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Appendix 10. Effect of Backscattered Electrons on the Geometric Factors of the LEMS30 Telescope (Hong MS Thesis continued)

A10.3 Chapter 3 - Numerical Procedure

where DA'_ij is the effective area of DAi (see Figure A10-7) looking into the solid angle of DW_ij. DW_ij is the solid angle of the jth escaped electron trajectory; n is the total number of finite areas DA_i, and npas is the total number of escaped trajectories for DA_i (see Section A10-6 for detailed derivations). To calculate the W_ij one can trace an electron starting from the open aperture with a particular energy and see if it can reach the area DA_i of the detector. We also can do it time-reversed, starting at the center of DA_i and trace its trajectory to see if it escapes the open aperture. In this project we use the second procedure. Since we would compare the results of the effect of backscattering electrons on geometric factors of LEMS 30 to Buckley's non-scattering results, it is important to use his model of calculation. The procedure is outlined as follows:

1) The telescope has a complex geometric shape. In order to determine the fate of a particle's trajectory (check if the particle should continue its trajectory) we need a simulation of the geometry of the telescope. Buckley modeled the whole LEMS 30 subsystem with plane polygons (see Figures A10-4 - A10-5). A complete listing of the coordinates of each vertex and coefficients of each plane can be found in Section A9.5.

Figure A10-4 View angle q=0°, f=90°

Figure A10-5 View angle q=-20°, f=80°

2) The approximation of the magnetic field generated by the magnets is also needed. Buckley adapted Shodhan's model (Shodhan, S., Masters Thesis, Univ. of Kansas, 1988) to do the calculation for the LEMS 30 telescope.

3) Divide the detector sensor into 21 small areas of DA_i (see Figure A10-6). Choose a DA_i. (For the coordinates of the center and area of each DA_i see section 9.6.)

Figure A10-6 The detector is divided into 21 small DA_i.

4) Start from the center of the small area DA_i, choose initial q₀ and f₀ with a particular energy for a small time interval (Dt_j). During the time interval the particle moves from (x_j, y_j, z_j) to (x_j+1, y_j+1, z_j+1) under the influence of the magnetic field.

5) Determine whether the line segment formed by (x_j, y_j, z_j) and (x_j+1, y_j+1, z_j+1) has an intersection with any plane of the polygons of 1.

• If there is no intersection then calculate the trajectory of the electron for the time interval (Dt_j+1); repeat step 5.
• If there is an intersection between the line segment and a polygon then check to determine if the polygon is the open aperture polygon.
  1. If it is open aperture, then terminate the trajectory because the electron escapes the open aperture.
  2. If it is not, then determine if the trajectory has any earlier impact with the telescope.
     1. If there is no earlier impact, use a particular backscattering model (see Section A10.2) to reschedule the trajectory after the impact.
     2. If there was an earlier impact, then terminate the trajectory and the electron fails to be detected.

6) Repeat step 4 with change of Dq and Df until it exhausts all possible combinations of q and f.

7) Pick another DA_i and repeat step 4. Notice that because of the symmetry of the detector we only need to pick DA_i in a half of the detector. The results of the other half should be the same (see Figure A10-7).

Figure A10-7 The number of escapes at each DA_i for energy of 50 keV (Radzimski's h is used).

Figure A10-7 shows the number of escapes for energy of 50 keV for the electrons starting from the center of each DA_i when Radzimski's backscattering coefficients are used. The results include elastic specular single backscattering electrons. More results can be found in Section A10.9 for both Radzimski's and Neubert's h and other energies.

8) One can use the results for the particular energy of the electrons to calculate the geometric factor by using the following formula:

where n is total number of DA_i's on the detector, DA_i is the area of the i-th element on the detector, Dq_j is the interval at which the polar angle is chosen to scan through the whole solid angles, q_j and f_j are the polar and azimuthal angles at which the j-th electron starts from the center of the area which escapes the open aperture, and npas is the number of trajectories which reach the open aperture. The 25º that appears in the argument of cosine is needed because the detector B surface tilts in this coordinate system. See Section A10.6 for more details.

One can repeat all the above procedures to obtain the geometric factors for different energies. Shown in Figure A10-8 are the geometric factors vs. energies for including elastic specular single backscattering when Radsimski's h is used.

For details of the programming and codes see Section A10.8 and Section A10.11.

Figure A10-8 The geometric factor including specular backscattering; Radzimski's h is used.

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