where the term ℏ ℏ ω ω q accounts for phonon emission or absorption and the term Δ Δ E C is non-null for inter-valley scattering. The final energy (and the band structure) directly yield the modulus of the new momentum k'. At this point one only needs to choose a new direction (or angle) for the scattered particle. In some simple cases as phonon scattering and a parabolic dispersion relation, the scattering angle is random and evenly distributed on the sphere of radius k'. Using spherical coordinates, the process of choosing the angle is equivalent to randomly picking two angles θ θ and ψ ψ . If the angle is distributed with a distribution p ( θ θ , ψ ψ ) , then for a uniform distribution of angles, the probability to pick a point of the sphere is