Why Splitting and Joining Nuclei Both Release Energy

A nucleus packs positively charged protons into an incredibly small space, where their electric repulsion is enormous. Something stronger holds them together — and when nuclei split or join, a little of their mass turns into a great deal of energy.

This part covers the size and composition of nuclei, mass defect and binding energy, the binding energy curve, and fission and fusion.

**A nucleus contains $Z$ protons and $N$ neutrons, with mass number $A=Z+N$, and its radius grows as $R=R_0{A}^{1/3}$ with $R_0\approx 1.2\times 10^{-15}$ m, so all nuclei have about the same density.

Composition and notation:

- Protons and neutrons together are called nucleons
- Isotopes** share $Z$; isobars share $A$; isotones share $N$
- Atomic mass unit: $1$ u is one-twelfth of the mass of a carbon-12 atom, $1.66\times 10^{-27}$ kg

Worked example. For aluminium-27 and copper-64:

$$R_{Al}=1.2\times 10^{-15}\times 27^{1/3}=3.6\times 10^{-15}\text{ m},R_{Cu}=1.2\times 10^{-15}\times 4=4.8\times 10^{-15}\text{ m}$$

The density, since mass and volume both grow with $A$, is

$$\rho =\frac{1.66\times 10^{-27}}{\frac{4}{3}\pi (1.2\times 10^{-15}{)}^3}\approx 2.3\times 10^{17}{\text{ kg m}}^{-3}$$

An everyday example. A teaspoon of nuclear matter, about $5$ mL, would have a mass of roughly $10^{12}$ kg — more than a billion tonnes.

The substance. **Nuclear density does not depend on $A$** — volume grows in step with the number of nucleons, like packing more marbles into a bigger bag.

**A nucleus has less mass than its separate protons and neutrons; this mass defect $\Delta M$, converted using $E=\Delta M{c}^2$, is the binding energy needed to pull the nucleus apart into free nucleons.**

$$\Delta M=[Z{m}_p+(A-Z)m_n]-M,E_b=\Delta M{c}^2$$

With masses in u, $E_b=\Delta M\times 931.5$ MeV.

Worked example. A helium-4 nucleus has $2$ protons and $2$ neutrons. With $m_p=1.00728$ u, $m_n=1.00867$ u and a nuclear mass of $4.00151$ u:

$$\Delta M=2(1.00728)+2(1.00867)-4.00151=0.03039\text{ u}$$

$$E_b=0.03039\times 931.5\approx 28.3\text{ MeV},\frac{E_b}{A}\approx 7.1\text{ MeV per nucleon}$$

Checking the conversion. $1$ u $\times c^2=1.66\times 10^{-27}\times (3\times 10^8{)}^2\approx 1.49\times 10^{-10}$ J, about $930$ MeV.

An everyday example. Two magnets that snap together give out energy as they click, and you must do work to pull them apart again — binding energy works the same way.

The substance. Binding energy is not stored inside the nucleus — it is the energy released when the nucleus formed, and it must be supplied to break it apart.

**Binding energy per nucleon rises steeply for light nuclei, peaks at about $8.8$ MeV near mass number $56$, and falls slowly for heavy nuclei, so the most stable nuclei lie in the middle; light nuclei release energy by fusing and heavy ones by splitting.

Features of the curve:

- Low for very light nuclei** — about $1.1$ MeV per nucleon for deuterium
- **Peaks near $A=56$**, about $8.8$ MeV per nucleon — the region of greatest stability
- Nearly flat for middle mass numbers, around $8$ MeV per nucleon
- **Falls to about $7.6$ MeV per nucleon for uranium-238

Why it matters. Energy is released whenever nucleons end up in nuclei with higher binding energy per nucleon.

Worked example.** A nucleus with $A=240$ and $7.6$ MeV per nucleon splits into two nuclei of $A=120$ with $8.5$ MeV per nucleon:

$$Q=240\times (8.5-7.6)=240\times 0.9=216\text{ MeV}$$

An everyday example. Nuclear power stations in India use heavy uranium nuclei from the falling end of the curve, so splitting them releases energy.

The substance. The nearly constant binding energy per nucleon shows the nuclear force is short-ranged — each nucleon attracts only its nearest neighbours.

The nuclear force is a very strong, short-range attraction between nucleons that hardly depends on charge, and both fission — splitting a heavy nucleus — and fusion — joining light nuclei — release energy because the products have higher binding energy per nucleon.

Nuclear force:

- Much stronger than the electric repulsion between protons at nuclear distances
- Short range — almost zero beyond a few femtometres
- Charge independent — about the same for proton–proton, neutron–neutron and proton–neutron pairs
- Repulsive at very small separations

Fission. A slow neutron splits uranium-235:

$${}_{92}^{235}\text{U}+{}_0^1\text{n}\to {}_{56}^{141}\text{Ba}+{}_{36}^{92}\text{Kr}+3{}_0^1\text{n}$$

releasing about $200$ MeV; the extra neutrons can sustain a chain reaction.

Fusion. Light nuclei such as deuterium and tritium join at extremely high temperatures, releasing $17.6$ MeV per reaction and forming a helium-4 nucleus and a neutron.

Worked example. Fission of $1$ g of uranium-235:

$$N=\frac{6.02\times 10^{23}}{235}\approx 2.56\times 10^{21},E=2.56\times 10^{21}\times 200\times 1.6\times 10^{-13}\approx 8.2\times 10^{10}\text{ J}$$

An everyday example. The Sun shines by fusion in its hot core, while nuclear reactors generate electricity by controlled fission.

The substance. Fusion needs enormous temperatures because nuclei must overcome their electric repulsion before the short-range nuclear force can act.