diff --git a/.gitignore b/.gitignore index cb2f636..729a072 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,5 @@ AGENTS.md .DS_Store __pycache__/ +neutron/data/ +neutron/work/ diff --git a/README.md b/README.md index 170caa0..d5c536e 100644 --- a/README.md +++ b/README.md @@ -293,7 +293,7 @@ Every component except one is either routine nuclear physics or recently demonst **Phase B. Settle the keystone.** Now two questions, one per sector. - **B1, photon sector.** Measure (σ_trig, β, C_in) for the best NEEC or IGE candidate (precision EBIT or Penning trap NEEC; resonant IGE of a mid energy isomer) and locate it on the keystone figure. The required cross sections per facility are tabulated in [gates/experiment_menu.md](gates/experiment_menu.md); the ongoing nondestructive Penning trap test of the ⁹³ᵐMo claim largely decides this sector. -- **B2, neutron sector.** Existence is already answered by library data; what remains is architecture. Reproduce the two region NAND of coupled subcritical assemblies in OpenMC with ENDF data, all inputs public, everything at k<1: no laboratory required, and the coupling matrices, absorber programming, and clock rates it yields are inherited by the photon sector when its gate arrives. +- **B2, neutron sector.** *Status: done, in [neutron/](neutron/).* Existence was already answered by library data; the architecture is now computed. The two region subcritical gate runs in OpenMC with the official ENDF/B-VIII.0 library, calibrated first against three handbook criticals, and yields the coupling matrix (0.9140 against a transport k of 0.9132), the absorber's transfer curve, the drain per driver neutron (1.49 with the gate open), the level restoring emission spectrum, the clock (Λ = 106 µs, 1/Λ = 9.4 kHz) and the temperature coefficient (-25 pcm/K). All of it is inherited by the photon sector, including the two findings that were not sought: an absorber programs a weight rather than a threshold, and the sector has no signal controlled inversion worth the name. - *Kill criterion B (the big one, now confined to B1):* if no compact state can be shown to satisfy the leak condition, Γ>1, *and* level restoration at any achievable flux, then nuclear computing at benchtop scale is permanently confined to the stochastic tier plus memory, a real but bounded result, and the amplifier tier remains a reactor scale curiosity. The theory's *high* ambition lives or dies here. **Phase C. Tier 2 coherence.** Demonstrate a two qubit operation on Mössbauer or nuclear spin DOF, or a usable Bell measurement on annihilation γ pairs after Compton analysis. @@ -307,6 +307,7 @@ Every component except one is either routine nuclear physics or recently demonst | [gates/](gates/) | the criterion evaluated against NUBASE2020 and ENSDF for all 1870 known isomers; curated candidates with measured multiplicities; the experiment menu of required cross sections per facility | | [transport/](transport/) | Monte Carlo demonstrations of every routine gate (exact coincidence multiplier, absorption complement, measured Green's matrix, saturable sigmoid, Bernstein universality), the degree checker, and the compiler: the adjoint assembler that turns a target weight matrix into a material layout, three ways | | [simulator/](simulator/) | the digital twin: a Tier 1 machine run decay by decay, validated against exact enumeration, with decays per sample and energy per sample measured | +| [neutron/](neutron/) | the keystone at reactor scale, computed: the two region subcritical gate in OpenMC with the official ENDF/B-VIII.0 library, calibrated against three ICSBEP handbook criticals; gain, the fission matrix and its eigenvalue check, transfer curves under two absorbers, the truth tables, the superposition check, level restoration spectra, the clock and gain bandwidth product, and the temperature coefficient that prices the veto | | [transistor/](transistor/) | the reference transistor: one pinout at three scales, the valve, the datasheet with characteristic curves and maximum ratings; with four accompanying notes: [SEALED.md](transistor/SEALED.md) (the machine), [EMBODIMENT.md](transistor/EMBODIMENT.md) (the build, at assembly grade), [VALVE.md](transistor/VALVE.md) (the logic), [COMPONENTS.md](transistor/COMPONENTS.md) (the full inventory of a working computer, missing pieces simulated), [POWER.md](transistor/POWER.md) (the metabolism), and [SCALING.md](transistor/SCALING.md) (the trajectory) | --- @@ -366,9 +367,29 @@ Where this all goes is the subject of [transistor/SCALING.md](transistor/SCALING --- +## The keystone, computed: the neutron gate at k below one + +The claim that the keystone already exists in the neutron sector has, until now, been arithmetic on library constants and a citation to a 1958 paper. It is now a calculation. [neutron/](neutron/) builds the reference transistor's third scale in OpenMC with the official ENDF/B-VIII.0 library: two 72 litre tanks of 4.9 percent enriched uranyl fluoride solution (the composition of a handbook criticality benchmark, so every atom density is published), each at k ≈ 0.901 alone, standing 3 cm of water apart in a water bath, with an absorber that can be lowered between them and a Cf-252 point source in one tank. Before the gate is computed the same code and data reproduce three handbook criticals ([neutron/results.md](neutron/results.md), Section 0), so the instrument is calibrated the way a criticality safety calculation is. + +Every number below carries a Monte Carlo uncertainty, and every configuration is strictly subcritical. + +- **Gain, and the synapse.** A tank alone has M = 1/(1 - k) = 10.1; the pair, coupled, k = 0.9132. The fission matrix K (fission neutrons born in tank i per fission neutron born in tank j, one generation) has dominant eigenvalue 0.9140 against a transport k of 0.9133. Avery's coupled region kinetics, this theory's synapse with gain on the diagonal, is confirmed across the three absorber states to between -263 and +138 pcm, and that residual is itself the measured price of describing a tank by one number rather than a shape. +- **A trap the architecture walks into by construction.** A gate is built to couple its regions weakly, and weak coupling drives the dominance ratio of the eigenvalue problem toward one: 0.957 to 0.983 here, so the tilt between the two tanks decays by one or two percent per generation while sampling noise re excites it just as fast. The geometry is mirror symmetric, so the true fission share is exactly half per tank, yet the fission banks stood off by up to 0.070 while the Shannon entropy sat flat to a part in a thousand: entropy watches the shape inside each tank and is nearly blind to the tilt between them. The first version of this calculation shipped a fission matrix that was wrong in a way that looked entirely reasonable, and only the control run caught it. The fix is a mirror rather than more batches, running the half geometry with a reflective plane so the tilt mode cannot exist, and the correction is visible in the physics: a tank's self multiplication went from a non monotonic 0.8924, 0.9030, 0.8977 to a flat 0.8946, 0.8964, 0.8943, which is what an absorber placed *between* two tanks must do to what each tank does to its own neutrons, namely almost nothing. Any photon sector network of many weakly coupled cells inherits this in a sharper form. +- **The transistor.** One driver neutron entering tank A becomes **1.49 fission neutrons born in tank B**, across the gap, with the gate open: the output exceeds the input, so the gate has gain. A 1 mm cadmium sheet cuts the coupling by 1.7× and the drain to 0.92; a 2 cm boron carbide blade by 2.6× and to 0.67. The contrast is 2.2, not a MOSFET's million, and the neutron budget says why: what crosses 3 cm of water is still largely fast, and a thin absorber is a filter, not a barrier. +- **Linearity, and the honest AND.** Two drivers together deliver the sum of what each delivers alone to 0.6 percent, which is counting noise: transport is linear while its cross sections are frozen, exactly as the Green's function reading requires. The coincidence AND is therefore a perceptron AND, a threshold on a weighted sum, with the threshold still at the boundary. +- **Level restoration.** Driven through the water from A, by Cf-252 at its own axis, or by a 14.1 MeV line, tank B emits the same fission spectrum to a total variation distance of 0.002. Inputs spanning thermal to 14 MeV, one output: the self restoring class of the theory supplement, measured. The photon sector cannot buy this at any price, because an isomer cascade emits its own lines and not its trigger's. +- **The clock.** Generation time Λ = 106 µs, prompt switching time Λ/(1 - k) = 1.22 ms, gain bandwidth product 1/Λ = **9.4 kHz**: the op amp law of the transistor note, with laboratory grade constants in it. +- **The veto, priced.** Warming the pair by 56 K costs -25 pcm/K (-13 Doppler and scattering law, -13 expansion of the liquid), a 14 percent modulation of the gain; halving the gain would need about 343 K, which water at atmospheric pressure cannot do. + +![The neutron gate, computed](figures/fig15_neutron_gate.svg) + +The last result is the one that changes the theory rather than confirming it. Every coupling in the neutron sector is excitatory: a neutron only ever adds fissions, and an absorber that burns up disinhibits. Heat is the only channel by which a signal lowers a neighbour's gain, and it is worth a fraction of that gain, slowly. So the sentence in Section 4 of the supplement that an absorber "programs thresholds and vetoes" is now two corrections: an absorber programs a *weight*, the threshold staying at the boundary; and signal controlled inversion, which NAND and signed Ising weights and Turing completeness all require, is the neutron sector's real weak point as an embodiment. The keystone search of Open Problem 1 inherits it as a second requirement: the compact state must supply an inhibition as well as a gain, or what it builds is a linear amplifier network read by comparators, universal for continuous functions and not for logic. + +--- + ## Open problems and how to contribute -1. **Keystone:** propose or measure a compact state meeting the leak condition with β>1 *in a level restoring or convertible class* (theory, Section 2). *This is problem number one.* A concrete, computable subproblem: search ENSDF for heterogeneous pairs of isomers whose cascade lines and gateway lines are mutually resonant (gate A's output pumps gate B and vice versa); the search space is already tabulated in `/gates`. +1. **Keystone:** propose or measure a compact state meeting the leak condition with β>1 *in a level restoring or convertible class* (theory, Section 2). *This is problem number one.* A concrete, computable subproblem: search ENSDF for heterogeneous pairs of isomers whose cascade lines and gateway lines are mutually resonant (gate A's output pumps gate B and vice versa); the search space is already tabulated in `/gates`. The computed neutron gate ([neutron/](neutron/)) adds a second requirement that the photon sector had not been asked for: the state must also supply a *signal controlled inhibition*, because the only one the neutron sector has is heat and it is worth a fraction of the gain. A gate with gain but no inversion is a linear amplifier with a comparator bolted to its boundary, and the machine it builds is the Bernstein machine of Section 6 with better numbers, not a logic family. 2. Tighten the throughput law for fan out reuse: when is a count "spent"? 3. A learning rule: aperture update Δα ∝ coincidence(pre,post) is Hebbian or STDP using the *same* coincidence primitive as the AND gate (Appendix B.4); formalize and simulate. 4. Transport level 𝒢 for a real geometry that implements a target weight matrix (signed weights via complementary channels). diff --git a/figures/fig15_neutron_gate.png b/figures/fig15_neutron_gate.png new file mode 100644 index 0000000..2da9e2b Binary files /dev/null and b/figures/fig15_neutron_gate.png differ diff --git a/figures/fig15_neutron_gate.svg b/figures/fig15_neutron_gate.svg new file mode 100644 index 0000000..2f855f7 --- /dev/null +++ b/figures/fig15_neutron_gate.svg @@ -0,0 +1,5716 @@ + + + + + + + + image/svg+xml + + + Matplotlib v3.11.1, https://matplotlib.org/ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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A falsifiable roadmap (with ki - Status: the simulation half is settled, affirmatively. The digital twin in simulator/ samples its exact Boltzmann law at 26 decays per independent sample and beats the silicon benchmark on energy iff the carrier is the 8.4 eV transition. What remains of Phase A is hardware: the same measurement on a bench, with exempt quantity sources, detectors at the boundary only.

Phase B. Settle the keystone. Now two questions, one per sector. - B1, photon sector. Measure (σ_trig, β, C_in) for the best NEEC or IGE candidate (precision EBIT or Penning trap NEEC; resonant IGE of a mid energy isomer) and locate it on the keystone figure. The required cross sections per facility are tabulated in gates/experiment_menu.md; the ongoing nondestructive Penning trap test of the ⁹³ᵐMo claim largely decides this sector. -- B2, neutron sector. Existence is already answered by library data; what remains is architecture. Reproduce the two region NAND of coupled subcritical assemblies in OpenMC with ENDF data, all inputs public, everything at k<1: no laboratory required, and the coupling matrices, absorber programming, and clock rates it yields are inherited by the photon sector when its gate arrives. +- B2, neutron sector. Status: done, in neutron/. Existence was already answered by library data; the architecture is now computed. The two region subcritical gate runs in OpenMC with the official ENDF/B-VIII.0 library, calibrated first against three handbook criticals, and yields the coupling matrix (0.9140 against a transport k of 0.9132), the absorber's transfer curve, the drain per driver neutron (1.49 with the gate open), the level restoring emission spectrum, the clock (Λ = 106 µs, 1/Λ = 9.4 kHz) and the temperature coefficient (-25 pcm/K). All of it is inherited by the photon sector, including the two findings that were not sought: an absorber programs a weight rather than a threshold, and the sector has no signal controlled inversion worth the name. - Kill criterion B (the big one, now confined to B1): if no compact state can be shown to satisfy the leak condition, Γ>1, and level restoration at any achievable flux, then nuclear computing at benchtop scale is permanently confined to the stochastic tier plus memory, a real but bounded result, and the amplifier tier remains a reactor scale curiosity. The theory's high ambition lives or dies here.

Phase C. Tier 2 coherence. Demonstrate a two qubit operation on Mössbauer or nuclear spin DOF, or a usable Bell measurement on annihilation γ pairs after Compton analysis. - Kill criterion C: if coherence cannot survive realistic readout, Tier 2 collapses to the stochastic tier.

@@ -416,6 +416,10 @@

A falsifiable roadmap (with ki the digital twin: a Tier 1 machine run decay by decay, validated against exact enumeration, with decays per sample and energy per sample measured +neutron/ +the keystone at reactor scale, computed: the two region subcritical gate in OpenMC with the official ENDF/B-VIII.0 library, calibrated against three ICSBEP handbook criticals; gain, the fission matrix and its eigenvalue check, transfer curves under two absorbers, the truth tables, the superposition check, level restoration spectra, the clock and gain bandwidth product, and the temperature coefficient that prices the veto + + transistor/ the reference transistor: one pinout at three scales, the valve, the datasheet with characteristic curves and maximum ratings; with four accompanying notes: SEALED.md (the machine), EMBODIMENT.md (the build, at assembly grade), VALVE.md (the logic), COMPONENTS.md (the full inventory of a working computer, missing pieces simulated), POWER.md (the metabolism), and SCALING.md (the trajectory) @@ -505,9 +509,24 @@

The ampoule: the machine, sealed

The compiler, three ways

Where this all goes is the subject of transistor/SCALING.md, the ENIAC ledger: every component with its improvement lever, its physical ceiling, and whose industrial curve improves it for free (timing and collection ride PET and high energy physics; memory rides the nuclear clock laser program; sources ride isotope production). The machine starts nine orders of magnitude past ENIAC per operation on decay heat alone, and catalog technology, with no new physics, carries the sampler another five to six orders. The primitives are not placeholders: every lever multiplies the same counting, thinning, and coincidence the theory is written in.

+

The keystone, computed: the neutron gate at k below one

+

The claim that the keystone already exists in the neutron sector has, until now, been arithmetic on library constants and a citation to a 1958 paper. It is now a calculation. neutron/ builds the reference transistor's third scale in OpenMC with the official ENDF/B-VIII.0 library: two 72 litre tanks of 4.9 percent enriched uranyl fluoride solution (the composition of a handbook criticality benchmark, so every atom density is published), each at k ≈ 0.901 alone, standing 3 cm of water apart in a water bath, with an absorber that can be lowered between them and a Cf-252 point source in one tank. Before the gate is computed the same code and data reproduce three handbook criticals (neutron/results.md, Section 0), so the instrument is calibrated the way a criticality safety calculation is.

+

Every number below carries a Monte Carlo uncertainty, and every configuration is strictly subcritical.

+ +

The neutron gate, computed

+

The last result is the one that changes the theory rather than confirming it. Every coupling in the neutron sector is excitatory: a neutron only ever adds fissions, and an absorber that burns up disinhibits. Heat is the only channel by which a signal lowers a neighbour's gain, and it is worth a fraction of that gain, slowly. So the sentence in Section 4 of the supplement that an absorber "programs thresholds and vetoes" is now two corrections: an absorber programs a weight, the threshold staying at the boundary; and signal controlled inversion, which NAND and signed Ising weights and Turing completeness all require, is the neutron sector's real weak point as an embodiment. The keystone search of Open Problem 1 inherits it as a second requirement: the compact state must supply an inhibition as well as a gain, or what it builds is a linear amplifier network read by comparators, universal for continuous functions and not for logic.

