Circumstantial evidence for a non-Maxwellian plasma from femtosecond laser-matter interaction
aa r X i v : . [ phy s i c s . p l a s m - ph ] S e p Circumstantial evidence for a non-Maxwellian plasmafrom femtosecond laser-matter interaction
Sachie Kimura and Aldo Bonasera
INFN-LNS, via Santa Sofia, 62, 95123 Catania, Italy andCyclotron Institute, Texas A&M University, College Station TX 77843-3366, USA (Dated: November 13, 2018)We study ion acceleration mechanisms in laser-plasma interactions using neutron spectroscopy.We consider different types of ion-collision mechanisms in the plasma, which cause the angularanisotropy of the observed neutron spectra. These include the collisions between an ion in theplasma and an ion in the target, and the collisions between two ions in the hot plasma. By analyzingthe proton spectra, we suggest that the laser-generated plasma consists of at least two components,one of which collectively accelerated and can also produce anisotropy in the angular distribution offusion neutrons.
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I. INTRODUCTION
The development of small-scale high-intensity laser-systems with the chirped-pulse amplification (CPA) tech-nique has opened new research fields of the laser-plasmainteraction [1]. One of the promising applications of thesestudies is the ion-beam generation from laser irradiationon solid targets [2, 3, 4]. It is reported that protonsin Mylar (H C O ) target irradiation are acceleratedmore effectively than in foil target irradiation. Under-standing the ion acceleration mechanism in the laser-generated plasma is essential for applications. In thisconnection, nuclear reactions induced by laser-irradiationgive a unique clue in understanding the ion-accelerationmechanism [5]. By replacing the protons in the plasticCH n target by deuterons (CD n ), a plasma of deuteriumions is generated. In the plasma the reaction D( d, n ) Hewith a Q -value of 3.26 MeV is induced [6, 7, 8] and pro-duces monochromatic neutrons. The angular distribu-tion of the neutrons shows peculiar anisotropy not onlyon CD-plastic target [7, 8, 9] but also on D -gas jet [10]and on both D and CD clusters [11, 12]. The observedneutron angular distribution gives a direct hint to un-derstand ion acceleration mechanisms in aneutronic re-actions driven by laser as well [13, 14, 15].In a recent paper [8], Habara et al. discuss the re-sults of an experimental analysis of the neutron spectrain nuclear reactions induced by a laser-irradiation on aplastic CD target 50 µ m thick. They observed that neu-tron counts at I =1 × W/cm is larger and moreanisotropic than that at I =2 × W/cm . This is at-tributed to the fact that higher intensity laser-pulse canaccelerate ions more efficiently. The ion temperaturesof 70 and 300 keV at I =2 × and 1 × W/cm ,respectively, with a similar number of accelerated ions( N i =10 ) for both intensities are deduced, using a three-dimensional Monte Carlo (3D MC) code. They simulateion acceleration processes under different assumptionsand conclude that the deuterium ions are accelerated intothe target and cause the nuclear reaction in the target.The directionality of the plasma beam is deduced from the comparison to the differential cross section data ofthe reaction D( d, n ) He in conventional laboratory beam-target experiments [16, 17, 18, 19, 20, 21]. In this paperwe consider different types of ion-collision mechanisms inplasma, which cause the angular anisotropy of the ob-served neutron spectra, including collisions between anion in the hot plasma and an ion in the target (HT),and collisions between two ions in the hot plasma (HH),using at first the total number of accelerated ions andthe plasma temperature given in Ref. [8]. This assump-tion results in the overestimate of the absolute value ofthe neutron yield, if fusion for collisions between ionsin the hot plasma are properly included. This compo-nent was ignored in Ref. [8]. Using the SRIM code [22]we estimate the HT component which suggests a smallernumber of plasma ions compared to Ref. [8]. This, inturn, reduces the number of fusion originating from thecollisions among hot ions in the plasma reconciling to theexperimental observation. However, this is not the onlypossible explanation for the observed angular anisotropyin the neutron data. In fact the angular anisotropy in theneutron spectra can be observed, if a part of the plasmais collectively accelerated and even in the absence of HTmechanism. In order to shed some light on this pointwe study the plasma distribution reported in [3, 4] andshow that indeed such a collective component is observed.We mention that the origin and the mechanism of accel-erated ions have been discussed in detail, for a reviewRef. [5, 23], and by now it is known the existence of atleast two types of ion acceleration mechanisms, i.e., fromthe target front side into the target and from the tar-get rear side to the vacuum. One of these mechanismbecomes predominant depending on the target materialand thickness or laser parameters. The later can be acandidate for the collectively accelerated plasma. Finally,we stress the importance of knowing both characteristicsof fusion product, i.e., the spectra of neutron yield andthe spectra of plasma ions, under common experimen-tal conditions. At present those data are available onlyseparately.
