Final state interactions in nuclear pion photoproduction over the (3, 3) resonance

Volume 61B, number 5

PHYSICS LETTERS

26 Apr~ 1976

FINAL STATE INTERACTIONS IN NUCLEAR PION PHOTOPRODUCTION O V E R T H E (3, 3) R E S O N A N C E N. FREED Department of Physics, The Pehnsylvania State University, University Park, Pennsylvania 16802, USA * and Department of Nuclear Physics, Lund Institute of Technology, Lund, Sweden and

P. OSTRANDER Fayette Campus, The Pennsylvania State University, Uniontown, Pennsylvania 15401, USA Received 19 January 1976 An impulse approximation calculation is made on the reaction Sl V(% ~r+)sl Ti from the region just above photopion threshold to the tail of the (3, 3) resonancer Pion finn state distortions are taken into account through realistic optical potentials. Results are compared to those for plane-waveand surface-produced pions and to recent experiment. Comparison is made between surface-produced pions and pions distorted by the s-wavecompbnent of the optical p0teutlaL

The process of nuclear pion photoproduction possesses a number of features that make it very well suited as a probe of nuclear structure: For example, the sensitivity of photoproduction to particle-hole admixtures can beutilized [1 ] to provide information on ground-state correlations. In a different vein, the ~" e nature of the threshold amplitude (or = nucleon spin, e = photon polarization) leads to the possibility [2] of investigating the spin-flip strength of isospin analogs of giant resonance states, which are not easily excited otherwise, Since the photoproduced pion can be created deep within the nuclear interior, these reactions also provide a potentially powerful tool for the study of 7r-nucleus interactions over a wide range of pion energy. A great deal of attention has recently been focussed on the threshold region [7] where soft-pion theorems [3, 4] provide a guide to the single nuclear production amplitude and where n-nucleus final state interactions (f.s.i.) can be included through optical potentials whose parameters are obtainable [5] from mesic atom data. At these low energies, however, the nucleus is almost transparent to pions, the major part of the final state scattering for charged pions arising from the Coulomb potential [6]. At higher energies, in particular near the (3, 3) resonance, there is a strong pion reabs6rption in the nuclear interior. In the past, this reabsorption has been either ignored or taken into account via phenomenological models [8] whose connection to the basic n-nucleus interaction is not entirely clear. Now, however, with the availability of new, more reliable high energy photoproduction data [9], it becomes increasingly important to carry out a careful treatment of final state effects over a wide range of pion energy. Here we are motivated to carry out such a program by recent experimental work of the Lund-DESY collaboration [9] on the reaction 51V(7, 7r+)SlTi. In these experiments, total cross sections to bound final states were measured .from threshold to the (3, 3) resonance tail. These new results differ significantly from earlier experimental work on the same reaction [ 10] chiefly because of the greater care exercised in subtracting out the large (n, p) background. Furthermore, we have recently carried out a detailed analysis [11 ] of this reaction in which comparisons were made between plane wave and s u r face-produced [8] pions. The results of the present calculation will be compared with experiment and with our earlier calculation. * Permanent address. 449

Volume 61B, number 5

PHYSICS LETTERS

26 April 1976

We will work within the DWIA framework and incorporate the f.s.i, through an optical potential. We write the off-energy-shell transition matrix [12] (h = c = 1) Tfi(p ) = ~ f

d

t

~

t

t

p f (p, p )(•f, exp {i(k - p )'r a } ta( p ) ~Oi).

