excitation of nuclear collective states by heavy ions within the ... - JINR

¨¸Ó³ ¢ —Ÿ. 2006. ’. 3, º 6(135). ‘. 105Ä112“„Š 539.17.01EXCITATION OF NUCLEAR COLLECTIVE STATESBY HEAVY IONS WITHIN THE MODELOF SEMIMICROSCOPIC OPTICAL POTENTIALK. M. Hanna a , K. V. Lukyanov b , V. K. Lukyanov b ,Z. Metawei c ,B.Slowinski d, e , E. V. Zemlyanaya ba Ma**the**matical and Theoretical Physics Department, NRC, Atomic Energy Authority, Cairo, Egyptb Joint Institute for Nuclear Research, Dubnac Physics Department, Faculty **of** Science, Cairo University, Giza, Egyptd Faculty **of** Physics, Warsaw University **of** Technology, Warsaw, Polande Institute **of** Atomic Energy, Otwock-Swierk, PolandThe (semi)microscopic double-folding nucleusÄnucleus optical potentials are suggested for consideration**of** inelastic scattering with **excitation** **of** **collective** **nuclear** **states** **by** using **the** adiabatic approachand **the** elastic scattering amplitude in **the** high-energy approximation. The analytical expression forinelastic scattering amplitude is obtained keeping **the** ˇrst-order terms in **the** deformation parameter **of** apotential. Calculat**ions** **of** inelastic cross sect**ions** for **the** 17 O **heavy** **ions** scattered on different nuclei atabout hundred MeV/nucleon are made, and **the** acceptable qualitative agreement with **the** experimentaldata is obtained without introducing free parameters. The prospect **of** **the** method for fur**the**r applicat**ions**is discussed.„²Ö · ¸¸³μÉ·¥´¨Ö ´¥Ê·Ê£μ£μ · ¸¸¥Ö´¨Ö ¸ ¢μ§¡Ê¦¤¥´¨¥³ ±μ²²¥±É¨¢´ÒÌ ¸μ¸ÉμÖ´¨° Ö¤¥· ·¥¤² -£ ¥É¸Ö ¨¸μ²Ó§μ¢ ÉÓ (μ²Ê)³¨±·μ¸±μ¨Î¥¸±¨° Ö¤·μ-Ö¤¥·´Ò° μÉ¨Î¥¸±¨° μÉ¥´Í¨ ², ¤¨ ¡ É¨Î¥¸±¨°μ¤Ìμ¤ ¨ ³²¨ÉÊ¤Ê Ê·Ê£μ£μ · ¸¸¥Ö´¨Ö ¢ ¢Ò¸μ±μÔ´¥·£¥É¨Î¥¸±μ³ ·¨¡²¨¦¥´¨¨. μ²ÊÎ¥´μ ´ ²¨É¨-Î¥¸±μ¥ ¢Ò· ¦¥´¨¥ ¤²Ö ³²¨ÉÊ¤Ò ´¥Ê·Ê£μ£μ · ¸¸¥Ö´¨Ö, £¤¥ Ê¤¥·¦¨¢ ÕÉ¸Ö Î²¥´Ò ¥·¢μ£μ μ·Ö¤±¢ · §²μ¦¥´¨¨ μÉ¥´Í¨ ² μ · ³¥É·Ê ¤¥Ëμ·³ Í¨¨. ‚Òμ²´¥´Ò · ¸Î¥ÉÒ ´¥Ê·Ê£¨Ì ¸¥Î¥´¨° · ¸-¸¥Ö´¨Ö ÉÖ¦¥²ÒÌ ¨μ´μ¢ 17 O · §´Ò³¨ Ö¤· ³¨ ·¨ Ô´¥·£¨ÖÌ μ±μ²μ 100 ŒÔ‚/´Ê±²μ´, ¨ μ²ÊÎ¥´μÊ¤μ¢²¥É¢μ·¨É¥²Ó´μ¥ ¸μ£² ¸¨¥ ¸ Ô±¸¥·¨³¥´É ²Ó´Ò³¨ ¤ ´´Ò³¨ ¡¥§ ¢¢¥¤¥´¨Ö ¸¢μ¡μ¤´ÒÌ · ³¥É·μ¢.¡¸Ê¦¤ ÕÉ¸Ö ¢μ§³μ¦´μ¸É¨ ·¥¤²μ¦¥´´μ£μ μ¤Ìμ¤ ¤²Ö ¤ ²Ó´¥°Ï¨Ì ·¨²μ¦¥´¨°.INTRODUCTIONThe **the**ory **of** **excitation**s **of** **nuclear** **collective** **states** in peripheral **nuclear** collis**ions** isbased on **the** elastic scattering optical potential U(r) =V (r) +iW (r). Thislatterisusedto obtain **the** transition potential U int = U (N)int+ U (C)intfor inelastic channel. Recently, in [1],**the** nucleusÄnucleus inelastic scattering with **excitation** **of** 2 + rotational **states** was consideredin **the** framework **of** **the** high-energy approximation utilizing **the** phenomenological WoodsÄSaxon type potential. The **collective** variables {α λμ }, which characterize **the** deformation **of****the** surface **of** a potential, were introduced through **the** radiusR = R + δR,δR = R ∑ λμα λμ Y λμ (θ, φ). (1)

