View of the atom at the beginning of XX century
After the discovery of the electron and the measurement of its charge and mass, J. J. Thomson proposed so-called "Plum Pudding Moddel", in which electrons - elementary particles - are suspended in a pudding-like positively charged substance that contains most of atom's mass.
Rutherford was at the time busy studying alpha particles, and in 1907 at McGill by accident discovered that alpha particles are slightly deflected when they pass through matter. That gave him an idea to systematically study the structure of the atom by bombarding it with a beam of heavy alpha particles and exhamine how they scatter, and learn something about the structure of the atom. If J. J. Thomson was right, then alphas would be scattered only slightly and from the distribution of scattering angles, one can derive the density of "pudding" in 3D space. Rutherford and his assistant, Hans Geiger (of later Geiger-Muller counter fame) discovered that a small fraction of alpha particles scattered back, as if hitting something point-like and very heavy. The immediate hypothesis was that all positive charge of the atom was concentrated in one point -- the nucleus.
(Read
more about the historical backdrop of Rutherford scattering experiment.)
Rutherford's experiment
The main goal of this experiment is to confirm or disprove this hypothesis. Experimentally, our tool is the same as Rutherford's: we bombard a thin foil of gold with a beam of alpha particles, and count the rates of those that scatter in various points of space. In data analysis, our main weapon is the spatial distribution of the rate of scattered particles -- if the shape of that distribution matches the shape expected from a scattering from a point-like positive source of the electrostatic field, we have proved that Rutherford's conclusion was right.
First we need to find the equations of motion of the point-like particle
in the point-like electrostatic (or gravitational) field; for it we will
then derive the `scattering cross-section' -- the differential probability
for scattering into a given direction.
Motion of bodies in a central gravitational or electrostatic field
Since the time of Johannes Kepler's brilliant fit to Tycho Brache's
precise measurements of the motion of planets, it was known that planets'
orbits around the sun are ellipses. Newton's brilliantly proved that the
eliptical orbits are a consequence of the attractive gravitational force 
,
and went a step further and established that the motion of heavenly bodies
(such as comets) in the field of central attractive force with 
dependence (such as gravitational field of the such) is always a conical
section, depending on the initial conditions: a hyperbolla (body has sufficient
kinetic energy to avoid capture by the gravitational field), an ellipse
(the body is captured), and a parabolla (a limiting case between the two).
Exercise 1: retrace Newton's steps and prove that the
orbits of bodies in the gravitational field of an attractive force
are always conical sections.
It is vital to note that comets that enter the solar system with enough energy to avoid capture will be *scattered* by the field of the sun. And the orbit will be a hyperbolla.
We now switch to scattering in the electrostatic field: essentially
everything is the same as in the case of gravity, except that the force
can be both attractive and repulsive, the latter being the case for alpha
particles and a positively charged nucleus. Below is a sketch of
this process with the relevant kinematic variables, the impact parameter
b
and
the scattering angle 
:
Exercise 2: Prove that the equations of motions
of the alpha particle in the electric field of the nucleus, expressed in
terms of b , ,
total energy E (which is conserved since the scattering is
elastic) and charge of the nucleus e' is:
Cross section
For a beam of particles hitting a target, the cross section is defined as
The product of cross section and the incident flux tells us the total number of particles that will be scatterered by the target. The cross section is a property of the physical process itself and does not depend on the incident flux (which is an external input). For this reason the cross section is the preferred type of quantity reported by experimenters in atomic, nuclear and particle physics, and the preferred type of quantity calculated by theorists in those disciplines.
Obviously, the unit of cross section is area, so according to SI system
it must be meter-squared. However a more commonly used unit for cross-section
is barn, which
stands for 
.
(For example, the total cross-section to produce a pair of top and anti-top
quarks at Tevatron is about 5 pico-barns.)
Task 3: A uniform flux of particles is hitting
a heavy ball of radius R, and elastically bounces off the
ball's surface. Calculate the cross section of the ball using the
above definition. It will obviously be 
,
but it is nonetheless useful to go through mechanics and convince yourself
of this. The scattering cross section is just a extension of our
everyday intiuition of the cross section, generalized for objects with
`fuzzy' edges.
We now derive the cross section from .
This expression does not depend on the azimuthal angle 
,
and thus the probability for a particle to scatter into a unit of the azimuthal
angle d
is 1/2
.
The
differential cross section for scattering into a direction given by angles
and is then
So the key is to obtain 
,
which we can obtain by transforming variables from P(b) db,
which is easy to calculate since in a uniform flux the probability for
an alpha particle to have impact parameter b is 
.
We therefore get:
where in the third step we replaced db/dfrom
the above equation of motion, and in the fourth step we recognized that
sindis
a unit of solid angle. Putting everything together, we get
Here in the second step the charges of incoming particle and the target
e
and e' were respectively replaced by Ze and
Z'e,
the energy of the alpha particle was expressed as 
and we used the expression for the momentum transfer between the alpha
particle and the nucleus 
.
Thus we have obtained the well-known Rutherford formula for cross section
of elastic scattering.
Experimental setup
The setup consists of a cyllindrical hermetically closed chamber with a source and a detector, and a position for the golden foil on the axis of the chamber. Both bases of the cyllindar are transparent, and one contains the scale for the measurement of the scattering angle. The other base can be taken off when the chamber is not evacuated.
