The proton's charge density is roughly exponential,

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Nuclear Physics · Nuclear constituents, size and binding

The Nucleus Resolved, Ch How big a nucleus is, and what the mass surface knows

Problem

\rho_{\rm ch}(r) = \rho_{\rm ch}(0)\,e^{-r/b}

, with total charge

e

. ((a)) Do the angular part of

F(q^2) = \frac{1}{e}\int\rho_{\rm ch}(\vec r)\,e^{i\vec q\cdot\vec r}\,\mathrm{d}^3r

and show that

F(q^2) = 1/(1+q^2b^2)^2

. ((b)) Expand

F

to order

q^2

and read off

\langle r^2\rangle

. Check it by computing

\langle r^2\rangle

directly from the density. ((c)) The proton's rms charge radius is

0.841\,\mathrm{fm}

. Find

b

, and the momentum transfer at which

F

has fallen to one half.