Pion scattering on medium-weight nuclei and distortion effects in double charge exchange reactions

Volume 51B, number 2

PHYSICS LETTERS

22 July 1974

PION SCATTERING ON MEDIUM-WEIGHT NUCLEI AND DISTORTION EFFECTS IN DOUBLE CHARGE EXCHANGE REACTIONS S.A. GURVITZ and A.S. RINAT (Reiner) Weizmann Institute o f Science, Rehovot, Israel

Received 29 May 1974 An eikonal version of the DWIAis used to compute absorption and dispersion effects on the cross section for double charge exchange to its double analogue. Lacking information on the ~r-Slv optical potential, recent data on n-4°Ca elastic scattering have been analyzed in the Glauber approximation and the extracted elastic phase function has been scaled up to correspond to 51V. The result is used in a computation of the DWIAamplitude. Large reduction factors are obtained in agreement with experimental upper bounds. Discrepancies with previous estimates are discussed.

With the operation of meson factories, new data may be expected on intrinsically unique, pion double charge exchange (DCEX) reactions. Till today there is only scanty information on these reactions in general and on double charge exchange to the double analogue of the groundstate in particular [1]. Except for the lightest nuclei, only upper limits have been given for the near forward cross section for these reactions on 51V and 90Zr at energies E~r ~ 200 MeV [2]. Calculations performed in the impulse approximations (IA), produce cross sections at least a factor ~ 30 in excess of the observed upper bound [3]. A correct description of attenuation through initial, intermediate and final state scattering, absent in an impulse approximation (IA), is clearly crucial. Since double charge exchange to a double analogue state is a weak process a distorted-wave description should be adequate. Such a calculation requires as input distorted pion waves generated by a pion-nucleus optical potential F°~pt Contrary to nucleon optical potentials, which have been parametrized using abundant elastic nucleon scattering data, there is hardly similar phenomenological information on 1/~pt. In the following note we apply a DWlA description to the double charge exchange reactions on 51V. To be sure, no elastic data on 51V exist and the heaviest nucleus with extant elastic distributions around ETr ~ 200 MeV is 40Ca. However, parameters of optical potentials should approximately be related to nuclear dimensions, and we shall assume that 40Ca and 5 I v have sufficiently close mass num-

bers in order to enable a reliable scaling. Our first concern is thus the analysis of the recent rt-40Ca data [4]. These have been taken for E~r = 205,215 MeV and incidentally, differ by a factor "~2 at corresponding angles. Sucha difference cannot be produced by any simple theory, not even in the resonance region where the n.N amplitude varies strongly. We actually surmise that absolute normalizations are in error. The only description of the above data known to us is by Landau [5], who used a separable rr-N interaction for each partial wave [6] in order to construct a lowest order ~ p t for 40Ca. Solution of the LippmannSchwinger equation produces reasonable agreement for the 215 MeV data and consequently underestimates the 205 MeV data by a factor ~ 2. Landau's optical potential is not suitable for the simple analysis of DCEX reactions we have in mind. Alternatively we used the Glauber prescription for the elastic amplitude in the impact parameter representation [7]

~fexp(iq-b)

[1 - exp(ix~(E, b))] d2b.

(1)

Eq. (1) defines the elastic (optical) phase function, which is related to the optical potential Vopt(b, z) integrated over its path z for fixed b. We chose a modification of the linear approximation containing the density 19 and its Laplacian [8, 9]

109

Volume 51B, number 2

PHYSICS LETTERS

22 July 1974

oo

i

x(b)=T

o

•

--oo

The amplitudes u o, oo in (2) are the ,.standard combinations of the I = 1/2, 3/2 parts [9] u o = ~ [u(½) + 2u(~)]; with

oo = ~ [v(½) + 2v(~)l,

~

I

t (o)l

I

I

l

l

(Ib)'_

"

7r'4°Co elastic

Ir-4°Ca elastic

E ~b = 205 MeV

E ~b = 215 MeV

(3)

(aF = exp(i6/_.)sinfl+_ )

klab.

