SCIENCE · INVESTIGATE AND EXPLAIN

Invert a nonlinear count response before subtracting an independent background

A supplied nonparalyzable detector model has steady mean indicated rate m=r/(1+tau r), where r is the true interaction rate and tau is an independently calibrated dead time. Each recorded event excludes a fixed interval from further recording; incoming events in that interval do not extend it. The inverse is r=m/(1−tau m), valid only when tau m is below one.

Learning outcome: Invert total and background means independently and show the failed subtract-first calculation. Fit the first two corrected source means, freeze the third indicated response and compare it with the supplied fresh record. State why unknown efficiency, dead time or a changing background would weaken the inference.

Starting recap

Review finite sums, exponential and logarithmic functions, radial integration measures, first-order coupled equations and mass-energy units. Keep state counts, occupations, calibration records and unused tests distinct.

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Try

For true source100/s, background20/s and tau=.002 s, the initial total indication is120/1.24≈96.774194/s. The independent background indicates20/1.04≈19.230769/s. Inverting these separately gives120 and20/s, hence source100/s. Directly subtracting the indications instead gives about77.543424/s, which is not the true source rate.

Open the experiment

Predict

Which sequence correctly extracts the source rate under the stated model?

Your prediction and reason

Experiment

Invert total and background means independently and show the failed subtract-first calculation. Fit the first two corrected source means, freeze the third indicated response and compare it with the supplied fresh record. State why unknown efficiency, dead time or a changing background would weaken the inference.

Correct total detector response before subtracting background

Worked teaching model · synthetic values · Synthetic separately steady-rate replicas represent nominal ages0, 2, 4 s with true source interaction rates 100 exp(−lambda t) plus constant background 20/s. Each acquisition is steady and sufficiently long for mean-rate response m=r/(1+tau r); these are not exact finite windows of one continuously decaying source. Independent background run shares calibrated tau. Ages0, 2 fit amplitude/lambda; 4 remains fresh. No actual detector or Poisson count uncertainty is claimed after dead time.

Generating exponential rate parameter: 0.2 per nominal second Independently calibrated nonparalyzable dead time: 0.002 s

Reset keeps your prediction, captures and practice history.

Nonlinear response must be applied to the total rate

Graph axes rescale independently; read units and values, not screen angles. 1: two-record fitted response prediction (solid).2: fresh nominal-age-four record (open-circle marked location). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Nonlinear response must be applied to the total rate

two-record fitted response prediction

nominal age of steady replica (s)	mean indicated rate (1/s)
0	96.77419
0.06666667	95.91095
0.1333333	95.05551
0.2	94.20785
0.2666667	93.36792
0.3333333	92.53571
0.4	91.71117
0.4666667	90.89428
0.5333333	90.08499
0.6	89.28328
0.6666667	88.48911
0.7333333	87.70244
0.8	86.92323
0.8666667	86.15145
0.9333333	85.38707
1	84.63003
1.066667	83.88031
1.133333	83.13786
1.2	82.40264
1.266667	81.67461
1.333333	80.95373
1.4	80.23996
1.466667	79.53326
1.533333	78.83358
1.6	78.14088
1.666667	77.45512
1.733333	76.77626
1.8	76.10424
1.866667	75.43903
1.933333	74.78058
2	74.12884
2.066667	73.48378
2.133333	72.84534
2.2	72.21348
2.266667	71.58816
2.333333	70.96932
2.4	70.35692
2.466667	69.75092
2.533333	69.15127
2.6	68.55792
2.666667	67.97082
2.733333	67.38994
2.8	66.81521
2.866667	66.24659
2.933333	65.68404
3	65.12752
3.066667	64.57696
3.133333	64.03232
3.2	63.49356
3.266667	62.96064
3.333333	62.43349
3.4	61.91207
3.466667	61.39635
3.533333	60.88626
3.6	60.38176
3.666667	59.88281
3.733333	59.38936
3.8	58.90135
3.866667	58.41875
3.933333	57.9415
4	57.46957

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 nominal-age-four record

nominal age of steady replica (s)	mean indicated rate (1/s)
4	57.46957

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.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Independent background mean indicated rate	19.23077	per second
Total indicated mean rate at nominal age 0	96.77419	per second
Total indicated mean rate at nominal age 2	74.12884	per second
Total indicated mean rate at nominal age 4	57.46957	per second
Background corrected before subtraction	20	per second
Fitted exponential rate parameter	0.2	per nominal second
Frozen indicated rate at nominal age four	57.46957	per second
Fresh indicated rate at nominal age four	57.46957	per second
Naive rate from background-subtracted indicated means	0.1726799	per nominal second
Largest occupied dead-time fraction	0.1935484	ratio

Invert the total and independent background responses separately, then subtract true interaction rates.

For a steady nonparalyzable response m=r/(1+tau r), inversion is r=m/(1−tau m), valid only below saturation. Nonlinearity means inverse(mTotal−mBackground) is not the source rate. Correct the first two total means, subtract corrected independent background, fit lambda and amplitude, then pass the frozen third true-rate prediction through the detector law. Unknown tau or background adds inverse ambiguity. Dead-time output counts are not generally Poisson; no sqrt(count) confidence claim is made.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Invert the total and independent background responses separately, then subtract true interaction rates.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Define the state and evidence

Derive the quantitative relation

The original evidence packet uses separately steady-rate replicas indexed by nominal ages0,2,4 s. Their source rates are A exp(−lambda t), plus a common background. Each replica is held steady throughout its sufficiently long mean-rate acquisition. This deliberately avoids treating the steady formula as an exact finite-window count law for one continuously decaying specimen. Reported means are synthetic, not integer realized counts.

Test a changed model or boundary

An independent background-only run uses the same calibrated detector. Invert each total rate and the background rate separately, then subtract the corrected background. Nonlinearity means correcting the difference of indicated rates is generally wrong. Use only the corrected source means at nominal ages0 and2 to fit lambda=ln(s0/s2)/2 and amplitude s0; freeze the age4 total indicated prediction before revealing it.

Limit the inference

Unknown dead time or background would add inverse ambiguity; additional independent calibration is needed rather than silently borrowing generator values. Dead-time-modified output counts are not generally Poisson. The workbench does not attach a square-root-count confidence interval, claim an actual activity without efficiency, or provide detector/radiation operating instructions.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

With calibrated tau=.001 s and indicated steady mean rate200/s, find the total true interaction rate per second.

Numerical result in the requested units

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.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Which independent calibration, state-counting or physical-domain premise is needed, and which fresh observation could challenge the inference?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

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