Fundamental TechnologiesVoyager LECP Pages 
An Analysis of the Performance of the Magnetic
Deflection System
in the Voyager Low Energy Charged Particle Experiment
by Sheela Shodhan
5.1 Calculations
The geometric factor G of a detector for a particular energy is the sum over all the sampling areas of the detector of the product of the sampling area of the detector and the solid angle spanned by the open trajectories that connect the sampling area and the open aperture, i.e.:

where n is total number of sampling areas on the detector,
DA_{i} is the area of the ith. element on the detector,
DW_{i} is the solid angle spanned by the open trajectories
» å_{j = 1}^{npas}Dq_{j} Df_{j} sin( q_{j}+[(Dq_{j})/2])
Dq,Df are the intervals at which the polar and the azimuthal angles are scanned respectively and npas is the number of trajectories connecting DA_{i} and the open aperture.
5.1.1 Determination of DA_{i}
Each of the circular detector surfaces is divided into several sampling areas DA_{i}. Each one of these circular surfaces is fitted into a square which is then further divided into smaller squares. The centres of each of these smaller squares which fall within the circular detectors have been considered as to sample that particular area element. Those squares that contain the centre point but are cut off by the circular boundary of the detector have been approximated as trapezoids.
Then starting from a particular center point on the detector, for a particular energy, the solid angle is determined.
Figure 5.1. The detector surface divided into several sampling areas. The dotted points are the initial points of the trajectory.
5.1.2 Determination of the solid angle
For a given energy, starting from a particular point on the detector and with a particular set of polar and azimuthal angles, the motion of this particle is observed inside the sensor subsystem. The coordinates of these points are included in Appendix D. The equations of motion:
dX/dt = V
M dV/dt = q/c V × B
are solved by Hamming's modified double precision PredictorCorrector method in a "time reversed sense." "Time reversed sense" because the starting point on the detector surface in this calculation is the endpoint of the real particle. At every stage of the trajectory calculation the fate of the particle is determined as discussed later. If this particle's trajectory passes through the open aperture of the sensor subsystem then it is considered as escaped. The polar and the azimuthal angles at the aperture are computed and npas is incremented by one. However, if the particle hits any of the surfaces of the sensor subsystem then it is considered lost and thus ignored since the real particle would also not be able to hit this point on the detector. Then a new set of polar and azimuthal angles is chosen and this process is repeated. Thus, polar and azimuthal are scanned at specific intervals to determine whether the particle escapes or not. A specific range of these for which the particles escape the sensor subsystem forms the set of trajectories that connect the sampling area and the open aperture. Then the solid angle spanned by this set of trajectories is computed using the above equation.
5.1.3 Determination of the fate of the particle
Given the linesegment formed by the points (x_{i},y_{i},z_{i}) and (x_{i+1},y_{i+1},z_{i+1}) of the trajectory to determine whether the particle is lost or not, the process is as follows:
For each of the steps above from (1) to (5) the relevant equations used are included in Appendix C.
In this way, the trajectory calculation continues until a decision is made as to whether the particle hits any of the surfaces and is lost or it passes through the opening aperture and escapes.
5.2 Discussion of the Results
Since at every stage in the particle's trajectory its curvature is determined by the local relation,
r = mcv/Bq (in c.g.s. units)
the deflection of the low energy electrons should be more than that of the high energy ones. This explains the fact that the Beta detector, which is closer to the opening aperture than Gamma, primarily collects low energy electrons while the Gamma detector collects higher energy electrons.
On examining the trajectories of the particles of different energies emanating either from the Beta detector or the Gamma detector, we do observe that the low energy electrons are indeed deflected more than the high energy ones. This also manifests itself in the shift of the azimuthal angles for which the particles escape, from lower angles to higher angles, as the energy increases. For example, for the energy E=480 keV the range of azimuthal angles for a point on the Gamma detector is 167 to 227 deg. while for the same point for energy E=720 keV the range has shifted to 181 to 228 deg.
Also, for a particular energy, due to the inhomogeneity of the magnetic field, the curvature of the trajectory of the particle depends upon its position in the deflection system. In this time reversed calculation, the particles entering the deflection system into the region where the field is higher have higher curvatures (less radii) whilst those particles that enter into the region of lower field have lower curvatures (high radii).
Together with this field inhomogeneity which shapes the trajectory of the particles, the surfaces of the sensor subsystem restrict the values of the angles for which the particles escape to a finite range instead of all the allowable values.
