E.1.7 (HL)—Nuclear radius

Use the radius law

Nuclear radius $R$ grows as the cube root of nucleon number $A$ . The constant $R_0=1.20\times10^{-15}\,\mathrm{m}$ represents the scale of a single-nucleon nucleus in this model.

R=R_0A^{1/3}

Worked example — gold-197

For $A=197$ , $R=(1.20\times10^{-15})(197)^{1/3}=6.98\times10^{-15}\,\mathrm{m}$ . Since volume is proportional to $R^3\propto A$ while nuclear mass is also approximately proportional to $A$ , the model predicts approximately constant nuclear density.

Infer the density

Nuclear volume scales as $R^3$ , so $V\propto A$ . Since nuclear mass is approximately proportional to $A$ , the mass per unit volume is approximately constant across nuclei.

Scale carefully

If $A$ changes by a factor of $k$ , radius changes by $k^{1/3}$ , not by $k$ . The density remains approximately unchanged in this model.

Common trap

Do not assume a nucleus with eight times the nucleon number has eight times the radius. It has twice the radius and approximately the same density.

How it is assessed

Questions calculate A from a measured radius or compare radius and density when nucleon number changes.

Command terms

Determine / Compare

What earns marks

Apply cube-root scaling to radius, then use volume proportional to R^3 to justify constant density.

Watch for

Scaling radius directly with A or changing density when the model implies mass and volume grow proportionally.

Retrieve the HL extensions

Use $R=R_0A^{1/3}$ for nuclear scale, recognise when high-energy scattering exceeds the electrostatic model, and use energy conservation for head-on closest approach.

Retrieve the Bohr model

Hydrogen levels obey $E_n=-13.6/n^2\,\mathrm{eV}$ , and allowed angular momentum $mvr=nh/(2\pi)$ produces discrete orbits and energies.

E.1.7 (HL)—Nuclear radius

Model Nuclear Radius

E.1.7 (HL) Exam Analysis

Assessment in practice

Retrieve the HL Atomic Model