# Application of the Multi-Doorway Continuum Shell Model to the Magnetic Dipole Strength Distribution in 58Ni.

код для вставкиСкачатьA N N A L E N D E R PHYSIK 7. Folge. Band 47. 1990. Heft 8, S. 599-678 Application of the Multi-Doorway Continuum Shell Model to the Magnetic Dipole Strength Distribution in "Ni By H. SPANGENBERGER, F. BECKand A. RICIITER Institut fur Kernphysik, Technische Hochschule Darmstadt, Dnrmstadt, FRG Gewidvnet d e n Anncclen der Physik aus Anlab ilzres 200jiihrigen Bestehens. A b s t r a c t . The usual continuum shell model is extended so as to include a stnt'istical treatment of multi-doorway processes. The total confignrat,ion space of the nuclear reaction problem is subdivided into the primary doorway states which are coupled by the initial excitation t,o the nuclear ground state and the secondary doorway states which represent the complicated nature of multi-step rcactions. The latter are evaluated within the esciton model which gives the conpliiig widths between the various finestrucbure subspaces. This coupling is determined by a statistical factor related to the exciton model and a dynamical factor given by the interaction matrix element,^ of the interacting excitons. The whole structure defines the multi-doorway continuum shell model. In this work it is applied to the highly fragmented magnetic dipole strength in j*Ni observed in high resolution electron scattering. Anwendung des Multi-Doorway-Kontinuum-Sehalenniodells anf die Verteilung der magnetisehen Dipolstarke yon GSNi Iiihal t s u b e r s i c h t. Das Iiontinuum-Sclialenmodell wurde so erweitert, dial3 auch statistisclie ~~lulti-I)oorway-Prozesse berucksichtigt werdrn konnen. Hierzu wird der Konfigurationsraum unterteilt in den Raum der primaren Doorway-Zusthde, die dirckt i t u s dem Griindzustand angeregt werden, und den der sekundzren Doorway-Zustlnde, die die komplizierte Striiktur der Multi-StepReaktionen reprasentieren. Wahrend die primareii I ~ o o r w t ~ y - Z u s t ~ n inclusive de ihrer Aiiregungen mittels iiblicher Schalenmodellmethoden beschrieben werdm ktjnncm, werden die sekundzren Doorwr.ny-Zust,andesowie ihre vcrschiedencn liopplnngcri im Ralimen des Exciton-Modells beh;rndelt. I h s e Kopplungen sind durrli rincn a m drm Exciton-Modcll rosultierenden Paktor soaie durch ninen dynamisrhcn Yaktor bestimmt, tier sich iius dem ,\Zt~trisrlemrntder \~echselwirkendenExcitonrn berechnet. Die Struktur drr liopplungen definiert das l\lulti-l)oorw;iy-IConti~iu~im-Scli~tlenmodell, da8 hier auf dic Beschreibting drr st,ark fragmentiertm niagnctischcn TXpolstiiirke in 5*Ki trngewendet n-ird. I . Introduction The description of a iiriclear reaction within a niodel is equivalent to the, representnt i o n of the eigenvaliie problem i n a basis of model states. The particular choice of the basis states depends on the given nuclear reaction problem, the iiatiire of the imderlying cscitatioii, the numerical complexity and the computational inetliod 13 hich is used. GOO Ann. Physik Leipzig 47 (1990) 8 Due to the very complicated nature of the many body problem all methods have in common some approximations which replace the full A-particle system by a numerical manageable smaller one. There are two approximation strategies : The restriction of the configuration space and a statistical description of the complicated many-particle many-hole configurations. The restriction of the configuration space is usually done in shell model calculations where only a “small” number of shell model states1) are taken into account. Typical examples of those nuclear structure calculations are the oscillator shell model [l,21, and the continuum shell model [3, 41. Both types of models have in commoii the use of restricted sets of individual model states which are excited by the response to an external excitation operator from the ground state and interact via the residual nuclear interactions. The statistical description of nuclear reactions, in contrast to the shell model, works with representatives of statistical ensembles of states which i n general are large in number so that statistical assumptions are valid. Representatives for these models are the preequilibrium, or multi-step, or overlapping-doorway models [5 - 161. An intermediate concept between the pure nuclear structure calculations and the statistical models is the doorway model. Here the configuration space is separated into a small set of individual states, the doorway states, and a large number of complicated states which can be treated by statisticalmethods. I n the standard application of doorway models the doorway parameters are determined by a n adjustment to the experimental finestructure data [17, 181. Such a doorway analysis of experimental data allows a decision on the existence or non-existence of an underlying doorway mechanism and gives a first estimate for the ratio of the coupling strengths between the different channels. A deeper insight in the fragmentation mechanism, described within a doorway model, needs, however, a microscopic calculation of the coupling between the various configurations. To describe the coupling between bound state configurations on the same footing as the coupling to continuum states, a description within a continuum shell model basis is the appropriate choice. This leads to a combination of the doorway model concept with the microscopic continuum shell model description of nuclear reactions which is called the multi-doorway continuum shell model. The main concept of the model is presented in chapters 2 and 3, while the details of the calculation are published elsewhere. As an application of the multi-doorway continuum shell model the distribution of the magnetic dipole strength in 5sNi obtained from high-resolution inelastic electron scattering [ 191 has been analysed. ltesults are given in chapter 4. A summary follows in chapter 5. 