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Efficiency calculations for a circular detector viewing a circular radiator
N U C L E A R I N S T R U M E N T S A N D M E T H O D S 36 (I965) 302-308;
© N O R T H - H O L L A N D P U B L I S H I N G CO.
EFFICIENCY C A L C U L A T I O N S F O R A CIRCULAR D E T E C T O R V I E W I N G A CIRCULAR R A D I A T O R J. KONIJN*, A. LAUBER and B. TOLLANDER
AB Atomenergi, Studsvik, Sweden Received 21 April 1965 Consider the following geometrical configuration. Neutrons emanating from an isotropic point source hit a circular target. The secondary particles produced are counted by a circular detector. The planes of the radiator and the detector are both per~,endicular to the line through their centers and through the poii,t source. In this article the resulting total and differential detection
efficiencies are investigated for different types of angular emission distributions of the secondary particles. Both infinitely thin radiators and radiators of finite thickness are considered. Numerical results have been obtained for different values of the radiator thickness, radiator diameter, detector diameter, targetdetector distance and target-neutron source distance.
1. Introduction
the problem reduces to a purely geometrical one and can be treated in a general manner provided that the angular distribution function of the emitted particles is known. In this paper we consider three angular emission distributions in the lab system: the isotropic distribution, the cos ~l distribution where r/, the angle of emission in the lab system, can have all values from 0 to ~z, and the cos q distribution where q can have all values from 0 to ½~z.The last case corresponds to n-p scattering in a hydrogenous target: the angular emission function, being isotropic in the centre of mass system for incident neutron energies less than 5, possibly 10 MeV6), will be of the cos r/ type in the lab system. The scattering kinematics limits the maximum emission angle to flTz. If the radiator thickness t, even though negligible when compared to the other dimensions of the geometrical configuration, is of the same order of magnitude as Re, the range of the emitted particles in the target material, one must apply a correction for the self-absorption in the foil. Re depends on the energy of the released particle. This energy is determined by several factors: incident neutron energy, reaction Q value, atomic number of target and reaction products, angle of emission. We have treated the problem for two special cases of practical interest: 1. (n, p) and (n, e) reactions in medium-weight nuclei. Here the energy E¢ of the emitted particle and thus its range Re are practically independent of the angle of emission: Ee = E, + Q, E. being the energy of the incident neutron. Furthermore the lab system angular emission distribution will practically coincide with the center of mass system angular emission distribution. The angular emission distribution may be of any kind, but only the isotropic and the cos ~/ angular emission distribution are considered here. 2. n-p scattering in an hydrogenous target. Here the
A basic problem in particle detection is to compute the detection efficiency of a counter. The possibility of using solid state detectors for the measurement of reaction cross sections and neutron fluxes l'z) made it desirable to calculate the efficiencies for the corresponding geometrical configurations. In this paper the efficiency formulae are derived and some results are given in diagram form. Detailed numerical results in tabular form may be found elsewhere a- 5).
2. Geometrical configurations considered Neutrons emanate from an isotropic point s o u r c e this is an idealization of many experimental arrangements. The neutrons will knock on a circular radiator and, by nuclear reaction or scattering, produce secondary particles which are detected by a circular disc (the detector). The planes of the radiator and the de1
TI ~ f o N E U S O U R C E ~" T A R G E T a: incident neutron
D E T E C T O R
b: light reaction product
Fig. l. General geometrical configuration.
tector are perpendicular to the line through their centers and through the point source (fig. 1). The assumption is made that all particles reaching the detector surface are detected. I f the target thickness is negligible compared to the range of the emitted particles in the target material, * Now at the Institute for Nuclear Research, Amsterdam, The Netherlands.
302
EFFICIENCY
CALCULATIONS
FOR A CIRCULAR s
energy Ee of the scattered proton may be expressed as E c = E n c o s 2 r/,
303
DETECTOR G
(1)
where E n is the energy of the incident neutron and r/is the angle of emission in the lab system. In the highenergy region the dependence of the range on the energy can be fairly well expressed by the relation SOURCE
R = ctE~-,
(2) w : siR
where c( is a constant. The particle range in the target will then be Re = Rn COS 3 T], (3) where R, is the range in the target material of a proton with the energy En of the incident neutron. As discussed earlier, the maximum emission angle is ½;z. The angular emission distribution is of the cos r/ type in the lab system. Finally, we have considered the differential detection efficiency as a function of the proton energy in the case of n-p scattering in a hydrogenous target. Both infinitely thin targets and targets, thin in the sense considered above, have been studied. The proton energy is considered at the moment when the proton ][eaves the target, which in the case of a target of finite 1Lhickness evidently differs from the energy at the moment of generation. Steinberg 7) has given a series expansion method for calculating the total efficiency in the case of an infinitely thin circular radiator viewed by a circular detector, the angular emission distribution being isotropic (radio-active source). The response of surfacetype detectors to circular Lambert's-law radiators has been investigated by several authors s' 9). The total and differential efficiencies in the case of n-p scattering in a hydrogenous radiator of finite thickness exposed to a parallel neutron t~eam have been calculated by Rossi and Staub 1°) for a 2~ geometry and approximately by Friedland et al. ll) for a circular radiator viewed by a circular detector. Parker et al.a2) have calculated the differential efficiencies for a circular surface barrier detector when half of the detector is covered by a semi-circular hydrogenous radiator, the neutron source being at infinite distance. 3. Total effieiencies for an infinitely thin radiator
The geometrical configuration is shown in fig. 2. Rx and R are the radii of target and detector respectively, s is the distance from the neutron source to the target and a the distance from target to detector. We introduce the following dimensionless parameters p = RI/R;
q = a/R;
w = s/R;
r = p/R.
