T-l Circular Radiator and a Circular Detector
26
AE-101 o T-l Solid Angle Computations for a Circular Radiator and a Circular Detector J. Konijn and B. Tollander AKTIEBOLAGET ATOMENERGI STOCKHOLM SWEDEN 1963
Transcript of T-l Circular Radiator and a Circular Detector
AE-101
oT - l Solid Angle Computations for a
Circular Radiator and a Circular
Detector
J. Konijn and B. Tollander
AKTIEBOLAGET ATOMENERGISTOCKHOLM SWEDEN 1963
AE--101
SOLID ANGLE COMPUTATIONS FOR A CIRCULAR RADIATOR AND
A CIRCULAR DETECTOR
J. Konijn and B. Tollander
Summary:
The problem of particle detection, when using an isotropic neutron point
source at different distances from a circular target or a radioactive source
as seen by a circular detector, e. g. a solid state counter, is dealt with.
Tables are given for different distances of the source» when the reaction at
the target has an isotropic or a cosine angular distribution in the laboratory
system.
Printed and distributed in February 1963.
CONTENTS
Page
Introduction 3
I. THE NEUTRON SOURCE AT INFINITE DISTANCE 3
A. Isotropic Angular Distribution 3
B. The Nonisotropic Angular Distribution 5
II. THE NEUTRON SOURCE AT FINITE DISTANCE 6
A. Isotropic Angular Distribution 6 •
B . Nonisotropic Angular Distribution 7
Commentary 8
References 9
Figures 10
Tables 11
- 3 -
Introduction
A basic problem in particle detection is to compute the solid angle subtended
by a detector with respect to the source. A counter of finite efficiency must
accept a range of particle angles. In this paper the following problem will*) 1)be dealt with. Similar calculations have been performed by Bame et al. '
Neutrons emanate from an isotropic point source. This is an idealization
of most experimental arrangements. The neutrons will knock on a circular
radiator and produce secondary particles, which are detected by a circular disc
(the detector). The plane of the circular radiator and detector are both normal
to the line through their centers and through the point source.
The neutrons will through nuclear reaction or scattering produce secondary
particles in the target which in fact is' a radiator. Each point of the radiator
emits thus particles of which the angular distribution may be isotropic or
not. It is evident that the number of particles that reach the detector will be
a function of the angular distribution function of the emitted particles at the
target (see Fig» 1).
In this paper both the case of an isotropic angular distribution and a
nonisotropic angular distribution in the laboratory system will be considered,
the first case representing a radioactive source, the second an isotropic
nuclear reaction in the center of mass system. All angles discussed below
are taken in the laboratory system.
I. THE NEUTRON SOURCE AT INFINITE DISTANCE
A. Isotropic Angular Distribution
In this particular case one only has to consider two circular discs. In
Fig. 2 the particles from the element of area dAj of the target may enter a
detector element dA and the angle 0 depends on the positions of the elements.
The resulting geometric solid angle, A, is found by an integration over both
radiator and detector:
dAdA (1)l
- 4 -
The following quantities are introduced:
R
P =l
d A d A i =
so that one obtains the solid angle from
IQ =
ir pin sterrad
(2)
(3)
The integral I will be of the form
R x R ZfT ZTT
- \ drj \ dpi \ dtp \ d 9jt>2l 2R a
The following transformations are done
a r
(4)
= Rp 0 ^ p < 1R
r , = Rr
e i =
0 > r Si
<P
R
^ 3 1? 2TT -
= P
and this results in
P 1 2Tr 2TT -<P
I = \ rd r \ p dp V d ^ \ d 3
(5)
o [ ? ? 2 1
qc + r + pc - 2pr cos 9J
3 /2(6)
This can after two integrations be rewritten as
P 2Tf
I = Ztr q V rdr \ —j J » ' + ̂ -
o o '
2+ » - a
(7)
with = rcosA 2 2 , 2 . 2 .P = q + r sin 3
2 «2 2 , 2a + p = q + r
(8)
p>2 + (1 - a ) 2 = q2 + r 2 + 1 - 2 r cos I
- 5 -
As cos 3 is an even function the integral will be
p TT
I = f ( r i 3 ) d 3 = I(p,q) (9)
where f(r ,3) is the above function in a and |J (eq. 7).
The integralform I(p, q) has been coded for the Ferranti "Mercury"
Computer and been evaluated for different values of p and q. The results are
presented in table I. The parameter p has only been varied from 0. 01 - 1.0
as it is easy to compute the integral for values of p > 1 by the relation
q) = P 2 I ( ^ •£•)
Furthermore the integral I(p, q) will converge to different values at
q = 0 depending on the value of p. This value is easily found, when is realized
that Cl must then either have the value 2irp or Zlf depending on p ^ 1 (target
larger than detector) or p %. 1 (target smaller than detector)
2
I(p, 0) =
Q(P, 0) =
2n
2irZpZ
277/p2
2TT
P> 1
P t 1
P •= 1
P = 1
(II)
(12)
B. The Nonisotropic Angular Distribution
When a nuclear reaction in the center of mass system has an isotropic
angular distribution, the angular distribution in the laboratory system will be
a cosine. The angular distribution of any nuclear reaction may be written as
a sum of t e rms , consisting of raising powers of cosif' in the laboratory system:
W (13)
If one knows the constants A each term of this ser ies can be integrated
separately.
-6 -
Owing to the fact that s ~ oo (see Fig. 1) vji in this case is equal to the angle 9.
For ;i cos-distribution one therefore obtains the formula
P 1 2iT ITT - <p n ^ , sr C C C q ti o
I T W d r V pdp V <l<p\ (14)
o o o d--•-?> [q i r + P - 2p r cos3J
This integral can easily be integrated and then gives
/ + q2 + I ) 2 - 4 p 2 j (15)
This formula has also been coded for the Mercury computer and the result
from this programme is presented in Table 11.
Due to the fact that a cosine has been introduced in the above integral the
normalization gives a slightly changed expression for the subtended solid angle
in this case
Q - 2 j in sterrad (16)Tf P
11. THE NEUTRON SOURCE AT FINITE DISTANCE
A. (sotropic Angular Distribution
If s i.s finite it iti easy to see that the number of particles emanated from
different places of the target is proportional constant to
(.7)2 Z " 2 , 2s •* r . w ^ r
where w := s/R and r --- rj /R. (18)
The integral I (eq. 9) then is slightly changed into
p IT
1 -- 41Tqw2 \ 2r d r \ f ( r i 3 ) d a (19)+ r
o
The r e s u l t f rom the M e r c u r y C o m p u t e r in this- c a s e \a seen in T a b l e IiJ.
- 7 -
From the formula above it is easy to see that
2I * I , >
w = oo w £00
w I2 2 w = 00
w + p
(20)
which means that I , „ must differ from I by approximately less than2 2 w too w -00 ' '2 2
P /w . Here ii s
a =i rp
/ _2\1In 1 +
2.\ w /J
in sterrad» (20 a)
B. Nonisotropic Angular Distribution
In this case after two integrations one gets
IT
a 47Tqwp
C 2 r d r
J [w + r 3 /2 g(r. (21)
where
A— arctg
2 2q + r
1-ttTI
Bx + C2 2 2
L (a + x )
(22)
with a = r cos d
a =
A = a.
2 2q + rwq + q
- *2
[2= a a -
C = - a [••2 | » 2
(23)
The result when using this formula is presented in Table IV. Here is
a = 2 1 (4)In 1 +
in sterrad. (23 a)
- 8 -
Commentary
All the above formulas are easily given by using
P i irdr C dp \ h ( r , a ) d3 (24)
o o o
where
h ( r S) - w 2 p_r_[qw- r 2 + prcosa]1 1 , ,Iw +r a +P +r — ZprcosSI ^ ' '
Here n means a cos -distribution in the laboratory system. It is easy to prove
that for w - oo one can take the limit of h(r, 3) and integrate this limit.
The above formulas are all valid for the case of reactions on heavy nuclei
as no limits are set to the value of the angle vj; (see Fig. 1). For a heavy
scattering nucleus, A » l the scattering angle in the lab. system then
approaches that in the center of mass system. Consequently, if scattering
for relatively heavy nuclei is spherically symmetric in the latter system,
it will also be so in the former.
The important case of neutron-proton scattering will be handled in a
separate paper. In this special case the angle *f is limited to values smaller
than V/2, due to the kinematics of this nuclear reaction,.
- 9 -
References
1) Bame Jr, S. J. , E. Haddad, J. E. Perry Jr, and R. K- Smith
Rev. Sci. Instr. _28 (1957) 997.
*)Bame et al. have only considered the case of a cosine distribution in the
lab. system. Their parameter choice is somewhat different:
l/p» <l/p a n <^ arctg p/w and covers the region p = 1, 1/2, 1/3; w = 1 - 30;
q = 1 - 14.
