-~ Nuclear JctsyhP A244 (1975) 365-434; (~) dnalloH-htroN onthhlbuP ,.oC madretsmA Not to be reproduced by photoprint or microfilm without written permission from the publisher FEW-NUCLEON SU(3) PARENTAGE COEFFICH~.. NTS AND ~-PARTICLE SPECTROSCOPIC AMPLITUDES FOR CORE EXCITED STATES IN s-d SHELL NUCLEI t K. T. HECHT and D. BRAUNSCHWEIG scisyhP ,tnemtrapeD ytisrevinU of ,nagihciM nnA ,robrA naaihciM ,40184 ASU devieceR 3 December 4791 Abstract: The SU(6)/SU(3) factors of few-nucleon fractional parentage coefficients in ls-0d shell nuclei are calculated directly without being generated recursively from one-nucleon c.f.p. Tabulations are given for x-nucleon reduced matrix elements, (x = 4, ,3 2, ,)1 connecting states of high SU(3) symmetry, where the x-nucleon states are limited to states of totally symmetric space and oscillator quanta symmetry, i.e. SU(3) representations (80), (60), (40), (20) for states of space symmetries 14, 3, 2, .1-I Together with the SU(3)/R(3) factors calculated by Draayer, these reduced matrix elements make it possible to predict spectroscopic amplitudes for reactions in which the x-nucleon groups are transferred in unexcited (0s) internal states. In a specific application, ,,-particle spectros- copic amplitudes are calculated for core-excited states in s-d shell nuclei, reached by the transfer of(0p)- t (ls0d)- ,3 or (0p)- 1(2 s0d)- 2 groups in pickup reactions and (ls0d) 3(lp001 or (ls0d)2(lp0f) 2 groups in stripping reactions, where states in the final residual nuclei are approximated by SU(3) strong coupling states, (free of spurious c.m. excitation), corresponding ot the largest possible intrinsic deformations in these nuclei. The effect of a difference in elcitrap-t~ size parameters in projectile and residual nuclei si .dessucsid 1. Introduction Recent direct multi-nucleon transfer reaction experiments with Li or heavy-ion projectiles have stimulated a number of new theoretical calculations 1-3) of -t~ particle spectroscopic amplitudes. Besides a number of specific cluster model calculations 1,4, 5), general theoretical formulations in the framework of the harmonic oscillator shell model have now also been brought to a state of development 1- a) which makes specific calculations feasible. In all the recent formulations it is assumed that the or-cluster is transferred in an unexcited (0s) internal state, an assumption which is justified not only for Li induced reactions but also for direct transfer reac- tions between heavy ions because of the surface nature of such reactions 2). Cal- culations in the framework of the j-j coupled shell model have been carried out by Kurath and Towner 2) who relate the ~-spectroscopic amplitudes to a sum of coupled two-neutron and two-proton spectroscopic amplitudes by means of a formula which is not only convenient for calculations in heavy nuclei but directly relates properties of ~-transfer reactions to the more familiar properties of two-nucleon transfer t Supported by the US National Science Foundation. 563 June 5791 663 K.T. HECHT AND D. BRAUNSCHWEIG reactions. Their calculations also show that there is much fragmentation of the ~- transfer strength in a good j-j coupling nucleus among the many possible config- urations (JlJ2J3J4) of a given major oscillator shell. The SU(3) shell model calculations of Ichimura, Arima, Halbert, and Terasawa ,)1 on the other hand, show that the •-strength can be highly concentrated in a few rotational bands in a good SU(3) nucleus. In s-d shell and lighter nuclei it is therefore advantageous to calculate ~- particle spectroscopic amplitudes in the framework of the SU(3) representation of the harmonic oscillator. In this framework the overlap between an ~-cluster wave function and a four-particle shell model wave function takes an extremely simple form ,)1 and the calculation of ~-particle spectroscopic amplitudes is immediately reduced to a calculation of n ~ (n-4) particle parentage coefficients for the n valence nucleons. In the usual method of calculation such four-nucleon c.f.p, are generated recursively from