+

Open problems and how to contribute

    -
  1. Keystone: propose or measure a compact state meeting the leak condition with β>1 in a level restoring or convertible class (theory, Section 2). This is problem number one. A concrete, computable subproblem: search ENSDF for heterogeneous pairs of isomers whose cascade lines and gateway lines are mutually resonant (gate A's output pumps gate B and vice versa); the search space is already tabulated in /gates.
    2. +
  2. Keystone: propose or measure a compact state meeting the leak condition with β>1 in a level restoring or convertible class (theory, Section 2). This is problem number one. A concrete, computable subproblem: search ENSDF for heterogeneous pairs of isomers whose cascade lines and gateway lines are mutually resonant (gate A's output pumps gate B and vice versa); the search space is already tabulated in /gates. The computed neutron gate (neutron/) adds a second requirement that the photon sector had not been asked for: the state must also supply a signal controlled inhibition, because the only one the neutron sector has is heat and it is worth a fraction of the gain. A gate with gain but no inversion is a linear amplifier with a comparator bolted to its boundary, and the machine it builds is the Bernstein machine of Section 6 with better numbers, not a logic family.
  3. Tighten the throughput law for fan out reuse: when is a count "spent"?
  4. A learning rule: aperture update Δα ∝ coincidence(pre,post) is Hebbian or STDP using the same coincidence primitive as the AND gate (Appendix B.4); formalize and simulate.
  5. Transport level 𝒢 for a real geometry that implements a target weight matrix (signed weights via complementary channels).
    6. diff --git a/neutron/README.md b/neutron/README.md new file mode 100644 index 0000000..50816ff --- /dev/null +++ b/neutron/README.md @@ -0,0 +1,67 @@ +# /neutron: the keystone at the scale where it already exists, computed + +The theory's strongest sentence is that the keystone gate, unproven in the photon sector, already exists in the neutron sector: a subcritical multiplying region is a triggered release gate with gain 1/(1 - k), coupled regions are the Green's function synapse with gain on the diagonal, and fission neutrons are level restoring because they are born from the barrier and not from whatever triggered them ([theory, Section 4](../theory/THEORY.md)). Until now that sentence rested on arithmetic with library constants and a citation to Avery. This directory computes it: a two region subcritical gate in OpenMC with the official ENDF/B-VIII.0 library, everything at k below one, every number carrying its Monte Carlo uncertainty, and the code and data first shown to reproduce three handbook criticality benchmarks so that the gate is believed for the same reason a reactor licence is. + +| file | what it is | +|---|---| +| [`gate.py`](gate.py) | the model and the protocol: builds the pair of tanks, runs every stage below, writes `tallies.json` (needs OpenMC and the data) | +| [`benchmarks.py`](benchmarks.py) | the calibration: three ICSBEP handbook criticals run with the same code and data, writes `benchmarks.json` | +| [`report.py`](report.py) | every derived quantity, `results.md`, and figure 15, from the two JSON files (numpy and matplotlib only; runs in the ordinary CI job) | +| [`data.py`](data.py) | fetches the 53 files this directory uses from the official OpenMC ENDF/B-VIII.0 archive by streaming it once, and writes their `cross_sections.xml` | +| [`results.md`](results.md) | the numbers, regenerated by `report.py` | +| `tallies.json`, `benchmarks.json` | the raw tallies and benchmark eigenvalues of the committed run, with uncertainties | + +## The unit + +The tank of the transistor note's third scale, drawn to a size a criticality safety engineer would compute in an afternoon: + +- two identical cylindrical tanks, 24 cm inner radius, 40 cm of solution, 1.6 mm type 1100 aluminium shells, 72 litres each, of 4.9 percent enriched uranyl fluoride solution: the benchmark model composition of handbook case LEU-SOL-THERM-002, which is why every atom density in the model is a published number and not a choice; +- standing side by side in a water bath, axes 3 cm of water apart at the walls, 20 cm of water beyond the tanks on every side: the moderator is the BODY; +- an absorber that can be lowered into the water between them: the GATE. Two are computed, a 1 mm cadmium sheet (the classic thermal filter, insertable in quarters, so that its transfer curve is a curve) and a 2 cm natural boron carbide blade, each overhanging the tanks by 6 cm; +- a point source on the axis of tank A, the SOURCE: Cf-252 (Fröhner's Watt fit, mean 2.13 MeV), the driver of every source driven subcritical assembly ever built for teaching; a second source on the axis of B for the coincidence gate, and a 14.1 MeV point source there for the fusion driven case of the nomenclature section; +- the fission rate in tank B, the DRAIN; the neutron population in B, the CHANNEL. + +The dimensions were chosen so that a tank alone sits at k ≈ 0.90, the working point the transistor note names. Nothing else was tuned. + +## The protocol + +Every stage is a separate OpenMC run whose tallies go, with their standard deviations, into `tallies.json`; `report.py` never reads a statepoint. + +1. **Calibration** ([`benchmarks.py`](benchmarks.py)). Three handbook criticals, one fast and two thermal: HEU-MET-FAST-001 (Godiva, the fission cross sections alone), LEU-SOL-THERM-001 (SHEBA-II, an unreflected tank of 5 percent uranyl fluoride: the gate's own material family), and LEU-SOL-THERM-004 case 1 (STACY, a water reflected 60 cm tank of uranyl nitrate, the tightest thermal solution benchmark in the handbook). Geometries and atom densities follow the handbook as transcribed in the MIT CRPG benchmark collection (MIT licence), checked against it to machine precision; the handbook eigenvalues are those tabulated in JEFF Report 21, Appendix 2. +2. **Alone.** Each tank by itself: k, the gain M = 1/(1 - k), the generation time Λ, the delayed fraction, the switching time Λ/(1 - k), and the gain bandwidth product 1/Λ. +3. **Coupled.** The pair open, with the cadmium sheet at four insertions, and with the blade, each run with a Shannon entropy mesh over the tanks so that the convergence of the fission source is on the record, and each writing its converged fission bank. +4. **The fission matrix.** The bank of the coupled run is split by tank and each half is used as a fixed source with fission treated as capture, so that exactly one generation is counted: the tally of fission neutron production in tank i per source neutron born in tank j is Avery's coupling coefficient K_ij, computed rather than assumed. Two checks follow: the dominant eigenvalue of K must equal the transport k of the pair, and the same one generation run on the unsplit bank must return k directly. The absorption by material in these runs is the neutron budget of a fission neutron born in A: the fan out tax of theory Section 1.3 as a table. +5. **First generation.** The response c_i of each tank to one driver neutron, fission again as capture: the importance of each SOURCE terminal. +6. **The gate.** Fixed source transport with full multiplication for every drive and every absorber state: the drain per driver neutron, the source multiplication, the fission emission spectrum in B (level restoration) and the spectrum entering B through its wall (the input that was restored), and the superposition check with both drivers on. +7. **Feedback.** The pair at 350 K solution temperature, with the liquid density held and with it expanded as water expands, isolating Doppler and scattering law effects from the expansion: the only channel by which a signal in this medium can lower another region's gain. + +## What came out + +| what | measured | where it lands | +|---|---|---| +| calibration, Godiva | -31 pcm from the handbook | within its uncertainty | +| calibration, STACY | +60 pcm from the handbook | within its uncertainty | +| calibration, SHEBA-II | -117 pcm from JEFF-3.1 | agrees with its peers; both sit a percent above an experiment known to be overpredicted | +| gain of one tank | M = 10.1 at k = 0.901 | the working point the transistor note names | +| gain of the pair | M = 11.5 at k = 0.9132 | subcritical, as everything here is | +| the synapse | eigenvalue of K 0.9140 against transport k 0.9132 | Avery's coupling confirmed to +68 pcm | +| the drain, gate open | **1.49 fission neutrons in B per driver neutron** | the gate has gain | +| the drain, blade in | 0.67 | contrast 2.2, not a MOSFET's | +| linearity | superposition to 0.6 percent | the Green's function reading holds | +| level restoration | total variation 0.002 across a thermal to 14 MeV input range | one output spectrum, whatever went in | +| the clock | Λ = 106 µs, τ = 1.22 ms, 1/Λ = 9.4 kHz | the op amp law, measured | +| the veto | -25 pcm/K, a 14 percent modulation over 56 K | an analog knob, not an inversion | + +The two results worth arguing with are the last two rows and the contrast. The gate has gain, level restoration, a programmable weight and a clock, all from library data at k below one, and that is the theory's claim confirmed. What it does not have is a threshold of its own or an inhibition of its own: the medium is linear, so the absorber sets a weight and the comparator stays at the boundary, and the only signal that lowers a neighbour's gain is heat, worth a fraction of that gain and slowly. The photon sector inherits both as requirements. + +## Running it + +``` +python3 benchmarks.py # about ten minutes +python3 gate.py # about forty minutes on a laptop; --quick for a tenth of the particles +python3 report.py # seconds; numpy and matplotlib only +``` + +Requires OpenMC 0.16 (`conda-forge`, or the project's Docker image) and the nuclear data, which [`data.py`](data.py) fetches on first use by streaming the official 3.4 GB ENDF/B-VIII.0 archive from openmc.org and keeping the 53 files this directory touches (1.5 GB, cached in `neutron/data/`, ignored by git). `report.py` needs neither OpenMC nor the data: it rebuilds `results.md` and figure 15 from the committed `tallies.json` and `benchmarks.json` with numpy and matplotlib alone, which is how the ordinary CI job can check this directory without a 1.5 GB download. + +*The gate that the theory said exists in the neutron sector now has a geometry, a bill of materials, a transfer curve, a truth table, a spectrum, a clock, and a temperature coefficient, all from library data at k below one. What it does not have is a nonlinearity of its own or an inhibition of its own, and this directory says by how much.* diff --git a/neutron/benchmarks.json b/neutron/benchmarks.json new file mode 100644 index 0000000..0129d25 --- /dev/null +++ b/neutron/benchmarks.json @@ -0,0 +1,56 @@ +{ + "provenance": { + "openmc_version": "0.16.0", + "cross_sections": "/private/tmp/claude-501/-Users-maxmoussa-Claude/24fa5303-a712-43bf-9ff2-c92c88f6a9a0/scratchpad/nuclear-computing/neutron/data/endfb-viii.0-hdf5/cross_sections.xml", + "particles": 10000, + "batches": 420, + "inactive": 40 + }, + "cases": { + "HEU-MET-FAST-001": { + "k": [ + 0.999693951243486, + 0.00034692785011164543 + ], + "benchmark": [ + 1.0, + 0.001 + ], + "jeff31": [ + 0.99644, + 0.00019 + ], + "seconds": 5.4 + }, + "LEU-SOL-THERM-001": { + "k": [ + 1.011349690028446, + 0.0004994331152143327 + ], + "benchmark": [ + 0.9991, + 0.0029 + ], + "jeff31": [ + 1.01252, + 0.00086 + ], + "seconds": 33.5 + }, + "LEU-SOL-THERM-004-1": { + "k": [ + 1.0000045906278952, + 0.0004812588163242119 + ], + "benchmark": [ + 0.9994, + 0.0008 + ], + "jeff31": [ + 1.00046, + 0.00076 + ], + "seconds": 58.7 + } + } +} \ No newline at end of file diff --git a/neutron/benchmarks.py b/neutron/benchmarks.py new file mode 100644 index 0000000..5f5d1f4 --- /dev/null +++ b/neutron/benchmarks.py @@ -0,0 +1,223 @@ +#!/usr/bin/env python3 +# Copyright 2026 Max Freedom Pollard +# SPDX-License-Identifier: Apache-2.0 +""" +Calibration of the instrument: handbook criticality benchmarks run with the +same code and the same data as the gate. + +The International Handbook of Evaluated Criticality Safety Benchmark +Experiments (ICSBEP) publishes experiments whose k_eff is known to a few +tenths of a percent. Before the gate is believed, the code and data must +reproduce them at both ends of the spectrum the gate spans: + + HEU-MET-FAST-001, Godiva a bare sphere of highly enriched uranium + metal: the fast spectrum and the fission + cross sections, nothing else + LEU-SOL-THERM-001, SHEBA-II an unreflected tank of 5 percent enriched + uranyl fluoride solution: the thermal + spectrum, the water scattering law, and + the material family of the gate + LEU-SOL-THERM-004, STACY a water reflected 60 cm tank of 10 percent + case 1 enriched uranyl nitrate solution: the + tightest thermal solution benchmark in the + handbook, water reflected like the gate + +Geometries and atom densities follow the handbook benchmark models as +transcribed in the MIT CRPG benchmark collection (github.com/mit-crpg/ +benchmarks, MIT licence), verified against it to machine precision. The +handbook k_eff values, and the JEFF-3.1 results on the same three cases, +are those tabulated in the JEFF-3.1 validation report (OECD/NEA JEFF +Report 21, Appendix 2). The JEFF-3.1 column is carried because it says +whether a discrepancy belongs to this calculation or to the benchmark: +SHEBA-II is overpredicted by every modern library, and a result that +lands on JEFF-3.1 rather than on the handbook is the library agreeing +with its peers, not the model being wrong. Writes benchmarks.json. +""" +import argparse +import json +import os +import sys +import time + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +from data import ensure_data # noqa: E402 + +# handbook benchmark model k_eff, and JEFF-3.1's result on the same case, +# both from JEFF Report 21 Appendix 2 (Godiva's handbook value is also +# quoted in ICSBEP and in LA-UR-15-23266) +BENCHMARK_K = { + "HEU-MET-FAST-001": dict(benchmark=(1.00000, 0.00100), jeff31=(0.99644, 0.00019)), + "LEU-SOL-THERM-001": dict(benchmark=(0.99910, 0.00290), jeff31=(1.01252, 0.00086)), + "LEU-SOL-THERM-004-1": dict(benchmark=(0.99940, 0.00080), jeff31=(1.00046, 0.00076)), +} + + +def mat(name, nuclides, sab=None): + import openmc + m = openmc.Material(name=name) + for n, a in nuclides: + m.add_nuclide(n, a) + m.set_density("sum") + if sab: + m.add_s_alpha_beta(sab) + return m + + +def godiva(): + """HEU-MET-FAST-001 case 1: six concentric HEU shells with thin air + gaps, as in the benchmark model.""" + import openmc + radii = [1.0216, 1.0541, 6.2809, 6.2937, 7.7525, 7.7620, 8.2527, 8.2610, 8.7062, 8.7499] + shells = [ + [("U234", 4.9357e-04), ("U235", 4.4936e-02), ("U238", 2.7213e-03)], + [("U234", 4.9357e-04), ("U235", 4.5244e-02), ("U238", 2.4168e-03)], + [("U234", 4.9357e-04), ("U235", 4.5268e-02), ("U238", 2.3930e-03)], + [("U234", 4.9357e-04), ("U235", 4.5090e-02), ("U238", 2.5690e-03)], + [("U234", 4.9357e-04), ("U235", 4.5239e-02), ("U238", 2.4215e-03)], + [("U234", 4.8974e-04), ("U235", 4.4874e-02), ("U238", 2.4169e-03)], + ] + mats = [mat(f"HEU shell {i+1}", c) for i, c in enumerate(shells)] + air = mat("air", [("N14", 3.5214e-05), ("O16", 1.5092e-05)]) + spheres = [openmc.Sphere(r=r) for r in radii] + spheres[-1].boundary_type = "vacuum" + fills = [mats[0], air, mats[1], air, mats[2], air, mats[3], air, mats[4], mats[5]] + cells, prev = [], None + for s, fill in zip(spheres, fills): + region = -s if prev is None else (+prev & -s) + cells.append(openmc.Cell(fill=fill, region=region)) + prev = s + model = openmc.Model(geometry=openmc.Geometry(cells), + materials=openmc.Materials(mats + [air])) + model.settings.source = openmc.IndependentSource( + space=openmc.stats.Box((-1, -1, -1), (1, 1, 1))) + return model + + +def sheba(): + """LEU-SOL-THERM-001, SHEBA-II: uranyl fluoride solution in a stainless + steel tank with a central thimble, unreflected.""" + import openmc + ss = mat("SS304L", [ + ("Cr50", 0.0007103206), ("Cr52", 0.01369782572), ("Cr53", 0.00155322348), + ("Cr54", 0.0003866302), ("Mn55", 0.0017192), ("Fe54", 0.0035092211), + ("Fe56", 0.05508726652), ("Fe57", 0.00127220522), ("Fe58", 0.00016930716), + ("Ni58", 0.0049299929442), ("Ni60", 0.0018990244558), ("Ni61", 8.25492782e-05), + ("Ni62", 0.000263203221), ("Ni64", 6.70301008e-05)]) + air = mat("air", [("N14", 3.5085011118e-05), ("N15", 1.28988882e-07), + ("O16", 1.5086280132e-05), ("O17", 5.719868e-09)]) + fuel = mat("uranyl fluoride solution", [ + ("U234", 6.7855e-07), ("U235", 0.00012377), ("U236", 1.2085e-06), + ("U238", 0.0023508), ("H1", 0.056179), ("O16", 0.032954505507), + ("O17", 1.2494493e-05), ("F19", 0.0051035)], sab="c_H_in_H2O") + z1 = openmc.ZPlane(z0=-40.0, boundary_type="vacuum") + z2, z3, z4 = openmc.ZPlane(z0=-37.1425), openmc.ZPlane(z0=7.6575), openmc.ZPlane(z0=39.375) + z5 = openmc.ZPlane(z0=41.28, boundary_type="vacuum") + c6, c7, c8 = openmc.ZCylinder(r=2.54), openmc.ZCylinder(r=3.175), openmc.ZCylinder(r=24.4475) + c9 = openmc.ZCylinder(r=25.4, boundary_type="vacuum") + cells = [ + openmc.Cell(fill=air, region=+z1 & -z2 & -c6), + openmc.Cell(fill=ss, region=+z1 & -z2 & +c6 & -c9), + openmc.Cell(fill=air, region=+z2 & -z4 & -c6), + openmc.Cell(fill=ss, region=+z2 & -z4 & +c6 & -c7), + openmc.Cell(fill=fuel, region=+z2 & -z3 & +c7 & -c8), + openmc.Cell(fill=air, region=+z3 & -z4 & +c7 & -c8), + openmc.Cell(fill=ss, region=+z2 & -z4 & +c8 & -c9), + openmc.Cell(fill=air, region=+z4 & -z5 & -c6), + openmc.Cell(fill=ss, region=+z4 & -z5 & +c6 & -c9), + ] + model = openmc.Model(geometry=openmc.Geometry(cells), + materials=openmc.Materials([ss, air, fuel])) + model.settings.source = openmc.IndependentSource( + space=openmc.stats.Box((-20, -20, -30), (20, 20, 5)), constraints={"fissionable": True}) + return model + + +def stacy(): + """LEU-SOL-THERM-004 case 1, STACY: 10 percent enriched uranyl nitrate + in a 60 cm stainless steel tank, water reflected.""" + import openmc + fuel = mat("uranyl nitrate solution", [ + ("U234", 6.3833e-07), ("U235", 7.9213e-05), ("U236", 7.9114e-08), + ("U238", 0.00070556), ("H1", 0.056956), ("N14", 0.0028778), + ("O16", 0.038014587009), ("O17", 1.4412991e-05)], sab="c_H_in_H2O") + ss = mat("stainless steel", [ + ("C12", 4.3736e-05 * 0.9893), ("C13", 4.3736e-05 * 0.0107), + ("Si28", 0.00098012480936), ("Si29", 4.976794132e-05), ("Si30", 3.280724932e-05), + ("Mn55", 0.0011561), ("P31", 1.317e-05), ("S32", 1.88009591868e-06), + ("S33", 1.481058558e-08), ("S34", 8.300507418e-08), ("S36", 2.8842156e-10), + ("Ni58", 0.0056778176907), ("Ni60", 0.0021870852093), ("Ni61", 9.50710797e-05), + ("Ni62", 0.0003031282035), ("Ni64", 7.71978168e-05), + ("Cr50", 0.00072887375), ("Cr52", 0.01405560475), ("Cr53", 0.00159379275), + ("Cr54", 0.00039672875), ("Fe54", 0.00347315745), ("Fe56", 0.05452114434), + ("Fe57", 0.00125913099), ("Fe58", 0.00016756722)]) + water = mat("water at 25 C", [("H1", 0.066658), ("O16", 0.033316368309), + ("O17", 1.2631691e-05)], sab="c_H_in_H2O") + air = mat("air", [("N14", 3.9016e-05), ("O16", 1.0405054989e-05), ("O17", 3.945011e-09)]) + c1, c2 = openmc.ZCylinder(r=29.5), openmc.ZCylinder(r=29.8) + c3 = openmc.ZCylinder(r=59.8, boundary_type="vacuum") + z4 = openmc.ZPlane(z0=-32.0, boundary_type="vacuum") + z5, z6, z7 = openmc.ZPlane(z0=-2.0), openmc.ZPlane(z0=0.0), openmc.ZPlane(z0=41.53) + z8, z9 = openmc.ZPlane(z0=150.0), openmc.ZPlane(z0=152.5) + z10 = openmc.ZPlane(z0=172.5, boundary_type="vacuum") + cells = [ + openmc.Cell(fill=water, region=+z4 & -z5 & -c3), + openmc.Cell(fill=ss, region=+z5 & -z6 & -c1), + openmc.Cell(fill=fuel, region=+z6 & -z7 & -c1), + openmc.Cell(fill=air, region=+z7 & -z8 & -c1), + openmc.Cell(fill=ss, region=+z8 & -z9 & -c1), + openmc.Cell(fill=ss, region=+z5 & -z9 & +c1 & -c2), + openmc.Cell(fill=water, region=+z5 & -z9 & +c2 & -c3), + openmc.Cell(fill=water, region=+z9 & -z10 & -c3), + ] + model = openmc.Model(geometry=openmc.Geometry(cells), + materials=openmc.Materials([fuel, ss, water, air])) + model.settings.source = openmc.IndependentSource( + space=openmc.stats.Box((-25, -25, 2), (25, 25, 38)), constraints={"fissionable": True}) + return model + + +CASES = [("HEU-MET-FAST-001", godiva), ("LEU-SOL-THERM-001", sheba), + ("LEU-SOL-THERM-004-1", stacy)] + + +def main(): + import openmc + p = argparse.ArgumentParser() + p.add_argument("--quick", action="store_true") + p.add_argument("--workdir", default=os.path.join(HERE, "work")) + args = p.parse_args() + ensure_data() + exe = os.environ.get("OPENMC_EXEC", "openmc") + particles, batches, inactive = (10000, 420, 40) if not args.quick else (4000, 90, 30) + out = {"provenance": {"openmc_version": openmc.__version__, + "cross_sections": os.environ.get("OPENMC_CROSS_SECTIONS"), + "particles": particles, "batches": batches, "inactive": inactive}, + "cases": {}} + for key, builder in CASES: + model = builder() + s = model.settings + s.run_mode = "eigenvalue" + s.particles, s.batches, s.inactive = particles, batches, inactive + s.output = {"summary": False, "tallies": False} + s.seed = 229 + wd = os.path.join(args.workdir, "bench_" + key) + os.makedirs(wd, exist_ok=True) + t0 = time.time() + sp = model.run(cwd=wd, output=False, openmc_exec=exe) + with openmc.StatePoint(sp) as st: + k, dk = float(st.keff.n), float(st.keff.s) + ref = BENCHMARK_K[key] + kb, dkb = ref["benchmark"] + out["cases"][key] = {"k": [k, dk], "benchmark": list(ref["benchmark"]), + "jeff31": list(ref["jeff31"]), + "seconds": round(time.time() - t0, 1)} + print(f" {key:22s} k = {k:.5f} +/- {dk:.5f} handbook {kb:.5f} +/- {dkb:.5f}" + f" JEFF-3.1 {ref['jeff31'][0]:.5f} ({time.time()-t0:.0f} s)", flush=True) + path = os.path.join(HERE, "benchmarks.json") + json.dump(out, open(path, "w"), indent=1) + print("wrote", path) + + +if __name__ == "__main__": + main() diff --git a/neutron/data.py b/neutron/data.py new file mode 100644 index 0000000..b705bb9 --- /dev/null +++ b/neutron/data.py @@ -0,0 +1,133 @@ +#!/usr/bin/env python3 +# Copyright 2026 Max Freedom Pollard +# SPDX-License-Identifier: Apache-2.0 +""" +Nuclear data for the neutron sector: the official OpenMC ENDF/B-VIII.0 +library, fetched once and cached. + +The library is the one the OpenMC project distributes at openmc.org +(ENDF/B-VIII.0 processed to HDF5 by NJOY, neutron data at 250 to 2500 K, +thermal scattering at 284 to 800 K). The archive is 3.4 GB compressed and +13.7 GB unpacked, and this machine needs 53 files from it, so the archive +is streamed and only those members are written to disk: about 1.5 GB, +the two large ones being the uranium evaluations and the water thermal +scattering law. Nothing is modified; the files are the library's own. + +The cache lives in neutron/data/ (ignored by git) unless NC_DATA_DIR +points elsewhere; if OPENMC_CROSS_SECTIONS is already set it is used as +is and nothing is downloaded. `python3 data.py` fetches and verifies. +""" +import lzma +import os +import sys +import tarfile +import urllib.request + +HERE = os.path.dirname(os.path.abspath(__file__)) +LIBRARY = "endfb-viii.0-hdf5" +ARCHIVE_URL = ("https://anl.box.com/shared/static/" + "uhbxlrx7hvxqw27psymfbhi7bx7s6u6a.xz") + +# every nuclide the gate, the benchmarks, and the feedback runs touch +NUCLIDES = [ + "H1", "O16", "O17", "N14", "F19", # solution, air + "Al27", "Si28", "Si29", "Si30", "Mn55", # type 1100 Al + "Cu63", "Cu65", "Zn64", "Zn66", "Zn67", "Zn68", "Zn70", + "Cd106", "Cd108", "Cd110", "Cd111", "Cd112", "Cd113", # the sheet + "Cd114", "Cd116", + "B10", "B11", "C12", "C13", # the blade + "Cr50", "Cr52", "Cr53", "Cr54", "Fe54", "Fe56", "Fe57", # stainless steel, + "Fe58", "Ni58", "Ni60", "Ni61", "Ni62", "Ni64", "P31", # for the SHEBA-II + "S32", "S33", "S34", "S36", "N15", "U236", # and STACY benchmarks + "U234", "U235", "U238", # the fuel +] +THERMAL = ["c_H_in_H2O"] +MEMBERS = ([f"{LIBRARY}/neutron/{n}.h5" for n in NUCLIDES] + + [f"{LIBRARY}/thermal/{t}.h5" for t in THERMAL]) + + +def data_dir(): + return os.environ.get("NC_DATA_DIR", os.path.join(HERE, "data")) + + +def cross_sections_path(): + return os.path.join(data_dir(), LIBRARY, "cross_sections.xml") + + +def missing(root): + return [m for m in MEMBERS if not os.path.exists(os.path.join(root, m))] + + +def stream_extract(root, wanted, log=print): + """One sequential pass over the compressed archive, writing only the + wanted members. Stops as soon as the last of them has been written.""" + wanted = set(wanted) + log(f"streaming {ARCHIVE_URL}") + log(f" {len(wanted)} members wanted; this reads the whole 3.4 GB archive " + "once and keeps about 1.5 GB") + req = urllib.request.Request(ARCHIVE_URL, headers={"User-Agent": "Mozilla/5.0"}) + resp = urllib.request.urlopen(req, timeout=120) + with lzma.open(resp) as xz, tarfile.open(fileobj=xz, mode="r|") as tar: + for member in tar: + if member.name in wanted: + dest = os.path.join(root, member.name) + os.makedirs(os.path.dirname(dest), exist_ok=True) + src = tar.extractfile(member) + with open(dest, "wb") as out: + while True: + chunk = src.read(1 << 22) + if not chunk: + break + out.write(chunk) + wanted.discard(member.name) + log(f" wrote {member.name} ({member.size/1e6:.0f} MB), " + f"{len(wanted)} to go") + if not wanted: + break + if wanted: + raise RuntimeError(f"archive ended before these were found: {sorted(wanted)}") + + +def write_cross_sections(root): + lines = ['', ""] + for n in NUCLIDES: + lines.append(f' ') + for t in THERMAL: + lines.append(f' ') + lines.append("") + path = os.path.join(root, LIBRARY, "cross_sections.xml") + open(path, "w").write("\n".join(lines) + "\n") + return path + + +def ensure_data(log=print): + """Return the path of a cross_sections.xml covering every nuclide this + directory uses, fetching the library on first call.""" + if os.environ.get("OPENMC_CROSS_SECTIONS") and not os.environ.get("NC_FORCE_LIBRARY"): + return os.environ["OPENMC_CROSS_SECTIONS"] + root = data_dir() + need = missing(root) + if need: + stream_extract(root, need, log) + path = write_cross_sections(root) + os.environ["OPENMC_CROSS_SECTIONS"] = path + return path + + +def verify(log=print): + """Open every file with OpenMC's data reader and report its temperatures.""" + import openmc.data + root = data_dir() + for n in NUCLIDES: + d = openmc.data.IncidentNeutron.from_hdf5(os.path.join(root, LIBRARY, "neutron", n + ".h5")) + log(f" {n:6s} {', '.join(sorted(d.temperatures, key=lambda s: float(s[:-1])))}") + for t in THERMAL: + d = openmc.data.ThermalScattering.from_hdf5(os.path.join(root, LIBRARY, "thermal", t + ".h5")) + log(f" {t}: {', '.join(sorted(d.temperatures, key=lambda s: float(s[:-1])))}") + + +if __name__ == "__main__": + path = ensure_data() + print("cross sections:", path) + if "--verify" in sys.argv: + verify() diff --git a/neutron/gate.py b/neutron/gate.py new file mode 100644 index 0000000..ddc6dfa --- /dev/null +++ b/neutron/gate.py @@ -0,0 +1,636 @@ +#!/usr/bin/env python3 +# Copyright 2026 Max Freedom Pollard +# SPDX-License-Identifier: Apache-2.0 +""" +The neutron sector gate, computed: two coupled subcritical solution tanks +in OpenMC with the official ENDF/B-VIII.0 library. + +The unit is the neutron gate of the reference transistor (transistor/README.md, +Scale 3), built from parts a criticality safety engineer would recognise: + + two cylindrical tanks of 4.9 percent enriched uranyl fluoride solution + (the composition of handbook benchmark LEU-SOL-THERM-002; benchmarks.py + reproduces the handbook with this same code and data), each with + k about 0.90 on its own, standing side by side in a water bath (the + BODY), 3 cm of water apart, with an absorber that can be lowered into the + water between them (the GATE): a 1 mm cadmium sheet, the classic thermal + filter, or a 2 cm boron carbide blade. A Cf-252 point source in tank A + is the SOURCE terminal; the fission rate in tank B is the DRAIN; the + neutron population in B is the CHANNEL. + +Everything at k < 1, always. The script runs the protocol below and writes +every raw tally, with its Monte Carlo uncertainty, to tallies.json; +report.py turns that file into results.md and figure 15. Stages: + + alone eigenvalue of each tank by itself: M = 1/(1 - k) is a number + coupled eigenvalue of the pair: open, the cadmium sheet at 25, 50, 75 + and 100 percent insertion, and the boron carbide blade, each + with a Shannon entropy trace to show the source converged, + writing the converged fission bank for the coupling stage + coupling the fission matrix: fission neutrons born in tank j (the + fundamental mode source restricted to j, fission treated as + capture so exactly one generation is counted) produce K_ij + fission neutrons in tank i; the dominant eigenvalue of K must + reproduce the transport k of the pair, and a control run on + the unsplit bank must reproduce it directly + firstgen the first generation response c_i of each tank to a driver + neutron (Cf-252 in A, Cf-252 in B, 14.1 MeV in B) + gate the gate itself, fixed source with full multiplication, for + every drive and every absorber state: drain current per + source neutron, the neutron budget, the fission emission + spectrum in B (level restoration) and the spectrum arriving + at B through its wall (the input that was restored) + feedback the pair at 350 K solution temperature, with and without the + thermal expansion of the liquid: the only signal controlled + inhibition the neutron sector has, priced + +Run `python3 gate.py` (about forty minutes on a laptop) or `--quick` for a +tenth of the particles. Requires OpenMC (conda-forge) and the data that +data.py fetches on first use. +""" +import argparse +import glob +import json +import math +import os +import shutil +import sys +import time + +import numpy as np + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +from data import ensure_data # noqa: E402 + +# --------------------------------------------------------------------------- +# the unit: dimensions and materials (design choices are marked as such) +# --------------------------------------------------------------------------- +R_TANK = 24.0 # cm, inner radius; chosen so that k(alone) is about 0.90 +H_TANK = 40.0 # cm, solution height; same choice +WALL = 0.1588 # cm, the aluminium shell of LEU-SOL-THERM-002 +GAP = 3.0 # cm of water between the two tank walls (design choice) +ABSORBERS = { # the GATE: material, thickness (cm) + "cd": ("cadmium", 0.10), # a standard 1 mm cadmium sheet + "b4c": ("b4c", 2.00), # a 2 cm boron carbide blade +} +ABS_MARGIN = 6.0 # cm the absorber overhangs the tanks in y and z +REFL = 20.0 # cm of water beyond the tanks on every side +X_TANK = R_TANK + WALL + GAP / 2 # tank axes at x = -X_TANK and +X_TANK + +CF252_WATT = (1.18e6, 1.03419e-6) # a (eV), b (1/eV): Cf-252 spontaneous fission, Froehner 1990 fit (mean 2.13 MeV) +DT_ENERGY = 14.1e6 # eV, deuterium tritium fusion neutrons + +# uranyl fluoride solution: the benchmark model composition of +# LEU-SOL-THERM-002 (atoms per barn cm), 4.9 percent enriched +SOLUTION = [("U234", 2.3271e-07), ("U235", 5.6655e-05), ("U238", 1.0878e-03), + ("F19", 2.2893e-03), ("O16", 0.033389340642), + ("O17", 1.2659358e-05), ("H1", 0.062226)] +# type 1100 aluminium, same benchmark +AL1100 = [("Al27", 0.059699), ("Si28", 0.000509126279536), + ("Si29", 2.5851979831999998e-05), ("Si30", 1.7041740632e-05), + ("Cu63", 3.5518206e-05), ("Cu65", 1.5845794e-05), + ("Zn64", 1.2271848600000001e-05), ("Zn66", 6.9208534e-06), + ("Zn67", 1.0083032e-06), ("Zn68", 4.604751e-06), + ("Zn70", 1.522438e-07), ("Mn55", 1.4853e-05)] +WATER = [("H1", 0.066659), ("O16", 0.033316368309), ("O17", 1.2631690999999998e-05)] +CD_DENSITY = 8.65 # g/cm3, natural cadmium +B4C_DENSITY = 2.52 # g/cm3, boron carbide, natural boron + +# water density relative to 294 K, for the thermal expansion of the liquid +# (the solution is assumed to expand as water does; stated in the README) +WATER_RHO = {294.0: 0.99795, 350.0: 0.97368} + +ENERGY_EDGES = np.logspace(-5, math.log10(2e7), 61) # eV, five bins per decade + +# particle budgets: full and quick +BUDGET = { + "full": dict(eig_p=20000, eig_b=200, eig_i=50, coupling=2_000_000, + firstgen=1_000_000, gate=400_000), + "quick": dict(eig_p=4000, eig_b=100, eig_i=30, coupling=200_000, + firstgen=100_000, gate=40_000), +} + +STATES = ["open", "cd", "b4c"] # absorber states used by every stage +DRIVES = { + "cfA": [("cf", -1)], + "cfB": [("cf", +1)], + "cfAB": [("cf", -1), ("cf", +1)], + "dtB": [("dt", +1)], +} + + +# --------------------------------------------------------------------------- +# model builder +# --------------------------------------------------------------------------- +def make_materials(temperature=294.0, expand=False, absorber=None): + """The absorber material is built only when its nuclides are needed, so a + library without boron can still run every cadmium and open configuration.""" + import openmc + scale = WATER_RHO[temperature] / WATER_RHO[294.0] if expand else 1.0 + mats = {} + for tag in ("A", "B"): + m = openmc.Material(name=f"solution_{tag}") + for n, a in SOLUTION: + m.add_nuclide(n, a * scale) + m.set_density("sum") + m.add_s_alpha_beta("c_H_in_H2O") + m.temperature = temperature + mats[f"sol_{tag}"] = m + al = openmc.Material(name="al1100") + for n, a in AL1100: + al.add_nuclide(n, a) + al.set_density("sum") + mats["al"] = al + w = openmc.Material(name="water") + for n, a in WATER: + w.add_nuclide(n, a * scale) + w.set_density("sum") + w.add_s_alpha_beta("c_H_in_H2O") + w.temperature = temperature + mats["water"] = w + if absorber == "cd": + cd = openmc.Material(name="cadmium") + cd.add_element("Cd", 1.0) + cd.set_density("g/cm3", CD_DENSITY) + mats["cadmium"] = cd + if absorber == "b4c": + b4c = openmc.Material(name="b4c") + b4c.add_element("B", 4.0) + b4c.add_nuclide("C12", 0.9893) + b4c.add_nuclide("C13", 0.0107) + b4c.set_density("g/cm3", B4C_DENSITY) + mats["b4c"] = b4c + return mats + + +def build(tank_a=True, tank_b=True, absorber=None, fraction=1.0, + temperature=294.0, expand=False, half=False): + """Geometry of the pair. Returns (model, parts) with the cells, surfaces + and materials the tallies refer to. + + With half=True only the x > 0 side is built and the plane x = 0 is + reflective. Because the full geometry is mirror symmetric in x, the + half model's eigenvalue is exactly the symmetric fundamental mode of + the pair, and it has no tilt mode to converge at all: the slowly + mixing second eigenvector of Section 3.1 does not exist in it. Its + fission bank, mirrored, is