Phys 622 Problems Chapter 6

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1 1 Problem 1 Elastic scattering Phys 622 Problems Chapter 6 A heavy scatterer interacts with a fast electron with a potential V (r) = V e r/r. (a) Find the differential cross section dσ dω = f(θ) 2 in the first Born approximation and sketch its angular dependence. (b) What is the dependence of σ tot on the energy of the incident electron in the limit kr >> 1, where k is the magnitude of the electron momentum? Compare the result to the dependence σ tot 1 of E 2 the Rutherford scattering. Problem 2 Scattering of identical particles In the previous problem you did not have to consider the identity of particles. that the incident and the target particle are identical. Now assume (a) There are several modifications that have to be made even in the classical case if the particles are identical. First, if the scatterer is much heavier than the incident particle, the center-of-mass is practically the same as the laboratory coordinate system. This is no longer the case if the particles are identical. Show that, θ = 2θ in this case, where θ and θ are the scattering angles in the center-of-mass and laboratory frames. (b) With this change of angle, the solid angle will also change dω dω. angle dω in the laboratory frame? What is the solid (c) Finally, consider that the angle in the laboratory frame between the two particles after scattering is always π/2 in classical mechanics and that both particles can be detected. If we assume the Rutherford scattering case, how would the cross-section dσ = ( Z dω ka ) 4 1 have to be modified to sin 4 (θ/2) describe scattering in the laboratory frame, accounting for all the modifications mentioned? Hint: in the classical case we add differential cross-sections dσ dω. (d) In the quantum-mechanical case we add the scattering amplitudes. Moreover, the way in which they are added is determined by the symmetry of the wave function, ( which in ) turn is determined by the spin of the particles. For unpolarized spin 1/2 particles ρ =, and the total 1/4 3/4 scattering cross-section is dσ dσ = T r(ρ ) = 1 f(θ) + f(π dω dω 4 θ) f(θ) f(π 4 θ) 2. Assume that the two identical particles have spin 1 and are both polarized up. How would the expressions for ρ and dσ have to be changed? dω (e) Same for the case when the particles have spin 1, the incident particle is polarized up and the target is unpolarized. (f) Same for the case when the particles have spin 1 and are unpolarized.

2 2 (g) Same for the case when the particles have spin. Problem 3 Scattering by a δ function shell Consider the scattering potential V (r) = αδ(r R). (a) Find the scattering amplitude f(θ) in the 1st order Born approximation. Write down in detail all the steps, from the definition of the scattering amplitude to the final result. (b) Find the differential and total cross section in the limiting case of small incident energies E. Sketch the azimuthal dependence of the differential cross section and give a physical interpretation of the result. (c) Same as above in the limiting case of large incident energies E. Problem 4 One-dimensional scattering Consider a one-dimensional problem with a delta-function potential V (x) = Ω δ(x) (this form is m used so that Ω has units of momentum) and a particle incident on this potential from the left. (a) The traditional way to solve this problem is by solving the Schrodinger equation for a wavefunction x ψ L = ψ L (x) = e ikx + re ikx and x ψ R = te ikx to the left and right of x =, respectively. Set the continuity condition ψ L (x = ) = ψ R (x = ) and a discontinuity in the derivative ψ R(x=) x ψ L(x=) x = 2Ωψ() to obtain a system of two equations. Solve this system to show that the reflectivity coefficient is r = 1 ik/ω 1. All scattering methods started with the Schrodinger equation for energy E > and should give an answer consistent with this solution. For instance, we showed in class how the 1st order Born approximation gave the 1st order term in the expansion of r in powers of k/ω (large k correspond to the high energies of the Born approximation). This problem illustrates the application of partial wave scattering. (b) There is no angular momentum in 1D, so we cannot use the usual Elm basis set of partial waves. Show that [Ĥ, ˆp], so that the basis set made of two waves travelling to the left and to the right of the traditional solution, although intuitive, is also not good. Because of this, for instance we have to represent the same wavefunction twice in this basis set: as ψ =. ( ) 1r for x < and as ψ =. ( ) for x > in the traditional solution. t Show that, in contrast, [Ĥ, ˆπ] =, where ˆπ is the parity operator. Therefore, we can use a basis set made of states with even and odd parity, or E, ±. We now have two parallel scattering channels : for states of even parity and for states of odd parity. Scattering can be described with

