PHY138Y Nuclear & Radiation Section. Suggested Extra Problems: © A.W. Key
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PHY138Y – Physics for the Life Sciences
Nuclear and Radiation Section
Suggested Extra Problems
(for discussion in tutorials)
By popular demand, I offer these additional problems for your enjoyment. If you can
attempt them before your tutorial, you will have a better chance of dealing with
difficulties. I will not post solutions, although occasionally I may attach an answer; ask
your tutor if you need help in solving them. Care! These problems have not been
exhaustively proofread, so be on the lookout for errors or misprints; if a question is not
clear, or you think an answer may be wrong, check with your tutor.
For Supplementary Notes I
1. Fermi Problem. Do you think that there are more or less atoms in a grain of sand than
there are grains of sand in a typical beach? Check your guess. (Question phoned in to the
CBC Radio Programme, Quirks and Quarks, Saturdays, 99.1FM).
2. Size of Nucleus. If your head represents the size of the nucleus, how far East do you
have to go to encounter the electron in the first Bohr orbit? (You may need to consult a
map!).
3. Number of Nucleons in Body. Approximately how many nucleons do you have in your
body? State your assumptions clearly.
4. The Bohr Model. Using classical mechanics and Bohr’s hypothesis of quantized
angular momentum for the electron in its orbit, show that the energy of the nth circular
electronic orbit is inversely proportional to n2.
Problems from Knight
Chapter 24 – 1,2,3,8,9,10, 19(a,b), 20(a,b),21,22,23,24
Chapter 42 – 12,13,14, 42
For Supplementary Notes II
1. A Conceptual Question. Prove that there must be at least two gamma rays emitted
from the annihilation of a free electron with a positron (Hint: ‘free’ means that the
electron is not bound in an atom; go to the frame in which the momentum is zero).
2. Another. Electrons cannot produce bremsstrahlung radiation in a vacuum. Why not?
3. The Heating of Radiation. Radiation delivers energy to the tissue it penetrates. It
might be thought that the damage it causes could be caused by heating. To test this
hypothesis, consider the following. An X-ray dose of 2 Gy can be lethal. If the
equivalent energy were absorbed as heat, what would be the rise in body temperature?
(The specific heat of tissue may be assumed to be that of water: 4200 J.kg-1.K-1 ) (Ans. 5
x 10-4K)
4. Compton Scattering. An X-ray photon of wavelength 4.9 pm enters tissue and
undergoes a Compton scattering. The scattered photon has a wavelength of 6.4 pm.
What is the energy of the electron produced in this process? (59 keV)
5. X-ray Image Density. A measure of the brightness of images on exposed film is the
'optical density'; an optical density of n means that 10-n of light incident on the film is
transmitted. In order that a diagnostician can see reasonable images on an X-ray film, we
might demand an optical density of 1 - meaning that 10% of the incident light is
transmitted - when the X-rays traverse body tissue. (Of course, even less of the light will
be transmitted when the X-rays traverse muscle and bone). In order for X-ray film to
PHY138Y Nuclear & Radiation Section. Suggested Extra Problems: © A.W. Key
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yield this density, an exposure of approximately 20 mR must be delivered to the film
during exposure. A) For 60 keV X-rays, what is the minimum exposure (in R) needed to
reach the required optical density of the exposed film? (Obtain the required value of the
attenuation coefficient from the table in SNII. Take the width of an 'average' person to be
25cm. You will also need to make a fairly obvious assumption about the reduction of the
exposure as the X-rays traverse the bodily tissue.) B) What would be the required
minimum exposure for 40 keV X-rays? C) Without doing the calculation, at which
energy would you expect the absorbed dose to the patient to be greater? (Ans. 2.15R,
5.84R, 40 keV).
Problems from Knight.
Chapter 24 – 21, 22, 23, 24 (the last will be useful for later problems).
For Supplementary Notes III
1. Radioactive dating. A piece of charcoal of mass 25.0g is found in the ruins of an
ancient city. The sample shows a 14C activity of 250 decays per minute. How long has
the tree from which the charcoal came been dead? T½ = 5730 years. The ratio of 14C to
12
C atoms in the atmosphere is 1.30*10-12 (Ans. 3.33*103 years)
2. Radioactive Decay. A dose of 99mTc serum for a lung scan had an activity of 180 MBq
in a volume of 3.5 ml when it was prepared at 11:30h. If you wished to inject 23 MBq
from this dose into a patient at 16:30h, what volume would you administer? (The half-life
of the radiopharmaceutical is 6 hours.)
