ANALYSIS OF QUASIRESONANT REACTIONS IN A HYBRID ANGULAR MOMENTUM SCHEME (II)

Transcription

1 ANALYSIS OF QUASIRESONANT REACTIONS IN A HYBRID ANGULAR MOMENTUM SCHEME (II) M. S. ATA Nuclear Research Centre, Atomic Energy Establishment, Cairo, Egypt H. COMISEL, C. HATEGAN Institute of Atomic Physics, Bucharest, Romania N. A. SLIAKHOV Physico-Technical Institute, Kharkov, Academy of Sciences, Ukraine Received March 30, 006 The deuteron-stripping threshold anomalies in reactions on A 90 mass target nuclei have been numerically analyzed on basis of the hybrid angular momentum scheme. 1. INTRODUCTION A method suitable for the numerical analysis of quasiresonant processes was established, in a previous paper, [1], in the framework of a hybrid angular momentum scheme. The quasiresonant process consists in direct transitions, preceded or followed by a single channel resonance, []. Direct transitions are approached in the transfer angular momentum coupling scheme while the resonant ones are usually represented in the total angular momentum scheme. A hybrid scheme for the both angular momentum couplings was proposed [1] for processing the direct and resonant transition amplitude elements. In this work, an extended numerical study of the deuteron-stripping threshold anomalies, as quasiresonant processes, has been performed for (d, p) reactions on A 90 mass target nuclei. The direct interaction was approached by using Distorted Waves in Born Approximation, (DWBA), [3, 4], while the anomal effect originating in the 3-p neutron single particle state at zero energy was described according to phenomenologic Lane Model, [5]. The phenomenologic Lane Model is outlined in Chapter ; the threshold compression factor of the 3-p 3/ resonant wave function was reviewed and similar calculation has been performed for the 3-p 1/ single particle state too. The IAEA Fellow. Formerly with Institute of Atomic Physics in Bucharest, Romania. Rom. Journ. Phys., Vol. 51, Nos. 7 8, P , Bucharest, 006

2 74 M. S. Ata et al. denominator of the quasi-resonant term of the collision matrix was written according to Lane parameterization for a p-wave single particle state. The transition amplitude elements were reevaluated in Chapter 3 for both direct and quasiresonant transitions. Three different methods for adding the two elements in the total transition amplitude have been implemented in the framework of a DWBA computer code. In Chapter 4 there were studied computational aspects for describing the threshold anomaly in the cross-section and analyzing power excitation functions. The input angular momenta partitions from deuteron entrance channel populating the p-wave proton channel have been analyzed for a d 5/ angular momentum transfer. The developed numerical methods and procedures have been applied in Chapter 5 for analyzing 3-p wave threshold effects reported in the (d, p) reactions on A 90 mass nuclei.. PHYSICS ASPECTS AND THEORETICAL MODELING OF DEUTERON-STRIPPING THRESHOLD ANOMALY The deuteron-stripping threshold anomaly is an example of quasiresonant process. It stands for a 3-p -wave neutron single particle resonance at zero energy, (peculiar feature for A 90 mass region nuclei), followed by a direct isospin interaction between the neutron threshold channel and the open analogue proton one. First evidences of deuteron-threshold anomaly did reveal that the anomaly is related to the opening of a neutron channel; the new open neutron channel is not any one but rather the isobar analogue channel coupled by isospin interaction to the proton observed one. C. F. Moore et al., [6], have observed an anomaly in excitation function of cross-section for the reaction 90 Zr(d, p) 91 Zr(g.s.) at the deuteron energy 7.05 MeV. This is the threshold energy of the neutron analogue channel. The exit reaction channels 91Nb + n = T 91Zr + t+ p and 91 Zr + p are, by definition, analogue channels. The analog channels are coupled by the isospin potential t T which contains the terms t + T and tt + (t, T isospins of emergent particles and residual nuclei). This fact has demonstrated that the isospin coupling of observed proton and threshold neutron channels is essential in producing the observed threshold anomaly. Consequently the first reaction model, [7], proposed for this anomaly was the Coupled Channel Born Approximation (CCBA). The calculations using CCBA model did evince a strong variation of the p 3/ -wave neutron wave functions near zero energy, [8]. One has concluded there is a correlation of the anomaly with the 3-p wave neutron single particle state at zero (threshold) energy for A 90 mass nuclei. The main physical aspects of 3-p wave threshold anomaly are, (a) isospin coupling t T of the analog (d, p) and

3 3 Analysis of quasiresonant reactions 743 ( d, n ) channels, (b) there exists a p-wave neutron state at zero energy for A 90 mass nuclei and (c) the interplay (coincidence) of 3-p wave neutron state with neutron threshold, [9]. The cross-section and analyzing power measurements have evinced the strongest threshold effect occurs in 88 Sr(d, p) 89 Sr reaction, see e.g. [5, 10]. The CCBA model reproduced the anomaly in cross-section but did not reproduce the anomaly in analyzing power, see e.g. [11]. In addition to this shortcoming the CCBA model has some other weak points, e.g. the threshold anomaly dependence with the Coulomb channel radius r C. Starting with conclusions of CCBA approach, Lane has proposed a phenomenologic model, [5], based on the interplay of the p-wave neutron single particle resonance with the threshold. In Lane s model, the S-matrix element of a (d, p) reaction is completed, just for the p-wave proton channel, by an additional term originating in the neutron single particle resonance, S j αdp dp Sdp + δl1 ε j (1) The Lane parameters α fix the strength of the isospin coupling between neutron and proton analogue channels, while the denominator, ( ε j E 0 Ej E i/ Γ), where the resonance position, E j, and total width, Γ, are energy dependent functions) has a peculiar energy dependence as it is outlined below. The resonance term is added to the background S-matrix only for the p-wave exit proton channel (l, j are orbital and total angular momenta of the emergent particle). The (quasi)resonant denominator is derived using the features of a neutron single particle state near zero energy. According to Lane and Thomas, [1], the wave function describing a bound state ascending to zero energy has to be renormalized. The renormalization factor (compression factor), is defined in R-Matrix theory as, where πn β ( E) = 1 = γ / 1 θ / ( ) πn ds de ds d ρ γ is the neutron reduced width, S(E) Shift Function, θ = 1 πn ma = γ ( / ) neutron dimensionless reduced width, h / ma Wigner width, m reduced mass, a channel radius, ρ = kr, k neutron wave number. The quantity, ds / d( ρ ), is negative for bound states and varies rapidly near zero energy; therefore a compression effect of the energetic scale for the 3-p wave neutron single particle state is occurred. ()

