Seawater Tritium Measurement with Dual-Diameter Scintillating Fiber Arrays Under Variable Radionuclide Interference

1. Introduction

Tritium (³H), the only radioactive isotope among the three naturally occurring hydrogen isotopes, is monitored in nuclear-facility discharges. Marine tritium derives from cosmogenic production, historical weapons testing and nuclear activities, whose relative contributions vary in space and time [1,2,3]. It commonly occurs as tritiated water and follows water transport. Online monitoring can track concentration changes in effluents and receiving waters during a release, provided that the detector distinguishes tritium from other radionuclides entering the measurement cell.

Nie et al. investigated coastal tritium dispersion in Hangzhou Bay [4], and Qu et al. examined coupled transport in water and the atmosphere [5]. Fukushima observations likewise reveal variable marine tritium distributions [6,7]. These findings motivate timely measurements without assuming proportional changes in tritium and interfering radionuclides.

Liquid scintillation counting provides sensitive laboratory measurements of aqueous tritium [8]. With electrolytic enrichment, Lin et al. reported a minimum detectable activity concentration (MDA) of 0.10 Bq L⁻¹ for 350 mL seawater samples [9]. Zhang et al. developed a 700 mL enrichment method with a reported detection limit of 0.45 tritium units and applied it to the East China Sea [10]. These values depend on each study’s preparation conditions and statistical conventions. Sample preparation, enrichment and counting delay the availability of results. Automated liquid scintillation enables online analysis but requires reagents, sample conditioning and waste handling [8].

Solid scintillators allow repeated measurements without mixing each sample with a liquid scintillation cocktail. Granulated flow cells and plastic scintillation microspheres have been investigated [11,12]. The TRITIUM project developed modular fiber detectors through design, transport studies and field installation [13,14]. Sanada et al. combined a pellet flow cell with coincidence measurement, a veto detector and shielding, using spectral information to identify non-tritium interference qualitatively [15]. Interference assessment is thus an established requirement for scintillation-based tritium instruments.

Cheng et al. developed a plastic scintillating fiber array with a measured efficiency of 1.6 × 10⁻³ and an unshielded MDA of 3165 Bq L⁻¹ over 1600 s [16]. Subsequent work examined array geometry [17] and simulated geometric optimization with suppression of external γ rays and cosmic muons [18]. These studies motivate an internal reference channel. Their source, shielding and readout conditions differ from those used here: published MDAs provide application context, whereas the present structural comparison uses common conditions.

Dissolved interference remains inside the measurement cell even with effective external shielding. Surveys around Chinese coastal nuclear facilities identify natural ⁴⁰K and ²²⁶Ra and artificial ⁹⁰Sr and ¹³⁷Cs alongside tritium [19,20]. Coastal ⁹⁰Sr also reflects transport and groundwater inputs [21,22], while observations off Japan show changing radiostrontium-to-radiocesium ratios [23,24]. Energetic electrons and photon-generated secondary electrons can cross a thin sensing fiber while depositing energy within the tritium window. A low-energy counting window therefore cannot identify the radionuclide responsible for every accepted event.

The proposed structure combines 1 mm sensing fibers with larger, independently read out fibers. Fine fibers provide the water–scintillator interface for short-range tritium electrons; larger fibers collect energy from penetrating interference. Coincident reference signals reject candidate events, while reference counts estimate the residual sensing-window background. We test whether this arrangement improves tritium measurement when sample composition is withheld, while retaining most of the fine-fiber response. Both structures use simultaneous background information: the conventional array retains its above-endpoint reference.

We examine this structural concept through radiation-transport simulation and matched performance comparisons. Seven layouts share a source library and readout model. Literature-based seawater compositions test calibration transfer as interfering activity concentrations and radionuclide ratios change. The estimator receives channel counts but no sample-specific interference composition. Concentration bias and root mean squared error (RMSE) quantify model-predicted measurement error; composition-calibrated MDA characterizes retained counting sensitivity. We compare conventional and dual-diameter arrays, then assess reference diameter, placement and optical collection to identify a suitable configuration for integrated tritium monitoring under variable interference.

2. Materials and Methods

2.1. Array Geometry

Figure 1 compares the conventional array with Inner and Outer arrays containing 5 mm reference fibers at the center and periphery, respectively. The cylindrical measurement chamber has a 25 mm inner radius; the chamber and fibers are 250 mm long. The 1 mm sensing fibers follow a hexagonal lattice with 1.3 mm pitch. The conventional array, Old, contains 1285 sensing fibers. Inner arrays contain seven large fibers; the illustrated Outer 5 mm array contains 26. These layouts differ in both reference-fiber placement and number. The transport model represents the fibers as homogeneous polystyrene in direct contact with water.

The single-fiber simulations in Figure 2 compare mean energy deposition from monoenergetic electrons and photons at diameters of 1, 2, 3, 5, 7.5 and 10 mm. Tritium-energy electron deposits vary little with diameter. The 546 keV response approaches a plateau, and the 1310 keV response shows diminishing gains near 7.5–10 mm, whereas responses to 2280 keV electrons and to photons continue to increase. Larger reference fibers therefore improve energy collection from penetrating radiation, with gains that depend on particle type and energy. The 5, 7.5 and 10 mm candidates span these response regimes and are evaluated in complete arrays to balance interference response against the loss of sensing-fiber area.

