Computing a differential from plotted data

[alex3]

I have a differential that depends only on $\cos{\theta}$

$$\frac{\operatorname{d}\sigma}{\operatorname{d} \Omega} = f(\cos{\theta})$$

I am numerically solving this differential equation for $\sigma$, which physically is a cross section, for $0 \leq \theta \leq \pi$. The differential contains a parameter $s$. I am solving the differential for a given value of $s$, then incrementing this parameter and solving the differential again.

I am solving the differential over a range of $s$ (3 to 200), in increments of 0.1 ($s$ is, for the curious, my particle accelerator energy in GeV).

My problem is that I'm asked to now plot the transverse momentum

$$p_{T} = \lvert p_{f} \rvert \sin{\theta}$$

I've been told that I am effectively trying to plot

$$\frac{\operatorname{d} \sigma}{\operatorname{d} p_{T}}$$

and that I'll have to do some sort of histogram/binning to make my plot.

I'm not sure what I should binning; the differentials, the final integrated cross section?

So, in essence, my questions is how can I plot $\frac{\operatorname{d} \sigma}{\operatorname{d} p_{T}}$ given that I have numerical data for $\frac{\operatorname{d}\sigma}{\operatorname{d} \Omega}$ (and $\sigma$)?

 

[diazona]

Normally posts which don't use the homework template tend to be at low priority, so keep that in mind for the future... but anyway: $\mathrm{d}\sigma/\mathrm{d}p_T$ can be physically interpreted as the "amount" of cross section which falls into a certain (small) range of $p_T$, divided by the size of that range. When you are plotting this differential cross section, it makes sense to plot it versus $p_T$.

 

[alex3]

Normally posts which don't use the homework template tend to be at low priority, so keep that in mind for the future

Apologies, I had trouble fitting the question into the format. I shall try harder next time :)

$\mathrm{d}\sigma/\mathrm{d}p_T$ can be physically interpreted as the "amount" of cross section which falls into a certain (small) range of $p_T$, divided by the size of that range. When you are plotting this differential cross section, it makes sense to plot it versus $p_T$.

Do you mean I should plot $\operatorname{d}\sigma/\operatorname{d} p_T$ against $p_T$? I know I need to do that, my problem is trying to find the differential values. i.e. How could I figure out "the 'amount' of cross section which falls into a certain (small) range of $p_T$"?

Would I calculate a small range in $p_T$ and then see what values of $\sigma$ (the integrated cross section) have a value in this range?

 

[diazona]

Do you mean I should plot $\operatorname{d}\sigma/\operatorname{d} p_T$ against $p_T$? I know I need to do that, my problem is trying to find the differential values. i.e. How could I figure out "the 'amount' of cross section which falls into a certain (small) range of $p_T$"?

Would I calculate a small range in $p_T$ and then see what values of $\sigma$ (the integrated cross section) have a value in this range?

Yeah, that sounds about right. The details vary somewhat depending on exactly what kind of data you have, so it's kind of hard for me to be specific without actually seeing your numbers (or inventing an example); unfortunately I don't have time for that right now but if you will still be working on this in a couple days, perhaps I can get back to it).

 

[paranormal]

I understand your confusion and can provide some guidance on how to approach this problem. First, let's clarify the definitions of the variables involved. The differential \frac{\operatorname{d}\sigma}{\operatorname{d} \Omega} represents the rate of change of the cross section \sigma with respect to the solid angle \Omega. This means that for a given value of s, you can calculate the cross section for different values of \theta using the equation \sigma = \int \frac{\operatorname{d}\sigma}{\operatorname{d} \Omega} \operatorname{d} \Omega.

Now, you are interested in plotting the transverse momentum p_{T} = \lvert p_{f} \rvert \sin{\theta}, which is a function of both \theta and the final momentum p_{f}. To do this, you will need to use the chain rule to calculate \frac{\operatorname{d} \sigma}{\operatorname{d} p_{T}}. This involves taking the derivative of \frac{\operatorname{d}\sigma}{\operatorname{d} \Omega} with respect to p_{T}, which can be done numerically using your existing data for \frac{\operatorname{d}\sigma}{\operatorname{d} \Omega}.

Next, you will need to bin your data in order to plot \frac{\operatorname{d} \sigma}{\operatorname{d} p_{T}}. This means dividing your data into intervals or "bins" and calculating the average value of \frac{\operatorname{d}\sigma}{\operatorname{d} \Omega} within each bin. This will give you a set of data points that you can then plot on a graph.

In terms of choosing the binning, it will depend on your specific data and the range of values for p_{T}. You may need to experiment with different bin sizes to find the best representation of your data. In general, it is recommended to have at least 10-20 bins to accurately capture the shape of the data.

In summary, to plot \frac{\operatorname{d} \sigma}{\operatorname{d} p_{T}}, you will need to use the chain rule to calculate the derivative of \frac{\operatorname{d}\sigma}{\operatorname{d} \Omega} with respect to