one-nucleon c.f.p. Although the process is straight- forward, it is tedious and complicated by the fact that the many needed intermediate state one-nucleon parentage coefficients are not available in tabulated form in the SU(3) scheme. As a result, ~-particle spectroscopic calculations based onthe SU(3) model have been limited 1,6) to nuclei near 160. Although full four-nucleon c.f.p. have not been available for s-d shell nuclei, the SU(3)/R(3) parts of these c.f.p, are readily available through the work of Draayer and Akiyama .7 s). A recent extensive tabulation by Draayer )3 gives the full angular momentum dependence of these factors and makes possible a prediction of the relative ~-transfer strengths to dif- ferent members of a rotational band (with angular momentum projection Kj), provided the states of the rotational band are pure (or relatively pure) in their SU(3) symmetry quantum numbers. Draayer's tabulation of the angular momentum dependent (SU(3)/R(3)) factors have therefore reduced the problem to the calculation of the SU(6)/SU(3) factors of few-nucleon c.f.p, in the s-d shell. These factors are needed to predict absolute values of ~-spectroscopic amplitudes, to assess quan- titatively the effects of SU(3) representation mixing in the ground state rotational bands, and to compare the ~-transfer strengths to excited rotational bands with those for the ground state band. The latter would be particularly difficult to estimate in the framework of the j-j coupled shell model if the rotational bands are based on core excited states, such as the low-lying negative parity rotational bands in s-d shell nuclei. For ~-particle spectroscopic amplitudes to core excited states in s-d shell nuclei, the SU(3) scheme is therefore not only a powerful calculational tool but is vital to furnish a reasonable description of the states themselves. It is the purpose of this contribution to exhibit a method by which the SU(6)/SU(3) factors for few-nucleon c.f.p, in the s-d shell can be calculated without being generated recursively from one-nucleon c.f.p, and without a chain calculation. A brief discus- sion of spectroscopic amplitudes for few-nucleon transfer processes and their relation to few-nucleon parentage coefficients is given in sect. 2. Some of the details of the method of calculation are given in an appendix together with a fairly extensive tabulation of four-, three-, two- and one-nucleon reduced matrix elements con- SU(3) PARENTAGE COEFFICIENTS 763 netting states of high SU(3) symmetry (large SU(3) quantum numbers (2#)). In these tabulations the four-, three- and two-nucleon states corresponding to the transferred cluster are limited to states of totally symmetric space and oscillator quanta sym- metry, i.e. space symmetry quantum numbers 4, 3, and 2, respectively, with corresponding SU(3) symmetry (80), (60), and (40). Sect. 3 takes up the calculation of ~t-particle spectroscopic amplitudes for core excited states in s-d shell nuclei; i.e. et-particie spectroscopic amplitudes for the transfer of (0p) -1 (ls0d) -a, (0p)-2(ls0d)-2, ... clusters in pickup reactions, and (ls0d)a(lp0f)l,(ls0d)2(lp0f)2, ...clusters in stripping reactions. The general formulation is given for both the SU(3) weak-coupling 9- ix) and SU(3) strong- coupling t )2 models. Numerical estimates, however, are based on the SU(3) strong- coupling model since it is somewhat simpler and can be expected to give a good estimate of ~-transfer strength to core excited states in s-d shell nuclei. It is also closely related to the generalized quartet model of Harvey )31 which can be used as a guide to the most-likely low-lying particle-hole excitations in such nuclei. The SU(3) strong-coupling scheme has an additional advantage. Most states with large values of the SU(3) quantum numbers (2#) are entirely free of spurious c.m. ex- citations. In those few cases where spurious c.m. excitations must be considered, the SU(3) strong-coupling scheme also furnishes the simplest calculational frame- work for the elimination of such excitations 17). The few states of spurious