(4)
TARGET
q = a/R
DETECTOR
p = RI/R r
=
Y/R
Fig. 2. Geometrical configuration for calculation the total efficiency in "the case of a n infinitely thin radiator.
Consider all the particles emitted at an angle r/from a point P of the radiator. Their trajectories will form a cone with the direction of the incident neutron as axis. The cone intersects the detector plane, giving a conical section. The part of the intersection inside the detector area is projected on a unit sphere around the point P of the target. The arc of circle so obtained, 2q~,divided by 2n gives the fractional detection efficiency for the particles emitted at the point P at an angle q. Letting P move over the whole radiator and dividing by the target area gives us the fractional efficiency for all particles emitted at an angle r/. For finite w, the neutrons emitted from the neutron source at an angle ~k cause a neutron flux, incident on the radiator at a distance p, that is w 2] ( w 2 + r 2) times the neutron flux when ~b = 0 . This factor must be introduced into the integral. For ~o always equal to n the efficiency is 1. A normalization factor has thus to be included, namely p2
No =
(5)
w 2 In (1 + pZ/w2)"
Finally, integrating over all angles q and dividing by 4rr gives the total detection efficiency F. F can be expressed by the general formula (1 + n )
['"
F = r~ln(l+p2/w2)Jo&lSinq
Icos"r/
f"
r~0 o - - d+r ' w r22
(6)
Here n = 0 corresponds to the lsotropic angular emission function and n = 1 , 2 , 3 . . . to the cosq, cos 2 q, cos 3 q... emission functions respectively. As there cannot be any negative contribution to the detector efficiency, only the absolute value of the cosine is of interest. For infinite w (parallel neutron beam) the integral reduces to
304
J. KON|JN
F~ - (1 + n) /[p2
dq sin q I cos" r/I 0
r e dr,
et al. 1 [ J
(7)
o
[
I
I
I
Os
where tp=O,
ifr>
I
:ti J
l+qtgr/
(1- r2 - q2 tgZ q) q~ = arc cos
---2rq t-g ~
'
(8)
ifl-qtgfl
, j
ifr < 1-qtgq
t
0~
0t
I
005~--
In the case of a target lying upon the detector (q = 0) the efficiency F will have different values depending on whether the target is larger than the detector or vice versa (p > 1 or p < 1). F = 1/2p 2 f o r p > l
and F = ½ f o r p < l .
=-
_4
f~"
.
~P
(lo) 11
_
~
~
r
!
j
f
05
+
O~
i
i
J
i
O~
05
I
2 -q
r~0
reln(l+p2/wZ)"]° dqsmrlcoSqJ owZ+r2 dr.
b
i
(9) 0ot
n-p
-'----
002
In the case o f n-p scattering, one obtains instead o f (6) F
°
,
[
~
Fig. 4. F as a function o f q in the case o f a cosinus angular emission d i ~ r i b u t i o n , p and w values are the same as in fig. 3.
The efficiency as a function o f q for different values
ofpandwisgiveninfig. 3-5fortheisotropic, cosq and n-p scattering angular emission distribution respectively. For numerical values ( 0 . 0 1 < q < 1 0 0 , 0.15 < p < 1, 0.5 < w < oo), ref. 3,4).
jJ
1 o
0o~ I
01
. 02
.
.
0s
I
. 1
0 05
2
~, q
Fig. 3. F as a f u n c t i o n o f q in the case o f a n i s o t r o p i c a n g u l a r emission distribution. C u r v e l : p = 0.15 w = 0.5-oo C u r v e 2: p = 1 w = 0.5 Curve3:p = 1 w = oo C u r v e 4: p = 3 w = 0.5 Curve 5:p= 3 w= 1 C u r v e 6: p = 3 w = 2.5 C u r v e 7: p = 3 w = oo
0 02 t-
001
I 01
02
05
1
2
" q F i g . 5. F a s a f u n c t i o n ofq in the case o f n - p scattering, p a n d ~, v a l u e s are the s a m e as in fig. 3.
EFFICIENCY
CALCULATIONS
4. Total effieiencies for a radiator of finite thickness Besides the finite target thickness, the geometrical configuration is similar to that of fig. 2. The dimensionless parameters p, q and w are the same as before. We introduce t, the target thickness and R~, the range in the target material of the emitted particles. We consider only the case when t < R, R~, s and a.
FOR A CIRCULAR
DETECTOR
305
Introducing the dimensionless variable u = obtain R e I t/Re C~ = --du.
H(O)
5 - Jo
=
Nt
(13)
pZlw2)
w 2 In (I +
(14)
Finally, integrating over all angles r/ and dividing by 4n gives the total detection efficiency G. G can be expressed by the general formula
\ t
NEUTRON SOURCE
we
Letting P move over the whole target surface and dividing by the target area gives the fractional detection efficiency for all particles emitted at an angle r/. As before, the factor w2/(w2 + r 2) has to be introduced into the integral for finite values o f w, giving rise to a normalization factor p2
H
\
x/Re,
G =
(1 + n) Re n In (1 + p2/w2) t
0
dr/sinr/[cos"r/[ x
TARGET
Fig. 6. Geometrical configuration for calculating the total efficiency in the case of a radiator of finite thickness.
Consider all the particles emitted at an angle r/from a point P inside the radiator (fig. 6). The trajectories form a cone with the direction of the incident neutron as axis. To the range R~ there corresponds a circle on the cone surface. The plane of the circle is perpendicular to the cone axis and intersects the target surface. In many cases a part of the circle will lie outside the target. q'he corresponding arc of circle 2v, divided by 2n, gives the fraction of particles emitted at the point P with an angle r/ escaping from the target and thus having a chance to be detected. Whether they are detected or not depends on the value of the angle q) defined in sect. 3. If t can be neglected in comparison with R, R~, s and a, as assumed here, ~0 will depend on p, q, w and r only and be independent o f the position of the point P on the line HC. It can then be shown that the fraction of particles detected is given by c~/n, where
(¥/Re--cosr/cos )' sinqs[n¢
c~ = v = arc cos \
(11)
Letting P move from H to C and dividing by H C gives the fraction of particles along HC with an angle r/ lhat are detected. Because of circular symmetry, this fraction of detected particles will be the same for all neutrons emitted at an angle O and can be written
G~ =
-t-J0or/sinr/[cos"r/]
do g
t
(12)
c~du.
