Hadron Cross Section Calculation

[benjammin]

Hi all,

I'm trying to understand the solution to a problem from a textbook. We're asked to find the proportionality constant for $$\frac{d\sigma_{\rm elastic}}{dq^2}\propto e^{bt},$$ where q is the momentum transfer in a scattering process, defined in elastic scattering as $$q=2p\sin(\theta/2).$$ So the solution says to make the following argument: $$\frac{d\sigma_{\rm elastic}}{dq^2}=\frac{d\sigma_{\rm elastic}}{d\Omega}\frac{d\Omega}{dq^2}=\lambda/2\frac{d\sigma_{\rm elastic}}{d\Omega}.$$ Which seems to imply that $$\frac{d\Omega}{dq^2}=\lambda/2.$$ I just don't understand that step. Does anyone have any ideas?

Thanks!

 

[AndyW]

Hi there,

I can understand your confusion with this step. The proportionality constant in this case, denoted by $\lambda$, is actually a combination of different factors that are involved in the scattering process. It includes the strength of the interaction between the particles, the target density, and the number of particles involved in the scattering. Therefore, it is not a constant value, but rather a variable that depends on the specific scattering process being studied.

In this case, the argument being made is that the proportionality constant, $\lambda$, is related to the differential cross section, $\frac{d\sigma_{\rm elastic}}{d\Omega}$, which is a measure of the probability of particles scattering into a specific solid angle. This makes sense, as the proportionality constant would affect the overall magnitude of the differential cross section.

To further understand this, we can look at the units of each term in the equation. The differential cross section has units of area, while the solid angle has units of steradians (a measure of solid angle). Therefore, the proportionality constant must have units of inverse solid angle in order for the equation to be dimensionally consistent. This further supports the idea that it is related to the differential cross section.

I hope this helps clarify the step in the solution. Let me know if you have any other questions. Keep up the good work in understanding the material!