Fig I
SOURCE
TARGET DETECTOR
•ssffi- a
Target
dA
at OK
-11-
W = I N F I N I T Y
F o. oi
i*9542»1*8753,1*7775»1.681 1,I .5868 ,I .4952 ,1*3218,
l*l°39»8.9814,7.4080,5.7813,
2.0839,1.or 29,5.8936,
Isotropic Distribution
Table Ii
2.6857,1.9839,1*5243»9*79^2,4*3719»2.4028,
3*9467»9.8690,
0.025
-3 1. 221 4, -2"3 1.1721, -2•3 1.1109, -2-3 1.0506, -2"3 9*917°, -3*3 9*3442, -3•3 8. 2607, ~3"3 7*2735, -3-4 5.6137, -3•4 4 .6295, -3"4 3.6139, -3-4 3.0718, -3•4 1 . 3 0 3 3 , - 3
'4 °*33°7* ~4"5 3*OÖ34» ~4•5 2.3957, -4•5 1.6786, -4
•5 1*2399» -4"5 9*5267, -5-6 6. r226, -5"6 3.7324, -5-6 1.5392, -5-7 2.4667, -6-8 6. 1681, -7
o. 05
4.8854,4.6Ö8I,
4*4433»4.2021,3.9662,3*?369»3*3°34»2.9OÖ4,
I.85IO,r . -5 445»3 . 2 8 4 3 ,5.2081,
2*5319»I.4732»9.5824,6.7140,
4*9595»3.8106,
2.449°,I.0930,6.1569,9.8668,
2.4^73,
-2-2-2•2-2•3
-2-3"2-3-2
-3"3"3"3"4•4.4
-4"4"4•5-6-6
o. 10
I .9541, -11.8750, ~i1.7768,1.6800,i*5ö54»i*493°*I.3199,I . I 6 J 8 ,8.9616,
7*391°»5.7684,
-1-1
-1-1-3- 2- 2
3 3 9 4 »2.0813, ~2I .OI22, -2
3 .8321 , -33.6852, -31.9836, -3
9*7955» - 44.3718, -42.4627, - 43.0466, -59.8693, -6
W » I N F I N I T Y
o* 15 O. 20 O.25 30
0. 010.050. IO0.150.200.25
°*35°*450.630. 80
1. 00
1.502. OO3.OO4. 005.006. 007.008. 0010.015.020.050. 01 0 0
4*3963»4.2176,3*9957»3.7771,3*5634»3*35DO»3.9O42,2.6odo,2.0107,1.6581,1.2942,7.4297,4*6754»2.2756,1.3248,8.6193,6.0403,4.4623,3.4388,2.2038,9.83ÖI,5*5 41^»Ö.8800,2 . 3JJO6 ,
""I-I-I-I
-x"I-I™" I
-I"X""I-2
- 2
-3-3
-3""3-3-3—3-4-4-5
*~5
7.8156,7*4954»7.0982,6.7073,6.3353,5*9547»5.2558,4.6215,3.5006,3*9357»3.2917,
1.3168,8.3935,4.0406,3*3535»I* 53x6,1. 0734,7. 9309,6.0944,3*9173»1-7485,9.8504,1.5787,3*9476*
~l-I-1~i
-1-1-l-1-1~l-1-I—2-2
-a""3
-a-3-3-3""3-4-4-5
I.22II,I.I706,I.IO8O,
I.0464,9.8632,9.2807,8.1836,7.1906,5*535°»4.5628,3.5625,3.0494,1.2922,
°*3°3°»3.6738,2.391°»1.6765,1.2388,9.5200,
6» u 98,2.7319*1*5391,3.4666,6.1683,
0
0
0
0
-I-I-1-1
~l""I-1-1-I-a""2-2- 2—2
~3"3"3"3-4-5
1.7580,1.6847,i*S936»1. 5040,I. 4167»I.3322,1*1733»1.0300,7.9200,6.5278,5*°979»2.9369,1.8543,9.0600,5.2842,3.4412,2 . 4.1 zii,1*7831,i*3?°4»8.8107,3*9336»2 . 316 I ,3.552O,8.8833,
0
0
0
0
0
0
00
-1-1
-1
-1
-I-3-3- 2— 3-2-2
~"3"3-3-4"5
-12-
Isotropic Distribution
Table I
u =VP
0,01o. o^O. IO
0.15o. 30o. 25°*35o. 450.65o. 3o1.001.502 . OO
3 • 004 . o o5. 006. 007.008.0010. o15.020.050.0
I 3O
o. 40
3 .12a8 ,2»99OI,2 .8235 ,2.66OI,2 . 5 O I i f2.3480,2.06 i 3 ,
I**O57»1.3850,I . 1 4 1 t ,8 .9185,
3 .2676,I . 6 0 2 9 ,9.3666,6 .1058 ,4 .2836,
2 * 4 3 4 4 ,
6.9914,
3*9393»
oooooo 'o'ooo
~x
~2- a"2
"" 2
-2
- 3- 3
I . 579I, - 4
0.50
4.8 7 5 o, o4.6610, o4.3904, o4.1257, o3.8656, o3.6240, o
2 .7677 , o2 . I I 7 O , O1.7441, o1.3649, o
7*9343, ""i
2 .4892 , -11.4580, - 19.5164, -26 .6812 , ~2
4*9415» ~ 3
3*7999» ~22*4445» "21.0921, - 26 . 1 5 4 1 , -39.8600, —42 * 4 6 7 3 » ""4
0.60
7.0177,6 . 6 8 9 1 ,6 . 2 7 8 1 ,5.8786,
5* 4949»5.1302,4 » ^643»3.8854,2. 9644,2* 4432,1.9158,I . 121 I ,7*1719,3 * 5 5 / 3 »2.0900,1.3662,9.6000,7.1042,5* 4 65°,3*5173,
8* 36o 1,1. 4207,
3*553° ,
oooooooooooo
-I-I-I-2-2-2-2-2
-3"3"4
o. 70
9*5570,9.0586,8.4574,7.87 93,7*3312,6.8178,5.8988»5*i*7O»3.8978,3 . 2164,2 ,
1 •
9.6079,4» 8004,2 .8295,1.8529,
i*3°33»9.6513,
7*4275»4.7829,2. 1388,1.2057,i*9336,
. oooooooooooo
- I- I- I
- I
- 2
- 2
- 34.8357, -4
W = INFINITY
o. 01
o. 100.15
o. 20
o. 25
°*35°*45
o. 80I . OO1.502 . OO3 . 004. 005. 006. 00
7. 008. 0010. o15.020» o50.0
IOO
o . 80
1 .2496 ,
1*17 3 5»I . O Ö O d ,
I .OO52,9*2977,0 . 6 0 7 3 ,7 . 4 0 5 1 ,6 . 4 0 8 3 ,4 . 8 8 3 5 ,
3 • ^ 9 9 *1 .9008 ,1 .2324,6 . 2 0 7 4 ,3 . 6 7 2 9 ,2 . 4 I O J ,
I . 2 5 7 8 ,9 . 6 8 4 8 ,6 . 2 4 0 1 ,2 . 7 9 2 1 ,
2 . 5 2 5 4 ,6 . 3 1 6 4 ,
r1x1ooooooooo
-I
•I
-I
•2•2
• 2
•3"4
O, 90
I . 5 8 2 3 ,I . 4618 ,
1*3357,1 . 2 2 3 7 ,1 . 1 2 4 9 ,
1*0373»8 . 8 9 4 4 ,
7 * 6 9 5 7 ,5 .8^ ,53 ,4.8070,3.8800,2*33 ; j7»I . 5 2 8 6 ,7 . 7ÖCS6,4 .6 164,
III1I1ooooooo
2 . I 4 I O ,I . 5 8 8 0 ,I . 2 2 3 4 ,7 . 8 8 7 9 , ~
3* 5 3 1 3 "X • ; 9 X 3 ,3 . 1 9 6 1, - 3
-1- I-X-x- I- 2
2- 2
I . OO
r . 9 2 4 3 ,
I . 5 4 8 6 ,
1. 41 1 '3,1*295-*,1 . 1 9 5 1 ,1 . 0 2 6 9 ,8 . 9 1 5 1 ,6 . 8690 ,
5 * 7 3 5 1 ,4*5<337»2*7993»1 . 8 4 5 7 ,9* 47^7»5 * 6 5 5 6 ,3 , 7 3 9 0 ,2 . 6 3 3 5 ,1 .9550 ,!•5O7I»9*7246,4*3 57 5»2 . 4 5 8 2 ,3*945 5»
111ii11oooooo
- I"* I- I- I""" 2""• 2
— 2
~39.8685, -4
-13-
Cosine Distribution
Table IIi
\ p
O. 01o. 050. 100.15o, 200.25
o*350.450.650. 801.001. 502. OO3.OO4. 005.006. 007. 008. 00IQ. O15.030. 050.0100
W =
•X0. oz0. 050. ro0.15o. 200. 25
°*350.450.650. 80z.001. 502. OO3.OO4.OO5.OO6»oo7. 008. 0010.015.020. 050. 0100
i HF1 r-i 1 T Y
0. 01
9.8686,9.8450,9*771-9»9.6524,9.4900,9. 2890,
