{ "openapi": "3.1.1", "info": { "title": "Half-Life, Activity \u0026 Decay Chain Calculator API", "version": "1.0.0", "summary": "Radioactive decay: remaining amount and activity, time to a level, half-life fits and chains.", "description": "Every endpoint performs the same calculation as the human page and returns inputs, result, units, assumptions, sources, warnings and computedAt. MCP endpoint: /mcp." }, "servers": [ { "url": "https://prooflinehalflife.usefulweb.app" } ], "paths": { "/api/v1/calculate": { "post": { "operationId": "calculate", "summary": "Radioactive decay: remaining amount and activity, time to a level, half-life fits and chains.", "description": "Work out radioactive decay the way textbooks and labs do: how much of a nuclide is left after a time and what its activity is (Bq or Ci, from a percentage, an activity, a mass or a number of atoms), how long it takes to fall to a given level, which half-life fits your measured count rates, and how the members of a decay chain such as Mo-99 → Tc-99m or radon and its progeny grow in and decay (Bateman equations with branching). Nineteen preset nuclides carry evaluated half-lives from ENSDF/NUBASE2020. Part of the Proofline math notebook.\n\nMethodology:\n- N(t) = N₀·2^(−t/T½) = N₀·e^(−λt) with λ = ln 2 / T½; activity A = λ·N in becquerel (1 Ci = 3.7 × 10¹⁰ Bq); atoms from a mass via the molar mass of the nuclide (AME2020 where Proofline’s chemistry engine has it, otherwise the mass number) and the exact Avogadro constant. Time to a level: t = ln(N₀/N)/λ.\n- Half-life fits: least squares of ln(value) against time with Proofline’s QR fitter gives λ and its standard error; two points give the half-life directly.\n- Chains: the Bateman equations Nₖ(t) = N₁(0)·Πbᵢλᵢ·Σ e^(−λᵢt)/Π(λⱼ − λᵢ) with branching fractions bᵢ; the time of the daughter maximum is ln(λ₂/λ₁)/(λ₂ − λ₁).\n- Verified: 1 g of Ra-226 has 0.989 Ci (the historical curie), C-14 has a specific activity of 166 GBq/g, the Tc-99m activity of a Mo-99 generator peaks after 22.85 h, and radon progeny reach secular equilibrium within hours.\n\nDecay is a statistical law: small samples fluctuate around these averages. No dose, shielding or exposure calculation is made. Nothing here is advice on handling chemicals or radioactive material safely.", "tags": [ "apps" ], "x-mcp-tool": "proofline_half_life", "x-owner-app": "proofline", "requestBody": { "required": true, "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProoflineHalfLifeInput" }, "examples": { "example1": { "summary": "Iodine-131: 100 MBq after 30 days", "value": { "mode": "decay", "nuclide": "i131", "initial": "100 MBq", "elapsed": "30 d" } }, "example2": { "summary": "Carbon-14: how long until 25 % is left?", "value": { "mode": "time", "nuclide": "c14", "initial": "100 %", "target": "25 %" } }, "example3": { "summary": "Activity of 1 g of radium-226 (about 1 curie)", "value": { "mode": "decay", "nuclide": "ra226", "initial": "1 g", "elapsed": "0 y" } }, "example4": { "summary": "Half-life from count rates", "value": { "mode": "fit", "measurements": "0 h 1000\n6 h 501\n12 h 250\n18 h 126\n24 h 62" } }, "example5": { "summary": "Mo-99 → Tc-99m generator over a week", "value": { "mode": "chain", "chainPreset": "mo99-tc99m", "initial": "10 GBq" } }, "example6": { "summary": "Radon-222 and its progeny reach equilibrium", "value": { "mode": "chain", "chainPreset": "rn222", "initial": "1000 Bq" } } } } } }, "responses": { "200": { "description": "Calculation result envelope.", "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProoflineHalfLifeResult" } } } }, "400": { "description": "Malformed JSON body.", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "422": { "description": "Invalid input (see errors[]).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "429": { "description": "Rate limit exceeded (see Retry-After).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } } } } }, "/api/v1/tool": { "get": { "operationId": "getToolDefinition", "summary": "Inputs, output schema and examples for this tool", "responses": { "200": { "description": "Tool definition.", "content": { "application/json": { "schema": { "type": "object" } } } } } } } }, "components": { "schemas": { "Problem": { "type": "object", "description": "RFC 9457 problem details. 