Exercises: Technological and medical applications

In diagnostic nuclear medicine, tiny amounts of radiopharmaceuticals—substances labelled with radioactive isotopes that accumulate in the organ under study—are administered to the patient.

1. a) Explain what advantage is offered by using a radioactive isotope to study the functioning of an organ from outside the body.
2. b) Reason why it is desirable for the radiopharmaceutical to have a half-life of hours rather than years.

Radiotherapy uses ionising radiation to destroy tumours.

1. a) Explain why ionising radiation mainly damages rapidly dividing cells, such as cancerous ones.
2. b) State the difference between the radiation dose used in a diagnostic test and that used in radiotherapy.
3. c) Mention a logical precaution to protect the healthy tissues surrounding the tumour during treatment.

A nuclear fission power plant is essentially a large boiler: the heat generated in the reactor heats water whose steam drives turbines coupled to an electrical generator.

1. a) State which physical process provides the heat in the reactor.
2. b) Explain the function of the moderator (water or graphite) surrounding the fuel rods.
3. c) Describe the energy transformation chain from the uranium nucleus to the electrical grid.

In a reactor, each uranium-235 fission releases between two and three neutrons, which can induce further fissions in neighbouring nuclei, as shown in the figure.

1. a) Explain what a chain reaction is.
2. b) State the function of the cadmium or boron control rods inserted between the fuel.
3. c) Reason what would happen if too many rods were removed and the number of neutrons grew without limit.

The international ITER project, under construction in the south of France, aims to demonstrate that nuclear fusion can be a viable source of electricity.

1. a) State which two isotopes of hydrogen are intended to be fused in ITER.
2. b) Point out two advantages of fusion over fission as an energy source.
3. c) Explain why a fusion reactor cannot undergo an uncontrolled chain reaction like that of a fission reactor.

Radioactive waste is classified according to its activity and the time it takes to lose it.

1. a) Give an example of low-level waste from a hospital and another of high-level waste from a nuclear power plant.
2. b) Explain why high-level waste requires monitored storage for a much longer time.
3. c) Reason why waste is "confined" rather than destroyed: can the radioactivity of a material be "switched off" or "burnt"?

The biological effect of radiation is measured in sieverts ($\text{Sv}$), and we continuously receive natural doses: from radon in the ground, cosmic rays, food, and even our own body.

1. a) Order these typical effective doses from lowest to highest: chest X-ray ($0.1\ \text{mSv}$), transatlantic flight ($0.1\ \text{mSv}$), and natural background received in a year ($2.4\ \text{mSv}$).
2. b) Reason why aircraft crews receive a higher annual dose than the population living at ground level.

Radioactive sources have many applications outside medicine and electricity generation.

1. a) In pipeline construction, welds are "X-rayed" using gamma sources: explain the principle behind this quality control technique.
2. b) In the paper industry, beta sources are used to monitor sheet thickness continuously: explain how this method works.
3. c) Disposable surgical equipment is sterilised using gamma radiation: state why this procedure is preferred over heat.

Technetium-99m (half-life $6\ \text{h}$) cannot be transported over long distances, so hospitals obtain it on site from a molybdenum-99 "generator" (half-life $66\ \text{h}$), which decays to produce it.

1. a) If a sample of technetium-99m with an activity of $1000\ \text{MBq}$ is extracted from the generator, calculate the activity available $12\ \text{h}$ later.
2. b) Calculate what fraction of the initial activity remains after $18\ \text{h}$.
3. c) Reason why the hospital cannot order technetium-99m from a distant factory, but can receive a molybdenum-99 generator every week.

In a PET scan, fluorine-18 is administered, which emits positrons. Each positron annihilates with an electron in the body, and the mass of both is converted into two identical gamma photons travelling in opposite directions, as shown in the figure. Data: electron mass $m_e = 9.11 \times 10^{-31}\ \text{kg}$, $c = 3 \times 10^{8}\ \text{m/s}$ and $1\ \text{eV} = 1.6 \times 10^{-19}\ \text{J}$.

1. a) Calculate the total energy released in the annihilation of an electron and a positron (which has the same mass) applying $E = m \times c^2$.
2. b) Calculate the energy of each photon and express it in $\text{keV}$.
3. c) Explain why the detectors in the scanner ring register both photons almost simultaneously and what information this coincidence provides about the point where the annihilation occurred.

A nuclear reactor delivers a thermal power output of $3000\ \text{MW}$. Each fission of uranium-235 releases about $200\ \text{MeV}$. Data: $1\ \text{eV} = 1.6 \times 10^{-19}\ \text{J}$ and $1\ \text{MW} = 10^{6}\ \text{W}$.

1. a) Express the energy released in one fission in joules.
2. b) Calculate how many fissions must occur each second to sustain this thermal power.
3. c) If the efficiency of the power plant in converting heat into electricity is $33\ \%$, calculate the electrical power delivered to the grid.

