Description
Sample validated practice cases for Varshni Bandgap Calculator from an explicit candidate bank, returning separate prompts and calculator answers with repeat policy and replay metadata.
A worked-example generator selects educational cases from a finite bank supplied by the user. It validates those cases with an existing calculator and returns prompts separately from calculated answers. Sampling changes which eligible cases appear; it does not create a new scientific model or invent measurements.
Semiconductor calculations combine physical units with empirical material and device models. Field, doping density, temperature and energy conventions are essential, and a finite mathematical answer does not establish that a model fits a device or operating regime.
Underlying calculator: Evaluate Eg0−alpha×T²/(T+beta) in eV using a stable logarithmic form. T and beta are positive kelvins, Eg0 positive eV, and alpha nonnegative eV/K; defaults are 1.17 eV, 4.73×10⁻⁴ eV/K and 636 K. Extrapolation can produce nonphysical bandgaps.1
When to use this tool
Use this generator when you already have candidate Varshni Bandgap Calculator inputs and want reproducible practice prompts with separately available calculator answers.
How to use the tool and how it works
- Prepare candidatesJson as distinct {id,input} records using the calculator fields and units below. Choose a replay seed, exercise count and repeat policy.
- Run the generator and read result.status and result.rejected before using the exercise list. Correct or deliberately exclude rejected candidates.
- Read each exercise’s prompt and separate answer. Keep result.model and result.replay with exported examples so units, source calculator and repeat policy remain explicit.
Validate every supplied {id,input} candidate through semiconductor/bandgap-temperature-varshni, then check its published input and output schemas. Calculator mechanics: Evaluate Eg0−alpha×T²/(T+beta) in eV using a stable logarithmic form. T and beta are positive kelvins, Eg0 positive eV, and alpha nonnegative eV/K; defaults are 1.17 eV, 4.73×10⁻⁴ eV/K and 636 K. Extrapolation can produce nonphysical bandgaps. Retain successful prompts/answers and structured rejection records. Draw eligible IDs with the chosen repeat policy using one advancing random source; equal answers do not merge different candidate IDs.2
| Input field | Meaning and units | Accepted input |
|---|---|---|
| seed | Nonblank replay seed, at most 256 UTF-16 code units; an injected source overrides seeded draws. | Text, at most 8,192 UTF-16 code units; narrower grammar/range rules also apply. |
| count | Requested exercise count, integer 1–32. | Finite safe integer satisfying the stated bounds. |
| withoutReplacement | True selects distinct eligible candidate IDs; false allows an eligible ID to recur. | Boolean true or false. |
| candidatesJson | JSON text for at most 128 exact {id,input} records. id is a nonempty distinct string; input is an object for the linked calculator using the candidate fields below. Equal inputs/answers may have different IDs. | Text, at most 8,192 UTF-16 code units; narrower grammar/range rules also apply. |
Every candidate is evaluated before selection. count is 1–32; candidatesJson contains at most 128 records with exactly id and input keys. Duplicate or empty IDs, malformed bank records and invalid outer fields cause an input error. A base-calculator failure is retained in rejected rather than silently discarded.
If no candidate succeeds, status is empty-eligible-bank. If distinct selection requests more cases than succeed, status is insufficient-eligible-bank. Both return no exercises. Otherwise status is ok. Every selected answer is the actual output object of the linked calculator, not a fabricated answer inferred from a prompt.
| Candidate input field | Meaning | Required / type and units |
|---|---|---|
| tempK | Lattice temperature. Must be a finite number greater than 0 K. | Required. number; unit: K; ≥5e-324 |
| egZeroEv | Bandgap at 0 K. Must be a finite number greater than 0 eV; silicon default 1.17 eV. | Optional under the source contract. number; unit: eV; ≥5e-324; default 1.17 |
| alphaVarshni | Varshni alpha coefficient. Must be a finite number at least 0 eV/K; silicon default 4.73e-4 eV/K. | Optional under the source contract. number; unit: eV/K; ≥0; default 0.000473 |
| betaVarshni | Varshni beta coefficient. Must be a finite number greater than 0 K; silicon default 636 K. | Optional under the source contract. number; unit: K; ≥5e-324; default 636 |
| Answer field | Meaning | Declared type and units |
|---|---|---|
| bandgapEv | Bandgap at temperature T (eV). Finite; may be negative if T far outside fitted range. | number; unit: eV |
These decoded structures show the examples inside JSON text fields. Supply their serialized JSON strings as input.