3 3 two phase shifts only: δ S and δ A. (c) An incident wave x ψ inc = e ikx and an scattered wave x ψ L,scatt = re ikx and x ψ R,scatt = te ikx are used in the traditional solution. Show that these waves do not have any specific parity symmetry. Combine a right-travelling with a left-travelling incident wave to define a new incident wave of even parity. Obtain the outgoing wave by combining the corresponding outgoing waves of the traditional solution (you may consider that r and t are known). Apply the definition of the phase shift x ψ out = e 2iδ S x ψ inc to find δ S. Give an interpretation of its limit when k. (d) Same for an incident wave of odd parity: find the odd parity channel phase shift δ A and give an interpretation of its limit when k. Problem 5 Scattering off a flat-top potential Consider a typical nuclear physics flat-top potential V = V for r < R and V = for r > R. Define k 2 = 2mE and k 2 = 2mV. (a) Find the phase shift δ of the l = partial wave for E V and confirm that the total cross-section σ reduces to the hard-sphere result 4πR 2 when V. (b) Find the δ phase shift in the E V limit. (c) Show that when R is the smallest parameter in the problem, or k, k 1, the l = cross-section R is the same and σ R 6 in both (a) and (b) cases. (d) Solve with the Born approximation in the E V limit and confirm that the differential cross-section dσ/dω is not isotropic, in contrast to the result in part (b). Note: the l = partial wave result from part (b), which was obtained in the same high-energy limit, gives an incomplete description of the scattering. We have to include more partial waves (with l > ) to better describe the scattering with partial waves in this limit. 1 V/V 1 2 r/r

4 4 Problem 6 Scattering off an exponential potential Representing a state in the partial wave basis set is a general method, applicable to scattering off all spherically-symmetric potentials. It must therefore be possible to obtain the partial wave phase shifts δ l from the scattering amplitude f(θ) of the Born approximation. This problem illustrates how to obtain the l = partial wave phase shift δ when the Born approximation cross section is known. This can be useful when the potential does not have a well-defined boundary, beyond which V (r) =, and the method used for the flat-top (problem 1) or hard-sphere potential cannot be applied. However, because we start with the high-energy Born approximation, there will be no resonances. e Take a potential with an exponential tail V (r) = V αr, where α = 1/R is the inverse of a αr characteristic length R. (a) Find f(θ) with the Born approximation for E V. Hint: use the expression for f(θ) for spherically-symmetric potentials and sin(x) = eix e ix 2i. (b) Apply the relation f(θ) = l= (2l + 1)P l(cosθ)f l to find f and sketch its dependence on k. Hint: multiply by P l (cosθ) and integrate over dω. Use the orthogonality of spherical harmonics dωy m l (θ, φ)yl m (θ, φ) = δ ll δ mm for the particular case m = m = and Yl m= 2l+1 = P 4π l(cosθ). The integral can be done by hand. (c) Obtain σ from f and show that σ R 6 for k, k 1 R. Hint: ln(1 + x) x for small x. V/V r/r (d) We obtained σ (2mV ) 2 R 6 in the limit of small R for two very different potentials, using both partial wave and Born approximation methods. Can you show why this should be so?

5 5 Hint: you may use the relation between rate w and dσ and the DOS ρ(e dω f) that was derived for core-level X-ray photoemission of electrons into free final states. Note: the dependence σ R 6 is is the same as that of Rayleigh light scattering cross-section. However, while Rayleigh light scattering has a classical dipolar angular dependence because of the additional light polarization that has to be considered, here we have a scalar field ψ and the angular distribution is isotropic at low energies.