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Effects of antisymmetrization in nuclear direct reactions
IZ.F:2.Nj
Ntsclear Physics A348 (1980) 321- 338; @ ~orfh-~o~la~
P~l~shjn~ Co., Amsterdam
Not to be reproduced by photoprint or microfilm withont written permission from the publisher
EFFECTS OF ANTIS~~ET~ZATION IN NUCLEAR DIRECT REACTIONS ’ M. LEMERE ++and Y. C. TANG School qf Physics and Astronomy, University
of Minnesota,Minneapolis,
Minnesota 55455, USA
and E. J. KANELLOPOULOS
and W. SUNKEL
Institut fir Theoretische Physik der Universitlit Ttibingen, D-7400 Tcbingen, BRD Received 28 April 1980 Abstract: The effects of antisymmetrization in direct reactions are studied by examining the general properties of the coupling-hamiltonian kernel function occurring in a coupled-channel resonatinggroup formulation. Together with a previous investigation of the coupling-normalization kernel function, one finds from this study interesting systematics concerning the general behaviour of direct-reaction processes and some justification for the use of simple three-body models in phenomenological analyses. In addition, it is shown that certain coupling terms make comparatively insigni~cant contributions; these terms may, therefore, be frequently omitted m a r~onating-group calculation, thus reducing to a considerable extent the computational effort required in such a calculation.
1. Introduction In a recent inv~tigation I), we have studied the effects of antisymmetr~ation in direct reactions by examining in detail the stucture of the coupling-normalization kernel (CNK) occurring in a resonating-group formulation ‘, 3). From this investigation, we obtained useful information not only about the likelihood of multi-nucleon transfer processes, but also concerning the connection between nucleon-exchange terms in the many-nu~l~n wave function of a microscopic fo~ulation and various phenomenological, macroscopic mechanisms. In addition, it was also found that there is some justification for the utility of simple three-body models commonly employed in conventional analyses 4-7). Even though such models, by their macroscopic nature, are necessarily rather crude, they do contain, by adjusting appropriate parameters, some of the important features inherent in resonating-group calcuiations. + This research was supported in part by the US Department of Energy and the Bundesministerium &r Forschung und Technologie, BRD. ” Present address: Department of Microbiology, University of Southern California, Los Angeles, California 90033, USA. 321
M. LeMere et al. / Antisymmetvization
322
Although it seems reasonable to expect that the basic features of antisymmetrization in direct reactions may be learned by studying the CNK alone, one may still justifiably argue that there might exist features which are specifically associated with the short-range nature of the nucleon-nucl~n potential. Indeed, in the scattering case *), it has been found that a careful examination of the exchange-hamiltonian kernel did yield additional information concerning the energy dependence of exchange processes. Thus, in the present investigation, we shall undertake to examine the structure of the ~upling-hamiltonian kernel (CHK). As will be seen below, this will certainly be a considerably more complicated task. However, with the use of the complex-generator-coordinate technique 9-13) (CGCT), a general expression for the important exponential factors occurring in the CHK can be readily derived and, thereby, a detailed analysis leading to definitive results can in fact be achieved. In the next section, we give a concise description of the coupled-channel resonatinggroup formulation and discuss the basic structure of the CHK. Sect. 3 is devoted to a brief review of important results obtained from the CNK study I) and to a discussion of the general features of the equivalent local coupling potentials which are constructed to yield the same Born reaction amplitudes as the various nucleon-exchange terms in this kernel function. Explicit resonating-group calculations in the a(p, d)3He case, where contributions from individual exchange terms are investigated, are then presented in sect. 4, the purpose being to demonstrate, in a realistic example, the validity of the conclusions reached from the simple plane-wave Born-approximation (PWBA) study. Finally, in sect. 5, we summarize the results and discuss the implications of the findings of this investigation. 2. Resonating-group formulation and coupling kernel fusions 2.1. RESONATING-GROUP
FORMULATlON
OF DIRECT-REACTION
PROCESSES
Consider a direst-reaction process A(a b)B, which involves in the incident channel (channel fj two clusters a and A having nucleon numbers N, and N,, and in the outgoing channel (channel g) two clusters b and B having nucleon numbers N, and N,. For definiteness, we adopt the convention that N, < N.4, N, < N,,
(1)
N,sN,, Also, for simplicity in discussion, we shall assume that all these clusters have spin zero and are described by translationally invariant shell-model functions in harmonicoscillator wells having a common width parameter CLIn the simplest resonatinggroup formulation of such a process 2), the trial wave function If/ is then written as II/ = *f-t&,
(2)
M. L&few
323
et al. j Antisyrmnetrizatian
where ll/r = JJ’[&%(L.
*.,N,)&(N,-I-1,. **,N,$_N,;N,+N,3-l,...,N,3-N,) x wwwLII.)I~
I+$= s$‘[&(l)...)
N,;N,+l,...,
N,+N,)&(N,+N,+l)...)
(3)
JJ,+N,+NlJ x Wg)Z(L,.)fi
(4)
with SB’ being an intercl~ter ~tisymmetrization operator, &,, 4A, &, and $, being antisymmetrized cluster internal functions, and Z(R,.,,) being a normahzable function describing the motion of the total c.m. The relative-distance vectors R, and Rg are defined as Nn+NA Rr=k $ rt-& .=C ri, (5) A I N,+l a i-l Rg=+
$ +b
i-l
.;iN” B I
ri,
(6)
Nb+l
and the quantity N, is the number of nucleons in the transferred cluster t, given by fvt = Nb--I?,.
(7)
Also, it is noted that, in eqs. (3) and (4), the arguments of the internal functions of the clusters indicate which nucteons are involved in these clusters; such a labelling is permissible, because the wave functions are totally antisymmetrized. The functions F(R,) and G(R,) describe the relative motions between the clusters in the incident and outgoing channels, respectively. They are obtained by solving the projection equation
= 0,
(8)
where ET is the total energy of the system, given by E, = E,+E,+E, = E,+E,+E,,
(9)
with Ef and E, being the relative energies of the clusters in channels f and g, respectively, and E,, E,, E, and E, being cluster internal energies obtained by computing the expectation values of the cluster hamiltonians with respect to the assumed internal functions. The hamiltonian H in eq. (8) is a galilean-invariant operator which has the form H = i T+ 5 Ejj-T,,,,, i
(10) Z&,,, being the
M. LeMere et al. / Ant~y~~etr~atio~
324
kinetic-energy operator of the total cm., and vj being a nucleon-nucleon potential chosen to fit adequately the two-nucleon scattering data especially in the low-energy region. In order to make our subsequent analysis especially transparent, we shall choose vj to be as simple as possible; thus we adopt the central nucleon-nucleon potential + given by eq. (8) of ref. 14), which has a spatial form factor oij = exp (- M$),
(11)
and in which the Majorana space-exchange operator is not explicitly contained (i.e., the exchange operator -ejPTj, instead of the Majorana operator P& is employed). By employing the procedure described in ref. ‘), one obtains the following coupled integrodifferential equations which F(R;) and G(Rh) satisfy:
1 s
- & V& + I/,,(R;) - E, F(R;) + c
K&R;, R;)F(R;)dR;
+K&t;,
R;)G(R;)d&' = 0,
s
+
s
K&R;, R;‘)F(R;‘)dR;= 0,
(12)
(13)
where $r and 4 are, respectively, the reduced masses of the clusters in the incident and outgoing channels. Also, I$, and I+,, represent the direct potentials, and K,, and K, represent the exchange kernel functions in channels f and g, respectively, while K,, and K, represent the coupling kernel functions. From the above equations, it is easily seen that the transition from channei f to channel g is effected by the coupling kernel I(,, which has the form I&JR;, R;‘) = &CR;, R;‘)- E,&(R;, where H, is the coupling-hamiltoni~ H&R;> q)
= ($&$(Rg
and N, is the coupling-normalization
R;‘),
(14)
kernel (CHK) given by - R;)WW’[$a&~(R,
- Rf’Pl),
(15)
kernel (CNK) given by
&(R;, R;‘) = <&,$r$(Rg - R$W’[$,$$(Rr
- R;‘)z]).
(16)
In a previous investigation ‘), we have examined the effects of antisymmetrization in direct reactions by studying the properties of the simpler CNK function. Here we shall try to extend our understanding by further concentrating on the more complicated CHK fiction. t For the same reason mentioned above, the proton charge will be assumed as infinitesimally small.
hf. LeMere et al. / Antisymmetrization
325
Upon performing a partial-wave expansion, one obtains by solving the resultant coupled integrodifferential equations in each l-state the desired S-matrix which has diagonal elements Sir and Stp, and off-diagonal elements S& and Sir. Using these matrix elements, one can then calculate straightforwardly various differential scattering and reaction cross sections 15).
