ANALYSIS OF QUASIRESONANT REACTIONS IN A ...ANALYSIS OF QUASIRESONANT REACTIONS IN A HYBRID ANGULAR MOMENTUM SCHEME (II) M. S. ATA Nuclear Research Centre, Atomic Energy Establishment,

ANALYSIS OF QUASIRESONANT REACTIONSIN 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, 2006

The deuteron-stripping threshold anomalies in reactions on A ∼ 90 mass targetnuclei have been numerically analyzed on basis of the hybrid angular momentumscheme.

1. INTRODUCTION

A method suitable for the numerical analysis of quasiresonant processeswas established, in a previous paper, [1], in the framework of a hybrid angularmomentum scheme. The quasiresonant process consists in direct transitions,preceded or followed by a single channel resonance, [2]. Direct transitions areapproached in the transfer angular momentum coupling scheme while theresonant ones are usually represented in the total angular momentum scheme. Ahybrid scheme for the both angular momentum couplings was proposed [1] forprocessing the direct and resonant transition amplitude elements.

In this work, an extended numerical study of the deuteron-strippingthreshold anomalies, as quasiresonant processes, has been performed for (d, p)reactions on A ∼ 90 mass target nuclei. The direct interaction was approached byusing Distorted Waves in Born Approximation, (DWBA), [3, 4], while theanomal effect originating in the 3-p neutron single particle state at zero energywas described according to phenomenologic Lane Model, [5].

The phenomenologic Lane Model is outlined in Chapter 2; the thresholdcompression factor of the 3-p3/2 resonant wave function was reviewed andsimilar calculation has been performed for the 3-p1/2 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. 741–767, Bucharest, 2006

742 M. S. Ata et al. 2

denominator of the quasi-resonant term of the collision matrix was writtenaccording to Lane parameterization for a p-wave single particle state. Thetransition amplitude elements were reevaluated in Chapter 3 for both direct andquasiresonant transitions. Three different methods for adding the two elements inthe total transition amplitude have been implemented in the framework of aDWBA computer code. In Chapter 4 there were studied computational aspectsfor describing the threshold anomaly in the cross-section and analyzing powerexcitation functions. The input angular momenta partitions from deuteron entrancechannel populating the p-wave proton channel have been analyzed for a d5/2

angular momentum transfer. The developed numerical methods and procedureshave been applied in Chapter 5 for analyzing 3-p wave threshold effects reportedin the (d, p) reactions on A ∼ 90 mass nuclei.

2. PHYSICS ASPECTS AND THEORETICAL MODELINGOF DEUTERON-STRIPPING THRESHOLD ANOMALY

The deuteron-stripping threshold anomaly is an example of quasiresonantprocess. 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 isospininteraction between the neutron threshold channel and the open analogue protonone.

First evidences of deuteron-threshold anomaly did reveal that the anomalyis related to the opening of a neutron channel; the new open neutron channel isnot any one but rather the isobar analogue channel coupled by isospin interactionto the proton observed one. C. F. Moore et al., [6], have observed an anomaly inexcitation function of cross-section for the reaction 90Zr(d, p)91Zr(g.s.) at thedeuteron energy 7.05 MeV. This is the threshold energy of the neutron analoguechannel. The exit reaction channels 91 91Nb n T Zr t p− ++ = + and 91Zr p+ are,by definition, analogue channels. The analog channels are coupled by the isospinpotential t T⋅ which contains the terms t T+ − and t T− + (t, T – isospins ofemergent particles and residual nuclei). This fact has demonstrated that theisospin coupling of observed proton and threshold neutron channels is essentialin producing the observed threshold anomaly. Consequently the first reactionmodel, [7], proposed for this anomaly was the Coupled Channel BornApproximation (CCBA).

The calculations using CCBA model did evince a strong variation of thep3/2-wave neutron wave functions near zero energy, [8]. One has concluded thereis a correlation of the anomaly with the 3-p wave neutron single particle state atzero (threshold) energy for A ≈ 90 mass nuclei. The main physical aspects of 3-pwave threshold anomaly are, (a) isospin coupling t T⋅ of the analog (d, p) and

3 Analysis of quasiresonant reactions 743

( , )d n channels, (b) there exists a p-wave neutron state at zero energy for A ≈ 90mass nuclei and (c) the interplay (coincidence) of 3-p wave neutron state withneutron threshold, [9].

The cross-section and analyzing power measurements have evinced thestrongest threshold effect occurs in 88Sr(d, p)89Sr reaction, see e.g. [5, 10]. TheCCBA model reproduced the anomaly in cross-section but did not reproduce theanomaly in analyzing power, see e.g. [11]. In addition to this shortcoming theCCBA model has some other weak points, e.g. the threshold anomaly dependencewith the Coulomb channel radius rC.

Starting with conclusions of CCBA approach, Lane has proposed aphenomenologic model, [5], based on the interplay of the p-wave neutron singleparticle 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 additionalterm originating in the neutron single particle resonance,

1

jdp

dp dp lj

S Sα

→ + δε

(1)

The Lane parameters α fix the strength of the isospin coupling between neutronand proton analogue channels, while the denominator, 0( / 2 ),j E jE E i→ε ≈ − − Γ

where the resonance position, ,jE and total width, Γ, are energy dependent

functions) has a peculiar energy dependence as it is outlined below. Theresonance term is added to the background S-matrix only for the p-wave exitproton channel (l, j are orbital and total angular momenta of the emergentparticle).

The (quasi)resonant denominator is derived using the features of a neutronsingle particle state near zero energy. According to Lane and Thomas, [12], thewave function describing a bound state ascending to zero energy has to berenormalized. The renormalization factor (compression factor), is defined inR-Matrix theory as,

2 221 1( )

1 / 1 2 / ( )n

EdS dE dS dπ

β = =+ − θγ ρ

(2)

where 2nπγ is the neutron reduced width, S(E) – Shift Function, 2 =θ

12 22( / )n ma−

π= γ – neutron dimensionless reduced width, 2 2/h ma – Wigner

width, m – reduced mass, a – channel radius, ρ = kr, k – neutron wave number.The quantity, 2/ ( ),dS d ρ is negative for bound states and varies rapidly near zeroenergy; therefore a compression effect of the energetic scale for the 3-p – waveneutron single particle state is occurred.


