Introductory Particle Physics - Form factor, charge distribution?

[jeebs]

Here is the problem. I've been messing around with it for a while but I'm not sure if what I'm trying to do is the right way to approach this.

The form factor F(q) = \int\rho(\vec{r})e^{i\vec{q}.\vec{r}/\hbar}d^{3}\vec{r} is the 3D Fourier Transform of the normalised charge distribution \rho(\vec{r}).

For a simplified model of a proton's charge distribution, \rho(r)\propto (e^{-r/R})/r.

R can be considered as some characteristic "size" of the proton, setting the rate at which the charge dies away, but does not constitute a hard edge to the proton.

i) Find the constant of proportionality required to normalise \rho correctly.
ii) something else that presumably needs the answer to i) first.

I am new to all this particle physics business, so I am in unfamiliar territory and I'm not sure how to approach this question. I've so far just aimlessly waded into this and ended up with a couple of sides of mindless mathematical messing around. This could be a simple question or a complicated one for all I know, so I thought i'd post it here before I bother my busy lecturer...

Anyway, my closest attempted solution:

-what I thought was that I should assume the proton has its highest charge density at its centre and it gradually fades away, uniformly in all directions.

-I'm also thinking that r must be the distance from the centre of the proton, so that as r tends to infinity, the charge density \rho(r) approaches zero.

-I'm trying to find some constant of proportionality here, let's call it A, so that \rho(r) = A(e^{-r/R})/r.


-I'm thinking that if I do \int \rho(r) dV = 1 then I can solve for A, but this is as far as I have got, I'm struggling with how to take this integral any further.

Am I on the right lines, has anyone got any suggestions that would make my life easier?

thanks.

 

[gabbagabbahey]

Anyway, my closest attempted solution:

-what I thought was that I should assume the proton has its highest charge density at its centre and it gradually fades away, uniformly in all directions.

-I'm also thinking that r must be the distance from the centre of the proton, so that as r tends to infinity, the charge density \rho(r) approaches zero.

-I'm trying to find some constant of proportionality here, let's call it A, so that \rho(r) = A(e^{-r/R})/r.


-I'm thinking that if I do \int \rho(r) dV = 1 then I can solve for A, but this is as far as I have got, I'm struggling with how to take this integral any further.

Am I on the right lines, has anyone got any suggestions that would make my life easier?

thanks.

Looks fine so far, now just compute the volume integral using spherical coordinates, centered on the proton's center (r=0)...you should know how to do that.