spherical surface, say ABC (Fig. I), whose angles are. A, B and C. If the radius of the sphere is h, the area of the spherical triangle is given by (Korn and Korn,.
Radial. Phys. Chem. Vol. 44, No. l/2, pp. l-4, 1994 Elsevier Science Ltd. Printed in Great Britain 0969-806X/94 $6.00+ 0.00
Pergamon
SOURCE-DETECTOR GEOMETRICAL EFFICIENCY YOUNIS Department
S. SELIM
and
MAHMOUD
of Physics, Faculty of Science,
I. ABBAS
Alexandria,
Egypt
Abstract-By the use of simple analytical spherical-trigonometry a mathematical expression of a solid angle subtended by a point to a plane triangle is deduced. Generalizing, solid angles subtended by a non-axial point source to a circular disk detector and a circular disk source to a circular disk detector are deduced into rigid mathematical expressions very easily computed, for the first time. Results were compared with previous tabulations. By this method, with some mathematical manipulations, solid angles could be deduced for detectors of different eeometrical shapes, e.g. rectangular, elliptical, spherical and cylindrical.
INTRODUCTION Calculations
of source
and detector
A = (A + B + C - n)h*. solid angles
The solid angle Q of the triangle definition is:
have
been treated by several authors over a long period of time (Knoll, 1989). Direct mathematical expressions have been given only for very special cases of detector shapes. Masket (1957) treated the problem of a point source and virtually any subtended object by the use of contour integrals. Results for a circle and a cylindrical volume are presented. He gave also the solid angle for an axial point and a subtended elliptical detector. Rowlands (1961) treated the problem by the use of the electrostatic field theory. He thus obtained the solid angle for a point source and rectangular shapes. The best tabulations to our knowledge so far, among others are of Masket et al. (1956) for a point source radiating to a circular disk and to a cylindrical detector. Also for a circle to a circle, Gardner and Verghese (1971) developed a computation program assuming the detector as a set of adjacent trapezoids. In the present work we started to consider an isotropic source point radiating to a plane triangle, hence the problem reduces to the calculation of the solid angle of the vertex of an oblique trihedral pyramid, a problem which is easy to tackle.
MATHEMATICAL
Q=A+B+C-x.
by
(2)
In order to treat the problem analytically we choose the principle coordinates X, Y and Z as shown in Fig. 2. Considering a plane bi-lateral triangle of side length R, its vertex is the origin 0, and the bisector of the vertex angle $ is the extension of the Y-axis. Let the source point be N with coordinates (I, m, h). We draw a reference spherical shell around N with radius h (not shown). The coordinates of the corners of the oblique trihedral pyramid are immediately determined. By simple algebra, the absolute lengths of the edges are determined, and by applying the cosine law for each side, the vertex angles a, b and c are deduced.
VIEW POINTS
The intersections of generatrices arising from the vertex (source) to the sides of the triangle base (detector) with the reference sphere surface whose center is the vertex will form a spherical triangle. Its sides fortunately are parts of major circles on the spherical surface, say ABC (Fig. I), whose angles are A, B and C. If the radius of the sphere is h, the area of the spherical triangle is given by (Korn and Korn, 1968): RPc44-I?--8
at the vertex,
(1)
Fig. I. Spherical I
triangle
ABC
YOUNIS S. SELIM and MAHMOUD 1. ABBAS Z t
N (I,m,h)
P i (R sin v/2.-R
0 f (O,O,O) Q i (-R sin v/2,-R
cos v/2,
0)
cos ~012, 0)
Fig. 2. Point to bilateral triangle.
Thus the sides of the spherical triangle are determined. On the other hand, applying the law of cosines for the sides of the spherical triangle (Korn and Korn, 1968) the corner angles A, B and C are also determined. After straightforward but lengthy calcu-
lations we can find these angles which have the form of inverse trigonometrical functions. In order to calculate the solid angle for a circular disk, we take the limits of $ with the appropriate expansion.
-
‘(’7
I1
I II
5
P Fig. 3. Solid angle for a non-axial point to circles of different radii (present work and Masket CI ul.. 1956).
II
Along major axi\.
