OCR A-Level Physics: Nuclear, Particle and Medical Physics

The graph of binding energy per nucleon against nucleon (mass) number rises steeply from hydrogen, reaches a broad maximum at iron ($^{56}_{26}\text{Fe}$2656​Fe, about $8.8 \ \text{MeV per nucleon}$8.8 MeV per nucleon), and then falls slowly towards the heaviest nuclei such as uranium.

Explain what is meant by binding energy per nucleon, and use the shape of this curve to explain why energy is released both when light nuclei fuse and when heavy nuclei undergo fission. You should make clear, in each case, why the products are more stable than the original nuclei.

(6 marks)

A sealed cobalt-60 source ($^{60}_{27}\text{Co}$2760​Co) is used in a radiotherapy machine. When new, its activity is measured as $3.2 \times 10^{4} \ \text{Bq}$3.2×104 Bq. Cobalt-60 has a half-life of 5.27 years.

Quantity	Value
Initial activity, $A_0$A0​	$3.2 \times 10^{4} \ \text{Bq}$3.2×104 Bq
Half-life, $t_{1/2}$t1/2​	5.27 years
Time in service, $t$t	12.0 years
1 year	$3.156 \times 10^{7} \ \text{s}$3.156×107 s

(a) Calculate the decay constant $\lambda$λ of cobalt-60, in $\text{year}^{-1}$year−1. (2 marks)

(b) Calculate the activity of the source after it has been in service for 12.0 years. (2 marks)

(c) Calculate the number of cobalt-60 nuclei present in the source when it was new. (2 marks)

A student measures the count rate from a freshly prepared radioactive source using a Geiger-Muller tube, recording a reading every 2.0 minutes. Before adding the source she measures the background count rate as a steady 20 counts per 2 minutes.

Time / min	0	2.0	4.0	6.0	8.0
Measured count / (counts per 2 min)	500	359	260	190	140

(a) Correct the readings for background and show that the corrected count rate falls by a constant ratio over each 2.0-minute interval, confirming the decay is exponential. (2 marks)

(b) Use the corrected data to determine the half-life of the source. (2 marks)

(c) Hence calculate the decay constant $\lambda$λ of the source, in $\text{min}^{-1}$min−1. (1 mark)

One reaction studied in experimental fusion reactors is the fusion of deuterium and tritium:

$^{2}_{1}\text{H} + {}^{3}_{1}\text{H} \to {}^{4}_{2}\text{He} + {}^{1}_{0}\text{n}$12​H+13​H→24​He+01​n

The rest masses of the particles are:

Particle	Rest mass / u
Deuterium, $^{2}_{1}\text{H}$12​H	2.01410
Tritium, $^{3}_{1}\text{H}$13​H	3.01605
Helium-4, $^{4}_{2}\text{He}$24​He	4.00260
Neutron, $^{1}_{0}\text{n}$01​n	1.00867

Use $1 \ \text{u} = 1.66 \times 10^{-27} \ \text{kg}$1 u=1.66×10−27 kg and $c = 3.00 \times 10^{8} \ \text{m s}^{-1}$c=3.00×108 m s−1.

(a) Calculate the mass defect for this reaction, in atomic mass units (u). (2 marks)

(b) Hence calculate the energy released, in joules, when one such reaction occurs. (2 marks)

(c) State this energy in MeV, given $1 \ \text{u}$1 u is equivalent to $931.5 \ \text{MeV}/c^2$931.5 MeV/c2. (1 mark)

In a diagnostic X-ray, a parallel beam passes through a thickness of soft tissue. For the photon energy used, the tissue has a linear attenuation coefficient $\mu = 0.20 \ \text{cm}^{-1}$μ=0.20 cm−1. The beam enters the tissue with an intensity $I_0 = 4.0 \times 10^{4} \ \text{W m}^{-2}$I0​=4.0×104 W m−2 and travels through 5.0 cm of tissue before reaching the detector.

The intensity is reduced according to $I = I_0 e^{-\mu x}$I=I0​e−μx.

(a) Calculate the intensity of the beam reaching the detector after passing through the 5.0 cm of tissue. (2 marks)

(b) Calculate the half-value thickness of this tissue (the thickness that reduces the beam intensity by half). (2 marks)

In the Standard Model, the proton and neutron are baryons, each built from three quarks. The up quark has charge $+\frac{2}{3}e$+32​e and the down quark has charge $-\frac{1}{3}e$−31​e.

(a) State the quark composition of a proton and of a neutron. (1 mark)

(b) Beta-minus ($\beta^-$β−) decay can be described at the quark level. Write the quark transformation that occurs, and name the two other particles emitted. (1 mark)

(c) Show that electric charge is conserved in the quark-level process you wrote in part (b). (1 mark)