(1)

O~

The lab momenta of the pion and photon are denoted by p and k, respectively, ~i and ~bf are the initial and final nuclear wavefunctions. The free 77rN vertex t~ is evaluated for the a nucleons (assumed at rest) participating in the transition With all momentum-dependent c0ntributions'in t c~retained [ 11, 12]. The term

if(v, Y) =(2r)-3/2 f dr exp(ip"r) ~b-(r, p)*

(2)

is the Fourier transform of the wavefunction appropriate to the scattering state of the 7r-nucleus system. The normalization is such that if q~-(r, p) = (27r)-3/2 exp(ip-r) (no f.s.i.), the usual PWlA result [11 ], TP'W'(B) = ~ (Of, exp {i(k - p)"r a } ta(V) ¢i),

(3)

Ot

is obtained. The single-nucleon amplitude ta(p') has been shown [13] to vary slowly as a function of p. Using this result to replace ra(p') by its on-energy-shell equivalent ta(p) and expanding the photon and pion wavefunctions in partial waves we find, after averaging over photon polarizations and nuclear spin, i rfi(p)12 = ~

2(2s i +1)

E

1~ 2K + 1

qdL ,?IL ,,t.nK • ×In~Mm (--)X+IFn•(,u)il+L[(2l+l)( 27+ 1)] 1]2C~oooC_mmoCr~MYl_m(P)(J f ~6~l(Pr~)lL(kra)SynKT~]]Ji)l

.,

L,~ The F_nx~u) are combinations of single particle amplitudes [ 11, 13] which are taken from the multipole analysis of Berends et al; [14] Svn k is the tensor product [Y~(r)® on ] K with o 0 = 1 and o I = n. The ~l(pr) are the distorted pion partial waves described below. The nuclear aspects of the problem are contained in the reduced matrix elements and have been described in detail in ref. [i 1 ] . Briefly, an inert 48Ca core is assumed, the 51V ground state is written [Ji) = 1(f7/2) 3; 7/2-), and a detailed shell model calculation is carried out on 51Ti to fired those final states which are stable against particle emission. Only these states will contribute to the experimental cross sections which are measured by radioactivation techniques, The sure'in the reduced matrix element will include only valence nucleons since core-excited configurations correspond to states lying above particle threshold. The pion wavefunctions 6~l(pr) are taken to be solutions of a Klein-Gordon equation for a potential having CouIomb and strong interaction parts. The Coulomb interaction is that for a uniform sphere of charge of radius 4.63 fro. For the 7r-nucleus optical potential we chose three commonly used forms: 1) standard Kisslinger [15] ; 2) local Laplacian [16] ; 3) modified Kisslinger [17]. For all three potentials we took a Woods-Saxon nuclear matter density for the assumed equivalent proton and neutron distributions. The half-density radius e = 3.94 fm and skin thickness t = 2.22 fro, both parameters coming from an analysis [18] of elastic electron scattering on 51V. The s- and p-wave scattering amplitudes were obtained, for T,r t>50 MeV, from Sternheim and Auerbach [ 19] who Fermi-averaged the 1968 CERN phase shift data to allow for zero-point motion of the nucleons in the nucleus. Higher-order partial waves were included through the s-wave amplitude. We have averaged the results of [19] over neutrons and protons. For T~ < 50 MeV, the amplitudes were extrapolated by requiring that they fit smoothly onto the values obtained [5] from mesic atom data. The actual numerical integration of the Klein-Gordon equation was carried out using the program PIRK [20], kindly sent to us by G.A. Miller. 450

Volume 61B, number 5

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26 April 1976

The decomposition of the reduced matrix element follows along the same lines as that of our earlier calculation. We note, however, the replacement of the radial integral

Rlnl3 =fRln(r) Jl (qr) Rf(r)

r 2 dr

(5)

which could be evaluated analytically [11 ] as a function of the m o m e n t u m transfer q = Ik - pl, by the integral

RlnlL 3 =f Rln(r) 611(Pr)JL(kr) Rf(r) r2 dr,

(6)

which must be done numerically as a function of photon and pion momenta. Finally, integrating over pion angles and including nuclear recoil contributions, we find for the total cross section

m + 2k r °= 2Ji~+ 1 ~ Jdp~/p2.

P -- ~ M~(p, ui) 1 2MfEf+2Mfk-2Mfx/p2 +#2nhKrn 2K+

32rr3Mf

+~-2x/m 2 -

X l~Li L I~/(2l + 1)(27 + 1) C~cloLc~_~rnoC~mK exp {i(8l + oi)} Yl_m(Ui, 0) X Orf ll ~ 6tI (pr~)]L (kr~) S,ynK rg IIJi>

1 (7)

2.