All layouts share the chamber envelope. Large fibers change the sensing-fiber count, exposed lateral area and water volume (Table 1). Comparisons hold activity concentration and counting time constant while retaining each layout’s water volume. Before optical readout, tritium count-rate sensitivity per unit concentration is G = V_w ε_H, where V_w is water volume and ε_H the acceptance probability per tritium decay.

2.2. Source Compositions and Radiation Transport

The emission library comprises tritium β decay and eight interference cases (Table 2). The ⁴⁰K β and γ branches represent the abundant natural seawater component; ⁹⁰Sr–⁹⁰Y represents penetrating anthropogenic β interference, and the ¹³⁷Cs daughter γ line represents radiocesium-induced secondary electrons. Selected ²²⁶Ra, ²¹⁴Pb and ²¹⁴Bi γ lines represent natural-series interference documented in coastal surveys [19,20,23]. Each history contains one primary emission and its transport-generated secondaries. Emission yields are applied once to the corresponding parent activity concentration.

Calibration composition L uses Ningde regional mean activity concentrations of ⁴⁰K, ²²⁶Ra, ⁹⁰Sr and ¹³⁷Cs [19]. Transfer samples use literature Sr/Cs pairs. All 27 surface samples at 2 m in Casacuberta et al.’s Table 1 [23] were screened for matches in the updated cesium dataset by station, local sampling date and nominal depth [25]. Coordinates were converted to decimal degrees and required to agree within 0.001° in each coordinate, consistent with the reported cesium-position precision. This yielded 19 unique pairs: two records lacked corresponding cesium samples and six failed the coordinate criterion. Selection used neither activity thresholds nor detector performance. Cesium activity concentrations were corrected from 6 April 2011 to each sampling date using a 30.018 y half-life [26], converted from Bq m⁻³ to Bq L⁻¹ and rounded to three significant digits. The modeled pairs span 0.0011–0.085 Bq L^{−1 90}Sr and 0.00226–3.46 Bq L^{−1 137}Cs (Table 3).

All 19 pairs are combined with the Yantai natural matrix: 9.99 Bq L^{−1 40}K and 0.00930 Bq L^{−1 226}Ra [20]. These controlled computational compositions do not reconstruct the original field samples. Identifiers follow the KOK stations; H29 denotes station 29, with the largest cesium input. The near-calibration sample M_ref combines the Fuqing natural matrix [19] with 0.0037 Bq L^{−1 90}Sr and 0.100 Bq L^{−1 137}Cs from a separate cruise at station 20 on 26 August 2011 [24]. Together, L, M_ref and the 19 station-based compositions give 21 modeled conditions. Five stations summarize the cesium-sorted range through the minimum, nearest-rank quartiles, median and maximum; all stations remain in the comparison.

Equal activity concentrations are assigned within ⁹⁰Sr–⁹⁰Y and ²²⁶Ra–²¹⁴Pb–²¹⁴Bi. These equilibrium assumptions give consistent daughter contributions across layouts. All comparisons use the selected emissions in Table 2, rather than complete decay schemes.

Radiation transport is simulated in Geant4 11.2.2 [27,28] with G4EmStandardPhysics_option4. Production range cuts are 0.5 μm for electrons and 1 μm for photons and positrons. The modeled seawater comprises 96.5% water and 3.5% NaCl by mass and has a density of 1.025 g cm⁻³. Polystyrene and PTFE use the corresponding Geant4 NIST materials. Primary positions are uniformly distributed throughout the water volume, with isotropic emission directions. Each of the 63 layout–emission combinations contains 10⁶ histories.

For complete-array transport, electron energies follow the approximate allowed β spectrum

$${\mathrm{f}}_{\mathrm{\beta }}\left(\mathrm{T}\right)\propto \mathrm{p}\mathrm{W}{({\mathrm{T}}_0-\mathrm{T})}^2\mathrm{F}(\mathrm{Z},\mathrm{W})$$

(1)

Here T is kinetic energy, T₀ the endpoint, W = T + m_ec² and pc = [T(T + 2m_ec²)]^1/2. The Coulomb factor is F = x/[1 − exp(−x)], with x = 2π α_fs Z W/(pc); α_fs is the fine-structure constant and Z the daughter atomic number. The approximation omits transition-specific forbidden shape factors [29]. Photon cases use the discrete energies and yields in Table 2. Source definitions are identical across geometries.

2.3. Deposited Energy and SiPM Response

Let E_s and E_l denote the energy deposited during one history, summed over all fibers in the small- and large-fiber channels, respectively. The raw small-fiber window is 0 < E_s ≤ 18.6 keV. In dual-diameter arrays, accepted channel A additionally requires E_l < 18.6 keV; reference channel C includes every history with E_l ≥ 18.6 keV, regardless of E_s. Old uses the raw window as A and E_s > 18.6 keV as C:

$$\mathrm{O}\mathrm{l}\mathrm{d}:\mathrm{A}:0<{\mathrm{E}}_{\mathrm{s}}\le 18.6\mathrm{k}\mathrm{e}\mathrm{V};\mathrm{C}:{\mathrm{E}}_{\mathrm{s}}>18.6\mathrm{k}\mathrm{e}\mathrm{V}$$

$$\mathrm{D}\mathrm{u}\mathrm{a}\mathrm{l}:\mathrm{A}:0<{\mathrm{E}}_{\mathrm{s}}\le 18.6\mathrm{k}\mathrm{e}\mathrm{V},{\mathrm{E}}_{\mathrm{l}}<18.6\mathrm{k}\mathrm{e}\mathrm{V}$$

$$\mathrm{D}\mathrm{u}\mathrm{a}\mathrm{l}:\mathrm{C}:{\mathrm{E}}_{\mathrm{l}}\ge 18.6\mathrm{k}\mathrm{e}\mathrm{V}\left(\mathrm{a}\mathrm{n}\mathrm{y}{\mathrm{E}}_{\mathrm{s}}\right)$$

(2)

For Old, reference counts equal the number of histories with positive small-fiber energy deposition minus the raw-window count. Both channel pairs are disjoint; neither deposited-energy reference accepts tritium events.