(15)
r~,odrqS(r/). (16)
where
(t/R e, cost/)
= Min
(17)
and ~o is given by eq. (8). In the case o f n - p scattering R~ is a function oft/, the emission angle of the scattered proton: Re = R,
COS 3
r/
where R,, is the range in the target material of a proton with the energy E, of the incident neutron. Introducing u' = u cos 3 r/, we obtain instead of eq. (15)
4
R. ( ~
n In (I + p2/w2)
--
t
x
j o fP
dr/sinr/cost/ x
r dr ( '/n~ 0c du'
o w~ + r 2 .Jo
(18)
where c~ = v = arc cos
H(~b) = f ' ~ - - ~ - .
r dr [ ''/R~
o W[ + r 2 .Jo
Here n = 0 corresponds to the isotropic angular emission function and n = 1, 2, 3... to the cos r/, cos2 r/, c os3 r/.-. emission functions respectively. For infinite w (parallel neutron beam) the integral reduces to
G..p =
= ~o otherwise.
fe
×
[ u' -
cos 4 r/cos
t
\cos 3 r/sin r/sin ~ ! ' c~ = q0 otherwise.
ifv<~0
(19)
306
J. KONIJN
et al.
In analogy with (16) this integral for infinite w reduces to
6
A
Gn_p,~ -
7Cp2 t
o
dq sinq cos tl
o
r e dr00/),
(t/R., cos4r/)
f
~---
W=I
05
0~ (21)
and ~o is given by eq. (8). The efficiency as a function o f t/R e and t/R, for w = 1 and different values o f q a n d p is given in fig. 7 9 for the isotropic, cos q and n-p scattering angular em;ssion distributions respectively. For numerical values ( 0 . 1 < q <=3 ; = 0 . 1 5 < p < =3 ' = , 0 . 5 < w < o o ' = , 0.01 <=t/Ro <=1), ref. 5).
061
-
(20)
where 0 = Min
-
?
03 i
021
01~-i j G@
I
I I
0[__ 0ol
......
003
01
!
03 -
I
"L'
Re
F i g . 8. G as a f u n c t i o n o f t / R e in t h e c a s e o f a c o s i n u s a n g u l a r e m i s s i o n d i s t r i b u t i o n , w, q a n d p v a l u e s a r e t h e s a m e a s in fig. 7.
~ I
C_D
05i-
W=I
P(Ee/E.) =
2
/P
zc I n (1 + p a / w 2 )
rp
dr. (22)
o w z q- r 2
04
For infinite w (parallel neutron beam) the integral reduces to
03
T
02 ~
P(Ee/E,)=~
or~pdr,
(23)
b
0 L 001
~t" ~) ~-} ~ L 00,3 01 .
Fig. 7. G as a f u n c t i o n o f t / R e emission C u r v e 1: w = 1 Curve2:w= 1 C u r v e 3: w = 1 C u r v e 4: w = 1 C u r v e 5: w = 1 Curve 6:w = I
.
)
03 .
.
L/~ e
in t h e c a s e o f a n i s o t r o p i c a n g u l a r distribution. q = 0.15 p =- 0 . 1 5 q=0.15 p= 1 q = 0.50 p = 0.15 q = 0.50 p = 1 q = 1 p = 0.15 q = 1 p = 1
5. Differential ettieiencies for n-p scattering
5.1. INFINITELYTHIN RADIATOR In this case the energy of the proton leaving the target is identical with Ee, the energy at the m o m e n t o f generation. Eo is a function of the emission angle q, i.e. Ee =
E. cos 2 q
where E, is the energy of the incident neutron. It will thus be convenient to consider the fractional efficiency, as a function o f q. Incidentally, this has been done in sect. 3 when calculating the total efficiencies. It can then easily be shown that the differential detection efficiency as a function of proton energy can be written
where ~0 is given by eq. (8). The differential efficiency as a function o f Ee/En for p = q = 1 and some values of w is given in fig. 10. For numerical values ( 0 . 2 0 < q < l , 0.15
In this case, the target being of small but finite thickness t, the energy E of the proton when leaving the target will in general differ from the energy E e at the m o m e n t of generation. We consider the problem for infinite w (parallel neutron beam). 1 ......
,. . . . .
T
-,-
Z
05
0 o~
0 03
O~
0.5
1
F i g . 9. G a s a f u n c t i o n o f I / R e in t h e c a s e o f n - p s c a t t e r i n g , w, q a n d p v a l u e s a r e t h e s a m e as in fig. 7.
EFFICIENCY CALCULATIONS FOR A CIRCULAR DETECTOR T
T
r - - ~
Otherwise, e
7
p=q=l
08
[Se~(max) ~t-- o-'t~ l t -- Rn t'e(max)] --~ /~.
