12.2 Nuclear Physics

Show that the decay constant is related to the half-life by the expression 

Uranium-238 has a half-life of 4.47 × 10⁹ years and decays to thorium-234. The thorium decays (by a series of further nuclear processes with short half-lives) to lead.

Assuming that a rock was originally entirely uranium and that at present, 1.5% of the nuclei are now lead, calculate the age of the rock. Give your answer in years to 2 significant figures.

The ionisation current I produced by α-particles emitted in the decay of radon can be measured experimentally. The logarithmic graph shows how current, ln I, varies with time, t.

Using the graph, determine the half-life of radon.

An electron beam of energy 1.3 × 10⁻¹⁰ J is used to study the nuclear radius of beryllium-9. The beam is directed from the left at a thin sample of beryllium-9. A detector is placed at an angle θ relative to the direction of the incident beam.

The radius of a beryllium-9 nucleus is 2.9 × 10⁻¹⁵ m. The beryllium-9 nuclei behave like a diffraction grating. 

Sketch the expected variation of electron intensity against the angle from the horizontal.

The isotope beryllium-10 is formed when a nucleus of deuterium collides with a nucleus of beryllium-9 . The radius of a deuterium nucleus is 1.5 fm.

[1]

The nucleus of beryllium-9 is replaced by a nucleus of gold-197.

Suggest the change, if any, to the following: 

Unstable uranium-238 has various nuclear decay modes to become stable thorium-234. The total amount of energy released when it decays is measured to be 210 keV. 

Outline, without calculation, the intermediate decay modes between the unstable uranium-238 to the stable thorium-234. 

A possible decay chain for uranium-238 is: 

Calculate the total amount of energy, in joules, carried away as gamma radiation in this decay chain. 

Deduce an alternative decay chain from unstable uranium-238 to stable thorium-234 which releases the same amount of energy in the form of gamma radiation as in part (b). 

Justify your answer with a calculation. 

The half-life of uranium-238 is so long in comparison to any of the isotopes in its decay chain that we can assume the number of lead-206 nuclei,  at any time is equal to the number of uranium-238 that have decayed. 

The number of uranium-238 nuclei  at time t is given by the equation: 

Where  is the number of uranium-238 nuclei at t = 0.

Show that the ratio of to  is given by:

Enriched uranium fuel is a mixture of the fissionable uranium-235 with the more naturally abundant uranium-238. Mixtures of radioactive nuclides such as this are very common in the nuclear power industry. 

Two samples of radioactive nuclides X and Y each have an activity of A₀ at t = 0. They are subsequently mixed together. 

The half-lives of X and Y are 16 and 8 years respectively. 

Show that the total activity of the mixture at time t = 48 years is equal to:

Iodine-131  has a half-life of 8.02 days.

Calculate the decay constant of .

The initial activity of the sample of Iodine−131 is 6.5 × 10⁴ Bq.

Determine the activity after 16 days.

Determine the mass of the iodine−131 in the sample after 16 days.

Iodine−131 decays through a number of decay modes. Two of these are decay of 606 keV and gamma emission of 364 keV. The product of the  decay is . 

Sketch a nuclear energy level diagram to represent these decays. 

Ernest Rutherford was able to deduce a relationship for the size of the nucleus using the Rutherford scattering experiment shown below:

A radioisotope has a nuclear radius of 7.41 fm

Determine the nucleon number of the isotope.

A beam of high-energy neutrons is directed at the nucleus and a pattern is formed on a detector screen.

Explain the pattern which is observed.

The neutrons are accelerated to a speed of 2.88 × 10⁸ m s⁻¹.

Determine the angle of the first minimum.

The decay constant of the isotope is 9.72 × 10⁻¹⁰ yr⁻¹. The mass of a sample of this isotope is 600 g.

Determine the activity of the sample.

A nucleus of sodium−24 decays into a stable nucleus of magnesium−24. It decays by β^– emission followed by the emission of γ-radiation as the magnesium−24 nucleus de-excites into its ground state.

The sodium−24 nucleus can decay to one of three excited states of the magnesium−24 nucleus. This is shown in the diagram below:

The energies of the excited states are shown relative to the ground state.

Calculate the maximum possible speed of the emitted beta particle in MeV.

The excited magnesium nucleus de-excites through production of gamma radiation of discrete wavelengths.

Calculate the shortest wavelength of emitted radiation.

The graph shows the activity of a sample of sodium−24 with time.

Use the graph to calculate the decay constant of sodium−24.

The detector in this experiment measures 4% of the activity from the sample.

Determine the activity of sample after 27 hours from the start of the recording,

Americium-241 has a half-life of 432 years. A small sample is held in a school for use in experiments.

The teacher uses a Geiger-Müller counter to measure the count rate at close range. The relationship between activity and count rate is a ratio of 6:1. Over 5 minutes, the count is 13 600. 

Determine the activity of the sample.

Determine the activity of the americium sample after 748 years.

Americium−241 decays through a series of decays to uranium-233.

The energies from each decay path are recorded. 

Explain the differences between the energy profiles for the alpha decays and the beta decays.

The beta decay of protactinium−233 to uranium−233 has a half−life of 27 days. A sample of protactinium has an activity of 3879 Bq.

Determine the number of protactinium−233 nuclei in the sample.