For emission i, P_A,i and P_C,i are the accepted and reference counts divided by the number of generated histories. The dual-diameter veto fraction is η_v,i = 1 − P_A,i/P_R,i, where P_R,i is the raw-window probability. This fraction quantifies coincident-candidate rejection, separately from the reference-channel efficiency used for statistical correction.

For an energy deposit E in MeV at axial position z, the mean number of primary photoelectrons at end e is

$${\mathrm{\mu }}_{\mathrm{e}}(\mathrm{E},\mathrm{z})={\mathrm{Y}}_{\mathrm{\gamma }}\mathrm{Q}\mathrm{E}{\mathrm{f}}_{\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{p},\mathrm{e}}\mathrm{e}\mathrm{x}\mathrm{p}(-{\mathcal{l}}_{\mathrm{e}}/\mathrm{\Lambda }){\mathrm{c}}_{\mathrm{e}}\overline{\mathrm{P}\mathrm{D}\mathrm{E}}$$

(3)

Yγ is scintillation yield, Q relative low-energy light yield, f_trap,e the fraction guided toward end e, ℓ_e propagation distance, Λ attenuation length, and c_e effective coupling and photosensor coverage. The overbar denotes spectrum-averaged photon-detection efficiency (PDE). SP101 parameters follow Cheng et al.’s optimization study, published online in 2025 and assigned to the 2026 journal volume [18]: Yγ = 9500 photons MeV⁻¹, refractive index n_PS = 1.58, emission peak 423 nm and Λ = 2.10 m. Q = 1 defines proportional light production. The axial coordinate spans 0 ≤ z ≤ L, where L is the fiber length, ℓ₁ = z and ℓ₂ = L − z.

Mean SiPM PDE is set to 0.50 following the optical study [18]; the S14161-4050HS-06 datasheet specifies 0.50 at 450 nm and 2.7 V overvoltage [30]. Effective coupling and photosensor coverage are represented by c_e = 0.50, with 0.75 used to test improved collection. This factor describes delivery to the active photosensor area, separately from PDE. For the bare polystyrene–water interface, n_water = 1.34 and f_trap,e = (1 − n_water/n_PS)/2 in the meridional approximation. Trapping, attenuation, coupling and detection remain separate factors, with identical settings and channel cuts across layouts.

With Poisson scintillation, independent photon detection and low SiPM occupancy, the two end counts are independent conditional on deposited energy and axial position. Requiring at least one primary photoelectron per end gives the signal-only trigger probability

$${\mathrm{P}}_2(\mathrm{E},\mathrm{z})=(1-{\mathrm{e}}^{-{\mathrm{\mu }}_1})(1-{\mathrm{e}}^{-{\mathrm{\mu }}_2})$$

(4)

For general thresholds, trigger probabilities follow from Poisson tails. The summed primary-photoelectron distribution is averaged over deposited energy and axial position. SiPM gain converts avalanches to charge; optical crosstalk and afterpulsing modify the avalanche distribution [31]. The baseline excludes correlated avalanches and describes the radiation-only primary-photoelectron response without a measured electronics model.

Let f_i(x) describe the joint deposited-energy and axial-position distribution for emission i, and R(n|x) the conditional readout response, with n the vector of end-specific primary-photoelectron counts. For disjoint readout selections Ω_A and Ω_C, source probabilities are

$${{\mathrm{P}}_{\mathrm{j},\mathrm{i}}}^{\mathrm{P}\mathrm{E}}=\int {\mathrm{f}}_{\mathrm{i}}(\mathrm{x}){\sum }_{\mathrm{n}\in {\mathrm{\Omega }}_{\mathrm{j}}}\mathrm{R}\left(\mathrm{n}\right|\mathrm{x})\mathrm{d}\mathrm{x}$$

(5)

The distribution f_i covers all primary histories, including zero-deposit histories. N_s and N_l are the primary-photoelectron counts summed over both ends of the small- and large-fiber channels, respectively. The small-channel window requires 2 ≤ N_s ≤ 6 PE with both ends triggered. Old C requires N_s ≥ 7 PE and both ends triggered. Dual-diameter C requires N_l ≥ 7 PE with both large-channel ends triggered; A excludes C. The 6 PE upper boundary is the baseline mid-fiber mean response at 18.6 keV rounded down; it remains fixed across layouts and coupling tests. These selections include interference migration into A and tritium leakage into C.

