Separate energy calibration from source identification
A measured spectrum first needs a response model. This fictional instrument indicates y=gE+b for true transition energy E, common gain g and baseline b. Independently stipulated reference energies2 and4 eV provide two equations for two response parameters. Their difference removes b, giving g=(y4−y2)/2; then b=y2−2g. A reference must be known independently of the instrument being calibrated, otherwise the algebra merely restates its own output.
Learning outcome: Vary gain and offset separately, using only the first two references. Freeze the third prediction before checking. Describe one independent calibration failure that a same-generator perfect fit cannot exclude.
Starting recap
Separate per-event energy, expectation rate and measurement indication. Check particle identities, conservation, threshold branches and calibration roles before making a source or detector claim.
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For gain 1.05 and offset 0.1 eV, the2 and4 eV references indicate2.2 and4.3 eV. Their difference recovers gain 1.05 and offset 0.1. The unused6 eV reference should indicate6.4 eV. An unknown indication3.25 eV corresponds to 3 eV, not3.25 eV.
Open the experimentPredict
After fitting gain and offset from independently known2 and4 eV references, which is an unused response test?
Experiment
Vary gain and offset separately, using only the first two references. Freeze the third prediction before checking. Describe one independent calibration failure that a same-generator perfect fit cannot exclude.
Calibrate a line-energy indication before inferring an unknown
Worked teaching model · synthetic values · Fictitious reference transitions have independently stipulated energies 2 and 4 eV. Indicated energy y=gE+b. Use those two only for calibration; a fresh 6 eV reference tests the affine response. An unknown source has true 3 eV in this revealed generator, not a real element identification.
Reset keeps your prediction, captures and practice history.
Reference fit and a genuinely unused energy record
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: fit from 2 and 4 eV references (solid).2: fresh 6 eV reference (filled markers).3: unknown after inverse inference, not an independent standard (filled markers). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Reference fit and a genuinely unused energy record
fit from 2 and 4 eV references
| independently stipulated energy (eV) | indicated energy (eV) |
|---|---|
| 0 | 0.1 |
| 0.075 | 0.17875 |
| 0.15 | 0.2575 |
| 0.225 | 0.33625 |
| 0.3 | 0.415 |
| 0.375 | 0.49375 |
| 0.45 | 0.5725 |
| 0.525 | 0.65125 |
| 0.6 | 0.73 |
| 0.675 | 0.80875 |
| 0.75 | 0.8875 |
| 0.825 | 0.96625 |
| 0.9 | 1.045 |
| 0.975 | 1.12375 |
| 1.05 | 1.2025 |
| 1.125 | 1.28125 |
| 1.2 | 1.36 |
| 1.275 | 1.43875 |
| 1.35 | 1.5175 |
| 1.425 | 1.59625 |
| 1.5 | 1.675 |
| 1.575 | 1.75375 |
| 1.65 | 1.8325 |
| 1.725 | 1.91125 |
| 1.8 | 1.99 |
| 1.875 | 2.06875 |
| 1.95 | 2.1475 |
| 2.025 | 2.22625 |
| 2.1 | 2.305 |
| 2.175 | 2.38375 |
| 2.25 | 2.4625 |
| 2.325 | 2.54125 |
| 2.4 | 2.62 |
| 2.475 | 2.69875 |
| 2.55 | 2.7775 |
| 2.625 | 2.85625 |
| 2.7 | 2.935 |
| 2.775 | 3.01375 |
| 2.85 | 3.0925 |
| 2.925 | 3.17125 |
| 3 | 3.25 |
| 3.075 | 3.32875 |
| 3.15 | 3.4075 |
| 3.225 | 3.48625 |
| 3.3 | 3.565 |
| 3.375 | 3.64375 |
| 3.45 | 3.7225 |
| 3.525 | 3.80125 |
| 3.6 | 3.88 |
| 3.675 | 3.95875 |
| 3.75 | 4.0375 |
| 3.825 | 4.11625 |
| 3.9 | 4.195 |
| 3.975 | 4.27375 |
| 4.05 | 4.3525 |
| 4.125 | 4.43125 |
| 4.2 | 4.51 |
| 4.275 | 4.58875 |
| 4.35 | 4.6675 |
| 4.425 | 4.74625 |
| 4.5 | 4.825 |
| 4.575 | 4.90375 |
| 4.65 | 4.9825 |
| 4.725 | 5.06125 |
| 4.8 | 5.14 |
| 4.875 | 5.21875 |
| 4.95 | 5.2975 |
| 5.025 | 5.37625 |
| 5.1 | 5.455 |
| 5.175 | 5.53375 |
| 5.25 | 5.6125 |
| 5.325 | 5.69125 |
| 5.4 | 5.77 |
| 5.475 | 5.84875 |
| 5.55 | 5.9275 |
| 5.625 | 6.00625 |
| 5.7 | 6.085 |
| 5.775 | 6.16375 |
| 5.85 | 6.2425 |
| 5.925 | 6.32125 |
| 6 | 6.4 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
fresh 6 eV reference
| independently stipulated energy (eV) | indicated energy (eV) |
|---|---|
| 6 | 6.4 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
unknown after inverse inference, not an independent standard
| independently stipulated energy (eV) | indicated energy (eV) |
|---|---|
| 3 | 3.25 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| First reference indication | 2.2 | eV indicated |
| Second reference indication | 4.3 | eV indicated |
| Fitted gain | 1.05 | ratio |
| Fitted offset | 0.1 | eV indicated |
| Fresh six-eV reference indication | 6.4 | eV indicated |
| Frozen six-eV prediction | 6.4 | eV indicated |
| Unknown source indication | 3.25 | eV indicated |
| Inferred unknown energy | 3 | eV |
| Unknown wavelength using supplied hc | 413.333 | nm |
Two known references identify an affine response, but only independent unused evidence tests that assumption.
Differences remove the offset: gain=(y4−y2)/(4−2). Then offset=y2−2gain and unknown energy=(yunknown−offset)/gain. The third known line was not used to calibrate. The unknown plotted at its inferred energy is not an additional calibration check. A shared bias in reference energies or nonlinearity outside their span remains a possible failure; agreement with the revealed synthetic generator is not a real instrument certification.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Two known references identify an affine response, but only independent unused evidence tests that assumption.
Differences remove the offset: gain=(y4−y2)/(4−2). Then offset=y2−2gain and unknown energy=(yunknown−offset)/gain. The third known line was not used to calibrate. The unknown plotted at its inferred energy is not an additional calibration check. A shared bias in reference energies or nonlinearity outside their span remains a possible failure; agreement with the revealed synthetic generator is not a real instrument certification.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
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Which independent observation would reject the selected model or calibration premise, and which quantity is not identified by these records?
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