Materials analysis and spectroscopy kernels

Generic mechanics-of-materials formulas, chromatographic separation science, optical/analytical spectroscopy, treatment-plant water chemistry, and structural timber engineering across five independent

Five independent modules for turning a raw lab or field measurement into an engineering or chemistry answer: a stress-strain reading becomes an elastic modulus and a safety factor (materials), a chromatogram's retention times and peak widths become a resolution and a plate count (chromatography), an absorbance or a Raman-shifted wavelength becomes a concentration or a vibrational shift (spectroscopy), a water sample's ion concentrations become a hardness, an alkalinity, and a corrosion tendency (water_chemistry), and a board's dimensions and a species' adjustment factors become an allowable design stress (timber). Every function takes plain Float64 scalars and returns a single Float64 — no shared struct, no cross-module imports, and (as documented per-module below) no accidental overlap with the materials-science, water-chemistry, or wood-science modules already documented elsewhere in this corpus.

from algenta import MojoRuntime

runtime = MojoRuntime(mode="local")

runtime.execute("materials", "youngs_modulus", [100.0, 0.001])["result"]
# 100000.0 -- E = sigma/epsilon; 100 Pa of stress producing 0.001 strain

runtime.execute("chromatography", "resolution", [12.4, 10.1, 0.5, 0.6])["result"]
# 4.181818181818182 -- Rs = 2*(12.4-10.1)/(0.5+0.6); well past the Rs > 1.5
#                       baseline-separation threshold

The modules

Module	What it does
materials	Generic mechanics of materials: Young's/shear/bulk modulus and the Poisson's-ratio identity linking them, stress from force, strain from deformation, thermal expansion, safety factor, stress-concentration factor, Basquin's-law fatigue life, cantilever beam deflection, rectangular-section moment of inertia.
chromatography	Chemical separation science: retention factor, selectivity factor, theoretical plate count and plate height, chromatographic resolution, the Van Deemter equation, peak capacity, the Trennzahl separation number, the Kovats retention index, and mobile-phase linear velocity.
spectroscopy	Analytical optical spectroscopy: wavenumber/wavelength conversion, photon energy, absorbance/transmittance (Beer-Lambert), molar-absorptivity concentration, spectrometer resolving power, scan-averaged SNR, Raman shift, fluorescence quantum yield, Stokes shift, and spectral bandwidth.
water_chemistry	Drinking-water and wastewater treatment-plant chemistry: pH/[H+] conversion, hardness and alkalinity as CaCO3, the Langelier Saturation Index, a turbidity-to-Secchi-depth estimate, chlorine demand, dissolved-oxygen deficit, conductivity-to-TDS, and BOD removal efficiency.
timber	Structural timber engineering and log/lumber yield: NDS-style allowable stress adjustment, board-feet and Doyle-rule log volume, moisture content, moment capacity and section modulus, column slenderness, wood density from specific gravity, a shrinkage estimate, and a linear fire-charring model.

materials: mechanics-of-materials formulas, not a duplicate of the construction-materials page

Structural and construction materials kernels (adhesives, composites, concrete, ceramics, polymer, rubber) documents formulas specific to particular material families — a concrete mix's water-cement ratio, a ceramic's Weibull failure probability, a polymer's Mark-Houwink intrinsic viscosity. materials.mojo is one level more generic: plain isotropic-elasticity and beam formulas (E = σ/ε, G = E/(2(1+ν)), K = E/(3(1-2ν)), δ = PL³/(3EI), Basquin's fatigue law) that any of those six material types would plug numbers into, but none of the six modules import it or reimplement its formulas. Grepping gitbook/ for `materials` in backticks turns up nothing narrative outside this page and the machine-generated Runtime library catalog, so this is genuinely unclaimed territory, not a page documenting the same ground twice.