II)
5
IS
P Fig. 4. Solid angle for a point to ellipses of different lengths (present work).
axial
Sourcedetector
geometrical
3
efficiency
Table la. Ctrcular disk to circular disk solid angle values for various values of R/h and S/h. Present work
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.1
0.03097
0.03029
0.02925
0.02792
0.02640
0.02476
0.02309
0.02144
0.01986
0.01836
0.2
0.12118
0.11870
0.11481
0.10980
0.10401
0.09776
0.09134
0.08496
0.07xx0
0.07294
0.3
0.26330
0.25832
0.25045
0.24023
0.22829
0.21526
0.20172
0.18815
0.17492
0.16226
0.4
0.44682
0.43921
0.42708
0.41114
0.39228
0.37143
0.34946
0.32716
0.30515
0.28387
0.5
0.65999
0.65007
0.63414
0.61294
0.58750
0.55894
0.52838
0.49687
0.46532
0.43445
0.6
0.89149
0.87986
0.86104
0.83571
0.80487
0.76971
0.73141
0.69126
0.65039
0.60980
0.7
I.13156
I.11891
1.09828
1.07022
1.03562
0.99553
0.95113
0.90377
0.85472
0.80518
0.8
1.37248
1.35943
1.33802
1.30864
1.27198
1.22891
1.18044
1.12783
1.07237
1.01536
0.9
1.60862
1.59567
1.57432
1.54481
1.50763
1.46338
1.41290
1.35721
1.29749
1.23503
1.64321
1.58649
1.52472
1.45900
1.96798
1.90198
2.37672
2.31633
Rlh
1.0
1.83615
1.82367
1.80299
1.77421
1.69389
I.73777
I.2
2.25714
2.24617
2.22792
2.20233
2.16946
2.12937
2.08221
2.02826
I.4
2.62810
2.61887
2.60346
2.58179
2.55316
2.51931
2.47830
2.43075
1.6
2.95058
2.94297
2.93026
2.91234
2.88909
2.86038
2.82603
2.78588
2.73981
268767
I.8
3.22974
3.22351
3.21312
3.19844
3.17937
3.15580
3.12750
3.09430
3.05603
3.01243
2.0
3.47158
3.46648
3.45799
3.44598
3.43038
3.41110
3.38790
3.36069
3.32925
3.29333
2.2
3.68181
3.67762
3.67065
3.66080
3.64801
3.63220
3.61320
3.59089
3.56511
3.53563
2.4
3.86543
3.86197
3.85622
3.84810
3.83756
3.82453
3.80890
3.79054
3.76935
3.74513
2.6
4.02670
4.02383
4.01906
4.01231
4.00356
3.99276
3.97980
3.96461
3.94709
3.92708
2.8
4.16913
4.16672
4.16272
4.15708
4.14976
4.14075
4.12994
4.11727
4.10269
4.08605
3.0
4.29560
4.29357
4.29019
4.28543
4.21927
4.27169
4.26260
4.25197
4.23975
4.22582
Table lb. Circular disk to circular disk solid angle values for various values of
R/h and S/h. Present work
S/h
R/h
I.1
1.2
1.3
1.4
1.5
1.6
I.8
19
20
0.1
0.01696
0.01567
0.01449
0.01341
0.01243
0.01153
0.01071
0.00997
0.00929
0.00868
1.7
0.2
0.06747
0.06239
0.05772
0.05344
0.04955
0.04598
0.04274
0.03980
0.03711
0.03466
0.3
0.15034
0.13924
0.12898
0.11954
0.11091
0.10301
0.09581
0.08925
0.08326
0.07780
0.4
0.26367
0.24470
0.22705
0.21075
0.19577
0.18202
0.16945
0.15795
0.14744
0.13784
0.5
0.40477
0.37665
0.35027
0.32573
0.30304
0.28214
0.26293
0.24532
0.22918
0.21440
0.6
0.57026
0.53234
0.49643
0.46273
0.43134
0.40224
0.37538
0.35064
0.32789
0.30699
0.7
0.75617
0.70853
0.66287
0.61958
0.57890
0.54093
0.50565
0.47299
0.44284
0.41503
0.8
0.95804
0.90146
0.84647
0.79373
0.74364
0.69648
0.65234
0.61123
0.57307
0.53773
0.9
I.17112
1.10700
1.04373
0.98222
0.92312
0.86689
0.81382
0.76401
0.71749
0.67418
1.0
1.39061
1.32082
1.25085
I.18182
1.1145x
1.04987
0.98817