2.2. COUPLING-NORMALIZATION
AND COUPLING-HAMILTONIAN
KERNEL FUNCTIONS
The CNK may be decomposed into the form
where N:(R;, R/) = P,, exp ( - AxNq2 - C,&’
*Rb - B,,Rz),
(18)
with P,. being a polynomial in R;r2, Rf’. Ri and Kg2. The quantity x denotes the number of nucleons interchanged between the group of nucleons labelled by (1,. . ., NJ in cluster a and the group of nucleons labelled by (N, + N, + 1, . . ., N, + NJ in cluster A [see eq. (3)]. Its minimum and maximum values are equal to 0 and N,, respectively. By using the CGCT described in ref. 2), one can derive general expressions for the coefficients AxN, B,, and C,,. These expressions are
(19)
(21) where
r, = ~tb,+Nt)+A~,
(22)
rd = ~t(llt+N,)-A~,
(~3)~
with
(24)
(25)
M. LeMere et al. / Antisymmetrization
326
A = x-j+
(27)
Here the important point to note is that rd is positive for all values of x and C,, has a positive or negative value depending upon whether x is larger or smaller than pl. The expression for the CHK is more complicated. It has the form J&4&, R;‘) = 1 c G,(q x 4
4%
(4 = a, b, c, d, 4,
(28)
- C,&’ . Rb - B,$Zf),
(29)
where H;,(Rb, R;‘) = P,, exp (-Ax&”
with P,, (q = a, b, c, d, e) being again a polynomial in R;“, Rf’ . Rh and RL’. Here one sees that, in contrast to the CNK, there appear now five types of exponential factors for each value of x (in the x = 0 and x = N, cases, there are only four types; this will be explained below). To explain briefly the origin of these five types of coupling terms, we make use of the diagrammatic representation introduced by LeMere et al. *). This is shown in
a
b
C
Fig. 1. Diagrammatical representation
d
e
of direct-reaction coupling types.
327
M. LeMere et al. / Antisymmetrization
fig. 1, where representative diagrams for each type are depicted +. As has been explained in ref. *), each dot on the three horizontal lines denotes a set of all occupied single-nucleon spatial states with the same spin and isospin. The upper line represents the cluster a in the incident channel, while the lower two lines represent collectively the larger cluster A with the middle line representing the transferred cluster t. Also, the dashed line is used to call attention to the fact that nucleons i andj are interacting, while the solid-line exchange loop is used to indicate the nucleons which are interchanged. Furthermore, we have adopted the convention that diagrams shown in fig. 1 are considered to represent implicitly all other diagrams which contain additional exchange loops not involving the interacting nucleons. The type-a coupling term arises from both the kinetic-energy and potential-energy operators in the hamiltonian H, while coupling terms of other types come only from the potential-energy operator. Also, it is easily inferred from these diagrams that there is no x = N, contribution for type b and no x = 0 contribution for type c. General expressions for the coefficients Axq, B,, and C,, can again be derived by using the CGCT. The results are (30)
(31)
(32) where
;1=L
(33)
a+21c ’
TABLE 1 for A”,, xdqi &
Expressions Coupling
type
‘Kq
Xi,
8,4
8,,,
en, and c,, Riq
C”,
0
01
cidq
4 0 -/~,+24 -k--24 k -I4
0
0
-pLp-2A -pL,+2A
14,,+2A /.%-2d
-4,-24 -P(,+2A
-& -I4
-Y Pf
-k -I4
+ For simplicity in presentation, we assume that the nucleon-nucleon potential character. Considerations with the operators cj, Pi, and - P;P; can be similarly
0 -&-U -/~(,+24
-1 0 0
has a purely discussed.
-pLs -k
Wigner
328
M. LeMere
ef al. ,I ~nf~~~~efr~z~fion
and the constants An4, ;?+, &, &,
c,,* and ed, are tabulated in table I, with i+3= Etf+l$
(34)
&n =
(35)
Pr-P**
From this table and eqs. (30H32), one sees that the type-a term and the CNK have exactly the same exponential factor; therefore, for each value of x, these two kernels have rather similar structure. 3. Equivalent coupling twenties 3.1. CONSTRUCTION OF EQUIVALENT BORN APPROXIMATION (PWBA)
COUPLING
POTENTiALS
IN THE PLANE-WAVE
The structure of the CHK is rather complicated. Thus, to determine, at relatively high energies, the relative importance of the various nucleon-exchange terms in the five types of coupling functions, we adopt the following simplifying procedure. For a coupling kernel function of the form
&.(Fg,R;‘)
= PC exp ( - aq2 - cRf’ * Kg - bRh),
(36)
with PC being a polynomial in R;r’, K. RL and Ri2, we first compute the PWBA reaction amplitude, given by B = e exp (- ik, . R#T,(R~, R;‘) exp (ik, . R;‘)dR;dR;’
- &(aki+ck;
k,+bkf)
1 ,
(37)
where & is a proportionality constant unimportant for our present consideration, k, and k, are wave vectors in the incident and outgoing channels, respecti+ely, and PC is a polynomial in ki, k, *k, and kf. Then, we construct, in the manner indicated below, an equivalent local energy-dependent coupling potential which leads, also in the PWBA, to the same reaction ampitude as that given by &. The information we seek can then be obtained by examining the spatial and energy dependence of this coupling potential. Quite clearly, this is a rather crude procedure to study the properties of the coupling kernel; however, it is, in our opinion, the simplest way to extract semi-quantitatively information about antisymmetrization effects from the complicated CHK function. Two cases have to be separately considered. These are as follows: (i) c < 0. In this case, the PWBA reaction amplitude is forward-peaked and the equivalent coupling potential has the form &(R$S(RI:-R;‘). By using the criterion mentioned above and the fact that, at the relatively high energies under consideration
M. LeMere et al. / Antisymmetrization
in this investigation, the relative energies E, and E, satisfy the approximate E f M E, = E,
329
relation (38)
one easily finds that
&.(Q = PCexp( -E/E,) exp[ - (RhlRd2],
(39)
with P, being a polynomial in E and Ra. This equivalent potential is characterized by a characteristic range R, and a characteristic energy E, which have the forms (40)
(41) with M being the nucleon mass. (ii) c > 0. Here the PWBA reaction amplitude is backward-peaked and the equivalent coupling potential has the form ~c(Z$)G(R;:+&‘), with I$&) given again by eqs. (39H41). 3.2. REVIEW OF RESULTS FROM THE CNK STUDY
In a previous investigation ‘), the features of the CNK have been carefully studied by using the procedure described in the preceding subsection. The results obtained were as follows: (i) The critical quantity in the analysis is xN = pLt.For x < xN, the PWBA reaction amplitude is forward-peaked, while for x > xN, the PWBA reaction amplitude is backward-peaked. (ii) For x < xN, both the characteristic range R,, and the characteristic energy E,, decrease monotonically with x and have largest values when x = 0. Since direct reactions occur predominantly in the peripheral region, it is clear that longer-ranged equivalent potentials will have larger influence. Therefore, this shows that the x = 0 term (no-exchange term), with characteristic range R,, and characteristic energy EON, is the most important one among all exchange terms with x < xN. (iii) For x > xN, both the characteristic range R,, and the characteristic energy E, increase monotonically with x and have largest values when x = N,. This indicates that the x = N, term (maximum-exchange term), with characteristic range R,, and characteristic energy EmN, is the dominant one among all exchange terms with x > xN. (iv) The no-exchange and maximum-exchange terms correspond to light-particle and heavy-particle pickup processes commonly employed in phenomenological analyses 16,“) . In a macroscopic description, these processes can be accounted for by the use of three-body models. Thus, the finding here that these terms yield the
330
M. LeMere
et al. / Antisymmetrlzation
dominant contributions provides some justification for the adoption of three-body simplifications in phenomenological studies. (v) By studying the properties of the no-exchange term, it was shown that there is a general tendency in favour of the pickup of a light cluster containing a relatively small number of nucleons. We should mention, however, that this conclusion was reached without taking into account the effect arising from the formation of nucleon clusters, such as a-clusters, in the various nuclei involved. A proper consideration of this latter effect will definitely modify the above statement to a certain extent. (vi) From a careful investigation of the maximum-exchange term, we obtained the interesting information that back-angle reaction cross sections will have large values only if N, and N, do not greatly differ. In a systematic experiment of (p, a) reaction on light nuclei at E, = 38 MeV [ref. ‘“)I, it was indeed found that the ratio of backward-angle to forward-angle cross section does become progressively smaller as the target nucleus becomes heavier. In the following subsections, we shall further examine the CHK with a similar procedure in order to see whether or not these findings need to be substantially modified. 3.3. STUDY OF COUPLING-HAMILTONIAN
KERNEL (CHK)
The study of the CHK is conducted in exactly the same way as in the CNK case. For each coupling type (q = a, b, c, d, e), one determines and then examines the properties of the equivalent coupling potential which contains a polynomial factor pX4and is characterized by a characteristic range R,, and a characteristic energy E,,. By using eqs. (40) and (41) we find, in a straightforward manner, the general expressions for R,, and E,,. These expressions are
h2c!
