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Effects of exchange and final-state interactions in the reactions 2H(3He, 3H p)p and 2H(3He, 3He p)n
) l.B: 2.B
1
Nuclear Physics A221 (1974) 593 - 607; @ Not to
EFFECTS
North-Holland
Publishing
Co., Amsterdam
be reproduced by photoprint or microfilm without written permission from the publisher
OF EXCHANGE
IN THE REACTIONS
AND FINAL-STATE
INTERACTIONS
2H(3He, 3H p)p AND 2H(3He, 3He p)n
R. E. WARNER t and S. A. GOTTLIEB it Oberlin
College,
Oberlin,
Ohio 44074,
USA
and G. C. BALL, W. G. DAVIES, A. J. FERGUSON
and J. S. FORSTER
Atomic Energy of Canada Ltd., Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada KOJ IJO
Received 17 December 1973 Abstract:
Absolute coincidence cross sections for the 2H(3He, 3He p)n and 2H(3He, 3H p)p reactions were measured at EHc = 35.9 MeV. Spectra dominated by the nucleon-nucleon final-state interaction (FSI) are fitted by a fully antisymmetrized PWBA theory which includes the effects of FSI in all its matrix elements. Previously reported 26.8 MeV data showing both FSI and quasi-elastic scattering (both with and without charge exchange) are also fitted by the theory, which qualitatively describes the shapes of all these spectra and the ratios of the cross sections for the various processes. Predictions of Watson-Migdal theory are fitted to the FSI spectra and differences between the two theories are analyzed.
E
NUCLEAR
REACTIONS ZH(3He, jHe p)n and ZH(3He, 3H p)p, E = 35.9 MeV; measured absolute coincidence a(El , Of, e2). Enriched target.
1. Introduction Final-state interactions (FSI) are observed when reactions between elementary particles or nuclei produce multi-body final states and two strongly interacting particles emerge from the reaction with very low relative velocity. For example, the cross section for the reaction 2H+ 3He + 3H+p+p is strongly inhibited by the Coulomb repulsion whenever any two final particles exit nearly together. In contrast, when a neutron and proton emerge from the ‘H + 3He -+ 3He + p + n reaction with low relative velocity, the yield is enhanced by the attractive n-p interaction. Originally FSI effects were studied to gain information about systems which could not be studied through conventional scattering experiments; e.g. the n-n scattering length was measured in the reactions n- f2H -+ y+n+n [ref. ‘)I and 2H+3H -+ 3He +n+n [ref. “)I. There is now a parallel interest in the mechanisms which govern such reactions. It is of interest to know whether they can be adequately understood through the Watson-Migdal (WM) theory 3, 4), which only takes account of the scattering wave function of the two strongly interacting particles, or whether more detailed theories are needed. If one uses a more detailed theory such as the t Research supported by the National Science Foundation under grant GP-35985. +t Present address: Physics Department,
Princeton University, 593
Princeton,
NJ 08540, USA.
594
R. E. WARNER
et al.
modified PWBA ‘) or DWBA 6), is it sufficient to include only a few simple mechanisms such as charge-exchange and pickup “) or must the full effects of complete antisymmetrization ‘) be taken into account? In this paper we present measurements of the absolute coincidence cross sections for the reactions 2H(3He, 3He p)n and 2H(3He, 3H p)p at 35.9 MeV helion (3Hef+) bombarding energy, at several geometries where the two final nucleons can emerge nearly together and thus exhibit FSI. These data are fitted by the predictions of three theories: a fully antisymmetrized PWBA theory ‘) with final-state interaction effects included in all matrix elements; a DWBA theory “) which considers only two reaction mechanisms, charge-exchange and pickup; and the Watson-Migdal theory 3, “). Fully antisymmetrized PWBA calculations ‘) of the cross sections for these two reactions have recently been reported. The effects of exchange were taken fully into account, but plane waves were used to represent the relative motions of all unbound particles; initial- and final-state interactions between nuclei were thus ignored. The original purpose of these calculations was to interpret coincidence experiments “) showing intense spectator peaks for both reactions. Although the magnitude of the calculated cross sections was too high, the calculations reproduced rather well not only the shapes of the observed spectator peaks but the ratios of the measured cross sections for the two reactions. Some discrepancies between predicted and observed peak shapes were noted where the final nucleon-nucleon relative energy sNN was low, and were attributed to nucleon-nucleon FSI. The use of a HulthCn wave function “) for the deuteron prevented us from including FSI effects since most of the seven matrix elements of the antisymmetrized theory could not be evaluated. Since then, the Van Oers-Slaus deuteron wave function 2, lo), a sum of three Gaussian functions closely approximating a Hulthen function, has been brought to our attention. It permits FSI effects to be included in all matrix elements of the theory (hereafter called PWBAFSI) which should describe structure in the observed cross sections resulting from both spectator poles and FSI. Thus in this paper we compare the PWBA-FSI predictions not only with the new 35.9 MeV data showing p-p and p-n FSI, but with our earlier 26.8 MeV p-p FSI “) and spectator pole data as well. Since the WM theory is the very simplest theory of FSI, it is of interest to compare its predictions with those of more detailed theories. We examine the assumptions of the WM theory with particular reference to an earlier study by Haybron ‘i) which concluded that the WM theory was more reliable for reactions proceeding by pickup than by charge-exchange. We disagree with his conclusion, and identify factors of kinematic origin omitted from Haybron’s analysis.
2. Experimental
procedure
Our experimental procedures will be briefly summarized here since they have been described in greater detail elsewhere i3). A CD, target was bombarded with helions (3Hef+) whose energy at the target centre was 35.9 MeV. Coincident charged
‘HC3He,3H p)p, 2H(3He, 3He p)n
595
particles were detected by two AE-E counter telescopes placed on opposite sides of, and coplanar with, the beam. (Such geometries will be designated by (e,, t?,,), where x and y specify the particles detected at the two angles.) Both telescopes employed 0.345 cm diam. defining apertures and 2000 pm E-counters; the AE counter thicknesses were 100 pm and 150 ,um in the proton and t-3He telescopes, respectively. Pulse heights and time intervals between AE pulses were recorded event by event on magnetic tape. During analysis each particle was identified by referring its AE and E to a triton range table “). Absolute cross sections for the reactions ZH(3He, 3He p)n and 2H(3He, 3H p)p were deduced from the yield of events which obeyed the correct three-body kinematic relationship EHe versus Ep or Et versus E,,. Loss of deuterium from radiation damage in the target was monitored by integrating the beam current and measuring the yield of helions elastically scattered from both ‘H and “C at 15”. The deuterium densities found from the 2H/Q and ‘H/l’C ratios (assuming no loss of carbon from the target) were consistent within 5%. The normalization of our data is believed to be accurate within lo%, and only statistical uncertainties are shown in our results which are presented in figs. 1 and 3. 3. Theories describing final-state interactions In this section we briefly describe existing theories that have been used to interpret spectra dominated by final-state interactions (FSI). In the “post” form of formal scattering theory i4), the matrix element for a rearrangement collision is given by the equation: T,i = (~r-‘lW~lll/i+‘).
(I)
The initial-state wave function $I” obeys outgo’m g-wave boundary conditions and is an eigenfunction of the total Hamiltonian H, while the final-state wave function (6:-’ obeys incoming-wave boundary conditions and is an eigenfunction of HI where W, = H-H, is the cause of the transition. In practice one cannot obtain exact wave functions J/I’) and 4i-j for a system as large as A = 5, even assuming that all the interactions are known. The wave functions are therefore replaced by various approximations that stress physically important aspects of the reactions being studied. The most drastic approximation of all is the plane wave Born approximation (PWBA) in which particles not bound to each other are assumed not to interact with each other at all; i.e. their relative motion is described by plane waves. This approximation may nevertheless have qualitative success in predicting those features of nuclear reactions that are primarily influenced by boundstate wave functions which appear as factors in $i” and&-‘. For instance, the angular distributions of nucleons from deuteron stripping reactions 1“) and the energy distributions in three-body final states following quasi-elastic deuteron breakup are largely determined by the internal wave function of the initial deuteron. The PWBA cannot, by definition, account for the FSI effects observed here when
R. E. WARNER
596
et al.
the two final nucleons emerge with low relative velocity: namely, ‘H(3He, 3He p)n spectra caused by the attractive p-n interaction, in the ZH(3He, 3H p)p spectra resulting from the Coulomb protons. A common ingredient of all theories that seek to interpret the inclusion of a nucleon-nucleon scattering wave function as a 3.1. THE
ANTISYMMETRIZED
PWBA
The general approach, first taken account of the FSI between the two ing wave function &), rather than wave function. The spatial parts of q,i +)
=
THEORY
WITH
FSI
the maxima in the and the minima repulsion between these FSI effects is factor in &-‘.
