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Fast neutron isochronous flight path spectrometry
NUCLEAR
INSTRUMENTS
AND
METHODS
24 (1963)
290--300,
NORTH-HOLLAND
PUBLISHING
CO.
FAST NEUTRON ISOCHRONOUS FLIGHT PATH SPECTROMETRY* M. L. R O U S H t , A. S. F I G U E R A and W. F. H O R N Y A K t t
Department o/ Physics and Astronomy, University o/ Maryland, College Park, Maryland Received 4 March 1963
A time-of-flight fast n e u t r o n spectrometer for continuous beams of neutrons is described. The spectrometer utilizes the isochronous principle t h a t monoenergetic neutrons scattered from a hydrogenous m e d i u m arrive a t a s u i t a b l y oriented spherical surface after a fixed t r a n s i t time. The scatterer ( " s t a r t " detector) is a cylinder of NE-102 p l a s t i c s c i n t i l l a t o r 3" long and ½" diameter. The " s t o p " detector, also of NE-102, is a segment of a spherical shell with t a p e r i n g thickness selected to o b t a i n optim u m resolution. This scintillator has a cap diameter of 18 inches
1. Introduction
The measurement of neutron energies by the time-of-flight method involves the determination of the elapsed time required by the neutron to traverse a prescribed flight path. All variants of this technique derive a " s t o p " signal or time marker signifying the arrival of the neutron at the end of its flight path simply through the use of a suitably placed neutron detector. However, either one of two methods are generally employed to obtain the " s t a r t " signal or zero time reference marker. In one method this is accomplished by detecting the occurrence of an event which is simultaneous with the emission of the neutron (e.g., the emission of an associated particle1), or g a m m a ray 2, 3). While this technique permits the use of a continuous accelerator beam in generating the neutrons, it is necessarily limited to the class of reactions yielding coincident events. Alternatively, a variety of pulsed beam techniques may be employed to generate a suitable zero reference time4). These systems operate by allowing the initiating beam to strike the target in bursts of appropriately short duration. The required zero time marker is derived from * Research supported in p a r t by U.S. Atomic E n e r g y Commission. N.S.F. G r a d u a t e Cooperative Fellow. ** Presently on leave a t Universit6 de Paris, Orsay (Seine-etOise), France. 290
with the radius of c u r v a t u r e of the spherical surface being 30 inches. A c o n v e n t i o n a l time-to-pulse-height converter is used. E n e r g y resolution of 9--14% has been obtained for the energy range 1.8 .~ E n <~ 10 MeV. The efficiency, a t 5 MeV, is estimated to be 1 X 10 -3 per n e u t r o n i n c i d e n t on the scatterer. The u n i t discussed is ideally suited to a n g u l a r d i s t r i b u t i o n measurements for reactions resulting in numerous n e u t r o n groups. Other a p p l i c a t i o n s of the isochronous principle are proposed.
either a signal originating in the beam pulser or from the actual beam pulse striking the target. These methods have the advantage of not requiring an associated event so they m a y employ neutrons from a wide class of reactions. However, this method in its simplest form must contend with an unfavorable d u t y cycle or resort to relatively sophisticated bunching devices. The method is also beset with considerable background and counting rate problems. This article describes a time-of-flight fast neutron spectrometer which operates with a continuous accelerator beam and does not require an associated event. Previously, neutron spectrometers have been designed employing the scattering of neutrons from a primary hydrogenous detector to a secondary 1) G. K. O'Neill, Phys. Rev. 95 (1954) 1235. P. L. Okhuys6n, E. W. Bennett, J. B. Ashe and W. E. Millett, Rev. Sci. Instr. 29 (1958) 982. 2) G. C. Neilson, W. IK. Dawson a n d F. A. Johnson, Rev. Sci. Instr. 30 (1959) 963. J. B. Garg, Nucl. Instr. and Meth. 6 (1960) 72. 3) G. C. Neilson a n d D. B. James, Rev. Sci. Instr. 26 (1955) 1018. ~) J. H. Neiler a n d W. M. Good, in Fast Neutron Physics, J. B. Marion and J. L. Fowler, Eds. (Interscience, New York, 1960), P a r t I, Chapter V.A. L. D. Chisholm, W. E. Dance, D. C. R a l p h and R. C. Mobley, Bull. Amer. Phys. Soc., Ser. l I , 6 (1961) 240. J. Lowe, J. V. Kane and M. A. E l - \ ¥ a h a b , Bull. Amer. Phys. Soc., Ser. II, 6 (196i) 253.
FAST NEUTRON
ISOCHRONOUS
detector at a fixed angle 3' 5). The recoil scintillation pulse height spectrum of the primary detector was utilized when coincidence gated at a suitable delay to remove 7-ray effects. These spectrometers achieved only limited resolution due to the poor intrinsic resolution of the scintillator and the recoil energy variation resulting from the angular spread subtended by the detectors. They further suffered from relatively low inherent efficiency. While the present spectrometer is also based upon the scattering of neutrons from a primary hydrogeneous detector to a secondary detector, it, however, makes use of a time-of-flight technique incorporating the notion of isochronous surfaces6). This allows the use of a secondary counter which subtends a large solid angle and a relatively long primary scatterer producing a considerable increase in efficiency. Careful consideration of the geometrical dimensions results in a spectrometer with good energy resolution and relatively large detection efficiency in those applications which inherently require the energy analysis of neutrons restricted to a relatively small range of emission angles from the target, as for example in angular distribution studies.
2. Idealized Isochronous Flight Path Spectrometer When the generalized isochronous flight path principle discussed in appendix A is applied to n-p scattering, an idealized time-of-flight neutron spectrometer m a y be visualized as illustrated in fig. 1. It is imagined that a specific reaction is under study that gives rise to the emission of various enerIDEAL SPECTROMETER
• Accelerator
Beam
Z nO
~TarQet
Fig. 1. Idealized isochronous time of-flight spectrometer, shown as it m i g h t be used in a n a n g u l a r d i s t r i b u t i o n s t u d y .
FLIGHT
PATH
SPECTROMETRY
291
gy neutron groups (target and beam configuration shown). The spectrometer is used to measure the angular distribution of the various groups by rotating the spectrometer proper about the target (i.e., varying O). The spectrometer determines the neutron energy by measuring the flight time of the neutron scattered from the hydrogeneous " s t a r t " detector to the " s t o p " detector shell contoured to match an isochronous surface. By definition the nature of this surface is such that the neutron transit time is independent of the scattering angle 0. Two favorable practical realizations, discussed in sections 2.1 and 2.2, make such a spectrometer an effective research tool. 2.1. I N T E R C E P T I O N
ANGULAR RANGE
The interception efficiency over the isochronous surface is not constant but varies with 0 in a manner proportional to 42 cos 0 daL(O)tl(n' ). The factor 42 cos 0 is the slant thickness of the stop detector shell (i.e., thickness seen by scattered neutron). The factor deE(0) is the differential cross section for n-p scattering in the laboratory system at the start scatterer; daL(0) ~ (ar/x) Cos 0 dOE = 2aT sin0 COS 0 dO, for E , < 15 MeV. The factor rl(n') is the biased efficiency per unit slant thickness of the shell for the scattered neutron. For plastic scintillator plates 1" to 2" thick (the appropriate range for present purposes), integrally biased to accept scintillation pulses greater than 300-500 keV proton energy equivalent, ~/varies as 1Iv to good empirical accuracy in the range 2 < E n, < 15 MeV (r/ = (11 + 1)/~/E, per cent/cm with E , i n MeV, rather than the lIE,, variation of a T 7 ) ) . Thus, ~/(n') is proportional to 1/v = llv 1 cos 0 and just cancels the cos 0 variation of the shell slant thickness, leaving the angular dependence of the interception efficiency to v a r y as deE. Hence it is appropriate to speak of the shell intercepting a certain fraction of the neutrons scattered at the start detector which 6) p. R. C h a g u a n , G. E. O w e n a n d L. M a d a n s k y , R e v . Sci. I n s t r . 26 (1955) 1165. 6) W. F. H o r n y a k a n d L. F. Chase, Jr., Bull. Amer. P h y s . Soc., Ser. I I , 7 (1962) 21. M. L. Roush, A. S. F i g u e r a a n d W. F. H o r n y a k , Bull. A m e r . P h y s . Soc., Ser. I I , 8 (1963) 40. 7) H. GrSssler a n d K. Tesch, Nucl. Instr. a n d Meth. 10 (1961) 353.
M. L. R O U S H et al.
292
are t h e n detected with relatively c o n s t a n t efficiency over the shell. This fraction f i n t e r c e p t e d in the range 0~ -< 0 --- 02 is:
I I ~ aT
fo~ aT COS 0 2~ sin 0 d0.
(1)
0t 7~
Since the i n t e g r a n d is s y m m e t r i c a l in 0 a b o u t its m a x i m u m value at 0 = 45 °, it is e v i d e n t t h a t 0, a n d 02 should be selected such t h a t 01 = 45 ° a n d 02 = 45 ° + fl, w h e n e q u a t i o n (1) gives / = sin 2fl.
(2)
Thus, if for practical reasons the spherical shell is restricted to t h e range 01 = 30 ° a n d 02 = 60 ° (i.e., fl = 15°), one-half of all the n e u t r o n s s c a t t e r e d at the s t a r t detector would still be intercepted.
photoelectric efficiency w h e n used in conjunction w i t h existing photomultipliers. F o r this reason the value of fl was reduced to 8.8 ° a n d t h e resulting surface was s u b d i v i d e d into sections which it was felt could be optically coupled in a satisfactory m a n n e r to existing fast photomnltipliers.
1 klght-pipe~ NE-102Scinfllotor
Polyvinylto~uene
1
/
/
.