06
O~r O
307
O~ O~ 03
d,,
O~ Os @ O~ O~ 4 ....... E,/E~ Fig. 10. P as a function of EdEnin the case of n-p scattering for p=q=l. Introducing the dimensionless variables ~e = E d E , ;
e = E/E,,
The integral (27) has been calculated with P(e~) values corresponding to b o t h finite and infinite w. In b o t h cases the foil thickness correction as calculated above has been used. As this correction was derived under the assumption of infinite ~ , the,calculations will be exact for infinite w only. The e r r o r introduced will be discussed in § 6. F o r finite w, the differential efficiency can be expressed as Q@ =
(24)
7t In (1 +
f
p2/w2)
ee(max)
8
it can easily be shown that the fractional detection efficiency as a function of the energy of the p r o t o n when leaving the target can be written
i',
3 R , et Q~(~) = p~ 5 - ~
where P ~ ( ~ ) is equivalent to
(dee)e~
rq~(ee)dr (30) 0
where ~0 is given by eq. (8).
e(eo/l~.)
=
- 2 -
np 2
f P rcp dr o
'.as in eq. (23). If the range o f the protons is assumed to be proportional to Ei~, El being their energy at any m o m e n t , x satisfies the equation
X = Rnee(~; e
--
8~),
O~b ~ 03 . . . .
ff
~½
'Se~ P~(ee) dec.
(27)
,
\
i
0
O~ 02
Os 04
O~ 06
Ow 08 09 1 . . . . £,,£~
Fig. 12. Q as a function of E/En in the case of n-p scattering for p=q=w=l.
The lim'ts of the integral are
~e(min) ~ C
~
U
01
~e(mln)
e,(m,x) = 1 if (1 - t / R , ) ~ < e,
q
....
(26)
where R, is the range in the radiator of protons with the energy E , of the incident neutrons. Eq. (25) can then be written ==
(29)
For infinite w eq. (29) reduces to
.... 1
O~(e)
d,-
x 3o w 2 + r 2
(28)
~RECOIL PI~OTON
NCtDiNT
NEUTITON TARGET Fig. 11. Geometrical configuration for calculating the differential efficiency in the case of a radiator of finite thickness.
The differential efficiency as a function of e = E/E, for p = q -- w = 1 and different values of t / R , is given in fig. 12. For numerical values (0.1 < q < 3 ; 0.1 < p < 3; 0.5 - w < oo; 0.01 __
Discussion
The calculated total and differential efficiencies for an infinitely thin target can be considered as exact. In the case o f targets o f finite thickness, however, approximations had to be introduced in order to make the calculations possible. F o r m u l a (15), giving the efficiency in the case of (n,p) and (n,cQ reactions in medium-weight nuclei, rests on two assumptions: first that the energy and thus the range o f the emitted par-
308
J. KONIJN e t a/.
ticle is i n d e p e n d e n t o f the angle o f emission, second t h a t the lab system a n g u l a r emission d i s t r i b u t i o n is identical with the center o f mass system a n g u l a r emission distribution. The a s s u m p t i o n s are true if the masses o f the incident particle a n d o f one o f the reaction p r o d u c t s are negligible c o m p a r e d to the masses o f the target and the o t h e r reaction p r o d u c t . The degree o f exactness in each case can easily be tested by means o f the c o r r e s p o n d i n g reaction kinetic formulae. F o r m u l a e (18), (20), (29) and (30), giving the total a n d partial efficiencies in the n-p scattering case, rest on the a s s u m p t i o n that eq. (2) a n d eq. (3) are valid for all energies. This is not true for p r o t o n energies less than ca. 0.4 MeV, ref. 12,13). It can be shown, however, that provided that the energy o f the incident n e u t r o n s is > 2 MeV, the error i n t r o d u c e d will not exceed 2 ~ . F o r finite values o f w a n o t h e r e r r o r is i n t r o d u c e d into f o r m u l a (29) owing to the use o f the foil thickness correction for infinite w. A conservative e s t i m a t i o n shows that the resulting e r r o r will never exceed 5 ~ if c o n s i d e r a t i o n is restricted to cases where p < ½w. The p and w values considered in 5) c o n f o r m this requirement. The c o m p u t e r p r o g r a m s used to d e t e r m i n e the total and differential detection efficiencies in the case o f a r a d i a t o r o f finite thickness become i n a p p r o p r i a t e for large values o f p and q: the c o m p u t i n g time necessary to
keep c o m p u t i n g errors within the limits prescribed becomes excessively long. As it is easy in these cases to find r a p i d l y converging series expansions, the efficiency calculations in 5) have been restricted to p and q values in the range 0.1-3, i.e. the geometrical configurations in which r a d i a t o r diameter, d e t e c t o r d i a m e t e r and r a d i a t o r - d e t e c t o r distance are o f the same o r d e r o f magnitude.
References l) G. Dearnaley and A. T. G. Ferguson, Nucleonics 20 (1962) 84. 2) j. Konijn and A. Lauber, Nucl. Phys. 48 (1963) 191. 3) j. Konijn and B. Tollander, AB Atomenergi, Report no. AE-101 (Febr. 1963). 4) j. Konijn, A. Lauber and B. Tollander, AB Atomenergi, Report no. AE-116 (August 1963). 5) A. Lauber and B. Tollander, AB Atomenergi Internal Report no. SSI-138 (August 1964). 6) j. E. Perry, Jr., in Fast Neutron Physics (ed. J. B. Marion and J. L. Fowler, Interscience, New York, 1960) Part I, p. 630. 7) E. P. Steinberg, Argonne Nat. Lab., ANL-5622 (1956). 8) For a review, see A. I. Mahan and W. F. Malmhorg, J. Opt. Soc. Am. 44 (1954) 644. 9) S. J. Bame Jr., E. Haddad, J. E. Perry Jr. and R. K, Smith, Rev. Sci. Instr. 28 (1957) 997. 10) B, B. Rossi and H. H. Staub, lonization chambers and counters (McGraw Hill, New York 1949) p. 147. ll) S. S. Friedland, F. P. Ziemba and E. L. Zimmerman, AD294324 (NP-12226) (1962). 12) j. B. Parker, P. H. White and R. J. Webster, Nucl. Instr. and Meth. 23 (1963) 61. 13) Ref. 1o)p. 137.
