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The momentum distribution of the neutron in 2H and 3He as determined by quasi-elastic scattering
Nuclear Physcs A395 (1983) 364-388 @ North-Holland Publishing Company
THE
MOMENTUM ‘He
DISTRIBUTION
AS DETERMINED
OF THE
NEUTRON
BY QUASI-ELASTIC
IN *H AND
SCATTERING
GARY F. KREBS*, C. COX**, J.M. DANIELS, T. GAJDICAR***, P. KIRKBY**** and A.D. MAY Dept. of Physics, University of Toronto, Toronto, Ontario, Canada M5S lA7 Received 29 March 1982 (Revised 17 September 1982) Abstract: The momentum distribution of the neutron in ‘H was determined in the range 0 to 365 MeV/c and in 3He in the range O-295 MeV/c, by quasi-elastic scattering of 667 MeV/c protons. The measurements were analyzed by the plane wave impulse approximation and a correction was made for multiple scattering in the target nucleons. The results for ‘H fit the HulthCn-Yamaguchi wave function quite well but show deviations similar to those found by other workers. For ‘He there are no previously published measurements for comparison, but the results fit several expressions for the wave function, proposed for reasons of plausibility and analytical tractability, with parameters close to those obtained by electron scattering. The momentum distribution in 3He has a small but significant dip around 100 MeV/c which is not found in any of the theoretical expressions.
E
NUCLEAR REACTIONS 3He(p, pn), *H(p, pn), relative a(E,, E,, ep, &, On,4,); deduced momentum and for p in ‘H.
(p, 2p), E = 667 MeV; measured distributions for n in 3He and ‘H
1. Introduction ‘Quasi-elastic extensively momentum
scattering
(QES) of protons
off very light muclei
has been studied
in recent years since Serber ‘) suggested that it would reveal the internal of the nucleons within the nucleus. By measuring the momentum of
the scattered particle one may then deduce the momentum-space wave function of the ejected nucleon within the target nucleus. A great deal of data has been collected around incident energies of 150 MeV [refs. ‘-‘)I for the *H(p, 2p)n reaction. In addition, there are measurements at the higher energies of 460 MeV [ ref. 9)], 600 MeV [ref. *‘)I, 660 MeV [ref. “)I, 970 MeV [refs. 12,13)], 1000 MeV [ref. 14)], and at 585 MeV [ref. “)I. The reaction *H(p, pn)‘H has been studied far less extensively than the *H(p, 2p)n reaction due to the inherent difficulty of * This work is part of a thesis by GFK, current address: Lawrence Berkeley Laboratory, University of California, Berkeley, California, USA 94720. ** Current address: Advanced Product Operations, SingeriLink, Sunnyvale, Calif., USA 94086. *** Current address: Computing Devices Co., Ottawa, Ontario, Canada KlG 3M9. **** Current address: Research Division, Ontario Hydro, Toronto, Ontario, Canada M8Z 5S4. 364
365
G.F. Krebs et al. / M o m e n t u m distribution
detecting the neutron. All existing measurements were made with proton energies below 200 MeV [ref. 8)] except for the study at 585 MeV and 800 MeV [ref. ~5)]. The reaction 3He(p, 2p) was studied at 590 MeV [ref. 25)] but only up to a Fermi momentum of 75 MeV/c, and no measurements have yet been published for the momentum distribution of the neutron in 3He. This paper describes a kinematically complete asymmetric experiment and presents unnormalized momentum distributions for the proton and neutron in 2H up to 365 MeV/c using the reaction 2H (p, pn) and up to 325 MeV/c using the reaction 2H(p, 2p), as well as the unnormalized momentum distribution for the neutron in 3He up to target nucleon momentum of 295 MeV/c. In describing the (p, 2p) or (p, pn) reactions the plane wave impulse approximation is usually used. In this simple description one assumes the incident projectile interacts elastically with only one of the target nucleons. Modifications of this simple description have been made to take into account effects due to multiple scattering and final-state interactions 17,18). We have used the plane wave impulse approximation to evaluate our data, but we have taken account of shadowing and rescattering in a different way from the way this is usually done. A report of the preliminary analysis of this work was presented at the Fifth International Conference on High Energy Physics and Nuclear Structure ~9).
2. Apparatus This experiment was carried out at the Lawrence Berkeley Laboratory using a 667 MeV proton beam from the 184 inch synchrocyclotron. The experimental arrangement is shown in fig. 1. The beam was incident on a cylindrical liquid deuterium target with dimensions 2.5 cm diameter and 5.1 cm long. The liquid 3He C2 Y- DIRECTION
~
~RO'1"O~4 ~' r(~_m_~-~\\\i~~ •
RING- - r,, X-DIRECTION OOUNTER (BV)
-
[
~
~'SPARK ~ CHAMBERS
C3
!!!~o
M2
V~9 SPARK"
~
MII M3
II I-"---,,p,#
S ,,T,O I CHAMBER
6 M5
~..
~ /VETO
COUNTERS(VI-V6)
~"~-I ~ " ~ x\x NEUTRON I J [ | .~' \ \ \ \ ' - - " COUNTERS
Ii IiI1|~~ , , ' " Fig. 1. Plan view of the experiment.
ll2inl
366
G.F. Krebs et al. / M o m e n t u m distribution
cylindrical target was 8.9 cm diameter and 15.2 cm long. Both targets were constructed of kapton and were centered at the beam's final focus with the long axis oriented vertically. Located just in front of the rarget was an N E 102 plastic scintillator veto counter (BV) which had a 5.1 cm diamter hole in its center and was placed so that the incident beam passed through this hole. This served to remove from analysis stray events not originating from the beam. A spark chamberbending magnet proton spectrometer was used to measure the m o m e n t u m and angle of the "fast" scattered proton. The knocked out neutron in the (p, pn) reaction (or proton in the (p, 2p) reaction) was detected in a spectrometer consisting of a bank of Pilot Y plastic scintillator bars. Beam monitoring was carried out with plastic scintillation counters M 1 - M 6 placed 6.1 m downstream behind a 0.64 cm Cu secondary target. The beam was centered on the target by monitoring a split-ion chamber in front of the Cu target. Typically the (p, 2p) and (p, pn) data were collected with incident beam intensities of 1 0 7 and 10 s protons/sec, respectively.
2.1. THE PROTON SPECTROMETER The proton spectrometer consisted of 6 entrance and 6 exit spark chambers before and after a 1 6 t o n C-magnet. In addition the spectrometer had three scintillation counters C1, C2 and C3. Each chamber had horizontal and vertical wires spaced 0.16 cm apart. The windows were constructed of kapton, and a gas mixture of 90% neon and 10% argon was circulated through the chambers. The aperture of the proton spectrometer was defined by the first spark chamber and had a value of ±8 ° horizontally and +2 ° vertically. The chambers gave digitized horizontal and vertical positions for the scattered proton using magnetostrictive readout followed by an A D C . The resolution of the chambers was found to be ±0.6 mm horizontally and +1.0 mm vertically. Scintillation counters C1, C2 and C3 were slabs of N E 102 plastic. Protons passing through the spectrometer were signalled by a coincidence in these counters. The vertical magnetic field component of the C-magnet was measured at 2.5 cm intervals in 6 horizontal p l a n e s - t h e average magnetic field strength was 7.5 kG. The two remaining mutually perpendicular components of the magnetic field were determined using Maxwell's equations 20).
2.2. THE NEUTRON SPECTROMETER The neutron counters measured the position and m o m e n t u m of the knocked out neutron by the time-of-ffight (TOF) technique. The neutron counter bank consisted of 30 bars of Pilot Y scintillation plastic with dimensions 2.5 cm wide x 15.2 cm deep × 96.7 cm high with photomultiplier tubes ( R C A 7585) at each end. These were arranged in three rows of ten as shown in fig. 1. The front surface of the rows were located at 144.7 cm, 165.1 cm and 185.4 cm from the center of the target.
G.F. Krebs et al. / Momentum distribution
367
Six veto counters, V 1 - V 6 , covered the front of the bank of neutron counters to detect and, if desired, to reject charged particles. Two levels of discrimination were used on the output pulses from the neutron counter photomultiplier tubes; a low level to give timing pulses relatively free of jitter and a higher level to give a pulse suitable for particle identification. The photomultiplier voltages and discriminator levels were set using the scintillations produced by a 6°Co y-ray source.
2.3. LOGIC A possible Q E S event was first signalled by (C1 • C 2 . C3 • BV), a coincidence in the pulses from counters C1, C2 and C3 of the proton spectrometer and no coincident pulse from the b e a m veto counter. The logic circuits then opened a 100 ns gate in which to detect a neutron in the neutron counter bank. The leading edge of the pulse from C1 started a four-channel time-to-digital converter (TDC) which was stopped by signals from the neutron counter bank. The four channels of the T D C were stopped, respectively, by signals from the b o t t o m photomultipliers and from the top photomultipliers of each of the three rows of ten neutron counters. H a r d wired logic identified which counter had stopped the T D C , and also identified ambiguous or invalid events. Such ambiguous or invalid events might be signalled by a pulse from the veto counters V 1 - V 6 indicating that the pulse detected by the neutron counter was a charged particle, or a pulse from two neutron counters in the same row. When the conditions for an apparantly valid event were met, the high voltage was applied to the spark chamber planes. For each event the spark coordinates and the fiducial positions of the 12 spark chambers, and two timing m e a s u r e m e n t s from either end of a neutron counter were recorded by an on-line P D P - 9 computer. Typically 8 events/sec were recorded on magnetic tape.
3. Experimental details 3,1. OPERATION AND CALIABRATION OF THE PROTON SPECTROMETER The positions of the spark chambers were determined by surveying. However, it was found during analysis that there were small errors in these positions. For example, the apparent position of the spark in one chamber might be systematically off the line defined by the sparks in the other chambers. Small corrections were made to the surveyed positions to eliminate these errors. The efficiency of the chambers was determined in two ways. First, the n u m b e r of sparks was c o m p a r e d with the n u m b e r of pulses from a small plastic scintillator positioned just in front of the chamber; the efficiency of a plastic scintillator for the detection of 660 M e V protons was taken to be unity. Secondly, the efficiency was found by counting the n u m b e r of sparks missing from a track. Using the criteria that a track must be defined by at least three collinear sparks, and at least one from each end of the
368
G.F. Krebs et al. / Momentum distribution
bank of six chambers, the overall efficiency was found to be 98%, to be slightly dependent on position and to be independent of proton energy. The computer p r o g r a m S P E C T O 21) was used to determine the m o m e n t u m of the scattered proton. The overall resolution of the proton spectrometer was found to be 1%. The acceptance of the proton spectrometer was determined by a Monte Carlo simulation. Fast scattered protons were generated with different directions and m o m e n t a and different origins within the target volume. The trajectory through the proton spectrometer was determined from the known magnetic field. Acceptance occurred when the 6 exit spark chambers were able to record the resulting trajectory. Fig. 2 shows the acceptance region for protons scattered from the target center and indicates h o m o g e n e o u s acceptance by the proton spectrometer between scattering angles of 12.5 ° to 17.5 ° when the m o m e n t u m is greater than 950 M e V / c . An artificial entrance window, smaller than the physical window, was imposed on the proton spectrometer to ensure that all protons within this window would be counted with known efficiency without vignetting.
3.2. THE NEUTRON COUNTERS The horizontal scattering angle was determined to within +0.5 ° by noting the neutron counter in which the neutron was detected. The T O F and the vertical scattering angle were obtained from the times tu and t~ at which the pulses from the u p p e r and lower photomultiplier tubes stopped the T D C . We can put tu = T - D l v p + l u / v
tl =
T-D/vp+
+ Tu,
ll/v+ TI,
1400
/
//
12007-/ / MOMENTUMOF - ' ~ 7 O III
I/I
IO00r-i I 111
I I I
REGION OF / / ~ ACCEPTANCE~ / / ~ ~
;-//
i/i 80o 6/ / / I
IO 12 14 16 18 20 SCATTERING ANGLE- DEGREES ( 81 ) Fig. 2. Result of calculations of tracks through the proton spectrometer. Protons whose entrance coordinates fall in the shaded region do not pass all the way through the spectrometer.
