1What exactly is computed

The quantity computed is the personal dose-equivalent rate for skin, Ḣ_p(0.07) — an operational quantity defined at a depth of 0.07 mm (70 µm, the basal layer of the epidermis) and averaged over an area of 1 cm². This is the quantity against which skin limits are set. The dose comes from beta particles of purified uranium (dominated by ^234mPa, E_max = 2.27 MeV); the gamma contribution is included in the result and is below 0.3 %.

The geometry is beads resting directly on the skin, with no air gap and no clothing. This is the conservative (worst-case) direct-contact situation.

2The model

The calculation is built on a point surface kernel h(r) — the dose rate in the basal layer at a point of skin located a distance r from a point beta source on the surface:

h(r) = C · exp(−(r/a)b) / (r² + r0²)s

A bead is treated as an emitting face of radius d/2 with a radial emission profile q(ρ) ∝ (1 − (ρ/(d/2))²)m, which reflects the varying thickness of a rocaille bead. The dose rate at a point is the sum over all beads; the result is then averaged over a disc of the selected area. The parameters C, a, b, s, r0, m are fitted so that the model reproduces the Geant4 calculation (see § 3).

Scaling with the input parameters:

• Uranium content enters linearly: the ²³⁸U specific activity of the glass is proportional to the uranium mass fraction (1.2356·10⁷ Bq per kg of uranium).
• Bead size enters in two ways: through the area of the emitting face (integration over the disc d/2) and through saturation with thickness. Beta particles escape only from a surface layer of order 1/(ν·ρ) ≈ 0.8 mm, so a bead's contribution grows roughly as its area rather than its volume; the saturation factor is 1 − exp(−ν·ρ·h) with ν = 5.19 cm²/g.
• Glass density acts only through that saturation factor: the surface emission from a saturated layer is practically independent of density, because the layer thickness is inversely proportional to it.
• Packing pitch and number of beads enter through the summation — no separate fitting is needed for a new geometry.

3Verification: model against Geant4

Below is the comparison with the Monte Carlo reference points used to calibrate the kernel. The values are computed by this page right now, with the same code that produces the calculator result.

4Limits of applicability

• The calibration was performed for a rocaille bead of Ø 4.1 mm packed at a pitch of 4.1 mm. Diameters of roughly 2…8 mm should be regarded as interpolation; beyond that range the result is an extrapolation, and the page warns about it explicitly.
• Purified uranium is assumed, with the ²³⁸U chain broken at ²²⁶Ra: the emitters are ²³⁴Th, ^234mPa, ²³⁵U and ²³¹Th. For ore or unpurified material the decay-product activity would be higher.
• Contact is assumed tight and without any gap. Any interlayer — a glove, fabric, or an unusually thick stratum corneum — reduces the dose, and reduces it strongly: the model does not account for this and is therefore conservative.
• Not accounted for: intake, glass fragments entering a wound, hand contamination and transfer of activity. These are separate and potentially more significant exposure pathways.
• The stated uncertainty of ±10 % is the uncertainty of the underlying Geant4 calculation (mainly glass density and composition). The model's own deviation from Geant4 (0.7 %) is contained within it, whereas the systematic effects of bead shape and of the contact assumption are not included.

The kernel is calibrated against the data of "Calculation of the personal skin dose-equivalent rate Ḣ_p(0.07) from beta and gamma radiation of uranium glass beads" (Geant4 11.2.1, G4EmStandardPhysics_option4 + G4RadioactiveDecay). The method, input data, the independent classical cross-check and the full list of references are given in the main paper.