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

5.2.1 · Chemistry › Nuclear & Radiochemistry

1. The N/Z ratio

KYA HAI: $N$ = neutrons ki sankhya, $Z$ = protons ki sankhya. Ratio $N / Z$ batata hai ki nucleus mein neutron "padding" sahi hai ya nahi.

BAND UPAR KYO CURVE KARTA HAI (heavy nuclei ke liye N/Z > 1):

Strong force sirf nearest neighbours tak pahunchta hai (range ~ 1–2 fm). Yeh saturate karta hai.
Coulomb repulsion har ek pair of protons ke beech act karta hai, isliye yeh roughly $Z^{2}$ ke saath badhta hai — long range.
Jaise $Z$ badhta hai, total repulsion glue se faster badhti hai. Compensate karne ke liye, heavy nuclei ko extra neutrons chahiye jo attractive strong force add karte hain bina koi charge add kiye.

2. Magic numbers

KYUN (shell-model analogy):

Jaise electrons atomic shells fill karte hain aur ek full shell (noble gas) chemical stability deta hai, waise nucleons nuclear energy shells fill karte hain.
Ek closed nuclear shell matlab hai ki agla nucleon bahut zyada energy level par baithega — ek bada gap. Isliye magic nuclei tightly bound hote hain aur react karne se reluctant rehte hain.

3. Mass defect and Binding Energy

BINDING ENERGY KAISE NIKALTE HAIN — first principles se: Einstein: energy aur mass ek hi cheez hai, $E = m c^{2}$ . Jo mass gayab hui woh binding energy ban gayi:

Binding energy per nucleon (asli stability metric):

4. The Binding-Energy-per-Nucleon Curve

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

Flashcards

Heavy stable nuclei ko N/Z > 1 kyun chahiye?

Coulomb repulsion ~Z² ke saath badhta hai (long range, har proton pair) jabki strong force saturate karta hai (short range); extra neutrons bina charge add kiye glue add karte hain.

Ek nucleus band of stability ke upar hai — kaisa decay hoga?

β⁻ decay (

n \to p + e^{-} + \overset{ν}{ˉ}

), jo N/Z ko lower karta hai.

Ek nucleus band of stability ke neeche hai — kaisa decay hoga?

β⁺ decay ya electron capture (

p \to n + e^{+} + ν

), jo N/Z ko raise karta hai.

Magic numbers list karo.

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

Doubly magic nucleus kya hota hai? Ek example do.

Z aur N dono mein magic; jaise

^{4}

He,

^{16}

^{208}

Pb.

Mass defect define karo.

Δ m = (Z m_{p} + N m_{n}) - M_{n u c l e u s}

; parts bound nucleus se zyada wazan karte hain.

1 u se energy ka conversion?

1 u = 931.5

MeV.

Binding energy ko A se divide kyun karte hain?

Yeh compare karne ke liye ki har nucleon kitna tightly held hai; per-nucleon value stability reflect karta hai, sirf size nahi.

Sabse zyada BE per nucleon kis nucleus ka hai?

^{56}

Fe (~8.8 MeV/nucleon), sabse stable.

Fusion aur fission DONO energy kyun release karte hain?

Dono nuclei ko Fe peak ki taraf move karte hain, BE per nucleon increase karta hai.

Kis Z ke upar koi stable nucleus nahi hai?

Z = 83 (Bismuth ke upar).

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho nucleus ek ball pit hai jisme do tarah ke bachche bhare hain: "plus" bachche jo sabko door dhakelte hain, aur "zero" bachche jo chupchap apne paas wale se haath milate hain. Agar bahut zyada plus bachche hain aur zero bachche kam hain, toh pit blast ho jaata hai — isliye badi bheed mein beech mein bahut saare shant zero bachche chahiye. Kuch crowd sizes (2, 8, 20, 28, 50, 82, 126) bilkul perfectly fit hote hain jaise ek fully packed lunchbox — woh super sturdy hote hain. Aur yeh magic trick hai: jab bachche haath milate hain toh woh thoda halke ho jaate hain, aur woh lost weight wahi "stickiness" ban jaata hai jo unhe saath pakde rakhti hai. Iron-56 sabse comfortable, sabse settled crowd hai.

Connections

Radioactive decay — alpha, beta, gamma (decay type N/Z position se predict hota hai)
Nuclear fission and fusion (BE/nucleon curve se energy release)
Radiocarbon dating ( $^{14}$ C β⁻ example)
Shell model of the nucleus (magic numbers ki origin)
Einstein mass-energy equivalence (binding energy ke peeche $E = m c^{2}$ )
Half-life and decay kinetics (unstable nuclei kitni tezi se decay karte hain)