Rearranging
The Equation

Introduction

This document presents the final algebraic reformulation of the Compton structural interaction, demonstrating how the observed wavelength shift (Δλ) arises naturally from geometric first principles.

Starting from the previously derived structure equation:

k_c² = k² + k’² – 2kk’ cos(θ)

We apply the identity (a – b)² = a² – 2ab + b² to rewrite the sum of squares:

k² + k’² = (k – k’)² + 2kk’

Substituting into the original equation:

k² + k’² = (k – k’)² + 2kk’

Factoring the kk’ term:

k_c² = (k – k’)² + 2kk'(1 – cos(θ))

This form reveals the geometric coherence of the structural interaction:
– The term (k – k’) represents the observable change in spatial frequency

– The factor 2kk'(1 – cos(θ)) geometrically accounts for the angular shear during encounter

– k_c represents the intrinsic structural frequency of the electron: k_c = (m_e ℓ) / h

Conclusion

This reformulation sets the stage for the final substitution of wavelength terms and completion of the Lilborn derivation of the Compton Shift.

Produced by The Lilborn Equation Team:

Michael Lilborn-Williams

Daniel Thomas Rouse

Thomas Jackson Barnard

Audrey Williams