Threshold anomaly in non-central forces

Phymcs Letters B 300 (1993) 303-307 North-Holland PHYSICS LETTERS B Threshold anomaly in non-central forces J. G6mez-Camacho and M.A. Nagarajan ' Departamento de Fiswa At6mwa. Molecular y Nuclear. UntversMad de Sevtlla. 4pdo 1065. E-41OSO Sevtlla. Spare Recewed 9 November 1992 The behavmur of the threshold anomaly for non-central potentials, wh,ch account for collectwe excltat,ons m heavy-~on colhs~ons. ~s investigated. It ~s shown thai the non-central potentmls should exhibit an energ5 dependence at energies m the vicinity of the Coulomb barrier Th~s energy dependence is, however, different from that of the elastkc optical potenual, occurring at lower energies It ff further shown thai there are correctmns to the tradmonal collectwc model such that. ff the trans~tmn potential ~s expressed as the derivative of the optical potential, the corresponding deformation length will be complex and energy-dependent Simple model calculauons are presented Durmg recent years, evidence of the energy-dependence of the real (the threshold anomaly) and imaginary parts of nucleus-nucleus optical potentials has been found by careful analyses of the elastxc scattering at energies m the vtcmlty of the Coulomb barrter [ 1-5] This energy dependence has been attributed to the couphng of non-clasttc channels to the elastic channel tn this energy region [6.7]. The threshold anomaly has also been related to the rapid mcrease of the surface imaginary potential as the energy ts mcreased above the Coulomb barrier and the consequent correction to the real potential through a dispersion relation [8-10]. The optical potential can thus be written as U(E)= V o+AI'(E) +tH(E), ( I ) where go is an energy-independent potential (double-folded potential, for example), I,I'(E) ts the tmagmary potential and the dispersive real potenual AI'(E) ts defined as AV(E)= P f I4'(E') ~dE', (2) where P denotes a prmclpal value mtegral. Generahzlng this to the case of a set of strong-coupled channels, it was suggested by Satchler [ 1 1 ] that Permanent address SERC Daresbury Laborator3., Daresburs, Warrmgton WA4 4AD, UK each element of the potential matrix, which defines the coupled channels equation, should be expected to exhtbtt a threshold anomaly. In such a case, the energy dependence of the different diagonal and off-diagonal elements of the potenttal matrix will be attributed to couphngs to other channels not mcluded m the subset of coupled channels. However. a priori, there ts no reason for all the terms of the potential matrix to have the same energy dependence. Satchler [ 1 1 ] argued that, m the case of collective excitations, one would expect a similar energy dependence m the coupling mteractlon (transition potential) and the elastic optical potenttal, though one could expect a shift m the threshold of the coupling potential by approxtmatel,, half of the excitation energy. He further concludes that tfone uses an optical potential that ts consistent with the dispersion relatton but still needs an energy-dependent deformation length, then either the optical model itself or the nuclear structure model. or both, are madequate The excitation functions of the first excited 3 - state of 2°8Pb by ~60 at energies around the Coulomb barrter have been measured and analysed [ 12,13] and more recently detailed angular dtstrtbutions for the same system were measured at energies from above to below the Coulomb barrier [14]. These angular distributions were analyzed and the results hinted at a posstbthty that the transition potential may have an 0370-2693/93/$ 06 00 © 1993 Elsevier Science Pubhshers B V All rights reserved 303 Volume 300, number 4 PHYSICS LETTERS B 18 February 1993 energy dependence which is different from that of the elastic opucal potentml [ 14 ]. We present here a sxmple model which shows that, for collective excitatmns, the central and couphng potentmls, although consistent separately with dispersion relations, could exhibit different energy dependences Note that coupling potentmls include the case of rconentauon potentials for the case of elastic scattering of nuclm wqth 1>/I Let us consider the usual collective model deserlptmn of inelastic exc~- tatmn. The optical potentml ~_" ~s considered as a functmn of the collective variable 6, which is a measure of the change m distance of the surfaces of the colhdmg nuclm. The real and mlagmary parts of the potential can be expanded m a Taylor series m powers of(~, and to the first order md. dl(~ dll. I(,J)=l-~7 " , I~'(6)=~- dr o. (3) If the potentmls are exponential in the regmn of mterest, then l'(d)=V(l+~/a), tt'(,5)=H'(l+d/a), (4) where a is the diffuseness parameter, whlch has been assumed to be the same for the real and imaginary potentials. Note that, when expressing the ~magmary potenual as a function of& we are imphc~tl~ assuming that the collectwe variable changes slow b, compared wHh the processes that contribute to the ~magmary potentml That implies also that the energy of the barrier B will be a function old. Approximately. we can write B(d)..