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

This page builds every letter, symbol and idea the parent Fusion note uses, starting from nothing. If a symbol appears in that note, it is earned here first, with a plain meaning and a picture.

1. What a nucleus is made of — protons, neutrons, nucleons

The picture (Figure s01): imagine a bag of two colours of marbles clumped tight. Yellow marbles (protons) push each other away because like charges repel; grey marbles (neutrons) are neutral peacemakers holding the clump together.

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

2. Counting the pieces — $Z$ , $N$ , $A$

Now we name the counts. These letters appear all over the parent note.

3. Why nuclei don't just fall apart — the two forces

Here is the tug-of-war that the whole curve is built on.

The picture (Figure s02): two magnets that snap together when close (strong force = velcro) but two same-pole magnets that push apart even from a distance (Coulomb = the shove you feel before contact).

4. Mass defect — where the energy hides ( $Δ m$ )

Before the formula, meet its three masses and how anyone actually gets them:

Here is the strange, central fact of nuclear physics:

The picture: put loose bricks on a scale — heavy. Glue them into a wall and weigh again — lighter. The missing weight didn't vanish; it left as energy when the glue set.

5. Mass becomes energy — $E = m c^{2}$

See Mass-Energy Equivalence E=mc^2 for the deep story. For us the takeaway is: the missing mass $Δ m$ from §4 becomes energy $Δ m c^{2}$ .

6. Binding energy $B$ and the number that runs everything: $B / A$

But raw $B$ isn't fair to compare — a big nucleus has more of everything. So we divide by the number of nucleons:

Why this ratio and not $B$ itself? Because fusion moves nucleons from a loosely-glued arrangement to a tightly-glued one. Only the per-nucleon value tells you the direction of downhill. See the Binding Energy per Nucleon Curve for its shape.

7. The Q-value — the scoreboard of a reaction

This is just §4 and §5 applied to a whole reaction: weigh the reactants, weigh the products, the missing mass $\times c^{2}$ is the payout.

8. Charge, positrons, neutrinos, gammas — the solar cast

The Sun's p–p chain introduces four more symbols. Meet them.

9. Temperature, plasma, tunnelling — starting the fire

Intuition Where this formula comes from and why it goes like

$1/ r$ Two charges push each other with a force that gets weaker with distance — that's Coulomb's law. The energy stored in bringing them from far away to a separation $r$ is that force added up over the whole approach; doing that sum gives an energy that grows like $1/ r$ : the closer you push, the more stored-up repulsion energy you've paid for.

The pieces $Z_{1} Z_{2} e^{2}$ say: charge times charge. Each nucleus carries charge $Z e$ , so their product sets how hard they shove.
The $1/ r_{0}$ says: the closest approach (nuclear range $r_{0}$ ) is where the wall is tallest — you must pay the most energy right at the moment before the strong force takes over.
$\frac{1}{4 π ε _{0}}$ is just the fixed "strength dial" of electricity. The picture (Figure s04): the red barrier is exactly this $1/ r$ hill; a nucleus arriving from the right must have enough energy to reach the top at $r_{0}$ before it can drop into the deep strong-force well.

Intuition Why

multiply these three — the break-even bookkeeping A fusion plasma is a leaky bucket: it earns energy from fusion reactions and loses energy by radiating and conducting heat away. To win, earnings must beat losses over the time the plasma is held.

How often do nuclei collide and fuse? More crowded plasma ⇒ more collisions, so the fusion power scales with density $n$ (and it climbs with temperature $T$ , since hotter nuclei fuse more readily). So the energy earned grows with both $n$ and $T$ , and with how long you hold it, $τ_{E}$ .
Roughly, energy earned $\propto n \times (reaction vigour set by T) \times τ_{E}$ , while energy lost is fixed by the plasma's own heat content. Requiring earned ≥ lost collapses to a single threshold on the product $n T τ_{E}$ — that's why all three multiply, not add. Weakening any one factor can be repaid by boosting another. Benchmark ("big enough"): for D–T fuel you need $n T τ_{E} ≳ 3 \times 1 0^{21} keV\cdots\cdotm^{- 3} .$ A tokamak reaches it with large $τ_{E}$ (hold it long); ICF reaches it with huge $n$ (crush it dense). Same threshold, opposite strategy.

Prerequisite map

Equipment checklist

Read each question, answer in your head, then reveal.

_{Z}^{A} X

, which number counts protons and which counts all nucleons?

The small bottom number

Z

= protons; the big top number

A

= all nucleons; neutrons are

N = A - Z

Why must

A = Z + N

Every nucleon is either a proton or a neutron and nothing else, so summing the two buckets recovers the total — like total beads = red beads + blue beads.

What does

Δ

mean in

Δ m

"The change in" — here, the difference between the mass of the loose pieces and the mass of the assembled nucleus.

How are nuclear masses actually measured?

With a mass spectrometer — charged particles bend in magnetic/electric fields, and the bend reveals the mass.

Why is a built nucleus lighter than its separate parts?

The missing mass left as binding energy when the nucleons stuck together — mass converted to energy via

E = m c^{2}

Why is the conversion factor in

E = m c^{2}

equal to

c^{2}

(not

c

Energy has units of mass×speed², so the only mass-and-light combination with energy units is

m c^{2}

;

c

is the natural exchange rate and

c^{2}

is huge.

Where does the number

931.5

come from?

It is

E = m c^{2}

applied to exactly one atomic mass unit:

1 u \times c^{2} = 931.5

MeV, so

1 u = 931.5

MeV/

c^{2}

What is the difference between

B

and

B / A

, and why do we plot

B / A

B

is total glue energy;

B / A

is glue per nucleon. Only the per-nucleon value shows the direction of "downhill," so it reveals whether fusion or fission releases energy.

Why does weighing reactants minus products (times

c^{2}

) give the energy released?

Total energy is conserved; lost rest-mass energy cannot vanish, so it reappears as kinetic energy of the products.

Which two forces fight inside a nucleus, and how do their ranges differ?

Strong force = short-range attraction (only neighbours); Coulomb force = long-range repulsion between all protons.

Why does the Coulomb barrier

U_{C}

scale like

1/ r

and like

Z_{1} Z_{2}

Adding up Coulomb's force over the approach gives energy

\propto 1/ r

(tallest at closest approach

r_{0}

); the charges

Z_{1} e

and

Z_{2} e

multiply, so the wall grows with

Z_{1} Z_{2}

Name the three symbols in the Lawson triple product and why they multiply.

n

= density,

T

= temperature,

τ_{E}

= confinement time; fusion energy earned scales with all three, so break-even is a threshold on their product (

≳ 3 \times 1 0^{21}

keV·s·m⁻³ for D–T).

What does

γ

mean in this topic?

The Greek letter gamma, standing for a gamma ray — a high-energy packet of light (photon).

Why does the Sun produce positrons and neutrinos but the Earth D–T reaction does not?

The Sun must convert protons into neutrons (a weak-interaction beta process), which emits

e^{+}

and

ν_{e}

; D–T already has its neutrons supplied by tritium.