Bound-pion absorption followed by emission of two nucleons

2.A .1 : 2B

Nuchrar Physics A298 (1978) 367-381 ; © North-f~folland Pwblishlng Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written peemiwton from the publisher

BOUND-PION ABSORPTION FOLLOWED BY EMISSION OF TWO NUCLEONS R . S . BHALERAO and Y . R . WAGHMARE

Department of Physics, Indian'Institute of Technology, Kanptv-208016, Itidia Received 4 May 1977 (Revised 5 September 1977) Abstract : The bound-pion absorption reaction, oiz. '~C(n - , NN), is studied using I-Iartree-Fcek (HF) wave functions obtained with the unitary-model-operator approach starting with the hard-core nucleon-nucleon (NIA interaction . The inequality of the energies of the two outgoing nucleons is treated exactly and calculations are done using the "1 N model" for rz-absorption . Other effects taken into account are : NN scattering in the final state, contributions of all excited states of `°B and '°He with E < 5 MeV, and etTects of the strong a-nucleus interaction and the finite nuclear siu on the bound-a wave function. Branching ratios and angular distributions of absorption rates are in better agreement with experimental data . The correct order of magnitude of the total absorption rate is reproduced . Whatever the effects of short-range wrrelations present in the HF wave functions, they are not masked by the NN final-state interaction . The contribution of excited states in '°B and 1 °Be is found to be quite Iarge . Absorption rates obtained with the HF and oscillator wave functions ditFer significantly both in siu and shape .

)< . I~trOdaCl300

The importance of the (~c-, NN) reaction in the study of nuclear structure is well known t). Though this reaction has now been studied for more than fifteen years, results ofan experiment in which the final state was determined "wmpletely", became available only in 1972 [ref. z)] . Here "completely" is used in the sense that, unlike the earlier experiments 3' 4), charged particles emitted in the reaction were identified and energies of both the emitted particles and the opening angles between them were measured. Identification ofthe charged particles in the final state is important because it is known that even with coincident neutrons there is considerable emission of deuterons and tritons in this reaction Z). The angle and energy resolution in this experiment z) was also better than that achieved previously a " s.) . These recent measurements on the t2C(~-, pn) reaction, when compared with some of the important calculations e-lt), gave poor agreement t. These theories could reproduce neither the observed angular distributions of the absorption rate nor the nn/rap branching ratios with sufficient accuracy . Experimentally, the branching ratio was found to be 8.8 f 1 .3 [ref. Z)] or 10.0 f 2.0 [ref. s)]. The earlier ~ There is an error in the calculation of ref. i~ as explained in sect . 3. 367

368

R . S . BHALERAO AND Y . R . WAGHMARE

experiments 3 " a) gave the somewhat lower values 5.0 f 1 .5 and 2.5 f 1 .0, respectively, but this was presumably because these measurements suffered from contamination of the heavier hydrogen isotopes that are emitted along with nucleons. Calculations e-s) which employed the Jastrow-0orrelated wave functions, took into account the final-state NN interaction in a variety of ways and neglected the final-state nucleon-nucleus interaction, all predicted the branching ratio to be less than one. The two 1 N model calculations '~" is) which took into account, besides other effects, the final-state nucleon-nucleus interaction, became available a little later. In one of them ' 3) the branching ratio was not calculated. In the other 1 ~) it was found to be x 2.5, in disagreement with the recent experimental values. The second important point is the treatment of short-range correlations (SRC) in this problem. Most authors who employ the 1N model for ~-absorption, use the Jastrow ansatz to get a correlated wave function . Naturally, the origin of the correlations, viz. the NN interaction, does not enter the calculation explicitly and nothing much can be learnt about it. Koltun and Reitan ' a) have tackled this problem by solving the Schrödinger equation for the relative motion of the NN pair using the Hama;:.-Johnston potential. The only calculations 1 sb . °) which uses the correlation function derived from Brueckner theory has some serious limitations, as discussed in sect. 2. The earlier calculation 1~ based on the unitary-model-operator approach (UMOA) was approximate and erroneous as explained in sect. 3. In the Eckstein-model ' e) calculations the SRC are not introduced explicitly and once again it is difficult to get any direct information about the nuclear forces with this approach . The unitary-model-operator approach to handle the short-range NN correlations was first suggested by Villars l ') and later on developed by Mittelstaedt is), Da Providencia et al. 1~ and Shakin et al. 2~. A connection has been established between the reaction-matrix theories and the UMOA. In particular, it has been shown 21) that it is possible to find a unitary transformation for which the transformed two-body potential in the UMOA can be written as ~}{K+K+), K being the reaction matrix. In refs. l' " i e) it is shown that by using various possible definitions of the model operator, one can relate this method to the Brueckner-BetheGoldstone approach . Solutions of the generalized Hartree-Fock (HF) equations based on the UMOA and involving realistic interactions, take into account some effects of the short-range nucleon-nucleon correlations and hence are used in this work to represent the pioncapturing pair of nucleons. Earlier these wave functions had provided satisfactory results for static properties of some closed-shell and closed-subshell nuclei Z~ . Recently we have tested these wave functions for high~nergy elastic electron scattering on some spherical as well as deformed by shell nuclei sz) . For 1ZC(e, e)'ZC excellent agreement with the experimental data was obtained for values of q, the momentum transferred to the nucleus, up to 370 MeV/c. For q between 370 and 550 MeV/c, however, the agreement was not as good, apparently because these wave functions do not contain enough high-momentum components .