In brachytherapy, radioactive sources are placed next to or even inside the tumour. For certain prostate tumours, permanent iodine-125 "seeds" with a half-life of $59$ days are implanted.

1. a) Explain what advantage irradiating the tumour from the inside has compared to irradiating it from the outside with an external beam.
2. b) Calculate what fraction of the initial activity of the seeds remains after $236$ days.
3. c) Reason why it is desirable for the implanted isotope to have a half-life of weeks or months rather than years.

Inside ITER, the deuterium-tritium plasma must reach around $1.5 \times 10^{8}\ \text{K}$, about ten times the temperature at the centre of the Sun. Observe the diagram of the tokamak:

1. a) Explain why no material can serve as a "vessel" touching the plasma, and what the magnetic confinement solution in tokamak-type reactors consists of.
2. b) Reason why fusion is stably sustained in the Sun despite a much lower core temperature, considering its enormous mass.
3. c) The goal of ITER is to produce ten times more fusion power than is injected to heat the plasma: explain what this goal means and why it would represent a historic milestone.

A point gamma source emits in all directions, so the dose received decreases with the square of the distance. At $1\ \text{m}$ from a certain industrial source, the dose rate is $8\ \mu\text{Sv/h}$. The figure shows this relationship.

1. a) Calculate the dose rate at $2\ \text{m}$ and at $4\ \text{m}$ from the source.
2. b) Calculate at what distance the rate is reduced to one hundredth of the value measured at $1\ \text{m}$.
3. c) Radiation protection is summarised in three rules: time, distance, and shielding. Relate each of these to the previous results and explain how they reduce the dose received.

Radon-222 is a naturally occurring radioactive gas, arising from the decay chain of uranium-238 present in granitic soils, which seeps into homes and accumulates in poorly ventilated basements. Its half-life is $3.8$ days.

1. a) Explain why, even though each radon atom disappears within a few days, the radon in a house built on granitic soil is never "depleted".
2. b) If air containing a certain amount of radon is sealed tightly, calculate what fraction remains after $15.2$ days.
3. c) State a simple measure to reduce the concentration of radon in a home and justify it physically.

In some processing plants, spices and other foodstuffs are irradiated with gamma radiation from cobalt-60 to eliminate bacteria, moulds, and insects.

1. a) Explain how gamma radiation manages to eliminate microorganisms.
2. b) Reason why irradiated food does not become radioactive, considering that at the energies used, the radiation does not alter the atomic nuclei of the food.
3. c) When the source is not in use, it is submerged in a water pool or kept behind thick concrete walls: justify this measure using the penetration properties of gamma radiation.

In Ireland, nuclear power plants supply around $20\ \%$ of the electricity with virtually zero $\text{CO}_2$ emissions, but they generate long-lived waste and their phase-out is planned.

1. a) Present two technical arguments in favour of retaining nuclear power plants within the energy mix.
2. b) Present two technical arguments against retaining them.
3. c) Argue, using physics criteria (emissions, waste, security of supply, risks), which position seems most reasonable to you: scheduled phase-out, retention, or long-term commitment to fusion.

A nuclear power plant with an electrical power output of $1\ \text{GW}$ has an efficiency of $33\ \%$. In each fission of uranium-235, $200\ \text{MeV}$ are released. Data: $1\ \text{eV} = 1.6 \times 10^{-19}\ \text{J}$, $N_A = 6.02 \times 10^{23}\ \text{mol}^{-1}$, molar mass of uranium-235: $235\ \text{g/mol}$ and one year $\approx 3.15 \times 10^{7}\ \text{s}$.

1. a) Calculate the thermal power of the reactor.
2. b) Calculate the number of fissions occurring each second and the thermal energy generated in a year.
3. c) Calculate the mass of uranium-235 that undergoes fission in that year, express it in tonnes, and comment on the result by comparing it with the millions of tonnes of coal that an equivalent thermal power plant would burn.

The deuterium-tritium reaction releases $17.6\ \text{MeV}$ for each pair of fused nuclei. Data: $1\ \text{eV} = 1.6 \times 10^{-19}\ \text{J}$, $N_A = 6.02 \times 10^{23}\ \text{mol}^{-1}$, molar masses: deuterium $2\ \text{g/mol}$ and tritium $3\ \text{g/mol}$, and energy content of coal: $3 \times 10^{7}\ \text{J/kg}$.

1. a) Calculate the energy released in each reaction, in joules.
2. b) Calculate the energy released when $5\ \text{g}$ of deuterium-tritium mixture (one mole of pairs) undergoes complete fusion.
3. c) Calculate how many kilograms of coal would be required to obtain the same energy, and draw a conclusion about the energy density of fusion fuel.
4. d) Despite this advantage, commercial fusion power plants do not yet exist: mention two technical difficulties that account for this.