2. Multi-Doorway Continuum Shell Model 2.1. Multi-Doorway Concept I n the usual continuum shell model only two typos of states are distinguished, namely the bound states lqi > with A bound particles and the continuum states Ipp;E > with ( A - 1)bound particles plus one particle in the continuum with energy E . The scattering aiid bound states form two orthogonal subspaces 9’ and 2, respectively, in which the eigenvalue problem has to be solved. In each of the two subspaces 9’ and 9 there are states with different degrees of complexity, beginning with simple excited states of the 1-particle 1-hole type up to highly excited n-particle n-hole configurations ( n 5 A ) . The use of complexity (e.g. the number of excited particles and holes) as an order paraI ) Small means in this context that only a restricted number of shells are involved which nevertheless could lead t o matrices of dimensionality 2 600 x 2 500 [l]. H.SPANGENBERGERe t al., Multi-Doorway Continuum Shell Model GO1 meter in the subspaces 8 and 2 leads to a subdivision of the subspaces into an increasing sequence of finestructure subspaces 8, and 2, with g==Ugn7 n 2= v 2n, n where n indicates in general the various orthogonal finestructure subspaces. In the exciton model [20] n counts the number of excited particles and holes in the particular finestructure subspace 9%and 2%. Introducing projection operators Pn and Qn which project onto the finestructure subspaces g n and 2,, respectively, the eigenvalue problem ( E - H ) ly ( E ) > = 0 can be written as N N 2 ( E Q m L n - Q m H Q n ) ] Y ( E )> == n=2’ n=l QmHPn 1 N 2 ( E P m d m n - PmHPn) I Y ( E ) > n= 1 (1) A’ = n= Iy(E) > > PmHQn 1 I v(-@)> 9 where rn runs from m = 1 to m = N with N as maximum number of finestructure subspaces in each subspace. The projectors are defined in such a way that holds. I n the coupled system of 2N equations (eq. ( I ) ) the original A-particle problem is still present (with the basic restriction of the continuum shell model) in the coupling terms QmHQn, PmHP,, QmHP, and P J € Q n which represent the coupling between the various states in the total configuration space. A systematic way of approximations is needed which replaces this full system by a smaller one without loss of the main physical information characterizing the particular nuclear excitations. This is given in the multi-doorway-model by a restricted coupling scheme : Thus, only neighbouring finestructure subspaces are coupled directly (see Big. 1). Due t o this coupling structure the states in the finestructure subspace of rank n act as doorway states for the states of rank n 1 which itself acts as doorways for the states in 2 etc. up to the states of rank N 5 A . For the the finestructure subspace of rank n coupled system of equations (eq. ( I ) ) this means that t h e summation over n is restricted t o n = m and n = rn & 1. The whole structure represents the multi-doorway continuum shell model [15, 21, 221 + + C n=m,m& 1 (EQmdmn - C (E:Pm6m, n=?n,?n&1 > = n= mC ,m*l Q m H Q n ) Iy(EJ’) - PmHPw) Iy(B) > = QmHPn Iy(E)> > 2’ P m H Q n ly(E)>. n=m,mil (4) Since only the states of rank n = 1 are coupled to the groundstate by the external interaction (see Fig. I), we call these states primary doorway states whereas t h e states Ann. Physik Leipzig 47 (1990) 8 60% ... ... ... ... ... ... ... 4 ... - u Fig. 1. Coupling in the multi-doorway model: Shown are the restricted couplings of the finestructure subspaces 9%and 2%'respectively, in the multi-doorway model; 3 denotes the space of the groundstate in the following finestructure subspaces are denoted as secondary doorway states [231. The primary doorways carry the initial excitation strength, which is localized in energy. The coupling between the primary and secondary doorways leads to a spreading of the excitation strength. The latter displays the influence of the complicated structure. While in most of the cases only a small number of primary doorways are important which have to be treated explicitly, the secondary doorway states are large in number and justify statistical approximations. To make this difference explicit in the coupled system of equations it is efficient to project out all finestructure subspace components with n 2 2. This is equivalent to a modified resolvent method and leads to a coupled system of equations in the space of the primary doorway states The influence of the secondary doorway states is contained in the effective operators whose explicit forms are and hrl as well as in the effective potentials w?;: and given by the recursion formulas w?2 H. SPANGENBERGER et al., Multi-Doorway Continuum Shell Model 603 The projected propagators in the finestructure subspace 2k and Pk, respectively, are given by Gf := Qk[E- h2kl-l Qk, (9) Gf := Pk[E - k&]-'Pk. Iiisertion of the recursion formula (6) for hf,k into the projected propagator Gf shows:) that V&, ,G,", V f T has the meaning of an effective potential [ 1 51 Pf,&:= Vf,ktlGf+dtv$+,,k. Introdncing f$,k (10) into the equation for Gf leads to the continued fraction 1 1 1 1 which displays the whole iteration structure of the effective operators G f . 3 ) With the effective potentials f f k and f [ ,(eq. (10)) the effective operators hf,n,and hGk (eq. (6)) can be written as where the first term acts in one finestructure subspace only, namely gk and $Pk,respectively, and the second term contains the contributions of the complicated multi-doorway configurations which are described above. In analogy to the effective hamiltonian h,& and hCk the effective potentials wf,{ and v[$ (eq. (7)), which describe the effective coupling between the 9-and 9-spaces, could also be represented as a sum of two terms Q,P %,n = vQJ' + k,rL = f 0 1' k,'n vf,$+ q g , with a similar meaning of the two terms as in eq. (12). 2) 3, Analog equations follow for G:. The same holds for GC. 604 Ann. Physik Leipzig 47 (1990) 8 With these notations for the effective hamiltonians (eq. (12)) and the effective potentials (eq. (13)) the coupled system of equations (eq. ( 5 ) ) can be reduced to the form where the contributions from the multi-doorway structure are given by the right hand side of the above equations. The left hand side is up to the couplings V f , f and Vf,$ identical with the usual continuum shell model, evaluated in the restricted space 9, and PIof the primary doorway-states. The explicit form of the projectors is where I pp)> denotes the eigeristittes of HO with complexity n, and I pf) ; E > represents the Slaterdeterminants of the ( A - 1)-particle residual nucleus eigenstates of Ho and the one-particle continuum eigenstate with energy E . After multiplication of the first E' I equation in eq. (14) from left by < py)I and the second equation from left by < the coupled system of equations becomes ~9); + < p p ; I vc2 Ipp >]a?)(E) = 0, E' with amplitudes aP)(E)= < p p I y ( E ) >, aj;Z)(E, E ) = < p p ; & jy(E) >. The above coupled system of equations, written for the components ]pi1)> and IpF);E >, represents the key set of equations of the multi-doorway continuum shell model. The further evaluation of the model needs the calculation of the matrix elements between primary and secondary doorway states and the introduction of statistical assumptions for the latter. 