1. Jun 6, 2009

### EliotHijano

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.

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

I calculate $$\nu _{o}$$ Doing the following:

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

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

For T=300K we have:

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

SO:

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

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

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.