8.7924,8.2075,6.9381,6.0179,
4*9347»3.0367,1*9739»9.8695,5.8056,3.7960,2.6675,.1*9739»I*5I84,9*7719»4*3Ö7*.2* 4612,
3*9463»9.8686,
J NF1N|TY
o. 1 5
2.2204,2.2150,2. 1983,2.1707,
. 2.1334*2.0874,
1.9739»1.8408,
1*5537»1.3468,i.1041,6.8001,
4* 4254,3.21Ö3,I.3046,8«5339*5.9982,4*4394»3*4152»2.19Ö3,9.8350,5*5375»8.8790,2*2204,
-4-4-4-4-4"*4"*4-4""4~4-4-4—4-5"5-5.-5-5-5-6-6-6-7-8
-I
-I-I-I
-I-I-I
-I-I-I-I-2
~3-2 '- 2
*~3"*3-3~3-3~4~4"*5-5
o. 035
6, 1679,6.153 t,6.1074,6.0327,5*9311»5.8054,
5*495°»5.1293,4*335d»3.7607,3.0838,1.8 977»1.2336,6.1682,3.6284,2.3724,1.6671,
r.2337,9.4899,6.1074,2.7294,
I.5383,2* 4664,6.1679,
o. 20
3*9474*3*937°»3.9073,
3*0575»3.7902,
3*7O7i»3.5039,3.3643,3.7518,2.3843,
I.9542,L2O44,7.8454,3*9337»2.3171»1.5162,1.0659,
7.8895,6.0699,3.9073,
I.7465»9.8440,r.5785,3*9474»
-3-3-3-3-3-3""3-3~3""3-3-3"3-4"4-4-4-4-5-5-5-5-6-7
-I-I-I-I-I-I— I
-I-I-I""I-1—2- 2
-3-3-2
-3-3~3-3 ."*4-4-5
0. 05
2,4672,2* 4612,2.4429,2.4130,2.3723,2.3219,2.1976,2.0512,
i*7336»1.5036,1.2329,
7.5880,4.9328,2.4668,1*4512,
9.4891,6.6682,4*9346»3*7959»2.4429,I.O918,
6.1531,9.8656,2.4672,
0.25
6. 1678,6.1521,6.1034,6.0341,
5.9167,5.7845,5.4604,5.0832,4.2788,3*7O49»3*°3°i»1.8739,1.2215,6.1340,3.6160,2.3670,
1.6644,1.2322,9.4810,6.1037,2.7287,
I.538O,2.46Ö4,6. 1678,
- 2~3- 2
~2"2- 2
-2-2
-2-2
-2
~3-3-3-3-4-4-4-4-4-4"5-6-6
-i-1
-i-i
-t
-r-1-1
-1-i
• 7 1
"** X
T-2-2
- 2- 2- 2
"~3-3-3-3~4-5
0. 10
9.8686,9.8447,9.7709,9.6503,9.4865,9.2839,8.7839,8.i960,6.9237,6.0037,4.9225,
3*O3O3»I.9708,9.8607,5.8034,
3*7946»2.6668,i»9735»1*5182,
9.7709,4.3669,2* 4612,3.9463,9.8686,
0. 30
8.8817,8.8583,8.7863,8.6688,8.5101,
8.3151,7*8394.7.2889,6.1241,5.2989,4*3414.2.68l3,1.7513,8.8lI3,5.I093,
3*4O5I,2.395O,
1*7734»I*3647»8.7869,3.9288,2.2146,
3*5515»8.8817,
-2
— 3-2-3-2
*""3-2-2- 2
-3
- 2
-2"*2
~3-3~3"3-3-3-4"*4-4-5-6
— I-I*•* I
-I"* I-I-I-IT""" I
-r-1-1- 2- 2
-2-3-3
- 2
-3-3-3-4— 5
- 1 4 -
Cosine Distr ibution
Table II2
W = INFINITY
4 0 o. 50 0.60 o. 70
o. o~0.050. ro0.150. 20
0. 25
°*35
0. 650. 801. 00
1.502. 003.00
4. OO
5.006. 00
7. 008. 00
10. 0
15.020. 0
50.01 0 0
W =
0. 01
0. 050, 10
0.150. 20
o« 25
<>• 350. 4 5o« 050. 3o1. 00
1.502. 003.00
4. 005. 006. 007. 00
8.0010. 0
15.020. 0
50. 01 0 0
1.5739,t»5745»1.5606,I.5381,i»5°79»i.4711,1.3022,1.2810,
2.0714,9*255i»7.5804,4.6964,3.0790,
. 3C*5567»- 9*2°74»6.0379,4.2501,3.1484,2.4236,1*5611,6..9824,3*93°4»6.3136»I.579O,
1HFINI TY
0. 80
6.3148,6.2735,6.1531,5»975Ö»5.7636,5-53C-4»5.0490,4.5836,3*7554»3-3357*2.6653,1.6990.,
1*143 4»5.9705,3.5882,2.3732,1.6789,1.2477,9.6244,6.3150,2.7871,1*5727*2.5250,6.3155,
0
0
0
0
0
0
0
0
0
-i~ i
- 1
- 1
-1*~2
"™ 2
~ 2
- 3
- 2
—'2
-3"~3~4"4
0
. 0
0
0
0
0
0
0
0
0
0
0
0— v
*"*!
-I-I-I- 2
~2"" 2
~ 2
-3-4
2.4671,2.4592»2.4351,2.3962,2*3 444»3.2819,2* 1339»l.9692,1.6378,1*4125,I..1569,7.1998,4*7432,2.4130,1.4316,9* 4°3°»6,6351,4.9107,3*73l7»2.4370,i.0906,6.1493,9.8647,2.4671,
0. 90
7.9902,7*3957*7. 6535,7*3 475»7.0212,
6.6941,6.0661,5*4S95»4.4999»3.8882,3.2185,2.0767,I.4107,7*4473»4.5003,2.9852,3 . 11 5 6 ,
1-573 9»1.3150,7.0529,3-5a/;i>»1.9896,
3*^954»7.9929,
0
0
0
0
0
0
0
0
0
0 '
0
-1
~l- 1
-1~ 2É**'2
- 2
~2— 2- 2
~3~4~*4
0
0
0
0
0
60
0
0
0
0
0
0
~l
-x-" 1
•* 1
_ j
- 2
- 2
-3-4
3*5525»3*5393,3*4989»3*43 45»3*35°2»3.2502,3.0200,
2.7724,2.2919,i»97'39*1*6179,1.0130,
6,. 7141,3.4412,2.0491,1.3486,9.5138,7*0563»5*4366»3*5°55*1.5697»8.8536,1.4205,3•5536,
r. 00
9-7714»9.3883,8.9308,.8.4960,8.0832,
- 7.6914,6.9673,6.3165,5.2090,4 * 5340,3.7699,2.4674»1.6934,9:0478,5.5001,3.6604,2.5991,1.9360,i* 4957"»9.6770,4*3479»2-4551»3-9447»9.8676,
0
0
0
0
0
0
0
0
0
0
0
0
r*i-I-1
-I- 2
"*"2— 2- 2
- 2
-3~3-4
O
O
O
O
O
O
O
O
O
O
O
O
O
~ I~I-t-I"™I
-I"™ 2t
"•*2
-2
-3"~4
.4*6;J52»4.8126,4*7447»4*6391,4.5046,4*35°°»4.0090,3*6588,3.0081,2.5894,2.1262,1*3419»8.9570,4.6309,2.7695»i.8269,1.2904,9.5802,7.3853»4-7653»2*1353*1.3045,1-9333»4-S354,
O0
0
0
0
0
0
0
0
0
0
-.1
-i
~.i-i
~ 1
- 2
— 2-2
- 2
-3-4
W =
o. xoo. 15o. 200.25o*35
0.65o. 80J.OO
1*502 . OO3 . O O4* 005.006 . 007. 008. 0010. o15.030. o50.9
100
0.5
0.15
j3.0166,3.4132,3.2136,
3.4974,1.9355,
7* U46»4* 4771»2.1790,I .3686,8.2533»5*7837,4.37 Ä3,3.3833,3 . I I O 3 ,
9.4184»5*3°57»8.5039,2 . I3Ö2»
-1•i•i•I•i•1-I-i•r-3•2-2-2
•3
'3"3"3'3•4
-4•5'5
-15-
Isotropic Distribution
Table ini
°*3
l*36 l4»1.3850,1.2105,
1.0039,8.8053,6.7719,5.5816,4.3588,3.5*°5»1.5847,7.7408,4*5143»2.9396,3 . O 6 l I,
I .533I,I . I / 0 6 ,7*5^57»3*3593,1.8929,
ooooo
-I
-I
-I
-I
-1
-2
-3-3-3"2-2
-3
3 * ° 4 7 7 »2.8663,2.6905,2.5216,3 . 3 O 3 3 ,I . 9 3 0 1 ,1*4777»1*2175»
5*5374» -
i.7388,
1*0119,6.6016,
•3"4
7.5866, -5
3.4366,3.6347,1.6948,7*57°4»4.2659,6.Ö307,1.7102,
ooooooooII
-I-t
-3-3-2
"2
*3-3
- 4
O.?