106 Hanna K. M. et al.Here θ, φ are spherical coordinates **of** a space vector r in **the** laboratory system. The wavefunct**ions** **of** rotational **states** and **collective** variables {α λμ } are given as follows:√2I +1|IM〉 =8π 2 D (I)M0 (Θ i), α 2 μ = β 2 D (2) ∗μ0 (Θ i), (2)where β 2 is **the** deformation parameter and {Θ i } are **the** intrinsic axis rotational angles.In [1], suggesting small β 2 ≪ 1, **the** transition potential was obtained as **the** derivative **of**U(r, R), and **the** inelastic scattering amplitude was derived in adiabatic approximationf IM (q) =〈IM| f(q, {α λμ })|00〉, (3)where q = 2k sin (ϑ/2) is **the** momentum transfer; k is **the** relative momentum, and ϑ,**the** angle **of** scattering. The elastic scattering amplitude f(q, {α λμ }) was taken in **the** highenergyapproximation (HEA) with **the** ®frozen¯ coordinates **of** **collective** motion {α λμ }. Then,inelastic cross sect**ions** for **the** 17 O **heavy** **ions** scattered on different nuclei at about hundredMeV/nucleon were calculated, and an acceptable agreement with **the** experimental data wasreceived. So, **the** conclus**ions** were made on applicability **of** HEA to study **the** nucleusÄnucleusinelastic processes.The aim **of** this paper is to apply not phenomenological but microscopic potentials forcalculating an inelastic scattering amplitude. The matter **of** fact is that **the** phenomenologicalpotentials, used for inelastic scattering, must be specially ˇtted in **the** corresponding elasticchannel at **the** same energy and for **the** same couple **of** scattered nuclei as **the**y are in inelasticchannel. O**the**rwise, at present **the**re are no tables **of** global optical potentials for **the** **heavy**ionelastic scattering at different energies and kinds **of** colliding nuclei. Moreover, **the**reexists **the** problem **of** ambiguity **of** parameters **of** phenomenological potentials (see, e.g., [2])since **the** ˇt needs a large amount **of** data, and thus any additional information, involved intoconsideration, in particular, **the** data **of** inelastic scattering, is very desirable.On **the** o**the**r hand, in **the** last two decades **of** years **the** double-folding (DF) microscopicnucleusÄnucleus potentials occur ra**the**r popular. They are calculated using **the** followingexpression:∫V DF (r) =V D (E,r) +V EX (E,r) = d 3 r 1 d 3 r 2 ρ 1 (r 1 ) ρ 2 (r 2 ) v D (ρ, E, r 12 )+∫[ ]+ d 3 r 1 d 3 r 2 ρ 1 (r 1 , r 1 + r 12 )ρ 2 (r 2 , r 2 − r 12 )v EX ik(r)r12(ρ, E, r 12 )exp, (4)Mwhere r 12 = r + r 2 − r 1 is **the** distance between nucleons **of** colliding nuclei; k(r) is **the** localmomentum **of** relative motion **of** nuclei, and M = A 1 A 2 /(A 1 +A 2 ). (For details see, e.g., [3,4].) These DF potentials apply **the** **nuclear** density distribut**ions** ρ(r) and matrices ρ(x, y), andalso include **the** effective nucleonÄnucleon potentials v D and v EX . In principle, all **of** **the**sequantities are known from independent experimental studies. Dependence **of** NN potentials onkinetic energy and on **the** matter density in overlapping region **of** nuclei was also established.DF potentials take into account **the** antisymmetrization **of** **the** system **by** accounting for **the**knock-on effects (interchange **of** nucleons 1 and 2) and describe sufˇciently well **the** shape**of** **the** peripheral region **of** potentials, very important in format**ions** **of** both **the** differentialand total cross sect**ions**. For a period **of** years, in comparison with experimental data, **the**se

