Lecture: Scattering theory 30.05.2012 SS2012: Introduction to Nuclear and Particle Physics, Part 2 2 1
Part I: Scattering theory: Classical trajectoriest and cross-sections Quantum Scattering 2
I. Scattering experiments Scattering experiment: A beam of incident scatterers with a given flux or intensity (number of particles per unit area da per unit time dt ) impinges on the target (described by a scattering potential); the flux can be written as The number of particles per unit time which are detected in a small region of the solid angle, dω, located at a given angular deflection specified by (θ, φ), can be counted as 3
Scattering cross section The differential cross-section for scattering is defined as the number of particles scattered into an element of solid angle dω in the direction (θ,φ) per unit time : [dimensions of an area] dσ ( ϑ, φ ) dω = 1 J inc dn sc dω J inc - incident flux (1.1) The total cross-section corresponds to scatterings through any scattering angle: (1.2) Most scattering experiments are carried out in the laboratory (Lab) frame in which the target is initially at rest while the projectiles are moving. Calculations of the cross sections are generally easier to perform within the center-of-mass (CM) frame in which the center of mass of the projectiles target system is at rest (before and after collision) one has to know how to transform the cross sections from one frame into the other. Note: the total cross section σ is the same in both frames, since the total number of collisions that take place does not depend on the frame in which the observation is carried out. However, the differential cross sections dσ/ σ/dω are not the same in both frames, since the scattering angles (θ,φ) are frame dependent. 4
Connecting the angles in the Lab and CM frames Elastic scattering of two structureless particles in the Lab and CM frames: Note: consider for simplicity NON-relativistic kinematics! r V r CM V 1 L To find the connection between the Lab and CM cross sections, we need first to find how the scattering angles in one frame are related to their counterparts in the other. If and if we have denote the position of m 1 in the Lab and CM frames, respectively, denotes the position of the center of mass with respect to the Lab frame, A time derivative of this relation leads to (1.3) where are the velocities of m 1 in the Lab and CM frames before collision and is the velocity of the CM with respect to the Lab frame. Similarly, the velocity of m 1 after collision is (1.4) 5
Connecting the angles in the Lab and CM frames Since r V r CM V 1 L (1.5) (1.6) Dividing (1.6) by (1.5), we end up with (1.7) Use that CM momenta r p CM ( m 1 = + r p m 1 L 2 + r )V r p CM 2 L = m 1 r V 1 L + m 2 r V (1.8) 2 L Since from (1.8) (1.9) or from (1.9) (1.10) (1.11) 6
Connecting the angles in the Lab and CM frames Since the center of mass is at rest in the CM frame, the total momenta before and after r r r r r collisions are separately zero: v pc = p1c + p 2 C = p C = p 1C + p 2 C = (1.12) (1.13) Since the kinetic energy is conserved: (1.14) (1.15) 0 In the case of elastic collisions, the speeds of the particles in the CM frame are the same before and after the collision; From (1.11) (1.16) Dividing (1.9) by (1.16) (1.17) 7
Connecting the angles in the Lab and CM frames Finally, a substitution of (1.17) into (1.7) yields (1.17) and using we obtain (1.18) Note: In a similar way we can establish a connection between θ 2 and θ. From (1.4) we have + using the x and y components of this relation are (1.19) (1.20) 8
Connecting the Lab and CM Cross Sections The connection between the differential cross sections in the Lab and CM frames can be obtained from the fact that the number of scattered particles passing through an infinitesimal cross section is the same in both frames: What differs is the solid angle dω : in the Lab frame: in the CM frame: (1.21) (1.22) Since there is a cylindrical symmetry around the direction of the incident beam (1.23) From (1.18) (1.24) 9
Connecting the Lab and CM Cross Sections Thus : (1.25) From (1.23) and (1.20) (1.26) Limiting cases: 1) the Lab and CM results are the same, since (1.17) leads to θ 1 =θ (1.27) 2) then from (1.20) from (1.25) (1.28) 10
From classical to quantum scattering theory 1. Classical theory: - scattering of hard spheres - individual well-defined trajectories 2. Quantum theory: - scattering of wave pakeges wave particle duality - probabilistic origin of scattering process 11
II. Classical trajectoriest and cross-sections Scattering trajectories, corresponding to different impact parameters, b give different scattering angles θ. All of the particles in the beam in the hatched region of area dσ = 2π bdb are scattered into the angular region (θ, θ + dθ) The equations of motion for incident particle: (Newton s law) + initial conditions: since initial kinetic energy defines the trajectory (2.1) (2.2) (2.3) For a particle obeying classical mechanics: the trajectory for the unbound motion, corresponding to a scattering event, is deterministically predictable, given by the interaction potential and the initial conditions; the path of any scatterer in the incident beam can be followed, and its angular deflection is determined as precisely as required 12
Classical trajectoriest and cross-sections The number of particles scattered per unit time into the angular region (θ, θ+dθ) with any value of φ can be written as (2.4) (2.5) The knowledge of b(θ), obtained directly from Newton s laws or other methods, is then sufficient to calculate the scattering cross-section. For the scattering from nontrivial central forces, the trajectory can be obtained from the equations of motion by using energy- and angular momentum- conservation methods E.g., one can rewrite L - angular momentum in the form (2.6) 13