J, - (x - x4)2
Exq =zf C~~-lx-x,l][~,-Ix-x,l]+K,’
(43)
where the constants x,, J,, and K, are listed in table 2. For the type-a term, the characteristic values are the same as those of the CNK; this is so, because, as has been mentioned previously, these two kernel functions possess the same exponential factor. The quantity xq is determined by setting C,, equal to zero. It is again a critical quantity in the analysis, because the PWBA reaction amplitude corresponding to the type-q coupling term is either forward or backward peaked depending upon whether x is smaller or larger than x4. From table 2, one notes that xq differs from xN only for the coupling types b and c. In fact, even for these two types, the difference, being equal to A, is rather small. The reason for this is that, in a realistic situation, JCis approximately equal to c1and, hence,
M. LeMere et al. / Antisymmetrization
331
TABLE2 Expressions for x4, Jq and K,, Coupling type q
%
I has a value around 3 which is substantially smaller than the value of xN = p, in almost all cases of interest +. An examination of the general properties of the characteristic quantities R,, and E,, is tedious but can be straightforwardly carried out. Here again, one finds that for x < x4 (4 = a, b, d, e), R,, and E,, have largest values when x = 0, and for x > xq (4 = a, c, d, e), R,, and E,, have largest values when x = N,. Therefore, similar to the finding in the CNK case, the present CHK study indicates that, among all exchange terms, the no-exchange term has the largest influence when x < xq and the maximum-exchange term has the largest influence when x > xq. Since there are no x = 0 type-c and x = N, type-b contributions, we have also carefully examined the nature of the x = 1 type-c term and the x = N,- 1 type-b term. For both of these terms, the result shows that the characteristic ranges and energies are appreciably smaller than those of the relevant no-exchange and maximum-exchange terms of other coupling types. In the following discussion, we shall, therefore, no longer consider these two terms. 3.4. NO-EXCHANGE
AND MAXIMUM-EXCHANGE
COUPLING POTENTIALS
Since the no-exchange and maximum-exchange terms are found to be particularly important, we concentrate in this subsection on’ studying the properties of these terms. In the CNK case, the no-exchange and maximum-exchange characteristic ranges and energies are given by ‘)
f,
Ro, =
w
h20! N
-%I,=~ R mN =
(45)
N’ t
4
f
(46)
a(N, - N, - NJ 1 ’ t For heavy-ion reactions, p, is substantially larger than 1. Even in a rearrangement reaction initiated by an incident nucleon, the smallest possible value of p, is f.
M. LeMere
332
E mN =
et al. / Antisymmetrization
h2a N __ 2M N,-N,-N;
(47)
From these equations and the relation N, s %N,-NJ
(48)
which follows from eq. (l), we reached the conclusions listed as items (v) and (vi) in subsect. 3.2. In the CHK case, we perform the analysis by first defining no-exchange factors foq, goq (q = a, b, d, e) and maximum-exchange factors fm4, gm4 (q = a, c, d, e) in the following way: (49) f0q = ROq/RON, (50)
9 oq = Eo,/EoN> fmq
=
9 mq =
Rmq/RmN,
(51)
E,qiE,N’
(52)
These factors are functions of 1, N,, N, and N,. For example, in the case of type d, we find gOd=f&2=l+;l
++; (
9
a
) t>
.
Since i has a value around f, it is easily seen that all the type-d factors have values close to 1. Similarly, we can obtain the expressions for the f- and g-factors of all other.coupling types by using eqs. (42) and (43). A careful examination of these expressions then reveals that, for 1 z 3, all f-factors are smaller than or equal to 1, while all g-factors are not very different from 1. This indicates, therefore, that the conclusions which can be reached by studying the CHK are in fact substantially the same as those obtained from the CNK study. In fig. 2, we illustrate the behaviour of the f- and g-factors as functions of 1 in two examples, namely, the “Ne(a, ,6Li)1‘F and 24Mg( “C, 160)“Ne reactions. Here the solid and dashed curves represent the values of these factors in the noexchange and maximum-exchange cases, respectively. As is seen, the f-factor of the type-b term is comparatively small. Therefore, because of the peripheral nature of direct reactions, this particular coupling type may be relatively unimportant. In fact, one further sees from this figure that even the type-c term may be omitted in certain reactions. On the other hand, the type a, d, and e coupling terms do have comparable characteristic ranges and energies, and, thus, must generally be taken into consideration in a direct-reaction calculation.
333
;; 0.9 t z? , 0.8 Y0.7
0.6 1.4
1.3
1.2 i
I.1
? &
I.0 0.9 0.8
0.7
0
0.1
0.2
0.3 x
0.4
0.5
0.1
0.2
0.3
0.4
0.5
x
Fig. 2. Thef- and g-factors for the ‘“Ne(a, ‘Li)18Fand 24Mg(“C, LbO)LuNereactions.
The finding that the CNK and CHK studies yield substantially similar conclusions is, in fact, not too surprising. From eqs. (19)-(21) and eqs. (30)-(32), one readily notes that the exponential factors in the CHK and CNK become the same in the limit case where the nucleon-nucleon potential has a range approaching infinity {i.e., fc + 0). Since nucleon-exchange processes occur predomin~tly when the interacting nuclei are in close proximity, one may plausibly expect that the range of the nucleonnucleon potential will not qualitatively influence antisymmetrization effects to a large extent. 4. Explicit study of the a(p, d)3He reaction As a demonstration in a practical example, we discuss now the coupled-channel resonating-group study ofthe cc(p,d)3He reaction. Although this example is somewhat too simple and cannot serve to illustrate all the detailed findings described in the
334
M. LeMere et al. / Anfisymmefrtation
preceding section, it does give us some general feelings about the importance of and the roles played by the various coupling terms. The r~(p,d)3He resonating-group calculation is described in ref. “). Since, in our present consideration, the main concern is to study antisymmetrization effects, we shall simplify the calculation by neglecting Coulomb effects and assuming that the internal spatial structures of the IX,3He, and d-clusters are represented by singlegaussian functions with width parameters of 0.514,0.36 and 0.2 fmB2, respectively. For the nucleon-nucleon potential, we use again the one given in ref. 14) with IC= 0.46 fm- ’ [Isee eq. (1 l)] and adopt a Serber exchange mixture. For a qualitative understanding, we first compute the characteristic ranges and characteristic energies of the various equivalent coupling potentials. In order to utilize the formulae given in sect. 3, we assume in this computation that all internal width parameters take on the average value of CI= 0.36 fm-‘. This is a tolerable assumption in view of the fact that il is rather insensitive to CIand we are only conducting a qualitative examination of the basic features of this problem. The results are given in table 3, where the values of R, and E, for various coupling types with x = 0 and 1 are listed. These values will be useful as an aid in our analysis to extract useful information about antisymmetr~ation effects from the resonating-group calculation. TABLE3 Values of R, and Exp calculated with c( = 0.36 fm-’
0
E d e a
1
i e
3.33 1.24 2.54 2.74
37 32 64 55
2.36 0.50 1.90 2.02
19 18 29 26
From studying table 3 and using the fact that the value of xq is between 0 and 1 for all coupling types, one can make the following statements: (i) The PWBA reaction amplitudes corresponding to x = 0 and 1 coupling terms are, respectively, peaked in the forward and backward directions. (ii) The characteristic ranges of the direct potentials in channels f and g are equal to 2.07 and 2.33 fm, respectively [see eq. (74) of ref. “)I_ Comparing with these values, we note that for types a, d and e, the values of the characteristic ranges are either similar or somewhat larger. This indicates that, for the calculation of the differential reaction cross section, these three coupling types must be taken into consideration.
