Pion absorption followed by two-nucleon emission

1.D.I:

Nuclear Physics A144 (1970) 441 --448; (~) North-HollandPublishing Co., Amsterdam

8.A.8

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

P I O N A B S O R P T I O N F O L L O W E D BY T W O - N U C L E O N E M I S S I O N W. E L S A E S S E R a n d J. M. E I S E N B E R G

Department of Physics, University of Virginia, Charlottesville, Virginia t Received 11 A u g u s t 1969

Abstract: The process is considered in which a negative pion in a 2P atomic orbital is captured on a correlated pair of nucleons. Mutual final-state scattering of the emitted nucleons is taken into account. The absorption process is treated in perturbation theory using a nonrelativistic pionnucleon interaction. Asymptotic radial wave functions of the emitted nucleon pair, used in earlier works, are replaced by radial wave functions in a realistic nucleon-nucleon potential. The results are significantly affected by this substitution. Numerical results, including angular and energy distributions and absorption rates, are presented for pair emission from the ls~ and lpsk subshells.

1. Introduction W h e n a negative pion, b o u n d in an atomic orbital, is captured by the nucleus the final state which is most often produced is that in which two nucleons emerge from the nucleus in a more or less b a c k - t o - b a c k direction 1,2). This is because the absorpt i o n of the b o u n d pion introduces a great deal of energy into the nucleus, b u t n o mom e n t u m . The nuclear well does n o t provide m o m e n t u m c o m p o n e n t s of sufficient m a g n i t u d e to allow for a b s o r p t i o n o n a single nucleon. The t w o - n u c l e o n a b s o r p t i o n mode, on the other h a n d , involves a final state of high energy, b u t small total m o m e n tum. This mode also has very high relative m o m e n t u m for the two nucleons. It thus probes quite small values of the separation distance for the relative wave f u n c t i o n of the t w o - n u c l e o n system. As a consequence, it may be able to yield i n f o r m a t i o n on n u c l e o n - n u c l e o n correlations in the initial g r o u n d state wave f u n c t i o n for the nucleus [refs. 1,3)]. U n f o r t u n a t e l y , the possibility of extracting a description of the initial state correlations is very critically c o n t i n g e n t on whether or n o t the final state effects can be adequately h a n d l e d in the theory of this pion a b s o r p t i o n process. In the final state, the exiting nucleons scatter offeach other a n d offthe residual nucleus. The former effect, in particular, has a very i m p o r t a n t influence o n the a b s o r p t i o n tt. Moreover, the theoretical predictions for the t w o - n u c l e o n emission are very sensitive to the details of the description of the final state. M a n y of the previous calculations 4 - 7 ) have simulated the n u c l e o n - n u c l e o n scattering in the outgoing state by m e a n s of a wave t W o r k s u p p o r t e d in part by the N a t i o n a l Science F o u n d a t i o n .

tt See refs. 4.5) here referred to as I and II. These contain extensive references to earlier work in this area. 441

442

w . ELSAESSER AND J. M. EISENBERG

function which contained the experimental scattering phase parameters, and used for its radial dependence the asymptotic radial wave function, which is of course strictly valid only outside the interaction region. This procedure had been shown to be a reasonable one in calculations 8) of the photodisintegration of the deuteron. It was then taken over for use 9) in the i 6 0 ( ' ~ , np)14N quasi-deuteron reaction, and finally was applied to pion absorption with the emission of two nucleons 4-7). More recently, some results have been reported ~o) which use for the radial wave function of the outgoing pair of nucleons the actual solution of the radial Schr6dinger equation with a realistic nucleon-nucleon potential. The present paper extends these results in a number of ways, the most important of which is that it considers the totally differential transition rate, i.e. the rate as a function both of pair opening angle and of the energy division. This is clearly quite important in order to facilitate the comparison with experiments. The radial wave functions which are used here are the solutions of the nucleon-nucleon scattering problem for a hard-core Hamada-Johnston potential [ref. 11)] which was refitted lz) to the Yale phase parameters 13) for pp scattering data t. The replacement of the asymptotic wave functions with these Yale wave functions has led to a number of significant changes in the calculational results for pion absorption with two-nucleon emission.

2. R e v i e w o f the f o r m a l i s m

The formalism for the present theory of pion absorption was presented in some detail in I, and reviewed in I[. We shall here indicate only such elements of the theory as are essential for defining its nature, and for specifying the ways in which the present calculation differs from those in I and II. We choose for the interaction operator (h = c - -

1)

H(12) = - - f m ~=t

a'V,~z'~b---z'~ba'VN

,

(1)

m

where the summation refers to the two nucleons which make up the capturing pair. Here f is the pion-nucleon coupling constant, m and M are the pion and nucleon masses, q~ is the pion field, and ~ and z are the nucleon spin and isospin operators. The subscripts ~z and N on the gradient operators indicate that they act on ~ and on the nuclear wave functions, respectively. The transition rate is written in terms of eq. (1) as

dw -

2~p(E) ~ ~ I(K, k; S'M's, T'M~.IH(12)lll Jl , 12 J2; JM, TMT)[ 2, 1 + c~tlt26j~i z S'M'sT" JMT

(2)

t The a u t h o r s are deeply i n d e b t e d to Professor G. Breit a n d his g r o u p at Yale for s u p p l y i n g these functions.

