Question

9. An the scattering of particles of energy E 2k/2m by a nucleus, an experimenter finds a differential cross section dơ d(0.86 2.55 cos 2.77 cos0) What partial waves are contributing to the scattering, and what are their phase shifts at the given energy? (a) (b) What is the total cross section?

Answer 1

Solution:

Given that

Given that (a) The scattering Amplitude is given as: f(theta) = 1/k Sum_i = 0^infinity (2l + 1) e^i delta l pl (cos theta) sin delta l___(i) Now By differential cross section d sigma/d Ohm = vert l (theta) vert^2 rightwardsdoublearrow 1/k^2 (0.86 + 2.55 cos theta + 2.77 cos^2 theta)___(ii) Here cos^2 theta term can be achieve by considering l = 0 and l = 1 it can be proved also as we can see f(theta) = 1/K Sum_l = 0, 1 (2l + 1) e^i delta l pl(cos theta) sin delta l rightwardsdoublearrow 1/K [e^i delta_0 p_0 (cos theta) sin delta_0 + 3e^i delta_1 p_1 (cos theta) sin delta_1] \r\n vert f(theta) vert^2 = 1/K^2 [e^-i delta_0 p_0 (cos theta) sin delta_0 + 3e^-i delta_1 p_1 (cos theta) sin delta_1] [e^i delta_0 p_0 (cos theta) sin delta_0 + 3e^-i delta_1 p_1 (cos theta) sin delta_1] rightwardsdoublearrow 1/K^2 [(p_0 (cos theta))^2 sin^2 delta_0 + 9(p_1 (cos theta))^2 sin^2 delta_1^2 + p_1 (cos theta) p_0 (cos theta) sin delta_0 sin delta_1 6 cos (delta_0 - delta_1)]