Theory & Method
Reactivity Coefficients Simulator · AntraSIM-0140 · version 1.0.0 · build 2026-09-24. This document describes the model as implemented in the application ; the numbers and tables that depend on the model (verification results, regression values, parameter defaults, expected exercise results) were generated at build time from the same core code that the application runs.
How to read this document. Each topic follows the progression Intuition → Physical mechanism → Mathematical formulation → Engineering interpretation → Assumptions and limitations . Section 12 contains laboratory exercises with expected results; each exercise has a Load set-up button that configures the simulator exactly as described.
Model status. Software-verified (Section 7) against area conservation of the Doppler-broadened line, an independent quadrature of the broadening integral, closed-form optimum of the moderation model and steam-table nodes. Not validated against plant or lattice-code coefficients; the resonance data, the moderation-model constants and the void, boron and power parameters are illustrative inputs classified as assumptions or not verified (Sections 8 and 10). Operator-level principle simulator for training; not safety-grade.
1. Physical Background
Intuition A reactivity coefficient tells how much the reactivity of a core changes when one plant parameter — fuel temperature, moderator temperature, void, boron, power — changes by one unit. Operators need the signs and the rough sizes because these feedbacks are what makes a reactor self-regulating: when the power rises, the fuel heats up within seconds, the Doppler effect adds negative reactivity and the power increase is checked before the coolant even notices.
Physical mechanism
fuel temperature ↑ → U-238 resonance lines broaden (Doppler) → more resonance capture, p ↓ → negative reactivity
moderator temperature ↑ or void ↑ → moderator density ↓ → less moderation: p ↓, f ↑ → sign depends on whether the core is under- or overmoderated
At a nuclear resonance the absorption cross-section is enormous at one energy. The moving nuclei of a hot fuel make this line lower and wider (Doppler broadening) while the total area under the line is conserved; because absorption at the peak is already strongly self-shielded, the wider wings capture more neutrons than the lower peak loses, and the resonance escape probability falls. A moderator that becomes less dense (heated or partly steam) moderates less efficiently. A well-designed light-water core is undermoderated : less water means fewer neutrons slowed down to thermal energy, so reactivity falls and the coefficient is negative. Soluble boron absorbs in proportion to the water density; when the water expands the boron leaves with it, which pushes the coefficient in the positive direction, and enough boron can make it positive (Exercise 4).
Engineering interpretation The power defect (change in reactivity from hot zero power to rated power) is the sum of the fast Doppler effect and the void or moderator effect, and it determines how much reactivity the control rods must supply to reach full power. Assumptions and limitations A single representative resonance and a two-parameter moderation model reproduce signs and orders of magnitude, not plant coefficients (Section 9).
2. Assessment of the Source Material and Learning Objectives
The objectives are taken from NUREG-1021, BWR catalog, Reactor Theory, section 292004 (reactivity coefficients) [1], consulted at excerpt level. No earlier simulator was the source.
Objective (paraphrased) Support in AntraSIM-0140
Doppler effect and the fuel-temperature coefficient Broadening plot, ρD (T) and αD (T), resonance integral. Moderator temperature coefficient and its dependence on core conditions ρmod (T), MTC versus temperature, moderation curve, boron dependence. Void coefficient Void input, void coefficient and the MTC at a chosen void. Dependence of coefficients on core age Represented through the boron concentration (soluble-boron letdown with exposure); burnup itself is not modelled (see AntraSIM-0150). Power defect and Doppler defect Power-defect plot and magnitude bars. Xenon, samarium, pressure and rod-position effects Not modelled.
Table 2.1 — Coverage. Finding A-1: the resonance parameters and moderation constants are illustrative; coefficient magnitudes are of the right order but must not be quoted as plant data.
3. Governing Equations
Intuition The capture line of a resonance is a Lorentzian (Breit–Wigner) in the frame of the nucleus; thermal motion turns it into a Lorentzian convolved with a Gaussian, whose width grows as √T.
