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Nuclear radius and density questions
Nuclear radii can be estimated from the closest approach of an alpha particle and measured by diffracting electrons off the nucleus. The two methods agree, and they give R = R₀A^(1/3), from which it follows that medium and heavy nuclei share very nearly one density.
18 original questions · 49 marks · the nuclear radius and density notes · Nuclear physics
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State the equation relating nuclear radius to nucleon number, and give the approximate value of the constant R0.
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R = R0A1/3, where A is the nucleon number (1); R0 ≈ 1.2 × 10−15 m (1.2 fm) (1).Calculate the radius of a nucleus with nucleon number A = 64.
R0 = 1.2 × 10−15 mMark scheme
R = R0A1/3 = 1.2 × 10−15 × 641/3 = 1.2 × 10−15 × 4 (1)
R = 4.8 × 10−15 m (1)State what the relationship R = R0A1/3 implies about the density of nuclear matter.
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Volume ∝ R3 ∝ A, and mass ∝ A (1); so the density (mass/volume) is approximately the same for all nuclei (1).State typical values for the radius of an atom and for the radius of a nucleus.
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Atomic radius: about 1 × 10−10 m (1). Nuclear radius: a few femtometres, of order 10−15 m to 10−14 m (1).A student has measured nuclear radii R for nuclides covering a wide range of nucleon numbers A. State the graph the student should plot to obtain a straight line through the origin, and state what the gradient of that line represents.
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Plot R against A1/3 (1); the gradient of the line is the constant R0, about 1.2 fm (1).Calculate the radius of an aluminium nucleus, for which A = 27.
R0 = 1.2 × 10−15 mMark scheme
R = R0A1/3 = 1.2 × 10−15 × 271/3 = 1.2 × 10−15 × 3 (1)
R = 3.6 × 10−15 m (1)Taking the mass of a nucleon as 1.66 × 10−27 kg and the volume per nucleon as that of a sphere of radius R0, estimate the density of nuclear matter.
R0 = 1.2 × 10−15 mMark scheme
ρ = u/((4/3)πR03) (1)
= (1.66 × 10−27)/((4/3)π × (1.2 × 10−15)3) (1)
ρ = 2.29 × 1017 kg m−3 (1)Calculate the ratio of the radius of a nucleus with A = 125 to the radius of a nucleus with A = 8.
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R125/R8 = (125/8)1/3 (1)
ratio = 2.5 (1)Explain how the near-constant density of nuclei supports the idea that nucleons are closely packed.
Electron diffraction shows that a nuclide has a nuclear radius of 6.0 × 10−15 m. Determine the nucleon number of the nuclide.
R0 = 1.2 × 10−15 mThe radius of a gold atom is 1.35 × 10−10 m and the radius of a gold nucleus is 7.0 × 10−15 m. Calculate the fraction of the volume of the atom occupied by the nucleus, and comment on your answer.
Explain why high-energy electrons are a suitable probe for measuring the radius of a nucleus.
An alpha particle with kinetic energy 5.0 MeV is fired head-on at a gold nucleus (Z = 79). Estimate the distance of closest approach.
k = 8.99 × 109 N m2 C−2, e = 1.60 × 10−19 CCalculate the density of a carbon-12 nucleus (A = 12), taking the nucleon mass as 1.66 × 10−27 kg and R0 = 1.2 × 10−15 m. Compare your value with the density of nuclear matter in general, 2.3 × 1017 kg m−3.
Describe how electron diffraction can be used to determine the radius of a nucleus.
Determine the minimum initial kinetic energy, in MeV, that an alpha particle needs in order to approach within 4.0 × 10−14 m of the centre of a gold nucleus (Z = 79) in a head-on collision.
k = 8.99 × 109 N m2 C−2, e = 1.60 × 10−19 C, 1 MeV = 1.60 × 10−13 JFor one nuclide, the closest-approach method with 5 MeV alpha particles gives a nuclear radius estimate of 4.5 × 10−14 m, while electron diffraction gives 7.2 × 10−15 m. Explain why the two values differ, and deduce which is the better estimate of the nuclear radius.
A neutron star can be modelled as a sphere with the density of nuclear matter, 2.3 × 1017 kg m−3. Estimate the radius of a neutron star of mass 4.0 × 1030 kg.
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