-B( I-~/R) . (5) where B is the height of the undeformed bamer and R ~s the corresponding radms. To explore the ~mphcartons of the effect of~ on both the geometr.~ of the potentml as well as on the bamer, we consider a factonzable form of It'(E) H'(I..') = 14"l-'(.v). (`6) where It" is a functmn of the radms while F(x) expresses the energy dependence m terms of the paranaeter x= (B-E)~.1. F(x) ~s assumed to be a Fermi functmn F(x)= [ 1 + exp(.v) ]-~. The parameter J is a measure of the energy range over wh~ch the ,maglnary potentml increases. According to eqs (4) and (5), both W and t"(x) are dependent on the collecrive variable 6. With the ansatz for It'(E) ofeq. (6), the dispersive potenual AI '(E) ofeq. ( I ) becomes aXI'(E) = It'G(.v), (7) where G (.v) is related to F(x) through the dispersion relatmn G(.x)=- p i ---=-F(V)dy. (8) 7/' I'--.V -x The optical potential can be written as /,,'(E. d)= I'(E. d)+11t'(E, d). (9) To first order in d, one obtains I'(E, d)= I'o+ lt'(/(.r) d + - [ I'+ It'G(x)+ It'eG'(.v)I, (10) a W(E. 6) = Jtl.(x) d + - [ liT(x)+ IfiF'(.\)] , (l 1 ) a where we have introduced the parameter c = aB/RA and used eq. (4). In these equauons, the diagonal potentml U0(E) ts the term independent on d and the transmon potenual r_',(E) is the term linear in ~. We can wine th~s exphcltl~ as Uo(E) = I ~, + l!'G(x) +ll!'F(x). ( 12 ) t,( E) d(dU°(E)ar t ) =- \---+a- ~l'(G'(.v)+d:'(x)) . (13) The first term on the right of eq. (13) ~s the usual prescription of the collective model where the transmon potentml ~s related to the derivative of the central potemml. The new feature of this model is the second term on the right which arises from the shift of the barrier due to deformauon. Eqs ( 12 ) and ( 13 ) suggest that, while the central terms will exhibit the threshold anomaly around an energy B, the non-central terms will exhibit it around an energy B( 1 -a/ R ). Besides, all the non-central terms will exhibit the threshold anomaly at the same energy, independently of the mult~polarlty or the couphng strength 304 Volume 300, number 4 PHYSICS LETTERS B 18 February 1993 (provided that a first order treatment ts valid) It is of mterest to point out that exactly the same expressions ( 12 ) and ( 13 ) are obtamcd for the ccntral and transstton potenttals if one considers the adiabatic hmlt when the coupled equattons can be decoupled mto elgenchannels. In the case of including the ground state and a one-phonon state, the two c~- gcnchannels describe the scattering by potentials of radix R +fi, with the corresponding barriers B( I g fi/ R). The transition potential is half the difference of the two cxgenchannel potentials and up to order d yields exactly eq. ( 13 ). If one stdl defines the coupling potential by _ dU(E) Ut(E)=-6¢fr ~-r " (14) the ratio of the cffectlve to the static deformation length of the potential becomes complex and energy dependent: G' (A) "tI/~" (X) R(E)= =l-c l)~/W+G(x)+tF(x)" (15) Eq. (15) shows separately the contrlbuttons to the effective dcformatton of the static deformation and of the dispersive effect due to the change in the barrler energies. R (E) acquires an i magmary part in the VlClntty of the barrier, of the order ofe W/4I'(B). The real part of R (E) has an energy dependence around the bamer that is consistent wtth the tmagmary part through dispersion relations. The size of the correction to the conventional model depends cructally on Ihc parameter e, that is determined primarily by the range A over which the tmaglnary potential rises (for E=B-A, the imaginary potential is 27%, while for E=B+A, it is 73% of the maximum value). Depending on the value of~, one obtains different shapes of the energy depcndcnce of the imaginary part of the transition potential. For < 1, the ~maginary transmon potential rises monotomcally to a constant value 6d W/dr. For ~ > 1. one observes a characteristic maximum in the imaginary transttton potential just above the barrier, and thcn a gradual decrease to the value dd W/dr at larger cnergies. Wc tllustrate this in figs. 1 and 2, where wc show both the elastic (central) and transition potentials for the inelastic scattering of ~60 on z°8pb to the first 3state of the target. Calculatmns have been done taking B=80 MeV, a=0 6 fm and R= 12 fm. Two val- .... I Fig I Central (full hne) and non-central (dashed hne) potenttals versus the energy for A=2 5 MeV The real parts are the upper hnes, and the imaginary parts are the lower hnes ~,.? ?