The Quark Model — Question 8

Question 8

Resolving a proton requires a sufficiently large momentum transfer

Use the reduced spatial-resolution estimate $\ell=\hbar c/Q$ with $\hbar c=.1973$ GeV fm. An experiment reaches momentum-transfer energy Q=4.0 GeV. Take a characteristic proton radius .84 fm.

1. Calculate the resolution and ratio of proton radius to resolution.

2. Find Q needed to resolve .010 fm and distinguish Q from the incident beam energy.

Question 8 - Solution

Resolving a proton requires a sufficiently large momentum transfer

(a) Method and calculation

$$\boxed{\ell=.1973/4.0=.049325\ {\mathrm{fm}}}.$$$$\boxed{R/\ell=.84/.049325=17.0299}.$$

(b) Interpretation and further calculation

$$\boxed{Q_{\mathrm{needed}}=.1973/.010=19.73\ {\mathrm{GeV}}}.$$$$Q^2=-(p_{\mathrm{in}}-p_{\mathrm{out}})^2c^2,\quad Q\ne E_{\mathrm{beam}}\ \text{in general}.$$

Physical check

This is an order-of-scale resolving criterion using reduced wavelength. Increasing beam energy helps only if substantial momentum is transferred.