What is the error in the Boltzmann approximation if the crystal size is not infinite but contains 102, 103, 104, 105, 106, 109 atoms? This is a very real situation in submicron transistors In which an active region has a very small volume and one must choose a smaller differential volume dxdydz In order to give many volume elements in a finite difference analysis of the problem.
For example, in an Si volume of 0.1x1x1μm3=107A3 =10-17cm3, such as the base of a BIT or channel of a MOST, there are only 10-17x2x1022=2x105 Si atoms.
This base or channel layer must be partitioned into 102 or more volume elements in order to accurately calculate the transistor characteristics using numerical techniques. Then, each volume would have only 2000 Si atoms.