Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Problem with Doppler Broadening

  1. Jun 6, 2009 #1
    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]


    [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:
    $Line$ & $\Delta \nu \ (GHz)$ \\ \hline\hline
    $\alpha \ LYMAN$ & $ 30.4685$ \\ \hline
    $\alpha \ BALMER$ & $ 5.64236$ \\ \hline
    $\alpha \ PASCHEN$ & $ 1.974826$ \\ \hline

    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?

  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted