One of the first physically based approaches to estimate implantation
profiles was performed by Lindhard, Scharff and Schiøtt
(LSS-theory) [Lin63]. To get a solution of the BTE (2.1-1) they
calculate the moments of the implantation profile by integro-differential
equations. Using spherical coordinates, we define a probability
that an ion with energy 
will come to rest at a distance
and angle 
to its current position and direction. If the ion
moves a small (vector) distance 
, the differential scattering
cross section 
describes the probability that a collision will
occur resulting in an energy loss 
. If the maximum possible energy
transfer in a collision is 
and there are 
target atoms per unit
volume, we get (2.1-2).
Recurrence relations for the moments of 
are found by expanding in
terms of Legendre polynomials, equating polynomial coefficients and taking
the moments. The results of moments' calculation are available in form of
tables for various ion/target combinations in the book of Gibbons
[Gib75].
Only the first two moments can be calculated with reasonable accuracy.
Therefore, the potential of the LSS-theory is exhausted by calculating the
projected range 
and the vertical and the lateral standard deviations
(
and 
, respectively). A slightly different approach
about ion range theory, the Projected Range ALgorithm PRAL, has been
proposed by Biersack [Bie81], [Bie82].