M. LeMere et al. / Antisymmetrization
335
(iii) The characteristic ranges of type b and c terms are significantly smaller than those of the direct potentials; therefore, these coupling types are expected to make only minor contributions to the c~(p,d)3He direct-reaction process +. (iv) The type-d coupling term has largest values for the characteristic energies. For this type, it is noted in particular that the value for x = 0 is more than twice as large as that for x = 1. Because of this, one expects that the reaction cross section in the backward direction will decrease with energy at a considerably faster rate than that in the forward direction. (v) The value of (N,-N,NJ is fairly small in this case; consequently, the x = 1 type-d and e characteristic energies acquire rather large values around 30 MeV. Therefore, since the energy dependence of the depth of the equivalent coupling potential comes from the factor exp (-E/E,.,), it is anticipated that even at the relatively high energy of E, = 130 MeV (Eg FZ 105 MeV) which we are mainly concerned with here, there should still be a noticeable rise in reaction cross section in the largeangle region. In tig. 3, we show by solid dots the resonating-group reaction cross sections
a
(p,d13He
E,=l30
0.001
0
MeV
....
r-g
-
x=0
----
x=
\ \ ,\
,
I
I
20
40
60
80
only
I only
I
100
I
120
140
160
180
B(deg) Fig. 3. Angular distributions for the or(p, d)3He reaction at Er = 130 MeV. The solid dots represent the results of the full resonating-group calculation. The solid and dashed curves show, respectively, the results obtained when x = 0 coupling terms alone and x = 1 coupling terms alone are taken into consideration.
t The a(p, d)3He resonating-group study described here was performed in 1973 when there was little understanding concerning the roles played by the various coupling types. Aa s consequence, items (ii) and (iii) have, unfortunately, not been examined.
336
M. LeMere et al. / Antisymmetrization
calculated at E, = 130 MeV. From this figure, one notes that, in the back-angle region, there is indeed a significant rise in the reaction cross section. On the other hand, because of the relatively high energy involved, the cross section at 180” is not too large, which is evidenced by the fact that the backward-to-forward ratio, i.e., o(180”)/a(0”), has a rather small value equal to 0.024. At lower energies of E, = 70 and 100 MeV, the backward-to-forward ratios are equal to 0.15 and 0.049, respectively. Together with the value of 0.024 at 130 MeV, we find that this ratio does decrease quite rapidly with energy, in agreement with the qualitative finding mentioned in statement (iv) of a preceding paragraph. From fig. 3, it is also noted that the full resonating-group result in the forward angular region can be reproduced very well by a calculation employing only x = 0 coupling terms (solid curve), while that in the backward angular region can be represented satisfactorily by a calculation employing only x = 1 coupling terms (dashed curve). Based on the qualitative discussion given above, this is entirely to be expected. To demonstrate more clearly the different characters of the x = 0 and x = 1 equivalent coupling potentials, we depict in fig. 4 the behaviour of the S-matrix element Sir as a function of 1 at E, = 130 MeV. In this figure, the solid line connects all Stf points calculated with x = 0 coupling terms only (i.e., S:,(O) points), while the dashed lines connect all even-l and all odd-l Sff points calculated with x = 1 coupling terms only (i.e., Sf,(l) points). Here one sees that, although the S;,(O) points vary rather smoothly with 1, the S:,(l) points show a distinct odd-even l-dependent or parity-dependent behaviour. This latter behaviour is, of course, a consequence of the fact that, in the present case, x = 1 is larger than the values of xq for all coupling types. In addition, one can also see from fig. 4 that the x = 0 equivalent coupling
E, = 130 MeV
* odd
R t
-0.04
Fig. 4. The behaviour of S:,(x) as a function of I at Ef = 130 MeV.
M. LeMere et al. /Antisymmetrization
337
potential is longer-ranged that the x = 1 equivalent coupling potential. Thus, while I,!&(l)/ is already vanishingly small for 1x 7, &(O)l has still an appreciable magnitude even for 1 x 8. By examining the R, values given in table 3, this is again easily understandable. The ~(p, d)3He example illustrates some of the CNK and CHK findings described in subsects. 3.2 and 3.3. For a thorough demonstration, one needs to analyze a more complicated case, such as the “Ne(cc, 6Li)18F or the 24Mg(12C, i60)20Ne reaction. Unfortunately, however, resonating-group calculations for these reactions are not easy to perform and, indeed, are not available at the present moment. 5. Conclusion In this investigation, we have studied the effects of intercluster antisymmetrization in direct reactions by examining the structure of the coupling-hamiltonian kernel (CHK) function occurring in a resonating-group coupled-channel formulation. The results show that, except for some modifications in quantitative details, the conclusions reached in this way are essentially similar to those obtained by studying the structure of the simpler coupling-normalization kernel (CNK) function. Thus, the main features of antisymmetrization are generally those summarized by items ($0-(i) listed in subsect. 3.2. These items deal with the general behaviour of direct reactions and should be very useful as a qualitative aid in interpreting experimental results. From the present study, we learn in addition that certain coupling types may contribute rather insignificantly to the direct-reaction process. These particular types may, therefore, be omitted from the corresponding coupled-channel resonatinggroup calculation. In view of the fact that resonating-group calculations of this kind are frequently rather tedious to perform, the possibility of omitting some coupling terms will usually result in considerable saving in computational effort. Recently, Teplov et al. ‘) have found some systematic trends in direct reactions by analyzing a large variety of experimental results. In comparison, it is noted that there is a general agreement between our conclusions and their findings. We should mention, however, that these authors have performed their analysis mainly at relatively low energies where o.ur present consideration is expected to have only qualitative significance. For our purpose, it would be much more interesting to check against experimental systematics at higher energies of about 20 to 50 MeV/nucleon. Finally, we must emphasize that our present effort represents just our initial attempt to study the effects of antisymmetrization in direct reactions. Many interesting aspects have yet to be examined. For example, it will certainly be useful to investigate the influence on reaction systematics which comes from the possibility of the formation of nucleon clusters in the various nuclei involved. Within the present formulation, one can achieve this by studying the general properties of the polynomial factors appearing in the coupling kernel functions. Quite obviously, this will be a
338
M. LeMere et ai. / ~ntjs_v~etr~~tj~
major and difficult task, but is certainly worth carrying out for the purpose of obtaining a better understanding of the important role played by the Pauli principle in nuclear direct reactions. This work was completed while one of us (Y.C.T.) was a visitor at the Institut fur Theoretische Physik der Universitlt Tubingen for the Humboldt Senior US Scientist Award. He wishes to thank Professor K. Wildermuth for the kind hospitality and the Alexander von Humboldt-Stiftung for financial assistance.
1) M. LeMere, E. J. Kanellopoulos, W. Siinkel and Y. C, Tang, Phys. Lett. 87B (1979) 31 i 2) Y. C. Tang, M. LeMere and D. R. Thompson, Phys. Reports 47 (1978) 167 3) K. Wildermuth and Y. C. Tang, A unified theory of the nucleus (Vieweg, Braunschweig, Germany, 1977) 4) M. B. Greenfield, M. F. Werby and R. J. Philpott, Phys. Rev. Cl0 (1974) 564 5) I. B. Teplov, N. S. Zelenskaya, V. M. Levedev and A. V. Spasskif, Sov. J. Part. Nucl. 8 (1977) 310 6) N. Astern, Direct nuclear reaction theories (Wiley-~ters~ience* New York, 1970); S. R. Catanch and C. M. Vincent, Phys. Rev. Cl4 (1976) I739 7) D. F. Jackson, Nuclear reactions (Methuen, London, 1970) 8) M. LeMere, D. J. Stubeda, H. Horiuchi and Y. C. Tang, Nucl. Phys. A320 (1979) 449 9) W. Stinkel and K. Wildermuth, Phys. Lett. 418 (1972) 439 IO) H. Horiuchi, Prog. Theor. Phys. 47 (1972) 3058 II) D. R. Thompson and Y. C. Tang, Phys. Rev. Cl2 (1975) 1432; Cl3 (1976) 2597 12) D. R. Thompson, M. LeMere and Y. C. Tang, Phys. Lett. 69B (1977) 1 13) M. LeMere, Ph.D. Thesis, University of Minnesota (1977) 14) D. R. Thompson and Y. C. Tang, Phys. Rev. C4 (1971) 306 15) F. S. Chwieroth, Y. C. Tang and D. R. Thompson, Phys. Rev. C9 (1974) 56 16) R. Hub, D. Clement and K. Wildermuth, 2, Phys. 252 (1972) 324 17) F. S. Chwieroth, Y. C. Tang and D. R. Thompson, Fizika {Suppl. 2) 9 (1977) 13 18) G. Gambadni, I. Jori, S. Micheletti, N. Molho, M. Pignanelli and G. Tagliaferri, Nucl. Phys. Al26 f 1969) 562
Ntsclear Physics A348 (1980) 321- 338; @ ~orfh-~o~la~
P~l~shjn~ Co., Amsterdam
Not to be reproduced by photoprint or microfilm withont written permission from the publisher
EFFECTS OF ANTIS~~ET~ZATION IN NUCLEAR DIRECT REACTIONS ’ M. LEMERE ++and Y. C. TANG School qf Physics and Astronomy, University
of Minnesota,Minneapolis,
Minnesota 55455, USA
and E. J. KANELLOPOULOS
and W. SUNKEL
Institut fir Theoretische Physik der Universitlit Ttibingen, D-7400 Tcbingen, BRD Received 28 April 1980 Abstract: The effects of antisymmetrization in direct reactions are studied by examining the general properties of the coupling-hamiltonian kernel function occurring in a coupled-channel resonatinggroup formulation. Together with a previous investigation of the coupling-normalization kernel function, one finds from this study interesting systematics concerning the general behaviour of direct-reaction processes and some justification for the use of simple three-body models in phenomenological analyses. In addition, it is shown that certain coupling terms make comparatively insigni~cant contributions; these terms may, therefore, be frequently omitted m a r~onating-group calculation, thus reducing to a considerable extent the computational effort required in such a calculation.
