# Deep Inelastic Scattering, peak of cross section, width?

1. Feb 21, 2015

### binbagsss

I'm looking at deep inelastic scattering of a low-energy inelastic electron scattering from a stationary proton target . I am given $E$ and $\theta$ where $\theta$ is the scaterring angle.
$E=4.879 GeV , \theta=10^{0}$

I am given a figure of cross section $\frac{d^{2}\sigma}{d\Omega dE'}$

The question says the peak at $E'=4.14GeV$ is due to the production of the $\Delta^{+}$ resonance. Calculate the mass and width of the baryon?

Solution:

Mass: to get the mass we use the invariant mass of the hadronic system, $W$, where $W=W(E')$ has already been attained in another part of the question, so this is fine , I just plug in.

Width: I'm really stuck on this one, the textbook I'm using doesn't give any formula . It just says $W=1.18Gev$ therefore the width is $110MeV$

I have no idea what to do. All I can think of is the uncertainty principle, but I'm a bit confused using this in natural units. Everything is measured in energy , my textbook related $W$ to the lifetime , so surely $W$ can't be relateed to some other quantity too?

I know that $p=E'$, $p$ the momentum of the scattered electron, as the question says electron mass can be neglected.

I'm new to this topic and struggling , your assistance is greatly appreciated !

2. Feb 21, 2015

### fhc6791

Hmm. You could treat the problem classically. There is quite a bit of material out there on Rutherford scattering by charged particles, the Symon textbook comes to mind. The electron's perihelion distance (if you consider it as an orbit with the target as a foci) should be the baryon's radius, which you could double to find the width.

This is only a suggestion, there may be a more direct approach, but since you are given the scattering angle and energy, orbits were the first thing that came to mind when I read this.

Good luck!

3. Feb 27, 2015

### binbagsss

thanks for your reply. i'm pretty sure the method you refer to has not been covered in our course.
And whilst the textbook uses an uncertainty relation between W and the lifetime, it also has a table stating lifetime and width are related by the uncertainty principle.