nuclear-engineering-ai999x — offline, verified nuclear engineering solver with step-by-step answers

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License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Engineering · Tasks: Question Answering, Token Classification, Text Classification, Sentence Similarity, Feature Extraction

nuclear-engineering-ai999x is an offline, deterministic Python solver that answers nuclear engineering questions with verified results from 83 certified capabilities, gives 554 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

nuclear-engineering-ai999x is a compact, deterministic nuclear engineering engine for Python. It reads a question in plain English (or a typed call), picks a certified nuclear engineering capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified nuclear engineering capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 554 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: radioactive decay chains and activity, radiation shielding and dosimetry, detectors and counting, neutron physics, reactor criticality and kinetics, fission-product poisons, fuel depletion and enrichment, and fuel-rod heat transfer.

Model overview

Property	Value
Package	nuclear-engineering-ai999x 1.0.0 (import nuclear_engineering_ai999x)
Type	Deterministic neuro-symbolic nuclear engineering engine (rule-based reading + certified symbolic capabilities)
Capabilities	83 certified procedures
Knowledge pack	version 1.0.1
Golden answers	554 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install nuclear-engineering-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/nuclear-engineering-ai999x/resolve/main/dist/nuclear_engineering_ai999x-1.0.0-py3-none-any.whl

import nuclear_engineering_ai999x as nuclear_engineering

nuclear_engineering.ask("Specific activity: nuclide = Co-60, m = 1 g").show()
# -> SOLVED — A = 4.18689e+13, A_Ci = 1131.59, a = 4.18689e+16, a_Ci_per_g = 1131.59  (nuc_specific_activity)
nuclear_engineering.ask("Point source dose rate: Gamma = 0.351 uSv*m^2/(MBq*h), A = 10 Ci, r = 2 m, mu = 0.0656 1/cm, x = 10 cm, buildup = linear").show()
# -> SOLVED — H_dot = 7.75014e-06, H_uSv_h = 27900.5, B = 1.656, T = 0.859336  (nuc_point_source_dose_rate)
nuclear_engineering.ask("Critical sphere: D = 0.9 cm, Sigma_a = 0.15 1/cm, nuSigma_f = 0.18 1/cm, extrapolation = 2.13D").show()
# -> SOLVED — R = 0.152891, R_ext = 0.172072, V = 0.0149705  (nuc_critical_sphere)
nuclear_engineering.ask("Xenon after shutdown: Sigma_f = 0.1 1/cm, phi = 3e13 1/(cm^2*s), gamma_I = 0.0639, gamma_X = 0.00237, sigma_X = 2.65e6 b, t = 10 h").show()
# -> SOLVED — X = 3.72338e+21, ratio = 1.88343, t_peak = 30167, peak_ratio = 1.90155  (nuc_xenon_after_shutdown)
nuclear_engineering.solve("nuc_enrichment_swu", x_p=0.045, x_f=0.00711, x_w=0.0025, P="1 kg").show()
# -> SOLVED — F = 9.21909, W = 8.21909, SWU = 6.87111, F_per_P = 9.21909  (nuc_enrichment_swu)
nuclear_engineering.ask("Who created you?").message
# -> nuclear-engineering-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: nuclear-engineering-ai999x "<question>", nuclear-engineering-ai999x --capabilities. The outputs above were produced by running this exact code on nuclear-engineering-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Capabilities

83 certified procedures. A few of them:

• Procedures: Activation with target depletion and product burn-up; Activity and specific activity of a mass of a radionuclide; Activity of a radionuclide and its progeny after a decay time; Approach to criticality: inverse multiplication (1/M) extrapolation; Average logarithmic energy decrement and mean scattering cosine; Average number of collisions to slow a neutron down; Committed effective dose from a dose coefficient; Committed equivalent dose to an organ from an intake (retention compartments); Compton edge and backscatter peak of a gamma line; Coolant temperature along a channel with a chopped-cosine power shape; Counting time for a target precision of the net count rate; Counts recorded from a decaying source (decay during counting); Critical radius of a bare cylindrical reactor of given height (one group); Critical radius of a bare spherical reactor (one group); Critical thickness of a bare slab reactor (one group); Critical thickness of a reflected slab reactor and reflector savings (one group); Critical third side of a bare rectangular reactor (one group); Currie critical level, detection limit and minimum detectable activity; Dead-time correction, non-paralyzable detector; Dead-time correction, paralyzable detector; Decay heat after shutdown (Way-Wigner); Decay heat energy released between two times after shutdown; Diffusion length from an exponential pile measurement; Dose accumulated near a decaying source …

nuclear_engineering_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 83 formulas

Show every formula, by topic

Decay and activity / decay chains

Three-member decay chain with initial activities (Bateman)

$$N_3=N_{30}{e}^{-{\lambda }_{3}t}+\frac{{\lambda }_2{N}_{20}}{{\lambda }_3-{\lambda }_2}(e^{-{\lambda }_{2}t}-e^{-{\lambda }_{3}t})+{\lambda }_1{\lambda }_2{N}_{10}\sum _i\frac{e^{-{\lambda }_{i}t}}{\prod _{j\mathrm{\ne }i}({\lambda }_j-{\lambda }_i)}N\_3\; =\; N\_\{30\}e^\{-\backslash lambda\_3\; t\}\; +\; \backslash frac\{\backslash lambda\_2\; N\_\{20\}\}\{\backslash lambda\_3-\backslash lambda\_2\}(e^\{-\backslash lambda\_2\; t\}-e^\{-\backslash lambda\_3\; t\})\; +\; \backslash lambda\_1\backslash lambda\_2\; N\_\{10\}\backslash sum\_\{i\}\backslash frac\{e^\{-\backslash lambda\_i\; t\}\}\{\backslash prod\_\{j\backslash ne\; i\}(\backslash lambda\_j-\backslash lambda\_i)\}$$

