Validation of the work planning calculator

A-U-01 — The inverse square law from solid angle

In the code: inverseSquare in src/engine/alara.ts

1. A point source emits the same number of particles per second in every direction.
2. At distance d those particles are spread over a sphere of area 4πd². Nothing is created or destroyed in between, so the number crossing unit area falls as 1/d².
3. Dose rate is proportional to that fluence rate, so it falls the same way: Ḋ(d₂) = Ḋ(d₁)·(d₁/d₂)².
4. No property of the source, the radiation or the material enters. That is why this is the one relation on the screen that needs no data at all.
5. It is also why it is the first control a radiation protection plan reaches for: doubling the distance costs nothing and removes three quarters of the dose.

A-U-02 — What the point-source assumption costs, as a function of distance

In the code: the scope of inverseSquare

1. The derivation above needs the source to be a point. Real sources have extent, and the honest question is not whether the assumption is exact — it never is — but how far away it stops mattering.
2. That question has a closed-form answer for a uniform line source of length L. Integrating 1/(d² + l²) along the line gives a dose rate proportional to (2/(L·d))·arctan(L/(2d)) at perpendicular distance d from the midpoint.
3. Dividing by the point-source expression A/d² leaves arctan(x)/x with x = L/(2d), which is less than one for every finite distance and tends to one as d grows.
4. So the point approximation always OVERESTIMATES the dose rate from an extended source. For planning that is the safe direction, which is worth knowing as well.
5. The report tabulates the overestimate against d/L and solves for the distance at which it falls below 10%, 5%, 1% and 0.1%. That turns a caveat into a number a planner can use.
6. Two things this does not cover: air attenuation, which reduces the rate further at long distance, and scatter from floors and walls, which increases it. Neither is in this calculator.

A-U-03 — Stay time, and the difference between zero and unknown

In the code: stayTime in src/engine/alara.ts

1. A dose budget divided by a dose rate is a time. There is nothing more to it when the rate is a positive number.
2. The interesting cases are the others. A rate of exactly zero means the budget is never spent, so the stay time is unbounded — that answer is correct and useful.
3. A rate that is missing, not-a-number, negative or infinite means something different: there is no answer. Reporting unbounded there would tell a worker they may stay indefinitely because the field was left blank.
4. The screen must therefore distinguish three outcomes, not two, and the engine returns a not-a-number for the third so that the screen can.
5. This case exists because the distinction was got wrong twice — see the refusals.

A-U-04 — Collective dose adds, and must not cancel

In the code: collectiveDose in src/engine/alara.ts

1. Collective dose is the sum over tasks of workers × hours × dose rate, in person-millisieverts.
2. It is a plain sum, so it is exactly additive: splitting a task into two halves changes nothing.
3. What matters is what happens to a bad row. A negative number of workers, or a negative rate, produces a negative contribution that silently cancels part of a legitimate task, and the total still looks like a total.
4. So a single invalid row makes the whole sum unanswerable rather than being absorbed into it.

A-U-05 — Half-value and tenth-value layers are definitions

In the code: hvlFromMu and tvlFromMu in src/engine/alara.ts

1. Narrow-beam attenuation is I = I₀·e^(−µx). Setting I/I₀ = 1/2 gives x = ln2/µ, and 1/10 gives x = ln10/µ.
2. These follow from the exponential and nothing else, so they are exact for any µ the user supplies.
3. Their ratio is therefore ln10/ln2 = log₂10 = 3.3219…, independent of µ and of the material — which is checked as an identity.
4. A µ of zero has no half-value layer: the beam is not attenuated at all. Returning an infinite thickness reads as “a very thick shield works”, when the truth is that no thickness of that material works.

A-W-01 — Backing off from one metre to four

1. The reference is 2 mSv/h at 1 m.
2. Ḋ(4 m) = 2 × (1/4)² = 2/16 = 0.125 mSv/h.
3. Three metres of floor removed 93.75% of the dose rate and cost nothing.
4. If the source is a line 1 m long, the point approximation overstates this by about 2% at 4 m — see A-E-01.

A-W-02 — How far to reach a working limit

1. From 2 mSv/h at 1 m, the distance at which the rate is 0.02 mSv/h is d = 1 × √(2/0.02).
2. √100 = 10, so d = 10 m.
3. Check: 2 × (1/10)² = 0.02 mSv/h. The two expressions agree, which is identity A-ID-02.
4. A hundredfold reduction in rate needs a tenfold increase in distance — the square root is why distance stops being cheap eventually.

A-W-03 — Stay time on a one-millisievert budget

1. At the 4 m position from A-W-01 the rate is 0.125 mSv/h.
2. A task budget of 1 mSv gives 1/0.125 = 8 hours.
3. At 1 m it would have been 0.5 hours. The same job, the same budget, sixteen times the working time.
4. This is a task budget set by the planner, not a regulatory limit; the calculator does not supply dose limits.

A-W-04 — Collective dose for a two-task job

1. Task one: 3 workers × 2 h × 0.5 mSv/h = 3 person·mSv.
2. Task two: 2 workers × 4 h × 0.25 mSv/h = 2 person·mSv.
3. Total 5 person·mSv.
4. The two tasks contribute unequally despite the second taking more worker-hours, because the rate differs — which is the comparison the quantity is for.

A-W-05 — Half-value layer from a measured attenuation coefficient

1. A measured linear attenuation coefficient of 1.2 per centimetre.
2. HVL = ln2/µ = 0.693 147 180 6 / 1.2 = 0.577 622 650 47 cm.
3. This is narrow-beam: a real broad beam has scatter reaching the detector, so the observed thickness for half is larger.

A-W-06 — Tenth-value layer from the same coefficient

1. TVL = ln10/µ = 2.302 585 093 / 1.2 = 1.918 820 910 8 cm.
2. TVL/HVL = 1.918 820 910 8 / 0.577 622 650 47 = 3.321 928 094 9, which is log₂10.
3. The ratio carries no µ, so it is the same for every material and every energy — identity A-ID-04.