/,/
2.2. S T A R T D E T E C T O R L E N G T H
As discussed in a p p e n d i x B, u n d e r the contribution of the scatterer length to line width, t h e effect of p a t h length differences from the front a n d b a c k portions of t h e scatterer are largely cancelled, near 0 = 45 °, b y the changes in n e u t r o n velocity caused b y t h e cerresponding changes in scattering angle. Thus, in the present spectrometer a 3" long scatterer which is 10 % of the radius R = 30", produces a line w i d t h of only ¼% (FWHM) in time, discounting the effects of multiple scattering above E . = 2 MeV. Such a scatterer will s c a t t e r 54% of t h e i n c i d e n t n e u t r o n s at E t = 2 MeV a n d 2 1 % at E~ = 12 MeV. The effects of scattering from t h e c a r b o n i n the s t a r t detector is largely eliminated b y the discrimin a t o r level used. I t should be noted, however, t h a t a n y r e m a n e n t effects would not be expected to produce narrow lines since the scatterer would be improperly located for isochronous s c a t t e r i n g from c a r b o n (i.e., S = R/12 for carbon).
3. The Actual Spectrometer I n t h e idealized s p e c t r o m e t e r of section 2 the shell stop detector would be s u r r o u n d e d b y a n optical reflector a n d the scintillation light would be collected, with c o n s t a n t time delay, near t h e central region C of fig. 1. While the fabrication of the shell described in section 2 (fl = 15 °) is e n t i r e l y feasible, all light coUection systems envisioned failed b y one order of m a g n i t u d e in p r o v i d i n g a d e q u a t e total
/ Fig. 2. The spectrometer section a c t u a l l y used. For increased efficiency e i g h t a d d i t i o n a l such sections m a y be used arranged a z i m u t h a l l y w i t h respect to target-scatterer axis.
Fig. 2 shows t h e s p e c t r o m e t e r a c t u a l l y used. The radius of c u r v a t u r e of t h e shell was selected to be R = 30" as a reasonable c o m p r o m i s e b e t w e e n : (1) t h e desire to keep fl large w h i c h w i t h the requirem e n t t h a t t h e " c a p " diameter, AA' in fig. 2, be k e p t from b e c o m i n g too large to provide a d e q u a t e light collection requires R small, a n d (2) t h e desire to keep t h e intrinsic electronic time spread contribution to line shape ({2 a n d ~3 of a p p e n d i x B) acc e p t a b l y small for 2-15-MeV n e u t r o n s w h i c h requires R large. The shell thickness was selected u s i n g the optim i z a t i o n procedure discussed in a p p e n d i x C. W i t h R = 30" a n d selecting t h e line w i d t h c o n t r i b u t i o n of t h e g e o m e t r y (~i) to be 3 to 4 per cent i n t i m e gives (in t h e n o t a t i o n of the appendix) ~ - 1~". T h u s the interior surface of the shell has a radius of c u r v a t u r e R = 30", while the exterior surface has a radius of c u r v a t u r e R + dR = R + 22 = 30{". T h e resulting tapered thickness of t h e shell gives a c o n s t a n t energy resolution for t h e n e u t r o n s scattered at different angles (see a p p e n d i x C). T h e shell,
FAST NEUTRON
ISOCHRONOUS
having a cap diameter of 18", was made of NE-102 scintillator and was cemented to a polyvinyltoluene light pipe which was in turn optically coupled to a 58AVP photomultiplier*. The 18" cap diameter was estimated to be the m a x i m u m size that could be coupled to a single photomultiplier and provide satisfactory light collection and thus keep timing dispersion to acceptable values. Up to eight additional identical sections m a y be added azimuthally to more closely approximate the complete spherical shell. Each section will intercept 2.3 % of the scattered neutrons. For 5-MeV neutrons incident on the scatterer, the shell has an estimated intrinsic detection efficiency of approximately 15 % for the intercepted neutron. The scatterer is an NE-102 scintillator 3" in length and ½" in diameter coupled to a 56AVP photomultiplier. This scatterer diameter with = ~" and the target-to-scatterer distance of d = 15" gives a line width contribution :of approximately 4 % in time (FWHM) (see appendix B).
DYNODE
FLIGHT
PATH
SPECTROMETRY
293
than a 10% intercepted contribution of multiply scattered neutrons at 2-MeV incident energy. For the present scatterer and shell resulting spectrometer efficiency is approximately 1 x 10-3 for 5-MeV incident neutrons. Current pulses from the anode of the photomultipliers are coupled directly to the grid of 6688 limiter tubes normally carrying 22 ma. By the use of delay-line** clipping, 100-ns rectangular pulses are obtained having an amplitude of 2.7 volts after transmission through a cathode follower (5842 tube) and 100 feet of RG63/U cable. Fig. 3 shows the block diagram of the electronics. The limiter signals are fed to the time-to-pulse-height converter which measures their time overlap. The converter uses a 6BN6 tube operated at a screen grid voltage of 10 volts and plate voltage of 20 volts, as suggested by Green and BellS). The output of the time converter is coupled to a multichannel analyzer through a White cathode follower. To improve the ratio of true to accidental counts,
N I TEGRAF L
ANODE
ANODE I I00ns
.--I
iLIAINTO~REI C,EL "STOP" ~1
ANODE
GOU NTERI~5BAVPI--'~DY --" NO F'D -'lE --q
CON I CD I ENCE
hCOINCiDENCE
I00 ns DELAY
=
2Fs
IAMPLINDTsE(~RR'L ~--
Fig.3. Block d i a g r a m of t h e electronics e m p l o y i n g a time-to-pulse-height Multiple scattering, if present, could result in neutrons with incorrect flight times and is most important, of course, for low neutron energies. The present diameter scatterer is estimated to give less * T h i s scintillator a l r e a d y m a c h i n e d a n d p e r m a n e n t l y cotmented to t h e l i g h t pipe w a s o b t a i n e d f r o m N u c l e a r E n t e r -
prises, Inc., Winnipeg, Canada. ** 330-ohm d e l a y line, t y p e Z3M, furnished b y Telcon W o r k s , Greenwich, L o n d o n S.E. 10, England.
converter.
a 6BN6 fast coincidence " p r i o r i t y " circuit is added. By delaying the input " s t o p " signals 100 ns, the priority circuit output signal can be used as an anticoincidence gate for the multichannel analyzer, thereby eliminating events in which the stop signal precedes the start signal. The resolution of the electronics was measured b y a) R. E. Green a n d R. B. Bell, Nucl. I n s t r . 3 (1958) 127.
294
M.L.
R O U S H et
feeding test signals from a fast mercury-switch pulser directly to each limiter. The s p e c t r u m obt a i n e d is shown in fig. 4. A second peak is introduced for calibration, b y inserting a 7.0-ns delay in
al.
Fig. 5 shows the results of a t i m i n g calibration using a Na 22 7_ray source. F r o m the shift in position of t h e peak when a 30.0-ns delay line is inserted, the value of 0.388 ns per c h a n n e l is obtained.
4. Experimental Results Fig. 6 displays a typical time-of-flight spectrum. The Beg(~,n)C 12 reaction yields two n e u t r o n
COUNTS ZOn sec 4,~ C-I¥,~-NNEL5 [
I" 500(
I d
2000
~l~nCHANNEL5
=.
sec
1000
300
¢J E 200 i
-2.8 CHANNELS •=020 nsec
e,~l =,,lkeV d2T~rge! ~,=60 E==20tc,ev Q=45OO~Cotlom~ o,,=o" Shel0~c=055 l MeV,Proton energy n= En'7.71MeV E.-293MeV
(J I00 ~" 140
230 240 CHANNELNUMBER
LSO"
250
CHANNEL NUMBER
~g.--
Fig. 4. The intrinsic line width of the time-to-pulse-height converter using high gain amplification following the converter. The line displaced by the use of a 7.0-ns delay line is also shown.
Fig. 6. A typical neutron spectrum showing the results with the Bed(e, n)C TM reaction. The line labelled y results from Compton scattering of 7rays.
t h e signal from one of the limiters. A portion of the resulting line w i d t h is a t t r i b u t a b l e to the multic h a n n e l analyzer so t h a t the resolution of t h e electronics following the photomultipliers is proba b l y b e t t e r t h a n 0.2 ns. The s t a b i l i t y of the system has been found to correspond to a drift of less t h a n 0.5 ns over a 16-hour period using a stabilized source of a.c. power.
groups leading to t h e g r o u n d s t a t e a n d first excited s t a t e of C 12. The peak observed in c h a n n e l 342 is due to the g a m m a radiation scattered from the first scintillator to the spherical shell a n d serves as a convenient reference marker. Fig. 7 shows the
COUNTS 500
Iq
400300200 IO0
se¢ %30.0 77.3n CHANNELSII
i'SS8 ' n see/ch°nnel ~
o ,_.2
,F
/
z
E~ • l.S MeV •
£s" t.2 MeV
~
F
~
,
120 150 140 v 190 200 210 220 CHANNEL NUMBER
Fig. 5. Time calibration of electronics i n c l u d i n g the p h o t o m u l t i pliers, e m p l o y i n g a delay line shifted secondary line.
2O CHANNEL ~ R Fig. 7. Time spectra showing separation of two n e u t r o n groups differing in energy b y 400 keY.
FAST NEUTRON
ISOCHRONOUS
relative separation of two low energy n e u t r o n groups. Resolution a n d line shape m e a s u r e m e n t s h a v e been m a d e using t h e T(p, n), T(d, n), Be9(a, n), a n d
R-30*
NEUTRON ENERGY(MeV)
FLIGHT
PATH
SPECTROMETRY
295
The shell properly contoured for R = 30", w h e n used at R = 45", (the cap rim set on the R = 45" sphere, with its center at 45 ° to the i n c i d e n t b e a m as m e a s u r e d from the scatterer) will give b e t t e r resolution b o t h because of t h e increased flight p a t h a n d because the resulting sagittal flight p a t h error will reduce t h e time delay i n t r o d u c e d b y the longer light p a t h from the scintillator rim to p h o t o multiplier. For R = 45" this cancellation is exact at Eo ~ 4 MeV; At 2 (see a p p e n d i x B) is a p p r o p r i a t e l y reduced to 1.60 - 0.90/E, ns. The resulting theoretical curve is seen to fit t h e R = 45" d a t a of fig. 9 r e a s o n a b l y well. Fig. 10 shows for En = 8 MeV, the v a r i a t i o n of energy resolution with shell discriminator setting.
