 1}}{4}\) with the factors \({{(\beta + 1)}^{{ - \frac{1}{{n{\kern 1pt} - {\kern 1pt} 1}}{\kern 1pt} - {\kern 1pt} f}}}.\) In the particular case of n = 3, formulas for Im ν and γ are somewhat cumbersome and they are presented here only for zero velocity.

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Kochanov, V.P. Simplified Theory of Line Broadening: Dependence of Spectral Line Parameters on Velocity and Temperature. Atmos Ocean Opt 34, 528–541 (2021). https://doi.org/10.1134/S1024856021060166

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Keywords:

  • line broadening constants
  • interaction potential
  • scattering amplitudes