G.F. Krebs et al. / M o m e n t u m distribution
369
where T is the TOF, D is the distance from the target to the counter C1, and Vp is the velocity of the fast proton. The distance from the point where the scintillation is produced to the top and bottom of the plastic bar are, respectively, lu and l~, and v is the effective velocity of light in the plastic. Tu and T~ are times which represent the delay in the signals passing through the system and include a common quantity which is the delay time of the pulse from C1 to the TDC. Rearranging these we find 2 ( T - D / v p ) = (q + t ~ ) - (11+ lu)/V - T 1 - T~ = (q + t~) - T~ , 1 - (l,1)
lu) = ( t , -
tu) - ( T , - T u ) = ( q -
G) - Td,
where Ts = T~+ Tu and Ta = T ~ - Tu. Thus, in order to find the T O F T and the vertical position ½(l~-l,) where the neutron is detected, we need three calibration constants, Ts, Td and v, for each neutron counter. Two small scintillators T1 and T2, 2.5 cm x 1.3 cm x 0.64 cm, were placed in front of the centre of a neutron counter (lu = 10. Protons scattered from a polythene target were counted by the neutron counters, the logic being triggered by a coincidence in T1 and T2, and the T D C being started by the leading edge of T1. Then the difference in tu and t~ gave the calibration constant Td. By moving the counters T1 and T2 up and down the bar the effective velocity of light in the plastic
o3 Z 8
_
6
-.-~4 20
-40
I
I
-20
I
0 Z
20
I
40
(cm)
"-
Fig. 3. Measurement of the times tu and fi of arrival of a pulse from the upper and lower photomultiplier tubes of a typical neutron counter as a function of the height z above the center of the scintillator. From the slope of these lines the effective velocity of light in the scintillator was determined to be 13.02 cm/ns. Error bars are statistical in origin.
was determined to be ( 1 . 3 0 2 + 0 . 0 1 3 ) x 101°cm/sec (See fig. 3). By leading the pulse from T1 along the same cable as was used for C1, the value of Ts was given as Ts-- - ( q + tu). From the scatter in the values of tu and td the standard deviations of the derived quantities and T were found; these are shown in fig. 4, and are typically 2.3 cm for the vertical position z and 1.3 ns for T for events in the centre
G.F. Krebs et al. / Momentum distribution
370
~3 I
o 2
Q)
~,4 J -40
I -20
1 0
Z n
I b'
-40
G
,
-20
I 20
I 40
- (cm) Ou
u
I
t
I
0
20
40
" (cm) Fig. 4. R e s o l u t i o n in the d e t e r m i n a t i o n of the vertical p o s i t i o n z a n d the time-of-flight T of a scintillation. T h e s c a t t e r was f o u n d to b e g a u s s i a n ; ~z and O'T are the r e s p e c t i v e s t a n d a r d d e v i a t i o n s .
part of the neutron counters. This calibration was later refined during the analysis of the results of the experiments. W h e n protons are scattered from a polythene target this scattering is coplanar; in this way the zero of z was related to the nominally horizontal plane defined by the b e a m and the proton spectrometer. The T O F calibration was refined by analysing the 2H(p, pn) measurements. Assuming that Ts is not known, then in this reaction the energy and absolute value of the m o m e n t u m of the scattered neutron are unknown as is the m o m e n t u m of the spectator proton and the m o m e n t u m q of the scattered neutron before the collision, a total of nine unknown quantities. In addition to the four equations of conservation of energy and m o m e n t u m , the m o m e n t u m of the spectator proton is - q giving three equations. Two m o r e equations of the form E 2 = p 2C 2 _4_m 2C4 relate the energy and m o m e n t u m of the spectoator proton and the scattered neutron. Thus the event can be solved completely and Ts determined. T h e neutron counter efficiency was determined using the computer program of Stanton 22). The results for our discriminator bias setting and geometry are shown in fig. 5. In addition, the neutron counter efficiency was checked at several energies by comparing the counting rates between the 2H(p, 2p) and 2H(p, pn) reactions. The ratio of these counting rates is equal to the product of the ratio of the scattering cross sections, which are known for free nucleons, and the ratio of counter efficiencies. Assuming the value of the efficiency for proton detection to be unity, the results obtained from the experiment were found to be consistent to within 10% of the calculations shown in fig. 5. A further check of the neutron counter efficiency was obtained from the analysis of the 2H(p, p n ) l H and 3He(p, pn)p,p reactions and will be explained in sect. 4.
G.F. Krebs et al. / Momentum distribution
371
02-
0
I 2
J I I 5 4 5
I I0
NEUTRON
I I 20 50
I 50
I I I I00 200
ENERGY
~
I 500
1 I000
(MeV)
Fig. 5. The efficiency of a typical neutron counter for our discriminator bias as determined by the Monte Carlo program of Kurz-Stanton 23.
I000 800 600 Kz 400 - 4 ~ 20C 0 -40
-20
0 Kz
20 J-
ToooF
-20
40
0
20
40
MeVlc
T
]
I-z 200 hi
0
1272
F302 13:52 Kx ~ MeV/c
1287
1327
T -40
-20
0 Ky
20 ~
40 -20 MeV/c
0
20
40
Fig. 6. The distribution of the components of b e a m m o m e n t u m , Kx, K,. and K~: (a) as measured experimentally, (b) results of the Monte Carlo calculation to determine instrumental broadening, and (c) values for the beam with the instrumental broadening removed.
372
G.F. Krebs et al. / Momentum distribution
3.3. THE BEAM The distribution of b e a m m o m e n t u m was determined from the reaction 1H(p, p ) l H using a liquid hydrogen target. The scattering angle and energy as determined by the proton spectrometer, along with the scattering angle determined from the neutron counter, are sufficient to calculate the three mutually perpendicular b e a m m o m e n t u m components. The curves labelled " a " in fig. 6 show the distribution of m o m e n t u m for the c o m p o n e n t s K~, Ky and Kz which have directions along the b e a m line, perpendicular to the b e a m line in the horizontal plane, and vertical, respectively. Folded into these distributions are the effects of the unknown exact interaction points in the target and neutron counter. This instrumental broadening effect was calculated by a Monte Carlo program. Events were generated by choosing an interaction point at r a n d o m in the target volume. The point of interaction in the neutron counter was chosen on the horizontal centre line across the face of a neutron counter and was then m o v e d in a vertical direction according to the uncertainty in the determination of the z-coordinate derived from the timing calibration. The b e a m was taken to be monoenergetic and with no divergence. The event was then analysed by the kinematic reconstruction program which assumes that the scattering takes place at the point where the fast proton track intersects the vertical plane through the b e a m axis, and that the interaction with the neutron counter is at the center of the front face (for protons). The components of b e a m m o m e n t u m resulting from this analysis are shown as curve (b) in fig. 6. The true distribution of b e a m m o m e n t u m components after unfolding are shown as curves (c) in fig. 6. They have standard deviations of 23 M e V / c for K x , 44 M e V / c for Kv and 7 M e V / c for K~. 3.4. BACKGROUND The principle c o m p o n e n t of the background was due to stray particles being detected by the neutron counters while the 100 ns gate was open. This background was measured with the target empty and with the timing gate open all the time. T h e background rate was found to be proportional to the b e a m current, confirming its origin. Typically, in an actual run, each neutron counter registered about 14 background counts per hour.
4. Data analysis 4.1. CALCULATION PROCEDURES Approximately 70% of the events collected were rejected; the approximate percentages rejected for various reasons were: (i) Approximately 5% because more than one neutron counter detected these events leading to an irresolvable ambiguity.
G.F. Krebs et al. / M o m e n t u m distribution
373
(ii) A p p r o x i m a t e l y 50% because the proton track was outside the window of the proton spectrometer. (iii) A p p r o x i m a t e l y 5% because the trajectory of the fast proton did not intersect with the target volume. (iv) A p p r o x i m a t e l y 10% because the track through the proton spectrometer could not be fitted. (v) Approximately 3% because they were identified as elastic scattering without b r e a k u p of the target nucleus. Those events not rejected for the first four reasons were analyzed kinematically. The analysis procedure assumed that the b e a m was h o m o g e n e o u s and directed along the x-axis, that the scintillation produced by the neutron occurred at the center of a horizontal cross-section of the neutron counter, and that the point of scattering was where the track of the fast proton intersected the vertical plane through the b e a m axis provided this point lay in the target volume. The laws of conservation of m o m e n t u m were applied to the collision, assuming this to be an elastic scattering of a free neutron, to find the m o m e n t u m q of this neutron before the collision. T h e values obtained by this calculation are not the true ones because the analysis assumes an ideal experiment; deviations from ideal conditions mentioned earlier are responsible for the values of q so obtained being scattered about their true values. This scattering, or instrumental broadening, was determined by a Monte Carlo calculation, and its effects were r e m o v e d from the final result by a standard deconvolution process. Events were generated assuming some fixed value of q but choosing the point of scattering, the m o m e n t u m of the beam, and the point of interaction of the neutron in the neutron counter at r a n d o m according to the distribution of these quantities already determined. Those events were then analyzed by the analysis routine to find q. It was found that the scatter in the values of q was independent of q and could be described by a trivariant gaussian distribution in which the standard deviations were 23 M e V / c , 44 M e V / c and 7 M e V / c in the x-, y- and z-directions, respectively. The principle axes of this distribution were coincident with the x-, y- and z-axes within experimental error. It was obvious that still about 3% of the events recorded were spurious. Some of these events were identified by the circumstance that the kinematic analysis, which requires the solution of a quadratic equation, gave no real values for IK21, the m o m e n t u m of the scattered neutron. In other cases, they could be distinguished because the distribution of fast proton m o m e n t a , Igll was doubled humped. It was not possible to reject events which fell in the second peak of IKll because in many cases this p e a k was not resolved from the main peak and, in m a n y cases, sat on the top of a significant tail of the main peak. However, calculated values of qx and qy also showed double peaks. We therefore required that all acceptable events had to fall within generously assigned ranges of [K2[, qx and qy. This removed all the spurious events without cutting off the tail of the distribution of desired events.
374
G.F. Krebs et al. / Momentum distribution
Trials with different ranges of acceptance of [K21, q~ and qv showed that these could be chosen in a way which separated out cleanly the unwanted events. The identification of the origin of these events is peripheral to the object of our investigation. However, we notice that they are caused by some sort of scattering since they produce a peak in the fast proton distribution, and that they are quite rare, being observed only at large "neutron scattering angles", 0,, where the cross section for the reaction p(2H, pn)p is very low, of the order of one thousandth of the greatest value we were able to observe. The extra peak is kinematically consistent with elastic scattering of protons off deuterons with no breakup of the deuteron. We believe that these events are produced by a fast proton from 2H(p, p)2H scattering in chance coincidence with a background deuteron from some other scattering event. The method of determining the m o m e n t u m distribution P (q) is basically to count the number of events with a given value of q. However, in the number of events which are to be counted there are in addition to P (q) several other factors, some of which are unknown. The method of analysis was designed to eliminate these factors. First, the absolute value of the efficiency of the neutron counters was not well known. We therefore selected the center portion ( - 1 0 cm~
G.F. Krebs et al. / M o m e n t u m distribution
375
qy(MeV/c)
-E~OI/ I D( { I( (R''~'~ '!1~/ BEAMI ECTION-~' 8()-(MeV/c)qx
28* Fig. 7. The phase space in the x-y plane seen by the neutron counter at -70 °, with the proton spectrometer entrance path centered at 15°. Lines of constant c.m. angle 0* are shown. The two shaded areas represent the two parts of phase space visible to a given neutron counter with a given 0* bite for which the momentum of the target neutron has the same absolute value.
calculations of n e u t r o n c o u n t e r efficiency d e p i c t e d in fig. 5 are c o n s i s t e n t to within a few p e r c e n t - 10% in the worst case - with these e x p e r i m e n t a l l y d e t e r m i n e d ratios. A m o m e n t u m d i s t r i b u t i o n p (q) was o b t a i n e d for each n e u t r o n c o u n t e r a n d each c.m. angle by t a k i n g the n u m b e r of satisfactory events recorded, s u b t r a c t i n g the b a c k g r o u n d , weighting according to the efficiencies of the p r o t o n s p e c t r o m e t e r a n d the n e u t r o n c o u n t e r s , a n d dividing this result by the v o l u m e of the visible phase space. T h e s e d i s t r i b u t i o n s did n o t each cover the full r a n g e of q, n o r did they all cover the s a m e range. T h e y were pieced t o g e t h e r to form a final e x p e r i m e n t a l d i s t r i b u t i o n in the following way. T h e partial d i s t r i b u t i o n s f r o m all the n e u t r o n c o u n t e r s for a given c.m. angle were c o m b i n e d to give a full r a n g e d i s t r i b u t i o n d e r i v e d from scattering events with that c.m. angle. In this process, the a b s o l u t e values of the n e u t r o n c o u n t e r efficiencies (i.e, the vertical scale of the curves like fig. 5) were n o t k n o w n . T h e y were f o u n d by d e m a n d i n g that the d i s t r i b u t i o n s f r o m a n y two n e u t r o n c o u n t e r s should be the s a m e over the o v e r l a p p i n g r a n g e of q. T h e
376
G.F. Krebs et al. / Momentum distribution
partial d i s t r i b u t i o n s f r o m different c o u n t e r s were m u l t i p l i e d by these scale factors for n e u t r o n c o u n t e r efficiency a n d were c o m b i n e d , weighted according to their statistical accuracy. Finally, the partial d i s t r i b u t i o n s for all the different values of c.m. angle were a d d e d to give the e x p e r i m e n t a l d i s t r i b u t i o n .