© N O R T H - H O L L A N D P U B L I S H I N G CO.
EFFICIENCY C A L C U L A T I O N S F O R A CIRCULAR D E T E C T O R V I E W I N G A CIRCULAR R A D I A T O R J. KONIJN*, A. LAUBER and B. TOLLANDER
AB Atomenergi, Studsvik, Sweden Received 21 April 1965 Consider the following geometrical configuration. Neutrons emanating from an isotropic point source hit a circular target. The secondary particles produced are counted by a circular detector. The planes of the radiator and the detector are both per~,endicular to the line through their centers and through the poii,t source. In this article the resulting total and differential detection
efficiencies are investigated for different types of angular emission distributions of the secondary particles. Both infinitely thin radiators and radiators of finite thickness are considered. Numerical results have been obtained for different values of the radiator thickness, radiator diameter, detector diameter, targetdetector distance and target-neutron source distance.
1. Introduction
the problem reduces to a purely geometrical one and can be treated in a general manner provided that the angular distribution function of the emitted particles is known. In this paper we consider three angular emission distributions in the lab system: the isotropic distribution, the cos ~l distribution where r/, the angle of emission in the lab system, can have all values from 0 to ~z, and the cos q distribution where q can have all values from 0 to ½~z.The last case corresponds to n-p scattering in a hydrogenous target: the angular emission function, being isotropic in the centre of mass system for incident neutron energies less than 5, possibly 10 MeV6), will be of the cos r/ type in the lab system. The scattering kinematics limits the maximum emission angle to flTz. If the radiator thickness t, even though negligible when compared to the other dimensions of the geometrical configuration, is of the same order of magnitude as Re, the range of the emitted particles in the target material, one must apply a correction for the self-absorption in the foil. Re depends on the energy of the released particle. This energy is determined by several factors: incident neutron energy, reaction Q value, atomic number of target and reaction products, angle of emission. We have treated the problem for two special cases of practical interest: 1. (n, p) and (n, e) reactions in medium-weight nuclei. Here the energy E¢ of the emitted particle and thus its range Re are practically independent of the angle of emission: Ee = E, + Q, E. being the energy of the incident neutron. Furthermore the lab system angular emission distribution will practically coincide with the center of mass system angular emission distribution. The angular emission distribution may be of any kind, but only the isotropic and the cos ~/ angular emission distribution are considered here. 2. n-p scattering in an hydrogenous target. Here the
A basic problem in particle detection is to compute the detection efficiency of a counter. The possibility of using solid state detectors for the measurement of reaction cross sections and neutron fluxes l'z) made it desirable to calculate the efficiencies for the corresponding geometrical configurations. In this paper the efficiency formulae are derived and some results are given in diagram form. Detailed numerical results in tabular form may be found elsewhere a- 5).
2. Geometrical configurations considered Neutrons emanate from an isotropic point s o u r c e this is an idealization of many experimental arrangements. The neutrons will knock on a circular radiator and, by nuclear reaction or scattering, produce secondary particles which are detected by a circular disc (the detector). The planes of the radiator and the de1
TI ~ f o N E U S O U R C E ~" T A R G E T a: incident neutron
D E T E C T O R
b: light reaction product
Fig. l. General geometrical configuration.
tector are perpendicular to the line through their centers and through the point source (fig. 1). The assumption is made that all particles reaching the detector surface are detected. I f the target thickness is negligible compared to the range of the emitted particles in the target material, * Now at the Institute for Nuclear Research, Amsterdam, The Netherlands.
302
EFFICIENCY
CALCULATIONS
FOR A CIRCULAR s
energy Ee of the scattered proton may be expressed as E c = E n c o s 2 r/,
303
DETECTOR G
(1)
where E n is the energy of the incident neutron and r/is the angle of emission in the lab system. In the highenergy region the dependence of the range on the energy can be fairly well expressed by the relation SOURCE
R = ctE~-,
(2) w : siR
where c( is a constant. The particle range in the target will then be Re = Rn COS 3 T], (3) where R, is the range in the target material of a proton with the energy En of the incident neutron. As discussed earlier, the maximum emission angle is ½;z. The angular emission distribution is of the cos r/ type in the lab system. Finally, we have considered the differential detection efficiency as a function of the proton energy in the case of n-p scattering in a hydrogenous target. Both infinitely thin targets and targets, thin in the sense considered above, have been studied. The proton energy is considered at the moment when the proton ][eaves the target, which in the case of a target of finite 1Lhickness evidently differs from the energy at the moment of generation. Steinberg 7) has given a series expansion method for calculating the total efficiency in the case of an infinitely thin circular radiator viewed by a circular detector, the angular emission distribution being isotropic (radio-active source). The response of surfacetype detectors to circular Lambert's-law radiators has been investigated by several authors s' 9). The total and differential efficiencies in the case of n-p scattering in a hydrogenous radiator of finite thickness exposed to a parallel neutron t~eam have been calculated by Rossi and Staub 1°) for a 2~ geometry and approximately by Friedland et al. ll) for a circular radiator viewed by a circular detector. Parker et al.a2) have calculated the differential efficiencies for a circular surface barrier detector when half of the detector is covered by a semi-circular hydrogenous radiator, the neutron source being at infinite distance. 3. Total effieiencies for an infinitely thin radiator
The geometrical configuration is shown in fig. 2. Rx and R are the radii of target and detector respectively, s is the distance from the neutron source to the target and a the distance from target to detector. We introduce the following dimensionless parameters p = RI/R;
q = a/R;
w = s/R;
r = p/R.
(4)
TARGET
q = a/R
DETECTOR
p = RI/R r
=
Y/R
Fig. 2. Geometrical configuration for calculation the total efficiency in "the case of a n infinitely thin radiator.
