# E.4 The nuclear atom

> IB Physics HL · Theme E: Nuclear and quantum physics
> Source: https://www.owlsprep.com/study/ib-physics-hl-u5-e-4-the-nuclear-atom/

This sub-topic covers the experimental discovery of the nuclear atom, key properties of atomic nuclei, and definitions of fundamental nuclear quantities. You will learn how experimental evidence overturned early models of the atom.

**Prerequisites:** Conservation of energy and momentum; Coulomb's law for electrostatic force

## Learning objectives

- Describe Rutherford's alpha scattering experiment and its key conclusions
- Calculate nuclear radius and recall properties of nuclear density
- Distinguish between atomic number, mass number and neutron number
- Identify isotopes and describe their properties
- State key properties of the strong nuclear force

## Rutherford Alpha Scattering Experiment

Before 1911, the accepted model of the atom was J.J. Thomson's 'plum pudding model', which described the atom as a uniform sphere of positive charge with electrons embedded evenly throughout. Geiger and Marsden, working under Rutherford's direction, tested this model by firing fast alpha particles at a thin gold foil.

**Rutherford Alpha Scattering** — An experiment that measured the deflection of alpha particles passing through thin metal foil to probe internal atomic structure

*Example:* Most alpha particles passed undeflected, while ~1 in 8000 were deflected by more than 90°

**Worked example:** State three key observations from Rutherford's experiment and one conclusion from each.

1. Observation 1: Most alpha particles pass straight through the foil undeflected
2. Conclusion 1: The atom is mostly empty space
3. Observation 2: A small number of alpha particles are deflected through angles > 90°
4. Conclusion 2: All positive charge and almost all mass of the atom are concentrated in a very small central nucleus
5. Observation 3: Deflection rate decreases with increasing deflection angle
6. Conclusion 3: The nucleus follows Coulomb's inverse-square law of electrostatic repulsion

> **exam_tip**
>
> Always link each observation directly to its conclusion in exam responses; marks are awarded for explicit connections.

## Nuclear Size and Density

Alpha scattering can be used to estimate the radius of a nucleus. The closest approach of an alpha particle to the nucleus gives an upper bound for the nuclear radius. Precise scattering experiments have confirmed an empirical relationship between nuclear radius and mass number:

$$r = r_0 A^{1/3}$$

Where $r_0 \approx 1.2 \times 10^{-15} \text{ m}$ and $A$ is the mass number. Since the volume of a spherical nucleus is $V = \frac{4}{3}\pi r^3$, substituting gives $V \propto A$, meaning volume is proportional to the number of nucleons. All nuclei therefore have approximately the same constant density.

**Worked example:** Calculate the radius of an oxygen-16 nucleus, given $r_0 = 1.2 \times 10^{-15} \text{ m}$.

1. 1. Identify the mass number: $A = 16$
2. 2. Substitute into the nuclear radius formula:
3. $$r = (1.2 \times 10^{-15}) \times (16)^{1/3}$$
4. 3. Calculate $16^{1/3} \approx 2.52$
5. 4. Final result:
6. $$r \approx 3.0 \times 10^{-15} \text{ m}$$

> **info**
>
> Nuclear density is approximately constant across all elements at ~$2 \times 10^{17} \text{ kg m}^{-3}$, which is 14 orders of magnitude larger than typical atomic density.

## Nuclear Composition and Isotopes

**Nuclear Composition** — Atomic number $Z$ = number of protons, Mass number $A$ = total number of nucleons ($Z + N$), Neutron number $N = A - Z$ = number of neutrons

*Notation:* $^A_Z\text{X}$ (standard nuclide notation, X = chemical symbol)

*Example:* Helium-4: $^4_2\text{He}$, 2 protons, 2 neutrons, $N=2$

Isotopes are nuclides of the same element that share the same atomic number $Z$ but have different mass numbers $A$ and different neutron numbers $N$. They have identical chemical properties because chemical reactions depend on electron configuration, which is determined by $Z$, but different nuclear properties.

**Worked example:** State the number of protons, neutrons and nucleons for uranium-235 and uranium-238, and write their standard notation.