The compression effect of a single particle resonance can be approached byusing Optical Model (OM – for deriving the scattering wave functions), andShell Model (SM – for determination of the eigenvalues and eigenfunctions of asingle particle bound state). The compression factor can be written, in fact, as theratio between a normalized bound state wave function, u(r), and its extension toinfinity, f(r), [5]. Lane and Thomas have proved that in case of a quasibound

nuclear state, the integral 2

0( )f r dr

∞

∫ can be performed not to the infinity but

rather to the outer turning point at, [12],

2

0

2 2 2 2

0

( )( ) / ( )

t

a

a a

a

u drE

u dr u a f a f drβ =

+

∫∫ ∫

(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 theircorresponding energies for the 3-p3/2 and 3-p1/2 neutron single particle stateshave been determined. The calculations have been performed using a DWUCK4routine, dedicated to numerical analysis of stripping reactions on unbound states,[13], which has been adapted to fit for our purpose. One has used Ross opticalpotential and channel radii Lynn has determined in the systematic of singleparticle states, see Ref. [14]. The optical model parameters for the Woods-Saxonnuclear potential used in present calculations were: the radius parameterr0 = 1.3 fm, the neutron well depth V0 = –42.8 fm, the diffuseness d = 0.69 fmand the spin-orbit coupling λ = 39.5. The compression factors have beencomputed by using equation (3) for each of the considered nuclear radius and theresults are presented in Table 1. One could remark from the presented data, thevalues obtained for the 3-p3/2 single particle state did reproduce the compressionfactor derived by Lane. One should notice a similar energy variation of β(E)function for the 3-p1/2 neutron single particle state, see Fig. 1.

The rapid energy variation of the renormalization factor near zero energycould be related to the special behaviour of the 3-p3/2 neutron wave functionobserved in OM calculations for the A ≈ 90 mass region, [8]. A similar effectshould be observed also for the 3-p1/2 component, in the mass region A ≈ 110. A

weak or even not observed threshold effect induced by 3-p1/2 -wave in A ≈ 110mass region can not be explained by a nonsignificant amplitude of p1/2 wave,[15]; it should have other origins.


Table 1

The compression factor β for 3-p3/2 and 3-p1/2 single particle neutron states for different p-waveenergy eigenvalues (a – channel radius). The factors β(E) have been determined within SMcalculations for A ~ 90 mass target nuclei by using a Woods-Saxon potential with appropriated

half-way well radii R1/2 (same as a channel radii) and energy eigenvalues given in Ref. [14]

a [fm] E3/2 [MeV] β(E3/2) β(E3/2)(a) a [fm] E1/2 [MeV] β(E1/2)

6.186.256.326.5756.8

0–0.15–0.42–1.57–2.8

0.130.210.270.390.45

0.110.210.270.390.45

6.376.426.476.566.656.847.0

0–0.11–0.24–0.61–1.03–1.95–2.92

0.140.210.240.310.360.540.53

(a) From Ref. [5].

Fig. 1 – The Optical Model β(E) compressionfactor for jp = 3/2 (circles) and jp = 1/2(diamonds) p-wave neutron state. The dashedand continuous lines represent fitted curvesusing theoretical approach in the frameworkof 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-matrixterms as,

12

1 1

1( ) / 2j

j nE E S iP b i−

π

=ε′− − + − − Γγ

(4)

where j = 3/2, 1/2 is the total spin of the proton p-wave channel, Ej is the energyof the resonance (corresponding to b boundary conditions at a channel radius),Sl = 1 and Pl = 1 are the Shift and Penetration functions respectively, calculated

according to R-matrix theory, 2nπγ and Γ′ – reduced width and total width of the

p-wave neutron single particle resonance. As we already discussed, the Shift andPenetration functions have a very rapid behaviour near zero energy. This featuredoes distinguish Lane term from an usual Breit-Wigner form. It also provides agood description of the experimental characteristics of 3-p -wave thresholdanomaly (a typical 100–200 KeV energy shift from the neutron analoguethreshold and a half-width around 700 KeV).


One can rewrite the energy variation of the resonant denominator εj(E) interms of the dimensionless reduced width θ2 and the channel radius a. UsingLane parameterization, the resonant denominator could be written as,

12 2 2

0

11/ 2 ( ) ( )j

jx S S i y P− =ε

ρ − − θ − + θ∓(5)

One has defined, 2 2 2 11[ ( (0) ) ]( / )j j nx E S b ma −

π= − − γ – energy shift terms,2 2 2 2 1( ) 2 ( / ) ,nk a E Q ma −ρ = = − 2 2 11/ 2 ( / ) ,y ma −′= Γ kn – neutron wave

number and Q represents the Q-value of the analogue-isobar neutron thresholdchannel.

The (S – S0)θ2 and Pθ2 terms have been obtained by using R-matrix valuesof the Shift and Penetration functions, [12], and a linear expansion of the Shiftfunction near zero energy. The resonant denominator for a p-wave single particlestate could be consequently written for E < 0, as

12 2 2

11/ 2 /(1 )j

jx iy−ε =

− ρ − ρ θ + ρ −(6)

while for E > 0,

12 2 2 3 2 2

11/ 2 /(1 ) [ /(1 )]j

jx i y−ε =

+ ρ θ + ρ − + ρ θ + ρ(7)

These formulae of the resonant denominator will be used in the following for thenumerical description of deuteron-stripping threshold anomaly.

Lane has estimated [5] the dimensionless reduced width θ2 = 4 for the 3-p3/2

neutron single particle state for a certain channel radius, a = 8 fm, by comparingthe R-Matrix compression factor derived for negative energies,

2 21( )

1 ( 2) /( 1)Eβ =

+ θ ρ + ρ +(8)

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 reducedwidth. Vogt has obtained an empirical formula describing the variation of thereduced width versus diffuseness radius, [16]. By taken into account a diffusenessof 0.69 fm, one obtains a reduced width close to Lane’s one. Other studies of theresonance’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 thecompression factor of the 3-p3/2 single particle state. Additionally calculations


have been performed for the 3-p1/2 state too. In Fig. 1, the compressioncoefficients β(E) derived within Shell Model calculations (see Table 1; filledcircles for 3-p3/2 state and diamonds for 3-p1/2, respectively) were fitted by usingEq. (8) and represented with solid and dotted lines. The following reduced widthand channel radius parameters for the jp = 3/2 and jp = 1/2 neutron single particleresonances have been obtained: θ2 = 4, a = 8 fm and θ2 = 3.7 and a = 7.6 fm,respectively. The cross section threshold effect becomes about 10% weakerwhen our values for θ2 and a are used in case of a 3-p1/2 resonance.

