1. Theory of Scattering Lecture 3

2. Free Particle: Energy, In Cartesian and spherical Coordinates. Wave function: (plane waves in Cartesian system) (spherical waves in spherical system)

3. Plane waves can be expressed in terms of spherical wave states When wave vector k is along z-axis and m = 0,

4. Partial Wave Analysis: Consider that the incident plane wave is travelling along z-axis, we may express it as ----------(1) We can write the above plane wave as superposition of angular momentum Eigen states each with definite angular momentum number ℓ, so we write ---------(2) So we expressed in above Eq. incident plane wave as sum of partial waves which will be distorted by scattering potential. Rayleigh’s Formula

5. We know write the solution to the Schrodinger Eq -----------(3) As we know the solution of above Eq. can be written using separable variable technique. So we write ----------(4) Foe central potential the system is symmetrical about z-axis and hence rotational invariant and the scattered wave will not depend upon azimuthal angle φ and thus m = 0. Also -----------(5)

6. The radial wave function in Eq. (4) obeys following Eq. --------(6) Also as we have discussed the total wave function after Scattering is written by superposition of incident and Scattered wave i.e. we write -----(7) Using (2) in (7), we get -----------(8)

7. Note that in Eq. (8) we have no dependence on φ. Note that Eq. (4) and (8) are two ways of writing the total scattered wave. In scattering experiments the distance of detector from the scattering center is large as compared to size of target. We now utilize this fact to find the scattering amplitude and differential cross-section.

8. For large value of r, the Bessel function can be approximated as, ---------(9) Using (9) in (8), we get the asymptotic form

9. --------------(10)

10. For large value of r, the radial wave equation (6) can be written as (Radiation Zone) -----------(11) The general solution of above equation consist of linear combination of spherical Bessel and Neumann functions ------------(12)

11. Bessel’s function of first kind Bessel’s function of 2nd kind Known as Neumann function

12. In asymptotic limit the Neumann function is written as -----(13) Using Eq. (9) and (13) in Eq. (12), we get -----(14) If V(r) = 0 for all r, solution It means at r = 0, R kl should vanish. Now Neumann functions have property that they diverge when argument is zero i.e. at r = 0 they will diverge. Hence in this case we cannot consider contribution of Neumann function to solution.

13. We introduce the phase shift to achieve the regular solution near the origin by rewriting where

14. Asymptotic form of radial function is written as -----(15) Note that when in Eq. (15) the R kl reduces to Bessel function which is finite at r = 0. The becomes zero in absence of scattering potential. It is called phase shift.

16. The phase shift measure the degree to which is different from. Remember that was the Radial wave function in absence of scattering. We now use Eq (15) in Eq. (4) and write the asymtotic form Of solution as, -------------(17) The wave function defined in above Eq. represent the distorted plane wave because it contain the phase shift which comes into picture because of scattering.

17. Using Equation ----------(17) We write Eq. (16) as -------(18) Above Eq. Gives us the asymptotic form of Eq. (4). Also we have obtained earlier the Eq. (10) as asymptotic form of Eq. (7), which is -------(19)

18. Comparing coefficients of Eq. (18) and (19), we get Or ----------(20) Using Eq. (20) in Eq. (18), we get -------------(21)

19. Comparing coefficients of in Eq. (21) with Eq. (10), we get, ---------(22) Using above Eq. becomes -------------(23) is known as partial wave amplitude.

20. Using (23), the Differential cross-section is written as ---------(24) Total cross-section is written as ------------(25) Using We get ----------(26)

21. gives us the partial cross-section corresponding to scattering of particles in various angular momentum states. For the case of scattering between particles which are in S-state, ℓ = 0 (low energy scattering). The scattering amplitude using (23) can be written as, -----------(27) Differential and total cross-section, using (27) are -----------(28)

22. For forward scattering Forward scattering amplitude is written as, -----(29) Using Eq. (26) and (29), we get -------(30) Above Eq. is known as optical theorem.

23. The physical origin of this theorem is the conservation of particles (or probability): the beam emerging (after scattering) along the incident direction (θ=0) contains less particles than the incident beam, since a number of particles have scattered in various other directions. This decrease in the number of particles is measured by the total cross section σ; that is, the number of particles removed from the incident beam along the incident direction is proportional to σ or, equivalently, to the imaginary part of f (0).