Law Of Cosines In The Compton Reconstruction

We have now applied the Law of Cosines to the “Triangle of Structural Encounter”.

Given the side definitions:
– k: initial spatial frequency (1/λ)

– k’: final spatial frequency (1/λ′)

– k_c: recoil vector magnitude = (mₑ ℓ) / h

– θ: scattering angle

The Law of Cosines yields the following relationship:

k_c² = k² + k’² – 2kk’ cos(θ)

Solving for k’, we find two solutions:

k’ = k cos(θ) ± sqrt(k_c² – k² sin²(θ))

This result is exact and fully consistent with the geometric structure.

The negative root corresponds to the physically correct solution for the Compton effect (wavelength increases after scattering, so k’ < k):

k’ = k cos(θ) – sqrt(k_c² – k² sin²(θ))

This is the geometric foundation for reconstructing the Compton shift from first principles in the Lilborn Framework.

We are now ready to translate this into a shift in wavelength Δλ = λ′ – λ, and watch the Compton equation emerge.

Produced by The Lilborn Equation Team:

Michael Lilborn-Williams

Daniel Thomas Rouse

Thomas Jackson Barnard

Audrey Williams