BOUND-PION ABSORPTION

369

In the (rz - , NN) reaction on 12C, with rz~apture by a (Os~)Z nucleon-pair, it can be shown that the relative momentum (k) of the two nucleons in the final state varies from 0 to 270 MeV/c. Hence 1/k takes values between 0.73 fm and oo, thereby indicating roughly the necessity and nature of correlations required to be present in the wave function of the rz-capturing nucleon pair. ["fhe wrresponding figures for rz-capture by a (Op~)2 pair are 325 MeV/c and 0.61 fm.] Of course it is observed experimentally that the two nucleons are more likely to emerge back-to-back with nearly equal energies, and hence the short-range wrrelations are more important than the long-range ones in this reaction . However, experimentally s) k(Os t)2 is found to be peaked at x 235 MeV/c and k(Op~.)Z at x 295 MeV/c. Secondly, the momentum transferred to the nucleus which is same as the recoil momentum (~ of the nucleus in this reaction, varies experimentally between 0 and x 200 MeV/c with a peak at S 100 MeV/c [ref. °`)] . Hence the use of the HF wave functions is not unjustified . In many calculations the kinematics of the three-body final state has been treated quitecrudely by assuming theenergies ofthe two outgoing nucleons to be equal ' ° " 1 s) or integrating the absorption rate over energy on a fairly coarse grid with an accompanying error as large as 20 ~ [refs. e . ')]. Experimentally z), however, the single-nucleon energy spectrum, though peaked at equal energy sharing (~ 55 MeV), is found to be about 40 MeV wide and hence the equal energy sharing approximation may not be a good one . In the present work the inequality of the energies of the two outgoing nucleons has been taken into account exactly, unlike refs. 6 " ' " lo" 's). Contributions of all excited states (E < 5 MeV) of the two residual nuclei are studied systematically in a model to be described later. We have calculated for the reaction 12C(rz -, NN) angular distributions for the emitted pair, branching ratios and the total absorption rate . We use the 1N model for rz-absorption and neglect restettering of the rz before its absorption (see sect . 5). Wave functions used for the bound pion take into account the strong rz-nucleus interaction and the finite nuclear size 1~. The NN interaction in the final state (FSn is taken into account, while that between each of the two nucleons and the residual nucleus is neglected (see sect. 5). More important, however, is the use of wave functions baséd on the UMOA. It is interesting to compare results obtained with these wave functions with equivalent results obtained with wave functions based on Jastrow or Brueckner correlations. It must be stressed at this point that these wave functions contain no'free adjustable parameter anywhere. In the next section we survey the recent experimental and theoretical work in a little more detail . The theory is presented in sect . 3 followed by calculational details in sect. 4. Results are presented and discussed in sect. 5 which is followed by the conclusions in sect. 6.