INCLUDED
by Henley rt al. ‘) (hereafter called HRY), took final free nucleons by incorporating their scatterthe corresponding plane wave, into the final-state the intial and final wave functions thus become: +d
+He
,iX.
(he -Rd),
(2) (3)
The subscript T denotes the final trion (either 3H or 3He) and NN denotes either the n-p or p-p pair. The trion wave function & is a symmetric Gaussian function with characteristic length y- ’ = 2.78 fm. The momenta hK, hk and hq refer to the motion of the initial 3He relative to the deuteron, the final trion relative to the nucleonnucleon c.m., and the final nucleons relative to each other, respectively. The relationship between nucleon-nucleon relative energy, eNN, and momentum is h2q2 = MpcNN. The major difference between the present work and that of HRY is that we include all exchange effects by antisymmetrizing the wave functions. The Van Oers-Slaus deuteron wave function
(4
i=l
which closely approximates the Hulthtn function is used; the constants given in ref. ’ “). Finally, the two body transition potential V(P) = --*vc(l
-PoP,)e-szrZ
Cj and Mi are
(5)
is taken to have a depth V, = 72.5 MeV and a characteristic length 6-r = 1.46 fm. This particular combination of central and exchange interactions, operating between fully antisymmetrized wave functions, is a Serber force. Tobocman + has shown how total antisymmetrization (antisymmetry of the wave functions for all nucleon exchanges) is achieved for rearrangement collisions. In general, the effect of antisymmetrization is to establish coherence between the amplitudes of various reaction mechanisms (such as pickup and charge-exchange) and to introduce other exchange amplitudes that are normally neglected. Seven basic matrix elements are found to determine the two reactions under discussion. These t Ref. Is), p. 100.
ZH(3He, 3H p)p, ZH(3He, 3He p)n
597
include one breakup, four knock-out, and two pickup amplitudes. These broad categories denote whether 3, 2 or 1 nucleons are common to the initial and final trions; they are subdivided according to the origin and destination of the two nucleons which interact, through eq. (6). Thus & denotes breakup; the first subscript says that one interacting nucleon, which ends in the trion, began in the initial 3He; the second subscript says that the other interacting nucleon, which becomes a free nucleon, came from the initial deuteron. The matrix elements are further defined in ref. ‘). They can all be written in the general form:
x
dsz
. Zed,,X.Zed,,J. Z d3U ,-&,u*,‘&r s
,-a.zzeik,
s
where the constants up, k, and 6,, vary from repeated use of the formula
one matrix
uedyuy ued,,,:’ Y, element
to another.
s
(6) By
(7)
ewnr
each term in the sum over i reduces to a constant
times a one-dimensional integral containing the nucleon-nucleon wave function x6-j. The p-p scattering wave functions that describe the FSI effects in the 2H(3He, 3H p)p reaction were computed for a point-charge Coulomb potential and a real central Woods-Saxon nuclear potential with V,, = 33.5 MeV, R, = 1.477 fm and a = 0.5 fm. Gaussian potentials with V, = 41.0 MeV, I/, = 72.5 MeV, rl = 1.59 fm, and Y3= 1.47 fm (with subscripts denoting singlet or triplet states) were used to compute the n-p scattering wave functions for the 2H(3He, 3He p)n reaction. We have found that these potentials reproduce the appropriate nucleon-nucleon scattering lengths and effective ranges; the p-n triplet potential has also been given by Preston 16). Taking appropriate sums and averages ‘) of the matrix elements leads to the following formulae for the cross sections for the two reactions:
d30 dQ,dQ,dE; ;
IW+,+,=
tie+,+, = (i-&.$M: +b%M:
+M:
&12M:
(8)
IMI”, if
+M;12+&M;12,
(9)
+M,~2)s=o
+-M: +3M:12+&-2M: M:
M:
- 27cp(E,)z - ho
= 12(P;
-PL
= - 6( B,‘, + K;) - 12(P; M:
+M;
+12M;
+3M,12)s=1,
+K;), + K*$ + K&) - 24K,;,
= 6(B:, + K:).
(10) (11) (12) (13)
R. E. WARNER
598
et al.
The cross section applies to the detection of particle 1 with energy E, in solid angle dQ,, in coincidence with particle 2 in solid angle d&; we consider only the Iarger of the two energies Ez consistent with a given E1. The laboratory velocity of the incident particle is U, and the density of final states is p(E,). The k superscripts denote the parity of the two final free nucleons, and the subscripts to the curly brackets in eq. (10) indicate whether singlet or triplet n-p wave functions enter into the enclosed matrix elements. 3.2. DWBA
TREATMENT
OF THE
ZH(3He, 3H
p)p REACTION
In this approach 6), which is also a straightforward extension of the modified PWBA treatment of HRY ‘), we consider only charge-exchange and pick-up processes, as do HRY. The initial 3He-d interaction is now taken into account by replacing the plane wave in eq. (2) by an appropriate distorted wave. Likewise, the wave function for relative motion in the final state is the product of a p-p scattering wave function (for the potential described in subsect. 3.1) and a distorted wave for the t-(pp) relative motion, assuming that their interaction depends only upon the separation of their centres of mass. DWBA fits to our 26.8 MeV data have already been reported “)_ Good fits to the shapes of the p-p FSI spectra were obtained, while the magnitudes were over-predicted by factors of 1 to 4. It was subsequen~y pointed out to us that the nuclear potentials used for the 3He-d and t-(pp) distorted waves were unrealistically strong 1‘). Therefore in the present work we use V = 35 MeV and W = 6 MeV for the real and imaginary depths of the triton-(pp) and 3He-d interaction, and R = r,AS = 3.16 fm for their ranges. Thus the strength Wri is 163 MeV * fm2, about twice the accepted value for elastic nucleon scattering. 3.3. WATSON-MIGDAL
THEORY
The basic assumptions of the Watson-Migdal (WM) theory are that when the two strongly interacting nucleons have sufficiently low relative energy, em,,, their scattering wave functions can be factored into parts depending only upon enN and their separation, respectively, and that all othnr quantities that appear in the reaction amplitudes are constants. An equivalent statement about the wave function is that its &-dependence (or q-dependence) must be the same both inside and outside the range of nuclear forces. Watson “> and Haybron I*) show why this is true for short-range interactions. Phillips I*) does so for the Coulomb interaction as well. Thus the energy dependence of the matrix element is just that of the scattering wave function, and we have for the n-p FSI matrix element and cross section: eis sin 6 Tfi ~~~ 4 d3a x @J [sin’ ~5~1-3sin’ li3 ] . dQ, dS’&dE, q2
2H(3He, 3H
599
p)p, 2H(3He, 3He p)n
For the p-p case, where there is only singlet s-wave scattering, the matrix element is
Definitions of the nuclear phase shifts 6 (from effective range theory) and of the Coulomb penetration factor Co and phase shift ~~ appear in refs. *, 16). 4. Comparison of experimental data with predictions 4.1. THE
p-p FINAL-STATE
INTERACTION
In fig. 1, we show absolute cross sections for the reaction 2H(3He, 3H p)p at E, = 35.9 MeV, at three geometries where the p-p relative energy goes as low as a few keV. These are plotted against both ePp and the lab energy Ep of the detected proton. Although the counter telescope cutoff for protons is nominally 4 MeV, two spectra have been extended to about 2 MeV. This was done by including both events which satisfy normal acceptance criteria and those where a triton was identified in the forward telescope in coincidence with a particle stopping in the AE counter of the second telescope. This leads to artificial minima in the spectra at Ep = 3.6 MeV, and artificial maxima at 2.8 MeV, caused by protons whose energy loss in the second telescope’s E-counter are below the discriminator threshold. Since such particles are counted but assigned an energy about 1 MeV too low, our averaged cross sections in this energy region should be reliable even though artificial structure appears. P-P
ENERGY,
cpp
5 I *a ,I .5 I. ~~~1. /--- ‘T_.7_77 1-----r
PROTON
LAB
ENERGY,
,
MeV
.rmzr 5 1. 2. I I /
MeV
Fig. 1. Absolute cross sections, and predictions of the PWBA-FSI, DWBA and WM theories, for the ZH(3He, 3H p)p reaction at 35.9 MeV incident energy at three geometries (BP, 0,) where low p-p relative energies were obtained. Predictions of the PWBA-FSI theory were normalized by factors between 0.04 and 0.17, those of the DWBA theory by 0.6 to 3.6, and WM predictions were normalized arbitrarily. Arrows mark the expected location of peaks from sequential decay of the 21.3 MeV second excited state of 4He into the triton and undetected proton.
