{"uid":"cap_zIn-9519XyfR1van6-Igp","slug":"debye-continuous-phonon-spectrum-thermal-state-functions-calculator-b20fb38d","name":"Debye Continuous Phonon Spectrum Thermal State Functions Calculator","description":"Compute the thermal state functions of an explicit continuous Debye phonon spectrum.","url":"https://detect.halowerk.com/v1/F-TD2000-GEO-INDUSTRY-B009/debye-continuous-phonon-density-energy-heat-capacity-entropy?utm_source=zero.xyz","method":"POST","headers":{},"bodySchema":{"type":"object","properties":{"artifact":{"type":"object","required":["debyeHeat"],"properties":{"debyeHeat":{"type":"object","required":["data","claim"],"properties":{"data":{"type":"object","required":["debye_temperature_kelvin","temperatures_kelvin"],"properties":{"temperatures_kelvin":{"type":"array","items":{"type":"number","maximum":10000,"minimum":0.1},"maxItems":32,"minItems":1},"debye_temperature_kelvin":{"type":"number","maximum":5000,"minimum":0.1}},"additionalProperties":false},"claim":{}},"additionalProperties":false}},"additionalProperties":false}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.003","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.003/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_uYzaxw4pWLS-igRMQM8Fo","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.003","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Computes thermal state functions (energy, heat capacity, entropy) of an explicit continuous Debye phonon spectrum given a Debye temperature and a set of sample temperatures.","exampleAgentPrompt":"Compute the Debye thermal state functions — energy, heat capacity, and entropy — for a material with a Debye temperature of 345 K at the following sample temperatures: 10, 50, 100, 200, 300, and 400 Kelvin.","exampleUseCases":[{"title":"Low-temperature specific heat modeling","prompt":"For a material with a Debye temperature of 470 K, calculate the heat capacity, internal energy, and entropy at temperatures 5, 10, 20, 50, and 100 Kelvin using the continuous Debye phonon model."},{"title":"Room-temperature thermodynamic benchmarking","prompt":"I need the full Debye thermal state functions — energy, heat capacity, and entropy — for copper (Debye temperature 343 K) evaluated at 50, 100, 150, 200, 250, 300, and 350 Kelvin."},{"title":"High-temperature phonon entropy sweep","prompt":"Compute the Debye entropy and heat capacity for a refractory ceramic with a Debye temperature of 1050 K across a temperature sweep of 300, 500, 800, 1200, 1500, and 2000 Kelvin."}],"resultDescription":"Returns computed thermodynamic quantities derived from the continuous Debye phonon density of states: internal (phonon) energy, constant-volume heat capacity, and entropy — each evaluated at every requested sample temperature, grounded in the specified Debye temperature.","failureModes":["temperatures_kelvin array is empty or exceeds 32 items — schema validation error","Debye temperature or any sample temperature outside allowed range (0.1–5000 K or 0.1–10000 K) — validation failure","Missing required fields debye_temperature_kelvin or temperatures_kelvin — 400 Bad Request","Payment not provided or insufficient USDC — 402 Payment Required","Numerical instability at extreme low temperatures approaching absolute zero"],"whenToPreferThis":"Use this endpoint when you need analytically rigorous thermodynamic state functions (energy, Cv, entropy) under the continuous Debye phonon model for a single material characterized by its Debye temperature, evaluated over a custom temperature grid of up to 32 points. Prefer this over Einsten-model or discrete-spectrum endpoints when the continuous Debye approximation is physically appropriate (most elemental solids and simple compounds).","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-11-04T06:38:35.635Z","isFirstParty":false,"canonicalSlug":"debye-continuous-phonon-spectrum-thermal-state-functions-calculator-b20fb38d"}