Lecture 6 Scattering theory Partial Wave Analysis. SS2011: Introduction to Nuclear and Particle Physics, Part 2 2

Transcription

1 Lecture 6 Scattering theory Partial Wave Analysis SS2011: Introduction to Nuclear and Particle Physics, Part 2 2 1

2 The Born approximation for the differential cross section is valid if the interaction between the projectile particle and the scattering potential V(r ) is considered to be small compared with the energy of the incident particle (cf. Lecture 5). Let s obtain the cross section without imposing any limitation on the strength of V(r ). We assume here the potential to be spherically symmetric. The angular momentum of the incident particle will therefore be conserved, a particle scattering from a central potential will have the same angular momentum before and after the collision. Assuming that the incident plane wave is in the z-direction and hence (1) we may express it in terms of a superposition of angular momentum eigenstates, each with a definite angular momentum number l : (2) We can then examine how each of the partial waves is distorted by V(r ) after the particle scatters from the potential. 2

3 Consider the Schrödinger equation in CM frame: (3) The most general solution of the Schrödinger equation (3) is Since V(r ) is central, the system is symmetrical (rotationally invariant) about the z-axis. The scattered wave function must not then depend on the azimuthal angle ϕ; m =0. Thus, as Y l0 (θ,ϕ) P l (cosθ), the scattered wave function (4) becomes (4) (5) where R kl (r ) obeys the following radial equation 2 2µE ( here k = 2 h ) (6) Each term in (5), which is known as a partial wave, is a joint eigenfunction of L 2 and L Z. 3

4 A substitution of (2) into with ϕ=0 (and k=k 0 for elastic scattering) gives (7) (8) The scattered wave function is given, on the one hand, by (5) and, on the other hand, by (8). Consider the limit 1) Since in almost all scattering experiments detectors are located at distances from the target that are much larger than the size of the target itself. The limit of the Bessel function j l (kr) for large values of r is given by the asymptotic form of (8) is given by r (9) (10) 4

5 Since because one can write (10) as (11) (12) 2) To find the asymptotic form of (5), we need first to determine the asymptotic form of the radial function R kl (r ). At large values of r, the scattering potential is effectively zero radial equation (6) becomes (13) The general solution of this equation is given by a linear combination of the spherical Bessel and Neumann functions (14) where the asymptotic form of the Neumann function is (15) 5

6 Inserting (9) and (15) into (14), we obtain the asymptotic form of the radial function: (16) If V(r)=0 for all r (free particles), the solution of the radial equation (6), rr kl (r ), must vanish at r=0; thus R kl (r ) must be finite at the origin (at r=0). Since the Neumann function diverges at r=0, the cosine term in (16) does not represent a physically acceptable solution; one has to introduce the phase shift δ l to achieve the regular solution near the origin by rewriting (14) R ( r ) = C [ cos( δ ) j ( kr) sin( δ ) n ( kr) ] in the form (17) kl l l l l l where we have 6

7 Thus, the asymptotic form of the radial function (16) can be written as R kl ( r ) r C l cos lπ 2 kr ( δ ) sin kr sin( δ ) l l lπ cos kr 2 (18) With δ l =0, the radial function R kl (r ) of (18) is finite at r =0, since R kl (r ) in (17) reduces to j l (kr). So δ l is a real angle which vanishes for all values of l in the absence of the scattering potential (i.e., V =0); δ l is called the phase shift of the l th partial wave. The phase shift δ l measures the distortion of R kl (r ) from the free solution j l (kr) due to the presence of the potential V(r ) Attractive (repulsive) potentials imply that δ l > 0 (δ l < 0) corresponding to the wave being pulled in ( pushed out ) by the scattering center resulting in a phase delay (advance). 7

8 Using (17) we can write the asymptotic limit of the scattered wave function (5) as (19) This wave function (19) is known as a distorted plane wave, which differs from a plane wave by the phase shifts δ l. Since one can rewrite (19) as (20) Compare (20) and (12): (12) We obtain: (21) 8

9 Substituting (21) into (20) and this time equating the coefficient of expression with that of (12), we have in the resulting from (12) from (20) (22) which by combining with leads to (23) where f l (θ ) is denoted as the partial wave amplitude. 9