R. E. U’ARNER
600
RELATIVE
PROTON
Fig. 2. The
same
as fig.
P-P
LAB
et 01.
ENERGY,
cppr
MeV
-------
WATSON-MiGDAL
ENERGY,
MeV
I, at 26.8 MeV incident energy. Normalizations PWBA-FSI, and 1.0 to 1.9 for DWBA.
4He:,.3-
were
‘5
0.09
to 0.52 for
Predictions of the PWBA-FSI, DWBA, and WM theories are compared with our 35.9 MeV data in fig. 1, and our earlier 26.8 MeV data “) in fig. 2. Pronounced dips in the cross section are observed in figs. 1 and 2 whenever spp goes below about 0.1 MeV as a result of the Coulomb repulsion between the emerging protons. For spp s 0.5 MeV the three theories fit the data acceptably and in fact the predicted shapes are nearly identical. The latter is expected since, near the minimum, the p-p relative momentum varies much more rapidly with E, than does the t-(pp) relative momentum. Thus most of the Ep dependence of the matrix elements is contained in the p-p scattering function, which appears in all three theories. Away from the minima, broad peaks are observed in five spectra where the relative energy E,,, of the triton and undetected proton corresponds to the excitation energy of the 21.3 MeV second excited state of 4He (which decays yielding these two particles). None of the three theories attempts to include the effects of this state in 4He. One of the worst fits appears in fig. Ic where the excess of observed over predicted yield occurs over a much greater energy range than can be accounted for by the width (r z 2 MeV) of the 4He state. Between the p-p FSI minimum and the 4He peak the lab energy of the undetected proton reaches a minimum value of about 0.8 MeV. Although this is a rather high energy for a spectator pole effect, the possibility exists that the exceptionally poor fit results from interference of the three different processes.
2H(3He. “H p)p, ZH(3He, 3He p)n
601
The DWBA fits are slightly superior to those of the other two theories for the 26.8 MeV data though probably not at 35.9 MeV. The DWBA normalization was more uniform at the lower energy where the required normalization factors varied from 1.0 to 1.9 as opposed to 0.6 to 3.6 at 35.9 MeV. The DWBA fits shown were obtained with I’ = 35 MeV, but the quality of the fits did not vary appreciably as I/ was changed from 25 to 50 MeV with Vri fixed at 163 MeV * fm’. The PWBA-FSI predictions were all normalized by factors between 0.07 and 0.13, except that factors of about 0.5 were used for the two spectra with Bt = 35” (the largest triton emergence angle studied) in fig. 2. At 6, = 35” the charge-exchange amplitudes are smaller, and the pickup amplitudes larger, than at smaller 0. The transfer of two nucleons within the triton may be more the result of higher order than first order processes, and hence less well described by the PWBA-FSI theory.
RELATIVE .l.cm.1
.5
I.
p-n
ENERGY,
2.
------
_I 02 .I
cpn
, MeV
.5
2.
I.
WATSON-MIGDAL
r-m
2.1.5.lD15.I
.p
15l
(52;.
i
4
6
2.
i
26.5=,,,)
16
12 PROTON
I.
LAB
ENERGY,
MeV
Fig. 3. Absolute cross sections and PWBA-FSI and WM predictions for the *H(3He, 3He p)n reaction at 26.8 and 35.9 MeV. PWBA-FSI normalizations were 0.04 to 0.058. Arrows show expected peak location for sequential decay of 4He (21.3 MeV) into 3He+n.
602
R. E. WARNER
4.2. THE p-n FINAL-STATE
et al.
INTERACTION
Fig. 3 shows data for the 2H(3He, 3He p)n reaction at 35.9 and 26.8 MeV incident energy, for four geometries where epn g oes below 0.1 MeV. As is expected, the attractive p-n interaction produces strong broad peaks where this occurs. (In two cases, the lower half of the peak is below the proton telescope cutoff.) The p-n FSI peaks are well fitted by both the WM and PWBA-FSI theories. In particular, both theories predict a mixture of singlet and triplet contributions that agrees with the data. The PWBA-FSI theory has no free parameters and the ratio of singlet to triplet states is not adjustable. Thus its success may be less fortuitous than that of the WM theory, which includes only the n-p wave function while omitting the overlap integrals and isospin factors which would appear in a more detailed theory and which would differ for the singlet and triplet states. Enhancements from the sequential decay of 4He” (21.3 MeV) into 3He+n appear in the three spectra where this is kinematically possible. Only half peaks are seen since the proton energies corresponding to this excitation are near the upper kinematic limits. Parker et al. 19) have also observed that this state decays by both proton and neutron emission. 4.3. SPECTATOR
POLES
Fig. 4 shows the fits to the previously reported spectator-pole data “) for these two reactions. These fits are not much better than those from the earlier fully antisymmetrized PWBA theory ‘). FSI effects were included partly in the hope of fitting the high-energy end of the (15;, , 4.4;) spectrum, where a p-n FSI enhancement is observed. A rise in the PWBA-FSI curve is obtained, but the minimum between FSI and spectator peaks is too shallow. The predicted width of the (1.5&, 2.5;) peak is too great, and presumably that of the (15:) 25;) peak would be as well if the enhancements due to 4He” (21.3 MeV) sequential decay could be subtracted from the experimental data. Previously it had been found that the HulthCn function often overpredicted the width of spectator peaks; this difficulty was overcome 20) by smoothly cutting-off the wave function at small separations, which also attenuated its high-momentum components. The justification for such a procedure was that, when a projectile collides with a deuteron whose nucleons are close together, reactions other than quasi-free scattering will probably occur. It is disappointing therefore that the present PWBA-FSI theory, which is fully antisymmetrized and should account for all reaction mechanisms and predict all final states in their proper ratio, should also predict excessive widths. Again the explanation may be that high-momentum transfers lead preferentially to higher-order processes, so that the PWBA fails. One hope for both the earlier theory and the present one was that, since they are explicitly charge-independent, they would correctly predict the ratio of the 3He+p+n and t +p+p cross sections. The present theory does so within a factor of about 1.5; the normalizations for these two final states were about 0.12 and 0.08, respectively. This is slightly better than the earlier theory 7), which gave a ratio of about 2.
ZH(3He, 3H p)p, ZH(3He, 3He p)n
2H(3He,3He
603
p)n 30-
20!-
IO-
O-
‘He
LAB
ENERGY,
FneV
Fig. 4. Absolute cross sections for the two reactions at 26.8 MeV for geometries where the spectator nucleon lab energy (right hand scale) could be near zero. PWBA-FSI predictions for 2H(3He, 3H p)p (upper panels) were normalized by about 0.075 and those for 2H(3He, 3He p)n by about 0.115.
