Absorbed dose, equivalent dose, and shielding

Absorbed dose is D = dε̄/dm: the mean energy imparted by ionising radiation to a small element of matter, divided by its mass. Its unit, the gray, is 1 J kg⁻¹, and a dose always belongs to a material, because air, water and bone absorb the same beam differently. By thermal standards the energies are minute. A whole-body dose of about 5 Gy, fatal to roughly half of those exposed without treatment, deposits 5 J in each kilogram and warms water by 5 ÷ 4186 = 1.2 × 10⁻³ K. The harm is not heat. Radiation spends its energy in ionisations of roughly 30 eV each, over a thousand times kT at body temperature and several times the few electronvolts that hold a chemical bond, strung along the tracks of charged particles. One gray puts about 5.6 × 10⁴ of them into a cell nucleus 8 µm across. The gray counts the joules; the biology depends on where they land.

A 5 MeV alpha particle loses energy in tissue at around 100 keV µm⁻¹; the electrons set moving by cobalt-60 gamma rays lose about 0.3 keV µm⁻¹. At the same absorbed dose, the alpha energy is packed into a few dense tracks that break DNA in clusters that cells repair badly, so a gray of alpha particles does more harm. Protection practice folds this into a radiation weighting factor, H_T = w_R D_T, summed over the radiations present, with w_R = 1 for photons, electrons and muons, 2 for protons, 20 for α particles and heavy ions, and a smooth function of energy for neutrons, from 2.5 for thermal neutrons to about 20 near 1 MeV. The result is the equivalent dose, in sieverts. The sievert has the gray's dimensions, J kg⁻¹, yet no instrument measures an equivalent dose: survey meters and badges read operational estimates calibrated in sieverts, while the quantity itself is an absorbed dose multiplied by a rounded convention chosen from low-dose biological effectiveness. So 2.0 mGy of alpha particles to the lung is 40 mSv to the lung, while 2.0 mGy of gamma rays is 2.0 mSv.

Few exposures are uniform: a dental radiograph irradiates the jaw, inhaled radon progeny the airways. The effective dose E = Σ_T w_T H_T weights each tissue's equivalent dose by its share of the total stochastic detriment: 0.12 for red marrow, colon, lung, stomach, breast and a remainder group; 0.08 for the gonads; 0.04 each for bladder, oesophagus, liver and thyroid; 0.01 each for bone surface, brain, salivary glands and skin. The weights sum to 1 like a probability distribution, and that normalisation is the point: if every tissue receives H, then E = H. So 20 mSv to the lung alone gives E = 0.12 × 20 mSv = 2.4 mSv, nominally as risky as 2.4 mSv spread over the whole body, and one year of world-average natural background. That background, about half of it from radon, is the baseline every other figure is read against: a chest radiograph, about 0.02 mSv, is three days of it; an abdominal CT scan, about 10 mSv, is four years.

For a small gamma source, energy conservation does the work. A source of activity A emitting y photons of energy E_γ per decay sends y A E_γ joules per second through every sphere drawn around it, so at distance r the energy fluence rate is y A E_γ/(4π r²). Tissue retains a fraction μₑₙ/ρ of it per unit areal density, so Ḋ = y A E_γ (μₑₙ/ρ)/(4π r²). For caesium-137, y = 0.851, E_γ = 662 keV and μₑₙ/ρ = 3.26 × 10⁻³ m² kg⁻¹ give 84 µGy h⁻¹ at 1 m for each GBq. Doubling the distance quarters the rate; stepping back by a factor of √2 halves it. The law fails close to an extended source, whose size is then comparable with r (near the middle of a long line source the rate falls only as 1/r), and it cannot see radiation scattered back by walls, floor and air. Alpha and beta emitters are a different case: their particles have finite ranges, a few centimetres of air for alphas and up to several metres for betas, or about 40 µm and a few millimetres of tissue, so distance protects against them abruptly, not as a power law.

A photon crossing a slab has the same probability μ dx of interacting in each thin layer dx, however far it has already travelled. For N photons in a narrow beam that gives dN/dx = −μN, so N(x) = N₀ e^(−μx): each photon crosses uncollided with probability e^(−μx), and its mean free path is 1/μ. Each half-value thickness x₁/₂ = ln 2/μ halves the beam. For 662 keV photons in lead μ ≈ 1.26 cm⁻¹, so x₁/₂ ≈ 5.5 mm, and 25 mm of lead, about 4.5 half-value thicknesses, transmits e^(−3.15) = 4.3%; water, with μ ≈ 0.086 cm⁻¹, needs 8.1 cm per halving. The narrow-beam law is optimistic for a real shield. Compton-scattered photons leave the pencil beam yet can still reach you, so a broad beam transmits B e^(−μx), with a buildup factor B ≥ 1 that grows with thickness; practical tables for caesium-137 in lead quote nearer 6.5 mm per halving. Charged particles do not attenuate exponentially at all: they stop at a range.

A dose is the mass average of a random process, and at low dose the randomness shows. One milligray of gamma rays sends, on average, about one electron track through each cell nucleus. Track numbers are Poisson distributed, so a fraction e⁽⁻¹⁾ = 37% of nuclei receive none while others receive two or three: one milligray describes the average, not any cell. Risk inherits that structure. For protection at low dose, ICRP assumes a linear, no-threshold nominal risk of about 5 × 10⁻² per sievert, averaged over ages and both sexes and close to the chance of a fatal cancer, so an effective dose of 10 mSv adds about 5 × 10⁻⁴ to that chance, beside a lifetime baseline near 20% of dying of cancer. That is an expectation across many people, and E itself is computed for a reference phantom, not for anyone's organs. At high doses, above roughly 0.5 to 1 Gy, tissue reactions with thresholds take over, and doses are then quoted in grays to the organ, not in sieverts.