1. Introduction In a recent inv~tigation I), we have studied the effects of antisymmetr~ation in direct reactions by examining in detail the stucture of the coupling-normalization kernel (CNK) occurring in a resonating-group formulation ‘, 3). From this investigation, we obtained useful information not only about the likelihood of multi-nucleon transfer processes, but also concerning the connection between nucleon-exchange terms in the many-nu~l~n wave function of a microscopic fo~ulation and various phenomenological, macroscopic mechanisms. In addition, it was also found that there is some justification for the utility of simple three-body models commonly employed in conventional analyses 4-7). Even though such models, by their macroscopic nature, are necessarily rather crude, they do contain, by adjusting appropriate parameters, some of the important features inherent in resonating-group calcuiations. + This research was supported in part by the US Department of Energy and the Bundesministerium &r Forschung und Technologie, BRD. ” Present address: Department of Microbiology, University of Southern California, Los Angeles, California 90033, USA. 321
M. LeMere et al. / Antisymmetvization
322
Although it seems reasonable to expect that the basic features of antisymmetrization in direct reactions may be learned by studying the CNK alone, one may still justifiably argue that there might exist features which are specifically associated with the short-range nature of the nucleon-nucl~n potential. Indeed, in the scattering case *), it has been found that a careful examination of the exchange-hamiltonian kernel did yield additional information concerning the energy dependence of exchange processes. Thus, in the present investigation, we shall undertake to examine the structure of the ~upling-hamiltonian kernel (CHK). As will be seen below, this will certainly be a considerably more complicated task. However, with the use of the complex-generator-coordinate technique 9-13) (CGCT), a general expression for the important exponential factors occurring in the CHK can be readily derived and, thereby, a detailed analysis leading to definitive results can in fact be achieved. In the next section, we give a concise description of the coupled-channel resonatinggroup formulation and discuss the basic structure of the CHK. Sect. 3 is devoted to a brief review of important results obtained from the CNK study I) and to a discussion of the general features of the equivalent local coupling potentials which are constructed to yield the same Born reaction amplitudes as the various nucleon-exchange terms in this kernel function. Explicit resonating-group calculations in the a(p, d)3He case, where contributions from individual exchange terms are investigated, are then presented in sect. 4, the purpose being to demonstrate, in a realistic example, the validity of the conclusions reached from the simple plane-wave Born-approximation (PWBA) study. Finally, in sect. 5, we summarize the results and discuss the implications of the findings of this investigation. 2. Resonating-group formulation and coupling kernel fusions 2.1. RESONATING-GROUP
FORMULATlON
OF DIRECT-REACTION
PROCESSES
Consider a direst-reaction process A(a b)B, which involves in the incident channel (channel fj two clusters a and A having nucleon numbers N, and N,, and in the outgoing channel (channel g) two clusters b and B having nucleon numbers N, and N,. For definiteness, we adopt the convention that N, < N.4, N, < N,,
(1)
N,sN,, Also, for simplicity in discussion, we shall assume that all these clusters have spin zero and are described by translationally invariant shell-model functions in harmonicoscillator wells having a common width parameter CLIn the simplest resonatinggroup formulation of such a process 2), the trial wave function If/ is then written as II/ = *f-t&,
(2)
M. L&few
323
et al. j Antisyrmnetrizatian
where ll/r = JJ’[&%(L.
*.,N,)&(N,-I-1,. **,N,$_N,;N,+N,3-l,...,N,3-N,) x wwwLII.)I~
I+$= s$‘[&(l)...)
N,;N,+l,...,
N,+N,)&(N,+N,+l)...)
(3)
JJ,+N,+NlJ x Wg)Z(L,.)fi
(4)
with SB’ being an intercl~ter ~tisymmetrization operator, &,, 4A, &, and $, being antisymmetrized cluster internal functions, and Z(R,.,,) being a normahzable function describing the motion of the total c.m. The relative-distance vectors R, and Rg are defined as Nn+NA Rr=k $ rt-& .=C ri, (5) A I N,+l a i-l Rg=+
$ +b
i-l
.;iN” B I
ri,
(6)
Nb+l
and the quantity N, is the number of nucleons in the transferred cluster t, given by fvt = Nb--I?,.
(7)
Also, it is noted that, in eqs. (3) and (4), the arguments of the internal functions of the clusters indicate which nucteons are involved in these clusters; such a labelling is permissible, because the wave functions are totally antisymmetrized. The functions F(R,) and G(R,) describe the relative motions between the clusters in the incident and outgoing channels, respectively. They are obtained by solving the projection equation
= 0,
(8)
where ET is the total energy of the system, given by E, = E,+E,+E, = E,+E,+E,,
(9)
with Ef and E, being the relative energies of the clusters in channels f and g, respectively, and E,, E,, E, and E, being cluster internal energies obtained by computing the expectation values of the cluster hamiltonians with respect to the assumed internal functions. The hamiltonian H in eq. (8) is a galilean-invariant operator which has the form H = i T+ 5 Ejj-T,,,,, i
(10) Z&,,, being the
M. LeMere et al. / Ant~y~~etr~atio~
324
kinetic-energy operator of the total cm., and vj being a nucleon-nucleon potential chosen to fit adequately the two-nucleon scattering data especially in the low-energy region. In order to make our subsequent analysis especially transparent, we shall choose vj to be as simple as possible; thus we adopt the central nucleon-nucleon potential + given by eq. (8) of ref. 14), which has a spatial form factor oij = exp (- M$),
(11)
and in which the Majorana space-exchange operator is not explicitly contained (i.e., the exchange operator -ejPTj, instead of the Majorana operator P& is employed). By employing the procedure described in ref. ‘), one obtains the following coupled integrodifferential equations which F(R;) and G(Rh) satisfy:
1 s
- & V& + I/,,(R;) - E, F(R;) + c
K&R;, R;)F(R;)dR;
+K&t;,
R;)G(R;)d&' = 0,
s
+
s
K&R;, R;‘)F(R;‘)dR;= 0,
(12)
(13)
where $r and 4 are, respectively, the reduced masses of the clusters in the incident and outgoing channels. Also, I$, and I+,, represent the direct potentials, and K,, and K, represent the exchange kernel functions in channels f and g, respectively, while K,, and K, represent the coupling kernel functions. From the above equations, it is easily seen that the transition from channei f to channel g is effected by the coupling kernel I(,, which has the form I&JR;, R;‘) = &CR;, R;‘)- E,&(R;, where H, is the coupling-hamiltoni~ H&R;> q)
= ($&$(Rg
and N, is the coupling-normalization
R;‘),
(14)
kernel (CHK) given by - R;)WW’[$a&~(R,
- Rf’Pl),
(15)
kernel (CNK) given by
&(R;, R;‘) = <&,$r$(Rg - R$W’[$,$$(Rr
- R;‘)z]).
(16)
In a previous investigation ‘), we have examined the effects of antisymmetrization in direct reactions by studying the properties of the simpler CNK function. Here we shall try to extend our understanding by further concentrating on the more complicated CHK fiction. t For the same reason mentioned above, the proton charge will be assumed as infinitesimally small.
hf. LeMere et al. / Antisymmetrization
325
Upon performing a partial-wave expansion, one obtains by solving the resultant coupled integrodifferential equations in each l-state the desired S-matrix which has diagonal elements Sir and Stp, and off-diagonal elements S& and Sir. Using these matrix elements, one can then calculate straightforwardly various differential scattering and reaction cross sections 15).
