Keep a band-gap interpretation conditional on transport assumptions

Define the state and evidence

A supplied intrinsic-carrier transport family has log conductivity L(T)=A+p ln(T/300)−Theta/T, where Theta=Eg/(2kB). The logarithm is of conductivity relative to a declared reference unit. The exponent p describes a stipulated mobility or prefactor temperature dependence. A is unknown to the inverse analyst even though the teaching generator reveals A=0.

Derive the quantitative relation

With p independently fixed, subtract p ln(T/300) and fit A and Theta from only the 300 K and 400 K records. Their difference gives Theta=[L400−L300−p ln(4/3)]/[1/300−1/400]. Freeze the 500 K prediction before revealing its record. Ignoring the mobility term produces an apparent activation parameter shifted by p ln(4/3)/(1/300−1/400).

Test a changed model or boundary

Compare an equally calibrated contact-limited alternative R(T)=C+D/T using the same two conductivity records converted to resistance. This family also has two free coefficients. Its unused-temperature resistance must remain positive before taking a logarithm. A negative extrapolation rejects that family at that temperature; a positive but inaccurate prediction can be assessed against supplied error bounds.

Limit the inference

A successful activation curve does not uniquely establish a band gap. Doping, contact effects, a different mobility exponent, temperature readout error or a varying response gain can alter the interpretation. The workbench reports Eg only conditional on its intrinsic-carrier law and fixed p. No fabrication, heating or electrical measurement is requested.

Worked original case

For Theta=1000 K and p=1, L300=−10/3 and L400=ln(4/3)−2.5. Removing the known mobility term recovers 1000 K, while ignoring it gives about 1345.218487 K. With kB=8.617333262×10^−5 eV/K, the conditional gap is about 0.172346665 eV. The unused 500 K log response is ln(5/3)−2.

Investigation

Fit both two-parameter families with the same two records, keeping the mobility exponent independent. Freeze their third-temperature predictions and check positivity before logarithms. Change the assumed exponent and identify which inferred parameter shifts. Retain an independent-contact or temperature-calibration test.

Changed-case practice

Try the changed case before feedback

With Theta=750 K and supplied kB=8.617333262e−5 eV/K, find the conditional gap Eg in eV.

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original derivations and virtual evidence checked 31 October 2026 against the primary explanation. All questions and numerical packets are original. Supplied theoretical distributions and synthetic records are not actual laboratory measurements. No nuclear, electrical, heating, beam or material-handling procedures are requested. Retain first reasoning, actual criterion critique, revision and a separately assigned unfamiliar task in the course evidence workspace; public keys do not authenticate mastery.