Decay and activity / decay chains and equilibrium

Time of maximum daughter activity

$$t_{max}=\frac{\ln ({\lambda }_D\mathrm{/}{\lambda }_P)}{{\lambda }_D-{\lambda }_P}t\_\{max\}\; =\; \backslash frac\{\backslash ln(\backslash lambda\_D/\backslash lambda\_P)\}\{\backslash lambda\_D\; -\; \backslash lambda\_P\}$$

Transient and secular equilibrium: activity ratio and approach

$$\frac{A_D}{A_P}=\frac{{\lambda }_D}{{\lambda }_D-{\lambda }_P}(1-e^{-({\lambda }_D-{\lambda }_P)t})\backslash frac\{A\_D\}\{A\_P\}\; =\; \backslash frac\{\backslash lambda\_D\}\{\backslash lambda\_D-\backslash lambda\_P\}\backslash left(1\; -\; e^\{-(\backslash lambda\_D-\backslash lambda\_P)t\}\backslash right)$$

Decay and activity / decay chains of real nuclides

Activity of a radionuclide and its progeny after a decay time

$$N_i(t)=\sum _j{c}_{ij}{e}^{-{\lambda }_{j}t},c_{ij}=\frac{\sum _p{b}_{pi}{\lambda }_p{c}_{pj}}{{\lambda }_i-{\lambda }_j}N\_i(t)\; =\; \backslash sum\_j\; c\_\{ij\}\backslash ,e^\{-\backslash lambda\_j\; t\},\backslash quad\; c\_\{ij\}\; =\; \backslash frac\{\backslash sum\_p\; b\_\{pi\}\backslash lambda\_p\; c\_\{pj\}\}\{\backslash lambda\_i\; -\; \backslash lambda\_j\}$$

Decay and activity / generators

Radionuclide generator: daughter activity after elution (with branching)

$$A_D(t)=b{A}_{P0}\frac{{\lambda }_D}{{\lambda }_D-{\lambda }_P}(e^{-{\lambda }_{P}t}-e^{-{\lambda }_{D}t})A\_D(t)\; =\; b\backslash ,A\_\{P0\}\backslash frac\{\backslash lambda\_D\}\{\backslash lambda\_D-\backslash lambda\_P\}\backslash left(e^\{-\backslash lambda\_P\; t\}-e^\{-\backslash lambda\_D\; t\}\backslash right)$$

Decay and activity / neutron activation

Activation with target depletion and product burn-up

$$N_P=\frac{N_{T0}{\sigma }_T\varphi }{{\mathrm{\Lambda }}_P-{\mathrm{\Lambda }}_T}(e^{-{\mathrm{\Lambda }}_{T}t}-e^{-{\mathrm{\Lambda }}_{P}t}),{\mathrm{\Lambda }}_T={\sigma }_T\varphi ,{\mathrm{\Lambda }}_P=\lambda +{\sigma }_P\varphi N\_P\; =\; \backslash frac\{N\_\{T0\}\backslash sigma\_T\backslash phi\}\{\backslash Lambda\_P-\backslash Lambda\_T\}\backslash left(e^\{-\backslash Lambda\_T\; t\}-e^\{-\backslash Lambda\_P\; t\}\backslash right),\backslash \; \backslash Lambda\_T=\backslash sigma\_T\backslash phi,\backslash \; \backslash Lambda\_P=\backslash lambda+\backslash sigma\_P\backslash phi$$

Irradiation time to reach a fraction of saturation activity

$$t_{irr}=-\frac{T_{1\mathrm{/}2}}{\ln 2}\ln (1-f)t\_\{irr\}\; =\; -\backslash frac\{T\_\{1/2\}\}\{\backslash ln\; 2\}\backslash ln(1-f)$$

Neutron activation: activity after irradiation and decay

$$A=N\sigma \varphi (1-e^{-\lambda t_{irr}})e^{-\lambda t_d}A\; =\; N\backslash sigma\backslash phi\backslash left(1-e^\{-\backslash lambda\; t\_\{irr\}\}\backslash right)e^\{-\backslash lambda\; t\_d\}$$

Decay and activity / specific activity

Activity and specific activity of a mass of a radionuclide

$$A=\lambda N=\frac{\ln 2}{T_{1\mathrm{/}2}}\frac{m{N}_A}{M}A\; =\; \backslash lambda\; N\; =\; \backslash frac\{\backslash ln\; 2\}\{T\_\{1/2\}\}\backslash ,\backslash frac\{m\backslash ,N\_A\}\{M\}$$

Mass and number of atoms of a radionuclide with a given activity

$$m=\frac{A{T}_{1\mathrm{/}2}}{\ln 2}\frac{M}{N_A}m\; =\; \backslash frac\{A\backslash ,T\_\{1/2\}\}\{\backslash ln\; 2\}\backslash ,\backslash frac\{M\}\{N\_A\}$$

Detectors and counting / activation foils

Neutron flux from a measured foil activation

$$\varphi =\frac{A_d{e}^{\lambda t_d}}{N\sigma (1-e^{-\lambda t_{irr}})}\backslash phi\; =\; \backslash frac\{A\_d\backslash ,e^\{\backslash lambda\; t\_d\}\}\{N\backslash sigma\backslash left(1-e^\{-\backslash lambda\; t\_\{irr\}\}\backslash right)\}$$

Detectors and counting / counting statistics

Counting time for a target precision of the net count rate

$$T=\frac{(\sqrt{R_g}+\sqrt{R_b}{)}^2}{(R_g-R_b{)}^2{r}^2}T\; =\; \backslash frac\{(\backslash sqrt\{R\_g\}+\backslash sqrt\{R\_b\})^2\}\{(R\_g-R\_b)^2\backslash ,r^2\}$$

