Question bank — Radioactive decay — alpha, beta, gamma — mechanisms

True — alpha 2 protons + 2 neutrons le jaata hai, isliye nucleon aur charge conservation se $A\to A-4$ aur $Z\to Z-2$, koi exception nahi.

True — ek neutron us hi nucleus ke andar proton ban jaata hai, isliye total nucleon count unchanged rehta hai; sirf $Z$ shift hota hai $+1$ se.

False — $\gamma$ emission sirf energy shed karta hai; $A$ aur $Z$ dono unchanged rehte hain, isliye ye same element ka same isotope hai.

False — $Q<0$ ka matlab hai products parent se bhaari hain, isliye decay mein energy lagti hai aur spontaneously forbidden hai; time us energy ko supply nahi kar sakta.

False — electron usi waqt create hota hai jab $n\to p+e^{-}+{\bar{\nu }}_e$; nucleus mein pehle se koi electron nahi baitha hota.

Zyaadatar true hai lekin subtle hai — kinematics ek single central $K{E}_{\alpha }$ fix karta hai (ek discrete line, figure s03 dekho), lekin real lines mein emitting atoms ke recoil broadening aur thermal Doppler motion se thodi finite width hoti hai.

False — momentum conserve karne ke liye daughter ko recoil karna hota hai, isliye $K{E}_{\alpha }=Q{m}_Y/(m_Y+m_{\alpha })$, jo $Q$ se thoda kam hai.

False — zyaada alpha energy ka matlab Coulomb barrier patla/chhota hai jisme tunnel karna aasaan hai, jo bahut chhoti half-life deta hai (Geiger–Nuttall).

False — $\gamma$ fixed nuclear levels ke beech two-body de-excitation hai, isliye $E_{\gamma }=E_i-E_f$ discrete lines deta hai.

False — ${\beta }^{+}$ mein ek proton neutron ban jaata hai, isliye $Z\to Z-1$; $Z$ badhata hai ${\beta }^{-}$ se.

False — electron capture (K-capture) bhi same kaam karta hai $Z\to Z-1$ ek inner orbital electron niglake, $p+e^{-}\to n+{\nu }_e$, aur ye akela option hai jab $Q<2m_e{c}^2$ ho (positron create karne ke liye energy kaafi nahi).

Galat particle — ${\beta }^{-}$ mein antineutrino ${\bar{\nu }}_e$ emit hota hai; ${\nu }_e$ (matter neutrino) ${\beta }^{+}$ ke saath hota hai, jaise lepton-number bookkeeping mein zaroori hai.

Error — $Z$ 2 se ghatta hai, isliye Thorium ka $Z=90$ hai; alpha 2 units of charge le gaya.

Motive ki galti — ye observed continuous beta spectrum se forced tha, jo two-body kinematics produce nahi kar sakta; ye energy, momentum aur angular-momentum conservation bhi restore karta hai.

Ulta hai — $\gamma$ tab emit hota hai jab excited daughter $Y^{\ast }$ ek lower level par aake energy khoata hai, $E_{\gamma }=E_i-E_f>0$.

Error — ${}_2^4$He clump ki binding energy unusually high hai (~7.07 MeV/nucleon), aur yahi cheez $Q$ ko positive banati hai; akela proton eject karne se usually $Q<0$ milta hai.

Error — create hone wala proton nucleus mein $+e$ add karta hai, jo electron ke $-e$ ko exactly cancel karta hai; net charge conserved rahta hai.

Ulta hai — bhaari aur doubly charged hone ki wajah se alphas strongly ionise karte hain aur turant ruk jaate hain (paper rok leta hai); $\gamma$ sabse gehre penetrate karta hai.

Error — wo instead apni energy directly ek inner orbital electron ko de sakta hai, use eject karke (internal conversion); energy ek monoenergetic conversion electron ke roop mein jaati hai, photon ke roop mein nahi.

Beta ek three-body decay hai (daughter + electron + ${\bar{\nu }}_e$), isliye released energy kisi bhi proportion mein share ho sakti hai; alpha two-body hai, isliye kinematics ek fixed energy pin karta hai (figure s02 vs s03).