In order to evaluate the resonant denominator εj, Lane has calculated y = 2while xj coefficients may vary in the [–3, 3] interval. These values were obtained

from the following estimations. Using a = 8 fm, the Wigner unit width 2 2/ mabecomes ≈ 2/3 while for a total spreading width Γ′ of about 3 MeV ones gety = 2. A spin-orbit splitting of E1/2 – E3/2 = 4/3 MeV means x1/2 – x3/2 = 2.Assuming [–4/3, 4/3] MeV interval for the variation of the averaged resonanceenergy, it results a variation range of [–3, 3] for the xj coefficients.

Fig. 2 – Energy dependence of Lanedenominator for the 3-p single particleresonance. The curves are drawn from leftto right for different resonance energy positions, xj = –3, xj = –1, xj = 1 and xj = 3.

Using the above parameters, one has analyzed the energy dependence of2

1/ jε term near the analogue neutron channel threshold for xj coefficients

values in the considered [–3, 3] range, see Fig. 2. Note that for the largestadmitted x1/2 = 3 value, its magnitude is decreased by a factor of 2. However, themagnitude of 1/εj term preserves a nearly same value for the other allowed shiftcoefficients.

3. TRANSITION AMPLITUDE

The anomal transition amplitude element corresponding to R-matrixelement was derived within a compatible DWBA angular momentum couplingscheme, [1],


, ,,

0

1( )(2 1)(2 1) 2

0

p

d d p p p Bd a p B

d d d p

d d d d d d A d A

jJpl j l j m M md pm m m m d

l l Jj j

p p p p p p B B B

l s m j m j I m m JM

R dT l l k

l s m j m j I M m m JM

π−

< ⎪ >< ⎪ >

θ= + +

< ⎪ >< − ⎪ >

∑ ∑ (9)

where md, mA, mp and mB denote the magnetic quantum numbers of deuteron,target nucleus, proton and residual target respectively, M is the projection of

total angular momentum J and , ( )p

p B

jpm M md − θ is the rotational matrix at pθ

scattering angle.According to Lane model, the reaction matrix element used to describe a

p-wave single particle resonance is,

2 2; 1, 1/ 2,3 / 2

, 21 11/ 2,3 / 2

( / )

( ) / 2d d p p

d d p p

p

l j l jJl j l j

jj n

maR

E E S iP b i

= =

= π

α=

′− − + − − Γγ∑ (10)

where (ld, jd) denotes the values of orbital and total deuteron angular momenta inthe entrance channel corresponding to (lp = 1, jp = 1/2, 3/2) proton one while J isthe total angular momentum and the α Lane parameters fix the strength of thethreshold anomaly.

The direct interaction process is approached by using Distorted WavesBorn Approximation (DWBA), [3, 4]. According to DWUCK code, [18], thedirect transition amplitude term is,

4 2 1 2 1 p d

B p A d

mm mA A B A B B lsj lsjm m m m

d p lsj

BT j I jm m m I m l A tk k A

β π= + < − ⎪ > +∑ (11)

where l, s, j and m are the orbital angular momentum, spin, total spin and itsprojection for the transferred neutron, A, B are the masses of the target andresidual nuclei, respectively, Alsj is the spectroscopic amplitude of the residualnuclear state while the ‘reduced’ transition amplitude is defined by

,,

0 0 ,

00 0 (2 1) (2 1)(2 1)(2 1) ( )

p d

d p

d d p p

p p d d p

p d

mm mlsj

l l ld d d d d p p p p p p d d d

j l j l

p p pl j l j j

p d p d p d plsj m m m

d d d

t

l s m j m i l s m j m j jm m m j m

l s j

l l l l s j l l s j X d

l s j

− −

−

=

= < ⎪ > < ⎪ >< − ⎪ >

⎛ ⎞⎜ ⎟

< ⎪ > + + + + θ⎜ ⎟⎜ ⎟⎝ ⎠

∑ (12)

The quantities ,p p d dl j l jlsjX denote for the radial integrals,


( ), , ( ) ( , )p p d d p d

p d

l j l j j jA p d lsj d d dlsj l l

A AX dr k r f r k rB B

= χ χ∫ (13)

Here, p

p

jlχ and d

d

jlχ are the proton and deuteron radial distorted waves and flsj is

the nuclear form factor. The radial integrals do contain the energy dependence ofthe transition amplitude element in the entrance and exit channels. These areresponsible for channel energy dependences in a same way as the penetrationfactor does in R-Matrix Theories.

The methods regarding the sum of the above transition amplitude elementshave been outlined in our previous paper, [1]. One has reviewed, however, themain procedures by emphasizing some computational aspects regarding thepresent numerical analysis.

The total transition amplitude is given by the sum,

T = Tβ + Tπ (14)

This is straightforward to be accomplished for a zero spin target nucleus. In sucha case, IA = 0 results in J ≡ jd and M ≡ md. One may notice that Clebsch-Gordancoefficients multiplied by the rotational matrix from expression (9) becomeidentical with those from relation (11). The anomal reaction matrix element willbe added to the DWBA radial integrals, [19, 20],

, ,,(1 )p p d d p p d d

d d p p

l j l j l j l j Jlsj lsj l j l jX X R→ + (15)

whether the α coefficients from ,d d p p

Jl j l jR term are replaced by new α′ quantities

defined by, [21],

,; , ; ,~ p p d d

d d p p d d p p

p p pl j l j

l j l j l j l jlsj

d d d

l s j

l s j X

l s j

⎛ ⎞⎜ ⎟ ′α α⎜ ⎟⎜ ⎟⎝ ⎠

(16)

From computational point of view, this method is a straightforward choice foradding the two amplitude transition elements. However, the above 9-j Wignersymbol forces the total angular momentum coupling scheme to coincide with thetransferred angular momentum scheme, describing a resonant reaction and adirect mechanism, respectively. Because the two coupling schemes are notidentical, some transitions from entrance channel (ld, jd) to the final channel(lp, jp) will be consequently lost, see e.g. Ref [1]. Besides this shortcoming, theabove addition method implies larger computer timings if a minimizationprocedure is used for searching the α parameters.

