- Email: [email protected]
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Nuclear Physics 74 (1965) 261--280; (~) North-Holland Publishing Co., Amsterdam
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be reproduced by photoprint or microfilm without written permission from the publisher
THE DEUTERON-DEUTERON INTERACTION AT 270 TO 507 MeV/c J. E. A. LYS t and L. L Y O N S
Nuclear Physics Laboratory, Oxford Received 19 March 1965 A bubble chamber experiment has been performed to study the deuteron-deuteron interaction in the m o m e n t u m range 270 to 507 MeV/c. Results are presented for elastic scattering, the stripping reactions and the inelastic reactions. The elastic scattering differential cross-section for the incident m o m e n t u m range 433 to 507 MeV/c is compared with impulse approximation calculations.
Abstract:
E
I I
N U C L E A R R E A C T I O N S ~H(d, d), (d, d'), (d, n), (d, p), E = 20-67 MeV; measured ~(E), ~(E; 0) for (d, d).
1. Introduction Over the past two decades, the nature of the forces in the two-nucleon system has been the subject of intensive study. We are at last beginning to obtain some degree of understanding of these forces, and we can expect that in the near future a considerable amount of effort will be spent in seeing how well the properties of larger nuclear systems can be explained in terms of the basic two-nucleon interaction. In particular, it will not be possible to obtain information on the existence of specifically many-body forces or on the nature of off-the-energy-shell effects if attention is confined to systems containing only two nucleons. It would seem that, as part of this programme of studying nuclear systems, the few-nucleon system might be the simplest with which to start. With this end in view, the Oxford deuterium bubble chamber group has been studying proton-deuteron and deuteron-deuteron interactions. The results of the former experiment 2) and a preliminary account of the deuteron-deuteron experiment 2) have already been published. This article gives a more complete account of the results of the second experiment. In the m o m e n t u m range below the meson-production threshold, the possible reactions involving only strongly interacting particles are: D+D ~ D+D
elastic scattering,
D + D --, P + T D + D ~ 3He + N
stripping reactions,
D + D --* D + P + N D + D ~ P+ N + P+ N
inelastic scattering.
t N o w at Nuclear Physics Laboratory, University o f Liverpool, England. 261
(1) (2)
(3) (4) (5)
262
J. E. A. LYS A N D L. L Y O N S
These reactions have been studied at a large number of momenta below 230 MeV/c. Above 230 MeV/c, the inelastic reactions 3) have been investigated at 265 MeV/c, elastic scattering and the stripping reactions 4) at 270 MeV/c and all 5) the above reactions at 310 MeV/c. The tritium production differential cross-section has been measured 6) at 865 MeV/c, and the helium production at a few angles at 865 MeV/c [ref. 7)] at 1.3 GeV/c and at 1.7 GeV/c [refs. s, 9)]. Thus there is a large m o m e n t u m range through which very little data exist. This experiment was intended to provide a survey of reactions (1)-(5) in this region, the bubble chamber being a good tool for such a study. The Oxford bubble chamber was filled with deuterium and exposed to a 540 MeV/c deuteron beam. Experimental factors limited the range of incident momenta over which interactions were measured to 270 to 507 MeV/c. These values of m o m e n t u m correspond to kinetic energies of 20 to 67 MeV for the incident deuterons. Some theoretical studies of deuteron-deuteron elastic scattering have recently been made lo-12). These have all been based on the impulse approximation 13) (henceforth abbreviated at IA) in which the basic interaction is assumed to be between one nucleon from each of the colliding deuterons. At least in first order IA, the other two nucleons are assumed not to take part in the collision, and will thus have the same velocities after the interaction as they possessed within the initial state deuterons before the collision. Then the cross-section for elastic scattering is simply related to the probability that the momenta of the four final state nucleons are consistent with those of the nucleons within two deuterons. In this way the elastic scattering is related in a fairly direct way to the two-nucleon interaction. In sect. 2 we describe the details of the experimental procedure of obtaining the photographs, and the method of analysing the events on them. Sect. 3 gives an account of the way in which the events were divided between the different possible reactions, and the results of this analysis are presented in sect. 4. Finally in sect. 5, the elastic scattering differential cross-section that we obtain for incident deuterons in the m o m e n t u m range 433 to 507 MeV/c is compared with some impulse approximation calculations.
2. Experimental Procedure 2.1. THE BEAM The deuteron beam was produced by placing a thick graphite target in the internal beam of the Harwell 2.8 m synchrocyclotron. The circulating protons were degraded in passing through the target to too low a m o m e n t u m to escape through the beam exit channel. In a proton-carbon collision, however, a pick-up reaction can occur with the deuteron so formed having a higher m o m e n t u m than that of the incident proton. Such deuterons would then be degraded when travelling through the target, and could emerge from the trailing edge of the target with the correct m o m e n t u m (about 550 MeV/c) to be accepted by the beam channel.
D-D
263
INTERACTION
A m o r e c o m p l e t e description o f the p r o d u c t i o n o f the d e u t e r o n b e a m a n d o f its p r o p e r t i e s m a y be f o u n d in ref. t4). 2.2. BEAM TRANSPORT The b e a m was t r a n s p o r t e d to the b u b b l e c h a m b e r b y the system shown in fig. 1. A t the entrance to the b u b b l e c h a m b e r , the b e a m was focussed to an image 2 c m high a n d 4 m m wide. The b e n d i n g magnets a n d c o l l i m a t o r s were used to achieve m o m e n tuna selection on the d e u t e r o n b e a m .
- - ---C 2
/~/C
APPFa,OX SCALE (m }
BC
Fig. 1. Beam transport system. H.C. is the Harwell cyclotron, with graphite target T. The deuteron beam is transported by two bending magnets BM1 and BM2 through 3 collimators C1, C2 and C 3. The quadrupole pair Q1 and Q2 focusses the beam at the entrance to the bubble chamber B.C; W is a concrete shielding wall.
The m o m e n t u m s p r e a d in the b e a m was m e a s u r e d b y taking a r a n g e - s p e c t r u m o f the b e a m . F o r this p u r p o s e a triple coincidence scintillation c o u n t e r telescope with a l u m i n i u m a b s o r b e r s was used when the b e a m was being set up, a n d the b u b b l e c h a m b e r itself was used during the actual experiment. I n this way, the b e a m was d e t e r m i n e d to have a m o m e n t u m o f 540 M e V / c a n d an r.m.s, s p r e a d o f 2.5 M e V / c at the entrance to the b u b b l e chamber. T h e n u m b e r o f p r o t o n s in the b e a m was m e a s u r e d as being 4 per 1000 deuterons. I t is to be n o t e d t h a t the range o f these p r o t o n s was seven times as great as those o f
264
S. E. A. LYS AND L. LYONS
the deuterons; this facilitated the estimation of the proton background from the range curve. The intensity of the beam at the chamber was 4 deuterons per pulse. 2.3. THE BUBBLE CHAMBER The bubble chamber assembly was very similar to that used for the proton-deuteron experiment and previously described in ref. 15). The major modification was that a thin beam entry window was required to allow the deuterons to enter the chamber without being degraded unnecessarily. To do this, it was necessary to rebuild the body of the chamber itself. This was made out of a single piece of a-brass. Onto the body, the beam window was connected by a flanged joint, sealed by a double indium O-ring. The window was 2 cm diam. stainless steel, 0.30 g/cm 2 thick. The liquid volume within the chamber was cylindrical in shape, being 12 cm in diameter and 10 cm deep. The deuterium used in this experiment had a protium content of 1.1 ~ and was cleaned by liquefaction and by passing it through a charcoal trap cooled in liquid nitrogen. The density of this deuterium at operating conditions was determined by measuring the apparent depth of the chamber when empty and when full of liquid. Then the density p was obtained from the refractive index n by means of the relationship 16) n2 - 1 p oc n 2 + 2 , the constant of proportionality being determined from the known 1 7 ) values of p and n for liquid deuterium at 20 ° K. After correcting for the protium contamination in the liquid, a value of 134.2 4- 1.3 g/1 was obtained for use in the calculation of the cross-sections of the various reactions. The bubble chamber had no magnetic field, the energy of tracks being obtained from their observed range in the deuterium. Thus if a track interacted in flight to give a zero-prong event (i.e. both charged prongs were too slow to be visible), the track would be ascribed too low an energy, and the event may have been misclassified; all cross-sections were subsequently corrected to allow fol this. Note that the energy of a track will also depend on the nature of the particle producing the track. For instance for a given range, the energy of a track assumed to be a deuteron would be 1.37 times larger than if the track were that of a proton. 2.4. THE EXPERIMENTAL EXPOSURE The chamber was operated at a temperature of 31.7 ° K with a volume expansion ratio of 1.4 ~ . The cyclotron's cycling rate was 100/sec. The chamber was expanded once every 1.7 sec, the timing sequence of the expansion of the chamber and the illuminating flash being such that only one beam pulse was photographed per expansion.
D-D INTERACTION
265
A total of 8 000 stereo pairs of photographs was subsequently scanned for events. These photographs were taken at two beam momenta, viz. the full beam momentum, and a degraded beam of about 460 MeV/c. These momenta were chosen to facilitate the kinematic analysis of the events. The deuterons slowed down in crossing the chamber with the result that interactions over the incident momentum range 433-507 MeV/c were obtained with the full momentum beam, and over the range 270-442 MeV/c with the degraded beam. The lower limit is here set by the increased momentum spread of a degraded beam, and by the short range of low-energy deuterons; the degraded deuteron beam in fact stopped in the chamber liquid. A total of 3 000 interactions was obtained in the higher momentum interval, and 2200 in the lower. 2.5. D A T A A N A L Y S I S
The pictures were first examined on a scanning table, where they were projected to four times life size. All beam tracks undergoing interactions within a predefined acceptance region in the bubble chamber were recorded; these interactions appear in the bubble chamber as two-, one- or zero-prong events. The number of beam tracks was counted on every fifth frame so that absolute cross-sections could be determined. These events were measured on a digitised measuring machine which had a least count of 1 pro. In fact the setting accuracy on a point on the film was typically of the order of 10 pm, which corresponds to about 50 gm in the bubble chamber space. As the tracks were straight, it required only two points to be measured at their ends to define them. Two fiducial marks were also measured on each view, thereby enabling the event to be located in the bubble chamber co-ordinate system. At the end of an event, the first fiducial was remeasured, thus providing a check against the malfunctioning of the measuring machine's digitisation system. Thus for a one- or twoprong event, six or seven points respectively were measured on each view. These events were measured at the rate of 8 per hour. The output from the measuring machines was in the form of punched paper tape. This was used as input to a single computer programme specially written for this experiment, and run on the Oxford Mercury computer. The programme made checks on the input data, reconstructed the events in space, calculated various kinematic correlation functions and their errors, and made histograms of some of the quantities of interest. The events were processed at the rate of 6 per rain.