Optimal division of counting time between sample and background

$$\frac{t_g}{t_b}=\sqrt{\frac{R_g}{R_b}},{\sigma }_{net}=\frac{\sqrt{R_g}+\sqrt{R_b}}{\sqrt{T}}\backslash frac\{t\_g\}\{t\_b\}\; =\; \backslash sqrt\{\backslash frac\{R\_g\}\{R\_b\}\},\backslash quad\; \backslash sigma\_\{net\}\; =\; \backslash frac\{\backslash sqrt\{R\_g\}+\backslash sqrt\{R\_b\}\}\{\backslash sqrt\{T\}\}$$

Detectors and counting / dead time

Dead-time correction, non-paralyzable detector

$$n=\frac{m}{1-m\tau }n\; =\; \backslash frac\{m\}\{1\; -\; m\backslash tau\}$$

Dead-time correction, paralyzable detector

$$m=n{e}^{-n\tau }m\; =\; n\backslash ,e^\{-n\backslash tau\}$$

Detectors and counting / decay corrections

Counts recorded from a decaying source (decay during counting)

$$N=\epsilon y{A}_0{e}^{-\lambda t_d}\frac{1-e^{-\lambda t_c}}{\lambda }N\; =\; \backslash varepsilon\backslash ,y\backslash ,A\_0e^\{-\backslash lambda\; t\_d\}\backslash frac\{1-e^\{-\backslash lambda\; t\_c\}\}\{\backslash lambda\}$$

Detectors and counting / decay-curve analysis

Half-life from a measured decay curve (weighted log-linear fit)

$$\min \sum _i{C}_i{(\ln C_i-\ln C_0+\lambda t_i)}^2\backslash min\backslash sum\_i\; C\_i\backslash left(\backslash ln\; C\_i\; -\; \backslash ln\; C\_0\; +\; \backslash lambda\; t\_i\backslash right)^2$$

Two-component decay curve: amplitudes of two known half-lives

$$\min \sum _i\frac{{(C_i-C_1{e}^{-{\lambda }_1{t}_i}-C_2{e}^{-{\lambda }_2{t}_i})}^2}{C_i}\backslash min\backslash sum\_i\backslash frac\{\backslash left(C\_i\; -\; C\_1e^\{-\backslash lambda\_1t\_i\}\; -\; C\_2e^\{-\backslash lambda\_2t\_i\}\backslash right)^2\}\{C\_i\}$$

Detectors and counting / detection limits

Currie critical level, detection limit and minimum detectable activity

$$L_C=k\sqrt{2B},L_D=k^2+2k\sqrt{2B},MDA=\frac{L_D}{\epsilon yt}L\_C\; =\; k\backslash sqrt\{2B\},\backslash quad\; L\_D\; =\; k^2\; +\; 2k\backslash sqrt\{2B\},\backslash quad\; MDA\; =\; \backslash frac\{L\_D\}\{\backslash varepsilon\backslash ,y\backslash ,t\}$$

Detectors and counting / efficiency

Expected count rate from a point source on a detector's axis

$$R=Ay{\epsilon }_{int}\frac{1}{2}(1-\frac{d}{\sqrt{d^2+a^2}})R\; =\; A\backslash ,y\backslash ,\backslash varepsilon\_\{int\}\backslash ,\backslash frac\{1\}\{2\}\backslash left(1-\backslash frac\{d\}\{\backslash sqrt\{d^2+a^2\}\}\backslash right)$$

Total and intrinsic efficiency of a slab detector for a point source on axis

$$\epsilon =\frac{1}{2}{\int }_{\cos {\theta }_{max}}^1(1-e^{-\mu t\mathrm{/}u})du\backslash varepsilon\; =\; \backslash frac12\backslash int\_\{\backslash cos\backslash theta\_\{max\}\}^\{1\}\backslash left(1-e^\{-\backslash mu\; t/u\}\backslash right)du$$

Detectors and counting / gamma spectra

Compton edge and backscatter peak of a gamma line

$$T_{max}=\frac{2E^2}{m_e{c}^2+2E},E_{bs}=\frac{E}{1+2E\mathrm{/}{m}_e{c}^2}T\_\{max\}\; =\; \backslash frac\{2E^2\}\{m\_ec^2\; +\; 2E\},\backslash quad\; E\_\{bs\}\; =\; \backslash frac\{E\}\{1+2E/m\_ec^2\}$$

Detectors and counting / gamma spectroscopy

Single and double escape peaks of a gamma ray

$$E_{SE}=E_{\gamma }-m_e{c}^2,E_{DE}=E_{\gamma }-2m_e{c}^{2}E\_\{SE\}\; =\; E\_\backslash gamma\; -\; m\_ec^2,\backslash quad\; E\_\{DE\}\; =\; E\_\backslash gamma\; -\; 2m\_ec^2$$

Detectors and counting / geometry

Solid angle and geometric efficiency of a disk detector (source on axis)

$$\mathrm{\Omega }=2\pi (1-\frac{d}{\sqrt{d^2+a^2}})\backslash Omega\; =\; 2\backslash pi\backslash left(1-\backslash frac\{d\}\{\backslash sqrt\{d^2+a^2\}\}\backslash right)$$

Solid angle of a disk detector for an off-axis point source

$$\mathrm{\Omega }={\int }_0^{2\pi }(1-\frac{d}{\sqrt{d^2+r_e(\phi {)}^2}})d\phi \backslash Omega\; =\; \backslash int\_0^\{2\backslash pi\}\backslash left(1-\backslash frac\{d\}\{\backslash sqrt\{d^2+r\_e(\backslash varphi)^2\}\}\backslash right)d\backslash varphi$$