Consider all the particles emitted at an angle r/from a point P of the radiator. Their trajectories will form a cone with the direction of the incident neutron as axis. The cone intersects the detector plane, giving a conical section. The part of the intersection inside the detector area is projected on a unit sphere around the point P of the target. The arc of circle so obtained, 2q~,divided by 2n gives the fractional detection efficiency for the particles emitted at the point P at an angle q. Letting P move over the whole radiator and dividing by the target area gives us the fractional efficiency for all particles emitted at an angle r/. For finite w, the neutrons emitted from the neutron source at an angle ~k cause a neutron flux, incident on the radiator at a distance p, that is w 2] ( w 2 + r 2) times the neutron flux when ~b = 0 . This factor must be introduced into the integral. For ~o always equal to n the efficiency is 1. A normalization factor has thus to be included, namely p2
No =
(5)
w 2 In (1 + pZ/w2)"
Finally, integrating over all angles q and dividing by 4rr gives the total detection efficiency F. F can be expressed by the general formula (1 + n )
['"
F = r~ln(l+p2/w2)Jo&lSinq
Icos"r/
f"
r~0 o - - d+r ' w r22
(6)
Here n = 0 corresponds to the lsotropic angular emission function and n = 1 , 2 , 3 . . . to the cosq, cos 2 q, cos 3 q... emission functions respectively. As there cannot be any negative contribution to the detector efficiency, only the absolute value of the cosine is of interest. For infinite w (parallel neutron beam) the integral reduces to
304
J. KON|JN
F~ - (1 + n) /[p2
dq sin q I cos" r/I 0
r e dr,
et al. 1 [ J
(7)
o
[
I
I
I
Os
where tp=O,
ifr>
I
:ti J
l+qtgr/
(1- r2 - q2 tgZ q) q~ = arc cos
---2rq t-g ~
'
(8)
ifl-qtgfl
, j
ifr < 1-qtgq
t
0~
0t
I
005~--
In the case of a target lying upon the detector (q = 0) the efficiency F will have different values depending on whether the target is larger than the detector or vice versa (p > 1 or p < 1). F = 1/2p 2 f o r p > l
and F = ½ f o r p < l .
=-
_4
f~"
.
~P
(lo) 11
_
~
~
r
!
j
f
05
+
O~
i
i
J
i
O~
05
I
2 -q
r~0
reln(l+p2/wZ)"]° dqsmrlcoSqJ owZ+r2 dr.
b
i
(9) 0ot
n-p
-'----
002
In the case o f n-p scattering, one obtains instead o f (6) F
°
,
[
~
Fig. 4. F as a function o f q in the case o f a cosinus angular emission d i ~ r i b u t i o n , p and w values are the same as in fig. 3.
The efficiency as a function o f q for different values
ofpandwisgiveninfig. 3-5fortheisotropic, cosq and n-p scattering angular emission distribution respectively. For numerical values ( 0 . 0 1 < q < 1 0 0 , 0.15 < p < 1, 0.5 < w < oo), ref. 3,4).
jJ
1 o
0o~ I
01
. 02
.
.
0s
I
. 1
0 05
2
~, q
Fig. 3. F as a f u n c t i o n o f q in the case o f a n i s o t r o p i c a n g u l a r emission distribution. C u r v e l : p = 0.15 w = 0.5-oo C u r v e 2: p = 1 w = 0.5 Curve3:p = 1 w = oo C u r v e 4: p = 3 w = 0.5 Curve 5:p= 3 w= 1 C u r v e 6: p = 3 w = 2.5 C u r v e 7: p = 3 w = oo
0 02 t-
001
I 01
02
05
1
2
" q F i g . 5. F a s a f u n c t i o n ofq in the case o f n - p scattering, p a n d ~, v a l u e s are the s a m e as in fig. 3.
EFFICIENCY
CALCULATIONS
4. Total effieiencies for a radiator of finite thickness Besides the finite target thickness, the geometrical configuration is similar to that of fig. 2. The dimensionless parameters p, q and w are the same as before. We introduce t, the target thickness and R~, the range in the target material of the emitted particles. We consider only the case when t < R, R~, s and a.
FOR A CIRCULAR
DETECTOR
305
Introducing the dimensionless variable u = obtain R e I t/Re C~ = --du.
H(O)
5 - Jo
=
Nt
(13)
pZlw2)
w 2 In (I +
(14)
Finally, integrating over all angles r/ and dividing by 4n gives the total detection efficiency G. G can be expressed by the general formula
\ t
NEUTRON SOURCE
we
Letting P move over the whole target surface and dividing by the target area gives the fractional detection efficiency for all particles emitted at an angle r/. As before, the factor w2/(w2 + r 2) has to be introduced into the integral for finite values o f w, giving rise to a normalization factor p2
H
\
x/Re,
G =
(1 + n) Re n In (1 + p2/w2) t
0
dr/sinr/[cos"r/[ x
TARGET
Fig. 6. Geometrical configuration for calculating the total efficiency in the case of a radiator of finite thickness.
Consider all the particles emitted at an angle r/from a point P inside the radiator (fig. 6). The trajectories form a cone with the direction of the incident neutron as axis. To the range R~ there corresponds a circle on the cone surface. The plane of the circle is perpendicular to the cone axis and intersects the target surface. In many cases a part of the circle will lie outside the target. q'he corresponding arc of circle 2v, divided by 2n, gives the fraction of particles emitted at the point P with an angle r/ escaping from the target and thus having a chance to be detected. Whether they are detected or not depends on the value of the angle q) defined in sect. 3. If t can be neglected in comparison with R, R~, s and a, as assumed here, ~0 will depend on p, q, w and r only and be independent o f the position of the point P on the line HC. It can then be shown that the fraction of particles detected is given by c~/n, where
(¥/Re--cosr/cos )' sinqs[n¢
c~ = v = arc cos \
(11)
Letting P move from H to C and dividing by H C gives the fraction of particles along HC with an angle r/ lhat are detected. Because of circular symmetry, this fraction of detected particles will be the same for all neutrons emitted at an angle O and can be written
G~ =
-t-J0or/sinr/[cos"r/]
do g
t
(12)
c~du.
