Fusion — D-T reaction, solar fusion (p-p chain), tokamak - ICF

WHY does fusion release energy?

WHY a curve? Plot $B/A$ (binding energy per nucleon) vs mass number $A$ . It rises steeply for light nuclei, peaks near $A \approx 56$ (iron/nickel, $B/A \approx 8.8$ MeV), then slowly falls.

Moving toward the peak from the left (fusing light nuclei) → energy out → fusion.
Moving toward the peak from the right (splitting heavy nuclei) → energy out → fission.

Figure — Fusion — D-T reaction, solar fusion (p-p chain), tokamak - ICF

Deriving the energy released (Q-value) from scratch

Worked Example 1 — The D–T reaction

$^2_1\text{H} + ^3_1\text{H} \rightarrow ^4_2\text{He} + ^1_0\text{n}$

Masses (atomic mass units, $1\,u = 931.5$ MeV/ $c^2$ ):

$m(^2\text{H}) = 2.014102\,u$
$m(^3\text{H}) = 3.016049\,u$
$m(^4\text{He}) = 4.002602\,u$
$m(n) = 1.008665\,u$

Step 1 — sum reactants. $2.014102 + 3.016049 = 5.030151\,u$ . Why this step? Q depends only on initial vs final total mass.

Step 2 — sum products. $4.002602 + 1.008665 = 5.011267\,u$ .

Step 3 — mass defect. $\Delta m = 5.030151 - 5.011267 = 0.018884\,u$ . Why this step? The vanished mass becomes energy.

Step 4 — convert. $Q = 0.018884 \times 931.5 \approx \mathbf{17.6\ MeV}$ .

Step 5 — split the energy. Momentum conservation: the lighter neutron carries more KE. $E_n : E_{\alpha} = m_\alpha : m_n \approx 4 : 1 \;\Rightarrow\; E_n \approx 14.1\,\text{MeV}, \quad E_\alpha \approx 3.5\,\text{MeV}$ Why this step? In a reaction starting near rest, $p_n = p_\alpha$ , so $E = p^2/2m$ gives KE inversely proportional to mass. The 14 MeV neutron is what breeds tritium and heats the blanket; the 3.5 MeV $\alpha$ stays to keep the plasma hot.

Solar fusion — the proton–proton (p–p) chain

The Sun is too cool for the easy D–T reaction; it must fuse plain protons. Net result: $4\,\mathrm{^1_1H} \to \mathrm{^4_2He} + 2\,e^+ + 2\,\nu_e + \text{energy} \;(\approx 26.7\ \text{MeV})$

Worked Example 2 — Net solar Q-value

$Q = \big[4\,m(\text{}^1\text{H}) - m(\text{}^4\text{He})\big]c^2 - (\text{positron annihilation accounting})$ Using $4(1.007825) - 4.002602 = 0.028698\,u$ , $Q \approx 0.028698 \times 931.5 \approx 26.7$ MeV. Why this step? Neutrinos escape carrying ~2% away; the rest (~26.7 MeV per helium) is what we feel as sunlight.

HOW do we make fusion happen on Earth? The Coulomb barrier

Two engineering routes

Common mistake Steel-manning the common errors

"Fusion happens because nuclei attract each other, so it's easy." Why it feels right: the strong force IS attractive at short range, and fusion does release energy. The fix: before the strong force can grab, the positive charges repel (Coulomb barrier). You need huge KE/temperature just to get close enough — that's the hard part. Energy-favourable ≠ easy to start.

"Iron can release energy by fusion too." Why it feels right: fusion sounds like it always gives energy. The fix: $\text{Fe}$ sits at the peak of $B/A$ . Fusing past iron costs energy. Fusion only releases energy for $A$ below the peak.

"The Sun runs the D–T reaction." Why it feels right: D–T is the famous "fusion reaction." The fix: the Sun starts from ordinary protons (p–p chain). D–T is our Earth-lab choice because it has the largest cross-section at achievable temperatures.

"All 17.6 MeV from D–T can be captured as useful charged-particle energy." Why it feels right: total Q is 17.6 MeV. The fix: ~14.1 MeV leaves as a neutral neutron (no magnetic confinement) — it must be slowed in a blanket; only the 3.5 MeV $\alpha$ heats the plasma directly.

Recall Feynman: explain to a 12-year-old

Imagine tiny balls that really want to stick together and, when they do, they fling out a burst of energy — like two strong magnets snapping shut. The catch: the balls also have a force-field that pushes them apart, so you have to throw them at each other crazy-fast (super hot) to get past the "no entry" zone. The Sun does this with hydrogen because it's gigantic and patient. On Earth we either build a magnetic doughnut to hold the hot gas (tokamak) or crush a fuel pellet with lasers (ICF) to force the balls together for an instant.

Flashcards

What property of the binding-energy-per-nucleon curve makes fusion exothermic?

B/A

rises toward the peak at

A\approx56

; light nuclei fusing move "uphill" in binding, releasing the difference as energy.