Fig. 8. T h e m e a s u r e d line w i d t h s ( F W H M ) in t h e e n e r g y r a n g e 2 -< E n < 10 MeV. T h e c u r v e shown r e p r e s e n t s t h e t h e o r e t i c a l estimate.
D(d, n) reactions as sources of neutrons. T h e actual e x p e r i m e n t a l results are c o m p a r e d in figs. 8 a n d 9 w i t h t h e theoretical prediction (see a p p e n d i x ]3). These m e a s u r e m e n t s were m a d e w i t h the sidec h a n n e l d i s c r i m i n a t o r for the spherical shell set at a n effective p r o t o n recoil energy of ~ 500 keV. Fig. 8 shows a s o m e w h a t faster observed rise t h a n predicted in At at the lower energies possibly ind i c a t i n g the effects of multiple scattering. This effect is also evident for A E / E as m a y be seen in fig. 9. 2G
/~
o R-30" x R-45"
IB
o I4
~'~E"+~]'2
,o
.0.4 Fn=B MeV , R-36"
o
~
,~
;
-02
~l,, ~'5
DISER]MINATOR SETTING(PHOTO-CATI'IOOEELECTRONS}
Fig. 10. T h e v a r i a t i o n of yield a n d r e s o l u t i o n w i t h shell discrimin a t o r setting.
Acknowledgement The authors are grateful to Lloyd F. Chase, Jr., a n d staff m e m b e r s of the Nuclear Physics Division of Lockheed Missile a n d Space D e v e l o p m e n t Labor a t o r y for m a n y s t i m u l a t i n g discussions. We are also grateful to J. B. H e a t h of Nuclear Enterprises, Inc., for m a k i n g possible t h e fabrication of the shell detector.
Appendix A CALCULATION OF THE ISOCHRONOUS SURFACE FOR A GENERAL REACTION
'
;.
~,
~
,E, . . . .
N~utr~ Energy (MeV)
~o
Fig. 9. A c o m p a r i s o n of e x p e r i m e n t a l a n d t h e o r e t i c a l e n e r g y r e s o l u t i o n for b o t h R 30" a n d R 45", in t h e e n e r g y r a n g e 2< E n< 20MeV.
Fig. 11 illustrates a general reaction involving an accelerated particle of mass M 1 striking a s t a t i o n a r y t a r g e t nucleus M 2 resulting in f r a g m e n t s M s a n d M , at a certain reaction Q. The i n c i d e n t particle
296
M.L.
R O U S H e~ al.
P
Q = O, M 1 = M 3, M 2 = M 4 )
equation (A4) simply
reduces to :
,' Mt,Vt,E I
/
"
O~-C M4" 0C-$ CP-
R
Fig. 11. Reaction kinematics illustrating t h e isochronous surface t r a c e d b y point P.
laboratory velocity and energy are v t and E l , the laboratory and center-of-mass velocity of fragment M3 is v3 and v3c respectively, while v.... is the velocity of the center-of-mass. A trivial application of the kinematic conservation laws gives: z Vc.m.
It is worth remarking that any arbitrarily shaped area detection system for fragment M 3 if moved along the isochronous spherical surface (i.e., rotated about C at radius R) while subtending different laboratory solid angles at the target position O depending on 0L, has a constant c.m. solid angle. In addition to the application of the above isochronous principle to the determination of the angular distribution of various neutron groups originating in some reaction under investigation, the subject of the main body of the present paper, two additional embodiments of the principle are worth citing. Fig. 12 illustrates applications suitable in angular correlation and neutron polarization experiments.
(AI)
2MtEI/(M 1 + M2) z ,
ISOCHRONOUS / SL~FACE
v2¢ = 229I4 [M2E t + (M1 +
M2)Q]/M3(Mt +
As is already clear from elementary considerations, equations (AI) (A2) show t h a t v.... and v3© are independent of the emission angle 0c in the c.m. system. It then follows that the fragment M 3 originating its flight at the target position 0 will arrive at P, a point on a spherical surface with its center at C, in a fixed time independent of 0c. If the distances OC and CP are designated as S and R respectively, it is only required that R/S = v3Jv .....
(A3)
+ (M 1 + M2)Q/EI]/MtM 3
(A4)
or
[M 2
for the spherical surface of radius R to become an isochronous surface for fragment M 3. To keep the isochronous time single valued requires the usual restriction to reaction energetics involving E t and Q, that v3c > v¢.=.. The flight time of fragment M 3 from 0 to P is clearly: t =S/v .... = S(M 1 + M2)/Mtv 1 .
"STO~'
M2) 2 . (A2)
R2/S 2 = M,
(A6)
R/S = M2]M t .
(A5)
For the special case of elastic scattering (i.e.,
/
~
I
, O-TARGET POSITION \
/
~
-
//
~
/ //~
/
~
~"
X "START'SONTILLATOR\ \ ~
\
~
BeI(d,n)~°" (3.5BMeV)~ Ed=2.50 MeV J (a)
/SCINTILLATOR
I
~"I
. . . . . R=7.88 $
/ /~x"SURFACE , LEFT"STO~"
\
\ nv/4 ~ d
~/8 • ~
-
", \
Id2, n
-
\/ /\
HELIUM=S.AR=~/
/
'\ /
/
RIGHT"ST(~"
~qWa~.~d ~ -
SONTILLATOR "
~TARGET pOSITION R-4S
(b)
Fig. 12. Two suggested e m b o d i m e n t s of the isochronous principle. I n (a) the application is to an a n g u l a r correlation study, d r a w n to scale for the Beg(d, n)B l°* (3.58 MeV) reaction. I n (b) the application is to a n e u t r o n polarization s t u d y using a helium scatterer.
For the sake of definiteness let the situation in fig. 12a refer to the reaction Be9(d,n)B t°* at E o = 2.50 MeV. Reference is then made to a possible angular correlation study to measure the E2-M1 mixture in the 2+(3.58-MeV) level in B t°* decay branch to the 1 + (2.15-MeV) level. Here, as in m a n y other cascade rich reactions, a positive identification (even if possible) of the y-ray energy (1.43 MeV) is insufficient to determine uniquely the neutron group populating the initiating level. Thus,
FAST
NEUTRON
ISOCHRONOUS
a neutron energy determination is necessary. In the arrangement shown the neutron group is to be identified by a time-of-flight measurement using a fast " s t a r t " scintillator to detect the ? ray (using only a very broad range pulse height selection) with the " s t o p " detector being a thin shell scintillator formed to correspond to the appropriate isochronous spherical surface. Here the center of the sphere is displaced a distance S from the Be target such that R = 7.88S. I t should be noted that each neutron group (i.e., Q) requires a different value of R / S , thus for each geometry only one group is truly isochronous over the entire surface P. This fact is particularly important when the stop detector subtends a relatively large solid angle. In particular, it is seen that the usual practice of using a thin flat plate stop detector oriented perpendicular to a radius vector from the target will introduce a larger timing spread than warranted by the plate thickness relative to the flight path and the electronics. If a flat plate is to be used, it should at least be oriented tangent to the appropriate isochronous sphere. If in obtaining the correlation function the neutron stop detector is rotated on a goniometer arm centered at C, the value of 0c can be directly read from the goniometer protractor and the c.m. solid angle subtended by the neutron detector will be held a constant. Further, since the flight time is then constant, as the detector is rotated, a relatively narrow time display window can be used reducing the dead time losses. The arrangement illustrated in fig. 12b is intended to measure the polarization of various neutron groups, for example in the Be9(d, n)B 1°* reaction, making use of the elastic scattering from a high pressure helium scintillator cell. Here, the isochronous principle is applied to the secondary reaction [i.e., He*(n, n)He 4] to identify the neutron groups using a time-of-flight measurement. Only a very broad range pulse height selection is suggested for the helium cell start detector. In this case R = 4S, which in view of the relatively large range in 0 permissible to the left and right stop detectors by nature of the polarizing properties of the He*(n, n)He* reaction at backward angles, makes the use of properly contoured stop detectors essen-
FLIGHT
PATH
SPECTROMETRY
297
tial. (The azimuthal angle extension of the stop detectors (not shown) must be determined with the ensuing loss of polarization in mind.). Appendix
B
CALCULATION
OF SPECTROMETER
RESOLUTION
The energy resolution of the spectrometer is the result of the combined effect of a number of geometrical and statistical factors. A number of geometrical factors are directly related and are added as coherent sources of line width. The spectrometer resolution is conveniently expressed as ~T, the ratio of line width (FWHM) in time units to the time of flight of the neutron between the two scintillators, with ¢r2 = ¢~ + ~2 + ~ + ~], (A7) where ~t is the spread due to shell thickness, scatterer radius, and beam spot size, ~2 is the spread due to the combined effect of phosphor decay time, light collection efficiency, and light transit time spread, ~3 is the spread introduced by the photomultipliers and electronics, ~, is the spread introduced by the length of the scatterer. The dependence of flight time upon neutron energy is given by t ----72.3d/~JE n
(A8)
where t is in nanoseconds, if d is any specified flight path in meters and E n is in MeV. The isochronous flight time t in appendix A, eq. (A5) reduces for n-p scattering to t = 2Sly 1 = 2R[v t since S = R. For the present spectrometer D = 2R = 60"; thus, a given uncertainty in time At (in nanoseconds) results in a fractional line width of '-= A t / t = At~/En/IIO.2.
