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Decay During Transport and Daughter Nuclides

This page explains how radioactive decay between the stack and the receptor is handled, how daughter nuclides enter the airborne dose pathways, and which data the ENSI-G14 Beilage (Stand 28 May 2025) does not give, so that we had to add it.

Deviations from ENSI-G14

Two choices on this page are not in G-14: decay is applied inside the plume at the real travel time (switchable, see below), and the branching ratios \(f_{D_i}\) come from ICRP 107, because the Beilage refers to Table 4.3 for them but no Table 4.3 contains them.

1. Why we decay the plume

What G-14 does

The submersion (A3.1.1) and inhalation (A3.2.1) formulae decay the emitted parent over a fixed flight time \(T_{fz}\):

\[ E_{sub} = \frac{Q}{k_C}\,\chi_{\{K;L\},S}\,k_s \left[e^{-\lambda T_{fz}}\, e_{imm,M}\, k_{SPE,M} + \sum_i f_{D_i}\,\frac{\lambda_{D_i}}{\lambda_{D_i}-\lambda} \left(e^{-\lambda T_{fz}} - e^{-\lambda_{D_i} T_{fz}}\right) e_{imm,D_i}\, k_{SPE,D_i}\right] \]

The package uses \(T_{fz} = 6.3\cdot10^{-6}\) a (3.3 min) for short-term and \(1.6\cdot10^{-5}\) a (8.4 min) for long-term scenarios (data/other_constants.yaml), whatever the distance to the receptor.

Why this is not enough

The real travel time is \(t = x/U\). That ranges from seconds next to the stack to hours at 10 km in light wind (10 km at 1 m/s is 2.8 h). A single \(T_{fz}\) is wrong at both ends:

  • Near field, short-lived nuclides. The fixed \(T_{fz}\) removes activity that has not decayed yet. For O-15 (\(T_{1/2}\) = 2 min) at 30 m from a stack, G-14 removes 94 % of the activity, while the plume has lost about 10 %: the dose is underestimated by a factor of 17 (see Near Field, Short Stacks).
  • Far field. The parent has decayed more than \(e^{-\lambda T_{fz}}\), and the daughters have grown in much more than the formula gives (section 2).

What we do

The averager weights each plume component of each weather bin by a time function evaluated at that component's own travel time, \(t = x/U_{H_a}\) for the elevated component and \(t = x/U_0\) for the ground-level one (dispersion/plume_decay.py):

\[ g_M(t) = e^{-\lambda t}, \qquad g_{D_i}(t) = \frac{\lambda_{D_i}}{\lambda_{D_i}-\lambda}\left(e^{-\lambda t} - e^{-\lambda_{D_i} t}\right) \]

These are the G-14 factors with \(T_{fz} \to x/U\). The result for one nuclide is a DecayedDispersion, holding \(\chi\) for the parent and one \(\chi\) per daughter. The dose functions accept it in place of \(\chi\) and then refuse a T_fz. When every path has the same travel time \(t\), the result equals G-14 with \(T_{fz} = t\) exactly. This is tested in tests/test_plume_decay.py.

  • Switch: ScenarioEvaluator.decay_in_plume and LongTermEvaluator.decay_in_plume. The default is True; set it to False to get the G-14 \(T_{fz}\).
  • Deposition is decayed the same way only for nuclides without daughters. For decay chains, the ground-shine and ingestion coefficients of Table 4.3 count the daughters per deposited parent. Decaying the deposited parent would drop the daughters grown in the air, so for chains deposition stays undecayed, as in G-14.

2. What this changes for the daughters

In-growth depends strongly on the travel time. At the G-14 flight times it is small for most chains; at travel times of an hour it can dominate. The table gives the daughter dose divided by the surviving parent's dose (adult inhalation; submersion includes \(k_{SPE}\)), with the branching ratios of section 3:

Parent Pathway 3.3 min (G-14 short) 8.4 min (G-14 long) 30 min 1 h 3 h
Te-132 → I-132 submersion 0.15 0.37 1.3 2.3 5.4
Te-133m → Te-133, I-133 inhalation 0.04 0.10 0.38 0.89 6.1
Te-133m → Te-133, I-133 submersion 0.02 0.05 0.13 0.17 0.29
Te-129m → Te-129, I-129 submersion 0.04 0.10 0.31 0.53 0.99
Te-131m → Te-131, I-131 inhalation 0.002 0.005 0.018 0.034 0.094
Sb-126m → Sb-126 inhalation 0.003 0.008 0.04 0.17 15

A ratio above 1 means the daughter contributes more than the parent. For Sb-126m (\(T_{1/2}\) = 19 min) the large value at 3 h reflects a parent that has almost disappeared.