II. NUMBER OF PLASMA IONS DERIVEDFROM THE OBSERVED NEUTRON YIELD
In practice we consider the following two types ofmechanisms for neutron generation in high intensity laserirradiation.(A) Collisions between two ions in the laser-heatedplasma. Both ions are moving with thermal ve-locity. Under this assumption the direction ofthe incident reaction channel is random, hence theangular distribution of reaction products will beisotropic [7]. The contribution to the neutron yieldfrom this mechanism is called “HH” in this paper.(B) Collisions between an accelerated ion in the laser-produced plasma and a cold nucleus in the bulkof the target. Under this second assumption theangular distribution of reaction products is possiblyanisotropic. The contribution from this mechanismis called “HT” component.We stress the importance of considering both mechanismsmentioned above, comprehensively, because either mech-anism might be predominant, depending on the char-acteristics of the target. As an example in the case ofneutron yield observation from laser pulses irradiationon deuterated clusters [12] both mechanisms play a keyrole.If we assume that neutrons are produced by the colli-sion of the ions in the hot plasma component (HH), interms of the number of the accelerated ions N i , the num-ber of fusion per solid angle, or reaction rate [24], is givenby N ( HH ) f π = 14 π N i n cr τ Z σ ( v ) vφ ( v ) dv , (1)where n cr =10 / λ is the plasma critical density [25, 26]; τ is the laser pulse duration; σ ( v ) and v are the reactioncross section and the relative velocity of the colliding ions.In general the reaction cross section σ ( v ) is given as afunction of the incident energy, instead of the velocity,but here we have written it as a function of the velocityto keep the consistency in the velocity integral. Later σ will be represented as a function of the incident energy.We stress that assuming the critical density, which is thelowest limit to the real density reached in the experiment,the fusion yield given by Eq.(1) is underestimated. Thedensity profile of the plasma simulated in Ref [8], usingthe PIC code, has an exponential shape which varies from4 n cr to 0.1 n cr . φ ( v ) is the relative velocity spectrum ofa pair of ions and is given by a Maxwellian-distributionat the temperature kT HH =70 or 300 keV: φ ( v ) = (cid:18) µ πkT HH (cid:19) exp (cid:18) − µv kT HH (cid:19) , (2)where µ is the reduced mass of ions. Eq. (1) gives 1. × and 4. × , per solid angle, at the temperature of 70 and of 300 keV, respectively, see Tab.I. The estimatedyield is comparable with the neutron spectra in Fig.s 3and 5 in Ref. [8]. The contribution from HH componentis, therefore, expected to be seen in the figure as a peakat the neutron energy 2.45 MeV and to be isotropic [7].In order to compare this to their energy distribution, wecan roughly assume that the neutron distribution is aGaussian distribution with a center-of-mass (CM) energyat 2.45 MeV and a width given by the temperature ofthe plasma. This gives an estimate at all angles of 1.4 × and 1.3 × ion/MeV/sr respectively, which isseen neither in figures 3 nor 5 in Ref. [8]. In the figures,if anything, one sees a clear angular dependence of theneutron yield and shifts of the observed peaks from theexpected energy 2.45 MeV. This implies that either theirestimated temperature or the number of accelerated ionsgiven is too large. In other words the authors of Ref. [8]should have estimated the number of neutrons comingfrom the HH component, and show that this componentis negligible compared to the HT contribution. As wewill show in the following discussion, the contributionfrom the HT component, which has the correct angulardependence of the observed neutron spectra, is indeeddominant but with a smaller number of plasma ions.If we consider the collisions between the ions in theplasma and the almost stable nuclei in the target (HT),the angular distribution of reaction products is expectedto be anisotropic. To estimate the neutron yield from theHT component, one should take into account that oneof the colliding ions is at rest in the laboratory frame.Therefore in the reaction rate per pair of colliding ions,the velocity spectrum Eq.(2) is modified as: φ HT ( v ) = (cid:18) m πkT HH (cid:19) exp (cid:18) − m v kT HH (cid:19) , (3)where m is the mass of ions in plasma, instead of thereduced mass. One can define an effective temperatureas, kT effHT = ( µ/m ) kT HH . (4)Now we can use the effective temperature to estimatethe number of fusion. For simplicity we estimate themost probable energy of the plasma ions that cause thenuclear reaction given by the Gamow peak energy ( E G )[24]. The Gamow energy can be found using the saddlepoint method, i.e.: ddE (cid:18) EkT + bE − (cid:19) = 0 , (5)where b = 31 . Z Z M (keV ), denoting the atomicnumber of the colliding nuclei Z , Z and the reducedmass number M . We remind that M = A A / ( A + A ),where A and A are the mass numbers of the collidingions, respectively, is different from the reduced mass µ .The temperature kT is replaced by the plasma tempera-ture, i.e., the temperature of the HH component in con-ventional discussions, but for the present case of the HTcomponent, kT is replaced by the effective temperature.Then, E G = (cid:16) bkT effHT / (cid:17) . (6)We specially mention that Eq. (5) is valid in the case ofsub-barrier reactions, because this condition comes fromthe product of a Maxwellian and the