3. Radioactivity in the Body. Before nuclear weapons tests were outlawed, people had
absorbed radioactive 137Cs resulting in an activity of about 40 Bq. What mass of 137Cs
does this correspond to?
4. Alpha Decay. Thorium 232 decays by emitting an alpha particle. A) What is the
nucleus that remains after the decay (the daughter nucleus)? B) Calculate the energy, in
MeV, released in this decay (the disintegration energy). C) If you can neglect the recoil
of the daughter nucleus, what is the energy of the alpha particle?
5. Relativity. Can we use Newtonian mechanics for alpha particles? If we define a nonrelativistic particle (i.e. we do not need to use relativistic calculations) as one for which
the ratio of its speed to that of the speed of light is less than 0.1, is a 5 MeV particle nonrelativistic?
6. Correct Calculation of Alpha Energy. A parent nucleus, P, decays to a daughter, D,
and an alpha particle. Show that the kinetic energy of the alpha particle is given by
QmD/(mD+ mα), where mD is the mass of the daughter, mα is the alpha mass, and Q is the
disintegration energy of the decay.
7. Radioactive Decay Formula. Show that the number of radioactive nuclei at time t,
N(t) can be written as N(t) = N(0)( ½ )t/τ where τ is the half-life. (See Knight, equation
42.19)
Problems from Knight
Chapter 42: Most problems in this chapter are relevant. I suggest 6,14,26,30,59.
For Supplementary Notes IV
1. Equivalent dose. An X-ray beam has an intensity of 0.4 W.m-2. 0.072 m2 of a patient’s
chest is exposed for 0.3 s. The radiation is totally absorbed by 3.5 kg of tissue. If RBE is
1.1 determine the biologically equivalent dose received by the patient (0.27 rem).
2. Dose from Partner. Suppose you spend a night sleeping next to another human being.
Estimate the radiation dose you receive and compare it to the average environmental
load. (Hints: use the values given in SNIV. Which isotopes might be important?
PHY138Y Nuclear & Radiation Section. Suggested Extra Problems: © A.W. Key
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Calculate the activity in pCi for an assumed weight of your bed partner; guess their halfwidth, density, and average mass energy coefficient. Calculate the amount of energy in
the radiation that escapes from their body. Is absorption in the air between you
important? Guesstimate the % of the total emitted from your partner that hits you. Make
an assumption about how much of this energy is deposited in you. Thus calculate the
effective dose).
3. Dental X-Rays. An X-ray tube used for dental diagnosis, operating at a maximum tube
voltage of 60 keV (i.e. average energy of about 20 keV) delivers 2.0 R to a 2x3cm2 area
of a patient’s skin. The mass energy absorption coefficient of air at this energy is 0.148
cm2.g-1. A) calculate the dose in air. B) calculate the energy fluence in air. The tooth to
be examined is located in the patient’s mouth (where else?), 1.2 cm distant from the point
of entry of the X-rays. C) If the mass energy absorption coefficient of the tissue between
the point of entry and the tooth is 0.150 cm2.g-1 and its density is 0.96 g.cm-3, what is the
energy fluence at the location of the tooth? D) If the mass energy absorption coefficient
of the tooth is 2.51 cm2.g-1, what is the absorbed dose in the tooth, if its depth is 0.5 cm.,
and its density is 1.85 g.cm-3? E) What is the equivalent dose absorbed by the tooth? F)
Do you think this exposure is a threat to the patient’s health? Make clear your argument,
and state any assumptions you make.
4. Dose from Gamma Rays. You are 3.5 metres from a source of 1.5 MeV gamma
radiation whose strength is 3 Ci. Neglecting absorption in the air, A) what is the energy
fluence rate at your body? B) What is the energy fluence rate emerging from your body?