3. Event Assignment Each event was checked against the various possible reactions for consistency (within the calculated errors) with energy and momentum conservation. 3.1. E L A S T I C S C A T T E R I N G
In general an elastic scatter produced a two-prong event. For small angle scatters, however, the recoil deuteron was usually too short to leave a track in which case only a single prong was seen.
266
J. E. A. LYS A N D L . L Y O N S
3.1.1. Two-prong events M a n y o f the t w o - p r o n g events h a d the longer p r o n g leaving the i l l u m i n a t e d region o f the c h a m b e r while the shorter p r o n g s t o p p e d , in which case the four conservation equations were used to calculate the u n k n o w n energy o f the n o n - s t o p p i n g t r a c k t a n d also to p r o v i d e three i n d e p e n d e n t checks on whether the d a t a were consistent with elastic scattering. There also was the r e q u i r e m e n t that the o b s e r v e d t r a c k length o f the n o n - s t o p p i n g p r o n g s h o u l d n o t be longer t h a n the range required for the a b o v e
IN
~ Fj S
(a)
l
k.
r
t
'
,
i -4.0
~
0
4.0
2.0
f 2 4.0
g~ F,
4
:-I-
[ _1
LLL
4k
'--L~
-4:cq~-- ~ 0
6
4.0
---2T5 r~3 q~
,/G 4
so
Fig. 2. Histograms of deviation functions for the elastic scattering hypothesis. (a) The angularcorrelation deviation function. (b) The first angle-energy deviation function. (c) The coplanarity deviation function. (d) The square root of the selection function Gd.
calculated energy. The f o r m o f the three checks was such as to minimize the weight given to the least accurately d e t e r m i n e d parts o f the data, viz. the direction o f the s h o r t e r prong. F o r those events where b o t h p r o n g s s t o p p e d within the c h a m b e r a f o u r t h check was possible. Details on the exact form o f the correlation functions used to check the d a t a are given in a p p e n d i x 1. T h e expected error in each c o r r e l a t i o n function was also calculated. This error included c o n t r i b u t i o n s f r o m the m e a s u r e m e n t errors, f r o m multiple scattering o f the tracks by the d e u t e r i u m in the b u b b l e c h a m b e r , a n d f r o m the u n c e r t a i n t y in the incident b e a m m o m e n t u m . O t h e r sources o f error were shown to be u n i m p o r t a n t . A p p e n d i x 2 gives m o r e details o f the w a y in which the various errors were calculated. H i s t o g r a m s o f the correlation functions divided by the c o r r e s p o n d i n g error (henceforth this ratio is referred to as the deviation function) are shown for the elastic scatt It is to be noted that in all cases the energy of the incident deuteron at the interaction was calculated from the known beam energy at the entrance to the bubble chamber and from the measured path length in deuterium up to the interaction vertex.
D-D INTERACTION
267
tering assignment in fig. 2; there are clear peaks centred on zero for the elastic events, with a background caused by other processes. The method of selecting elastic events was to define a function Ge~ as the sum of the squares of the three deviation functions. Events with only one stopping prong and with Gel less than 20 were taken as elastic. Those with Gel in the range 20-100 were re-examined on the scanning table. In some cases simple tests could be performed which made the event assignment unambiguous. Otherwise the events were remeasured, and those which had the new value of Ge~ less than 20 were classified as elastic. For those events where four checks were possible, the extra condition that the square of the fourth deviation function be iess than four was imposed. The number of events with Gc~ less than 20 which failed to satisfy this fourth condition was small, and was consistent with the number expected from a calculation of the probability of the longer prong undergoing a zero-prong interaction (and hence being assigned too low an energy). This confirms that for those events where one of the prongs does not stop, the loss of information about the energy of that prong does not introduce any bias into the assignment of events. For 2 ~o of the events, the energy of neither prong was known, in which case only two checks could be performed, and the above criteria were suitably modified. Those two-prong events satisfying the elastic criteria but with scattering angle * less than 12½° were re-analysed as one-prong events (see sect. 3.1.2 below) because the measurement error on the shorter prong was so large that the two-prong analysis was unreliable. Similarly, events consistent with the P-T reaction but with proton scattering angles less than 20 ° at the higher incident momentum or 30 ° at the lower incident momentum were also classified as one-prong events for further analysis. With these limits threre were no two-prong events which were consistent with both the P - T reaction and elastic scattering. 3.1.2. One-pron9 ecents For the higher incident momenta, the small angle elastically scattered deuterons did not stop in the chamber but at the lower momenta they generally did. In the latter case, only one check was possible (the second angle-energy correlation described in appendix 1) since three of the four conservation equations were required to calculate the momentum of the unseen recoil. Events were taken as elastic when the square of the deviation function was less than 4. This criterion was obtained from a study of two-prong elastic scatters. There was also the condition that the calculated energy of the recoil deuteron should be small enough ( ~ 3 MeV) for it not to leave a visible track in the chamber. When even the single prong had unknown energy, the angle-energy correlation was replaced by an inequality that the observed track length should be at least as great as that required from the calculated energy of the prong. In general, this conAll angles referred to in the text are in the laboratory system, unless specific mention is made to the contrary.
268
J. E. A. L Y S A N D
L. L Y O N S
dition was not sufficient to separate elastic scatters from the other reactions in an unambiguous manner *, and it was necessary to remove the contamination of nonelastic events statistically as follows. The number of P - T events with forward protons was estimated by demanding symmetry about 90 ° of the P - T production cross-section in the centre-of-mass ** Finally an analysis of the confirmed inelastic events (see subsects 3.4 below) enabled an extrapolation to be performed into the kinematic region where they would be confused with elastic scatters. This procedure led to 13 ~o of the high incident momentum one-prong events at less than 121° and none of the low incident momentum events being taken as inelastic. 3.2. T H E P - T R E A C T I O N
For the two-prong events, the analysis was similar to that described above for the elastic events. The form of the correlations are again given in appendix 1. For this reaction, however, it was necessary to calculate the selection function GpT twice for each event, since in general it was not obvious which prong was produced by which of the non-identical final state particles. Exactly the same criteria were applied to the lower value of GpT as in the elastic case. The kinematics of the production reaction are such that the events were seen as twoprongs, except for forward scattered protons, when the triton recoil was unseen if its energy was less than about 5 MeV. Then, however, the proton energy was such that it did not stop within the chamber, and hence all the one-prong P-T events were of the sort where no exact kinematic check was possible. The number of these which were P - T events was estimated from the requirement that the differential cross-section for the reaction D + D ~ P + T must be symmetric about 90 ° in the centre-of-mass. The contamination in the other channels from this source was thus estimated. 3.3. T H E a H e - N R E A C T I O N
As the 3He particles are doubly charged, this reaction resulted in short one-prong events. Besides the requirement that there be no visible recoil, only one check could be performed (the second angle-energy correlation mentioned in appendix 1). A histogram of the relevant deviation function for possible helium production events with helium centre-of-mass angles less than 90 ° showed a considerable background from other processes. When the helium production angle was larger than 90 ° (c.m.), its energy was usually so low that the track was too short to be visible and the event appeared as a zero-prong interaction. Again the requirement of symmetry in the centre-of-mass system was used to obtain the total number of 3He-N events. * T h e ~ H e - N final state could, however, be separated u n a m b i g u o u s l y since the doubly charged aHe always h a d a range m u c h shorter t h a n those o f the elastically scattered f o r w a r d deuterons. ~* Events with f o r w a r d tritons always p r o d u c e d a visible recoil p r o t o n a n d so these events could be selected u n a m b i g u o u s l y f r o m the t w o - p r o n g events (see subsect. 3.2 below).
D-D INTERACTION
269
3.4. INELASTIC REACTIONS Two-prong events not classifed as elastic scatters or as tritium production were assume to be break-up events. In principle a separation between the two inelastic reactions D + D -~ P + N + D , (4) D + D ---, P + N + P + N ,
(5)
is possible if both prongs stopped within the chamber. This is based on just one check (since there are three unknown kinematic variables associated with the neutron in reaction (4)) and there is also a two-fold ambiguity because the final state particles are not identical. A preliminary analysis showed that the separation was difficult, and because of the large class of events in which the separation was not even in principle possible, no further attempt was made to divide the inelastic events between the two reactions. It is possible that a two-prong inelastic event could accidentally have a track configuration resembling that of an elastic scatter, and hence have been misclassified. To investigate the magnitude of this effect, events with the longer prong scattered through 13 ° were examined. (The problem of identification becomes much less serious at larger scattering angles.) Those inelastic events with the length of the shorter prong approximately equal to that required for an elastic scatter were found to have a distribution of the shorter prong's laboratory angle and azimuth with respect to the longer prong which was consistent with isotropy. Thus the probability that the short prong would lie in the required direction (within typical errors) for an elastic scatter was calculated. In this way it was found that perhaps half an event had been misclassified, and thus this source of contamination was negligible. When one of the charged particles in reactions (4) or (5) had low energy, the event would appear to be single-pronged. Then the event was assumed to be inelastic if it was inconsistent with reactions (1)-(3). At the lower incident momenta, this proved to be a satisfactory separation technique; an extrapolation of a histogram of prong length of inelastic events indicated that no events were expected to have a prong length as long as that required for reactions (1) or (2), and the cross-section for reaction (3) was too small to provide any appreciable effect. On the other hand, at the higher incident momenta, the track of the one-prong inelastic events frequently left the chamber, and so essentially all the small angle elastic and tritium production reactions (which also lead to small angle single prongs leaving the chamber) could have been inelastic events. A statistical separation was then made as follows. Histograms were made of the energy of the shorter prong for those twoprong inelastic events with angle of scatter of the longer prong in a number of angular ranges below 12½° . These were extrapolated to the limit of the short track being of too low an energy to be visible. This led to a total of 13 ~ of the one-prong events below 12½° being taken as inelastic. The extrapolation procedure was checked to give the correct numbers at lower incident momenta where the assignment of events was simpler.