PION ABSORPTION

443

where the ket refers to an antisymmetrized state of two nucleons in (lljl)and (12j2) subshells of a harmonic oscillator shell model. This state is assumed to be correlated by means of a functional form introduced by Dabrowski 14), namely

f(r) =

0 1-exp

r < r c,

{-fl[(r/rc)2-1]}

(3)

r > rc.

The pion absorption leads to the ejection of these two nucleons, and a residual nucleus of spin J (projection M) and isospin T (projection MT). In eq. (2), p(E) is the density of finat states. The final state wave functions are given, in terms of phase shifts 6ssr' by I'+S'

]K, k; S'M's, T'M~.) = 47ze'K'R

E

~ irC(l'S'J'; 2'M'sM')

/ ' = 0 d ' = l t ' - S ' l M"s

x exp ( -- ifs'sr,,')gSf'(kr) YI~,(k)C(I'S'J'; x Yv, a,+~,s-M,,s(O)lS'M) ',

2'+ M's - M's', M's', 2'+ M's)

T'M'T),

(4)

where we have used relative coordinates and c.m. coordinates for the outgoing pair, r = r l - - r 2,

R=½(rl+r2),

(5)

and correspondingly we have introduced the relative and total momenta k = ½(kl-k2),

K =

kl+k2.

(6)

In earlier work, the radial dependence of the pair wave function in eq. (4) was taken to be

g~,'f,'(kr) = cos 6~'sT,~ir(kr)-sin 6s'f'nr(kr),

(7)

where jr(kr) and nr(kr) are spherical Bessel and N e u m a n n functions, respectively. Eq. (7) is now replaced by the Yale wave functions for a hard-core Hamada-Johnston potential with parameters fit to the Y L A M phase shifts 13). As in I and II, we continue to suppress the effects of the tensor force in admixing states of different orbital angular momenta; thus the Yale wave functions were uncoupled by performing a transformation dictated by the mixing parameter es. Since es is less than about 3 ° in all cases but one, we felt this to be an adequate procedure for our present calculational purposes. 3. Results of the calculation

The outgoing two-nucleon state considered in the present work included all partial waves with orbital and total angular m o m e n t u m less than or equal to two. On the basis of estimates using the asymptotic radial wave functions, one can verify that the remaining partial waves contribute less than about 1%. Using the Yale wave

444

W. ELSAESSERAND J. M. EISENBERG

functions, the radial integrations were performed numerically. We note here that, in evaluating the transition matrix elements, we have subtracted spurious contributions which may occur when neither initial state correlations nor final state scattering are present; these spurious effects arise because our initial and final state wave functions are not taken in the same potential, and therefore are not orthogonal. TABLE 1 A s s u m e d p r o t o n separation energies and harmonic oscillator range parameters Nucleus

1s~_ separation energy (MeV)

1p~ separation energy (MeV)

12C

33.00

15.96

160

34.00

18.45

1p~ separation energy (MeV)

Harmonic oscillator range parameter 0.645 f m - x

12.13

0.629 fro- ~

The shell-model parameters which we require in the present calculation are shown in table 1 for lZc and ~60. These include nucleon separation energies and harmonic oscillator parameters. It will be noted that these quantities do not differ appreciably for the two nuclei. We have here used the values for 160 in order to facilitate comparison with the results of I and II. At the same time, we shall make occasional use of the present calculation to compare with data for various subshells in 12C. Throughout our numerical work the mean excitation energy of the residual nucleus was held fixed at 10 MeV. Lastly, since the transition rate proved not to be strongly sensitive to initial state correlations in the present formalism, we have taken f(r) = 1 for the function of eq. (3), except where otherwise noted. In fig. 1 are plotted the asymptotic pair wave function of eq. (7) and the Yale wave function for the 3S~ partial wave at 86.5 MeV. At slightly less than 2 fm the Yale wave functions have, for all practical purposes, assumed their asymptotic form for the partial wave exhibited here. The curves of fig. 2 present a comparison between transition rates for the Ip~ subshell, as functions of opening angle, from the present formulation and those from II. Several significant features are the increased transition rates and the relatively large back-to-back scattering obtained in the present work as compared to II. Fig. 3 exhibits transition rates for nucleon emission from the (ls~, lp~_) and (l s½)z subshells. There is considerable enhancement in transition rates for the case of np emission from t h e (1S~) 2 subshells in the back-to-back direction. Figs. 4 and 5 show the energy sharing of pairs emitted from the (lp~) 2, (ls~, lp~) and (ls~) z subshells respectively at 180 ° opening angles. The dashed curves of fig. 4 are from II. It is significant to note that, for nn emission from the lp~ subshell the dip near equal energy sharing, predicted in II, does not appear in the present work. However, the flatness of the peak is in marked disagreement with the experimental aZC

I I | I I t

~(r) d2w

"-. b / f " \

dEdcosO 4

/

//~.

