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

This page assumes nothing. If the parent note used a symbol, wrote a formula, or expected you to "just know" something, we build it here from the ground up, in an order where each idea leans only on the ones before it.

0. What a nuclear reaction even is

We write this shorthand: $a+X⟶Y+b\text{compact form}X(a,b)Y$

Read the compact form left-to-right as: target $X$, hit by $a$, spits out $b$, becomes $Y$.

Look at the figure: the burnt-orange dart is the projectile $a$, the teal disc is the target $X$. After the collision the disc has changed (it's now $Y$) and a small plum particle $b$ shoots away. Nothing is created from nothing — the same nucleons are just re-sorted.

1. Mass — measured in "u", the atom's own ruler

Why invent a new unit? Because in these reactions the tiny difference between two masses is what matters, and writing that difference in kilograms would drown it in zeros. In $u$, a proton is about $1.007$ and a neutron about $1.009$ — differences of a thousandth of a $u$ are the whole story.

2. Energy, kinetic energy, and the symbol $T$

We track $T$ separately for every particle: $T_a$ for the projectile, $T_X$ for the target (often zero, target at rest), $T_Y$ and $T_b$ for the products.

3. The bridge: $E=m{c}^2$ and the number 931.5

Here is the keystone that lets mass and energy talk to each other.

Because we measure mass in $u$ and energy in MeV, we bake $c^2$ into one handy conversion:

The figure is a see-saw: mass on the left pan, motion (kinetic energy) on the right. When products are lighter than reactants, the missing mass slides across the see-saw and reappears as extra speed. That sliding is the Q-value.

4. Binding energy $B$ — how tightly glued a nucleus is

Why the parent note cares: it rewrites the Q-value as $Q=(B_Y+B_b)-(B_X+B_a),$ so a reaction gives out energy when the products are more tightly bound than the reactants. Which nuclei are most tightly bound? That is exactly the shape of the Binding energy per nucleon curve — the reason both Nuclear fission and Nuclear fusion release energy: both climb toward the curve's peak.

5. Conservation laws — the two rules that never break

These are the engine behind the whole page. Energy conservation defines the Q-value. Momentum conservation is the subtle one — it is why an energy-absorbing reaction needs more than $|Q|$ to get going, because the products are forced to keep drifting forward and can't spend all the energy on rearranging. (Full treatment: Conservation laws in collisions.)

The figure shows the momentum arrows. Before: only the projectile has an arrow (target at rest). After: the total arrow must be the same length and direction — so the products cannot both stand still. That leftover forward motion is energy that can't be used for the reaction, and it is why the threshold energy exceeds $|Q|$.

6. The cross-section picture and the letters $\sigma ,n,\varphi ,I$

Now the "how likely" side. The parent note introduces several new letters at once; here they are, one at a time.

Now the supporting cast of symbols:

With these, the parent's two headline results read plainly:

• Attenuation $I=I_0{e}^{-n\sigma x}$ — a beam thins out exponentially as it bores through thickness $x$, because each thin slice removes a fixed fraction.
• Reaction rate $R=\sigma \varphi$ — (area to hit) × (how fast particles arrive).

7. The exponential $e^{-n\sigma x}$ — why that shape

Why it appears here: in each thin slice $dx$ the beam loses the same fraction $n\sigma dx$ of whatever is left — not a fixed amount. "Same fraction per step" is the exact fingerprint of the exponential. The identical logic drives Radioactive decay kinetics, where it's time that ticks by instead of thickness.

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