Drawing No. EH–NR–016 // Nuclear Engineering & Radiation
Radiation Protection Formula Sheet
A compact health-physics reference for the core relations behind activity, external exposure control, shielding and protection quantities. The page keeps activity, absorbed dose, equivalent dose and effective dose separate because they describe different physical and protection concepts.
Fast reference, with engineering context
Use the equations directly for screening calculations, then open the linked EngineerHub tools for input handling and unit conversion. Formula applicability and major limitations are stated beside each relation.
Reference conventions
Time, distance and shielding
01 // Radioactive decay and activity
For one radionuclide with no production term, the number of atoms and activity decrease exponentially.
| Symbol | Meaning | SI units | US customary |
|---|---|---|---|
| A | Activity | Bq = s⁻¹ | Ci or Bq |
| A₀ | Initial activity | Bq | Ci or Bq |
| λ | Decay constant | s⁻¹, d⁻¹ | s⁻¹, d⁻¹ |
| T½ | Physical half-life | s, h, d, y | s, h, d, y |
| N | Number of radioactive atoms | atoms | atoms |
Worked example — three half-lives
Given: A₀ = 100 MBq, physical half-life = 8 days, elapsed time = 24 days.
Biological clearance is not included in physical radioactive decay.
For internal activity, an effective half-life can be useful when physical decay and biological removal are both approximately first order: 1/Teff = 1/Tphys + 1/Tbio.
02 // Dose from a known dose rate
If dose rate is approximately constant over the exposure interval, accumulated dose is rate times time.
| Symbol | Meaning | SI units | US customary |
|---|---|---|---|
| D | Dose quantity being accumulated | Gy or Sv as appropriate | rad/rem as appropriate |
| Ḋ | Corresponding dose rate | Gy/h, Sv/h | rad/h, rem/h |
| t | Exposure time | h, s | h, min |
Worked example — 20 minutes in a field
Given: dose-equivalent rate = 0.50 mSv/h for 20 min.
US check: ≈0.0167 rem = 16.7 mrem.
Only multiply quantities of the same type; do not mix absorbed-dose rate with effective dose without a valid conversion basis.
Time reduction is a practical exposure-control method, but job planning must also consider source changes, occupancy, task constraints and uncertainty.
03 // Inverse-square scaling for a point source
For an isotropic point source in free space with negligible attenuation and scatter, fluence and dose rate scale approximately with 1/r².
| Symbol | Meaning | SI units | US customary |
|---|---|---|---|
| Ḋ | Dose-rate quantity | Gy/h or Sv/h | rad/h or rem/h |
| r | Distance from effective point source | m | ft |
Worked example — moving away from a point source
Given: Dose rate = 2.0 mSv/h at 1.0 m; new distance = 3.0 m.
Do not use the point-source inverse-square law in the near field of an extended source without geometry correction.
Large area/line sources, contact measurements, shielded sources and strong scatter fields can depart substantially from ideal 1/r² behavior.
04 // Exponential attenuation, HVL and TVL
For a narrow monoenergetic photon beam in a homogeneous absorber, uncollided intensity decreases exponentially.
| Symbol | Meaning | SI units | US customary |
|---|---|---|---|
| I,I₀ | Transmitted / incident intensity or uncollided fluence | consistent units | consistent units |
| μ | Linear attenuation coefficient | m⁻¹, cm⁻¹ | in⁻¹ |
| x | Shield thickness | m, cm | in |
| HVL | Half-value layer | m, cm | in |
| TVL | Tenth-value layer | m, cm | in |
Worked example — two half-value layers
Given: HVL = 6.0 cm, so μ = ln2/6 = 0.1155 cm⁻¹; shield x = 12 cm.
Broad-beam shielding often requires a buildup factor; μ is energy- and material-dependent.
Exponential attenuation is exact for the ideal uncollided component. Real protection calculations may include scatter, secondary radiation, source spectrum and geometry.
05 // Absorbed dose
Absorbed dose is energy imparted by ionizing radiation per unit mass.
| Symbol | Meaning | SI units | US customary |
|---|---|---|---|
| D | Absorbed dose | Gy = J/kg | rad |
| dε̄ | Mean energy imparted | J | erg or J |
| dm | Mass receiving energy | kg | g or kg |
Worked example — energy deposited in tissue
Given: 0.020 J is imparted to 2.0 kg of material.