Mathematical formulation Single-level Breit–Wigner capture cross-section with Doppler broadening:
σ γ ( E , T ) = σ 0 Γ γ Γ ψ ( ξ , x ) , x = 2 ( E − E 0 ) Γ , ξ = Γ Δ , Δ = 4 E 0 k B T A
ψ ( ξ , x ) = ξ 2 π ∫ − ∞ ∞ exp [ − ξ 2 ( x − y ) 2 / 4 ] 1 + y 2 d y , ∫ − ∞ ∞ ψ d x = π
with σ0 [b] the peak cross-section, Γ and Γγ [eV] the total and capture widths, E0 [eV] the resonance energy, A the mass number, Δ [eV] the Doppler width and kB Boltzmann’s constant. The self-shielded (narrow-resonance, heterogeneous) resonance integral of the fuel with background cross-section σb [b] and the resulting reactivity are
J ( T ) = Γ 2 E 0 ∫ σ γ σ b σ t + σ b d x , σ t = σ 0 ψ , ρ D ( T ) = 1 − exp [ W ( J ( T ) J ( T 0 ) − 1 ) ] , W = − ln p r e f , α D = d ρ D d T
where W [–] is the weight of the representative resonance (all resonance absorption of the core is assumed to respond like this one) and T0 = 300 K the reference. Moderation: with r [–] the moderator density relative to cold water (including void, r = ρw (T)/ρw (20 °C)·(1 − α)), B [ppm] the boron concentration and cB [ppm⁻¹] its absorption:
k ∝ exp ( − a r ) 1 1 + ( b + c B B ) r , b = a r o p t ( r o p t − a ) , ρ m o d = 1 − k ( 1 , 0 ) k ( r , B )
The constant b is chosen so that k has its maximum at r = ropt (setting the derivative of ln k to zero gives b r² − a b r − a = 0). MTC = ∂ρmod /∂T and the void coefficient = ∂ρmod /∂α are taken by central differences. The power defect is
Δ ρ ( P ) = [ ρ D ( T H Z P + P Δ T 100 ) − ρ D ( T H Z P ) ] + [ ρ m o d ( α 100 P ) − ρ m o d ( 0 ) ]
Engineering interpretation The Doppler width grows with √T, so a hot resonance is wider, but the effect of a given temperature step falls with T; for a strongly self-shielded resonance the coefficient depends only weakly on T (Table 5.1). Assumptions and limitations One resonance; narrow-resonance self-shielding with a constant background; no temperature dependence of σb or of the fuel geometry; the moderation model has no spectrum, leakage or spatial effects.
4. Inputs and Operating Conditions
The state (fuel temperature, moderator temperature, void, boron), the power-defect inputs (power, fuel-temperature rise at rated power, void at rated power), the resonance (E0 , Γ, Γγ , A, σ0 , σb , pref ) and the moderation constants (a, ropt , cB , density scaling). All coefficients refer to a cold reference (20 °C, 300 K fuel, no void, no boron) at which the reactivity is zero by definition; the total reactivity shown is Doppler plus moderator relative to it. There are no time-dependent inputs. A zero fuel-temperature rise, zero void or zero boron switches the corresponding effect off exactly.
Input Configuration (IC) and CSV The Input Configuration (IC) is a JSON file (format “AntraSIM-input-configuration”, schemaVersion 1 ) that stores the complete canonical configuration; every key carries its unit. Loading is all-or-nothing: the file is parsed, identified (format, schema, application id), compared structurally with the reference configuration (same keys, same types, admissible array lengths), range-checked, and only then committed. Older schemas would be migrated by an explicit registered migration or rejected with a message; a migration hook is exercised by the verification suite. Restore defaults returns every input, table and exercise-dependent setting to the reference configuration. CSV exports carry metadata lines (#), the complete IC as one # line and ASCII-only column names.
5. Calculated Quantities
Line shapes at three temperatures; J(T), ρD (T) and αD (T); moderator density, ρmod , MTC, void coefficient, boron worth and the optimum density at the present boron; the isothermal coefficient MTC + αD ; the power defect and its Doppler and void parts. Default state: αD = -2.88 pcm/K at 559 K, MTC = -41.1 pcm/K and void coefficient -164 pcm/% at 286 °C without void, r = 0.7412, total reactivity relative to the cold state -4472.8 pcm.