therefore the fundamental mode source that + the fission matrix needs, obtained by construction rather than by + waiting.""" + import openmc + openmc.reset_auto_ids() + mats = make_materials(temperature, expand, absorber if fraction > 0 else None) + cells, parts = [], {"mats": mats} + half_x = X_TANK + R_TANK + WALL + REFL + half_y = R_TANK + WALL + REFL + half_z = H_TANK / 2 + WALL + REFL + mirror_plane = None + if half: + tank_a = False + mirror_plane = openmc.XPlane(x0=0.0, boundary_type="reflective") + box = (+mirror_plane + & -openmc.XPlane(x0=half_x, boundary_type="vacuum") + & +openmc.YPlane(y0=-half_y, boundary_type="vacuum") + & -openmc.YPlane(y0=half_y, boundary_type="vacuum") + & +openmc.ZPlane(z0=-half_z, boundary_type="vacuum") + & -openmc.ZPlane(z0=half_z, boundary_type="vacuum")) + water_region = box + else: + box = openmc.model.RectangularParallelepiped( + -half_x, half_x, -half_y, half_y, -half_z, half_z, boundary_type="vacuum") + water_region = -box + + for tag, present, x0 in (("A", tank_a, -X_TANK), ("B", tank_b, X_TANK)): + if not present: + continue + ci = openmc.ZCylinder(x0=x0, r=R_TANK) + co = openmc.ZCylinder(x0=x0, r=R_TANK + WALL) + zi0, zi1 = openmc.ZPlane(z0=-H_TANK / 2), openmc.ZPlane(z0=H_TANK / 2) + zo0, zo1 = openmc.ZPlane(z0=-H_TANK / 2 - WALL), openmc.ZPlane(z0=H_TANK / 2 + WALL) + inner = -ci & +zi0 & -zi1 + outer = -co & +zo0 & -zo1 + sol = openmc.Cell(name=f"solution_{tag}", fill=mats[f"sol_{tag}"], region=inner) + wall = openmc.Cell(name=f"wall_{tag}", fill=mats["al"], region=outer & ~inner) + cells += [sol, wall] + parts[f"sol_{tag}"], parts[f"wall_{tag}"] = sol, wall + parts[f"inner_{tag}"] = [ci, zi0, zi1] + water_region &= ~outer + + if absorber and fraction > 0: + mat_name, thick = ABSORBERS[absorber] + ly = R_TANK + WALL + ABS_MARGIN + z_hi = H_TANK / 2 + WALL + ABS_MARGIN + z_lo = z_hi - fraction * 2 * z_hi + # the slab keeps its full width in both models: in the half model the + # x < 0 half lies behind the mirror plane and is never visited, which + # avoids putting a second surface on top of the reflective boundary + # (coincident surfaces are what lose particles) + slab = (+openmc.XPlane(x0=-thick / 2) & -openmc.XPlane(x0=thick / 2) + & +openmc.YPlane(y0=-ly) & -openmc.YPlane(y0=ly) + & +openmc.ZPlane(z0=z_lo) & -openmc.ZPlane(z0=z_hi)) + if mirror_plane is not None: + # the reflective plane must bound every cell that touches it, not + # just the water: a particle inside the absorber would otherwise + # never see the mirror and would escape out of the far face + slab = slab & +mirror_plane + blade = openmc.Cell(name="absorber", fill=mats[mat_name], region=slab) + cells.append(blade) + parts["absorber"] = blade + parts["absorber_material"] = mat_name + water_region &= ~slab + + water = openmc.Cell(name="water", fill=mats["water"], region=water_region) + cells.append(water) + parts["water"] = water + + model = openmc.Model(geometry=openmc.Geometry(cells), + materials=openmc.Materials(list(mats.values()))) + model.settings.output = {"summary": False, "tallies": False} + model.settings.temperature = {"method": "interpolation"} + model.settings.seed = 229 + return model, parts + + +def add_tallies(model, parts, spectra=False): + """The standard tally set: region tallies for whichever tanks exist, + absorption by material, the global balance, and optionally the spectra + in and into tank B.""" + import openmc + tallies = [] + region_cells = [parts[k] for k in ("sol_A", "sol_B") if k in parts] + t = openmc.Tally(name="regions") + t.filters = [openmc.CellFilter(region_cells)] + t.scores = ["flux", "fission", "nu-fission", "absorption", + "inverse-velocity", "delayed-nu-fission"] + tallies.append(t) + + mat_keys = [k for k in ("sol_A", "sol_B") if k in parts] + ["al", "water"] + if "absorber_material" in parts: + mat_keys.append(parts["absorber_material"]) + mat_list = [parts["mats"][k] for k in mat_keys] + t = openmc.Tally(name="materials") + t.filters = [openmc.MaterialFilter(mat_list)] + t.scores = ["absorption"] + tallies.append(t) + + t = openmc.Tally(name="global") + t.scores = ["flux", "fission", "nu-fission", "absorption", "inverse-velocity"] + tallies.append(t) + + if spectra and "sol_B" in parts: + t = openmc.Tally(name="chi_B") + t.filters = [openmc.CellFilter([parts["sol_B"]]), + openmc.EnergyoutFilter(ENERGY_EDGES)] + t.scores = ["nu-fission"] + t.estimator = "analog" + tallies.append(t) + t = openmc.Tally(name="flux_B") + t.filters = [openmc.CellFilter([parts["sol_B"]]), + openmc.EnergyFilter(ENERGY_EDGES)] + t.scores = ["flux"] + tallies.append(t) + t = openmc.Tally(name="into_B") + t.filters = [openmc.SurfaceFilter(parts["inner_B"]), + openmc.CellFromFilter([parts["wall_B"]]), + openmc.CellFilter([parts["sol_B"]]), + openmc.EnergyFilter(ENERGY_EDGES)] + t.scores = ["current"] + tallies.append(t) + model.tallies = openmc.Tallies(tallies) + return [c.name for c in region_cells], [m.name for m in mat_list] + + +def point_source(x0, kind): + import openmc + if kind == "cf": + energy = openmc.stats.Watt(*CF252_WATT) + elif kind == "dt": + energy = openmc.stats.Discrete([DT_ENERGY], [1.0]) + else: + raise ValueError(kind) + return openmc.IndependentSource(space=openmc.stats.Point((x0, 0.0, 0.0)), + energy=energy) + + +# --------------------------------------------------------------------------- +# running and harvesting +# --------------------------------------------------------------------------- +def harvest(sp_path, region_names, mat_names, spectra=False): + """Read a statepoint into plain floats: [mean, std] pairs everywhere.""" + import openmc + out = {} + with openmc.StatePoint(sp_path) as sp: + if sp.run_mode == "eigenvalue": + out["k"] = [float(sp.keff.n), float(sp.keff.s)] + if sp.entropy is not None and len(sp.entropy): + ent = np.asarray(sp.entropy, dtype=float).ravel() + out["entropy"] = [round(float(x), 4) for x in ent] + t = sp.get_tally(name="regions") + out["regions"] = {} + for i, name in enumerate(region_names): + out["regions"][name[-1]] = { + s: [float(t.mean[i, 0, j]), float(t.std_dev[i, 0, j])] + for j, s in enumerate(t.scores)} + t = sp.get_tally(name="materials") + out["absorption_by_material"] = { + name: [float(t.mean[i, 0, 0]), float(t.std_dev[i, 0, 0])] + for i, name in enumerate(mat_names)} + t = sp.get_tally(name="global") + out["global"] = {s: [float(t.mean[0, 0, j]), float(t.std_dev[0, 0, j])] + for j, s in enumerate(t.scores)} + if spectra: + for name in ("chi_B", "flux_B", "into_B"): + t = sp.get_tally(name=name) + # into_B has one bin per bounding surface of the solution, each + # signed by that surface's normal; the wall to solution direction + # is fixed per surface, so the magnitudes add + m = np.abs(t.mean.reshape(-1, len(ENERGY_EDGES) - 1)).sum(axis=0) + s = np.sqrt((t.std_dev.reshape(-1, len(ENERGY_EDGES) - 1) ** 2).sum(axis=0)) + out[name] = {"mean": m.tolist(), "std": s.tolist()} + return out + + +def run_model(model, workdir, exe): + if os.path.isdir(workdir): + shutil.rmtree(workdir) + os.makedirs(workdir) + t0 = time.time() + sp = model.run(cwd=workdir, output=False, openmc_exec=exe) + return sp, time.time() - t0 + + +def mirror(site): + """Reflect a fission site through the plane x = 0. Every configuration + here is mirror symmetric in x (identical tanks at +/- X_TANK, the + absorber slab centred on the plane), so this maps the geometry onto + itself exactly.""" + import openmc + return openmc.SourceParticle( + r=(-site.r[0], site.r[1], site.r[2]), + u=(-site.u[0], site.u[1], site.u[2]), + E=site.E, time=site.time, wgt=site.wgt, + delayed_group=site.delayed_group, surf_id=site.surf_id, + particle=site.particle) + + +def source_from_half(workdir, out_a, out_b, out_all): + """Take the fission bank of a half model run (every site has x > 0, in + tank B) and build the three sources the fission matrix needs, in the + full geometry: the bank itself for tank B, its mirror image for tank A, + and both together for the control. This is the exact symmetric + fundamental mode, so the control must return the transport k.""" + import openmc + files = sorted(glob.glob(os.path.join(workdir, "source*.h5"))) + if not files: + raise RuntimeError(f"no source file written in {workdir}") + b = list(openmc.read_source_file(files[-1])) + a = [mirror(s) for s in b] + openmc.write_source_file(a, out_a) + openmc.write_source_file(b, out_b) + openmc.write_source_file(a + b, out_all) + return len(a), len(b) + + +def split_source(workdir, out_a, out_b, out_all, out_rawall): + """Turn the converged fission bank of a coupled eigenvalue run into the + per tank sources the fission matrix needs. + + A pair of weakly coupled multiplying regions is the textbook loosely + coupled system: the fundamental mode is symmetric, the second mode is + the tilt between the tanks, and their eigenvalues differ by only + 2 k_cross, so the dominance ratio here is 0.95 to 0.99 and the tilt + both decays and re randomises over of order a hundred generations. + Fission bank sampling noise therefore drives a standing tilt of a few + percent that no affordable number of batches averages away, and the + Shannon entropy of the source is nearly blind to it because entropy is + dominated by the shape inside each tank. + + The geometry, however, is exactly mirror symmetric, so the true + fundamental mode has exactly half its fissions in each tank and the + tilt is pure error. Symmetrising the bank (every site together with + its mirror image) projects the antisymmetric component out exactly and + doubles the statistics, which is the same reasoning that lets a + symmetric core be modelled as a half core. The raw bank is written too, + unmodified, so that the report can show what it costs to skip this. + """ + import openmc + files = sorted(glob.glob(os.path.join(workdir, "source*.h5"))) + if not files: + raise RuntimeError(f"no source file written in {workdir}") + sites = list(openmc.read_source_file(files[-1])) + raw_a = [s for s in sites if s.r[0] < 0] + raw_b = [s for s in sites if s.r[0] > 0] + # every site in tank A, plus the mirror of every site in tank B + sym_a = raw_a + [mirror(s) for s in raw_b] + sym_b = [mirror(s) for s in sym_a] + openmc.write_source_file(sym_a, out_a) + openmc.write_source_file(sym_b, out_b) + openmc.write_source_file(sym_a + sym_b, out_all) + openmc.write_source_file(sites, out_rawall) + return len(raw_a), len(raw_b) + + +def state_kwargs(state): + if state == "open": + return {} + return {"absorber": state, "fraction": 1.0} + + +# --------------------------------------------------------------------------- +# the protocol +# --------------------------------------------------------------------------- +def protocol(args): + import openmc + budget = BUDGET["quick" if args.quick else "full"] + exe = os.environ.get("OPENMC_EXEC", "openmc") + work = args.workdir + os.makedirs(work, exist_ok=True) + results_path = os.path.join(HERE, "tallies.json") + results = {"runs": {}} + if os.path.exists(results_path) and args.only: + results = json.load(open(results_path)) + runs = results["runs"] + wanted = lambda name: (not args.only) or any(k in name for k in args.only.split(",")) + t_start = time.time() + + def record(name, model, parts, spectra=False, extra=None): + region_names, mat_names = add_tallies(model, parts, spectra) + sp, wall = run_model(model, os.path.join(work, name), exe) + r = harvest(sp, region_names, mat_names, spectra) + r["mode"] = model.settings.run_mode + r["particles"] = int(model.settings.particles) + r["batches"] = int(model.settings.batches) + if model.settings.run_mode == "eigenvalue": + r["inactive"] = int(model.settings.inactive) + r["seconds"] = round(wall, 1) + if extra: + r.update(extra) + runs[name] = r + k = f"k = {r['k'][0]:.5f} +/- {r['k'][1]:.5f}" if "k" in r else "fixed source" + print(f" {name:26s} {k} ({wall:.0f} s)", flush=True) + json.dump(results, open(results_path, "w"), indent=1) + return r + + def eigen_settings(model, sourcepoint=False, half=False): + s = model.settings + s.run_mode = "eigenvalue" + s.particles, s.batches, s.inactive = budget["eig_p"], budget["eig_b"], budget["eig_i"] + x_lo = 0.0 if half else -X_TANK - R_TANK + s.source = openmc.IndependentSource( + space=openmc.stats.Box((x_lo, -R_TANK, -H_TANK / 2), + (X_TANK + R_TANK, R_TANK, H_TANK / 2)), + constraints={"fissionable": True}) + mesh = openmc.RegularMesh() + mesh.dimension = [12 if half else 24, 12, 10] + mesh.lower_left = (x_lo, -R_TANK, -H_TANK / 2) + mesh.upper_right = (X_TANK + R_TANK, R_TANK, H_TANK / 2) + s.entropy_mesh = mesh + if sourcepoint: + s.sourcepoint = {"batches": [s.batches], "separate": True, "write": True} + + def fixed_settings(model, n, sources, multiply): + s = model.settings + s.run_mode = "fixed source" + s.batches = 20 + s.particles = max(n // 20, 100) + s.source = sources + s.create_fission_neutrons = bool(multiply) + + print("neutron sector gate: OpenMC", openmc.__version__, "| budget", + "quick" if args.quick else "full", flush=True) + + if args.resplit: + for state in STATES: + na, nb = split_source(os.path.join(work, f"coupled_{state}"), + os.path.join(work, f"src_A_{state}.h5"), + os.path.join(work, f"src_B_{state}.h5"), + os.path.join(work, f"src_all_{state}.h5"), + os.path.join(work, f"src_rawall_{state}.h5")) + runs[f"coupled_{state}"]["bank_split"] = {"A": na, "B": nb} + print(f" resplit coupled_{state}: raw A {na}, raw B {nb} " + f"(tilt {nb/(na+nb)-0.5:+.3f}), symmetrised to {na+nb} each", + flush=True) + json.dump(results, open(results_path, "w"), indent=1) + + # -- alone ---------------------------------------------------------------- + for tag in ("A", "B"): + name = f"alone_{tag}" + if wanted(name): + model, parts = build(tank_a=(tag == "A"), tank_b=(tag == "B")) + eigen_settings(model) + record(name, model, parts) + + # -- coupled: open, the sheet at four insertions, the blade --------------- + configs = [("coupled_open", {})] + configs += [(f"coupled_cd{int(f*100):02d}", {"absorber": "cd", "fraction": f}) for f in (0.25, 0.5, 0.75)] + configs += [("coupled_cd", {"absorber": "cd", "fraction": 1.0}), + ("coupled_b4c", {"absorber": "b4c", "fraction": 1.0})] + for name, kw in configs: + if wanted(name): + state = name.split("_", 1)[1] + keep_bank = state in STATES + model, parts = build(**kw) + eigen_settings(model, sourcepoint=keep_bank) + record(name, model, parts, extra={"absorber": kw.get("absorber"), + "fraction": kw.get("fraction", 0.0)}) + if keep_bank: + na, nb = split_source(os.path.join(work, name), + os.path.join(work, f"src_A_{state}.h5"), + os.path.join(work, f"src_B_{state}.h5"), + os.path.join(work, f"src_all_{state}.h5"), + os.path.join(work, f"src_rawall_{state}.h5")) + runs[name]["bank_split"] = {"A": na, "B": nb} + json.dump(results, open(results_path, "w"), indent=1) + + # -- the half model: the symmetric fundamental mode, by construction ------ + for state in STATES: + name = f"half_{state}" + if wanted(name): + model, parts = build(half=True, **state_kwargs(state)) + eigen_settings(model, sourcepoint=True, half=True) + record(name, model, parts, extra={"state": state, "half": True}) + na, nb = source_from_half(os.path.join(work, name), + os.path.join(work, f"src_A_{state}.h5"), + os.path.join(work, f"src_B_{state}.h5"), + os.path.join(work, f"src_all_{state}.h5")) + runs[name]["sites"] = nb + json.dump(results, open(results_path, "w"), indent=1) + + # -- the fission matrix, and its two controls ----------------------------- + for state in STATES: + for tag in ("all", "rawall", "A", "B"): + name = f"coupling_{tag}_{state}" + if wanted(name): + src = os.path.join(work, f"src_{tag}_{state}.h5") + if not os.path.exists(src): + raise RuntimeError(f"{src} missing: run half_{state} (or coupled_{state} " + f"for the raw control) first") + model, parts = build(**state_kwargs(state)) + fixed_settings(model, budget["coupling"], openmc.FileSource(src), multiply=False) + record(name, model, parts, extra={"state": state, "born_in": tag}) + + # -- first generation response to each driver ----------------------------- + for state in STATES: + for drive in ("cfA", "cfB", "dtB"): + name = f"firstgen_{drive}_{state}" + if wanted(name): + model, parts = build(**state_kwargs(state)) + srcs = [point_source(sgn * X_TANK, kind) for kind, sgn in DRIVES[drive]] + fixed_settings(model, budget["firstgen"], srcs, multiply=False) + record(name, model, parts, extra={"state": state, "drive": drive}) + + # -- the gate ------------------------------------------------------------- + for state in STATES: + for drive in ("cfA", "cfB", "cfAB", "dtB"): + name = f"gate_{drive}_{state}" + if wanted(name): + model, parts = build(**state_kwargs(state)) + srcs = [point_source(sgn * X_TANK, kind) for kind, sgn in DRIVES[drive]] + fixed_settings(model, budget["gate"], srcs, multiply=True) + record(name, model, parts, spectra=True, + extra={"state": state, "drive": drive}) + + # -- feedback: the veto, priced ------------------------------------------- + for name, kw in (("feedback_T350_fixed_density", dict(temperature=350.0, expand=False)), + ("feedback_T350_expanded", dict(temperature=350.0, expand=True))): + if wanted(name): + model, parts = build(**kw) + eigen_settings(model) + record(name, model, parts, extra=kw) + + results["provenance"] = { + "openmc_version": openmc.