270
J.
E.
A.
LYS
AND
L.
LYONS
4. Corrections and Results Distance averaged cross-sections were calculated in six separate incident momentum ranges from the formula
NT/No) NTpL
Ni loge(1 O"i
=
--
where Ni is the number of events of the particular process whose cross-section is ai, ArT is the total number of events, No is the number of incident deuterons, p is the density of deuterium, and L is the length of the region in which events were accepted. It was necessary to apply corrections to some of the above quantities. 4.1. BEAM TRACKS The number of incident deuterons was estimated by counting the beam tracks on every fifth frame. This number was then corrected for protons in the beam, for tracks entering the chamber but not meeting the beam track requirements, and for low momentum deuterons in the beam. The beam contained a proton contamination of 0.4 %, and the number of beam tracks was reduced to allow for this. The interactions produced by these protons would have been incorrectly classified as inelastic; their number was estimated from the known P - D total cross-section, and the D - D inelastic cross-section was reduced accordingly. Interactions were accepted only if the incident track entered the chamber through the beam entry window and lay within 7.5 ° of the average beam direction. By measuring a sample of the beam tracks, it was found that 3 % of them failed to meet the above criteria. An estimate of the number of low momentum deuterons in the beam was obtained by counting the beam tracks which appeared to stop within the chamber and had a range shorter than that of the major part of the beam. These could alternatively be zero-prong events produced by the full momentum beam. Thus 3He-N events with the neutron produced at less than 5° or more than 150 ° could give rise to a zero-prong event; their number was obtained from the requirement of symmetry of the cross-section about 90 ° (c.m.) and by assuming a reasonable value of the forward crosssection, the exact value of which was not critical for this calculation. Zero-prong events were also produced by inelastic reactions in which the two protons have very little energy. By charge symmetry, the number of such events is equal to the number of inelastic events in which the protons carry away almost all the available energy, which number is experimentally available. In the above manner it was found that 4 % of the 540 MeV/c beam had momenta between 325 and 525 MeV/c. Most of their interactions would have been classified as inelastic, and were subtracted from the total number of that type.
D-D INTERACTION
271
4.2. L I Q U I D D E U T E R I U M
The 1.1 ~ p r o t o n impurity in the deuterium has a similar effect to the p r o t o n contamination in the beam. The liquid deuterium density was corrected for this effect, and the estimated n u m b e r of interactions on the protons was removed f r o m the total number o f inelastic events. 4.3. S C A N N I N G
LOSSES
There are two different effects which cause events to be missed when the film is being scanned. One is that events situated in unfavourable azimuthal planes can be exceedingly difficult to see on the scanning table, and there is a systematic loss of such events; and the other is a r a n d o m loss which occurs because o f the h u m a n fallibilities of scanners. Histograms o f the number of events as a function o f azimuthal plane showed that there was a loss of events at unfavourable azimuths when the scattering angle was less t h a n 12½° . In calculating cross-sections, only the events in the azimuthal range 40 ° to 90 ° were used when the scattering angle was below 10½°, and the azimuthal range 20 ° to 90 ° was used for scattering angles 10½° to 12½°. The only exception to this was for the two-prong inelastic events, for which no azimuthal loss was observed, and hence all these events were used in calculating the cross-section. The r a n d o m scanning loss was determined by comparing the numbers of events in the favourable azimuthal ranges seen in two independent scans of the film. A value o f 98.5 ~o was derived for the scanning efficiency. 4.4. E R R O R
ESTIMATES
The following sources o f error were taken into account in calculating the errors on the cross-sections given below: (i) the statistical error on the n u m b e r o f events o f the particular category being considered, (ii) the statistical error on the n u m b e r o f beam tracks, (iii) the uncertainty in the density of deuterium, (iv) the estimated error involved in classifying the small angle one-prong events as elastic scatters or as inelastic events, (v) the estimated error in determining the number o f zero-prong inelastic events (and hence the low m o m e n t u m tail o f the incident beam). In the differential cross-section for elastic scattering, only the contributions from (i) and (iv) above were used to obtain relative errors for the various angular ranges. All other cross-sections are given with total errors. It is to be noted, however, that in all cases the major source o f error was the statistical error on the n u m b e r o f events, and so the differences between relative and absolute errors are insignificant.
272
J. E. A. L Y S A N D TABLE
1
L. LYONS
(a)
TABLE 1 (b)
Elastic scattering differential cross-section da/d.Q (mb/sr)
O¢.m. (degrees)
Elastic scattering differential cross-section at 433-507 MeV/c
310-377 MeV/c
377-430 MeV/c
763 4-178 584±80 380±46 280 ± 28 150i19 80i l 1 45.1 4-7.7 33.74-6.4 28.3±5.7 46.5~7.5
5 4 7 ! 133 ~4±59 425±44 183 4- 20 111i14 41.54-7.0 19.6~4.5 14.04-3.6 15.04-3.6 30.44-5.4
5-9 9-17 17-25 25-33 33-41 41-51 51-61 61-71 71-81 81-90
Oe.m. (degrees) 5-9 9-13 13-17 17-21 21-25 25-29 29-33 33 37 37-41 41-45 45-49 49-57 57-65 65-73 73-81 81-90
da/d~Q (mb/sr) 3944-66 4265-56 367±44 311£37 255--25 173 ± 14 112:1_ 10 97.0±9.3 44.34-6.0 41.64-4-5.6 18.04-3.5 7 . 9 ± 1.6 7.3 4-1.5 8.54-1.5 8.6:t: 1.5 7.94-1.4
(rw)
~OCC
IOOO
oo t:<
500
~OO
I OZ
SOL
50
I
r 2L
I
+ t 3~
OCM
~b
9b
36 @cM
6'0
90
Fig. 3. Elastic scattering differential cross-section (a) Filled circles: incident m o m e n t u m 377-430 MeV/c, Open circles: incident m o m e n t u m 310-377 MeV/c. The solid curve is taken from ref. 5). (b) The experimental points are for the incident m o m e n t u m range 433 to 507 MeV/c. Curve A is calculated from the theory of Tubis and Chern, and curve B is that derived by Queen.
D - D INTERACTION
273
4.5. ELASTIC SCATTERING Because the initial a n d final states b o t h involve two identical particles, the elastic scattering cross-section is automatically symmetric a b o u t 90 ° in the centre-of-mass a n d hence we present results for the range below 90 ° only.
120C
.....
t ......
~00
400
300
40O
i
i
~.
i
500 i
i
(Mov/c)
Fig. 4. Total and elastic cross-sections as a function of momentum. Total cross-section is denoted by - -i~-- and elastic scattering cross section by - -[I]-" The incident momentum is P0.
TABLE 2
Cross-sections (in mb) Incident momentum (MeV/c)
270-310
310-377
377427
427--442
433M-72
472-507
Total
1280:L 123
10234-48
8644-40
7394-69
707±24
6534-23
Elastic
752 ~91
630±36
451~26
3584-48
3324-19
2764-16
354-8
42___7
P-T
16.0422.2
3He_N Inelastic
18.24-2.9 424±65
323±24
330±23
337~48
342±18
The differential cross-section for elastic scattering is presented in table 1(a) for the m o m e n t u m intervals 310-377 MeV/c a n d 377-430 MeV/c, a n d in table l ( b ) for the higher m o m e n t u m range 433-507 MeV/c. These data are shown in figs. 3(a) a n d 3(b),
274
J.E.A.
LYS A N D L , L Y O N S
respectively. The 433-507 MeV/c results are compared with two theoretical curves which are based on the IA and are described in more detail in sect. 5 below. The variation with m o m e n t u m of the cross-section for elastic scattering through angles larger than 10 ° (c.m.) is given in table 2 and is shown in fig. 4. It is seen that elastic scattering accounts for approximately half of the total cross-section, becoming increasingly important at lower incident momenta. 4.6. STRIPPING REACTIONS
As previously mentioned in subsects. 3.2 and 3.3, it was not always possible to observe and separate these reactions throughout the complete angular range, and it was necessary to invoke the requirement that the differential cross-sections be symmetric about 90 ° in the centre-of-mass in order to obtain their total cross-sections. ENERGY @ a V )
INC~ENT '
+
'
'I
'
'
'
'
0.1
I
'
I'0
'
'
I
'
'
'
'
I0
I
IO0
100
4tO
I/ Io
I
I
I
I
I
Z0
Jo
5,0
~OO
2oO
INCIDENT
I 3o0
I 50o
MOMENTUM Q M e @
Fig. 5. Cross-section for the stripping reactions. The curve from ref. 21) and the 310 MeV/c point 5) are for the reaction D ÷ D -~ 3He+N. The results of this experiment (shown as + ) and the 865 MeV/c cross section 6) are for the reaction D + D --> P + T .
Apart from the corrections already mentioned, it was necessary to remove a contamination of 20 ~ from the number of 3 H e - N events at the higher incident momenta. This was estimated from the magnitude of the background of inelastic events outside the peak of the deviation function which was used to select these events. Furthermore, at the lower incident momenta, the background was so large that no estimate of this particular cross-section could be made. A correction was also applied for the very forward produced 3He; these appeared to be low momentum beam tracks. The cross-section for the P - T final state was not calculated for the incident momentum intervals 270-310 MeV/c and 427-442 MeV/c because of the small numbers of events in these ranges. Table 2 again contains the cross-sections. For incident momenta, 433-507 MeV/c
D-D INTERACTION
275
the ratio of the two stripping reactions is 0.88 + 0.18, which is consistent with the value of unity expected from charge symmetry. Fig. 5 shows the P - T cross-section compared with previous measurements, mainly at lower incident momenta. 4.7. INELASTIC EVENTS The cross-section for the sum of the two inelastic processes is given in table 2. It is seen that the inelastic cross-section is fairly constant as a function of momentum. 4.8. TOTAL CROSS-SECTION The total cross-section was obtained from the observed number of events of all types, corrected for all the effects mentioned above. Again elastic scatters at less than 10 ° in the centre-of-mass were excluded. The cross-section is given as a function of the incident m o m e n t u m in table 2 and in fig. 4.