For readout integration, the (E_s, E_l) grid distribution minimizes relative entropy subject to transport constraints [32]:

$${\mathrm{p}}^{\mathrm{*}}={\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}_{\mathrm{p}}\sum _{\mathrm{m}}{\mathrm{p}}_{\mathrm{m}}\mathrm{l}\mathrm{n}\left({\displaystyle \frac{{\mathrm{p}}_{\mathrm{m}}}{{\mathrm{w}}_{\mathrm{m}}}}\right)$$

$$\mathrm{A}\mathrm{p}=\mathrm{b},\mathrm{p}\ge 0,\sum _{\mathrm{m}}{\mathrm{p}}_{\mathrm{m}}=1$$

$${\sum }_{\mathrm{m}:{\mathrm{E}}_{\mathrm{s},\mathrm{m}}+{\mathrm{E}}_{\mathrm{l},\mathrm{m}}\ge \mathrm{u}}{\mathrm{p}}_{\mathrm{m}}\le \mathrm{P}(\mathrm{T}\ge \mathrm{u})(\mathrm{\beta }\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}\mathrm{s})$$

(6)

Here p_m and w_m are cell probabilities and reference weights; A p = b encodes the tallies. Constraints preserve normalization, zero/positive-deposit probabilities, accepted and reference fractions, seven joint probabilities P(0 < E_s ≤ 18.6 keV, E_l < T_k), with T_k = 1, 5, 10, 18.6, 30, 50 and 100 keV, and the first two energy moments. The support satisfies E_s + E_l ≤ E₀, where E₀ is the β endpoint or photon energy; β-source inequalities P(E_s + E_l ≥ u) ≤ P(T ≥ u) also constrain the grid. These constraints retain the supplied transport information while leaving the unrecorded energy distribution model dependent. The baseline uses a uniform measure over positive-energy cells with explicit zero-deposit atoms and a grid of 74 linear intervals below 18.6 keV and 66 logarithmic intervals above, including all selection thresholds. Half-resolution grids and the reference measure dE/(E + 0.1 keV) test discretization and shape sensitivity. Baseline deposits share a uniformly distributed axial position; independent channel positions are tested separately. This constrained distribution supports readout integration without generating additional Geant4 histories. Variation across assumptions is reported as model sensitivity, not a confidence interval.

2.4. Tritium Measurement with Fixed Calibration

In either response domain, the expected background count and total mean count in channel j are

$${\mathrm{B}}_{\mathrm{j}}=\mathrm{t}{\mathrm{V}}_{\mathrm{w}}\sum _{\mathrm{i}}{\mathrm{a}}_{\mathrm{i}}{\mathrm{b}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{j},\mathrm{i}}+\mathrm{t}{\mathrm{r}}_{\mathrm{j}}$$

$${\mathrm{\mu }}_{\mathrm{j}}={\mathrm{B}}_{\mathrm{j}}+\mathrm{t}{\mathrm{V}}_{\mathrm{w}}{\mathrm{A}}_{\mathrm{H}}{\mathrm{P}}_{\mathrm{j},\mathrm{H}},\mathrm{j}\in \{\mathrm{A},\mathrm{C}\}$$

(7)

Here j denotes A or C, t is acquisition time in seconds, V_w is water volume in litres, a_i is parent activity concentration in Bq L⁻¹, and b_i is the selected emission yield per decay. A_H is tritium concentration and P_j,H its channel response probability; the interference sum excludes tritium. Each calculation consistently uses either deposited-energy or PE probabilities. Residual rates r_j (s⁻¹) represent additional backgrounds after channel selection. They are zero in both baseline domains; specified nonzero rates test background sensitivity.

The background conversion coefficient is κ = B_A/B_C. In the absence of additional backgrounds, scaling all interfering activity concentrations by the same factor leaves κ unchanged. With the calibration coefficient κ₀ held fixed, the effective tritium concentration sensitivity is G = V_w(P_A,H − κ₀P_C,H). For G > 0 and independent Poisson channel counts, the tritium concentration estimator and its counting variance are:

$${\mathrm{Â}}_{\mathrm{H}}={\displaystyle \frac{{\mathrm{N}}_{\mathrm{A}}-{\mathrm{\kappa }}_0{\mathrm{N}}_{\mathrm{C}}}{\mathrm{t}\mathrm{G}}}$$

$$\mathrm{V}\mathrm{a}\mathrm{r}({\mathrm{Â}}_{\mathrm{H}})={\displaystyle \frac{{\mathrm{\mu }}_{\mathrm{A}}+{{\mathrm{\kappa }}_0}^2{\mathrm{\mu }}_{\mathrm{C}}}{{\left(\mathrm{t}\mathrm{G}\right)}^2}}$$

(8)

The count difference is not truncated. In the deposited-energy benchmark, P_C,H = 0 and G = V_w ε_H; the PE model includes tritium leakage into the reference channel in both G and μ_C. For each layout and readout setting, κ₀ and G are determined from the modeled L calibration and then held fixed. The estimator uses only A and C counts, acquisition time and these parameters. Within the specified response library, test compositions, true background ratios and true tritium concentrations are used to define the computational samples and evaluate performance; they are withheld from the estimator. L tests calibration consistency, whereas the other compositions test calibration transfer.

A change in composition introduces the following systematic bias in the tritium concentration result:

$$\mathrm{B}\mathrm{i}\mathrm{a}\mathrm{s}({\mathrm{Â}}_{\mathrm{H}})=\mathrm{E}[{\mathrm{Â}}_{\mathrm{H}}]-{\mathrm{A}}_{\mathrm{H}}={\displaystyle \frac{{\mathrm{B}}_{\mathrm{A}}-{\mathrm{\kappa }}_0{\mathrm{B}}_{\mathrm{C}}}{\mathrm{t}\mathrm{G}}}$$

(9)