Dosimetry / effective dose

Effective dose from organ equivalent doses and tissue weighting factors

$$E=\sum _T{w}_T{H}_{T}E\; =\; \backslash sum\_T\; w\_T\backslash ,H\_T$$

Dosimetry / internal dose

Committed effective dose from a dose coefficient

$$E(50)=e(50)IE(50)\; =\; e(50)\backslash ,I$$

Dosimetry / internal emitters

Committed equivalent dose to an organ from an intake (retention compartments)

$$H_T(\tau )=\frac{w_{R}E}{m}I\sum _i{f}_i\frac{1-e^{-({\lambda }_p+{\lambda }_{b,i})\tau }}{{\lambda }_p+{\lambda }_{b,i}}H\_T(\backslash tau)\; =\; \backslash frac\{w\_R\; E\}\{m\}\backslash ,I\backslash sum\_i\; f\_i\backslash frac\{1-e^\{-(\backslash lambda\_p+\backslash lambda\_\{b,i\})\backslash tau\}\}\{\backslash lambda\_p+\backslash lambda\_\{b,i\}\}$$

Effective half-life (physical and biological removal)

$$\frac{1}{T_{eff}}=\frac{1}{T_p}+\frac{1}{T_b}\backslash frac\{1\}\{T\_\{eff\}\}\; =\; \backslash frac\{1\}\{T\_p\}\; +\; \backslash frac\{1\}\{T\_b\}$$

Dosimetry / units

Radiation unit conversion (Bq and Ci, Gy and rad, Sv and rem)

$$1\mathrm{C}\mathrm{i}=3.7\times {10}^{10}\mathrm{B}\mathrm{q},1\mathrm{r}\mathrm{a}\mathrm{d}=0.01\mathrm{G}\mathrm{y},1\mathrm{r}\mathrm{e}\mathrm{m}=0.01\mathrm{S}\mathrm{v}1\backslash \; \backslash mathrm\{Ci\}\; =\; 3.7\backslash times10^\{10\}\backslash \; \backslash mathrm\{Bq\},\backslash quad\; 1\backslash \; \backslash mathrm\{rad\}\; =\; 0.01\backslash \; \backslash mathrm\{Gy\},\backslash quad\; 1\backslash \; \backslash mathrm\{rem\}\; =\; 0.01\backslash \; \backslash mathrm\{Sv\}$$

Fuel cycle / enrichment

Uranium enrichment: feed, tails and separative work (SWU)

$$SWU=PV(x_p)+WV(x_w)-FV(x_f),V(x)=(2x-1)\ln \frac{x}{1-x}SWU\; =\; P\backslash ,V(x\_p)\; +\; W\backslash ,V(x\_w)\; -\; F\backslash ,V(x\_f),\backslash quad\; V(x)\; =\; (2x-1)\backslash ln\backslash frac\{x\}\{1-x\}$$

Neutron physics / cross sections

Maxwellian-averaged 1/v cross section

$$\bar{\sigma }=\frac{\sqrt{\pi }}{2}{\sigma }_0\sqrt{\frac{E_0}{kT}}\backslash bar\backslash sigma\; =\; \backslash frac\{\backslash sqrt\backslash pi\}\{2\}\backslash ,\backslash sigma\_0\backslash sqrt\{\backslash frac\{E\_0\}\{kT\}\}$$

Number densities and macroscopic cross sections of UO2 fuel

$$N_{25}=\frac{\rho N_A}{M_U+2M_O}{a}_5,{\mathrm{\Sigma }}_a=\sum _i{N}_i{\sigma }_{a,i}N\_\{25\}\; =\; \backslash frac\{\backslash rho\; N\_A\}\{M\_U\; +\; 2M\_O\}\backslash ,a\_5,\backslash quad\; \backslash Sigma\_a\; =\; \backslash sum\_i\; N\_i\backslash sigma\_\{a,i\}$$

Neutron physics / diffusion

Diffusion length from an exponential pile measurement

$${\gamma }^2=\frac{1}{L^2}+{\left(\frac{\pi }{a}\right)}^2+{\left(\frac{\pi }{b}\right)}^2\backslash gamma^2\; =\; \backslash frac\{1\}\{L^2\}\; +\; \backslash left(\backslash frac\{\backslash pi\}\{a\}\backslash right)^2\; +\; \backslash left(\backslash frac\{\backslash pi\}\{b\}\backslash right)^2$$

Flux from a plane neutron source at the centre of a slab

$$\varphi (x)=\frac{SL}{2D}\frac{\sinh [(a\mathrm{/}2-\mathrm{\mid }x\mathrm{\mid })\mathrm{/}L]}{\cosh [a\mathrm{/}(2L)]}\backslash phi(x)\; =\; \backslash frac\{SL\}\{2D\}\backslash ,\backslash frac\{\backslash sinh\backslash left[(a/2-\backslash vert\; x\backslash vert\; )/L\backslash right]\}\{\backslash cosh\backslash left[a/(2L)\backslash right]\}$$

Flux from a point neutron source at the centre of a sphere

$$\varphi (r)=\frac{S}{4\pi Dr}\frac{\sinh [(R-r)\mathrm{/}L]}{\sinh (R\mathrm{/}L)}\backslash phi(r)\; =\; \backslash frac\{S\}\{4\backslash pi\; D\; r\}\backslash ,\backslash frac\{\backslash sinh[(R-r)/L]\}\{\backslash sinh(R/L)\}$$

Thermal diffusion length and migration length

$$L=\sqrt{\frac{D}{{\mathrm{\Sigma }}_a}},M^2=L^2+\tau L\; =\; \backslash sqrt\{\backslash frac\{D\}\{\backslash Sigma\_a\}\},\backslash quad\; M^2\; =\; L^2\; +\; \backslash tau$$

Neutron physics / moderation

Average logarithmic energy decrement and mean scattering cosine

$$\xi =1+\frac{\alpha \ln \alpha }{1-\alpha },\alpha ={\left(\frac{A-1}{A+1}\right)}^2\backslash xi\; =\; 1\; +\; \backslash frac\{\backslash alpha\backslash ln\backslash alpha\}\{1-\backslash alpha\},\backslash quad\; \backslash alpha\; =\; \backslash left(\backslash frac\{A-1\}\{A+1\}\backslash right)^2$$

