Chapter 8: Quantum and nuclear physics

The chapter problem: the hospital tracer

A hospital prepares a radioactive tracer for a scan. The physics team checks the tracer and the equipment before the patient arrives.

These are the facts to use all the way through the chapter.

• The tracer contains fluorine-18. Its half-life is 110 minutes, and it decays by emitting a positron.
• At the start of the morning the tracer has an activity of 8.0 × 10⁸ Bq.
• The scan needs an activity of at least 2.0 × 10⁸ Bq when it starts.
• Each positron meets an electron in the body. The two particles annihilate, and their mass becomes two gamma-ray photons. Each photon has an energy of 511 keV.
• The gamma detector has a photocathode with a work function of 2.00 eV. The team tests it with ultraviolet light of wavelength 250 nm.
• The team also calibrates a teaching electron gun in the training laboratory. It accelerates electrons through a potential difference of 150 V.
• The Planck constant is $6.63 \times 10^{-34}\,\text{J}\,\text{s}$. The speed of light is $3.00 \times 10^{8}\,\text{m}\,\text{s}^{-1}$. The charge of the electron is $1.60 \times 10^{-19}\,\text{C}$. The rest energy of an electron is 0.511 MeV.

By the end of the chapter you will work out the energy and frequency of the gamma photons and check that the ultraviolet light releases electrons from the detector. You will find the de Broglie wavelength of the electrons in the gun. You will follow the activity of the tracer through the morning, and find the energy released when a positron annihilates with an electron. At the end you will decide whether the tracer is still strong enough to use, and explain your reasoning.

First contact: no working yet. Predict. The activity of the tracer halves every 110 minutes. The morning lasts 330 minutes. Will the activity have fallen to about half, about an eighth, or about a thousandth of its starting value? Write down your answer and one sentence of reasoning.

By the end of this chapter you will be able to

• describe light as a stream of photons, and use $E = hf = \dfrac{hc}{\lambda}$;
• explain the photoelectric effect, and define the work function and the threshold frequency;
• use the photoelectric equation $hf = \phi + E_{k(\text{max})}$;
• explain why the maximum kinetic energy depends on frequency, not intensity;
• find the stopping potential from the maximum kinetic energy;
• describe the de Broglie hypothesis, and use $\lambda = \dfrac{h}{p}$;
• explain how electron diffraction shows that particles have wave properties;
• explain why the wave nature of a large object is not observed;
• define activity and half-life, and use $A = \lambda N$ and $N = N_0 e^{-\lambda t}$;
• find the decay constant from the half-life, and read a decay curve;
• write a balanced nuclear equation for alpha, beta and gamma decay;
• explain the random nature of radioactive decay;
• define mass defect and binding energy, and use $E = mc^2$;
• calculate the energy released in a nuclear reaction or an annihilation;
• explain why fission and fusion both release energy.

Where it is used

A hospital uses a positron emitter for a scan, and works out the activity at the time of the scan. A particle physicist identifies a nucleus from the energy of the gamma rays it emits. An astronomer reads the composition of a star from its spectrum. A power engineer follows the binding energy of a nucleus to judge whether a reaction releases energy. An electron microscope uses the de Broglie wavelength of electrons to see details far smaller than a light microscope can show. The same ideas explain why a smoke alarm, a radiotherapy beam and the Sun all work.

Your turn to connect it

Think of a light or a radiation you have met that can affect a material. It could be sunlight on a solar cell, ultraviolet light on a fluorescent label, a smoke alarm, or a hospital X-ray. Write down two examples. For one of them, say what you think the light or the radiation does to the material.

Roadmap

1. 8.1 Photons and the photoelectric effect
2. 8.2 Wave-particle duality
3. 8.3 Radioactive decay and half-life
4. 8.4 Nuclear energy and E = mc²
5. Review