Besides, we also expect the symmetry about the z = 0.0 plane in the magnetic field and in the modelled sensor subsystem, to be reflected in the angular distributions of the escaping particles at the detectors and at the apertures.
The above observations support the validity and the correctness of the magnetic field model.
5.2.1 Variation of the geometric factors with energy
DETECTOR  ENERGY keV 
GEOMETRIC FACTOR cm^{2} sr 

Beta  40  0.010177 
80  0.025129  
160  0.026672  
320  0.003406  
480  0.000803  
Gamma  250  0.019426 
500  0.038626  
720  0.024262  
1440  0.005594  
2880  0.000992 
Table 5.1. Geometric factors for different energies for the Beta and the Gamma detectors
5.2.2 Tables of the number of particles that pass
Figure 5.2. Variation of the geometric factors with energy for the Gamma detector.
Figure 5.3. Variation of the geometric factors with energy for the Beta detector.
Figure 5.4. Points on the Gamma detector which are the initial points of the trajectories.
2  4  6  8  10  
2  499  526  499  
4  498  528  531  528  498 
6  528  557  576  557  528 
8  552  576  576  576  552 
10  568  564  568 
Table 5.2. Number of particles that pass for different points on the Gamma detector for E=500 keV. Note: The scanning intervals Dq, Df = 2 deg.
2  4  6  8  10  
2  245  265  245  
4  286  301  314  301  286 
6  331  346  1402*  346  331 
8  361  387  379  387  361 
10  419  404  419 
Table 5.3. Number of particles that pass for different points on the Gamma detector for E=720 keV. Note: The scanning intervals Dq, Df = 1 deg for point marked by *, for all others, Dq,Df = 2 deg.
2  4  6  8  10  
2  32  37  32  
4  31  54  64  54  31 
6  60  83  350*  83  60 
8  93  106  114  106  93 
10  125  137  125 
Table 5.4. Number of particles that pass for different points on the Gamma detector for E=1440 keV. Note: The scanning intervals Dq, Df = 2 deg and for points marked *, Dq, Df = 2 deg.
2  4  6  8  10  
2  0  10  0  
4  0  13  37  13  0 
6  10  53  74  53  10 
8  45  93  120  93  45 
10  151  162  151 
Table 5.5. Number of particles that pass for different points on the Gamma detector for E=2880 keV. Note: The scanning intervals Dq, Df = 1 deg.
Figure 5.5. Points on the Beta detector which are the initial points of the trajectories.
2  4  6  8  10  
2  124  140  124  
4  54*  67*  70*  67*  54* 
6  107  129  142  129  107 
8  97  123  66*  123  97 
10  100  54*  100 
Table 5.6. Number of particles that pass for different points on the Beta detector for E=40 keV. Note: The scanning intervals Dq = 2 deg. For points marked by *, Df = 4 deg, for others Df = 2 deg.
2  4  6  8  10  
2  131  137  131  
4  258*  292*  150  292*  258* 
6  142  155  164  155  142 
8  145  160  164  160  145 
10  163  165  163 
Table 5.7. Number of particles that pass for different points on the Beta detector for E=80 keV. Note: The scanning intervals Dq = 2 deg. For points marked by *, Df = 2 deg, for others Df = 4 deg.
2  4  6  8  10  
2  140  149  140  
4  136  153  168  153  136 
6  141  166  172  166  141 
8  156  171  176  171  156 
10  173  177  173 
Table 5.8. Number of particles that pass for different points on the Beta detector for E=160 keV. Note: The scanning intervals Dq = 2 deg. and Df = 4 deg.
2  4  6  8  10  
2  18  28  18  
4  11  48  95  48  11 
6  72  153  204  153  72 
8  198  310  360  310  198 
10  532  566  532 
Table 5.9. Number of particles that pass for different points on the Beta detector for E=320 keV. Note: The scanning intervals Dq,Df = 1 deg.
2  4  6  8  10  
2  0  0  0  
4  0  0  0  0  0 
6  0  7  29  7  0 
8  16  66  97  66  16 
10  158  188  158 
Table 5.10. Number of particles that pass for different points on the Beta detector for E=480 keV. Note: The scanning intervals Dq,Df = 1 deg.
Additional Figures:
Appendix A
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Last modified 12/9/02, Tizby HuntWard
tizby@ftecs.com