(3 0

60

O-o

120

180

Fig. 1 Fig. 2 Fig. 1. Pair r a d i a l wave functions p l o t t e d a g a i n s t nucl e on s e p a r a t i o n distance. The solid line refers to the Y a l e wav e function used in the present calculation, a nd the da s he d line to the a s y m p t o t i c fu nction used in I a n d II. The curve p l o t t e d is for the aS 1 p a r t i a l wave at 86.5 MeV.

Fig. 2. T r a n s i t i o n rates as a f u n c t i o n o f o p e n i n g angle for two nucleons e m i t t e d from the lp~_ subshell, w i t h one nucleon h a v i n g 40 MeV kinetic energy a n d the ot he r one h a v i n g a b o u t 50 MeV. The curves labeled a refer to a b s o r p t i o n o f a 2p p i o n w i t h e mi s s i on of an np pair, a n d those l a be l e d b are for nn. The solid lines refer to the p r e s e n t c a l c u l a t i o n a nd the da s he d lines to the results of II. The units are l 0 s4 s e c - 1 . M e V - s for the 2P(np) cases, a n d 10 s3 s e c - s . M e V - s for the 2P(nn) cases.

.....

16

d2w

//

d Ed cosO

/// 0

b

,

,

60

////

///tt

d2w-I2 dEdcosO

l

t

//'/ ,

,

8°

,

t20

,

,

180

0

20

4O 6O E{MeV}

8O

I00

Fig. 3 Fig. 4Fig. 3. T w o - n u c l e o n t r a n s i t i o n rates as a f u n c t i o n of o p e n i n g angle at a p p r o x i m a t e l y e qua l energy sharing. The solid lines are for (ls~, lp~_) -1 emission w i t h a p p r o x i m a t e l y 40 MeV for each nucleon a n d the d a s h e d curves for (1 s½)-2 e m i s s i o n w i t h a p p r o x i m a t e l y 30 Me V for each nucleon. The curves m a r k e d a refer to a b s o r p t i o n o f a 2P p i o n w i t h emissio n o f a n np pair, a n d t hos e labeled b are for nn. The units are l 0 s* sec - s • MeV - s .

Fig. 4. T w o - n u c l e o n t r a n s i t i o n rates as a f u n c t i o n of the energy of one nuc l e on for b a c k - t o - b a c k e m i s s i o n f r o m the l p k subshell. The solid lines refer to the pre s e nt w o r k a n d the da s he d lines to the results o f II. The curves labeled a refer to a b s o r p t i o n o f a 2P p i o n w i t h e m i s s i o n of an n p pair, a n d those l a b e l e d b are for nn. The units of the o r d i n a t e are 10 s4 s e c - 1 . M e V - s for the curves m a r k e d a a n d 1013 sec - s • MeV -1 for those m a r k e d b. The e x p e r i m e n t a l points, f r o m ref. ss) b u t n o r m a l i z e d to the area u n d e r the theoretical curve, are for nn e m i s s i o n from the l p k subshell of s2C.

446

W . ELSAESSER A N D J. M. E I S E N B E R G

data ~ 5) shown here but normalized to the area under the theoretical ~60 curve for nn emission. In fig. 5 are shown experimental points from ref. as) for nn emission from the ls~ subshell of t2C, adjusted by the same normalization as for the lp~ data. The results of the present formulation predict that, for a given energy, the ratio of

I

i I ~\

I /

6

\\a\ \ \

d2 w dEdcosO

\

4.

iI'

~~

I ///

0

N

20

40

80

60

E(MeV)

Fig. 5. Two-nucleon transition rates as a function of the energy o f one nucleon for back-to-back emission. The solid curves are for (1 s~, lp k ) - t emission, and the dashed curves for (ls~r)-2 emission. The curves labeled a refer to absorption of a 2P p i o n with emission of an n p pair, and those labeled b are for nn. The units of the ordinate are l014 sec -1 • MeV -1. The experimental points, from ref. is) give transition rates for nn emission from the lsx subshell of x2C. The experimental transition rates were adjusted by the same normalization as the experimental rates s h o w n in fig. 4.

80

T,4

~oi dv,~ dccsOl

40

~ J

i "I

i

/ "/

i.