5. Discussion
and conclusions
Since the shapes of the PWBA-FSI and WM predictions are so similar, there arises the question of whether it is really necessary and advantageous to use the former, much more detailed, theory rather than the simpler one. One might also ask whether the PWBA-FSI theory is an improvement over its immediate predecessor, the modified PWBA theory of HRY “). To answer the latter question first, we show in fig. 5 PWBA predictions for at 45$ 31:) and 35.9 MeV, with only two matrix elements included: 2H(3He, 3H P)P ( Btd which is identical to the charge-exchange matrix element of HRY and P,t a pickup matrix element similar to theirs. (The particular pickup amplitude used by HRY does not appear in a treatment, such as the PWBA-FSI, employing “post” interactions.) For these calculations, and throughout this section, we use a Hulthen function rather than a Van Oers-Slaus function for the initial deuteron. The amplitudes Btd and Ptt (the only ones which can be evaluated for the Hulthen
604
R. E. WARNER
p-p
RELATIVE
I. 15
.5 r
*H(‘He,
c
ENERGY,
et 01.
lpp ,
MeV
.I .OOl -.T--~ r~
tp)p
4
I 12
8
PROTON
LAB
ENERGY,
____I 16
MeV
Fig. 5. Predictions of the PWBA-FSI theory using only the non-rearrangement and pickup matrix elements B,, and P,, . both separately and together, are compared with one p-p FSI spectrum and the WM prediction.
function) then reduce to single terms rather than the sum of three terms mentioned in subsect. 3.1. The PWBA fit to the data, when only these two amplitudes are included, is very poor, and in particular the minimum is much too broad. Fig. 5 shows that the (arbitrarily normalized) cross sections for charge-exchange or pickup alone equal to zero) are quite different from (obtained by setting Ptt, or &,, resp:ctively, each other and from the prediction with both included. The poor PWBA fit results mainly from destructive interference between these two amplitudes, and the importance of including all amplitudes that coherently determine the cross section is thus emphasized. To see whether the close agreement between the PWBA and WM theories is accidental, we return to the basic assumptions of the WM theory. The first of these is that the shape of the nucleon-nucleon scattering wave function is energyindependent. Fig. 6, in which ~6,) is normalized by the energy-dependent factor of eq. (16), shows that this is true for nucleon separations up to 4 or 5 fm. Thus the first assumption will be least valid for those matrix elements in which the integrals containing ~6,’ have the slowest cutoffs. These are, in fact, the three spectator pole matrix elements B,, , K,, and Kdt for which the cutoff parameters a, (see eq. 6) are just the Cli of the deuteron wave function; all other matrix elements are cut off more rapidly by the trion and/or nuclear force range parameters y and 6. However, fig. 5 shows that the charge-exchange cross section (labelled Btd) follows the WM
ZH(3He, 3H p)p, 2H(3He, 3He p)n
I 0
I
I
I
5
IO
15
PROTON SEPARATION,
605
I
r , fm
Fig. 6. Wave functions for p-p scattering in a Woods-Saxon well multiplied by the factor sin &I-’ which, according to WM theory, removes their dependence upon the p-p relative energy Ebb.
qC,,[ei(m+W
RELATIVE
p-p
ENERGY,
epp,
MeV
Fig. 7. Dependence upon ePDof factors in the matrix elements Btd and P,,. Upper panels show the integrals containing the p-p scattering wave function, multiplied by qCo[eWm+W sin I&,]-~. Lower panels show the kinematic coefficient of the integrals. All four quantities are divided by their values at ePB= 0; all four would be straight horizontal lines if WM theory were valid.
606
R. E. WARNER
et rrl.
prediction much more faithfully than does the pickup cross section (P,,). This can be understood by reference to fig. 7, where the two factors which determine these two matrix elements arc plotted against Ear. The upper panels show the integrals containing the p-p scattering wave function, normalized by the right-hand side of cq. (16); the lower panels show the coefficients which multiply these integrals and which, as discussed in subsect. 3. I, contain various kinematic factors. All four curves are further normalized to their values at cpp = 0; according to the WM theory, all should be straight horizontal lines. The charge-exchange integral In does in fact vary somewhat more rapidly with c,,,, than does its pickup counterpart I!,, as is expected because of its slower cutoff but the pickup kinematic coefficient T has by far the largest energy variation, and is the principal cause of the large difference between the pickup and WM cross sections. The WM and charge-cxchangc cross sections agree rather closely both because the energy variations of Ia and S partially compensate each other, and because the contributions to the integral Ia at large nucleon separations, where the WM assumptions about the scattering wave function break down, are reduced by the oscillations of exp (i k, . x) see cq. (6). The pickup amplitudes, which are poorly represented by WM theory, are in general rather small at the geometries here investigated. Thus various accidental factors lead to closer agreement between the WM and PWBA-FSI thcorics than one can expect in general. Haybron ‘l) concluded that the WM theory better represents reactions proceeding by pickup than by charge-exchange. Fig. 5 shows that the opposite is true. In this earlier study, only the integrals I* and I,, were examined. It was further concluded that WM theory works better for pickup since the two nucleons that interact strongly in the final state were initially located in a small volume; i.e. within the trion. However, the pickup coefficient Tcontains a factor exp( - IR+ $k1’/4yZ) which is related to the probability of finding the picked-up nucleon in the initial helion with its final momentum. This quantity varies rather rapidly with cpp partly because y2 is so larg: or, equivalently, because the trion is so small. Thus the compactness of the trion is partially responsible for the large difference between WM and PWBA pickup cross sections. The criterion that the two strongly interacting nucleons b: confined initially to a small spatial volume is not sufficient for th: validity of WM theory. Other effects, rooted in nuclear structure or details of the nuclear fort: but related to experimental kinematics, can bc substantial. The principal achievement of the fully antisymm:trized PWBA-FSI theory is that it reasonably well describes the shape ofcoincidcncz energy spectra from three body final states when a single process (n-p or p-p FSI, ordinary or charge-exchange quasi-free scattering) dominates these spectra. The theory, which is explicitly charge-independent and has no free parameters, also qualitatively predicts the ratios of the cross sections for these different processes (i.e. normalizing factors between 0.04 and 0.17 were applied to all predictions shown in sect. 4, with the two exceptions noted in subsect. 4.1). Incorporation of distorted waves into the theory, to describe the initial 3He-d and final trion-dinucleon interactions, may provide better absolute normalization than
2H(3He, 3H p)p, ZH(3He, 3He p)n
the factor of 10 by which the PWBA-FSI
overpredicts
absolute cross sections.
607
However,
discrepancies in the detailed shapes of spectra may still remain. For example, since the trion-dinucleon relative energy varies slowly at the high energy end of the spectator peak spectra (fig. 4) it is not obvious that distorted waves could produce the drastic changes needed there without spoiling the fit elsewhere. A more accurate description of these spectra in regions where two or more processes coexist may have to await a Faddeev treatment or some other fundamental improvement. We are indebted to Professors Olexa Bilaniuk and Erich Vogt for their interest and encouragement to undertake this work, R. L. Brown for expert assistance with electronics, and George Gorsline and the Oberlin College Computer Centre staff for computational assistance. References 1) R. P. Haddock, R. M. Salter, M. Zeller, J. B. Czirr and D. R. Nygren, Phys. Rev. Lett. 14 (1965) 318 2) H. T. Larson, A. D. Bather, K. Nagatani and T. A. Tombrello, Nucl. Phys. Al49 (1970) 161 3) K. M. Watson, Phys. Rev. 88 (1952) 1163 4) A. B. Migdal, ZhETF (USSR) 28 (1955) 3; [JETP (Sov. Phys.) 1 (1955) 21 5) E. M. Henley, F. C. Richards and D. Y. L. Yu, Nucl. Phys. A103 (1967) 361 6) L. A. Charlton, S. M. Kelso and R. E. Warner, Nucl. Phys. Al78 (1971) 39 7) R. E. Warner and E. W. Vogt, Nucl. Phys. A204 (1973) 433 8) R. E. Warner, G. C. Ball, W. G. Davies, A. J. Ferguson and J. S. Forster, Phys. Rev. Lett. 27 (1971) 961 9) M. J. Moravcsik, Nucl. Phys. 7 (1958) 113 10) W. T. H. van Oers and I. Slaus, Phys. Rev. 160 (1967) 853 11) R. M. Haybron, Nucl. Phys. All2 (1968) 594 12) B. Hird and R. W. Ollerhead, Nucl. Jnstr. 71 (1969) 231 13) R. E. Warner, G. R. Flier], W. G. Davies, G. C. Ball, A. J. Ferguson, J. S. Forster and S. A. Gottlieb, Nucl. Phys. Al92 (1972) 341 14) A. Messiah, Quantum mechanics (North-Holland, Amsterdam, 1966) p. 824 15) W. Tobocman, Theory of direct nuclear reactions (Oxford University Press, Oxford, 1961) p. 13 16) M. A. Preston, Physics of the nucleus (Addison-Wesley, Reading, Mass., 1962) p. 25 17) E. W. Vogt, private communication 18) R. J. N. Phillips, Nucl. Phys. 53 (1964) 650 19) P. D. Parker, P. F. Donovan, J. V. Kane and J. F. Mollenauer, Phys. Rev. Lett. 14 (1965) 15 20) A. McIntyre, P. H. Beatty, J. D. Bronson, R. J. Hastings, J. G. Rogers and M. S. Shaw, Phys. Rev. C5 (1972) 1796
1
Nuclear Physics A221 (1974) 593 - 607; @ Not to
EFFECTS
North-Holland
Publishing
Co., Amsterdam
be reproduced by photoprint or microfilm without written permission from the publisher
OF EXCHANGE
IN THE REACTIONS
AND FINAL-STATE
INTERACTIONS
2H(3He, 3H p)p AND 2H(3He, 3He p)n
R. E. WARNER t and S. A. GOTTLIEB it Oberlin
College,
Oberlin,
Ohio 44074,
USA
and G. C. BALL, W. G. DAVIES, A. J. FERGUSON
and J. S. FORSTER
Atomic Energy of Canada Ltd., Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada KOJ IJO
Received 17 December 1973 Abstract:
Absolute coincidence cross sections for the 2H(3He, 3He p)n and 2H(3He, 3H p)p reactions were measured at EHc = 35.9 MeV. Spectra dominated by the nucleon-nucleon final-state interaction (FSI) are fitted by a fully antisymmetrized PWBA theory which includes the effects of FSI in all its matrix elements. Previously reported 26.8 MeV data showing both FSI and quasi-elastic scattering (both with and without charge exchange) are also fitted by the theory, which qualitatively describes the shapes of all these spectra and the ratios of the cross sections for the various processes. Predictions of Watson-Migdal theory are fitted to the FSI spectra and differences between the two theories are analyzed.
