Rutherford Scattering in Landau book

[roeb]

I'm reading Landau and Lifgarbagez's Mechanics book and am having a hard time proving the following:

On page 53, they present theta_0 = arccos( ... ). As described on page 48 eqn 18.2 the integral should produce this theta_0. However, I am not quite sure what r_min is? On page 48, they say 'It should be recalled that r_min is a zero of the radicand. I *think* that means rmin is a turning point, yes?

On page 36, they integrate eqn 18.2 (I assume with the same limits: rmin to inf), however notice there is an extra term. (Also they integrated it for U(r) = - alpha/r, but that is just a sign change).

Furthermore, eqn 15.14 on page 38 (evaluated for a repulsive potential U = +a/r) shows:
p/r = -1 + e cos(phi). However, if I am not mistaken, for Rutherford's problem, cos(phi) = -1/e, where does the p/r term go?

Here's a summary of what I am asking if it didn't make sense:
1) I am trying to find phi_0 for a repulsive potential U(r) = a/r
2) Using Kepler's problem with the signs changed, phi_0 on page 53 and phi on the top of page 36 don't seem to match (different limits of integration?) I did of course convert to E = 1/2 m vinf^2 and M = mv_inf p, but that still doesn't appear to yield the same result.

 

[Bob S]

(I don't have a copy of L & L) The variable r min is usually used to connotate the distance of closest approach of the alpha particle in Rutherford scattering. The scattering, being elastic, has a well defined differential cross ection in a central Coulomb field (point source). If r-min is too small, the alpha particle hits the nucleus, and the differential cross section is modified.

 

[astrokreng]

Hello,

Thank you for bringing this issue to my attention. I understand your confusion and I will try to clarify the concepts for you.

Firstly, let's start with the concept of r_min. Yes, you are correct that r_min is a turning point. It is the minimum distance between the particle and the scattering center in the Rutherford scattering problem. In other words, it is the point where the particle reaches its closest approach to the scattering center before being deflected.

Moving on to the equations, the extra term that you noticed in equation 18.2 on page 36 is due to the use of a different potential function, U(r) = -alpha/r, which changes the integral slightly. This is why the result may seem different from the one on page 53, where U(r) = a/r.

As for the issue with the p/r term in equation 15.14, you are correct that it should be there for the Rutherford scattering problem. This term represents the angular momentum of the particle and its inclusion is necessary for the correct solution. It may have been omitted in the book for simplicity or to focus on other aspects of the problem.

Lastly, to find phi_0 for a repulsive potential U(r) = a/r, you can follow the same approach as in the book, but with the appropriate potential function. Keep in mind that the limits of integration may change accordingly.

I hope this helps clarify your doubts. If you have any further questions, please let me know.