6000 4000 2000 -
~x
IOOC O0 I-Z W > I W rr
t
>I-co z w
o x •o
60C 400
o~x
q,
200 IOC 6C
T.
40 20 IO 6 4
2 I
i
I
1
I
I00
I
[
1
200
,
,
,
300
Iql --*(MeV/c) Fig. 8. Momentum distribution of the neutron in 2H determined from the reaction 2H(p, pn) for different c.m. scattering angles O*. Error bars are statistical in origin. • - 42° ± 1°, © - 44° ± 1% × - 46° ± 1° c.m. angle. The vertical scale is not the same for each c.m. angle.
M a n y tests of consistency were p e r f o r m e d to check that there were n o systematic errors in this a s s e m b l y of the final d i s t r i b u t i o n curve. F o r example, figs. 8 a n d 9 show partial d i s t r i b u t i o n s for 2H a n d 3He respectively for t h r e e different c.m. angles, a n d it is a p p a r e n t that t h e r e are n o n o t i c e a b l e systematic differences in the shapes of the distributions. Similarly, m o m e n t u m d i s t r i b u t i o n curves were c o n s t r u c t e d using different c o m b i n a t i o n s of n e u t r o n c o u n t e r s a n d c.m. angles, a n d n o significant difference was seen b e t w e e n any of these curves. T h e final e x p e r i m e n t a l l y d e t e r m i n e d d i s t r i b u t i o n s are given in figs. 1 0 - 1 2 ; these d i s t r i b u t i o n s are, of course, u n n o r m a l i z e d . Fig. 13 shows the n u m b e r of events which c o n t r i b u t e d to each p o i n t in the final d i s t r i b u t i o n .
G.F. Krebs et al. / Momentum distribution
377
IO00C 500C
0
09 I 0 0 0 1-500 1.13 >
.'o.
.__1 w rr
~00
I
5o
•
•
x G) Z U.I a
~0
T~
•
~T~
I0
I 50
1 I00
I 150
Iql--
1 200
1 250
I 300
1 550
(MeV/c)
Fig. 9. M o m e n t u m distribution of the neutron in 3He as determined from the reaction 3He(p, pn)p, p for different c.m. scattering angles 0". Error bars are statistical in origin. 0 - 4 2 ° + 1°, 0 - 4 4 ° + 1°, × - 4 6 ° + 1° c.m. angle. T h e vertical scale is not the same for each c.m. angle.
4.2. C O R R E C T I O N S F O R I N S T R U M E N T A L B R O A D E N I N G A N D D O U B L E S C A T T E R I N G
There are two corrections which must be made to the experimentally determined distributions. The first is to remove the instrumental broadening. To do this accurately using the trivariate gaussian distribution would be very difficult, however, the experiment is mainly confined to a horizontal plane, and the scatter in q~ is of very little significance. Next, the scatter in q~ and qy are not too different, with qy being the most significant. We therefore approximated the instrumental broadening by a trivariate gaussian distribution with spherical symmetry and a standard deviation of 40 M e V / c . The effect of taking other reasonable values of this standard deviation was negligible. Then the effect of instrumental broadening is represented by the equation qf(q)
x/2--~ 2
~ O f ( Q ) e -(°-Q~2/2'~2 d O ,
G.F. Krebs et al. / Momentum distribution
378
600Or-_
%',,,
600
?\
200
Y> ~
\°~o
I00_
6o-
\
f 40: >z
eo
\\
20
\
\
't' k
,o
64
\ \ \ \ 5O
100
150
Iql
2OO 250
500
,- (MeV/c)
Fig. 10. Momentum distribution of the proton in 2H determined by the reaction 2H(p, 2p)n not corrected for instrumental broadening. The solid curve is the square of the Hulth6n-Yamaguchi wave function and the dashed curve is the same function convoluted with the instrumental broadening function. Error bars are statistical in origin.
w h e r e F ( q ) is t h e t r u e d i s t r i b u t i o n as a f u n c t i o n of n e u t r o n m o m e n t u m q, a n d f ( Q ) is t h e m e a s u r e d d i s t r i b u t i o n as a f u n c t i o n of e x p e r i m e n t a l l y d e t e r m i n e d n e u t r o n m o m e n t u m Q. I n s t r u m e n t a l b r o a d e n i n g was r e m o v e d b y t h e well k n o w n F o u r i e r transform method. T h e o t h e r c o r r e c t i o n is for s h a d o w i n g a n d r e s c a t t e r i n g . In the i n d e p e n d e n t p a r t i c l e m o d e l , t h e effect of s h a d o w i n g is to c h a n g e t h e c h a n c e that an i n c i d e n t p r o t o n will a c t u a l l y r e a c h the n e u t r o n , a n d so it c h a n g e s t h e s c a t t e r i n g cross s e c t i o n of t h e n e u t r o n inside t h e nucleus b y s o m e c o n s t a n t b u t u n k n o w n f a c t o r w h o s e origin is p r i c i p a l l y g e o m e t r i c a l . In o u r analysis this s c a t t e r i n g cross s e c t i o n is u n k n o w n a n d as has b e e n e x p l a i n e d e a r l i e r , t h e analysis has b e e n c a r r i e d o u t in such a w a y t h a t this cross s e c t i o n n e e d n o t b e k n o w n . T h u s s h a d o w i n g can b e i g n o r e d . C o n s i s t e n t with t h e i n d e p e n d e n t p a r t i c l e ( Q E S ) m o d e l , we c o n s i d e r e d t h a t the classical p i c t u r e of a s e c o n d s c a t t e r i n g f r o m a r e a l p a r t i c l e is a valid m e a n s of t r e a t i n g this p r o b l e m . Such a t r e a t m e n t has b e e n u s e d successfully for light nuclei 23) a n d is p r e f e r a b l e to t h e d i s t o r t e d w a v e i m p u l s e a p p r o x i m a t i o n ( D W I A ) which t r e a t s t h e rest of the n u c l e u s as a h o m o g e n e o u s fluid 24); t h e D W I A is m o r e a p p r o p r i a t e
G.F. Krebs et al. / M o m e n t u m distribution
379
.x,
% '~ ~
• 'i.\Oo
200 -_
\
T 6o40>_ _
09
z
U_l
ee O
\\ ee~i \
-
zo
~4~4i'
k
,t
\
I0 64 --
IT ~
\ \
~ \
\
\
Moo
,50
2o0
zso
30o
35o
I ql -,'- (MeV/c) Fig. 11. M o m e n t u m of the neutron in 2H determined by the reaction 2H(p, pn)p not corrected for instrumental broadening. T h e solid curve is the square of the H u l t h 6 n - Y a m a g u c h i wave function, and the dashed curve is the same function convoluted with the instrumental broadening function. The error bars are statistical in origin.
for heavier nuclei with a large n u m b e r of nucleons which can be treated statistically and where there is often no b r e a k u p of the residual nucleus. The probability of a second interaction by the fast proton is negligible; the probability of a scattering of the neutron on its way out was estimated, from the work of Haracz and Lim 23), t o be 0.07 for 2H and 0.16 for 3He. This is a relatively small correction, and approsimations which permit mathematical simplification are acceptable. We assumed that the second scattering is that of a neutron with its average energy ( = 6 0 MeV) off a proton at rest (which it is on the average). At these energies, there is only S-wave scattering which is isotropic in the c.m. frame. This results in a false m o m e n t u m c o m p o n e n t being added to the m o m e n t u m which the neutron had after it was first scattered and which is always perpendicular to the line joining the target position to the neutron counter. Thus a fraction k of the neutrons, which should be represented by a point in q-space, appears to be represented by a different point, and because the scattering is isotropic in the c.m. frame, rescattering distributes these points uniformly over a disc of radius qm where qm is the average m o m e n t u m of the scattered neutron. The true distribution ~b(q) and the observed distribution 6(q) are thus related by
G.F. Krebs et al. / M o m e n t u m distribution
380
1000019"o•
5ooo~-~ '~o~,
o,~
500
f >" 1-
"~ • ~
200
I00
N
•
5O 2C 0
i
I 50
I
l i I00
L i I t I i I 150 200 250 300
Iq I----~ (MeV/c) Fig. 12. Momentum distribution of the neutron in 3He determined from the reaction 3He(p, pn)p,p. A-measurements not corrected for instrumental broadening nor for rescattering; O - c o r r e c t e d for instrumental broadening; © - c o r r e c t e d for instrumental broadening and rescattering. Error bars are statistical in origin.
IQQOC oogag
6000 4000 200C
" _:~
"°°'~lt
~ AO0
l
100C
u~ Iz ILl > w
60C o 40C
03
tJJ 0
~o°Ooo &o
"°~al°°o
e
213(: 21~ ° e e
ICX2
~2 o- I o
6C 40
oo •
• o
AA~o0o 2C IC
AS'
I
1 50
I
I t I J I I I 100 150 20(3 250
I"1--"
t I i 300
•
°l°°
350
CM v/ I
Fig. 13. Summary of the number of events used to determine the momentum distributions.
G.F. Krebs et al. / Momentum distribution
381
the integral equation 6 ( q ) = (1
-k)6(q)+k
40(0)0 dO.
The first term on the right-hand side is the part of the momentum distribution which is not rescattered, and the second term is the "average" over the disc in q-space. The upper limit of the integral has been put equal to oo instead of qm to make this equation easily soluble; this is physically permissible, since qm is about 400 MeV/c, and the integrand Q4~(Q) is negligible even at these values of q. The lower limit is put equal to q, since any neutron counter sees only neutrons whose initial momentum q is close to being parpallel to the line joining the target to that counter, and the difference, if any, between q and the true lower limit is of second order. This equation, written in its canonical form, ~0(q) ~b(q)- 1 - k
2k I~ (1-k)q 2 ¢b(O)Q dO,
is well known as Volterra's equation. It is solved by first differentiating with respect to q to get a first-order linear differential equation for d~(q): d~b (q) dq
2k 1 d~b (q) ( 1 - k ) q ~ qd~(q) ( l - k ) dq
for which an integrating factor is exp ( - k q 2/(1 -- k)q 2 ). The integral which contains d&(q)/dq is converted into an integral containing 4~(q) by integration by parts, and the arbitrary constant is found by comparing the terms of an expansion, in powers of k for small k, of the original integral equation and the solution of the differential equation. Finally, the solution of the integral equation is found as
O(q) ~ 2 k e q2k/(1-k)q~ foe ~b(q) ( l - k ) (1-k)q2m J~
I[I(U)Ue -u2k/(l-k)q~ du
5. Results 5.1. D E U T E R I U M
The result from the momentum distribution of the proton in 2H as determined from the reaction 2H(p, 2p) is shown in fig. 10. The momentum distribution of the neutron in 2H from the 2H(p, pn) reaction is shown in fig. 11. These results have not been corrected for instrumental broadening nor for rescattering. In these two figures, the square modulus of the Hulth6n-Yamaguchi wave function 23) is plotted, as is also this same wave function broadened by the instrumental broadening function. It is easily seen that instrumental broadening is significant.
G.F. Krebs et al. / Momentum distribution
382 5C
~-
2,./,,p,ep J
IC
0
l ~ ,~'~
n-
0080
o
1
O.
O.lo i
I
5o
i
I
,oo
i
I
l
I
t
I
i
I
3oo
i
I
Iql Fig. 14. ( 3 - the ratio of the measurements of the m o m e n t u m distribution in 2H, as determined from the reaction 2H(p, 2p)n, to the broadened square of the Hulth6n-Yamaguchi wave function. Other points are the ratio of the m o m e n t u m distribution to the square of the Hulth6n-Yamaguchi wave function as reported, • - by Witten t7), and • - By Perdrisat u~). Error bars are statistical in origin.