The second addition method consisted in a separate computing of theanomal transition amplitude term. The result will be added at the level of thereduced transition amplitude in the DWBA term (11),

750 M. S. Ata et al. 10

p d p d p dmm m mm m mm mt t tβ π= + (17)

where we have omitted the indices (l, s, j) from the background term. One has todefine a reduced resonant transition amplitude similar to the DWBA by startingfrom the definition of the reaction cross section for a resonance process, [21],

,0

1 14 2 1p d d a p B

d p dmm m m m m m

p

E E k At Tk l B D

π π=π +

(18)

where D0 = –123.5 MeV Fm3/2 is the product of the neutron-proton potentialtimes the deuteron internal function in zero range approximation. This choicerequests a special routine devoted for numerical calculation of the anomaltransition amplitude which has to match to the DWUCK routines. The validationof the results has been performed by using the following numerical procedure.

The Lane coefficients should be multiplied by corresponding DWBA radialintegrals. In this way, the computed cross sections should be identically up to afactor with the ones obtained within the first addition method. We shall considerthis procedure, in the following, as the third method used in this work toconstruct the total transition amplitude term.

On the other hand, the multiplication of Lane parameters by radial integralshas also a physical meaning. If the resonance would be shifted far away fromneutron threshold, the resonant-like term should behave as proton p-wavebackground, [22]. In that limit, Lane parameter should have an energydependence as (d, p) DWBA radial integrals corresponding to lp = 1 protonpartial wave.

In case of considering a minimization algorithm to be applied for searchingthe Lane parameters, the last two methods are more suitable than the first one.These methods act in a straightforward and fast way for computing thetheoretical function provided by Lane anomal term only, while initially DWBAcalculations have supplied in advance for the direct reduced amplitude transitionelements.

The 3-p threshold anomaly was observed mainly for deuteron-strippingreactions on zero spin target nuclei from the A ∼ 90 mass region. The strippingreactions on non zero spin target nuclei yielding residual nuclei with zero spincan be also approached using the above addition procedures on basis of thereciprocity theorem for the DWBA inverse reaction cross section. Even when thetarget as well as the residual nuclei have both non zero spins, one still can usethe first method on basis of the correspondence between the DWBA radialintegrals and the collision matrix, [4, 21]. An exact calculation of the totaltransition amplitude, i.e. of the excitation cross section functions, supposes a re-evaluation of DWUCK numerical routines for the determination of DWBA crosssection. Now the sum of the transition amplitudes could no longer be done at the


level of the reduced transition amplitude elements. One has to derive the entireexpressions for the (9) and (11) terms. The DWBA cross section,

2

2 21 1

2 1 2 1(2 ) B p A d

A B d p

a b bm m m m

a A a m m m m

kd Td k J s

μ μσ =Ω + +π ∑ (19)

is computed in DWUCK code by using the orthogonality property of Clebsch-Gordan coefficients when these are added for the mA and mB magnetic indices.Basically, the sum is performed upon the squares of reduced amplitude transitionelements,

( )2

22 1 4 1 2 12 1 2 1

p d

d p

mm maBlsj lsj

A d p b mm m ls

kJd B l A td J E E k A j

+σ π= +Ω + +∑ ∑ (20)

Here, Ed and Ep are the center of mass energies for the entrance and exitchannels, respectively. In this case, the background transition amplitude,

,B p A dm m m mTβ 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 isconstrained by the angular momentum transferred coupling scheme while theothers two did compute almost separately the direct and anomal resonanttransition amplitudes. However, the numerical results, i.e. the computed excitationfunctions 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-dipresonance shapes can be then numerically achieved in agreement to theexperimental 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 residualtarget is 5/2+. The p3/2 and p1/2-wave proton channel is consequently relatedaccording to the total angular momentum coupling scheme to the following(ld, jd) angular momentum partitions in the deuteron entrance channel: (1, 1);(1, 2); (3, 2); (3, 3); (3, 4); (5, 4) and (1, 2); ( 3, 2); (3, 3), respectively. In orderto describe the characteristics of the anomaly, the Lane coefficients,

; 1, 1/ 2,3 / 2 ,d d p pl j l j= =α should be replaced by their strength and phase numerical

values. Each of them is related to a given partition in the deuteron-entrance andproton-exit channels.

752 M. S. Ata et al. 12

In first step of the present numerical approach, one has concerned to aseparate analysis of the behaviour of the excitation functions for each of theabove considered (ld, jd) partitions in the deuteron entrance channel.

A straightforward study of the anomal cross-section shape is obtainedstarting from the cross-section expansion of a quasiresonant process in terms of abackground and an anomal S-matrix element,

2 2 *~ 2 Re( )dp dp dp dp dp dpS S S S Sσ + Δ ≈ + Δ (21)

One has neglected the square of the anomal S-matrix element, 2

~ 0.dpSΔ The

shape of the deuteron-stripping anomaly will be consequently described by theinterference of the background and the anomal terms,

** *

* 22

/ 2~ Re( ) ~ Re ~ Re

( ) 1/ 4

jdp dp dp dp

j j

E E iS S S S

E E

⎛ ⎞⎛ ⎞ − − Γα ⎜ ⎟Δσ Δ α⎜ ⎟⎜ ⎟ ⎜ ⎟ε − + Γ⎝ ⎠ ⎝ ⎠(22)

where one has assumed the energy dependence of the resonant denominator as,~ / 2 .j jE E iε − − Γ The interplay between the phases of Sdp background term

and α* coefficients should determine the shape of the anomaly. Four typicalshapes could be attended while changing the relative phase Φ within [0, 2π]range: a reversed S-like form (Φ(Sdpα*) ~ 0), a resonance form (Φ(Sdpα*) ~ π/2),a S-like form (Φ(Sdpα*) ~ π) and a resonant dip form (Φ(Sdpα*) ~ 3π/2). A genericexample is presented in Figs. 3 using a single (ld = 3, jd = 4) deuteron partition inthe entrance channel.

