Nuclear reactions — Q-value, cross-section

5.2.5 · Chemistry › Nuclear & Radiochemistry

Ek generic reaction aise likhi jaati hai: $a + X ⟶ Y + b or compactly X (a, b) Y$ jahan $a$ = projectile, $X$ = target, $Y$ = product nucleus, $b$ = ejected particle.

1. The Q-value — energy bookkeeping

WHY this form?

KYA conserve hota hai: total energy (rest + kinetic) aur momentum.

Pehle ki total energy = baad ki total energy likho: $rest before (m_{a} + m_{X}) c^{2} + KE before (T_{a} + T_{X}) = rest after (m_{Y} + m_{b}) c^{2} + KE after (T_{Y} + T_{b})$

Kinetic energy mein change isolate karne ke liye rearrange karo: $Δ T (T_{Y} + T_{b}) - (T_{a} + T_{X}) = [(m_{a} + m_{X}) - (m_{Y} + m_{b})] c^{2} \equiv Q$

2. Threshold energy — endoergic reactions ko minimum kick kyon chahiye

$Q < 0$ ke liye, tumhara andaza ho sakta hai ki bas $∣ Q ∣$ supply karo. Galat — momentum bhi conserve hona chahiye, isliye products ko chalta rehna padega; kuch incoming energy center-of-mass motion ke roop mein "lock" ho jaati hai aur use nahi ki ja sakti.

3. Cross-section $σ$ — reaction ki probability

Deriving the attenuation law

Setup: intensity $I$ ki beam (particles/area/time) ek slab mein enter karti hai. $d x$ thickness ke thin slice mein jo $n$ nuclei per unit volume hai, beam-area per unit "targets" ki sankhya $n d x$ hai, har ek blocking area $σ$ ke saath.

$d x$ mein fraction jo hata diya jaaye: $\frac{- d I}{I} = n σ d x (Why? covered-area fraction = n d x \cdot σ)$

$0$ se $x$ tak integrate karo: $\int_{I_{0}}^{I} \frac{d I}{I} = - nσ \int_{0}^{x} d x \Rightarrow ln \frac{I}{I _{0}} = - nσ x$

Reaction rate

Figure — Nuclear reactions — Q-value, cross-section

4. Worked examples

5. Common mistakes (steel-manned)

6. Active recall

Q-value definition (mass form)

Q = [(m_{a} + m_{X}) - (m_{Y} + m_{b})] c^{2}

; energy released = lost rest mass.

Sign of Q for exoergic vs endoergic

Q>0 exoergic (mass lost, energy out); Q<0 endoergic (mass gained, energy in).

Q in terms of binding energy

Q = (B_{Y} + B_{b}) - (B_{X} + B_{a})

; exoergic if products more tightly bound.

Threshold energy formula

E_{t h} = - Q (m_{a} + m_{X}) / m_{X}

(target at rest).

Why threshold exceeds |Q|

Momentum conservation locks some energy into CM motion, unavailable for reaction.

Conversion factor mass→energy

1 u = 931.5 MeV/c².

Cross-section unit

1 barn = 10⁻²⁸ m² = 10⁻²⁴ cm².

Physical meaning of σ

Effective target area = probability measure for the reaction (not literal size).

Beam attenuation law

I = I_{0} e^{- nσ x}

, with n = nuclei per volume.

Macroscopic cross-section & mean free path

Σ = nσ

(cm⁻¹); mean free path = 1/Σ.

Reaction rate per nucleus

R = σ ϕ

where φ = flux = nv.

Compact reaction notation X(a,b)Y means

target X + projectile a → product Y + ejected b.

Recall Feynman: 12-saal ke bacche ko samjhao

Socho balls target pe phenkna. Q-value ek piggy bank ki tarah hai: jab balls aur targets milke ek sundar shape banate hain, kuch "mass" energy mein badal jaati hai aur bahar aati hai (woh hai tumhara paisa). Kabhi kabhi nayi shape bhaari hoti hai, toh tumhe energy daalni padti hai — aur thoda extra dena padta hai kyunki baad mein tukde udte rehte hain aur poori tarah ruk nahi sakte. Cross-section bas yeh hai ki target kitna bada dikhta hai: bada target hit karna aasaan hai, toh reaction zyada hoti hai. Kuch targets apni real size se bahut bade dikhte hain ek quantum "magnet" effect ki wajah se.

Connections

Binding energy per nucleon curve — $Q$ ka sign curve ke peak ki taraf move karne se follow karta hai.
Mass defect & E=mc² — har Q-value ke peeche yahi engine hai.
Nuclear fission aur Nuclear fusion — dono $Q > 0$ exploit karte hain.
Neutron flux & reactor physics — $Σ = nσ$ aur $R = σ ϕ$ use karta hai.
Conservation laws in collisions — CM frame ke zariye threshold deta hai.
Radioactive decay kinetics — contrast: decay spontaneous hoti hai, reactions ko projectile chahiye.