Radiation calculator: half-life, distance and shielding
Radioactive decay
- Formula: A = A₀ × e^(−λt), where λ = 0.693 ÷ half-life
- Decay constant: λ = 0.693 ÷ 8 = 0.08663 per time unit
- Exponent: −0.08663 × 24 = −2.079
- A = 100 × e^(−2.079) = 100 × 0.1251
- Answer: 12.51 (that is 3 half-lives)
Inverse square law
- Formula: I₂ = I₁ × (d₁ ÷ d₂)²
- Put in the numbers: 100 × (2 ÷ 6)²
- Square the ratio: (0.3333)² = 0.1111
- Answer: 11.11
Shielding with half-value layers
- Formula: I = I₀ × e^(−μx), where μ = 0.693 ÷ HVL
- μ = 0.693 ÷ 0.5 = 1.386 per unit of thickness
- Exponent: −1.386 × 1.5 = −2.079
- I = 80 × 0.1251
- Answer: 10 (3 half-value layers)
Formulas and source
- A = A₀ × e^(−λt), λ = 0.693 ÷ half-life
- I₂ = I₁ × (d₁ ÷ d₂)²; d₂ = d₁ × √(I₁ ÷ I₂)
- I = I₀ × e^(−μx), μ = 0.693 ÷ HVL; x = HVL × log₂(I₀ ÷ I)
The formula comes from
OSHA, Ionizing Radiation: Introduction to Health Physics handout(opens osha.gov in a new tab)“A t = A o e (- λ x t) = Activity after time t ... λ= 0.693/T ½ = disintegration constant”
The formula comes from
OSHA, Ionizing Radiation: Introduction to Health Physics handout(opens osha.gov in a new tab)“Estimating Attenuation in Air Using Inverse Square Law I 2 = I 1 [ d 1 / d 2 ] 2”
The formula comes from
OSHA, Ionizing Radiation: Introduction to Health Physics handout(opens osha.gov in a new tab)“I = I o Be -µx = intensity on other side of shield ... Half-Value Layer (HVL): The shield thickness necessary to reduce intensity by half. HVL = 0.693 / µ.”
Formulas and calculators on the BCSP exams
- BCSP gives you no formula sheet, and you can't bring one.
- When a question needs a formula, the formula is written into the question, unless the question is testing whether you know that formula.
- You get an on-screen calculator that works like the TI-30XS, plus a whiteboard. You can't bring your own calculator.
- On the ASP, Mathematical Calculations is its own domain, worth 10% of the exam (ASP11 blueprint).
BCSP on formulas
BCSP, What to Expect When Sitting for a BCSP Exam (item 7)(opens bcsp.org in a new tab)“For any question requiring a mathematical formula, the formula will be included within the exam question (unless the question is specifically testing your knowledge of a formula). You will not be provided or permitted to bring a formula sheet.”
BCSP on calculators
BCSP, What to Expect When Sitting for a BCSP Exam (item 6)(opens bcsp.org in a new tab)“You will be provided an on-screen calculator, emulating the TI-30XS scientific calculator. You will also have access to a white board and dry erase marker. The testing center will not provide a physical calculator or permit candidates to bring their own.”
FAQ
How do you calculate radioactive decay from a half-life?
A = A₀ × e^(−0.693t ÷ half-life). Each half-life cuts the activity in half, so 100 mCi with an 8-day half-life is about 12.5 mCi after 24 days (three half-lives).
What is the inverse square law?
Radiation from a point source falls with the square of the distance: I₂ = I₁ × (d₁ ÷ d₂)². Double the distance and the rate drops to a quarter. 100 mR/h at 2 ft is 100 × (2 ÷ 6)² = 11.1 mR/h at 6 ft.
How far away do I need to be to reach a target rate?
Rearrange the inverse square law: d₂ = d₁ × √(I₁ ÷ I₂). To go from 400 mR/h at 1 ft down to 100 mR/h, you need 1 × √4 = 2 ft.
What is a half-value layer?
The thickness of a material that cuts the radiation in half. Two half-value layers cut it to a quarter, three to an eighth. 80 mR/h behind three half-value layers is 10 mR/h.
How thick does a shield need to be?
Count the halvings you need, n = log₂(I₀ ÷ I), and multiply by the half-value layer. Going from 80 to 10 mR/h is 3 halvings, so with a 0.5 in half-value layer you need 1.5 in.
The shielding formula leaves out the buildup factor (B) in OSHA's handout, as exam questions usually do, so real readings behind thick shields can be higher. This tool is for study, not for planning radiation work.