605 H. SPANGENBERGIR ct al., Multi-Doorway C o n t i n ~ ~ uShell m Model 2.2. Statist,icalProperties of the Effective Potentials As a result of the multi-doorway concept and the projector formalism the complicated pc17@$ nature of the A-particle system is summarized in the effective potentials and as well as in the effective propagators GP and G: (cf. eqs. (10-13)), where all quantities are defined by recursion formulas which represent the iteration of the interaction through the sequence of finestructure subspaces of increasing complexity. With the assumption that there exist finestructrire subspaces of highest rank N , the iteration (eq. for starts with ?f,, vc$ With the solution of the eigenvalue problem4) in the finestructure subspaces 22A, @N;i - a , N ) IYN;?> = 07 The next element in the chain is The corresponding eigenvalue problem in the space 3?,\, . is given by (EAr-l,j 4 - HN--l,N-l Miiltiplication from left by - -0 VN-l,Ar-l) < qN- ] l$?N-l;f > = 0- leads to With the assumption of random phases for the matrix elements of V N - l , x and VN,N-l as well as a large number of states IpN;{),the last equation can be approximated by The matrix element of F$-l,N- I , depends on energy. With the assumption, that SN contains a large number of states per energy interval (high level density), the summation >. The averaging is perover i can be replaced by an averaging over the states formed by introducing a complex energy denominator E -+ I3 < I N , where I N DN represents the averaging interval which should be large in comparison to the mean level spacing DN of the states in the finestructure subspace 2lX. The result can be expressed by + + with the real part 4 ) The eigenstates of H27,N are denoted by eigenstates /p(?) > of HO. 1 1 ~ ) >~ which ; ~ should be distinguished from the 606 Ann. Pliysik Leipzig 4 i (1990) 8 and the imaginary part (spreading-width) The eigenralue problem in 21N-l thus becomes [EAl-, ] - - ( ~ % l , Y - l ) I N l lRv-l,7 > = 0 , -a1-LX-1 [EL?,-,7 - (HW-i,N--1- ( v A“(2~ - - 1 , ~ ~ - - l ) i A T ) + I I V ~ 1,7 > = 0 , ~ i t the h imaginary part of EAT-1,, 1 Irn EAT-l, ( E ) = -l’$N - 1 . p ) . (25) (26) Due t o the complex areraged effective potential ( ? ~ T - l , , , - l ) ~ x there exists a non-hermitian eigenvalue problem which leads to a diagonalization in a biorthogonal basis, indicated by the states 1qA-,,> and / ~ p ~ , >. , ~ After the averaging over the states in the varies only on the scale of space 21x, the effective potential ( @ - l , N - l ) I N This in turn leads to 9 variation of E A r - , ,on J the same scale5). For the average decay F 21A7 and the mean level distance Dx-l of the states in g N u 1 the , relation width TAT -- r, + D-\.-~, (27) miist be fulfilled, because only then there are enough states Iqs 1,1 > in the interval to diagonalize ( f l ~ r ~ l ,(p$ AT l,N --l, ) I A’ ). Following this line in the recursion chain from k = N , N - I, 21’ - 2 , .._tip to the m-th member in the chain the spreading width of the states in the finestrricture subspace 21m is given by the imaginary part rN + Tm&= 2Im m- - < ~ m , I7V 2 , m / q m , 7 > I Introdiicing Vz,, leads to - x < ~ m ~ - l ; s j T i , + l , m ] ~ m>. ;i The calculation of r2 is performed by the averaging over the states 1qmij >.The replaceleads ment of the summation over Ic by an averaging over the states l y m + l ; k >€ SmL1 to the “mean spreading width” [14] in the finestructure subspace From the above argument it becomes clear, that the mean spreading width of the states in 2m-l, acts as averaging interval for the effective pcteritial @ m in the space 2m.This introduces a self-averaging process with the result, that ?$,m varies 0 1 1 the scale of I‘$&.Thus in tiirn, only the coupling of the states in 21m to a representative i n gm must be known to determine the effective potential Vg,,. With the same assumptions introduced above, the effective potential 5, This means EN- ;i is independent of the energy in the interval T, = 21N . H. SPANGEXBERGER et al., Multi-Doorway Continuum Shell Model 60 7 can be calculated with the result vf,f = n+liN 2 71=11-i>Z p-1 (I7 Z’ E , m + ~ aE@m m=l +1 Jd~/~rn+l;p(&)> (29) 1. X - < Y ? n - + l ; p ( E ) I) v;g. E - Em~+pp(&) With the dynamical self-averaging of the effective potentials and propagators, the corresponding matrix elements in the coupled system of equations (16) can be calculated. 2.3. The Matrix Elements of the Effective Potentials The matrix elements of the effective potentials ?y,l and p{:t can be expressed by the effective propagators G2&and Gp, respectively. If the self-averaging assumption, made in the preceeding section, is valid for the states in the finestructure spaces 2, and 8,, then Fy,l is given by Re (@ I ff11#)) E’ - Ic2;k 1 =c 2 (E’ -E2;$)2 + (?Ti) (qPl ~ ? . z l P 2 ; d A I ( Y 2 ; k @,l ivy’) ? kE.’22 1 VF,l I#)) =rg E 2 E92 Im <&) 1 (&) + (+2) I vX21 pZ;d( -i M ~ ; ~ t ~ f ”.) ( E - EZ;,;.)2 Replacing the summation over k E 2zby a n averaging over the states val Tj,then, in the limes N$! -+ co,it follows6) (P?) and - I C ~ ~in; ~the) inter- I 7 ? , 2 IY 2 , k ) (V2,n; 1 w,: 1 Y i l ) ) (<(pi (1) I yo1,2 I % ; k > < y J 2 ; k I vp, I#)>)k, \ Epa - E k + 1 ; 2 I N 0:. Thus, for the real and imaginary parts of the matrix elements one obtains Re (y;” I Vfl 1 qi’))= 0 , 52 I m <Y!ll)l q l l q j l ) ) =-<<q$’)l @ q 2 1yJ2;L> - 1 ri;i,i. I n the same way 8) flf ?‘il is given by denotes the nnmber of states in the interval Ti. CyJ2;kI vp, l L p > ) k j (30) Ann. Physik Leipzig 47 (1990) 8 608 $92 The value of the matrix elements f?;: and depends on the correlation between the matrix elements V f 2 , V z 3 , ..., V2and V$$ which couple the finestructure subspaces 91, &, 23,. . .,9r,-land 2 ,with 9%. If the number of states in each finestructure subspace is large, as assumed in the previous section, then there are many terms which must be summed up. With the assumption of random phases of the matrix elements of V$,m !. this leads to a vanishing contribution down to a critical complexity n > ncrit <pp)I Fez I pI;L)(E)) -o for n > ncrit. (32) If the above condition holds already for ncrit= 1, then the matrix element of approximated by is With analogous methods one obtains The coupled system of equations is completely determined by the matrix elements beand the couplingwidthsrl;i,i, lyP,&, d) tween the finestructure subspaces9, ,PI and The latter represent the various couplings between the primary and secondary doorway states. While the matrix elements between the primary doorway states can be calculated by the usual continuum shell model methods’), the calculation of the coupling widths needs a model or an assumption about the statistical distribution of the states in the spaces of the secondary doorway states. r:’;$,i(&‘). r&&), The calculation of the matrix elements is usually performed in a one-particle one-hole continuum shell model basis. H. SPANGENBERCER et al., Multi-Doorway Continuum Shell Modcl GO9 3. Exciton Model The calculation of the matrix elements between the complicated secondary doorway states in the finestructure subspaces 9%and Pn withn > 1is performed within an exciton model [16, 201. This means, that each finestructure subspace is characterized by a particular exciton number which counts the number of particles and holes in it. The h = 2n. The space Pn has space 9n has n particles ( p ) and n holes (h) with N = p ( n - I) bound particles, n holes and one particle in the continuum. The physical concept of the exciton model is illustrated in Fig. 2. The ground-state (left hand side) is excited by an electromagnetic interaction into a 1-particle 1-hole state (center). The residual nuclear interaction allows three types of further excitation : + 1. The excited particle could interact with another bound particle forming a new particle-hole pair (right hand side). This gives a 2-particle 2-hole state. The reaction is characterized by an exciton pair creation. 