2.2. COUPLING-NORMALIZATION
AND COUPLING-HAMILTONIAN
KERNEL FUNCTIONS
The CNK may be decomposed into the form
where N:(R;, R/) = P,, exp ( - AxNq2 - C,&’
*Rb - B,,Rz),
(18)
with P,. being a polynomial in R;r2, Rf’. Ri and Kg2. The quantity x denotes the number of nucleons interchanged between the group of nucleons labelled by (1,. . ., NJ in cluster a and the group of nucleons labelled by (N, + N, + 1, . . ., N, + NJ in cluster A [see eq. (3)]. Its minimum and maximum values are equal to 0 and N,, respectively. By using the CGCT described in ref. 2), one can derive general expressions for the coefficients AxN, B,, and C,,. These expressions are
(19)
(21) where
r, = ~tb,+Nt)+A~,
(22)
rd = ~t(llt+N,)-A~,
(~3)~
with
(24)
(25)
M. LeMere et al. / Antisymmetrization
326
A = x-j+
(27)
Here the important point to note is that rd is positive for all values of x and C,, has a positive or negative value depending upon whether x is larger or smaller than pl. The expression for the CHK is more complicated. It has the form J&4&, R;‘) = 1 c G,(q x 4
4%
(4 = a, b, c, d, 4,
(28)
- C,&’ . Rb - B,$Zf),
(29)
where H;,(Rb, R;‘) = P,, exp (-Ax&”
with P,, (q = a, b, c, d, e) being again a polynomial in R;“, Rf’ . Rh and RL’. Here one sees that, in contrast to the CNK, there appear now five types of exponential factors for each value of x (in the x = 0 and x = N, cases, there are only four types; this will be explained below). To explain briefly the origin of these five types of coupling terms, we make use of the diagrammatic representation introduced by LeMere et al. *). This is shown in
a
b
C
Fig. 1. Diagrammatical representation
d
e
of direct-reaction coupling types.
327
M. LeMere et al. / Antisymmetrization
fig. 1, where representative diagrams for each type are depicted +. As has been explained in ref. *), each dot on the three horizontal lines denotes a set of all occupied single-nucleon spatial states with the same spin and isospin. The upper line represents the cluster a in the incident channel, while the lower two lines represent collectively the larger cluster A with the middle line representing the transferred cluster t. Also, the dashed line is used to call attention to the fact that nucleons i andj are interacting, while the solid-line exchange loop is used to indicate the nucleons which are interchanged. Furthermore, we have adopted the convention that diagrams shown in fig. 1 are considered to represent implicitly all other diagrams which contain additional exchange loops not involving the interacting nucleons. The type-a coupling term arises from both the kinetic-energy and potential-energy operators in the hamiltonian H, while coupling terms of other types come only from the potential-energy operator. Also, it is easily inferred from these diagrams that there is no x = N, contribution for type b and no x = 0 contribution for type c. General expressions for the coefficients Axq, B,, and C,, can again be derived by using the CGCT. The results are (30)
(31)
(32) where
;1=L
(33)
a+21c ’
TABLE 1 for A”,, xdqi &
Expressions Coupling
type
‘Kq
Xi,
8,4
8,,,
en, and c,, Riq
C”,
0
01
cidq
4 0 -/~,+24 -k--24 k -I4
0
0
-pLp-2A -pL,+2A
14,,+2A /.%-2d
-4,-24 -P(,+2A
-& -I4
-Y Pf
-k -I4
+ For simplicity in presentation, we assume that the nucleon-nucleon potential character. Considerations with the operators cj, Pi, and - P;P; can be similarly
0 -&-U -/~(,+24
-1 0 0
has a purely discussed.
-pLs -k
Wigner
328
M. LeMere
ef al. ,I ~nf~~~~efr~z~fion
and the constants An4, ;?+, &, &,
c,,* and ed, are tabulated in table I, with i+3= Etf+l$
(34)
&n =
(35)
Pr-P**
From this table and eqs. (30H32), one sees that the type-a term and the CNK have exactly the same exponential factor; therefore, for each value of x, these two kernels have rather similar structure. 3. Equivalent coupling twenties 3.1. CONSTRUCTION OF EQUIVALENT BORN APPROXIMATION (PWBA)
COUPLING
POTENTiALS
IN THE PLANE-WAVE
The structure of the CHK is rather complicated. Thus, to determine, at relatively high energies, the relative importance of the various nucleon-exchange terms in the five types of coupling functions, we adopt the following simplifying procedure. For a coupling kernel function of the form
&.(Fg,R;‘)
= PC exp ( - aq2 - cRf’ * Kg - bRh),
(36)
with PC being a polynomial in R;r’, K. RL and Ri2, we first compute the PWBA reaction amplitude, given by B = e exp (- ik, . R#T,(R~, R;‘) exp (ik, . R;‘)dR;dR;’
- &(aki+ck;
k,+bkf)
1 ,
(37)
where & is a proportionality constant unimportant for our present consideration, k, and k, are wave vectors in the incident and outgoing channels, respecti+ely, and PC is a polynomial in ki, k, *k, and kf. Then, we construct, in the manner indicated below, an equivalent local energy-dependent coupling potential which leads, also in the PWBA, to the same reaction ampitude as that given by &. The information we seek can then be obtained by examining the spatial and energy dependence of this coupling potential. Quite clearly, this is a rather crude procedure to study the properties of the coupling kernel; however, it is, in our opinion, the simplest way to extract semi-quantitatively information about antisymmetrization effects from the complicated CHK function. Two cases have to be separately considered. These are as follows: (i) c < 0. In this case, the PWBA reaction amplitude is forward-peaked and the equivalent coupling potential has the form &(R$S(RI:-R;‘). By using the criterion mentioned above and the fact that, at the relatively high energies under consideration
M. LeMere et al. / Antisymmetrization
in this investigation, the relative energies E, and E, satisfy the approximate E f M E, = E,
329
relation (38)
one easily finds that
&.(Q = PCexp( -E/E,) exp[ - (RhlRd2],
(39)
with P, being a polynomial in E and Ra. This equivalent potential is characterized by a characteristic range R, and a characteristic energy E, which have the forms (40)
(41) with M being the nucleon mass. (ii) c > 0. Here the PWBA reaction amplitude is backward-peaked and the equivalent coupling potential has the form ~c(Z$)G(R;:+&‘), with I$&) given again by eqs. (39H41). 3.2. REVIEW OF RESULTS FROM THE CNK STUDY
In a previous investigation ‘), the features of the CNK have been carefully studied by using the procedure described in the preceding subsection. The results obtained were as follows: (i) The critical quantity in the analysis is xN = pLt.For x < xN, the PWBA reaction amplitude is forward-peaked, while for x > xN, the PWBA reaction amplitude is backward-peaked. (ii) For x < xN, both the characteristic range R,, and the characteristic energy E,, decrease monotonically with x and have largest values when x = 0. Since direct reactions occur predominantly in the peripheral region, it is clear that longer-ranged equivalent potentials will have larger influence. Therefore, this shows that the x = 0 term (no-exchange term), with characteristic range R,, and characteristic energy EON, is the most important one among all exchange terms with x < xN. (iii) For x > xN, both the characteristic range R,, and the characteristic energy E, increase monotonically with x and have largest values when x = N,. This indicates that the x = N, term (maximum-exchange term), with characteristic range R,, and characteristic energy EmN, is the dominant one among all exchange terms with x > xN. (iv) The no-exchange and maximum-exchange terms correspond to light-particle and heavy-particle pickup processes commonly employed in phenomenological analyses 16,“) . In a macroscopic description, these processes can be accounted for by the use of three-body models. Thus, the finding here that these terms yield the
330
M. LeMere
et al. / Antisymmetrlzation
dominant contributions provides some justification for the adoption of three-body simplifications in phenomenological studies. (v) By studying the properties of the no-exchange term, it was shown that there is a general tendency in favour of the pickup of a light cluster containing a relatively small number of nucleons. We should mention, however, that this conclusion was reached without taking into account the effect arising from the formation of nucleon clusters, such as a-clusters, in the various nuclei involved. A proper consideration of this latter effect will definitely modify the above statement to a certain extent. (vi) From a careful investigation of the maximum-exchange term, we obtained the interesting information that back-angle reaction cross sections will have large values only if N, and N, do not greatly differ. In a systematic experiment of (p, a) reaction on light nuclei at E, = 38 MeV [ref. ‘“)I, it was indeed found that the ratio of backward-angle to forward-angle cross section does become progressively smaller as the target nucleus becomes heavier. In the following subsections, we shall further examine the CHK with a similar procedure in order to see whether or not these findings need to be substantially modified. 3.3. STUDY OF COUPLING-HAMILTONIAN
KERNEL (CHK)
The study of the CHK is conducted in exactly the same way as in the CNK case. For each coupling type (q = a, b, c, d, e), one determines and then examines the properties of the equivalent coupling potential which contains a polynomial factor pX4and is characterized by a characteristic range R,, and a characteristic energy E,,. By using eqs. (40) and (41) we find, in a straightforward manner, the general expressions for R,, and E,,. These expressions are
h2c!
