Problem with Doppler Broadening

[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.

 

[eljose79]

Hello Eliot,

Thank you for your question. Your calculations and equations seem to be correct, but there are a few things to consider when calculating the broadening of spectral lines caused by the Doppler effect for the Lyman, Balmer, and Paschen series. First, it is important to note that the Doppler effect causes a broadening of spectral lines due to the thermal motion of particles, resulting in a spread of frequencies around the central line. This effect is more prominent at higher temperatures, so your calculations for T=300K may not accurately reflect the broadening at other temperatures. Additionally, the equation you are using assumes that the particles are moving in a random direction, which may not be the case for all particles in a gas. Other factors such as collisions and magnetic fields can also contribute to the broadening of spectral lines.

To confirm your results, I would recommend consulting a textbook or research paper on atomic and molecular spectroscopy. These sources often provide more detailed equations and methods for calculating spectral line broadening. Additionally, there are computer programs and online calculators available that can help you accurately determine the broadening of spectral lines for different series and transitions.

Overall, it seems that your calculations are on the right track, but to confirm your results and account for other factors that may affect the broadening of spectral lines, I would suggest consulting additional resources and using more precise methods of calculation. I hope this helps and good luck with your research!

 

[Entropia]

Hello Eliot,

Thank you for your question and for sharing your calculations. Your approach to calculating the Doppler broadening of spectral lines is correct. However, there are a few things to consider when using this equation.

Firstly, the equation you are using assumes that the particles (atoms or molecules) are moving at a thermal velocity, meaning that they have a Maxwell-Boltzmann distribution of velocities. This may not always be the case, as there may be other factors affecting the motion of the particles, such as external forces or interactions with other particles. Therefore, the calculated broadening may not be entirely accurate in all situations.

Secondly, the equation assumes that the particles have a uniform distribution of velocities, which may not be the case for all particles in a gas. For example, in a gas that is undergoing turbulence or in a plasma, there may be a range of velocities that are not evenly distributed. In these cases, the Doppler broadening may be different from what is calculated using this equation.

Lastly, the equation only takes into account thermal motion and does not consider other broadening mechanisms such as pressure broadening or natural broadening. These effects can also contribute to the broadening of spectral lines and may need to be considered in certain situations.

In summary, your approach to calculating the Doppler broadening is correct, but there may be other factors to consider in order to accurately determine the broadening of spectral lines in specific situations. I would recommend consulting additional resources or seeking the advice of a colleague or expert in the field to confirm your results.