7 EF J ................ N=O a ................. a ................ a ~ N=2 N=4 t Y E I.' .................. 1 ............... ................ a N=O N=l N=3 Fig. 2. Exciton model without (upper part) and with (lower part) nucleon emission. The energy of the continuum particle is denoted by E , E , represents the binding energy and U is the excitation energy Ann. Physik Leipzig 47 (1990) 8 6 10 2. The excited particle interacts due to the residual interaction with a bound particle which leads to an absorption and reemission of an excited particle. There is still a 3 particle 1-hole state, the reaction process is characterized by an exciton scattering. 3. The initially excited particle recombines with the hole state which leads to a N = 0 exciton state. This reaction is characterized by an exciton pair destruction. As indicated i n the lower part of Fig. 2 , the excitation as well as the exciton pair creation could lead to an emission of a particle into the continuum. The probability for the three processes with A N = 0, & 2 ( A N denotes the change in the number of excitons) depends on the density of the final states which could be reached from the initial state. For a level density which increases with N this leads to a predominance of the exciton pair creation process over the other two types of reactions. The main assumption in the exciton model is the statistical equivalence of all states inside an exciton configuration. This in turn allows the replacement of matrix elements, calculated inside and between different exciton configurations, respectively, by matrix elements which are calculated between representatives of the particular exciton configurations. The residual states in each exciton configuration contribute only as combinatorical factors to the level density. The various couplings between the finestructure subspaces (as shown in Fig. 3) correspond to the three types of reactions : on+n--l - b,&+ 2a- : excitori pair destruction - 9,& + 9, - : exciton pair destruction with nucleon emission on+n - 9,--f .?la:exciton scattering - J!?&+ P r Lexciton : scattering with nucleon emission on+n+l - -2, + &+ - 9,+ Pa : exciton pair creation : excitori pair creation with nucleon emission Equivalent transitions start in the .P,-space, where always one particle is in the continuum which can be absorbed into the .%space or makes a 8-space scattering. The value of a matrix element in the exciton model depends on four factors: 1. the factor &$, which denotes the number of realizations to pick tip ( p - ( A ) particlcs and ( h - 6 ) holes from a p-particle h-hole configuration, 2. the factor y!,d, which represents the density of final states for the c-particle d-hole configuration of the residual exciton system, 3. the probability P$(U/E), that after removing ( p - a)particles and ( h - b ) holes from the initial p-particle h-hole configuration with total energy E, the residual aparticle b-hole confignration has an energy Ea,b U , where U denotes the excitation energy of the exciton system, arid < 4. the matrix element of the residual interaction between the representatives of the exciton-system. This term, as it is shown later on, depends on the angular momentum coupling of the interacting excitons among themselves and to the spectator excitons. While the first factor g$$ is a purely combinatorial factor, the second factor g!,d depends on the density of final states. The latter is determined by the single particle density and the energy of the excitons returning to the exciton system after interaction. The FtA G11 H. SY-4NGENBERQER et al., Multi-Doorway Continuum Shell Model ....... m A..(l ) (1 ) 1 V ......... ......... Fig. 3. Coupling between the finestructure subspaces in the exciton model. The particle graphs are indicated, where the thick line stands for the continuum particle. Spectator excitons are shown as double line probability Pi;:(U / E )is given by ~ $ ( U / E= ) 7 x [fielo(m] 1 2 P!h!%,,AE) 0 w ds? ... dEg 1 ds'; ... dEi n I1 (36) e01(4)] P x b 6 E- 2 s : - ( i= 1 ;s!;)@(U- ~ E S -X E ! ) , j= 1 i= 1 j=1 the single-particle and hole density is denoted by elo(sf)and pot(&!;), respectively [15]. The &function in P$$(U / E )guarantees the conservation of energy, while the @-function restricts the available excitation energy U up to the total energy E of the exciton system. These factors determine significantly the energy dependence of the matrix elements calculated within the exciton model. The three statistical factors $$, g!,d and Pi::(U / E ) >(( ) ( )---( ......... active particle-pair active particle- hole-pair 7 ..... active hole-pair Fig. 4. Active exciton pair in thr N-exciton systcms: denotes the pxticlc line, tho Iiolc line and 1 I spectator excitons 612 Ann. Physik Leipzig 47 (1990) 8 can be p u t together in G pa b:c d~(UI'E)= &k&&$WlE). (37) I n the case of a delta-force, which will be assumed further on, the two-particle residual interact>ionis invariant under particle-hole conjungation after averaging over the complicated configurations. Thus, terms which are equal up to particle-hole conjugation can be summed up. This in turn leads to the twelve terms characterizing the excitori model interactions between the finestructure subspaces in the multi-doorway continuum shell model. The resulting equations are obtained for an energy-independent singleparticle density Q . The following statistical factors determine together with the matrix elements of the two-particle residual interaction of the active excitons (see Fig. 4) the (g) N = e2E 1 = -ph(p 4E - (39) 1 ) ( N - 1) ( N 2 ) u( 4 N-3 = 2E ( N - 1)(N- (1 U -z), H. SPANGENBERGER e t al., Multi-Doorway Continuum Shell Model 613 Here GZh;2,1, for example, represents a graph, where 2 excitons (2 particles or 1 particle and 1hole) are removed from the p h = N-exciton system t o interact while 1 particle indicates ” or hole returns to the residual exciton system after interaction. The index ‘‘9 that