Analytical Mechanics - Exercise 5 - Scattering.

Analytical Mechanics - Exercise 5 Scattering.
1. Consider a perfectly elastic scattering from an ellipsoid formed by rotating
an ellipse about the z-axis:
y2
z2
x2
+
+
=1
A2
A2
B2
A beam of point like particles is traveling along the z-axis and scatters o
the ellipsoid. In this problem you are asked to calculate the dierential cross
section
a.
σ(θ)
and to compare it to the cross section from a hard sphere.
Since the ellipsoid is rotationally symmetric, all the geometry can be
worked out in the x-z plane.
Make a sketch of the ellipse and indicate the
scattering angle and the impact parameter. Write the relationship between the
scattering angle
θ
dx
dz .
and the scattering
and the slop of the ellipse at the impact point
b. Write the relationship between the impact parameter b
angle
θ. Calculate the dierential cross section. You may use the relation
1 d(b2 ) db 2 dθ = b dθ to simplify the calculation.
c. Set A = B = R and verify that the dierential cross section does not
R2
depend on θ and it is equal to
4 where R is the radius of the sphere.
d. Consider an ellipsoid that is shot in the Z direction. Assume that A is
kept constant and B = αA where α < 1. Write the cross section in the forward
direction at θ = 0 in terms of α and A. Is it smaller or larger than that of a
hard sphere?
e. What is the total cross section of scattering from the ellipsoid? Hint: Do
not do any calculations other than those done in elementary geometry.
2.
Examine the scattering produced by a repulsive central force
f = kr−3 .
Show that the dierential cross section is given by
σ(θ)dθ =
where
3.
x
is the ratio
Particles with mass
θ
π and
m
E
k
(1 − x)dx
2
2E x (2 − x)2 sin(πx)
is the energy.
are scattered on lenses according to the laws of
geometric optics. The radius (hight) of the lens is
1
R
and its focal length is
f.
Figure 1: Illustration
a.
What is the dierential cross section for scattering (assume the lens is
dσ
dΩ ?
b. Assume the lenses has a mass that is equal to the particles mass (the lens
stationary)
can move but cannot rotate). Find the dierential cross section for the particles
and the lenses.
c. Find the total cross section for scattering.
4. Find the dierential cross sections from the following surfaces of revolution:
a.
z
r(z) = Bsin( ); 0 ≤ z ≤ πa
a
b.
z
r(z) = Btan−1 ( ); 0 ≤ z
a
2