SCIENCE · INVESTIGATE AND EXPLAIN

Separate spatial attenuation, counting spread and dose claims

A spatial attenuation model and a radioactive survival law can both be exponential while describing different independent variables. Here a fixed narrow beam has source expectation800×2^(−x/H) counts/min after thicknessx, whereH is a supplied half-value thickness in centimetres. Its source strength is not decaying with elapsed time in this model. The equivalent coefficient is μ=ln2/H with inverse-length units; it is not a decay constant with inverse-time units.

Learning outcome: Compare thickness 0,H,2H with baseline retained and removed. Separately calculate one-minute gross and background-subtracted counting spreads. Explain why thickness, observation duration, source half-life and biological dose cannot be interchanged.

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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Atx=H=2 cm, source rate is400 counts/min and gross rate 420. A one-minute gross count has Poisson standard deviation sqrt420≈20.4939 counts. Subtracting an independent one-minute background sample gives net standard deviation sqrt440≈20.9762 counts, not sqrt400.

Open the experiment

Predict

For independent equal-duration gross and background Poisson counts, the variance of their difference is:

Your prediction and reason

Experiment

Compare thickness 0,H,2H with baseline retained and removed. Separately calculate one-minute gross and background-subtracted counting spreads. Explain why thickness, observation duration, source half-life and biological dose cannot be interchanged.

Distinguish spatial attenuation from decay and counting noise

Worked teaching model · synthetic values · Fixed source and narrow-beam geometry with expectation rate 800*2^(−x/H)+20 counts/min. No buildup, energy dependence, dead time or changing geometry. A stipulated one-minute counting interval gives a Poisson count model; no real radiation source, shielding prescription, dose estimate or health advice.

Absorber thickness: 2 cm Supplied half-value thickness: 2 cm

Reset keeps your prediction, captures and practice history.

Spatial attenuation with a nonzero counting baseline

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: gross indicated rate (solid).2: background-subtracted source contribution (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Spatial attenuation with a nonzero counting baseline

gross indicated rate

thickness (cm)	expected rate (counts/min)
0	820
0.075	799.474
0.15	779.474
0.225	759.987
0.3	741
0.375	722.501
0.45	704.476
0.525	686.914
0.6	669.802
0.675	653.129
0.75	636.884
0.825	621.056
0.9	605.634
0.975	590.608
1.05	575.967
1.125	561.702
1.2	547.803
1.275	534.261
1.35	521.066
1.425	508.209
1.5	495.683
1.575	483.478
1.65	471.586
1.725	459.999
1.8	448.709
1.875	437.71
1.95	426.992
2.025	416.549
2.1	406.375
2.175	396.461
2.25	386.802
2.325	377.39
2.4	368.22
2.475	359.286
2.55	350.58
2.625	342.098
2.7	333.834
2.775	325.781
2.85	317.935
2.925	310.291
3	302.843
3.075	295.586
3.15	288.515
3.225	281.625
3.3	274.912
3.375	268.372
3.45	261.999
3.525	255.79
3.6	249.74
3.675	243.845
3.75	238.102
3.825	232.505
3.9	227.053
3.975	221.74
4.05	216.564
4.125	211.521
4.2	206.607
4.275	201.819
4.35	197.154
4.425	192.608
4.5	188.179
4.575	183.864
4.65	179.66
4.725	175.563
4.8	171.572
4.875	167.683
4.95	163.893
5.025	160.201
5.1	156.604
5.175	153.099
5.25	149.684
5.325	146.357
5.4	143.114
5.475	139.956
5.55	136.878
5.625	133.879
5.7	130.957
5.775	128.11
5.85	125.336
5.925	122.633
6	120

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.

