Description
Question 1 (3 marks)
A particle of mass m moves in a spherically symmetric potential,
V (r) = −C e
−αr
r
, (1)
with C, α > 0.
(a) Derive an effective one-dimensional potential that governs the qualitative motion of the particle. Sketch
your potential.
(b) Using your solution to (a), discuss and classify the expected motion of the particle as a function of the
initial energy, E0.
Question 2 (3 marks)
A typical potential describing the interaction between particles is very complicated and the analytic computation of scattering cross sections is difficult. However, we can often gain qualitative insight by approximating
the precise form of the potential.
One common treatment of intermolecular scattering is the Lennard-Jones potential (which is in itself
already an approximation):
VLJ = V0
a
r
12
−
a
r
6
. (2)
It features a divergent repulsive contribution at short ranges (r a) and a long attractive tail (r a), with
the minimum of the potential occurring at r = 21/6a.
(a) When the short-range repulsive contribution is expected to dominate the physics we can adopt a
so-called “hard-core” interaction,
Vhc(r) = (
∞ if r ≤ a0
0 if r > a0
, (3)
where a0 characterizes the length scale of the interaction (a0 6= a).
This potential is equivalent to the
scattering of a particle of an impenetrable sphere. Compute the scattering cross section σ(Θ) and total
cross section σT for this potential. Discuss and interpret both results.
(b) When the attractive part of the interaction is important we instead adopt a so-called “soft-core”
interaction,
Vsc(r) = (
−V0 if r ≤ a0
0 if r > a0
. (4)
Show that the impact parameter associated with this potential can be written as,
s =
a sin (Θ/2)
q
1 + 1
n2 −
2
n
cos (Θ/2)
(5)
where n =
p
1 + V0/E and E is the energy of the incident particle. Hint: You might find it useful to
consider energy and momentum conservation, and consider the associated expressions for r ≤ a0 and
r > a0 separately.
Question 3 (3 marks)
Consider central force field of the form F(r) = k/r3
.
(a) Using the formula,
Θ(s, E) = π − 2
Z umax
0
sdu
q
1 −
V (u)
E − s
2u
2
, (6)
with u = 1/r, show that,
Θ(s, E) = π
”
1 −
s
√
2E
√
k + 2Es2
#
. (7)
In principle, one can obtain the scattering cross section from Eq. (7). However, we shall pursue a different
route for this example. The motion of a particle in the central force can be written in terms of u = 1/r as,
d
2u
dθ2
+ u = −
m
l
2
dV (u)
du . (8)
(b) Show that a parametrization of the motion is,
u(θ) = α cos(γθ) + β sin(γθ),
γ =
r
1 +
mk
l
2
.
(9)
(c) Use that the particle approaches from an initial angle θi = π to show that: i) α = −β tan(γπ) and ii)
γ = π/(Θ − π).
(d) Finally, defining x = Θ/π, show that the cross section can be obtained as,
σ(Θ)dΘ = k
E
(1 − x)dx
x
2(2 − x)
2 sin(πx)
. (10)
Question 4 (1 marks)
Read Secs. I and II of the article “Elastic scattering by a paraboloid of revolution” by Evan James at
https://doi.org/10.1119/1.15593 [Am. J. Phys. 56, 423 (1988)]. Write a brief summary of the article,
discussing the main conclusions and results of the work.