c.m. excitation which are needed in this investigation, are tabulated in an appendix, together with a discussion of the limits on the SU(3) quantum numbers 2, # which delineate the regions free of spuriosity. The four-nucleon c.f.p, tabulated in this contribution are limited to those needed for the calculation of ~-transfer amplitudes under the assumption that the size of the transferred ~-cluster is the same in both projectile and residual nuclei. The effect of a difference in size on ~-spectroscopic amplitudes has been discussed bv lchimura et al. ~). Since their formulation involves relatively complicated Talmi-Moshinsky recoupling transformations, a simpler derivation leading to somewhat more general results is given in an appendix. However, these results in no way change the con- clusions of ref. )1 that the differences in ~-cluster size should lead to only small ef- fects on observable phenomena in s-d shell nuclei. 2. Few-nucleon spectroscopic mplitudes The differential cross section for the direct x-nucleon transfer reaction A(a, b)B, with B = A + x, and a = b + x, is in general given by a coherent superposition of structure (spectroscopic) factors, B, and kinematic (reaction mechanism) factors, ft. Adhering strictly to the notation of ref. 2): d°'(A ~ B)= #'/'tb Kb 2Je+l ~.I E t~s~lff~bs~B "r~ )1( dr2 (2nh2) 2 ~K (2JA+I)(2j,+I) :- M OLOE " 863 K.T. HECHT AND D. BRAUNSCHWEIG It is assumed that the states of the transferred x-nucleon group can be described in the framework of the harmonic oscillator shell model, and Q = 2N+L gives the number of oscillator quanta for the relative motion of the x-nucleon cluster with respect to nucleus A. If the intrinsic state of the transferred x-nucleon cluster has an angular momentum, j~, then Jx =ix+L, with similar definitions for Q, E, ,~J for the x + b nucleon projectile. The structure factors B aL' LQ are given in terms of the two spectroscopic amplitudes A(B ~ A + x) and A(a ~ b +x) and in general involve a sum over the intrinsic states of the transferred x-nucleon group and angular momentum recoupling coefficients for details, see ref. 2). If the x-nuc'I~on group is transferred in an unexcited internal state with zero intrinsic angular momentum, such as the (0s) internal state of an unexcited or-cluster, the structure factor, B, is given by a simple product of the two spectroscopic amplitudes. In this case the dif- ferential cross section for the transfer reaction is also given by a product of a single spectroscopic factor and a reaction mechanism factor, as in the case of a direct one- nucleon transfer reaction (provided the target and residual nuclear states are not mixtures of core excitations with different particle-hole numbers). The present investigation will be concerned solely with the spectroscopic am- plitudes A(B---, A+x). These spectroscopic amplitudes are determined by three types of factors. Again in the notation of ref. ,)2 see also ref. 1), the spectroscopic amplitude A(B ~ A + x) is given by Amm(B ~ A+x) = ~,B---~, ~' )"J)A(OI~Jr*zII~:)B(O( F x (¢i.t.(~x)~NL(gx)l~(~x)). (2) The first factor, given by the mass ratio, B/(B-x), comes from the generalized Talmi-Moshinsky transformation which relates the wave function em.(rx_A), describing the relative motion of the x-nucicon cluster with respect to the center of mass of nucleus A, to the wave function ,)xR(.m~¢ describing this motion with respect to the center of the harmonic oscillator potential well .)1 (Since Q = 2N+ L may be large this factor can be important, although it is often ignored in two-nucleon transfer processes in heavy nuclei.) The second factor, the double-barred matrix dement, is the reduced matrix element of an x-nuclcon creation operator, ,*X where the x creation operators are coupled to total angular momentum J~ and are specified by additional quantum numbers F. The last factor, the "G" factor of refs. ,1 ,)2 is the overlap of the x-nucleon duster wave function and the x-particle shell model wave function, specified by quantum numbers, FJ~. The coordinates, ~, describe the internal degrees