The Boltzmann transport equation (BTE) describes the scattering processes of
the ions in the target by changes in the statistical momentum's distribution
. The number of particles with energies and angles in the
momentum interval 
that go through a unit-area element at depth
normal to the surface is given by 
. With
, the angle between the direction of ion motion and the -axis,
and 
the probability of a collision, the spatial evolution of this
momentum is given by the BTE (2.1-1). The ions can be scattered from
state 
into a final state 
or they can be scattered out
of into .
Monte Carlo simulations described above are one way to fulfil the BTE. Here
we sketch the BTE method [Chr80], [Tak83], [Gil86], a
numerical algorithm for solving (2.1-1). The momentum distribution
function is represented as a two-dimensional matrix 
corresponding
to particles of energy 
(
) moving in direction
(
) (see Figure 2.1-3).
In the initial state the momentum distribution in the surface plane is a
-function, given by the ion's energy 
and flight
direction. The momentum can be substituted by energy 
and
angle 
. At each step in the calculation, the redistribution of the
particles in the momentum space at depth in a depth element 
is
obtained from (2.1-1). While ions with an energy below some
threshold 
are stopped and stored as concentration at depth ,
the others continue until they have lost enough energy to come to rest (see
Figure 2.1-4).
Both, Monte Carlo simulations and the BTE method are very time consuming. Profiles predicted for amorphous targets may differ rather strongly from measurements in crystalline materials. The BTE method usually gives smoother dopant profiles in less CPU-time [Chr80] in one space dimension. Both methods are based on first principles and allow the treatment of arbitrary target structures and parasitic effects like damage, back scattering or knock-in effects. Today, Monte Carlo simulations in crystalline targets are more or less standard in cutting-edge scientific research, whereas the BTE method is less frequently used.