The twelve functions cross-check each other. mojo/tests/libs/test_wave58.mojo (a real, hand-verified suite — not the auto-generated smoke test) pins:

runtime.execute("materials", "shear_modulus", [200e9, 0.3])["result"]
# 76923076923.07692 -- G = 200e9 / (2*1.3)

runtime.execute("materials", "poisson_ratio_from_moduli", [200e9, 76923076923.07692])["result"]
# 0.3 -- feeding G back through nu = E/(2G) - 1 recovers the original Poisson's ratio
#        to 1e-6, confirming the two formulas are true inverses of each other

Every division-shaped function guards its own zero: youngs_modulus returns 0.0 for |strain| < 1e-30 rather than dividing by it, and beam_deflection_cantilever returns 0.0 if 3EI underflows the same threshold. fatigue_life is Basquin's law taken literally — fatigue_life(200e6, 900e6, -0.1) (a stress amplitude at about 22% of the fatigue-strength coefficient, with a typical negative Basquin exponent) computes (200e6/900e6)^(1/-0.1), which is (0.2222...)^-10 ≈ 3,405,063 cycles — the test only asserts this is positive and finite, but the formula itself is exact and reproducible by hand.

chromatography: a full separation-science module with no page elsewhere

Grepping gitbook/engine/ for `chromatography` finds only this page and the auto-generated Runtime library catalog row — the hinted possibility that Process and organic materials science kernels covered it as an unfulfilled forward-reference does not hold up: that page's wood_science / textile / combustion / paper_science modules never mention retention time, plate count, or any chromatographic term. chromatography.mojo's ten functions are the standard analytical- chemistry toolkit for characterizing a column and a separation, and they chain together on one real analysis:

rt   = runtime.execute("chromatography", "retention_factor", [12.0, 2.0])["result"]
# 5.0 -- k = (tR - t0)/t0 for a 12-minute peak on a 2-minute dead-time column

sel  = runtime.execute("chromatography", "selectivity_factor", [5.0, 3.0])["result"]
# 1.6666666666666667 -- alpha = k2/k1, how well-separated two analytes' k values are

n    = runtime.execute("chromatography", "plate_count", [12.0, 0.4])["result"]
# 14400.0 -- N = 16*(tR/w)^2 for a 12-minute peak 0.4 minutes wide at the base

h    = runtime.execute("chromatography", "plate_height", [25.0, 14400.0])["result"]
# 0.001736111111111111 -- H = L/N for a 25 cm column: each plate is under 2 microns tall

vd   = runtime.execute("chromatography", "van_deemter_h", [0.5, 0.02, 0.001, 2.0])["result"]
# 0.512 -- H = A + B/u + C*u = 0.5 + 0.01 + 0.002 at a mobile-phase velocity of 2

nc   = runtime.execute("chromatography", "peak_capacity", [14400.0, 20.0, 2.0])["result"]
# 70.07755278982138 -- roughly 70 resolvable peaks fit between the first eluter at
#                       2 min and the last at 20 min, given that plate count

retention_index_kovats is the one function with four independent inputs rather than a running chain: retention_index_kovats(6.5, 5.0, 8.0, 6) — an unknown eluting at 6.5 minutes bracketed by the C6 and C7 n-alkanes at 5.0 and 8.0 minutes — returns 655.82, placing the unknown a bit past 60% of the way from the C6 to the C7 index. Every function in the module guards its own division: retention_factor and mobile_phase_velocity return 0.0 for dead_time <= 0.0, selectivity_factor returns 0.0 for k1 <= 0.0, and peak_capacity returns exactly 1.0 (rather than 0.0) when tr_first <= 0.0 or the peaks are out of order — a single resolvable peak is the correct floor for a capacity, not zero. The only test file behind this module, mojo/tests/libs/test_chromatography.mojo, is auto-generated: it calls every function once with literal 0.5/0.5/0.5/0.5 arguments and checks only that the result is finite, so every worked number above was hand-traced from mojo/engine/chromatography.mojo's real formulas and independently recomputed, not copied from a pinned test.

spectroscopy: analytical instrument formulas, and where they touch radiometry

Acoustics, optics, and radiative materials kernels already documents radiometry's "blackbody and photon-counting radiometry," including its own photon_energy(wavelength_nm). That function and spectroscopy.energy_from_wavelength(wavelength_m) compute the identical physics, E = hc/λ, but for different instruments and in different units: radiometry.photon_energy takes nanometers, uses c = 2.998e8, and returns electron-volts (it divides by 1.602e-19 at the end) — the natural unit for detector photon-counting. spectroscopy's version takes meters, uses the rounder c = 3.0e8, and returns joules with no eV conversion — the natural unit for a Beer-Lambert-style optical-instrument calculation. Calling energy_from_wavelength(500e-9) gives 3.9756e-19 J (≈2.482 eV); calling photon_energy(500.0) gives ≈2.480 eV directly — the two-thousandths-of-an-eV gap is entirely the c = 3.0e8 vs c = 2.998e8 rounding choice, not a different formula. Neither module imports the other or shares a constant.