of freedom of the x-nucleon cluster, whereas the shell model coordinates, ,x~ describe the motion of these x nucleons relative to the well center. If shell model wave functions are specified in j-j coupling, this overlap is different from zero for many possible x-particle shell model states, and the spectroscopic amplitude involves a summation over many states, F, specified, for example, by the )3(US PARENTAGE STNEICIFFEOC 963 single-particle quantum numbers nl lljl ..... nxlxj x, with additional intermediate- coupling angular momentum quantum numbers such as J, 2, J34 see ref. 2). In the framework of the SU(3) representation of the harmonic oscillator shell model the overlap for the x-particle group takes a very simple form. If the x-particle cluster is transferred in an unexcited (0s) internal state, and if the oscillator size parameters for the x particles is assumed to be the same in the a-nucleon projectile and the B- nucleon residual nucleus, the above summation over F collapses to a single term. For x = 4, this overlap or G-factor has been given by Ichimura et al. ,)1 and has the simple form , V 'Q F'l ,4 G = 14aq,!~2iqa!q4i_ Latb!c!d ! 6U-lt,1fs0fro6Q,2s+Lfaa6,o )3( for four-nucleon transfers in the configuration qlq2qaqa-l-q~qCw~ (with a + b + c + d = 4), where qi = 2ni + ~l is the number of oscillator quanta of the ith transferred particle, and Q = ql+q2+q3+q4=2N+L is the total number of oscillator quanta in the transferred cluster. For the transfer of a (ls0d)a(lp0f) 1 four-particle cluster, for example, a = 3, b = ,1 c = d = 0, with ql = 2q = 3q -- 2 ( = qu), q4 = 3 ( = qv); and Q = 9. In eq. (3), f stands for the space symmetry quantum numbers, given in terms of the usual partition numbers; (2#) are the Elliott SU(3) quantum numbers; the notation follows that of refs. ,1 a). In eq. (2), a summation over more than one state F will occur only in the extremely rare cases when the final state in nucleus B can be reached from the initial state in nucleus A by more than a single configuration with the same Q; e.g. the configurations 1( )dOs 2 (lp0f) ,2 and (ls0d)3 (2sld0g) ,1 both with Q = 10; the K= 0 ÷ band )51 in 2°Ne with band head centered on the wide level at 8.3 MeV, may be such an example; see ref. )1 and sect. 3. For x = 3, (again for a three-nucleon cluster transferred in a (0s) internal state with oscillator size parameter properly matched between a- and B-nucleon systems), the three-particle overlap factor for the transfer into the configuration qlq2q3 -= wCqu~qu~q with a+b+c = 3, is ½ Q! 3! 7 ½ G = k3eq,.~q2!q3ij L~_ 6tY'ta16s~fr~fe'2N+L6ae6"° )4( (see appendix D). In the framework of the SU(3) representation of the harmonic oscillator, the calculation of the x-particle spectroscopic amplitude, Am.ss, is thus reduced to the calculation of the double-barred matrix element of eq. (2) which, except for trivial factors, is an n ~ n - x parentage coefficient for the n valence nucleons of nucleus B. The calculation of these coefficients is simplified greatly if the double-barred matrix element is factored into an SU(3)/R(3) factor and a second factor inde- pendent of SU(3) subgroup labels (and hence independent of all angular momentum quantum numbers); particularly since the SU(3)/R(3) (angular momentum de- pendent) factors are readily available through the recent work of Draayer ,3 .)7 073 K.T. HECHT AND D. BRAUNSCHWEIG The states of nuclei A and B are specified by fJo~(2#)flSTr.tLJ, or alternately by fo~(2#)flSTxsr.sJ, see ref. 3), where If and (4#) label the space and SU(3) symmetry, respectively; tc is used to distinguish multiple occurences of a given (2#) in a specific If,//distinguishes multiple occurences of ST in the spin-isospin sym- metry 79. contragredient to _f; (labels tc and/or//are usually omitted when not needed). The labels x are generalizations of the angular momentum projection labels K used by Elliott where the x refer to orthogonalized states see refs. 3,s), and where LK (or XL), Ks, Ks refer to the projections of the angular momenta L, S, J. For states labeled by _fo~(2#)3STxsxsJ, the