370

R . S . HHALERAO AND Y . R . WAGHMARE

2. Swey of experimental anti theoretical work A survey of the experimental work shows that for comparing the calculated angular distributions for x- capture on 12C, only two sets of experimental data are available: the old propane bubble~hamber data of Demidov et al. s3) and the recent data of Lee et al. Z). No measurements, however, are available for the angular distribution for isC(x-, nn). As mentioned earlier, the only calculation 1 sb, °) which used the correlation function derived from Brueckner theory has some limitations . They are as follows (i) the energies of the two outgoing nucleons are assumed to be equal, (ü) the use of pure hydrogenic wave functions for n -, (iii) the neglect of the N-nucleus FSI, (iv) the neglect of the NN FSI in ref. 1 sb) and (v) the branching ratio is in disagreement with the experimental data . We discuss below only some of the very recent theoretical work on the (x -, NN) reaction. For details of the earlier work one may refer to the review article by Kopaleishvili za). Recently Nyman ss) has determined the energies and angular correlations of pairs of nucleons emitted in the capture of x- using a simple two-body model without assuming any detailed knowledgeof nuclearcorrelations . He has investigated how the various aspects of the reaction depend on the bulk properties of the nucleus, such as its matter density and radius . In the latest of a series of papers by Moms and Weber 1 zb) a coupled-channel formalism has been developed and a throe-body partial-wave analysis has been carried out for the final-state scattering . `This includes a two-body residual interaction and an optical model potential. They found that no agreement with the experimental data was possible without including Jastrow-type short-range oorrelations (SRC) with dominant momentum components between 0.3 and 0.4 GeV/c. The branching ratio was in disagreement with the experimental data (sect. 1). Garcilazo et al. ' s) have proposed a theory of the three-body final state based on the Faddeev equations. For the best-fit correlation parameter (r~, they found the N-nucleus FSI to be slightly more important than the NN FSI. Clark and Ristig zb) have developed a unified and consistent approach for the extraction of information on SRC from reactions of the type (y, N), (y, 2N), (n, N), (x, 2N), (e, e'N), etc. They have developed the model transition amplitude in a well-behaved cluster expansion with terms arranged according to increasing order in a smallness parameter characterizing the correlations. Such an approach cannot, however, replace the treatment of SRC as in the microscopic theories . Kopleishvili etal. ~') have shown that in the 4He(x-, NN) reaction, the n-rescattering effect plays a decisive role at small momentum transfers, while the exchange between all nucleons in the final state plays a major role at large momentum transfers.

BOiJND-PION ABSORPTION

37l

3. Theory The absorption rate for the reaction X(n - , NN)Y in first-order perturbation theory is given by (te = c = 1 here and elsewhere),

where V is the volume of normalization, k and K are the relative and c.m . moments of the outgoing nucleon pair, M is the nucleon mass, m --- 2MMYl(2M+MY), MY is the mass of the residual nucleus, Ef is the total kinetic energy in the final state and for H we take 1 Z) the static part of the nonrelativistic reduction of the pseudoscalarpseudovector interaction. Now if9 is the angle between the moments of the two outgoing nucleons, the partial absorption rate can be shown to be d(cos B)

2)s

J

~~

~Hfr

I ZkK2

IK ô~_ . ~ I dz,

(2)

where Kz

ôK aY

I

s.i

x

ME

yZ(z2-~_) - ~+~Y[(1 -zZxl- YZZZ)]~

2

CyZ

l a? +z 2 ~~ -a+~

(3)

1- z2 ~~ 1-y2z2

MEf ~ K

-

s 4Ya+a - (1 - z2)~(1 - YZ Z Z)~f2(a + +a?YZ)~Y2 z 2 (á+MZ/m )

(4)

and where y =- cos 9,z =- oos m (w being the angle between k and ~, af - (á f Ml2m), and ~ is the azimuthal angle of k. Here ~~ Hfr ~z is a function of k, K and co alone. The inequality of energies of the two outgoing nucleons is treated exactly in the derivation of eq . (2). If the two nucleons are assumed to be ejected with equal energies, then u~ _ }n and the expression for the absorption rate can be simplifïed to dw Va kK -- ( s ~ ~ ~Htr~Z (a_K2 +MEr)s, ( Ë d(cos B) le,=s~ 2n) where all quantities on the right-hand side have to be evaluated at z = 0. A somewhat similar expression with kK replaced by k Z has been given by Cheon l s') and used by Ka.ushal et al. t~ . This expression is clearly incorrect and k 2 should be replaced ~ by kK. f Private communication with the author .

