Beta dose rate and shielding

Maximum energy and mean energy do different jobs

A beta spectrum is continuous. The decay energy is shared between the electron and a neutrino, so electrons emerge with every energy from zero up to a maximum set by the transition. The mean is roughly a third of the maximum, though the exact ratio depends on the shape of the spectrum and the nuclear charge.

Two numbers, two uses. Shielding follows the maximum, because a shield must stop the fastest electron that will ever be emitted, not the average one. Dose follows the mean, because dose is deposited energy and what is deposited is the average over many decays. Using the maximum for dose overestimates by about a factor of three; using the mean for shielding leaves a shield that most of the spectrum passes through.

Range, and why it is quoted per gram

Electron range is tabulated as a mass thickness in g/cm² rather than a distance, because to first order a given mass of material stops electrons regardless of what the material is. The Katz–Penfold relation, R = 0.412 E^{1.265 − 0.0954 ln E} for energies from 0.01 to 3 MeV, is the standard empirical fit and is accurate to roughly 10%. Dividing by density converts it to a thickness.

The consequence is that any material works if you use enough of it, and the practical choice is about convenience and about what comes out the other side. Y-90, the highest-energy common beta emitter at 2.28 MeV, needs about 11 mm of water, 9 mm of acrylic, 4 mm of aluminium or 1 mm of lead. Below the full range, transmission through an absorber is roughly exponential, but the exponential is only an approximation: past the range the transmission is exactly zero, and a naive exponential never reaches zero. This calculator truncates it there.

Why lead is the wrong beta shield

Lead stops beta particles in the least thickness, which makes it the obvious choice and the wrong one. Decelerating electrons radiate, and the fraction of the beta energy converted into bremsstrahlung X-rays rises with the atomic number of the absorber. The thick-target approximation, f ≈ 3.5 × 10⁻⁴ · Z · E_max, puts lead an order of magnitude worse than acrylic for the same beta.

The bremsstrahlung is a continuous X-ray spectrum reaching up to the full beta energy, with a mean around a third of it, and it is far more penetrating than the beta it came from. You can convert a shielding problem that a few millimetres of plastic would have solved into a photon problem that needs centimetres of lead. The correct order is to stop the beta in a low-Z absorber — acrylic, polyethylene, aluminium — and then, only if the residual bremsstrahlung matters, add a high-Z layer outside it.

The dose rate offered here, and the one that is not

The infinite-medium dose rate is exact, and it is exact for a reason worth stating: if activity is distributed uniformly through a volume large compared with the beta range, every emitted electron deposits its energy inside that volume. Energy conservation then gives the dose rate directly as the mean beta energy times the activity concentration. At a plane surface of a semi-infinite medium it is exactly half, because nothing above the surface returns energy.

Skin dose from surface contamination is deliberately not calculated. It needs a point kernel integrated over the actual source geometry, the air gap, any covering, and the depth of the basal layer at 7 mg/cm². The answer is extremely sensitive to all of them, and a single number presented without those inputs would be confidently wrong. Codes such as VARSKIN exist for this and should be used for it.