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SCIENCE · INVESTIGATE AND EXPLAIN

Close a photon recoil energy and momentum ledger

A photon has momentum magnitude E/c along its propagation direction. In the displayed plane the incident photon travels along positive x; the scattered photon leaves at angle θ. Conservation therefore gives electron recoil components px=E−Eprime cosθ and py=−Eprime sinθ when momentum is expressed in keV/c and photon energy in keV. Subtracting scalar momentum magnitudes would discard the direction that makes this account meaningful.

Learning outcome: Compare 0°,90°,180° at two incident energies. Reconstruct recoil from the vector difference before using its energy formula. Retain both residuals and quantify where the classical kinetic estimate disagrees.

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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Try

For E=100 keV and θ=90°, Eprime=51100/611≈83.633388 keV. Recoil components are (100,−83.633388) keV/c and its kinetic energy is about 16.366612 keV, matching photon loss. A nonrelativistic kinetic estimate is larger and does not close this exact account.

Open the experiment

Predict

In the stated initially-rest electron model, exactly forward photon scattering gives:

Experiment

Compare 0°,90°,180° at two incident energies. Reconstruct recoil from the vector difference before using its energy formula. Retain both residuals and quantify where the classical kinetic estimate disagrees.

Close a photon and recoil energy-momentum account

Worked teaching model · synthetic values · One photon scatters from one initially stationary free electron. Electron rest energy is supplied as 511 keV. Momentum components use keV/c; recoil kinetic energy uses the relativistic energy relation, supplied for algebraic checking rather than a full relativity course.

Reset keeps your prediction, captures and practice history.

Photon and electron momenta share one scale

momentum y (keV/c)momentum x (keV/c)010083.63340

Spatial axes use the same pixels per physical unit, preserving geometry. 1: incident photon momentum (solid).2: outgoing photon then recoil, head to tail (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Photon and electron momenta share one scale

incident photon momentum

momentum x (keV/c)momentum y (keV/c)
00
1000

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.

outgoing photon then recoil, head to tail

momentum x (keV/c)momentum y (keV/c)
00
083.6334
1000

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
QuantityValueUnit
Scattered photon energy83.6334keV
Photon energy loss16.3666keV
Electron recoil momentum x100keV/c
Electron recoil momentum y-83.6334keV/c
Relativistic recoil kinetic energy16.3666keV
Classical recoil estimate16.6287keV
Energy conservation residual3.55271e-15keV
Scattered-to-incident wavelength ratio1.19569ratio

The recoil closes both vector momentum and relativistic energy, not just a wavelength shift.

With the electron initially at rest, Eprime=E/[1+(E/511)(1−cos theta)]. Photon momentum magnitude in keV/c equals its energy in keV. Recoil is the vector difference, with kinetic energy sqrt(511²+p²)−511 in keV. The numerically stable equivalent p²/[sqrt(511²+p²)+511] avoids subtracting nearly equal values. A classical p²/(2m) estimate overstates recoil energy here; it is only a low-momentum approximation. Forward scattering has exactly zero loss and recoil.

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

The recoil closes both vector momentum and relativistic energy, not just a wavelength shift.

With the electron initially at rest, Eprime=E/[1+(E/511)(1−cos theta)]. Photon momentum magnitude in keV/c equals its energy in keV. Recoil is the vector difference, with kinetic energy sqrt(511²+p²)−511 in keV. The numerically stable equivalent p²/[sqrt(511²+p²)+511] avoids subtracting nearly equal values. A classical p²/(2m) estimate overstates recoil energy here; it is only a low-momentum approximation. Forward scattering has exactly zero loss and recoil.

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

Understand

State the event and measurement model

A photon has momentum magnitude E/c along its propagation direction. In the displayed plane the incident photon travels along positive x; the scattered photon leaves at angle θ. Conservation therefore gives electron recoil components px=E−Eprime cosθ and py=−Eprime sinθ when momentum is expressed in keV/c and photon energy in keV. Subtracting scalar momentum magnitudes would discard the direction that makes this account meaningful.

Reconcile the quantitative evidence

For one initially stationary free electron with rest energy M=511 keV, simultaneous energy and momentum conservation yields Eprime=E/[1+(E/M)(1−cosθ)]. The electron kinetic energy is sqrt(M²+p²)−M in the same energy units. This relativistic relation is supplied for algebraic consistency; it is not replaced by p²/(2M) when recoil is large. The latter is a low-momentum approximation that can be tested against the exact ledger rather than treated as another independent conservation law.

Preserve limits and alternative explanations

Forward scattering has θ=0, Eprime=E and zero recoil. At a nonzero angle, the photon loses energy and its wavelength grows. That energy loss must equal recoil kinetic energy under the stated initially-rest, single-electron assumptions. An observed packet that closes vector momentum but fails energy closure is not a physical event in this model. It could signal different initial motion, another participant or a measurement error, not permission to ignore one conservation equation. All displayed arrows share a common momentum scale.

Worked original case

For E=100 keV and θ=90°, Eprime=51100/611≈83.633388 keV. Recoil components are (100,−83.633388) keV/c and its kinetic energy is about 16.366612 keV, matching photon loss. A nonrelativistic kinetic estimate is larger and does not close this exact account.

Investigation

Compare 0°,90°,180° at two incident energies. Reconstruct recoil from the vector difference before using its energy formula. Retain both residuals and quantify where the classical kinetic estimate disagrees.

Changed-case practice

Try the changed case before feedback

For incident energy 50 keV and backscattering, use M=511 keV to find scattered photon energy in keV.

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.

Apply

In the stated initially-rest electron model, exactly forward photon scattering gives:

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?

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Separate photon supply from an emission thresholdNext: Predict matter-wave directions from accelerating energy
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