one particle goes into a continuum state. Analogous the term GZ,h;l,3describes a graph, where 1 exciton (particle or hole) from the N-exciton system interacts under an exciton pair creation and 3 excitons return after interaction t o the residual ( N - 1)exciton system. The index ‘‘9” indicates that no particles go into the continuum after the interaction. The statistical densities, summarized in eqs. (38-49), are shown in Figs. 5 and 6 for N = 2, 3, 4 , 6 as a function of U J E and E , respectively, where the Pauli corrected energy ( E - B ) with + 1 B = -( P i 4en + h: + Pn - 250 1 I I I I I 0.8 1.0 - 2 1.0 E 2 0.5 I 0. an (3 0.0 0.0 0.2 0.L 0.6 U/E 0.8 1.0 0.0 0.2 0.L 0.6 WE. Fig. 5. Statistical exciton densities as a function of lJ/E for N = 2, 3, 4, 6 and E = 10MeV. The particular reactions are indicated by a graph, where the thick arrows indicate the continuum-particle Ann. Physik Leipzig 47 (1990) 8 614 has been used. The latter leads to a multiplication of Pg$(U/E) by a factor ( E / U ) N - l which makes the N-independent densities G:/L;o,z,G,&,1,;o,3 and G$,h;,,2(see eqs. (43), (44) and (49)), due to the Pauli correction B , N-dependent. The densities, depicted in Figs. 5 and 6 are arranged in such a way, that densities which belong to reactions with an equal number of returning excitons are side-by-side. The upper row shows the reactions with exciton pair creation, in the center are the exciton scattering processes without change of the total exciton number and in the lower row are the reactions which belong to exciton pair destruction processes. In Fig. 5, left column, are the densities for the continuum-continuum scattering (9,+ 9,;m = n, n & 1) and to the right the reactions which represent the couplings 9,+ 8,; m = n,n & 1. I n Fig. 6, left, are the densities for 9,+ 9,;m = n, n & 1 and t o the right for 9,+ 9,;m = n, n & 1. In all figures the processes of exciton pair creation are one, respectively two orders of magnitude larger, for the selected values for N and E , as the reactions of exciton scattering and exciton pair destruction. This behaviour results from the increasing density of states with increasing exciton number which makes exciton pair creation processes more likely than other processes. Since this behaviour depends on E and N there is a value of N where the process of exciton pair destruction becomes more Q, E c )---A "i 30 20 a d 20 c3 10 15 20 25 30 E (MeV) 35 LO 01' 10 .._.. I I 1 I ' 1 15 20 25 30 35 LO 1 E (MeV) Fig. 6. Statistical exciton densities as a function of E , shown for N = 2, 3, 4, j.The particular reaction is indicated by a graph, wherc the thick line indicates the incoming continuum-particle likely than the process of exciton pair creation. This can be demonstrated for the density G$?ll;,,3,the density for exciton pair creation, which scales as 1 / ( N I), while the density Q$,ll,3,1 for exciton pair destruction scales as (Ar - 2). For the values Q = 4.4 MeV1 and E = 10 MeV both processes of exciton pair creation and esciton pair destruction become equally likely a t N 12. This behaviour, as demonstrated for G:,lL,l,3 and G$,,c,3,1, also holds for the other densities but for different values of N . The comparison of the densities G$,h;?,!withG[,,,,,, arid G$,l,;2,L with GEl,,2,1as well as G,&( 3,1 withGL,g;3,,,shows that the transitions d,t+ 9, are one order of magnitude larger than the transitions between A?,&-+ Ym(rn = n, n & 1). This leads to a dominaiice of the couplings between bound states in comparison to the coupling t o the open channels, a dominance which decreases asl lwell ,2,, with increasing N , which means that the ratios G$,ll,l,3/G[,, , , 2 and G ~ , l l , 2 , 2 / G ~ ~ as G$,h;3,1/G[h;3,0 become smaller than 1for a particular value of iy for a given energy E and a single particle density Q . The absolute value of the particular matrix element in the escitoii model depends on the statistical densities and the absolute value of the interaction matrix elements of the interacting excitons. The latter depends on the angular inonientuin coupling of the interacting excitons between themselves and between the spectator escitons (see Fig. 7 ) , where the particle-particle matrix element is classified by the angular momenta jl,j z of the incoming particles, the angular momenta j 3 , j4 of the outgoing particles, the total spin j5 of the spectator excitons and the angular momentuin coupling between them. The angular momentum coupling between the various angular momenta is different for bound particles and particles which are in the continuum. In the first case the interacting excitons are coupled to L and L', respectively, which are coupled with j 5 t o the conserved total spin J (see left hand side of Fig. 7 ) . h i the second case, the bound interacting particle (spin j4)is coupled with the spectator excitons (spin j5)to the spin Q which is coupled to angular momentum j 3 zz jL of the outgoing continuum particle to the total spin J (see right hand side of Fig. 7 ) . Here the angular momenta of the continuum particles are denoted by j l which represents the coupling of the orbital angular momentum 1 of the continuum particle and its spin s = 112. The calculation of such matrix elements, for a surface-delta-force [ 11 + - V(1, 2) = -4nli,,S[r(l) - r(2)lS [ r ( l )- R], - -+I (51) Lf I J . J Fig. 7. Angular momentum coupling in tlits cixcitoii i n ~ d e lfor :I particle-particle interaction with bound particles only (left hand side) and one outgoing continuum particle (riglit hand side) Ann. Physik Leipzig 45 (1990) 8 G16 where r ( l ) ,r(2)denote the coordinates of the interacting particles and R represents the nuclear radius, is performed by usual shell model methods. Since only the matrix elements averaged over the complicated states contribute to the coupling widths in the multi-doorway-model (cf. eqs. (30) ... (34)), they must be averaged over the quantum numbers characterizing the exciton states. With the averaging rules outlined in another paper, the averaged matrix elements for the coupling widths between the finestructure subspaces L?nand 2m,L?n and Pmas 1 are shown in Figs. 8-12. Conservation of energy and Pmfor m = n, n well as leads to E =E - E, -U, (52) where E denotes the energy of the continuum-particle. E is the total energy, E B represents the binding energy and U the excitation energy of the residual nucleus. The widths for the transitions 2, + Pmand Pm-+ Prnare calculated for E = 2E, = 20 MeV where an incoming or outgoing s-wave neutron has been assumed. For the transitions L?,%-+ 2mthe energy interval 10 MeV _< E 5 40 MeV, already used in Fig. 6, is shown. In all cases the value of V,, in the surface-delta-force (cf. eq. (51)) was fixed a t V , = 0.53 MeV [I]. 