When an electron in a periodic potential of lattice is accelerated by an electric field or magnetic field, then the mass of the electron is called effective mass. \[ \boxed{m^* = \frac{\hbar^2}{\frac{d^2E}{dk^2}}}\]
Considering the free electron as wave packet, the group velocity \(v_g \) corresponding to the particle's velocity can be written as
\[ v_g = \frac{dw}{dk} \tag{1}\] The energy of a particle is given by \[ E=hv = \frac{hw}{2\pi} = \hbar w \] \[ w = \frac{E}{\hbar} \tag{2}\] Sustitute \(w\) in Eqn 1, \[v_g = \frac{1}{\hbar} \frac{dE}{dk} \] Acceleration \[ a =\frac{dv_g}{dt} = \frac{1}{\hbar}\frac{d^2E}{dk^2} \frac{dk}{dt} \tag{3} \] We know \(p\) from debroglie wavelength, \[ p = \frac{h}{\lambda} = \frac{h \times 2\pi}{\lambda \times 2\pi}= \hbar k \tag{4} \] \[ \frac{dp}{dt} = \hbar \frac{dk}{dt} \] The force acting on the electron \[ F= m^*\times a \rightarrow m^*= \frac{F}{a} \tag{5} \] where \[ F = \frac{dp}{dt} = \hbar \frac{dk}{dt} \] Subsitute \(F\) and \(a\) values in Eqn (5) \[ m^* = \frac{\hbar \frac{dk}{dt}} {\frac{1}{\hbar}\frac{d^2E}{dk^2} \frac{dk}{dt}} \] \[ \boxed{m^* = \frac{\hbar^2} {\frac{d^2E}{dk^2}} } \tag{6} \] The equation indicates the effective mass of an electron moving under periodic potential.