The above predictions of the anomaly’s shape are not entirely reproducedfor each allowed deuteron partitions by running an adapted based DWUCKcode. Its additional routines derive the quasiresonant transition amplitudeelements, summing them to the DWBA background. The numerical tests didevince that for some partitions, e.g. (ld = 3, jd = 2), (ld = 3, jd = 3) or (ld = 5,jd = 4), the deuteron-stripping anomaly shape could not be acquired neither forthe cross-section nor for the analyzing power excitation functions, irrespectivewhat addition method has been used.

To understand the different manifestations regarding the partition selectionin the deuteron entrance channel, a deeper introspection of DWBA cross sectionterm has been performed. According to Eq. (17) and Eq. (20), the anomal crosssection could be approximated by a coherent sum of background DWBA andanomal reduced transition amplitude elements,

*~ Re( )p d p dmm m mm mt tβ πΔσ ∑ (23)


(a) (b)

Fig. 3 – (a) The computational single-dip cross-section derived withinmethod 1 at 160° angle using α(ld = 3, jd = 4, jp = 3/2) and the followingphases: 0, π/2, π and 3π/2. The calculations have been performed withxj = 2, Qdn = 7.3 MeV and optical model parameters corresponding to thedeuteron-stripping reaction on 88Sr target. In the figure there are listed therelative phases between Sdp and α* terms. (b) The energetic shift of theresonant dip cross-section for xj = –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 fouranalyzed shapes of the “computational” cross-section should be reproduced foreach element of Eq. (23) but not necessary for their sum.

A numerical example is illustrated in Fig. 4. The experimental cross-section for the 88Sr(d, p)89Sr deuteron-stripping reaction at 160° scattering angleand the computed cross-section obtained for two partitions of the deuteronangular momentum in the entrance channel, (ld = 1, jd = 2) and (ld = 3, jd = 2),are represented in Fig. 4 (a) by filled circles, solid line and dashed lines,respectively. As one can see, the experimental threshold anomaly is describedfor the (ld = 1, jd = 2) entrance channel. The two main components of the sumfrom Eq. (23), corresponding to mp = –1/2, md = 1, m = 1/2 and mp = 1/2, md = 0,m = 1/2 spin projections, have same relative phases, see Fig. 4 (b) – solid lines.The anomal effect is not reproduced for (ld = 3, jd = 2) partition due to theopposite phases of the two main components, see Figs. (4) – dashed lines.Numerical tests have proved that the destructive interference occurring for someangular momenta partitions in deuteron entrance channel is given by the behaviour

754 M. S. Ata et al. 14

Fig. 4 – (a). Differential cross section, experimental (filled circles) andcomputational [solid line for (ld, jd) = (1, 2) and dashed line for (ld, jd) = (3, 2)]of the 88Sr(d, p)89Sr deuteron stripping reaction at 160° scattering angle. (b).Constructive (solid lines) or destructive (dashed lines) interferences of the twomain transition amplitude elements corresponding to the magnetic indices sets,(mp = –1/2, md = 1, m = 1/2) and (mp = 1/2, md = 0, m = 1/2), result in a successful

or improper description of the deuteron-stripping threshold anomaly.

of the Clebsch-Gordan coefficients describing the quantum kinematical part ofthe transition amplitude elements. These kinematical factors are also responsiblefor the cancellation of the contributions to the anomal cross-sectioncorresponding to other angular momenta partitions, e.g. (ld = 3, jd = 3) and(ld = 5, jd = 4).

The cross-section features of the deuteron-stripping threshold anomaly canbe well described for the other partitions in the entrance channel, i.e. (ld = 1,jd = 1), (ld = 1, jd = 2), (ld = 3, jd = 4) and (ld = 1, jd = 2), (ld = 3, jd = 3)corresponding to p3/2 and p1/2-proton wave, respectively. One has investigatedthe phase values of Lane coefficients describing a resonant dip shape of theanomaly for each of the entrance deuteron angular momentum corresponding tothe p3/2 or p1/2-wave proton exit channel. The obtained phases of α coefficientsfor the three proposed adding methods are presented in Table 2. The difference

Table 2

The phases p

d d

jl jϕ of Lane coefficients generating a resonant anomal dip cross-section for stripping

reaction at 160° scattering angle. In the first line and first column of the table, there are given theangular momentum partitions and the three addition methods. The presented values provide only aqualitative description for the resonant dip shape of the anomaly. The obtained ones from fit of

experimental data will be listed in the following section

(ld jd) jp (1, 1) 3/2 (1, 2) 3/2 (3, 4) 3/2 (1, 2) 1/2 (3, 3) 1/2

ϕI

ϕII

ϕIII

π/2π

π/2

π/20

3π/2

ππ0

π/20

3π/2

ππ0


of the phases obtained using the third and the second adding methods, Δϕ = = ϕIII – ϕII, is about π/2 for ld = 1 and π for ld = 3 deuteron orbital momenta.These values correspond to the phases of the radial integrals multiplying theLane coefficients. In Fig. 5 one has represented the phases extracted fromDWBA code for the two considered deuteron angular momenta. One may alsoremark the difference phase for two different orbital momenta partitions isrelated to the relative phases of their corresponding radial integrals.

Fig. 5 – The phase of the 1 3/ 2, 1 22 1/ 2 5/ 2

p p d dl j l jl s jX = = = == = = and

1 3/ 2, 3 22 1/ 2 5/ 2p p d dl j l j

l sjX = = = == = = radial integrals, versus incident energy,

for a deuteron-stripping reaction on a 88Sr target at 160°scattering angle. The radial integrals correspond to l = 1and l = 3 deuteron orbital momenta in the entrance channel and to a p3/2 –wave in the proton exit channel, respectively.

