UM-P-80/65.
ISOSPIN EFFECTS IN PION-NUCLEAR SCATTERING
I. Morrison and K. Amos
School of Physics, University of Melbourne,
Parkville, Victoria 3052, Australia
Abstract: A simple isospin model of inelastic pion-nucleus scattering is discussed and quantitative results compared with recent experiments on light nuclei. 1. Introduction:
Intermediate energy pion bearcs, especially with incident energies near the (3,3) resonance, are now frequently used as sensitive probes of nuclear transition densities1-1*) • The (3,3) resonance energy region is particularly interesting, since by assuming (3,3) dominance and ignoring any charge splitting in the pion-nucleus T-matrix5^, the cross section ratio RLo(Tr~)/a(ir )] to a specific nuclear level tends to values of 1/9 and 9 for scattering fron bound protons and neutrons respectively. Therefore pion reaction data could be used to differentiate between proton and neutron transition densities in nuclei and to a higher degree than has been achieved by analyses of electromagnetic decay of (or inelastic hadron scattering to) discrete states of nuclei6} With this possibility in aind, Lee and Lawson7' used a general pion-nucleus D.W.I.A. scattering theory and analysed inelastic pion scattering data from the excitation of discrete, collective 2 states of 180. As with direct reaction hadron scattering to (and electromagnetic excitation of) such collective states, the effective charge concept had to be enployed in the restricted basis used, to explain the magnitudes of observed pion cross-sections. However strong state dependent effective charges7' change the ratio R, from that expected with a pure pion-nucleon t-matrix, in a complex manner. Clearly, therefore, transitions which are better determined by the usual model prescriptions are required to make good use of the pion reaction sensitivity. Transitions to states of stretched, unique, particle- hole excitation are candidates and especially if they are of unnatural parity
and therefore do not support a collective enhancement. •>
Data from inelastic pion scattering to such states has been8-10 reported recently and we analyse this data using a sinple isospin reaction model avoiding the complex numerics of a full DWIA analysis but nevertheless providing a gross test of the pion nuclear scattering reaction mechanism near the (3,3) resonance.
2. Description of the Model:
The isospin dependence of pion-nuclear scattering
matrices, T.f, is displayed by
Tif "<2VVV,t,1MriiT2V (1) where M_ and M_ are the initial and final state pion isospin
projections respectively and T2(Hr ) and T^CM- ) are the isospin quantum numbers of the initial and final nuclear states. Clearly, for inelastic (noncharge-exchange) pion scattering VL and M_ equate to M_ and H,, respectively. Then, if the pion-nucleus interaction is of one body form, by introducing the total isospin, T , we can reexpress E Tif= Aw V^VTJW in which <...!}.> are the nuclear, one body coefficients of fractional 3. parentage (cfp) connecting the initial (finaP mass A nuclear states to an intermediate, mass A-l, nuclear state with isospin T'. Assuming that the pion-nucleon interaction is independent of the angular momentum states and recoupling in isospin, we obtain T 1 if ^aBYT'Ts^o '!^ 3 1 s s s f ^ ] fl^ ] t( ) m 3 [rTjJ iT'ToT2J l > with S = (25+1) and in which tl ' is the reduced pion-nucleon scattering amplitude for isospin s(*5 or 3/2), namely t(s) = Clearly in this model we have suppressed all specific angular momentum dependence of the reaction, whereas in a full analysis, the complete angular momentum dependence of the nuclear states, the nucleons, and the pions (via a partial wave expansion) must be specified7^• Thus Tif = AET,T <1 T„ H, H, |ToMo> O 3 H <1 T2 H^ H^TM^ s T2 T% t(S)c(T,;T2T ) (5) (T.T0Tj(i'T0T2) '* with C(T»;T2T,J = IQ6Y being the isospin parentage overlap of the initial and final nuclear states. This overlap may be formally evaluated by using standard shell model techniques and can be expresses in terms of ihe .ir-o.