Nuclear Radius

Q7.

(a) Scattering experiments are used to investigate the nuclei of gold atoms.

In one experiment, alpha particles, all of the same energy (monoenergetic), are incident on a foil made from a single isotope of gold.

(i) State the main interaction when an alpha particle is scattered by a gold nucleus.

The electromagnetic (or electrostatic or Coulomb) interaction

[1 mark]

(ii) The gold foil is replaced with another foil of the same size made from a mixture of isotopes of gold. Nothing else in the experiment is changed.

Explain whether or not the scattering distribution of the monoenergetic alpha particles remains the same.

The mark requires a described distribution and the reason for it.

The scattering distribution remains the same because the alpha particles interact with a nucleus whose charge/proton number/atomic number remains the same and therefore the repulsive force remains the same.

The scattering distribution will change - it will become less distinct because there is a mixture of nuclear masses (which gives a mixture of nuclear recoils).

The reference must be made to the mass variation and not density or size variation for the mark to be awarded.

[1 mark]

(b) Data from alpha-particle scattering experiments using elements other than gold allow scientists to relate the radius R, of a nucleus, to its nucleon number, A.

The graph below shows the relationship from the data obtained from a scattering experiment. The data obtained obeys the relationship R = r₀ A^1/3

(i) Use information from the graph to show that r₀ is about 1.4 × 10^–15 m.

Any point on the line can be used to obtain corresponding values of R and A - they can then be inserted into the equation to calculate r₀.

The simplest is to read off the value for A = 175

r₀ = 8.0 x 10^-15 / 175^1/3

r₀ = 1.43 x 10^-15 m

The substitution and calculation must be shown for the mark.

[1 mark]

(ii) Show that the radius of a nucleus is about 5 × 10^–15 m.

R = r₀ A^1/3

R = 1.43 x 10^-15 x 51^1/3

R = 5.3 x 10^-15 m

(R = 5.2 x 10^-15 m from r₀ = 1.4 x 10^-15 m)

The first mark was awarded for putting the correct value for A into the equation, the second mark was for calculating the value to compare to the given one which must be 2 or more significant figures.

[2 marks]

(c) Calculate the density of a nucleus. State an appropriate unit for your answer.

density = mass / volume

m = (23 x 1.673 + 28 x 1.675) x 10^-27

m = 8.5 x 10^-26 kg

V = ⁴/₃ π r³

V = ⁴/₃ π (5.3 x 10 ^-15 )³

V = 6.24 x 10^-43 m³

 

ρ = m/V

ρ = 8.5 x 10^-26/6.24 x 10^-43= 1.37 x 10 ¹⁷

ρ = 1.4 x 10 ¹⁷kg m^-3

[3 marks]

(Total 8 marks)