4 744 M. S. Ata et al. 4 The compression effect of a single particle resonance can be approached by using Optical Model (OM for deriving the scattering wave functions), and Shell Model (SM for determination of the eigenvalues and eigenfunctions of a single particle bound state). The compression factor can be written, in fact, as the ratio between a normalized bound state wave function, u(r), and its extension to infinity, f(r), [5]. Lane and Thomas have proved that in case of a quasibound nuclear state, the integral rather to the outer turning point a t, [1], f () r dr can be performed not to the infinity but 0 β ( E) = a 0 a u dr 0 at u dr + u( a) / f( a) f dr a (3) Here, the channel wave function f(r) has an asymptotic behaviour as f(r) exp(ikr) for E > 0 and f(r) exp( kr) for E < 0. The wave functions for the internal and external regions as well as their corresponding energies for the 3-p 3/ and 3-p 1/ neutron single particle states have been determined. The calculations have been performed using a DWUCK4 routine, dedicated to numerical analysis of stripping reactions on unbound states, [13], which has been adapted to fit for our purpose. One has used Ross optical potential and channel radii Lynn has determined in the systematic of single particle states, see Ref. [14]. The optical model parameters for the Woods-Saxon nuclear potential used in present calculations were: the radius parameter r 0 = 1.3 fm, the neutron well depth V 0 = 4.8 fm, the diffuseness d = 0.69 fm and the spin-orbit coupling λ = The compression factors have been computed by using equation (3) for each of the considered nuclear radius and the results are presented in Table 1. One could remark from the presented data, the values obtained for the 3-p 3/ single particle state did reproduce the compression factor derived by Lane. One should notice a similar energy variation of β(e) function for the 3-p 1/ neutron single particle state, see Fig. 1. The rapid energy variation of the renormalization factor near zero energy could be related to the special behaviour of the 3-p 3/ neutron wave function observed in OM calculations for the A 90 mass region, [8]. A similar effect should be observed also for the 3-p 1/ component, in the mass region A 110. A weak or even not observed threshold effect induced by 3-p 1/ -wave in A 110 mass region can not be explained by a nonsignificant amplitude of p 1/ wave, [15]; it should have other origins.

5 5 Analysis of quasiresonant reactions 745 Table 1 The compression factor β for 3-p 3/ and 3-p 1/ single particle neutron states for different p-wave energy eigenvalues (a channel radius). The factors β(e) have been determined within SM calculations for A ~ 90 mass target nuclei by using a Woods-Saxon potential with appropriated half-way well radii R 1/ (same as a channel radii) and energy eigenvalues given in Ref. [14] a [fm] E 3/ [MeV] β(e 3/ ) β(e 3/ ) (a) a [fm] E 1/ [MeV] β(e 1/ ) (a) From Ref. [5] Fig. 1 The Optical Model β(e) compression factor for j p = 3/ (circles) and j p = 1/ (diamonds) p-wave neutron state. The dashed and continuous lines represent fitted curves using theoretical approach in the framework of R Matrix Theory for the energy derivatives of the S(E) Shift function. The denominator of the resonance S-matrix element is written in R-matrix terms as, ε 1 j = 1 E E ( S + ip b) γ i/γ j 1 1 where j = 3/, 1/ is the total spin of the proton p-wave channel, E j is the energy of the resonance (corresponding to b boundary conditions at a channel radius), S l = 1 and P l = 1 are the Shift and Penetration functions respectively, calculated according to R-matrix theory, γπn and Γ reduced width and total width of the p-wave neutron single particle resonance. As we already discussed, the Shift and Penetration functions have a very rapid behaviour near zero energy. This feature does distinguish Lane term from an usual Breit-Wigner form. It also provides a good description of the experimental characteristics of 3-p -wave threshold anomaly (a typical KeV energy shift from the neutron analogue threshold and a half-width around 700 KeV). πn (4)

6 746 M. S. Ata et al. 6 One can rewrite the energy variation of the resonant denominator ε j (E) in terms of the dimensionless reduced width θ and the channel radius a. Using Lane parameterization, the resonant denominator could be written as, One has defined, 1 1 ε j = x 1/ ρ ( S S ) θ i( y+ Pθ ) ka 1 n E Q ma j 0 1 j [ j ( 1(0) ) πn]( / ) x = E S b γ ma energy shift terms, ρ = ( ) = ( / ), y = 1/ Γ ( / ma) 1, k n neutron wave number and Q represents the Q-value of the analogue-isobar neutron threshold channel. The (S S 0 )θ and Pθ terms have been obtained by using R-matrix values of the Shift and Penetration functions, [1], and a linear expansion of the Shift function near zero energy. The resonant denominator for a p-wave single particle state could be consequently written for E < 0, as while for E > 0, 1 j ε = xj 1 j ε = xj 1 1/ ρ ρθ /(1 +ρ) iy 1 + 1/ ρθ /(1 +ρ) i[ y+ρ3θ /(1 +ρ)] These formulae of the resonant denominator will be used in the following for the numerical description of deuteron-stripping threshold anomaly. Lane has estimated [5] the dimensionless reduced width θ = 4 for the 3-p 3/ neutron single particle state for a certain channel radius, a = 8 fm, by comparing the R-Matrix compression factor derived for negative energies, β ( E) = 1 (8) 1 +θ ( ρ+ )/( ρ+ 1) to the numerical results obtained via Shell Model calculations. On the other hand, from neutron-scattering measurements it was established the influence the nuclear diffuseness is manifesting on the single particle reduced width. Vogt has obtained an empirical formula describing the variation of the reduced width versus diffuseness radius, [16]. By taken into account a diffuseness of 0.69 fm, one obtains a reduced width close to Lane s one. Other studies of the resonance s width behaviour near zero energy have been carried out, see e.g. Kadmensky et al., [17]. In this work one has derived energy dependence obtained by Lane for the compression factor of the 3-p 3/ single particle state. Additionally calculations (5) (6) (7)