The contributions of individual source changes can be expressed as

$$\mathrm{B}\mathrm{i}\mathrm{a}\mathrm{s}({\mathrm{Â}}_{\mathrm{H}})=\sum _{\mathrm{i}}\mathrm{\Delta }{\mathrm{a}}_{\mathrm{i}}{\mathrm{d}}_{\mathrm{i}}$$

$${\mathrm{d}}_{\mathrm{i}}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{w}}{\mathrm{b}}_{\mathrm{i}}({\mathrm{P}}_{\mathrm{A},\mathrm{i}}-{\mathrm{\kappa }}_0{\mathrm{P}}_{\mathrm{C},\mathrm{i}})}{\mathrm{G}}}$$

(10)

Here Δa_i is the change in activity concentration from calibration, and d_i is the tritium-equivalent response per unit source activity concentration. Branches of the same parent share its change in activity concentration. Assuming stable readout and residual background, the expression separates changes in radionuclide concentrations from geometry-dependent response ratios. Because each source contribution is weighted separately, different Sr/Cs pairs need not produce proportional biases. The expression also provides a consistency check on the multinuclide calculation.

2.5. Measurement Error and MDA

MDA is evaluated separately from concentration measurement error under fixed calibration. It denotes minimum detectable activity concentration in Bq L⁻¹, written A_D in the equations, rather than total activity in the cell. We use a nominal one-sided false-positive probability α = 0.05 and detection probability 0.95 [33]. Each composition’s correct background ratio defines a composition-calibrated sensitivity benchmark. For independent Poisson counts in the disjoint channels, conditioning on their total yields the binomial upper-tail test [34]

$${\mathrm{p}}_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}={\displaystyle \sum _{\mathrm{j}={\mathrm{N}}_{\mathrm{A}}}^{\mathrm{n}}}{\displaystyle \frac{\mathrm{n}!}{\mathrm{j}!(\mathrm{n}-\mathrm{j})!}}{\mathrm{\rho }}^{\mathrm{j}}{(1-\mathrm{\rho })}^{\mathrm{n}-\mathrm{j}}$$

$$\mathrm{n}={\mathrm{N}}_{\mathrm{A}}+{\mathrm{N}}_{\mathrm{C}},\mathrm{\rho }={\displaystyle \frac{\mathrm{\kappa }}{1+\mathrm{\kappa }}}$$

(11)

A detection is declared when the upper-tail probability is at most 0.05. Averaging the conditional rejection probability over the Poisson distribution of the total count gives the detection power, with omitted tails below 10⁻¹³; A_D corresponds to power 0.95. The 5% false-positive bound holds for independent Poisson null counts when the coefficient equals the true background ratio treated as known. The test is also conservative if the applied ratio is higher. With a transferred κ₀, actual false-positive probabilities are evaluated from each sample’s true background means rather than assumed to satisfy the nominal bound. These evaluation inputs remain withheld from the estimator. Detection-power calculations include tritium leakage into C.

Measurement performance is assessed by the expected measurement result E[Â_H], absolute bias, counting standard deviation and RMSE. Here E[Â_H] is the model-predicted mean of the statistical estimator over repeated acquisitions under the Poisson model, not an experimental reading. Absolute bias is |E[Â_H] − A_H|, with overestimation or underestimation stated explicitly. A mean closer to the true concentration need not imply smaller error in an individual acquisition. RMSE combines squared bias and counting variance:

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\mathrm{B}\mathrm{i}\mathrm{a}\mathrm{s}\left({\mathrm{Â}}_{\mathrm{H}}\right)}^2+\mathrm{V}\mathrm{a}\mathrm{r}\left({\mathrm{Â}}_{\mathrm{H}}\right)}$$

(12)

Primary calculations assess measurement performance at true tritium concentrations of 0 and 100 Bq L⁻¹ over matched acquisition times of 600, 3600 and 14400 s (10 min, 1 h and 4 h). A separate 4 h test uses approximately logarithmically spaced concentrations of 30, 100 and 300 Bq L⁻¹ with interference held fixed. These prescribed tritium levels vary independently of the literature-based interference activity concentrations. The time comparison tests when the reduction in squared bias outweighs the counting-variance penalty; 4 h is an evaluated interval, not a fitted optimum. Within each comparison, response parameters remain fixed, and test-sample truth is not used to adjust selections or calibration.

Wilson intervals characterize individual transport probabilities [35]. Performance rankings are point estimates with response parameters held fixed; each layout has hundreds to approximately one thousand accepted tritium histories. Grid refinement, alternative energy and position models, coupling changes and neighboring integer cuts assess model sensitivity. Calibration uncertainty is distinct from Poisson counting error [36]. For the deposited-energy estimator, an unbiased estimate of κ with variance u²(κ), independent of sample counts, adds u²(κ)(B_C² + B_C) to the null numerator variance. In the PE model, κ enters both G and the correction, so their calibration uncertainties are correlated. Reported RMSE includes counting variation and interference-transfer bias under fixed calibration; it does not represent the combined uncertainty from all response and calibration assumptions.

Design selection uses the equally weighted mean RMSE across the 19 matched station-based compositions, calculated separately for each readout setting, acquisition time and tritium concentration. The primary ranking uses 100 Bq L⁻¹ and 4 h, excluding L and the separate M_ref check. This criterion is considered alongside retained tritium response and composition-calibrated MDA in L and H29. It selects one fixed layout for the tested range, without using an individual sample’s unknown composition to choose the layout with the lowest error for that sample.

Post-processing was implemented in Python, and figures were rendered with Matplotlib 3.10.8 from the input and calculated numerical tables. ChatGPT (OpenAI; accessed September 2026; model version not recorded) assisted with preparation of the analysis and plotting code. The calculations follow the equations, selections and reconstruction constraints specified above; the numerical figures are deterministic visualizations of their underlying data.