Classical trajectoriest and cross-sections The angular momentum can be written via the angular velocity from (2.6) (2.7) the angle through which the particle moves between two radial distances r 1 and r 2 : (2.8) (2.9) Scattering trajectories in a central potential : r min - the distance of closest approach, Θ deflection angle: Use that the initial angular momentum and from (2.8), (2.9) (2.10) 14
Classical Coulomb scattering The potential corresponding to the scattering of charged particles via Coulomb s law can be written in the general form (2.11) where the constant point-like charges Z 1 e and Z 2 e corresponds to the electrostatic interaction of from (2.10) (2.12) Note that for large energies (E ) or vanishing charges (A 0), one has r min b Use that from (2.12) (2.13) (2.14) 15
Classical Coulomb scattering Use that (2.15) Rutherford formula - characteristic for Coulomb scattering from a point-like charge: (2.16) Note: If one attempts to evaluate the total cross-section using Eq. (2.16), one obtains an infinite result. This divergence is due to the infinite range of the 1/r potential, and the infinity in the integral comes from scatterings as θ 0 which corresponds to arbitrarily large impact parameters where the Coulomb potential causes a scatter through an arbitrarily small angle, which still contributes to the total cross-section. 16
III. Quantum Scattering The scattering amplitude of spinless, nonrelativistic particlesp articles: Consider the elastic scattering between two spinless, nonrelativistic particles of masses m 1 and m 2. During the scattering process, the particles interact with each other. If the interaction is time independent, we can describe the two-particle system with stationary states (3.1) where E T is the total energy and Schrödinger equation: is a solution of the time-independent (3.2) is the potential representing the interaction between the two particles. In the case where the interaction between m 1 and m 2 depends only on their relative distance one can reduce the eigenvalue problem (3.2) to two decoupled eigenvalue problems: one for the center of mass (CM), which moves like a free particle of mass M= m 1 +m 2 (which is of no concern to us here) and another for a fictitious particle with a reduced mass moves in the potential which (3.3) 17
Scattering of spinless particles In quantum mechanics the incident particle is described by means of a wave packet that interacts with the target. After scattering, the wave function consists of an unscattered part propagating in the forward direction and a scattered part that propagates along some direction (θ,ϕ) 18
Scattering of spinless particles One can consider (3.3) as representing the scattering of a particle of mass µ from a fixed scattering center described by V(r), where r is the distance from the particle µ to the center of V(r): (3.3) assume that V(r) has a finite range a: Vˆ ( r ) = = 0, if 0, if r r > a a Outside the range, r > a, the potential vanishes, V(r)= 0; the eigenvalue problem (3.3) then becomes (3.4) In this case µ behaves as a free particle before the collision and can be described by a plane wave (3.5) where is the wave vector associated with the incident particle; A is a normalization factor. Thus, prior to the interaction with the target, the particles of the incident beam are independent of each other; they move like free particles, each with a momentum 19
Scattering of spinless particles When the incident wave (3.5) collides or interacts with the target, an outgoing wave is scattered out. In the case of an isotropic scattering, the scattered wave is spherically symmetric, having the form In general, however, the scattered wave is not spherically symmetric; its amplitude depends on the direction (θ,ϕ) along which it is detected and hence (3.6) where f (θ,ϕ) is called the scattering amplitude, is the wave vector associated with the scattered particle, and θ is the angle between After the scattering has taken place, the total wave consists of a superposition of the incident plane wave (3.5) and the scattered wave (3.6): (3.7) where A is a normalization factor; since A has no effect on the cross section, as will be shown later, we will take it equal to one. 20
Scattering amplitude and differential cross section Introduce the incident and scattered flux densities: (3.8) (3.9) Inserting (3.5) into (3.8) and (3.6) into (3.9) and taking the magnitudes of the expressions thus obtained, we end up with (3.10) We recall that the number dn(θ,ϕ) of particles scattered into an element of solid angle dω in the direction (θ,ϕ) and passing through a surface element da= r 2 dω per unit time is given as follows: (3.11) Using (3.10) (3.12) 21
Scattering amplitude and differential cross section By inserting (3.12) and in (1.1), we obtain: (3.13) Since the normalization factor A does not contribute to the differential cross section, it will be taken to be equal to one. For elastic scattering k 0 is equal to k; so (3.13) reduces to (3.14) The problem of determining the differential cross section dσ/dω therefore reduces to that of obtaining the scattering amplitude f (θ,ϕ). 22
Part II: Scattering theory: Born Approximation 23
Scattering amplitude We are going to show here that we can obtain the differential cross section in the CM frame from an asymptotic form of the solution of the Schrödinger equation: (1.1) Let us first focus on the determination of the scattering amplitude f (θ, φ), it can be obtained from the solutions of (1.1), which in turn can be rewritten as where 2 2µE k = 2 h The general solution of the equation (1.2) consists of a sum of two components: (1.2) 1) a general solution to the homogeneous equation: (1.3) In (1.3) is the incident plane wave 2) and a particular solution of (1.2) with the interaction potential 24
