Photon recoil / Angular spectra

Compton Angular Spectrometer

How does a scattered photon's energy change with angle, and what does that reveal? The wider the angle, the less energy the photon keeps; the recoiling electron carries off the rest. A 662 keV gamma ray deflected by 90 degrees keeps about 288 keV. Energy and momentum conservation fix this curve through two numbers, the incoming energy and the electron's rest energy. One angle cannot separate them, but a scan across a wide range of angles pins down both.

Swing a calibrated energy detector around a scattering target and record the pulse height of each photopeak event. From the scan over angles, estimate the energy of the incoming photons and the rest energy of the electron.

The electrons are free and at rest, each photon scatters once, the photons are unpolarized, and a narrow slit in the meridional plane selects the angle. Only events that leave their full energy in the photopeak are recorded, with Gaussian energy resolution. There is no continuum, and electron binding, detector efficiency and absolute rates are not modeled. The angle and energy calibrations are taken as known. The simulated data test the estimation method; they do not reproduce the historical discovery or a QED calculation.

Incoming photons travel rightward. The detector faces the target at the selected angle. Paths and sizes are schematic; energy data appear below.

Calibrated pulse heights of the kept events

Turn the detector to build an energy curve.

Collimated photonPassive scattering targetRotating energy detector

A scan records the selected number of individual photopeak pulses at each of fifteen angles, from twenty through one hundred sixty degrees. These are conditional detected events, not a count-rate or cross-section measurement.

Ready. Predict, then acquire.

Raw detector records

Energy falls as the angle opens.

$\theta/\mathrm{deg},\quad E_\gamma/\mathrm{keV}$

Retained pulsesFitted meanSaved pulses

Thin gold: separate source mean, not acquired data. Every displayed point comes from a retained pulse. Dense scans can overlap.

Measured peak centroids and fit residuals

Check the angular response.

$\theta/\mathrm{deg},\quad (\overline E-\widehat E)/s_{\overline E}$

Measured photopeak at the current angle

Inspect the pulse-height population.

$E_\gamma/\mathrm{keV},\quad N_{\rm bin}$

This histogram rebins only the retained pulses at the current angle. Moving the detector after a scan selects another recorded population. Counts outside the displayed range remain in the export and fit.

Inference from the records alone

What the retained observations support

Retained events

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Inferred incident energy

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Inferred electron rest energy

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Centroid residual RMS

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Known energy calibration and slit geometry are required. The fit searches incident energy from 450 to 850 keV and rest energy from 300 to 750 keV. Boundary fits are flagged; residual RMS is not a parameter confidence interval.

Saved comparison, kept in memory

Save a run to compare or restore its settings and raw events.

Measure recoil in coincidence Read the historical evidence ORTEC apparatus reference Compton 1923 paper

Start with the essentials

Focus question: How does a scattered photon's energy change with angle, and what does that reveal?
One-sentence intuition: Keep what the simulation assumes about the source apart from what you estimate from the recorded events.

Core mathematical model

Energy of the scattered photon

E_\gamma(\theta)=\frac{E_0}{1+\frac{E_0}{m_ec^2}(1-\cos\theta)}

The angle is measured from the incoming photon direction; it is not the electron's recoil angle. Energy and momentum conservation together fix this relation.

Mean energy through a slit of finite width

\overline E_\gamma=\frac{\int_{\theta-a}^{\theta+a}E_\gamma(\vartheta)K(\vartheta)\,d\vartheta}{\int_{\theta-a}^{\theta+a}K(\vartheta)\,d\vartheta}

The meridional slit passes every angle in its range equally, and its height is negligible. Each angle is weighted by the relative unpolarized Klein–Nishina density. The slit covers one azimuth only, not a full ring around the beam.

Finding the two energy scales

\frac1{E_\gamma}=\frac1{E_0}+\frac{1-\cos\theta}{m_ec^2}

For a slit of zero width, this identity shows why both energies can be determined. The actual fit uses the mean energies through the finite slit; it does not take reciprocals of individual noisy pulses. The source values are not given to the fit.

Common difficulties

The simulation already contains the law

Typical misconceptionGetting the source parameter back independently confirms the law.

Better mental modelThe events were generated from that law, so getting the parameter back tests the measurement and estimation steps. The data are not historical observations, they do not give a confidence interval for the parameter, and they are not external scientific certification.