E
NUCLEAR
REACTIONS ZH(3He, jHe p)n and ZH(3He, 3H p)p, E = 35.9 MeV; measured absolute coincidence a(El , Of, e2). Enriched target.
1. Introduction Final-state interactions (FSI) are observed when reactions between elementary particles or nuclei produce multi-body final states and two strongly interacting particles emerge from the reaction with very low relative velocity. For example, the cross section for the reaction 2H+ 3He + 3H+p+p is strongly inhibited by the Coulomb repulsion whenever any two final particles exit nearly together. In contrast, when a neutron and proton emerge from the ‘H + 3He -+ 3He + p + n reaction with low relative velocity, the yield is enhanced by the attractive n-p interaction. Originally FSI effects were studied to gain information about systems which could not be studied through conventional scattering experiments; e.g. the n-n scattering length was measured in the reactions n- f2H -+ y+n+n [ref. ‘)I and 2H+3H -+ 3He +n+n [ref. “)I. There is now a parallel interest in the mechanisms which govern such reactions. It is of interest to know whether they can be adequately understood through the Watson-Migdal (WM) theory 3, 4), which only takes account of the scattering wave function of the two strongly interacting particles, or whether more detailed theories are needed. If one uses a more detailed theory such as the t Research supported by the National Science Foundation under grant GP-35985. +t Present address: Physics Department,
Princeton University, 593
Princeton,
NJ 08540, USA.
594
R. E. WARNER
et al.
modified PWBA ‘) or DWBA 6), is it sufficient to include only a few simple mechanisms such as charge-exchange and pickup “) or must the full effects of complete antisymmetrization ‘) be taken into account? In this paper we present measurements of the absolute coincidence cross sections for the reactions 2H(3He, 3He p)n and 2H(3He, 3H p)p at 35.9 MeV helion (3Hef+) bombarding energy, at several geometries where the two final nucleons can emerge nearly together and thus exhibit FSI. These data are fitted by the predictions of three theories: a fully antisymmetrized PWBA theory ‘) with final-state interaction effects included in all matrix elements; a DWBA theory “) which considers only two reaction mechanisms, charge-exchange and pickup; and the Watson-Migdal theory 3, “). Fully antisymmetrized PWBA calculations ‘) of the cross sections for these two reactions have recently been reported. The effects of exchange were taken fully into account, but plane waves were used to represent the relative motions of all unbound particles; initial- and final-state interactions between nuclei were thus ignored. The original purpose of these calculations was to interpret coincidence experiments “) showing intense spectator peaks for both reactions. Although the magnitude of the calculated cross sections was too high, the calculations reproduced rather well not only the shapes of the observed spectator peaks but the ratios of the measured cross sections for the two reactions. Some discrepancies between predicted and observed peak shapes were noted where the final nucleon-nucleon relative energy sNN was low, and were attributed to nucleon-nucleon FSI. The use of a HulthCn wave function “) for the deuteron prevented us from including FSI effects since most of the seven matrix elements of the antisymmetrized theory could not be evaluated. Since then, the Van Oers-Slaus deuteron wave function 2, lo), a sum of three Gaussian functions closely approximating a Hulthen function, has been brought to our attention. It permits FSI effects to be included in all matrix elements of the theory (hereafter called PWBAFSI) which should describe structure in the observed cross sections resulting from both spectator poles and FSI. Thus in this paper we compare the PWBA-FSI predictions not only with the new 35.9 MeV data showing p-p and p-n FSI, but with our earlier 26.8 MeV p-p FSI “) and spectator pole data as well. Since the WM theory is the very simplest theory of FSI, it is of interest to compare its predictions with those of more detailed theories. We examine the assumptions of the WM theory with particular reference to an earlier study by Haybron ‘i) which concluded that the WM theory was more reliable for reactions proceeding by pickup than by charge-exchange. We disagree with his conclusion, and identify factors of kinematic origin omitted from Haybron’s analysis.
2. Experimental
procedure
Our experimental procedures will be briefly summarized here since they have been described in greater detail elsewhere i3). A CD, target was bombarded with helions (3Hef+) whose energy at the target centre was 35.9 MeV. Coincident charged
‘HC3He,3H p)p, 2H(3He, 3He p)n
595
particles were detected by two AE-E counter telescopes placed on opposite sides of, and coplanar with, the beam. (Such geometries will be designated by (e,, t?,,), where x and y specify the particles detected at the two angles.) Both telescopes employed 0.345 cm diam. defining apertures and 2000 pm E-counters; the AE counter thicknesses were 100 pm and 150 ,um in the proton and t-3He telescopes, respectively. Pulse heights and time intervals between AE pulses were recorded event by event on magnetic tape. During analysis each particle was identified by referring its AE and E to a triton range table “). Absolute cross sections for the reactions ZH(3He, 3He p)n and 2H(3He, 3H p)p were deduced from the yield of events which obeyed the correct three-body kinematic relationship EHe versus Ep or Et versus E,,. Loss of deuterium from radiation damage in the target was monitored by integrating the beam current and measuring the yield of helions elastically scattered from both ‘H and “C at 15”. The deuterium densities found from the 2H/Q and ‘H/l’C ratios (assuming no loss of carbon from the target) were consistent within 5%. The normalization of our data is believed to be accurate within lo%, and only statistical uncertainties are shown in our results which are presented in figs. 1 and 3. 3. Theories describing final-state interactions In this section we briefly describe existing theories that have been used to interpret spectra dominated by final-state interactions (FSI). In the “post” form of formal scattering theory i4), the matrix element for a rearrangement collision is given by the equation: T,i = (~r-‘lW~lll/i+‘).
(I)
The initial-state wave function $I” obeys outgo’m g-wave boundary conditions and is an eigenfunction of the total Hamiltonian H, while the final-state wave function (6:-’ obeys incoming-wave boundary conditions and is an eigenfunction of HI where W, = H-H, is the cause of the transition. In practice one cannot obtain exact wave functions J/I’) and 4i-j for a system as large as A = 5, even assuming that all the interactions are known. The wave functions are therefore replaced by various approximations that stress physically important aspects of the reactions being studied. The most drastic approximation of all is the plane wave Born approximation (PWBA) in which particles not bound to each other are assumed not to interact with each other at all; i.e. their relative motion is described by plane waves. This approximation may nevertheless have qualitative success in predicting those features of nuclear reactions that are primarily influenced by boundstate wave functions which appear as factors in $i” and&-‘. For instance, the angular distributions of nucleons from deuteron stripping reactions 1“) and the energy distributions in three-body final states following quasi-elastic deuteron breakup are largely determined by the internal wave function of the initial deuteron. The PWBA cannot, by definition, account for the FSI effects observed here when
R. E. WARNER
596
et al.
the two final nucleons emerge with low relative velocity: namely, ‘H(3He, 3He p)n spectra caused by the attractive p-n interaction, in the ZH(3He, 3H p)p spectra resulting from the Coulomb protons. A common ingredient of all theories that seek to interpret the inclusion of a nucleon-nucleon scattering wave function as a 3.1. THE
ANTISYMMETRIZED
PWBA
The general approach, first taken account of the FSI between the two ing wave function &), rather than wave function. The spatial parts of q,i +)
=
THEORY
WITH
FSI
the maxima in the and the minima repulsion between these FSI effects is factor in &-‘.