(A9)
B.1. Calculation of¢1 (due to beam spot size, scatterer radius, and shell thickness)
An approximate calculation will be effected by assuming the length of the scatterer 2a ,~ d, a n d taking 0 = 45 ° as representative of the extended spherical shell. Thus, referring to fig. 13 let 0 = 45 °, tp =
45 ° -
a - - e, a n d ~ k
=
45 ° +
a +
~,where
298
M. L. R O U S H
~ (s + r)/d
and
e m (r + 2)lx/2L.
(A10)
Flight p a t h L t represents the extreme case when the p a t h is shortest and the neutron velocity is greatest. P a t h L 2 represents the opposite extreme. The flight time for the central p a t h is
t = L/v I cos 0 = a/2L/v x ,
(All)
where v 1 is the incident neutron velocity. For flight p a t h L tt _
LI _ L - ((r + 2)Ix/~ v 1 cos~o v 1 cos (45 ° - ~ -- ~)
(a12)
and similarly for p a t h L 2
tz
L2 L + ((r + 2)/x/2) v I c o s ~ - v t c o s ( 4 5 ° + ~ + ~)
.
(AI3)
Since the angles ~ a n d , are small cos (45 ° + ~ + ~) m (1 T ~-T- ~)/~/2-. (A14) From eqs. (All), (A12), (A13), and using eq. (A14), we obtain after some simplification (t 2 - t,)/t = 2(~ + e) + 2(r + 2)/~/2 L .(A15) Using the expressions in (A10) and noting t h a t L = ~/2R : (t 2 - t~)/t = 2(s + r)/d + 2(r + 2)/R.
(A16)
Expression (A16) gives the fractional full base width, the fractional w i d t h at half-maximum will be approximately one-half of this value:
et al.
¢1 = (s + r)/d + (r + 2 ) / R .
(A17)
For our particular spectrometer we have chosen s = ~ i n c h , r = ¼inch, d = 15inches, 2 = ~ i n c h , R = 30 inches, which results in ~x = 0.0396.
(A18)
A n o t h e r sort of angular error is i n t r o d u c e d b y multiple scattering in the scatterer. While this effect is difficult to calculate and leads to the modification of the line shape mainly b y the addition of a long delay tail, an order of magnitude estimate gives ~[ ~< 0.01 at E . = 2 MeV for the present case, the value at other energies varying inversely with E n. It has been assumed t h a t in the energy range E . > 2 MeV the effects of multiple scattering m a y be neglected.
B.2. Estimation of ~2 (due to the combined effect of phosphor decay t~me, light collection efficiency, and light transit t,me spread) The NE-102 plastic scintillator has a decay time of m 3 ns. This time constant combined with the light collection efficiency determines the time delay for the emission of a sufficient n u m b e r of electrons from the photomultiplier photocathode to subsequently cut off the limiter. A further correlated delay occurs due to the light collection transit time from the corners of the shell where the light collection efficiency is also poorest (in NE-102 an optical p a t h length of 7.5" corresponds to a 1-ns time delay). While the combined effect of these factors on the line shape is relatively complicated, requiring a resort to numerical methods, a greatly simplified approach leads to an a p p r o x i m a t e expression (limiting at 4 electrons, photo cathode equivalent, and integral discrinfination at 8 electrons) of At2 = 1 80 - 0.90/E, ns, which in t u r n gives: ~z = [1.63 ~/En - 0.82/~/E.~ x 10 -2
(A19)
B.3. Estimation of ~3 (due to transzt time spread in the photomultiplier and fluctuatzons in the total electronic system) Fig. 13. The illustration of the geometrical factors influencing the line width contribution ~1 (appendix ]3).
This effect i. due almost completely to the photo multlpFers since the foalowing electronics has a measured i~trmsic resolution better than 0 2 n .
FAST NEUTRON
ISOCHRONOUS
FLIGHT
PATH
SPECTROMETRY
T h e m a n u f a c t u r e r ' s specification of t r a n s i t t i m e s p r e a d for t h e 56AVP a n d 58AVP p h o t o m u l t i p l i e r s are 0.5 a n d 1.0 ns, r e s p e c t i v e l y . T h e factors c o m bine to give AQ = 1.14 ns, f r o m w h i c h we o b t a i n 43 = 0.0104 x/E~ .
299
~.~ ~*
p
_~2_
ISOGHRONOUS SPHERICAL
S.EL,
,-'///
-\
,Y2/ I
(A20)
\
B.4. Calculation of 44 (due to the scatterer length) R e f e r r i n g to fig. 14a, t h e f r a c t i o n a l s p r e a d i n flight t i m e for s c a t t e r e d n e u t r o n s to arrive at p o i n t P f r o m a p o i n t x along t h e s c a t t e r e r relative to t h a t f r o m t h e c e n t e r of t h e s c a t t e r e r (of l e n g t h 2a) is calculated. Clearly, l~ = l 2 + x 2 + 2xl cos 0 a n d 1t cos 0 = l c o s 0 + x a n d t h e flight t i m e s for t h e i n d i c a t e d p a t h s are
W~ o
W--~o (a) "t'2q
~'+~
t t1
= l/vt cos 0 = D/v t , =
(A21) -I'll
ll/v 1 cos ~b = 12x/vxll cos ~ .
Writing the fractional (t! - t)/t, t h e r e results
time
delay
as
At/t = x(lcos20 + x cosO)/DcosO(lcosO + x).
At/t =
(A22)
=0 ,
:3
~-
N o t i n g t h a t 0 = 45 ° + r, a n d t h a t in p r a c t i c a l cases 1 cos 0 >> x a n d f u r t h e r t h a t fl < 15 ~, (A22) r e d u c e s to At/t ~ 2x(x - D sin 2fl)/D 2 .
(A23)
Fig. 14b s h o w s At/t as a f u n c t i o n of X / D a n d ft. T h a t for s m a l l r, At/t is of o r d e r X 2 / D 2 results f r o m t h e fact t h a t while t h e flight p a t h l~ > l, t h e corr e s p o n d i n g n e u t r o n v e l o c i t y is also greater, since ¢ < 0, giving a c a n c e l l a t i o n of first o r d e r t e r m s in X / D . (For x n e g a t i v e l 1 < l a n d ~ > 0, a g a i n r e s u l t i n g in a c a n c e l l a t i o n to first order.) F o r geom e t r i e s n o t t o o d i s s i m i l a r f r o m t h a t u s e d here a n a p p r o x i m a t e value for 4~, c a n be g i v e n as 4', ,~ (a/D) sin 2tim,
(A24)
w h e r e t h e s p h e r i c a l s h e l l e x t e n d s from 0 = 45 ° + fl to 0 = 45 ° - tim, a n d fl~ > a/2D. A p p l y i n g equation (A24) to t h e p r e s e n t case, w i t h 2a = 3" gives 4~ -~ 0.0075.
(A25)
A more elaborate numerical calculation shows that (A25) is g o o d to ___ 10% accuracy.
',e--#o -20 (b) Fig. 14. T h e illustration in (a) of the geometrical factors influencing the line w i d t h c o n t r i b u t i o n ~2 (appendix ]3). I n (b) t h e fractional t r a n s i t t i m e delay a p p r o p r i a t e to (a) is shown.
T h e l e n g t h of t h e s c a t t e r e r also influences t h e r e s o l u t i o n t h r o u g h t h e t i m e s p r e a d due to o p t i c a l p a t h l e n g t h differences to t h e p h o t o m u l t i p l i e r . I n t h e p r e s e n t case we o b t a i n At = 0.19 ns. T h u s , t h e final t i m e d i s p e r s i o n due to t h e s c a t t e r e r l e n g t h can be e x p r e s s e d as 44 = (0.56 + 0.029E,) ~
x
10 - 2
(A26}
C o m b i n i n g ix, 42, 43, a n d 44 as i n d i c a t e d b y (A7} yields 4T = atT/t = (13.6 + 3.8E, + 0.7/E). ~ x 10 -2 (A27)
300
M.L.
ROUSH
This can be c o n v e r t e d to give t h e corresponding w i d t h s i n time a n d energy
AtT = (4.6 + 16.5/E n + 0.9/E2.)~:ns
(A28)
and
AE/E = (55 + 15.0E n + 3/E,) ~:percent.
(A29)
et al.
desired n e u t r o n energy. I t remains t h e n to select s, r, 2, a n d d so as to optimize the efficiency for a given ~1 a n d R. I n those applications where s can be m a d e negligibly small, it will be useful to introduce an efficiency figure of m e r i t F = 2r2/d 2. Solving equation (A17) (with s = 0) for r a n d s u b s t i t u t i n g it into t h e expression for F gives
Appendix C SHELL
THICKNESS
CONSIDERATIONS
F o r the fractional c o n t r i b u t i o n of t h e shell thickness to t h e line w i d t h to be i n d e p e n d e n t of 8 requires t h a t the scintillator m a t e r i a l composing the shell be confined between two isochronous surfaces h a v i n g radii R a n d R + AR, selected to give t h e desired value of AR/R. The thickness ~ s h o w n in fig. 13 is just ½AR. The two isochronous surfaces are of course t a n g e n t at t h e scatterer a n d h a v e t h e i r centers displaced b y a distance AR. Given a certain electronic capability, the selection of R will d e t e r m i n e ~2 a n d ~3- The usual compromise between resolution a n d efficiency would lead to a s p e c t r o m e t e r design giving ~ 1 (the largest geometrical source of line width) a p p r o x i m a t e l y equal to t h e c o m b i n e d effect of 42 a n d 43 at t h e
F = .~ \ ~ - ]
(A29)
I t follows trivially from t h e o p t i m i z a t i o n of F t h a t )" = J~R¢I
(A30)
d
INSTRUMENTS
AND
METHODS
24 (1963)
290--300,
NORTH-HOLLAND
PUBLISHING
CO.
