The importance of spin flip in pion charge exchange reactions

15 September 1975

PHYSICS LETTERS

Volume 58B, number 3

THE IMPORTANCE OF SPIN FLIP IN PION CHARGE EXCHANGE REACTIONS * D.A. SPARROW Nuclear Physics L.aboratory,Department of Physics and Astrophysics, Universityof Colorado, Boulder, Colorado 80302, USA Received 1 April 1975 (Revised version received 1 July 1975) Pion scattering from 3He is investigated in the context of a simple reaction model. It is shown that for energies below around 200 MeV the spin flip mode can dominate the 3He(n-, n0)3H differential cross-section at most angles, and contribute up to 50% of the total charge exchange cross-section.

The subject of pion scattering from nuclei has received considerable attention recently [ 11. In particular pion charge exchange (cx) reactions have been of interest because of discrepancies between theoretical predictions and experimental results for the 13C(n+,rr0)13N g.s. excitation function [2]. Study of the reaction 3He(n-, rr”)3H has beensuggested [3] as a means to help clarify these discrepancies and test the reaction models in more detail, since detection of the recoiling tritons would allow measurement of the angular.distribution, as well as the total cross-section for this reaction. We have investigated the importance of spin flip (sf) in pion reactions on a 3He target using a simple reaction model. The results show that the sf contributions to the charge exchange reaction are not negligible, and may even dominate. Further, the size of the charge exchange spin flip contribution depends dramatically on the symmetry properties of the target wave function. The model for a-3He reactions has been deliberately kept simple. The Coulomb interaction is neglected, and the Glauber multiple diffraction theory [4] was used to treat the scattering. Only the leading component of the 3He and 3H wave functions was retained [S]. The radial wave function was a Gaussian in relative coordinates [5] &I, ‘2, ‘3) = Cst exp {- 3 442

+ 43 + &)1

, (1)

where ‘ii = ri - ri .

* Work supported in part by the U.S. Energy Research and Development Administration.

The value of cr = 0.148 was chosen to fit the 3He r.m.s. charge radius, and the center-of-mass constraint is included in these calculations. The pion nucleon scattering amplitude was assumed to be of the form &-N(9) ‘fu(9) +&(9)1’ r (2) + c()(q)a*fi + c,(q)l*r

u-ii ,

where 9 is the momentum transfer,1 and z are the nucleon and pion isospin operators, e is the nucleon Pauli spin matrix and n is defined by n = 9 X i, t being the direction of the incident beam. If the scattering is dominated by a p-wave resonance, the momentum transfer dependence of the no spin flip (nsf) term may be represented for the forward hemisphere by the Gaussian f0J9)

=fuJ exp t-&12) .

(3)

This form is accurate within a few percent from O-60” if 0 is determined by setting exp (--&I~) = P1(cos e),

at 8 = 60” .

The values of foJ are chosen to give the forward scat-

tering amplitude in agreement with the phase shift parameterization of Roper et al. [6]. The sf term is taken to be coJ(4) = cuJ z exp (--&I~) .

(4)

The values of co and cx were chosen to give the correct scattering amplitude at 45’) again assuming p-wave dominance and using the phase shifts of ref. [6]. This 309

PHYSICS LETTERS

Volume 58B, number 3

IO 4-

--du/dil

+_

t-

15 September 1975

I

I 3He(n-,

I

I

I

I

7~0)

nsf

3He

Oj, , , ,“\, 0

20

40

60

80

100

120

I

6 cm

Fig. 1. n*B He scattering at 154 MeV. The calculation uses p = In (2)/k and the relative Gaussian wave function.

choice for normalizing the n-N spin flip amplitude gives a model amplitude which is reasonably accurate over the angular range from O-60“, but too low beyond these. Other parameterizations lead to somewhat larger cross-sections at low energies, but the qualitative nature of the results is not sensitive to the parameterization of the sf amplitude. It should be noted that use of a more realistic n-N amplitude within the context of the Glauber multiple scattering series gives predictions for the large angle cross-sections which depend strongly on the extrapolation into the unphysical region (4 B k) [7]. Use of a Gaussian avoids this ambiguity, however the large angle predictions are unrealistic. In evaluating the resulting multiple scattering series, double and triple spin flip terms were dropped. First, in this instance these terms are expected to be small compared with nsf and single sf terms. Further, it seems to be a questionable procedure to use an eikonal formalism for a multiple scattering series involving two or more scatterings with amplitudes which peak around 310

ecm Fig. 2. 3He(n-, n0)3H differential cross-sections, with (solid) and without (dashed) spin flip contributions. Inclusion of spin flip dramatically affects the angular distribution for energies up to 200 MeV.

90”. The multiple scattering series can be written in terms of six distinct functions of momentum transfer. (Including double and triple sf terms would give a total of twelve such functions.) The nsf and sf scattering amplitudesf,,&), and fsf(q) may be written [3,4]

(5)

The F’s contain the information about the radial wave function, and the A’s and B’s are completely determined from the spin-isospin part of the wave functions. For an unpolarized target the nsf and sf amplitudes are incoherent. Defining 7 = a/ 18, the F’s are given by

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Volume 58B, number 3

Fig. 3. 3He(n-, ITO)~Htotal cross-section (solid line) and nsf (dashed line) and sf (dotted line) contributions. Spin flip contributes around 50% of the total cross-section at energies below 200 MeV.

