Nuclear Radius & Density (OCR A Level Physics)

Nuclear Radius

• The radii of some nuclei are shown in the table below:

• In general, nuclear radii are of the order 10^–15 m or 1 fm

• The nuclear radius, R, varies with nucleon number, A as follows:

• The key features of this graph are:

  • The graph starts with a steep gradient at the origin

  • Then the gradient gradually decreases to almost horizontal

• This means that

  • As more nucleons are added to a nucleus, the nucleus gets bigger

  • However, the number of nucleons A is not proportional to its size r

Calculating the Nuclear Radius

• The radius of nuclei depends on the nucleon number, A of the atom

• This makes sense because as more nucleons are added to a nucleus, more space is occupied by the nucleus, hence giving it a larger radius

• The exact relationship between the radius and nucleon number can be determined from experimental data

• By doing this, physicists were able to deduce the following relationship:

R = r₀A^1/3

• Where:

  • R = nuclear radius (m)

  • A = nucleon / mass number

  • R₀ = constant of proportionality = 1.2 fm = 1.2 x 10⁻¹⁵ m (the radius of a proton)

Mean Densities of Atoms and Nuclei

Equation for Nuclear Density

• Assuming that the nucleus is spherical, its volume is equal to:

• Where R is the nuclear radius, which is related to mass number, A, by the equation:

• Where R₀ is a constant of proportionality

• Combining these equations gives:

• This shows that the nuclear volume, V, is proportional to the mass of the nucleus, A

• Mass (m), volume (V), and density (ρ) are related by the equation:

• The mass, m, of a nucleus is equal to:

m = Au

• Where:

  • A = the mass number

  • u = atomic mass unit

• Using the equations for mass and volume, nuclear density is equal to:

• Since the mass number A cancels out, the remaining quantities in the equation are all constant

• Therefore, this shows the density of the nucleus is:

  • Constant

  • Independent of the radius

• The fact that nuclear density is constant shows that nucleons are evenly separated throughout the nucleus regardless of their size