0.025 25 I 0.020 0.015 / ~ I ~ i ' ' ~ I - 7 0.010 2.3 L I - N=2 N=3 _ _ _ _ _ _ N=L ____.__ __._ N=5 .. . . - 0.005 0.000 I f I I I I ~ _ I . 0.020 >, E n--. L" 0.015 0.010 0.005 0.000 5 al 0.010 E -& 0.005 0.000 0.0 0.2 0.L 0.6 0.8 1.0 U/E Fig. 8. Coupling widths for 2%+.L!Pm: L-0 The widths are shown for E = 2 E B = 20 MeV, J = 1 and H. SPANaENBERGER et a]., Multi-Doorway Continuum Shell Model 617 The widths for 1%+ 8, are shown in Fig. 8. The rapid decrease for U J E - 112 results from the total energy E = 2E, and the dependence of the continuum wave function as a function of E (cf. eq. ( 5 2 ) ) .Here it should be noticed that according to E = 2E, the excitation energy U is restricted to U 5 1/2E. As in Fig. 5 there is a dominance of the exciton pair creation process compared t o exciton scattering and exciton pair destruction. The same holds for the transitions 02, + 9, and Pa + 9, presented in + 8, are restricted to UIE 5 112 Figs. 9 and 10, respectively, where the widths for 8% by the same reason as in the case 9%+ 8,. The direct comparison of the transitions 9%+ 8, and 9%+ 9, for m = n, n rf 1 and N = 2, 4, 6, 8 (see Figs. 11 and 12) illuminates the dependence of the processes exciton pair creation, exciton scattering and exciton pair destruction on the exciton number N and the energy E. While for N = 2 (one-particle one-hole states) the transitions to N = 4 configurations (exciton pair creation) is dominant, the exciton scattering exceeds the other processes for N = 4. For N = 8, however the width of the transition 14+ P3 becomes stronger than the widths of 94+ P5 and 94+ P4.A similar behaviour can be found for the transition widths for 9,+ 9, although the widths for 9,+ .;2n+l dominate for higher excitation This means, that energies much more than the corresponding widths for 9%+ 8%+1. the transitions 1,+ 9n+lare stronger in comparison with the transitons 9,+ 9, 1 E 0.3 w 0.2 - 56 L 0.1 0.00 I ' I I I I I I 10 15 20 25 30 35 LO E (MeV) Pig. 9. Coupling widths for 9,--* 02,: The widths are shown for J = 1 618 Ann. Physik Leipzig 47 (1990) 8 and 3!?&-+ as well as the transitions between the 1- and .P-space. The primary doorway-state width thus depends mainly on the coupling to the bound complicated states in the finestructure-suhspaces l,tfor n 2 2. O3 t N=2 ___ ........... ........... ..................... -- a53 0.05 L I I I' .,' o,oo F , , , I ' 4.- . I I : ,I I 0.10 ... o,oo..-- '.\ ----____ ............ ----___-....... , ..,........"i 0.0 0.2 ,, *3;. I 0.4 , I 0.6 0.8 \ ~ a I 1.0 WE Fig. 10. Coupling widths for [=('=0 9% Pm: The widths are shown for E = 2E, = 2OMeV, J = 1 and 4. Application of the Multi-doorway Continuum Shell Model Tho multi-doorway continuum shell model presented in the previous chapter is used to calculate the magnetic dipole strength distribution in WNi. The results are then compared with the experimental data [ 191 from high-resolution inelastic electron scattering at the Darmstadt electron linear accelerator (DALINAC). These data have already been employed previously in a doorway state analysis [18]. First, we recall some salient features of the data which make them ideal candidates for the application of the multidoorway continuum shell model developed here. At second, the necessary ingredients of the calculation are described and the results are discussed. The magnetic dipole strength distribution in determined from backward angle j nelastic electron scattering is shown in the upper part of Fig. 13. It is the most fragmented magnetic strength distribution that is known so far in any nucleus. The strength, however, is concentrated in two, perhaps even three, bumps near excitation energies around E , w 8.4, 11.4 and 13MeV. This gross structure superimposed onto the fine 619 H. SPANGENBERGER et a]., Multi-Doorway Continuum Shell Model 0 010 0 005 0 000 00 02 OL 06 08 10 U/E Fig. 11. Coupling strengths between 9-and 8-space: The widths for 9, + Yntl (full linc), 9,--f 8, (dashed-dotted line) and 9,+ .YnPl (dashed line) are shown for N = 2n = 2,4, 6,8 structure is brought out clearly in the lower part of Fig. 13 where the Lorentz averaged strength distribution is characterized by center-of-gravity energies of E, a 8.5, 10.5 and 13.6 MeV. As has been demonstrated in ref. [19] the two large bumps can be associated with t h e To + To and To -+ To 1 isospin components of the magnetic dipole giant resonance i n 5*Ni (Note that To = 1is the isospin of the BNi ground state). The shell mod6 1 structure of these states is that of [(2p3/a);,l(1f;zf l f 5 , 2 ) 1 , 1 J J ~ , T = 1 + ,and 1 configurations, respectively, that are the possible iff$ Lf5,2 If:; 1f5!,),,,]Jn,T= [( particle hole excitations of the (2p3/2):,1 ground state. The observed fragmentation of + Ann. Physik Lsipzig 47 (1990) 8 620 I I I I I I / / 0.8 0.6 0.4 0.2 0.0 -t 5 L . / 0.L 0.8 c Q3->Q,/ 1.o 0.5 10 15 20 25 30 35 LO E (MeV) Fig. 12. Coupling strengths in the 9-space: The widths for 9, + A?n+ (full line), 2% -+ (dashed line) are shown for N = 2% = 2,4, 6,s dotted line) and 9,,+ 9n-l .?&(dashed- strength can then be interpreted as the coupling of two doorway states into more complicated many-particle many-hole configurations Iq ) The fit mentioned above of the two Lorentzians to the two prominent peaks seen in the data supports the two-doorway state picture and provides a good description of the averaged data up to an excitation energy of E, m 12.5 MeV where the third bump shows up in the strength distribution [18]. H. SPANGENBERGER e t al., Multi-Doorway Continuum Shell Model GZ1 We recall here briefly the procedure and the main outcome of the previous twodoorway state analysis of ref. [18]. According to the Lorentzians fitted t o the averaged experimental strength distribution the measured B ( M 1 ) fstrength was divided into two sets (see lower part of Fig. 13). The doorway state analysis performed for each set in terms of a K-matrix fit yielded the energy positions of the complicated states and their coupling matrix elements to the particular doorway state. From these quantities the energy positions Ed of the doorway states, the energy-dependent shifts A: and spreading widths have been calculated which describe the averaged theoretical strength function of the particular doorway state. A comparison between the measured and calculated distribution of the fine structure parameters indeed indicated the realization of a twodoorway state mechanism in the experimental data where the two doorway states could be attributed to a AT = 0 arid AT' = 1 ( f 7 / ~ f s l z )spin-flip excitation in the 5 W i nucleus noted above. This result makes the 5sNi magnetic dipole strength data a good candidate for the analysis within the multi-doorway continuum shell model. 