FAST NEUTRON ISOCHRONOUS FLIGHT PATH SPECTROMETRY* M. L. R O U S H t , A. S. F I G U E R A and W. F. H O R N Y A K t t
Department o/ Physics and Astronomy, University o/ Maryland, College Park, Maryland Received 4 March 1963
A time-of-flight fast n e u t r o n spectrometer for continuous beams of neutrons is described. The spectrometer utilizes the isochronous principle t h a t monoenergetic neutrons scattered from a hydrogenous m e d i u m arrive a t a s u i t a b l y oriented spherical surface after a fixed t r a n s i t time. The scatterer ( " s t a r t " detector) is a cylinder of NE-102 p l a s t i c s c i n t i l l a t o r 3" long and ½" diameter. The " s t o p " detector, also of NE-102, is a segment of a spherical shell with t a p e r i n g thickness selected to o b t a i n optim u m resolution. This scintillator has a cap diameter of 18 inches
1. Introduction
The measurement of neutron energies by the time-of-flight method involves the determination of the elapsed time required by the neutron to traverse a prescribed flight path. All variants of this technique derive a " s t o p " signal or time marker signifying the arrival of the neutron at the end of its flight path simply through the use of a suitably placed neutron detector. However, either one of two methods are generally employed to obtain the " s t a r t " signal or zero time reference marker. In one method this is accomplished by detecting the occurrence of an event which is simultaneous with the emission of the neutron (e.g., the emission of an associated particle1), or g a m m a ray 2, 3). While this technique permits the use of a continuous accelerator beam in generating the neutrons, it is necessarily limited to the class of reactions yielding coincident events. Alternatively, a variety of pulsed beam techniques may be employed to generate a suitable zero reference time4). These systems operate by allowing the initiating beam to strike the target in bursts of appropriately short duration. The required zero time marker is derived from * Research supported in p a r t by U.S. Atomic E n e r g y Commission. N.S.F. G r a d u a t e Cooperative Fellow. ** Presently on leave a t Universit6 de Paris, Orsay (Seine-etOise), France. 290
with the radius of c u r v a t u r e of the spherical surface being 30 inches. A c o n v e n t i o n a l time-to-pulse-height converter is used. E n e r g y resolution of 9--14% has been obtained for the energy range 1.8 .~ E n <~ 10 MeV. The efficiency, a t 5 MeV, is estimated to be 1 X 10 -3 per n e u t r o n i n c i d e n t on the scatterer. The u n i t discussed is ideally suited to a n g u l a r d i s t r i b u t i o n measurements for reactions resulting in numerous n e u t r o n groups. Other a p p l i c a t i o n s of the isochronous principle are proposed.
either a signal originating in the beam pulser or from the actual beam pulse striking the target. These methods have the advantage of not requiring an associated event so they m a y employ neutrons from a wide class of reactions. However, this method in its simplest form must contend with an unfavorable d u t y cycle or resort to relatively sophisticated bunching devices. The method is also beset with considerable background and counting rate problems. This article describes a time-of-flight fast neutron spectrometer which operates with a continuous accelerator beam and does not require an associated event. Previously, neutron spectrometers have been designed employing the scattering of neutrons from a primary hydrogenous detector to a secondary 1) G. K. O'Neill, Phys. Rev. 95 (1954) 1235. P. L. Okhuys6n, E. W. Bennett, J. B. Ashe and W. E. Millett, Rev. Sci. Instr. 29 (1958) 982. 2) G. C. Neilson, W. IK. Dawson a n d F. A. Johnson, Rev. Sci. Instr. 30 (1959) 963. J. B. Garg, Nucl. Instr. and Meth. 6 (1960) 72. 3) G. C. Neilson a n d D. B. James, Rev. Sci. Instr. 26 (1955) 1018. ~) J. H. Neiler a n d W. M. Good, in Fast Neutron Physics, J. B. Marion and J. L. Fowler, Eds. (Interscience, New York, 1960), P a r t I, Chapter V.A. L. D. Chisholm, W. E. Dance, D. C. R a l p h and R. C. Mobley, Bull. Amer. Phys. Soc., Ser. l I , 6 (1961) 240. J. Lowe, J. V. Kane and M. A. E l - \ ¥ a h a b , Bull. Amer. Phys. Soc., Ser. II, 6 (196i) 253.
FAST NEUTRON
ISOCHRONOUS
detector at a fixed angle 3' 5). The recoil scintillation pulse height spectrum of the primary detector was utilized when coincidence gated at a suitable delay to remove 7-ray effects. These spectrometers achieved only limited resolution due to the poor intrinsic resolution of the scintillator and the recoil energy variation resulting from the angular spread subtended by the detectors. They further suffered from relatively low inherent efficiency. While the present spectrometer is also based upon the scattering of neutrons from a primary hydrogeneous detector to a secondary detector, it, however, makes use of a time-of-flight technique incorporating the notion of isochronous surfaces6). This allows the use of a secondary counter which subtends a large solid angle and a relatively long primary scatterer producing a considerable increase in efficiency. Careful consideration of the geometrical dimensions results in a spectrometer with good energy resolution and relatively large detection efficiency in those applications which inherently require the energy analysis of neutrons restricted to a relatively small range of emission angles from the target, as for example in angular distribution studies.
2. Idealized Isochronous Flight Path Spectrometer When the generalized isochronous flight path principle discussed in appendix A is applied to n-p scattering, an idealized time-of-flight neutron spectrometer m a y be visualized as illustrated in fig. 1. It is imagined that a specific reaction is under study that gives rise to the emission of various enerIDEAL SPECTROMETER
• Accelerator
Beam
Z nO
~TarQet
Fig. 1. Idealized isochronous time of-flight spectrometer, shown as it m i g h t be used in a n a n g u l a r d i s t r i b u t i o n s t u d y .
FLIGHT
PATH
SPECTROMETRY
291
gy neutron groups (target and beam configuration shown). The spectrometer is used to measure the angular distribution of the various groups by rotating the spectrometer proper about the target (i.e., varying O). The spectrometer determines the neutron energy by measuring the flight time of the neutron scattered from the hydrogeneous " s t a r t " detector to the " s t o p " detector shell contoured to match an isochronous surface. By definition the nature of this surface is such that the neutron transit time is independent of the scattering angle 0. Two favorable practical realizations, discussed in sections 2.1 and 2.2, make such a spectrometer an effective research tool. 2.1. I N T E R C E P T I O N
ANGULAR RANGE
The interception efficiency over the isochronous surface is not constant but varies with 0 in a manner proportional to 42 cos 0 daL(O)tl(n' ). The factor 42 cos 0 is the slant thickness of the stop detector shell (i.e., thickness seen by scattered neutron). The factor deE(0) is the differential cross section for n-p scattering in the laboratory system at the start scatterer; daL(0) ~ (ar/x) Cos 0 dOE = 2aT sin0 COS 0 dO, for E , < 15 MeV. The factor rl(n') is the biased efficiency per unit slant thickness of the shell for the scattered neutron. For plastic scintillator plates 1" to 2" thick (the appropriate range for present purposes), integrally biased to accept scintillation pulses greater than 300-500 keV proton energy equivalent, ~/varies as 1Iv to good empirical accuracy in the range 2 < E n, < 15 MeV (r/ = (11 + 1)/~/E, per cent/cm with E , i n MeV, rather than the lIE,, variation of a T 7 ) ) . Thus, ~/(n') is proportional to 1/v = llv 1 cos 0 and just cancels the cos 0 variation of the shell slant thickness, leaving the angular dependence of the interception efficiency to v a r y as deE. Hence it is appropriate to speak of the shell intercepting a certain fraction of the neutrons scattered at the start detector which 6) p. R. C h a g u a n , G. E. O w e n a n d L. M a d a n s k y , R e v . Sci. I n s t r . 26 (1955) 1165. 6) W. F. H o r n y a k a n d L. F. Chase, Jr., Bull. Amer. P h y s . Soc., Ser. I I , 7 (1962) 21. M. L. Roush, A. S. F i g u e r a a n d W. F. H o r n y a k , Bull. A m e r . P h y s . Soc., Ser. I I , 8 (1963) 40. 7) H. GrSssler a n d K. Tesch, Nucl. Instr. a n d Meth. 10 (1961) 353.
M. L. R O U S H et al.
292
are t h e n detected with relatively c o n s t a n t efficiency over the shell. This fraction f i n t e r c e p t e d in the range 0~ -< 0 --- 02 is:
I I ~ aT
fo~ aT COS 0 2~ sin 0 d0.
(1)
0t 7~
Since the i n t e g r a n d is s y m m e t r i c a l in 0 a b o u t its m a x i m u m value at 0 = 45 °, it is e v i d e n t t h a t 0, a n d 02 should be selected such t h a t 01 = 45 ° a n d 02 = 45 ° + fl, w h e n e q u a t i o n (1) gives / = sin 2fl.
(2)
Thus, if for practical reasons the spherical shell is restricted to t h e range 01 = 30 ° a n d 02 = 60 ° (i.e., fl = 15°), one-half of all the n e u t r o n s s c a t t e r e d at the s t a r t detector would still be intercepted.
photoelectric efficiency w h e n used in conjunction w i t h existing photomultipliers. F o r this reason the value of fl was reduced to 8.8 ° a n d t h e resulting surface was s u b d i v i d e d into sections which it was felt could be optically coupled in a satisfactory m a n n e r to existing fast photomnltipliers.
1 klght-pipe~ NE-102Scinfllotor
Polyvinylto~uene
1
/
/
.