Limitations, the same as in G-14:

  • In-growth uses two members, parent → daughter. A three-member chain such as Te-131m → Te-131 → I-131 is entered as two daughters of Te-131m. Section 3 explains how their weights are chosen.
  • The daughter's dose factor is its own Table 4.1 value, e.g. I-131_aer: the iodine formed from tellurium in the plume is taken to be aerosol-bound, as in the Beilage.
  • The ground-shine and ingestion coefficients of Table 4.3 are used as tabulated, once per row; they are not affected by the travel time.

3. What the Beilage leaves out, and what we added

"Slash" daughters

Four rows of Tables 4.3.1 to 4.3.7 list two names in the column "Berücksichtigte Tochterprodukte" ("daughter products accounted for"). These are the members of a decay chain. Each name is a separate nuclide of Table 4.1. The branching ratios are from ICRP 107:

Parent Entry Decay paths
Sb-127 (3.85 d) Te-127/Te-127m 82.3 % β⁻ → Te-127 (9.4 h); 17.7 % β⁻ → Te-127m (109 d) → Te-127
Te-129m (33.6 d) Te-129/I-129_aer 63 % IT → Te-129 (70 min) → I-129; 37 % β⁻ → I-129
Te-131m (30 h) Te-131/I-131_aer 22.2 % IT → Te-131 (25 min) → I-131; 77.8 % β⁻ → I-131
Te-133m (55 min) Te-133/I-133_aer 17.5 % IT → Te-133 (12.5 min) → I-133; 82.5 % β⁻ → I-133

They are not parent/daughter pairs merged into one entry, like Cs-137/Ba-137m or Te-132/I-132 in Table 4.1. Notes b and c before Table 4.3.1 allow that merging only for a daughter with \(T_{1/2}\) < 10 min (submersion, inhalation) or < 1 h (ground, ingestion), with a parent of at least 2 h or 12 h. For the airborne pathways none of the four qualifies. Also, merged pairs appear as parents in Table 4.1, never in the daughter column.

Earlier versions of the package looked up the whole string, e.g. Te-131/I-131_aer, as a single Table 4.1 nuclide, and raised NuclideNotFound for these four parents. The airborne pathways now split the entry on / (DoseTemplate._airborne_daughters). The ground-shine and ingestion coefficients of the row are the total over the chain and are still applied once.

Branching ratios \(f_{D_i}\)

G-14 defines \(f_{D_i}\) as the "Anteil der Mutteraktivität, die in die Tochter \(D_i\) zerfällt", and the Beilage's parameter list gives "Verzweigungsverhältnis in die Tochter \(D_i\) (branching fraction): vgl. Tab. 4.3". No Table 4.3 has such a column. The package used to set \(f_{D_i} = 1\) everywhere. The code comment claimed this was conservative and wrong only for the four slash parents; in fact it was also wrong for Nb-95m, Sb-125, Sb-126m and Cm-243.

We added data/daughter_branching.yaml, which lists only the pairs with \(f_{D_i} \neq 1\) (ICRP 107):

Parent Daughter: \(f_{D_i}\) Rest of the decays
Nb-95m Nb-95: 0.944 β⁻ to stable Mo-95
Sb-125 Te-125m: 0.2314 β⁻ to stable Te-125
Sb-126m Sb-126: 0.14 β⁻ to stable Te-126
Sb-127 Te-127: 0.8232, Te-127m: 0.1768
Te-129m Te-129: 0.63, I-129_aer: 1.0
Te-131m Te-131: 0.222, I-131_aer: 1.0
Te-133m Te-133: 0.175, I-133_aer: 1.0
Cm-243 Pu-239: 0.9976, Am-243: 0.0024

How chains are weighted: a daughter fed directly and through a shorter-lived intermediate gets the total fraction. For example, all of Te-131m ends as I-131, so I-131 gets 1.0 although only 77.8 % is direct. Treating the 25 min detour through Te-131 as direct speeds up the in-growth slightly, which is conservative. A path through a longer-lived intermediate, such as Sb-127 → Te-127m (109 d) → Te-127, adds nothing over a flight time and is dropped. With this convention the fractions of one parent can add up to more than 1.

