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

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¨¸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 Mathematical 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 considerationof 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. Calculations of inelastic cross sections 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 further applicationsis 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° μɨΥ¸±¨° μÉ¥´Í¨ ², ¤¨ ¡ ɨΥ¸±¨°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, £¤¥ ʤ¥·¦¨¢ ÕÉ¸Ö Î²¥´Ò ¥·¢μ£μ μ·Ö¤±¢ · §²μ¦¥´¨¨ μÉ¥´Í¨ ² μ · ³¥É·Ê ¤¥Ëμ·³ ͨ¨. ‚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 theory of excitations of nuclear collective states in peripheral nuclear collisions 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 ofthe 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 wavefunctions 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 ofU(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 sections 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 conclusions 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 they are in inelasticchannel. Otherwise, at present there are no tables of global optical potentials for the heavyionelastic scattering at different energies and kinds of colliding nuclei. Moreover, thereexists 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 other hand, in the last two decades of years the double-folding (DF) microscopicnucleusÄnucleus potentials occur rather 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 distributions ρ(r) and matrices ρ(x, y), andalso include the effective nucleonÄnucleon potentials v D and v EX . In principle, all of thesequantities 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 theknock-on effects (interchange of nucleons 1 and 2) and describe sufˇciently well the shapeof the peripheral region of potentials, very important in formations of both the differentialand total cross sections. For a period of years, in comparison with experimental data, these


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, thewhole 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 theory [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 ofthe 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, andthen to test the potential U(r) =N r V DF + iN im V DF (r), as well. These potentials werecalled the semimicroscopic ones because their 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 sections using forFig. 1. The double-folding potentials (dashed lines) and their 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 sections to the Rutherford 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 sections to the Rutherford 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 ,andtheircomparisons 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 regionof an applicability of HEA at θ √ 2/kR. So, these potentials can be applied further incalculations of inelastic scattering of the same nuclei at the same energy for comparisons withexistent experimental data on the 2 + state excitations [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 there thedependence 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 oftwo 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 theabove-considered cases. All of them 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, andtheeikonal 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 havingthe 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 accountthe Coulomb distortion of the straight-ahead trajectory situated in expressions of the highenergytheory. 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 ofthe 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 sections 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 nuclearand 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 onthe r plane, where the main transition takes place. The β (c)2 deformations are taken as theywere extracted in [9] using the known reduced electric transition probabilities B(E2 ↑) in thetarget nuclei. (For parameters see set 1 in the table.)Figure 3 shows these results by dashed lines. We see that the calculations performedwithout free parameters are in a qualitative agreement with the experimental data. The slopesof all curves are in coincidence with the behavior of the data. As to the absolute values ofcross sections, they can be slightly improved by increasing the deformation parameters. Anexception is seen at small angles (very peripheral collisions) for heavy nuclei 120 Sn, 208 Pb(large charges), where the multistep Coulomb excitation must give large contribution while

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