Nuclear stability — N - Z ratio, magic numbers, binding energy per nucleon

1. The N/Z ratio

WHAT: $N$ = number of neutrons, $Z$ = number of protons. The ratio $N/Z$ tells us whether a nucleus has the right neutron "padding".

WHY the band curves upward (N/Z > 1 for heavy nuclei):

The strong force only reaches its nearest neighbours (range ~ 1–2 fm). It saturates.
The Coulomb repulsion acts between every pair of protons, so it grows roughly as $Z^2$ — long range.
As $Z$ increases, total repulsion grows faster than the glue. To compensate, heavy nuclei need extra neutrons which add attractive strong force without adding any charge.

2. Magic numbers

WHY (the shell-model analogy):

Just like electrons fill atomic shells and a full shell (noble gas) gives chemical stability, nucleons fill nuclear energy shells.
A closed nuclear shell means the next nucleon would have to sit in a much higher energy level — a big gap. So magic nuclei are tightly bound and reluctant to react.

3. Mass defect and Binding Energy

HOW to get binding energy — from first principles: Einstein: energy and mass are the same thing, $E=mc^2$ . The mass that vanished became binding energy:

Binding energy per nucleon (the real stability metric):

Worked example Binding energy of

$^4_2\text{He}$ Use $m_p = 1.00728$ u, $m_n = 1.00866$ u, $M(^4\text{He nucleus}) = 4.00150$ u.

Mass of parts: $2(1.00728) + 2(1.00866) = 4.03188$ u. Why? sum of 2 protons + 2 neutrons.
$\Delta m = 4.03188 - 4.00150 = 0.03038$ u. Why? mass defect = parts − whole.
$E_B = 0.03038 \times 931.5 \approx 28.3$ MeV. Why? convert mass to energy.
$\bar B = 28.3/4 \approx 7.07$ MeV/nucleon. Why? per-nucleon to compare with others. This high value is why He-4 (the α particle) is so stable.

4. The Binding-Energy-per-Nucleon Curve

Figure — Nuclear stability — N - Z ratio, magic numbers, binding energy per nucleon

Flashcards

Why do heavy stable nuclei need N/Z > 1?

Coulomb repulsion grows as ~Z² (long range, every proton pair) while the strong force saturates (short range); extra neutrons add glue without adding charge.

A nucleus lies above the band of stability — what decay?

β⁻ decay (

n\to p+e^-+\bar\nu

), which lowers N/Z.

A nucleus lies below the band of stability — what decay?

β⁺ decay or electron capture (

p\to n+e^++\nu

), which raises N/Z.

List the magic numbers.

2, 8, 20, 28, 50, 82, 126.

What is a doubly magic nucleus? Give one example.

Magic in both Z and N; e.g.

^4

He,

^{16}

^{208}

Pb.

Define mass defect.

\Delta m = (Z m_p + N m_n) - M_{nucleus}

; the parts weigh more than the bound nucleus.

Conversion of 1 u to energy?

1\,\text{u} = 931.5

MeV.

Why divide binding energy by A?

To compare how tightly each nucleon is held; per-nucleon value reflects stability, not just size.

Which nucleus has the highest BE per nucleon?

^{56}

Fe (~8.8 MeV/nucleon), the most stable.

Why do BOTH fusion and fission release energy?

Both move nuclei toward the Fe peak, increasing BE per nucleon.

Above which Z is there no stable nucleus?

Z = 83 (above Bismuth).

Recall Feynman: explain to a 12-year-old

Imagine the nucleus is a ball pit packed with two kinds of kids: "plus" kids who push everyone away, and "zero" kids who quietly hold hands with whoever is next to them. If there are too many plus kids and not enough zero kids, the pit explodes apart — so big crowds need lots of calm zero kids in between. Some crowd sizes (2, 8, 20, 28, 50, 82, 126) just fit perfectly like a fully packed lunchbox — those are super sturdy. And here's the magic trick: when kids hold hands they actually become a little lighter, and that lost weight turned into the "stickiness" holding them together. Iron-56 is the comfiest, most settled crowd of all.

Connections

Radioactive decay — alpha, beta, gamma (decay type predicted by N/Z position)
Nuclear fission and fusion (energy release from the BE/nucleon curve)
Radiocarbon dating ( $^{14}$ C β⁻ example)
Shell model of the nucleus (origin of magic numbers)
Einstein mass-energy equivalence ( $E=mc^2$ behind binding energy)
Half-life and decay kinetics (how fast unstable nuclei decay)

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, nucleus ke andar do cheezein ladd rahi hain: protons jo ek doosre ko electric force se door dhakelte hain, aur strong nuclear force jo paas waale nucleons ko chipka ke rakhta hai. Strong force sirf nearest padosi tak kaam karta hai (saturate ho jaata hai), lekin proton-proton repulsion poore nucleus mein, har pair ke beech kaam karta hai aur Z² ke saath badhta hai. Isliye jaise-jaise nucleus bhaari hota hai, usse extra neutrons chahiye jo bina charge badhaaye glue add karein — yahi reason hai ki heavy stable nuclei ka N/Z ratio ~1.5 ho jaata hai jabki light nuclei ka ~1.

Stability ka "sweet spot" ek patli curve hoti hai jise band of stability kehte hain. Agar nucleus band ke upar hai (zyada neutrons) to β⁻ decay karega (neutron → proton), aur neeche hai (kam neutrons) to β⁺ ya electron capture karega. Z = 83 ke baad koi stable nucleus nahi bachta, wahan alpha decay hota hai. Kuch khaas counts — 2, 8, 20, 28, 50, 82, 126 (magic numbers) — pe shells full ho jaati hain, bilkul noble gas ki tarah, isliye woh nuclei extra stable hote hain. $^{208}$ Pb doubly magic hai, isiliye decay chains wahin ruk jaati hain.

Stability ko number mein measure karne ke liye binding energy per nucleon use karte hain. Bound nucleus apne alag-alag protons-neutrons se thoda halka hota hai — yeh "mass defect" $E=mc^2$ se energy ban jaata hai (1 u = 931.5 MeV). Per nucleon isliye divide karte hain taaki size ka effect hat jaaye aur pata chale har nucleon kitni tightly bound hai. Yeh curve $^{56}$ Fe pe peak karti hai — Iron sabse stable hai. Iske left mein nuclei jud ke (fusion) aur right mein nuclei toot ke (fission) energy release karte hain, dono peak ki taraf chadhte hain. Yahi reason hai ki suraj fusion se aur reactor fission se energy dete hain.

Test yourself — Nuclear & Radiochemistry

Connections