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.

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 $m_e = 9,11 \cdot 10^{-31}\ \text{kg}$, $c = 3 \cdot 10^{8}\ \text{m/s}$ and $1\ \text{eV} = 1,6 \cdot 10^{-19}\ \text{J}$.

1. a) Calculate the total energy released in the annihilation of an electron and a positron (which has the same mass) applying $E = m \cdot c^2$.
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.

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.

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.

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).

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.

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

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.

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.

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.

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.