4 . 7 I I 3 ,4.4026,4.1083,3*83°4»3*3374»3.8939 ,2.2083,1.6214,1.4299,8.3900,
2.6736, -3
l .0384,7.3385,5*35°4»4. 1164,2.6498,I.1045,6.6765,1.0706,2.6774,
ooooooooo
• I•I
• I- I
•2
•2
•2
•2
•3
•3
•4
I . O
6*559l»0.0451,5*5894»5.1803,4*4745»
2.9837,2.4774,i*9637»1*1774»7.6639,3.8778,3.3999,1.5111,i•0650,7.8960,6.0817,3.9302,1*7548»9*8955»1.5876,3.9703,
oooooooooo
•I
•I-I-I
•3•3
"3•3•4
W = 1 . o
o . 15 0 . 7 I . O
0.100.150. 200.25
°*350. 450.650. 80i. 00
I* 502. 003.004. 005. 006. 007. 008. 0010. 015.020.0
50.0
100
3*9514»
3*7353»3.5340,
3«3Jt»S,3.9314,3.5793,1.9885,1.6398,I.2799,
7*347 5»4.6337,2.2504,i*3 101,
d. 5238,5*9733»4. 4128,
3*3 9°'*»2.1793»9.7371,
5.4796,8.7816,
3.195 9.
-1
-i~* I-1-i
-i"" X-1-1-3
"2-3-2
~3-3-3-3
-4""4~5—5
r. 5360,1.4403,
1.3567,1.2758,1.1237,9.8650,
7*5Ö59»6.3525,
4.8828,3 . 8 12 0 ,
i*775ö»8.6760,5.0601,
3*295a>2.3104,
1*7074»r.3133,8.4366,3.*7OO5»3.1230,3. 401 t,8.5050,
0
0
0
0
0
-I-I
-I-I
~t
-r~3-3-2
"* 2-2-2
-3-3-3-4""5
3.9307,
3.6853,3*4574»3« 2387,3.3337,a*4753»1.8939,1.5603,j.2209,
7.0935,4.5116,3.2232,i. 3019,
8.4963,5*9645»4» 4113,
3*393°»3.1821,
9*7 479»5*4931»8.8062,
3.2O33,
O
OOO
O
OO
O0
-I-I-I-I"2-3—2*•* 3
"2
-3~3~4~4
0.8990,
6. 't3 5o,5.9940,
5*5796»4.8348,4. 1980,3.1998,
2.6399,2.0747,I.22l6,7.85IO,3*9153»2.3058,
i*5°93»1.0614,
7.8582,6.0470,
3*^93 4»1.7*408,9.8126,
i*5736»3*9354,
0
0
0
0
0
0
0
0
0
0
-i
-i
-1-i
-i-3- 2-2
-2
~3~3"*4
i.1009,1.0086,
9.2866,8.5815,7.3878,6.4139»4.9398,4.1047,3.3679,1.9791,1.2974,6.6166,
3*9387»2.5939,1.8395,1*3574»r.0460,6 . 7 46 3 ,3 . 0215,1. 7043,
3.7349»6.8404,
110
0
0
0
0
000
0
-1
-i
-1
-i-1
~x-2
-2*"* 2-3"*4
W = i. i
-16-
Isotropic Distribution
Table III
i 1
O. 10o. 15o.o,o.
202535
0.05o. öoi» 001.502. 003. 00
4. 005. 006. 007. 00ö. 0010. o15.020. o50. o100
0.15
3 »97S0»3*75Ö4»3*5457,3»3393,2.9495,
2*S95i»2.0008,1.6499,1.2878,
4 .~6 522,
2. 2<">43,1.3182,
ö*5?65»6.0102,4.4401,3*4117»2» i
8.8359,2.2095,
•i•1•1•1•1•1•i•r
•1
• 2
-z'3•3•3•3'3*4•4•5*5
t •
i*
i.
1*
i.
i*
7*6.4*2.X.8.
5*3*2*1*1.8.
3*2*
3*8.
5626,4748,3892»3OÖ3>1506,oici,766Ö,40x5,9992,8800,
1814,3742,3 659,7484,3 43Ö»63&:»8569»17 3°,4828,7092,
Oooooo-I•I
-I-I-I-2
-2-Z"2-2
-2
"3-3"3"4-5
4.3*3*3*3*2*2.I*
I •
7*4*2.I.9*6.4»3*2.
I*
5*9*2.
o. 5
1641,9136,6711,43Ö5»0078,6209,0096,
295 5»52.95»7900,
0250,3359,6860,6034,
3554, -
0.7
7.6605,7*I4I°»6.6480,6.1854,5*3556,4.6480,3*54i6»Z» (jZZZf2.2974,i*354i»8.7099,4*3477,..2.5616,
i*i795»8.7337»6.7210,4.3277,
i*935i»1.0908,
1.7494,4*375°,
oooooooooo-I-I-I-I-r"2-2-2-2-2
"3"4
III •I8.7*5*4*3*2*I.
7*43211.8.
2994,1878,0920,0081,
6699,5262,79°5»8273,8500,34°6»53«<>*8607,0912,
11i1ooooooo
-I-II
2.
3*8.
iO0°5* -*1815, -1,6190, -i-2478, -r0495,6060,0341,264545»1651,
•2
•2"2
'3"4
wp
o. 10
o. r $o. 20
0.25
°*35
0.650. 801. 001*502. OO3.OO4. OO
5* °°6.007. 008. 0010. o15.020. o50. o100
2*5
O. I <
9085,
3*5570,3*35°°»2.9589,3.6034,2.007 i,1.6552,1.2919,7.4164,4.6671,2.3715,1.3224,d.603d,6.0393,4*4543»3*4226,2.1998,9« 8184,
8.8640,2.2166,
-I-I-I•I
-I-1-I-I-I-2
"4-5"5
1. 5822,
- T S *t -J *
I.4O66,I.3227,I. I6CO,I.O227,7.863 9,6.4816,5.0618,2 . 91 (> 1,
-2 I.84II,-z 3.9955,-z 5.2466,-3 3*4*67,~3-3~3
2.3956, -
1.3607,8.7479,
3*9°55»2» 2003,3.5266,8.8189,
0.7 1.0
0
0
0
0
0
0
.t"III12
2
22
2
2
33345
4.3052,4.0459,3*7949»3*5542,3.1087,2.7147,2.0766,L7IO8,1.33 döt7.7820,
4*9513»2.4410,1.4297,9*3315,6.5512,4*8453,3*72.59,2.3969,I.0708,6.0342,9.6738,2 » 4192 8
0
0
0
0
0
0
0
0
0
-I~ i
-I-I-2
-z- 2
*"* 2— 3
-2
~3-4~4
8.1450,7.5908,7.0643,6.5707,5.6867,
*K • 7 *? *•* 8 •3•1015,2 . 4389,I.4384,9.257O,4.6234,2.7247,I.784I,I.2549,9.2924,-7.1512,4.6048,2.O59I,r.1607,1.8615,
4*6S55»
0
0
0
0
00
0
0
0
0
-1
-1
-1-1~ 2
-2
~ 2
-2
-z~3-4
1.4450,1.3187,z. 2112,1*117 4,9.6051,8.3380,6.4204,5*3572,4.2779,2.6078,1.7173»8«8o I ?,5.2516,3.4614,2.4441,r.0142,1.3984,9.0224,4.0425,2.2804,3.6600,9*1543»
t1
ft
t0
0
0
0
0
0
0
-i-1
-i- i
-1
-x- 2
- 2
- 2
-3-4
- 1 7 -
Isotropic Distribution
Table III3
w. p
1 1
0. IO
0.15O.2O
0. 25
°*350.45o.<>50. 80
1 • 00
1.502. 003.00
4. 00
5.006. 00
7.00
8. 0010. 015.020. 050. 0
1 0 0
= 3*5
0. 15
3.9920,
3-773^»3.5602,3*3539»2.0615,
2,6057,2.00Ö9,I.6506,
1.2930,
7.4229,4.6712,2.2735»l*323°»Ö.6114,6.0347,4.4582,3*4Z56>2.2018,
9.8271,5*53bo»8.8718,2.2165,
""i
-1— 1-1
-1- 1
"~i-1
•** 1
— 2
-2- 2
- 2
~3~3
-3~3~4-4-5-5
0.3
l.5878,1.4985,I-4H5»1-3273»1. 1691,I.0263,7-8913,6,5041,5.0794,2.9263,1.8475,9.0270,
5.2649,3.4287,2.4040,
1.7766,i*3654»8.7785»3.9192,2.20Ö0,
3* 5 3 9°»8* 0498,
0
0
0
0
0
0
-I*~ I
-I-I- 1
- 2
"~2
- 2
- 2
-2
- 2
-3-3-3-4~5
°*5
4*3464»4.0845,3*83*0,3.5880,3« I38i»2.7403,2.096 r,1,7369,
i»35i4»7.85.56,4.9984,2.4643,1.4434»9.4209,6.6140,4.8918,3*76*7»2.4199,1.0811,6.0922,
9.7667,2.4434,
0
0
0
0
0
0
0
0
0
-Imm J
-I—"I
•"• 2
-2
- 2
-2™*2
- 2
~3-4~4
0.7
