Closest approach with impact parameter

[ognik]

Homework Statement

A proton of mass m, charge +e and (asymptotic) momentum mv is incident on a Nucleus of charge +Ze at an impact parameter b. Consider only coulomb repulsion and classical mechanics, what is distance of closest approach - d?

Homework Equations

Haven't encountered impact parameter before so please check the following:
I found $b = \frac{kq_1q_2}{mv^2} \sqrt{ \frac{1+Cos \theta}{1-Cos \theta}}$
I gather if b = 0, the proton would be aimed straight at the center of the nucleus, let's say that approach is a vector $\vec{r_0}$
Then b is the distance between $\vec{r_0}$ and the actual path $\vec{r_i}$, parallel to $\vec{r_0}$.
And after closest approach, the proton moves off in direction $\vec{r_f}$, at an angle $\theta$ to $\vec{r_i}$?

Having said all that, I am unsure of how to apply b...

The Attempt at a Solution

I couldn't see using conservation of energy would help, the proton doesn't stop...
Conservation of momentum looks promising, especially as we don't have the initial distance the proton was when we measured v. Aligning the x-axis with $\vec{r_i}$ we have $\vec{p_i} = \hat{x}.mv_i$ and $\vec{p_f} = \hat{x}.mv_f Sin \theta + \hat{y}.mv_f Cos \theta$

I'm stuck here, can't see how to involve b - and anyway b is not d, d will be greater than b because the proton starts diverging from $\vec{r_i}$ at some distance along $\vec{r_i}$ before the nucleus - where the coulomb repulsion starts affecting the protons path.

I did wonder if I could make an approximation as to the points where it starts (and symmetrically stops) diverging, I could then approximate d by taking a straight line between those 2 points and minimizing the distance between that line and the nucleus?

 

[BvU]

No expert (any more), but: in the center of mass all initial kinetic energies are converted to coulomb energy at the distance of closest approach. Can you do something with that ? (I lookedhttp://ion-tracks.de/iontracktechnology/2/1/2.html)

[edit] are you allowed some comfortable approximations (non-relativistic, no target nucleus recoil), or do you think you need to plow through the general case -- which may include Z = 1, pp scattering ?

 

[ognik]

I am sure those approximations are OK. I also think I am looking for a simplistic solution - this is a question that appears in a 'maths for physicists' course, the main section is vectors, the sub-sections I am doing is dot and cross product. As I wrote that, I thought about angular momentum (in this case L = p X r) because it is amenable to the cross-product ...would the initial r be b? And then use conservation of L to find the value at r = d? I'm not sure how to approach that at the moment, how would I minimise r for the angular momentum at r = d?

 

[ognik]

Hi - was hoping someone could add to this, I can't think of anything new above what I have already thought of (above) - so some tips or hints would be good please

 

[phyzguy]

Try using conservation of angular momentum. What is the angular momentum when the proton is a large distance from the nucleus with an impact parameter b? What is it when it is at closest approach?