P0 Concentric-Sphere Analytical Benchmark

This project implements the frozen benchmark-only concentric-sphere model for P0 electrostatic, weighting-field, and Shockley–Ramo verification. It is a mathematical reference case for later cross-software verification; it is not a PTW 32002 geometry model.

Scope

Included:

• closed-form potential and radial electric field in a concentric spherical shell;
• inner-electrode weighting potential and weighting field;
• signed Shockley–Ramo current and endpoint induced charge;
• independent finite-difference, Gauss-law, capacitance, similarity, polarity, and trajectory checks;
• deterministic CSV, YAML, Markdown, and SHA-256 evidence artifacts.

Excluded:

• actual 32002 internal geometry or materials;
• COMSOL, Geant4, Garfield++, electronics, calibration, absolute air-kerma rate, or AI/ML;
• physical measurement uncertainty claims.

The numerical tolerances in this project are implementation-verification tolerances only. They are not measurement uncertainties.

Frozen case and conventions

The main case uses a = 0.01 m, b = 0.1 m, Va = 1 V, and Vb = 0 V. Outward radius and radial velocity are positive. The field definitions and signal convention are

E_r = -d(phi)/dr
Ew_r = -d(phi_w)/dr
i = q * v_r * Ew_r
Q_ind = q * [phi_w(r_start) - phi_w(r_end)]

q is signed: an electron has q < 0, while a positive ion has q > 0. The physical field depends on the applied conductor potentials. The weighting field is an auxiliary unit-boundary solution for the readout electrode and does not contain the physical bias voltage.

Public API and units

All internal calculations use SI units. Boundary evaluations at r = a and r = b are allowed; values outside the closed interval raise ValueError.

Function	Inputs	Output
potential(r, a, b, Va, Vb)	r, a, b in m; Va, Vb in V	physical potential in V
electric_field_radial(r, a, b, Va, Vb)	r, a, b in m; Va, Vb in V	outward radial field in V/m
weighting_potential(r, a, b)	r, a, b in m	dimensionless
weighting_field_radial(r, a, b)	r, a, b in m	outward radial weighting field in 1/m
induced_current(q, v_r, r, a, b)	q in C; v_r in m/s; radii in m	current in A
induced_charge(q, r_start, r_end, a, b)	q in C; radii in m	induced charge in C

CSV data dictionary

data/p0_benchmark.csv contains 201 points in the closed interval [a, b].

Column	Unit	Definition
r_m	m	radial coordinate
r_over_a	dimensionless	r/a
r_over_b	dimensionless	r/b
phi_V	V	physical potential
phi_over_deltaV	dimensionless	(phi - Vb)/(Va - Vb)
E_r_V_per_m	V/m	physical outward radial field
E_r_times_b_over_deltaV	dimensionless	E_r b/(Va - Vb)
phi_w	dimensionless	inner-electrode weighting potential
Ew_per_m	1/m	outward radial weighting field
Ew_times_b	dimensionless	Ew b
r2_E	V m	r^2 E_r
r2_Ew	m	r^2 Ew
gauss_flux	V m	4 pi r^2 E_r; multiply by vacuum permittivity to obtain enclosed charge in C

Reproduction

From this directory, using the verified environment:

& 'C:\ProgramData\anaconda3\python.exe' -W error -m pytest -q
& 'C:\ProgramData\anaconda3\python.exe' scripts\generate_benchmark.py

The generator recomputes all ten hard verification chains. It returns a nonzero process status if any chain fails. The source implementation is in src/concentric_sphere.py; evidence generation is in scripts/generate_benchmark.py; automated tests are in tests/.

Evidence artifacts

• data/p0_benchmark.csv
• manifests/P0_parameter_manifest.yaml
• reports/P0_Verification_Report.md
• manifests/P0_SHA256SUMS.txt

The checksum manifest excludes itself, Git internals, Python bytecode, and test caches. The parameter manifest records the frozen implementation-plan hash and the committed source HEAD used for generation. If no commit or repository exists, it records the reason explicitly.