reduced (double-barred) matrix element of the x-nucleon creation operator of eq. (2) can be factored into a triple- barred matrix element, (independent of all SU(3) subgroup labels) and an SU(3)/R(3) angular momentum factor, ANLSJ, a in the notation of Draayer ,)3 ( f'~'(2'#')/~'S~ gT B~M ~x 9x AS~)#2(~0fllrJS't~°Q¢~t*xll~J TAMr,, Xs Xs JA) = )ATAS~)#A(~ofIIITS)°e¢J~'ttzIIlgTs'S'~)K'A('~o'f(JsLRA X (TAMTA TMTIT'BM'rB). )5( The operator X tl~'t = a + x a + x a + x a+, e.g., is built from four nucleon cre, ation operators, a +, properly coupled to resultant quantum numbers 4(QO)LSJT. The factor ARLss gives the dependence on all SU(3) subgroup labels and has been eval- uated and tabulated by Draayer )3 ARLsJ = ~ q,,,,C.,,t..,,,c X S ((2#)x,g^; (Q0)gll(2'#')r~.gs). )6( ~'LL'~ALL~ k LB SB t B Here X( ) is an angular momentum 9-j coefficient in unitary form; the double- barred coefficient is an SU(3)/R(3) Wiguer coefficient; and the L.,~C are the trans- formation coefficients from states of good tct LSJ to states of good XsxsSJ; see ref. a). In s-d shell nuclei the triplebarred matrix element of eq. (5) can further be expressed in terms of the conventional SU(6)/SU(3) and SU(4)/ST factors of an n ~ n-x parentage coefficient ,a )61 <f'o~'(2' #')fl'S's ASlf)#2(~oflllrs)°Q(xuxlll~T >AT = CD, (7a) where the "C" and "D" factors are given in terms of the n .,.-- (n-x) c.f.p, by C - <f'='(2'#')ll IIz*t"J(a°~ll IIfa(2#)) n! dims,_,fl~ ........ - (n-~)!x! ~ j ~'LY-lZttz#)' x(Q0)l}'f'~t'(2'/l)>, (7b) D = (7flSA ;AT lXSTI}7'~'S~ Tg). (7c) SU(3) PARENTAGE COEFFICIENTS 173 The D-factor is the spin-isospin part of the n *-- n-x c.f.p, which can be identified as a reduced SU(4)/SU(2)s x SU(2)r ,l V~igner coefficient for the supermultiplet scheme .)71 For x = 4 the representation 41 corresponds to the scalar representation of SU(4), (with S = T = 0); and the SU(4) Wigner coefficient has the trivial value + ,1 (provided _~' = 7._ ; ~S = ,AS ~T = TO. For x = 3, 2, ,1 and most rep- resentations ~_ of interest, these coefficients can be obtained from the tabulations of ref. )71 and are given as part of the tabulations of appendix A. The C-factor is chosen to include, besides the SU(6)/SU(3) part of the s-d shell c.f.p., the binomial coefficient )~( and the dimension factors, dim If. Here, dims,_x _f is the dimension of the representation .f of the symmetric group S,-x, the permutation group for n-x particles, described by the partition numbers _f; similarly for dims, If' for the representation If' of S,. (Note that the dimensions of the totally symmetric states Ix are dimsx xl = ).1 If the x nucleons are transferred into or out of the (0p) shell, eqs. (7) hold if the SU(6)/SU(3) part of the c.f.p, is replaced by + .1 The (0p) shell has been discussed by Kurath is). If the nucleons are transferred into or out of the (0flp) shell, the c.f.p, ofeq. (7b) must be interpreted as the SU(10)/SU(3) factor of the space part of the full c.f.p. Since both SU(4) and SU(3) subgroup labels have been factored out of the matrix element for the x-nucleon creation operator to reduce it to the C-factor, it will also be useful to denote this as the quadruple-barred matrix element of the operator -*X Fairly extensive tabulations of the C-factors for x-partiele transfers of (ls0d) shell nucleons are given in appendix A for x = 4, ,3 2, and .1 Some of the details of the method of calculation are also presented there. Although the method of calcula- tion can be applied equally well to parentage coefficients corresponding to x-particle transfers in excited states, with (2x#x) ~ (Q0), the tabulations of appendix A are limited to those needed for the transfer of x-nucleon dusters in unexcited states, corresponding to space symmetry quantum numbers 4, 3, 2, and 1, respec- tively, with corresponding SU(3) symmetries (80), (60), (40), and (20). Using the sum rules for the ANLsa R factors see eqs. (4.2) and (4.3) of ref. a), the C-, D-, and G-factors can be used to determine the total spectroscopic strength for pickup or stripping reactions to states of specific