R. S. HHALERAO AND Y. R. WAGHMARE

We, however, do not make the equal energy approximation and unless specified otherwise all calculations described below make use of eq . (2) given above. The matrix element Hf ,, obtained in the impulse aproximation assuming n_ to be absorbed by a pair of nucleons is given in the appendix . The expression for ~:~ can be found in ref. t ~ and that for ~f~ in ref. za). 4. Calcalatíoaal details The matrix element Hf , [eq. (A.1)] when substituted in eq. (2) gives the absorption rate dw/d(oos ~. There are, however, many different cases possible depending on the choice of the initial and final states, and these cases are described in table 1 . The second column of this table gives the configuration of the pion-capturing nucleonpair in the initial state, while J and T in the third column stand for the total angular Twe~ 1

The'=C(a', Pn)' °He and "C(a - , nnf°B reactions, and the various cases possible Case

Conlig.

Allowed 1`(7~

1 .2 1 .3

(Opnn (~InOP3n)

0+(1), 2+(1) 1 (1), 2 (1)

2.3

(Osl/30P3n)

1 - (0), 2-(0)

3.2

(Opa /:)

0+(1), 2+(1)

States of residual nuclei with nonzero oontrib.

Residual nucleus

1, 2

',Be b

(except third level) 3

(third level)

Notation used in the text : e.g . 2.1 .4 means case 2.1 contributing to the fourth level of °B.

momentum and isospin of the pair. States of the residual nuclei which receive a nonzero contribution from these configurations are given in the fourth column. Since the isospin of the third level of ' °B is different from that of the rest, this level is treated separately . This table is consistent with the repeated experimental _ observation z ~ °) that the probability of n capture by a (Os~Op~.) pair is very small. In these experiments no peak was observed in the excitation spectrum at the energy correspondingto the (Os}) - t(OptJ - t two-hole state. This finding can also be explained on the basis of low correlation between Os~ and Op~ nucleons . All excited states ofthe two residual nuclei with energy less than 5 MeV were taken into consideration and their contributions studied systematically . These states are described in table 2 and here the notation is same as in the fourth column of table 1. The matrix element Hf,

BOUND-PION ABSORPTION

373

T~st .s 2

Excited states (E < 5 MeV) of the residual nuclei taken into oonaideration Level no.

1(g.a.)

2

i

E (MeV)

00+(1)

3.368 2+(1)

'°B

E (MeV) !"(~

0 3 + (0)

0.718 1 + (0)

3

4

5

6

1 .740 0+(1)

2.155 1 + (0)

3.590 2 + (0)

4.773 3 + (0)

T~Le 3 Expansion caefrxtients d"?, for'~C Ceq. (A.1)] obtained with the Yale interaction Occupied HF orbit Os,/z "Y3/3

Neutron states

Proton states

Coo

Coi

Coi

Coo

Coi

Coa

0.9357 0.9672

0.3140 0.1215

0.1608 0.2233

0.9379 0.9685

0.3083 0.0844

0.1594 0.2344

for the case 3.1 .3 was 'found to be zero and hence there remain ten different cases denoted by 1 .1 .1, etc. The expansion coefficients G~~ occurring in e4 . (A.1) are given in table 3. The NN scattering phase shifts at 142 MeV (which corresponds to the average value of k) were taken from ref. 2~ and all the approximations mentioned in this connection by Eisenberg et al. e) were made in this work also. This allows a direct comparison of our results with those in ref. 6). Since the ~cNN interaction used in this work is a sum of one-body operators, it cannot lift simultaneously two independent particles bound in a well and put them in two noninteracting unbound states i .e. its matrix element between uncorrelated initial and final states ought to be zero. But because of the various approximations made in writing . these states, this is not so and the spurious contribution must be subtracted . 5. Resalts xnd discaseion We have calculated absorption rates, for all the cases described in table 1, in three different situations (i) HF wave functions, FSI (i.e. interaction between the two outgoing nucleons) included . (ü) HO wave functions, FSI included. (üi) HO wave functions, FSI neglected. These results, some of which are shown in figs . 1 and 2, when summed with appropriate weight factors lead to figs . 3 and 4 for (~-, pn), figs. 5 and 6 for (~-, nn),

374

R . S. BHALERAO ,AND Y . R . WAGHMARE

o~0

20

40

BO

80

OPENLAG

100 t20

140

160

1B0

ANGLE (DEG .)