Assume that the mass energy absorption coefficient for your body at this gamma ray
energy is 0.021 cm2g-1. Your average density can be assumed to be 1.0 g.cm-3 (on a good
day!). C) What, approximately, is the dose rate to your body? (you will need to estimate
your surface area). D) if you stand in this position for a full minute, what is the dose you
receive? E) What is the equivalent dose? F) What is the effective dose?
Problems from Knight
Knight chap 42 – 35 to 38.
For Supplementary Notes V
1. Committed Dose from a Tracer. 20 mCi of 99mTc is administered to a patient. The
uptake in the liver, which weighs 1.8 kg, is known to be 70%. The average energy of the
gamma rays from the decay of the 99mTc is 140 keV and the nuclear half life of 99mTc is
6.0 hours. The biological half life in the liver is 6.0 days, and about 25% of the gamma
ray energy is deposited in the liver. A) Calculate the absorbed dose to the liver in mGy.
B) What is the equivalent dose received by the liver? C) What is the effective dose?
2. Effective Life – an analytic problem. A radioisotope, whose nuclear decay constant is
λn , is implanted into an organ and removed after a time T. The radioisotope has an
effective half-life in the organ of λeff. Let ND(T) be the number of radioactive nuclei that
have disappeared in time T. Some of these have decayed in the organ, contributing to the
dose, the rest have been excreted. A) What fraction, f, of these nuclei has decayed in the
organ, in terms of λn and λeff ? B) Write down the fraction, (1-f) that has been excreted
in this time.
3. HDR Treatment of Prostate Cancer. The prostate gland is a male organ about the size
of a walnut, with a mass of approximately 60 grams, located under the bladder. Cancer
of the prostate is the third most deadly form of male cancer in Canada (after lung and
colorectal cancer). It is sometimes treated by inserting needles containing Iridium-192
into the prostate for short times. 192Ir has a half-life of 74 days, and delivers gamma rays
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PHY138Y Nuclear & Radiation Section. Suggested Extra Problems: © A.W. Key
of average energy around 400 keV to the prostate. If a dose of 10 Gy is required, how
long must the 192Ir remain in the prostate if its activity is 10 Ci ?
4. Isotopic Dilution. Dilution techniques measure the size of what is called the
‘exchangeable pool’; this is that part of the material being measured that can exchange
with the environment through ingestion and excretion. 5 ml of chloride, containing 0.16
MBq of 38Cl, is injected into the body. 74 minutes later a sample of blood is found to
contain 3.2 mg of stable chlorine, and 1.1 Bq of 38Cl. Assuming thorough mixing,
calculate the amount of exchangeable chlorine. The half-life of 38Cl is 37 minutes. (Ans.
116g) .
Problems from Knight
Knight chap 42 - 59, 61, 66 (this is a good one, interesting and practical)
For Supplementary Notes VI
1. MRI – Larmor Frequencies. What is the energy of the photon that will be absorbed by
a 1H nucleus in a 1.5 Tesla magnetic field? How does this compare in energy to a 2x1019
Hz x-ray photon? What, approximately, is the ionization potential for a typical organic
molecule? Which of the two photons will ionize the molecule?
2. MRI – Relaxation Times. A sample has a T1 of 1.0 seconds. A)If the net
magnetization is set equal to zero, how long will it take for the net magnetization to
recover to 98% of its equilibrium value? B) A sample has a T2 of 100 ms. How long will
it take for any transverse magnetization to decay to 37% of its starting value?
S
Z
N
x
S
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G
N
A
L
3. MRI Frequency Spectrum (Exam 2006). A laboratory
sample used for MRI research, containing three capsules of
water (indicated by the three black dots in the diagram), is
placed in a magnetic field. The magnetic field is constant
and uniform in the +z direction; Bz (z) = B0. The field has a
gradient in the +x direction; it decreases in the +x direction
according to Bz(x) = B0 - xG, where G is a constant greater
than zero, as shown in the diagram to the left. The graphs
below show the result of the measurement of the Larmor
frequencies of the protons in each capsule. The values of
frequency are plotted along the x-axis, and the strength of the
signal is plotted on the y-axis. Which of the graphs most
closely approximates the frequency spectrum you would
expect to see?
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G
N
A
L
frequency
(A)
Problems from Knight
Knight chap 33 – 59
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G
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frequency
(B)
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G
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frequency
(C)
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frequency
(D)
frequency
(E)