3. Results

3.1. Tritium Measurement Performance Across Seawater Interference Compositions

In the baseline optical model at a true tritium concentration of 100 Bq L⁻¹, Inner 5 mm reduces absolute bias by 35.5–59.0% across all 19 station-based compositions (Table 4). For S18, the median-cesium case, the predicted mean concentrations are 102.10 Bq L⁻¹ for Old and 100.90 Bq L⁻¹ for Inner 5 mm. The corresponding values are 105.77 and 102.40 Bq L⁻¹ for the upper-quartile case S25, and 133.75 and 113.85 Bq L⁻¹ for H29. Both layouts predict a mean of 100.00 Bq L⁻¹ in L, where calibration and evaluation compositions coincide. Bias is therefore suppressed consistently across the Sr/Cs pairs, despite differences in the magnitude of the transfer error.

For Inner 10 mm, the predicted mean concentration in H29 is 99.00 Bq L⁻¹, closer to the true concentration than for either Old or Inner 5 mm. However, enlarging the reference cluster removes sensing fibers and increases counting uncertainty. Bias suppression must therefore be assessed alongside total measurement error. Across the 19 compositions, mean absolute bias decreases from 4.30 Bq L⁻¹ for Old to 1.79 for Inner 5 mm, while the equally weighted mean 4 h RMSE rises from 15.72 to 16.12 Bq L⁻¹. Inner 5 mm has lower 4 h RMSE only in S23 and H29; Old has lower RMSE in the other 17 compositions. The reduction is only 0.6% in S23 but substantial in H29. Reduced sensitivity to interference thus does not imply lower total error in every composition.

In H29 at 100 Bq L⁻¹ tritium, the 600 s RMSE is 81.8 Bq L⁻¹ for Old, 84.6 for Inner 5 mm and 102.1 for Inner 10 mm. At this duration, bias reduction does not offset the added counting uncertainty. After 3600 s, Inner 5 mm reduces RMSE from 45.4 to 36.8 Bq L⁻¹, or 19.1%. After 14400 s, the reduction is from 37.02 to 21.95 Bq L⁻¹, or 40.7%; the corresponding zero-tritium RMSE reduction is 44.8%. For this stationary composition at 100 Bq L⁻¹, the Old–Inner 5 mm RMSE crossover occurs at approximately 888 s. This behavior follows from a persistent bias and counting variance that decreases as 1/t.

Figure 3 compares performance across the full composition set and its time dependence in H29. Most station pairs produce much smaller transfer errors than H29, so extending integration to 4 h need not compensate for the loss of sensing response. In the separate near-calibration composition M_ref, biases are 0.97 Bq L⁻¹ for Old and 0.40 for Inner 5 mm, with Old retaining lower 4 h RMSE. Within the tested set, the dual-diameter geometry offers its greatest measurement benefit when composition changes cause appreciable bias under the retained calibration.

Varying tritium concentration at fixed interference composition provides a further test of modeled measurement performance (Table 5). In H29 at 4 h and c_e = 0.50, true concentrations of 30, 100 and 300 Bq L⁻¹ give predicted means of 43.85, 113.85 and 313.85 Bq L⁻¹ for Inner 5 mm, compared with 63.75, 133.75 and 333.75 Bq L⁻¹ for Old. Inner 5 mm reduces RMSE by 43.5%, 40.7% and 34.3%, respectively. The additive interference bias is independent of tritium concentration, whereas signal counting variance increases with concentration. RMSE relative to the true concentration therefore decreases as concentration rises, even though absolute RMSE increases. The benefit thus extends across the three prescribed tritium levels.

3.2. Interference Rejection and Background Reference Response

Table 6 gives emission-normalized channel probabilities. For Inner 5 mm, the ⁴⁰K β reference probability is 3.1019%, compared with 0.6701% in the accepted channel, yet the same-event veto removes only 1.41% of its raw ⁴⁰K β candidates. Outer 7.5 mm rejects 34.13% of ⁴⁰K γ candidates and 20.78% of ¹³⁷Cs γ candidates, but only 4.53% of ⁴⁰K β candidates. The independent reference provides background information even when a tritium-like deposit has no simultaneous large-fiber trigger (Figure 4). The measurement benefit therefore depends on statistical background tracking as well as event rejection.

Reference counts alone do not determine correction quality. In L at 600 s, the deposited-energy model predicts 1051.68 reference counts and 11.03 accepted background counts for Old, compared with 48.07 and 10.17 for Inner 5 mm. Old therefore has lower relative counting uncertainty in the reference channel. Correction quality also depends on how consistently each reference predicts the accepted background as radionuclide weights change.

The accepted-to-reference ratios explain the differing composition sensitivities. For Old, P_A/P_C is 0.01026 for ⁴⁰K β and 0.05819 for the selected ¹³⁷Cs γ line; Inner 10 mm gives 0.07346 and 0.08593. The closer ratios reduce sensitivity to changes in the relative contributions of these emissions. From L to H29, deposited-energy κ changes by 30.0% for Old, 14.7% for Inner 5 mm and 4.3% for Inner 10 mm (Figure 5). With the L calibration retained, H29 biases are 21.3, 12.2 and 4.2 Bq L⁻¹, respectively. These direct transport results support the interference mechanism identified in the PE calculation.