Fig . 1 . One of the cases described in table 1 . The absorption rate dw/d(oos~ (sac - ') is in units of a'G =N2 1 (= 2 .336 x 10 - ') MeV~ . HO : Rates with harmonic oscillator (HO) wave functions neglecting final-state nucleon-nucleon interaction (FSI) ; HO, FSI : Rates with HO wave functions taking into account FSI ; HF, FSI : Rates with Hartree-Fork wave functions taking into account FSI . Dashed curve : same as HF, FSI but energies of the two outgoing nucleons assumed to be equal .

and figs . 7 and 8 for (n - , hTI~. In all these figures our results are compared with the earlier work wherever available. We have compared the absorption rates obtained with and without the equal energy approximation [eqs. (5) and (2)]. As can be seen from fig. l the two curves are of somewhat similar shape, with the former results being smaller in magnitude. However, unlike the latter curve, the curve for the equal energy approximation suddenly drops to zero at 8 = 180° (not shown in fig. 1). This is because if El = EZ, K(B = 180°) vanishes, thereby making the right-hand side of eq . (5) zero . The following general pattern emerges from the study of angular distributions of absorption rates for the various cases. Cases 2.1 .2 and 2.1 .4, 2.2.1 and 2.2.6, 2.2.2 and 2.2.4 give similar results as expected. One interesting feature of all these curves (HF, FSn is that those corresponding to n~apture by the (Op~) z pair show a dip while those corresponding to n-capture by the (Os})2 pair increase monotonically.

BOUND-PION ABSORPTION

375

Fig . 2. See caption to fig . 1 .

120 n0 140 b0 160170180120 140 160180120140 160 iB0 Fig . 3 .

OPENNß

ANGLE (DEG.)

Fig. 4.

Fig. 3. Absorption rate ve~sua opening angle for'=C(x - , Pn)'°Be. The solid curve represents the present work and ie normalized to the highest experimental point (aa in all other figures) . The dotted carves which represent earlier calculations are normalized W 180° and are discussed in the text. (Ctirve GEL neglects all Penal-state interactions.) Experimental data : izC(x', Pn), Lee et al. _). Fig. 4. Absorption rate versus opening angb for '=C(x', pn) 1 °He for the two two-hole states . See caption to fig . 3 for other details.

37 6

R . S . BHALERAO AND Y . R . WAGHMARE

~5 á ë a 0 u

Y 3 v

40

60

80

100

OPENING ANGLE (DEG" )

120

160

180

Fig. 5 . Absorption rate versus opening angle for'~C(1t - , nn)'°B : (a) all possible excited states of the residual nucleus included, (b) only the ground state. Experimental data :' 6 0(R -; nn), Nordberg et al. 4). No experimental data available for 'sC(n-, nn). t2 10~

i 5

Or

0

'

40

80

120

160

OPENING

0 ~

40

~

80

120 ~ 160

ANGLE (DEG " )

Fig. 6 . Absorption rate versus opening angle for'=a>t - , nn)'°B for the two two-hole states. Ctirve (a) includes contribution from the ground state of'°B only . Other curves : all possible excited states . No experimental data available .

The contribution of excited states is found to be quite large. Comparison of the curves HO, FSI and HF, FSI shows that the latter are always about an order of magnitude larger except at low angles in some cases. In some cases (e.g. 1 .1 .1, 1 .2.1, etc.) the shapes of the two curves (HO, FSI and HF, FSn dif%r drastically from each other. Comparison of the curves HO and HO, FSI (fig. ~ shows that the NN FSI enhances the probability of back-to-back emission quite drastically, but otherwise leaves the absorption rates relatively unchanged. A similar tendency has also been

BOUND-PION AHSORP'fION

377

Fig . 7 . Grand total'=C(a - , NN) . Meaning of the solid curves as in figs . 1 and 2 . Dashed curve : HF, FSI with spurious contribution subtracud .

