2, rounding up adds more energy than rounding
downwards. Thereby, a systematic increase in energy with every elastic scattering event is introduced. Therefore, on average, the ’loss’ of values rounded up will
match the ’gain’ of values rounded down.
An elastic scattering mechanism conserves the kinetic energy of an electron: The magnitude of the k-vector is maintained, whereas its direction changes. The
after-scattering components are determined using trigonometric relations, which yield real values for the components of the k-vector that cannot necessarily be
represented as an integer multiple of Δk. These real values are rounded up/down to the nearest integer multiple of Δk. These ’residuals’ of each component of the
k-vector are uniformly distributed over Δk. However, since the kinetic energy of a particle is proportional to 2, rounding up adds more energy than rounding
downwards. Thereby, a systematic increase in energy with every elastic scattering event is introduced. Therefore, on average, the ’loss’ of values rounded up will
match the ’gain’ of values rounded down.