(15)
r~,odrqS(r/). (16)
where
(t/R e, cost/)
= Min
(17)
and ~o is given by eq. (8). In the case o f n - p scattering R~ is a function oft/, the emission angle of the scattered proton: Re = R,
COS 3
r/
where R,, is the range in the target material of a proton with the energy E, of the incident neutron. Introducing u' = u cos 3 r/, we obtain instead of eq. (15)
4
R. ( ~
n In (I + p2/w2)
--
t
x
j o fP
dr/sinr/cost/ x
r dr ( '/n~ 0c du'
o w~ + r 2 .Jo
(18)
where c~ = v = arc cos
H(~b) = f ' ~ - - ~ - .
r dr [ ''/R~
o W[ + r 2 .Jo
Here n = 0 corresponds to the isotropic angular emission function and n = 1, 2, 3... to the cos r/, cos2 r/, c os3 r/.-. emission functions respectively. For infinite w (parallel neutron beam) the integral reduces to
G..p =
= ~o otherwise.
fe
×
[ u' -
cos 4 r/cos
t
\cos 3 r/sin r/sin ~ ! ' c~ = q0 otherwise.
ifv<~0
(19)
306
J. KONIJN
et al.
In analogy with (16) this integral for infinite w reduces to
6
A
Gn_p,~ -
7Cp2 t
o
dq sinq cos tl
o
r e dr00/),
(t/R., cos4r/)
f
~---
W=I
05
0~ (21)
and ~o is given by eq. (8). The efficiency as a function o f t/R e and t/R, for w = 1 and different values o f q a n d p is given in fig. 7 9 for the isotropic, cos q and n-p scattering angular em;ssion distributions respectively. For numerical values ( 0 . 1 < q <=3 ; = 0 . 1 5 < p < =3 ' = , 0 . 5 < w < o o ' = , 0.01 <=t/Ro <=1), ref. 5).
061
-
(20)
where 0 = Min
-
?
03 i
021
01~-i j G@
I
I I
0[__ 0ol
......
003
01
!
03 -
I
"L'
Re
F i g . 8. G as a f u n c t i o n o f t / R e in t h e c a s e o f a c o s i n u s a n g u l a r e m i s s i o n d i s t r i b u t i o n , w, q a n d p v a l u e s a r e t h e s a m e a s in fig. 7.
~ I
C_D
05i-
W=I
P(Ee/E.) =
2
/P
zc I n (1 + p a / w 2 )
rp
dr. (22)
o w z q- r 2
04
For infinite w (parallel neutron beam) the integral reduces to
03
T
02 ~
P(Ee/E,)=~
or~pdr,
(23)
b
0 L 001
~t" ~) ~-} ~ L 00,3 01 .
Fig. 7. G as a f u n c t i o n o f t / R e emission C u r v e 1: w = 1 Curve2:w= 1 C u r v e 3: w = 1 C u r v e 4: w = 1 C u r v e 5: w = 1 Curve 6:w = I
.
)
03 .
.
L/~ e
in t h e c a s e o f a n i s o t r o p i c a n g u l a r distribution. q = 0.15 p =- 0 . 1 5 q=0.15 p= 1 q = 0.50 p = 0.15 q = 0.50 p = 1 q = 1 p = 0.15 q = 1 p = 1
5. Differential ettieiencies for n-p scattering
5.1. INFINITELYTHIN RADIATOR In this case the energy of the proton leaving the target is identical with Ee, the energy at the m o m e n t o f generation. Eo is a function of the emission angle q, i.e. Ee =
E. cos 2 q
where E, is the energy of the incident neutron. It will thus be convenient to consider the fractional efficiency, as a function o f q. Incidentally, this has been done in sect. 3 when calculating the total efficiencies. It can then easily be shown that the differential detection efficiency as a function of proton energy can be written
where ~0 is given by eq. (8). The differential efficiency as a function o f Ee/En for p = q = 1 and some values of w is given in fig. 10. For numerical values ( 0 . 2 0 < q < l , 0.15
In this case, the target being of small but finite thickness t, the energy E of the proton when leaving the target will in general differ from the energy E e at the m o m e n t of generation. We consider the problem for infinite w (parallel neutron beam). 1 ......
,. . . . .
T
-,-
Z
05
0 o~
0 03
O~
0.5
1
F i g . 9. G a s a f u n c t i o n o f I / R e in t h e c a s e o f n - p s c a t t e r i n g , w, q a n d p v a l u e s a r e t h e s a m e as in fig. 7.
EFFICIENCY CALCULATIONS FOR A CIRCULAR DETECTOR T
T
r - - ~
Otherwise, e
7
p=q=l
08
[Se~(max) ~t-- o-'t~ l t -- Rn t'e(max)] --~ /~.
06
O~r O
307
O~ O~ 03
d,,
O~ Os @ O~ O~ 4 ....... E,/E~ Fig. 10. P as a function of EdEnin the case of n-p scattering for p=q=l. Introducing the dimensionless variables ~e = E d E , ;
e = E/E,,
The integral (27) has been calculated with P(e~) values corresponding to b o t h finite and infinite w. In b o t h cases the foil thickness correction as calculated above has been used. As this correction was derived under the assumption of infinite ~ , the,calculations will be exact for infinite w only. The e r r o r introduced will be discussed in § 6. F o r finite w, the differential efficiency can be expressed as Q@ =
(24)
7t In (1 +
f
p2/w2)
ee(max)
8
it can easily be shown that the fractional detection efficiency as a function of the energy of the p r o t o n when leaving the target can be written
i',
3 R , et Q~(~) = p~ 5 - ~
where P ~ ( ~ ) is equivalent to
(dee)e~
rq~(ee)dr (30) 0
where ~0 is given by eq. (8).
e(eo/l~.)
