Nuclear reactions — Q-value, cross-section — Chemistry notes

A generic reaction is written: $a + X \longrightarrow Y + b \qquad \text{or compactly} \qquad X(a,b)Y$ where $a$ = projectile, $X$ = target, $Y$ = product nucleus, $b$ = ejected particle.

1. The Q-value — energy bookkeeping

WHY this form?

WHAT we conserve: total energy (rest + kinetic) and momentum.

Write total energy before = total energy after: $\underbrace{(m_a+m_X)c^2}_{\text{rest before}} + \underbrace{(T_a+T_X)}_{\text{KE before}} = \underbrace{(m_Y+m_b)c^2}_{\text{rest after}} + \underbrace{(T_Y+T_b)}_{\text{KE after}}$

Rearrange to isolate the change in kinetic energy: $\underbrace{(T_Y+T_b)-(T_a+T_X)}_{\Delta T} = \big[(m_a+m_X)-(m_Y+m_b)\big]c^2 \equiv Q$

2. Threshold energy — why endoergic reactions need a minimum kick

For $Q<0$ , you might guess you just need to supply $|Q|$ . Wrong — momentum must also be conserved, so the products must keep moving; some incoming energy is "locked up" as center-of-mass motion and is unavailable.

3. Cross-section $\sigma$ — the probability of reaction

Deriving the attenuation law

Setup: a beam of intensity $I$ (particles/area/time) enters a slab. In a thin slice of thickness $dx$ with $n$ nuclei per unit volume, the number of "targets" per unit beam-area is $n\,dx$ , each blocking area $\sigma$ .

Fraction removed in $dx$ : $\frac{-dI}{I} = n\,\sigma\,dx \quad\text{(Why? covered-area fraction = }n\,dx\cdot\sigma\text{)}$

Integrate from $0$ to $x$ : $\int_{I_0}^{I}\frac{dI}{I} = -n\sigma\int_0^x dx \;\Rightarrow\; \ln\frac{I}{I_0} = -n\sigma x$

Reaction rate

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

4. Worked examples

Worked example (a) Q-value of

$^{14}\text{N}(\alpha,p)^{17}\text{O}$ Masses (u): $^{14}\text{N}=14.003074$ , $\alpha=4.002603$ , $^{17}\text{O}=16.999132$ , $p=1.007825$ . $\Delta m = (14.003074+4.002603)-(16.999132+1.007825) = -0.001280\ \text{u}$ Why negative? Products are heavier → mass gained → endoergic. $Q = -0.001280 \times 931.5\ \tfrac{\text{MeV}}{\text{u}} = -1.19\ \text{MeV}$ This was Rutherford's first artificial transmutation — and it's endoergic, so $\alpha$ 's needed enough energy.

Worked example (c) Cross-section → attenuation

Thermal neutrons hit boron, $\sigma=755$ b for $^{10}\text{B}(n,\alpha)$ . Number density $n=1.3\times10^{23}\ \text{cm}^{-3}$ . Find thickness halving the beam. $\Sigma = n\sigma = 1.3\times10^{23}\times755\times10^{-24} = 98.2\ \text{cm}^{-1}$ $I/I_0 = \tfrac12 = e^{-\Sigma x}\Rightarrow x = \frac{\ln 2}{\Sigma}=\frac{0.693}{98.2}=7.1\times10^{-3}\ \text{cm}.$ Why $\ln 2$ ? Halving means $e^{-\Sigma x}=\tfrac12$ . A huge $\sigma$ → tiny half-thickness → great neutron absorber.

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_{th}=-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^{-n\sigma x}

, with n = nuclei per volume.

Macroscopic cross-section & mean free path

\Sigma=n\sigma

(cm⁻¹); mean free path = 1/Σ.

Reaction rate per nucleus

R=\sigma\phi

where φ = flux = nv.

Compact reaction notation X(a,b)Y means

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

Recall Feynman: explain to a 12-year-old

Throwing balls at targets. The Q-value is like a piggy bank: when the balls and targets snap together into a tidier shape, some "mass" turns into energy and pops out (that's the money you get). Sometimes the new shape is heavier, so you must pay in energy — and you have to pay extra because the pieces fly off afterwards and can't fully stop. The cross-section is just how big the target looks: a bigger target is easier to hit, so the reaction happens more often. Some targets look way bigger than they really are because of a quantum "magnet" effect.

Connections

Binding energy per nucleon curve — sign of $Q$ follows from moving toward the peak.
Mass defect & E=mc² — the engine behind every Q-value.
Nuclear fission and Nuclear fusion — both exploit $Q>0$ .
Neutron flux & reactor physics — uses $\Sigma=n\sigma$ and $R=\sigma\phi$ .
Conservation laws in collisions — gives threshold via CM frame.
Radioactive decay kinetics — contrast: decay is spontaneous, reactions need a projectile.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, nuclear reaction matlab do nuclei ka collision jisme particles rearrange ho jaate hain, jaise $a+X \to Y+b$ . Yahan do cheezein important hain. Pehli — Q-value: ye batata hai ki reaction me energy nikli (release hui) ya lagi (absorb hui). Formula simple hai: reactants ka total mass minus products ka total mass, into $c^2$ . Agar mass kam ho gaya (products halke), to wo missing mass energy ban ke nikal aati hai — ye exoergic, $Q>0$ . Agar products heavy ho gaye, to humein energy deni padti hai — endoergic, $Q<0$ . Yaad rakho: 1 u = 931.5 MeV.

Ab ek twist: endoergic reaction me students sochte hain ki bas $|Q|$ jitni energy de do. Galat! Momentum bhi conserve karna padta hai, isliye reaction ke baad products aage move karte rehte hain, aur thodi energy uss "getaway" me chali jaati hai. Isliye actual threshold energy thodi zyada hoti hai: $E_{th}=|Q|\,(m_a+m_X)/m_X$ . Isko "tax on the escape" samjho.

Dusri cheez — cross-section $\sigma$ : ye reaction hone ki probability ka measure hai, units me area (barn, $10^{-24}\,cm^2$ ). Socho target nucleus ek dartboard hai; jitna bada bullseye dikhega, utni asaani se dart lagega. Lekin yahan quantum effect ki wajah se kabhi target apne real size se hazaar guna bada bhi dikh sakta hai (resonance). Jab beam ek slab me ghusti hai to intensity exponentially kam hoti hai: $I=I_0 e^{-n\sigma x}$ . Ye reactor physics me bahut kaam aata hai — boron jaise high- $\sigma$ material neutrons ko turant rok dete hain. Bas ye samajh lo: Q = energy ka hisaab, $\sigma$ = chance ka hisaab.

Nuclear reactions — Q-value, cross-section

1. The Q-value — energy bookkeeping

WHY this form?

2. Threshold energy — why endoergic reactions need a minimum kick

3. Cross-section $\sigma$ — the probability of reaction

Deriving the attenuation law

Reaction rate

4. Worked examples

5. Common mistakes (steel-manned)

6. Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Nuclear & Radiochemistry

Connections