General solution of Schrödinger eq. in terms of Green s function The general solution of (1.2) can be expressed in terms of Green s function. (1.4) is the Green s function corresponding to the operator on the left side of eq.(1.3) The Green s function is obtained by solving the point source equation: (1.5) (1.6) (1.7) 25
Green s function A substitution of (1.6) and (1.7) into (1.5) leads to (1.8) The expression for can be obtained by inserting (1.8) into (1.6) (1.9) (1.10) To integrate over angle in (1.10) we need to make the variable change x=cosθ (1.11) 26
Method of residues Thus, (1.9) becomes (1.12) (1.13) The integral in (1.13) can be evaluated by the method of residues by closing the contour in the upper half of the q-plane: The integral is equal to 2π i times the residue of the integrand at the poles. 27
Green s functions Since there are two poles, q =+k, the integral has two possible values: the value corresponding to the pole at q =k, which lies inside the contour of integration in Figure 1a, is given by (1.14) the value corresponding to the pole at q =-k, Figure 1b, is (1.15) Green s function represents an outgoing spherical wave emitted from r and the function corresponds to an incoming wave that converges onto r. Since the scattered waves are outgoing waves, only is of interest to us. 28
Born series Inserting (1.14) into (1.4) we obtain for the total scattered wave function: (1.16) This is an integral equation. All we have done is to rewrite the Schrödinger (differential) equation (1.1) in an integral form (1.16), which is more suitable for scattering theory. Note that (1.16) can be solved approximately by means of a series of successive or iterative approximations, known as the Born series. the zero-order solution is given by the first-order solution is obtained by inserting into the integral of (1.16): (1.17) 29
Born series the second order solution is obtained by inserting into (1.16): (1.18) the n th order approximation for the wave function is a series which can be obtained by analogy to (1.18). continuing in this way, we can obtain to any desired order; 30
Asymptotic limit of the wave function In a scattering experiment, since the detector is located at distances (away from the target) that are much larger than the size of the target, we have r>>r, where r represents the distance from the target to the detector and r the size of the target. If r >> r we may approximate: (1.19) 31
Asymptotic limit of the wave function Substitute (1.19) to (1.16): for r>>r ikr ikr ' r µ e e r r 3 = φ inc ( ) V ( ) ( )d r 2 h ψ π r rr (1.16) From the previous two approximations (1.19), we may write the asymptotic form of (1.16) as follows: (1.20) (1.21) where ia a plane wave and k is the wave vector of scattered wave; the integration variable r extends over the spacial degrees of freedom of the target. The differential cross section is given by (1.22) 32
The first Born approximation If the potential V(r) is weak enough, it will distort only slightly the incident plane wave. The first Born approximation consists then of approximating the scattered wave function Ψ(r ) by a plane wave. This approximation corresponds to the first iteration in the Born series of (1.16): (1.16) that is, Ψ(r ) is given by (1.17): (1.23) Thus, using (1.21) we can write the scattering amplitude in the first Born approximation as follows: (1.24) 33
The first Born approximation Using (1.23), we can write the differential cross section in the first Born approximation as follows: (1.25) where is the momentum transfer; are the linear momenta of the incident and scattered particles, respectively. In elastic scattering, the magnitudes of are equal (Figure 1); hence (1.26) Figure 1: Momentum transfer for elastic scattering: 34
The first Born approximation If the potential is spherically symmetric, and choosing the z-axis along q (Figure 1), then and therefore (1.27) Inserting (1.27) into (1.24) and (1.25) we obtain (1.28) (1.29) In summary, we have shown that by solving the Schrödinger equation (1.1) in first-order Born approximation (where the potential V(r ) is weak enough that the scattered wave function is only slightly different from the incident plane wave), the differential cross section is given by equation (1.29) for a spherically symmetric potential. 35
Validity of the firts Born approximation The first Born approximation is valid whenever the wave function Ψ(r ) is only slightly different from the incident plane wave; that is, whenever the second term in (1.23) is very small compared to the first: (1.23) (1.30) Since we have (1.31) 36
Validity of the firts Born approximation In elastic scattering k 0 = k and assuming that the scattering potential is largest near r=0, we have (1.32) (1.33) Since the energy of the incident particle is proportional to k (it is purely kinetic, ) we infer from (1.33) that the Born approximation is valid for large incident energies and weak scattering potentials. That is, when the average interaction energy between the incident particle and the scattering potential is much smaller than the particle s incident kinetic energy, the scattered wave can be considered to be a plane wave. 37
Born approximation for Coulomb potential Let s calculate the differential cross section in the first Born approximation for a Coulomb potential (1.34) where Z1e and Z1e are the charges of the projectile and target particles, respectively. In a case of Coulomb potential, eq.(1.29): (1.35) becomes (1.36) (1.37) 38
Born approximation for Coulomb potential Now, since an insertion of (1.37) into (1.36) leads to (1.38) where is the kinetic energy of the incident particle. Eq. (1.38) is Rutherford formula Note: (1.38) is identical to the purely classical case! cf. Part I, eq.(2.16) 39