6.4.1 - The nuclear atom, radius and density

Scattering evidence for the nucleus

Before Rutherford's scattering experiment, a plausible model was that positive charge and mass were spread through the atom, with electrons embedded in that spread-out structure. The alpha-particle scattering experiment tested this model by firing a narrow beam of fast, positively charged alpha particles at very thin gold foil and detecting the angles through which they were scattered.

The observations have three important consequences.

Most alpha particles passed straight through the foil. That means the atom is mostly empty space on the scale of the alpha particle path. If the atom were full of spread-out positive matter, many alpha particles would be slowed or deflected slightly by that material.

A small fraction of alpha particles were deflected through large angles, and a very small number were scattered backwards. Because alpha particles are positively charged, this requires a strong electrostatic repulsion from a concentrated positive charge. The fact that large deflections are rare tells us that this positive charge occupies a very small fraction of the atom's volume.

The alpha particles are much more massive than electrons, so electrons cannot explain the rare large deflections. The evidence points to a small, positive, dense nucleus containing most of the atom's mass, with electrons outside it.

OCR questions often give scattering data rather than asking for the story from memory. For example, a table might show many counts at small scattering angles and very few counts at large scattering angles. The physics link is the same: high count at small angle means most particles are undeflected; low but non-zero count at large angle means rare close encounters with a small positive nucleus.

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Nuclear model and notation

The simple nuclear model separates the atom into a central nucleus and the surrounding electrons.

Protons have positive charge, neutrons have no charge, and electrons have negative charge. In a neutral atom, the number of electrons is equal to the number of protons. This lesson is about the structure and scale of the atom; decay equations and particle transformations belong to later nuclear physics lessons.

Two numbers describe a nucleus.

The neutron number is therefore:

Nuclear notation writes a nucleus as:

Here $X$ is the chemical symbol, $Z$ is the proton number, and $A$ is the nucleon number. The proton number identifies the element. If $Z = 6$, the element is carbon, whatever the neutron number is.

For example, $^{12}_{6}\text{C}$ and $^{14}_{6}\text{C}$ are isotopes of carbon. Both have 6 protons. Carbon-12 has 6 neutrons, while carbon-14 has 8 neutrons. The isotope changes the nucleon number, not the element.

Worked example:

For $^{59}_{27}\text{Co}$:

• proton number $Z = 27$, so the nucleus contains 27 protons
• nucleon number $A = 59$, so the nucleus contains 59 protons plus neutrons
• neutron number $= A - Z = 59 - 27 = 32$
• a neutral cobalt atom has 27 electrons

Size scale and strong force

The nucleus is tiny compared with the atom. A typical atomic radius is about $10^{-10}\,\text{m}$. A typical nuclear radius is about $10^{-15}\,\text{m}$ to $10^{-14}\,\text{m}$.

Compare these sizes using powers of ten. If an atom has radius $1.0\times10^{-10}\,\mathrm{m}$ and its nucleus has radius $1.0\times10^{-15}\,\mathrm{m}$, then:

So the atom's radius is about $100\,000$ times larger than the nuclear radius. The volume comparison is even more dramatic because volume depends on radius cubed.

The scale unit used for nuclei is the femtometre:

The nucleus should fly apart if only electrostatic forces acted, because protons repel other protons. The fact that nuclei exist shows that there is another force between nucleons.

[DIAGRAM: asset_name: 6.4.1 - The nuclear atom, radius and density - diagram 2; asset_slug: 6_04_1_nuclear_atom_radius_and_density__diagram_02; recommended_method: drawn_physics; description: Schematic signed strong force: repulsive below 0.5 fm, attractive to 3 fm, zero beyond; axes and sign explanation explicit.; caption: The strong nuclear force is repulsive below about 0.5 fm, attractive from about 0.5 to 3 fm and negligible beyond.]