Beta dose rate and shielding
Infinite-medium dose rate, Katz–Penfold range, transmission through absorbers, and bremsstrahlung yield by atomic number.
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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 E1.265 − 0.0954 ln E g/cm² from 0.01 MeV up to a branch point at 2.5 MeV, above which R = 0.530 E − 0.106, is the standard empirical fit. Dividing by density converts it to a thickness. It is a fit to absorption measurements, so the thickness it gives is a depth of penetration rather than the length of the electron's path, which is longer. How far below the path length it sits, and what this site could and could not establish about the fit, are set out in the validation report.
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 · Emax, puts lead an order of magnitude worse than acrylic for the same beta. Checked against the radiation yields NIST tabulates, that expression understates lead's disadvantage rather than overstating it — the measured comparison is in the validation report.
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.
Shield materials for beta emitters works that order through with the nuclides carried here: the thickness each endpoint needs, the fraction converted in acrylic against lead, and the cases where the endpoint to size on is not the one the nuclide is named for.
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.
Worked examples
A shield for a yttrium-90 syringe
A syringe of Y-90 is drawn up behind a shield, and the question is how thick the acrylic sleeve must be. Choose Y-90, the “Shielding” mode and acrylic, then try 5 mm and 10 mm.
| Result | Value |
|---|---|
| Range, g/cm² | 1.09 |
| Acrylic to stop it, mm | 9.2 |
| Transmission at 5 mm | 1.91% |
| Transmission at 10 mm | 0 |
| To bremsstrahlung, acrylic | 0.52% |
| To bremsstrahlung, lead | 6.5% |
The range is a mass thickness, so it is divided by the density of acrylic, 1.19 g/cm³, to give 9.2 mm. A 10 mm sleeve stops every beta; a 5 mm sleeve lets about one in fifty through, which at the fingertips of someone handling gigabecquerels is not a small dose. Lead would stop the betas in under a millimetre, but it converts about 13 times more of their energy into bremsstrahlung than acrylic does. That is why the sleeve is acrylic first, with a thin lead or tungsten layer outside it only if the photon dose rate measured at the surface still calls for it.
Absorbed dose from yttrium-90 spread through tissue
Y-90 microspheres deposit their energy within a few millimetres of where they lodge, so a lobe of liver that holds them uniformly is close to the infinite-medium case. Choose Y-90 and the “Dose rate” mode, and enter first 1 GBq in 1 kg, then 2 GBq in 1.6 kg.
| Case | Start, mGy/h | Total, Gy |
|---|---|---|
| 1 GBq in 1 kg | 538 | 49.7 |
| 2 GBq in 1.6 kg | 672 | 62.1 |
The calculator gives the starting dose rate. Because the activity decays away completely, the total dose is that rate multiplied by the mean life, T½/ln 2, which for Y-90 is 92.4 hours. The first row gives 49.7 Gy for each GBq per kg, which is the factor of about 50 used in microsphere treatment planning. The plan itself also corrects for activity shunted to the lungs and for uneven uptake, neither of which this calculation knows about.
FAQ
How thick does a beta shield need to be?
At least the maximum range of the betas, which depends on the endpoint energy and is nearly the same in g/cm² for any light material. Divide by the density to get a thickness: Y-90 needs 1.09 g/cm², or 9.2 mm of acrylic. The range used here is the Katz–Penfold fit to aluminium absorption data.
Why not use lead to shield beta emitters?
Lead stops betas in a very thin layer, but the fraction of their energy turned into bremsstrahlung X-rays rises with atomic number. For Y-90 it is about 6.5% in lead against 0.52% in acrylic, and those photons are far more penetrating than the betas. Use a low-Z material first and add lead outside it only if the photon dose rate needs it.
What is the difference between maximum and mean beta energy?
Beta decay shares its energy with a neutrino, so the electrons come out with a spectrum from zero up to the endpoint, Emax. The endpoint sets the range and therefore the shield thickness; the mean energy, typically between a third and two-fifths of Emax, sets the dose because it is the energy actually deposited per decay.
Can this calculator give skin dose from a contamination spot?
No. Skin dose depends on the source size, any air gap or covering, and the depth of the basal layer at 7 mg/cm², and a single number without those would be misleading. Use a code built for it, such as VARSKIN. The calculator gives only the exact infinite-medium dose rate and its half at a plane surface.
How do I get the total dose from a beta emitter that decays away?
Multiply the starting dose rate by the mean life, T½/ln 2, which is the integral of the decay curve. For Y-90 the mean life is 92.4 hours, so 1 GBq spread uniformly through 1 kg of tissue delivers 49.7 Gy in total.