A m o r e sensitive p r e s e n t a t i o n of the results is m a d e in figs, 14 and 15. In fig. 14 w e h a v e plotted the ratio of our data to the b r o a d e n e d square of the H u l t h 6 n Y a m a g u c h i w a v e function along with this ratio for the results of Perdrisat et al. lo) and W i t t e n et al. 17). T h e s e ratios are also s h o w n in fig. 14 for the 2H(p, 2p)n 40-
2H
2O I0
6
~ 2F
o12 ~ J
I
1 .... ~t
l
i
I
J
i
I
i
I
i
I
i
I
(MeV/c) Fig. 15. O - the ratio of the measurements of the m o m e n t u m distribution in 2H as determined from the reaction 2H(p, pn)p to the broadened square of the Hulth6n-Yamaguchi wave function. • - t h e ratio of the m o m e n t u m distribution to the square of the Hulth6n-Yamaguchi wave function as reported by Felder ls~. Error bars are statistical in origin.
G.F. Krebs et al. / Momentum distribution
383
reaction. W e have verified that the rescattering correction is quite negligible for 2H. Our results differ m o r e from the H u l t h 6 n - Y a m a g u c h i wave function than those of Witten's but are in closer agreement with the results of Perdrisat at large values of m o m e n t u m . The results from this experiment show m o r e structure at lower values of m o m e n t u m particularly from 50 M e V / c to 100 M e V / c . Witten's results show an abrupt change from the H u l t h 6 n - Y a m a g u c h i curve at about 260 M e V / c whereas our results as well as Perdrisat's show a gradual change starting from about 135 M e V / c . With the 2H(p, p n ) l H reaction much less data is available for comparison. In fig. 15 only the ratio from Felder's 15) results are shown for comparison with the results of this experiment. O u r results are in fairly close agreement with those of Felder's. Again we see m o r e structure at low values of m o m e n t u m in the results of our experiment. The higher densities at large m o m e n t u m in Felder's results may be due to a persistent neutron background 15). I0000 5OO0 (13 I--
0ZXd. 88~
2OOO
D lad
..-I LIJ rlr"
T
>. t..-
I000 50O
e~
2OO I00
Z UJ a
5O
2O 0
i
t
i
[
I
Ioo
J
i
I
200
i
i
t
I
300
lql ~(MeV/c) Fig. 16. T h e m o m e n t u m distribution of the neutron in 3He, showing the effect of deconvolution and rescattering. A-original m e a s u r e m e n t s , O - m e a s u r e m e n t s deconvoluted to remove the instrumental broadening, and C) - after correction for rescattering.
5.2. 3He
The results for the m o m e n t u m distribution of the neutron in 3He are shown in fig. 16 along with the result after removing the instrumental broadening by the usual deconvolution process. In the deconvolution procedure only the first six harmonics were retained; using more than six caused the result to "blow u p " from noise. The deconvoluted m o m e n t u m distribution was then corrected for rescattering. The effect of rescattering is also shown in fig. 16 and is seen to be a small
G.F. Krebs et al. / Momentum distribution
384
TABLE 1 The momentum density ( M e V / c ) 3 of the neutron in 3He; A, original measurments; B, corrected for instrumental broadening; C, corrected also for rescattering; and D, normalized to unit density at momentum q - 0 q
A
B
C
D
5 15 25 35 45 55 65 75 85 95 105 115 125 135 145 155 165 175 185 195 205 215 225 235 245 255 265 275 285 295
139193 459121 720567 768547 586993 541830 507215 464761 417048 367177 318372 286433 240423 234344 199614 169711 151022 128346 117638 122540 94074 80856 68617 65588 49824 35461 28411 27046 20856 19085
953379 927934 879246 811429 729963 641009 550675 464353 386227 318992 263814 220496 187782 163753 146208 133000 122276 112609 103047 93072 82534 71551 60414 49505 39226 29932 21894 15262 10044 6198
1111300 1081389 1024161 944454 848720 744213 638126 536806 445186 366440 301946 251470 213535 185873 165885 151029 139107 128426 117843 106737 94914 82501 69837 57366 45560 34846 25549 17855 11784 7190
0.953379 0.927718 0.878622 0.810242 0.728113 0.638457 0.547445 0.460524 0.381923 0.314367 0.259038 0.215735 0.183191 0.159460 0.142313 0.129567 0.119340 0.110176 0.101098 0.091569 0.081428 0.070777 0.059910 0.049814 0.039086 0.029894 0.021918 0.015318 0.010110 0.006109
effect. The values of the momentum distributions for the results shown in fig. 16 are given in table 1. Note that the dip at zero momentum in the original data is an instrumental effect which is removed by deconvolution. A comparison was made between three trial S-wave wave functions and the momentum distribution corrected for instrumental broadening and rescattering. The square of the trial wave functions are gaussian: Irving:
I~(q)l 2 = N l
e
2q2/9c~2
,
I~b(q)12= N 2 / ( a 2 + q2) 11/2 ,
Irving-Gunn:
1~ (q)l 2 = N3/(0¢ 2 / 4 + q 2 ) 7 / 2 ,
385
G.F. Krebs et al. / Momentum distribution
TABLE 2 Parameters for the fitting of three proposed wave functions to the experimentally determined momentum distribution gaussian N o~ c~'
X2 (reduced)
Irving-Gunn
0.67 + 0.04 0.30+0.01fm a 0 . 3 8 5 fm a
88
2.27 ± 0.4 0.89+0.03fm 0.77 fm 1
37
Irving
1
0.54 + 0.2 1.22±0.04fm 1.26 fm 1
1
52
T h e p a r a m e t e r s N a n d a are d e f i n e d in the text; a ' is the p a r a m e t e r a d e t e r m i n e d by e l e c t r o n s c a t t e r i n g 27).
where N1, N2, N3 are normalization constants. A t w o - p a r a m e t e r fit was carried out with each of these functions to our corrected distribution and the results are shown in table 2. T h e y indicate the best fit with the I r v i n g - G u n n function. Fig. 17 shows the m o m e n t u m distribution of the neutron in 3He after corrections for instrumental broadening and rescattering, along with the squared modulus of the Irving-Gunn wave function. The ratios of our corrected m o m e n t u m distribution to the three functions above are shown in fig. 18. Shown along with our data are three data points - normalized to our data - from the work of Kitching 25) on the reaction 3He(p, 2p)2H.
6. Conclusion We have described an experiment in which the m o m e n t u m distribution of the neutron in 2H and 3He was measured using the quasi-elastic scattering of fast protons. This is the square modulus of the neutron wave function in m o m e n t u m space. In the case of 2H, much theoretical and experimental work has been done on this problem. Our m e a s u r e m e n t s agree very well with other m e a s u r e m e n t s at m o m e n t u m values greater than 100 M e V / c , but are significantly lower around 50 M e V / c . In all cases, there is a discrepancy between the theoretical wave function and the experimental results. In the case of 3He, there are no other experimental results to compare ours with. Further, there is no soundly based theoretical expression for this m o m e n t u m distribution. Those which have been proposed have been no more than plausible expressions chosen for their analytical tractability. All the same, our m e a s u r e m e n t s fit the proposed analytic expressions with p a r a m t e r s very close to those determined by Collard et al. 26) by electron scattering, and by Kitching et al. 25) for the wave function of the proton in 3He from the reaction 3He(p, 2p)2H. The agreement with these analytic expressions is, however, not close. A noticeable feature of our results is a significant dip at m o m e n t a around 100 M e V / c , and a hint of this trend can be
386
G.F. Krebs et al. / Momentum distribution
1.0 06
•°oOO
0.3 -
•l•
o~ --
O.
•O
6 I
O.C~ -
f
>-
i-
0.03
(.t) z
hi d21 0.01 0.006 0.007
1
I
1
1
I
1
50
I00
150
200
250
300
Iql - - ~
(M~V/c)
Fig. 17. The m o m e n t u m distribution of the neutron in 3He and the best fit of the square of the Irving-Gunn wave function. Error bars are statistical in origin.
G~USSIAN ***~.
30120~
.-*
llJoi
[ 0
iIl
I
--°-*',-ooeeeoo*
. . . .
. . . . . . . . . .
2.0.---
t
"B-~-~
, o
0
or"
IHVING- G U N N
20
..**tetl~
1
I0
t
0.6
L
I
I
1
tO0
Iql
,
I
,
1
t
200
I
~
I
300
-(MeV/c)
Fig. 18, The ratio of the m o m e n t u m distribution of the neutron in 3He to the best fit of the square of various wave functions proposed for the system• The measurements by Kitching et al. 28) using the • 3 reaction - He(p, 2p)p,n are shown as open circles. The error bars are statistical in origin.
G.F. Krebs et al. / Momentum distribution
387
seen in the m e a s u r e m e n t s of K i t c h i n g et al. 25). W e can at p r e s e n t offer n o e x p l a n a tion of this a n d regard it as a n o t h e r i n t e r e s t i n g p r o b l e m in theoretical n u c l e a r physics to be e x p l a i n e d at a later data. T h e financial s u p p o r t of the N a t i o n a l Sciences a n d E n g i n e e r i n g R e s e a r c h C o u n c i l of C a n a d a is gratefully a c k n o w l e d g e d . O n e of us ( G F K ) was the recipient of a T o r o n t o O p e n F e l l o w s h i p a n d the J.C. Stevens A w a r d . W e are i n d e b t e d to m a n y p e o p l e o n the staff of the L a w r e n c e B e r k e l e y L a b o r a t o r y for facilitating this e x p e r i m e n t a n d especially to Dr. O. C h a m b e r l a i n who e n c o u r a g e d us to do this e x p e r i m e n t a n d lent us the n e u t r o n counters. T h e advice a n d help of m a n y others is a c k n o w l e d g e d with thanks, in paricular, Dr. K. Crowe a n d J.F. Vale, the staff of the 184 in. s y n c h r o c y c l o t r o n , a n d D. N e l s o n of the m a g n e t group. This e x p e r i m e n t w o u l d not have b e e n possible w i t h o u t extensive c o n t r i b u t i o n s f r o m the technical staff, a n d others, of the e x p e r i m e n t a l team. In p a r t i c u l a r we w o u l d like to express o u r t h a n k s to H. C o o m b e s , E.I. D e n n i g , Dr. S. F r i e d l a n d e r , W. Hilger, K . U . Kaires, Professor R. Kreps, A . K . C . Kiang, Professor J. M c A n d r e w , Professor J.D. P r e n t i c e a n d Dr. R.R. W h i t n e y .