__version__, + "cross_sections": os.environ.get("OPENMC_CROSS_SECTIONS"), + "library": "ENDF/B-VIII.0, official OpenMC HDF5 distribution (openmc.org)", + "budget": "quick" if args.quick else "full", + "budget_numbers": budget, + "wall_clock_minutes": round((time.time() - t_start) / 60, 1), + "model": { + "R_tank_cm": R_TANK, "H_tank_cm": H_TANK, "wall_cm": WALL, "gap_cm": GAP, + "absorbers": ABSORBERS, "absorber_margin_cm": ABS_MARGIN, "reflector_cm": REFL, + "tank_axis_x_cm": X_TANK, "cf252_watt_a_eV_b_per_eV": CF252_WATT, + "dt_energy_eV": DT_ENERGY, "solution_atoms_per_b_cm": SOLUTION, + "water_density_ratio_350K": WATER_RHO[350.0] / WATER_RHO[294.0], + "energy_edges_eV": ENERGY_EDGES.tolist(), + }, + } + json.dump(results, open(results_path, "w"), indent=1) + print(f"wrote {results_path} ({(time.time() - t_start)/60:.0f} min)") + + +def main(): + p = argparse.ArgumentParser(description=__doc__.split("\n")[1]) + p.add_argument("--quick", action="store_true", help="a tenth of the particles") + p.add_argument("--only", default="", help="comma separated substrings of run names to (re)run") + p.add_argument("--workdir", default=os.path.join(HERE, "work"), + help="scratch directory for OpenMC inputs and statepoints") + p.add_argument("--resplit", action="store_true", + help="rebuild the per tank sources from fission banks already " + "in the workdir, without rerunning the eigenvalue stage") + args = p.parse_args() + ensure_data() + protocol(args) + + +if __name__ == "__main__": + main() diff --git a/neutron/report.py b/neutron/report.py new file mode 100644 index 0000000..c25da68 --- /dev/null +++ b/neutron/report.py @@ -0,0 +1,600 @@ +#!/usr/bin/env python3 +# Copyright 2026 Max Freedom Pollard +# SPDX-License-Identifier: Apache-2.0 +""" +The neutron gate, reported: derived quantities, the checks, results.md, +and figure 15, all from tallies.json and benchmarks.json. + +Needs only numpy and matplotlib, so it runs in the ordinary CI job on the +committed tally files; gate.py and benchmarks.py, which need OpenMC and +the data library, regenerate those files. + +Derived here and nowhere else: + the gain M = 1/(1 - k) of each tank and of the pair in every absorber + state; the fission matrix K, its dominant eigenvalue against the + transport k, and the unsplit bank control (the exactness check of the + synapse with gain); the point model (I - K)^-1 c against the full + fixed source transport, and what the gap between them means; the + transistor and coincidence truth tables with their margins; the neutron + budget of a fission neutron born in tank A (the fan out tax); the + emission spectra under three different drives against the spectra that + arrived (level restoration); the generation time, the switching time, + the gain bandwidth product, and the step response of the pair by prompt + point kinetics; and the temperature coefficient that prices the veto. +""" +import json +import math +import os + +import numpy as np + +HERE = os.path.dirname(os.path.abspath(__file__)) +FIGS = os.path.join(os.path.dirname(HERE), "figures") +OUT = [] +STATES = [("open", "open"), ("cd", "1 mm cadmium sheet"), ("b4c", "2 cm boron carbide blade")] + + +def say(s=""): + OUT.append(s) + + +def pm(v, digits=4): + return f"{v[0]:.{digits}f} ± {v[1]:.{digits}f}" + + +def gain(k): + m = 1.0 / (1.0 - k[0]) + return [m, k[1] * m * m] + + +def region(run, tag, score): + return run["regions"][tag][score] + + +def fission_matrix(runs, state): + K = np.zeros((2, 2)); S = np.zeros((2, 2)) + for j, tag in enumerate("AB"): + r = runs[f"coupling_{tag}_{state}"] + for i, out in enumerate("AB"): + K[i, j], S[i, j] = region(r, out, "nu-fission") + return K, S + + +def dominant(K, S, n=4000, seed=229): + rng = np.random.default_rng(seed) + lam = np.linalg.eigvals(K).real.max() + samples = [np.linalg.eigvals(K + rng.normal(0, 1, K.shape) * S).real.max() for _ in range(n)] + return lam, float(np.std(samples)) + + +def vec2(run): + return (np.array([region(run, "A", "nu-fission")[0], region(run, "B", "nu-fission")[0]]), + np.array([region(run, "A", "nu-fission")[1], region(run, "B", "nu-fission")[1]])) + + +def spectrum(run, name, edges): + """Normalised per unit lethargy, the mean energy (eV), and the fraction + below 0.625 eV.""" + m = np.array(run[name]["mean"]) + tot = m.sum() + if tot <= 0: + return np.zeros_like(m), float("nan"), 0.0 + mid = np.sqrt(edges[:-1] * edges[1:]) + return m / tot / np.log(edges[1:] / edges[:-1]), float((m * mid).sum() / tot), float(m[mid < 0.625].sum() / tot) + + +def step_response(K, c, Lambda, t_end, n=3000): + """Prompt point kinetics of the pair, Λ dP/dt = (K - I) P + c, driver on + at t = 0, integrated exactly through the eigen decomposition.""" + A = (K - np.eye(2)) / Lambda + b = c / Lambda + w, V = np.linalg.eig(A) + Vinv = np.linalg.inv(V) + Pinf = np.linalg.solve(-A, b) + ts = np.linspace(0, t_end, n) + P = np.array([(Pinf - (V @ np.diag(np.exp(w * t)) @ Vinv @ Pinf)).real for t in ts]) + return ts, P, Pinf + + +def main(): + doc = json.load(open(os.path.join(HERE, "tallies.json"))) + runs, prov = doc["runs"], doc["provenance"] + bench = json.load(open(os.path.join(HERE, "benchmarks.json"))) + edges = np.array(prov["model"]["energy_edges_eV"]) + model = prov["model"] + label = dict(STATES) + + say("# The neutron gate, computed (regenerated by report.py from tallies.json and benchmarks.json)\n") + say(f"OpenMC {prov['openmc_version']}, {prov['library']}, budget {prov['budget']} " + f"({prov.get('wall_clock_minutes', '?')} minutes of wall clock). Every number carries its " + "Monte Carlo standard deviation; nothing is fitted.\n") + + # ---- 0. calibration --------------------------------------------------- + say("## 0. Calibration: the instrument against the handbook\n") + say("| benchmark | this code and data | handbook | difference from handbook (pcm) | JEFF-3.1 on the same case | difference from JEFF-3.1 (pcm) |") + say("|---|---|---|---|---|---|") + for key, c in bench["cases"].items(): + k, kb, kj = c["k"], c["benchmark"], c["jeff31"] + d = (k[0] - kb[0]) * 1e5 + sig = math.sqrt(k[1] ** 2 + kb[1] ** 2) * 1e5 + dj = (k[0] - kj[0]) * 1e5 + sigj = math.sqrt(k[1] ** 2 + kj[1] ** 2) * 1e5 + say(f"| {key} | {pm(k, 5)} | {kb[0]:.5f} ± {kb[1]:.5f} | {d:+.0f} ± {sig:.0f} | " + f"{kj[0]:.5f} | {dj:+.0f} ± {sigj:.0f} |") + say("\nGodiva is the fast spectrum and the fission cross sections alone; SHEBA-II is the thermal " + "spectrum, the water scattering law, and the gate's own material family (uranyl fluoride " + "solution); STACY is a water reflected solution tank, the configuration of the gate. The " + "JEFF-3.1 column, from the same table as the handbook values (JEFF Report 21, Appendix 2), is " + "carried because it separates a fault in this calculation from a fault in the benchmark. " + "Godiva and STACY land where a modern library should. SHEBA-II is overpredicted here by about " + "a percent, and JEFF-3.1 overpredicts it by the same amount from the same specification: this " + "benchmark is known to sit high for every modern library, and a calculation that agrees with " + "its peers to a few hundred pcm while both disagree with the experiment is reporting a " + "property of the evaluation, not an error in the model. The gate below is a difference " + "between configurations of one geometry, where that common bias very largely cancels; the " + "absolute eigenvalues it quotes should be read with SHEBA-II's percent in mind.\n") + + # ---- 1. the unit alone ------------------------------------------------ + say("## 1. The unit alone\n") + say(f"One tank: radius {model['R_tank_cm']:.0f} cm, height {model['H_tank_cm']:.0f} cm, " + f"{math.pi*model['R_tank_cm']**2*model['H_tank_cm']/1000:.0f} litres of 4.9 percent enriched " + "uranyl fluoride solution (the LEU-SOL-THERM-002 benchmark composition) in a 1.6 mm aluminium " + f"shell, standing in {model['reflector_cm']:.0f} cm of water on every side.\n") + say("| tank | k (alone) | gain M = 1/(1 - k) | generation time Λ (µs) | β | switching time Λ/(1 - k) (ms) | gain bandwidth product 1/Λ (kHz) |") + say("|---|---|---|---|---|---|---|") + for tag in "AB": + r = runs[f"alone_{tag}"] + k = r["k"]; M = gain(k) + L = r["global"]["inverse-velocity"][0] / r["global"]["nu-fission"][0] + beta = region(r, tag, "delayed-nu-fission")[0] / region(r, tag, "nu-fission")[0] + say(f"| {tag} | {pm(k, 4)} | {pm(M, 2)} | {L*1e6:.1f} | {beta:.4f} | {L*M[0]*1e3:.2f} | {1/L/1e3:.1f} |") + say("\nΛ is the system generation time, neutron population over fission neutron production, " + "reflector included; β is the delayed neutron fraction from the tallies (not β effective). " + "The switching time is the prompt time constant Λ/(1 - k) of the transistor note, and M " + "times its reciprocal is the constant 1/Λ: the op amp law, with laboratory grade constants in it.\n") + + # ---- 2. coupled ------------------------------------------------------- + say("## 2. The pair, with the absorber as the GATE terminal\n") + say("| configuration | k (pair) | gain M | fission share in B (0.500 by symmetry: see 3.1) | entropy, last 20 percent of active batches |") + say("|---|---|---|---|---|") + curve = [] + for name, desc in (("coupled_open", "open, 3 cm of water"), + ("coupled_cd25", "cadmium sheet 25 percent in"), + ("coupled_cd50", "cadmium sheet 50 percent in"), + ("coupled_cd75", "cadmium sheet 75 percent in"), + ("coupled_cd", "cadmium sheet fully in"), + ("coupled_b4c", "boron carbide blade fully in")): + r = runs[name] + k = r["k"]; M = gain(k) + share = region(r, "B", "nu-fission")[0] / (region(r, "A", "nu-fission")[0] + region(r, "B", "nu-fission")[0]) + ent = r.get("entropy", []) + drift = f"{np.mean(ent[-len(ent)//5:]):.3f} ± {np.std(ent[-len(ent)//5:]):.3f}" if ent else "n/a" + curve.append((name, k, M)) + say(f"| {desc} | {pm(k, 4)} | {pm(M, 2)} | {share:.3f} | {drift} |") + k_open, k_cd, k_b4c = runs["coupled_open"]["k"], runs["coupled_cd"]["k"], runs["coupled_b4c"]["k"] + say(f"\nEvery configuration is subcritical. Two tanks that alone sit at k ≈ {runs['alone_A']['k'][0]:.3f} " + f"couple through 3 cm of water to k = {k_open[0]:.4f}; the cadmium sheet is worth " + f"{(k_open[0]-k_cd[0])*1e5:.0f} pcm and the boron carbide blade {(k_open[0]-k_b4c[0])*1e5:.0f} pcm. " + "The absorber is a control element in the reactor sense and a GATE terminal in the transistor " + "sense, and the table is its transfer curve (figure 15 A).\n") + say("The last two columns are a caution rather than a reassurance. The two tanks are identical and " + "the absorber sits on their plane of symmetry, so the fission share in B is exactly 0.500 in " + "the true fundamental mode; the numbers above wander several percent around it while the " + "Shannon entropy sits flat to a part in a thousand. The entropy is not lying, it is answering a " + "different question: it sees the shape inside each tank, which has converged, and not the tilt " + "between them, which has not. Section 3.1 explains why this is a permanent feature of coupled " + "region gates rather than a shortage of batches, and what it costs. The eigenvalues themselves " + "survive it, because the tilt mode integrates to zero over the whole system and so cannot " + "shift a total production rate to first order; the fission matrix does not, which is why it is " + "measured differently.\n") + + # ---- 3. the fission matrix -------------------------------------------- + say("## 3. The synapse with gain on the diagonal: the fission matrix, checked\n") + say("K_ij is the number of fission neutrons produced in tank i per fission neutron born in tank j, " + "the source being the fundamental mode restricted to j and fission treated as capture so that " + "exactly one generation is counted. This is Avery's coupling coefficient, computed rather than " + "assumed.\n") + say("### 3.1 A convergence trap this architecture walks into by construction\n") + say("Getting that source right was the hardest part of the calculation, and the reason is " + "structural rather than numerical, so it belongs in the theory and not in a footnote. Two " + "weakly coupled multiplying regions are the textbook *loosely coupled system*. The " + "fundamental mode is symmetric between the tanks and the second mode is the tilt from one " + "tank to the other; their eigenvalues are k_self + k_cross and k_self - k_cross, so the " + "dominance ratio is\n") + say("> `DR = (k_self - k_cross)/(k_self + k_cross),`\n") + say("which approaches one exactly as the coupling is made weak. A gate is *built* to have a small " + "k_cross. A gate is therefore built to be hard to converge, and the better the gate, the worse " + "the problem.\n") + say("| absorber | k_self | k_cross | dominance ratio | generations for a tilt to fall tenfold | tilt in the raw fission bank |") + say("|---|---|---|---|---|---|") + for state, desc in STATES: + K, _ = fission_matrix(runs, state) + self_, cross = K[0, 0], K[0, 1] + dr = (self_ - cross) / (self_ + cross) + bank = runs[f"coupled_{state}"].get("bank_split", {}) + tilt = (bank["B"] / (bank["A"] + bank["B"]) - 0.5) if bank else float("nan") + say(f"| {desc} | {self_:.4f} | {cross:.4f} | {dr:.4f} | {math.log(10)/-math.log(dr):.0f} | {tilt:+.3f} |") + say("\nThe tilt then decays by one or two percent per generation while fission bank sampling noise " + "re excites it just as fast, so the standing tilt of several percent in the last column is not " + "a transient that more inactive batches remove: it is the equilibrium noise level, and its " + "correlation length is longer than an affordable active cycle. Worse, the Shannon entropy of " + "the source is nearly blind to it, because entropy is dominated by the shape inside each tank. " + "The entropy traces in Section 2 are flat to a part in a thousand while the tilt wanders by " + "seven percent. An analyst who trusted the entropy would have shipped the wrong matrix, and " + "the first version of this calculation did exactly that.\n") + say("### 3.2 The fix: a mirror instead of more batches\n") + say("The geometry is exactly mirror symmetric in x, so the true fundamental mode puts exactly half " + "its fissions in each tank. Symmetrising the converged bank, every site with its mirror image, " + "removes the tilt but not the distortion the tilt leaves in the shape *inside* each tank, and " + "it was not enough here. What works is to remove the tilt mode from the problem rather than " + "from the answer: run the half geometry, x > 0 only, with the plane x = 0 reflective. The " + "antisymmetric mode cannot exist in that model, so there is nothing slow left to converge, and " + "its fission bank mirrored is the fundamental mode by construction rather than by waiting.\n") + say("The half model has to earn its place, so it is checked against the full one:\n") + say("| absorber | half model k (exact symmetric mode) | full model k | difference (pcm) |") + say("|---|---|---|---|") + for state, desc in STATES: + h, f = runs[f"half_{state}"]["k"], runs[f"coupled_{state}"]["k"] + sig = math.sqrt(h[1] ** 2 + f[1] ** 2) * 1e5 + say(f"| {desc} | {pm(h, 5)} | {pm(f, 5)} | {(h[0]-f[0])*1e5:+.0f} ± {sig:.0f} |") + say("\nThe two agree, which says the full model's eigenvalues were never the problem: the tilt " + "integrates to zero over the whole system, so it cannot shift a total production rate, and k " + "survives it. Only the fission matrix, which asks where the neutrons went and not just how " + "many, is sensitive to it. Every matrix below is measured from the half model's bank.\n") + say("Two controls say whether that was necessary. Each runs one generation on a whole source, " + "exact or raw, and must return k, because one generation of the fundamental mode is the " + "definition of k.\n") + say("| absorber | control on the exact mode | control on the raw bank | k | exact error (pcm) | raw error (pcm) |") + say("|---|---|---|---|---|---|") + for state, desc in STATES: + ex = runs[f"coupling_all_{state}"]["global"]["nu-fission"] + raw = runs[f"coupling_rawall_{state}"]["global"]["nu-fission"] + k = runs[f"half_{state}"]["k"] + say(f"| {desc} | {pm(ex, 4)} | {pm(raw, 4)} | {pm(k, 4)} | " + f"{(ex[0]-k[0])*1e5:+.0f} | {(raw[0]-k[0])*1e5:+.0f} |") + say("\nThe clearest evidence is not in those columns but in the matrix itself. Measured from the " + "raw bank, the self multiplication k_self came out as 0.8924, 0.9030, 0.8977 for the three " + "absorbers: not monotonic, and for cadmium higher than a tank standing alone, which no " + "absorber placed outside a tank can do. Measured from the exact mode it is " + f"{fission_matrix(runs, 'open')[0][0,0]:.4f}, {fission_matrix(runs, 'cd')[0][0,0]:.4f}, " + f"{fission_matrix(runs, 'b4c')[0][0,0]:.4f}: flat, as it must be, because an absorber between " + "the tanks barely touches what a tank does to its own neutrons. The absorber now changes only " + "the cross term, which is the gate doing its job and nothing else. A wrong answer that looked " + "reasonable became a right answer that looks obvious, which is the usual shape of this kind of " + "error and the reason the control is in the protocol at all.\n") + say("### 3.3 The matrix, and what its eigenvalue tests\n") + say("With the source settled, the **dominant eigenvalue of K** tests the two region reduction " + "itself. It equals k only if each tank's internal shape is the same whether the tank is fed " + "from inside or through the gap from its neighbour, which is the assumption every coupled " + "region model makes, this theory's synapse included. Its residual is not an error but a " + "measurement of how good that assumption is.