8.2946 ,7.7285,7»I9I7,6.6887,5*7879»5.0213,3.8252,3«I564»2.4822,
I. 4642,9» 4^45»4.7079,2.7747,I.8169,I.2780,9*4637»7.2831,4.6898,2.0971,I. 1822,1.8959,4.7415»
0
0
0
0
0
0
0
0
0
0
-I-I-I-I— I-2
-2
-2-2~ 2
"*3-4
I.O
I.* 4 933»1.3622,1.2507,X'i536»9»9i5°»8.6072,6.6296,5*5333»4.4204,2.6970,1.7771,9«il39»
5*4395»3*5859»2.5322,1.8797»I.449°»9*3489*4.1890,2*363i»3*7937»9* 4864,
1
1
1I0
0
0
0
0
0
0
-1-1
—"I
-I-1
-1-2-2- 2
-3-4
to =
15 0 . 3 O . 7 1 . 0
O. IO
o. i 5
o. 200.25
°*35o«450.650. 80I. 001.50
2. 00
3.004. 005. 006. 007.008. 0010.0
15.03O. O50.01 0 0
3*993 9»3*7754»3.5618,3*3545»2.962 y,2. 6069,2.OO98,I.6574,x.2936,7.4264,4*6733»2.2746,1.3242,
8.6154,6*°375»4*4603,3.4272,2.2028,
9*831?»5.5385,8.8760,2.2196,
-1
"-X~ 1-1- 1
- 1
- 1
~ 1- 1
- 2
—2
"*• 3
-3"3~3-3-3~4~4-5"5
1.5907,1.5013,1.4142,1.3298,I.1712,i.0282,
7»9°59»0.5161,
5.0888,2.93x7,1.8510,9. 0438,5.2747»3*435°»2.4085,1*7799»1.3680,8.7949,3.9265,2.2x22,
3* 5456»8.8063,
0
0
0
0
0
0
"*X
~x"* X"™I
~I- 2
-2
*"• 2
- 2
- 2
""2
"*3"3-3~4-5
4.3687,4* *°54»3.8506,3.6062,
3.1540,2.7542,2. 1067,l*7356,I.3582,7*8955»5*°239»2. 4769,1.4 508,9.46 93,6.64Ö0,4.9170,3.78x0,2 • 4324,1.0867,6» 1235,9.8170,3*455°»
0
0
0
0
0
0
0
0
0
~ X•"i
- 1
~x*** 2-2
- 2
- 2
"*2
- 2
""3~4-4
8*37 65,7.8044,7.2620,
6*753 7»5*8437»5.0695,3.8617,3. 1866,2.5060,
1.4785,9* 5169,4*7544»2.8023,
1.8350,1.2908,
9*558i,7*3558,4*7367,2.1181,
1.1940,
1*9149»4.7809,
0
0
0
0
0
0
0
0
0
0
-x~x-1
-x-I
- 2
- 2
-2
-2
"*2
-3-4
x.5208,I.3869,I.2732,I.I743»I.0091,
8.7604,6.7487,5*°337»4.5016,2.7478,1.81x2,9* 2922,5.5469,3*657°»2.5825,1.917 x,1*4779»9*5356»4.2727,2.4103,
3.8686,9.6763,
1
IIX10
0
0
0
0
0
-i-1
-i-1
-x-i- 2- 2
— 2
-3-4
-18-
Isotropic Distribution
Table III4
W = i o . o
. P o. t °*3 0 . 7 1. o
o. ioo. 150.200. 25
°* Z50. 450.650. 801. 001.502. 003. 004. 005.00
6. 007. 008. 0010. 015.020. 0
50. 0
I 00
3*9952,3.7767,3.5630,
3*3556»2.9639»2.6078,2.0105,
1.6579,1.2941,7.4289,4.6749,
2*2753»
8.6183,6.0395,4.4618,3.4284,2.203 5,9.8350,5.5404,8.8790,2.2203,
-I
"* I-I-I-I-I-I
-I
~ I-2"~ 2
-2
-2
~5"~3~3~3~3-4"*4-5— c
1*5929»I.5033,r.4161,1.3316,1.1728,I.0295,7. 916 5,6.5249,5.0956,2.9356,1»8535,9, 0 560,5.2818,
3*4397»2.4118,
I* 7823,1.3658,d.8068,
• 3*9310,2.2151,
3*55°4»8.8783,
0
0
0
0
0
0
~ 1-1
-i~i
-i-2- 2— 2w 3
"*" 2
""-3
"3-3-3-4"5
4*3849»4.1206,
3.8648,3 • 6 x96,
3*l656»2.7643,2.1144,1.7420,
i*3632»7.9245,5.0425,2.4861,1.4562,9.5046,6.6728,
4*9353»3*7951»2 • ii] *j X *J 9
1.0907,6.1464,
9*8537»2* 4642,
00
0
0
0
0
000
-i-i-1"• 2
- 2
*"*2-2-2
- 2-2
"3-4~4
8.4370,7.8604,7*3137.6.8016,
5.8849,5.1050,3.8887,3.2089,2.5236,I.4890,9.5849,4.7888,2.8226,I.8484,I•3002,9.6277,7.4094,4.7712,
3.I336»I.2O27,I.9289,4.8239,
OOOOO
OO
O
0O
-I-I-I-I
"*X-2-2
"Z-2-2
-3-4
1*54X5»
i»4°55»1.2901,1.1898,1.0224,
8.8757»6.8383,5.7092,
4.5628,2.7862,I» 8369,9.4267,5.6279,3.7IO6,2.6205,
1*9454»1.4997»9.6764,4*3359»2.4460,
3*9259»9.8195,
i
il
XX0
0
0
0
00
~I-I
-I-I-I-1-2~ 2~ 2
~3-4
-19-
Cosine DistributionTable IV
i
o .o .o .0 .0 .
o .• o.o .X .X*3 .
3»4»5*• 3 .
7*S .
v;
xor-S•?.o
"53545^ »Oooo
5°oooooooooooooo
xo«o^5 • o3 0« Ö
5?;
ii
•'o0 0
« o.5
o, 15
2.0556,3.0377,2.9909,X , 94-') x ,x . ^ 3 7 3 ,r . 7 X i o ,1 . 4 4 x 5 ,x« 34-^7,X.0335,6.3090,4.xrx2,2 . 0 6 3 4,X.3l64>7.-9543,5*^0x5,4.247O,5 . 2 9 3 1 ,3»0556,9*2945*K • X •* d. 0 t
»J w T •#
"•3X7-%2.O r t0'5,
" I 0 0
-z—x-x— T
—• T
—•T
-x-x*~3
- 3
~*3- 2
" 3- 3" 3mm ,y
- 3
~ 4*~5-5
6 .6 .
6 .6 ,5*4»
' 4 .3»3 .I .
tf.4 .2 .X .X .X .fi.3»X .3 .
7 .
O*3
77 5 - '6350,
45^5*O43J.59o6,^ 7 5»0397,
3993*O445»343O,
°374**> 53 5»07xo,3*79,0695,-999,O934*74^*-072,0360,
""X— T
" I™x— T
~x~x
""X*~2"*" x
~ 2~"3~*3~ 2~ 3~ 2" • • :
- 3
—• 4T
-5
x .X •
X .I .
x«I «0'' •7*6*3»3 .X .
7»5.3 -w ft
2 .X .
6 .3 ,5-x .
0 . 5
39 3 - ,3 555>3138,266?,1 6 5'^,0636,4 '* i J *
4 9 ^ ,-3^4»?535*5C"o,3 3 ' 4.99C2,3979»755*.79 57*1603,39^9»301^-,5^54.754-.4433,
0.00060
""I— T
— T
*"X
~x
~2"*2*"2— 3
— 3
- 2
""3~3"*4
•T
O . 7
I.93O9.
1 • 7'5aö,1, -17 56 ,r .5O75*.^ • 3 5 3 4 .i » O 9 3 7 ,
9*3745.7.7035,4»93 46*3.3560,x . 7 ® 0 7 ,X.09X9,
7»3O43»5.21x3,3»^979»3.0333,1.9663,8.9113,5*0563,8.3O3I,
OO
O
O
O
O
O
~x~~z~x-x""X-1"Z
-z- 2~*2""2
- 3- 3- 4"4
x o o
2.203X,3.O353,X»S9328X.769X,X.5603,i . 3 #9 3 ,I.X237»9-7095.S.O364a
5.4004,3*80x4,2 . X 2 3 7 ,1*33^4»9.0589,^•5*3^5*4*92*5,3•*43"»
. 2 . 5 2 2 1 ,i»i545*6.5832,2.0770,2.7207,
00
0
00
• 0
0"*Z
"X* I" I~X*"2™*2
- 2
r s" 2- 2
*"3""3~ 4
o. 15 0 . 3 O . 7 I . O
o.xo0.150» 300.25
°«3 5°»45
O«oOx.oo=•503 . OO3 . 004» 00>oo6. 007» 00Cooxo. 0- 5* 030. 050.0
xoo
3.1593.3 » X '•• X 3 ,
2» 093 5,3.O473,2 . 9 3 4 4 .x. '^027,x . 5 3 0 2 ,2.3-73.Z.0797,Ö.653-,4*3337,3 .1732 ,^•»3797*'^•3745*5.88:8a,4*3590,•3»3540,"* - j V T »
5*4437.C. 7300,3.2^35,
—*• x
~*~— T
""X
~x- 2" 3*"3**3
~3— •>
~3—.1
~3T
*r
j
""5
• - • •
- w »
7*7»7-
5»4 .3*2 ft
r •' - • .
