CYCLOTRON PROBLEM
[PARTICLES]
[TOPICS]
In a cyclotron, particles are accelerated to energies of a few MeV. The particles are
usually positive ions.The dees are metal shells which shield out electric fields but not
magnetic ones. There is a constant magnetic field throughout the vacuum tank and an
alternating electric field in the small gap between the dees.
Here is a schematic diagram of such an arrangement:
If the field is in phase with the ions, they receive a kick from it every half cycle. These
kicks increase the kinetic energy of the particle and hence the radius of the orbit of the
ions until they exit through the window.
QUESTIONS
Q1) Explain why the dees must shield the particles from electric fields. What would
be a suitable metal to make them from ?
Q2) Why must the dees be placed inside a vacuum tank ?
Q3) Derive a formula for the frequency at which the ions circle the cyclotron
(a) If relativistic increase in mass is ignored.
(b) If relativistic increase in mass is accounted for.
Hence deduce that the answer is independent of the kinetic energy of the ions
if v << c.
Q4) Evaluate (a) for protons if the magnetic field strength B is 1.5 Tesla. Would it
matter which way the apparatus was oriented in the Earth's magnetic field ? Explain.
Q5) Explain as fully as you can why the ions are forced to spiral outwards.
Q6) For protons show that the maximum speed in a 60cm diameter cyclotron is
approximately 4.3x107 ms-1
Q7 (a) Show from first principles that the kinetic energy of a particle of rest mass mo
is given by
(b) Show that this is roughly 9.8 Mev for a proton. What fraction of its rest mass /
energy is this?
Q8)(a) If the protons redeive a kick of 105 volts each time they pass through a gap,
then calculate the number of revolutions they make before leaving through the
window.
(b)If the relativistic increase in mass is ignored then calculate the time it takes for a
proton to exit the cyclotron.
Q9)(Hard) (a) If the increase in mass is taken into account show thatif the energy
given to the particle on each cycle is DK, then the time spent inthe first half cycle is
given by the equation
(b) Hence deduce that the total time spent in the cyclotron is given by:
and evaluate this for the proton.
Q10) Comment on the significance of this correction
CYCLOTRON
The American physicist Ernest O. Lawrence won the 1939 Nobel Prize in physics for a
breakthrough in accelerator design in the early 1930s. He developed the cyclotron, the first
circular accelerator. A cyclotron is somewhat like a linac wrapped into a tight spiral. Instead of
many tubes, the machine has only two hollow vacuum chambers, called dees, that are shaped like
capital letter Ds back to back .A magnetic field, produced by a powerful electromagnet, keeps
the particles moving in a circle. Each time the charged particles pass through the gap between the
dees, they are accelerated. As the particles gain energy, they spiral out toward the edge of the
accelerator until they gain enough energy to exit the accelerator.
Q1) Explain why the particles are accelerated as the move across the gap but move at constant
radius in either dee.
Q2) Show that the maximum speed a proton could have in a dee of radius R and strength B is
given by (ignoring relativistic effects.)
vm = BeR / mp
Q3)Evaluate this for protons in a 1.20m diameter cyclotron of field strength 0.50T. Hence show
that the frequency of the alternating p.d must be 7.61 MHz.
When nuclear particles in a cyclotron gain an energy of 20 MeV or more, they become
appreciably more massive, as predicted by the theory of relativity. This tends to slow them down
and throws the acceleration pulses at the gaps between the dees out of phase. The solution, the
synchrocyclotron, is sometimes called the frequency modulated cyclotron. In this instrument,
the oscillator (radio-frequency generator) that accelerates the particles around the dees is
automatically adjusted to stay in step with the accelerated particles; as the particles gain mass,
the frequency of accelerations is lowered slightly to keep in step with them. As the maximum
energy of a synchrocyclotron increases, so must its size, for the particles must have more space
in which to spiral.
(3)BETATRON
When electrons are accelerated, they undergo a large increase in mass at a relatively low energy.
At 1 MeV energy, an electron weighs two and one-half times as much as an electron at rest.
Synchrocyclotrons cannot be adapted to make allowance for such large increases in mass.
Q4) Using the relativistic formula for mass, calculate the speed of an electron that has a mass
two and a half times its rest mass.
Therefore, another type of cyclic accelerator, the betatron, is employed to accelerate electrons.
The betatron consists of a doughnut-shaped evacuated chamber placed between the poles of an
electromagnet. The electrons are kept in a circular path by a magnetic field called a guide field.
By applying an alternating current to the electromagnet, the electromotive force induced by the
changing magnetic flux through the circular orbit accelerates the electrons. During operation,
both the guide field and the magnetic flux are varied to keep the radius of the orbit of the
electrons constant.
(4)SYNCHROTRON
The synchrotron is the most recent and most powerful member of the accelerator family. A
synchrotron consists of a tube in the shape of a large ring through which the particles travel; the
tube is surrounded by magnets that keep the particles moving through the center of the tube. The
particles enter the tube after already having been accelerated to several million electron volts.
Particles are accelerated at one or more points on the ring each time the particles make a
complete circle around the accelerator. To keep the particles in a rigid orbit, the strengths of the
magnets in the ring are increased as the particles gain energy. In a few seconds, the particles
reach energies greater than 1 GeV and are ejected, either directly into experiments or toward
targets that produce a variety of elementary particles when struck by the accelerated particles.
