Math 1320 - Lab 2
Name:
Today we will consider a series of particle physics type problems. Answer the questions below
and be sure explain your reasoning behind your approach to these questions with appropriate
diagrams and complete sentences. In your writing, try to be complete enough so that someone
who has completed Math 1310 would understand what you are doing and why you are doing it,
i.e. you don’t have to explain what an integral is, but you do have to explain why you chose a
particular form for an integral that you use.
1.
(10 points) Bubble chambers are (were) used in experiments to detect charged high energy
particles. The chamber is filled with superheated liquid hydrogen and when charged particles
enter, they leave an ionization track which forms a line of bubbles which can be visualized.
Kinks in the track represent where a decay event occured (a high energy particle decayed into
another particle). Assume a high energy particle enters the chamber and undergoes one decay
event before decaying further. It (and its descendent) leaves a track that can be represented in
parametric form by
(
0.5 cos(t), 0 ≤ t ≤ 3π
4
x(t) =
3π
0.5 cos(3π/4.) + 0.1 cos(t − 3π/4.),
4 ≤ t ≤ 6π
(
0.5 sin(t), 0 ≤ t ≤ 3π
4
y(t) =
3π
0.5 sin(3π/4.) + 0.1 sin(t − 3π/4.),
4 ≤ t ≤ 6π
(
t, 0 ≤ t ≤ 3π
4
z(t) =
3π
3π/4. + t/2.,
4 ≤ t ≤ 6π
If the particle and its descendent travel at 0.8c (c is the speed of light), how much time does the
particle survive for? What about the particle and its descendent? You can leave your answers
in terms of c.
2.
(10 points) An experimental physicist is able to confine two charged particles in such a way
that they are always between 1 and 2 angstroms apart and measures the average potential energy
between them to be 720 (eV). They are considering using two model equations for the potential
energy between the particles, the Lennard-Jones potential model and the Born-Huggins-Mayer
potential model. Equations for each potenital (in eV) as a function of separation distance (in
angstroms) is given below. Which model does a better job at matching the experiment (if we
assume all separation distances are equally likely). Let σ = 2 angstroms.
σ 12 σ 6
−
VLJ (r) = 4 ∗ (0.5) ∗
r
r
VBHM (r) = e8.75(σ−r) −
50 10
− 8
r6
r
3.
(10 points) Consider the bond energy between salt ions Na+ and Cl− . Positive and negative
charges form an attractive force fa proportional to the inverse square of the distance r between
the charges, which can cause the sodium and chloride to combine in a solid crystal
fa (r) = −
A
,
r2
where A is a positive constant and the negative sign − indicates that the force fa pulls the two
opposing charges together. For large distances r, the force is weak, but nearby the force gets
stronger, much like the attractive force of a magnet. The two atoms also have a repulsive force
that prevents the atoms from fusing infinitesimally close to each other.
fb (r) =
B
,
rn
where B is a positive constant and the force goes with the inverse nth power of distance (the
power is typically anywhere between the 7th and 13th power, depending on the particular atoms
involved).
The net force fnet on the two ions is
fnet (r) = fb (r) + fa (r)
The ions are in a stationary equilibrium when their position r∗ is such that the net force is zero:
fnet = 0.
(a) Solve for r∗ .
(b) From the equilibrium point, compute the bonding energy of the two molecules by computing
the work involved to move the two ions from r∗ , to a position infinitely far apart. That is,
compute the work done against the force fnet (r) as we pull apart the ions by making r go to
infinity to verify that the work required has a form rC∗ where C is a positive constant that
depends on n and A.