4«3»3 .
x.• i .
•-> •
3*3 .3»• ^ »
I9f*4»°'^9- .9049,
' * - r C* - •
v 2 j 7 -
g<5ov1,4^39,6x39,I5O4,^337,xS44-,23^7,
6518,37 31 ,
^993..^7-3»07x0,j - ->•>*
32O4,
— j "
~ i
- i
• • T
~ i
-x~ I- 5- I*"X"*3- 3~ 2"Z~ 2- 2
""3- 3- 4-5
3 »
' • *
-t *
X ." *X .X .X*
9*5*3»X .
x«7 .5*4*3*2 .
9*5*••j *
2 .
O3 v 4 ,9 " j 3 *9 2 C C ,
^673,7303,5-59,3oCv5,x 259,3x90,75O2,VzCK,96S7,X7^5*7559.49.0C,o3o3,14-3»0345*"393*2-632,3*20,o3x3 ,
0
0
00
0
0
0
0- 2
-x-x— i
~x-z— 2
"Z- 3~ 3—.;
•i•t
- 4
3*3937.3.2733,3*=393»3« OOO4,3 . 7 3 X ^ y
3.4575,I . "j\)•/<->,
X.72 6 4 ,2.410',
6.oHx9,3.3074,X.9434*1*393 4»9.X936,^••^57 4*5*3054,3.44x0,1*5537,^ .790- ,x.4303,
3*5599*
00
00
0
0
0
0
0
***x~'X •
""X
——
-z""2- 3- 3""2
~*3~3~4
4*7473,4*4^53,4.X3Ii f3.SC.02,3.4520. '3*O953.2.5280,3 « x 93 ̂ ,2.Ö346»X.2230,0,$C6z t
4*73O4*2.5411,x.9^*r,2*4276,2.0733,S*33f3,5.4462,3.4789»l e 4O9X,2.2912,5.7567„
00
000
00
c00
- 2
~r* * A
**i
-x••"•"*~2*"3
•»2
*"3- 4
u =
- 2 0 -
Cosine Distribution
Table IV2
0*15 O.7 I . O
O . I O
o.x 5O. 3OO.35*-3 5o. 450,650.80I . 0 0
J .5*2,003»**4. co5.006,007.008.00XO.O2 5 . 020.0 •
5 * - *: o o
s . *•»
0 , ; 00.25.0, 2 0*• A»50-35*»450,65o.ftoT . O O
x*5 °2.OO3« **4.005.00S.oo7.00
i o. 0I j . O
20, 0
T O O
3,l8o3,3.X523,:3,II45,2.o6^2,1.9 54'"-»
I .53^9,I .3319,I . 0 9 1 " ,6.73^3,4.3794,3.194%I.3937,f>*45 r t3»5» 94"''4»4,4017,3.3867,3,l8o2,9.7470,5*4943»8» 8117,2.3038,
- 2, 5
o.x 5
3.^913,3.163 4,3 , T 5 5 8 ,.1, 0795 ,* • ? 6 5 %1,^337,I,5463,L3403,I.0986,6,7678,4-4*55»2,2072,x,2998,8.5036,5*9777»4. | 3 4 6 ,3.4043,
9» 7?.5%5.52x4,8,^544,i . sx44 ,
-z~ I- " T
""I- I
" I~*I
""I
~ I"™ 3
— - >
~"3
~ * •
"3—-j
~ 4~"4""5- 5
~ I- I— T
- T
-x" I
~x- I
"*2
- 3
~ 3
~3~3- 3- 3""3
-4~4
j
*~ j
8.5045»8.37^4,%2i34,8.0:47,7.538%6.9966,5*864??,5.0705,4 . 2 5 3 3 »3 • 5 ° •' 3»T.^05,0 . 4" 00,5 . o * 5 - .3,2874,2.3144,1.714%r . 3 2 0 2 ,^.5068,3.8072,3.1471,3,4464,°,62X4,
* . 3
0 .̂ •- ei 1
8,5534,^•38°7*8.1900,7.7107,7.1613,
5.19^5,
4*^5^7,2.6309,1,72*2,8.6706,5.*>;3l,3.3573,3.J(f'36,T »75OT,L 3 4 7 3 ,
-j • " -'-'3»3.1890,> JX34,O _ rt - ..
• 7 • 5 J *
"̂ X"T- I~X""I
~x- I_ -™*I
- I
~z- 3
""3
"" —
- 3
"3- 3~3""4- 5
- 1
—x- i
~ i
- 1
- i
- 1
- 1
-x- I
- x""* Z
- 3~"Z-2
~ 2- 3
" 3- 3~"S"4"5
3.2379»2 . X n 2 %3 , 1 3 6 5 ,3 . O-53O,i . 9 1 6 1 ,2*7599»I•4556,* - 53 3»t « 0 2 C 3 ,6.4XX3»4.3446,2 .1763 ,x . 3 5 7 7 »8.55-5*- .0403 ,4.4-5-»3.4586,
X . O O 2 O ,
5.5572»9**973»2,2771,
*»5
2.35*%3,3 06 7,3 . 2 5 T T ,2,1^60,3 . 0 3 6 5 ,
z « ° 7 3 %•* •« r# / i ; A,
* * ̂ j - V »1.3379»I .*957,6.S3 51»4.5167,2.3088,2.574J»9.0442,6.3816,4*7354»3*^497»2»3547,»•O5S4»5.9552»9* 5689,3.3944»
00
00
0
0
0
0
0
- I- I"1~~l
"*3
"*" 3"°3- 2
**" "̂
~4"*4
0
0
0
0
0
0
0
0 '0
~x- 1- 1""i- 2
~ 2
- 3
- 2
- 3
- 3"4- 4
3-9973»3.8700,3«7-*5*.3»57*5»3.253*»2.945%2,4031 ,3,0653,1.6972,X.O7 5-*1-,7 . 2 7 3 8 ,3 , " i 4 4 ,2 . 3 0 3 7 ,i .53*6,x.0845.".0790,6,2440,4«*437*1.82 09»X»0398,1.6605,4.1597»
0 . 7
• 4.4302,4.2951,4*T 475»3.9^62,3*^469»
3.7095»
t.9I43»1.3X44,8 . X 5 3 7 ,4.2540,3.5592»i,6949,1.2005,8.93^5*6.2963,4-4599»2,0046,1,1326,1,8231,4.5*543,
0
00
00
0
0
0
0
0
*~z-x-x
~x- 2"2
- 2- 3
""2
"i- 4
0
O
00
0
0
0
0 ,0
0- I- I- X
-x- 2- 3
~ 2*"3
~ 2
- 3- 4
6,2362,5.8288,5-4753,5-1594»4.6127,4.X 501,3.4036,2.9562,2.4737,I.6437»I . J464 ,6.2831,3,8852,2,6x6r,1.8733»1,4040,1,0900,7.XO25»3.22x4,1,8279,3.9634,7.4354*
I . O
7»557*.7 . i 038,6,7020,6,3367,5.6935,5.I37X.4,2236,3,67x0,3,0683,2.0307,1,4091,7.6578,4-7*77.3.X570,3,2539,1.6857,1.3*64,8.49*3*3.8382,2.1741,
3.5x32,8.8054,
00
00
0
0
0
0
0
0
0
- 1
~xT~X- I
-x~ 3- 2
~ 2
"3"4
00
0
0
0
0
0
0
0
0
0
~x- 1-X
~x~x—I
~ 2—2
"2
- 3~4
v
o, i oo . T 5O«2O0.3 5
0.' 45Oa 650,^0I , OOx , 503a OO3 . OO4.OO5*OO6 , 007*oo% 0010.0
15.020.050,0- •
z 00
TT
•i >
O.IO0,150.200.350.350.450.^50,^0I a OO1,502,003 , OO •
4» 005*°o6,007*008,00XO,0X 5 . O2O.O50.O
ZOO
3*5
0.15
3.1944»3.x 66%,3,1392,-V . -r -' - V j ,
X.9692,1.^36 0,
1*5493»
i .1008,6.7807,4.4136,2 . 2 1 1 O,I . 3 O I 9 ,8 , 5 1 6 9 , •5.^869,4.4313*3.4092,3.1945,9.8094,5.5390,8. ^663.,3.3X74,
~ 5»o
0.15
3.1963,3,1685,3 .X3IX,3 . O 8 4 9 ,X » 9 7 X 3 ,t . f 3 ^ 0 ,X , 5 5 1 I ,
*«3444»I a I O 3 I ,
6.7^85,4 .4 -84 ,3.2132,x« 3031 ,-.5345.5.9930,4.4350,3 a 4X3 0,3,1962,9.^X69,5.5331»8.2727,3.2189,
- I- I
~x-x- I
~x~x- i
~x~ 3- 2- 2• " 2
" 3"~3""3~3- 3- 4- 4"5~5
- 1• " T
~ I""X""X
-x- I
-x- I-2- 2~ 3- 3
~3- 3~*3"3- 3" 4" 4~5~5
Cosine
0 . 3
.8.7263,8,6043,8.44X 5,8,2434,,7, 7640,7*2*3*»6.0543,5.3366,4.2899,2.*509,-.7338,8.7394,5 . x 5 5 X ,3.37*1,3.3770,1.7605,I.355I*8.7377»3*9040,2,30IX,3*53*3,8.S333,
0 . 3
%7543*^•6334,8.4717.3.3743»7.7955»7,3440,6.0820,5.2672,4.310.:,2. ^-630,1.7403,ft.7639,5.1743,3*3900,3.3^51»1.7*504,**3595»**7555»3*9159*2,2077,3-54^3*- . " 5 7 1 ,
-21-Distribution
Table IV"5
- I
- x- X- I*•* x
-x- x- I
- x- x
*-*- 3~ 2- 2-*2
~3~3~*3- 4- 5
- I""X- I""I- I
t-t
- T
-X""I
-x- 2- 3""2~*2~ 2- 3
—3~3" 3"*4-5
0 . 5
3.3^^4»3.345(?»3,2907,2,2259,2.0756,Ia?XX 4 ,I,5858,1.3^6.
