Scattering

Definitions

Problem:

A target of Aluminum (A = 27) with an aerial density of 1 mg/cm2 is positioned perpendicular to a 0.5 mA beam of deuterons (Z = 1).  If the cross section for producing protons at an angle of 30o with respect to the beam direction is 15 mb/sr, (1 mb = 10-27 cm2), how many protons per second will be incident on a 1 cm2 detector facing the target at 30o and 10 cm from the target?

Solution:


Hard sphere scattering

Problem:

Calculate the differential and total scattering cross section for scattering a particle off a fixed, "hard" sphere of radius R.

Solution:

Problem:

A steel disk A with radius R moves with speed v = 10 m/s when it collides with a second identical disk B at rest.  The collision is elastic and has an impact parameter “b”.  After the collision, the speed of disk A is equal to 5 m/s.  What is the value of the impact parameter “b”?  Neglect friction. 

Solution:

Problem:

Consider the perfectly elastic scattering of a hard sphere of mass m and initial velocity v0 against a stationary hard sphere of mass M.  The sum of the radii of the two spheres is D.
(a)  Compute the energy lost by the small mass in the collision in the lab frame as a function of the scattering angle θ in the center of mass frame.
(b)  Compute the probability P(θ)dθ of scattering through an angle θ in the center of mass frame, assuming that the particles m in the incident beam are uniformly distributed across a cross section of the beam.  Find the differential scattering cross section σ(θ), and show that your expression integrates to the known total hard sphere cross section πD2.
(c)  Use the results of parts (a) and (b) to compute the average energy loss per collision.  For what value of m/M does the average energy loss maximize?

Solution:


Scattering by a 1/rn potential

Problem:

A particle of mass m moves under a central repulsive force F(r)  =  km/r3.   At its distance of closest approach r0 it has speed v0.
(a)  Find the orbital equation r(θ) for the particle motion, evaluating constants in terms of r0 and v0.
(b)  Find the impact parameter and the total angular deflection, assuming the particle approaches from large r.
(c)  Sketch the particle trajectory, indicating the impact parameter and total deflection calculated in part (b).

Solution:

Problem:

A fixed force center scatters a particle of mass m and initial velocity u0 according to the force law f(r) = k/r3.  Determine the differential scattering cross section.

Solution:

Problem:

Two particles of mass m are moving in the x-y plane.  The inter-particle potential energy is U(r1 - r2) = U0/|r1 - r2|2 with U0 > 0.  The initial conditions are
r1(t = 0) = (-∞, -y0),  r2(t = 0) = (∞, y0), and p1(t = 0) = (p0, 0),  p2(t = 0) = (-p0, 0),
with p0 > 0.  What is the distance of closest approach of the two particles?

Solution:

Problem:

A point like comet of mass m approaches a sun with mass M and Radius R with speed v > 0.  What is the total cross section for the comet to crash on the sun?

Solution: