Nuclear Radius & Density (OCR A Level Physics)

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Nuclear Radius

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

Nuclear Radii Table, downloadable AS & A Level Physics revision notes

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

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

Nuclear Radius Graph, downloadable AS & A Level Physics revision notes

  • 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 = r0A1/3

  • 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

V proportional to 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

Worked example

Calculate the approximate density of a lithium nucleus.

Assume the atomic mass of lithium to be 7u.

Step 1: Write down the equations: 

  • Density: d e n s i t y space equals space fraction numerator m a s s over denominator v o l u m e end fraction equals fraction numerator A u over denominator bevelled 4 over 3 space pi space R cubed end fraction
  • Volume of a sphere: V space equals space 4 over 3 pi space R cubed
  • From the data booklet: nuclear radius, R space equals space r subscript 0 space A to the power of bevelled 1 third end exponent
    • Where:
      • r= constant = 1.2 x 10-15 
      • A = mass number = 7 for lithium

Step 2: Combine equations: 

  • V4 over 3π R 4 over 3π (rA1/3)34 over 3π ro3 A

Step 3: Calculate the volume of the nucleus: 

  • V = 4 over 3πro3 A = 4 over 3π x (1.2 x 10-15)x 7 = 5.07 x 10-44 m

Step 4: Calculate mass of lithium nucleus: 

  • Mass = Au = 7 x (1.661 x 10-27) = 1.1627 x 10-26 kg

Step 5: Calculate the density: 

  • Density = mass over volumefraction numerator 1.1627 cross times 10 to the power of negative 26 end exponent over denominator 5.07 cross times 10 to the power of negative 44 end exponent end fraction = 2.29 x 1017 kg m-3

Step 6: Finalise your answer: 

  • The density of a lithium nucleus is 2.3 x 1017 kg m-3 (2 s.f.)

Exam Tip

Don't let all the powers and letters confuse you. Work through each step of a question one by one. It is just mass/volume to get the density with a little bit of substitution! 

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Katie M

Author: Katie M

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.

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