SHAPE is a code to build-up x-ray fluorescence (XRF) intensity spectra by computing the intensity contributions from multiple-collision chains with recourse to the computation of analytical solutions deduced in the frame of photon transport theory. Such a theoretical model describes the diffusion of photons in an homogeneous target of infinite thickness comprising different kinds of isolated atoms. It is assumed that the excitation is produced with an unpolarized or linearly polarized, monochromatic and collimated source, and that the emitted spectrum is collected into a sharply collimated solid angle using a Si or Ge solid state detector.

Recently, SHAPE has been modified to include the effects of the photon polarization as much as the scattering interactions Rayleigh and Compton change the state of polarization of the radiation undergoing these kinds of collisions. Thus, the multiple scattering intensity results modified by these polarization changes. This happens even if the source emits unpolarized photons, and much more when it is linearly polarized as can be done with this version of SHAPE.

The physical variables defining the problem are the source energy, the directions of incidence (two angles) and take-off (two angles), and the sample composition (up to 10 different especies of atoms). In addition, SHAPE allows us to keep only the interactions of interest, making possible the study of selected groups of interactions that modify isolated portions of the spectrum.

The interactions are classified by the number and the type of the collisions. With SHAPE you can calculate all the contributions due to 1 and 2 collisions of any combination of the photoelectric, Compton and Rayleigh effects, and those due to 3 collisions of pure photolectric type.

Both directions -incidence and take-off- define the scattering angle. Due to the symmetry of the problem, we only need only three of the four angles: the two polar angles (about the normal to the sample) and the take-off azimuthal angle (assuming the incidence azimuthal angle is 0). So, the take-off azimuthal angle is always measured from the projection of the incidence direction on the sample surface.

With this information SHAPE computes first the discrete and continuous parts of the backscattered spectrum (in wavelength). For the interactions photoelectric, Rayleigh and Compton, it uses the kernels described in the bibliography below. These kernels are sufficiently precise as to give an energy detailed spectrum.

SHAPE converts the wavelength spectra to the energy domain and simulates the response of either an intrinsic Ge or a Si solid state detectors to render the output closer to that of an experimental spectrum. It also simulates the spectrum modification as a consequence of the attenuation in the air paths between the SOURCE and the TARGET, and between the TARGET and the DETECTOR, which are denoted as d1 and d2, respectively.

To convert the discrete components of the spectrum, the line width of the monochromatic excitation source must be introduced interactively. The natural widths of the characteristic lines (much narrower than the breadth due to the detection process) are included in the data base. The line width of Compton peak, the so called Compton profile -dependent of the momentum distribution of the electrons in the atoms of the target- is usually greater to the detection width, and is also included in the data base. The Rayleigh width should be defined accurately to fit well the experimental height in the region of the Rayleigh and Compton peaks. The interactive input of this width can be skipped by calling SHAPE in batch mode.

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ALMA MATER STUDIORUM - Università  di Bologna
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