Read acceleration from the curvature tensor

Ignore Berry curvature for this conventional local acceleration calculation and take a band isolated enough for semiclassical motion. Differentiating the band velocity while using hbar times k velocity equal to force gives acceleration as the energy Hessian divided by hbar squared acting on force. This defines the inverse effective-mass tensor. The Hessian is symmetric for a smooth scalar band, and its principal axes identify directions where acceleration and force are parallel.

At a nondegenerate parabolic minimum all principal curvatures are positive. At a band maximum they are negative, so electron acceleration can oppose the applied force without implying a negative fundamental rest mass. Describing an almost filled band in terms of missing electrons leads to positively charged holes with a positive mass near a simple maximum. Near inflection points the local curvature mass may diverge even when the band velocity is finite, limiting a constant-mass approximation.

For a two-dimensional parabolic band with principal masses mx and my, contours of constant energy are ellipses in wavevector space. In a perpendicular magnetic field, the Lorentz force moves k around such a contour while doing no work. Combining the two component equations gives a frequency proportional to magnetic field divided by the geometric mean of the principal masses. The orbit is closed only in the appropriate band region; open Fermi-surface trajectories need not have this simple periodic motion.

The cyclotron mass may also be obtained from the derivative of reciprocal orbit area with respect to energy. For an ellipse that derivative gives precisely the geometric mean, providing an independent check. In a general nonparabolic band the cyclotron mass depends on energy and field direction, and need not equal a density-of-states mass or any one Hessian eigenvalue. Quantum oscillations require further quantization conditions, which should not be inferred from a classical orbit alone.

A curvature extracted from a wide energy fit can differ from the local band-edge curvature that controls low-energy acceleration. Conversely, an extremely narrow finite-difference estimate can amplify numerical eigenvalue noise. Check the fit window, wavevector resolution, units, and whether nearby bands change character. A tensor fit must retain mixed derivatives when the chosen coordinates are not principal axes; setting them to zero by habit can invent a false alignment of force and acceleration.

The function lab illustrates the sign of curvature in one coordinate. Reset and set coefficient to 1 and position to 1; predict value 1 and tangent slope 2. Reset and set coefficient to −1 and position to 1; predict value −1 and slope −2. The selected parabolas have opposite curvature, analogous to opposite conventional inverse-mass signs. This dimensionless function model contains no hbar, tensor fitting, magnetic orbit, or cyclotron-frequency calculation.