Excitation **of** Nuclear Collective States **by** Heavy Ions **within** **the** Model 107real DF potentials were supplemented **by** **the** phenomenological imaginary potentials W P (r)having three (or more) free parameters. By doing so, it was shown that one needs to diminishslightly **the** calculated real part **by** introducing a renormalization coefˇcient N r , and thus, **the**whole potential U(r) =N r V DF + iW P (r) has four (or more) free parameters.However, recently in [5], it was demonstrated that **the** imaginary part can also be calculatedmicroscopically **by** transforming **the** eikonal phase **of** **the** high-energy microscopical **the**ory [6,7] **of** scattering **of** complex systems. It was shown in [5] that this imaginary potential W Hcontains **the** folding integral which corresponds to **the** integral **of** only **the** direct part V D **of****the** DF potential (4). This optical potential has **the** form U(r) =N R V DF (r)+iN im W H (r).In addition, it was reasonable to generalize this form to include **the** exchange term, too, and**the**n to test **the** potential U(r) =N r V DF + iN im V DF (r), as well. These potentials werecalled **the** semimicroscopic ones because **the**ir basic forms W H and V DF were calculatedmicroscopically, without introducing free parameters, and only two parameters N r and N immust be adjusted to experimental data.Figure 1 shows **by** dashed lines **the** double-folding potentials V DF calculated in [8] forscattering **of** **the** 17 O **heavy** **ions** on different nuclei at E lab = 1435 MeV. The respectiveoptical potentials were adjusted to **the** elastic scattering differential cross sect**ions** using forFig. 1. The double-folding potentials (dashed lines) and **the**ir derivatives (solid lines) calculated fordifferent couples **of** nuclei at E lab = 1435 MeV

108 Hanna K. M. et al.Fig. 2. The ratio **of** **the** elastic scattering differential cross sect**ions** to **the** Ru**the**rford one (solid lines)calculated using **the** semimicroscopic optical potentials N rV DF + iN imV DF at E lab = 1435 MeV andcompared with **the** experimental data from [9]imaginary terms **the** forms N im W H and N im V DF . In Fig. 2 we reproduce **the** ratios **of** elasticcross sect**ions** to **the** Ru**the**rford one calculated in [8] in **the** framework **of** **the** high-energyapproximation using **the** microscopic potentials U(r) =N r V DF (r) +iN im V DF ,and**the**ircomparisons with **the** experimental data from [9]. The adjusted normalization coefˇcientsN r and N im were obtained as 0.6 and 0.6 for 60 Ni, 0.6 and 0.5 for 90 Zr, 0.5 and 0.5 for120 Sr, and 0.5 and 0.8 for 208 Pb. One sees fairly well agreement with **the** data in **the** region**of** an applicability **of** HEA at θ √ 2/kR. So, **the**se potentials can be applied fur**the**r incalculat**ions** **of** inelastic scattering **of** **the** same nuclei at **the** same energy for comparisons wi**the**xistent experimental data on **the** 2 + state **excitation**s [9].

