The form factor of a radial gaussian charge distribution

[wdednam]

Homework Statement

Compute the form factor of rho(r) = rho0 * exp[-ln2*r^2/R^2]

where rho0 and R are constants.

Homework Equations

F(q) = 4pi/q $$\int$$sin(q*r)rho(r)rdr

where the limits of integration run from 0 to $$\infty$$

The Attempt at a Solution

This is probably more of a math problem, and I've tried everything mathematical I could think of to evaluate this integral:

Substitution, Integration by parts, taylor approximation to turn the exponential into a polynomial in r, writing sin(q*r) in exponentials... But all of this got me nowhere fast.

Then I tried looking up the integral in a table, but couldn't come across it in 30 pages worth of standard integrals. (Schaum series book)

As a last resort, I tried numerical integration in excel using set values for R and q, but of course the answer depends very much on the values of R and q, so this offered me no insight.

Thanks in advance for any help.

Wynand.

 

[malawi_glenn]

what if you instead of this sin-thing use the more general form of the Form factor, that it is the fourier-transform of the charge distribution?

I can have a closer look later, I know I have done this problem a couple of years ago

 

[wdednam]

I think that might just work, thank you.

Let me try and see where it gets me, and then I'll get back to you here to let you know how it went.

Thanks again!

Wynand.

 

[wdednam]

So, I could do the integral based on your suggestion. Here is the answer I got:

F(q) = (pi/ln2)^(3/2)*R^3*rho0*exp{-q^2*R^2/(4ln2)}

Thanks a lot again.

wdednam.

 

[malawi_glenn]

It looks ok to me :-)

Well done