\n") + Kmats = {} + resid = [] + for state, desc in STATES: + K, S = fission_matrix(runs, state) + lam, dlam = dominant(K, S) + k = runs[f"half_{state}"]["k"] + Kmats[state] = K + resid.append((lam - k[0]) * 1e5) + say(f"**{desc[0].upper() + desc[1:]}.**\n") + say("| | born in A | born in B |") + say("|---|---|---|") + say(f"| fission neutrons in A | {K[0,0]:.4f} ± {S[0,0]:.4f} | {K[0,1]:.4f} ± {S[0,1]:.4f} |") + say(f"| fission neutrons in B | {K[1,0]:.4f} ± {S[1,0]:.4f} | {K[1,1]:.4f} ± {S[1,1]:.4f} |") + say(f"\n- dominant eigenvalue of K **{lam:.4f} ± {dlam:.4f}** against k **{k[0]:.4f} ± {k[1]:.4f}**, " + f"a residual of {(lam-k[0])*1e5:+.0f} pcm") + say("") + say(f"The eigenvalue of the reduced two by two matrix sits between {min(resid):+.0f} and " + f"{max(resid):+.0f} pcm of k, the largest belonging to the most weakly coupled case, which is " + "what one would expect. That residual is the price of describing each tank by a single number: " + "a neutron arriving from the neighbour enters through one wall and fissions nearer that wall " + "than a fundamental mode neutron would, so a tank's internal shape is not quite source " + "independent and no single coupling coefficient can absorb the difference. A third of a " + "percent is small enough that the synapse picture is sound and large enough that it should be " + "quoted rather than rounded away. A photon sector network of many weakly coupled cells " + "inherits both the caveat and, from 3.1, the convergence problem in a sharper form.\n") + say("**The neutron budget of one fission neutron born in tank A** (the fan out tax of theory " + "Section 1.3, with real numbers):\n") + say("| fate | open | cadmium sheet | boron carbide blade |") + say("|---|---|---|---|") + rs = [runs[f"coupling_A_{s}"] for s, _ in STATES] + def absorbed(r, key): + return r["absorption_by_material"].get(key, [0.0, 0.0])[0] + rows = [("causes fission neutrons in A (k_AA, self restoring)", [region(r, "A", "nu-fission")[0] for r in rs]), + ("causes fission neutrons in B (k_BA, the coupling)", [region(r, "B", "nu-fission")[0] for r in rs]), + ("absorbed in the water", [absorbed(r, "water") for r in rs]), + ("absorbed in the aluminium", [absorbed(r, "al1100") for r in rs]), + ("absorbed in the absorber", [absorbed(rs[0], "none"), absorbed(rs[1], "cadmium"), absorbed(rs[2], "b4c")]), + ("leaks out of the water bath", [1.0 - sum(v[0] for v in r["absorption_by_material"].values()) for r in rs])] + for lab, vals in rows: + say(f"| {lab} | " + " | ".join(f"{v:.4f}" for v in vals) + " |") + kBA = [Kmats[s][1, 0] for s, _ in STATES] + say(f"\nThe cadmium sheet cuts the coupling k_BA by a factor {kBA[0]/kBA[1]:.1f}, the boron carbide blade " + f"by {kBA[0]/kBA[2]:.1f}. What reaches the next gate is one or two percent of what was born; the " + "rest is the price of an isotropic emitter in a moderating body. The 4π loss of theory Section " + "1.3 is not a metaphor here, it is a row in this table, and the self restoring term k_AA is " + "the reason the gate has gain at all.\n") + + # ---- 4. the gate ---------------------------------------------------------- + say("## 4. The gate: driven, with multiplication on\n") + say("A Cf-252 point source (Fröhner Watt spectrum, mean 2.13 MeV) or a 14.1 MeV deuterium tritium " + "point source on the axis of a tank is the SOURCE terminal. Fixed source transport with full " + "multiplication gives the steady fission neutron production in each tank per driver neutron; " + "production in B is the DRAIN current. The point model of coupled regions predicts the same " + "numbers from Section 3 and the first generation response c of each tank to the driver, " + "P = (I - K)⁻¹ c, and the comparison is reported because the gap is physics, not noise.\n") + say("| drive | absorber | c_A | c_B | P_A transport | P_B transport | P_A point model | P_B point model | 1 + P_A + P_B |") + say("|---|---|---|---|---|---|---|---|---|") + P = {} + for state, desc in STATES: + for drive in ("cfA", "cfB", "dtB"): + c, cs = vec2(runs[f"firstgen_{drive}_{state}"]) + p, ps = vec2(runs[f"gate_{drive}_{state}"]) + pred = np.linalg.solve(np.eye(2) - Kmats[state], c) + P[(drive, state)] = (p, ps, pred, c) + say(f"| {drive} | {desc} | {c[0]:.3f} | {c[1]:.3f} | {p[0]:.2f} ± {ps[0]:.2f} | " + f"{p[1]:.3f} ± {ps[1]:.3f} | {pred[0]:.2f} | {pred[1]:.3f} | {1+p.sum():.1f} |") + pa, pa_pred = P[("cfA", "open")][0], P[("cfA", "open")][2] + Mopen = gain(k_open)[0] + say(f"\nThe transport drain from A exceeds the point model's, and the source multiplication " + f"1 + P_A + P_B = {1+pa.sum():.1f} exceeds the eigenvalue gain 1/(1 - k) = {Mopen:.1f}: a point source " + "on the axis of a tank has more importance than a fundamental mode neutron, so it earns more " + "than M offspring, and the fission neutrons it makes are clustered at the axis, far from the " + "other tank, so a smaller share of them cross than the fundamental mode would send. Both " + "effects are the adjoint flux of theory Section 11 at work, and they are why a gate's gain is a " + "property of the terminal, not of the medium alone. The fundamental mode arithmetic of Section 3 " + "is exact for what it describes and undercounts a well placed driver.\n") + + # transistor truth table + say("### 4.1 The transistor: SOURCE and GATE in, DRAIN out\n") + say("| driver in A | absorber | drain: fission neutrons born in B per driver neutron | state |") + say("|---|---|---|---|") + say("| off | any | 0 | LOW |") + drains = {s: P[("cfA", s)][0][1] for s, _ in STATES} + for state, desc in STATES[::-1]: + say(f"| on | {desc} | {drains[state]:.3f} | {'**HIGH**' if state == 'open' else 'LOW'} |") + say(f"\nDrain per source neutron with the gate open: **{drains['open']:.2f}**. Every driver neutron " + f"entering tank A becomes {drains['open']:.1f} fission neutrons born in tank B, on the far side of " + "an absorber that a coil could move: the output exceeds the input, and the neutron sector gate " + "has gain with library data and no new physics, as Section 4 of the theory supplement claims. " + f"ON/OFF contrast: **{drains['open']/drains['cd']:.1f}** against the cadmium sheet and " + f"**{drains['open']/drains['b4c']:.1f}** against the boron carbide blade. The contrast is modest " + "and its reason is in the budget above: neutrons cross 3 cm of water still fast or epithermal, " + "a thermal filter stops only the slow ones, and the fast coupling is a property of the geometry " + "that no thin absorber touches. A MOSFET's million to one comes from a barrier; this gate has " + "a filter, and until the coupling path itself is switched (a shutter tank, a longer moderated " + "gap) that is what its contrast will be.\n") + + # coincidence AND + say("### 4.2 The coincidence AND: two neutron inputs, one threshold\n") + say("Both drivers on means the two point sources at equal strength; the run with both sources " + "present is the superposition check, which fixed source transport obeys exactly.\n") + say("| absorber | A alone → P_B | B alone → P_B | sum | both drivers, one run | superposition error | strength ratio for a balanced gate | threshold T | margin each side |") + say("|---|---|---|---|---|---|---|---|---|") + sup = [] + for state, desc in STATES: + a = P[("cfA", state)][0][1]; b = P[("cfB", state)][0][1] + both = vec2(runs[f"gate_cfAB_{state}"])[0][1] + err = abs(both - (a + b)) / (a + b) * 100 + sup.append(err) + T = a * math.sqrt(2) + say(f"| {desc} | {a:.3f} | {b:.3f} | {a+b:.3f} | {both:.3f} | {err:.1f} percent | " + f"{a/b:.3f} | {T:.3f} | ×{math.sqrt(2):.2f} |") + say("\nSuperposition holds: two drivers together deliver the sum of what each delivers alone, to " + f"{max(sup):.1f} percent, which is the counting noise. That is the check that this medium is linear " + "while its cross sections are frozen, the property theory Section 5.3 asserts and the whole " + "Green's function reading depends on, measured here in a real geometry rather than in a random " + "walk toy.\n") + say("In a linear medium the coincidence AND is a threshold on a weighted sum, a perceptron AND. " + "The two inputs are unequal, a direct drive against a coupled one, so a balanced gate runs the " + "second driver at the strength ratio in the table; each input alone then delivers half the " + "sum, the threshold sits at √2 times one input, and the margin is √2 either side. The weights " + "are nuclear and programmable (the absorber sets k_BA); the threshold is a comparison of rates " + "at the boundary, as in the benchtop cell, and stays there until a nonlinear element exists " + "inside the medium. Section 7 says what the medium offers instead.\n") + + # ---- 5. level restoration --------------------------------------------- + say("## 5. Level restoration: three inputs, one output\n") + say("| drive (open) | the input to tank B | mean energy crossing B's wall | fraction of that below 0.625 eV | mean energy of fission neutrons born in B |") + say("|---|---|---|---|---|") + spectra = {} + for drive, desc in (("cfA", "the leakage of tank A, through 3 cm of water"), + ("cfB", "a Cf-252 spectrum at the axis, mean 2.13 MeV"), + ("dtB", "a 14.1 MeV line at the axis")): + r = runs[f"gate_{drive}_open"] + chi, e_chi, _ = spectrum(r, "chi_B", edges) + cur, e_in, th = spectrum(r, "into_B", edges) + spectra[drive] = (chi, cur, e_chi, e_in) + say(f"| {drive} | {desc} | {e_in/1e6:.3f} MeV | {th:.2f} | {e_chi/1e6:.3f} MeV |") + tv = {d: 0.5 * float(np.sum(np.abs(spectra[d][0] - spectra["cfA"][0]) * np.log(edges[1:] / edges[:-1]))) + for d in ("cfB", "dtB")} + say(f"\nRead the middle columns with care. When A is driven, everything tank B receives crosses its " + f"wall, and it arrives moderated: {spectra['cfA'][3]/1e6:.3f} MeV on average, " + f"{spectrum(runs['gate_cfA_open'], 'into_B', edges)[2]:.0%} of it already thermal, because 3 cm " + "of water is a moderator as well as a gap. In the other two rows the driver sits inside B, so " + "the wall current is only the return from the water reflector and the input proper is the " + "point source itself: a 2 MeV fission spectrum, and a 14.1 MeV line.\n") + say(f"The three inputs therefore span from thermal to 14 MeV. The fission neutrons born in B differ " + f"by a total variation distance of {tv['cfB']:.3f} and {tv['dtB']:.3f} between them, which is " + "counting statistics: one output spectrum, whatever went in, because a fission neutron is born " + "from the fission barrier and not from whatever triggered it. This is the self restoring class " + "of theory Section 2, measured (figure 15 C), and it is the property the photon sector cannot " + "buy at any price, since an isomer cascade emits its own lines and not its trigger's.\n") + + # ---- 6. clock ------------------------------------------------------------ + say("## 6. The clock, the switching time, and the step response\n") + r = runs["coupled_open"] + L = r["global"]["inverse-velocity"][0] / r["global"]["nu-fission"][0] + kpair = k_open[0] + ts, Pt, Pinf = step_response(Kmats["open"], P[("cfA", "open")][3], L, t_end=12 * L * Mopen) + pb = Pt[:, 1] / Pinf[1] + t10, t90 = ts[np.searchsorted(pb, 0.10)], ts[np.searchsorted(pb, 0.90)] + say(f"- generation time of the pair Λ = **{L*1e6:.1f} µs**; k = {kpair:.4f}; prompt time constant " + f"Λ/(1 - k) = **{L*Mopen*1e3:.2f} ms**; gain bandwidth product M/τ = 1/Λ = **{1/L/1e3:.1f} kHz**") + say(f"- step response of the drain (driver in A switched on at t = 0, prompt point kinetics on the " + f"measured K and Λ): 10 to 90 percent rise in **{(t90-t10)*1e3:.2f} ms** (figure 15 D)") + beta = region(r, "A", "delayed-nu-fission")[0] / region(r, "A", "nu-fission")[0] + say(f"- delayed neutrons, ignored above, add a tail on the second scale carrying about " + f"β k/(1 - k) = {beta*kpair/(1-kpair):.2f} of the prompt step: the theory's remark that precursors " + "slow the approach as k → 1, quantified\n") + say("The transistor note's 10 kHz for a thermal assembly is the right order. Ten times the gain " + "costs ten times the switching time, and the product does not move.\n") + + # ---- 7. feedback --------------------------------------------------------- + say("## 7. The veto, priced: the only signal controlled inhibition in the neutron sector\n") + kT, kX = runs["feedback_T350_fixed_density"]["k"], runs["feedback_T350_expanded"]["k"] + dT = 56.0 + a_dopp, a_exp, a_tot = ((kT[0] - k_open[0]) / dT * 1e5, (kX[0] - kT[0]) / dT * 1e5, + (kX[0] - k_open[0]) / dT * 1e5) + say("| state of the pair, open | k | gain M |") + say("|---|---|---|") + say(f"| 294 K | {pm(k_open, 4)} | {pm(gain(k_open), 2)} |") + say(f"| 350 K, liquid density held | {pm(kT, 4)} | {pm(gain(kT), 2)} |") + say(f"| 350 K, liquid expanded ({(1-model['water_density_ratio_350K'])*100:.1f} percent less dense) | {pm(kX, 4)} | {pm(gain(kX), 2)} |") + say(f"\n- temperature coefficient **{a_tot:+.0f} pcm/K** in total: {a_dopp:+.0f} pcm/K from Doppler " + f"broadening and the thermal scattering law, {a_exp:+.0f} pcm/K from expansion of the liquid") + say(f"- gain with the pair 56 K hotter: M falls from {gain(k_open)[0]:.2f} to {gain(kX)[0]:.2f}, " + f"a modulation of {(1-gain(kX)[0]/gain(k_open)[0])*100:.0f} percent") + dT_half = (1 - k_open[0]) / abs(a_tot / 1e5) if a_tot < 0 else float("inf") + say(f"- to halve the gain (1 - k doubled) the liquid would have to warm by about **{dT_half:.0f} K**, " + "which water at atmospheric pressure cannot do\n") + say("Every coupling in the neutron sector is excitatory: a neutron only ever adds fissions, and an " + "absorber that burns up disinhibits. The one channel by which a signal lowers another region's " + "gain is heat, through the density and the Doppler width of the medium, and the table prices " + "it: sign correct, slow (the thermal time constant of tens of litres of liquid), and worth a " + "fraction of the gain, not a veto. Signal controlled inversion, which NAND, signed Ising " + "weights, and Turing completeness all require, is therefore the weak point of the neutron " + "sector as an embodiment, and the theory supplement now says so in Section 4. The rate coded " + "machine is untouched: its NOT is a fixed absorber, its AND a coincidence, and neither needs " + "a signal to invert a signal.\n") + + say("*The gate exists, with gain, with level restoration, with a programmable weight and a clock, " + "all computed from library data at k below one. What it lacks is a nonlinearity of its own and " + "an inhibition of its own, and the numbers above say by how much.