Gamma Shielding Calculator: Dose Rate and Radioactive Decay

Scope of buildup: mixed-material GP dose results are blocked. The live GP calculation combines optical depths only for identical active materials; it does not use Broder at interfaces. The uncollided diagnostic model remains available for mixed stacks.

The source model is a point source in free air. Physical decay is $A(t)=A_0\exp(-\lambda t)$, with $\lambda=\ln(2)/T_{1/2}$. The CNSC coefficients give anterior-posterior effective dose, not air kerma or a personal dose equivalent. Numerically, the tabulated gamma constant is evaluated at the 1 m reference distance, then scaled by $(1\,\mathrm{m}/r)^2$.

For each retained photon line $i$, interpolate NIST total mass attenuation $\mu/\rho$ log-log in energy and multiply by nominal density. The optical depth is $X_i=\sum_j\mu_{ij}t_j$. Relative line weights are $w_i\propto y_i E_i(\mu_{en}/\rho)_{air,i}$, normalized to one. They are an air-kerma weighting proxy for transmission of the tabulated effective-dose field. The uncollided transmission is $T_0=\sum_i w_i\exp(-X_i)$. Low-energy X-rays and very weak lines are omitted; this approximation matters for deep shielding.

The homogeneous buildup fit uses Harima's GP expression:

The $K=1$ limit is $1+(b-1)x$. We independently fitted the restricted form $d=0$ to the EGS4 columns of Hirayama & Shin (1998), Tables 1–3, attached s3 PDF pp2–4. At 0.1, 0.3, 0.6, 1 and 3 MeV, $b$ is the published 1-mfp value; $c,a$ minimize log-space residuals at 1,3,4,7,10 mfp. The maximum fitted-sample residual is 2.8%. Coefficients are interpolated linearly in log energy. For $0\le x\le1$, we use the explicit interpolation $B=b^x$. No ANSI/ANS-6.4.3 table or coefficient is reproduced. The iron 0.3-MeV table repeats 4.23 at 2 and 3 mfp; the suspect 2-mfp entry is excluded rather than fitted.

The separate benchmark audit reproduces the rejected two-layer Broder expression, $B_1(x_1)+B_2(x_1+x_2)-B_2(x_1)$. Hirayama & Shin Table 4(d), PDF p6 / printed p821, reports $B=4.11$ for 1 MeV photons through 5 mfp water followed by 2 mfp lead. The approximation gives 10.7612, or 2.62 times the published value. Reversing the materials while retaining 5 mfp first and 2 mfp second gives published 7.91 versus 8.32428 from Broder. These are independent mixed-material comparison data, not validation of the rejected approximation. The table's depth column is total depth: row 7 of the 5-mfp-first-layer block means 2 mfp in the second material. Both ordered cases have regression checks on the published data and on rejection of live mixed-material GP output.

For supported homogeneous stacks, $B_i=B_{GP}(X_i)$ at the sum of the active optical depths, and $T_B=\sum_i w_i B_i\exp(-X_i)$. Splitting one material among several layers leaves the result unchanged. For mixed stacks, or depths outside the fit, this transmission is intentionally unavailable; it is not silently replaced by the uncollided value.

The lead 3-MeV fit was recomputed using the correct EGS4 sample $B(7)=4.09$, rather than the transcribed 4.01. Its new maximum sample residual is 1.959568%; all raw fitting samples are stored and every reported residual is independently recomputed by a property check.

The shielding sweep is capped by 3 times the entered stack, the 10-mfp range, and 99.9% of the available source-to-observer distance. The numerical margin keeps the plotted shield inside the geometry boundary.

The reported HVL and TVL solve $\sum_i w_i\exp(-\mu_{i1}t)=0.5$ and $0.1$ for layer 1 material alone, without buildup. The dose-budget stay time is $H_{budget}/\dot H_{now}$ with the rate held constant. Planned-stay dose integrates decay, $H=\dot H_{now}(1-\exp(-\lambda\Delta t))/\lambda$, with a series expansion near zero.

The fixed NRC reference reproduces NUREG-1556 Vol.9 Rev.2, Appendix U Example 1: 60 mCi I-131, physical half-life 8.04 d, occupancy 0.125 at 1 m gives 4.59 mSv using its conventional exposure-to-dose approximation. It intentionally uses the NRC example's constants rather than the CNSC effective-dose coefficient.

• An isotropic point source, with distance measured from its centre and large compared with its physical size. No air attenuation, container attenuation, source self-shielding or room scatter.
• Physical decay only. No biological clearance, replenishment or general daughter ingrowth; Cs-137 assumes equilibrium with Ba-137m.
• Each shield covers the direct source-to-observer ray with no gaps. Inputs are actual path lengths. Fixed nominal densities: lead 11.35, elemental iron 7.874 and water 1.000 g/cm³.
• Retained photon energies and yields are listed in the tool tables. Energy-absorption weighting approximates effective-dose transmission; spectral hardening is evaluated over those lines. Missing low-energy and weak lines are not restored by gamma-constant normalization.
• GP fits represent point-isotropic exposure buildup in an effectively infinite homogeneous medium. All active layers must have the same material for GP dose results. Mixed-material Broder values appear only in the benchmark audit; they never enter live dose or budget-time results. Finite barriers and backscatter are not resolved.
• Continuous presence during the planned stay, fixed position and shield arrangement. The budget is supplied by the user and is not a prescribed legal allowance. 1 mCi = 37 MBq; 1 mrem = 10 µSv. Conventional activity and dose-rate results are included because the runtime has no built-in radiological-unit conversion.

Mixed-material buildup is unavailable. Selecting GP for different active materials raises an error and withholds the dose results. In uncollided mode, the mixed stack can be explored, but a prominent warning explains that omitted scattered dose can make the calculated budget time too long for an allowed stay. Use a validated transport model or measured dose for that decision.

Use for education and preliminary comparisons, not final radiation-protection decisions. Photon attenuation and buildup interpolation are confined to 0.1–3 MeV. For GP, every retained line must have total optical depth no greater than 10 mfp. Uncollided calculations can extend beyond this bound, with no GP comparison shown; internal safeguards never make a GP result valid outside its range. The fit residual at tabulated samples does not establish accuracy at intermediate energies, for real slabs or for mixed layers. Tc-99m in lead is particularly sensitive to the coarse low-energy buildup interpolation and fluorescence. Above 8 mfp, weak omitted lines can dominate.

HVL/TVL outputs are uncollided spectral values; do not substitute them for published broad-beam first or equilibrium shielding layers. Infinite-medium buildup can sometimes predict a value above the vacuum free field; the tool flags that outcome rather than concealing it.

No skin dose, internal contamination, beta/positron shielding, beta-generated bremsstrahlung, neutron dose, penetrations, streaming or finite-source geometry is modeled. Even with correct numerical results, final shielding design and its installation verification belong to a qualified radiation-protection expert.

The I-131 calculation is explicitly a historical January 2008 NRC reference, not a patient-release assessment or a threshold test. Patient release requires an authorized professional, patient-specific retention and occupancy, assessment of contacts and internal exposure pathways, appropriate instructions, and the jurisdiction's rules in force. NUREG-1556 Vol.9 Rev.3 (September 2019) is cited for context; publication status reviewed 2026-10-28.