OPENING

ANGLE (DEG .)

Fig. 8 . Absorption rate versus opening angle for ' =C(a - , NN) : (a) all possible excited states and both (Os,~l) - = and (fiPa,~ -2 configurations, (b) all possible excited states but oniy (Oaln ) - = configuration. Bubble-chamber data of Demidov er al. _~) .

37 8

R . S . BHALERAO AND Y . R . WAGHMARE

observed by Garcilazo et al. ' 3). This shows that, contrary to the observation of Kopaleishvili 3 ~, Eisenberg et al. e) and Guy et al.'), the effects of short-range oorrelations (SRC) present in the initial wave function are not masked by the NN FSI. Morris and Weber ' z) also observed that the NN FSI acts as a small perturbation compared to the SRC for 0.2 S k S 0.4 GeV/c. Hence the possibility of this reaction throwing light on short-range correlations need not be ruled out on these grounds. This statement, of course, has to stand the scrutiny of a more complete theory which incorporates N-nucleus FSI and n-restettering, which are neglected in this work . It is not easy to estimate the effect of the nucleon-nucleus FSI (in the presence of the NN FSI) on the calculated results for the following reason . The kinetic energy ofeach of the two outgoing nucleons varies over a wide range of 0-100 MeV with a broad peak at x 55 MeV. This alters the optical well parameters to a significant extent . One indication of the importance of the N-nucleus FSI is the relative magnitude of the spurious contribution to the absorption rate. As can be seen from fig. 7, the curve with the spurious contribution subtracted differs from the uncorrected curve by about 10 ~. The correction to the one-body absorption operator due to restettering of a pion is known s') to be important in the reactions ZH(n - , nn) and °He(~ - , NIA. We, however, neglect this effect because the aim of the present work is only to study the efficacy of the Hartree-Fock wave functions based on a realistic NN interaction in a first-order calculation of the (n - , NIA reaction on 12 C, and to compare the, results thus obtained with equivalent results in the literature, e.g. ref. 6). For additional discussion on the neglect of pion-restettering, ref. 1 z) may be referred to. According to ref. iz), pion restettering may possibly enhance the absorption rates by a factor 2, without introducing any new qualitative features. In comparing theoretical results with the experimental data, more importance should be given to the values at large B (> 125°), which are more likely to correspond to the two-nucleon absorption . In fact, the slight disagreement of our results with the experimental data at low angles (figs. 3 and 4) could be due to absorption on clusters or to the evaporation mechanism. Lee et al. Z) have not given experimental data for B < 115°. Usually experimental data are also not given for B ~ 170° and hence in this work we have normalized all our results to the highest experimental points available. The curve K in fig. 3 represents an a-cluster calculation by Kolybasov ' 1) and the curve EE is from Elsaesser et al. 8). In the latter calculation the N-nucleus FSI is neglected and the radial wave function of the emitted pair is obtained by using a realistic NN potential. The curve GEL represents a calculation ~ which uses the formalism of ref. ') but neglects all FSL Both'EE and GEL take into account SRC in the initial state using the Jastrow ansatz . Besides the angular distribution of the nucleon pair emitted, other equally

HOUND-PION ABSORPTION.