2.II93»6.9774,4.6068,3.35^7,3.39*4»9aI988,
6,488x,4.8129,3.70*6,3.3919,1.07x6,6,0460,9,7094,3,4393,
0 . 5
3,4098,2,3681,3.3*39*3.3495,3.0995,r*934^9X a 6 0 6 T,
x,3845,
7,0652,4,661?,3.3773*2.4137,9.2887,
^•5494*4«?573»3.743*,2,4129,i • 08 07,6,0960,9.7865,3,4482,
000
0
0
0
0
0
0— T
~ I
-x-x-2~"2
""3- 2~ 2— 3
~4" 4
00
0
0
0
0
0
0
0
~x-x"*X- I~ 3- 2~ 2- 3- 3"*2
~3~4- 4
0 * 7
4.5603,4*43^^»4.3937,4.13x0,
3.7^70,3*4437,2 , ^ 2 X 1 ,3,4367,
1*9934»1 . 2 6 2 8 ,
8. 4656,4.4059,3.6463,x , 7 5 0 7 ,* . 339 * »9.2x29,7aI I04 ,
4.5955.2aO638,I.1655,1.8746,4*^9*9»
0 , 7
4,6426,4,5250,
4.38x1,4- ' - ' 03 ,3 ,^749,3.5373»3.^926,2.48^8,3.O44I,I.2936,%66o9,4.4989»3.6987,1,7^38,1.26x8,9.377-»7*3353,4*6739»2,0976,x.1842,*.9O35*4.7<>33,
0000000000
- 1
- x- 1
- x- I- 2- 3- 3
- 3- 3
~*3" 4
0000000000
~*x~x-x- x- I- 3- 3- 3- 2~ 3
- 3""4
X*O
8.0766,7,6X52,7,2003.
6,8x936.X397.5.54*O>4 , 5"O x js3•908 5»3*3*50»2,1881,2.5*43»8.x 931,5*O33Oj,^ , ^ 6 X 1 ,3,396* $I.79OX,x«3§62,8,997s,4,0609,2,2983,3*7083,9,2893,
x,o
8.4x46,7*9534»7.5335»7**442*6.4437.5,8285,4a-OO3,4aT7I4,3,4^34,3.2934,1*5*34»5,5375.5.2208,3*4885,2,4840,£.8543,,**4349*9.3O5*»4**942*3,3733,3.8236,9.5719»
00
0
0
0
0
0
0
0
0
- xt-t
-t-t
• ~ r
~x- 2
-2~°2
- 3~4
0
0
0
0
0
0
0
00
0
0~ j
- I- i
- x
- z- 2- 3"2
~3"*4
V - xo.e
0.15
-22-Cosine Distribution
Table IV4
0 . 3O
7. >o
1 5 , 0
0 . 7 x.o
3.227$»3.X7OO,3.1336,3.0*65»2*972.)»**'**397»2.5536,2.345%1.1033,*.7955.4«.;33%•3. ?x 50 ,
1.3041»**• *3°'%
4*4379»3*4142,3.x 976»9.** 2 26,5*53^3»•%*774»2.2301,
-x-X
- x- I- I
-x—x- 1
- 2
-2
~3
~3" 3- 3"4- 4- 5" 5
***77^*x, ~*X•^.^5^9, *•!<^.<9'!>6, -»x%3OO3, - t7.^227, -17.37x3, "i">.TO7X, - I
4*33*^7, "X-•^739» " r
X.74^9, ~t^•7934» -25.1903, ~33*399'*» ""2•'•39^7» "*5
X.77XX, -3X.3^30, -3^•7771, -33.9250, —32.2x26, -33.54*3, -4*.*755 , -»5
*•42*0»i.j^*^»
2.3334»3.369%3.1305,?*?554»X . * 3 49 ,X.4OO9,
x . x 4.74,7.X452,4,71x0,5.3095»1,4247,"'•3^3°»/>.5993»4.^930,3,76^5,2.42=94*x•0^76,6.2340,9»**43^»2.4633,
00000<>000
- I- 1- I
- x-2
-z- 3~S"*3- 3
- 3~4" 4
4.7503,4*SO*2*4»45-3»4..»999,3» ;553 ,3.6050,3.9600,3*5474»2,0930,X.332X,
S.^3^9,4.5^07,3.7437»I.*II7,X.3^o6,9«5t2ff>»7,3363,4*73*4»3.T24O,j # i 9 ^ 7 .1.9353,4.°x67,
0000000000
- X
- x- I- I~ I- 2- 2- 2- 3- 3
"3- 4
8.7377»S.2748»7.^553,7.46X5»6.7443,6.IO73,5*0334*4*3739»3.6476,3.39S2»X.^4^9»S.85XX »5*39*9»3*5992»»•5592.2.90*3,2.4756,0.55^0,4*30x4,3.430$,3*912 5,9.7897,
00000000000
**2"*X«*•#- 2m*X*•!- 3- 3- 3
~3• 4
LIST OF PUBLISHED AE-REPORTS
1—29. (See the back cover of earlier reports.)
30. Metallographic study of the isothermal transformation of beta phase inzircaloy-2. By G. Östberg. 1960. 47 p. Sw. cr. 6:—.
31. Calculation of the reactivity equivalence of control rods in the secondcharge of HBWR. By P. Weissglas. 1961. 21 p. Sw. cr. 6:—.
32. Structure investigations of some beryllium materials. By I. Fäldt and G.Lagerberg. 1960. 15 p. Sw. cr. 6:—.
33. An emergency dosimeter for neutrons. By J. Braun and R. Nilsson. 1960.32 p. Sw. cr. 6:—.
34. Theoretical calculation of the effect on lattice parameters of emptyingthe coolant channels in a DiO-moderated and cooled natural uraniumreactor. By P. Weissglas. 1960. 20 p. Sw. cr. 6:—.
35. The multigroup neutron diffusion equations/1 space, dimension. By S.Linde. 1960. 41 p. Sw. cr. 6:—.
36. Geochemical prospecting of a uraniferous bog deposit at Masugnsbyn,Northern Sweden. By G. Armands. 1961. 48 p. Sw. cr. 6:—.
37. Spectrophotometric determination of thorium in low grade minerals andores. By A.-L. Arnfelt and I. Edmundsson. 1960. 14 p. Sw. cr. 6:—.
38. Kinetics of pressurized water reactors with hot or cold moderators. ByO. Norinder. 1960. 24 p. Sw. cr. 6J—.
39. The dependence of the resonance on the Doppler effect. By J. Rosén.1960. 1? p. Sw. cr. 6:—.
40. Measurements of the fast fission factor (£) in UO^elements. By O. Ny-lund. 1961. Sw. cr. 6:—.
44. Hand monitor for simultaneous measurement of alpha and beta conta-mination. By I. D . Andersson, J. Braun and B. Söderlund. 2nd rev. ed.1961. Sw. cr. 6:—.
45. Measurement of radioactivity in the human body. By I. D. Anderssonand I. Nilsson. 1961. 16 p. Sw. cr. 6:—.
46. The magnetisation of MnB and its variation with temperature. By N.Lundquist and H. P. Myers. 1960. 19 p. Sw. cr. 6:—.
47. An experimental study of the scattering of slow neutrons from H:O andD2O. By K. E. Larsson, S. Holmryd and K. Olnes. 1960. 29 p. Sw. er. 61—.
