The evaluated cross-section data constitute an array of points (Ei,σi). Taking into account that the points are dense enough on energy for the cross-section to be smoothed by the spreading caused by a spectroscopy system resolution the energy spectrum for a uniform thick target is constructed in a following way.
First the energy Ei is assigned to be equal to the energy E1 which a projectile of the initial energy E0 penetrating a sample possesses at the depth xi, where the interaction characterized by the cross-section σi occurs. Then the depth xi and the corresponding energy E3i registered by a detector are calculated. In order to speed up the calculations stopping power was approximated by a following function
which can be integrated analytically. The corresponding formulas are as follows:
where θ is a scattering angle, E2 is the energy of the outgoing particle immediately after an interaction and k is a kinematical factor.
The interaction yield Y1(E1i) at the depth xi is obtained by a convolution of the cross-section with the beam spreading function which is assumed to be represented by Bohr’s straggling theory:
with Zt and Zp standing for the charges of the target nucleus and the projectile respectively, and C is atomic concentration.
The yield of the registered particles Y3(E3i) is calculated as a convolution of the yield Y1(E1i) with the Gaussian spreading function, the variance including both straggling on the way out and the detector resolution:
The yield per a MCA channel of width ΔE3 is
where Q and Ω are the number of projectiles and the detector solid angle respectively, and
The stopping power approximation used in the calculations is accurate up to tenths of percent in the energy range of interest as can be seen from the following figure. It is worth noting that the discrepancy between different systematics of the stopping power data is much greater.