/,/
2.2. S T A R T D E T E C T O R L E N G T H
As discussed in a p p e n d i x B, u n d e r the contribution of the scatterer length to line width, t h e effect of p a t h length differences from the front a n d b a c k portions of t h e scatterer are largely cancelled, near 0 = 45 °, b y the changes in n e u t r o n velocity caused b y t h e cerresponding changes in scattering angle. Thus, in the present spectrometer a 3" long scatterer which is 10 % of the radius R = 30", produces a line w i d t h of only ¼% (FWHM) in time, discounting the effects of multiple scattering above E . = 2 MeV. Such a scatterer will s c a t t e r 54% of t h e i n c i d e n t n e u t r o n s at E t = 2 MeV a n d 2 1 % at E~ = 12 MeV. The effects of scattering from t h e c a r b o n i n the s t a r t detector is largely eliminated b y the discrimin a t o r level used. I t should be noted, however, t h a t a n y r e m a n e n t effects would not be expected to produce narrow lines since the scatterer would be improperly located for isochronous s c a t t e r i n g from c a r b o n (i.e., S = R/12 for carbon).
3. The Actual Spectrometer I n t h e idealized s p e c t r o m e t e r of section 2 the shell stop detector would be s u r r o u n d e d b y a n optical reflector a n d the scintillation light would be collected, with c o n s t a n t time delay, near t h e central region C of fig. 1. While the fabrication of the shell described in section 2 (fl = 15 °) is e n t i r e l y feasible, all light coUection systems envisioned failed b y one order of m a g n i t u d e in p r o v i d i n g a d e q u a t e total
/ Fig. 2. The spectrometer section a c t u a l l y used. For increased efficiency e i g h t a d d i t i o n a l such sections m a y be used arranged a z i m u t h a l l y w i t h respect to target-scatterer axis.
Fig. 2 shows t h e s p e c t r o m e t e r a c t u a l l y used. The radius of c u r v a t u r e of t h e shell was selected to be R = 30" as a reasonable c o m p r o m i s e b e t w e e n : (1) t h e desire to keep fl large w h i c h w i t h the requirem e n t t h a t t h e " c a p " diameter, AA' in fig. 2, be k e p t from b e c o m i n g too large to provide a d e q u a t e light collection requires R small, a n d (2) t h e desire to keep t h e intrinsic electronic time spread contribution to line shape ({2 a n d ~3 of a p p e n d i x B) acc e p t a b l y small for 2-15-MeV n e u t r o n s w h i c h requires R large. The shell thickness was selected u s i n g the optim i z a t i o n procedure discussed in a p p e n d i x C. W i t h R = 30" a n d selecting t h e line w i d t h c o n t r i b u t i o n of t h e g e o m e t r y (~i) to be 3 to 4 per cent i n t i m e gives (in t h e n o t a t i o n of the appendix) ~ - 1~". T h u s the interior surface of the shell has a radius of c u r v a t u r e R = 30", while the exterior surface has a radius of c u r v a t u r e R + dR = R + 22 = 30{". T h e resulting tapered thickness of t h e shell gives a c o n s t a n t energy resolution for t h e n e u t r o n s scattered at different angles (see a p p e n d i x C). T h e shell,
FAST NEUTRON
ISOCHRONOUS
having a cap diameter of 18", was made of NE-102 scintillator and was cemented to a polyvinyltoluene light pipe which was in turn optically coupled to a 58AVP photomultiplier*. The 18" cap diameter was estimated to be the m a x i m u m size that could be coupled to a single photomultiplier and provide satisfactory light collection and thus keep timing dispersion to acceptable values. Up to eight additional identical sections m a y be added azimuthally to more closely approximate the complete spherical shell. Each section will intercept 2.3 % of the scattered neutrons. For 5-MeV neutrons incident on the scatterer, the shell has an estimated intrinsic detection efficiency of approximately 15 % for the intercepted neutron. The scatterer is an NE-102 scintillator 3" in length and ½" in diameter coupled to a 56AVP photomultiplier. This scatterer diameter with = ~" and the target-to-scatterer distance of d = 15" gives a line width contribution :of approximately 4 % in time (FWHM) (see appendix B).
DYNODE
FLIGHT
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SPECTROMETRY
293
than a 10% intercepted contribution of multiply scattered neutrons at 2-MeV incident energy. For the present scatterer and shell resulting spectrometer efficiency is approximately 1 x 10-3 for 5-MeV incident neutrons. Current pulses from the anode of the photomultipliers are coupled directly to the grid of 6688 limiter tubes normally carrying 22 ma. By the use of delay-line** clipping, 100-ns rectangular pulses are obtained having an amplitude of 2.7 volts after transmission through a cathode follower (5842 tube) and 100 feet of RG63/U cable. Fig. 3 shows the block diagram of the electronics. The limiter signals are fed to the time-to-pulse-height converter which measures their time overlap. The converter uses a 6BN6 tube operated at a screen grid voltage of 10 volts and plate voltage of 20 volts, as suggested by Green and BellS). The output of the time converter is coupled to a multichannel analyzer through a White cathode follower. To improve the ratio of true to accidental counts,
N I TEGRAF L
ANODE
ANODE I I00ns
.--I
iLIAINTO~REI C,EL "STOP" ~1
ANODE
GOU NTERI~5BAVPI--'~DY --" NO F'D -'lE --q
CON I CD I ENCE
hCOINCiDENCE
I00 ns DELAY
=
2Fs
IAMPLINDTsE(~RR'L ~--
Fig.3. Block d i a g r a m of t h e electronics e m p l o y i n g a time-to-pulse-height Multiple scattering, if present, could result in neutrons with incorrect flight times and is most important, of course, for low neutron energies. The present diameter scatterer is estimated to give less * T h i s scintillator a l r e a d y m a c h i n e d a n d p e r m a n e n t l y cotmented to t h e l i g h t pipe w a s o b t a i n e d f r o m N u c l e a r E n t e r -
prises, Inc., Winnipeg, Canada. ** 330-ohm d e l a y line, t y p e Z3M, furnished b y Telcon W o r k s , Greenwich, L o n d o n S.E. 10, England.
converter.
a 6BN6 fast coincidence " p r i o r i t y " circuit is added. By delaying the input " s t o p " signals 100 ns, the priority circuit output signal can be used as an anticoincidence gate for the multichannel analyzer, thereby eliminating events in which the stop signal precedes the start signal. The resolution of the electronics was measured b y a) R. E. Green a n d R. B. Bell, Nucl. I n s t r . 3 (1958) 127.
294
M.L.
R O U S H et
feeding test signals from a fast mercury-switch pulser directly to each limiter. The s p e c t r u m obt a i n e d is shown in fig. 4. A second peak is introduced for calibration, b y inserting a 7.0-ns delay in
al.
Fig. 5 shows the results of a t i m i n g calibration using a Na 22 7_ray source. F r o m the shift in position of t h e peak when a 30.0-ns delay line is inserted, the value of 0.388 ns per c h a n n e l is obtained.
4. Experimental Results Fig. 6 displays a typical time-of-flight spectrum. The Beg(~,n)C 12 reaction yields two n e u t r o n
COUNTS ZOn sec 4,~ C-I¥,~-NNEL5 [
I" 500(
I d
2000
~l~nCHANNEL5
=.
sec
1000
300
¢J E 200 i
-2.8 CHANNELS •=020 nsec
e,~l =,,lkeV d2T~rge! ~,=60 E==20tc,ev Q=45OO~Cotlom~ o,,=o" Shel0~c=055 l MeV,Proton energy n= En'7.71MeV E.-293MeV
(J I00 ~" 140
230 240 CHANNELNUMBER
LSO"
250
CHANNEL NUMBER
~g.--
Fig. 4. The intrinsic line width of the time-to-pulse-height converter using high gain amplification following the converter. The line displaced by the use of a 7.0-ns delay line is also shown.
Fig. 6. A typical neutron spectrum showing the results with the Bed(e, n)C TM reaction. The line labelled y results from Compton scattering of 7rays.
t h e signal from one of the limiters. A portion of the resulting line w i d t h is a t t r i b u t a b l e to the multic h a n n e l analyzer so t h a t the resolution of t h e electronics following the photomultipliers is proba b l y b e t t e r t h a n 0.2 ns. The s t a b i l i t y of the system has been found to correspond to a drift of less t h a n 0.5 ns over a 16-hour period using a stabilized source of a.c. power.
groups leading to t h e g r o u n d s t a t e a n d first excited s t a t e of C 12. The peak observed in c h a n n e l 342 is due to the g a m m a radiation scattered from the first scintillator to the spherical shell a n d serves as a convenient reference marker. Fig. 7 shows the
COUNTS 500
Iq
400300200 IO0
se¢ %30.0 77.3n CHANNELSII
i'SS8 ' n see/ch°nnel ~
o ,_.2
,F
/
z
E~ • l.S MeV •
£s" t.2 MeV
~
F
~
,
120 150 140 v 190 200 210 220 CHANNEL NUMBER
Fig. 5. Time calibration of electronics i n c l u d i n g the p h o t o m u l t i pliers, e m p l o y i n g a delay line shifted secondary line.
2O CHANNEL ~ R Fig. 7. Time spectra showing separation of two n e u t r o n groups differing in energy b y 400 keY.
FAST NEUTRON
ISOCHRONOUS
relative separation of two low energy n e u t r o n groups. Resolution a n d line shape m e a s u r e m e n t s h a v e been m a d e using t h e T(p, n), T(d, n), Be9(a, n), a n d
R-30*
NEUTRON ENERGY(MeV)
FLIGHT
PATH
SPECTROMETRY
295
The shell properly contoured for R = 30", w h e n used at R = 45", (the cap rim set on the R = 45" sphere, with its center at 45 ° to the i n c i d e n t b e a m as m e a s u r e d from the scatterer) will give b e t t e r resolution b o t h because of t h e increased flight p a t h a n d because the resulting sagittal flight p a t h error will reduce t h e time delay i n t r o d u c e d b y the longer light p a t h from the scintillator rim to p h o t o multiplier. For R = 45" this cancellation is exact at Eo ~ 4 MeV; At 2 (see a p p e n d i x B) is a p p r o p r i a t e l y reduced to 1.60 - 0.90/E, ns. The resulting theoretical curve is seen to fit t h e R = 45" d a t a of fig. 9 r e a s o n a b l y well. Fig. 10 shows for En = 8 MeV, the v a r i a t i o n of energy resolution with shell discriminator setting.