^calar and isovector transition densities between the two nuclear states7). In factoring out the parentage overlap we have neglected the specific angular momentum dependence (a,B) of the scattering amplitude, t .However if the transition density is specified (or dominated) by a single (a,f$) pair, as is usually the case in the excitation of high spin, stretched particle-hole configuration, then this factorization and therefore Equ . (5) is "exact" to within an overall normalisation. Thus our model is particularly suited for an analysis of recent inelastic pion scattering experiments, on 28 10 13c 5,8)f 160 9) and Si ), involving scattering to the stretched particle-hole states 9/2*(9.5 MeV), 4"(circa 18 MeV), and 6~(11.58 and 14.36 MeV) respectively. The unnatural parity transitions are of particular interest in that hadron scattering analyses^J indicate that collective enhancement (effective charge) of transition strengths are not required and therefore the ratio R should be defined via Equ . (5) by isospin arguments alone. 5. The form of Equ . (5) is such that the scattering matrix for inelastic scattering from self conjugate nuclei (T2 = 0), assuning (3,3) dominance, is •X3/2) Tif - t^/3 £«T >M,/26T ^ C7) + For excitation of states with pure isospin, therefore, the u~ cross-sections are the same with the isoscalar transitions being fourfold the isovector ones. The experimental ratios from both the 4" states in 160 and from the 6~ states 28Si are however smaller than 4, with the reduction in the 160 data being interpreted by Holtkamp et al.12^ as a reflection of isospin mixing amongst the 4" states. Barker et al.13) have shown, however, that a fully consistent description of these 4" excitations requires not only isospin mixing of the states but also non-negligible s=h. admixtures in the transition amplitudes for pion nucleon scattering. For self conjugate nuclei, the intermediate isospin in our model, T', has the value of h implying, in the absence of isospin mixing in the final nuclear states, all transitions are of purely isoscalar or of isovector type. For non self-conjugate nuclei T2 and T4 are non zero hence a sum over T' values of 1z±h must be made in general. The precise values of the cfp factor C(T'=0,1) then affect results. For the case of 13C T' is 0 or 1 and with the incoming (and outgoing) pion isospin projection now defined as M , the scattering amplitudes to Ti» = h final states are 6. for T' = 0 and TJJ? = AC(1;^0 [(l+M^H/S t(J*)+8t(3/2)/54 •(2-M^) (4/2 t(3d+5t(3/2))/54] (9) for T' = 1. These simplify if (3,3) dominance is assumed to give (T2=T„=y T if » A[c(0;¥d(2-M¥)/6 •C(l ;y0 [ (1+Mn) 8+(2-M^) 53/S4 J 3/2 *t< > (10) The difference between TT ,;il.loving strength to a ?;iven level depends not only upon the pion charge but also upon the cfp overlaps, C(T'). Specifically, the asymmetry in pion scattering defined by A = [a(Tr)-aU+)]/[a(Tr")+o(TT+)] (H) 2 in which O(TT) are proportional to |Tf. | , give the results A = +0.80 C(0) = 1 C(l) = 0 A = -0.324 C(0) = 0 C(l) = 1, (12) with the value of A being very sensitive to relative changes in the values of C(T') and biased towards any admixtures of C(l). / Finally we reiterate that xf many (aB) pairs contribute to scattering amplitudes for the excitation of low lying collective states, our nodel is approximate, as it neglects the details of the relative contribution of these pairs. Our model should therefore be used with a '"collective" state only as a qualitative guide to explain any isospin dependence. Nevertheless this may be a useful technique as the more complex DWIA analyses have problems due to the collectivity of such transitions7'. 3. Results: The asymmetries, defined by Equn. (11), for transitions to low lying states in 13C have been recently measuredi>8>and are reproduced in figure 1. The limiting values of our model assuming (5,5) dominance are also shown. Clearly positive parity excitations are synonomous with positive pion asymmetries and vice versa. This property is anticipated as the pertinent shell nodel calculations11*) reveal significant parentage T' = 1 for the negative parity states in 13C below 12 MeV excitation while the positive parity states result from the weak coupling of an extra core (s-d shell) neutron with the low lying T = 0 levels in 12C. Hence the positive parity levels have dominant T1 = 0 parentage and therefore positive asymmetries, and vice-versa for negative parity states with large T' = 1 parentage amplitudes. Of all the transitions, only the 9/2+ (9.3 MeV) excitation agrees with our simple model prediction. Of all the [ 8. states given, this state is the only example of a stretched (single a,B) transition. The transitions to other levels involve Multiple o,B pairs, for which our model is only an approximation. Nevertheless the qualitative agreement is good in view of the sensitivity of the asymmetries to