7 7 Analysis of quasiresonant reactions 747 have been performed for the 3-p 1/ state too. In Fig. 1, the compression coefficients β(e) derived within Shell Model calculations (see Table 1; filled circles for 3-p 3/ state and diamonds for 3-p 1/, respectively) were fitted by using Eq. (8) and represented with solid and dotted lines. The following reduced width and channel radius parameters for the j p = 3/ and j p = 1/ neutron single particle resonances have been obtained: θ = 4, a = 8 fm and θ = 3.7 and a = 7.6 fm, respectively. The cross section threshold effect becomes about 10% weaker when our values for θ and a are used in case of a 3-p 1/ resonance. In order to evaluate the resonant denominator ε j, Lane has calculated y = while x j coefficients may vary in the [ 3, 3] interval. These values were obtained from the following estimations. Using a = 8 fm, the Wigner unit width / ma becomes /3 while for a total spreading width Γ of about 3 MeV ones get y =. A spin-orbit splitting of E 1/ E 3/ = 4/3 MeV means x 1/ x 3/ =. Assuming [ 4/3, 4/3] MeV interval for the variation of the averaged resonance energy, it results a variation range of [ 3, 3] for the x j coefficients. Fig. Energy dependence of Lane denominator for the 3-p single particle resonance. The curves are drawn from left to right for different resonance energy positions, x j = 3, x j = 1, x j = 1 and x j = 3. Using the above parameters, one has analyzed the energy dependence of 1/ ε j term near the analogue neutron channel threshold for x j coefficients values in the considered [ 3, 3] range, see Fig.. Note that for the largest admitted x 1/ = 3 value, its magnitude is decreased by a factor of. However, the magnitude of 1/ε j term preserves a nearly same value for the other allowed shift coefficients. 3. TRANSITION AMPLITUDE The anomal transition amplitude element corresponding to R-matrix element was derived within a compatible DWBA angular momentum coupling scheme, [1],

8 748 M. S. Ata et al. 8 π mm, mm d a p B < ls d d0md jm d d>< ji d Amm d A JM> 1 J jp ( 1)( 1),, ( ) d p Rldjdlpjd p p mpm m θ (9) B kd ll d d Jjdjp < ls p p0mp jm p p >< ji p BM mm B B JM> T = l + l + where m d, m A, m p and m B denote the magnetic quantum numbers of deuteron, target nucleus, proton and residual target respectively, M is the projection of total angular momentum J and d jp m, ( p ) p M m θ is the rotational matrix at θ p B scattering angle. According to Lane model, the reaction matrix element used to describe a p-wave single particle resonance is, R α J ldjd; lp= 1, jp= 1/,3/ ma ldjd, lpj = p j = 1/,3/ Ej E S1 ip1 b γπn i p ( / ) ( + ) Γ / (10) where (l d, j d ) denotes the values of orbital and total deuteron angular momenta in the entrance channel corresponding to (l p = 1, j p = 1/, 3/) proton one while J is the total angular momentum and the α Lane parameters fix the strength of the threshold anomaly. The direct interaction process is approached by using Distorted Waves Born Approximation (DWBA), [3, 4]. According to DWUCK code, [18], the direct transition amplitude term is, β 4 mmpmd Tmmmm = π B j 1 IAjmAmB ma IBmB l 1Alsjt B p A d kk A + < > + lsj (11) d p lsj where l, s, j and m are the orbital angular momentum, spin, total spin and its projection for the transferred neutron, A, B are the masses of the target and residual nuclei, respectively, A lsj is the spectroscopic amplitude of the residual nuclear state while the reduced transition amplitude is defined by mmpm lsj t d = d p = < ls0m jm > i < ls0 m jm >< j jm mmjm, > jl jl d d p p l l l d d d d d p p p p p p d d d ls j p p p lpjp, ldjd jp p d p d p d lsj m, m m p < ll00 l0 > (l + 1) (s + 1)( j + 1)(l + 1) lsj X d ( θ ) p d ls d dj d (1) The quantities X lpjp, ldjd lsj denote for the radial integrals,

9 9 Analysis of quasiresonant reactions 749 j A ( ) lpjp, ldjd jp jd Xlsj = A dra l kp, rd flsj( rd) ( d, d) p l k r B χ χ (13) B d j p d Here, χ l and χ p l are the proton and deuteron radial distorted waves and f d lsj is the nuclear form factor. The radial integrals do contain the energy dependence of the transition amplitude element in the entrance and exit channels. These are responsible for channel energy dependences in a same way as the penetration factor does in R-Matrix Theories. The methods regarding the sum of the above transition amplitude elements have been outlined in our previous paper, [1]. One has reviewed, however, the main procedures by emphasizing some computational aspects regarding the present numerical analysis. The total transition amplitude is given by the sum, T = T β + T π (14) This is straightforward to be accomplished for a zero spin target nucleus. In such a case, I A = 0 results in J j d and M m d. One may notice that Clebsch-Gordan coefficients multiplied by the rotational matrix from expression (9) become identical with those from relation (11). The anomal reaction matrix element will be added to the DWBA radial integrals, [19, 0], lpjp, ldjd lpjp, ldjd J lsj lsj l j, l j X X (1 + R ) (15) J d d p p whether the α coefficients from R ldjd, lpj term are replaced by new α quantities p defined by, [1], ls p pjp lpjp, ldjd αl ;, ~ djdlpjp ls j Xlsj α ldjd; lp, j (16) p ls d dj d From computational point of view, this method is a straightforward choice for adding the two amplitude transition elements. However, the above 9-j Wigner symbol forces the total angular momentum coupling scheme to coincide with the transferred angular momentum scheme, describing a resonant reaction and a direct mechanism, respectively. Because the two coupling schemes are not identical, some transitions from entrance channel (l d, j d ) to the final channel (l p, j p ) will be consequently lost, see e.g. Ref [1]. Besides this shortcoming, the above addition method implies larger computer timings if a minimization procedure is used for searching the α parameters. The second addition method consisted in a separate computing of the anomal transition amplitude term. The result will be added at the level of the reduced transition amplitude in the DWBA term (11),