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
13)
14) 15)
R. Serber, Phys. Rev. 72 (1947) 1114 A.F. Kuckes, R. Wilson and P.F. Cooper Jr., Ann. of Phys. 15 (1961) 193 D.G. Stairs, R. Wilson and P.F. Cooper Jr., Phys. Rev. 129 (1963) 1672 M. Morlet, R. Frascaria, B. Geoffrion, N. Marty, B. Tatischeff and A. Willis, Nucl. Phys. A129 (1969) 177 F. Takeutchi, T. Yuasa, K. Kuroda and Y. Sakamoto, Nucl. Phys. A152 (1970) 434 F. Takeutchi, Y. Sakamoto and Y. Yuasa, Phys. Lett. 35B (1971) 498 M. Morlet, R. Frascaria, N. Marty and A. Willis, Nucl. Phys. A191 (1972) 385 J.P. Didelez, H. Nakamura, F. Reide, T. Yuasa, I.D. Goldman and E. Hournay, Phys. Rev. C10 (1974) 529 H. Tyr6n, S. Kullander, O. Sunberg, R. Ramachandran, P. Isacsson and T. Berggren, Nucl. Phys. 79 (1966) 321 C.F. Perdrisat, L.W. Swenson, P.C. Gugelot, E.T. Boschitz, W.K. Roberts, J.S. Vincent and J.R. Priest, Phys. Rev. 187 (1969) 1201 G.A. Leskin, JETP (Sov. Phys.) 5 (1957) 371 B.S. Aladashvili, B. Badelek, V.V. Glagolev, R.M. Lebedev, J. Nassalski, M.S. Nioradze, I.S. Saitov, A. Sandacz, T. Siemiarczuk, J. Stepaniak, V.N. Streltsov and P. Zielinski, Nucl. Phys. B86 (1975) 461 B.S. Aladashvili, B, Badelek, V.V. Glagolev, R.M. Lebedev, J. Nassalski, M.S. Nioradze, G. Odyniec, I.S. Satov, A. Sandacz, T. Siemiarczuk, J. Stepaniak, V.N. Streltsov and P. Zielinski, Nucl. Phys. B92 (1975) 189 W.D. Simpson, J.L. Friedes, H. Palevski, R,J. Sutter, G.W. Bennet, B. Gotschalk, G. Igo, R.L. Stearns, N.S. Wall, D.M. Corley and G.C. Phillips, Nucl. Phys. A140 (1970) 201 R.D. Felder, T.R. Witten, T.M. Williams, M. Furic, F.S. Mutchler, N,D. Gabitzsch, J. HudomaljGabitzsch, J.M. Clement Jr., G.C. Phillips, E.V. Hungerford, L.Y. Lee, M. Warneke, B.W. Mayes and J.C. Allred, Nucl. Phys. A264 (1976) 397
388
G.F. Krebs et al. / M o m e n t u m distribution
16) V.K.C. Cheng and P.G. Roos, Nucl. Phys. A225 (1974) 397 17) T.R. Witten, M. Furic, G.S. Mutchler, C.R. Fletcher, N.D. Gabitzsch, G.C. Phillips, J. Hudomalj, L.Y Lee, B.W. Mayes, J. Allred and C. Goodman, Nucl. Phys. A254 (1975) 269 18) J.M. Wallace, Phys. Rev. C5 (1972) 609 19) P. Kirkby, J.M. Daniels, T. Gajdicar, S. Friedlander, G.F. Krebs, H. Coombes, E.I. Dennig and K. Kaires, 5th Int. Conf. on high energy physics and nuclear structure, Uppsala 1973, ed. G. Tibell (North-Holland, Amsterdam, 1974) 383 20) B. Fagerstr6m, A. Johansson, G. Tibell, G. Landaud, F. Lemeilleur, J. Yonnet, A. Cordaillat, S. Kullander and P.U. Renberg, Nucl. Instr. 100 (1972) 237 21) SPECTO, Lawrence Radiation Laboratory Computer Library 22) R.M. Edelstein, J.S. Russ, R.C. Thatcher, M. Elfield, E.L. Miller, N.W. Reay, N.R. Stanton, M.A. Abolins, M.T. Lin, K.W. Edwards and D.R. Gill, Nucl. Instr. 100 (1972) 355 23) R.D. Haracz and T.K. Lim, Phys. Rev. Lett. 31 (1973) 1263 24) G. Jacob and Th.A.J. Maris, Rev. Mod. Phys. 45 (1973) 6 25) P. Kitching, G.A. Moss, W.C. Olsen, W.J. Roberts, J.C. Alder, W. Dollhopt, W.J. Kossler, C.F. Perdrisat, D.R. Lehman and J.R. Priest, Phys. Rev. C6 (1972) 769 26) H. Collard, R. Hofstadter, A. Johnsson, R. Parks, M. Ryneveld, A. Walker, M.R. Yearian, R.B. Dayan and R.T. Wagner, Phys. Rev. Lett. 11 (1963) 132 27) R. Frascaria, Nucl. Phys. A178 (1971) 307
THE
MOMENTUM ‘He
DISTRIBUTION
AS DETERMINED
OF THE
NEUTRON
BY QUASI-ELASTIC
IN *H AND
SCATTERING
GARY F. KREBS*, C. COX**, J.M. DANIELS, T. GAJDICAR***, P. KIRKBY**** and A.D. MAY Dept. of Physics, University of Toronto, Toronto, Ontario, Canada M5S lA7 Received 29 March 1982 (Revised 17 September 1982) Abstract: The momentum distribution of the neutron in ‘H was determined in the range 0 to 365 MeV/c and in 3He in the range O-295 MeV/c, by quasi-elastic scattering of 667 MeV/c protons. The measurements were analyzed by the plane wave impulse approximation and a correction was made for multiple scattering in the target nucleons. The results for ‘H fit the HulthCn-Yamaguchi wave function quite well but show deviations similar to those found by other workers. For ‘He there are no previously published measurements for comparison, but the results fit several expressions for the wave function, proposed for reasons of plausibility and analytical tractability, with parameters close to those obtained by electron scattering. The momentum distribution in 3He has a small but significant dip around 100 MeV/c which is not found in any of the theoretical expressions.
E
NUCLEAR REACTIONS 3He(p, pn), *H(p, pn), relative a(E,, E,, ep, &, On,4,); deduced momentum and for p in ‘H.
(p, 2p), E = 667 MeV; measured distributions for n in 3He and ‘H
1. Introduction ‘Quasi-elastic extensively momentum
scattering
(QES) of protons
off very light muclei
has been studied
in recent years since Serber ‘) suggested that it would reveal the internal of the nucleons within the nucleus. By measuring the momentum of
the scattered particle one may then deduce the momentum-space wave function of the ejected nucleon within the target nucleus. A great deal of data has been collected around incident energies of 150 MeV [refs. ‘-‘)I for the *H(p, 2p)n reaction. In addition, there are measurements at the higher energies of 460 MeV [ ref. 9)], 600 MeV [ref. *‘)I, 660 MeV [ref. “)I, 970 MeV [refs. 12,13)], 1000 MeV [ref. 14)], and at 585 MeV [ref. “)I. The reaction *H(p, pn)‘H has been studied far less extensively than the *H(p, 2p)n reaction due to the inherent difficulty of * This work is part of a thesis by GFK, current address: Lawrence Berkeley Laboratory, University of California, Berkeley, California, USA 94720. ** Current address: Advanced Product Operations, SingeriLink, Sunnyvale, Calif., USA 94086. *** Current address: Computing Devices Co., Ottawa, Ontario, Canada KlG 3M9. **** Current address: Research Division, Ontario Hydro, Toronto, Ontario, Canada M8Z 5S4. 364
365
G.F. Krebs et al. / M o m e n t u m distribution
detecting the neutron. All existing measurements were made with proton energies below 200 MeV [ref. 8)] except for the study at 585 MeV and 800 MeV [ref. ~5)]. The reaction 3He(p, 2p) was studied at 590 MeV [ref. 25)] but only up to a Fermi momentum of 75 MeV/c, and no measurements have yet been published for the momentum distribution of the neutron in 3He. This paper describes a kinematically complete asymmetric experiment and presents unnormalized momentum distributions for the proton and neutron in 2H up to 365 MeV/c using the reaction 2H (p, pn) and up to 325 MeV/c using the reaction 2H(p, 2p), as well as the unnormalized momentum distribution for the neutron in 3He up to target nucleon momentum of 295 MeV/c. In describing the (p, 2p) or (p, pn) reactions the plane wave impulse approximation is usually used. In this simple description one assumes the incident projectile interacts elastically with only one of the target nucleons. Modifications of this simple description have been made to take into account effects due to multiple scattering and final-state interactions 17,18). We have used the plane wave impulse approximation to evaluate our data, but we have taken account of shadowing and rescattering in a different way from the way this is usually done. A report of the preliminary analysis of this work was presented at the Fifth International Conference on High Energy Physics and Nuclear Structure ~9).
2. Apparatus This experiment was carried out at the Lawrence Berkeley Laboratory using a 667 MeV proton beam from the 184 inch synchrocyclotron. The experimental arrangement is shown in fig. 1. The beam was incident on a cylindrical liquid deuterium target with dimensions 2.5 cm diameter and 5.1 cm long. The liquid 3He C2 Y- DIRECTION
~
~RO'1"O~4 ~' r(~_m_~-~\\\i~~ •
RING- - r,, X-DIRECTION OOUNTER (BV)
-
[
~
~'SPARK ~ CHAMBERS
C3
!!!~o
M2
V~9 SPARK"
~
MII M3
II I-"---,,p,#
S ,,T,O I CHAMBER
6 M5
~..
~ /VETO
COUNTERS(VI-V6)
~"~-I ~ " ~ x\x NEUTRON I J [ | .~' \ \ \ \ ' - - " COUNTERS
Ii IiI1|~~ , , ' " Fig. 1. Plan view of the experiment.
ll2inl
366
G.F. Krebs et al. / M o m e n t u m distribution
cylindrical target was 8.9 cm diameter and 15.2 cm long. Both targets were constructed of kapton and were centered at the beam's final focus with the long axis oriented vertically. Located just in front of the rarget was an N E 102 plastic scintillator veto counter (BV) which had a 5.1 cm diamter hole in its center and was placed so that the incident beam passed through this hole. This served to remove from analysis stray events not originating from the beam. A spark chamberbending magnet proton spectrometer was used to measure the m o m e n t u m and angle of the "fast" scattered proton. The knocked out neutron in the (p, pn) reaction (or proton in the (p, 2p) reaction) was detected in a spectrometer consisting of a bank of Pilot Y plastic scintillator bars. Beam monitoring was carried out with plastic scintillation counters M 1 - M 6 placed 6.1 m downstream behind a 0.64 cm Cu secondary target. The beam was centered on the target by monitoring a split-ion chamber in front of the Cu target. Typically the (p, 2p) and (p, pn) data were collected with incident beam intensities of 1 0 7 and 10 s protons/sec, respectively.
2.1. THE PROTON SPECTROMETER The proton spectrometer consisted of 6 entrance and 6 exit spark chambers before and after a 1 6 t o n C-magnet. In addition the spectrometer had three scintillation counters C1, C2 and C3. Each chamber had horizontal and vertical wires spaced 0.16 cm apart. The windows were constructed of kapton, and a gas mixture of 90% neon and 10% argon was circulated through the chambers. The aperture of the proton spectrometer was defined by the first spark chamber and had a value of ±8 ° horizontally and +2 ° vertically. The chambers gave digitized horizontal and vertical positions for the scattered proton using magnetostrictive readout followed by an A D C . The resolution of the chambers was found to be ±0.6 mm horizontally and +1.0 mm vertically. Scintillation counters C1, C2 and C3 were slabs of N E 102 plastic. Protons passing through the spectrometer were signalled by a coincidence in these counters. The vertical magnetic field component of the C-magnet was measured at 2.5 cm intervals in 6 horizontal p l a n e s - t h e average magnetic field strength was 7.5 kG. The two remaining mutually perpendicular components of the magnetic field were determined using Maxwell's equations 20).
2.2. THE NEUTRON SPECTROMETER The neutron counters measured the position and m o m e n t u m of the knocked out neutron by the time-of-ffight (TOF) technique. The neutron counter bank consisted of 30 bars of Pilot Y scintillation plastic with dimensions 2.5 cm wide x 15.2 cm deep × 96.7 cm high with photomultiplier tubes ( R C A 7585) at each end. These were arranged in three rows of ten as shown in fig. 1. The front surface of the rows were located at 144.7 cm, 165.1 cm and 185.4 cm from the center of the target.
G.F. Krebs et al. / Momentum distribution
367
Six veto counters, V 1 - V 6 , covered the front of the bank of neutron counters to detect and, if desired, to reject charged particles. Two levels of discrimination were used on the output pulses from the neutron counter photomultiplier tubes; a low level to give timing pulses relatively free of jitter and a higher level to give a pulse suitable for particle identification. The photomultiplier voltages and discriminator levels were set using the scintillations produced by a 6°Co y-ray source.
2.3. LOGIC A possible Q E S event was first signalled by (C1 • C 2 . C3 • BV), a coincidence in the pulses from counters C1, C2 and C3 of the proton spectrometer and no coincident pulse from the b e a m veto counter. The logic circuits then opened a 100 ns gate in which to detect a neutron in the neutron counter bank. The leading edge of the pulse from C1 started a four-channel time-to-digital converter (TDC) which was stopped by signals from the neutron counter bank. The four channels of the T D C were stopped, respectively, by signals from the b o t t o m photomultipliers and from the top photomultipliers of each of the three rows of ten neutron counters. H a r d wired logic identified which counter had stopped the T D C , and also identified ambiguous or invalid events. Such ambiguous or invalid events might be signalled by a pulse from the veto counters V 1 - V 6 indicating that the pulse detected by the neutron counter was a charged particle, or a pulse from two neutron counters in the same row. When the conditions for an apparantly valid event were met, the high voltage was applied to the spark chamber planes. For each event the spark coordinates and the fiducial positions of the 12 spark chambers, and two timing m e a s u r e m e n t s from either end of a neutron counter were recorded by an on-line P D P - 9 computer. Typically 8 events/sec were recorded on magnetic tape.