5. Comparison of Results with Impulse Approximation Calculations The I A considers deuteron-deuteron scattering as the superposition of the scatterings of the quasi-free nucleons of the incident deuteron on those of the target deuteron. Beyond a simple superposition, various effects can be considered. These include multiple scattering, the effect of the binding forces and off-the-energy-shell effects caused by the internal motion of the nucleons within the deuteron; all these are expected to become increasingly important at lower incident momenta and at larger scattering angles. The above complications also apply to the nucleon-deuteron scattering problem, but is has been found 18) that the simple IA (i.e. with neglect of the three above effects) gives an adequate description of the 275 MeV/c differential cross-section out to about 30 °. Thus one might reasonably compare an IA calculation with at least the 433 to 507 MeV/c D - D elastic scattering data. Tubis and Chern aa) have used simple IA to study D - D elastic scattering. They have neglected many exchange terms which are not expected to contribute appreciably at high incident momenta and at small scattering angles, and they omitted any deuteron D state and the Coulomb interaction in order to obtain an explicit expression for the elastic scattering in terms of nucleon-nucleon amplitudes and a deuteron formfactor. To apply their expression to 465 MeV/c deuteron-deuteron scattering, the nucleon-nucleon scattering matrix coefficients * are required at 232 MeV/c and the deuteron-deuteron and nucleon-nucleon centre-of-mass scattering angles 0d and 0 n are related by the expression sin ½0, = 2 sin ½0~, • As these are not readily available we have obtained the value of the quantity I (used in the equation in the appendix of ref. 11)) by extrapolating the values given in ref. 19) for the scattering matrix coefficients at incident momenta P0 of 275 MeV/c and. 421 MeV/c, assuming at each angle that Ix P 2 0 varies linearly with P20.
276
J.
E.
A.
LYS A N D
L.
LYONS
which results in equal momentum transfers in the two cases. We have used Moravcsik's approximation II deuteron wave function 2o) to calculate the deuteron form factor, and then obtain the curve A shown in fig. (3b) for the elastic scattering crosssection. Queen 12) has considered the deuteron-deuteron interaction as the scattering of one deuteron on the two quasi-free nucleons of the second deuteron. He has used his IA calculation of nucleon-deuteron scattering at 247 MeV/c, which includes off-theenergy-shell and multiple scattering corrections, to calculate deuteron-deuteron elastic scattering at 494 MeV/c. The more important multiple scattering terms for the deuteron-deuteron collision are also included. The calculated cross-section is shown as curve B in fig. 3(b). The lack of agreement between the experimental points and curve A, even at small angles, shows that the simple IA is not adequate at these momenta. The differences between curve B and the data, however, could be due to the various approximations which were needed to reduce the theoretical computation to manageable proportions. Clearly more complete calculations would be required before it will be possible even to begin a study of the possible influences of many-body forces or off-the-energy-shell effects in such an experiment. On the other hand, the fact that curve B is a better fit than curve A to our data suggests that in this momentum range, multiple scattering effects may be important. We wish to thank Professor D. H. Wilkinson and Dr. D. F. Shaw for their continued interest throughout the experiment. We are indebted to Dr. H. W. K. Hopkins for the part he played in redesigning the bubble chamber and obtaining the pictures and to Mr. B. Rose for his advice concerning the deuteron beam. The experiment would not have been possible without the patient and careful work of our scanning and measuring team of Frances Duff, Jan Hawkins and Anne Pether.
Appendix 1 THE CORRELATION FUNCTIONS For a two-body final state, let the angles of the particles be 0~ and 08, and their energies (which may or may not be known, depending on whether the particle stops within the chamber) be E~ and E~. Then the following checks were applied to test whether the event in question was consistent with a particular Ieaction: (i) coplanarity; this is defined as the triple scalar product of the three unit vectors along the directions of the incident and the two outgoing tracks and it is to be noted that all two-body final states should be coplanar, (ii) an angular correlation 0~(0,)- 0~; here 0~ is the expected value of 0~, calculated from the observed value of 08;
D-D INTERACTION
277
(iii) an angle-energy correlation E~(O~)- E,, the quantity E2 is calculated as a function of the angle of the other track, and this latter angle is used since for events with only one prong stopping, it is the angle of the other (and usually longer) prong which is in general determined with smaller error, and hence tends to provide a more sensitive check on a particular hypothesis; (iv) a second angle-energy correlation E~(Oo)-Eo, this is used for events in which both prongs stop in the chamber. The correlations (ii)-(iv) were evaluated non-relativistically; this introduces negligible error for the momenta involved in this experiment. ELASTIC SCATTERING
Here the forms of the correlations reduce to particularly simple expressions (ii)
0~(0~) = ½7c-0p,
(iii)
E~(O~) = Eo sin 2 0p,
(iv)
E~(Oa) = Eo cos z 0~,
where Eo is the incident energy. STRIPPING REACTIONS
We assume that the two particles have masses rn, and m~, m d being the mass of the deuteron. The energy released in the reaction is Q. The correlations (ii) to (iv) are obtained as follows: For correlation (ii) the centre-of-mass scattering angle O~ of particle fl can be obtained from 0~ by solving the equation tg Op -=
m~Vsin O~ Vo + m~ V cos Ot~
where V and V0 are given in terms of the incident energy by Vz =
½Eo + Q m,m:(m~ + m:) '
V2o = Eo/4m d.
Then the calculated value of O allows us to obtain 0~ from the equation tg 0, =
m~Vsin O~ V 0 - m~ Vcos O~"
In the P - T reaction, fl was taken as the proton, since then O~ is obtained as a singlevalued function of 0p. The angle-energy correlation (iii) is obtained from the one given in (iv) via the relationship E~(O#) = Eo + Q - E;(O#).
278
J.E.A.
F o r correlation (iv) w e set
E~(O') -
LYS A N D L. LYONS
mJma =
~ and
mJm,
=/3. Then
~Q+s-1) +~~)2 {2 c°s2 Ot~ ~~+-/ ~ (Eo
c~+~ T h e f o r m s o f the a b o v e correlations for D + D --, P + T in fig. 6.
at 390 M e V / c are s h o w n
40
Or
\
20
\
60
120
180
Op
(© 40
40
ENERGy ENE~~'f 20
20
O "~
60 °
120 °
180 °
O°
20 °
40:
PROTON LAB A N G L E TRITON
LAB
ANGLE
Fig. 6. Correlations for D + D ~ P-FT. (a) The angular correlation, which is relatively insensitive to the incident momentum. (b) The proton and triton energies as a function of the proton angle. (c) The proton and triton energies and a function of the triton angle. Both (b) and (c) are derived for an incident momentum of 390 MeV/c.
D-D
INTERACTION
279
Appendix 2 CALCULATION OF ERRORS IN CORRELATION FUNCTIONS The calculated error dE on each correlation function F was obtained by adding in quadrature the contributions AF1, AF2 and AF3 from the measurement error, from the momentum spread in the incident beam and from multiple scattering effects. The measurement error for each event was calculated from the formula
where A~ are the 20 or 24 measured film co-ordinates for a one- or two-prong event (made up of three of four track points plus two fiducials on each view). The setting errors AA~ were determined by making several measurements of a few events. A value of 10/zm was used for all AA~, except for the y co-ordinates of the two points on the incident track (the incident tracks lay approximately parallel to the x-axis), for which the error was taken as 5/~m. Explicit expressions for OF/aA~ were derived for each of the correlation functions. The momentum spread of the deuteron beam caused an uncertainty AP in the deuteron momentum at the interaction vertex *. Because of the form of the rangemomentum relationship, AP increases as the deuterons slow down. Thus while the momentum spread at 540 MeV/c was 0.5 %, by 270 MeV/c it had increased to 5 %. The resultant error in each correlation function was calculated as
AE2 = de__AP. aP An expression for OF/OP was derived for each correlation function, and AP was calculated for each event. Errors due to multiple scattering of the tracks by the liquid deuterium were in some cases comparable with the measurement errors. The contribution from this source was calculated as i= i
\Oni!
Here nl are the appropriately projected angles of the incident and outgoing tracks. The r.m.s, multiple scattering errors Ani were calculated from the Gaussian approximation to Moli~re scattering.
t The uncertainty in P due to straggling as the deuterons slowed down was always negligible.
280
J. E. A. LYS A N D L. L Y O N S
References 1) 2) 3) 4) 5)
6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)
M. Davison et al., Nuclear Physics 45 (1963) 423 L. Lyons, J. E. A. Lys and H. W. K. Hopkins, Phys. Lett. 3 (1963) 359 B. V. Rybakov, V. A. Sidorov and N. A. Vlasov, Nuclear Physics 23 (1961) 491 R. G. Freemantle, thesis, Birmingham University (1964) unpublished W. T. H. van Oers, H. Arnold and K. W. Brockman Jr., Nuclear Physics 46 (1963) 611; W. T. H. van Oers and K. W. Brockman Jr., Nuclear Physics 48 (1963) 625; W. T. H. van Oers, private communication C. S. Godfrey, Phys. Rev. 96 (1954) 493 C. S. Godfrey, Phys. Rev. 96 (1954) 1621 Yu. K. Akimov, O. V. Savchenko and L. M. Soroko, JETP (Soviet Physics) 14 (1962) 512 K. R. Chapman et al., Nuclear Physics 57 (1964) 499 O. Brander, Nuclear Physics 36 (1962) 82 A. Tubis and B. Chern, Phys. Rev. 128 (1962) 1352 N. M. Queen, Phys. Lett. 13 (1964) 236 G. F. Chew, Phys. Rev. 80 (1950) 196 G. Giacomelli and B. Rose, A E R E report R3981 (1962) unpublished H. W. K. Hopkins etal., Nucl. Instr. 12 (1961) 323 J. H. Van Vleck, The theory of electric and magnetic susceptibilities (Oxford University Press, London, 1959) p. 83 D. B. Chelton and D. B. Mann, U C R L 3421 (1956) L. Castillejo and L. S. Sing,h, Nuovo Cim. 11 (1959) 131 A. K. Kerman, H. McManus and R. M. Thaler, Ann. of Phys. 8 (1959) 551 M. J. Moravcsik, Nuclear Physics 7 (1958) 113 T. C. Griffith and E. A. Power, Nuclear forces and the few nucleon problem (Pergamon Press, London, 1960) p. 527
2.A.l:
[
Nuclear Physics 74 (1965) 261--280; (~) North-Holland Publishing Co., Amsterdam
2.L
I
Not to
be reproduced by photoprint or microfilm without written permission from the publisher
THE DEUTERON-DEUTERON INTERACTION AT 270 TO 507 MeV/c J. E. A. LYS t and L. L Y O N S
Nuclear Physics Laboratory, Oxford Received 19 March 1965 A bubble chamber experiment has been performed to study the deuteron-deuteron interaction in the m o m e n t u m range 270 to 507 MeV/c. Results are presented for elastic scattering, the stripping reactions and the inelastic reactions. The elastic scattering differential cross-section for the incident m o m e n t u m range 433 to 507 MeV/c is compared with impulse approximation calculations.