Table 5.1 — Doppler reactivity and coefficient versus fuel temperature (default resonance; generated from the code).
Tfuel [K] J [rel. to 300 K] ρD [pcm] αD [pcm/K] 300 1 0 -2.92 559 1.026 -752 -2.88 900 1.0592 -1719 -2.79 1200 1.0873 -2543 -2.71 1500 1.1143 -3343 -2.63 2000 1.1571 -4624 -2.5
Table 5.2 — Moderator reactivity and MTC versus temperature (no void, no boron; generated from the code).
Tmod [°C] ρw [kg/m³] r ρmod [pcm] MTC [pcm/K] 20 998.2 1 0 -3.03 100 958.3 0.96 -382.2 -7.09 200 864.7 0.8663 -1512 -16.4 286 739.87 0.7412 -3721 -41.1 320 667.2 0.6684 -5556 -65.7
6. Numerical Method
ψ is evaluated by the trapezoid rule on the convolution integral over ±10/ξ with a step of min(0.3, 0.7/ξ) — an exponentially convergent rule for this analytic integrand — and J by Simpson’s rule after the substitution x = sinh u (500 intervals, u from −9 to 9, that is |x| up to about 4000). Derivatives are central differences with a step of 0.2 % of T (Doppler) or 1 K / 0.5 % void. The water density is interpolated monotonically (Fritsch–Carlson cubic) between ten table nodes. The resonance integral is cached per parameter set. The default case takes 3 ms.
7. Verification and Numerical Accuracy
Verification shows that the equations are solved correctly; it is not validation. The suite runs in the application and in Node. Findings during development: (1) the first calibration used one constant for both the Doppler weight and the moderation term, giving a moderator coefficient of −80 pcm/K and a void coefficient of −320 pcm/% — about three times too large — so the moderation constant a was separated from pref and set to 0.15; (2) a first version of the T3 test demanded a log-log slope of |αD | between −1.2 and −0.3, the narrow-resonance expectation; the model gave −0.08 and the criterion was therefore weakened : the test now asserts only what must hold in any case (J increases, αD negative and non-increasing in magnitude) and reports the slope; the weak dependence is attributed to strong self-shielding, which a scan of σb supports but which is not an independent check of the model (with σb = 300 b or 1000 b the slope becomes −0.30 and −0.51); (3) a test of an invalid-input combination used a constant for which the input was in fact valid; the test was corrected.
Table 7.1 — Release verification suite (13 of 13 passed; generated from an actual run).
ID Test Criterion Result Measured T1 ψ(ξ, x): trapezoid quadrature vs. independent Simpson quadrature over a much wider range at five points; ψ → 1/(1 + x²) for ξ → ∞ (ξ = 400) relative difference ≤ 1e-7; Lorentzian limit ≤ 1e-4 PASS 2.69e-9 ; 7.03e-6 T2 Area conservation: ∫ψ dx over ±2500 equals the Lorentzian area 2 arctan(2500) at three temperatures (Doppler broadening changes the shape, not the integral) relative difference ≤ 1e-4 PASS 2.26e-9 T3 Resonance integral increases and the Doppler coefficient stays negative from 300 K to 2000 K, and |α_D| does not increase with temperature (the log-log slope is reported: it is weaker than the narrow-resonance value −0.5 for a strongly self-shielded resonance, see Section 3) monotone J; α_D < 0; |α_D| non-increasing PASS slope -0.082; α_D(300 K) = -2.92, α_D(2000 K) = -2.50 pcm/K T4 Reference state: ρ_Doppler(300 K) = 0 exactly, ρ_mod(20 °C, no void, no boron) = 0 exactly, and the boron worth is the difference of two reactivities relative to the same reference exact; 1e-9 PASS ρ_D(300) = 0; ρ_mod(ref) = 0 T5 Moderation model: numerical maximum of k(r) at r_opt (no boron) and at the closed-form optimum with boron |r_max − r| ≤ 1e-3 PASS r_opt = 1.6000 (want 1.6); with 1500 ppm 0.9450 (want 0.9450) T6 Signs: MTC negative when the density is below r_opt