=
- 2 -
np 2
f P rcp dr o
'.as in eq. (23). If the range o f the protons is assumed to be proportional to Ei~, El being their energy at any m o m e n t , x satisfies the equation
X = Rnee(~; e
--
8~),
O~b ~ 03 . . . .
ff
~½
'Se~ P~(ee) dec.
(27)
,
\
i
0
O~ 02
Os 04
O~ 06
Ow 08 09 1 . . . . £,,£~
Fig. 12. Q as a function of E/En in the case of n-p scattering for p=q=w=l.
The lim'ts of the integral are
~e(min) ~ C
~
U
01
~e(mln)
e,(m,x) = 1 if (1 - t / R , ) ~ < e,
q
....
(26)
where R, is the range in the radiator of protons with the energy E , of the incident neutrons. Eq. (25) can then be written ==
(29)
For infinite w eq. (29) reduces to
.... 1
O~(e)
d,-
x 3o w 2 + r 2
(28)
~RECOIL PI~OTON
NCtDiNT
NEUTITON TARGET Fig. 11. Geometrical configuration for calculating the differential efficiency in the case of a radiator of finite thickness.
The differential efficiency as a function of e = E/E, for p = q -- w = 1 and different values of t / R , is given in fig. 12. For numerical values (0.1 < q < 3 ; 0.1 < p < 3; 0.5 - w < oo; 0.01 __
Discussion
The calculated total and differential efficiencies for an infinitely thin target can be considered as exact. In the case o f targets o f finite thickness, however, approximations had to be introduced in order to make the calculations possible. F o r m u l a (15), giving the efficiency in the case of (n,p) and (n,cQ reactions in medium-weight nuclei, rests on two assumptions: first that the energy and thus the range o f the emitted par-
308
J. KONIJN e t a/.
ticle is i n d e p e n d e n t o f the angle o f emission, second t h a t the lab system a n g u l a r emission d i s t r i b u t i o n is identical with the center o f mass system a n g u l a r emission distribution. The a s s u m p t i o n s are true if the masses o f the incident particle a n d o f one o f the reaction p r o d u c t s are negligible c o m p a r e d to the masses o f the target and the o t h e r reaction p r o d u c t . The degree o f exactness in each case can easily be tested by means o f the c o r r e s p o n d i n g reaction kinetic formulae. F o r m u l a e (18), (20), (29) and (30), giving the total a n d partial efficiencies in the n-p scattering case, rest on the a s s u m p t i o n that eq. (2) a n d eq. (3) are valid for all energies. This is not true for p r o t o n energies less than ca. 0.4 MeV, ref. 12,13). It can be shown, however, that provided that the energy o f the incident n e u t r o n s is > 2 MeV, the error i n t r o d u c e d will not exceed 2 ~ . F o r finite values o f w a n o t h e r e r r o r is i n t r o d u c e d into f o r m u l a (29) owing to the use o f the foil thickness correction for infinite w. A conservative e s t i m a t i o n shows that the resulting e r r o r will never exceed 5 ~ if c o n s i d e r a t i o n is restricted to cases where p < ½w. The p and w values considered in 5) c o n f o r m this requirement. The c o m p u t e r p r o g r a m s used to d e t e r m i n e the total and differential detection efficiencies in the case o f a r a d i a t o r o f finite thickness become i n a p p r o p r i a t e for large values o f p and q: the c o m p u t i n g time necessary to
keep c o m p u t i n g errors within the limits prescribed becomes excessively long. As it is easy in these cases to find r a p i d l y converging series expansions, the efficiency calculations in 5) have been restricted to p and q values in the range 0.1-3, i.e. the geometrical configurations in which r a d i a t o r diameter, d e t e c t o r d i a m e t e r and r a d i a t o r - d e t e c t o r distance are o f the same o r d e r o f magnitude.
References l) G. Dearnaley and A. T. G. Ferguson, Nucleonics 20 (1962) 84. 2) j. Konijn and A. Lauber, Nucl. Phys. 48 (1963) 191. 3) j. Konijn and B. Tollander, AB Atomenergi, Report no. AE-101 (Febr. 1963). 4) j. Konijn, A. Lauber and B. Tollander, AB Atomenergi, Report no. AE-116 (August 1963). 5) A. Lauber and B. Tollander, AB Atomenergi Internal Report no. SSI-138 (August 1964). 6) j. E. Perry, Jr., in Fast Neutron Physics (ed. J. B. Marion and J. L. Fowler, Interscience, New York, 1960) Part I, p. 630. 7) E. P. Steinberg, Argonne Nat. Lab., ANL-5622 (1956). 8) For a review, see A. I. Mahan and W. F. Malmhorg, J. Opt. Soc. Am. 44 (1954) 644. 9) S. J. Bame Jr., E. Haddad, J. E. Perry Jr. and R. K, Smith, Rev. Sci. Instr. 28 (1957) 997. 10) B, B. Rossi and H. H. Staub, lonization chambers and counters (McGraw Hill, New York 1949) p. 147. ll) S. S. Friedland, F. P. Ziemba and E. L. Zimmerman, AD294324 (NP-12226) (1962). 12) j. B. Parker, P. H. White and R. J. Webster, Nucl. Instr. and Meth. 23 (1963) 61. 13) Ref. 1o)p. 137.