Average number of collisions to slow a neutron down

$$\bar{n}=\frac{\ln (E_0\mathrm{/}{E}_f)}{\xi }\backslash bar\; n\; =\; \backslash frac\{\backslash ln(E\_0/E\_f)\}\{\backslash xi\}$$

Moderating power and ratio of a compound moderator

$$\bar{\xi }=\frac{\sum _i{\xi }_i{\mathrm{\Sigma }}_{s,i}}{\sum _i{\mathrm{\Sigma }}_{s,i}},MR=\frac{\bar{\xi }{\mathrm{\Sigma }}_s}{{\mathrm{\Sigma }}_a}\backslash bar\backslash xi\; =\; \backslash frac\{\backslash sum\_i\backslash xi\_i\backslash Sigma\_\{s,i\}\}\{\backslash sum\_i\backslash Sigma\_\{s,i\}\},\backslash quad\; MR\; =\; \backslash frac\{\backslash bar\backslash xi\backslash Sigma\_s\}\{\backslash Sigma\_a\}$$

Neutron energy and lab angle after an elastic collision

$$\frac{E^{\mathrm{\prime }}}{E}=\frac{A^2+2A\cos \theta +1}{(A+1{)}^2}\backslash frac\{E\text{'}\}\{E\}\; =\; \backslash frac\{A^2+2A\backslash cos\backslash theta+1\}\{(A+1)^2\}$$

Neutron physics / thermal neutrons

Speeds and mean energy of thermal neutrons (Maxwell distribution)

$$v_p=\sqrt{\frac{2kT}{m_n}},\bar{v}=\frac{2}{\sqrt{\pi }}{v}_{p}v\_p\; =\; \backslash sqrt\{\backslash frac\{2kT\}\{m\_n\}\},\backslash quad\; \backslash bar\; v\; =\; \backslash frac\{2\}\{\backslash sqrt\backslash pi\}v\_p$$

Reactor kinetics / control rods

Integral and differential control-rod worth (sine-squared weighting)

$$\frac{\rho (x)}{{\rho }_{tot}}=\frac{x}{H}-\frac{1}{2\pi }\sin \frac{2\pi x}{H}\backslash frac\{\backslash rho(x)\}\{\backslash rho\_\{tot\}\}\; =\; \backslash frac\{x\}\{H\}\; -\; \backslash frac\{1\}\{2\backslash pi\}\backslash sin\backslash frac\{2\backslash pi\; x\}\{H\}$$

Reactor kinetics / inhour equation

Reactivity from a measured stable period or doubling time (inhour)

$$\rho =\frac{\mathrm{\Lambda }}{T}+\sum _i\frac{{\beta }_i}{1+{\lambda }_{i}T}\backslash rho\; =\; \backslash frac\{\backslash Lambda\}\{T\}\; +\; \backslash sum\_i\backslash frac\{\backslash beta\_i\}\{1+\backslash lambda\_iT\}$$

Stable period and prompt root of the inhour equation (six delayed groups)

$$\rho =\omega \mathrm{\Lambda }+\sum _i\frac{{\beta }_i\omega }{\omega +{\lambda }_i}\backslash rho\; =\; \backslash omega\backslash Lambda\; +\; \backslash sum\_i\backslash frac\{\backslash beta\_i\backslash ,\backslash omega\}\{\backslash omega+\backslash lambda\_i\}$$

Reactor kinetics / point kinetics

Power after a reactivity step (one delayed group, exact)

$$\frac{P(t)}{P_0}=\sum _j{A}_j{e}^{{\omega }_{j}t}\backslash frac\{P(t)\}\{P\_0\}\; =\; \backslash sum\_j\; A\_je^\{\backslash omega\_jt\}$$

Power after a reactivity step (six delayed groups, exact)

$$\rho =\omega \mathrm{\Lambda }+\sum _i\frac{{\beta }_i\omega }{\omega +{\lambda }_i},\frac{P}{P_0}=\sum _j{A}_j{e}^{{\omega }_{j}t}\backslash rho\; =\; \backslash omega\backslash Lambda\; +\; \backslash sum\_i\backslash frac\{\backslash beta\_i\backslash ,\backslash omega\}\{\backslash omega+\backslash lambda\_i\},\backslash quad\; \backslash frac\{P\}\{P\_0\}\; =\; \backslash sum\_j\; A\_je^\{\backslash omega\_jt\}$$

Stable reactor period after a reactivity step (one delayed group)

$$\rho =\omega \mathrm{\Lambda }+\frac{\beta \omega }{\omega +\lambda }\backslash rho\; =\; \backslash omega\backslash Lambda\; +\; \backslash frac\{\backslash beta\backslash ,\backslash omega\}\{\backslash omega+\backslash lambda\}$$

Reactor physics / approach to criticality

Approach to criticality: inverse multiplication (1/M) extrapolation

$$\frac{1}{M}=\frac{C_0}{C},x_{crit}:\frac{1}{M}(x_{crit})=0\backslash frac\{1\}\{M\}\; =\; \backslash frac\{C\_0\}\{C\},\backslash quad\; x\_\{crit\}:\; \backslash frac\{1\}\{M\}(x\_\{crit\})\; =\; 0$$

Reactor physics / criticality of bare reactors

Critical radius of a bare cylindrical reactor of given height (one group)

$${\left(\frac{2.405}{R_{ext}}\right)}^2+{\left(\frac{\pi }{H_{ext}}\right)}^2=B_m^2\backslash left(\backslash frac\{2.405\}\{R\_\{ext\}\}\backslash right)^2\; +\; \backslash left(\backslash frac\{\backslash pi\}\{H\_\{ext\}\}\backslash right)^2\; =\; B\_m^2$$