U=~M(a°k2 + 2al++at-);

klab

O=9-~-(2al+

(4) b

Tile amplitudes Uo, oo (u, v) have been constructed from the most recent phase shift analysis [ 10]. For the density p in (2) we choose a Fermi distribution with the following values for the parameters [11, 12] I

4°Ca: SlV:

half density distance c;

skin thickness t

3.650 fm 3.94 fm

2.280 fm 2.22 fm

j,

(5)

These and the amplitudes above provide a phase function x(b) without free parameters. Figs. la, b contain the data [4], together with Landau's [5] and our prediction. Since the latter agree fairly well amongst them and with the 215 MeV data, we trust these more than the 205 MeV data. Further we take the agreement as an indication that, without too much insight why, the Glauber amplitude (1) apparently gives an adequate description of the scattering and that consequently the phase function (2) should also be reliable. As a further, and not too stringent test, we computed the *r-51Vtotal cross section to be 1559 mb in perfect agreement with a A 2/3 extrapolation (1560 mb) of total cross sections for elements between 4He and 32S [131. As argued above we now postulate that the phase function for the ground state of 51V, its single and double analogues 51Cr* and 51Mn* should be obtained by simple scaling of 0 in (2) using the parameters (5). It is this phase which describes pion absorption and dispersion in a DWlA amplitude for double charge exchange and which we also calculate in an eikonal approximation [9, 14]

110

i

16'1

I

I

I

1.0

I

2.0

I

I

3.0

I

I

1.0

I

I

2.0

I

3.0

-t (fr~ 21 Fig. 1. a,b. Experimental differential cross section of pions on 4°Ca (En = 205,215 Me'v) [4] together with the Glauber prediction ( - - ) and the results of Landau [5] (---).

fDCEX(Ecq)

(6)

=~---~f exp(iq'b) xDCEX(E,b)exp(iXoo(E,b ))d2b. For XDcEx we take the ,simple two-step transition density xDCEX(E, b) = ~ (7)

i
in terms of the single charge exchange profile functions, as follows related to the corresponding amplitude [7] "[CEX(E,b) = (2rrik) -1

fexp(iq.b).t'CEX(E,b)d2q. (8)

In (7) figure si, the usual transverse components of nucleon coordinates.

Volume 51B, number 2

PHYSICS LETTERS

22 July 1974

f~nV(0) = a(~) 1/2 klab/k2M

I0 ~

I .

%

I

51 V (IT¢,lr)51Mfl OAR

%

\

IC

%

do(E,

DCEX

--a-n--] s, v

.o I.tJ

8 -*~'~ b

I 10

|1 20

I 30

I 40

~

I .50

i, 60

e lob

Fig. 2. The predicted DWIA cross section for DCEX on Sly (E = 200 MeV) leading to its double analogue (--) against the IA result of Brown and Barshay [3] scaled from 4°Ca to SIV (---). At #lab = 8.5 we entered the experimental upper limit [2] and the limit~ calculated by Barmy and Vagradov [15].

As a last step a matrix element between two intial neutrons and final protons is involved in (7). We assume wave functions for both to be identical (harmonic oscillator and Saxon-Woods wave functions produced nearly the same results) and to have a range which exceeds the range o f the charge exchange amplitude. Thus, 7CEX(E, b-s)

(11)

with Koltun's estimate a ~ ½ for E . ~ 200 MeV [1]. In fig. 2 we show the differential cross section

ET = 200 MeV

\

,

I

-- ~k

fcrx(0)~ 2(b-s)

(9)

One than obtains, xDCEX (~, b)

(10)

~ ( - ~ ) 2 [f~NEX(0)[2 f 1~7/2(bz1 ; bz2)12dzi dz 2 , --oo

for a pair of f7/2 nucleons. F o r f c E x (0) we took its value at resonance averaged over the Fermi motion

_ kf l i v e , .DCEX . . . .

]

2

(12)

where we denote by f / t h e number o f excess neutrons (N(51V) = 5). Also shown is the IA prediction [3], the measured upper bound [2] and the result o f a calculation by Barshay and Vagradov [15] which we Wish to compare now ¢1 . Also Barshay and Vagradov focussed on the strong reduction o f the IA amplitude due to initial and final scattering. Lacking direct information on V°pt, the latter has been constructed in a model where around ETr ~ 200 MeV the medium is polarized by the pion through resonance-nucleon-hole formation. The thus constructed V °pt is then used in a calculation of the expected attenuation, which amounts to 0.06-0.27, definitely insufficient to account for the experimental upper bound and which is a factor/> 15 offtheir estimate (cf. fig. 2). Since the calculation o f Barshay and Vagradov differs from ours in two components each has to be compared in order to trace the different outcome. Regarding V°pt no direct comparison has been made between a known angular distribution o f elastic scattering and a prediction based on V °pt*2. An indirect test regards the spatial dependence o f l/°pt as computed in two calculations. Extrapolating the results o f Bj6rnenak et al. [16] for 51V and 90Zr to 120Sn one notices close agreement for Im V °pt as calculated by Barshay, Rostoker and Vagradov [19]. There is a factor ~ 2.5 *3 difference in Re V~ pt, but ,1 Our calculation is quite close to one presented by Bj~rnenak et al. [16], with tiny difference in the phase (2), which moreover could not have been compared with experiment as we did. One will observe a difference of a factor ~ 10 in the two predictions. We understand that an unfortunate computational mistake entered the results of ref. [16] (see ref. [17]). ,2 Agreement has been reported for total cross sections [ 18] but as already mentioned, these are quite insensitive to details of the input. ¢3 From comparisons of fig. 6 and 9 in ref. [t9] we concluded that the energy scale of the former is in error by a factor 10. 111