small changes in the ir~ scattering cross sections. The excitation of high-spin unnatural parity states in self conjugate nuclei, such as in the recent experiments on 160 *' and 28Si10), is of more direct interest in that our model is "exact" so that analyses of the ••-olevar.t dat? w> M tost the (3,7,) .l>»ii: :ice • assumption, namely that for good isospin, the isovector to isoscalar transition ratios are | ± ± R+ = o(ir ,AT=0)/o(Tr ,AT=l) = 4 (13) The 4" states in 160 lie close together near 18 MeV excitation and some isospin mixing occurs to affect their use in such a test13', but the two 6 states in ,8Si are separated by some 2.4 MeV and therefore 10 are expected to be very pure. As the measured values ' for R+ for the 6~ states (11.58 MeV isoscalar, 14.36 MeV isovector) are 1.5 and 1.7 (± 0.4) for R+ and R respectively, we anticipate a significant breaking of the (3,3) dominance assumption. The specific forms of the 28Si 6" scattering amplitudes 9. are (,j) (3/2) Tif = A/3[(/2t *2t )*T||0 • N (-/StW*t(V2))«_ .] (14) If we assume that the two 6~ states have pure isospin and allow a breaking of the (3,3) dominance assumption by including t^-aW*. (15) then the ratios are predicted by i9 2 i6 2 R+ = |/2oe *2! /|-/2ae *l| , (16) hence 2 2 R+=R =R= (4+2a +4/2aCos8)/(l*2a -2/2aCose) (17) We note that the result R =R is independent of the specific values of (a,6) which is supported within experimental error. But the magnitude R is very dependent on (a,;) and a ratio of 4 is obtained, not only with a=0 [the (3,3) dominance assumption] but also if a has the value 2/2 Cos9. However, using ar 'empirical' value of R = 1.5, the mixing amplitude 'a' is then determined by the quadratic equation 2cx2-14/2aCos8-S=0 which, for the physical constraint |a| so that the tv^ amplitudes are required to be at least 251 of those assumed in a (3,3) dominance picture. A similar result was found in the analysis of the excitation data for pion scattering to the 4~ states in 160 allowing for isospin (and configuration) mixing in the state description13)• We now consider isospin eixing in the 28Si 6~ states by using |*6-(12 MeV)> = Cos(e)J6~;T=0>*Sin(e)|6~;T=l> |»6-(14 MeV)>= -Sin(e)|6~;T=0>*Cos(e)|6~;T=l> (18) implying T+(12 MeV) = Cos(c)T+(T=0>Sin(e)T+(T=l) and T±(14 MeV) =-Sin(e)T+(T=0)*Cos(e)T(T=l), (19) using Equ . (14) (y (3/2) T±(12)=/2[Cos(e)*Sin(E)3t *[2CosCc}±Sir.(£)]t , (20) and (!i (3/2) T±(14)=/2[-Sin(e)iCos(e)]t ^[-2Sin(e)±Cos(€)]t . (21) The (3,3) dominance assumption then gives the ratios 2 ? R+ = {2Cos(e)+Sin(e)} /(2Sin(e)*Cos(c)} and R„ = 4 for pure isospin (e=0) as known from our earlier discussion. If we consider the difference between the ratios, independent of their actual individual values, *>v using the fraction difference n 2 2 f = 2(R_-RJ/(R »R+) = -40Sin(c)Cos(c)/[25Cos (c)Sin (c)*4], a variation with isospin nixing ancle results that is shown in figure 3. The experimental fraction difference lies in the range -0.33 to 0.47 which coincides with an isospin nixing angle range of 1.92 to -2.72 degrees. Thus the possible difference in the ratios, R, under the (3,3) dominance assumption needs little isospin nixing between the two 6~ states in 28Si, although the individual ratios will still be in the region of 4. If we do not aake the (3,3) doninance assumption but again use the relation LEqu". (l»)j between the t^** and t1""' amplitudes, the isospin nixing transition amplitudes are ie T+(12) = {/2[Cos(E)*Sin(e)3ae *[2Cos(e)±Sin(c)])X, and l, T+(14) = {/2l-Sin(c)H:osi«:)]oe '*[-2Sin(c)±Cos(e)3}X (24) Thus the cross-section ratios provide a pair of constraint equations from which two of the three variables a, 9 and c car. be defined from the third. Using ratios, R>t of 1.5 and 1.7 respectively and demanding |i| to be less than 1 for 3 varying fron 0° to 60°, we find a can vary between -0.23 and -0.43 while the isospin nixing angle t is only -1 to -2 . Fixing 0 as zero and taking extrene limits for the ratios, R+, of 1.3 and 2.0, a remains at a value of -0.23 while L increases to a value of -3 . Therefore the pion scattering data to the ?9Si 6* states are consistent with very pure isospin states in concurrence with the hadron scatterir.g data .inalvses11. The reduction in R (from the value 4) and 12. differences between the ratios, R+, are attributed in the model to substantial t amplitudes. We now correlate these results with the 13C reaction analyses. No longer using the (3,3) dominance