48. The resonance integral of thorium metal rods. By E. Hellstrand and. J.Weilman. 1961. 32 p. Sw. cr. 6:—.
49. Pressure tube and pressure vessels reactors; certain comparisons. By P.H. Margen, P. E. Ahlslröm and B. Pershagen. 1961. 42 p. Sw. cr. 6:—.
50. Phase transformations in a uranium-zirconium alloy containing 2 weightper cent zirconium. By G. Lagerberg 1961. 39 p. Sw. cr. 6:—.
51. Activation analysis of aluminium. By D. Brune. 1961. 8 p. Sw. cr. 61—.52. Thermo-technical data for DjO. By E. Axblom. 1961. 14 p. Sw. cr. 61—.53. Neutron damage in steels containing small amounts of boron. By H. P.
Myers. 1961. 23 p. Sw. er. 6:—.54. A chemical eight group separation method for routine use In gamma
spectrometric analysis. I. Ion exchange experiments. By K. Samsahi.1961. 13 p. Sw. cr. 6s—.
55. The Swedish zero power reactor R0. By Olof Landergärd, Kaj Cavallinand Georg Jonsson. 1961. 31 p. Sw. cr. 6:—.
56. A chemical eight group separation method for routine use in gammaspectrometric analysis. I I . Detailed analytical schema. By K. Samsahi.18 p. 1961. Sw. cr. 6:—.
57. Heterogeneous two-group diffusion theory for a finite cylindrical reactor.By Alf Jonsson and Göran Näslund. 1961. 20 p. Sw. cr. 6:—.
58. Q-values for (n, p) and (n, a ) reactions. By J. Koni[n. 1961. 29 p. Sw. cr.6:—.
59. Studies of the effective total and resonance absorption cross sections forzircaloy 2 and zirconium. By E. Hellstrand, G. Lindahl and G. Lundgren.1961. 26 p. Sw. cr. 61—.
60. Determination of elements in normal and leukemic human whole bloodby neutron activation analysis. By D. Brune, B. Frykberg, K. Samsahi andP. O. Wester. 1961. 16 p. Sw. cr. <s—.
61. Comparative and absolute measurements of 11 inorganic constituents of38 human tooth samples with gamma-ray spectrometry. By K. Samsahiand R. Söremark. 19 p. 1961. Sw. cr. 61—.
62. A Monte Carlo sampling technique for multi-phonon processes. By ThureHögberg. 10 p. 1961. Sw. cr. 6t—.
63. Numerical integration of the transport equation for infinite homogeneousmedia. By Rune Håkansson. 1962. 15 p. Sw. cr. 6:—.
64. Modified Sucksmith balances for ferromagnetic and paramagnetic mea-surements. By N. Lundquist and H. P. Myers. 1962. 9 p. Sw. cr. 6:—.
65. Irradiation effects in strain aged pressure vessel steel. By M. Grounesand H. P. Myers. 1962. 8 p. Sw. cr. 61—.
66. Critical and exponential experiments on 19-rod clusters (R3-fuel) In heavywater. By R. Persson, C-E. Wikdahl and Z. Zadwörski. 1962. 34 p. Sw. er.6 s—.
67. On the calibration and accuracy of the Guinier camera for the deter-mination of inlerplanar spacings. By M. Möller. 1962. 21 p. Sw. cr. 6:—.
68. Quantitative determination of pole figures with a texture goniometer bythe reflection method. By M. Möller. 1962. 16 p. Sw. cr. 6:—.
69. An experimental study of pressure gradients for flow of boiling water ina vertical round duct. Part I. By K. M. Becker, G. Hernborg and M. Bode.1962. 46 p. Sw. cr. 6:—.
70. An experimental study of pressure gradients for flow of boiling water ina vertical round duct. Part I I . By K. M. Becker, G. Hernborg and M. Bode.1962. 32 p. Sw. cr. 6:—.
71. The space-, time- and energy-distribution of neutrons from a pulsedplane source. By A. Claesson. 1962. 16 p. Sw. cr. 6t—.
72. One-group perturbation theory applied to substitution measurements withvoid. By R. Persson. 1962. 21 p. Sw. cr. 6:—.
73. Conversion factors. By A. Amberntson and S-E. Larsson 1962. 15 p. Sw.cr. 10s—.
74. Burnout conditions for flow of boiling water in vertical rod clusters.By Kurt M. Becker 1962. 44 p. Sw. cr. 6s—.
75. Two-group current-equivalent parameters for control rod cells. Autocodeprogramme CRCC. By O. Norinder and K. Nyman. 1962. 18 p. Sw. cr.6J—.
76. On the electronic structure of MnB. By N . Lundquist. 1962. 16 p. Sw. cr.6:—.
77. The resonance absorption of uranium metal and oxide. By E. Hellstrandand G. Lundgren. 1962. 17 p. Sw. cr. 6:—.
78. Half-life measurements of 'He, i«N, »O, *>F, »Al. "Se™ and "«Ag. By J.Konijn and S. Malmskog. 1962. 34 p. Sw. cr. 6:—.
79. Progress report for period ending December 1961. Department for ReactoiPhysics. 1962. 53 p. Sw. cr. 6:—.
80. Investigation of the 800 keV peak in the gamma spectrum of SwedishLaplanders. By I. O. Andersson, I. Nilsson and K. Eckerstig. 1962. 8 p..Sw. cr. 6:—. i
81. The resonance integral of niobium. By E. Hellstrand and G. Lundgren,1962. 14 p. Sw. cr. 6:—.
82. Some chemical group separations of radioactive trace elements. By K.JSamsahi. 1962. 18 p. Sw. cr. 6t—. *
83. Void measurement by the (y, n) reactions. By S. Z. Rouhani. 1962. 17 p.Sw. cr. 6:—.
84. Investigation of the pulse height distribution of boron trifluoride pro-portional counters. By I. D . Andersson and S. Malmskog. 1962. 16 p.Sw. cr. 6:—.
85. An experimental study of pressure gradients for flow of boiling waterin vertical round ducts. (Part 3). By K. M. Becker, G. Hernborg and M.Bode. 1962. 29 p. Sw. cr. 6 s—.
86. An experimental study of pressure gradients for flow of boiling waterin vertical round ducts. (Part 4). By K. M. Becker, G. Hernborg and M.Bode. 1962. 19 p. Sw. cr. 61—.
87. Measurements of burnout conditions for flow of boiling water in verticalround ducts. By K. M. Becker. 1962. 38 p. Sw. cr. 6s—.
88. Cross sections for neutron inelastic scattering and (n, 2n) processes. ByM. Leimdörfer, E. Bock and L. Arkeryd. 1962. 225 p. Sw. cr. 10s—.
89. On the solution of the neutron transport equation. By S. Depken. 1962.43 p. Sw. cr. 6:—.
90. Swedish studies on irradiation effects in structural materials. By M.Grounes and H. P. Myers. 1962. 11 p. Sw. cr. 6s—.
91. The energy variation of the sensitivity of a polyethylene moderated BF3proportional counter. By R. Fräki, M. Leimdörfer and S. Malmskog. 1962.12 p. Sw. cr. 6s—.
92. The backscattering of gamma radiation from plane concrete walls. ByM. Leimdörfer. 1962. 20 p. Sw. er. 6s—.
93 The backscatfering of gamma radiation from spherical concrete walls. ByM. Leimdörfer. 1962. 16 p. Sw. cr. 6:—.
94. Multiple scattering of gamma radiation in a spherical concrete wallroom. By M. Leimdörfer. 1962. 18 p. Sw. cr. 6:—.
95. The paramagnétism of Mn dissolved in a and /? brasses. By H. P. Myers.;and R. Westin. 1962. 13 p. Sw. cr. 6s—.
96. Isomorfic substitutions of calcium by strontium in calcium hydroxy-apatite. By H. Christensen. 1962. 9 p. Sw. cr. 6s—.
97. A fast time-to-pulse height converter. By O. Aspelund. 1962. 21 p. Sw. cr*6:—.
98. Neutron streaming in D2O pipes. By J. Braun and K. Randen. 1962.41 p. Sw. cr. 6 s—.
99. The effective resonance integral of thorium oxide rods. By J. Weifman.1962. 41 p. Sw. cr. 6s—.
100. Measurements of burnout conditions for flow of boiling water in verticalannuli. By K. M. Becker and G. Hernborg. 1962. 41 p. Sw. cr. 6s—.
101. Solid angle computations for a circular radiator and a circular detector.By J. Konijn and B. Tollander. 1963. 6 p. Sw. cr. 8s—.
Förteckning över publicerade AES-rapporter
1. Analys medelst gamma-spektrometri. Av Dag Brune. 1961. 10 s. Kr 6:—.
2. Beslrälningsförändringar och neutronatmosfär i reaktortrycktankar —några synpunkter. Av M. Grounes. 1962. 33 s. Kr 6;—.
Additional copies available at the library of AB Atomenergi, Studsvik, Nykö-ping, Sweden. Transparent microcards of the reports are obtainable throughthe International Documentation Center, Tumba, Sweden.
EOS-tryckeriema, Stockholm 1963