Volume 51B, number 2

PHYSICS LETTERS

this may not be too important in the region ETr 200 MeV where Im V°pt dominates. Since the (corrected) calculations o f Bj~rnenak et al. [16] are close to ours, the cause of, the discrepancy in the reduction factors of the medium must be traced in the way these have been calculated. Barshay and Vagradov [15] account for absorption reducing the amplitude by a factor exp(-l/X), where X denotes the pion mean free path ), ~ Ap o o~oNtal (Po = 3A/47rR3; ~(51V) ~ 1.5 fro) and l the traversed path element, The factor for q = 0 is averaged over the volume ~2 leading to an attenuation [15,20] 2 rn°n'dyn

R

g2A;oetoNt fo d2b

X [exp{-APoa~oNtal(R2-b2) 1/2) -1]

12 .

(13)

In the DWIA, however, the approximate elastic phase

(2) Xel(b )

~ iA ^vo o~rN/R 2 t o~,t

-ut'2~l/2 J

(2")

is weighted by the transition density xDCEX(b) for every b (cf. (6)), and thus yields for q ~ 0

fR xDCEX(E, b)exp(ixel(E,b ))dEb 2 rDWIA (g) ~

~ v i ~ C~X i E ~

" (14)

It should be stressed that the transition density due to f7/2 nucleons has its peak well inside the nucleus and decreases exponentially. Neglect o f dynamical weighting with XDCEx in ref. [15], consequently leads to a sizable underestimate o f the reduction factor r. By the same token one may raise the question whether in making a choice for a target yielding a maximal cross section for double charge exchange to a double analogue much weight should be given to a large neutron excess. This is usually the case for

112

22 July 1974

neutrons with large spin value entailing correspondingly large absorption. The less pronounced absorption o f low-/" neutrons may well offset a larger e n . hancement factor ( N ) .. The authors are grateful to R. Landau and A. Reitan for provided information.

References [ 1] For a general review see D.S. Koltun, Advances in Nuclear Sciences, Vol. 3 (1969) 71. [2] P.E. Boynton et al., Phys. Rev. 174 (1968) 1083. [3] S. Barshay and G.E. Brown, Phys. Lett. 6 (1965) 165. [4] R.C. Bercaw et al., contribution to 5th Int. Conf. on High energy Physicsand nuclear structure, Uppsala, 1973. [5] R.H. Landau, private communication. [6] R.H. Landau, S.C. Pathak and F. Tabakin, Ann. of Phys. 70 (1973) 299. [7] R.J. Glauber, Lectures in Theoretical Physics Interscience, Vol. 1 (1959) 315. [8] See for instance C. Wilkin, Proc. Spring School in Pion interactions 1971, CERN publ. 71-14. [9] G. F~'ldt, Phys. Rev. C5 (1972) 400 and references quoted. [10] J. Carter et al., Nucl. Phys. B58 (1973) 378. [ 11 ] R.A. Eisenstein, thesis Yale University, 1968. [12] G.A. Peterson, Phys. Rev. C7 (1973) 1028. [13] A.S. Clough et al., Phys. Letters B43 (1973) 475; C. Wilkin et al., Nucl. Phys. B62 (1973) 61. [14] Y. Alexander and A.S. Rinat, Ann. of Physics 82 (1974) 301. [15] S. Barshay and G. Vagradov, Phys. Letters 41B (1973) 131. [16] K. Bjernenak et al., Nucl. Phys. B20 (1970) 327. [17] A. Reitan, private communication. [18] S. Barshay, V. Rostokin and G. Vagradov, Phys. Letters 43B (1973) 271. [19] S. Barshay, V. Rostokin and G. Vagradov, Nucl. Phys. B59 (1973) 189. [20] F. Becker and Yu.A. Batusov, Revista del Nuovo Cimento 1 (1970) 309.