# Correcting results to fit Rutherford's Scattering Formula

• Fabris
In summary, In order to recreate the Rutherford Scattering experiment, you need to correct for the cone-shaped scattering that happens in reality. You need to know the distance to the receiver, the number of atoms in the target, and the angle off-axis at which you measured the scattering. Additionally, you need to know how the particles scatter off of a hard spherical target.
Fabris
I'm a Physics undergraduate at University and in my labs module I had to recreate the Rutherford Scattering experiment. The basic setup was similar to this:

My problem is that this setup only records data in one plane and not 3D. In reality the particles are scattered in a sort of cone similar to this:

One of the tasks is to correct for this so that I can compare my results to Rutherford's scattering formula. I have no idea how to do it and after looking all over the internet all I could come up with talks about solid angle which I haven't even covered in University. I think I may be over-complicating this issue and I'm hoping someone can help me.

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Fabris

Thank you for the answer but my lab work is no where near as advanced as what you have written. The goal of the experiment was to collect data at varying angles and to validate Rutherford's formula:

All the constant are taken out and put into one (A) to give:

The collected data has to be plotted next onto this theoretical curve and then the values of B and A are to be determined graphically (trial and error for the best match). The main reason for this experiment is to become proficient at data analysis and to get used to writing scientific reports. Now my problem is I don't know how to correct for the fact that the data was collected in only one plane.

I didn't collect data at 0 degrees (going straight through) as the instructions specifically requested to omit this reading. Also I don't understand how that reading could be of any use when Rutherford's scattering formula has an asymptote at 0 degrees.

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It looks like all they are looking for from you is to try to curve fit ## N(\theta)= A/sin^4(\theta/2) ## for a range where you got reasonably good data. At ## \theta=0 ## this number goes up the y-axis, but you could try a graph of ## N(\theta) ## vs. ## \theta ## from ## \theta=180 ## (back-scattered) to some angle ## \theta= 45 ## degrees, etc. i.e. compare theoretical to experimental and choose ## A ## for the best fit to your experiment. ## \\ ## Editing... If you measured the angle in the laboratory, the ## B ## in your formula should be equal to zero. I'd enjoying seeing a few of your data points of ## N (\theta) ## vs. ## \theta ##. ## \\ ## Incidentally, it was with Rutherford's experiment that they came to conclude there were actually nuclei in atoms. Up until this time, they thought solids were pretty much uniformly dense, like jello, and they never expected that a piece of gold foil would be able to deflect one of these alpha particles appreciably.

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Fabris
It looks like all they are looking for from you is to try to curve fit ## N(\theta)= A/sin^4(\theta/2) ## for a range where you got reasonably good data. At ## \theta=0 ## this number goes up the y-axis, but you could try a graph of ## N(\theta) ## vs. ## \theta ## from ## \theta=180 ## (back-scattered) to some angle ## \theta= 45 ## degrees, etc. i.e. compare theoretical to experimental and choose ## A ## for the best fit to your experiment. ## \\ ## Editing... If you measured the angle in the laboratory, the ## B ## in your formula should be equal to zero. I'd enjoying seeing a few of your data points of ## N (\theta) ## vs. ## \theta ##.

Does that mean that I do not need to correct the data in any way? The lab script says "In order to compare your results with “Rutherford’s scattering formula”, you must first make a correction for the fact that we have only sampled the scattering in the horizontal plane, whereas the α-particles are actually scattered in a 3D cone." and then "Consider what factors make a correction to the angular scale necessary.". I am perfectly fine with all other aspects of the experiment and didn't even think I would need to change the data in any way, however, these two sentences threw me off-guard.

And yes I am aware that B should be 0, however, in the apparatus used the source and detector were not aligned perfectly and therefore the actual peak was around 2-3 degrees.

Edit: What you said about fitting my data onto the curve is exactly what they are asking.

Fabris said:
Does that mean that I do not need to correct the data in any way? The lab script says "In order to compare your results with “Rutherford’s scattering formula”, you must first make a correction for the fact that we have only sampled the scattering in the horizontal plane, whereas the α-particles are actually scattered in a 3D cone." and then "Consider what factors make a correction to the angular scale necessary.". I am perfectly fine with all other aspects of the experiment and didn't even think I would need to change the data in any way, however, these two sentences threw me off-guard.

And yes I am aware that B should be 0, however, in the apparatus used the source and detector were not aligned perfectly and therefore the actual peak was around 2-3 degrees.
The angle ## \theta ## is the polar angle. Had you collected data in 3-D, you would have found a complete ## \phi ## symmetry. That is why ## \phi ## does not appear in the Rutherford formula=it's independent of ## \phi ##. Your laboratory angle is precisely ## \theta ##, neglecting any misalignments, etc.

Fabris
The angle ## \theta ## is the polar angle. Had you collected data in 3-D, you would have found a complete ## \phi ## symmetry. That is why ## \phi ## does not appear in the Rutherford formula=it's independent of ## \phi ##. Your laboratory angle is precisely ## \theta ##, neglecting any misalignments, etc.

Ok, thanks a lot. :)

One additional comment: You may have taken data for angles ## \theta ## on both sides of the peak at ## \theta=0 ## degrees. In the polar coordinate system of interest, these are actually the same angle for ## \theta ## on both sides of the peak, but the azimuthal angle ## \phi ## differs by ## \pi ## radians or 180 degrees at these two locations.

## 1. Why is it important to correct results to fit Rutherford's Scattering Formula?

Correcting results to fit Rutherford's Scattering Formula allows for more accurate and precise measurements of the scattering of particles. This formula is based on the fundamental laws of physics and has been extensively tested and verified, making it a reliable tool for analyzing experimental data.

## 2. How is Rutherford's Scattering Formula used to correct results?

Rutherford's Scattering Formula is used to calculate the expected scattering pattern based on the properties of the particles and the experimental conditions. By comparing the calculated results to the actual measurements, any discrepancies can be identified and corrected for, leading to more accurate data analysis.

## 3. What are some common sources of errors in scattering experiments?

Some common sources of errors in scattering experiments include experimental equipment malfunctions, environmental factors such as temperature and pressure changes, and human error in data collection and analysis. These errors can lead to deviations from the expected results and may require correction using Rutherford's Scattering Formula.

## 4. Do all particles follow Rutherford's Scattering Formula?

No, Rutherford's Scattering Formula is specifically designed for scattering experiments involving small, charged particles such as protons or alpha particles. Other types of particles, such as electrons or photons, may follow different scattering patterns and would require different formulas for correction.

## 5. Can Rutherford's Scattering Formula be used for all types of scattering experiments?

No, Rutherford's Scattering Formula is only applicable to experiments involving small, charged particles scattering off of a target nucleus. Other types of scattering, such as elastic or inelastic scattering of larger particles, would require different formulas for correction.

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