Energy share karne ke tarike (phase space) range ke across badhte phir girte hain, aur yeh statistical weighting — Fermi function se multiply hoke, jo ek Coulomb correction hai jo emitted electrons ko wapas kheenchta / positrons ko door dhakelta hai — observed curve sculpt karta hai.

Classically alpha ke paas Coulomb barrier (figure s01) chadne ki energy nahi hai; quantum mechanics use wall ke andar se paas hone ki thodi probability deta hai, aur yahi poora mechanism hai.

Parent ke rest mein hone par, momentum conservation deta hai $0=p_{\alpha }-p_Y$, isliye $p_{\alpha }=p_Y\equiv p$; kyunki $KE=p^2/2m$ aur dono same $p$ share karte hain, kinetic energy $1/m$ ke saath scale karti hai, isliye bhaari daughter ($m_Y\gg m_{\alpha }$) chhota hissa leti hai jabki $K{E}_{\alpha }=Q{m}_Y/(m_Y+m_{\alpha })$.

Beta decay mein ek quark flavour convert hota hai (neutron↔proton) aur leptons create hote hain — ye processes sirf weak force mediate karta hai; alpha nuclear + electromagnetic forces se intact clump par governed hai.

Daughter aksar ground state mein nahi balki excited nuclear level mein land karta hai, phir leftover energy photon ke roop mein shed karke de-excite hota hai.

$Q=(m_{\text{parent}}-m_{\text{products}})c^2$ jahan $c$ speed of light hai; sirf tabhi jab parent bhaari ho, freed rest-mass-energy kinetic energy ke roop mein appear ho sakti hai — warna nature ko energy supply karni padti.

${}^{14}\text{C}$ neutron-rich hai, isliye ek neutron ko proton mein convert karna (${\beta }^{-}$, $Z:6\to 7$) use stability ki line ki taraf le jaata hai.

Nahi — zero freed energy se products ke liye koi kinetic energy nahi hai, isliye transition threshold par hai aur spontaneous nahi hai.

Koi daughter nahi hogi — aise "decay" mein kuch peeche nahi bachega, isliye ye alpha decay nahi hai; alpha decay ke liye ek residual ${}_{Z-2}^{A-4}Y$ nucleus chahiye.

Endpoint $\approx Q$ hai, wo limiting case mein reach hota hai jab antineutrino essentially zero energy le jaata hai.

Sirf electron capture, ${\beta }^{+}$ nahi: positron banana kam se kam $2m_e{c}^2$ rest energy maangta hai (daughter mein electron create + positron emit), isliye us threshold ke neeche nucleus ko instead ek $K$-shell electron absorb karna padta hai, $p+e^{-}\to n+{\nu }_e$.

Ek inner-shell vacancy reh jaati hai, isliye outer electrons cascade down karte hain aur characteristic X-rays (ya Auger electrons) emit karte hain — jo EC ko ${\beta }^{+}$ se alag karne wala fingerprint hai.

Haan — momentum conservation ek tiny recoil force karta hai, isliye $E_{\gamma }=(E_i-E_f)$ minus ek small recoil term; recoil usually negligible hota hai lekin exactly zero kabhi nahi.

Agar kaafi energy available ho toh wo ek nucleon ya chhota cluster bhi emit kar sakta hai, lekin ordinary excited daughters ke liye competition $\gamma$-emission aur internal conversion ke beech hoti hai (dono $A,Z$ unchanged chodh dete hain).

Wo break ho jaata — electron ka lepton number $+1$ hai bina kisi partner ke, isliye ${\bar{\nu }}_e$ (lepton number $-1$) zaroori hai total zero rakhne ke liye.

Nahi — ek free neutron ${\beta }^{-}$-decay karta hai ($n\to p+e^{-}+{\bar{\nu }}_e$) $Q>0$ ke saath aur ~15-minute mean life ke saath, bilkul same mechanism jaise bound-neutron beta decay mein hota hai.