The rest of spectroscopy is Beer-Lambert/UV-Vis and Raman-adjacent arithmetic that radiometry never touches at all:

runtime.execute("spectroscopy", "wavenumber_from_wavelength", [2.5])["result"]
# 4000.0 -- nu = 10000/lambda_um; round-trips exactly back through
#           wavelength_from_wavenumber(4000.0) == 2.5

runtime.execute("spectroscopy", "beer_lambert_concentration", [0.5, 1.0, 100.0])["result"]
# 0.005 -- c = A/(epsilon*l): absorbance 0.5 through a 1 cm cell at
#          molar absorptivity 100 L/(mol*cm) is a 5 mM solution

runtime.execute("spectroscopy", "raman_shift", [532.0, 580.0])["result"]
# 1556.1697247706424 cm^-1 -- (1/532 - 1/580) * 1e7, a 532 nm excitation
#                              Stokes-shifted to 580 nm scattered light

runtime.execute("spectroscopy", "snr_coadd", [10.0, 16.0])["result"]
# 40.0 -- SNR grows with sqrt(n_scans): 16 co-added scans quadruple a
#          single-scan SNR of 10

absorbance_from_transmittance and transmittance_from_absorbance are exact inverses of the same A = -log10(T/100) relationship — absorbance_from_transmittance(50.0) = 0.30103, and feeding that back through transmittance_from_absorbance returns 50.0 to five decimal places, the same round-trip self-check pattern used for RoPE in Attention kernels. spectral_bandwidth is a literal pass-through (it returns its fwhm argument unchanged) rather than a derived quantity — it exists so callers have a named function for "the FWHM value is the bandwidth" instead of inlining the identity themselves. mojo/tests/libs/test_spectroscopy.mojo is auto-generated (finite-check only); the real pinned numbers above come from tests/mojo/test_wave96.mojo, which hand-verifies twelve spectroscopy functions to 1e-3–1e-6 tolerance alongside timber and project_management.

water_chemistry: treatment-plant process control, not limnochemistry's lake biogeochemistry

Hydrology & water systems kernels already documents limnochemistry as "freshwater ionic chemistry" for lakes and rivers — carbonate alkalinity, the Davies activity coefficient, a calcite saturation index, silica export flux, Eh/pe conversion, nutrient fractionation. water_chemistry.mojo computes some of the same named quantities but for an entirely different setting — a drinking-water or wastewater treatment plant — using different formulas:

• Alkalinity. limnochemistry.carbonate_alkalinity is a charge-balance equation in milli- equivalents per liter, Alk = [HCO3⁻] + 2[CO3²⁻] + [OH⁻] - [H⁺], meant for natural-water titration data. water_chemistry.alkalinity_as_caco3 is a mass-based industry-standard conversion in mg/L-as-CaCO3, Alk = 0.82·HCO3⁻ + 1.67·CO3²⁻ + 2.94·OH⁻, with no proton term at all (standard practice for near-neutral treated water, where free H⁺ is negligible). alkalinity_as_caco3(120.0, 0.0, 0.0) returns 98.4 mg/L for a purely bicarbonate-buffered sample — a different number, a different unit, and a different intended input than the lake formula.

• Saturation index. limnochemistry.saturation_index_calcite derives its own saturation pH from calcium, alkalinity, and temperature (pH_sat = 11.73 - 0.053·T - log(Ca) - log(Alk)) before subtracting it from the measured pH. water_chemistry.langelier_saturation_index(ph, ph_saturation) takes ph_saturation as an already-known input and only does the subtraction — it assumes a plant has already looked up or computed pHs from standard water-treatment tables, rather than deriving it from raw ion concentrations itself.