10 From (23) we obtain the differential cross sections (24) and the total cross sections reads: (25) Using the relation we obtain from (25): (26) where σ l are denoted as the partial cross sections corresponding to the scattering of particles in various angular momentum states. 10

11 The differential cross section (24) consists of a superposition of terms with different angular momenta; this gives rise to interference patterns between different partial waves corresponding to different values of l. The interference terms go away in the total cross section when the integral over θ is carried out. Note that when V=0 everywhere, all the phase shifts δ l vanish, and hence the partial and total cross sections, (24) and (26), are zero. In the case of low energy scattering between particles, that are in their respective s states, i.e. l=0, the scattering amplitude (23) becomes where we have used Since f 0 does not depend on θ, the differential and total cross sections in the CM frame are given by the following simple relations: (27) (28) 2 2µE ( here k = 2 h ) 11

12 Optical theorem The total cross section in CM frame can be related to the forward scattering amplitude f (0). Since for θ=0, eq. (23) leads to which - combined with (26) - yields the connection between f (0) and σ : (29) (30) This relation is known as the optical theorem. 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). Note: although (30) was derived for elastic scattering, the optical theorem (as will be shown later) is also valid for inelastic scattering. 12

13 Partial wave analysis for inelastic scattering The scattering amplitude (23) can be rewritten as (31) where (32) with (33) In the case where there is no flux loss, we must have. However, this requirement is not valid whenever there is absorption of the incident beam. In this case of flux loss, S l (k) is redefined by with, then (33) and (31) become (34) (35) (36) 13

14 Total elastic and inelastic cross sections The total elastic scattering cross section is given by (37) The total inelastic scattering cross section, which describes the loss of flux, is given by (38) Thus, if η l (k)= 1 there is no inelastic scattering, but if η l (k)= 0 we have total absorption, although there is still elastic scattering in this partial wave. The sum of (37) and (38) gives the total cross section: (39) Using (31) and (35) we get: A comparison of (40) and (39) gives the optical theorem relation Note that the optical theorem is also valid for inelastic scattering! (40) (41) 14

15 High-energy scattering from a black disk Consider the example: a black disk is totally absorbing, i.e., η l (k)= 0. Assuming the values of angular momentum l do not exceed a maximum value l max (l < l max ) and that k is large (high-energy scattering), we have l max =ka where a is the radius of the disk. Since η l = 0, equations (37) and (38) lead to (42) the total cross section then reads σ = σ + σ = 2πa tot el inel 2 (43) Classically, the total cross section is a disk equal to πa 2. The factor 2 in (43) is due to purely quantum effects, since in the high-energy limit there are two kinds of scattering: one corresponding to waves that hit the disk, where the cross section is equal to the classical cross section πa 2, and the other to waves that are diffracted - also of size πa 2. 15

16 Scattering of identical bosons Let s consider the scattering of two identical bosons in their center of mass frame. Classically, the cross section for the scattering of two identical particles whose interaction potential is central is given by (44) In quantum mechanics there is no way of distinguishing between the particle that scatters at an angle θ from the one that scatters at (π-θ ). Thus, the scattered wave function must be symmetric: (45) and also the scattering amplitude: (46) 16

17 Scattering of identical bosons Therefore, the differential cross section is (47) interference term - not in the classical case! For - quantum case - classical case (48) If the particles are distinguishable, the differential cross section will be four times smaller: (49) 17

18 Scattering of identical fermions Consider now the scattering of two identical spin 1/2 particles. E.g.: electron electron or proton proton scattering. The wave function of a two spin 1/2 particle system is either symmetric or antisymmetric: when the spatial wave function is symmetric, that is the two particles are in a spin singlet state, the differential cross section is given by (50) when the two particles are in a spin triplet state, the spatial wave function is antisymmetric, and then (51) If the incident particles are unpolarized, the various spin states will be equally likely, so the triplet state will be three times more likely as the singlet: (52) 18

19 Scattering of identical fermions if the incident particles are unpolarized: for quantum case (53) classical case (53) this quantum differential cross section for the scattering of identical fermions is half the classical expression, and four times smaller than the quantum differential cross section for the scattering of two identical bosons (48) - Note that, in the case of partial wave analysis for elastic scattering, using the relations and inserting them into (23) leads to: (23) We can write (54) (55) 19