background-subtracted source contribution

thickness (cm)	expected rate (counts/min)
0	800
0.075	779.474
0.15	759.474
0.225	739.987
0.3	721
0.375	702.501
0.45	684.476
0.525	666.914
0.6	649.802
0.675	633.129
0.75	616.884
0.825	601.056
0.9	585.634
0.975	570.608
1.05	555.967
1.125	541.702
1.2	527.803
1.275	514.261
1.35	501.066
1.425	488.209
1.5	475.683
1.575	463.478
1.65	451.586
1.725	439.999
1.8	428.709
1.875	417.71
1.95	406.992
2.025	396.549
2.1	386.375
2.175	376.461
2.25	366.802
2.325	357.39
2.4	348.22
2.475	339.286
2.55	330.58
2.625	322.098
2.7	313.834
2.775	305.781
2.85	297.935
2.925	290.291
3	282.843
3.075	275.586
3.15	268.515
3.225	261.625
3.3	254.912
3.375	248.372
3.45	241.999
3.525	235.79
3.6	229.74
3.675	223.845
3.75	218.102
3.825	212.505
3.9	207.053
3.975	201.74
4.05	196.564
4.125	191.521
4.2	186.607
4.275	181.819
4.35	177.154
4.425	172.608
4.5	168.179
4.575	163.864
4.65	159.66
4.725	155.563
4.8	151.572
4.875	147.683
4.95	143.893
5.025	140.201
5.1	136.604
5.175	133.099
5.25	129.684
5.325	126.357
5.4	123.114
5.475	119.956
5.55	116.878
5.625	113.879
5.7	110.957
5.775	108.11
5.85	105.336
5.925	102.633
6	100

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
Source expectation rate after attenuation	400	counts/min
Gross expectation rate including background	420	counts/min
Background-subtracted transmission	0.5	ratio
Linear attenuation coefficient	0.346574	cm⁻¹
Expected one-minute gross count	420	counts
Poisson standard deviation of gross count	20.4939	counts
Net-count standard deviation with independent one-minute background	20.9762	counts
Fractional gross-count standard deviation	0.048795	ratio

A half-value thickness changes transmission with distance; it is not the source half-life.

The narrow-beam model gives I/I0=2^(−x/H)=exp(−mu x), with mu=ln2/H. Background must be removed before taking a transmission ratio. For independent Poisson gross and equal-duration background counts, variances add in their difference, so net variance has expectation gross mean plus background mean. These are stochastic spread estimates, not hard bounds or a promise that observed counts equal their means. Detector counts alone do not identify activity, deposited energy, absorbed dose or biological consequence.

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

A half-value thickness changes transmission with distance; it is not the source half-life.

Observation to attach to a capture

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

Understand

State the event and measurement model

Reconcile the quantitative evidence

The instrument adds a constant20 counts/min background. Subtract that baseline before forming transmission: atx=H the source contribution halves, but the gross indication does not. The assumptions exclude scattered buildup, energy dependence, detector dead time and changing geometry, so the formula is not a practical shielding design or a material property measurement. A transmission at one energy cannot establish safety or spectral behavior for a different source.

Preserve limits and alternative explanations

For a stipulated independent Poisson one-minute count, mean count equals the numerical rate in counts/min and variance equals that mean. Thus the standard deviation is its square root, not a hard bound on every observation. If an independent one-minute background count is subtracted, its variance adds rather than cancels; with gross meanG and background meanB, net-count variance has expectationG+B. Longer counting can reduce relative random spread while leaving systematic efficiency or baseline errors unresolved. Detector counts alone do not identify deposited energy, absorbed dose or biological effects; this is not personal medical advice or an exposure protocol.

Worked original case

Investigation

Changed-case practice

Try the changed case before feedback

With initial source rate 800 counts/min and half-value thickness 2 cm, find source rate after4 cm, excluding background.

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 cases checked 31 October 2026 against the primary explanation. These are bounded virtual models, not instructions to illuminate, accelerate, collect or expose real material. No real laboratory participation, detector certification, personal medical conclusion or independent mastery is claimed. Keep first work, critique, revision and genuinely new assigned evidence separately from public practice.

Supplemental source checks, 31 October 2026: NIST narrow-beam attenuation and NIST Poisson statistics. Numerical cases remain original supplied models.

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

Which independent observation would reject the selected model or calibration premise, and which quantity is not identified by these records?

Your explanation and limitations

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