INCLUDED
by Henley rt al. ‘) (hereafter called HRY), took final free nucleons by incorporating their scatterthe corresponding plane wave, into the final-state the intial and final wave functions thus become: +d
+He
,iX.
(he -Rd),
(2) (3)
The subscript T denotes the final trion (either 3H or 3He) and NN denotes either the n-p or p-p pair. The trion wave function & is a symmetric Gaussian function with characteristic length y- ’ = 2.78 fm. The momenta hK, hk and hq refer to the motion of the initial 3He relative to the deuteron, the final trion relative to the nucleonnucleon c.m., and the final nucleons relative to each other, respectively. The relationship between nucleon-nucleon relative energy, eNN, and momentum is h2q2 = MpcNN. The major difference between the present work and that of HRY is that we include all exchange effects by antisymmetrizing the wave functions. The Van Oers-Slaus deuteron wave function
(4
i=l
which closely approximates the Hulthtn function is used; the constants given in ref. ’ “). Finally, the two body transition potential V(P) = --*vc(l
-PoP,)e-szrZ
Cj and Mi are
(5)
is taken to have a depth V, = 72.5 MeV and a characteristic length 6-r = 1.46 fm. This particular combination of central and exchange interactions, operating between fully antisymmetrized wave functions, is a Serber force. Tobocman + has shown how total antisymmetrization (antisymmetry of the wave functions for all nucleon exchanges) is achieved for rearrangement collisions. In general, the effect of antisymmetrization is to establish coherence between the amplitudes of various reaction mechanisms (such as pickup and charge-exchange) and to introduce other exchange amplitudes that are normally neglected. Seven basic matrix elements are found to determine the two reactions under discussion. These t Ref. Is), p. 100.
ZH(3He, 3H p)p, ZH(3He, 3He p)n
597
include one breakup, four knock-out, and two pickup amplitudes. These broad categories denote whether 3, 2 or 1 nucleons are common to the initial and final trions; they are subdivided according to the origin and destination of the two nucleons which interact, through eq. (6). Thus & denotes breakup; the first subscript says that one interacting nucleon, which ends in the trion, began in the initial 3He; the second subscript says that the other interacting nucleon, which becomes a free nucleon, came from the initial deuteron. The matrix elements are further defined in ref. ‘). They can all be written in the general form:
x
dsz
. Zed,,X.Zed,,J. Z d3U ,-&,u*,‘&r s
,-a.zzeik,
s
where the constants up, k, and 6,, vary from repeated use of the formula
one matrix
uedyuy ued,,,:’ Y, element
to another.
s
(6) By
(7)
ewnr
each term in the sum over i reduces to a constant
times a one-dimensional integral containing the nucleon-nucleon wave function x6-j. The p-p scattering wave functions that describe the FSI effects in the 2H(3He, 3H p)p reaction were computed for a point-charge Coulomb potential and a real central Woods-Saxon nuclear potential with V,, = 33.5 MeV, R, = 1.477 fm and a = 0.5 fm. Gaussian potentials with V, = 41.0 MeV, I/, = 72.5 MeV, rl = 1.59 fm, and Y3= 1.47 fm (with subscripts denoting singlet or triplet states) were used to compute the n-p scattering wave functions for the 2H(3He, 3He p)n reaction. We have found that these potentials reproduce the appropriate nucleon-nucleon scattering lengths and effective ranges; the p-n triplet potential has also been given by Preston 16). Taking appropriate sums and averages ‘) of the matrix elements leads to the following formulae for the cross sections for the two reactions:
d30 dQ,dQ,dE; ;
IW+,+,=
tie+,+, = (i-&.$M: +b%M:
+M:
&12M:
(8)
IMI”, if
+M;12+&M;12,
(9)
+M,~2)s=o
+-M: +3M:12+&-2M: M:
M:
- 27cp(E,)z - ho
= 12(P;
-PL
= - 6( B,‘, + K;) - 12(P; M:
+M;
+12M;
+3M,12)s=1,
+K;), + K*$ + K&) - 24K,;,
= 6(B:, + K:).
(10) (11) (12) (13)
R. E. WARNER
598
et al.
The cross section applies to the detection of particle 1 with energy E, in solid angle dQ,, in coincidence with particle 2 in solid angle d&; we consider only the Iarger of the two energies Ez consistent with a given E1. The laboratory velocity of the incident particle is U, and the density of final states is p(E,). The k superscripts denote the parity of the two final free nucleons, and the subscripts to the curly brackets in eq. (10) indicate whether singlet or triplet n-p wave functions enter into the enclosed matrix elements. 3.2. DWBA
TREATMENT
OF THE
ZH(3He, 3H
p)p REACTION
In this approach 6), which is also a straightforward extension of the modified PWBA treatment of HRY ‘), we consider only charge-exchange and pick-up processes, as do HRY. The initial 3He-d interaction is now taken into account by replacing the plane wave in eq. (2) by an appropriate distorted wave. Likewise, the wave function for relative motion in the final state is the product of a p-p scattering wave function (for the potential described in subsect. 3.1) and a distorted wave for the t-(pp) relative motion, assuming that their interaction depends only upon the separation of their centres of mass. DWBA fits to our 26.8 MeV data have already been reported “)_ Good fits to the shapes of the p-p FSI spectra were obtained, while the magnitudes were over-predicted by factors of 1 to 4. It was subsequen~y pointed out to us that the nuclear potentials used for the 3He-d and t-(pp) distorted waves were unrealistically strong 1‘). Therefore in the present work we use V = 35 MeV and W = 6 MeV for the real and imaginary depths of the triton-(pp) and 3He-d interaction, and R = r,AS = 3.16 fm for their ranges. Thus the strength Wri is 163 MeV * fm2, about twice the accepted value for elastic nucleon scattering. 3.3. WATSON-MIGDAL
THEORY
The basic assumptions of the Watson-Migdal (WM) theory are that when the two strongly interacting nucleons have sufficiently low relative energy, em,,, their scattering wave functions can be factored into parts depending only upon enN and their separation, respectively, and that all othnr quantities that appear in the reaction amplitudes are constants. An equivalent statement about the wave function is that its &-dependence (or q-dependence) must be the same both inside and outside the range of nuclear forces. Watson “> and Haybron I*) show why this is true for short-range interactions. Phillips I*) does so for the Coulomb interaction as well. Thus the energy dependence of the matrix element is just that of the scattering wave function, and we have for the n-p FSI matrix element and cross section: eis sin 6 Tfi ~~~ 4 d3a x @J [sin’ ~5~1-3sin’ li3 ] . dQ, dS’&dE, q2
2H(3He, 3H
599
p)p, 2H(3He, 3He p)n
For the p-p case, where there is only singlet s-wave scattering, the matrix element is
Definitions of the nuclear phase shifts 6 (from effective range theory) and of the Coulomb penetration factor Co and phase shift ~~ appear in refs. *, 16). 4. Comparison of experimental data with predictions 4.1. THE
p-p FINAL-STATE
INTERACTION
In fig. 1, we show absolute cross sections for the reaction 2H(3He, 3H p)p at E, = 35.9 MeV, at three geometries where the p-p relative energy goes as low as a few keV. These are plotted against both ePp and the lab energy Ep of the detected proton. Although the counter telescope cutoff for protons is nominally 4 MeV, two spectra have been extended to about 2 MeV. This was done by including both events which satisfy normal acceptance criteria and those where a triton was identified in the forward telescope in coincidence with a particle stopping in the AE counter of the second telescope. This leads to artificial minima in the spectra at Ep = 3.6 MeV, and artificial maxima at 2.8 MeV, caused by protons whose energy loss in the second telescope’s E-counter are below the discriminator threshold. Since such particles are counted but assigned an energy about 1 MeV too low, our averaged cross sections in this energy region should be reliable even though artificial structure appears. P-P
ENERGY,
cpp
5 I *a ,I .5 I. ~~~1. /--- ‘T_.7_77 1-----r
PROTON
LAB
ENERGY,
,
MeV
.rmzr 5 1. 2. I I /
MeV
Fig. 1. Absolute cross sections, and predictions of the PWBA-FSI, DWBA and WM theories, for the ZH(3He, 3H p)p reaction at 35.9 MeV incident energy at three geometries (BP, 0,) where low p-p relative energies were obtained. Predictions of the PWBA-FSI theory were normalized by factors between 0.04 and 0.17, those of the DWBA theory by 0.6 to 3.6, and WM predictions were normalized arbitrarily. Arrows mark the expected location of peaks from sequential decay of the 21.3 MeV second excited state of 4He into the triton and undetected proton.
