The calculations of heavy-ion fusion cross sections in the above and
sub-barrier regions are presented. A barrier penetration model with matrix
method for the calculations of transmission coefficients through real
one-dimensional barrier of nucleus-nucleus potential is used. The renormalization
parameter Ns of Satchler-Love M3Y nuclear potential and
critical radius Rei are the two parameters of the model. A very good
description of the experimental fusion cross sections for fp-shell colliding
ions was obtained for the full measured energy range. It was found that
both parameters Ns and reduced critical radius r„ are very similar for all
systems considered and are-energy independent

Czakański, Jacek

Kisiel, Jan

Kostrubiec, M.

Zipper, Wiktor

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Penetration of realistic potential barrier of H-I fusion  Author: Jan Kisiel, Jacek Czakański, M. Kostrubiec, Wiktor Zipper  Citation style: Kisiel Jan, Czakański Jacek, Kostrubiec M., Zipper Wiktor. (1992). Penetration of realistic potential barrier of H-I fusion. "Acta Physica Polonica. B" (1992, no. 1, s. 73-81). PEN ETR A TIO N  OF REALISTIC  PO TEN TIA L BA R R IER  OF H -I FUSION*J . K lS I E L , J . C Z A K A N S K I, M . K O S T R U B IE C  A N DW. Z i p p e rInstitute of Physics, University of Silesia Uniwersytecka 4, 40-007 Katowice, Poland(Received September ZS, 1991; revised version received December 27, 1991)The calculations of heavy-ion fusion cross sections in the above and sub-barrier regions are presented. A barrier penetration model with ma­trix method for the calculations of transmission coefficients through real one-dimensional barrier of nucleus-nucleus potential is used. The renor­malization parameter Ns of Satchler-Love M3Y nuclear potential and critical radius Rei are the two parameters of the model. A very good description of the experimental fusion cross sections for fp-shell colliding ions was obtained for the full measured energy range. It was found that both parameters Ns and reduced critical radius r„ are very similar for all systems considered and are-energy independent.PACS numbers: 25.70.Ij1. In tro d u c tio nThe barrier penetration model (BPM ) of the fusion of two nuclei is based on the assumption that fusion take place when the potential barrier of colliding ions is passed [1]. There are some practical applications o f the BPM  in which the different approximations of the real effective nucleus- nucleus potential have been done. The crucial point in the BPM  are the calculations of the transmission coefficients T[ns(E). The usually used ap­proach is the m ethod proposed by Hill and Wheeler [2]. They obtained the* This work was partly supported by the Polish Ministry of National Education under contracts P/04/233/90 and KBN No 201719101analytic expression for the Tfn*(E) by approximating the effective potential in the neighbourhood of the barrier maximum by parabola. Haider and Malik added a Coulomb tail to the parabolic shape of the nucleus-nucleus potential [3]. The double barrier m ethod was proposed by Descouvemont et al. [4]. These calculations can not explain the experimental fusion ex­citation function in the full measured energy range. The only exception is the work of Descouvemont et al. [4] in which the experimental data was fitted by using the shallow Saxon-W oods shape of the nuclear potential. However it is known from recent paper o f Wada and Horiuchi [5] that for 160 + 160  system  the equivalent local interaction should take the form of a deep potential which is consistent with the Pauli principle.It can be generally concluded that crfu,(E )  is not satisfactory well re­produced by the BPM  model, in both energy regions, i.e. sub- and above- the Coulomb barrier, simultaneously.The purpose of the present paper is to show that one-dimensional BPM  gives satisfactory results in <rfUh 7) description if: (i) the potential barrier is well approximated and (it) the nuclear potential is realistic. We applied the matrix m ethod [6] to  calculate the T\U*(E) through the realistic potential barrier. It is assumed