422 responses include per-field `errors`.", "required": [ "type", "title", "status", "detail" ], "properties": { "type": { "type": "string" }, "title": { "type": "string" }, "status": { "type": "integer" }, "detail": { "type": "string" }, "code": { "type": "string" }, "errors": { "type": "array", "items": { "type": "object", "required": [ "field", "message" ], "properties": { "field": { "type": "string" }, "message": { "type": "string" } } } } } }, "ProoflineHalfLifeInput": { "type": "object", "additionalProperties": false, "properties": { "mode": { "title": "Calculation", "type": "string", "enum": [ "decay", "time", "fit", "chain" ], "default": "decay" }, "nuclide": { "title": "Nuclide", "type": "string", "enum": [ "custom", "h3", "c14", "f18", "p32", "k40", "co60", "sr90", "mo99", "tc99m", "i131", "cs137", "po210", "rn222", "ra226", "th232", "u235", "u238", "pu239", "am241" ], "default": "custom" }, "halfLife": { "title": "Half-life (own nuclide)", "description": "A number with a time unit: s, min, h, d or y (365.2422 days).", "type": "string", "maxLength": 40, "default": "" }, "initial": { "title": "Starting amount", "description": "A percentage, an activity (Bq, kBq, MBq, GBq, Ci, mCi, µCi), a mass (g, mg; preset nuclides only) or a number of atoms. Empty: 100 % (chains: 1 MBq).", "type": "string", "maxLength": 60, "default": "" }, "elapsed": { "title": "Elapsed time", "type": "string", "maxLength": 40, "default": "" }, "target": { "title": "Remaining level (time mode)", "description": "A percentage, or an activity or mass in the same kind of unit as the start.", "type": "string", "maxLength": 60, "default": "" }, "measurements": { "title": "Measurements (fit mode)", "description": "One measurement per line: the time with its unit and the measured count rate or activity.", "type": "string", "maxLength": 20000, "default": "" }, "chainPreset": { "title": "Decay chain", "type": "string", "enum": [ "custom", "mo99-tc99m", "sr90-y90", "rn222", "ra226" ], "default": "custom" }, "chain": { "title": "Chain members (own chain)", "description": "One member per line: name, half-life (or “stable” for the last) and optionally the share of the previous member’s decays that feeds it.", "type": "string", "maxLength": 2000, "default": "" }, "duration": { "title": "Follow the chain for", "description": "Empty: the preset chain’s own period.", "type": "string", "maxLength": 40, "default": "" } }, "required": [] }, "ProoflineHalfLifeResult": { "type": "object", "additionalProperties": false, "required": [ "schemaVersion", "tool", "computedAt", "inputs", "summary", "result", "units", "assumptions", "sources", "warnings", "presentation", "links" ], "properties": { "schemaVersion": { "type": "string", "const": "1.0" }, "tool": { "type": "object", "additionalProperties": false, "required": [ "id", "name", "version" ], "properties": { "id": { "type": "string" }, "name": { "type": "string" }, "version": { "type": "string" } } }, "computedAt": { "type": "string", "format": "date-time" }, "inputs": { "type": "object", "description": "Normalised inputs actually used, including applied defaults." }, "summary": { "type": "string" }, "result": { "type": "object", "additionalProperties": false, "required": [ "mode", "nuclide", "halfLifeS", "lambda", "meanLifeS", "timeS", "halfLives", "start", "now", "molarMass", "specificActivityBqPerG", "chain", "fit" ], "properties": { "mode": { "type": "string", "enum": [ "decay", "time", "fit", "chain" ] }, "nuclide": { "type": [ "string", "null" ] }, "halfLifeS": { "type": [ "number", "null" ] }, "lambda": { "type": "number" }, "meanLifeS": { "type": "number" }, "timeS": { "type": [ "number", "null" ] }, "halfLives": { "type": [ "number", "null" ] }, "start": { "type": [ "object", "null" ] }, "now": { "type": [ "object", "null" ] }, "molarMass": { "type": [ "number", "null" ] }, "specificActivityBqPerG": { "type": [ "number", "null" ] }, "chain": { "type": [ "object", "null" ] }, "fit": { "type": [ "object", "null" ] } } }, "units": { "type": "object", "additionalProperties": { "type": "string" } }, "assumptions": { "type": "array", "items": { "type": "string" } }, "sources": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "title" ], "properties": { "title": { "type": "string" }, "url": { "type": "string" }, "publisher": { "type": "string" }, "retrievedAt": { "type": "string" }, "note": { "type": "string" } } } }, "warnings": { "type": "array", "items": { "type": "string" } }, "presentation": { "type": "array", "items": { "type": "object" }, "description": "Display blocks used by the human page (same data)." }, "links": { "type": "object", "additionalProperties": false, "required": [ "human", "api", "openapi", "methodology" ], "properties": { "human": { "type": "string", "format": "uri" }, "api": { "type": "string", "format": "uri" }, "openapi": { "type": "string", "format": "uri" }, "methodology": { "type": "string", "format": "uri" } } } } } } } }