10 750 M. S. Ata et al. 10 β π mm m mm m mm m t = t + t (17) p d p d p d where we have omitted the indices (l, s, j) from the background term. One has to define a reduced resonant transition amplitude similar to the DWBA by starting from the definition of the reaction cross section for a resonance process, [1], π mm m t = EE d p kd 1 A 1 T 4π k l+ 1 B D 0 π m m, m m p d d a p B p (18) where D 0 = 13.5 MeV Fm 3/ is the product of the neutron-proton potential times the deuteron internal function in zero range approximation. This choice requests a special routine devoted for numerical calculation of the anomal transition amplitude which has to match to the DWUCK routines. The validation of the results has been performed by using the following numerical procedure. The Lane coefficients should be multiplied by corresponding DWBA radial integrals. In this way, the computed cross sections should be identically up to a factor with the ones obtained within the first addition method. We shall consider this procedure, in the following, as the third method used in this work to construct the total transition amplitude term. On the other hand, the multiplication of Lane parameters by radial integrals has also a physical meaning. If the resonance would be shifted far away from neutron threshold, the resonant-like term should behave as proton p-wave background, []. In that limit, Lane parameter should have an energy dependence as (d, p) DWBA radial integrals corresponding to l p = 1 proton partial wave. In case of considering a minimization algorithm to be applied for searching the Lane parameters, the last two methods are more suitable than the first one. These methods act in a straightforward and fast way for computing the theoretical function provided by Lane anomal term only, while initially DWBA calculations have supplied in advance for the direct reduced amplitude transition elements. The 3-p threshold anomaly was observed mainly for deuteron-stripping reactions on zero spin target nuclei from the A 90 mass region. The stripping reactions on non zero spin target nuclei yielding residual nuclei with zero spin can be also approached using the above addition procedures on basis of the reciprocity theorem for the DWBA inverse reaction cross section. Even when the target as well as the residual nuclei have both non zero spins, one still can use the first method on basis of the correspondence between the DWBA radial integrals and the collision matrix, [4, 1]. An exact calculation of the total transition amplitude, i.e. of the excitation cross section functions, supposes a reevaluation of DWUCK numerical routines for the determination of DWBA cross section. Now the sum of the transition amplitudes could no longer be done at the

11 11 Analysis of quasiresonant reactions 751 level of the reduced transition amplitude elements. One has to derive the entire expressions for the (9) and (11) terms. The DWBA cross section, dσ μμ k = 1 1 Tmmmm dω B p A d ( π ) k J + s + a b b a A 1 a 1 mmmm A B d p (19) is computed in DWUCK code by using the orthogonality property of Clebsch- Gordan coefficients when these are added for the m A and m B magnetic indices. Basically, the sum is performed upon the squares of reduced amplitude transition elements, B ( ) 1 dσ JB + 4 ka = π 1 l+ 1A t dω J + 1E E k A j+ 1 A d p b mm m ls mmpmd lsj lsj (0) d p Here, E d and E p are the center of mass energies for the entrance and exit channels, respectively. In this case, the background transition amplitude, T β mmmm B p A d, should be supplementary determined in DWUCK code and added to its correspondent resonant term in Eq. (14). One may choose, consequently, within three methods for adding the transition amplitude terms. The first one is constrained by the angular momentum transferred coupling scheme while the others two did compute almost separately the direct and anomal resonant transition amplitudes. However, the numerical results, i.e. the computed excitation functions of cross section or analyzing power describing a quasiresonant effect, should be almost the same, irrespective what addition method has been used. 4. NUMERICAL APPROACH The 3-p threshold anomaly results from the interference of the (d, p) background and the threshold resonance terms; single-dip or double-dip resonance shapes can be then numerically achieved in agreement to the experimental evidence. For most of the reported deuteron-stripping anomalies, the spin and parity of the target nuclei is 0 + while the ground state of the residual target is 5/ +. The p 3/ and p 1/ -wave proton channel is consequently related according to the total angular momentum coupling scheme to the following (l d, j d ) angular momentum partitions in the deuteron entrance channel: (1, 1); (1, ); (3, ); (3, 3); (3, 4); (5, 4) and (1, ); ( 3, ); (3, 3), respectively. In order to describe the characteristics of the anomaly, the Lane coefficients, α ldjd; lp= 1, jp= 1/,3/, should be replaced by their strength and phase numerical values. Each of them is related to a given partition in the deuteron-entrance and proton-exit channels.

