= Bothe-Geiger coincidence experiment
{c}
In 1924–1925, <Walther Bothe> and <Hans Geiger> tested event-level <Compton scattering> correlations against the statistical conservation proposed by <BKS theory>.<The Coincidence Method by Walther Bothe>[The coincidence method, pp. 271–273]{ref}
Needle <Geiger counters> shared a wall. <X-rays> passed beside it through <hydrogen>; recoil <Electrons> registered in one counter and scattered <photons> triggered <Electron> release in the other. Photographic records resolved approximately $10^{-4}\,\mathrm s$.<The Coincidence Method by Walther Bothe>[p. 272: apparatus and timing]{ref}
\Image[/bothe-geiger-needle-counter-geometry.png]
{title=Two detection channels in the <Bothe-Geiger coincidence experiment>}
{id=image-bothe-geiger-needle-counter-geometry}
{description=Original functional schematic based on Bothe's account; dimensions, angles and trace marks are illustrative. The common wall separates the counter regions; the direct beam avoids it. The drawing specifies neither wall material nor chamber shape. <Photon> detection requires <Electron> release. Both records share a time coordinate.}
The coincidence excess matched paired-emission expectations. Bothe distinguishes a later directional test from this timing experiment.<The Coincidence Method by Walther Bothe>[p. 273: paired emission and later directional test]{ref}
The two channels have different physical roles. The first responds to the local recoil <Electron>'s <ionization>. The second converts a scattered <photon> into an <Electron> response, as in the <photoelectric effect>. A large counter signal is an amplified indication of an interaction, not a direct measurement of the incoming <photon>'s <energy>. The <detection efficiency> and accepted directions determine how many genuine partners can be seen. Consequently, failure to record every partner is compatible with paired production.
The relevant distinction is between an average balance and a balance in each event. For an initially stationary free <Electron>, <Compton recoil kinematics> gives
$$
E=E'+T_e,\qquad
\frac{E}{c}\hat{\mathbf n}=\frac{E'}{c}\hat{\mathbf n}'+\mathbf p_e.
$$
Here $E,E'$ are incident and scattered <photon> <energies>, $T_e$ is recoil <kinetic energy>, and $\mathbf p_e$ is <Electron> <momentum>. These are paired constraints, not just equations between averages over unrelated detections. Summing correctly paired events preserves the equations; throwing away the pair identities loses information. Two samples can have identical separate <energy> distributions while differing in their <joint probability distribution>.
The experiment therefore supplied evidence for the event association required by <photon> recoil, rather than weighing both outgoing <energies> in every event. A statistically established excess above <accidental coincidences> matters; one near-simultaneous pair cannot establish the association. It also cannot by itself settle every question about <wave-particle duality>.
Preparation, readout and background checks address different failure modes. The direct beam must avoid the second counter, and the counter boundary must prevent a recoil <Electron> from masquerading as its own <photon> partner. Separate-channel rates constrain <accidental coincidences>; common disturbances or electrical coupling could instead create a correlated instrumental background. A useful diagnostic is to displace one record in time: under a <stationary process> assumption, unrelated-pair expectations persist while prompt partners separate. This is the explanatory control modeled below, not an assertion that Bothe performed that particular record-shifting procedure. A background estimate and a statistical comparison are essential to the inference.
The existing <coincidence circuit> article explains electronic selection, and <accidental coincidence> derives the accepted-window background. Here pairing can be understood directly from the two recorded time sequences, without imposing a modern <detector trigger> on the historical readout.
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