Accidental Purity Floor Three-Row Scan - Detector Physics: Timing and Spatial Reconstruction Step by Step

Accidental purity measures the fraction of coincidence counts that are true pairs rather than random background, so higher singles rates push purity below a minimum floor. The three rows give purities 4/8 = 1/2, 4/12 = 1/3, and 4/16 = 1/4 against floor 1/3, yielding pass, pass, and fail exactly for the stated toy rate ledger; real detectors have rate instability, background estimation error, timing resolution, and counting statistics.

Accidental rate grows the denominator and lowers coincidence purity. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Small accidental product stays above the floor

Rates 1 hertz and 2 hertz make accidental rate 4 hertz.

p=4/8=1/2,\quad p_{\min}=1/3,\quad\text{pass}

Middle accidental product lands on the floor

Rates 2 hertz and 2 hertz make accidental rate 8 hertz.

p=4/12=1/3,\quad p_{\min}=1/3,\quad\text{pass}

Larger accidental product drops below the floor

Rates 2 hertz and 3 hertz make accidental rate 12 hertz.

p=4/16=1/4,\quad p_{\min}=1/3,\quad\text{fail}