12 75 M. S. Ata et al. 1 In first step of the present numerical approach, one has concerned to a separate analysis of the behaviour of the excitation functions for each of the above considered (l d, j d ) partitions in the deuteron entrance channel. A straightforward study of the anomal cross-section shape is obtained starting from the cross-section expansion of a quasiresonant process in terms of a background and an anomal S-matrix element, * dp Sdp Sdp Sdp Sdp Sdp σ ~ +Δ + Re( Δ ) (1) One has neglected the square of the anomal S-matrix element, Δ S dp ~0. The shape of the deuteron-stripping anomaly will be consequently described by the interference of the background and the anomal terms, * E / ~Re( * )~Re ~Re * j E i Γ Δσ SdpΔSdp S α dp S * dpα ε j ( Ej E) + 1/4 Γ () where one has assumed the energy dependence of the resonant denominator as, εj ~ Ej E i/ Γ. The interplay between the phases of S dp background term and α * coefficients should determine the shape of the anomaly. Four typical shapes could be attended while changing the relative phase Φ within [0, π] range: a reversed S-like form (Φ(S dp α * ) ~ 0), a resonance form (Φ(S dp α * ) ~ π/), a S-like form (Φ(S dp α * ) ~ π) and a resonant dip form (Φ(S dp α * ) ~ 3π/). A generic example is presented in Figs. 3 using a single (l d = 3, j d = 4) deuteron partition in the entrance channel. The above predictions of the anomaly s shape are not entirely reproduced for each allowed deuteron partitions by running an adapted based DWUCK code. Its additional routines derive the quasiresonant transition amplitude elements, summing them to the DWBA background. The numerical tests did evince that for some partitions, e.g. (l d = 3, j d = ), (l d = 3, j d = 3) or (l d = 5, j d = 4), the deuteron-stripping anomaly shape could not be acquired neither for the cross-section nor for the analyzing power excitation functions, irrespective what addition method has been used. To understand the different manifestations regarding the partition selection in the deuteron entrance channel, a deeper introspection of DWBA cross section term has been performed. According to Eq. (17) and Eq. (0), the anomal cross section could be approximated by a coherent sum of background DWBA and anomal reduced transition amplitude elements, β Δσ ~ Re( t t ) (3) mm m π* mm m p d p d

13 13 Analysis of quasiresonant reactions 753 (a) (b) Fig. 3 (a) The computational single-dip cross-section derived within method 1 at 160 angle using α(l d = 3, j d = 4, j p = 3/) and the following phases: 0, π/, π and 3π/. The calculations have been performed with x j =, Q dn = 7.3 MeV and optical model parameters corresponding to the deuteron-stripping reaction on 88 Sr target. In the figure there are listed the relative phases between S dp and α * terms. (b) The energetic shift of the resonant dip cross-section for x j = 3, 1, 1, 3. The dashed lines in both figures represent the DWBA background cross-section. implying constructive or destructive interferences between its terms. The four analyzed shapes of the computational cross-section should be reproduced for each element of Eq. (3) but not necessary for their sum. A numerical example is illustrated in Fig. 4. The experimental crosssection for the 88 Sr(d, p) 89 Sr deuteron-stripping reaction at 160 scattering angle and the computed cross-section obtained for two partitions of the deuteron angular momentum in the entrance channel, (l d = 1, j d = ) and (l d = 3, j d = ), are represented in Fig. 4 (a) by filled circles, solid line and dashed lines, respectively. As one can see, the experimental threshold anomaly is described for the (l d = 1, j d = ) entrance channel. The two main components of the sum from Eq. (3), corresponding to m p = 1/, m d = 1, m = 1/ and m p = 1/, m d = 0, m = 1/ spin projections, have same relative phases, see Fig. 4 (b) solid lines. The anomal effect is not reproduced for (l d = 3, j d = ) partition due to the opposite phases of the two main components, see Figs. (4) dashed lines. Numerical tests have proved that the destructive interference occurring for some angular momenta partitions in deuteron entrance channel is given by the behaviour

14 754 M. S. Ata et al. 14 Fig. 4 (a). Differential cross section, experimental (filled circles) and computational [solid line for (l d, j d ) = (1, ) and dashed line for (l d, j d ) = (3, )] of the 88 Sr(d, p) 89 Sr deuteron stripping reaction at 160 scattering angle. (b). Constructive (solid lines) or destructive (dashed lines) interferences of the two main transition amplitude elements corresponding to the magnetic indices sets, (m p = 1/, m d = 1, m = 1/) and (m p = 1/, m d = 0, m = 1/), result in a successful or improper description of the deuteron-stripping threshold anomaly. of the Clebsch-Gordan coefficients describing the quantum kinematical part of the transition amplitude elements. These kinematical factors are also responsible for the cancellation of the contributions to the anomal cross-section corresponding to other angular momenta partitions, e.g. (l d = 3, j d = 3) and (l d = 5, j d = 4). The cross-section features of the deuteron-stripping threshold anomaly can be well described for the other partitions in the entrance channel, i.e. (l d = 1, j d = 1), (l d = 1, j d = ), (l d = 3, j d = 4) and (l d = 1, j d = ), (l d = 3, j d = 3) corresponding to p 3/ and p 1/ -proton wave, respectively. One has investigated the phase values of Lane coefficients describing a resonant dip shape of the anomaly for each of the entrance deuteron angular momentum corresponding to the p 3/ or p 1/ -wave proton exit channel. The obtained phases of α coefficients for the three proposed adding methods are presented in Table. The difference j Table p The phases ϕ ld j of Lane coefficients generating a resonant anomal dip cross-section for stripping d reaction at 160 scattering angle. In the first line and first column of the table, there are given the angular momentum partitions and the three addition methods. The presented values provide only a qualitative description for the resonant dip shape of the anomaly. The obtained ones from fit of experimental data will be listed in the following section (l d j d ) j p (1, 1) 3/ (1, ) 3/ (3, 4) 3/ (1, ) 1/ (3, 3) 1/ ϕ I ϕ II ϕ III π/ π π/ π/ 0 3π/ π π 0 π/ 0 3π/ π π 0

