Connect a kinetic-energy inference to an allowed diffraction direction

Define the state, source and evidence

De Broglie wavelength is lambda=h/p, where momentum must match the kinetic-energy regime. For an electron with kinetic energy K, total energy is K+mc². Relativistic energy-momentum conservation gives pc=sqrt[K(K+2mc²)], whereas the nonrelativistic approximation is pc≈sqrt(2mc²K). The omitted K² term is a controlled approximation, not a separate particle identity.

Derive the quantitative relation

Use rounded mc²=511000 eV and hc=1239.841984 eV nm. Divide hc by either momentum-times-c expression to compare wavelengths at the same kinetic energy. The nonrelativistic wavelength is larger by the relative factor sqrt[1+K/(2mc²)]−1. This comparison separates approximation error from an energy-readout calibration error.

Check normalization, limits and inference

For an ideal uniform single aperture of width d, the first far-field zero obeys d sin(theta)=lambda. If lambda/d exceeds one, no propagating first-zero direction exists; clipping to 90 degrees would fabricate a result. This is not a crystal Bragg equation, whose spacing and angle conventions differ.

Compare a changed premise

An actual electron diffraction system also needs an emission distribution, coherence, aperture profile, fields and a calibrated observation geometry. The model claims none of these have been measured. Its speed-to-c ratio is reported so the chosen momentum approximation can be assessed explicitly, with no beam-operation instructions.

Worked original case

At K=100 eV, pc=sqrt(100×1022100) eV and wavelength is about 0.122636471 nm. With aperture width 0.2 nm, the first-zero sine is about 0.613182354 and a real direction exists. With width 0.05 nm, the same wavelength would require a sine greater than one and the first-zero prediction is rejected.

Investigation

Compare exact and nonrelativistic wavelengths at two energies. Hold energy fixed and change only aperture width until the first-zero existence condition changes. Retain the candidate sine even when the angle is withheld, then identify a separate energy or geometry calibration alternative.

Changed-case practice

Try the changed case before feedback

In a changed ideal aperture packet lambda=0.12 nm and width d=0.24 nm. Find the first-zero angle in degrees.

Numerical result in the requested unitsCompare this result

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 derivations and virtual evidence checked 31 October 2026 against the primary explanation. All questions, numerical packets and diagrams are original. These models do not establish actual beam, radiation, vacuum, optical or electrical laboratory performance. Public feedback is practice, not authenticated independent mastery. Preserve first reasoning, criterion critique, revision and a genuinely new assigned packet in the course evidence workspace.