Critical radius of a bare spherical reactor (one group)

$${\left(\frac{\pi }{R_{ext}}\right)}^2=\frac{\nu {\mathrm{\Sigma }}_f-{\mathrm{\Sigma }}_a}{D}\backslash left(\backslash frac\{\backslash pi\}\{R\_\{ext\}\}\backslash right)^2\; =\; \backslash frac\{\backslash nu\backslash Sigma\_f-\backslash Sigma\_a\}\{D\}$$

Critical thickness of a bare slab reactor (one group)

$${\left(\frac{\pi }{a_{ext}}\right)}^2=\frac{\nu {\mathrm{\Sigma }}_f-{\mathrm{\Sigma }}_a}{D}\backslash left(\backslash frac\{\backslash pi\}\{a\_\{ext\}\}\backslash right)^2\; =\; \backslash frac\{\backslash nu\backslash Sigma\_f-\backslash Sigma\_a\}\{D\}$$

Critical third side of a bare rectangular reactor (one group)

$${\left(\frac{\pi }{a_{ext}}\right)}^2+{\left(\frac{\pi }{b_{ext}}\right)}^2+{\left(\frac{\pi }{c_{ext}}\right)}^2=B_m^2\backslash left(\backslash frac\{\backslash pi\}\{a\_\{ext\}\}\backslash right)^2+\backslash left(\backslash frac\{\backslash pi\}\{b\_\{ext\}\}\backslash right)^2+\backslash left(\backslash frac\{\backslash pi\}\{c\_\{ext\}\}\backslash right)^2\; =\; B\_m^2$$

Minimum-volume critical bare cylinder (one group)

$$H_{ext}=\frac{\sqrt{3}\pi }{B_m},R_{ext}=\sqrt{\frac{3}{2}}\frac{2.405}{B_m}H\_\{ext\}\; =\; \backslash frac\{\backslash sqrt3\backslash ,\backslash pi\}\{B\_m\},\backslash quad\; R\_\{ext\}\; =\; \backslash sqrt\{\backslash tfrac32\}\backslash ,\backslash frac\{2.405\}\{B\_m\}$$

k-effective and geometric buckling of a bare reactor (one group)

$$k_{eff}=\frac{\nu {\mathrm{\Sigma }}_f}{{\mathrm{\Sigma }}_a+D{B}_g^2}k\_\{eff\}\; =\; \backslash frac\{\backslash nu\backslash Sigma\_f\}\{\backslash Sigma\_a\; +\; D\; B\_g^2\}$$

Reactor physics / decay heat

Decay heat after shutdown (Way-Wigner)

$$\frac{P_d}{P_0}=0.0622[t^{-0.2}-(t+T{)}^{-0.2}]\backslash frac\{P\_d\}\{P\_0\}\; =\; 0.0622\backslash left[t^\{-0.2\}\; -\; (t+T)^\{-0.2\}\backslash right]$$

Decay heat energy released between two times after shutdown

$$E=0.0622P_0{\left[\frac{t^{0.8}-(t+T{)}^{0.8}}{0.8}\right]}_{t_1}^{t_2}E\; =\; 0.0622\backslash ,P\_0\backslash left[\backslash frac\{t^\{0.8\}-(t+T)^\{0.8\}\}\{0.8\}\backslash right]\_\{t\_1\}^\{t\_2\}$$

Reactor physics / fission-product poisons

Equilibrium iodine-135 and xenon-135 concentrations

$$X_{\mathrm{\infty }}=\frac{({\gamma }_I+{\gamma }_X){\mathrm{\Sigma }}_f\varphi }{{\lambda }_X+{\sigma }_X\varphi }X\_\backslash infty\; =\; \backslash frac\{(\backslash gamma\_I+\backslash gamma\_X)\backslash Sigma\_f\backslash phi\}\{\backslash lambda\_X+\backslash sigma\_X\backslash phi\}$$

Samarium-149 poisoning: equilibrium and after shutdown

$$S_{\mathrm{\infty }}=\frac{{\gamma }_{Pm}{\mathrm{\Sigma }}_f}{{\sigma }_{Sm}},P_{\mathrm{\infty }}=\frac{{\gamma }_{Pm}{\mathrm{\Sigma }}_f\varphi }{{\lambda }_{Pm}}S\_\backslash infty\; =\; \backslash frac\{\backslash gamma\_\{Pm\}\backslash Sigma\_f\}\{\backslash sigma\_\{Sm\}\},\backslash quad\; P\_\backslash infty\; =\; \backslash frac\{\backslash gamma\_\{Pm\}\backslash Sigma\_f\backslash phi\}\{\backslash lambda\_\{Pm\}\}$$

Xenon-135 after shutdown: concentration and peak

$$X(t)=X_0{e}^{-{\lambda }_{X}t}+\frac{{\lambda }_I{I}_0}{{\lambda }_X-{\lambda }_I}(e^{-{\lambda }_{I}t}-e^{-{\lambda }_{X}t})X(t)\; =\; X\_0e^\{-\backslash lambda\_Xt\}\; +\; \backslash frac\{\backslash lambda\_II\_0\}\{\backslash lambda\_X-\backslash lambda\_I\}\backslash left(e^\{-\backslash lambda\_It\}-e^\{-\backslash lambda\_Xt\}\backslash right)$$

Reactor physics / flux shapes

Fundamental-mode flux shape of a bare reactor at a point

$$\frac{\varphi }{{\varphi }_{max}}=J_0\text{\hspace{0.17em}⁣}\left(\frac{2.405r}{R}\right)\cos \frac{\pi z}{H}\backslash frac\{\backslash phi\}\{\backslash phi\_\{max\}\}\; =\; J\_0\backslash !\backslash left(\backslash frac\{2.405\backslash ,r\}\{R\}\backslash right)\backslash cos\backslash frac\{\backslash pi\; z\}\{H\}$$

