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

[wdednam]

Homework Statement

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

Homework 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 = k_f - k_i
and p_f = h_bar * k_f
p_i = h_bar * k_i

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)

E_i = 1/2*m*v_i

E_r0 = J₀^2/(2*m*r₀^2) + V(r₀) or V´(r₀) according as r > R or r < R.

J₀ = J_i = m*v_i*b

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

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.

 

[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.

 

[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 realized 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.

 

[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.

 

[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?

 

[phys_student1]

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