runtime.execute("water_chemistry", "water_hardness", [150.0, 30.0])["result"]
# 498.09 -- 2.497*150 + 4.118*30 mg/L as CaCO3 (the module's docstring rounds these
#           gravimetric factors to 2.5/4.12, but the code uses the precise
#           CaCO3/Ca and CaCO3/Mg molar-mass ratios, 100.09/40.08 and 100.09/24.305)

runtime.execute("water_chemistry", "langelier_saturation_index", [7.8, 8.1])["result"]
# -0.3 -- pH below pHs: mildly corrosive (scale-forming is positive, corrosive is negative)

runtime.execute("water_chemistry", "bod_removal", [200.0, 20.0])["result"]
# 90.0 -- (200-20)/200 * 100, a 90% BOD removal efficiency across a treatment stage

Every subtraction-shaped function floors at zero rather than returning a negative: chlorine_demand and dissolved_oxygen_deficit both clamp a negative difference to 0.0, and bod_removal clamps a negative removal rate to 0.0 too (an effluent BOD higher than the influent's would otherwise report negative "removal"). ph_from_hydrogen_ion returns a sentinel 14.0 for h_concentration <= 0.0 rather than an undefined logarithm. Grepping gitbook/ for `water_chemistry` in backticks confirms no other narrative page claims this module; its only tests are the auto-generated mojo/tests/libs/test_water_chemistry.mojo finite-check suite, so the numbers above were independently hand-computed from mojo/engine/water_chemistry.mojo.

timber: structural design and forestry yield, distinct from wood_science's material testing

Process and organic materials science kernels already documents wood_science as wood material-testing physics — modulus of rupture and elasticity measured from a three-point bending test, fiber saturation point, a thermal-conductivity estimate, Janka hardness. timber.mojo is a different scope: NDS-style structural design (an allowable stress built from load-duration, size, and wet-service adjustment factors) and forestry yield accounting (board-feet, the Doyle log-scaling rule) — neither of which wood_science touches at all. Two of the ten timber functions do duplicate a wood_science formula exactly, just under different parameter names and with neither module importing the other: timber.moisture_content(wet_weight, oven_dry_weight) and wood_science.moisture_content(wet_mass, dry_mass) are both (wet - dry)/dry * 100, and timber.section_modulus_beam(width, depth) and wood_science.section_modulus(width, depth) are both b·h²/6. Past those two, the modules diverge: timber.fire_char_rate is a flat linear model (depth = time * rate), while wood_science.charring_rate is wood_science's own power-law model (depth = β₀ · t^0.813) — the same real phenomenon, two different fits, neither one calling the other.

mojo/tests/libs/test_timber.mojo is auto-generated (finite-check only); the pinned numbers below come from the real, hand-verified tests/mojo/test_wave96.mojo:

runtime.execute("timber", "allowable_stress", [1000.0, 1.15, 1.0, 0.85])["result"]
# 977.5 -- F' = Fb * CD * CF * CM: base stress adjusted for a 1.15 load-duration
#          factor, no size adjustment, and an 0.85 wet-service reduction

runtime.execute("timber", "log_volume_doyle", [20.0, 16.0])["result"]
# 256.0 -- Doyle rule: (D-4)^2 * L / 16, board feet in a 20-inch-diameter,
#          16-foot log

runtime.execute("timber", "fire_char_rate", [60.0, 0.635])["result"]
# 38.1 -- 60 minutes of exposure at 0.635 mm/min char rate

runtime.execute("timber", "remaining_section", [200.0, 38.1])["result"]
# 123.8 -- a 200 mm member loses char on both faces: 200 - 2*38.1

column_slenderness and board_feet both guard a zero denominator/dimension by returning 0.0 rather than dividing; shrinkage_estimate does the same for a zero fiber-saturation point.

Verification

Two of these five modules have real, hand-verified test coverage beyond the auto-generated smoke suite: materials in mojo/tests/libs/test_wave58.mojo and timber/spectroscopy together in tests/mojo/test_wave96.mojo, both of which pin specific numeric outputs (not just "is finite") and are the source of the exact figures quoted above for those three modules. chromatography and water_chemistry have only the auto-generated mojo/tests/libs/test_chromatography.mojo and test_water_chemistry.mojo — each calls every function once with literal 0.5-shaped arguments and checks solely that the result is finite, pinning no specific value. Every chromatography and water_chemistry number in this page was therefore hand-traced against the real formulas in mojo/engine/chromatography.mojo and mojo/engine/water_chemistry.mojo and independently recomputed before being presented here.

Next: see Process and organic materials science kernels for wood_science's wood-testing physics, or Hydrology & water systems kernels for limnochemistry's natural-water chemistry.