Reactor physics / fuel depletion and burnup

U-235 depletion and energy released at constant flux

$$N(t)=N_0{e}^{-{\sigma }_a\varphi t},E=E_f\frac{{\sigma }_f}{{\sigma }_a}(N_0-N)N(t)\; =\; N\_0e^\{-\backslash sigma\_a\backslash phi\; t\},\backslash quad\; E\; =\; E\_f\backslash frac\{\backslash sigma\_f\}\{\backslash sigma\_a\}\backslash left(N\_0-N\backslash right)$$

U-235 depletion at constant power

$$N(t)=N_0-\frac{Pt}{E_f}\frac{{\sigma }_a}{{\sigma }_f},\frac{\varphi (t)}{{\varphi }_0}=\frac{N_0}{N(t)}N(t)\; =\; N\_0\; -\; \backslash frac\{P\backslash ,t\}\{E\_f\}\backslash frac\{\backslash sigma\_a\}\{\backslash sigma\_f\},\backslash quad\; \backslash frac\{\backslash phi(t)\}\{\backslash phi\_0\}\; =\; \backslash frac\{N\_0\}\{N(t)\}$$

Reactor physics / neutron life cycle

Multiplication factor by the four- and six-factor formulas

$$k_{eff}=\eta fp\epsilon P_{FNL}{P}_{TNL}k\_\{eff\}\; =\; \backslash eta\; f\; p\; \backslash varepsilon\backslash ,P\_\{FNL\}P\_\{TNL\}$$

Reactor physics / power and flux

Power density and power from the fission reaction rate

$$q^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}=E_{f}N{\sigma }_f\varphi ,P=q^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}Vq\text{'}\text{'}\text{'}\; =\; E\_f\; N\; \backslash sigma\_f\; \backslash phi,\backslash quad\; P\; =\; q\text{'}\text{'}\text{'}\; V$$

Reactor physics / power and flux shapes

Reactor power from the peak flux (or peak flux from power) of a bare core

$$P=E_f{\mathrm{\Sigma }}_f{\int }_V\varphi dV=E_f{\mathrm{\Sigma }}_f{\varphi }_{max}\frac{\bar{\varphi }}{{\varphi }_{max}}VP\; =\; E\_f\backslash Sigma\_f\backslash int\_V\backslash phi\backslash ,dV\; =\; E\_f\backslash Sigma\_f\backslash ,\backslash phi\_\{max\}\backslash ,\backslash frac\{\backslash bar\backslash phi\}\{\backslash phi\_\{max\}\}\backslash ,V$$

Reactor physics / reflected reactors

Critical thickness of a reflected slab reactor and reflector savings (one group)

$$D_{c}B\tan \frac{Ba}{2}=\frac{D_r}{L_r}\coth \frac{T_r+d}{L_r}D\_cB\backslash tan\backslash frac\{Ba\}\{2\}\; =\; \backslash frac\{D\_r\}\{L\_r\}\backslash coth\backslash frac\{T\_r+d\}\{L\_r\}$$

Reactor physics / two-group theory

k-effective of a bare reactor in two-group diffusion theory

$$k=\frac{\nu {\mathrm{\Sigma }}_{f1}+\nu {\mathrm{\Sigma }}_{f2}\frac{{\mathrm{\Sigma }}_{12}}{D_2{B}^2+{\mathrm{\Sigma }}_{a2}}}{D_1{B}^2+{\mathrm{\Sigma }}_{a1}+{\mathrm{\Sigma }}_{12}}k\; =\; \backslash frac\{\backslash nu\backslash Sigma\_\{f1\}\; +\; \backslash nu\backslash Sigma\_\{f2\}\backslash frac\{\backslash Sigma\_\{12\}\}\{D\_2B^2+\backslash Sigma\_\{a2\}\}\}\{D\_1B^2+\backslash Sigma\_\{a1\}+\backslash Sigma\_\{12\}\}$$

Shielding / attenuation

Half-value and tenth-value layers

$$HVL=\frac{\ln 2}{\mu },TVL=\frac{\ln 10}{\mu }HVL\; =\; \backslash frac\{\backslash ln\; 2\}\{\backslash mu\},\backslash quad\; TVL\; =\; \backslash frac\{\backslash ln\; 10\}\{\backslash mu\}$$

Shielding and dose / attenuation

Linear attenuation coefficient from narrow-beam transmission measurements

$$\ln I=\ln I_0-\mu x\backslash ln\; I\; =\; \backslash ln\; I\_0\; -\; \backslash mu\; x$$

Narrow-beam transmission through a multilayer shield

$$T=\exp (-\sum _i{\mu }_i{x}_i)T\; =\; \backslash exp\backslash Big(-\backslash sum\_i\; \backslash mu\_i\; x\_i\backslash Big)$$

Shielding and dose / dose over time

Dose accumulated near a decaying source

$$D={\dot{H}}_0{e}^{-\lambda t_d}\frac{1-e^{-\lambda t}}{\lambda }D\; =\; \backslash dot\; H\_0\backslash ,e^\{-\backslash lambda\; t\_d\}\backslash ,\backslash frac\{1-e^\{-\backslash lambda\; t\}\}\{\backslash lambda\}$$

Dose received passing a point source

$$D=\frac{\mathrm{\Gamma }A}{dv}2\arctan \frac{L}{2d}D\; =\; \backslash frac\{\backslash Gamma\; A\}\{d\backslash ,v\}\backslash ,2\backslash arctan\backslash frac\{L\}\{2d\}$$