The synchrotron principle can be applied to either protons or electrons, although most of the
large machines are proton-synchrotrons.
Q5) Show that the radius of curvature of the path of particles of momentum p and
charge q in a synchrotron is given by the formula R = p / q B where B is the field strength.
Q6) A synchrotron of radius R has four straight sections of length L each. If the period of the
radio frequency oscillator corresponds to the time of one revolution,show that
(a) The speed of the particles must be
v = ( 2pR + 4L ) f
(b) by considering the relativistic momentum of particles of mass M , that the magneticfield
strength of the synchrotron is given by
In synchrotrons a computer is used to maintain this relation between magnetic field and oscillator
frequency.
The Strong Force
A moments thought indicates that the electrostatic repulsion due to a collection of protons in
close proximity would result in any potential nucleus flying apart. A new and fundamental force
is thus required to explain awhat holds a nucleus together against this repulsive force. The new
force is termed the Nuclear or Strong force. Although the details of this force are only crudely
understood, a potential energy diagram for two nucleons reveals much information:
The potential well is much deeper than the electrostatic repulsion which is shown for
comparison.Obviously this is only relevant for the proton-proton case. The strong force acts
equally for protons and neutrons so is charge independent. Particles on which the strong force
acts are termed
' Hadrons'. In the search for a theory of the strong interaction, Yukawa suggested the formula
R= h
2pcm
YUKAWA FORMULA
where R is the range of the force and m is the mass of the field quantum (vector boson).
Q1) (a)If force is defined as minus the rate of change of potential energy with distance or in
symbols F = - dE / dr , then sketch a force-separation curve for two nucleons , explaining clearly
your reasoning.
(b) If a deuteron is an ordinary hydrogen atom with an extra bound neutron, show that the
binding energy of the deuteronis about 2.2 MeV. Hence determine the least frequency of gamma
rays that could be used to split a deuteron into a free neutron and a proton.
Q2) (a)Calculate the mass of the field quanta of the strong force(in MeV). Look up the mass of
Pions
(Pi - Mesons) in the data pages and comment.
(b)Pions have a baryon number of zero and may be neutral(po) , negative (p-)or positively
charged (p+).If mesons consist of a quark/antiquark pair , deduce the quark structure of pions,
using quark data from the data page.
Q3) (a) If the nuclear radius R is given by the simple formula R = RoA 1/3 where A is the mass
number and Ro is a constant then show that the density of any nucleus is constant.
(b) If the value of Ro is 1.2 x 10-15 m then deduce the radius of a 12C atom and calculate its
density.Comment on this figure(the density of gold is 1.96 x 103 kgm-3.)
Q4) If the uncertainty principle is stated in the form DE.Dt > h / 2p and we assume that the
speed of pion travel is ~ c then deduce the Yukawa formula m = h / 2pRc . This will only apply
for an event in which energy DE is NOT conserved if the duration of the event is less than h /
2pDE
Clearly the pions are field quanta for the strong force so we can draw Feynman diagrams for
field interactions:
There are thus two types of field quanta associated with the strong force:
1. Gluons ,which hold quarks together to form nucleons - the quarks exchange gluons .They
are massless and travel at light speed - the theory of gluons requires that quarks have another
property called ' colour' as well as flavour, the exchange of gluons resulting in a change of quark
'colour'.
2. Pi - Mesons , which nucleons exchange to form bound atomic nucleii.These account for
hadron-hadron interactions and are (as shown above) , massiveand hence travel at sub-light
speed. The associated transfer of momentum as the pions are exchanged constitutes a force.
The Weak Interaction
The strong interaction is unable to account for beta decay. As far
as the structure of matter is concerned , another short range
interaction is responsible for the decay of nuclei that have topheavy neutron/proton ratios.This is termed the Weak interaction.
This force also affects non-nuclear particles, such as electrons
and neutrinos.
Q5) Use the Yukawa formula to estimate the mass of the field
quanta of the weak interaction if its range is 1 x 10-17 m. Give your answer in electronvolts.
Q6) Name the three field quanta for the weak force . They are collectively termed ' Intermediate
Vector Bosons'.
Q7) Draw a Feynman diagram for proton decay at the quark level if it is given by the process
p -> n + e+ + n.
NEUTRINOS
Neutrinos and antineutrinos are produced during weak interactions. Suppose a nucleus decays
into another nucleus by emitting an electron: A --> B + eWe apply the conservation of energy and momentum to this process, and show that energy does
not appear to be conserved:
Q7(a) If the relativistic increase in mass of a particle is given by
then write down formulae for (i) Total relativistic energy , (ii) Relativistic momentum. Hence
show that the relativistic equation for the energy of any particle in the weak interaction is given
by the equation:
.
where E is total energy, p is particle momentum and mo is the rest mass of the particle.
If the nucleus is at rest before the disintegration then the momentum of each part afterwards will
be equal and opposite.If we denote this as p, then by conservation of energy:
.
Q8) Explain the significance of each term in this equation.
Q9) Show that all electrons emitted by stationary A nuclei should have the same energy given
by:
a