Holding the natural matrix fixed across the 19 station-based compositions isolates variation in the Sr/Cs pair. Equation (10) weights each change in activity concentration by its tritium-equivalent response. Proportional increases in all background components leave κ unchanged; changes in their relative contributions cause the transfer error examined here. In L, accepted interference is dominated by ⁴⁰K β, whose small same-event veto fraction explains why background-reference tracking remains essential even when gamma-event rejection is effective.

Bias suppression in H29 is accompanied by lower false-positive probabilities. At c_e = 0.50, the actual probabilities after 600 s are 12.3%, 5.7% and 3.5% for Old, Inner 5 mm and Inner 10 mm; after 4 h, they are 88.2%, 24.7% and 4.1%. Table 7 lists the station-based compositions satisfying the 5% criterion at 4 h, with zero tritium and each layout’s L coefficient retained. Old meets the criterion in S14, S2, S17, S28 and S20. Inner 5 mm meets it in those five compositions plus S24, S9 and S11; Inner 10 mm meets it in all 19. L and M_ref are excluded from these totals. These finite-count results apply to the stated optical setting, duration and backgrounds. With composition-correct coefficients, all evaluated conditional tests satisfy the nominal bound under the assumptions in Section 2.5. Lower measurement RMSE and false-positive control remain distinct criteria.

3.3. Retained Tritium Response and MDA

Reference fibers displace part of the fine sensing array. Inner 5 mm retains 87.32% of Old’s small-fiber lateral area and 88.25% of its deposited-energy concentration sensitivity G. Inner 10 mm retains 57.06% of G, while Outer 7.5 mm retains 43.53%. Old has a tritium response probability of 0.1021%. Figure 2 relates these full-array responses to the single-fiber trends and geometric costs. The design objective is to limit signal loss while obtaining a reference less sensitive to changing interference.

At baseline coupling, the mid-fiber photoelectron yield is 0.1700 primary PE keV⁻¹ per end. A 5.7 keV deposit produces a mean total of 1.94 PE and a two-ended trigger probability of approximately 0.385. Integration over the deposited-energy distribution gives tritium acceptance probabilities of 0.03715%, 0.03332% and 0.02422% for Old, Inner 5 mm and Inner 10 mm (Table 8). Including reference leakage, their effective concentration sensitivities are in the ratio 1:0.8881:0.5763. The 5 mm cluster thus preserves considerably more tritium response than the larger-diameter alternative.

Inner 5 mm has the lowest composition-calibrated MDA among the six dual-diameter candidates. At 600 s, its deposited-energy MDA is 120.5 Bq L⁻¹ in L and 131.8 Bq L⁻¹ in H29 (Table 9), compared with 152.9 and 161.7 Bq L⁻¹ for Inner 10 mm and 97.7 and 105.9 Bq L⁻¹ for Old. The 5 mm central cluster therefore offers the best counting sensitivity within the tested dual-diameter family. Its lower interference-induced bias provides a separate measurement advantage over the conventional array.

The PE calculation preserves this MDA ordering (Table 8). At 600 s, Inner 5 mm gives 296.6 Bq L⁻¹ in L and 314.3 Bq L⁻¹ in H29, compared with 248.3 and 261.9 Bq L⁻¹ for Old: an increase of approximately 20%, with 88.8% of the effective response retained. At 4 h, the corresponding values are 47.3 and 50.6 Bq L⁻¹ for Inner 5 mm, compared with 40.7 and 43.3 Bq L⁻¹ for Old. The calculation includes photon fluctuations, two-ended selection and reference leakage, with the correct background coefficient applied to each composition. These values quantify composition-calibrated counting sensitivity, not detection limits for an uncorrected change in interference.

With PE cuts fixed, increasing common coupling c_e from 0.50 to 0.75 reduces the 600 s L MDA from 248.3 to 164.8 Bq L⁻¹ for Old, from 296.6 to 192.4 for Inner 5 mm, and from 377.8 to 252.2 for Inner 10 mm. Separately, an assumed accepted-channel residual rate of 0.01 s⁻¹, with r_C = 0 and recalibrated coefficients, raises baseline L MDA to 304.0, 389.2 and 508.3 Bq L⁻¹, respectively. The residual rate is a sensitivity-analysis input, not an experimentally measured background. Figure 6 compares MDA under the common readout conditions.

3.4. Readout Sensitivity and Design Selection

All 189 source–geometry reconstructions reproduce the transport probability constraints within 4.36 × 10⁻⁸. Grid refinement changes representative L MDA values by less than 0.2%. Replacing the uniform energy measure with a log-energy measure gives L MDA values of 257.9, 310.4 and 384.5 Bq L⁻¹ for Old, Inner 5 mm and Inner 10 mm, compared with baseline values of 248.3, 296.6 and 377.8 Bq L⁻¹. These checks distinguish discretization error from sensitivity to the assumed energy distribution. Figure 7 identifies the transport and readout stages and the common optical settings.

Measurement bias is more sensitive to energy reconstruction than calibrated MDA. In H29, the baseline PE model overestimates the concentration by 33.7 and 13.8 Bq L⁻¹ for Old and Inner 5 mm, and underestimates it by 1.0 Bq L⁻¹ for Inner 10 mm. With log-energy reconstruction, Old overestimates by 20.2 Bq L⁻¹, whereas Inner 5 mm and Inner 10 mm underestimate by 1.6 and 15.3 Bq L⁻¹. The near-zero baseline bias of Inner 10 mm therefore depends on cancellation among source contributions. At 4 h and 100 Bq L⁻¹, the log-energy model gives H29 RMSE values of 25.36 and 17.25 Bq L⁻¹ for Old and Inner 5 mm, retaining a 32.0% reduction. Among the three inner layouts evaluated under this reconstruction, Inner 5 mm has the lowest mean RMSE across the 19 compositions, although Old retains the lower ensemble mean.