379

stringent tests for any theory are its predictions about the branching ratios and the total (i.e. B-integrated) absorption rate w. Branching ratio : The branching ratios obtained in the present calculation are 1 .89 for the (Os.~)Z configuration, 4.14 for the (Op})s configuration and 3 .85 for the two configurations taken together, all for the angular range 170°-180°. These compare well with the data of Ozaki et al. 3), Nordberg et al. a) and Demidov et al. s3), but not with the recent data of Lee et al. s) or Calligaris et al. sb) . The branching ratios obtained by earlier workers have been mentioned in sect. 1 . Totalabsorption rate : This is found to be 2.34 x lOls sec- ' (without renormalization), in satisfactory agreementwith the recent experimental value 3 (3.9 t 1.3) x 101 s sec- 1 . A less recent experimental value is x 1 .55 x 101 s sec- 1 [ref s3)] . Both experimental and theoretical values have, however, their limitations: the experimental one because it includes channels other than (n - , NN) and the theoretical one because of various approximations made in the calculations . The two earlier calculations 12 .13) on the (~-, NN) reaction which employed the 1N model and took into account, besides other of%cts, N-nucleus FSI, were based on the Jastrow-type initial correlated states . One would like to have a calculation which takes into account this of%ct but which is based on the present approach . We have used, in this work, a nonrelativistic (NR) reduction of the pseudoscalarpseudovector interaction operator hiN`v x . An ambiguity in a careful NR reduction of the nN interaction was first pointed out by Barnhill 3°). For some of the most recent attempts to understand this problem see refs. 3s, 3s) . 6. Conclosion We have presented and discussed above, our results for the angular distributions, branching ratios and the total absorption rate for the reaction 12C(rz - , NN). The overall agreement with the shapes of the angular distributions of Lee et al. Z) and Demidov et al. s3) is very good. Though experimental data for 12C(n- , nn) angular distributions are not available, those for the reaction 160(rc- , nn) - in particular the increment in the absorption rates at B x 30°-40° (fig . 5) - are reproduced very well if the residual nucleus is assumed to be left in the ground state. The correct order of magnitude of the calculated total absorption rate strengthens confidence in the use of the Hartree-Fock (HF) wave functions based on the unitarymodel-operator approach (iJMOA), in reactions of the type (n -, NN). This is because, more than the angular distributions, the total absorption rate is sensitive to the short-range oorrelations (SRC). However, definite conclusions cannot be drawn until rescattering mechanisms and final-state nucleon-nucleus scattering are taken into account properly . The branching ratios, though larger than the earlier calculated values, are still smaller than the most recent data of Lee et al. 2). They, however, are not in disagreement with the less recent values from refs. 3 . a. 23) . At

380

R . S. BHALERAO AND Y. R. WAGHMARE

this point it must be stressed that there is no free adjustable parameter involved anywhere in the present calculations . Some other important conclusions are as follows: (i) The absorption rates obtained with oscillator wave functions and the HF wave functions based on the UMOA, differ significantly both in siu and shape. (ü) The absorption rates obtained with and without the equal energy approximation do not dif%r very much in shape, but the former are smaller in magnitude. (iü) The absorption ratescorresponding to the (p.ß) 2 configuration show a dip while those corresponding to the (sß)2 do not. (iv) The contribution ofthe excited states of 1°Band 1°Be isfound to be quite large. (v) Whatever the of%cts of the SRC present in the HF wave functions, they are not masked by the NN final-state interaction. We are thankful to Profs. H. S. Mani and T. Dass for some useful discussions. One of us (R.S.B.) acknowledges financial assistance during the course of this work from the Department of Atomic Energy, Government of India. Appeoùix For 2P pion absorption by 12 C the matrix element Hfi occurring in eq . (2) can be shown to be Hft - (4rz)ZG~~c

(- r~-xa+To-e,

(2J~+1x2T~+1)

h LS

j1

12

.i

i

S J

.12

r

~,e:

x ~ ~n1NL : ~i~n l hn 212 : ~ .~(1L.1 ; ( -M~-vxl(-M~-v)) ~

J'r . .

e1NL

( -i~+L

S .T .r . . x~ d'(IS'J'~(-Mc-v-v~+V/I)v~( - Ma-V+v")) L'S T~r -d~ X (IS'J' ;

exp ( 1aû T I )

(- M~-V)V"( -M~-v+v "))

x Y~_Ma_r_r,+~'>(~~

2L + 1 4~

s" ~ r'

INLF~i '

a.u~arr~a .ra.ráaroró+

where the notation is as in ref. 1~. The matrix element hTf , is a function of h, j 12 , j2, J, T, J~, T~ and k, K, S',~ v', T', ~l', J~, 7~ only . However. ~~~Hfi le carrbe shown to be a function of k, K and w alone.

BOUND-PION ABSORPTION

38 1

Refereac~es 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36)

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