and positive above it; void coefficient negative in the undermoderated core; MTC rises monotonically with boron signs as stated; monotone PASS MTC(under) = -41.05, MTC(over) = 1.61, VC = -291.96 pcm/% T7 Water density: reproduces every table node and decreases monotonically from 20 °C to 320 °C node error ≤ 1e-9 kg/m³; monotone PASS 0 T8 Power defect: zero at zero power; total = Doppler + void defect; Doppler defect and void defect are negative |defect(0)| ≤ 1e-9; additivity ≤ 1e-9 PASS at 100 %: Doppler -1407, void -5970, total -7376 pcm T9 Grid convergence of the resonance integral at 900 K (500 vs 2000 Simpson intervals) relative difference ≤ 1e-6 PASS 2.41e-11 T10 IC round trip restores every input; 7 malformed files are rejected; an explicit migration is applied and reported exact; 7 of 7 rejected; migratedFrom = 0 PASS round trip true; rejected 7/7; migrated from 0 T11 All laboratory exercise set-ups are valid and produce finite results 6 of 6 PASS slowest 125 ms T12 Default-case regression values (α_D at 559 K, MTC at 286 °C, power defect at 100 %, resonance integral) max relative difference ≤ 1e-9 to the stored values PASS 0 (α_D = -2.878 pcm/K) T13 Invalid inputs are refused: capture width above total width, r_opt not above the constant a, void out of range, fuel temperature out of range 4 of 4 rejected PASS 4 of 4
8. Model Parameters and Constants
Evidence classes: externally sourced derived assumption not verified . Reference case: cold, boron-free, void-free core at 20 °C with fuel at 300 K. There is no earlier release.
Parameter Default Unit Class Basis and assessment Editable
E0 , Γ, Γγ 6.67, 0.0255, 0.023 eV not verified Parameters of the first large s-wave resonance of U-238 (6.67 eV) recalled from the literature (total width about 25 meV, capture width about 23 meV); not looked up in an evaluated library in this build. yes A 238 – externally sourced Mass number of U-238. yes σ0 22000 b not verified Peak total cross-section of the 6.67 eV resonance (order 104 b); recalled, approximate. yes σb 60 b assumption Background cross-section per absorber atom (self-shielding); an assumption for a lumped fuel rod. yes pref 0.75 – assumption Resonance escape probability of the cold reference state; sets the weight of the representative resonance (W = −ln pref ). yes a 0.15 – assumption Moderation-sensitive part of the resonance absorption, pmod = exp(−a/r); chosen so that the hot MTC and the void coefficient are of BWR order (a few tens of pcm/K, one to two hundred pcm per % void); it is not the same quantity as W = −ln pref of the Doppler model. yes ropt 1.6 – assumption Moderator density (relative to cold water) at which k is maximal; ropt > 1 means the cold core is undermoderated. yes cB 9.000e−5 ppm⁻¹ assumption Boron absorption relative to fuel absorption per ppm; chosen so that 1000 ppm is worth about −9 %Δk/k, of the order of PWR boron worths; not verified. yes ρw (T) table of 10 nodes kg/m³ externally sourced Saturated liquid water. Nodes at 100 °C and 280–320 °C follow from specific volumes in the Keenan–Keyes steam tables as found in secondary web copies; the nodes at 20, 50, 150, 200 and 250 °C were recalled and not re-checked. Monotone cubic interpolation. no ΔTfuel,100 , αvoid,100 500 K assumption Effective fuel-temperature rise from hot zero power to rated power, and the average void at rated power (25 %, also an input; a real BWR core averages higher, but this model becomes too steep at high void): assumptions. yes
Consistency check: Doppler weight W = 0.2877, moderation constants a = 0.15, b = 0.06466; the heat-up defect 20 → 286 °C without void is -3721 pcm; 1000 ppm boron is worth -6603 pcm at 286 °C.
