Crystal Momentum and Effective

Effective MassEven though crystal momentum increases with applied force, it may well be that theparticle velocity is decreasing. To see why, consider the time-rate-of-change of the groupvelocity of a particle:a = ! !t v g = 1 !! !E!t !kThis can be rewritten as:a = ! !t v g = 1 !!k!t! 2 E!k 2But, since !k!t = F , we obtain:!"F = $#1 ! 2 E %! 2 !k 2 '&(1a = m * awherem * "= $#1 ! 2 ! 2 E!k 2%'&is the effective mass of the particle.(1The effective mass of a particle indicates how its velocity will change when a forceis applied. If the particle’s velocity increases in the direction of an applied force, then itseffective mass is positive. If, on the other hand, its velocity decreases in the direction ofan applied force, then its effective mass is negative. Contrary to common belief, aparticle with a negative effective mass doesn’t have a negative weight. Rather, it slowsdown when a force is applied, since its potential energy is increasing at the expense of itskinetic energy.