() 1(I¢'. I .'0 I 1~ l (g!W, : F~g 2 Sameasfig llorA=455MeV. ues ofd. which arc J=4.55 MeV, taken from ref. [9], and A= 2.50 MeV, have been used. In both cases, the optical potential is taken as 1.5 + 11.2 MeV for E= 150 MeV. Both calculations are consistent with the empirical data of the optical potential of 160+2°8Pb. These yield values of e of 0 88 and 1.6 respectively. The transition potential is normalized so that it coincides with the central potential at high energies. It is clearly seen that the energy dependence of the transition and central potentials are different in an energy region around the Coulomb barrier. Even though from figs. 1 and 2 it may appear that ff~< I, the traditional collective model, where the transltton poten305 Volume 300, number 4 PHYSICS LETTERS B 18 February 1993 tlal is taken as the product of a static deformation length and the dertvatwe of the complex potential, is valid, this is not strictly true In fig. 3 we show the real and imaginary parts of R(E) as a function of the energy for the two cases 3=4.55 McV and 3=2.5 MeV It is seen that even for ,f=4.55 MeV (~< I ), the deformation length becomes complex. It is evident, of course, that very high quaht3 inelastic data will be necessary to extract this complex deformation length at energies around the Coulomb barrier The parameter e can be understood m a time-dependent picture, r=,5//I has bccn described as a retardation time in the time-dependent p~cture of the 1magmas.' potential [ 15 ]. It is the characteristic time for the processes that contribute to the maagmary potential, that is, for the couphng to states not related to the collectwe variable d. aB/R is approxlmateb equal to I" Classically, the hamiltoman for a vlbraUonal collective variable d is II=P]/2M+Kd2/2l'cxp(d/a) , (16) where M and K are, respectwel.\., the inertia parameter and the harmonic constant of the collective variable We can define the characteristic time for collective excitation as dl~/dt I/T= -- (17) t',~ "7 f. :z'6(] --" t~(; 1(;(: : ~;" I 1(' ? I *d, ', : Fig 3 Real (upper hnes) and imaginary, (lov, er hnes) parts of lhe rauo of the effective and slallc deformatmn lengths as a runeIron of the energy The full hnes correspond to 3=2 5 MeV and the dashed hnes to A=4 55 MeV dP,,/dt call be obtained from eq. (16) using Hamilton's equation dP~ dH d I" - dd C + - a exp(:i/a). (18) dt The second term on the RHS dominates over the first in the adlabauc limit. On the other hand, as 6 will take values of the order of a, we can estimate the value of P,, as h/a. Thus, T=hR/aB is the characteristic time for collective excitations. Hence the parameter = r/T is the rat]o of the t~me for processes that take away flux from the coupled channels system to the tmae for the processes that couple vmhm the coupled channels system If r< 7" thc couphng to non-collectwe channels is fast compared to the couphng to collectwe channels, and then its effect on the diagonal and transition potentials is similar, producing a s~mflar energy dependence of both. If r> 7-, the couphng to non-collective channels Is slow compared to that of collective channels, and so a different energy dependence appears. Note that these condmons are not incompatible with the assumption that the collective variable changes slowly compared with processes that contribute to the imagmarv potential. The characteristic time of change old can be defined as dd/dt 1/7-- (19) d dd/dt can be obtained from eq. (16) using Hamilton's equation dd _ dll _ l',s . (20) dl dP~ .ll Hence. using the previous estimates for (5 and/~, one gets T'= 31a'/h. which can be larger than r if the inertia parameter ~s large enough. To conclude, we have shown that the transition potcntmls for collecm.e excitation have a different energy dependence to the diagonal potentials. The deformatmn length of the potentml becomes complex and energy-dependent around the Coulomb bamer. The size of this effect depends crucially on how fast the imaginary optical potential rises to its maximum value as the energy increases. These effects could be investigated measuring the scattering of polarized projectiles at energies around the Coulomb barrier. 306 Volume 300, number 4 PHYSICS LETTERS B 18 Fcbruar3' 1993 J.G.C. acknowledges partial support from the Spanish DGICYT under contract PB89-0636. 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