Increasing common coupling to c_e = 0.75 lowers MDA while keeping the selection cuts fixed (Table 10). Inner 5 mm retains 87.8% of Old’s effective response and gives composition-calibrated 4 h MDAs of 28.9 Bq L⁻¹ in L and 31.5 Bq L⁻¹ in H29. Its absolute bias in H29 is 10.5 Bq L⁻¹, compared with 19.8 Bq L⁻¹ for Old. At 100 Bq L⁻¹ tritium and 4 h, its RMSE is 16.12 Bq L⁻¹, compared with 22.72 Bq L⁻¹ for Old, a 29.1% reduction. Across the 30, 100 and 300 Bq L⁻¹ tests, the reductions are 33.5%, 29.1% and 20.6%. Improved collection thus lowers measurement RMSE while the reference cluster continues to suppress interference bias.

Among the six dual-diameter layouts, a 5 mm central reference cluster with c_e = 0.75 is preferred for the combined objectives of low ensemble-mean RMSE and retained counting sensitivity. At 100 Bq L⁻¹ and 4 h, its mean RMSE across the 19 compositions is 11.72 Bq L⁻¹, compared with 12.69 and 14.21 Bq L⁻¹ for Inner 7.5 and Inner 10 mm; the three Outer layouts give 14.18, 16.30 and 17.39 Bq L⁻¹. Inner 5 mm also has the lowest composition-calibrated MDA among the new layouts in L and H29. Old’s ensemble mean remains slightly lower at 11.27 Bq L⁻¹. For the sole objective of minimum H29 RMSE, Inner 7.5 mm gives 14.66 Bq L⁻¹, below Inner 5 mm at 16.12 Bq L⁻¹, but its calibrated MDA increases from 31.5 to 34.6 Bq L⁻¹. These comparisons use identical optical settings and counting times. The 5 mm choice does not establish 5% false-positive control for arbitrary compositions under frozen calibration. Composition-calibrated MDA and the actual false-positive probabilities in Section 3.2 concern different calibration conditions. Coupling and energy-reconstruction changes remain separate sensitivity tests.

4. Discussion

4.1. Structural Basis of Improved Tritium Measurement

The structural benefit arises from closer correspondence between the reference response and the interference entering the tritium window. An energetic electron can deposit little energy in a fine sensing fiber without triggering a coincident veto; the independent reference channel still estimates the remaining interference statistically. Performance therefore depends on both event rejection and the stability of the accepted-to-reference ratio. This explains why modest same-event beta rejection can accompany substantial bias suppression. The conventional array also uses simultaneous reference counts, so the comparison assesses the effect of geometry on reference response, rather than the presence or absence of background correction.

4.2. Interference Tolerance and Retained Sensitivity

The 5 mm central cluster retains approximately 89% of the effective tritium response while reducing sensitivity to composition changes. At baseline collection, its 600 s calibrated MDA is approximately 20% higher than Old’s, but the largest 4 h RMSE across the 19 station-based compositions at 100 Bq L⁻¹ falls from 37.02 to 21.95 Bq L⁻¹. The ensemble mean rises by 2.6%, from 15.72 to 16.12 Bq L⁻¹. A modest increase in mean RMSE therefore accompanies a substantially smaller error in the most perturbed case. Equal weighting describes the tested set, not environmental occurrence frequencies. Under the modeled conditions, this geometry is useful when tolerance of appreciable interference changes is needed alongside counting sensitivity, but offers little measurement benefit when calibration remains representative.

The intended application is seawater tritium measurement under variable radionuclide interference. The computational matrix represents seawater and the specified dissolved radionuclides, without covering the full range of water chemistry or optical turbidity. Absolute predictions remain conditional on the simplified emission library, constrained energy-response reconstruction and radiation-only primary-PE model. Prototype validation of the low-energy response, instrumental background and calibration transfer is needed to establish operational performance. The prescribed concentrations of 30, 100 and 300 Bq L⁻¹ test modeled measurement performance; they do not define the detector’s application range. Concentration results must be interpreted with their MDA, counting time and calibration conditions. Four-hour integration improves counting precision when composition and detector response remain approximately stationary. The acquisition interval should therefore reflect the acceptable concentration error and required update rate.

5. Conclusions

The dual-diameter design provides a separate interference reference while preserving the fine-fiber tritium interface. Within the specified simulation model, Inner 5 mm reduces concentration bias across all 19 literature-based Sr/Cs compositions. In the case with the greatest interference, it lowers 4 h RMSE by 40.7% at 100 Bq L⁻¹ and by 34.3–43.5% across the 30–300 Bq L⁻¹ tests. It retains 88.8% of the baseline effective response and has the lowest calibrated MDA among the new layouts. At a common coupling of 0.75, its 4 h MDAs are 28.9 Bq L⁻¹ in L and 31.5 Bq L⁻¹ in H29, and its ensemble-mean RMSE is the lowest among the tested dual-diameter layouts. At this coupling, the 7.5 mm central cluster minimizes H29 RMSE at 100 Bq L⁻¹ and 4 h, with a higher calibrated MDA. Old retains lower calibrated MDA and better precision for small composition changes. Geometry selection therefore depends on the measurement objective, with measurement error, counting sensitivity and false-positive control assessed under their respective calibration conditions.