Composition of the nucleus
A nucleus has two kinds of particles, together called nucleons:
- Protons: charge +e, mass 1.00728 u. Their number is the atomic number Z.
- Neutrons: no charge, mass 1.00866 u. Their number is N.
Mass number A = Z + N. We write a nucleus as ᴬ_Z X, for example ²³⁵₉₂U has 92 protons and 143 neutrons.
The atomic mass unit: 1 u = 1/12 of the mass of a carbon-12 atom = 1.6605 × 10⁻²⁷ kg.
Isotopes, isobars and isotones
- Isotopes: same Z, different N (¹H, ²H, ³H).
- Isobars: same A, different Z (³H and ³He).
- Isotones: same N, different Z (¹⁹⁸Hg and ¹⁹⁷Au both have N = 118).
The neutron was found by James Chadwick in 1932. Neutrons add nuclear pull without adding electric push, so they help keep heavy nuclei together.
Size and density of the nucleus
Scattering experiments show that the volume of a nucleus is proportional to A. So its radius follows
R = R₀ A^(1/3), with R₀ ≈ 1.2 fm (1 fm = 10⁻¹⁵ m, one femtometre or fermi).
Because volume ∝ A and mass ∝ A, the density is the same for all nuclei: about 2.3 × 10¹⁷ kg/m³. That is about 10¹⁴ times the density of water. A teaspoon of nuclear matter would weigh around a billion tonnes. Neutron stars are made of matter this dense.
Nuclear force
Protons push each other apart with a big electric force, yet the nucleus holds. So a stronger attraction must act. This is the nuclear force (strong force). Its features:
- Strongest force we know, much stronger than the electric force at nuclear distances.
- Short range: strong attraction from about 0.8 fm to 2.5 fm; almost zero beyond a few fm.
- Repulsive when nucleons come closer than about 0.8 fm, so the nucleus does not collapse.
- Charge independent: the pull is about the same for p–p, n–n and p–n.
- Saturates: a nucleon pulls only its nearest neighbours, not all nucleons.
The potential energy curve of two nucleons dips to a minimum near 0.8 fm: to the right of the dip the force pulls, to the left it pushes.
Mass–energy relation
Einstein showed that mass is a form of energy: E = mc², where c = 3 × 10⁸ m/s. Because c² is so large, a tiny mass is a huge energy: 1 g of mass equals 9 × 10¹³ J.
For nuclei we use a handy link: 1 u of mass = 931.5 MeV of energy. (1 MeV = 1.6 × 10⁻¹³ J.)
In every nuclear reaction the total of (mass + energy) stays the same. Mass can turn into energy and back.
Mass defect and binding energy
Add up the masses of Z free protons and N free neutrons. Now weigh the nucleus. The nucleus is always lighter. The difference is the mass defect:
Δm = [Z mₚ + (A − Z) mₙ] − M_nucleus
When nucleons join, this mass leaves as energy. To pull them apart again you must give back the same energy. This is the binding energy:
E_b = Δm c² = Δm (in u) × 931.5 MeV.
Tip: if atomic masses are given, use the mass of a hydrogen atom (1.007825 u) in place of mₚ, so the electrons cancel.
Binding energy per nucleon vs mass number
Binding energy per nucleon = E_b / A. It tells how tightly each nucleon is held; bigger means more stable.
Plot it against A and you get a curve with these features:
- Very low for the lightest nuclei (²H: 1.1 MeV), with bumps at ⁴He, ¹²C and ¹⁶O.
- Rises fast and becomes almost flat, about 8 MeV, from A ≈ 30 to 170. The flat part shows that the nuclear force saturates.
- Peak near A = 56 (iron): about 8.8 MeV. Iron is among the most stable nuclei.
- Falls slowly for heavy nuclei (²³⁸U: about 7.6 MeV) because the proton–proton electric push grows.
What it means: a heavy nucleus that breaks into two middle ones, or light nuclei that join into a bigger one, both move towards the peak. The products are more tightly bound, and the extra binding energy is released.
Nuclear fission
Fission: a heavy nucleus splits into two middle-sized nuclei. Example: a slow neutron hits ²³⁵U:
¹n + ²³⁵U → ²³⁶U* → ¹⁴¹Ba + ⁹²Kr + 3 ¹n + about 200 MeV
Why energy? BE/A of U is about 7.6 MeV; of the products about 8.5 MeV. Gain ≈ 0.9 MeV × 235 ≈ 200 MeV.
Chain reaction: the 2–3 new neutrons can split more uranium. In a nuclear reactor it is kept steady: a moderator (heavy water or graphite) slows neutrons, control rods (cadmium or boron) soak up extra neutrons, and a coolant carries heat to make steam for turbines. An uncontrolled chain reaction is the principle of an atom bomb.
Nuclear fusion
Fusion: two light nuclei join to make a heavier one. Example: ²H + ²H → ⁴He + about 23.8 MeV; or ²H + ³H → ⁴He + n + 17.6 MeV.
Both nuclei are positive and push each other away. To get close enough for the nuclear force to act, they must move very fast, so the gas must be at about 10⁷ K or more. That is why it is called thermonuclear fusion.