Here **, rn and Mf are the masses of the pion, nucleon~ and residual nucleus; Ef is the excitation energy of the final nucleus; 8 l and o l are the nuclear and Coulomb phase shifts. The single nucleon amplitudes are included in M~(p, ui) = } Y u=_l+ Fn-X~) Fn-x(g)* where the angle ui appearing in the F ' s [11] and spherical harmonic is

ui = (Mf/kp) {(p2 + k2)/2Mf + Ee _ ~ + V ~

+,,2}.

(8)

Before taking up the results, we comment briefly on the matter of convergence. With f.s.i, neglected, cross sections can be obtained analytically as integrals over momentum transfer [11 ]. In the present formulation this corresponds to summing over an infinite number of pion partial waves with the replacement ei(Sl+°l)611(Pr) -+/l(Pr). At low energy (k < 160 MeV) it was found that inclusion of only the I ~< 2 partial waves was sufficient to guarantee a less than 1% discrepancy between plane wave results based on eq. (7) and on ref. [11]. By k = 230 MeV this number had increased to l ~ 6 and by k = 400 MeV, to l <~ 12. It is therefore essential, especially at higher energies, to be certain that one is including a sufficient number of partial waves for good convergence. In fig. 1 total cross sections to all bound levels in 51Ti are depicted under various assumptions for the g-nucleus final state interactions. Although calculations were carried out for all three optical potentials, we illustrate in curve (d) the result using the Coulomb plus modified Kisstinger potential [17] (the "complete potential"). The result for the other two potentials were qualitatively similar. We note first that at low energy (k ~< 180 MeV) the complete potential gives results virtually identical to the predictions of the surface production model. In both cases, the attenuation factor o/o (plane wave pions) ~ 0.80, consistent with earlier findings at threshold [7, 21]. At higher energies, however, the two results diverge markedly. The surface production model yields an attenuation factor which remains relatively energy-independent from threshold to the tail of the (3, 3) resonance. There is no evidence of the strongly energy-dependent p-wave absorption which builds up rapidly [19] from threshold, peaking for the rrN case at photon energies ~290 MeV. By way of contrast, the cross section with the complete potential is very sensitive to this reabsorption effect. At T~r = 40 Me'Q, e.g., the imaginary part of the p-wave scattering amplitude cuts down the cross section by almost half its value for plane wave pions. At T~r = 60, the fraction is ~1/6. Curve (b) represents the cross section including only the s-wave component of the optical potential. The similarity between this curve and curve (c) for the surface production model is striking and indicates that the latter may provide a fairly good representation of final state distortions in the region just above threshold where p-wave 451

PHYSICS LETTERS

Volume 61B, number 5 o

F"

'1

I

F

(o)

I

-g

" I

,,-

I

I

I

I

I

-T'""|

.....

,,:: ................ , ..... ".,,

I

."

f-'

26 April 1976

/

"~.

"~ X',\

A.

o_..TIo

!

I 220

t

2 !~

I

i 300

i

1 , 340

I

I 380

Incident Photon Energy(MeV)

i

I 420

Fig. 1. Total cross sections for the reaction Slv(% ~r+)SlTi. Curve (a)plane wave pious [ 11] ; (b) pions distorted by swave component of t~ptieal potential; (c) pions calculated via phenomenologicalsurface production model [8, 11 ] ; (d) pions distorted by full Coulomb plus modified Kisslinger opficalpotential [!7] ; (e) experiment [9].