The Table 4.3 coefficients were computed with the same ratios

For a short-lived parent whose daughters decay within the exposure time, the ground-shine coefficient is approximately \(f_{D,BS} \approx k \sum_i f_{D_i}\, e_{BS,D_i} / \lambda\). Fitting \(k = 0.404\) on the Te-133 → I-133 row (2.63e-13) reproduces:

  • Te-133m: \(k\,(e_{BS,\text{I-133}} + 0.175\, e_{BS,\text{Te-133}})/\lambda\) = 1.565e-12; Table 4.3.1 gives 1.56e-12.
  • Sb-126m: with \(f\) = 0.14, 2.47e-13; Table 4.3.1 gives 2.36e-13. With \(f = 1\) it would be 1.8e-12.

ENSI therefore applied the ICRP 107 branching ratios inside Table 4.3. It only did not publish them for the airborne formulae, which need them explicitly.

Weighting the daughters does matter. Without it, the daughter terms of Sb-126m would be 7 times too large, and the Am-243 term of Cm-243 400 times too large.

To reproduce the values (the radioactivedecay package bundles ICRP 107):

import radioactivedecay as rd

nuclide = rd.Nuclide("Te-131m")
print(list(zip(nuclide.progeny(), nuclide.branching_fractions(), nuclide.decay_modes())))
# [('I-131', 0.778, 'β-'), ('Te-131', 0.222, 'IT')]

What the package uses for a given parent:

from ensi_g14_calculator.dose.short_term_dose import ShortTermDoseCalculator

calculator = ShortTermDoseCalculator()
print(calculator.decay_chain("Te-131m"))   # parent and daughter decay constants
print(calculator.f_Di["Te-131m"])          # {'Te-131': 0.222, 'I-131_aer': 1.0}

4. Further remarks

Transcription check

All seven Tables 4.3.x (30 rows each) were compared with the text of the Beilage PDF (Stand 28 May 2025). Names and all values are identical, so the observations below are about the Beilage itself.

Apparent errors in the Beilage

These are not corrected in the data: the CSV files stay identical to the Beilage.

Te-131m. In Table 4.3.1 every value of Te-131m equals the Te-131 row (resuspension 8.94e-15 ..., ground 3.06e-13). So do the leaf-deposition columns of Tables 4.3.2, 4.3.4, 4.3.6 and 4.3.7, and Table 4.3.3 is within 20 %. Physically the I-131 formed per Bq of deposited parent scales as \(1/\lambda_M\), so Te-131m should be about \(\lambda_{\text{Te-131}}/\lambda_{\text{Te-131m}} \approx 72\) times larger, plus the Te-131 branch:

  • the ground-shine estimate above gives 2.9e-11 Sv/(Bq/m²) instead of 3.06e-13;
  • the long-term Table 4.3.5 does have Te-131m about 93 times Te-131 for ground shine and 61 times for leaf ingestion, consistent with this.

The short-term rows of Te-131m look copied from Te-131. If so, the daughter contribution to short-term ground shine and ingestion of Te-131m is underestimated by about two orders of magnitude. For ground shine this is about a third of the parent's own contribution.

Cm-243 → Am-243. The Am-243 coefficients are 3 to 4 times the Pu-239 ones (Table 4.3.1). That matches \(f = 1\) for both rows: the ratio of the decay constants (3.3) times the ratio of the dose factors. With the EC branch of 0.24 %, Am-243 should be about 1 % of Pu-239. If confirmed, the error is conservative. We are less sure of this one than of Te-131m.

Pathways not affected

  • The photon point-kernel submersion (submersion_pki, see Submersion Models) has no daughter terms, by design.
  • Noble-gas daughters that deposit (note d before Table 4.3.1, e.g. Cs-137 formed from Xe-137) must still be entered as a separate release by the user.