Fig. 8. T h e m e a s u r e d line w i d t h s ( F W H M ) in t h e e n e r g y r a n g e 2 -< E n < 10 MeV. T h e c u r v e shown r e p r e s e n t s t h e t h e o r e t i c a l estimate.
D(d, n) reactions as sources of neutrons. T h e actual e x p e r i m e n t a l results are c o m p a r e d in figs. 8 a n d 9 w i t h t h e theoretical prediction (see a p p e n d i x ]3). These m e a s u r e m e n t s were m a d e w i t h the sidec h a n n e l d i s c r i m i n a t o r for the spherical shell set at a n effective p r o t o n recoil energy of ~ 500 keV. Fig. 8 shows a s o m e w h a t faster observed rise t h a n predicted in At at the lower energies possibly ind i c a t i n g the effects of multiple scattering. This effect is also evident for A E / E as m a y be seen in fig. 9. 2G
/~
o R-30" x R-45"
IB
o I4
~'~E"+~]'2
,o
.0.4 Fn=B MeV , R-36"
o
~
,~
;
-02
~l,, ~'5
DISER]MINATOR SETTING(PHOTO-CATI'IOOEELECTRONS}
Fig. 10. T h e v a r i a t i o n of yield a n d r e s o l u t i o n w i t h shell discrimin a t o r setting.
Acknowledgement The authors are grateful to Lloyd F. Chase, Jr., a n d staff m e m b e r s of the Nuclear Physics Division of Lockheed Missile a n d Space D e v e l o p m e n t Labor a t o r y for m a n y s t i m u l a t i n g discussions. We are also grateful to J. B. H e a t h of Nuclear Enterprises, Inc., for m a k i n g possible t h e fabrication of the shell detector.
Appendix A CALCULATION OF THE ISOCHRONOUS SURFACE FOR A GENERAL REACTION
'
;.
~,
~
,E, . . . .
N~utr~ Energy (MeV)
~o
Fig. 9. A c o m p a r i s o n of e x p e r i m e n t a l a n d t h e o r e t i c a l e n e r g y r e s o l u t i o n for b o t h R 30" a n d R 45", in t h e e n e r g y r a n g e 2< E n< 20MeV.
Fig. 11 illustrates a general reaction involving an accelerated particle of mass M 1 striking a s t a t i o n a r y t a r g e t nucleus M 2 resulting in f r a g m e n t s M s a n d M , at a certain reaction Q. The i n c i d e n t particle
296
M.L.
R O U S H e~ al.
P
Q = O, M 1 = M 3, M 2 = M 4 )
equation (A4) simply
reduces to :
,' Mt,Vt,E I
/
"
O~-C M4" 0C-$ CP-
R
Fig. 11. Reaction kinematics illustrating t h e isochronous surface t r a c e d b y point P.
laboratory velocity and energy are v t and E l , the laboratory and center-of-mass velocity of fragment M3 is v3 and v3c respectively, while v.... is the velocity of the center-of-mass. A trivial application of the kinematic conservation laws gives: z Vc.m.
It is worth remarking that any arbitrarily shaped area detection system for fragment M 3 if moved along the isochronous spherical surface (i.e., rotated about C at radius R) while subtending different laboratory solid angles at the target position O depending on 0L, has a constant c.m. solid angle. In addition to the application of the above isochronous principle to the determination of the angular distribution of various neutron groups originating in some reaction under investigation, the subject of the main body of the present paper, two additional embodiments of the principle are worth citing. Fig. 12 illustrates applications suitable in angular correlation and neutron polarization experiments.
(AI)
2MtEI/(M 1 + M2) z ,
ISOCHRONOUS / SL~FACE
v2¢ = 229I4 [M2E t + (M1 +
M2)Q]/M3(Mt +
As is already clear from elementary considerations, equations (AI) (A2) show t h a t v.... and v3© are independent of the emission angle 0c in the c.m. system. It then follows that the fragment M 3 originating its flight at the target position 0 will arrive at P, a point on a spherical surface with its center at C, in a fixed time independent of 0c. If the distances OC and CP are designated as S and R respectively, it is only required that R/S = v3Jv .....
(A3)
+ (M 1 + M2)Q/EI]/MtM 3
(A4)
or
[M 2
for the spherical surface of radius R to become an isochronous surface for fragment M 3. To keep the isochronous time single valued requires the usual restriction to reaction energetics involving E t and Q, that v3c > v¢.=.. The flight time of fragment M 3 from 0 to P is clearly: t =S/v .... = S(M 1 + M2)/Mtv 1 .
"STO~'
M2) 2 . (A2)
R2/S 2 = M,
(A6)
R/S = M2]M t .
(A5)
For the special case of elastic scattering (i.e.,
/
~
I
, O-TARGET POSITION \
/
~
-
//
~
/ //~
/
~
~"
X "START'SONTILLATOR\ \ ~
\
~
BeI(d,n)~°" (3.5BMeV)~ Ed=2.50 MeV J (a)
/SCINTILLATOR
I
~"I
. . . . . R=7.88 $
/ /~x"SURFACE , LEFT"STO~"
\
\ nv/4 ~ d
~/8 • ~
-
", \
Id2, n
-
\/ /\
HELIUM=S.AR=~/
/
'\ /
/
RIGHT"ST(~"
~qWa~.~d ~ -
SONTILLATOR "
~TARGET pOSITION R-4S
(b)
Fig. 12. Two suggested e m b o d i m e n t s of the isochronous principle. I n (a) the application is to an a n g u l a r correlation study, d r a w n to scale for the Beg(d, n)B l°* (3.58 MeV) reaction. I n (b) the application is to a n e u t r o n polarization s t u d y using a helium scatterer.
For the sake of definiteness let the situation in fig. 12a refer to the reaction Be9(d,n)B t°* at E o = 2.50 MeV. Reference is then made to a possible angular correlation study to measure the E2-M1 mixture in the 2+(3.58-MeV) level in B t°* decay branch to the 1 + (2.15-MeV) level. Here, as in m a n y other cascade rich reactions, a positive identification (even if possible) of the y-ray energy (1.43 MeV) is insufficient to determine uniquely the neutron group populating the initiating level. Thus,
FAST
NEUTRON
ISOCHRONOUS
a neutron energy determination is necessary. In the arrangement shown the neutron group is to be identified by a time-of-flight measurement using a fast " s t a r t " scintillator to detect the ? ray (using only a very broad range pulse height selection) with the " s t o p " detector being a thin shell scintillator formed to correspond to the appropriate isochronous spherical surface. Here the center of the sphere is displaced a distance S from the Be target such that R = 7.88S. I t should be noted that each neutron group (i.e., Q) requires a different value of R / S , thus for each geometry only one group is truly isochronous over the entire surface P. This fact is particularly important when the stop detector subtends a relatively large solid angle. In particular, it is seen that the usual practice of using a thin flat plate stop detector oriented perpendicular to a radius vector from the target will introduce a larger timing spread than warranted by the plate thickness relative to the flight path and the electronics. If a flat plate is to be used, it should at least be oriented tangent to the appropriate isochronous sphere. If in obtaining the correlation function the neutron stop detector is rotated on a goniometer arm centered at C, the value of 0c can be directly read from the goniometer protractor and the c.m. solid angle subtended by the neutron detector will be held a constant. Further, since the flight time is then constant, as the detector is rotated, a relatively narrow time display window can be used reducing the dead time losses. The arrangement illustrated in fig. 12b is intended to measure the polarization of various neutron groups, for example in the Be9(d, n)B 1°* reaction, making use of the elastic scattering from a high pressure helium scintillator cell. Here, the isochronous principle is applied to the secondary reaction [i.e., He*(n, n)He 4] to identify the neutron groups using a time-of-flight measurement. Only a very broad range pulse height selection is suggested for the helium cell start detector. In this case R = 4S, which in view of the relatively large range in 0 permissible to the left and right stop detectors by nature of the polarizing properties of the He*(n, n)He* reaction at backward angles, makes the use of properly contoured stop detectors essen-
FLIGHT
PATH
SPECTROMETRY
297
tial. (The azimuthal angle extension of the stop detectors (not shown) must be determined with the ensuing loss of polarization in mind.). Appendix
B
CALCULATION
OF SPECTROMETER
RESOLUTION
The energy resolution of the spectrometer is the result of the combined effect of a number of geometrical and statistical factors. A number of geometrical factors are directly related and are added as coherent sources of line width. The spectrometer resolution is conveniently expressed as ~T, the ratio of line width (FWHM) in time units to the time of flight of the neutron between the two scintillators, with ¢r2 = ¢~ + ~2 + ~ + ~], (A7) where ~t is the spread due to shell thickness, scatterer radius, and beam spot size, ~2 is the spread due to the combined effect of phosphor decay time, light collection efficiency, and light transit time spread, ~3 is the spread introduced by the photomultipliers and electronics, ~, is the spread introduced by the length of the scatterer. The dependence of flight time upon neutron energy is given by t ----72.3d/~JE n
(A8)
where t is in nanoseconds, if d is any specified flight path in meters and E n is in MeV. The isochronous flight time t in appendix A, eq. (A5) reduces for n-p scattering to t = 2Sly 1 = 2R[v t since S = R. For the present spectrometer D = 2R = 60"; thus, a given uncertainty in time At (in nanoseconds) results in a fractional line width of '-= A t / t = At~/En/IIO.2.