.'//

i 1

t ~

p1"

T

-

2O,~ f

,

20

60

I00

140

a

18

O9

Fig. 6. Total angular distribution of ejected pairs for 12C. The solid line refers to nn emission and the dashed line to np emission. The units are 1015 sec- i. The experimental points are from refs. t6, 17), with the arbitrary experimental units normalized to the theory at the largest opening angle. Those indicated by dots are for np emission and triangles are used for nn data.

PION ABSORPTION

447

(lp~) z to (ls6) z transitions is slightly less than 0.5 whereas experimental results 15) put this ratio at about 1.5. The total angular distributions o f ejected pairs from ( l s , ) 2, (ls~, lp~), and (lp~) 2 configurations are shown in fig. 6. D a t a o f Nordberg, Kinsey and B u r m a n ~6,17) for 12C are also shown in this figure. The arbitrary experimental units are normalized to the theory at the largest opening angle. TABLE 2 Total transition rates for 2P pion absorption Residual nucleus configuration (ls~) -z (Is.t_, lp~)- 1 (lp~) -z total

Present results np pair nn pair

Results from II np pair nn pair

20.6 36.3 24.1 81.0

12.14 22.41 15.07 49.62

7.2 15.2 10.2 32.6 113.6

4.09 7.60 5.53 17.22 66.84

The units are 1015 sec -1. Total transition rates for nucleons emitted from given subshells are shown in table 2. The corresponding results o f [[ are shown in the last two columns. The most salient features exhibited here are the increased transition rates (by approximately a factor o f 2) along with a slight increase in the nn/np ratio; now equal to 0.40 for 12C. Experiments by Nordberg, Kinsey and B u r m a n 16, ~7) resulted in an nn/np o f 2.5_+ 1.0 for t2C and 3.3 + 0 . 9 for the average of p-shell nuclei; the calculations presented in II for 160 gave a value o f 0.34 for this ratio. Finally, the sensitivity of the present formulation to changes in the correlation parameter fi of eq. (3) was tested by repeating several of the calculations for fl = 1.4 and 0.14 respectively with rc = 0.4 fm. The kinematic cases chosen for this test were nn emission from the lp~ and from the ls~ subshells at an opening angle of 180 ~ with each nucleon having approximately the same energy. F o r fl = 1.4 both cases gave approximately a 4 % increase in the transition rates c o m p u t e d from an uncorrelated initial state. F o r fl = 0.14, the transition rate from the lp~ subshell was approximately tripled while, for the ls~ case, the rate was more than doubled.

4. Discussion The use of Yale wave functions for the emitted nucleon pair in place o f the asymptotic radial functions used in I and II has the effect of enhancing the total nn and np angular distribution for t2C at large opening angles. This leads to somewhat better experimental agreement than given by the results o f II. On the other hand, there was almost no improvement in the nn/np ratio. Finally, transition rates in the present work

448

W. ELSAESSERAND J. M. EISENBERG

were consistently higher (by as m u c h as a factor of two at large opening angles) t h a n in the results of I a n d II, a n d two times higher t h a n experiment 18). Kopaleishvili 1o), taking into a c c o u n t the final state interaction between nucleons t h r o u g h a distorted wave approach, o b t a i n e d a n increase in the cross sections for double n u c l e o n emission following 1S pion a b s o r p t i o n by a b o u t a n order of m a g n i t u d e as c o m p a r e d to the values o b t a i n e d in the asymptotic a p p r o x i m a t i o n for the t w o - n u c l e o n wave function. The results of the present work strongly suggest that m u c h more adequate m e t h o d s must be used to treat the final state in this pion a b s o r p t i o n process. The potential well due to the residual nucleus must at least be taken into a c c o u n t * for its influence o n the effective energy at which the two-nucleon scattering takes place. Off-energy-shell effects in this scattering must also be included. The final state reached by the pion a b s o r p t i o n process should ultimately be h a n d l e d using techniques appropriate to the complete three-body system which is present in that state. The authors would like to express their gratitude to Professor Breit for supplying the n u c l e o n - n u c l e o n scattering wave functions used in this work, a n d to Drs. K. A. F r i e d m a n a n d R. E. Seamon for their work in o b t a i n i n g the n u c l e o n - n u c l e o n p o t e n tials, a n d finally, to Drs. R. E. Seamon a n d J. M. Holt, all of Yale University, for m a k i n g the c o m p u t e r runs. One of us (W.E.) would like to t h a n k the N a v a l W e a p o n s L a b o r a t o r y in Dahlgren, Virginia, for support during this project. A grant from the C o m p u t e r Science Center at the University of Virginia is acknowledged. t In this connection, a coupled channel formalism has been developed by Weber 19) for the treatment of the two outgoing nucleons in interaction with each other and with the residual nucleus.

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)

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