Fl,nsf(cl)

= exp {-(T

F2,nsf(4)

= j$ &I

F3,nsf(q)

=-

-1

+ kN21

9

ew {- 3 (y/2 + Pk21

1

,

ew (- 113 lQ2) ,

3k2 {2(y+P)12 (6)

F l,sf= i exp(i~~)%expI-(7+p)q2}

F2,sf

,

-1 1 =--l-exp(i@~)~exp{-f(~/2+/3)~2}, 2k 2(7+/3) 4 -i

F

1

lexp

(i$$z

exp (- l/3 k2)

.

3,sf=j&2(7+p))26 The angle +* is the azimuthal angle of the momentum transfer 4. The A’s for elastic and charge exchange scattering are given in ref. [3]. For elastic scattering of n* from 3He the coefficients Ei are B1= -(co 7 c,),

B, = -2(cofo

*co& T cxfo) ,

B3=-(cofo2~2cofof,~c,f~Tgc,f,2).

(7)

For the charge exchange reaction, similarly B,=fic,, B3=

B, = -2

.\/z(cxfo-fxco),

-~(c,f,2--f,cofo+3c,f~).

The importance of the spin flip contribution to the charge exchange reaction is directly related to the form

15 September 1975

of B,, the coefficient of the spin flip double scattering term. If only the P33 channel contributed to the interaction, this coefficient would identically vanish. In the region where the P33 resonance dominates, this term remains small. The interference of the two terms which occur in B2 can be interpreted in the following way. One of these two-step processes, charge exchange followed by spin flip (second term), is feeding the reaction at about the same rate that absorption (first term) is decreasing it. The importance of the spin flip contributions in the results presented below are a direct consequence of this cancellation. These results are insensitive to the reaction model and the radial wave function used. Large cx cross-sections may therefore be expected any time there exists a two-step process which competes with the usual absorption. It should be pointed out that this cancellations depends on the details of the spinisospin wave function. For example, the two terms contributing to B2 will add rather than cancel for the S’-state of the triton. In fig. 1,154MeV rr* -3 He elastic scattering crosssections are presented and compared with the experimental results [7]. Good agreement is obtained for the n+cross-section. The s--3He cross-section is qualitatively reproduced also. The rise in the experimental cross-section in the backward hemisphere can be .duplicated if a pion-nucleon amplitude generated from the phase shifts is used instead of a Gaussian form in the single scattering term. It should be noted that the sf contributions, though not negligible for n--3He, are generally small. For n+-3He the sf contribution to the differential cross-section is at most a few percent. In fig. 2 the n--3He charge exchange cross-section is plotted for energies of 160,200 and 240MeV, At 160 MeV the sf process contributes 50% of the total cx cross-section, and dominates the angular distribution except at very forward and very backward angles. Near the resonance, at 200 MeV, the situation is much the same, though not quite so dramatic. At energies above 200 MeV the sf contribution falls rapidly. Nevertheless, the effect on the angular distribution is still substantial, as the 240 MeV results show. The overall energy dependence of the spin flip and direct contributions to the charge exchange total crosssection is shown in fig. 3 for energies between 100-350 MeV. In this energy domain the Glauber approximation should be reliable. It is clear that spin flip will contribute substantially to the total charge exchange cross311

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PHYSICS LETTERS

section for energies below about 200 MeV. There are a number of conclusions to be drawn from these results. 1) Spin flip is a very Important mechanism for pion charge exchange on 3He at energies below around 200-250 MeV, and is probably important for many other nuclei also. 2) The presence of a large spin flip contribution should be readily determinable experimentally in this energy range since sf influences the angular distribution so dramatically. 3) The importance of the sf process depends upon the structure of the spin-isospin wave function, which in the case of 3He results in a partial cancellation of the usual absorption term. In summary, elastic and charge exchange scattering of pions from 3He has been calculated using the Glauber approximation, and simplified models for the 3He wave function and the pion-nucleon interaction. The results show that spin flip is a very important mechanism in the charge exchange reaction at energies below about 200-250 MeV, and significantly affects the total cross-section as well as the angular distribution. A 3He(rr-, TO)~H experiment, in which the recoils tritons were detected to determine the angular distribution of the scattered pions, would be very desirable. It

312

has recently come to our attention is being performed [lo].

15 September 1975

that this experiment

The author would like to thank E. Rost, J.J. Kraushaar and M. Iverson for many useful discussions. The author also acknowledges several valuable communications from W.R. Gibbs and J.M. Eisenberg.

References [l] J.M. Eisenberg, A review of pion-nucleus interactions, reprint of talk delivered at DNP meeting of APS, Bloomington, Indiana, Nov. 1973; M.M. Sternheim and R.R. SiIbar, Ann. Rev. of Nucl. Sci. 24 (1974) 249. [2] J. Alster et al., Bull. Am. Phys. Sot. 20 (1975) 84. [ 31 J.M. Eisenberg and V.B. Mandelzweig, Phys. Lett. 53B (1975) p. 405. There is an error in eq. (7b) of the above paper, which is corrected in eq. (6) here. [4] R.J. Glauber, in High energy physics and nuclear structure, ed. G. Alexander, North Holland,(1967) p. 311. [ 51 L.I. Schiff, Phys. Rev. 133 (1964) B802. [6] L.D. Roper, R.M. Wright and B.T. Feld, Phys. Rev. 138 (1965) B190. [ 71 K.-P. Lohs and V.B. Mandelzwelg, private communication. [8] I.V. Falomkin et al., Nuovo Cimento 24 (1974) 93. [9] A. Reitan, Nucl. Phys. B68 (1974) 387. [lo] J.E. Spencer, private communication.