Alpha Particle Spectroscopy - Is My Method Correct?

[Physy]

Homework Statement

Hey, first time poster here, I'm current writing up a lab report I just wanted to check my method is correct. I'm doing alpha particle spectroscopy using a cloud chamber setup. Most published reports seem to vary pressure (so the mass thickness varies). Our chamber let us vary pressure and distance, I decided to write my report explicitly around varying distance.

I've used the classical Beth-Bloch equation as the model. By plotting x vs alpha energy E I used function fitting to get a third order polynomial out, by taking the derivative of this I found dE/dx as a second order polynomial. Substituting the energy at each point I evaulated -dE/dx at each point and thus, -dX/dE. By plotting E vs -dX/dE I can integrate this to determine the range.

It gets complicated as there's a singularity at 160 KeV (From the log part of the BB equation). So far I overcame this by integrating to the limits of initial energy to the singularity point giving me an under estimate for the area (and hence range). I am not sure if this is the best approach, or how to quantify uncertainty. I used 2 integration methods, the trapezium rule and fitting the function as a high order polynomial and integrating that analytically, is that acceptable?

Homework Equations

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The Attempt at a Solution

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Thanks, sorry if this is unclear, feel free to point that out.

 

[mfb]

I don't understand what you did.

x is distance? How do you vary distance in a cloud chamber? Every track will have some distance, that is a measurement value.
How can you plot distance vs alpha energy? Do you have some emitters with known energy and plot the distribution of distances?

I've used the classical Beth-Bloch equation as the model.

That is a problematic model for typical energies of radioactive decays. You get $\beta \leq 0.05$, especially below 1 MeV the model won't work.

By plotting E vs -dX/dE I can integrate this to determine the range.

But that was a measurement before?

 

[Physy]

I don't understand what you did.

x is distance? How do you vary distance in a cloud chamber? Every track will have some distance, that is a measurement value.
How can you plot distance vs alpha energy? Do you have some emitters with known energy and plot the distribution of distances?

That is a problematic model for typical energies of radioactive decays. You get $\beta \leq 0.05$, especially below 1 MeV the model won't work.

But that was a measurement before?

Basically we used an MCA/MCB to measure the peak energy at various distances. http://www.cityu.edu.hk/ap/nru/pub_j96.pdf very similar to this report here.

By plotting the SRIM simulations and my model I saw that by approximating the equation as a third order polynomial doesn't work as below the low energy limit the BB model breaks down (the nuclear stopping power becomes the most prominent and overall stopping power increases) got some nice graphs that show this and outline why my method produces an underestimate.

A quick estimate for the range was taken by measuring the distance at which the peak energy is not distinguishable with background noise levels.

 

[mfb]

Ah, I see.

Why do you need Bethe-Bloch? As comparison, sure (without expecting a match for most of the energy range), but you have your own experimental values for dE/dx.

I used 2 integration methods, the trapezium rule and fitting the function as a high order polynomial and integrating that analytically, is that acceptable?

That depends on your data and the fit. Do you get a reasonable agreement between the methods, and with the measured range value?

 

[paranormal]

Your method seems sound and well thought out. It's great that you are using the classical Beth-Bloch equation as a model and that you have taken into account the singularity at 160 KeV. It's important to acknowledge and address any potential issues or limitations in your method.

Regarding your approach for handling the singularity, it may be helpful to consult with your instructor or a more experienced researcher to see if there are other techniques or methods that could be used to overcome this issue. It's also important to consider the potential impact of this singularity on your results and how it may affect the accuracy and precision of your measurements.

Using multiple integration methods is a good approach to validate your results and provide a better estimate of the range. However, it's important to quantify the uncertainty in your measurements and to discuss any potential sources of error in your experimental setup or data analysis.

Overall, your method seems appropriate and well-supported by theory. Just make sure to carefully document and explain your approach, any assumptions made, and any potential limitations or uncertainties in your results. Good luck with your lab report!