Abstract:
E
I I
N U C L E A R R E A C T I O N S ~H(d, d), (d, d'), (d, n), (d, p), E = 20-67 MeV; measured ~(E), ~(E; 0) for (d, d).
1. Introduction Over the past two decades, the nature of the forces in the two-nucleon system has been the subject of intensive study. We are at last beginning to obtain some degree of understanding of these forces, and we can expect that in the near future a considerable amount of effort will be spent in seeing how well the properties of larger nuclear systems can be explained in terms of the basic two-nucleon interaction. In particular, it will not be possible to obtain information on the existence of specifically many-body forces or on the nature of off-the-energy-shell effects if attention is confined to systems containing only two nucleons. It would seem that, as part of this programme of studying nuclear systems, the few-nucleon system might be the simplest with which to start. With this end in view, the Oxford deuterium bubble chamber group has been studying proton-deuteron and deuteron-deuteron interactions. The results of the former experiment 2) and a preliminary account of the deuteron-deuteron experiment 2) have already been published. This article gives a more complete account of the results of the second experiment. In the m o m e n t u m range below the meson-production threshold, the possible reactions involving only strongly interacting particles are: D+D ~ D+D
elastic scattering,
D + D --, P + T D + D ~ 3He + N
stripping reactions,
D + D --* D + P + N D + D ~ P+ N + P+ N
inelastic scattering.
t N o w at Nuclear Physics Laboratory, University o f Liverpool, England. 261
(1) (2)
(3) (4) (5)
262
J. E. A. LYS A N D L. L Y O N S
These reactions have been studied at a large number of momenta below 230 MeV/c. Above 230 MeV/c, the inelastic reactions 3) have been investigated at 265 MeV/c, elastic scattering and the stripping reactions 4) at 270 MeV/c and all 5) the above reactions at 310 MeV/c. The tritium production differential cross-section has been measured 6) at 865 MeV/c, and the helium production at a few angles at 865 MeV/c [ref. 7)] at 1.3 GeV/c and at 1.7 GeV/c [refs. s, 9)]. Thus there is a large m o m e n t u m range through which very little data exist. This experiment was intended to provide a survey of reactions (1)-(5) in this region, the bubble chamber being a good tool for such a study. The Oxford bubble chamber was filled with deuterium and exposed to a 540 MeV/c deuteron beam. Experimental factors limited the range of incident momenta over which interactions were measured to 270 to 507 MeV/c. These values of m o m e n t u m correspond to kinetic energies of 20 to 67 MeV for the incident deuterons. Some theoretical studies of deuteron-deuteron elastic scattering have recently been made lo-12). These have all been based on the impulse approximation 13) (henceforth abbreviated at IA) in which the basic interaction is assumed to be between one nucleon from each of the colliding deuterons. At least in first order IA, the other two nucleons are assumed not to take part in the collision, and will thus have the same velocities after the interaction as they possessed within the initial state deuterons before the collision. Then the cross-section for elastic scattering is simply related to the probability that the momenta of the four final state nucleons are consistent with those of the nucleons within two deuterons. In this way the elastic scattering is related in a fairly direct way to the two-nucleon interaction. In sect. 2 we describe the details of the experimental procedure of obtaining the photographs, and the method of analysing the events on them. Sect. 3 gives an account of the way in which the events were divided between the different possible reactions, and the results of this analysis are presented in sect. 4. Finally in sect. 5, the elastic scattering differential cross-section that we obtain for incident deuterons in the m o m e n t u m range 433 to 507 MeV/c is compared with some impulse approximation calculations.
2. Experimental Procedure 2.1. THE BEAM The deuteron beam was produced by placing a thick graphite target in the internal beam of the Harwell 2.8 m synchrocyclotron. The circulating protons were degraded in passing through the target to too low a m o m e n t u m to escape through the beam exit channel. In a proton-carbon collision, however, a pick-up reaction can occur with the deuteron so formed having a higher m o m e n t u m than that of the incident proton. Such deuterons would then be degraded when travelling through the target, and could emerge from the trailing edge of the target with the correct m o m e n t u m (about 550 MeV/c) to be accepted by the beam channel.
D-D
263
INTERACTION
A m o r e c o m p l e t e description o f the p r o d u c t i o n o f the d e u t e r o n b e a m a n d o f its p r o p e r t i e s m a y be f o u n d in ref. t4). 2.2. BEAM TRANSPORT The b e a m was t r a n s p o r t e d to the b u b b l e c h a m b e r b y the system shown in fig. 1. A t the entrance to the b u b b l e c h a m b e r , the b e a m was focussed to an image 2 c m high a n d 4 m m wide. The b e n d i n g magnets a n d c o l l i m a t o r s were used to achieve m o m e n tuna selection on the d e u t e r o n b e a m .
- - ---C 2
/~/C
APPFa,OX SCALE (m }
BC
Fig. 1. Beam transport system. H.C. is the Harwell cyclotron, with graphite target T. The deuteron beam is transported by two bending magnets BM1 and BM2 through 3 collimators C1, C2 and C 3. The quadrupole pair Q1 and Q2 focusses the beam at the entrance to the bubble chamber B.C; W is a concrete shielding wall.
The m o m e n t u m s p r e a d in the b e a m was m e a s u r e d b y taking a r a n g e - s p e c t r u m o f the b e a m . F o r this p u r p o s e a triple coincidence scintillation c o u n t e r telescope with a l u m i n i u m a b s o r b e r s was used when the b e a m was being set up, a n d the b u b b l e c h a m b e r itself was used during the actual experiment. I n this way, the b e a m was d e t e r m i n e d to have a m o m e n t u m o f 540 M e V / c a n d an r.m.s, s p r e a d o f 2.5 M e V / c at the entrance to the b u b b l e chamber. T h e n u m b e r o f p r o t o n s in the b e a m was m e a s u r e d as being 4 per 1000 deuterons. I t is to be n o t e d t h a t the range o f these p r o t o n s was seven times as great as those o f
264
S. E. A. LYS AND L. LYONS
the deuterons; this facilitated the estimation of the proton background from the range curve. The intensity of the beam at the chamber was 4 deuterons per pulse. 2.3. THE BUBBLE CHAMBER The bubble chamber assembly was very similar to that used for the proton-deuteron experiment and previously described in ref. 15). The major modification was that a thin beam entry window was required to allow the deuterons to enter the chamber without being degraded unnecessarily. To do this, it was necessary to rebuild the body of the chamber itself. This was made out of a single piece of a-brass. Onto the body, the beam window was connected by a flanged joint, sealed by a double indium O-ring. The window was 2 cm diam. stainless steel, 0.30 g/cm 2 thick. The liquid volume within the chamber was cylindrical in shape, being 12 cm in diameter and 10 cm deep. The deuterium used in this experiment had a protium content of 1.1 ~ and was cleaned by liquefaction and by passing it through a charcoal trap cooled in liquid nitrogen. The density of this deuterium at operating conditions was determined by measuring the apparent depth of the chamber when empty and when full of liquid. Then the density p was obtained from the refractive index n by means of the relationship 16) n2 - 1 p oc n 2 + 2 , the constant of proportionality being determined from the known 1 7 ) values of p and n for liquid deuterium at 20 ° K. After correcting for the protium contamination in the liquid, a value of 134.2 4- 1.3 g/1 was obtained for use in the calculation of the cross-sections of the various reactions. The bubble chamber had no magnetic field, the energy of tracks being obtained from their observed range in the deuterium. Thus if a track interacted in flight to give a zero-prong event (i.e. both charged prongs were too slow to be visible), the track would be ascribed too low an energy, and the event may have been misclassified; all cross-sections were subsequently corrected to allow fol this. Note that the energy of a track will also depend on the nature of the particle producing the track. For instance for a given range, the energy of a track assumed to be a deuteron would be 1.37 times larger than if the track were that of a proton. 2.4. THE EXPERIMENTAL EXPOSURE The chamber was operated at a temperature of 31.7 ° K with a volume expansion ratio of 1.4 ~ . The cyclotron's cycling rate was 100/sec. The chamber was expanded once every 1.7 sec, the timing sequence of the expansion of the chamber and the illuminating flash being such that only one beam pulse was photographed per expansion.
D-D INTERACTION
265
A total of 8 000 stereo pairs of photographs was subsequently scanned for events. These photographs were taken at two beam momenta, viz. the full beam momentum, and a degraded beam of about 460 MeV/c. These momenta were chosen to facilitate the kinematic analysis of the events. The deuterons slowed down in crossing the chamber with the result that interactions over the incident momentum range 433-507 MeV/c were obtained with the full momentum beam, and over the range 270-442 MeV/c with the degraded beam. The lower limit is here set by the increased momentum spread of a degraded beam, and by the short range of low-energy deuterons; the degraded deuteron beam in fact stopped in the chamber liquid. A total of 3 000 interactions was obtained in the higher momentum interval, and 2200 in the lower. 2.5. D A T A A N A L Y S I S
The pictures were first examined on a scanning table, where they were projected to four times life size. All beam tracks undergoing interactions within a predefined acceptance region in the bubble chamber were recorded; these interactions appear in the bubble chamber as two-, one- or zero-prong events. The number of beam tracks was counted on every fifth frame so that absolute cross-sections could be determined. These events were measured on a digitised measuring machine which had a least count of 1 pro. In fact the setting accuracy on a point on the film was typically of the order of 10 pm, which corresponds to about 50 gm in the bubble chamber space. As the tracks were straight, it required only two points to be measured at their ends to define them. Two fiducial marks were also measured on each view, thereby enabling the event to be located in the bubble chamber co-ordinate system. At the end of an event, the first fiducial was remeasured, thus providing a check against the malfunctioning of the measuring machine's digitisation system. Thus for a one- or twoprong event, six or seven points respectively were measured on each view. These events were measured at the rate of 8 per hour. The output from the measuring machines was in the form of punched paper tape. This was used as input to a single computer programme specially written for this experiment, and run on the Oxford Mercury computer. The programme made checks on the input data, reconstructed the events in space, calculated various kinematic correlation functions and their errors, and made histograms of some of the quantities of interest. The events were processed at the rate of 6 per rain.
