Relativistic Momentum: TRIUMF’s “Approaching the Speed of Light”
The story so far: In 1909 Rutherford developed the nuclear model of the atom by considering the results of
firing fast moving alpha particles at a gold foil. These alpha particles were accelerated naturally by
radioactive decay. However, when physicists want to probe inside the nucleus, they need faster particles.
In 1932 Ernest Lawrence built the first cyclotron to accelerate charged particles. The TRIUMF cyclotron in
Vancouver is the world’s largest cyclotron. In this exercise you will be looking at the particles that are
produced when protons accelerated by the cyclotron smash into a carbon nucleus.
1) What particles can be found in a nucleus?
a) protons, neutrons
b) protons, pions
c) neutrons, pions
d) all three
2) How many different elementary particles were produced when a proton smashes into a carbon nucleus
and some decay occurs?
a) 3
b) 6
c) 7
d) 9
3) Describe with words, diagrams and equations how the speeds of the particles were measured.
4) Describe with words, diagrams and equations how the momenta of thee particles were measured.
5) There are three peaks in the histogram for the time to travel 4.40 m. Why?
6) One histogram is shown below, two times. On the graph on the left, sketch what you would expect for
particles with twice the momentum and explain your reasoning. On the graph on the right, show what is
found and explain why this occurs.
Explain Predictions:
Explain Observations:
7) Sketch what you expect the results of graphing momentum vs. speed to look like and explain why.
p
v
8) The distance travelled was 4.40 m. Use your histogram to get the time and calculate the speed for each of
the three particles in units of c, the speed of light. Show your steps below.
a) Electron
b) Muon
c) Pion
9) The data forms a curve.
a) We’d like to graph the data so that it forms a straight line. What will straighten it? Try it.
b) What is the physical meaning of the slope of the straight line graph? Test it. What was your
percentage error? What were the sources of error?
c) How do you explain the curve and the new equation for momentum?
Relativistic Momentum: TRIUMF’s “Approaching the Speed of Light”: Teachers Guide
Assumed Preparation: This lesson assumes that the student is familiar with classical momentum, the force of
a constant magnetic field on a charge (F = qvB) and circular motion (a = v2/r). It does not require any
knowledge of special relativity.
Subversive Physics: This activity allows the students to do an experiment that uses special relativity. It also
sneaks in the fact that the nucleus is a bit more complicated than a collection of protons and neutrons and that
there are other ‘elementary’ particles such as positrons, pions and muons.
The Activity: The lesson is a resource from TRIUMF (Tri-University Meson Facility) in Vancouver, Canada.
It consists of a short video the gives you a brief description of TRIUMF, the experiment and an introduction to
pions and muons. View the video from 1.50 until 9.11 minutes. (The video after this part gives away too much
information that the students should be figuring out for themselves.) Data are provided for twenty different
momenta. The DVD contains a different worksheet that you might prefer and extra background information. It
also suggests how you could explore the particle physics a bit farther.
More Information: For more information about this resource and others involving modern physics go to
http://roberta.tevlin.ca. If you have any questions or suggestions email roberta@tevlin.ca
1) What particles can be found in a nucleus?
When a pion is exchanged, a proton may turn into a neutron or vice-versa, but the animation doesn’t
show this. The pions are called virtual particles because their creation violates the law of conservation of
energy. However, they do it so fast that the Heisenberg Uncertainty Principle prevents this from being
detected before they annihilate. A more fundamental description of the nucleus is one where the protons
and neutrons are not elementary but consist of up and down quarks. The proton is made of uud, the
neutron of udd and the pions consist of quark and antiquark pairs.
2) How many different elementary particles are produced when a proton smashes into a carbon nucleus and
some decay occurs?
There are three types of particles; electrons, muons and pions. However, the electrons and pions also
have antiparticles with opposite charge. (The antielectrons are often called positrons.) This brings the
count to five. Then there are three types of pions. The positive pion consists of an up and an antidown
quark. The negative pion consists of a down and an antidown quark. The neutral pion consists of an an up
and an antiup quark that that may be a down and an antidown quark. That brings the count to seven.
3) Describe with words, diagrams and equations how the speeds of the particles are measured.
v = 4.40/t
Plastic scintillators start and stop an electronic timer.
4) Describe with words, diagrams and equations how the momenta of thee particles are measured.
F = qvB and F = ma = mv2/r. Therefore mv = qBr = p. Only particles that curve by the right amount will
make it into the beam line. The magnetic field is increased to select particles of higher momentum.
5) There are three peaks in the histogram for the time to travel 4.40 m. Why?
The shortest time is for particles going fastest. All three types have the same momentum, so these must be
the lightest particles – the electrons. The middle peak is for the muons and the third peak is for the pions.
6) The histogram for one momentum is shown below. On the graph on the left, sketch what you would
expect for particles with twice the momentum and explain your reasoning. On the graph on the right,
show what occurs and explain why this occurs.
Twice the momentum suggests twice the speed, so you would expect peaks at half the time.
The time for the pion changes the most, but it is only reduced to two-thirds, not one half. The electron’s
time does not appear to change at all. This means that p = mv does not hold at really high speeds.
There a way fewer electrons and way more pions. At higher speeds, fewer of the pions have decayed into
muons and then electrons. This is for two reasons. They take less time getting from the carbon target to
the beam line. As well, there time is slowed relative to ours and so their half-life appears to increase.
7) Sketch what you expect the results of graphing momentum vs. speed to look like and explain why.
p
v
Now that we realize that p = mv does not hold at these speeds, we should
not expect a diagonal line with a slope equal to the mass. Many students
will realize that the graph will be showing speeds approaching c and they
will also know that the particles can’t go at that speed and will show an
asymptote there.
8) The distance travelled was 4.40 m. Use your histograms to get the time and calculate the speed for each
of the three particles in units of c, the speed of light. Show your steps below.
Find the time for the peak to the nearest tenth of a nanosecond. The speed in units of c will be
v = 4.40/0.3*t. (The speed of light times the conversion to seconds is 3 x 108 x 10-9 = 0.3.)
9) The graph forms a curve.
a) We’d like to graph the data so that it forms a straight line. What will straighten it? Try it.
Students will probably spend a lot of frustrated effort trying different powers of v. Eventually you may
have to nudge them and remind them of gamma. A graph of p vs. mv will yield a straight line.
b) What is the physical meaning of the slope of the straight line graph? Test it. What was your
percentage error? What were the likely sources of error?
The slope will be the mass for the particular particle. The momentum selection results in particles
with the same momentum, plus or minus some amount. That is why the histograms do not show three
simple columns, but show some significant spread. The masses of the three particles in MeV/c2 are
0.511, 106 and 140. You can’t use the electron’s data because the error is so great. You are not able to
measure the small increase in speed. The error for the pion will probably be less than for the muon.
c) How do you explain the curve and the new equation for momentum?
The new equation is p = mv. This is often explained by saying that the mass increases with speed,
i.e. m = mrest. What we think of as a particle’s mass is just its rest-mass. However, this interpretation
has some problems. The mass only increases in the direction of motion. How can mass increase in one
direction and not the others? Most particle physicists prefer to say that momentum is changed not
mass. Either way, the math is the same.