In the Sun, four hydrogen nuclei finally become one helium nucleus (the proton–proton cycle), releasing about 26.7 MeV. Per kilogram of fuel, fusion gives more energy than fission, and its fuel (hydrogen isotopes) is plentiful, but keeping such a hot gas in place on Earth is still being worked on (for example the ITER project, in which India is a partner).
| Fission | Fusion | |
|---|---|---|
| What happens | Heavy nucleus splits | Light nuclei join |
| Needs | Slow neutron | Very high temperature |
| Waste | Radioactive products | Mostly helium |
| Where | Power reactors | Sun and stars |
Try it: predict, then check
1. In free play, pick ²H, ⁴He, ⁵⁶Fe and ²³⁵U. Before each pick, guess its binding energy per nucleon. Which is the most tightly held?
2. Slide the two nucleons from 4 fm down to 0.5 fm. Write down where the arrow turns from nothing to pull, and from pull to push.
3. At home: take 12 marbles and stick them together with clay in a ball. Count how many neighbours an inside marble touches (about 12) and an outside marble touches (fewer). Surface nucleons are held less tightly, which is why small nuclei have lower binding energy per nucleon.
Key formulas and definitions
- A = Z + N
- R = R₀ A^(1/3), R₀ ≈ 1.2 fm; density ≈ 2.3 × 10¹⁷ kg/m³
- E = mc²; 1 u = 1.6605 × 10⁻²⁷ kg = 931.5 MeV
- Δm = Z mₚ + (A − Z) mₙ − M
- E_b = Δm × 931.5 MeV; BE per nucleon = E_b / A
- Q (energy released) = (mass before − mass after) × 931.5 MeV
Worked examples
1. Find the radius of the iron nucleus ⁵⁶Fe (R₀ = 1.2 fm).
Step 1: R = R₀ A^(1/3). Step 2: 56^(1/3) ≈ 3.83. Step 3: R = 1.2 × 3.83 ≈ 4.6 fm.
2. Find the ratio of the radii of ²⁷Al and ¹²⁵Te.
Step 1: R ∝ A^(1/3). Step 2: R_Al / R_Te = (27/125)^(1/3). Step 3: = 3/5. So the ratio is 3 : 5.
3. How much energy is locked in 1 g of mass?
Step 1: E = mc², m = 1 g = 10⁻³ kg. Step 2: E = 10⁻³ × (3 × 10⁸)² = 10⁻³ × 9 × 10¹⁶. Step 3: E = 9 × 10¹³ J.
4. Show that 1 u is about 931.5 MeV.
Step 1: 1 u = 1.6605 × 10⁻²⁷ kg. Step 2: E = mc² = 1.6605 × 10⁻²⁷ × (2.998 × 10⁸)² ≈ 1.4924 × 10⁻¹⁰ J. Step 3: Divide by 1.602 × 10⁻¹³ J per MeV: E ≈ 931.5 MeV.
5. Find the mass defect, binding energy and binding energy per nucleon of ⁴He. (mₚ = 1.00728 u, mₙ = 1.00866 u, He nucleus = 4.00151 u)
Step 1: Loose parts = 2(1.00728) + 2(1.00866) = 4.03188 u. Step 2: Δm = 4.03188 − 4.00151 = 0.03037 u. Step 3: E_b = 0.03037 × 931.5 ≈ 28.3 MeV. Step 4: Per nucleon = 28.3 / 4 ≈ 7.07 MeV.
6. Find the binding energy per nucleon of ¹⁶O. (atomic masses: ¹H = 1.007825 u, n = 1.008665 u, ¹⁶O = 15.994915 u)
Step 1: 8 ¹H + 8 n = 8.06260 + 8.06932 = 16.13192 u. Step 2: Δm = 16.13192 − 15.994915 = 0.13700 u. Step 3: E_b = 0.13700 × 931.5 ≈ 127.6 MeV. Step 4: Per nucleon = 127.6 / 16 ≈ 7.98 MeV.
7. Each fission of ²³⁵U gives 200 MeV. Find the energy from fission of 1 g of ²³⁵U.
Step 1: Number of nuclei = (1/235) × 6.022 × 10²³ ≈ 2.56 × 10²¹. Step 2: Energy = 2.56 × 10²¹ × 200 MeV = 5.12 × 10²³ MeV. Step 3: In joules = 5.12 × 10²³ × 1.6 × 10⁻¹³ ≈ 8.2 × 10¹⁰ J. (That is about the energy of burning 2.5 tonnes of coal.)
8. The binding energy of ²H is 2.22 MeV and of ⁴He is 28.3 MeV. Find the energy released in ²H + ²H → ⁴He.
Step 1: Energy released = BE of products − BE of reactants. Step 2: = 28.3 − 2 × 2.22 = 28.3 − 4.44. Step 3: = 23.86 MeV ≈ 23.9 MeV.
Common mistakes
- Thinking the nucleus is heavier than its parts. It is always lighter; the missing mass is the binding energy.
- Using R ∝ A instead of R ∝ A^(1/3). Eight times the nucleons make the radius only twice as big.
- Saying a nucleus with more total binding energy is always more stable. Compare binding energy PER NUCLEON, not the total.
- Mixing fission and fusion: fission is heavy splitting (needs a neutron), fusion is light joining (needs very high temperature).