3. Event Assignment Each event was checked against the various possible reactions for consistency (within the calculated errors) with energy and momentum conservation. 3.1. E L A S T I C S C A T T E R I N G
In general an elastic scatter produced a two-prong event. For small angle scatters, however, the recoil deuteron was usually too short to leave a track in which case only a single prong was seen.
266
J. E. A. LYS A N D L . L Y O N S
3.1.1. Two-prong events M a n y o f the t w o - p r o n g events h a d the longer p r o n g leaving the i l l u m i n a t e d region o f the c h a m b e r while the shorter p r o n g s t o p p e d , in which case the four conservation equations were used to calculate the u n k n o w n energy o f the n o n - s t o p p i n g t r a c k t a n d also to p r o v i d e three i n d e p e n d e n t checks on whether the d a t a were consistent with elastic scattering. There also was the r e q u i r e m e n t that the o b s e r v e d t r a c k length o f the n o n - s t o p p i n g p r o n g s h o u l d n o t be longer t h a n the range required for the a b o v e
IN
~ Fj S
(a)
l
k.
r
t
'
,
i -4.0
~
0
4.0
2.0
f 2 4.0
g~ F,
4
:-I-
[ _1
LLL
4k
'--L~
-4:cq~-- ~ 0
6
4.0
---2T5 r~3 q~
,/G 4
so
Fig. 2. Histograms of deviation functions for the elastic scattering hypothesis. (a) The angularcorrelation deviation function. (b) The first angle-energy deviation function. (c) The coplanarity deviation function. (d) The square root of the selection function Gd.
calculated energy. The f o r m o f the three checks was such as to minimize the weight given to the least accurately d e t e r m i n e d parts o f the data, viz. the direction o f the s h o r t e r prong. F o r those events where b o t h p r o n g s s t o p p e d within the c h a m b e r a f o u r t h check was possible. Details on the exact form o f the correlation functions used to check the d a t a are given in a p p e n d i x 1. T h e expected error in each c o r r e l a t i o n function was also calculated. This error included c o n t r i b u t i o n s f r o m the m e a s u r e m e n t errors, f r o m multiple scattering o f the tracks by the d e u t e r i u m in the b u b b l e c h a m b e r , a n d f r o m the u n c e r t a i n t y in the incident b e a m m o m e n t u m . O t h e r sources o f error were shown to be u n i m p o r t a n t . A p p e n d i x 2 gives m o r e details o f the w a y in which the various errors were calculated. H i s t o g r a m s o f the correlation functions divided by the c o r r e s p o n d i n g error (henceforth this ratio is referred to as the deviation function) are shown for the elastic scatt It is to be noted that in all cases the energy of the incident deuteron at the interaction was calculated from the known beam energy at the entrance to the bubble chamber and from the measured path length in deuterium up to the interaction vertex.
D-D INTERACTION
267
tering assignment in fig. 2; there are clear peaks centred on zero for the elastic events, with a background caused by other processes. The method of selecting elastic events was to define a function Ge~ as the sum of the squares of the three deviation functions. Events with only one stopping prong and with Gel less than 20 were taken as elastic. Those with Gel in the range 20-100 were re-examined on the scanning table. In some cases simple tests could be performed which made the event assignment unambiguous. Otherwise the events were remeasured, and those which had the new value of Ge~ less than 20 were classified as elastic. For those events where four checks were possible, the extra condition that the square of the fourth deviation function be iess than four was imposed. The number of events with Gc~ less than 20 which failed to satisfy this fourth condition was small, and was consistent with the number expected from a calculation of the probability of the longer prong undergoing a zero-prong interaction (and hence being assigned too low an energy). This confirms that for those events where one of the prongs does not stop, the loss of information about the energy of that prong does not introduce any bias into the assignment of events. For 2 ~o of the events, the energy of neither prong was known, in which case only two checks could be performed, and the above criteria were suitably modified. Those two-prong events satisfying the elastic criteria but with scattering angle * less than 12½° were re-analysed as one-prong events (see sect. 3.1.2 below) because the measurement error on the shorter prong was so large that the two-prong analysis was unreliable. Similarly, events consistent with the P-T reaction but with proton scattering angles less than 20 ° at the higher incident momentum or 30 ° at the lower incident momentum were also classified as one-prong events for further analysis. With these limits threre were no two-prong events which were consistent with both the P - T reaction and elastic scattering. 3.1.2. One-pron9 ecents For the higher incident momenta, the small angle elastically scattered deuterons did not stop in the chamber but at the lower momenta they generally did. In the latter case, only one check was possible (the second angle-energy correlation described in appendix 1) since three of the four conservation equations were required to calculate the momentum of the unseen recoil. Events were taken as elastic when the square of the deviation function was less than 4. This criterion was obtained from a study of two-prong elastic scatters. There was also the condition that the calculated energy of the recoil deuteron should be small enough ( ~ 3 MeV) for it not to leave a visible track in the chamber. When even the single prong had unknown energy, the angle-energy correlation was replaced by an inequality that the observed track length should be at least as great as that required from the calculated energy of the prong. In general, this conAll angles referred to in the text are in the laboratory system, unless specific mention is made to the contrary.
268
J. E. A. L Y S A N D
L. L Y O N S
dition was not sufficient to separate elastic scatters from the other reactions in an unambiguous manner *, and it was necessary to remove the contamination of nonelastic events statistically as follows. The number of P - T events with forward protons was estimated by demanding symmetry about 90 ° of the P - T production cross-section in the centre-of-mass ** Finally an analysis of the confirmed inelastic events (see subsects 3.4 below) enabled an extrapolation to be performed into the kinematic region where they would be confused with elastic scatters. This procedure led to 13 ~o of the high incident momentum one-prong events at less than 121° and none of the low incident momentum events being taken as inelastic. 3.2. T H E P - T R E A C T I O N
For the two-prong events, the analysis was similar to that described above for the elastic events. The form of the correlations are again given in appendix 1. For this reaction, however, it was necessary to calculate the selection function GpT twice for each event, since in general it was not obvious which prong was produced by which of the non-identical final state particles. Exactly the same criteria were applied to the lower value of GpT as in the elastic case. The kinematics of the production reaction are such that the events were seen as twoprongs, except for forward scattered protons, when the triton recoil was unseen if its energy was less than about 5 MeV. Then, however, the proton energy was such that it did not stop within the chamber, and hence all the one-prong P-T events were of the sort where no exact kinematic check was possible. The number of these which were P - T events was estimated from the requirement that the differential cross-section for the reaction D + D ~ P + T must be symmetric about 90 ° in the centre-of-mass. The contamination in the other channels from this source was thus estimated. 3.3. T H E a H e - N R E A C T I O N
As the 3He particles are doubly charged, this reaction resulted in short one-prong events. Besides the requirement that there be no visible recoil, only one check could be performed (the second angle-energy correlation mentioned in appendix 1). A histogram of the relevant deviation function for possible helium production events with helium centre-of-mass angles less than 90 ° showed a considerable background from other processes. When the helium production angle was larger than 90 ° (c.m.), its energy was usually so low that the track was too short to be visible and the event appeared as a zero-prong interaction. Again the requirement of symmetry in the centre-of-mass system was used to obtain the total number of 3He-N events. * T h e ~ H e - N final state could, however, be separated u n a m b i g u o u s l y since the doubly charged aHe always h a d a range m u c h shorter t h a n those o f the elastically scattered f o r w a r d deuterons. ~* Events with f o r w a r d tritons always p r o d u c e d a visible recoil p r o t o n a n d so these events could be selected u n a m b i g u o u s l y f r o m the t w o - p r o n g events (see subsect. 3.2 below).
D-D INTERACTION
269
3.4. INELASTIC REACTIONS Two-prong events not classifed as elastic scatters or as tritium production were assume to be break-up events. In principle a separation between the two inelastic reactions D + D -~ P + N + D , (4) D + D ---, P + N + P + N ,
(5)
is possible if both prongs stopped within the chamber. This is based on just one check (since there are three unknown kinematic variables associated with the neutron in reaction (4)) and there is also a two-fold ambiguity because the final state particles are not identical. A preliminary analysis showed that the separation was difficult, and because of the large class of events in which the separation was not even in principle possible, no further attempt was made to divide the inelastic events between the two reactions. It is possible that a two-prong inelastic event could accidentally have a track configuration resembling that of an elastic scatter, and hence have been misclassified. To investigate the magnitude of this effect, events with the longer prong scattered through 13 ° were examined. (The problem of identification becomes much less serious at larger scattering angles.) Those inelastic events with the length of the shorter prong approximately equal to that required for an elastic scatter were found to have a distribution of the shorter prong's laboratory angle and azimuth with respect to the longer prong which was consistent with isotropy. Thus the probability that the short prong would lie in the required direction (within typical errors) for an elastic scatter was calculated. In this way it was found that perhaps half an event had been misclassified, and thus this source of contamination was negligible. When one of the charged particles in reactions (4) or (5) had low energy, the event would appear to be single-pronged. Then the event was assumed to be inelastic if it was inconsistent with reactions (1)-(3). At the lower incident momenta, this proved to be a satisfactory separation technique; an extrapolation of a histogram of prong length of inelastic events indicated that no events were expected to have a prong length as long as that required for reactions (1) or (2), and the cross-section for reaction (3) was too small to provide any appreciable effect. On the other hand, at the higher incident momenta, the track of the one-prong inelastic events frequently left the chamber, and so essentially all the small angle elastic and tritium production reactions (which also lead to small angle single prongs leaving the chamber) could have been inelastic events. A statistical separation was then made as follows. Histograms were made of the energy of the shorter prong for those twoprong inelastic events with angle of scatter of the longer prong in a number of angular ranges below 12½° . These were extrapolated to the limit of the short track being of too low an energy to be visible. This led to a total of 13 ~ of the one-prong events below 12½° being taken as inelastic. The extrapolation procedure was checked to give the correct numbers at lower incident momenta where the assignment of events was simpler.