J, - (x - x4)2
Exq =zf C~~-lx-x,l][~,-Ix-x,l]+K,’
(43)
where the constants x,, J,, and K, are listed in table 2. For the type-a term, the characteristic values are the same as those of the CNK; this is so, because, as has been mentioned previously, these two kernel functions possess the same exponential factor. The quantity xq is determined by setting C,, equal to zero. It is again a critical quantity in the analysis, because the PWBA reaction amplitude corresponding to the type-q coupling term is either forward or backward peaked depending upon whether x is smaller or larger than x4. From table 2, one notes that xq differs from xN only for the coupling types b and c. In fact, even for these two types, the difference, being equal to A, is rather small. The reason for this is that, in a realistic situation, JCis approximately equal to c1and, hence,
M. LeMere et al. / Antisymmetrization
331
TABLE2 Expressions for x4, Jq and K,, Coupling type q
%
I has a value around 3 which is substantially smaller than the value of xN = p, in almost all cases of interest +. An examination of the general properties of the characteristic quantities R,, and E,, is tedious but can be straightforwardly carried out. Here again, one finds that for x < x4 (4 = a, b, d, e), R,, and E,, have largest values when x = 0, and for x > xq (4 = a, c, d, e), R,, and E,, have largest values when x = N,. Therefore, similar to the finding in the CNK case, the present CHK study indicates that, among all exchange terms, the no-exchange term has the largest influence when x < xq and the maximum-exchange term has the largest influence when x > xq. Since there are no x = 0 type-c and x = N, type-b contributions, we have also carefully examined the nature of the x = 1 type-c term and the x = N,- 1 type-b term. For both of these terms, the result shows that the characteristic ranges and energies are appreciably smaller than those of the relevant no-exchange and maximum-exchange terms of other coupling types. In the following discussion, we shall, therefore, no longer consider these two terms. 3.4. NO-EXCHANGE
AND MAXIMUM-EXCHANGE
COUPLING POTENTIALS
Since the no-exchange and maximum-exchange terms are found to be particularly important, we concentrate in this subsection on’ studying the properties of these terms. In the CNK case, the no-exchange and maximum-exchange characteristic ranges and energies are given by ‘)
f,
Ro, =
w
h20! N
-%I,=~ R mN =
(45)
N’ t
4
f
(46)
a(N, - N, - NJ 1 ’ t For heavy-ion reactions, p, is substantially larger than 1. Even in a rearrangement reaction initiated by an incident nucleon, the smallest possible value of p, is f.
M. LeMere
332
E mN =
et al. / Antisymmetrization
h2a N __ 2M N,-N,-N;
(47)
From these equations and the relation N, s %N,-NJ
(48)
which follows from eq. (l), we reached the conclusions listed as items (v) and (vi) in subsect. 3.2. In the CHK case, we perform the analysis by first defining no-exchange factors foq, goq (q = a, b, d, e) and maximum-exchange factors fm4, gm4 (q = a, c, d, e) in the following way: (49) f0q = ROq/RON, (50)
9 oq = Eo,/EoN> fmq
=
9 mq =
Rmq/RmN,
(51)
E,qiE,N’
(52)
These factors are functions of 1, N,, N, and N,. For example, in the case of type d, we find gOd=f&2=l+;l
++; (
9
a
) t>
.
Since i has a value around f, it is easily seen that all the type-d factors have values close to 1. Similarly, we can obtain the expressions for the f- and g-factors of all other.coupling types by using eqs. (42) and (43). A careful examination of these expressions then reveals that, for 1 z 3, all f-factors are smaller than or equal to 1, while all g-factors are not very different from 1. This indicates, therefore, that the conclusions which can be reached by studying the CHK are in fact substantially the same as those obtained from the CNK study. In fig. 2, we illustrate the behaviour of the f- and g-factors as functions of 1 in two examples, namely, the “Ne(a, ,6Li)1‘F and 24Mg( “C, 160)“Ne reactions. Here the solid and dashed curves represent the values of these factors in the noexchange and maximum-exchange cases, respectively. As is seen, the f-factor of the type-b term is comparatively small. Therefore, because of the peripheral nature of direct reactions, this particular coupling type may be relatively unimportant. In fact, one further sees from this figure that even the type-c term may be omitted in certain reactions. On the other hand, the type a, d, and e coupling terms do have comparable characteristic ranges and energies, and, thus, must generally be taken into consideration in a direct-reaction calculation.
333
;; 0.9 t z? , 0.8 Y0.7
0.6 1.4
1.3
1.2 i
I.1
? &
I.0 0.9 0.8
0.7
0
0.1
0.2
0.3 x
0.4
0.5
0.1
0.2
0.3
0.4
0.5
x
Fig. 2. Thef- and g-factors for the ‘“Ne(a, ‘Li)18Fand 24Mg(“C, LbO)LuNereactions.
The finding that the CNK and CHK studies yield substantially similar conclusions is, in fact, not too surprising. From eqs. (19)-(21) and eqs. (30)-(32), one readily notes that the exponential factors in the CHK and CNK become the same in the limit case where the nucleon-nucleon potential has a range approaching infinity {i.e., fc + 0). Since nucleon-exchange processes occur predomin~tly when the interacting nuclei are in close proximity, one may plausibly expect that the range of the nucleonnucleon potential will not qualitatively influence antisymmetrization effects to a large extent. 4. Explicit study of the a(p, d)3He reaction As a demonstration in a practical example, we discuss now the coupled-channel resonating-group study ofthe cc(p,d)3He reaction. Although this example is somewhat too simple and cannot serve to illustrate all the detailed findings described in the
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preceding section, it does give us some general feelings about the importance of and the roles played by the various coupling terms. The r~(p,d)3He resonating-group calculation is described in ref. “). Since, in our present consideration, the main concern is to study antisymmetrization effects, we shall simplify the calculation by neglecting Coulomb effects and assuming that the internal spatial structures of the IX,3He, and d-clusters are represented by singlegaussian functions with width parameters of 0.514,0.36 and 0.2 fmB2, respectively. For the nucleon-nucleon potential, we use again the one given in ref. 14) with IC= 0.46 fm- ’ [Isee eq. (1 l)] and adopt a Serber exchange mixture. For a qualitative understanding, we first compute the characteristic ranges and characteristic energies of the various equivalent coupling potentials. In order to utilize the formulae given in sect. 3, we assume in this computation that all internal width parameters take on the average value of CI= 0.36 fm-‘. This is a tolerable assumption in view of the fact that il is rather insensitive to CIand we are only conducting a qualitative examination of the basic features of this problem. The results are given in table 3, where the values of R, and E, for various coupling types with x = 0 and 1 are listed. These values will be useful as an aid in our analysis to extract useful information about antisymmetr~ation effects from the resonating-group calculation. TABLE3 Values of R, and Exp calculated with c( = 0.36 fm-’
0
E d e a
1
i e
3.33 1.24 2.54 2.74
37 32 64 55
2.36 0.50 1.90 2.02
19 18 29 26
From studying table 3 and using the fact that the value of xq is between 0 and 1 for all coupling types, one can make the following statements: (i) The PWBA reaction amplitudes corresponding to x = 0 and 1 coupling terms are, respectively, peaked in the forward and backward directions. (ii) The characteristic ranges of the direct potentials in channels f and g are equal to 2.07 and 2.33 fm, respectively [see eq. (74) of ref. “)I_ Comparing with these values, we note that for types a, d and e, the values of the characteristic ranges are either similar or somewhat larger. This indicates that, for the calculation of the differential reaction cross section, these three coupling types must be taken into consideration.