US check: 1 rad.
Absorbed dose is physical energy per mass; it does not by itself include radiation or tissue weighting.
For nonuniform irradiation, dose is spatially dependent. Organ/tissue averages used for protection quantities are defined by the relevant dosimetry framework.
06 // Equivalent dose to a tissue or organ
Equivalent dose weights absorbed dose in a tissue by radiation type using ICRP radiation weighting factors.
| Symbol | Meaning | SI units | US customary |
|---|---|---|---|
| HT | Equivalent dose in tissue T | Sv | rem |
| DT,R | Mean absorbed dose in tissue T from radiation R | Gy | rad |
| wR | Radiation weighting factor | dimensionless | dimensionless |
Worked example — illustrative alpha dose
Given: Mean absorbed dose to one tissue = 5 mGy from alpha particles; use wR = 20.
US check: 10 rem.
Equivalent dose is a protection quantity; do not use it to predict deterministic tissue reactions in an individual.
Radiation weighting factors are specified by the adopted protection framework. ICRP Publication 103 is the basis used here.
07 // Effective dose
Effective dose combines equivalent doses in tissues using tissue weighting factors to represent overall stochastic detriment for protection purposes.
| Symbol | Meaning | SI units | US customary |
|---|---|---|---|
| E | Effective dose | Sv | rem |
| HT | Equivalent dose to tissue T | Sv | rem |
| wT | Tissue weighting factor | dimensionless | dimensionless |
Worked example — two illustrative tissue contributions
Given: HT,1 = 20 mSv with wT,1 = 0.12; HT,2 = 10 mSv with wT,2 = 0.04.
Effective dose is intended for radiological protection, not individual medical risk prediction.
A complete effective-dose calculation sums all relevant tissue contributions under the adopted ICRP weighting scheme.
08 // Committed effective dose from intake
For a known radionuclide intake and an applicable committed effective dose coefficient, committed dose is intake times coefficient.
| Symbol | Meaning | SI units | US customary |
|---|---|---|---|
| I | Activity intake | Bq | µCi/Bq as needed |
| e(g) | Committed effective dose coefficient for nuclide, form, route and age | Sv/Bq | rem/Ci equivalent |
| E(τ) | Committed effective dose over integration period τ | Sv | rem |
Worked example — illustrative dose coefficient
Given: Intake = 5000 Bq; applicable committed effective dose coefficient = 2.0×10⁻⁸ Sv/Bq.
US check: 10 mrem.
Never reuse a coefficient across radionuclides, chemical forms, intake routes or age groups unless the reference explicitly permits it.
Internal dosimetry requires biokinetic and dosimetric models. Use current ICRP/IAEA or regulatory dose coefficients applicable to the radionuclide and exposure scenario.
09 // Quick formula summary
Compact print reference. Use the detailed sections above for definitions and limitations.
| Topic | Equation | Purpose | Tool |
|---|---|---|---|
| Decay | A=A₀e⁻λt | Activity over time | Dose guide |
| Time | D=Ḋt | Exposure duration | Dose guide |
| Distance | Ḋ∝1/r² | Point-source scaling | Shielding |
| Attenuation | I=I₀e⁻μx | Photon shielding | Shielding |
| Absorbed dose | D=dε/dm | Energy per mass | Dose guide |
| Equivalent dose | HT=ΣwRDT,R | Radiation weighting | Dose guide |
| Effective dose | E=ΣwTHT | Tissue weighting | Dose guide |
| Internal dose | E(τ)=I·e(g) | Intake × coefficient | NORM worker |
10 // Assumptions & limitations
Fundamental equations are only useful when their assumptions match the actual problem.
Activity (Bq), absorbed dose (Gy), equivalent dose (Sv) and effective dose (Sv) are not interchangeable. The same unit Sv can represent different protection quantities whose definitions must be stated.
Inverse square and simple exponential attenuation are idealized. Extended geometry, scatter, buildup and secondary radiation can dominate practical shielding problems.
Dose limits, dose constraints, reference levels and operational quantities are jurisdiction- and exposure-situation dependent. Use the applicable regulator and current IAEA/ICRP framework.
11 // Technical references
IAEA GSR Part 3 and ICRP Publication 103 provide the protection framework; NRC Part 20 is used here for SI/traditional dose-unit conversions. Regulatory requirements must be checked for the user's jurisdiction.