270
J.
E.
A.
LYS
AND
L.
LYONS
4. Corrections and Results Distance averaged cross-sections were calculated in six separate incident momentum ranges from the formula
NT/No) NTpL
Ni loge(1 O"i
=
--
where Ni is the number of events of the particular process whose cross-section is ai, ArT is the total number of events, No is the number of incident deuterons, p is the density of deuterium, and L is the length of the region in which events were accepted. It was necessary to apply corrections to some of the above quantities. 4.1. BEAM TRACKS The number of incident deuterons was estimated by counting the beam tracks on every fifth frame. This number was then corrected for protons in the beam, for tracks entering the chamber but not meeting the beam track requirements, and for low momentum deuterons in the beam. The beam contained a proton contamination of 0.4 %, and the number of beam tracks was reduced to allow for this. The interactions produced by these protons would have been incorrectly classified as inelastic; their number was estimated from the known P - D total cross-section, and the D - D inelastic cross-section was reduced accordingly. Interactions were accepted only if the incident track entered the chamber through the beam entry window and lay within 7.5 ° of the average beam direction. By measuring a sample of the beam tracks, it was found that 3 % of them failed to meet the above criteria. An estimate of the number of low momentum deuterons in the beam was obtained by counting the beam tracks which appeared to stop within the chamber and had a range shorter than that of the major part of the beam. These could alternatively be zero-prong events produced by the full momentum beam. Thus 3He-N events with the neutron produced at less than 5° or more than 150 ° could give rise to a zero-prong event; their number was obtained from the requirement of symmetry of the cross-section about 90 ° (c.m.) and by assuming a reasonable value of the forward crosssection, the exact value of which was not critical for this calculation. Zero-prong events were also produced by inelastic reactions in which the two protons have very little energy. By charge symmetry, the number of such events is equal to the number of inelastic events in which the protons carry away almost all the available energy, which number is experimentally available. In the above manner it was found that 4 % of the 540 MeV/c beam had momenta between 325 and 525 MeV/c. Most of their interactions would have been classified as inelastic, and were subtracted from the total number of that type.
D-D INTERACTION
271
4.2. L I Q U I D D E U T E R I U M
The 1.1 ~ p r o t o n impurity in the deuterium has a similar effect to the p r o t o n contamination in the beam. The liquid deuterium density was corrected for this effect, and the estimated n u m b e r of interactions on the protons was removed f r o m the total number o f inelastic events. 4.3. S C A N N I N G
LOSSES
There are two different effects which cause events to be missed when the film is being scanned. One is that events situated in unfavourable azimuthal planes can be exceedingly difficult to see on the scanning table, and there is a systematic loss of such events; and the other is a r a n d o m loss which occurs because o f the h u m a n fallibilities of scanners. Histograms o f the number of events as a function o f azimuthal plane showed that there was a loss of events at unfavourable azimuths when the scattering angle was less t h a n 12½° . In calculating cross-sections, only the events in the azimuthal range 40 ° to 90 ° were used when the scattering angle was below 10½°, and the azimuthal range 20 ° to 90 ° was used for scattering angles 10½° to 12½°. The only exception to this was for the two-prong inelastic events, for which no azimuthal loss was observed, and hence all these events were used in calculating the cross-section. The r a n d o m scanning loss was determined by comparing the numbers of events in the favourable azimuthal ranges seen in two independent scans of the film. A value o f 98.5 ~o was derived for the scanning efficiency. 4.4. E R R O R
ESTIMATES
The following sources o f error were taken into account in calculating the errors on the cross-sections given below: (i) the statistical error on the n u m b e r o f events o f the particular category being considered, (ii) the statistical error on the n u m b e r o f beam tracks, (iii) the uncertainty in the density of deuterium, (iv) the estimated error involved in classifying the small angle one-prong events as elastic scatters or as inelastic events, (v) the estimated error in determining the number o f zero-prong inelastic events (and hence the low m o m e n t u m tail o f the incident beam). In the differential cross-section for elastic scattering, only the contributions from (i) and (iv) above were used to obtain relative errors for the various angular ranges. All other cross-sections are given with total errors. It is to be noted, however, that in all cases the major source o f error was the statistical error on the n u m b e r o f events, and so the differences between relative and absolute errors are insignificant.
272
J. E. A. L Y S A N D TABLE
1
L. LYONS
(a)
TABLE 1 (b)
Elastic scattering differential cross-section da/d.Q (mb/sr)
O¢.m. (degrees)
Elastic scattering differential cross-section at 433-507 MeV/c
310-377 MeV/c
377-430 MeV/c
763 4-178 584±80 380±46 280 ± 28 150i19 80i l 1 45.1 4-7.7 33.74-6.4 28.3±5.7 46.5~7.5
5 4 7 ! 133 ~4±59 425±44 183 4- 20 111i14 41.54-7.0 19.6~4.5 14.04-3.6 15.04-3.6 30.44-5.4
5-9 9-17 17-25 25-33 33-41 41-51 51-61 61-71 71-81 81-90
Oe.m. (degrees) 5-9 9-13 13-17 17-21 21-25 25-29 29-33 33 37 37-41 41-45 45-49 49-57 57-65 65-73 73-81 81-90
da/d~Q (mb/sr) 3944-66 4265-56 367±44 311£37 255--25 173 ± 14 112:1_ 10 97.0±9.3 44.34-6.0 41.64-4-5.6 18.04-3.5 7 . 9 ± 1.6 7.3 4-1.5 8.54-1.5 8.6:t: 1.5 7.94-1.4
(rw)
~OCC
IOOO
oo t:<
500
~OO
I OZ
SOL
50
I
r 2L
I
+ t 3~
OCM
~b
9b
36 @cM
6'0
90
Fig. 3. Elastic scattering differential cross-section (a) Filled circles: incident m o m e n t u m 377-430 MeV/c, Open circles: incident m o m e n t u m 310-377 MeV/c. The solid curve is taken from ref. 5). (b) The experimental points are for the incident m o m e n t u m range 433 to 507 MeV/c. Curve A is calculated from the theory of Tubis and Chern, and curve B is that derived by Queen.
D - D INTERACTION
273
4.5. ELASTIC SCATTERING Because the initial a n d final states b o t h involve two identical particles, the elastic scattering cross-section is automatically symmetric a b o u t 90 ° in the centre-of-mass a n d hence we present results for the range below 90 ° only.
120C
.....
t ......
~00
400
300
40O
i
i
~.
i
500 i
i
(Mov/c)
Fig. 4. Total and elastic cross-sections as a function of momentum. Total cross-section is denoted by - -i~-- and elastic scattering cross section by - -[I]-" The incident momentum is P0.
TABLE 2
Cross-sections (in mb) Incident momentum (MeV/c)
270-310
310-377
377427
427--442
433M-72
472-507
Total
1280:L 123
10234-48
8644-40
7394-69
707±24
6534-23
Elastic
752 ~91
630±36
451~26
3584-48
3324-19
2764-16
354-8
42___7
P-T
16.0422.2
3He_N Inelastic
18.24-2.9 424±65
323±24
330±23
337~48
342±18
The differential cross-section for elastic scattering is presented in table 1(a) for the m o m e n t u m intervals 310-377 MeV/c a n d 377-430 MeV/c, a n d in table l ( b ) for the higher m o m e n t u m range 433-507 MeV/c. These data are shown in figs. 3(a) a n d 3(b),
274
J.E.A.
LYS A N D L , L Y O N S
respectively. The 433-507 MeV/c results are compared with two theoretical curves which are based on the IA and are described in more detail in sect. 5 below. The variation with m o m e n t u m of the cross-section for elastic scattering through angles larger than 10 ° (c.m.) is given in table 2 and is shown in fig. 4. It is seen that elastic scattering accounts for approximately half of the total cross-section, becoming increasingly important at lower incident momenta. 4.6. STRIPPING REACTIONS
As previously mentioned in subsects. 3.2 and 3.3, it was not always possible to observe and separate these reactions throughout the complete angular range, and it was necessary to invoke the requirement that the differential cross-sections be symmetric about 90 ° in the centre-of-mass in order to obtain their total cross-sections. ENERGY @ a V )
INC~ENT '
+
'
'I
'
'
'
'
0.1
I
'
I'0
'
'
I
'
'
'
'
I0
I
IO0
100
4tO
I/ Io
I
I
I
I
I
Z0
Jo
5,0
~OO
2oO
INCIDENT
I 3o0
I 50o
MOMENTUM Q M e @
Fig. 5. Cross-section for the stripping reactions. The curve from ref. 21) and the 310 MeV/c point 5) are for the reaction D ÷ D -~ 3He+N. The results of this experiment (shown as + ) and the 865 MeV/c cross section 6) are for the reaction D + D --> P + T .
Apart from the corrections already mentioned, it was necessary to remove a contamination of 20 ~ from the number of 3 H e - N events at the higher incident momenta. This was estimated from the magnitude of the background of inelastic events outside the peak of the deviation function which was used to select these events. Furthermore, at the lower incident momenta, the background was so large that no estimate of this particular cross-section could be made. A correction was also applied for the very forward produced 3He; these appeared to be low momentum beam tracks. The cross-section for the P - T final state was not calculated for the incident momentum intervals 270-310 MeV/c and 427-442 MeV/c because of the small numbers of events in these ranges. Table 2 again contains the cross-sections. For incident momenta, 433-507 MeV/c
D-D INTERACTION
275
the ratio of the two stripping reactions is 0.88 + 0.18, which is consistent with the value of unity expected from charge symmetry. Fig. 5 shows the P - T cross-section compared with previous measurements, mainly at lower incident momenta. 4.7. INELASTIC EVENTS The cross-section for the sum of the two inelastic processes is given in table 2. It is seen that the inelastic cross-section is fairly constant as a function of momentum. 4.8. TOTAL CROSS-SECTION The total cross-section was obtained from the observed number of events of all types, corrected for all the effects mentioned above. Again elastic scatters at less than 10 ° in the centre-of-mass were excluded. The cross-section is given as a function of the incident m o m e n t u m in table 2 and in fig. 4.
