Foundations — Radiation safety — units (Bq, Gy, Sv), shielding

This page assumes you have seen none of the notation. We build each symbol from plain words, tie it to a picture, and only then let it appear in a formula. When you are done, everything on the parent topic will read like sentences you already know.

0. What is a "nucleus" and why does it decay?

Picture a jar of popcorn kernels sitting on a hot plate. Each kernel might pop at any instant. It has no memory — a kernel that hasn't popped for a while is no more "ready" than a fresh one. The energy locked inside the nucleus is why it wants to change; the randomness is why we can only speak of probabilities.

We won't need the internal physics here — only this fact: each nucleus has a fixed chance of decaying in the next second, and it never ages.

1. Counting: $N$, "number of things"

Look at the jar in the figure. Each dot is one nucleus. $N$ is simply how many dots are still un-popped. As time passes dots disappear, so $N$ shrinks. That shrinking is the whole story of radioactivity — everything else measures how fast it shrinks.

2. Rate of change: $\frac{dN}{dt}$ and the minus sign

Because nuclei only ever leave the count (they don't come back), $N$ always goes down, so $\frac{dN}{dt}$ is always negative. That is why the parent writes a minus sign: $-\frac{dN}{dt}=(\text{how many decayed per second, as a positive number}).$ The minus flips the negative slope into a positive count of decays. Why bother? Because "number of decays per second" should feel positive — nobody says "minus 500 pops happened."

3. The decay constant $\lambda$ — "chance per second"

Picture: give every dot in the jar the same tiny chance $\lambda$ of vanishing this second. Then the total number vanishing this second is: $(\text{chance each})\times (\text{how many})=\lambda \times N.$ That product is the decay rate. Setting it equal to our positive decay count from §2: $-\frac{dN}{dt}=\lambda N.$ This is not a new law — it is the sentence "chance-per-nucleus times number-of-nuclei" written in symbols.

4. The exponential $e^{-\lambda t}$ — the shape that always appears

In the figure, notice the curve never touches zero — it just keeps halving. That "never quite gone" behaviour is why the parent says gamma rays are "never fully stopped, only attenuated." Same curve, different label.

5. Half-life $t_{1/2}$ and the log $\ln 2$

To find $t_{1/2}$: set the shrinking factor to one-half, $e^{-\lambda t_{1/2}}=\frac{1}{2}$, take $\ln$ of both sides (that's why we use $\ln$ — to free the exponent), and get $t_{1/2}=\frac{\ln 2}{\lambda }.$

See Half-life and Radioactive decay law for the full walkthrough — here we only needed to earn the symbols.

6. Activity $A$ — turning a count into a measurable rate

Because $N$ shrinks as $e^{-\lambda t}$ and $A$ is just $\lambda$ times $N$, activity rides the same exponential: $A=A_0{e}^{-\lambda t}$.

7. From physics to biology: $E$, $m$, $D$, $w_R$, $H$

Now the three deposit-and-harm symbols. Each is one honest division or multiplication.

8. Shielding & spreading: $\mu$, $x$, and $1/r^2$

The prerequisite map

Equipment checklist

Test yourself — cover the right side. If any line stumps you, re-read its section above before tackling the parent note.