3. Experimental details 3,1. OPERATION AND CALIABRATION OF THE PROTON SPECTROMETER The positions of the spark chambers were determined by surveying. However, it was found during analysis that there were small errors in these positions. For example, the apparent position of the spark in one chamber might be systematically off the line defined by the sparks in the other chambers. Small corrections were made to the surveyed positions to eliminate these errors. The efficiency of the chambers was determined in two ways. First, the n u m b e r of sparks was c o m p a r e d with the n u m b e r of pulses from a small plastic scintillator positioned just in front of the chamber; the efficiency of a plastic scintillator for the detection of 660 M e V protons was taken to be unity. Secondly, the efficiency was found by counting the n u m b e r of sparks missing from a track. Using the criteria that a track must be defined by at least three collinear sparks, and at least one from each end of the
368
G.F. Krebs et al. / Momentum distribution
bank of six chambers, the overall efficiency was found to be 98%, to be slightly dependent on position and to be independent of proton energy. The computer p r o g r a m S P E C T O 21) was used to determine the m o m e n t u m of the scattered proton. The overall resolution of the proton spectrometer was found to be 1%. The acceptance of the proton spectrometer was determined by a Monte Carlo simulation. Fast scattered protons were generated with different directions and m o m e n t a and different origins within the target volume. The trajectory through the proton spectrometer was determined from the known magnetic field. Acceptance occurred when the 6 exit spark chambers were able to record the resulting trajectory. Fig. 2 shows the acceptance region for protons scattered from the target center and indicates h o m o g e n e o u s acceptance by the proton spectrometer between scattering angles of 12.5 ° to 17.5 ° when the m o m e n t u m is greater than 950 M e V / c . An artificial entrance window, smaller than the physical window, was imposed on the proton spectrometer to ensure that all protons within this window would be counted with known efficiency without vignetting.
3.2. THE NEUTRON COUNTERS The horizontal scattering angle was determined to within +0.5 ° by noting the neutron counter in which the neutron was detected. The T O F and the vertical scattering angle were obtained from the times tu and t~ at which the pulses from the u p p e r and lower photomultiplier tubes stopped the T D C . We can put tu = T - D l v p + l u / v
tl =
T-D/vp+
+ Tu,
ll/v+ TI,
1400
/
//
12007-/ / MOMENTUMOF - ' ~ 7 O III
I/I
IO00r-i I 111
I I I
REGION OF / / ~ ACCEPTANCE~ / / ~ ~
;-//
i/i 80o 6/ / / I
IO 12 14 16 18 20 SCATTERING ANGLE- DEGREES ( 81 ) Fig. 2. Result of calculations of tracks through the proton spectrometer. Protons whose entrance coordinates fall in the shaded region do not pass all the way through the spectrometer.
G.F. Krebs et al. / M o m e n t u m distribution
369
where T is the TOF, D is the distance from the target to the counter C1, and Vp is the velocity of the fast proton. The distance from the point where the scintillation is produced to the top and bottom of the plastic bar are, respectively, lu and l~, and v is the effective velocity of light in the plastic. Tu and T~ are times which represent the delay in the signals passing through the system and include a common quantity which is the delay time of the pulse from C1 to the TDC. Rearranging these we find 2 ( T - D / v p ) = (q + t ~ ) - (11+ lu)/V - T 1 - T~ = (q + t~) - T~ , 1 - (l,1)
lu) = ( t , -
tu) - ( T , - T u ) = ( q -
G) - Td,
where Ts = T~+ Tu and Ta = T ~ - Tu. Thus, in order to find the T O F T and the vertical position ½(l~-l,) where the neutron is detected, we need three calibration constants, Ts, Td and v, for each neutron counter. Two small scintillators T1 and T2, 2.5 cm x 1.3 cm x 0.64 cm, were placed in front of the centre of a neutron counter (lu = 10. Protons scattered from a polythene target were counted by the neutron counters, the logic being triggered by a coincidence in T1 and T2, and the T D C being started by the leading edge of T1. Then the difference in tu and t~ gave the calibration constant Td. By moving the counters T1 and T2 up and down the bar the effective velocity of light in the plastic
o3 Z 8
_
6
-.-~4 20
-40
I
I
-20
I
0 Z
20
I
40
(cm)
"-
Fig. 3. Measurement of the times tu and fi of arrival of a pulse from the upper and lower photomultiplier tubes of a typical neutron counter as a function of the height z above the center of the scintillator. From the slope of these lines the effective velocity of light in the scintillator was determined to be 13.02 cm/ns. Error bars are statistical in origin.
was determined to be ( 1 . 3 0 2 + 0 . 0 1 3 ) x 101°cm/sec (See fig. 3). By leading the pulse from T1 along the same cable as was used for C1, the value of Ts was given as Ts-- - ( q + tu). From the scatter in the values of tu and td the standard deviations of the derived quantities and T were found; these are shown in fig. 4, and are typically 2.3 cm for the vertical position z and 1.3 ns for T for events in the centre
G.F. Krebs et al. / Momentum distribution
370
~3 I
o 2
Q)
~,4 J -40
I -20
1 0
Z n
I b'
-40
G
,
-20
I 20
I 40
- (cm) Ou
u
I
t
I
0
20
40
" (cm) Fig. 4. R e s o l u t i o n in the d e t e r m i n a t i o n of the vertical p o s i t i o n z a n d the time-of-flight T of a scintillation. T h e s c a t t e r was f o u n d to b e g a u s s i a n ; ~z and O'T are the r e s p e c t i v e s t a n d a r d d e v i a t i o n s .
part of the neutron counters. This calibration was later refined during the analysis of the results of the experiments. W h e n protons are scattered from a polythene target this scattering is coplanar; in this way the zero of z was related to the nominally horizontal plane defined by the b e a m and the proton spectrometer. The T O F calibration was refined by analysing the 2H(p, pn) measurements. Assuming that Ts is not known, then in this reaction the energy and absolute value of the m o m e n t u m of the scattered neutron are unknown as is the m o m e n t u m of the spectator proton and the m o m e n t u m q of the scattered neutron before the collision, a total of nine unknown quantities. In addition to the four equations of conservation of energy and m o m e n t u m , the m o m e n t u m of the spectator proton is - q giving three equations. Two m o r e equations of the form E 2 = p 2C 2 _4_m 2C4 relate the energy and m o m e n t u m of the spectoator proton and the scattered neutron. Thus the event can be solved completely and Ts determined. T h e neutron counter efficiency was determined using the computer program of Stanton 22). The results for our discriminator bias setting and geometry are shown in fig. 5. In addition, the neutron counter efficiency was checked at several energies by comparing the counting rates between the 2H(p, 2p) and 2H(p, pn) reactions. The ratio of these counting rates is equal to the product of the ratio of the scattering cross sections, which are known for free nucleons, and the ratio of counter efficiencies. Assuming the value of the efficiency for proton detection to be unity, the results obtained from the experiment were found to be consistent to within 10% of the calculations shown in fig. 5. A further check of the neutron counter efficiency was obtained from the analysis of the 2H(p, p n ) l H and 3He(p, pn)p,p reactions and will be explained in sect. 4.
G.F. Krebs et al. / Momentum distribution
371
02-
0
I 2
J I I 5 4 5
I I0
NEUTRON
I I 20 50
I 50
I I I I00 200
ENERGY
~
I 500
1 I000
(MeV)
Fig. 5. The efficiency of a typical neutron counter for our discriminator bias as determined by the Monte Carlo program of Kurz-Stanton 23.
I000 800 600 Kz 400 - 4 ~ 20C 0 -40
-20
0 Kz
20 J-
ToooF
-20
40
0
20
40
MeVlc
T
]
I-z 200 hi
0
1272
F302 13:52 Kx ~ MeV/c
1287
1327
T -40
-20
0 Ky
20 ~
40 -20 MeV/c
0
20
40
Fig. 6. The distribution of the components of b e a m m o m e n t u m , Kx, K,. and K~: (a) as measured experimentally, (b) results of the Monte Carlo calculation to determine instrumental broadening, and (c) values for the beam with the instrumental broadening removed.
372
G.F. Krebs et al. / Momentum distribution
3.3. THE BEAM The distribution of b e a m m o m e n t u m was determined from the reaction 1H(p, p ) l H using a liquid hydrogen target. The scattering angle and energy as determined by the proton spectrometer, along with the scattering angle determined from the neutron counter, are sufficient to calculate the three mutually perpendicular b e a m m o m e n t u m components. The curves labelled " a " in fig. 6 show the distribution of m o m e n t u m for the c o m p o n e n t s K~, Ky and Kz which have directions along the b e a m line, perpendicular to the b e a m line in the horizontal plane, and vertical, respectively. Folded into these distributions are the effects of the unknown exact interaction points in the target and neutron counter. This instrumental broadening effect was calculated by a Monte Carlo program. Events were generated by choosing an interaction point at r a n d o m in the target volume. The point of interaction in the neutron counter was chosen on the horizontal centre line across the face of a neutron counter and was then m o v e d in a vertical direction according to the uncertainty in the determination of the z-coordinate derived from the timing calibration. The b e a m was taken to be monoenergetic and with no divergence. The event was then analysed by the kinematic reconstruction program which assumes that the scattering takes place at the point where the fast proton track intersects the vertical plane through the b e a m axis, and that the interaction with the neutron counter is at the center of the front face (for protons). The components of b e a m m o m e n t u m resulting from this analysis are shown as curve (b) in fig. 6. The true distribution of b e a m m o m e n t u m components after unfolding are shown as curves (c) in fig. 6. They have standard deviations of 23 M e V / c for K x , 44 M e V / c for Kv and 7 M e V / c for K~. 3.4. BACKGROUND The principle c o m p o n e n t of the background was due to stray particles being detected by the neutron counters while the 100 ns gate was open. This background was measured with the target empty and with the timing gate open all the time. T h e background rate was found to be proportional to the b e a m current, confirming its origin. Typically, in an actual run, each neutron counter registered about 14 background counts per hour.
4. Data analysis 4.1. CALCULATION PROCEDURES Approximately 70% of the events collected were rejected; the approximate percentages rejected for various reasons were: (i) Approximately 5% because more than one neutron counter detected these events leading to an irresolvable ambiguity.
G.F. Krebs et al. / M o m e n t u m distribution
373
(ii) A p p r o x i m a t e l y 50% because the proton track was outside the window of the proton spectrometer. (iii) A p p r o x i m a t e l y 5% because the trajectory of the fast proton did not intersect with the target volume. (iv) A p p r o x i m a t e l y 10% because the track through the proton spectrometer could not be fitted. (v) Approximately 3% because they were identified as elastic scattering without b r e a k u p of the target nucleus. Those events not rejected for the first four reasons were analyzed kinematically. The analysis procedure assumed that the b e a m was h o m o g e n e o u s and directed along the x-axis, that the scintillation produced by the neutron occurred at the center of a horizontal cross-section of the neutron counter, and that the point of scattering was where the track of the fast proton intersected the vertical plane through the b e a m axis provided this point lay in the target volume. The laws of conservation of m o m e n t u m were applied to the collision, assuming this to be an elastic scattering of a free neutron, to find the m o m e n t u m q of this neutron before the collision. T h e values obtained by this calculation are not the true ones because the analysis assumes an ideal experiment; deviations from ideal conditions mentioned earlier are responsible for the values of q so obtained being scattered about their true values. This scattering, or instrumental broadening, was determined by a Monte Carlo calculation, and its effects were r e m o v e d from the final result by a standard deconvolution process. Events were generated assuming some fixed value of q but choosing the point of scattering, the m o m e n t u m of the beam, and the point of interaction of the neutron in the neutron counter at r a n d o m according to the distribution of these quantities already determined. Those events were then analyzed by the analysis routine to find q. It was found that the scatter in the values of q was independent of q and could be described by a trivariant gaussian distribution in which the standard deviations were 23 M e V / c , 44 M e V / c and 7 M e V / c in the x-, y- and z-directions, respectively. The principle axes of this distribution were coincident with the x-, y- and z-axes within experimental error. It was obvious that still about 3% of the events recorded were spurious. Some of these events were identified by the circumstance that the kinematic analysis, which requires the solution of a quadratic equation, gave no real values for IK21, the m o m e n t u m of the scattered neutron. In other cases, they could be distinguished because the distribution of fast proton m o m e n t a , Igll was doubled humped. It was not possible to reject events which fell in the second peak of IKll because in many cases this p e a k was not resolved from the main peak and, in m a n y cases, sat on the top of a significant tail of the main peak. However, calculated values of qx and qy also showed double peaks. We therefore required that all acceptable events had to fall within generously assigned ranges of [K2[, qx and qy. This removed all the spurious events without cutting off the tail of the distribution of desired events.