R. E. U’ARNER
600
RELATIVE
PROTON
Fig. 2. The
same
as fig.
P-P
LAB
et 01.
ENERGY,
cppr
MeV
-------
WATSON-MiGDAL
ENERGY,
MeV
I, at 26.8 MeV incident energy. Normalizations PWBA-FSI, and 1.0 to 1.9 for DWBA.
4He:,.3-
were
‘5
0.09
to 0.52 for
Predictions of the PWBA-FSI, DWBA, and WM theories are compared with our 35.9 MeV data in fig. 1, and our earlier 26.8 MeV data “) in fig. 2. Pronounced dips in the cross section are observed in figs. 1 and 2 whenever spp goes below about 0.1 MeV as a result of the Coulomb repulsion between the emerging protons. For spp s 0.5 MeV the three theories fit the data acceptably and in fact the predicted shapes are nearly identical. The latter is expected since, near the minimum, the p-p relative momentum varies much more rapidly with E, than does the t-(pp) relative momentum. Thus most of the Ep dependence of the matrix elements is contained in the p-p scattering function, which appears in all three theories. Away from the minima, broad peaks are observed in five spectra where the relative energy E,,, of the triton and undetected proton corresponds to the excitation energy of the 21.3 MeV second excited state of 4He (which decays yielding these two particles). None of the three theories attempts to include the effects of this state in 4He. One of the worst fits appears in fig. Ic where the excess of observed over predicted yield occurs over a much greater energy range than can be accounted for by the width (r z 2 MeV) of the 4He state. Between the p-p FSI minimum and the 4He peak the lab energy of the undetected proton reaches a minimum value of about 0.8 MeV. Although this is a rather high energy for a spectator pole effect, the possibility exists that the exceptionally poor fit results from interference of the three different processes.
2H(3He. “H p)p, ZH(3He, 3He p)n
601
The DWBA fits are slightly superior to those of the other two theories for the 26.8 MeV data though probably not at 35.9 MeV. The DWBA normalization was more uniform at the lower energy where the required normalization factors varied from 1.0 to 1.9 as opposed to 0.6 to 3.6 at 35.9 MeV. The DWBA fits shown were obtained with I’ = 35 MeV, but the quality of the fits did not vary appreciably as I/ was changed from 25 to 50 MeV with Vri fixed at 163 MeV * fm’. The PWBA-FSI predictions were all normalized by factors between 0.07 and 0.13, except that factors of about 0.5 were used for the two spectra with Bt = 35” (the largest triton emergence angle studied) in fig. 2. At 6, = 35” the charge-exchange amplitudes are smaller, and the pickup amplitudes larger, than at smaller 0. The transfer of two nucleons within the triton may be more the result of higher order than first order processes, and hence less well described by the PWBA-FSI theory.
RELATIVE .l.cm.1
.5
I.
p-n
ENERGY,
2.
------
_I 02 .I
cpn
, MeV
.5
2.
I.
WATSON-MIGDAL
r-m
2.1.5.lD15.I
.p
15l
(52;.
i
4
6
2.
i
26.5=,,,)
16
12 PROTON
I.
LAB
ENERGY,
MeV
Fig. 3. Absolute cross sections and PWBA-FSI and WM predictions for the *H(3He, 3He p)n reaction at 26.8 and 35.9 MeV. PWBA-FSI normalizations were 0.04 to 0.058. Arrows show expected peak location for sequential decay of 4He (21.3 MeV) into 3He+n.
602
R. E. WARNER
4.2. THE p-n FINAL-STATE
et al.
INTERACTION
Fig. 3 shows data for the 2H(3He, 3He p)n reaction at 35.9 and 26.8 MeV incident energy, for four geometries where epn g oes below 0.1 MeV. As is expected, the attractive p-n interaction produces strong broad peaks where this occurs. (In two cases, the lower half of the peak is below the proton telescope cutoff.) The p-n FSI peaks are well fitted by both the WM and PWBA-FSI theories. In particular, both theories predict a mixture of singlet and triplet contributions that agrees with the data. The PWBA-FSI theory has no free parameters and the ratio of singlet to triplet states is not adjustable. Thus its success may be less fortuitous than that of the WM theory, which includes only the n-p wave function while omitting the overlap integrals and isospin factors which would appear in a more detailed theory and which would differ for the singlet and triplet states. Enhancements from the sequential decay of 4He” (21.3 MeV) into 3He+n appear in the three spectra where this is kinematically possible. Only half peaks are seen since the proton energies corresponding to this excitation are near the upper kinematic limits. Parker et al. 19) have also observed that this state decays by both proton and neutron emission. 4.3. SPECTATOR
POLES
Fig. 4 shows the fits to the previously reported spectator-pole data “) for these two reactions. These fits are not much better than those from the earlier fully antisymmetrized PWBA theory ‘). FSI effects were included partly in the hope of fitting the high-energy end of the (15;, , 4.4;) spectrum, where a p-n FSI enhancement is observed. A rise in the PWBA-FSI curve is obtained, but the minimum between FSI and spectator peaks is too shallow. The predicted width of the (1.5&, 2.5;) peak is too great, and presumably that of the (15:) 25;) peak would be as well if the enhancements due to 4He” (21.3 MeV) sequential decay could be subtracted from the experimental data. Previously it had been found that the HulthCn function often overpredicted the width of spectator peaks; this difficulty was overcome 20) by smoothly cutting-off the wave function at small separations, which also attenuated its high-momentum components. The justification for such a procedure was that, when a projectile collides with a deuteron whose nucleons are close together, reactions other than quasi-free scattering will probably occur. It is disappointing therefore that the present PWBA-FSI theory, which is fully antisymmetrized and should account for all reaction mechanisms and predict all final states in their proper ratio, should also predict excessive widths. Again the explanation may be that high-momentum transfers lead preferentially to higher-order processes, so that the PWBA fails. One hope for both the earlier theory and the present one was that, since they are explicitly charge-independent, they would correctly predict the ratio of the 3He+p+n and t +p+p cross sections. The present theory does so within a factor of about 1.5; the normalizations for these two final states were about 0.12 and 0.08, respectively. This is slightly better than the earlier theory 7), which gave a ratio of about 2.
ZH(3He, 3H p)p, ZH(3He, 3He p)n
2H(3He,3He
603
p)n 30-
20!-
IO-
O-
‘He
LAB
ENERGY,
FneV
Fig. 4. Absolute cross sections for the two reactions at 26.8 MeV for geometries where the spectator nucleon lab energy (right hand scale) could be near zero. PWBA-FSI predictions for 2H(3He, 3H p)p (upper panels) were normalized by about 0.075 and those for 2H(3He, 3He p)n by about 0.115.