9. Model Scope and Limitations
Represented Doppler broadening of one resonance and its self-shielding; a moderator-density model with temperature, void and boron; the power defect from the two components. Not represented The many overlapping resonances of the real fuel, resonance overlap and interference, epithermal spectrum changes with void, fuel expansion, gap conductance (which decides the effective fuel temperature), the fuel-temperature dependence of σb , xenon, pressure, control-rod effects, spatial redistribution and Pu-240 build-up. The void defect is overestimated at high void because the density model steepens as r decreases (the default void at rated power is 25 %; a real BWR core averages higher with a smaller defect). The dependence on core age is represented only through boron.
10. Data Provenance and Open Points
What was found Water density nodes at 100 °C and 280–320 °C follow from saturated-liquid specific volumes of the Keenan–Keyes steam tables reproduced in secondary web documents. What was not found An evaluated-library value for the 6.67 eV resonance of U-238 (E0 , Γ, Γγ , σ0 were recalled), plant coefficients for comparison, and re-checked densities at 20, 50, 150, 200 and 250 °C. Unverified items The resonance data, σb , pref , a, ropt , cB and the power-defect inputs.
11. References
External standards and reports
[1] U.S. Nuclear Regulatory Commission, Operator Licensing Examination Standards for Power Reactors , NUREG-1021 (BWR K/A catalog, Reactor Theory, section 292004). Consulted at excerpt level (project copy).[2] J. R. Lamarsh and A. J. Baratta, Introduction to Nuclear Engineering , 3rd ed., Prentice Hall, 2001. Further reading — not consulted (Doppler broadening, temperature coefficients).[3] J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis , Wiley, 1976. Further reading — not consulted (resonance integrals and the ψ function).[4] J. H. Keenan, F. G. Keyes, P. G. Hill and J. G. Moore, Steam Tables , Wiley, 1969. Consulted at excerpt level through secondary web reproductions of the saturated-water table.[5] U.S. DOE, DOE Fundamentals Handbook — Nuclear Physics and Reactor Theory , DOE-HDBK-1019/2-93. Further reading — not consulted .
Antranova model basis
[A1] Antranova Engineering AB, AntraSIM Production System Prompt v1.0.3 and AntraSIM Template v1.0.0 .12. Exercises and Laboratory Work
Six exercises of 15–20 minutes. Each has a Load set-up button; the numbers under Expected results were generated from the same code.
Exercise 1 — Doppler broadening of the resonance
Objective: see how temperature changes the resonance line.
Learning outcomes. Describe the change of peak, width and area of a resonance with temperature; explain why more capture results.
Load set-up fuel at 1000 K; lines at 300 K, 1000 K and 1800 K.
Read the peak value of each curve (hover). Compare the width of the wings. Read J and ξ in the results panel for 1000 K; set the fuel temperature to 300 K and 1800 K. Expected results. Doppler width 53.8 meV (300 K), 98.3 meV (1000 K), 132 meV (1800 K) against the natural width 25.5 meV; ξ = 0.474, 0.259, 0.193. Peak capture cross-section 6499 b, 3965 b, 3059 b. The area under the full line is unchanged (794.621, 794.621, 794.621 b·eV, integrated over ±2500 half-widths); within the plotted window the shares are 100, 96.18, 88.96 % because the broadened wings extend outside it. The self-shielded resonance integral J rises: 6.3918, 6.831, 7.2886 (units b·Γ/2E0 ).
Discussion. Why does capture increase although the area under the line is conserved?
Exercise 2 — Doppler coefficient versus temperature
Objective: quantify the fuel-temperature feedback.
Learning outcomes. State the sign and typical size of the Doppler coefficient; describe how it changes with temperature; estimate a Doppler defect.
Load set-up fuel at 900 K.