374
G.F. Krebs et al. / Momentum distribution
Trials with different ranges of acceptance of [K21, q~ and qv showed that these could be chosen in a way which separated out cleanly the unwanted events. The identification of the origin of these events is peripheral to the object of our investigation. However, we notice that they are caused by some sort of scattering since they produce a peak in the fast proton distribution, and that they are quite rare, being observed only at large "neutron scattering angles", 0,, where the cross section for the reaction p(2H, pn)p is very low, of the order of one thousandth of the greatest value we were able to observe. The extra peak is kinematically consistent with elastic scattering of protons off deuterons with no breakup of the deuteron. We believe that these events are produced by a fast proton from 2H(p, p)2H scattering in chance coincidence with a background deuteron from some other scattering event. The method of determining the m o m e n t u m distribution P (q) is basically to count the number of events with a given value of q. However, in the number of events which are to be counted there are in addition to P (q) several other factors, some of which are unknown. The method of analysis was designed to eliminate these factors. First, the absolute value of the efficiency of the neutron counters was not well known. We therefore selected the center portion ( - 1 0 cm~
G.F. Krebs et al. / M o m e n t u m distribution
375
qy(MeV/c)
-E~OI/ I D( { I( (R''~'~ '!1~/ BEAMI ECTION-~' 8()-(MeV/c)qx
28* Fig. 7. The phase space in the x-y plane seen by the neutron counter at -70 °, with the proton spectrometer entrance path centered at 15°. Lines of constant c.m. angle 0* are shown. The two shaded areas represent the two parts of phase space visible to a given neutron counter with a given 0* bite for which the momentum of the target neutron has the same absolute value.
calculations of n e u t r o n c o u n t e r efficiency d e p i c t e d in fig. 5 are c o n s i s t e n t to within a few p e r c e n t - 10% in the worst case - with these e x p e r i m e n t a l l y d e t e r m i n e d ratios. A m o m e n t u m d i s t r i b u t i o n p (q) was o b t a i n e d for each n e u t r o n c o u n t e r a n d each c.m. angle by t a k i n g the n u m b e r of satisfactory events recorded, s u b t r a c t i n g the b a c k g r o u n d , weighting according to the efficiencies of the p r o t o n s p e c t r o m e t e r a n d the n e u t r o n c o u n t e r s , a n d dividing this result by the v o l u m e of the visible phase space. T h e s e d i s t r i b u t i o n s did n o t each cover the full r a n g e of q, n o r did they all cover the s a m e range. T h e y were pieced t o g e t h e r to form a final e x p e r i m e n t a l d i s t r i b u t i o n in the following way. T h e partial d i s t r i b u t i o n s f r o m all the n e u t r o n c o u n t e r s for a given c.m. angle were c o m b i n e d to give a full r a n g e d i s t r i b u t i o n d e r i v e d from scattering events with that c.m. angle. In this process, the a b s o l u t e values of the n e u t r o n c o u n t e r efficiencies (i.e, the vertical scale of the curves like fig. 5) were n o t k n o w n . T h e y were f o u n d by d e m a n d i n g that the d i s t r i b u t i o n s f r o m a n y two n e u t r o n c o u n t e r s should be the s a m e over the o v e r l a p p i n g r a n g e of q. T h e
376
G.F. Krebs et al. / Momentum distribution
partial d i s t r i b u t i o n s f r o m different c o u n t e r s were m u l t i p l i e d by these scale factors for n e u t r o n c o u n t e r efficiency a n d were c o m b i n e d , weighted according to their statistical accuracy. Finally, the partial d i s t r i b u t i o n s for all the different values of c.m. angle were a d d e d to give the e x p e r i m e n t a l d i s t r i b u t i o n .
6000 4000 2000 -
~x
IOOC O0 I-Z W > I W rr
t
>I-co z w
o x •o
60C 400
o~x
q,
200 IOC 6C
T.
40 20 IO 6 4
2 I
i
I
1
I
I00
I
[
1
200
,
,
,
300
Iql --*(MeV/c) Fig. 8. Momentum distribution of the neutron in 2H determined from the reaction 2H(p, pn) for different c.m. scattering angles O*. Error bars are statistical in origin. • - 42° ± 1°, © - 44° ± 1% × - 46° ± 1° c.m. angle. The vertical scale is not the same for each c.m. angle.
M a n y tests of consistency were p e r f o r m e d to check that there were n o systematic errors in this a s s e m b l y of the final d i s t r i b u t i o n curve. F o r example, figs. 8 a n d 9 show partial d i s t r i b u t i o n s for 2H a n d 3He respectively for t h r e e different c.m. angles, a n d it is a p p a r e n t that t h e r e are n o n o t i c e a b l e systematic differences in the shapes of the distributions. Similarly, m o m e n t u m d i s t r i b u t i o n curves were c o n s t r u c t e d using different c o m b i n a t i o n s of n e u t r o n c o u n t e r s a n d c.m. angles, a n d n o significant difference was seen b e t w e e n any of these curves. T h e final e x p e r i m e n t a l l y d e t e r m i n e d d i s t r i b u t i o n s are given in figs. 1 0 - 1 2 ; these d i s t r i b u t i o n s are, of course, u n n o r m a l i z e d . Fig. 13 shows the n u m b e r of events which c o n t r i b u t e d to each p o i n t in the final d i s t r i b u t i o n .
G.F. Krebs et al. / Momentum distribution
377
IO00C 500C
0
09 I 0 0 0 1-500 1.13 >
.'o.
.__1 w rr
~00
I
5o
•
•
x G) Z U.I a
~0
T~
•
~T~
I0
I 50
1 I00
I 150
Iql--
1 200
1 250
I 300
1 550
(MeV/c)
Fig. 9. M o m e n t u m distribution of the neutron in 3He as determined from the reaction 3He(p, pn)p, p for different c.m. scattering angles 0". Error bars are statistical in origin. 0 - 4 2 ° + 1°, 0 - 4 4 ° + 1°, × - 4 6 ° + 1° c.m. angle. T h e vertical scale is not the same for each c.m. angle.
4.2. C O R R E C T I O N S F O R I N S T R U M E N T A L B R O A D E N I N G A N D D O U B L E S C A T T E R I N G
There are two corrections which must be made to the experimentally determined distributions. The first is to remove the instrumental broadening. To do this accurately using the trivariate gaussian distribution would be very difficult, however, the experiment is mainly confined to a horizontal plane, and the scatter in q~ is of very little significance. Next, the scatter in q~ and qy are not too different, with qy being the most significant. We therefore approximated the instrumental broadening by a trivariate gaussian distribution with spherical symmetry and a standard deviation of 40 M e V / c . The effect of taking other reasonable values of this standard deviation was negligible. Then the effect of instrumental broadening is represented by the equation qf(q)
x/2--~ 2
~ O f ( Q ) e -(°-Q~2/2'~2 d O ,
G.F. Krebs et al. / Momentum distribution
378
600Or-_
%',,,
600
?\
200
Y> ~
\°~o
I00_
6o-
\
f 40: >z
eo
\\
20
\
\
't' k
,o
64
\ \ \ \ 5O
100
150
Iql
2OO 250
500
,- (MeV/c)
Fig. 10. Momentum distribution of the proton in 2H determined by the reaction 2H(p, 2p)n not corrected for instrumental broadening. The solid curve is the square of the Hulth6n-Yamaguchi wave function and the dashed curve is the same function convoluted with the instrumental broadening function. Error bars are statistical in origin.
w h e r e F ( q ) is t h e t r u e d i s t r i b u t i o n as a f u n c t i o n of n e u t r o n m o m e n t u m q, a n d f ( Q ) is t h e m e a s u r e d d i s t r i b u t i o n as a f u n c t i o n of e x p e r i m e n t a l l y d e t e r m i n e d n e u t r o n m o m e n t u m Q. I n s t r u m e n t a l b r o a d e n i n g was r e m o v e d b y t h e well k n o w n F o u r i e r transform method. T h e o t h e r c o r r e c t i o n is for s h a d o w i n g a n d r e s c a t t e r i n g . In the i n d e p e n d e n t p a r t i c l e m o d e l , t h e effect of s h a d o w i n g is to c h a n g e t h e c h a n c e that an i n c i d e n t p r o t o n will a c t u a l l y r e a c h the n e u t r o n , a n d so it c h a n g e s t h e s c a t t e r i n g cross s e c t i o n of t h e n e u t r o n inside t h e nucleus b y s o m e c o n s t a n t b u t u n k n o w n f a c t o r w h o s e origin is p r i c i p a l l y g e o m e t r i c a l . In o u r analysis this s c a t t e r i n g cross s e c t i o n is u n k n o w n a n d as has b e e n e x p l a i n e d e a r l i e r , t h e analysis has b e e n c a r r i e d o u t in such a w a y t h a t this cross s e c t i o n n e e d n o t b e k n o w n . T h u s s h a d o w i n g can b e i g n o r e d . C o n s i s t e n t with t h e i n d e p e n d e n t p a r t i c l e ( Q E S ) m o d e l , we c o n s i d e r e d t h a t the classical p i c t u r e of a s e c o n d s c a t t e r i n g f r o m a r e a l p a r t i c l e is a valid m e a n s of t r e a t i n g this p r o b l e m . Such a t r e a t m e n t has b e e n u s e d successfully for light nuclei 23) a n d is p r e f e r a b l e to t h e d i s t o r t e d w a v e i m p u l s e a p p r o x i m a t i o n ( D W I A ) which t r e a t s t h e rest of the n u c l e u s as a h o m o g e n e o u s fluid 24); t h e D W I A is m o r e a p p r o p r i a t e
G.F. Krebs et al. / M o m e n t u m distribution
379
.x,
% '~ ~
• 'i.\Oo
200 -_
\
T 6o40>_ _
09
z
U_l
ee O
\\ ee~i \
-
zo
~4~4i'
k
,t
\
I0 64 --
IT ~
\ \
~ \
\
\
Moo
,50
2o0
zso
30o
35o
I ql -,'- (MeV/c) Fig. 11. M o m e n t u m of the neutron in 2H determined by the reaction 2H(p, pn)p not corrected for instrumental broadening. T h e solid curve is the square of the H u l t h 6 n - Y a m a g u c h i wave function, and the dashed curve is the same function convoluted with the instrumental broadening function. The error bars are statistical in origin.
for heavier nuclei with a large n u m b e r of nucleons which can be treated statistically and where there is often no b r e a k u p of the residual nucleus. The probability of a second interaction by the fast proton is negligible; the probability of a scattering of the neutron on its way out was estimated, from the work of Haracz and Lim 23), t o be 0.07 for 2H and 0.16 for 3He. This is a relatively small correction, and approsimations which permit mathematical simplification are acceptable. We assumed that the second scattering is that of a neutron with its average energy ( = 6 0 MeV) off a proton at rest (which it is on the average). At these energies, there is only S-wave scattering which is isotropic in the c.m. frame. This results in a false m o m e n t u m c o m p o n e n t being added to the m o m e n t u m which the neutron had after it was first scattered and which is always perpendicular to the line joining the target position to the neutron counter. Thus a fraction k of the neutrons, which should be represented by a point in q-space, appears to be represented by a different point, and because the scattering is isotropic in the c.m. frame, rescattering distributes these points uniformly over a disc of radius qm where qm is the average m o m e n t u m of the scattered neutron. The true distribution ~b(q) and the observed distribution 6(q) are thus related by
G.F. Krebs et al. / M o m e n t u m distribution
380
1000019"o•
5ooo~-~ '~o~,
o,~
500
f >" 1-
"~ • ~
200
I00
N
•
5O 2C 0
i
I 50
I
l i I00
L i I t I i I 150 200 250 300
Iq I----~ (MeV/c) Fig. 12. Momentum distribution of the neutron in 3He determined from the reaction 3He(p, pn)p,p. A-measurements not corrected for instrumental broadening nor for rescattering; O - c o r r e c t e d for instrumental broadening; © - c o r r e c t e d for instrumental broadening and rescattering. Error bars are statistical in origin.
IQQOC oogag
6000 4000 200C
" _:~
"°°'~lt
~ AO0
l
100C
u~ Iz ILl > w
60C o 40C
03
tJJ 0
~o°Ooo &o
"°~al°°o
e
213(: 21~ ° e e
ICX2
~2 o- I o
6C 40
oo •
• o
AA~o0o 2C IC
AS'
I
1 50
I
I t I J I I I 100 150 20(3 250
I"1--"
t I i 300
•
°l°°
350
CM v/ I
Fig. 13. Summary of the number of events used to determine the momentum distributions.
G.F. Krebs et al. / Momentum distribution
381
the integral equation 6 ( q ) = (1
-k)6(q)+k
40(0)0 dO.
The first term on the right-hand side is the part of the momentum distribution which is not rescattered, and the second term is the "average" over the disc in q-space. The upper limit of the integral has been put equal to oo instead of qm to make this equation easily soluble; this is physically permissible, since qm is about 400 MeV/c, and the integrand Q4~(Q) is negligible even at these values of q. The lower limit is put equal to q, since any neutron counter sees only neutrons whose initial momentum q is close to being parpallel to the line joining the target to that counter, and the difference, if any, between q and the true lower limit is of second order. This equation, written in its canonical form, ~0(q) ~b(q)- 1 - k
2k I~ (1-k)q 2 ¢b(O)Q dO,
is well known as Volterra's equation. It is solved by first differentiating with respect to q to get a first-order linear differential equation for d~(q): d~b (q) dq
2k 1 d~b (q) ( 1 - k ) q ~ qd~(q) ( l - k ) dq
for which an integrating factor is exp ( - k q 2/(1 -- k)q 2 ). The integral which contains d&(q)/dq is converted into an integral containing 4~(q) by integration by parts, and the arbitrary constant is found by comparing the terms of an expansion, in powers of k for small k, of the original integral equation and the solution of the differential equation. Finally, the solution of the integral equation is found as
O(q) ~ 2 k e q2k/(1-k)q~ foe ~b(q) ( l - k ) (1-k)q2m J~
I[I(U)Ue -u2k/(l-k)q~ du
5. Results 5.1. D E U T E R I U M
The result from the momentum distribution of the proton in 2H as determined from the reaction 2H(p, 2p) is shown in fig. 10. The momentum distribution of the neutron in 2H from the 2H(p, pn) reaction is shown in fig. 11. These results have not been corrected for instrumental broadening nor for rescattering. In these two figures, the square modulus of the Hulth6n-Yamaguchi wave function 23) is plotted, as is also this same wave function broadened by the instrumental broadening function. It is easily seen that instrumental broadening is significant.
G.F. Krebs et al. / Momentum distribution
382 5C
~-
2,./,,p,ep J
IC
0
l ~ ,~'~
n-
0080
o
1
O.
O.lo i
I
5o
i
I
,oo
i
I
l
I
t
I
i
I
3oo
i
I
Iql Fig. 14. ( 3 - the ratio of the measurements of the m o m e n t u m distribution in 2H, as determined from the reaction 2H(p, 2p)n, to the broadened square of the Hulth6n-Yamaguchi wave function. Other points are the ratio of the m o m e n t u m distribution to the square of the Hulth6n-Yamaguchi wave function as reported, • - by Witten t7), and • - By Perdrisat u~). Error bars are statistical in origin.
A m o r e sensitive p r e s e n t a t i o n of the results is m a d e in figs, 14 and 15. In fig. 14 w e h a v e plotted the ratio of our data to the b r o a d e n e d square of the H u l t h 6 n Y a m a g u c h i w a v e function along with this ratio for the results of Perdrisat et al. lo) and W i t t e n et al. 17). T h e s e ratios are also s h o w n in fig. 14 for the 2H(p, 2p)n 40-
2H
2O I0
6
~ 2F
o12 ~ J
I
1 .... ~t
l
i
I
J
i
I
i
I
i
I
i
I
(MeV/c) Fig. 15. O - the ratio of the measurements of the m o m e n t u m distribution in 2H as determined from the reaction 2H(p, pn)p to the broadened square of the Hulth6n-Yamaguchi wave function. • - t h e ratio of the m o m e n t u m distribution to the square of the Hulth6n-Yamaguchi wave function as reported by Felder ls~. Error bars are statistical in origin.
G.F. Krebs et al. / Momentum distribution
383
reaction. W e have verified that the rescattering correction is quite negligible for 2H. Our results differ m o r e from the H u l t h 6 n - Y a m a g u c h i wave function than those of Witten's but are in closer agreement with the results of Perdrisat at large values of m o m e n t u m . The results from this experiment show m o r e structure at lower values of m o m e n t u m particularly from 50 M e V / c to 100 M e V / c . Witten's results show an abrupt change from the H u l t h 6 n - Y a m a g u c h i curve at about 260 M e V / c whereas our results as well as Perdrisat's show a gradual change starting from about 135 M e V / c . With the 2H(p, p n ) l H reaction much less data is available for comparison. In fig. 15 only the ratio from Felder's 15) results are shown for comparison with the results of this experiment. O u r results are in fairly close agreement with those of Felder's. Again we see m o r e structure at low values of m o m e n t u m in the results of our experiment. The higher densities at large m o m e n t u m in Felder's results may be due to a persistent neutron background 15). I0000 5OO0 (13 I--
0ZXd. 88~
2OOO
D lad
..-I LIJ rlr"
T
>. t..-
I000 50O
e~
2OO I00
Z UJ a
5O
2O 0
i
t
i
[
I
Ioo
J
i
I
200
i
i
t
I
300
lql ~(MeV/c) Fig. 16. T h e m o m e n t u m distribution of the neutron in 3He, showing the effect of deconvolution and rescattering. A-original m e a s u r e m e n t s , O - m e a s u r e m e n t s deconvoluted to remove the instrumental broadening, and C) - after correction for rescattering.
5.2. 3He
The results for the m o m e n t u m distribution of the neutron in 3He are shown in fig. 16 along with the result after removing the instrumental broadening by the usual deconvolution process. In the deconvolution procedure only the first six harmonics were retained; using more than six caused the result to "blow u p " from noise. The deconvoluted m o m e n t u m distribution was then corrected for rescattering. The effect of rescattering is also shown in fig. 16 and is seen to be a small
G.F. Krebs et al. / Momentum distribution
384
TABLE 1 The momentum density ( M e V / c ) 3 of the neutron in 3He; A, original measurments; B, corrected for instrumental broadening; C, corrected also for rescattering; and D, normalized to unit density at momentum q - 0 q
A
B
C
D
5 15 25 35 45 55 65 75 85 95 105 115 125 135 145 155 165 175 185 195 205 215 225 235 245 255 265 275 285 295
139193 459121 720567 768547 586993 541830 507215 464761 417048 367177 318372 286433 240423 234344 199614 169711 151022 128346 117638 122540 94074 80856 68617 65588 49824 35461 28411 27046 20856 19085
953379 927934 879246 811429 729963 641009 550675 464353 386227 318992 263814 220496 187782 163753 146208 133000 122276 112609 103047 93072 82534 71551 60414 49505 39226 29932 21894 15262 10044 6198
1111300 1081389 1024161 944454 848720 744213 638126 536806 445186 366440 301946 251470 213535 185873 165885 151029 139107 128426 117843 106737 94914 82501 69837 57366 45560 34846 25549 17855 11784 7190
0.953379 0.927718 0.878622 0.810242 0.728113 0.638457 0.547445 0.460524 0.381923 0.314367 0.259038 0.215735 0.183191 0.159460 0.142313 0.129567 0.119340 0.110176 0.101098 0.091569 0.081428 0.070777 0.059910 0.049814 0.039086 0.029894 0.021918 0.015318 0.010110 0.006109
effect. The values of the momentum distributions for the results shown in fig. 16 are given in table 1. Note that the dip at zero momentum in the original data is an instrumental effect which is removed by deconvolution. A comparison was made between three trial S-wave wave functions and the momentum distribution corrected for instrumental broadening and rescattering. The square of the trial wave functions are gaussian: Irving:
I~(q)l 2 = N l
e
2q2/9c~2
,
I~b(q)12= N 2 / ( a 2 + q2) 11/2 ,
Irving-Gunn:
1~ (q)l 2 = N3/(0¢ 2 / 4 + q 2 ) 7 / 2 ,
385
G.F. Krebs et al. / Momentum distribution
TABLE 2 Parameters for the fitting of three proposed wave functions to the experimentally determined momentum distribution gaussian N o~ c~'
X2 (reduced)
Irving-Gunn
0.67 + 0.04 0.30+0.01fm a 0 . 3 8 5 fm a
88
2.27 ± 0.4 0.89+0.03fm 0.77 fm 1
37
Irving
1
0.54 + 0.2 1.22±0.04fm 1.26 fm 1
1
52
T h e p a r a m e t e r s N a n d a are d e f i n e d in the text; a ' is the p a r a m e t e r a d e t e r m i n e d by e l e c t r o n s c a t t e r i n g 27).
where N1, N2, N3 are normalization constants. A t w o - p a r a m e t e r fit was carried out with each of these functions to our corrected distribution and the results are shown in table 2. T h e y indicate the best fit with the I r v i n g - G u n n function. Fig. 17 shows the m o m e n t u m distribution of the neutron in 3He after corrections for instrumental broadening and rescattering, along with the squared modulus of the Irving-Gunn wave function. The ratios of our corrected m o m e n t u m distribution to the three functions above are shown in fig. 18. Shown along with our data are three data points - normalized to our data - from the work of Kitching 25) on the reaction 3He(p, 2p)2H.
6. Conclusion We have described an experiment in which the m o m e n t u m distribution of the neutron in 2H and 3He was measured using the quasi-elastic scattering of fast protons. This is the square modulus of the neutron wave function in m o m e n t u m space. In the case of 2H, much theoretical and experimental work has been done on this problem. Our m e a s u r e m e n t s agree very well with other m e a s u r e m e n t s at m o m e n t u m values greater than 100 M e V / c , but are significantly lower around 50 M e V / c . In all cases, there is a discrepancy between the theoretical wave function and the experimental results. In the case of 3He, there are no other experimental results to compare ours with. Further, there is no soundly based theoretical expression for this m o m e n t u m distribution. Those which have been proposed have been no more than plausible expressions chosen for their analytical tractability. All the same, our m e a s u r e m e n t s fit the proposed analytic expressions with p a r a m t e r s very close to those determined by Collard et al. 26) by electron scattering, and by Kitching et al. 25) for the wave function of the proton in 3He from the reaction 3He(p, 2p)2H. The agreement with these analytic expressions is, however, not close. A noticeable feature of our results is a significant dip at m o m e n t a around 100 M e V / c , and a hint of this trend can be
386
G.F. Krebs et al. / Momentum distribution
1.0 06
•°oOO
0.3 -
•l•
o~ --
O.
•O
6 I
O.C~ -
f
>-
i-
0.03
(.t) z
hi d21 0.01 0.006 0.007
1
I
1
1
I
1
50
I00
150
200
250
300
Iql - - ~
(M~V/c)
Fig. 17. The m o m e n t u m distribution of the neutron in 3He and the best fit of the square of the Irving-Gunn wave function. Error bars are statistical in origin.
G~USSIAN ***~.
30120~
.-*
llJoi
[ 0
iIl
I
--°-*',-ooeeeoo*
. . . .
. . . . . . . . . .
2.0.---
t
"B-~-~
, o
0
or"
IHVING- G U N N
20
..**tetl~
1
I0
t
0.6
L
I
I
1
tO0
Iql
,
I
,
1
t
200
I
~
I
300
-(MeV/c)
Fig. 18, The ratio of the m o m e n t u m distribution of the neutron in 3He to the best fit of the square of various wave functions proposed for the system• The measurements by Kitching et al. 28) using the • 3 reaction - He(p, 2p)p,n are shown as open circles. The error bars are statistical in origin.
G.F. Krebs et al. / Momentum distribution
387
seen in the m e a s u r e m e n t s of K i t c h i n g et al. 25). W e can at p r e s e n t offer n o e x p l a n a tion of this a n d regard it as a n o t h e r i n t e r e s t i n g p r o b l e m in theoretical n u c l e a r physics to be e x p l a i n e d at a later data. T h e financial s u p p o r t of the N a t i o n a l Sciences a n d E n g i n e e r i n g R e s e a r c h C o u n c i l of C a n a d a is gratefully a c k n o w l e d g e d . O n e of us ( G F K ) was the recipient of a T o r o n t o O p e n F e l l o w s h i p a n d the J.C. Stevens A w a r d . W e are i n d e b t e d to m a n y p e o p l e o n the staff of the L a w r e n c e B e r k e l e y L a b o r a t o r y for facilitating this e x p e r i m e n t a n d especially to Dr. O. C h a m b e r l a i n who e n c o u r a g e d us to do this e x p e r i m e n t a n d lent us the n e u t r o n counters. T h e advice a n d help of m a n y others is a c k n o w l e d g e d with thanks, in paricular, Dr. K. Crowe a n d J.F. Vale, the staff of the 184 in. s y n c h r o c y c l o t r o n , a n d D. N e l s o n of the m a g n e t group. This e x p e r i m e n t w o u l d not have b e e n possible w i t h o u t extensive c o n t r i b u t i o n s f r o m the technical staff, a n d others, of the e x p e r i m e n t a l team. In p a r t i c u l a r we w o u l d like to express o u r t h a n k s to H. C o o m b e s , E.I. D e n n i g , Dr. S. F r i e d l a n d e r , W. Hilger, K . U . Kaires, Professor R. Kreps, A . K . C . Kiang, Professor J. M c A n d r e w , Professor J.D. P r e n t i c e a n d Dr. R.R. W h i t n e y .
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)
13)
14) 15)
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