8.3 Nuclear Instability & Radius

The radius of a gold-197 nucleus,  , is 6.87 × 10^–15 m.

Show that the density of this nucleus is about 2.4 × 10¹⁷kg m^–3.

Hence, or otherwise, calculate the radius of a chlorine–35 nucleus, .

Nuclear radii have been investigated using  particles in Rutherford scattering experiments and by using electrons in diffraction experiments. 

Make comparisons between these two methods of estimating the radius of a nucleus.
Detail of any apparatus used is not required.

For each method, your answer should contain:

• The principles on which each experiment is based including a reference to an appropriate equation. 
• An explanation of what may limit the accuracy of each method.
• A discussion of the advantages and disadvantages of each method. 

The quality of your written communication will be assessed in your answer. 

The results of electron scattering experiments using different target elements show that 

            R = 

where A is the nucleon number and   is a constant. 

Use this equation to show that the density of a nucleus is independent of its mass.

A beam of high-energy electrons is fired through a thin foil of beryllium-9 to determine its radius using electron scattering. The electrons produce a diffraction pattern on a fluorescent screen according to the relation: 

            sin θ = 

where θ is the angle of the first minimum, λ is the de Broglie wavelength of the electrons, and R is the radius of the nucleus. 

Figure 1 shows the variation of the electron intensity on the screen against the angle from the horizontal. 

Figure 1

Each electron in the beam has an energy of 1.55 × 10^–10 J. 

Calculate the radius of a beryllium-9 atom.

Calculate the density of nuclear material. 

Assume the mass of a nucleon = 1.0 u.

Figure 2 shows how the relative intensity of the scattered electrons varies with angle due to diffraction by an oxygen-16 nucleus. The angle is measured from the original direction of the beam. 

Figure 2

Each electron has an energy of 5.94 × 10^–11 J.

Calculate the ratio of 

Rutherford’s team estimated the radius of gold nuclei by calculating the distance of closest approach. 

In this method, an   particle with an initial kinetic energy of 8.1 MeV is directed towards the centre of a gold nucleus of radius R which contains 79 protons. The  particle is brought to rest at point X, a distance r from the centre of the nucleus as shown in Figure 1. 

Figure 1

Calculate the electric potential energy, in J, of the   particle at point X. 

Calculate r, the distance of closest approach of the  particle to the nucleus.

Determine the number of nucleons in the gold nucleus. 

R, radius of the gold nucleus = 7.16 × 10⁻¹⁵ m 

 = 1.23 × 10⁻¹⁵ m

The target nucleus is changed to one that has fewer protons. The   particle is given the same initial kinetic energy. 

Explain, without further calculation, any changes that occur to the distance r. 

Ignore any recoil effects.

Sketch on Figure 1a graph of neutron number, N, against proton number, Z, for stable nuclei over the range Z = 0 to Z = 80. 

Show suitable numerical values on the N axis. 

Figure 1

On Figure 1indicate, for each of the following, a possible position of a nuclide that may decay by: 

(i)          emission, labelling the position with A, 

(ii)        emission, labelling the position with B, 

(iii)        emission, labelling the position with C.

After Z = 20 the ratio of protons and neutrons changes. 

Explain why there is this imbalance between proton and neutron numbers by referring to the forces that operate within the nucleus. 

Your explanation should include the range of the forces and which particles are affected by the forces.

A particular nuclide is described as proton-rich. 

Discuss two ways in which the nuclide may decay. 

You may be awarded marks for the quality of written communication in your answer.

On Figure 1, sketch a graph to show how the radius, R, of a nucleus varies with its nucleon number, A. 

Figure 1

A student summarises how electron diffraction is used to investigate the radius of a nucleus as shown in Figure 2. Their physics teacher has identified four parts of the summary that need correcting. 

Figure 2

Identify the parts of the summary that need correcting and suggest an improvement to them in the space provided below. One has been done for you. 

Calculate the radius of auranium – 238 nucleus, . 

                  r₀ = 1.3 × 10^–15 m

Hence, show that the density of a uranium ­– 238 nucleus is approximately 2 × 10¹⁷ kg m^–3. 

Figure 1 shows a grid of neutron number against proton number. A nucleus   is marked. 

Figure 1

Draw arrows on Figure 1, each starting on  and ending on a daughter nucleus after the following transitions: 

(i)        emission              (label this arrow )
   neutron emission       (label this arrow n)
   electron capture         (label this arrow ) 

(ii)        Describe the process of electron capture. 

(iii)       Give the equation for electron capture by the nucleus  .

When decays to  by   decay, the daughter nucleus is produced in one of two possible excited states of metastable  . 

These two states are shown in Figure 2 together with their corresponding energies. 

Figure 2

Calculate the maximum possible kinetic energy, in MeV, which an emitted   particle can have.

The excited  nuclei then emits a  photon to become the nuclide . 

Calculate each of the three possible  photon energies in keV.

Figure 3 shows how cobalt-60 decays to nickel-60, which is a stable isotope, and Table 1 shows the energies of some of the emitted particles. 

Figure 3

                  Table 1

Using Table 1, determine the energies of: 

(i)         

(ii)       

Cobalt-60 has a half-life of 5.3 years, while technitium-99 has a half-life of 6 hours. 

Discuss, using data from Figure 2 and Figure 3, which radioactive isotope would be the most suitable for use as: 

(i)         A medical tracer 

(ii)        A treatment for cancer

The results of electron scattering experiments using different target elements show that the diameter, D, of a nucleus is related to its nucleon number, A, by 

            D  =

where   is a constant. 

Table 1 lists values of nuclear diameter for various elements: 

Table 1

 Complete the missing data in Table 1.

Plot a graph of diameter D against A^1/3. 

Use the graph to determine a value for the constant .

Assuming the mass of a nucleon is 1.67 × 10^–27 kg, it can be shown that the density of nuclear matter is around 2.8 × 10¹⁷ kg m^–3. 

State two further assumptions that have to be made when calculating the density of nuclear matter.