0.92978
0.87482
0.82334
1.2
I.83112
1.75641
1.67903
1.60021
I.52114
1.44295
1.36655
1.29271
1.22193
1.15459
I.4
2.24994
2.17806
2.10140
2.02089
1.93761
1.85273
1.76743
1.68280
1.59979
I.51918
I.6
2.62945
2.56524
2.49524
2.41989
2.33981
2.25580
2.16889
2.08017
1.99078
1.90180
I.8
2.96332
2.90856
2.84802
2.78173
2.70986
2.63270
2.55083
2.46499
2.37607
2.28513
2.0
3.25270
3.20716
3.15642
3.10031
3.03872
2.97154
2.89892
2.82114
2.73861
2.65203
2.2
3.50226
3.46478
3.42288
3.37633
3.32492
3.26833
3.20645
3.13924
3.06666
2.98897
2.4
3.71771
3.68693
3.65248
3.61415
3.57171
3.52483
3.47328
3.41685
3.35527
3.28847
2.6
3.90447
3.87910
3.85073
3.81918
3.78426
3.74563
3.70303
3.65635
3.60509
3.54911
2.9
4.06728
4.04625
4.02278
3.99670
3.96787
3.93600
3.90091
3.86237
3.82003
3.77367
3.0
4.21013
4.19257
4.17301
4.15132
4.12738
4.10097
4.07192
4.04008
4.00510
3.96680
Table Ic. Circular
disk to circular disk solid angle values for various
values of R/h and S/h. Present work
S/h
Rlh
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
0.1
0.00812
0.00761
0.00714
0.00671
0.00632
0.00596
0.00563
0.00532
0.00503
0.00478
0.2
0.03243
0.3
0.07282
0.06825
0.06408
0.06025
0.05673
0.05350
0.05053
0.04778
0.04524
0.04290
0.4
0.12906
0.12102
0.11365
0.10689
0.10068
0.09497
0.08970
0.08484
0.08035
0.07618
0.03039
0.02853
0.02682
0.02525
0.02381
0.02248
0.02126
0.02013
0.01908
0.5
0.20085
0.18843
0.17703
0.16656
0.15693
0.14806
0.13988
0.13233
0.12534
0.11887
0.6
0.28780
0.27017
0.25395
0.23903
0.22529
0.21263
0.20095
0.19013
0.18013
0.17087
0.7
0.38940
0.36580
0.34406
0.32402
0.30554
0.28848
0.21271
0.25812
0.24461
0.23208
0.8
0.50504
0.47483
0.44694
0.42117
0.39736
0.37534
0.35498
0.33611
0.31861
0.30236
0.9
0.63394
0.59664
0.56208
0.53006
0.50042
0.47295
0.44749
0.42388
0.40195
0.38158
1.0
0.71527
0.73051
0.68888
0.65020
0.61428
0.58094
0.54996
0.52119
0.49443
0.46954
I.2
1.09086
1.03084
0.97450
0.92173
0.87240
0.82632
0.78331
0.74319
0.70575
0.67082 0.90408
1.4
1.44153
1.36729
1.29669
1.22982
1.16673
1.10734
1.05154
0.99918
0.95008
I.6
I.81417
1.72872
1.64605
1.56660
1.49070
I.41848
1.34999
1.28524
1.22412
I.16651
I.8
2.19324
2.10148
2.01079
1.92198
1.83571
1.75246
1.67255
1.59619
1.52348
1.45444
2.0
2.56227
2.47026
2.37706
2.28368
2.19110
2.10010
2.01134
1.92537
1.84253
1.76305
2.2
2.90661
2.82011
2.73027
2.63804
2.54437
2.45027
2.35665
2.26434
2.17401
2.08616
2.4
3.21651
3.13945
3.05771
2.97185
2.88252
2.79063
2.69713
2.60293
2.50896
2.41600
2.6
3.48824
3.42223
3.35114
3.27514
3.19439
3.10948
3.02111
2.92997
2.83702
2.74324
2.8
3.72306
3.66780
3.60777
3.54287
3.47289
3.39804
3.31864
3.23495
3.14770
3.05770
3.0
3.92495
3.87918
3.82925
3.77499
3.71598
3.65217
3.58358
3.51005
3.43195
3.34979
YOUNIS S. SELIM and
4
In the limit, and to get the result for the circle, we have to rotate the X-Y-axes, around the Z-axis through a 27~ rotation, however keeping the source point fixed. Ultimately we get: (3) where h(h2+p2+Rp
G=
sint)
-cc
