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Scattering Theory Partial Wave Analysis LectureLecture 66 ScatteringScattering theorytheory PartialPartial WaveWave AnalysisAnalysis SS2011 : ‚Introduction to Nuclear and Particle Physics, Part 2 ‘ 1 PartialPartial wavewave analysisanalysis forfor elasticelastic scatteringscattering 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 PartialPartial wavewave analysisanalysis forfor elasticelastic scatteringscattering Consider the Schrödinger equation in CM frame: (3) The most general solution of the Schrödinger equation (3) is (4) 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 ϕϕϕ; m =0. Thus, as Yl0(θ,ϕ) ∼ Pl (cos θ), the scattered wave function (4) becomes (5) 2 µE where Rkl (r ) obeys the following radial equation ( here k 2 === ) h 2 (6) 2 Each term in (5), which is known as a partial wave , is a joint eigenfunction of L and LZ . 3 PartialPartial wavewave analysisanalysis forfor elasticelastic scatteringscattering A substitution of (2) into (7) with ϕϕϕ=0 (and k=k 0 for elastic scattering) gives (8) The scattered wave function is given, on the one hand, by (5) and, on the other hand, by (8). Consider the limit r →→→ ∞∞∞ 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 jl(kr ) for large values of r is given by (9) the asymptotic form of (8) is given by (10) 4 PartialPartial wavewave analysisanalysis forfor elasticelastic scatteringscattering Since (11) because one can write (10) as (12) 2) To find the asymptotic form of (5), we need first to determine the asymptotic form of the radial function Rkl (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 PartialPartial wavewave analysisanalysis forfor elasticelastic scatteringscattering 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 Rkl (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 δδδl to achieve the regular solution near the origin by rewriting (14) in the form Rkl )r( === Cl [[[cos (((δδδ l )))jl (((kr )))−−− sin (((δδδ l )))nl (((kr )))]]] (17) where we have 6 PartialPartial wavewave analysisanalysis forfor elasticelastic scatteringscattering Thus, the asymptotic form of the radial function (16) can be written as lπππ lπππ cos ((()(δδδ )))sin kr −−− −−− sin ((()(δδδ )))cos kr −−− l 2 l 2 R )r( →→→ C kl r→→→∞∞∞ l kr (18) With δδδl =0, the radial function Rkl (r ) of (18) is finite at r =0, since Rkl (r ) in (17) reduces to jl(kr). So δδδl is a real angle which vanishes for all values of l in the absence of the scattering potential (i.e., V =0); δδδl is called the phase shift of the l‘th partial wave . The phase shift δδδl measures the ‚distortion‘ of Rkl (r ) from the ‚free‘ solution jl(kr) due to the presence of the potential V(r ) Attractive (repulsive) potentials imply that δδδl > 0 ( δδδl < 0) corresponding to the wave being “pulled in” (“pushed out”) by the scattering center resulting in a phase delay (advance). 7 PartialPartial wavewave analysisanalysis forfor elasticelastic scatteringscattering 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 δδδl. Since one can rewrite (19) as (20) Compare (20) and (12): (12) We obtain: (21) 8 PartialPartial wavewave analysisanalysis forfor elasticelastic scatteringscattering Substituting (21) into (20) and this time equating the coefficient of in the resulting expression with that of (12), we have from (12) from (20) (22) which by combining with leads to (23) where fl(θ ) is denoted as the partial wave amplitude. 9 PartialPartial wavewave analysisanalysis forfor elasticelastic scatteringscattering 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 σσσl are denoted as the partial cross sections corresponding to the scattering of particles in various angular momentum states. 10 PartialPartial wavewave analysisanalysis forfor elasticelastic scatteringscattering 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 θθθ is carried out. Note that when V=0 everywhere, all the phase shifts δδδl 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 (27) where we have used Since f0 does not depend on θθθ , the differential and total cross sections in the CM frame are given by the following simple relations: (28) 2 µE ( here k 2 === ) 11 h 2 OpticalOptical theoremtheorem The total cross section in CM frame can be related to the forward scattering amplitude f (0). Since for θ=θ=θ= 0, eq. (23) leads to (29) which - combined with (26) - yields the connection between f (0) and σσσ : (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 (θθθ=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 σσσ; that is, the number of particles removed from the incident beam along the incident direction is proportional to σσσ 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 PartialPartial wavewave analysisanalysis forfor inelasticinelastic scatteringscattering 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 , Sl (k) is redefined by (34) with , then (33) and (31) become (35) (36) 13 TotalTotal elasticelastic andand inelasticinelastic crosscross sectionssections 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 ηηηl (k)= 1 there is no inelastic scattering , but if ηηηl (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: (40) A comparison of (40) and (39) gives the optical theorem relation (41) 14 Note that the optical theorem is also valid for inelastic scattering ! HighHigh --energyenergy scatteringscattering fromfrom aa blackblack diskdisk Consider the example: a black disk is totally absorbing , i.e., ηηηl (k)= 0. Assuming the values of angular momentum l do not exceed a maximum value lmax (l < lmax ) and that k is large (high-energy scattering), we have lmax =ka where a is the radius of the disk . Since ηηηl = 0, equations (37) and (38) lead to (42) the total cross section then reads 2 (43) σσσ tot === σσσ el +++ σσσ inel === 2πππa Classically , the total cross section is a disk equal to πππa2. 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 πππa2, and the other to waves that are diffracted - also of size πππa2. 15 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 (πππ-θθθ ). Thus, the scattered wave function must be symmetric: (45) and also the scattering amplitude: (46) 16 Scattering of identical bosons Therefore, the differential cross section is (47) interference term - not in the classical case ! For - quantum case (48) - classical case If the particles are distinguishable , the differential cross section will be four times smaller: (49) 17 Scattering of identical fermions Consider now the scattering of two identical spin 1/2 particles .
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