5. Comparison of Results with Impulse Approximation Calculations The I A considers deuteron-deuteron scattering as the superposition of the scatterings of the quasi-free nucleons of the incident deuteron on those of the target deuteron. Beyond a simple superposition, various effects can be considered. These include multiple scattering, the effect of the binding forces and off-the-energy-shell effects caused by the internal motion of the nucleons within the deuteron; all these are expected to become increasingly important at lower incident momenta and at larger scattering angles. The above complications also apply to the nucleon-deuteron scattering problem, but is has been found 18) that the simple IA (i.e. with neglect of the three above effects) gives an adequate description of the 275 MeV/c differential cross-section out to about 30 °. Thus one might reasonably compare an IA calculation with at least the 433 to 507 MeV/c D - D elastic scattering data. Tubis and Chern aa) have used simple IA to study D - D elastic scattering. They have neglected many exchange terms which are not expected to contribute appreciably at high incident momenta and at small scattering angles, and they omitted any deuteron D state and the Coulomb interaction in order to obtain an explicit expression for the elastic scattering in terms of nucleon-nucleon amplitudes and a deuteron formfactor. To apply their expression to 465 MeV/c deuteron-deuteron scattering, the nucleon-nucleon scattering matrix coefficients * are required at 232 MeV/c and the deuteron-deuteron and nucleon-nucleon centre-of-mass scattering angles 0d and 0 n are related by the expression sin ½0, = 2 sin ½0~, • As these are not readily available we have obtained the value of the quantity I (used in the equation in the appendix of ref. 11)) by extrapolating the values given in ref. 19) for the scattering matrix coefficients at incident momenta P0 of 275 MeV/c and. 421 MeV/c, assuming at each angle that Ix P 2 0 varies linearly with P20.
276
J.
E.
A.
LYS A N D
L.
LYONS
which results in equal momentum transfers in the two cases. We have used Moravcsik's approximation II deuteron wave function 2o) to calculate the deuteron form factor, and then obtain the curve A shown in fig. (3b) for the elastic scattering crosssection. Queen 12) has considered the deuteron-deuteron interaction as the scattering of one deuteron on the two quasi-free nucleons of the second deuteron. He has used his IA calculation of nucleon-deuteron scattering at 247 MeV/c, which includes off-theenergy-shell and multiple scattering corrections, to calculate deuteron-deuteron elastic scattering at 494 MeV/c. The more important multiple scattering terms for the deuteron-deuteron collision are also included. The calculated cross-section is shown as curve B in fig. 3(b). The lack of agreement between the experimental points and curve A, even at small angles, shows that the simple IA is not adequate at these momenta. The differences between curve B and the data, however, could be due to the various approximations which were needed to reduce the theoretical computation to manageable proportions. Clearly more complete calculations would be required before it will be possible even to begin a study of the possible influences of many-body forces or off-the-energy-shell effects in such an experiment. On the other hand, the fact that curve B is a better fit than curve A to our data suggests that in this momentum range, multiple scattering effects may be important. We wish to thank Professor D. H. Wilkinson and Dr. D. F. Shaw for their continued interest throughout the experiment. We are indebted to Dr. H. W. K. Hopkins for the part he played in redesigning the bubble chamber and obtaining the pictures and to Mr. B. Rose for his advice concerning the deuteron beam. The experiment would not have been possible without the patient and careful work of our scanning and measuring team of Frances Duff, Jan Hawkins and Anne Pether.
Appendix 1 THE CORRELATION FUNCTIONS For a two-body final state, let the angles of the particles be 0~ and 08, and their energies (which may or may not be known, depending on whether the particle stops within the chamber) be E~ and E~. Then the following checks were applied to test whether the event in question was consistent with a particular Ieaction: (i) coplanarity; this is defined as the triple scalar product of the three unit vectors along the directions of the incident and the two outgoing tracks and it is to be noted that all two-body final states should be coplanar, (ii) an angular correlation 0~(0,)- 0~; here 0~ is the expected value of 0~, calculated from the observed value of 08;
D-D INTERACTION
277
(iii) an angle-energy correlation E~(O~)- E,, the quantity E2 is calculated as a function of the angle of the other track, and this latter angle is used since for events with only one prong stopping, it is the angle of the other (and usually longer) prong which is in general determined with smaller error, and hence tends to provide a more sensitive check on a particular hypothesis; (iv) a second angle-energy correlation E~(Oo)-Eo, this is used for events in which both prongs stop in the chamber. The correlations (ii)-(iv) were evaluated non-relativistically; this introduces negligible error for the momenta involved in this experiment. ELASTIC SCATTERING
Here the forms of the correlations reduce to particularly simple expressions (ii)
0~(0~) = ½7c-0p,
(iii)
E~(O~) = Eo sin 2 0p,
(iv)
E~(Oa) = Eo cos z 0~,
where Eo is the incident energy. STRIPPING REACTIONS
We assume that the two particles have masses rn, and m~, m d being the mass of the deuteron. The energy released in the reaction is Q. The correlations (ii) to (iv) are obtained as follows: For correlation (ii) the centre-of-mass scattering angle O~ of particle fl can be obtained from 0~ by solving the equation tg Op -=
m~Vsin O~ Vo + m~ V cos Ot~
where V and V0 are given in terms of the incident energy by Vz =
½Eo + Q m,m:(m~ + m:) '
V2o = Eo/4m d.
Then the calculated value of O allows us to obtain 0~ from the equation tg 0, =
m~Vsin O~ V 0 - m~ Vcos O~"
In the P - T reaction, fl was taken as the proton, since then O~ is obtained as a singlevalued function of 0p. The angle-energy correlation (iii) is obtained from the one given in (iv) via the relationship E~(O#) = Eo + Q - E;(O#).
278
J.E.A.
F o r correlation (iv) w e set
E~(O') -
LYS A N D L. LYONS
mJma =
~ and
mJm,
=/3. Then
~Q+s-1) +~~)2 {2 c°s2 Ot~ ~~+-/ ~ (Eo
c~+~ T h e f o r m s o f the a b o v e correlations for D + D --, P + T in fig. 6.
at 390 M e V / c are s h o w n
40
Or
\
20
\
60
120
180
Op
(© 40
40
ENERGy ENE~~'f 20
20
O "~
60 °
120 °
180 °
O°
20 °
40:
PROTON LAB A N G L E TRITON
LAB
ANGLE
Fig. 6. Correlations for D + D ~ P-FT. (a) The angular correlation, which is relatively insensitive to the incident momentum. (b) The proton and triton energies as a function of the proton angle. (c) The proton and triton energies and a function of the triton angle. Both (b) and (c) are derived for an incident momentum of 390 MeV/c.
D-D
INTERACTION
279
Appendix 2 CALCULATION OF ERRORS IN CORRELATION FUNCTIONS The calculated error dE on each correlation function F was obtained by adding in quadrature the contributions AF1, AF2 and AF3 from the measurement error, from the momentum spread in the incident beam and from multiple scattering effects. The measurement error for each event was calculated from the formula
where A~ are the 20 or 24 measured film co-ordinates for a one- or two-prong event (made up of three of four track points plus two fiducials on each view). The setting errors AA~ were determined by making several measurements of a few events. A value of 10/zm was used for all AA~, except for the y co-ordinates of the two points on the incident track (the incident tracks lay approximately parallel to the x-axis), for which the error was taken as 5/~m. Explicit expressions for OF/aA~ were derived for each of the correlation functions. The momentum spread of the deuteron beam caused an uncertainty AP in the deuteron momentum at the interaction vertex *. Because of the form of the rangemomentum relationship, AP increases as the deuterons slow down. Thus while the momentum spread at 540 MeV/c was 0.5 %, by 270 MeV/c it had increased to 5 %. The resultant error in each correlation function was calculated as
AE2 = de__AP. aP An expression for OF/OP was derived for each correlation function, and AP was calculated for each event. Errors due to multiple scattering of the tracks by the liquid deuterium were in some cases comparable with the measurement errors. The contribution from this source was calculated as i= i
\Oni!
Here nl are the appropriately projected angles of the incident and outgoing tracks. The r.m.s, multiple scattering errors Ani were calculated from the Gaussian approximation to Moli~re scattering.
t The uncertainty in P due to straggling as the deuterons slowed down was always negligible.
280
J. E. A. LYS A N D L. L Y O N S
References 1) 2) 3) 4) 5)
6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)
M. Davison et al., Nuclear Physics 45 (1963) 423 L. Lyons, J. E. A. Lys and H. W. K. Hopkins, Phys. Lett. 3 (1963) 359 B. V. Rybakov, V. A. Sidorov and N. A. Vlasov, Nuclear Physics 23 (1961) 491 R. G. Freemantle, thesis, Birmingham University (1964) unpublished W. T. H. van Oers, H. Arnold and K. W. Brockman Jr., Nuclear Physics 46 (1963) 611; W. T. H. van Oers and K. W. Brockman Jr., Nuclear Physics 48 (1963) 625; W. T. H. van Oers, private communication C. S. Godfrey, Phys. Rev. 96 (1954) 493 C. S. Godfrey, Phys. Rev. 96 (1954) 1621 Yu. K. Akimov, O. V. Savchenko and L. M. Soroko, JETP (Soviet Physics) 14 (1962) 512 K. R. Chapman et al., Nuclear Physics 57 (1964) 499 O. Brander, Nuclear Physics 36 (1962) 82 A. Tubis and B. Chern, Phys. Rev. 128 (1962) 1352 N. M. Queen, Phys. Lett. 13 (1964) 236 G. F. Chew, Phys. Rev. 80 (1950) 196 G. Giacomelli and B. Rose, A E R E report R3981 (1962) unpublished H. W. K. Hopkins etal., Nucl. Instr. 12 (1961) 323 J. H. Van Vleck, The theory of electric and magnetic susceptibilities (Oxford University Press, London, 1959) p. 83 D. B. Chelton and D. B. Mann, U C R L 3421 (1956) L. Castillejo and L. S. Sing,h, Nuovo Cim. 11 (1959) 131 A. K. Kerman, H. McManus and R. M. Thaler, Ann. of Phys. 8 (1959) 551 M. J. Moravcsik, Nuclear Physics 7 (1958) 113 T. C. Griffith and E. A. Power, Nuclear forces and the few nucleon problem (Pergamon Press, London, 1960) p. 527