1.2 I 1 1.0 A mz II - 0.8 c A f 0.6 ?5 0.1 0.2 4 s z To -> To n 3 To -> 1, + 10 12 1 cuz I Y -2 A A w c 1 0 8 11 Ex (MeV) Fig. 13. Upper p x t : Af1 strength in "Ni extracted from mensnred (e,e') spectra for 47 transitions in thc excitntion energy range from G t o 16 MeV (from ref. [t9]). Lower part: TMrentz awraged experimental strength function (full line) obtained with an averaging interval of I = 0.4 MeV. The dashed lines show three Lorentzians fitted t o the experimental data with a summed strength indicated by the dashed-dotted line. The t w o large bumps arc labeled by the assumed isospin changcs (from ref. [tS]) G22 Ann. Physik Leipzig 47 (1990) 8 The calculation starts with the description of the W i nucleus in a one-particle onehole model (for details see [l]),The bound single-particle states as well as the single-particle scattering states are eigenstates to the single-particle hamiltonian where the singleparticle potential is given by V O ( r= ) V j o ( r ) Va0(r)la V c ( r )6mt,i/z. f o ( r )is taken as a Woods-Saxon potential fO(4 = [I e q J { ( r - R)/a)I-l with R = r,,A1/' (r4 m 1.27 fm) and a m 0.5 fm. For the spin-orbit potential a Thomas term is assumed with + + + The divergent behaviour a t r + 0 is eliminated by setting Vso(r)= 0 for r < rnlin, where rminis defined [24, 251 by the relative minimum of I Vso(r) I. The Coulomb potential V,(r) is taken as that of a homogeneous sphere. While R and a are kept fixed, the parameters Toand il are extracted from a fit of the single particle energies and scattering phases t o experimental data. The set of single particle states is given in Table 1, while the calculated single particle energies and the fitted parameter V , are listed in Table 2. Table 1. Scattering-, bound- and hole states in "Xi scattering states bound states hole states Table 2. Potential parameter V, and single particle energies for tlie single particle states in LWi. Tlic residnal potential parameters are fixed a t R = 4.92 fm, a = 0.54 fm and 1 = 16 MeV fni2 To(MeV) state lSl/!2 (s,/2) 1P3/2 (P3/2) lP1/2 l d 5 I2 (CZ~/~) t'S1/, (.>112 ) %/2 'fi/Z (.f7,2) kV3ja t'Pij2 IfY2 13912 %!,a (fjiZ) (Ya,a) (S.i/%) 63.80 51.60 hL.20 53.50 53.80 49.6C 51.00 51.60 51.20 51.30 51.0(' 53.30 51-00 &,,(MeV) -44.78 -34.86 -33.06 -27.64 -23.41 -21.01 - 16.04 -30.25 -9.16 - 9.48 - 6.8G -2.62 1.80 Fo(MeV) 63.80 51.60 51.20 53.50 53.80 49.60 51.00 51.60 51.20 51.30 51.00 53.50 61.00 &,(MeV) -33.i9 -24.48 -22.58 -17.74 - 13.6 1 -11.01 -4.38 -1.88 0.38 0.36 "81 .5.33 8.33 H. SPANQENBERQER e t al., Multi-Doorway Continuum Shell Model 623 The two-particle matrix elements of the residual interaction are calculated with a surface-delta-force [I] which was already used for the calculation of the coupling widths within the exciton model (cf. chapt. 3). With this residual interaction the coupled system of integral equations (eq. ( 3 5 ) )simplifies and can be solved by usual continuum shell model methods [24, 251. The results are depicted in figs. (14-16) for a simple energy independent single-particle density (equal spacing level density). For a more realistic single-particle density calculated within a self-consistent approximation [261 the calculated average magnetic dipole strength distribution is shown in Fig. 1 7 . I n Fig. 14 two different treatments of the continuum coupling are compared. The dashed curve shows the result with the bound particle and hole states of Table 1 only, i.e. no coupling to the continuum is included. The dot-dashed curve, in contrast, contains the full continuum coupling. For comparison the averaged experimental strength distribution is also included. Comparing the two theoretical curves shows very clearly that the main effect of broadening of the primary doorways stems from the coupling to the bound 2 p 2 h configurations (‘Gamma down’). This coupling is about one order of magnitude stronger than the continuum coupling (‘Gamma up’). Significant contributions from continuum states show up for higher excitation energies (Ez> 11MeV) only. Here the continuum adds strength and gives more structure to the high energy shoiilder of the upper doorway state. The influence of the density of states in the two-particle two-hole space is studied in Fig. 15. Results for = 1.1eg, 0.9 pa, 0.8 eo are compared with the curve (solid line) for e = eo. Once more the sensitive behaviour of the results on the situation in the 2p-2h configuration space can be seen. The individual ‘satellite’ doorways show up very clearly already for a ten percent reduction of the single particle level density. Increase of this density by the same amount, however, does not alter much. This indicates the stability of the pronoi~nceddoorway states a t 8.5 and 10.5 MeV with respect to moderate changes in the coupling to the complicated states. e I ~ I ~ I ~ l ~ / - I 6 I I 8 I I 10 I I 12 I I , 14 Ex (MeV) Fig. 14. M1 strength distribution for 58Nicalculated within the multi-doorway continuurn shell modcl with a n energy indepcndcnt single-particle level density. Tho dashed-dotted line shows the result for a calculation with continuum coupling while tho dashed line is calculated without continuum coupling. The averaged experimrntal strength distribution (see loucr part of Pig. 13 is givrn by the full line I 624 Ann. Physik Leipzig 47 (1990)8 6 8 10 12 11 Ex (MeV) Fig. 16. M1 strength distribution shown for different values of the energy independent single-partiole level density go. The results are shown in the upper part for 1.1 go, in the center for 0.9 go and in the lower part for 0.8 go, where eo = 0.49 MeV-’ was assumed. The result for po is indicated in all parts by the full linc I n order to separate out the two isospin transitions AT = 0, 1, and to study their mixing, we have performed a calculation which includes the primary spin-flip transitions (l/71z + 1/5/2)only, but retains the full 2 p 2 h couplings. The results are shown in Fig. 16. We reproduce the dominant contribution of the A T = 0, 1transitioris to the M I strength in the energy range 7 MeV < IT < 12 MeV which is already known from conventional shell model calculations [19]. The structure in the strength distribution around 14 MeV (cf. Fig. 14), however, can not be accounted for by the spin-flip transitions alone. The influence of isospin mixing is seen by comparing the summed strengths of the pure isospin transitions (broken line) with the calculation including the mixing of the two components via Coulomb interactions (dashed-dotted line). The main influence of the H. SPANGENBERGERe t al., Multi-Doorway Continuum Shell Model 625 mixing is a shift of strength towards higher excitations. Interference also leads to a slight depletion of strength in the lower, and a corresponding increase in the higher, of the two pronounced peaks. The results for the M1 strength distribution in W i show that the multi-doorway continuum shell model is able t o account for the main characteristics observed in the energy-averaged experimental strength function. Especially the grouping of strength into two prominent lines at 8.3 and 10.8 MeV and their broadening is well reproduced. 