{
"candidatesJson": [
{
"id": "canonical",
"input": {
"alphaVarshni": 0.000473,
"betaVarshni": 636,
"egZeroEv": 1.17,
"tempK": 300
}
}
]
}| Output field | Interpretation |
|---|---|
| result | Object containing the fields below. |
| result.status | ok, empty-eligible-bank or insufficient-eligible-bank. Non-ok banks return no exercises. |
| result.baseToolId | Canonical calculator ID used to validate and answer every candidate. |
| result.model | Underlying calculator description and actual input/output port definitions, including declared units. |
| result.replay | Seed, source kind, requested count/replacement policy and actual selection-source call count. |
| result.eligibleCandidateIds | All successfully validated candidate IDs in original candidate-bank order. |
| result.rejected | Original candidate ID/index and structured calculator or evidence-budget error. |
| result.exercises | Selected {candidateId,sourceIndex,prompt,answer} records in draw order; answers are actual calculator output objects. |
| result.practiceLimits | Additional per-field upper bounds imposed by this practice wrapper before calculator execution; an empty object means no additional search cap. |
{
"input": {
"seed": "batch57-example",
"count": 1,
"withoutReplacement": true,
"candidatesJson": "[{\"id\":\"canonical\",\"input\":{\"alphaVarshni\":0.000473,\"betaVarshni\":636,\"egZeroEv\":1.17,\"tempK\":300}}]"
},
"output": {
"result": {
"status": "ok",
"baseToolId": "semiconductor/bandgap-temperature-varshni",
"model": {
"description": "Evaluate the Varshni temperature dependence of a semiconductor bandgap: Eg(T)=Eg0 - alpha·T²/(T+beta), using a logarithmic evaluation that avoids overflowing T². The coefficients are material-specific empirical fit parameters; extrapolation beyond their fitted temperature range can produce nonphysical results.",
"inputs": [
{
"description": "Lattice temperature. Must be a finite number greater than 0 K.",
"id": "tempK",
"label": "Temperature",
"required": true,
"valueType": {
"unit": "K",
"type": "number"
}
},
{
"description": "Bandgap at 0 K. Must be a finite number greater than 0 eV; silicon default 1.17 eV.",
"id": "egZeroEv",
"label": "Bandgap at 0 K",
"required": false,
"valueType": {
"unit": "eV",
"type": "number"
}
},
{
"description": "Varshni alpha coefficient. Must be a finite number at least 0 eV/K; silicon default 4.73e-4 eV/K.",
"id": "alphaVarshni",
"label": "Varshni alpha",
"required": false,
"valueType": {
"unit": "eV/K",
"type": "number"
}
},
{
"description": "Varshni beta coefficient. Must be a finite number greater than 0 K; silicon default 636 K.",
"id": "betaVarshni",
"label": "Varshni beta",
"required": false,
"valueType": {
"unit": "K",
"type": "number"
}
}
],
"outputs": [
{
"description": "Bandgap at temperature T (eV). Finite; may be negative if T far outside fitted range.",
"id": "bandgapEv",
"label": "Bandgap",
"valueType": {
"unit": "eV",
"type": "number"
}
}
]
},
"replay": {
"seed": "batch57-example",
"source": "seeded",
"count": 1,
"withoutReplacement": true,
"draws": 1
},
"eligibleCandidateIds": [
"canonical"
],
"rejected": [],
"exercises": [
{
"candidateId": "canonical",
"sourceIndex": 0,
"prompt": {
"alphaVarshni": 0.000473,
"betaVarshni": 636,
"egZeroEv": 1.17,
"tempK": 300
},
"answer": {
"bandgapEv": 1.1245192307692307
}
}
],
"practiceLimits": {}
}
}
}Limitations and assumptions
- The exercise bank and its measurements are supplied by the caller. The generator selects validated cases; it does not establish the calculator model’s scientific validity, topic coverage or difficulty. Read status and rejected before using exercises.
- A nonblank seed of at most 256 UTF-16 code units is required. With ordinary use the seed initializes one deterministic advancing source; an application-injected random source takes precedence and the seed alone then cannot replay the result. Uniformity statements assume independent uniform source draws in [0,1). Source calls are bounded and degenerate injected sources can fail a rejection budget. This is not cryptographically secure.
- Candidate inputs and answers remain subject to the linked calculator’s model assumptions. The sampler tests its executable contract, not real-world truth, calibration, fitness for a decision or scientific validity.
- Candidates failing published source schemas are retained with practice_source_input_schema or practice_source_output_schema errors. The wrapper also reports practice_resource_limit for its extra integer-search caps and practice_evidence_budget_exceeded for oversized prompt/answer evidence. Check practiceLimits; the selected output has an additional aggregate evidence budget.
- Every declared outer field is required and unknown outer fields are rejected. Strings and JSON text have size limits, lists are bounded to 128 entries, and JSON nesting/node counts are bounded. Tighter operation-specific bounds stated above take precedence.
- Numeric results must be finite. Large physical magnitudes are allowed by the output evidence checks, while integer fields and source schemas retain their exactness rules. Source indexes are zero-based. Floating-point residues can remain even where an ideal formula gives an exact zero.
Alternative or Complementary methods
Use the linked calculator to inspect an individual candidate directly. Use this generator to select and package already supplied cases; build or review the exercise bank separately so topic coverage and difficulty reflect the teaching objective.
References
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