Past exam of the physics A course of the University of Cambridge / 2021 / Paper 1 / Section C / C8 / d

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- Solution d

Solution

For an independent count with a Poisson distribution, collected for exposure

T_{i}

at rate

R_{i}

, the expected count is

N_{i} = R_{i} T_{i}

and its standard deviation is

N_{i}

. The fractional rate uncertainty is consequently

\frac{σ _{R_{i}}}{R _{i}} ≃ \frac{1}{N _{i}} .

A target of

0.15

requires

N_{i} ≃ (0.15)^{- 2} = 44.44

counts at each proper-time setting. The exposure time for a Poisson rate at fixed fractional precision is therefore

T_{i} = 1/ (0.1 5^{2} R_{i})

t (ps) 0.25 0.75 1.25 1.75 2.25 2.75 T_{i} (days) 0.03319 0.12346 0.40404 1.43369 3.17460 14.81481

Since the settings are acquired independently rather than simultaneously,

T_{total} = \frac{1}{0.1 5 ^{2}} i \sum \frac{1}{R _{i}} ≃ 19.98 days ≃ 20 days .

Nearly three quarters of the exposure is spent at the latest, lowest-rate setting. Proper time in picoseconds labels the decay distribution; it is not the laboratory acquisition time in days. This estimate assumes counting statistics dominate, with no additional background-subtraction uncertainty or acquisition overhead.

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