# Nuclear Physics - mean-square charge radius of a uniformly charged sphere

1. Feb 18, 2009

### wdednam

1. The problem statement, all variables and given/known data

Show that the mean-square charge radius of a uniformly charged sphere (with radius R) is
< r^2 > = 3*R^2 / 5

2. Relevant equations

< r^2 > = $$\int$$ $$\varphi$$f* r^2 $$\varphi$$i dV

$$\varphi$$f* = exp(-i q dot r)

$$\varphi$$i = exp(i q dot r)

where q = kf - ki
and pf = h_bar * kf
pi = h_bar * ki

r > R: V(r) = -Ze^2 / (4*pi*e0) * 1/r^2
r < R: V´(r) = -Ze^2 / (4*pi*e0*R) * (3/2 - 1/2 * (r/R)^2)

Ei = 1/2*m*vi

Er0 = J0^2/(2*m*r0^2) + V(r0) or V´(r0) according as r > R or r < R.

J0 = Ji = m*vi*b

where b is the impact parameter and is b = Ze^2/ (4*pi*e0*m*vi^2) * cot(A/2) for hyperbolic orbits, Ei > 0

3. The attempt at a solution

I basically want to know if my approach is correct. I think that I have to find < r0^2 > where r0 is the distance of closest approach to the centre of the charge distribution either inside it (where the electron feels V´(r)) or outside it (where the electron feels V(r)).

I've done some calculations using the radial energy equation, trying to solve for r0^2, but the calculations are a mess and I get a feeling I'm going about it all wrong. The problem is from Krane's Introductory Nuclear Physics, Chapter 3, problem 3.1.

2. Feb 19, 2009

### malawi_glenn

You are mixing concepts here, you have tried to evaluate the Form factor, F, with the potential r^2 ..

you want to evaluate the mean square radius, then use formula for distributions:
$$1 = A\int \rho (\vec{r})d\vec{r}$$

this is from mathematical statistics. Now use that the distribution of charge is constant up radius R, and for radius larger than R it is zero. Angular integration gives you 4pi, so it is easy to find the normalization constant A.

Then you recall from statistics that the average value of some quantity Q is
$$<Q> = A\int Q\rho (\vec{r})d\vec{r}$$

Now you have what you need.
What you did wrong was to mix this up with the Form factor F.

3. Feb 19, 2009

### wdednam

Hi Malawi Glenn,

Thank you very much, I used the equations you suggested and got the required result.

I'm concurrently registered for Stat Mech, QM, Nuclear Physics and Solid State Phys this year and realised when I attempted this problem that I should have completed Stat Mech and QM before I took Nuc Phys and SS Phys, but what's done is done and I'll just have to cope.

Thanks again.

4. Feb 19, 2009

### malawi_glenn

well yes, QM is pre requirement for both SS phys and Nuc phys. But the math behind is just probability, continuous and discrete.

5. Oct 9, 2011

### phys_student1

I know it is too late but I have a question.

How could we find the normalization constant?

We have: ρ=3/(4.pi.R^3).

I know that we should also have A=4.pi, but the integration:

1= A ∫ρ dr

does not give that result. Any ideas?

6. Oct 11, 2011

up...