M. LeMere et al. / Antisymmetrization
335
(iii) The characteristic ranges of type b and c terms are significantly smaller than those of the direct potentials; therefore, these coupling types are expected to make only minor contributions to the c~(p,d)3He direct-reaction process +. (iv) The type-d coupling term has largest values for the characteristic energies. For this type, it is noted in particular that the value for x = 0 is more than twice as large as that for x = 1. Because of this, one expects that the reaction cross section in the backward direction will decrease with energy at a considerably faster rate than that in the forward direction. (v) The value of (N,-N,NJ is fairly small in this case; consequently, the x = 1 type-d and e characteristic energies acquire rather large values around 30 MeV. Therefore, since the energy dependence of the depth of the equivalent coupling potential comes from the factor exp (-E/E,.,), it is anticipated that even at the relatively high energy of E, = 130 MeV (Eg FZ 105 MeV) which we are mainly concerned with here, there should still be a noticeable rise in reaction cross section in the largeangle region. In tig. 3, we show by solid dots the resonating-group reaction cross sections
a
(p,d13He
E,=l30
0.001
0
MeV
....
r-g
-
x=0
----
x=
\ \ ,\
,
I
I
20
40
60
80
only
I only
I
100
I
120
140
160
180
B(deg) Fig. 3. Angular distributions for the or(p, d)3He reaction at Er = 130 MeV. The solid dots represent the results of the full resonating-group calculation. The solid and dashed curves show, respectively, the results obtained when x = 0 coupling terms alone and x = 1 coupling terms alone are taken into consideration.
t The a(p, d)3He resonating-group study described here was performed in 1973 when there was little understanding concerning the roles played by the various coupling types. Aa s consequence, items (ii) and (iii) have, unfortunately, not been examined.
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calculated at E, = 130 MeV. From this figure, one notes that, in the back-angle region, there is indeed a significant rise in the reaction cross section. On the other hand, because of the relatively high energy involved, the cross section at 180” is not too large, which is evidenced by the fact that the backward-to-forward ratio, i.e., o(180”)/a(0”), has a rather small value equal to 0.024. At lower energies of E, = 70 and 100 MeV, the backward-to-forward ratios are equal to 0.15 and 0.049, respectively. Together with the value of 0.024 at 130 MeV, we find that this ratio does decrease quite rapidly with energy, in agreement with the qualitative finding mentioned in statement (iv) of a preceding paragraph. From fig. 3, it is also noted that the full resonating-group result in the forward angular region can be reproduced very well by a calculation employing only x = 0 coupling terms (solid curve), while that in the backward angular region can be represented satisfactorily by a calculation employing only x = 1 coupling terms (dashed curve). Based on the qualitative discussion given above, this is entirely to be expected. To demonstrate more clearly the different characters of the x = 0 and x = 1 equivalent coupling potentials, we depict in fig. 4 the behaviour of the S-matrix element Sir as a function of 1 at E, = 130 MeV. In this figure, the solid line connects all Stf points calculated with x = 0 coupling terms only (i.e., S:,(O) points), while the dashed lines connect all even-l and all odd-l Sff points calculated with x = 1 coupling terms only (i.e., Sf,(l) points). Here one sees that, although the S;,(O) points vary rather smoothly with 1, the S:,(l) points show a distinct odd-even l-dependent or parity-dependent behaviour. This latter behaviour is, of course, a consequence of the fact that, in the present case, x = 1 is larger than the values of xq for all coupling types. In addition, one can also see from fig. 4 that the x = 0 equivalent coupling
E, = 130 MeV
* odd
R t
-0.04
Fig. 4. The behaviour of S:,(x) as a function of I at Ef = 130 MeV.
M. LeMere et al. /Antisymmetrization
337
potential is longer-ranged that the x = 1 equivalent coupling potential. Thus, while I,!&(l)/ is already vanishingly small for 1x 7, &(O)l has still an appreciable magnitude even for 1 x 8. By examining the R, values given in table 3, this is again easily understandable. The ~(p, d)3He example illustrates some of the CNK and CHK findings described in subsects. 3.2 and 3.3. For a thorough demonstration, one needs to analyze a more complicated case, such as the “Ne(cc, 6Li)18F or the 24Mg(12C, i60)20Ne reaction. Unfortunately, however, resonating-group calculations for these reactions are not easy to perform and, indeed, are not available at the present moment. 5. Conclusion In this investigation, we have studied the effects of intercluster antisymmetrization in direct reactions by examining the structure of the coupling-hamiltonian kernel (CHK) function occurring in a resonating-group coupled-channel formulation. The results show that, except for some modifications in quantitative details, the conclusions reached in this way are essentially similar to those obtained by studying the structure of the simpler coupling-normalization kernel (CNK) function. Thus, the main features of antisymmetrization are generally those summarized by items ($0-(i) listed in subsect. 3.2. These items deal with the general behaviour of direct reactions and should be very useful as a qualitative aid in interpreting experimental results. From the present study, we learn in addition that certain coupling types may contribute rather insignificantly to the direct-reaction process. These particular types may, therefore, be omitted from the corresponding coupled-channel resonatinggroup calculation. In view of the fact that resonating-group calculations of this kind are frequently rather tedious to perform, the possibility of omitting some coupling terms will usually result in considerable saving in computational effort. Recently, Teplov et al. ‘) have found some systematic trends in direct reactions by analyzing a large variety of experimental results. In comparison, it is noted that there is a general agreement between our conclusions and their findings. We should mention, however, that these authors have performed their analysis mainly at relatively low energies where o.ur present consideration is expected to have only qualitative significance. For our purpose, it would be much more interesting to check against experimental systematics at higher energies of about 20 to 50 MeV/nucleon. Finally, we must emphasize that our present effort represents just our initial attempt to study the effects of antisymmetrization in direct reactions. Many interesting aspects have yet to be examined. For example, it will certainly be useful to investigate the influence on reaction systematics which comes from the possibility of the formation of nucleon clusters in the various nuclei involved. Within the present formulation, one can achieve this by studying the general properties of the polynomial factors appearing in the coupling kernel functions. Quite obviously, this will be a
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M. LeMere et ai. / ~ntjs_v~etr~~tj~
major and difficult task, but is certainly worth carrying out for the purpose of obtaining a better understanding of the important role played by the Pauli principle in nuclear direct reactions. This work was completed while one of us (Y.C.T.) was a visitor at the Institut fur Theoretische Physik der Universitlt Tubingen for the Humboldt Senior US Scientist Award. He wishes to thank Professor K. Wildermuth for the kind hospitality and the Alexander von Humboldt-Stiftung for financial assistance.
1) M. LeMere, E. J. Kanellopoulos, W. Siinkel and Y. C, Tang, Phys. Lett. 87B (1979) 31 i 2) Y. C. Tang, M. LeMere and D. R. Thompson, Phys. Reports 47 (1978) 167 3) K. Wildermuth and Y. C. Tang, A unified theory of the nucleus (Vieweg, Braunschweig, Germany, 1977) 4) M. B. Greenfield, M. F. Werby and R. J. Philpott, Phys. Rev. Cl0 (1974) 564 5) I. B. Teplov, N. S. Zelenskaya, V. M. Levedev and A. V. Spasskif, Sov. J. Part. Nucl. 8 (1977) 310 6) N. Astern, Direct nuclear reaction theories (Wiley-~ters~ience* New York, 1970); S. R. Catanch and C. M. Vincent, Phys. Rev. Cl4 (1976) I739 7) D. F. Jackson, Nuclear reactions (Methuen, London, 1970) 8) M. LeMere, D. J. Stubeda, H. Horiuchi and Y. C. Tang, Nucl. Phys. A320 (1979) 449 9) W. Stinkel and K. Wildermuth, Phys. Lett. 418 (1972) 439 IO) H. Horiuchi, Prog. Theor. Phys. 47 (1972) 3058 II) D. R. Thompson and Y. C. Tang, Phys. Rev. Cl2 (1975) 1432; Cl3 (1976) 2597 12) D. R. Thompson, M. LeMere and Y. C. Tang, Phys. Lett. 69B (1977) 1 13) M. LeMere, Ph.D. Thesis, University of Minnesota (1977) 14) D. R. Thompson and Y. C. Tang, Phys. Rev. C4 (1971) 306 15) F. S. Chwieroth, Y. C. Tang and D. R. Thompson, Phys. Rev. C9 (1974) 56 16) R. Hub, D. Clement and K. Wildermuth, 2, Phys. 252 (1972) 324 17) F. S. Chwieroth, Y. C. Tang and D. R. Thompson, Fizika {Suppl. 2) 9 (1977) 13 18) G. Gambadni, I. Jori, S. Micheletti, N. Molho, M. Pignanelli and G. Tagliaferri, Nucl. Phys. Al26 f 1969) 562