effects are still small. Curve (b) is of interest f o r a second reason2 It is occasionally claimed that the calculation o f Saunders [13] on the analog reaction 88Sr(% I r ' ) 8 8 y provides evidence that final state distortions play a small role in charged pion photoproduction. Although that calculation did lead to attenuation factors of 0 . 7 - 0.8 at a photon energy of 230 Mev, the optical potential used [22] contained only s-wave contributions. Our s-wave resuits are completely consistent with Satmders' findings, but the complete potential paints a far different picture. A comparison of our results with experiment [9] shows that the p-wave absorption is too strong to allow for good agreement. We make three remarks concerning this. First, the measurements involve difficult background subtractions and the use of a smoothing program to obtain cross sections from the bremsstrahlung yield curves. Although this precludes the possibility of discerning details in the cross sections, the errors in curve (e) should b e no greater than +-20%. Second, the optical potentials we have used have been fairly reliable in predicting the results of pion scattering on light nuclei. Since there are no elastic data in the mass-50 region, we must employ the theoretical optical parameters determined from the 7rN scattering data and scale the potentials proportionately. This may tend to overestimate the strength of the optical potentials. Finally, we note that, given the f.s.i. = 0 predictions and the experimental curve, no realistic optical model will be likely to bridge the gap since the curves differ most in the low-energy region where the distortion effects should be small. • A more detailed analysis on this and other charged pion photoproduction reactions will be combined with the experimental results of the Lund-DESY groupand submitted elsewhere [23]. We are grateful to G. Rawitscher for an interesting discussion and to G. Miller and R. Eisenstein for help with the program PIRK. One o f us (N.F.) is indebted to Dr. Nits Robert Nilsson and the Nordic AcCelerator Committee for thesupport which made it possible for him to spend the summer at Lund working with B. Forkman, G. Jonsson, and other members of the photonuclear group.

References [1 ] K.S. Rao and V. Devanathan, Phys. Lett. 32B (1970) 578; Can. J. Phys. 53 (1975) 1292. [2] H. Uberall and A. Nagl, Phys. Lett. 45B (1973) 99. 452

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[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

PHYSICS LETTERS

26 April 1976

M. Ericson and M. Rho, Phys. Rpts. 5 (1972) 58. M. Mo1:eno, J. Pestieau and J. Urias, Phys. Rev. C12 (1975) 514. M. Krell and T.E.O. Ericson, Nuel. Phys. B l l (1969) 521. C. Tzara, Nucl. Phys. B18 (1970) 246. J.H. Koch and T.W. Donnelly, Nucl. Phys. B64 (1973) 478; F. Cannata et al., Can. J. Phys. 52 (1974) 1405; A. Nagl, F. Cannata and H. fQberall, Phys. Rev. C12 (1975) 1586. V. Devanathan et al., Nucl. Phys. B2 (1967) 329; V. Devanathan, G. Prasad and K. Rao, Phys. Rev. C8 (1973) 188. Lund-DESY collaboration: G.G. Jonsson, pvt. comm. G. Nydahl and B. Forkman, Nucl. Phys. B7 (1968) 97; I. Blomqvist, G. Nydahl and B. Forkman, Nucl. Phys. A162 (1971) 193. N. Freed and P. Ostrander, Phys. Rev. C l l (1975) 805. M.L. Goldberger and K.M. Watson, Collision theory (Wiley, New York, 1964). LM. Saunders, Nucl. Phys. B7 (1968) 293. F.A. Berends, A. Donnachie and D.L. Weaver, Nucl. Phys. B4 (1967) 54. 103. L.S. Kisslinger, Phys. Rev. 98 (1955) 761. H.K. Lee and H. McManus, Nucl. Phys. A167 (1971) 257. G.A. Miller, Phys. Rev. C10 (1974) 1242. G.A. Peterson et al., Phys. Rev. C7 (1973) 1028. M.M. Sternheim and E.H. Auerbach, Phys. Rev. C4 (1971) 1805. R.A. Eisenstein and G.A. Miller, Comp. Phys. Comm. 8 (1974) 130. E. Boric, H. Chandra and D. Drechsel, Nucl. Phys. A226 (1974) 58. R. Frank, J. Gammel and K.M. Watson, Phys. Rev. 101 (1956) 891. !. Blomqvist et al., to be published.

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