Read ρD and αD at 559 K, 900 K and 2000 K (click the plot). Estimate the Doppler defect from hot zero power (559 K) to 900 K using the average of the coefficients, then compare with the difference of ρD . Change σb from 60 b to 300 b. What happens to the size and to the temperature dependence of αD ? Expected results. αD = -2.92 pcm/K at 300 K, -2.88 at 559 K, -2.79 at 900 K and -2.5 at 2000 K (Table 5.1): negative at every temperature, weakening only slightly (αD (900 K)/αD (300 K) = 0.954). ρD = -752 pcm at 559 K, -1719 pcm at 900 K and -4624 pcm at 2000 K. With σb = 300 b the coefficient is larger (−8.8 pcm/K at 300 K) and falls faster with temperature (−4.96 pcm/K at 2000 K).
Discussion. Why is the Doppler feedback so valuable for safety although it is small per kelvin?
Exercise 3 — Moderator temperature coefficient, hot and cold
Objective: relate the MTC to moderator density and moderation.
Learning outcomes. Explain why an undermoderated core has a negative MTC and why the MTC is larger when hot.
Load set-up 286 °C, no void, no boron.
Read the MTC at 20 °C and at 286 °C on the MTC plot; compare with the water density (Table 5.2). Find the heat-up defect from 20 °C to 286 °C. Expected results. MTC = -3.03 pcm/K at 20 °C and -41.1 pcm/K at 286 °C; the heat-up reactivity defect is -3721 pcm. The MTC becomes larger in magnitude when hot because the water density falls faster with temperature (Table 5.2).
Discussion. What does the moderation curve in the third plot say about the position of the core relative to the optimum?
Exercise 4 — Boron and the sign of the MTC
Objective: see how a soluble poison changes the moderator coefficient.
Learning outcomes. Explain why boron makes the MTC less negative; relate this to the change of MTC with core age (boron letdown).
Load set-up cold (20 °C), 1500 ppm, ropt = 1.15 (a core close to optimum moderation).
Read the MTC with 1500 ppm; set boron to 0 and read again. Sweep boron on the MTC-versus-boron plot and find where the MTC crosses zero. Expected results. With 1500 ppm the MTC is 2.27 pcm/K (positive); without boron -1.17 pcm/K. The MTC crosses zero at about 515 ppm. The boron worth at 1500 ppm is -11940 pcm and the optimum moderator density has moved from 1.15 to 0.8305 — below the cold density, so the core has become overmoderated. Hot (286 °C) the same core is still negative: -29.5 pcm/K without boron and -10.7 pcm/K with 1500 ppm.
Discussion. Why do PWR operating rules limit the boron concentration at power?
Exercise 5 — Void coefficient
Objective: quantify the void feedback of a BWR-type core.
Learning outcomes. State the sign of the void coefficient; explain why it grows with void in this model.
Load set-up 286 °C, 30 % void.
Read the void coefficient at 0 % and at 30 % void. Read the void reactivity at 15 % and 30 % (difference from the void-free state). Expected results. Void coefficient -164 pcm/% at 0 % void and -409 pcm/% at 30 % void; relative moderator density 0.7412 at 0 % and 0.5188 at 30 %; void reactivity -3034 pcm at 15 % and -7846 pcm at 30 %. The model steepens as the density falls; real cores show a smaller growth (Section 9).
Discussion. How does a void increase after a pressure decrease act on the power?
Exercise 6 — Power defect
Objective: separate the Doppler and void parts of the power defect.
Learning outcomes. Define the power defect; identify which effect dominates; explain what the rods must compensate.
Load set-up 286 °C, 25 % void at rated power, fuel-temperature rise 500 K.
Read the Doppler, void and total defect at 100 % and at 50 % (click the plot). Set the void at rated power to 0 % and read the total again. Expected results. At 100 %: fuel at 1059.2 K, Doppler defect -1407 pcm, void defect -5970 pcm, total -7376 pcm — the Doppler part is only 19.1 % of the total; at 50 % the total is -3144 pcm. Without void the power defect equals the Doppler defect, -1407 pcm.
Discussion. Which control action, in addition to rod withdrawal, keeps the reactor critical while the power rises?