15 15 Analysis of quasiresonant reactions 755 of the phases obtained using the third and the second adding methods, Δϕ = = ϕ III ϕ II, is about π/ for l d = 1 and π for l d = 3 deuteron orbital momenta. These values correspond to the phases of the radial integrals multiplying the Lane coefficients. In Fig. 5 one has represented the phases extracted from DWBA code for the two considered deuteron angular momenta. One may also remark the difference phase for two different orbital momenta partitions is related to the relative phases of their corresponding radial integrals. Fig. 5 The phase of the lp 1jp 3/, ld 3jd X = = = = l 1/sj 5/ lp 1jp 3/, ld 1jd X = = = = l= s= 1/j= 5/ and = = = radial integrals, versus incident energy, for a deuteron-stripping reaction on a 88 Sr target at 160 scattering angle. The radial integrals correspond to l = 1 and l = 3 deuteron orbital momenta in the entrance channel and to a p 3/ wave in the proton exit channel, respectively. The excitation functions of the analyzing power, A y, should have a similar dependence on the relative phases, Φ(S dp α * ), as the cross-sections ones, by the following relation, * ΔAy ~Im S α dp * (4) ε Similar to the computed cross-section, the analyzing power excitation function behaves differently with the angular momenta partitions from the entrance channel. The numerical results obtained by using the partitions able to describe a resonant dip and the appropriated Lane coefficients are displayed in Figs. 6 (a), (b). One may notice the change of the shape of the anomal vector analyzing power from the S or reversed S form at backwards angles to the resonance or resonance dip form at forward scattering angles. The anomal effect is larger in analyzing power for (l d = 3, j d = 4) deuteron angular momenta while a nearly same depth is reproduced in cross-section for all the analyzed partitions. The analyzing power for the p 1/ proton wave has an opposite phase in respect to p 3/

16 756 M. S. Ata et al. 16 Fig. 6 Numerical analyses for the shape of analyzing power for different deuteron angular momentum partitions in the entrance channel. The excitation functions have been derived using α s phases generating a dip shape of the cross-section anomaly for 88 Sr ( d, p) 89 Sr reaction at 160 scattering angle. (a) The dotted-dashed, continuous and dotted lines represent the excitation functions for the p 3/ proton exit channel calculated for the following deuteron partitions: (l d = 1, j d = 1), (l d = 1, j d = ) and (l d = 3, j d = 4), respectively. (b) Similar representations obtained for the p 1/ proton exit channel with (l d = 1, j d = ) continuous line and (l d = 3, j d = 3) dotted line. component; this effect can be associated to the well known Yule-Haeberli rule, [3] established for the stripping reactions with polarized incident particles. The (d, p) stripping reaction at forward angles is controlled mainly by kinematical effects, e.g. transfer of linear momentum, [3, 8, 4]. Consequently the nuclear threshold effects are less discernable for this forward angle region. The nuclear effects become important outside of stripping peak, at backward angles (θ > 130 ). The cross-section could evince the nuclear threshold effects, even at forward angles, via spectroscopic factors, (θ 130 ). Indeed the spectroscopic factors should be independent on deuteron energy. However

17 17 Analysis of quasiresonant reactions 757 deviation from constant value of spectroscopic factors was found near the threshold, by analyzing the anomaly of 90 Zr(d, p) 91 Zr reaction, [5]. It is an indirect proof of the effect in cross-section at forward angles. On the other hand the threshold effect should be present in analyzing power at all measured angles. Indeed the A y is sensitive not to the magnitude of background but rather to interference effect and the anomaly is a genuine interference effect. Considering more than one angular momentum partitions in jp the entrance channel, the anomal resonant terms, T π ( α l ) dj will interfere as well. d Numerical calculations have been performed for separated or mixed angular momentum partitions in the deuteron entrance channel corresponding to the 3-p 3/ or 3-p 1/ -wave proton exit channel. The results will be discussed next section for all analyzed deuteron-stripping anomalies reported on A 90 mass region. Fig. 7 Resonance-dip (solid line) and double-dip (dotted line) shapes in the cross-section obtained for two input deuteron partitions corresponding to the p 3/ and p 1/ waves in proton channel. The analogue threshold energy is chosen here, Q dn = = 6.5 MeV, the resonance energetic shift, x j =. The dashed line represents the DWBA cross-section background calculated at θ = 160 scattering angle for a generic deuteron stripping reaction from the analyzed mass region. The Lane Model can reproduce also a double-dip shape of the anomaly, see [5]. According to Lane simulations, the double-dip anomaly is numerically obtained by the interference of the spin-orbit transition amplitude components corresponding to the p-wave resonance in the proton exit channel and the existing shift of the resonance E 3/ and E 1/ energies. In a first step, one has performed a resonance-dip shape of the anomaly by combining a resonance (for the exit p 3/ -wave) with a dip (for the exit p 1/ -wave), see Fig. 7, solid line. The analogue threshold energy does correspond for the ascendent slope of the background cross-section. According to Table, the phases of α coefficients (method I) are ϕ I (j p = 3/) = 0 and ϕ I (j p = 1/) = π. A similar result should be obtained for the descendent part of background cross-section by using reversed

18 758 M. S. Ata et al. 18 phases of α coefficients and an appropriated Q dn -value. A genuine double-dip shape has been obtained too, see dotted line in Fig. 7. However, one has to modify the total width parameter y from y = to y 0.5 in order to get the double-dip behaviour; this requires a very high compression factor. 5. ANALYSIS OF DEUTERON-STRIPPING THRESHOLD ANOMALY The analysis of 3-p threshold anomalies, both in analyzing power and cross-section excitation functions, have been performed for single dip crosssection shapes observed in deuteron-stripping reactions on the following target nuclei: 80 Se, [6], 86 Kr, [7], 88 Sr, [19, 8], 90 Zr, [9, 10, 4], 9 Zr, [30], 9 Mo, [10, 30, 94 Zr, [30], 94 Mo [30], and 106 Cd, [10, 0]. One has considered the description of the double-dip cross-section shape reported from 9 Zr(p, d) 91 Zr, [31], and 96 Zr(p, d) 95 Zr, [3], pick-up reactions too. One has to mention also that experimental analyzing power measurements are limited for only few deuteron stripping reactions listed above ( 88 Sr ( d, p) 89 Sr, 90 Zr ( d, p) 91 Zr, 9 Mo ( d, p) 93 Mo and 106 Cd (, d p) 107 Cd). The DWBA background was determined using a global set of optical model (OM) parameters, (averaged for the considered A 90 mass region), corresponding to deuteron and proton scattering in the entrance and exit channels, respectively. The OM parameters for the deuteron channel are the following, [33], V = 11.5 MeV, W = 5.8 MeV, V SO = 6.79 MeV, r = 1.06 fm, r W = fm, r SO = fm, r C = 1.3 fm, a = fm, a W = fm, a SO = fm. For the proton OM scattering one has used, [34], V = 58.5 MeV, W = 46.0 MeV, V SO = 6. MeV, r = 1.17 fm, r W = 1.3 fm, r SO = 1.01 fm, r C = 1.3 fm, a = 0.75 fm, a W = 0.6 fm, a SO = 0.75 fm. The OM parameters have been corrected for the energy dependence by using extrapolation relations, [34]. The spectroscopic factors were calculated for each of the analyzed deuteronstripping reaction, see Table 3. Some discrepancies regarding the angular distribution and excitation functions of cross section observed in the experimental data at backwards scattering angles are provided from different experimental measurements ( e.g. deuteron-stripping reactions on 90,9,94 Zr target nuclei). In description of the deuteron-stripping threshold anomaly by means of Coupled Channel Born Approximation formalism, a normalization coefficient was defined instead using the apriori known spectroscopic factor, see for example Refs. [7, 8]. The differences between the two values should have to lay out in the limits of experimental measurement errors. As a precautionary measure, before starting the numerical analysis of deuteron-stripping anomaly, the DWBA cross section was previously determined at backwards scattering angles