assumption in an analysis of the ir~ asymmetry of scattering to the 9/2'' ' (9.5 MeV) state and choosing an isoscalar (T'=0) cfp factor only, we find l6 TiA#C{(l+m7T)/3/2}ote +(2-m7t)/6] from which it may be deduced that A = [O(TT )-C(TT )]/[O(TT )+O(II )] i9 2 i8 2 = [9-|2/2ae +l| ]/[9+|2/2c.e +l| ] = [8-8a2-4/2oCos9]/[l0+8a2+4/2Cos03 (25) Clearly, an asymmetry of +0.8 results if 8a2 + 4/2aCos8 = 0 or, equivalently, that a vanishes [(3,3) dominance] or equals Cose/2. In the latter case, the 28Si and 13C results are consistent if a=-0.4 and 9=55 . For other values of A allowed by the experimental uncertainty5) for the 9/2+ transition, a spectrum of values is possible and is shown in the table for asymmetries in the range 0.7 to 1.0. 13. Using the +0.8 value for 13C, and the 28Si isovector (IA o to isoscalar 6~ ratio, a t amplitude being -0.4 exp(i55 ) of the t^ ' amplitude is the result which is also consistent with the 160, 4 transition analyses13) where a real scaling of -0.24 was deduced. Conclusions: A simple isospin model of the pion-nucleus T-matrix has been defined from which the asymmetries in inelastic pion scattering from a3C and isovector to isoscalar ratios of pion inelastic scattering to the 4 states in 160 and to the 6 states in 28Si can be explained in a simple but consistent manner. All the measured asymmetries in the 13C scattering are explained qualitatively by this model whilst quantitative results were obtained for the special case of excitation of the stretched particle hole 9/2* (9.5 MeV) state. To obtain the consistent agreement between an asymmetry +0.8 in the 9/2 excitation in 13C and the measured isoscalar to isovector transition ratios of 1.5 in 28Si, a significant component of isospin h effective pion-nucleon scattering amplitude is required in the reaction analyses. The required ratio [t /t ] is larger than would be obtained from a partial wave analysis of free pion-nucleon scattering15) and such enhancement we expect is due in part to off-shell effects in the pion-nucleus16) system and in part to kinematic-partial wave mixing in reaction amplitudes when the pion-nucleon t-matrixes are folded into the pion-nucleus frame. ' 14. Figure captions Fig. 1: The measured asymmetries, A, from pion inelastic scattering to low lying, discrete states in 13C. The (3,3) dominance model predictions for positive parity (isoscalar parentage) and negative parity (isovector parentage) are shown for comparison. Fig. 2: The t^ magnitude (a) and phase (9) values (relative to the t ' amplitudes) required to predict an isoscalar to isovector transition ratio of 1.5 for both it- inelastic scattering to the 6 states in 28Si. Fig. 3: The fraction difference 'f' in it" scattering ratios to the 6" states in 28Si, assuming (3,3) dominance and allowing isospin mixing, as a function of the isospin mixing angle, e. 15. Table 1: Asymmetries for 13C(Tr±,Tr~);9/2^ (9.5 MeV) 0° 30° 45° 60° -0.5 .96 .88 .80 .70 -0.4 1.00 .93 .86 .77 -0.3 .99 .95 .89 .89 -0.2 .95 .93 .89 .84 i-0.1 .89 .88 .86 .82 i° .80 .80 .80 .80 |0.1 .69 .70 .72 .74 ! 0.2 .57 .59 62 .6b 16. References 1) J.P. Egger et al., Phys. Rev. Letts. 39, 1608 (1977). 2) S. Iversen et al., Phys. Rev. Letts. 4£, 17 (1978). 3) S. Iversen et al., Phys. Letts. 82B, 51 (1979). 4) C. Lunke et al., Phys. Letts. 78B, 201 (1978). 5) D. Dehnhard et al., Phys. Rev. Letts. 4_3, 1091 (1979). 6) K. Aiuos, A. Faessler, I. Morrison, R. Smith and H. Muether, Nucl. Phys. A304, 191 (1978). 7) T.S.H. Lee and R.D. Lawson, Phys. Rev. C21, 679 (1980). 8) E. Schwarz et al., Phys. Rev. Letts. 4_3, 1578 (1979). 9) H.A. Thiessen, in 'Proceedings of the 8th International Conference on High Energy l'hysics and Nuclear Structure', Vancouver (1979) . 10) C. Olmer et al., Phys. Rev. Letts. /KS, 612 (1979). 11) K. Amos, J. Morton, I. Morrison and R. Smith, Aust. J. Physics 3_1_, 1 (1978). 12) D.B. Holtkamp et al., Phys. Rev. Letts, (to be published). 13) F. Barker, R. Smith, I. Morrison and K. Amos, J. Phys. G (to be published). 14) S. Cohen and D. Kurath, Nucl. Phys. 73_, 1 (1965). 15) V.S. Zidell, R.A. Arndt and L.D. Roper, VPI preprint (1979). 16) W.R. Gibbs, A.T. Hess and W.B. Kaufmann, Phys. Rev. C13, 1982 (1976). 17) T.S.H. Lee and F. Tabakin, Nucl. Phys. A226, 253 (1974). s s V s 0.8 s t V V s 0.6 5/ \ s s 04 s s % 012 s 0 s H5k s s y -0.2 - s 2 V N V h -CU y2 8 ,3 Ex. ( o (MeV) 20 40 60 80 d (degrees) 1.0 0 0.8 f 0.6 — / 0.4 — / 0.2 — / / I I I 2 4 6 8 -£ (degrees)