(A9)
B.1. Calculation of¢1 (due to beam spot size, scatterer radius, and shell thickness)
An approximate calculation will be effected by assuming the length of the scatterer 2a ,~ d, a n d taking 0 = 45 ° as representative of the extended spherical shell. Thus, referring to fig. 13 let 0 = 45 °, tp =
45 ° -
a - - e, a n d ~ k
=
45 ° +
a +
~,where
298
M. L. R O U S H
~ (s + r)/d
and
e m (r + 2)lx/2L.
(A10)
Flight p a t h L t represents the extreme case when the p a t h is shortest and the neutron velocity is greatest. P a t h L 2 represents the opposite extreme. The flight time for the central p a t h is
t = L/v I cos 0 = a/2L/v x ,
(All)
where v 1 is the incident neutron velocity. For flight p a t h L tt _
LI _ L - ((r + 2)Ix/~ v 1 cos~o v 1 cos (45 ° - ~ -- ~)
(a12)
and similarly for p a t h L 2
tz
L2 L + ((r + 2)/x/2) v I c o s ~ - v t c o s ( 4 5 ° + ~ + ~)
.
(AI3)
Since the angles ~ a n d , are small cos (45 ° + ~ + ~) m (1 T ~-T- ~)/~/2-. (A14) From eqs. (All), (A12), (A13), and using eq. (A14), we obtain after some simplification (t 2 - t,)/t = 2(~ + e) + 2(r + 2)/~/2 L .(A15) Using the expressions in (A10) and noting t h a t L = ~/2R : (t 2 - t~)/t = 2(s + r)/d + 2(r + 2)/R.
(A16)
Expression (A16) gives the fractional full base width, the fractional w i d t h at half-maximum will be approximately one-half of this value:
et al.
¢1 = (s + r)/d + (r + 2 ) / R .
(A17)
For our particular spectrometer we have chosen s = ~ i n c h , r = ¼inch, d = 15inches, 2 = ~ i n c h , R = 30 inches, which results in ~x = 0.0396.
(A18)
A n o t h e r sort of angular error is i n t r o d u c e d b y multiple scattering in the scatterer. While this effect is difficult to calculate and leads to the modification of the line shape mainly b y the addition of a long delay tail, an order of magnitude estimate gives ~[ ~< 0.01 at E . = 2 MeV for the present case, the value at other energies varying inversely with E n. It has been assumed t h a t in the energy range E . > 2 MeV the effects of multiple scattering m a y be neglected.
B.2. Estimation of ~2 (due to the combined effect of phosphor decay t~me, light collection efficiency, and light transit t,me spread) The NE-102 plastic scintillator has a decay time of m 3 ns. This time constant combined with the light collection efficiency determines the time delay for the emission of a sufficient n u m b e r of electrons from the photomultiplier photocathode to subsequently cut off the limiter. A further correlated delay occurs due to the light collection transit time from the corners of the shell where the light collection efficiency is also poorest (in NE-102 an optical p a t h length of 7.5" corresponds to a 1-ns time delay). While the combined effect of these factors on the line shape is relatively complicated, requiring a resort to numerical methods, a greatly simplified approach leads to an a p p r o x i m a t e expression (limiting at 4 electrons, photo cathode equivalent, and integral discrinfination at 8 electrons) of At2 = 1 80 - 0.90/E, ns, which in t u r n gives: ~z = [1.63 ~/En - 0.82/~/E.~ x 10 -2
(A19)
B.3. Estimation of ~3 (due to transzt time spread in the photomultiplier and fluctuatzons in the total electronic system) Fig. 13. The illustration of the geometrical factors influencing the line width contribution ~1 (appendix ]3).
This effect i. due almost completely to the photo multlpFers since the foalowing electronics has a measured i~trmsic resolution better than 0 2 n .
FAST NEUTRON
ISOCHRONOUS
FLIGHT
PATH
SPECTROMETRY
T h e m a n u f a c t u r e r ' s specification of t r a n s i t t i m e s p r e a d for t h e 56AVP a n d 58AVP p h o t o m u l t i p l i e r s are 0.5 a n d 1.0 ns, r e s p e c t i v e l y . T h e factors c o m bine to give AQ = 1.14 ns, f r o m w h i c h we o b t a i n 43 = 0.0104 x/E~ .
299
~.~ ~*
p
_~2_
ISOGHRONOUS SPHERICAL
S.EL,
,-'///
-\
,Y2/ I
(A20)
\
B.4. Calculation of 44 (due to the scatterer length) R e f e r r i n g to fig. 14a, t h e f r a c t i o n a l s p r e a d i n flight t i m e for s c a t t e r e d n e u t r o n s to arrive at p o i n t P f r o m a p o i n t x along t h e s c a t t e r e r relative to t h a t f r o m t h e c e n t e r of t h e s c a t t e r e r (of l e n g t h 2a) is calculated. Clearly, l~ = l 2 + x 2 + 2xl cos 0 a n d 1t cos 0 = l c o s 0 + x a n d t h e flight t i m e s for t h e i n d i c a t e d p a t h s are
W~ o
W--~o (a) "t'2q
~'+~
t t1
= l/vt cos 0 = D/v t , =
(A21) -I'll
ll/v 1 cos ~b = 12x/vxll cos ~ .
Writing the fractional (t! - t)/t, t h e r e results
time
delay
as
At/t = x(lcos20 + x cosO)/DcosO(lcosO + x).
At/t =
(A22)
=0 ,
:3
~-
N o t i n g t h a t 0 = 45 ° + r, a n d t h a t in p r a c t i c a l cases 1 cos 0 >> x a n d f u r t h e r t h a t fl < 15 ~, (A22) r e d u c e s to At/t ~ 2x(x - D sin 2fl)/D 2 .
(A23)
Fig. 14b s h o w s At/t as a f u n c t i o n of X / D a n d ft. T h a t for s m a l l r, At/t is of o r d e r X 2 / D 2 results f r o m t h e fact t h a t while t h e flight p a t h l~ > l, t h e corr e s p o n d i n g n e u t r o n v e l o c i t y is also greater, since ¢ < 0, giving a c a n c e l l a t i o n of first o r d e r t e r m s in X / D . (For x n e g a t i v e l 1 < l a n d ~ > 0, a g a i n r e s u l t i n g in a c a n c e l l a t i o n to first order.) F o r geom e t r i e s n o t t o o d i s s i m i l a r f r o m t h a t u s e d here a n a p p r o x i m a t e value for 4~, c a n be g i v e n as 4', ,~ (a/D) sin 2tim,
(A24)
w h e r e t h e s p h e r i c a l s h e l l e x t e n d s from 0 = 45 ° + fl to 0 = 45 ° - tim, a n d fl~ > a/2D. A p p l y i n g equation (A24) to t h e p r e s e n t case, w i t h 2a = 3" gives 4~ -~ 0.0075.
(A25)
A more elaborate numerical calculation shows that (A25) is g o o d to ___ 10% accuracy.
',e--#o -20 (b) Fig. 14. T h e illustration in (a) of the geometrical factors influencing the line w i d t h c o n t r i b u t i o n ~2 (appendix ]3). I n (b) t h e fractional t r a n s i t t i m e delay a p p r o p r i a t e to (a) is shown.
T h e l e n g t h of t h e s c a t t e r e r also influences t h e r e s o l u t i o n t h r o u g h t h e t i m e s p r e a d due to o p t i c a l p a t h l e n g t h differences to t h e p h o t o m u l t i p l i e r . I n t h e p r e s e n t case we o b t a i n At = 0.19 ns. T h u s , t h e final t i m e d i s p e r s i o n due to t h e s c a t t e r e r l e n g t h can be e x p r e s s e d as 44 = (0.56 + 0.029E,) ~
x
10 - 2
(A26}
C o m b i n i n g ix, 42, 43, a n d 44 as i n d i c a t e d b y (A7} yields 4T = atT/t = (13.6 + 3.8E, + 0.7/E). ~ x 10 -2 (A27)
300
M.L.
ROUSH
This can be c o n v e r t e d to give t h e corresponding w i d t h s i n time a n d energy
AtT = (4.6 + 16.5/E n + 0.9/E2.)~:ns
(A28)
and
AE/E = (55 + 15.0E n + 3/E,) ~:percent.
(A29)
et al.
desired n e u t r o n energy. I t remains t h e n to select s, r, 2, a n d d so as to optimize the efficiency for a given ~1 a n d R. I n those applications where s can be m a d e negligibly small, it will be useful to introduce an efficiency figure of m e r i t F = 2r2/d 2. Solving equation (A17) (with s = 0) for r a n d s u b s t i t u t i n g it into t h e expression for F gives
Appendix C SHELL
THICKNESS
CONSIDERATIONS
F o r the fractional c o n t r i b u t i o n of t h e shell thickness to t h e line w i d t h to be i n d e p e n d e n t of 8 requires t h a t the scintillator m a t e r i a l composing the shell be confined between two isochronous surfaces h a v i n g radii R a n d R + AR, selected to give t h e desired value of AR/R. The thickness ~ s h o w n in fig. 13 is just ½AR. The two isochronous surfaces are of course t a n g e n t at t h e scatterer a n d h a v e t h e i r centers displaced b y a distance AR. Given a certain electronic capability, the selection of R will d e t e r m i n e ~2 a n d ~3- The usual compromise between resolution a n d efficiency would lead to a s p e c t r o m e t e r design giving ~ 1 (the largest geometrical source of line width) a p p r o x i m a t e l y equal to t h e c o m b i n e d effect of 42 a n d 43 at t h e
F = .~ \ ~ - ]
(A29)
I t follows trivially from t h e o p t i m i z a t i o n of F t h a t )" = J~R¢I
(A30)
d