6 8 10 12 14 Ex (MeV) Fig. 16. The (lf7,2-1,fE,2) spin-flip contribution t o theM1 strength in 58Ni. Shown are theAT = 0 and A T = 1components (full lines) and the summed strength with (dashed-dotted line) and without (dashed line) isospin mixing \ z2 N W n w w l u1 8 10 12 14 E, (MeV) Fig. 17. MI Htrength distribution for j*Si cdlrulatrd within the multi-doorway continuum shell model nith a single-particle level density obtained in B self-ronslstent approximation [%I. The dasheddotted h e shows the result for a calculation with continuum coupling while the dashed line is cnlculated without continuurn coupling. The averaged experimental strength distribution (see lower part of Pig. 13 is given by the full line Ann. Physik Leipzig 47 (1990)8 62G The widths are to a large extent due to the coupling to more complicated states while the influence of the continuum shows up a t excitation energies essentially outside the domain of the two prominent spin-flip transitions in 5Wi. This continuum coupling, however, can be treated in a consistent way in the model also, and it is expected t o become an essential mechanism a t energies well above the one-nucleon escape thresholds. The discrepancies with experiment regard the exact positions and widths of the maxima which come out too much separated, and which are smaller and higher (about 10%) in the calculation compared with the experimental distribution. I n this respect it has to be emphasized that for the model calculation presented here no attempt was made to optimize the model parameters with respect to experiment, but have been taken over from shell model calculations in W i [l,19, 271. I n order to study the influence of the very crude approximation of the energy independent single particle density employed, we have performed a calculation with a single particle density resulting from a selfconsistent treatment [26] also. The result is shown in Fig. 17. As could already be expected, the increase of the density of states a t higher excitations leads to a shift of strength into that region which gives an overall improvement with experiment. 5. Conclusions The partition of the total configuration space into two sets of N orthogonal finestructure subspaces, one set of bound states and one set of states with one particle in the continuum, describes the basic structure of the model. The assumed restrictions in the couplings between neighbouring finestructure subspaces introduces a hierarchy concept into the model where each state of complexity n acts as doorway state for the states of complexity n 1. This leads to the multi-doorway continuum shell model as outlined in the previous chapters. The elimination of the complicated secondary doorway states by projector mcthods leads to a coupled system of equations in the space of the primary doorway states where the influence of the secondary doorway states are treated by effective potentials dt scrihing the coupling between the primary and Secondary doorway states. By assumptions related to the statistical properties of the effective potentials the latter could be approximated by dynamically averaged matrix elements which are equivalent to mean couplings widths between the various finestructure subspaces. The calculation of these coupling widths within an exciton model (cf. chapter 3) shows that each coupling width is given by two factors, a statistical factor which is related to the statistical properties of the exciton model and a dynamical factor which depends on the individual two-particle matrix element of the interacting exciton pair and the angular momentum coupling. The coupling widths between the finestructure-subspaccs 9,and 9m,9,and 8, as well as $?Pnand gm,calculated within the exciton model, show, that the coupling of the primary doorway states with n = 1 to the bound secondary doorway states with n = 2 dominates the other couplings with n 2 2. Thus the spreading width P.1 is the dominant width of the primary doorway states. After calculation of thc various coupling widths the coupled system of equations of the multi-doorway continuum shell model could be solved. For this the usual methods of the continuurn shell model can be used. Finally, an application of the model t o the distribution of magnetic dipole strength in 58Nihas shown that the salient features of the fragmentation of strength could be reasonably well accounted for. + A c k n o w l e d g e m e n t . We thank J.Bar-Touv for providing u s with his level density for 5Wi, calculated within a self-consistent approximation. This work has been supported by Deutsche Forschungsgemeinschaft and by the German Federal Minister of Research and Technology (BMFT) under the contract number 06 DA 184 I. H. SPSNGENBERGER e t d . , &dti-Doorway Continuum Shell Modcl (i27 References [l] BBUSSAARD, P. J.; GLAUDEMANS, P. IT.M.: Shell Model Application in Nuclear Spectrosropy. North Holland 1977. [2] BERTSCH, G. F.: The Practitioners Shell Model. North Holland 1972. [3] BLOCH,C.: In: Proceedings of the Int. School of Physics “Enrico Fermi”, Course XXXVI, ed. C. BLOCH,New York: Academic Press 1986, p. 143. 141 MAHAUX, C.; WEIDENM~LLER, H. A.: Shell Model Approach t o Nuclear Reactions. North Holland 1969. [5] AGASSI,D.; WEIDENMULLER, H. A.: Phys. Lett. 56 B (1975) 305. [6] MANTZOURANIS, G.; AGASSI,D.; WEIDENMULLER, H. A.: Phys. Lett. 57 B (1975) 220. L T ] AGASSI,,D.; M’EIDEhWULLER, H. A. ; &‘IANTZOURANIS,G. : PllyS. Rep. c 22 (1975) 146. 181 MANTZOURANIS, G. ; WEIDENMULLER, H. A. ; AGASSI,1).: Z. Phys. A 85 6 (19iG) 145. 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(PIT. Y.) 5 (1962) 287; FESHBACH, H.: I n : Documents on Nodern Physics; Reaction Dynamics. New York: Gordon and Breach Publisher 1973. [22] SPANGENBERGER, H. : Dissertation Technische Hochschule Darmstadt, Germany, 1987 (unpublished). [13] WANG,W. L.; SHAKIN,C. M.: Phys. Rev. C 5 (1972) 1898. [24] BIRKHOLZ, P. F. J.: Pliys. Lett. 34 R (1971) 1; Nuel. Phys. A 189 (1972) 385. r25] BIRKHOLZ, P. F. J.; HEIL,V.: Nucl. Phys. A 225 (1974) 429. J.; MORDECHAI, S.: Phys. Lett. B 207 (1988) 227. [26] BAR-TOUV, [27] VANHEES,A. 0. M.; GLAUDEMANS, P. W.M.; METSCH,B. G.: Z. Physik A 893 (1979) 327. Bei der Redaktion eingegangen am IS. Januar 1990. Ansehr. d. Verf. : Dr. H. SPANQENBERGER Prof. Dr. F. BECK Prof. Dr. A. RICHTER Institut fur Kernphysik Technische Hochschule Darmstadt Schlossgartenstrafle 9 W-6100 Darmstadt

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