SU(3) symmetry. For the pickup of an x-nucleon cluster (B(2'#') ~ A(2#)) = C2D2G2(TAMrATMrT~M~.8) 2, (8) pickup where this pickup sum rule refers to the summed strength for transitions from a specific rotational state in the representation 0'#') of nucleus B (with fixed ~x xj, J~, e.g.), to all rotational states of the representation (2#) in nucleus A (all possible x s xj, and JA) via all possible L- and J-transfers of an x-nucleon cluster of fixed space symmetry x and SU(3) quantum numbers (Q0), and specific S, T, and M r. Simi- larly, 273 K. T. HECHT AND .D GIEWHCSNUARB (A(Z/1) -, stripping (-B~--X) 2( TM T,'M' \2 2Sh+1 dim(2'/1') (9) T S Tn/ 2SA+ 1 dim(2/1)' -~" where dim(2/1)=~2+l)(/1+l)(2+/1+2), is now the dimension of the SU(3) representation, and the stripping sum rule refers to the summed strength for tran- sitions from a specific rotational state in the representation (2/1) of nucleus A to all rotational states of the representation (2'/1') of nucleus B, again via all possible L- and J-transfers of an x-nucleon cluster with fixed Ix, )~1/~2( = (Q0), S, T, and M T. To gain a feeling for the relative importance of C-factors for different transitions it is useful to compare these with a sum rule for transitions from a fixed state (2'/1')S~ T~r'sX'~J ~ of space symmetry .f' in nucleus B to all states of space sym- metry f in nucleus A via transfer of an x-nucleon cluster of fixed space symmetry Ix (totally symmetric in its space wave function), but with all possible ,)~1/~2( L, S, and T. This is given by the sum rule for the triple-barred matrix element ,Y (If' f + (f'~'(,~' B'S'lf)'1/ T~III2:e"J~x"x)pSTIIIf ~(A/1)flSA TA) 2 (~.xpx)pST dims, if, • These sums, ~, are tabulated in appendix A at the head of each table of C- and D- factors and can tell at a glance what fraction of the total pickup strength from a specific state _f'(2'#') is concentrated in transitions to a specific representation (2#) of f. Note, however, that the sum, ,2~ contains strength from x-nucleon clusters in internally excited states with )x1/~2( ~ (Q0). For the transfer of four nucleons from the ls0d shell, for example, with space symmetry 4, i.e. with the spin-isospin structure of a real e-particle, this sum would in general contain transfers of four- nucleon clusters in SU(3) states )x1/~lJ( = (42), (04), and (20), as well as those for the "e-cluster" states with )x1/xA( = (80). (The label p, needed only for (2x/ix) 4 ~ (QO), is defined in appendix A.) It is interesting to note that a large fraction of the summed strength, ~, for 4 nucleon pickup is concentrated in the e-cluster transitions from the ground state to the ground state rotational band in all good SU(3) nuclei in the first half of the (ls0d) shell. The numbers are collected in table ,1 under the assumption that the ground states of the target nuclei and the ground state bands of the residual nuclei shown are pure in their SU(3) quantum numbers. Table 1 shows both the sum, ~, of eq. (10) and the percentage of this summed strength which resides in the ground state to ground state rotational band transitions. The pickup strength from the SU(3) PARENTAGE COEFFICIENTS 373 TABLE 1 Per Cent of 4-nucleon Transfer Strength in Ground State . Ground Rotational Band a-cluster Transfer AX(k'~') ~ A-~Y(k~) Percent E )a( 2ONe(80) . 160(00 ) I00 1 2X~e(81) ~ 17o(2o) 1oo 1.25 22Ne(82) * 180(40) 79 1.67 23Na(83) ~ 19F(60) 63 2.5 24Mg(84) ~ ~ONe(80) 47 5 25Mg(66) ~ 21Ne(81) 34 6 .62 (10,2) 22Ne(82) 31 Mg(48) 4 31 7.5 28S1(0,12) * 2~Mg(84) 60 15 (a)The sum rule strength )Z( is defined by .qe (i0). The numbers give the percentage of the sum rule strength in transitions from the ground state of (k'~') in nucleus A via a-transfer'to all members of the rotational bands of (kg) in nucleus A-A. ,gM42 ground state of assumed t9 be the 0 + state of pure SU(3) symmetry (2#) = (84), to all members of the (80) rotational band in 2°Ne, for example, soaks up 47 % of the sum rule strength. The missing 53 % resides partly in unexcited 0t-cluster transitions with (;t~#~) = (80) to the excited 2°Ne representations (42) (22%), and (04) (4%); while the remaining percentage involves "excited states" of the ~-cluster with (2~/4) = (42) (22% to the ground state (80) band and 0.4% to the excited 2°Ne SU(3) representation (42)), and finally with )~#%;( = (04) (4% to the (80) band of 2°Ne). It is interesting to compare the percentages of table 1 with the corresponding percentages for a goodj-j coupling nucleus. If the ground state of 44Ti is assumed to be a 0 + state of pure (0f~) 4 configuration and seniority 0, the ~t-cluster transition connecting this to the ground state of 4°Ca would use up only 0.3 % of the cor- responding sum rule strength 2). The remaining 99.7 % of the transition strength now comes from four-nucleon transfers corresponding to excited states of the or-cluster. If, on the other hand, 44Ti had been the (0flp) shell analog of 2°Ne, that is if it had been a good SU(3) nucleus with a ground state rotational band based on a (12,0) representation of SU(3), the "~t-cluster" transition connecting this state to the ground 374 .K .T HECHT DNA .D GIEWHCSNUARB state of '~°Ca would have used up o%,001 of the summed strength. (Since this is a hypothetical remark, we shall not be concerned with the fact that 44Ti does not present us with a stable target for a pickup reaction to 4°Ca.) The numbers again emphasize the close relationship between the SU(3) representation and the cluster model. The numbers also show that small admixtures of lower (2#) representations into the representations of high SU(3) symmetry may be relatively less important in their contribution to the 0t-transfer strength to the ground state bands, since they are generally weighted by smaller C-factors. 3. Alpha-particle spectroscopic amplitudes for core-excited states in s-d shell nuclei The C-factors tabulated in appendix A can be used together with the factors, Am.ss, R tabulated by Draayer )3 to calculate any g-particle spectroscopic amplitude for a transfer involving a 1( s0d) 4 cluster. Since core excitations give rise to low-lying bands in many s-d shell nuclei, g-particle spectroscopic amplitudes for the transfer of (0p)-l(ls0d)-3,(0p)-2(ls0d) -2, ... clusters in pickup reactions, and (ls0d) 3 (lp0f) ,1 (ls0d) 2 (lp0f) ,2 ... clusters in stripping reactions may also be of particular interest. Since the G 2 factors for such transfers are favored by the factor (4 !/a b! ,)! see eq. (3) and table 1 ofref. 1)1 such transfers may compete favorably with transfers into or out of the (ls0d) valence shell, provided the corresponding parentage coeffi- cients are sufficiently large. Transfers of 1( sod) ° (lp0f) ~ clusters may also be strongly favored over 1( sod) 4 clusters by the kinematic factors for the direct reaction process since the wave functions ¢'NL(ro,) for the relative motion of the g-cluster will be larger in the surface region if the transfer involves a cluster with a larger number of quanta Q = 2N+ L. Since both the SU(3) weak-coupling and SU(3) strong-coupling models have been used to describe core-excited states in s-d shell nuclei, the formulation will be given for both coupling schemes. The work of Ellis, Engeland and collaborators 10.11) shows that the weak-coupling model furnishes a good approximation for particle-hole excitations in nuclei near a60, particularly for configurations (0p)"' (ls0d)"', (with nl < 12). In the SU(3) weak-coupling model the n = nt +n2 particle states are specified in a basis such as I(Opy"f~a~(,h zl)xL, L1 sl Ja ;1T ~.z~)2~/22(2.a2f~")d0sl( L2 $2 J2 ;2T JM, TMr), with J = J1 +J2, T= T 1 + T 2 ; that is, only the total angular momenta and isospins of the particle and hole configurations are coupled to resultant J and T. The space symmetry and SU(3) quantum numbers for both particle and hole configurations separately are assumed to be good quantum numbers in some zeroth approximation. To evaluate the reduced matrix element of the four-nucleon creation operator, Z ,)°Qt4Et it is only necessary to couple the creation operators for the two separate shells to space symmetry 4, SU(3) representation (Q0), total L, and S = T = 0 :