The excitation functions of the analyzing power, Ay, should have a similardependence on the relative phases, Φ(Sdpα*), as the cross-sections ones, by thefollowing relation,

*

*~ Imy dpA S⎛ ⎞αΔ ⎜ ⎟ε⎝ ⎠(24)

Similar to the computed cross-section, the analyzing power excitation functionbehaves differently with the angular momenta partitions from the entrancechannel. The numerical results obtained by using the partitions able to describe aresonant 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 analyzingpower from the S or reversed S form at backwards angles to the resonance orresonance dip form at forward scattering angles. The anomal effect is larger inanalyzing power for (ld = 3, jd = 4) deuteron angular momenta while a nearlysame depth is reproduced in cross-section for all the analyzed partitions. Theanalyzing power for the p1/2 proton wave has an opposite phase in respect to p3/2

756 M. S. Ata et al. 16

Fig. 6 – Numerical analyses for the shape of analyzing power for different deuteron angularmomentum 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 88Sr ( ,d p)89Sr reaction at 160°scattering angle. (a) The dotted-dashed, continuous and dotted lines represent the excitationfunctions for the p3/2 proton exit channel calculated for the following deuteron partitions: (ld = 1,jd = 1), (ld = 1, jd = 2) and (ld = 3, jd = 4), respectively. (b) Similar representations obtained for the

p1/2 proton exit channel with (ld = 1, jd = 2) – continuous line and (ld = 3, jd = 3) – dotted line.

component; this effect can be associated to the well known Yule-Haeberli rule,[23] established for the stripping reactions with polarized incident particles.

The (d, p) stripping reaction at forward angles is controlled mainly bykinematical effects, e.g. transfer of linear momentum, [3, 8, 24]. Consequentlythe nuclear threshold effects are less discernable for this forward angle region.The nuclear effects become important outside of stripping peak, at backwardangles (θ > 130°). The cross-section could evince the nuclear threshold effects,even at forward angles, via spectroscopic factors, (θ ≤ 130°). Indeed thespectroscopic factors should be independent on deuteron energy. However


deviation from constant value of spectroscopic factors was found near thethreshold, by analyzing the anomaly of 90Zr(d, p)91Zr reaction, [25]. It is anindirect proof of the effect in cross-section at forward angles.

On the other hand the threshold effect should be present in analyzing powerat all measured angles. Indeed the Ay is sensitive not to the magnitude ofbackground but rather to interference effect and the anomaly is a genuineinterference effect. Considering more than one angular momentum partitions in

the entrance channel, the anomal resonant terms, ( )p

d d

jl jT π α will interfere as well.

Numerical calculations have been performed for separated or mixed angularmomentum partitions in the deuteron entrance channel corresponding to the 3-p3/2

or 3-p1/2 -wave proton exit channel. The results will be discussed next section forall analyzed deuteron-stripping anomalies reported on A ∼ 90 mass region.

Fig. 7 – Resonance-dip (solidline) and double-dip (dotted line)shapes in the cross-sectionobtained for two input deuteronpartitions corresponding to thep3/2 and p1/2 waves in protonchannel. The analogue thresholdenergy is chosen here, Qdn == 6.5 MeV, the resonanceenergetic shift, xj = 2. Thedashed line represents theDWBA cross-section backgroundcalculated at θ = 160° scatteringangle for a generic deuteronstripping 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 numericallyobtained by the interference of the spin-orbit transition amplitude componentscorresponding to the p-wave resonance in the proton exit channel and theexisting shift of the resonance E3/2 and E1/2 energies. In a first step, one hasperformed a resonance-dip shape of the anomaly by combining a resonance (forthe exit p3/2-wave) with a dip (for the exit p1/2-wave), see Fig. 7, solid line. Theanalogue threshold energy does correspond for the ascendent slope of thebackground cross-section. According to Table 2, the phases of α coefficients(method I) are ϕI (jp = 3/2) = 0 and ϕI (jp = 1/2) = π. A similar result should beobtained for the descendent part of background cross-section by using reversed

758 M. S. Ata et al. 18

phases of α coefficients and an appropriated Qdn-value. A genuine double-dipshape has been obtained too, see dotted line in Fig. 7. However, one has tomodify the total width parameter y from y = 2 to y ∼ 0.5 in order to get thedouble-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 andcross-section excitation functions, have been performed for single dip cross-section shapes observed in deuteron-stripping reactions on the following targetnuclei: 80Se, [26], 86Kr, [27], 88Sr, [19, 28], 90Zr, [29, 10, 24], 92Zr, [30], 92Mo,[10, 30, 94Zr, [30], 94Mo [30], and 106Cd, [10, 20]. One has considered thedescription of the double-dip cross-section shape reported from 92Zr(p, d)91Zr,[31], and 96Zr(p, d)95Zr, [32], pick-up reactions too. One has to mention also thatexperimental analyzing power measurements are limited for only few deuteron

stripping reactions listed above (88Sr ( ,d p)89Sr, 90Zr ( ,d p)91Zr, 92Mo ( ,d p)93Mo

and 106Cd ( ,d p)107Cd).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 exitchannels, respectively. The OM parameters for the deuteron channel are thefollowing, [33], V = 112.5 MeV, W = 52.8 MeV, VSO = 6.79 MeV, r = 1.06 fm,rW = 1.391 fm, rSO = 0.766 fm, rC = 1.3 fm, a = 0.847 fm, aW = 0.719 fm,aSO = 0.441 fm. For the proton OM scattering one has used, [34], V = 58.5 MeV,W = 46.0 MeV, VSO = 6.2 MeV, r = 1.17 fm, rW = 1.32 fm, rSO = 1.01 fm,rC = 1.3 fm, a = 0.75 fm, aW = 0.6 fm, aSO = 0.75 fm. The OM parameters havebeen corrected for the energy dependence by using extrapolation relations, [34].