that the incoming particle penetrates the potential barrier and the fusion occurs when the critical distance R cz is reached. So the parameters o f the model are R „  and the parameters of nuclear interaction. The success of the microscopic double folded Satchler-Love M3Y potential [7] leads us to use it (with the suitable renormalization factor) as a nuclear part o f effective interaction of colliding nuclei.The results of the BPM  calculations o f the fusion cross section for the reactions: 12C +  12C, 14N +  12C, 12C +  ie 0 ,  and ie 0  +  160  are pre­sented. The calculations of the <Tfug(J5) were performed for the full measured energy range starting from the sub-barrier region up to several times the Coulomb barrier energy.The model is shortly outlined in Sect. 2. In Sect. 3 the results of calcu­lations and discussion are presented. The paper is concluded in Sect. 4.2. T h e  m o d e lIn this section we shall outline the framework of the model. The details have been described elsewhere [6].The one-dimensional, effective interaction potential VKg(r, I) of two col­liding nuclei is a  sum of nuclear k*f(r), Coulomb Vc (r) and centrifugal Vj(r, I) terms, i.e.Ve«(r, 0  =  Vfc(r) +  Vb(r) +  Fj(r, I). (1)It is assumed that the fusion occurs when the some critical distance R CT is reached by the incoming particle. Before that the projectile penetratesthe potential barrier, for a given angular momentum I, what is described by the transmission coefficients Tfus(E). The fusion cross section crias(E)  can be calculated, when T,iu*(J5)’s are known, according to the formula fromRef. [4]OO<r{ns(E) =  Y (Ti™ ( l ' E )= £  X  ( » +  Qj'+T)*21+1)T‘"“ (E>' (2>where k the wave number of the incoming particle, I  is its spin, and i  =  0 (£ =  1) for nonidentical (identical) nuclei. For the calculations of Tifus(E )’s the m atrix m ethod is proposed. It can be presented in the follow­ing steps (for the details see Ref. [6]).(i) The potential curve Fcff(r, I) for a given angular momentum I is divided in the region R CI < r < R Cut into N  equal parts (see Fig. 1 ). -Rcut is the distance beyond which the nuclear potential Lfo(r) becomes negligible (-Rcut =  15 fin in the calculations).(ii) Tjns(E)  for a given I can be expressed by the following formula:I f “ (E) = , (3)where: Z ~  (Z +) is the amplitude of the incoming (outgoing) wave function in r =  Rcut- It is expressed by the regular Fi and Gi Coulomb wave functions. A ~  (¿4+) is the amplitude o f the incoming (outgoing) wave function in r =  R CI.The amplitudes A~  and A + are obtained from the following matrix equationThe elements of the matrix Tf (i =  1, • • •, N )  depend on the height and width of the i-th  part of the potential barrier, and also on the energy of the projectile.(iii) In the lim it N  -* oo one obtains the staircase function which is a  good approximation o f the real potential barrier (see Fig. 1). The transmis­sion coefficients Tfua(E)  given by Eq. (3) tend towards the transmission coefficients through the real potential barrier.It should be stressed here that this m ethod can be applied for the cal­culations of transmission coefficients for any shape of the static nuclear potential barrier.Fig. 1. The shape of the effective potential Vtg(r,l) barrier for a given angular momentum t.3. R e su lts  an d  d iscu ss io nThe fusion cross section calculations were done for the following lp-shell systems: 12C +  12C, 14N +  12C, 12C +  160 ,  160  +  160 .  For these sys­tems the experimental data exist for wide energy range starting from below  Coulomb barrier region up to several times the Coulomb barrier energy.In the m odel presented there are the following parameters: the critical distance R ct and parameters o f the nuclear potential V n(t). In last decade the double folded M 3Y Satchler-Love potential [7] was used for both elastic scattering and fusion calculations with satisfactory results [7-9]. This lead us to use this potential with the suitable renormalizating factor A s as a nuclear part of Feff( r 5 0 * This reduces the number of fitted parameters of the model to two: R ct and N s,  which must be extracted from the comparison of the calculated <7fui (2?) with the experimental data.The fitting procedure was divided into two parts: energy region below and above the Coulomb barrier. It was found that the “best-fit” values of the reduced critical radius rCI (RCI =  rcr(Aj + A j ), where Aj and A 3 are mass numbers o f interacting nuclei) and Ns  do not depend on the energy ofcolliding ions. The results o f rcr, R CI and Ns  are presented in Table I. The calculated fusion excitation function <Xfus(2?) together with the experimental points are plotted in Figs 2 and 3 for the above and below Coulomb bar­rier energy regions, respectively. In all presented cases the good agreement between theory and experiment was obtained in the full measured energy range. Particularly in the most troublesome region for BPM  i.e., for the energies below the Coulomb barrier we acquired excellent agreement with experimental data. The oscillations which occur in <Tina(E)  for systems of identical, spin= 0  nuclei, i.e., 12C +  12C and 160  +  1 0  are also repro­duced by our model. The origin of the diffraction structure of CTfug(JS) is the “quantisation” of the angular momentum in units o f 2h [18,19]. It is distinctly seen for the above-Coulomb barrier energies (see Fig. 2) because the difference between two succeeding <T{UM(l,E )  is considerable. For sub­barrier energies the major part o f <TfUJ(F )  comes from <Tfu#(/ =  0 , 1?) what generates that erfug(I?) is a  smooth function of energy.TABLE IThe renormalizating factor Ns of Satchler-Love M3Y potential and r„ , Ret values for collisions between lp-shell nucleiRefs, for exp. dataReaction Z\ * Z2 NS rcr[fm] R „ Ifm] below aboveCou omb barrier12C +  12C 36 0.90 0.93 4.25 [10] [H]14N +  12C 42 1.05 0.95 4.46 [12] [13]12c  +  16o 48 0.99 0.98 4.71 [14] [15]16o  +  160 64 1.16 1.04 5.24 [16] [17]Haider and Malik [3] reproduced the general trend of the energy depen­dence of the fusion cross section in the sub-barrier region but the absolute magnitudes o f their cr{us(E)  values are unpredicted. They can improve the sub-barrier results by changing the “matching” radius R n but then the fits in the above-barrier region become poorer.It should be noted that Descouvemont et al. [4] obtained satisfactory description of the experimental <rjm (E)  for collisions between lp-shell nuclei, apart from the 160  +  160  system  for above Coulomb barrier energies, but they used the shallow nuclear potential. The depth of their Saxon-Wodtis shape nuclear potential was about 15 MeV. However from recent papers it is known that it should be much deeper. Kondo et al. [20] suggest that only the deep ( ~  300 MeV) nuclear potential is able to reproduce the res­onant structure of the elastic scattering excitation function for 16 0  +  16 0Fig. 2. Fusion cross sections <r{at(E) for the energies above the Coulomb barrier for collisions between lp-shell nuclei. The lines represent <T{U%{E) obtained with Eq. (2) with parameters from Table I. For experimental data refs, see Table I.system. Furthermore a unique deep potential for this system was found by analyzing nuclear rainbow data [21]. Also Wada and Horiuchi [5], by using the resonating group m ethod, reported that the equivalent local interac­tion for light heavy-ion systems like 160  +  160  is deep. The deep nuclear potential is consistent with the Pauli principle because it produces wave functions with proper number of radial nodes. It should be stressed here that Satchler-Love M3Y potential, used in this paper, fulfils this condition very well.It was assumed that the fusion occurs when the critical distance R CI is reached (J?CP =  rcr(A j^3 +  A ^ 3)), so it is expected that the value of the reduced critical radius rCT should be close to the overlap nuclear matter half density radius r 1/ 2 which has the standard value 1.0 ±  0.07 fm [22]. The obtained value of rcr satisfies this prediction very well (see Table I).The renormalizing factor Ns  of Satchler-Love M3Y potential for fusion takes the values from 0.90 for 12C +  12C system  to 1.16 for 160  +  160(MeV) (MeV)Fig. 3. Fusion cross sections <T{ut(E) for the energies below the Coulomb barrier for collisions between lp-shell nuclei. For details see Fig. 2system (see Table I). It is consistent with Ns  value obtained by Satch- ler and Love from elastic scattering data analysis. They reported [7] the N s = 1.11 ±  0.13 as a  result o f fitting procedure for considerable amount of combinations of colliding ions.4 . C o n clu sio n sIn this paper the description of the fusion cross section <rfns(E)  for some combinations of lp-shell ions is presented. For 12C +  12C, 14N +  12C, 12C +  160  and 160  +  160  systems the calculations of the <Xfus(JS) were performed for the full measured energy range including both below Coulomb barrier energies and energies up to several times Coulomb barrier energy. The BPM  with matrix method for the calculations of transmission coeffi­cients through real potential harrier was used. As a nuclear interaction the Satchler-Love double folded M3Y potential was taken.The remarkable agreement between the theoretical results and experi­mental data was obtained for the full measured energy range. It was foundthat the values of two parameters o f the model i.e., the reduced critical ra­dius rct and renormalizating factor Ns  o f  nuclear potential do not depend on the energy o f colliding ions and are very similar for all systems taken into account. The found of the fusion value of Ns  — 1.0 is consistent with Satchler and Love result obtained from elastic scattering analysis. Also the value of the second parameter o f the model rcr close to 1.0  fm is related to the overlap nuclear matter half density radius.The results o f this work show that the same nuclear potential, used as a real part o f the optical m odel potential, can reproduce satisfactory the elastic scattering [7] and also describes the energy behaviour o f the fusion cross section in the full measured energy range within one-dimensional BPM.We thank Prof. G.R. Satchler for his numerical double folded M3Y  potentials.REFERENCES[1] D. Glas, U. Mosel, Nucl. Phys. A237, 429 (1975).[2] D.L. Hill, J.A. Wheeler, Phys. Rev. 89, 1102 (1953).[3] Q. Haider, F.B. Malik, Phys. Rev. C26, 162 (1982).[4] P. Descouvemont, D. Baye, P.H. Heenzen, Z. Phys. A306, 79 (1982).[5] T. Wada, H. Horiuchi, Prog, Theor. Phys. 80, 488 (1988).[6] M. Lesiak, W. Zipper, J. Czakański, Acta Phys. Pol. BIO, 775 (1985).[7] G.R. Satchler, W.G. Love, Phys. Rev. 55, 183 (1979).[8] M. Hugi, L. Jarczyk, B. Kamys, J. Lang, R. Müller, J. Sromicki, A. Strzalkow­ski, E. Ungricht, G. Willim, Z. Wróbel, W. Zipper, J. Phys. G7, 1395 (1981).[9] W. Zipper, J. Czakański, J. Kisiel, P. von Brentano, Z. Phys. A337, 309 (1990).[10] M.D. High, B. Ciyec, Nucl. Phys. A282, 181 (1977).[11] M.N. Nambodiri, E.T. Chulick, J.B. Natowitz, Nucl. Phys. A203, 491 (1976).[12] Z.E. Świtkowski, R.G. Stokstadt, R.M. Wieland, Nucl. Phys. A279, 502 (1977).[13] J. Gomez del Campo, R.G. Stokstadt, J.A. Biggerstaif, R.A. Dayras, A.H. Snell, P.H. Stelson, Phys. Rev. C19, 2170 (1979).[14] B. Cqjec, C.A. Barnes, Nucl. Phys. A260, 461 (1976).[15] D.G. Kovar, D.F. Geesman, Т.Н. Briad, Y. Eisen, W. Henning, T.R. Ophel, M. Paul, K.E. Rahm, J.S. Sanders, P. Sperr, J.P. Schiffer, S.L. Tabor, S. Vigdor, B. Zeidman, F.W. Prosser, Phys. Rev. C20, 1305 (1979).[16] S. Wu, C. Barnes, Nucl. Phys. A422, 373 (1984); A. Kuronen, J. Keikonen, P. Tikkanen, Phys. Rev. C35, 591 (1987).[17] B. Fernandez, C. Gaarde, J.S. Larsen, S. Panotoppidau, F. Videbaek, Nucl. Phys. A306, 259 (1978).[18] N. Poffe, N. Rowley, R. Lindsay, Nucl. Phys. A410, 498 (1983).[19] A. Kabir, M. Kermode, N. Rowley, Nucl. Phys. A481, 94 (1988).[20] Y. Kondo, B.A. Robson, R. Smith, Phys. Lett. B227, 310 (1989).[21] Y. Kondo, F. Michel, G. Reidemeister, Phys. Lett. B242, 340 (1990).[22] J. Galin, D. Guerrau, M. Lefort, X. Tarrago, Phys. Rev. CO, 1018 (1974).

Penetration of realistic potential barrier of H-I fusion

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Penetration of realistic potential barrier of H-I fusion

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