1. 1. Uranium has atomic number $Z = 92$, so both isotopes have 92 protons and 92 nucleons from protons.
2. 2. For uranium-235: $A = 235$, $N = 235 - 92 = 143$ neutrons. Notation:
3. $$^{235}_{92}\text{U}$$
4. 3. For uranium-238: $A = 238$, $N = 238 - 92 = 146$ neutrons. Notation:
5. $$^{238}_{92}\text{U}$$

## The Strong Nuclear Force

Positively charged protons in the nucleus repel each other via the long-range electrostatic Coulomb force. There must be an attractive force stronger than this repulsion at short distances to hold the nucleus together: the strong nuclear force. Its key properties are:

- It is attractive between nucleons at separations of ~$0.5 \times 10^{-15} \text{ m}$ to $3 \times 10^{-15} \text{ m}$
- It becomes repulsive at separations less than ~$0.5 \times 10^{-15} \text{ m}$, preventing nuclear collapse
- It is very short range, and drops to zero at separations greater than ~$3 \times 10^{-15} \text{ m}$
- It acts equally between all nucleons, regardless of charge (proton-proton, proton-neutron, neutron-neutron)

> **tip**
>
> Large nuclei are unstable because the strong force is short range, while Coulomb repulsion is long range. Protons on opposite sides of a large nucleus only repel, with no net strong attraction.

## Common pitfalls

- **Wrong:** Confusing mass number with actual atomic mass in u
  - Why it fails: Mass number is a count of nucleons, not a measured mass. Binding energy reduces actual nuclear mass below $A$ u
  - Correct: State that mass number is the total number of nucleons, and actual mass is approximately $A$ u for rough calculations
- **Wrong:** Claiming large alpha deflection only proves the nucleus is positively charged
  - Why it fails: Deflection by positive charge was expected; the surprising result was deflection greater than 90°
  - Correct: Link large angle deflection to the conclusion that most mass is concentrated in a tiny nucleus
- **Wrong:** Writing the nuclear radius formula as $r = r_0 A^3$
  - Why it fails: Volume is proportional to $A$, not radius. Radius scales with the cube root of volume
  - Correct: Remember $V \propto r^3 \propto A \implies r \propto A^{1/3}$
- **Wrong:** Claiming isotopes have different chemical properties
  - Why it fails: Chemical properties depend on electron configuration, which is determined by atomic number $Z$, not mass number
  - Correct: State that isotopes have identical chemical properties but different nuclear properties
- **Wrong:** Stating the strong nuclear force is attractive at all distances
  - Why it fails: The strong force becomes repulsive at very short distances, which is required for nuclear stability
  - Correct: Describe the strong force as repulsive < 0.5 fm, attractive 0.5-3 fm, and zero beyond 3 fm

## Cheatsheet

| Quantity/Property | Symbol/Rule | Value/Definition |
| --- | --- | --- |
| Nuclear radius | $r = r_0 A^{1/3}$ | $r_0 = 1.2 \times 10^{-15}$ m |
| Atomic number | $Z$ | Number of protons |
| Mass number | $A$ | Total number of nucleons |
| Neutron number | $N$ | $N = A - Z$ |
| Nuclear density | $\rho$ | $\approx 2 \times 10^{17}$ kg m⁻³ (constant) |
| Strong force range |  | Repulsive < 0.5 fm, attractive 0.5-3 fm, zero > 3 fm |
| Rutherford key conclusion |  | Atom is mostly empty space, mass/charge in small nucleus |
| Isotopes |  | Same Z, different A, same chemical properties |

## What's next

Understanding the structure of the nuclear atom is the foundation for all further topics in nuclear physics, from radioactive decay to nuclear fission and fusion. The properties of the nucleus you have learned here, including nuclear size, composition, and the strong force, underpin explanations of nuclear instability and energy release in nuclear reactions. You will now build on this knowledge to explore how unstable nuclei decay, and how binding energy explains the energy released in fission and fusion reactions.

- [E.6 Fission and fusion (AHL)](https://www.owlsprep.com/study/ib-physics-hl-u5-e-6-fission-and-fusion/)

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