The spectroscopic factors were calculated for each of the analyzed deuteron-stripping reaction, see Table 3. Some discrepancies regarding the angulardistribution and excitation functions of cross section observed in theexperimental data at backwards scattering angles are provided from differentexperimental measurements ( e.g. deuteron-stripping reactions on 90,92,94Zr targetnuclei). In description of the deuteron-stripping threshold anomaly by means ofCoupled Channel Born Approximation formalism, a normalization coefficientwas defined instead using the apriori known spectroscopic factor, see forexample Refs. [27, 28]. The differences between the two values should have tolay out in the limits of experimental measurement errors. As a precautionarymeasure, before starting the numerical analysis of deuteron-stripping anomaly,the DWBA cross section was previously determined at backwards scattering angles


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. [42];(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 Factors81Se(1/2–) 0.31(a) 0.3687Kr(5/2+) 0.56(b) 0.6289Sr(5/2+) 0.79(c) , 1.03(d) 1.1791Zr(5/2+) 0.75(e) , 1.09(f) 0.9393Zr(5/2+) 0.64(g) 0.6495Zr(5/2+) 0.32(g) , 0.34(h) 0.41

93Mo(5/2+) 0.84(i) 0.8395Mo(5/2+) 0.59(i) 0.52107Cd(5/2+) 0.25(j) 0.30

and compared with the corresponding experimental excitation functions. In caseof no absolute differential cross-section data are provided, e.g. deuteron-strippingreactions on, 80Se [26] or 106Cd [10], target nuclei, a similar normalization factorwas carried out in the present calculations.

The DWBA direct interaction code DWUCK and additional routinesdevoted to the anomal resonant term have been adjusted to a minimizationlibrary code, MINUIT, [46]. This is a powerful numerical tool for estimating thebest fit parameter values and uncertainties including their correlations. Theobtained hybrid code, QUASIRES, is able to perform minimizations of thecross-section and analyzing power excitation functions for deuteron-strippingreactions using the three different numerical procedures for adding the direct andanomal transition amplitudes and to search for the best fit parameters including

p

d d

jl jα Lane coefficients. The Lane parameters will be replaced in the

minimization procedure by two fit parameters: a strength, amplitude likeparameter and a phase one. The spectroscopic factor of the involved residualnuclear state may be considered as additional fit parameter. However, itsvariation is constrained, as a rule, to some limitations, which can not exceed tento fifteen percents from the standard evaluated value. The xj energy shifts of the3-p3/2 and 3-p1/2 single particle resonances enter also in the minimization

procedure as fit parameters. Considering all partitions for a IA = 0+ → IB = 5/2+

transition, one has obtained 20 fit parameters (12 for 3 / 2pα and its phases, 6 for1/ 2 ,pα xj parameter and the spectroscopic factor).

760 M. S. Ata et al. 20

One has focused on the strongest threshold effect reported on88Sr ( ,d p)89Sr deuteron-stripping reaction. The minimization procedure usingfree fit parameters without any given constraints does result in correlationsbetween α parameters, both for their strengths or phases. Considering more thanone deuteron partition in the entrance channel, the anomal resonant terms,

( ),p

d d

jl jT π α will interfere each others. To avoid numerical correlations as well as

to predict the interferences between terms, one has to analyze the reactionexcitation functions using separated deuteron entrance partitions or mixed oneswith defined constraints rules for the parameters.

In Figs. 8, the experimental data of the analyzing-power (20°, 40°, 60°,90°, 120°, 140° and 160°) and cross-section (160°) excitation functions are fittedby using single incoming partitions, (ld = 1, jd = 2) – Fig. 8 (a), (ld = 3, jd = 4) –Fig. 8 (b), for p3/2-wave and (ld = 1, jd = 2) – Fig. 8 (c), (ld = 3, jd = 3) – Fig. 8(d), for p1/2-wave, respectively. The obtained strengths and phases have beenlisted in Table 4. One has to observe a large spread of the α’s strengths withrespect to the choice of selecting the input deuteron partitions. The cross-sectionand analyzing power excitation functions are both reproduced only for the p3/2-wave. One should remark however a less quantitative description of theexperimental analyzing power data. The functions are both reproduced only forthe p3/2-wave. One should remark however a less quantitative description of theexperimental analyzing power data. The computational analyzing powerfunctions in case of the p1/2-wave are in opposite phase in respect with theexperimental ones.

An approach for considering the mixing of different input deuteronpartitions 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 inthe deuteron entrance channel and it was used in former numerical analyzing ofthe deuteron-stripping anomaly in Refs. [10, 19, 20]. The partitions are definedonly by the deuteron orbital angular momentum, e.g. for the proton p3/2-wavethere are three partitions ld = 1 (jd = 1, 2), ld = 3 (jd = 2, 3, 4) and ld = 5 (jd = 4)and two ones for the p1/2-wave, ld = 1 (jd = 2) and ld = 3 (jd = 2, 3). The fitparameters obtained by using ld = 1 and ld = 3 orbital angular momenta forreproducing the anomaly cross-section for 88Sr(d, p)89Sr stripping reaction atθ = 160° scattering angle are given in Table 5.

Another numerical approach for describing the threshold anomaly usingmixed 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 αparameters only. This method is suitable to avoid the destructive interferencesbetween the various input deuteron angular momenta partitions. Within the


Fig. 8 – Experimental (filled circles) and computational (solid lines) excitation functions for 88Sr

( ,d p) 89Sr deuteron-stripping reaction: (a) – ld = 1, jd = 2, jp = 3/2; (b) – ld = 3, jd = 4, jp = 3/2;(c) – ld = 1, jd = 2, jp = 1/2; (d) – = 3, jd = 3, jp = 1/2.

numerical procedure, one has considered two parameters for describing thestrengths of α coefficients. These parameters correspond to the spin-orbitsplitting of the p-wave single particle resonance in the exit channel of the (d, p)stripping reaction, i.e. 3 / 2pα and 1/ 2 ,pα respectively, in a similar way to the spin-orbit splitting of the 3-p -wave Neutron Strength Function. The results obtainedusing all the input deuteron partitions are presented in Fig. 9. The experimentalanomaly 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

762 M. S. Ata et al. 22

Table 4

Numerical results for deuteron single partitions used in minimization of the experimental cross-section at θ = 160° scattering angle for the 88Sr (d, p) 89Sr stripping reaction

(ld, jd)jp

(1, 2)3/2

(3, 4)3/2

(1, 2)1/2

(3, 3)1/2

αI (rel. u.)ϕI (rad.)

12.4 2.39

4.273.57

22.5 2.81

22.8 3.64

αII (rel. u.)ϕII (rad.)

6.930.19

4.03.0

9.0–0.17

9.23.14

αIII (rel. u.)ϕIII (rad.)