19 19 Analysis of quasiresonant reactions 759 Table 3 Spectroscopic factors of the residual target nuclei ground states taken from the following references: (a) Ref. [35]; (b) Refs. [36, 37]; (c) Ref. [38]; (d) Ref. [39]; (e) Ref. [40]; (f) Ref. [41]; (g) Ref. [4]; (h) Ref. [43]; (i) Ref. [44]; (j) Ref. [45]. The corresponding values obtained by using our defined OM parameters are displayed in the third column Residual Target Nuclei Spectroscopic Factors 81 Se(1/ ) 0.31 (a) Kr(5/ + ) 0.56 (b) Sr(5/ + ) 0.79 (c), 1.03 (d) Zr(5/ + ) 0.75 (e), 1.09 (f) Zr(5/ + ) 0.64 (g) Zr(5/ + ) 0.3 (g), 0.34 (h) Mo(5/ + ) 0.84 (i) Mo(5/ + ) 0.59 (i) Cd(5/ + ) 0.5 (j) 0.30 and compared with the corresponding experimental excitation functions. In case of no absolute differential cross-section data are provided, e.g. deuteron-stripping reactions on, 80 Se [6] or 106 Cd [10], target nuclei, a similar normalization factor was carried out in the present calculations. The DWBA direct interaction code DWUCK and additional routines devoted to the anomal resonant term have been adjusted to a minimization library code, MINUIT, [46]. This is a powerful numerical tool for estimating the best fit parameter values and uncertainties including their correlations. The obtained hybrid code, QUASIRES, is able to perform minimizations of the cross-section and analyzing power excitation functions for deuteron-stripping reactions using the three different numerical procedures for adding the direct and anomal transition amplitudes and to search for the best fit parameters including j l j α Lane coefficients. The Lane parameters will be replaced in the p d d minimization procedure by two fit parameters: a strength, amplitude like parameter and a phase one. The spectroscopic factor of the involved residual nuclear state may be considered as additional fit parameter. However, its variation is constrained, as a rule, to some limitations, which can not exceed ten to fifteen percents from the standard evaluated value. The x j energy shifts of the 3-p 3/ and 3-p 1/ single particle resonances enter also in the minimization procedure as fit parameters. Considering all partitions for a I A = 0 + I B = 5/ + transition, one has obtained 0 fit parameters (1 for 3/ p α and its phases, 6 for p 1/, α x j parameter and the spectroscopic factor).

20 760 M. S. Ata et al. 0 One has focused on the strongest threshold effect reported on 88 Sr ( d, p) 89 Sr deuteron-stripping reaction. The minimization procedure using free fit parameters without any given constraints does result in correlations between α parameters, both for their strengths or phases. Considering more than one deuteron partition in the entrance channel, the anomal resonant terms, jp T π ( α l ), dj will interfere each others. To avoid numerical correlations as well as d to predict the interferences between terms, one has to analyze the reaction excitation functions using separated deuteron entrance partitions or mixed ones with defined constraints rules for the parameters. In Figs. 8, the experimental data of the analyzing-power (0, 40, 60, 90, 10, 140 and 160 ) and cross-section (160 ) excitation functions are fitted by using single incoming partitions, (l d = 1, j d = ) Fig. 8 (a), (l d = 3, j d = 4) Fig. 8 (b), for p 3/ -wave and (l d = 1, j d = ) Fig. 8 (c), (l d = 3, j d = 3) Fig. 8 (d), for p 1/ -wave, respectively. The obtained strengths and phases have been listed in Table 4. One has to observe a large spread of the α s strengths with respect to the choice of selecting the input deuteron partitions. The cross-section and analyzing power excitation functions are both reproduced only for the p 3/ - wave. One should remark however a less quantitative description of the experimental analyzing power data. The functions are both reproduced only for the p 3/ -wave. One should remark however a less quantitative description of the experimental analyzing power data. The computational analyzing power functions in case of the p 1/ -wave are in opposite phase in respect with the experimental ones. An approach for considering the mixing of different input deuteron partitions is to neglect the spin-orbit interaction in the input deuteron channel. This assumption may be related to the weakness of the spin-orbit interaction in the deuteron entrance channel and it was used in former numerical analyzing of the deuteron-stripping anomaly in Refs. [10, 19, 0]. The partitions are defined only by the deuteron orbital angular momentum, e.g. for the proton p 3/ -wave there are three partitions l d = 1 (j d = 1, ), l d = 3 (j d =, 3, 4) and l d = 5 (j d = 4) and two ones for the p 1/ -wave, l d = 1 (j d = ) and l d = 3 (j d =, 3). The fit parameters obtained by using l d = 1 and l d = 3 orbital angular momenta for reproducing the anomaly cross-section for 88 Sr(d, p) 89 Sr stripping reaction at θ = 160 scattering angle are given in Table 5. Another numerical approach for describing the threshold anomaly using mixed partitions is to maintain the phases from the previous single partition fits (or to allow a some given variance range) and to search for the strengths of the α parameters only. This method is suitable to avoid the destructive interferences between the various input deuteron angular momenta partitions. Within the