To treat thyroid cancer, iodine-131 (half-life $8$ days) is administered to the patient, which the body naturally concentrates in that organ.

1. a) Explain why iodine, whether radioactive or not, accumulates specifically in the thyroid, and what advantage this provides for treatment.
2. b) Calculate how many days must elapse for the activity of the administered iodine-131 to drop to $25\ \%$ and to $12.5\ \%$ of its initial value.
3. c) Following treatment, the patient must sleep alone and maintain distance from others for a few days: relate these guidelines to the results in the previous section and reason why the same isotope can be used both to diagnose (small doses) and to treat (large doses).

The Voyager probes, launched in $1977$, are too far from the Sun to use solar panels: they are powered by a radioisotope thermoelectric generator (RTG) containing plutonium-238, with a half-life of $87.7$ years.

1. a) Reason why a long half-life is essential to power a mission lasting decades.
2. b) Suppose the instruments require at least $25\ \%$ of the initial plutonium power: calculate how many half-lives the generator can last and how many years this corresponds to.
3. c) In practice, the Voyagers lose power primarily due to the ageing of their thermocouples, not because of plutonium decay: explain what this means and why the half-life only sets a theoretical limit on duration.

Following the accidents at Chernobyl ($1986$) and Fukushima ($2011$), the media spoke of a "nuclear explosion", but a power reactor cannot explode like an atomic bomb.

1. a) The fuel in power plants is enriched to only between $3\ \%$ and $5\ \%$ of uranium-235, whereas a bomb requires more than $90\ \%$: explain, using the concept of a chain reaction, why with $4\ \%$ enrichment the reaction cannot "run away" as in a bomb.
2. b) What, then, exploded at Chernobyl and Fukushima? Distinguish the two cases: at Chernobyl, relate it to the sudden power surge in the reactor during a test and the abrupt generation of steam; at Fukushima, to the decay heat still generated by fission products after shutting down the reactor, and to the accumulation of steam and hydrogen.
3. c) Explain what the defence-in-depth barriers (fuel cladding, reactor pressure vessel, and containment building) consist of and what function each one serves.

Two fission products abundant in waste are strontium-90 and caesium-137, both with a half-life of about $30$ years; plutonium-239, on the other hand, has a half-life of $24000$ years.

1. a) Calculate how many half-lives must pass for a sample to drop below $0.1\ \%$ of its initial activity (hint: $\left(\frac{1}{2}\right)^{10} = \frac{1}{1024}$).
2. b) Calculate the time this represents for caesium-137.
3. c) Make the same estimate for plutonium-239 and reason why deep geological disposal and research into transmutation techniques are proposed for long-lived waste.

Typical effective doses: chest X-ray, $0.1\ \text{mSv}$; abdominal CT scan, $10\ \text{mSv}$; transatlantic flight, $0.1\ \text{mSv}$; natural background received in one year, $2.4\ \text{mSv}$.

1. a) Calculate how many chest X-rays an abdominal CT scan is equivalent to in terms of dose.
2. b) Calculate how many years of natural background radiation this same CT scan corresponds to.
3. c) The ALARA principle ("as low as reasonably achievable") guides the practice of nuclear medicine and radiology: use it to argue when performing a CT scan is justified despite its dose, and what precautions should be required.

Unit challenge. How does a hospital manage the supply of radiopharmaceuticals with short half-lives? Act as a hospital medical physicist and analyse the logistics of $^{99\text{m}}\text{Tc}$ ($t_{1/2} = 6.0\text{ h}$):

1. Home experiment: Simulate exponential decay by tossing $N_0 = 80$ coins (or marked lentils). At each step $t$ (tosses 1 to 6), remove the decayed ones (heads). Record the surviving nuclei $N$ in a table with their uncertainty ($N \pm 1$), plot the graph $N(t)$, and estimate the decay constant $\lambda$.

2. Nuclear stability: Check the website of the Environmental Protection Agency (epa.ie) to find out how $^{99\text{m}}\text{Tc}$ is obtained from $^{99}\text{Mo}$. Justify the $\beta^-$ emission process by analysing the $N/Z$ ratio, the mass defect, and the binding energy per nucleon that determine its stability.

3. Technical proposal: Prepare a brief report: calculate the initial activity that must be prepared at $08:00\text{ h}$ to inject a diagnostic dose of $150\text{ MBq}$ at $14:00\text{ h}$, and evaluate the radiation protection protocols and the management of the waste generated.

Evaluation criteria:

• Rigour in processing home experimental data, table with uncertainties, and graphical fitting.

• Conceptual accuracy in explaining nuclear stability, mass defect, and radioactive processes.

• Feasibility of activity calculations and coherence in assessing the medical use of radioisotopes.