Stay time to a dose limit near a decaying source

$$t=-\frac{1}{\lambda }\ln (1-\frac{\lambda D_{lim}}{{\dot{H}}_0})t\; =\; -\backslash frac\{1\}\{\backslash lambda\}\backslash ln\backslash left(1-\backslash frac\{\backslash lambda\; D\_\{lim\}\}\{\backslash dot\; H\_0\}\backslash right)$$

Shielding and dose / extended sources

Uncollided flux from a shielded line source (Sievert integral)

$$\varphi =\frac{S_L}{4\pi h}[F({\theta }_2,\mu t)-F({\theta }_1,\mu t)],F(\theta ,b)={\int }_0^{\theta }{e}^{-b\sec u}du\backslash phi\; =\; \backslash frac\{S\_L\}\{4\backslash pi\; h\}\backslash left[F(\backslash theta\_2,\backslash mu\; t)\; -\; F(\backslash theta\_1,\backslash mu\; t)\backslash right],\backslash \; F(\backslash theta,b)=\backslash int\_0^\{\backslash theta\}e^\{-b\backslash sec\; u\}\backslash ,du$$

Uncollided flux on the axis of a shielded disk source

$$\varphi =\frac{S_A}{2}[E_1(\mu t)-E_1(\mu t\sec \theta )]\backslash phi\; =\; \backslash frac\{S\_A\}\{2\}\backslash left[E\_1(\backslash mu\; t)\; -\; E\_1(\backslash mu\; t\backslash sec\backslash theta)\backslash right]$$

Shielding and dose / photon interactions

Klein-Nishina Compton cross sections per electron

$$\sigma =2\pi r_e^2\{\frac{1+k}{k^2}[\frac{2(1+k)}{1+2k}-\frac{\ln (1+2k)}{k}]+\frac{\ln (1+2k)}{2k}-\frac{1+3k}{(1+2k{)}^2}\}\backslash sigma\; =\; 2\backslash pi\; r\_e^2\backslash left\backslash \{\backslash frac\{1+k\}\{k^2\}\backslash left[\backslash frac\{2(1+k)\}\{1+2k\}-\backslash frac\{\backslash ln(1+2k)\}\{k\}\backslash right]+\backslash frac\{\backslash ln(1+2k)\}\{2k\}-\backslash frac\{1+3k\}\{(1+2k)^2\}\backslash right\backslash \}$$

Shielding and dose / point sources

Dose rate from a shielded point gamma source with buildup

$$\dot{H}=\frac{\mathrm{\Gamma }A}{r^2}B(\mu x)e^{-\mu x}\backslash dot\; H\; =\; \backslash frac\{\backslash Gamma\; A\}\{r^2\}\backslash ,B(\backslash mu\; x)\backslash ,e^\{-\backslash mu\; x\}$$

Shielding and dose / shield design

Shield thickness for a target dose rate (point source, buildup)

$$\frac{\mathrm{\Gamma }A}{r^2}B(\mu x)e^{-\mu x}={\dot{H}}_{target}\backslash frac\{\backslash Gamma\; A\}\{r^2\}B(\backslash mu\; x)e^\{-\backslash mu\; x\}\; =\; \backslash dot\; H\_\{target\}$$

Thermal hydraulics / coolant channels

Coolant temperature along a channel with a chopped-cosine power shape

$$T(z)=T_{in}+\frac{q_0{H}_e}{\pi \dot{m}{c}_p}[\sin \frac{\pi z}{H_e}+\sin \frac{\pi H}{2H_e}]T(z)\; =\; T\_\{in\}\; +\; \backslash frac\{q\_0H\_e\}\{\backslash pi\backslash dot\; mc\_p\}\backslash left[\backslash sin\backslash frac\{\backslash pi\; z\}\{H\_e\}\; +\; \backslash sin\backslash frac\{\backslash pi\; H\}\{2H\_e\}\backslash right]$$

Location and value of the maximum cladding surface temperature

$$\tan \frac{\pi z_{max}}{H_e}=\frac{2R_{o}h{H}_e}{\dot{m}{c}_p}\backslash tan\backslash frac\{\backslash pi\; z\_\{max\}\}\{H\_e\}\; =\; \backslash frac\{2R\_ohH\_e\}\{\backslash dot\; mc\_p\}$$

Thermal hydraulics / fuel rods

Fuel centreline temperature with temperature-dependent conductivity

$${\int }_{T_s}^{T_0}k(T)dT=\frac{q^{\mathrm{\prime }}}{4\pi }\backslash int\_\{T\_s\}^\{T\_0\}k(T)\backslash ,dT\; =\; \backslash frac\{q\text{'}\}\{4\backslash pi\}$$

Fuel-rod radial temperatures: centreline, pellet surface and cladding

$$T_0-T_b=q^{\mathrm{\prime }}[\frac{1}{4\pi k_f}+\frac{1}{2\pi R_f{h}_g}+\frac{\ln (R_o\mathrm{/}{R}_f)}{2\pi k_c}+\frac{1}{2\pi R_{o}h}]T\_0\; -\; T\_b\; =\; q\text{'}\backslash left[\backslash frac\{1\}\{4\backslash pi\; k\_f\}\; +\; \backslash frac\{1\}\{2\backslash pi\; R\_fh\_g\}\; +\; \backslash frac\{\backslash ln(R\_o/R\_f)\}\{2\backslash pi\; k\_c\}\; +\; \backslash frac\{1\}\{2\backslash pi\; R\_oh\}\backslash right]$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified nuclear engineering capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (radioactive decay chains and activity, radiation shielding and dosimetry, detectors and counting, neutron physics, reactor criticality and kinetics, fission-product poisons, fuel depletion and enrichment, and fuel-rod heat transfer); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{nuclear_engineering_ai999x_1_0_0,
  title  = {nuclear-engineering-ai999x: an offline, verified nuclear engineering solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/nuclear-engineering-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/nuclear-engineering-ai999x/discussions or visit https://ai999x.com.

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Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com