The evacuation is performed by external vacuum pump. The pump should only be ran to evacuate the chamber, and then the vent should be shut and the pump switched off. (Otherwise the engine of the pump may interfere with the electronics and create spurious noise counts.)
We use a silicon detector to detect alpha particles. An external bias voltage causes the silicon to be depleted and there is no current through it. A passage of a charged particle creates electron-hole pairs which enable a pulse of the current proportional to the particle's energy lost in the silicon. In our case the alpha particles are slow enough that they are completely stopped by the layer of silicon and thus the collected charge is proportional to their total energy.
The silicon detector is connected to a discriminator preamplifier, which can operate in analog or digital mode. In the analog mode the pulse height is proportional to the total deposited charge, and in the digital mode the length of the square-wave pulse is proportional to the total deposited charge. Since in this experiment we are only interested in counting the rate of the alpha particles, we need the square-wave signal. The potentiometer on the discriminator sets the charge threshold for the beginning of the square-wave pulse.
Finally, the digital counter is
used to count the rising edges of the sqare-wave signal. Among a
multitude of features, only two are of interest: measuring the rate (in
counts per second or per minute), and in just keeping track of the number
of counts. When the rate is reasonably high, using digital counter
to measure the rate is sufficient. However when the rate is low,
it is safer to measure the number of counts and measure the time by a stopwatch.
That way one avoids the complications of low-statistics measurements.
Data analysis
In a typical counting experiment, the raw rate (the number of counts per unit time) consists of both the `signal' -- the counts caused by the physics phenomenon under study, and the `background' -- the counts caused by other sources. Obviously, we want the true signal, and thus the background rate has to be subtracted from the raw number of counts.
In the Rutherford scattering experiment, of interest are the following
rates as a function of scattering angle :
))
-- counts caused by alpha particles which scatter on anything other than
the gold foil: e.g., air (if the chamber has not been properly
evacuated), edges of the slit, etc.; this component can be obtained by
taking data without the gold foil.)
-- counts that are not caused by alpha particles: these are most commonly
due to cosmic rays, electronics noise, etc. Note that this rate is
included in R'().,
and then R'()
would have been subtracted from R()
to obtain the true `signal'.
However, in our case it happens that R'(),
ascompared to R(),
is negligible.
Task 1: Verify that the background sources are negligible
for two values of ,
especially at large angles (say 60 and 90 degrees). After it has been proven
that R'()
is small enough that it does not interfere with the measurement, you can
neglect it in the further analysis.
Task 2: Plot R()
and then fit it with the theoretical shape above. Use
a binned likelihood fit. A ROOT example of a very similar fit has
been provided as one
of the solutions to the sample midterm. Comment on the quality
of the fit and discuss whether the data support or refute the hypothesis
of an atomic nucleus.
Note 1: in a likelihood fit, it is vital to compare apples
and apples -- in this case, we need the number of counts that have all
been normalized to the same time interval. The likelihood
fit is important to properly treat the bins with low number of events (and
customarily by `low count' one means < 10), so it is natural to assume
that everything will be normalized to the bins with the lowest number of
counts.
Thus, the measurements should be planned with this in mind. For
example, the rate measurement for the small angles should take 5-10 minutes,
for the medium angles it should take 2 hours or so, and for the very large
angles 6-12 hours. (The details are left up to you.)
Finally, in the likelihood fit a measurement of zero counts is still a
valid data point, so please take this into account when deciding how to
collect data in certain regions.
Note 2: you should also keep in mind that you can skip
certain points (`bins') in the fit; the histogram that you'll be fitting
will contain zero entries, and thus the functional shape needs to explicitly
be set to zero in the extent of these whole bins. (If you are using
ROOT, do make sure that the function is exactly zero for the whole bin
-- ROOT integrates each function accross the bin, so if the fitting PDF
is non-zero the probability would `leak' into the empty bin, resulting
in a bias. (Note that such behavior in an fitting package is necessary
to fit binned data with rapidly varying functions.)
Post Scriptum
Rutherford's scattering experiment ushered the era of scattering as a powerful tool for studying physics at short distances. The scattering is an essential techique in atomic, nuclear and particle physics of today.
Essentially the same basic idea was applied in 1968 in so-called "deep inelastic scattering" experiments by Friedman, Kendall and Taylor in which very energetic electrons were scattered off protons. The energy was sufficient to probe distances shorter than the radius of the proton, and it was discovered that all the mass and charge of the proton was concentrated in smaller components, then called "partons". Partons were later identified with quarks, and Friedman, Kendall and Taylor pocketed the Nobel Prize in 1991.
In late 1990's, the data collected at Tevatron -- currently the world's highest energy accelerator which collides protons and antiprotons at the center-of-mass energy of 2 TeV -- showed the deviation in theory analogous to the deviation exhibited by the "deep inelastic scattering". While it generated a lot of excitement among both theorists and experimentalists -- raising hope that the quarks may be made up of even smaller elementary particles -- using more precise theoretical fits yielded a deviation from theory still within the limits of a combination of experimental and theoretical errors.
Tevatron experiments are taking data once again, so the story is far from over...