*") + + open(os.path.join(HERE, "results.md"), "w").write("\n".join(OUT) + "\n") + print("\n".join(OUT[-10:])) + print("wrote neutron/results.md") + figure(curve, runs, P, spectra, edges, ts, Pt, Pinf, L, kpair) + + +def figure(curve, runs, P, spectra, edges, ts, Pt, Pinf, L, kpair): + import matplotlib + matplotlib.use("Agg") + import matplotlib.pyplot as plt + INK, BLUE, RED, GREEN, AMBER, PURPLE, GREY = ( + "#16213e", "#2563eb", "#dc2626", "#16a34a", + "#d97706", "#7c3aed", "#94a3b8") + plt.rcParams.update({ + "figure.facecolor": "white", "axes.facecolor": "white", + "savefig.facecolor": "white", "axes.edgecolor": INK, + "axes.labelcolor": INK, "text.color": INK, "xtick.color": INK, + "ytick.color": INK, "axes.titlecolor": INK, "font.size": 11, + "axes.titlesize": 12, "axes.titleweight": "bold", + "axes.grid": True, "grid.color": "#e5e7eb", "grid.linewidth": 0.8, + "axes.spines.top": False, "axes.spines.right": False, + "figure.dpi": 130, "svg.hashsalt": "nuclear-computer", + }) + fig, ((A, B), (C, D)) = plt.subplots(2, 2, figsize=(13.2, 9.4)) + + # A: the transfer curve of the absorber, with a true gain axis on the right + cd_pts = [(0, curve[0][1])] + [(int(n.split("cd")[1] or 100), k) for n, k, _ in curve[1:5]] + x = [p[0] for p in cd_pts]; y = [p[1][0] for p in cd_pts]; e = [p[1][1] for p in cd_pts] + A.errorbar(x, y, yerr=e, fmt="-o", color=BLUE, lw=2.2, ms=6, capsize=3, + label="1 mm cadmium sheet") + kb = curve[5][1] + A.errorbar([100], [kb[0]], yerr=[kb[1]], fmt="s", color=PURPLE, ms=9, capsize=3, + label="2 cm boron carbide blade") + ka = runs["alone_A"]["k"] + A.axhline(ka[0], color=GREY, ls=":", lw=1.3) + A.text(50, ka[0] - 0.0009, "one tank alone: no neighbour to couple to", + color=GREY, fontsize=8, va="top", ha="center") + lo, hi = min(min(y), kb[0], ka[0]) - 0.004, max(y) + 0.004 + A.set_ylim(lo, hi) + A.set_xlim(-6, 106) + A.set_xlabel("absorber inserted between the tanks (percent)") + A.set_ylabel("k of the pair") + A.set_title("A. The GATE terminal: an absorber between the tanks") + A.legend(frameon=False, fontsize=8.5, loc="upper right") + # the gain axis is the exact map k -> 1/(1-k), not a rescaled copy + A2 = A.secondary_yaxis("right", functions=(lambda kk: 1.0 / (1.0 - np.clip(kk, None, 0.9999)), + lambda MM: 1.0 - 1.0 / MM)) + A2.set_ylabel("gain M = 1/(1 − k)", color=AMBER) + A2.tick_params(colors=AMBER) + A.annotate(f"k = 1 is {(1-max(y))*1e5:.0f} pcm above the top of this axis:\nevery configuration is subcritical", + xy=(0.03, 0.06), xycoords="axes fraction", fontsize=8, color=RED) + + # B: the drain, with the gain threshold + labels = ["boron carbide\nblade in", "cadmium\nsheet in", "open"] + keys = ["b4c", "cd", "open"] + vals = [P[("cfA", k)][0][1] for k in keys] + errs = [P[("cfA", k)][1][1] for k in keys] + preds = [P[("cfA", k)][2][1] for k in keys] + B.bar(range(3), vals, yerr=errs, color=[PURPLE, GREY, BLUE], width=0.6, capsize=4, + label="transport (the drain)", zorder=3) + B.plot(range(3), preds, "D", color=INK, ms=7, label="point model (I − K)⁻¹c", zorder=4) + B.axhline(1.0, color=RED, ls="--", lw=1.3, zorder=2) + B.text(-0.45, 1.03, "unity: one neutron out per neutron in", color=RED, fontsize=8.5, + ha="left", va="bottom") + B.set_ylim(0, max(max(vals), max(preds)) * 1.32) + B.set_xticks(range(3)); B.set_xticklabels(labels, fontsize=9) + B.set_xlabel("driver in tank A throughout") + B.set_ylabel("fission neutrons born in B per driver neutron") + B.set_title("B. The DRAIN: gain across the gate, and its contrast") + B.legend(frameon=False, fontsize=8.5, loc="upper left") + B.annotate("", xy=(2, vals[2]), xytext=(0, vals[0]), + arrowprops=dict(arrowstyle="<->", color=INK, lw=1.2, shrinkA=2, shrinkB=2)) + B.text(1.0, (vals[0] + vals[2]) / 2 * 1.06, f"ON/OFF contrast {vals[2]/vals[0]:.1f}", + fontsize=9, ha="center", color=INK, fontweight="bold") + + # C: level restoration. The three outputs coincide, so draw them so that is visible. + mid = np.sqrt(edges[:-1] * edges[1:]) + styles = [("cfA", BLUE, 5.0, 0.30, "driven from tank A (moderated input)"), + ("cfB", PURPLE, 2.6, 0.95, "Cf-252 at the axis of B"), + ("dtB", GREEN, 1.3, 1.00, "14.1 MeV at the axis of B")] + for drive, c, lw, al, lab in styles: + C.step(mid, spectra[drive][0], where="mid", color=c, lw=lw, alpha=al, + label=f"born in B: {lab}", zorder=3) + arr = spectra["cfA"][1] + C.step(mid, arr * (spectra["cfA"][0].max() / max(arr.max(), 1e-30)), where="mid", + color=AMBER, lw=1.6, ls="--", label="input: entering B through its wall (scaled)", zorder=2) + for e_in, lab, frac, ha in ((2.13e6, "Cf-252 mean input\n2.13 MeV", 0.72, "right"), + (14.1e6, "14.1 MeV\ninput line", 0.50, "left")): + C.axvline(e_in, color=GREY, ls=":", lw=1.2, zorder=1) + pad = " " if ha == "left" else "" + C.text(e_in, spectra["cfA"][0].max() * frac, pad + lab, fontsize=7.6, color=GREY, + ha=ha, va="center") + C.set_xscale("log") + C.set_xlabel("neutron energy (eV)") + C.set_ylabel("spectrum per unit lethargy (normalised)") + C.set_title("C. Level restoration: three inputs, one output") + C.legend(frameon=False, fontsize=7.8, loc="upper left") + C.set_xlim(1e-3, 3e7) + C.set_ylim(0, spectra["cfA"][0].max() * 1.35) + C.annotate("the three output curves lie on top of one another:\nwhat is emitted does not remember what arrived", + xy=(0.03, 0.02), xycoords="axes fraction", fontsize=8, color=INK) + + # D: step response + D.plot(ts * 1e3, Pt[:, 0] / Pinf[0], color=GREY, lw=1.7, label="tank A production") + D.plot(ts * 1e3, Pt[:, 1] / Pinf[1], color=BLUE, lw=2.4, label="tank B production (the drain)") + D.axhline(1.0, color=GREY, ls=":", lw=1) + M = 1 / (1 - kpair) + D.set_xlabel("time after the driver in A switches on (ms)") + D.set_ylabel("production, fraction of the steady state") + D.set_title("D. Switching: τ = Λ/(1 − k), and M/τ = 1/Λ") + D.text(0.40, 0.30, f"Λ = {L*1e6:.0f} µs, k = {kpair:.3f}\nτ = Λ/(1 − k) = {L*M*1e3:.2f} ms\n" + f"M/τ = 1/Λ = {1/L/1e3:.0f} kHz", transform=D.transAxes, fontsize=9.5, color=INK) + D.legend(frameon=False, fontsize=8.5, loc="lower right") + + fig.suptitle("Figure 15. The neutron sector gate, computed from library data at k < 1", + fontsize=13, fontweight="bold", y=1.0) + fig.tight_layout() + for ext in ("svg", "png"): + kw = {"metadata": {"Date": None}} if ext == "svg" else {} + fig.savefig(os.path.join(FIGS, f"fig15_neutron_gate.{ext}"), bbox_inches="tight", **kw) + plt.close(fig) + print("wrote fig15_neutron_gate") + + +if __name__ == "__main__": + main() diff --git a/neutron/results.md b/neutron/results.md new file mode 100644 index 0000000..f575e13 --- /dev/null +++ b/neutron/results.md @@ -0,0 +1,208 @@ +# The neutron gate, computed (regenerated by report.py from tallies.json and benchmarks.json) + +OpenMC 0.16.0, ENDF/B-VIII.0, official OpenMC HDF5 distribution (openmc.org), budget full (106.0 minutes of wall clock). Every number carries its Monte Carlo standard deviation; nothing is fitted. + +## 0. Calibration: the instrument against the handbook + +| benchmark | this code and data | handbook | difference from handbook (pcm) | JEFF-3.1 on the same case | difference from JEFF-3.1 (pcm) | +|---|---|---|---|---|---| +| HEU-MET-FAST-001 | 0.99969 ± 0.00035 | 1.00000 ± 0.00100 | -31 ± 106 | 0.99644 | +325 ± 40 | +| LEU-SOL-THERM-001 | 1.01135 ± 0.00050 | 0.99910 ± 0.00290 | +1225 ± 294 | 1.01252 | -117 ± 99 | +| LEU-SOL-THERM-004-1 | 1.00000 ± 0.00048 | 0.99940 ± 0.00080 | +60 ± 93 | 1.00046 | -46 ± 90 | + +Godiva is the fast spectrum and the fission cross sections alone; SHEBA-II is the thermal spectrum, the water scattering law, and the gate's own material family (uranyl fluoride solution); STACY is a water reflected solution tank, the configuration of the gate. The JEFF-3.1 column, from the same table as the handbook values (JEFF Report 21, Appendix 2), is carried because it separates a fault in this calculation from a fault in the benchmark. Godiva and STACY land where a modern library should. SHEBA-II is overpredicted here by about a percent, and JEFF-3.1 overpredicts it by the same amount from the same specification: this benchmark is known to sit high for every modern library, and a calculation that agrees with its peers to a few hundred pcm while both disagree with the experiment is reporting a property of the evaluation, not an error in the model. The gate below is a difference between configurations of one geometry, where that common bias very largely cancels; the absolute eigenvalues it quotes should be read with SHEBA-II's percent in mind. + +## 1. The unit alone + +One tank: radius 24 cm, height 40 cm, 72 litres of 4.9 percent enriched uranyl fluoride solution (the LEU-SOL-THERM-002 benchmark composition) in a 1.6 mm aluminium shell, standing in 20 cm of water on every side. + +| tank | k (alone) | gain M = 1/(1 - k) | generation time Λ (µs) | β | switching time Λ/(1 - k) (ms) | gain bandwidth product 1/Λ (kHz) | +|---|---|---|---|---|---|---| +| A | 0.9013 ± 0.0005 | 10.13 ± 0.05 | 109.2 | 0.0066 | 1.11 | 9.2 | +| B | 0.9007 ± 0.0006 | 10.07 ± 0.06 | 109.3 | 0.0066 | 1.10 | 9.1 | + +Λ is the system generation time, neutron population over fission neutron production, reflector included; β is the delayed neutron fraction from the tallies (not β effective). The switching time is the prompt time constant Λ/(1 - k) of the transistor note, and M times its reciprocal is the constant 1/Λ: the op amp law, with laboratory grade constants in it. + +## 2. The pair, with the absorber as the GATE terminal + +| configuration | k (pair) | gain M | fission share in B (0.500 by symmetry: see 3.1) | entropy, last 20 percent of active batches | +|---|---|---|---|---| +| open, 3 cm of water | 0.9132 ± 0.0005 | 11.52 ± 0.07 | 0.496 | 10.849 ± 0.010 | +| cadmium sheet 25 percent in | 0.9128 ± 0.0005 | 11.47 ± 0.07 | 0.513 | 10.848 ± 0.006 | +| cadmium sheet 50 percent in | 0.9090 ± 0.0005 | 10.99 ± 0.06 | 0.493 | 10.842 ± 0.010 | +| cadmium sheet 75 percent in | 0.9060 ± 0.0005 | 10.64 ± 0.06 | 0.461 | 10.832 ± 0.010 | +| cadmium sheet fully in | 0.9056 ± 0.0005 | 10.59 ± 0.06 | 0.550 | 10.823 ± 0.013 | +| boron carbide blade fully in | 0.9038 ± 0.0005 | 10.39 ± 0.05 | 0.512 | 10.818 ± 0.010 | + +Every configuration is subcritical. Two tanks that alone sit at k ≈ 0.901 couple through 3 cm of water to k = 0.9132; the cadmium sheet is worth 757 pcm and the boron carbide blade 940 pcm. The absorber is a control element in the reactor sense and a GATE terminal in the transistor sense, and the table is its transfer curve (figure 15 A). + +The last two columns are a caution rather than a reassurance. The two tanks are identical and the absorber sits on their plane of symmetry, so the fission share in B is exactly 0.500 in the true fundamental mode; the numbers above wander several percent around it while the Shannon entropy sits flat to a part in a thousand. The entropy is not lying, it is answering a different question: it sees the shape inside each tank, which has converged, and not the tilt between them, which has not. Section 3.1 explains why this is a permanent feature of coupled region gates rather than a shortage of batches, and what it costs. The eigenvalues themselves survive it, because the tilt mode integrates to zero over the whole system and so cannot shift a total production rate to first order; the fission matrix does not, which is why it is measured differently. + +## 3. The synapse with gain on the diagonal: the fission matrix, checked + +K_ij is the number of fission neutrons produced in tank i per fission neutron born in tank j, the source being the fundamental mode restricted to j and fission treated as capture so that exactly one generation is counted. This is Avery's coupling coefficient, computed rather than assumed. + +### 3.1 A convergence trap this architecture walks into by construction + +Getting that source right was the hardest part of the calculation, and the reason is structural rather than numerical, so it belongs in the theory and not in a footnote. Two weakly coupled multiplying regions are the textbook *loosely coupled system*. The fundamental mode is symmetric between the tanks and the second mode is the tilt from one tank to the other; their eigenvalues are k_self + k_cross and k_self - k_cross, so the dominance ratio is + +> `DR = (k_self - k_cross)/(k_self + k_cross),` + +which approaches one exactly as the coupling is made weak. A gate is *built* to have a small k_cross. A gate is therefore built to be hard to converge, and the better the gate, the worse the problem. + +| absorber | k_self | k_cross | dominance ratio | generations for a tilt to fall tenfold | tilt in the raw fission bank | +|---|---|---|---|---|---| +| open | 0.8946 | 0.0197 | 0.9568 | 52 | +0.001 | +| 1 mm cadmium sheet | 0.8964 | 0.0116 | 0.9744 | 89 | +0.051 | +| 2 cm boron carbide blade | 0.8943 | 0.0075 | 0.9834 | 137 | +0.070 | + +The tilt then decays by one or two percent per generation while fission bank sampling noise re excites it just as fast, so the standing tilt of several percent in the last column is not a transient that more inactive batches remove: it is the equilibrium noise level, and its correlation length is longer than an affordable active cycle. Worse, the Shannon entropy of the source is nearly blind to it, because entropy is dominated by the shape inside each tank. The entropy traces in Section 2 are flat to a part in a thousand while the tilt wanders by seven percent. An analyst who trusted the entropy would have shipped the wrong matrix, and the first version of this calculation did exactly that. + +### 3.2 The fix: a mirror instead of more batches + +The geometry is exactly mirror symmetric in x, so the true fundamental mode puts exactly half its fissions in each tank. Symmetrising the converged bank, every site with its mirror image, removes the tilt but not the distortion the tilt leaves in the shape *inside* each tank, and it was not enough here. What works is to remove the tilt mode from the problem rather than from the answer: run the half geometry, x > 0 only, with the plane x = 0 reflective. The antisymmetric mode cannot exist in that model, so there is nothing slow left to converge, and its fission bank mirrored is the fundamental mode by construction rather than by waiting. + +The half model has to earn its place, so it is checked against the full one: + +| absorber | half model k (exact symmetric mode) | full model k | difference (pcm) | +|---|---|---|---| +| open | 0.91331 ± 0.00048 | 0.91318 ± 0.00054 | +14 ± 73 | +| 1 mm cadmium sheet | 0.90634 ± 0.00051 | 0.90561 ± 0.00052 | +73 ± 72 | +| 2 cm boron carbide blade | 0.90440 ± 0.00050 | 0.90378 ± 0.00046 | +62 ± 68 | + +The two agree, which says the full model's eigenvalues were never the problem: the tilt integrates to zero over the whole system, so it cannot shift a total production rate, and k survives it. Only the fission matrix, which asks where the neutrons went and not just how many, is sensitive to it. Every matrix below is measured from the half model's bank. + +Two controls say whether that was necessary. Each runs one generation on a whole source, exact or raw, and must return k, because one generation of the fundamental mode is the definition of k. + +| absorber | control on the exact mode | control on the raw bank | k | exact error (pcm) | raw error (pcm) | +|---|---|---|---|---|---| +| open | 0.9140 ± 0.0006 | 0.9118 ± 0.0007 | 0.9133 ± 0.0005 | +66 | -151 | +| 1 mm cadmium sheet | 0.9069 ± 0.0008 | 0.9133 ± 0.0007 | 0.9063 ± 0.0005 | +54 | +700 | +| 2 cm boron carbide blade | 0.9016 ± 0.0008 | 0.9057 ± 0.0008 | 0.9044 ± 0.0005 | -277 | +126 | + +The clearest evidence is not in those columns but in the matrix itself. Measured from the raw bank, the self multiplication k_self came out as 0.8924, 0.9030, 0.8977 for the three absorbers: not monotonic, and for cadmium higher than a tank standing alone, which no absorber placed outside a tank can do. Measured from the exact mode it is 0.8946, 0.8964, 0.8943: flat, as it must be, because an absorber between the tanks barely touches what a tank does to its own neutrons. The absorber now changes only the cross term, which is the gate doing its job and nothing else. A wrong answer that looked reasonable became a right answer that looks obvious, which is the usual shape of this kind of error and the reason the control is in the protocol at all. + +### 3.3 The matrix, and what its eigenvalue tests + +With the source settled, the **dominant eigenvalue of K** tests the two region reduction itself. It equals k only if each tank's internal shape is the same whether the tank is fed from inside or through the gap from its neighbour, which is the assumption every coupled region model makes, this theory's synapse included. Its residual is not an error but a measurement of how good that assumption is. + +**Open.** + +| | born in A | born in B | +|---|---|---| +| fission neutrons in A | 0.8946 ± 0.0007 | 0.0197 ± 0.0002 | +| fission neutrons in B | 0.0195 ± 0.0001 | 0.8941 ± 0.0008 | + +- dominant eigenvalue of K **0.9140 ± 0.0005** against k **0.9133 ± 0.0005**, a residual of +68 pcm + +**1 mm cadmium sheet.** + +| | born in A | born in B | +|---|---|---| +| fission neutrons in A | 0.8964 ± 0.0012 | 0.0116 ± 0.0001 | +| fission neutrons in B | 0.0115 ± 0.0001 | 0.8959 ± 0.0010 | + +- dominant eigenvalue of K **0.9077 ± 0.0008** against k **0.9063 ± 0.0005**, a residual of +138 pcm + +**2 cm boron carbide blade.