21 1 Analysis of quasiresonant reactions 761 Fig. 8 Experimental (filled circles) and computational (solid lines) excitation functions for 88 Sr ( d, p) 89 Sr deuteron-stripping reaction: (a) l d = 1, j d =, j p = 3/; (b) l d = 3, j d = 4, j p = 3/; (c) l d = 1, j d =, j p = 1/; (d) = 3, j d = 3, j p = 1/. numerical procedure, one has considered two parameters for describing the strengths of α coefficients. These parameters correspond to the spin-orbit splitting of the p-wave single particle resonance in the exit channel of the (d, p) stripping reaction, i.e. α p3/ and α p1/, respectively, in a similar way to the spinorbit splitting of the 3-p -wave Neutron Strength Function. The results obtained using all the input deuteron partitions are presented in Fig. 9. The experimental anomaly cross-section is very well reproduced by the theoretical function. Because one has used an averaged set of optical model parameters, the computational analyzing power does reproduce the shape rather than the absolute

22 76 M. S. Ata et al. Table 4 Numerical results for deuteron single partitions used in minimization of the experimental crosssection at θ = 160 scattering angle for the 88 Sr (d, p) 89 Sr stripping reaction (l d, j d ) (1, ) j p 3/ (3, 4) 3/ (1, ) 1/ (3, 3) 1/ α I (rel. u.) ϕ I (rad.) α II (rel. u.) ϕ II (rad.) α III (rel. u.) ϕ III (rad.) Table 5 Numerical results for deuteron mixed partitions used in minimization of the experimental crosssection at θ = 160 scattering angle for the 88 Sr (d, p) 89 Sr stripping reaction j p 3/ 1/ l d 1 j d 1 α I (rel. u.) ϕ I (rad.) α II (rel. u.) ϕ II (rad.) α III (rel. u.) ϕ III (rad.) values of the the experimental data for all the analyzed scattering angles, especially at forwards angles. For the unfavorable cases, one has mainly focused on the shape of the anomaly rather than its amplitude. The improvement of reaction DWBA background by choosing other optical model parameters may increase the fit quality but certainly does affect the purpose of our work: a unitary description of the deuteron-stripping threshold anomalies in A 90 mass region. Peculiar optical model values may describe the threshold effect by using DWBA model only. From a physical insight, the deuteron stripping threshold anomaly should be described by α coefficients irrespective on the deuteron entrance channel related to the p-wave exit one, i.e. the deuteron-stripping anomaly should not depend on the entrance deuteron channel from the reaction process. The obtaining of similar values for Lane coefficient strengths in describing the deuteron-stripping anomaly cross-section, irrespective the deuteron entrance channel, comes to sustain the physical idea that the anomaly should not be dependent on the deuteron entrance channel, see for example Ref. [10]. However, our numerical results did not find the same strengths for the α parameters, irrespective the entrance deuteron partitions, as one may notice from Tables 4, 5.

23 3 Analysis of quasiresonant reactions 763 Fig. 9 Experimental (filled circles) and computational (solid lines) excitation functions for 88 Sr ( d, p) 89 Sr (left) and 90 Zr ( d, p) 91 Zr (right) deuteron-stripping reactions. A more plausible assumption should be reproducing of spectroscopic properties of α coefficients for entire analyzed mass region A ~ 90, namely the relationship of proportionality between the strengths of α parameters and the 3-p Neutron Strength Function, [47]. The strengths of the deuteron-stripping anomalies should follow the mass dependence of the Neutron Strength Function irrespective one has considered single or mixed partitions of the entrance deuteron angular momenta. The 3-p threshold anomaly observed in 90 Zr ( d, p) 91 Zr reaction crosssection, [6], is the next analyzed stripping reaction of the present work. Similarly to the case of the optical model parameters, the α s phases, resulting from the

24 764 M. S. Ata et al. 4 non diagonal terms of the background S dp matrix and from the fluctuation averaging, [47], should vary smoothly while changing the mass in the A 90 region. As a consequence, the phases of α parameters should be constrained to preserve their values. However, the relative phase between the direct and resonant transition amplitudes may depend from one stripping reaction to another, due to the energy variation of the radial integral phases, see e.g. Fig. 5. The α s phases from Table 4 established for 88 Sr ( d, p) 89 Sr reaction, should be not necessary identical with those describing an anomal dip for other deuteronstripping reactions. In Fig. 9 (right) one has represented the experimental analyzing power at 10, 140 and 160 scattering angles and the experimental cross-section at 155 for 90 Zr ( d, p) 91 Zr reaction, both with our computational results with circles and solid lines, respectively. A weaker effect was measured in the excitation functions of 9 Mo ( d, p) 9 Mo reaction, see Fig. 10. The set of average optical parameters did provide a poor description of the DWBA background for 160 scattering angle analyzing power. In Fig. 11 (a) and 11 (b), the excitation functions of 106 Cd ( d, p) 107 Cd stripping reaction are represented for the 10 and 160 scattering angles. The threshold effect for this reaction is quit small in the experimental analyzing power. The reproducing of analyzing power data was realized using both the p 3/ and p 1/ spin-orbit components of α parameters. The Yule-Haeberli rule could explain the destructive interference of the p 3/ and p 1/ contributions to the Fig. 10 Experimental (filled circles) and computational (solid lines) excitation functions for 9 Mo ( d, p) 93 Mo deuteron-stripping reaction at 10, 140 and 160 scattering angles.