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Problem with Doppler Broadening

  1. Jun 6, 2009 #1
    Hello,
    I would like to know how to calculate the broadening of the spectral lines caused by the Doppler effect for the Lyman, Balmer and Paschen series. To be more concrete, I would like to know the broadening of the alpha transitions.
    The equations I use are the following but i don't know if I am doing something wrong.

    [tex]\Delta \nu &=&2\frac{\nu _{o}}{c}\sqrt{\frac{2KT}{m}\ln \left( 2\right) }[/tex]

    I calculate [tex]\nu _{o}[/tex] Doing the following:

    [tex]E_{n}-E_{n^{\prime }} &=&\left[ \frac{1}{\left( n^{\prime }\right) ^{2}}-
    \frac{1}{\left( n\right) ^{2}}\right] \frac{Z^{2}e^{4}\mu }{2\left( 4\pi
    \varepsilon _{o}\right) ^{2}\hbar ^{2}}=-h\nu _{o}[/tex]

    [tex]\nu _{o} &=&-\frac{Z^{2}e^{4}\mu }{4\pi \left( 4\pi \varepsilon _{o}\right)
    ^{2}\hbar ^{3}}\left[ \frac{1}{\left( n^{\prime }\right) ^{2}}-\frac{1}{
    \left( n\right) ^{2}}\right][/tex]

    For T=300K we have:

    [tex]\nu _{o} &\approx &-\frac{e^{4}m_{e}}{4\pi \left( 4\pi \varepsilon
    _{o}\right) ^{2}\hbar ^{3}}\left[ \frac{1}{\left( n^{\prime }\right) ^{2}}-
    \frac{1}{\left( n\right) ^{2}}\right] \approx -3.288953357\cdot 10^{15}\left[
    \frac{1}{\left( n^{\prime }\right) ^{2}}-\frac{1}{\left( n\right) ^{2}}
    \right] \ \ Hz[/tex]

    SO:

    [tex]\Delta \nu &=&2\frac{\nu _{o}}{c}\sqrt{\frac{2KT}{m}\ln \left( 2\right) }
    \approx -0.000040625\cdot 10^{15}\left[ \frac{1}{\left( n^{\prime }\right)
    ^{2}}-\frac{1}{\left( n\right) ^{2}}\right] \ \ Hz[/tex]

    Finally, the following numbers are obtained:
    [tex]\begin{tabular}{|l|l|}
    \hline
    $Line$ & $\Delta \nu \ (GHz)$ \\ \hline\hline
    $\alpha \ LYMAN$ & $ 30.4685$ \\ \hline
    $\alpha \ BALMER$ & $ 5.64236$ \\ \hline
    $\alpha \ PASCHEN$ & $ 1.974826$ \\ \hline
    \end{tabular}[/tex]

    Unfortunatelly, I can't find any book to confirm this results, that is why I am posting this.
    What do you say? Am I doing anything wrong?

    Eliot.
     
  2. jcsd
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