952.6 5.0

550.6 0.2

1704.9 5.1

1474.6 0.2

Table 5

Numerical results for deuteron mixed partitions used in minimization of the experimental cross-section at θ = 160° scattering angle for the 88Sr (d, p) 89Sr stripping reaction

jp 3/2 1/2

ld

jd

11 2

32 3 4

12

32 3

αI (rel. u.)ϕI (rad.)

9.62.4 2.3

4.6–0.6 –0.8 3.5

22.52.8

22.83.7 3.7

αII (rel. u.)ϕII (rad.)

5.252.74 0.15

3.2–0.5 0.25 2.97

9.0–0.17

6.83.2 3.2

αIII (rel. u.)ϕIII (rad.)

952.65.0

441.3.7 3.1 0.28

1704.95.1

1064.0.4 0.2

values of the the experimental data for all the analyzed scattering angles,especially at forwards angles. For the unfavorable cases, one has mainly focusedon the shape of the anomaly rather than its amplitude. The improvement ofreaction DWBA background by choosing other optical model parameters mayincrease the fit quality but certainly does affect the purpose of our work: aunitary description of the deuteron-stripping threshold anomalies in A ∼ 90 massregion. Peculiar optical model values may describe the threshold effect by usingDWBA model only.

From a physical insight, the deuteron stripping threshold anomaly shouldbe described by α coefficients irrespective on the deuteron entrance channelrelated to the p-wave exit one, i.e. the deuteron-stripping anomaly should notdepend on the entrance deuteron channel from the reaction process. Theobtaining of similar values for Lane coefficient strengths in describing thedeuteron-stripping anomaly cross-section, irrespective the deuteron entrancechannel, comes to sustain the physical idea that the anomaly should not bedependent 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.


Fig. 9 – Experimental (filled circles) and computational (solid lines) excitation functions

for 88Sr ( ,d p) 89Sr (left) and 90Zr ( ,d p) 91Zr (right) deuteron-stripping reactions.

A more plausible assumption should be reproducing of spectroscopicproperties of α coefficients for entire analyzed mass region A ~ 90, namely therelationship of proportionality between the strengths of α parameters and the 3-pNeutron Strength Function, [47]. The strengths of the deuteron-strippinganomalies should follow the mass dependence of the Neutron Strength Functionirrespective one has considered single or mixed partitions of the entrancedeuteron angular momenta.

The 3-p threshold anomaly observed in 90Zr ( ,d p)91Zr reaction cross-section, [6], is the next analyzed stripping reaction of the present work. Similarlyto the case of the optical model parameters, the α’s phases, resulting from the

764 M. S. Ata et al. 24

non diagonal terms of the background Sdp matrix and from the fluctuationaveraging, [47], should vary smoothly while changing the mass in the A ∼ 90region. As a consequence, the phases of α parameters should be constrained topreserve their values. However, the relative phase between the direct andresonant transition amplitudes may depend from one stripping reaction toanother, due to the energy variation of the radial integral phases, see e.g. Fig. 5.

The α’s phases from Table 4 established for 88Sr ( ,d p)89Sr reaction, should benot necessary identical with those describing an anomal dip for other deuteron-stripping reactions. In Fig. 9 (right) one has represented the experimentalanalyzing power at 120°, 140° and 160° scattering angles and the experimental

cross-section at 155° for 90Zr ( ,d p)91Zr reaction, both with our computationalresults with circles and solid lines, respectively.

A weaker effect was measured in the excitation functions of 92Mo ( ,d p) 92Moreaction, see Fig. 10. The set of average optical parameters did provide a poordescription of the DWBA background for 160° scattering angle analyzing power.

In Fig. 11 (a) and 11 (b), the excitation functions of 106Cd ( ,d p)107Cd

stripping reaction are represented for the 120° and 160° scattering angles. Thethreshold effect for this reaction is quit small in the experimental analyzingpower. The reproducing of analyzing power data was realized using both the p3/2

and p1/2 spin-orbit components of α parameters. The Yule-Haeberli rule couldexplain the destructive interference of the p3/2 and p1/2 contributions to the

Fig. 10 – Experimental (filled circles) and computational (solid lines)

excitation functions for 92Mo ( ,d p) 93Mo deuteron-stripping reactionat 120°, 140° and 160° scattering angles.


Fig. 11 – Analyzing power at 120° – (a), 160° – (b), and cross-section at 160° – (c), for 106Cd ( ,d p)107Cd stripping reaction; experimental data (filled circle), DWBA background (dashed line) and

computational result (solid line).

Fig. 12 – (a)–(e). Differential cross section, experimental (filledcircles) and computational (solid lines), for the numerical analyzeddeuteron-stripping reactions on the following target nuclei: 80Se,86Kr, 92Zr, 94Zr and 94Mo. (f). Double-dip cross-section threshold

anomaly for 92Zr(p, d)91Zr pick-up reaction.

766 M. S. Ata et al. 26

analyzing power. The differential cross-section at 160° scattering angle isrepresented also in Fig. 11 (c). The threshold anomaly is weak in spite of thepresence of a large strength of the 3-p1/2 neutron strength function, [48]. Anexplanation of such behaviour is obtained in Ref. [15].

The single dip anomaly cross-sections have been also successfullyreproduced for the other studied deuteron stripping reactions, see Figs. 12 (a)–(e)for every allowed value of the angular momentum corresponding to the totalangular momentum coupling scheme and related to lp = 1 exit channel. InFig. 12 (f) one has represented the double-dip anomaly observed for the pick-upreaction, 92Zr(p, d)91Zr. The anomaly can be reproduced from the interplay ofthe p3/2 and p1/2 α coefficients only by modifying the total width y of theresonance (y = 2 → y ∼ 0.5). This result sustains the idea that double-dipanomaly originates rather in the microstructure of the neutron single particlethreshold state than in spin-orbit splitting, see e.g. Ref. [47].

6. CONCLUSIONS

In this work, the significant 3-p wave threshold anomalies reported indeuteron stripping reactions in A ∼ 90 mass region have been analyzed within theproposed hybrid angular momentum scheme by using a computer code based onDWBA formalism. The anomalies have been successfully described both incross-section and analyzing power excitation functions, irrespective the chosenmethod or the selection of input angular momenta partitions in deuteron entrancechannel. Physical aspects of the presented numerical results, [22, 49, 50, 15],will be developed in a forthcoming paper, [51].

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