Effective mass (solid-state physics) from both sides gives where is the th element of , is the th element of , and are the th and th elements of , respectively, and is the total energy of the particle according to the Planck–Einstein relation. The index is contracted by the use of Einstein notation (there is an implicit summation over ). Since Newton's second law uses the inertial mass (not the gravitational mass), we can identify the inverse of this mass in the equation above as the tensor This tensor expresses the change in group velocity due to a change in crystal momentum. Its
Group velocity amplitudes diminish as they approach the leading edge. The longer waves travel more slowly, and their amplitudes diminish as they emerge from the trailing boundary of the group. The group velocity is defined by the equation: where is the wave's angular frequency (usually expressed in radians per second), and is the angular wavenumber (usually expressed in radians per meter). The phase velocity is: . The function , which gives as a function of , is known as the dispersion relation. One derivation of the formula for group velocity is as follows. Consider a wave packet as a function of position