5. Discussion
and conclusions
Since the shapes of the PWBA-FSI and WM predictions are so similar, there arises the question of whether it is really necessary and advantageous to use the former, much more detailed, theory rather than the simpler one. One might also ask whether the PWBA-FSI theory is an improvement over its immediate predecessor, the modified PWBA theory of HRY “). To answer the latter question first, we show in fig. 5 PWBA predictions for at 45$ 31:) and 35.9 MeV, with only two matrix elements included: 2H(3He, 3H P)P ( Btd which is identical to the charge-exchange matrix element of HRY and P,t a pickup matrix element similar to theirs. (The particular pickup amplitude used by HRY does not appear in a treatment, such as the PWBA-FSI, employing “post” interactions.) For these calculations, and throughout this section, we use a Hulthen function rather than a Van Oers-Slaus function for the initial deuteron. The amplitudes Btd and Ptt (the only ones which can be evaluated for the Hulthen
604
R. E. WARNER
p-p
RELATIVE
I. 15
.5 r
*H(‘He,
c
ENERGY,
et 01.
lpp ,
MeV
.I .OOl -.T--~ r~
tp)p
4
I 12
8
PROTON
LAB
ENERGY,
____I 16
MeV
Fig. 5. Predictions of the PWBA-FSI theory using only the non-rearrangement and pickup matrix elements B,, and P,, . both separately and together, are compared with one p-p FSI spectrum and the WM prediction.
function) then reduce to single terms rather than the sum of three terms mentioned in subsect. 3.1. The PWBA fit to the data, when only these two amplitudes are included, is very poor, and in particular the minimum is much too broad. Fig. 5 shows that the (arbitrarily normalized) cross sections for charge-exchange or pickup alone equal to zero) are quite different from (obtained by setting Ptt, or &,, resp:ctively, each other and from the prediction with both included. The poor PWBA fit results mainly from destructive interference between these two amplitudes, and the importance of including all amplitudes that coherently determine the cross section is thus emphasized. To see whether the close agreement between the PWBA and WM theories is accidental, we return to the basic assumptions of the WM theory. The first of these is that the shape of the nucleon-nucleon scattering wave function is energyindependent. Fig. 6, in which ~6,) is normalized by the energy-dependent factor of eq. (16), shows that this is true for nucleon separations up to 4 or 5 fm. Thus the first assumption will be least valid for those matrix elements in which the integrals containing ~6,’ have the slowest cutoffs. These are, in fact, the three spectator pole matrix elements B,, , K,, and Kdt for which the cutoff parameters a, (see eq. 6) are just the Cli of the deuteron wave function; all other matrix elements are cut off more rapidly by the trion and/or nuclear force range parameters y and 6. However, fig. 5 shows that the charge-exchange cross section (labelled Btd) follows the WM
ZH(3He, 3H p)p, 2H(3He, 3He p)n
I 0
I
I
I
5
IO
15
PROTON SEPARATION,
605
I
r , fm
Fig. 6. Wave functions for p-p scattering in a Woods-Saxon well multiplied by the factor sin &I-’ which, according to WM theory, removes their dependence upon the p-p relative energy Ebb.
qC,,[ei(m+W
RELATIVE
p-p
ENERGY,
epp,
MeV
Fig. 7. Dependence upon ePDof factors in the matrix elements Btd and P,,. Upper panels show the integrals containing the p-p scattering wave function, multiplied by qCo[eWm+W sin I&,]-~. Lower panels show the kinematic coefficient of the integrals. All four quantities are divided by their values at ePB= 0; all four would be straight horizontal lines if WM theory were valid.
606
R. E. WARNER
et rrl.
prediction much more faithfully than does the pickup cross section (P,,). This can be understood by reference to fig. 7, where the two factors which determine these two matrix elements arc plotted against Ear. The upper panels show the integrals containing the p-p scattering wave function, normalized by the right-hand side of cq. (16); the lower panels show the coefficients which multiply these integrals and which, as discussed in subsect. 3. I, contain various kinematic factors. All four curves are further normalized to their values at cpp = 0; according to the WM theory, all should be straight horizontal lines. The charge-exchange integral In does in fact vary somewhat more rapidly with c,,,, than does its pickup counterpart I!,, as is expected because of its slower cutoff but the pickup kinematic coefficient T has by far the largest energy variation, and is the principal cause of the large difference between the pickup and WM cross sections. The WM and charge-cxchangc cross sections agree rather closely both because the energy variations of Ia and S partially compensate each other, and because the contributions to the integral Ia at large nucleon separations, where the WM assumptions about the scattering wave function break down, are reduced by the oscillations of exp (i k, . x) see cq. (6). The pickup amplitudes, which are poorly represented by WM theory, are in general rather small at the geometries here investigated. Thus various accidental factors lead to closer agreement between the WM and PWBA-FSI thcorics than one can expect in general. Haybron ‘l) concluded that the WM theory better represents reactions proceeding by pickup than by charge-exchange. Fig. 5 shows that the opposite is true. In this earlier study, only the integrals I* and I,, were examined. It was further concluded that WM theory works better for pickup since the two nucleons that interact strongly in the final state were initially located in a small volume; i.e. within the trion. However, the pickup coefficient Tcontains a factor exp( - IR+ $k1’/4yZ) which is related to the probability of finding the picked-up nucleon in the initial helion with its final momentum. This quantity varies rather rapidly with cpp partly because y2 is so larg: or, equivalently, because the trion is so small. Thus the compactness of the trion is partially responsible for the large difference between WM and PWBA pickup cross sections. The criterion that the two strongly interacting nucleons b: confined initially to a small spatial volume is not sufficient for th: validity of WM theory. Other effects, rooted in nuclear structure or details of the nuclear fort: but related to experimental kinematics, can bc substantial. The principal achievement of the fully antisymm:trized PWBA-FSI theory is that it reasonably well describes the shape ofcoincidcncz energy spectra from three body final states when a single process (n-p or p-p FSI, ordinary or charge-exchange quasi-free scattering) dominates these spectra. The theory, which is explicitly charge-independent and has no free parameters, also qualitatively predicts the ratios of the cross sections for these different processes (i.e. normalizing factors between 0.04 and 0.17 were applied to all predictions shown in sect. 4, with the two exceptions noted in subsect. 4.1). Incorporation of distorted waves into the theory, to describe the initial 3He-d and final trion-dinucleon interactions, may provide better absolute normalization than
2H(3He, 3H p)p, ZH(3He, 3He p)n
the factor of 10 by which the PWBA-FSI
overpredicts
absolute cross sections.
607
However,
discrepancies in the detailed shapes of spectra may still remain. For example, since the trion-dinucleon relative energy varies slowly at the high energy end of the spectator peak spectra (fig. 4) it is not obvious that distorted waves could produce the drastic changes needed there without spoiling the fit elsewhere. A more accurate description of these spectra in regions where two or more processes coexist may have to await a Faddeev treatment or some other fundamental improvement. We are indebted to Professors Olexa Bilaniuk and Erich Vogt for their interest and encouragement to undertake this work, R. L. Brown for expert assistance with electronics, and George Gorsline and the Oberlin College Computer Centre staff for computational assistance. References 1) R. P. Haddock, R. M. Salter, M. Zeller, J. B. Czirr and D. R. Nygren, Phys. Rev. Lett. 14 (1965) 318 2) H. T. Larson, A. D. Bather, K. Nagatani and T. A. Tombrello, Nucl. Phys. Al49 (1970) 161 3) K. M. Watson, Phys. Rev. 88 (1952) 1163 4) A. B. Migdal, ZhETF (USSR) 28 (1955) 3; [JETP (Sov. Phys.) 1 (1955) 21 5) E. M. Henley, F. C. Richards and D. Y. L. Yu, Nucl. Phys. A103 (1967) 361 6) L. A. Charlton, S. M. Kelso and R. E. Warner, Nucl. Phys. Al78 (1971) 39 7) R. E. Warner and E. W. Vogt, Nucl. Phys. A204 (1973) 433 8) R. E. Warner, G. C. Ball, W. G. Davies, A. J. Ferguson and J. S. Forster, Phys. Rev. Lett. 27 (1971) 961 9) M. J. Moravcsik, Nucl. Phys. 7 (1958) 113 10) W. T. H. van Oers and I. Slaus, Phys. Rev. 160 (1967) 853 11) R. M. Haybron, Nucl. Phys. All2 (1968) 594 12) B. Hird and R. W. Ollerhead, Nucl. Jnstr. 71 (1969) 231 13) R. E. Warner, G. R. Flier], W. G. Davies, G. C. Ball, A. J. Ferguson, J. S. Forster and S. A. Gottlieb, Nucl. Phys. Al92 (1972) 341 14) A. Messiah, Quantum mechanics (North-Holland, Amsterdam, 1966) p. 824 15) W. Tobocman, Theory of direct nuclear reactions (Oxford University Press, Oxford, 1961) p. 13 16) M. A. Preston, Physics of the nucleus (Addison-Wesley, Reading, Mass., 1962) p. 25 17) E. W. Vogt, private communication 18) R. J. N. Phillips, Nucl. Phys. 53 (1964) 650 19) P. D. Parker, P. F. Donovan, J. V. Kane and J. F. Mollenauer, Phys. Rev. Lett. 14 (1965) 15 20) A. McIntyre, P. H. Beatty, J. D. Bronson, R. J. Hastings, J. G. Rogers and M. S. Shaw, Phys. Rev. C5 (1972) 1796