Write the D–T fusion reaction.

{^2_1}\text{H} + {^3_1}\text{H} \rightarrow {^4_2}\text{He} + {^1_0}\text{n}

, Q ≈ 17.6 MeV.

How is the 17.6 MeV of D–T split between products?

~14.1 MeV to the neutron, 3.5 MeV to the $\alpha$ (KE inversely proportional to mass since momenta are equal). Why does the neutron carry most of the D–T energy? ::: Momentum conservation: $p_n = p_\alpha$ and $E = p^2/2m$ , so the lighter neutron gets KE in ratio $m_\alpha : m_n \approx 4:1$ . Net reaction of solar fusion? ::: $4\,\mathrm{^1H} \to \mathrm{^4He} + 2e^+ + 2\nu_e + \approx 26.7\,\mathrm{MeV}$ MeV. Which step of the p–p chain is the bottleneck and why? ::: $\text{p + p} \to \,^2\text{H} + \text{e}^+ + \nu$ — it needs a weak-interaction β-conversion, extremely slow, giving the Sun its long lifetime. What is the Coulomb barrier and why does it matter? ::: $U_C = \frac{Z_1Z_2e^2}{4\pi\varepsilon_0 r_0}$ ; nuclei must overcome this electrostatic repulsion (via high T + tunnelling) before the strong force binds them. State the Lawson/triple-product criterion. ::: $n\,T\,\tau_E$ must exceed a threshold (

3\times10^{21}

keV·s·m⁻³ for D–T) for net energy gain.

Tokamak vs ICF strategy in one phrase each?

Tokamak = low density, long confinement (magnetic doughnut); ICF = ultra-high density, ultra-short confinement (laser-imploded pellet).

Why is D–T preferred on Earth over p–p?

D–T has the largest cross-section at achievable temperatures; p–p's weak step is far too slow for a reactor.

Define mass defect.

\Delta m = (Zm_p + Nm_n) - m_{\text{nucleus}}

; the missing mass equals the binding energy via

B=\Delta m\,c^2

Connections

Binding Energy per Nucleon Curve
Nuclear Fission (other side of the same $B/A$ peak)
Mass-Energy Equivalence E=mc^2
Beta Decay & the Weak Interaction (drives p–p step 1)
Maxwell–Boltzmann Distribution (high-energy tail enables fusion)
Quantum Tunnelling (sub-barrier fusion)
Plasma Physics & Magnetic Confinement
Stellar Nucleosynthesis

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, fusion ka matlab hai do chhote nuclei ko jod kar ek bada nucleus banana. Magic ye hai ki binding energy per nucleon ka curve light nuclei se shuru hokar iron (Fe-56) tak upar chadhta hai. Iska matlab — jab hum hydrogen jaise halke nuclei ko jodte hain, product zyada tightly bound hota hai, aur bachi hui "mass" energy ban kar nikal jaati hai, $E=mc^2$ ke through. D–T reaction mein deuterium aur tritium milkar helium aur ek neutron dete hain, aur poora 17.6 MeV nikalta hai — usme se 14.1 MeV neutron le jaata hai aur 3.5 MeV alpha. Neutron zyada energy isliye le jaata hai kyunki wo halka hai (momentum same, par $E=p^2/2m$ ).

Suraj mein D–T nahi, balki saadhe protons fuse hote hain — p–p chain. Pehla step (proton + proton → deuterium) weak interaction se chalta hai, jo bohot slow hai. Isiliye Sun arbon saal tak jalta rehta hai: trigger slow hai par fuel itna zyada hai ki total power bohot bada banta hai. Net mein 4 hydrogen milkar 1 helium dete hain aur ~26.7 MeV.

Earth pe problem ye hai ki nuclei dono positive hote hain, to Coulomb repulsion (barrier) unhe paas aane nahi deta. Isko paar karne ke liye bohot high temperature chahiye (taaki kuch fast particles tunnel kar saken). Do tareeke hain: Tokamak — ek magnetic doughnut jisme garam plasma ko field lines pakad ke rakhti hain (low density, lambi confinement time); aur ICF — laser se ek chhoti pellet ko chaaron taraf se daba kar bohot density tak crush karna (high density, bohot short time). Dono ke liye Lawson criterion kehta hai ki $n \cdot T \cdot \tau_E$ ek threshold se upar hona chahiye, tabhi net energy gain milega. Yaad rakho: fusion energetically favourable hai, par start karna mushkil hai — yahi exam ka favourite trap hai.

Fusion — D-T reaction, solar fusion (p-p chain), tokamak - ICF

WHY does fusion release energy?

Deriving the energy released (Q-value) from scratch

Worked Example 1 — The D–T reaction

Solar fusion — the proton–proton (p–p) chain

Worked Example 2 — Net solar Q-value

HOW do we make fusion happen on Earth? The Coulomb barrier

Two engineering routes

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Nuclear & Radiochemistry

Connections