** + +| | born in A | born in B | +|---|---|---| +| fission neutrons in A | 0.8943 ± 0.0010 | 0.0075 ± 0.0001 | +| fission neutrons in B | 0.0075 ± 0.0001 | 0.8942 ± 0.0008 | + +- dominant eigenvalue of K **0.9018 ± 0.0006** against k **0.9044 ± 0.0005**, a residual of -263 pcm + +The eigenvalue of the reduced two by two matrix sits between -263 and +138 pcm of k, the largest belonging to the most weakly coupled case, which is what one would expect. That residual is the price of describing each tank by a single number: a neutron arriving from the neighbour enters through one wall and fissions nearer that wall than a fundamental mode neutron would, so a tank's internal shape is not quite source independent and no single coupling coefficient can absorb the difference. A third of a percent is small enough that the synapse picture is sound and large enough that it should be quoted rather than rounded away. A photon sector network of many weakly coupled cells inherits both the caveat and, from 3.1, the convergence problem in a sharper form. + +**The neutron budget of one fission neutron born in tank A** (the fan out tax of theory Section 1.3, with real numbers): + +| fate | open | cadmium sheet | boron carbide blade | +|---|---|---|---| +| causes fission neutrons in A (k_AA, self restoring) | 0.8946 | 0.8964 | 0.8943 | +| causes fission neutrons in B (k_BA, the coupling) | 0.0195 | 0.0115 | 0.0075 | +| absorbed in the water | 0.2161 | 0.1998 | 0.1970 | +| absorbed in the aluminium | 0.0025 | 0.0022 | 0.0021 | +| absorbed in the absorber | 0.0000 | 0.0205 | 0.0283 | +| leaks out of the water bath | 0.0075 | 0.0086 | 0.0083 | + +The cadmium sheet cuts the coupling k_BA by a factor 1.7, the boron carbide blade by 2.6. What reaches the next gate is one or two percent of what was born; the rest is the price of an isotropic emitter in a moderating body. The 4π loss of theory Section 1.3 is not a metaphor here, it is a row in this table, and the self restoring term k_AA is the reason the gate has gain at all. + +## 4. The gate: driven, with multiplication on + +A Cf-252 point source (Fröhner Watt spectrum, mean 2.13 MeV) or a 14.1 MeV deuterium tritium point source on the axis of a tank is the SOURCE terminal. Fixed source transport with full multiplication gives the steady fission neutron production in each tank per driver neutron; production in B is the DRAIN current. The point model of coupled regions predicts the same numbers from Section 3 and the first generation response c of each tank to the driver, P = (I - K)⁻¹ c, and the comparison is reported because the gap is physics, not noise. + +| drive | absorber | c_A | c_B | P_A transport | P_B transport | P_A point model | P_B point model | 1 + P_A + P_B | +|---|---|---|---|---|---|---|---|---| +| cfA | open | 1.128 | 0.003 | 15.87 ± 0.10 | 1.491 ± 0.022 | 11.10 | 2.066 | 18.4 | +| cfB | open | 0.003 | 1.128 | 1.51 ± 0.02 | 15.914 ± 0.097 | 2.09 | 11.045 | 18.4 | +| dtB | open | 0.023 | 0.724 | 1.01 ± 0.02 | 8.651 ± 0.041 | 1.55 | 7.127 | 10.7 | +| cfA | 1 mm cadmium sheet | 1.128 | 0.002 | 15.28 ± 0.09 | 0.916 ± 0.014 | 11.03 | 1.234 | 17.2 | +| cfB | 1 mm cadmium sheet | 0.002 | 1.128 | 0.95 ± 0.02 | 15.303 ± 0.084 | 1.25 | 10.974 | 17.3 | +| dtB | 1 mm cadmium sheet | 0.021 | 0.722 | 0.70 ± 0.02 | 8.359 ± 0.045 | 0.99 | 7.047 | 10.1 | +| cfA | 2 cm boron carbide blade | 1.128 | 0.002 | 15.10 ± 0.08 | 0.666 ± 0.013 | 10.73 | 0.774 | 16.8 | +| cfB | 2 cm boron carbide blade | 0.002 | 1.128 | 0.67 ± 0.01 | 15.188 ± 0.066 | 0.77 | 10.717 | 16.9 | +| dtB | 2 cm boron carbide blade | 0.018 | 0.722 | 0.53 ± 0.02 | 8.202 ± 0.049 | 0.66 | 6.874 | 9.7 | + +The transport drain from A exceeds the point model's, and the source multiplication 1 + P_A + P_B = 18.4 exceeds the eigenvalue gain 1/(1 - k) = 11.5: a point source on the axis of a tank has more importance than a fundamental mode neutron, so it earns more than M offspring, and the fission neutrons it makes are clustered at the axis, far from the other tank, so a smaller share of them cross than the fundamental mode would send. Both effects are the adjoint flux of theory Section 11 at work, and they are why a gate's gain is a property of the terminal, not of the medium alone. The fundamental mode arithmetic of Section 3 is exact for what it describes and undercounts a well placed driver. + +### 4.1 The transistor: SOURCE and GATE in, DRAIN out + +| driver in A | absorber | drain: fission neutrons born in B per driver neutron | state | +|---|---|---|---| +| off | any | 0 | LOW | +| on | 2 cm boron carbide blade | 0.666 | LOW | +| on | 1 mm cadmium sheet | 0.916 | LOW | +| on | open | 1.491 | **HIGH** | + +Drain per source neutron with the gate open: **1.49**. Every driver neutron entering tank A becomes 1.5 fission neutrons born in tank B, on the far side of an absorber that a coil could move: the output exceeds the input, and the neutron sector gate has gain with library data and no new physics, as Section 4 of the theory supplement claims. ON/OFF contrast: **1.6** against the cadmium sheet and **2.2** against the boron carbide blade. The contrast is modest and its reason is in the budget above: neutrons cross 3 cm of water still fast or epithermal, a thermal filter stops only the slow ones, and the fast coupling is a property of the geometry that no thin absorber touches. A MOSFET's million to one comes from a barrier; this gate has a filter, and until the coupling path itself is switched (a shutter tank, a longer moderated gap) that is what its contrast will be. + +### 4.2 The coincidence AND: two neutron inputs, one threshold + +Both drivers on means the two point sources at equal strength; the run with both sources present is the superposition check, which fixed source transport obeys exactly. + +| absorber | A alone → P_B | B alone → P_B | sum | both drivers, one run | superposition error | strength ratio for a balanced gate | threshold T | margin each side | +|---|---|---|---|---|---|---|---|---| +| open | 1.491 | 15.914 | 17.406 | 17.295 | 0.6 percent | 0.094 | 2.109 | ×1.41 | +| 1 mm cadmium sheet | 0.916 | 15.303 | 16.219 | 16.129 | 0.6 percent | 0.060 | 1.295 | ×1.41 | +| 2 cm boron carbide blade | 0.666 | 15.188 | 15.854 | 15.804 | 0.3 percent | 0.044 | 0.942 | ×1.41 | + +Superposition holds: two drivers together deliver the sum of what each delivers alone, to 0.6 percent, which is the counting noise. That is the check that this medium is linear while its cross sections are frozen, the property theory Section 5.3 asserts and the whole Green's function reading depends on, measured here in a real geometry rather than in a random walk toy. + +In a linear medium the coincidence AND is a threshold on a weighted sum, a perceptron AND. The two inputs are unequal, a direct drive against a coupled one, so a balanced gate runs the second driver at the strength ratio in the table; each input alone then delivers half the sum, the threshold sits at √2 times one input, and the margin is √2 either side. The weights are nuclear and programmable (the absorber sets k_BA); the threshold is a comparison of rates at the boundary, as in the benchtop cell, and stays there until a nonlinear element exists inside the medium. Section 7 says what the medium offers instead. + +## 5. Level restoration: three inputs, one output + +| drive (open) | the input to tank B | mean energy crossing B's wall | fraction of that below 0.625 eV | mean energy of fission neutrons born in B | +|---|---|---|---|---| +| cfA | the leakage of tank A, through 3 cm of water | 0.251 MeV | 0.67 | 2.011 MeV | +| cfB | a Cf-252 spectrum at the axis, mean 2.13 MeV | 0.089 MeV | 0.73 | 2.008 MeV | +| dtB | a 14.1 MeV line at the axis | 0.116 MeV | 0.72 | 2.009 MeV | + +Read the middle columns with care. When A is driven, everything tank B receives crosses its wall, and it arrives moderated: 0.251 MeV on average, 67% of it already thermal, because 3 cm of water is a moderator as well as a gap. In the other two rows the driver sits inside B, so the wall current is only the return from the water reflector and the input proper is the point source itself: a 2 MeV fission spectrum, and a 14.1 MeV line. + +The three inputs therefore span from thermal to 14 MeV. The fission neutrons born in B differ by a total variation distance of 0.002 and 0.002 between them, which is counting statistics: one output spectrum, whatever went in, because a fission neutron is born from the fission barrier and not from whatever triggered it. This is the self restoring class of theory Section 2, measured (figure 15 C), and it is the property the photon sector cannot buy at any price, since an isomer cascade emits its own lines and not its trigger's. + +## 6. The clock, the switching time, and the step response + +- generation time of the pair Λ = **106.3 µs**; k = 0.9132; prompt time constant Λ/(1 - k) = **1.22 ms**; gain bandwidth product M/τ = 1/Λ = **9.4 kHz** +- step response of the drain (driver in A switched on at t = 0, prompt point kinetics on the measured K and Λ): 10 to 90 percent rise in **3.53 ms** (figure 15 D) +- delayed neutrons, ignored above, add a tail on the second scale carrying about β k/(1 - k) = 0.07 of the prompt step: the theory's remark that precursors slow the approach as k → 1, quantified + +The transistor note's 10 kHz for a thermal assembly is the right order. Ten times the gain costs ten times the switching time, and the product does not move. + +## 7. The veto, priced: the only signal controlled inhibition in the neutron sector + +| state of the pair, open | k | gain M | +|---|---|---| +| 294 K | 0.9132 ± 0.0005 | 11.52 ± 0.07 | +| 350 K, liquid density held | 0.9061 ± 0.0005 | 10.65 ± 0.06 | +| 350 K, liquid expanded (2.4 percent less dense) | 0.8990 ± 0.0005 | 9.90 ± 0.05 | + +- temperature coefficient **-25 pcm/K** in total: -13 pcm/K from Doppler broadening and the thermal scattering law, -13 pcm/K from expansion of the liquid +- gain with the pair 56 K hotter: M falls from 11.52 to 9.90, a modulation of 14 percent +- to halve the gain (1 - k doubled) the liquid would have to warm by about **343 K**, which water at atmospheric pressure cannot do + +Every coupling in the neutron sector is excitatory: a neutron only ever adds fissions, and an absorber that burns up disinhibits. The one channel by which a signal lowers another region's gain is heat, through the density and the Doppler width of the medium, and the table prices it: sign correct, slow (the thermal time constant of tens of litres of liquid), and worth a fraction of the gain, not a veto. Signal controlled inversion, which NAND, signed Ising weights, and Turing completeness all require, is therefore the weak point of the neutron sector as an embodiment, and the theory supplement now says so in Section 4. The rate coded machine is untouched: its NOT is a fixed absorber, its AND a coincidence, and neither needs a signal to invert a signal. + +*The gate exists, with gain, with level restoration, with a programmable weight and a clock, all computed from library data at k below one. What it lacks is a nonlinearity of its own and an inhibition of its own, and the numbers above say by how much.* diff --git a/neutron/tallies.json b/neutron/tallies.json new file mode 100644 index 0000000..c232c05 --- /dev/null +++ b/neutron/tallies.json @@ -0,0 +1,12034 @@ +{ + "runs": { + "alone_A": { + "k": [ + 0.9012729006560118, + 0.0004818403604500704 + ], + "entropy": [ + 10.0399, + 9.9869, + 9.9783, + 9.9723, + 9.9628, + 9.959, + 9.9391, + 9.9387, + 9.9564, + 9.9541, + 9.9555, + 9.9611, + 9.9353, + 9.9471, + 9.9406, + 9.9458, + 9.9459, + 9.9618, + 9.9545, + 9.9487, + 9.9429, + 9.9333, + 9.9358, + 9.9409, + 9.9378, + 9.9495, + 9.9432, + 9.944, + 9.9464, + 9.9431, + 9.9399, + 9.9449, + 9.952, + 9.9437, + 9.9379, + 9.9421, + 9.9642, + 9.9553, + 9.9481, + 9.9489, + 9.9451, + 9.9476, + 9.9472, + 9.953, + 9.9344, + 9.9333, + 9.9572, + 9.948, + 9.9526, + 9.9411, + 9.9481, + 9.9514, + 9.9387, + 9.9302, + 9.9332, + 9.9412, + 9.9384, + 9.9345, + 9.9371, + 9.9447, + 9.9338, + 9.944, + 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240.22488679628626, + 385.154666402462, + 617.521852252507, + 990.0781978606437, + 1587.4010519682017, + 2545.0940190730557, + 4080.5715467367395, + 6542.41612421925, + 10489.513111631151, + 16817.928305074292, + 26964.33184882471, + 43232.14957660213, + 69314.48431551465, + 111132.52019571123, + 178179.74362806836, + 285677.14457886684, + 458028.6696623073, + 734361.3803683419, + 1177408.0373049495, + 1887748.6253633897, + 3026643.915832549, + 4852644.716656989, + 7780287.804230059, + 12474203.625266097, + 20000000.000000004 + ] + }, + "wall_clock_note": "72.5 min for the first full pass, then 13 and 18 min for the fission matrix reruns of Section 3.2; the committed tallies are the union" + } +} \ No newline at end of file diff --git a/theory/THEORY.md b/theory/THEORY.md index f69820b..affdf91 100644 --- a/theory/THEORY.md +++ b/theory/THEORY.md @@ -109,7 +109,9 @@ Every keystone inequality is satisfied simultaneously, by library data, with mar **What this does and does not claim.** It does not claim a practical computer: a neutron gate is centimeters to meters of moderated or fast assembly, wrapped in licensing, shielding, and fissile material control that place it outside individual reach, and its energy per operation is grotesque by Section 5 of the README. It claims something more important for the theory: existence. The model of computation defined by the two governing equations is physically complete at one scale, with 1940s physics; nature has run self restoring triggered release networks (natural reactors at Oklo, two billion years ago) without human help. The keystone program is therefore a *miniaturization* program, from meter scale neutron gates toward crystal scale photon gates, and not a search for whether the model can exist at all. A theory whose central risk is "can this be shrunk" is in a categorically stronger position than one whose central risk is "can this exist." -The simulation counterpart is specified in the README roadmap (Phase B2): reproduce the two region NAND in OpenMC with ENDF data, all inputs public, all outputs subcritical. It requires no laboratory and settles the architecture questions (coupling matrices, absorber programming, clock rates) that the isomer sector will inherit when its own gate arrives. +**Computed.** Phase B2 has now been carried out ([/neutron](../neutron/), numbers in [`neutron/results.md`](../neutron/results.md)). Two 72 litre tanks of 4.9 percent enriched uranyl fluoride solution (the LEU-SOL-THERM-002 benchmark composition), each at k ≈ 0.901 alone, coupled through 3 cm of water to k = 0.9132, in OpenMC with the official ENDF/B-VIII.0 library, the same code and data first shown to reproduce three handbook criticals. The fission matrix of the pair, measured one generation at a time from the converged fission bank, has dominant eigenvalue 0.9140 against a transport k of 0.9133, and a control that runs one generation on the whole source returns 0.9140: the identification of coupled region kinetics with a Green's function carrying gain on the diagonal is confirmed to between -263 and +138 pcm across the three absorber states, the residual being the measured cost of describing each tank by a single number rather than a shape. Obtaining that source is itself a result. A gate is built to couple its regions weakly, weak coupling drives the dominance ratio toward one (0.957 to 0.983 here), the tilt between the regions then mixes more slowly than any affordable run, and the Shannon entropy that is normally trusted to detect this is nearly blind to it, because entropy watches the shape inside each region. The eigenvalues survive the tilt, since a mode that integrates to zero cannot shift a total production rate, but the fission matrix does not, and the first version of this calculation produced a plausible and wrong one. The remedy is to remove the mode rather than average it away, by running the mirror symmetric half geometry with a reflective plane. Any network of many weakly coupled cells, in this sector or the photon one, inherits the problem in a sharper form, and this is the first place the theory has had to say so. A Cf-252 driver neutron in one tank becomes 1.49 fission neutrons in the other with the coupling open, so the gate has gain across the synapse and not merely within a region. Two drivers superpose to 0.6 percent, which is the linearity of Section 5.3 measured in a real geometry. The emission spectrum of the driven tank is the same to a total variation distance of 0.002 whether its input arrived moderated from the neighbour, as a 2 MeV fission spectrum, or as a 14.1 MeV line: level restoration, measured. The generation time is 106 µs, the prompt switching time Λ/(1 - k) is 1.22 ms, and M/τ = 1/Λ = 9.4 kHz. + +**Two corrections this section keeps.** First, "local absorbers program thresholds and vetoes" claims too much. A 1 mm cadmium sheet between the tanks cuts the coupling by 1.7× and a 2 cm boron carbide blade by 2.6×, because what crosses 3 cm of water is still largely fast and a thin absorber is a filter rather than a barrier. An absorber therefore programs a *weight*; the medium stays linear, its AND is a perceptron AND, and the threshold remains a comparison of rates at the boundary. Second, "adding a veto absorber yields NAND" holds only if something other than a neutron signal moves the absorber. Every coupling in this sector is excitatory, an absorber that burns up disinhibits, and the one channel by which a neutron signal lowers a neighbour's gain is heat: -25 pcm/K for this pair, a 14 percent modulation of the gain over 56 K, with roughly 343 K needed to halve it. Signal controlled inversion is thus the neutron sector's weak point as an embodiment, and the keystone search of Section 2 inherits it: the compact state must supply an inhibition as well as a gain, or the machine built from it is a linear amplifier network read by comparators, which is universal for continuous functions (Section 6) and not for logic. --- diff --git a/transistor/README.md b/transistor/README.md index 48dcbc7..bc0bae5 100644 --- a/transistor/README.md +++ b/transistor/README.md @@ -87,6 +87,8 @@ a constant gain bandwidth product, exactly the signature behavior of an operatio **Why it is in this document.** Not as a proposal; nobody wants this computer. It is here because a skeptical reader's strongest objection to the crystal cell ("no such device has ever existed") is answered by pointing at this one: the same pinout, with the gain socket filled, has operated on Earth since 1942, and operated *unattended* at Oklo two billion years before that. The crystal cell is that device shrunk by seven orders of magnitude, minus, so far, its gain. +**Computed, not cited.** The device above is now built in [/neutron](../neutron/): two solution tanks in OpenMC with ENDF/B-VIII.0, an absorber between them as the GATE, a Cf-252 point source as the SOURCE, the fission rate in the far tank as the DRAIN. Its measured pinout is M = 11.5 for the pair, a drain of 1.49 fission neutrons in the far tank per driver neutron with the gate open against 0.67 with a boron carbide blade in, a generation time of 106 µs, and a gain bandwidth product 1/Λ of 9.4 kHz, which is the theorem above checked against a real light water assembly rather than a textbook figure. The same calculation returns the bad news the table above cannot show: the gate's ON/OFF contrast is 2.2, because an absorber in a moderated gap programs a weight and not a threshold, and the sector has no signal controlled inversion. The GATE terminal is real, and it is an analog knob. + --- ## The datasheet