Applications — radiocarbon dating, medical (Tc-99m, I-131), RTG (Pu-238 in spacecraft): Foundations

This page assumes nothing. Before you can even read the parent note Applications, you must meet a handful of ideas. We introduce each in words and pictures before any of its shorthand symbols appear — nothing below is used before it is built. The order is: what a nucleus is → isotopes → how nuclei decay → how we count survivors → how fast they fall → the shape of that fall → half-life → activity → turning decay heat into power.

1. What is a nucleus? (the atom's core)

Figure 1 (below): a carbon nucleus drawn as marbles — 6 red protons and 8 grey neutrons packed together — beside the shorthand $_{6}^{14} C$ , with arrows pointing from each number to what it counts. Look at how the top number (14) and bottom number (6) attach to the symbol.

We write a nucleus as $_{Z}^{A} X$ :

$X$ = the element's letter (C, N, Tc...).
$Z$ = number of protons (bottom).
$A$ = mass number = protons + neutrons (top).

So $_{6}^{14} C$ means: carbon, 6 protons, and $14 - 6 = 8$ neutrons.

2. Isotopes and why some nuclei decay

The parent note needs isotopes because which isotope you pick decides the whole application: C-14 for dating, Tc-99m for imaging, Pu-238 for power.

3. The kinds of decay: $α$ , $β^{-}$ , $β^{+}$ , electron capture, $γ$

Definition The five decay modes

==Alpha ( $α$ )== — the nucleus spits out a small chunk: 2 protons + 2 neutrons (this chunk is $_{2}^{4} He$ ). Heavy and slow; stopped by a sheet of paper.
==Beta-minus ( $β^{-}$ )== — a neutron turns into a proton and flings out a fast electron ( $_{- 1}^{0} β$ ). Light and quick; stopped by aluminium foil.
==Beta-plus ( $β^{+}$ ), positron emission== — the opposite: a proton turns into a neutron and flings out a positron (an anti-electron, $_{+ 1}^{0} β$ ). Happens when a nucleus has too many protons.
Electron capture — instead of emitting a positron, the nucleus swallows one of its own inner electrons, again turning a proton into a neutron. Same net effect as $β^{+}$ .
==Gamma ( $γ$ )== — no particle, just a burst of pure energy (a high-energy light photon). Very penetrating; needs lead/concrete. It usually follows another decay that left the nucleus over-energised.

So there is not just one flavour of decay — the applications happen to use $α$ , $β^{-}$ and $γ$ , but $β^{+}$ and electron capture are real modes too. See the full treatment in Types of Radioactive Decay (alpha, beta, gamma).

Figure 2 (below): a single source at the left firing three rays into three barriers. Alpha stops at paper, beta stops at foil, and the red gamma ray — the key object — sails through all the way to the lead. This picture is the reason each application picks the radiation it does.

4. $N$ — counting what is left

5. Rate of change $\frac{d N}{d t}$ and the decay law

6. $λ$ — the decay constant

See Half-life and Decay Constant for how $λ$ ties to half-life.

7. Solving the law → $e^{- λ t}$ , the shape of the fall

Formula Separate the variables — WHY and WHAT at each step

Start from the law of §5: $\frac{d N}{d t} = - λ N$ Step 1 — gather $N$ on one side, $t$ on the other. Why? We want all the "how-many" on the left and all the "time" on the right so each side can be summed independently. $\frac{d N}{N} = - λ d t$ Step 2 — sum up (integrate) both sides from the start to now. Why? Each tiny slice $d N / N$ adds up to the total fractional change; on the right the constant $- λ$ just multiplies the elapsed time. The symbol $\int$ means "add up all these slices." $\int_{N_{0}}^{N} \frac{d N}{N} = - λ \int_{0}^{t} d t$ Step 3 — do the sums. Why $ln$ ? Adding up slices of $d N / N$ gives the natural logarithm $ln$ (this is its definition). The right side is just $- λ t$ . $ln N - ln N_{0} = - λ t \Rightarrow ln \frac{N}{N _{0}} = - λ t$ Step 4 — undo the $ln$ by raising $e$ to both sides. Why? $e^{()}$ is the exact opposite of $ln$ , so it frees $N$ : $\frac{N}{N _{0}} = e^{- λ t} \Rightarrow N = N_{0} e^{- λ t}$

Figure 3 (below): the survival curve $N / N_{0} = e^{- λ t}$ in red, with time measured in half-lives. Dashed black steps mark it dropping to 50%, then 25%, then 12.5% — each equal step of time chops the survivors in half. Watch how the curve flattens but never quite reaches zero.

8. $t_{1/2}$ — the half-life

9. Activity $A$ — what a detector actually reads

10. The Seebeck effect — heat into electricity

Prerequisite map

Everything upstream feeds into the parent topic Applications at the bottom.

Equipment checklist

Test yourself — cover the right side and answer each before revealing.

_{6}^{14} C

, how many neutrons?

A - Z = 14 - 6 = 8

neutrons.

What makes two nuclei isotopes?

Same number of protons

Z

, different number of neutrons (so different

A

Name all five decay modes.

Alpha, beta-minus, beta-plus (positron emission), electron capture, and gamma.

Which decay is stopped by a sheet of paper?

Alpha (

α

) — heavy, low penetration.

Which radiation is best for imaging inside the body and why?

Gamma (

γ

) — penetrating, so it escapes to a camera while depositing little dose.

What does

\frac{d N}{d t} = - λ N

say in plain words?

The rate of loss of nuclei is proportional to how many remain; the minus sign means the number is falling.

What does the decay constant

λ

physically represent?

The probability that one nucleus decays per unit time.

What is

e

, and why does it appear here?

The base of the natural logarithm (

\approx 2.718

); it solves the decay law because

e^{x}

has a slope equal to its own height.

Why is decay described by

e^{- λ t}

and not a straight line?

The loss rate is proportional to how many remain, and only the exponential's slope equals a constant times its own height.

Relation between

t_{1/2}

and

λ

, and where does it come from?

t_{1/2} = \frac{ln 2}{λ}

, from setting

N = N_{0} /2

N = N_{0} e^{- λ t}

and taking

ln

Fraction left after 3 half-lives?

(1/2)^{3} = 1/8 = 12.5%

Why can activity

A

stand in for atom count

N

Because

A = λ N

, so

A

is directly proportional to

N

and falls with the same

e^{- λ t}

What does the Seebeck effect convert, and why does an RTG use it?

A temperature difference into a voltage; the RTG uses decay heat vs cold space to make electricity with no moving parts.