Excitation **of** Nuclear Collective States **by** Heavy Ions **within** **the** Model 1091. SOME FORMULAE AND COMMENTSThe microscopic potentials have no obvious parameters something like **the** radius Rand diffuseness a **of** a WoodsÄSaxon potential. Therefore, in order to introduce **the**re **the**dependence on internal **collective** variables α λμ , we make, in analogy with (1), **the** respectivechanges **of** spatial coordinatesr ⇒ r + δr,δr = −r ∑ λμα λμ Y λμ (θ, φ). (5)Then, expanding **the** potential in δr we obtain **the** generalized optical potential consisting **of**two terms, **the** spherically symmetrical and deformed one:U (N) (r, {α λμ })=U (N) (r)+U (N)int (r, {α λμ}), (6)where **the** transition potential (its **nuclear** part) is as follows:U (N)int= −r d dr U(r) ∑ μα 2μ Y 2μ (θ, φ). (7)In Fig. 1 one can see **the** behavior **of** **the** derivatives **of** microscopic potentials V DF for **the**above-considered cases. All **of** **the**m have typical maxima in **the** surface region **of** a potential.The respective quadrupole part **of** **the** generalized Coulomb potential U (C) (r, {α λμ }) is obtainedas usually with **the** help **of** its deˇnition through **the** uniform charge density distributionhaving **the** radius R as in (1) with R C = r c (A 1/31 + A 1/32 ). This yields in [1]U (C)int= 3 5 U B[( r) 2Θ(R ( RC− r)+R C r) 3Θ(r ] ∑− R) α 2μ Y 2μ (θ, φ), (8)where U B = Z 1 Z 2 e 2 /R C .Then, we use **the** expression for high-energy amplitude **of** scatteringf(q) =i k ∫[ bdbdφ e iqb cos φ 1 − e iΦ] . (9)2πHere integration is performed over impact parameters b and on its azimuthal angle φ, and**the**eikonal phase is determined **by** **the** nucleusÄnucleus potentialΦ=− 1v∫ ∞−∞U(r + δr) dz, r = √ b 2 + z 2 , (10)where v is **the** relative velocity **of** colliding nuclei. Substituting here **the** total potential having**the** central and transition terms, one can writeΦ=Φ 0 (b)+Φ int (b, {α λμ },φ), (11)∑Φ int = β 2 G μ (b) D (2) ∗μ0 (Θ i)e iμφ , (12)μ=0,±2μ

110 Hanna K. M. et al.G μ (b) =− 2v∫ ∞0dz Y 2μ (arccos (z/r), 0) ×[× −r dU(r) + 3 [( r) 2Θ(R (dr 5 U RC) 3Θ(r ] ]B− r)+ − R) , (13)R C rwhere r = √ b 2 + z 2 . Substituting (11) in (9) and (3), and expanding **the** exponential functionexp (iΦ int ), we retain only a term **of** **the** ˇrst order in β 2 . Then, integration over rotationalangles Θ i can be performed, and one gets **the** inelastic scattering amplitudes f λμ (q) anddifferential cross section as follows [1]:f 20 (q) = √ k ∫∞β 2 bdbJ 0 (qb)G 0 (b)e iΦ0(b) , (14)5f 22 (q) =− k √5β 20∫0∞bdbJ 2 (qb)G 2 (b)e iΦ0(b) , (15)dσ indΩ = |f 20| 2 +2|f 22 (q)| 2 . (16)2. COMPARISON WITH EXPERIMENTAL DATA. SUMMARYWhen calculating **the** elastic and inelastic scattering amplitudes one has to take into account**the** Coulomb distortion **of** **the** straight-ahead trajectory situated in express**ions** **of** **the** highenergy**the**ory. This is made **by** exchanging, in **the** **nuclear** part **of** **the** phases Φ 0 (b) andΦ int (b), **the** impact parameter b **by** **the** distance **of** **the** turning point in **the** Coulomb ˇeld **of****the** point charge, i.e., b ⇒ b c =ā + √ b 2 +ā 2 ,whereā = Z 1 Z 2 e 2 /vk is a half **of** closestapproach distance at b =0.Firstly, we estimate inelastic cross sect**ions** **of** scattering **of** 17 O on different nuclei withoutintroducing any free parameters. To this end, we apply semimicroscopic optical potentialsU = N r V DF + iN im V DF calculated and adjusted in [8] to **the** experimental data on elasticscattering **of** **the** same nuclei [9]. The deformation parameters β (n)2 and β (c)2 for **nuclear**and Coulomb potentials, separately, were suggested to obey **the** relation β (c)2¯R C = β (n)2¯R n ,where ¯R are rms radii. Qualitatively, this relation supposes an equality **of** areas **of** rings on**the** r plane, where **the** main transition takes place. The β (c)2 deformat**ions** are taken as **the**ywere extracted in [9] using **the** known reduced electric transition probabilities B(E2 ↑) in **the**target nuclei. (For parameters see set 1 in **the** table.)Figure 3 shows **the**se results **by** dashed lines. We see that **the** calculat**ions** performedwithout free parameters are in a qualitative agreement with **the** experimental data. The slopes**of** all curves are in coincidence with **the** behavior **of** **the** data. As to **the** absolute values **of**cross sect**ions**, **the**y can be slightly improved **by** increasing **the** deformation parameters. Anexception is seen at small angles (very peripheral collis**ions**) for **heavy** nuclei 120 Sn, 208 Pb(large charges), where **the** multistep Coulomb **excitation** must give large contribution while