Coulomb barrierpenetrability [24]. The height of the Coulomb barrier forthe reaction D( d, n ) He is estimated to be about 470 keV.By approximating the energy of the accelerated ions bythe Gamow energy Eq. (6), the neutron yield per solidangle is written as: N ( HT ) f π = 14 π N i σ ( E G ) n T d, (7)where σ ( E G ) is the reaction cross section at the Gamowenergy. The reaction cross section data at this energyare taken from NACRE compilation [27] and given inTable I. The number density of the solid polyethylenetarget: n T = 1.2 × atoms/cm and d is the pro-jected range [22] of the accelerated ions in the target.Using the above listed numbers the neutron yield persolid angle is also reported in Tab. I. As a consequenceof this simple estimate, the yield per solid angle underthis assumption is a factor 10 higher than the yield fromthe collisions between two ions in the plasma. This isbecause the number density of the solid target is muchhigher than that of the laser-generated plasma. Howeverfrom eq. (4) the effective HT temperature is lower thanthe HH one, depending on the asymmetry in mass of thefusion ions. The higher temperature in the HH collisionmechanism has an advantage of the higher fusion crosssection. There is, therefore, a competition between twomechanism depending on the laser-intensity. This fea-ture might explain why at lower intensities (i.e. lower T)the neutron angular distribution is less anisotropic: Atthe lower intensity irradiation, the contribution from theHT component to the neutron yield is suppressed com-pared with the higher intensity irradiation. In fact thetemperature of the HT component is low and most re-actions happen below the Coulomb barrier. The nuclearreactions below the barrier are exponentially suppressed.Therefore the HH mechanism, which has higher temper-ature, contributes to the less anisotropic neutron yieldangular distribution. On the other hand, if the plasmatemperature is high enough, the corresponding Gamowenergy is above the Coulomb barrier. At energies abovethe Coulomb barrier, i.e. high T, to have higher densitiesgives higher fusion probabilities (above the barrier fusionprobabilities depend quadratically on densities and donot depend much on T) thus collisions between an ionfrom the plasma and an ion from the target becomesmore probable. We expect that in some temperature re-gion there is a transition from collisions occurring mainlyin the plasma to collisions occurring between the plasmaions and the target ones. An experimental detailed inves-tigation of this transition region would be very interesting ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
0 200 400 600 800 1000 1200 1400 1600 d N / d E d Ω pe r k e V S r E (keV)Mylar 6 µ mMylar 13 µ m FIG. 1: Experimental data of proton energy spectra for Mylartargets 6 (asterisks) and 13 (crosses) µ m-thick retrieved fromLee et al. [4]. The thick curves represent fitting of the protonspectra using Maxwellian distributions and distributions withextra accelerations. The thin curves are the different compo-nents and are summed into the thick curves. The obtainedfitting parameters are given in Tab. II. and instructive in order to understand the microscopicdynamics of fusion.The HT contribution to the neutron yield is a factor10 higher than experimentally observed data [8], as well.This again implies that either the estimated tempera-tures are too high or the estimated number of ions in theplasma is too large. From this simple estimate and theone above from the HH component we can argue that thenumber of ions in the plasma could be at least two ordersof magnitude smaller than that estimated in Ref. [8], i.e.,about 10 , instead of 10 . This reduction of the numberof accelerated ions results in a suppression of the neutronyield from the HH component which becomes negligible.A more involved calculation solving eq.(1) for the HTcomponent numerically gives results in agreement withour simple estimate and will be discussed in more detailin a following paper [28]. III. PLASMA TEMPERATURE SUGGESTEDBY THE PROTON SPECTRA FROM PLASTICTARGETS IRRADIATION
Another evidence of the smaller number of plasma ionsis shown by a direct observation of proton spectra in laserirradiation [3, 4]. Fig. 1 in Lee et al. [4] shows the protonenergy distribution at the laser intensity I = 2.2 × W/cm similar to [8], and the target material is Mylaror aluminum. We are especially interested in the resultsfrom the Mylar target, because the characteristics of theproduced plasma should be close to the characteristicsof the plasma from the deuterated plastic targets. We,therefore, selected two results from 6 and 13 µ m thickMylar target. A peculiar feature of the proton energydistribution is that it exhibits bumps in the higher energy TABLE I: Plasma temperature ( kT HH ) and neutron yield per solid angle ( N ( HH ) f /Sr ) from the HH component and the effectivetemperature ( kT effHT ) and N ( HT ) f /Sr from the HT component at the given laser intensities.I (W/cm ) kT HH (keV) N ( HH ) f /Sr kT effHT (keV) E G (keV) σ ( E G )(10 − cm ) d ( µ m) N ( HT ) f /Sr ×
70 1. ×
35 67 24 0.8 4. × ×
300 4. ×
150 176 66 2. 3. × TABLE II: Fitting parameters for proton energy spectra from6 and 13 µ m thick targets irradiation. Especially for 13 µ mthick target we have used three components, two of whichhave different relative velocities with respect to the hot-plasma component.T thick. c kT HH c kT E ( µ m) (10 ) (keV) (keV) (keV)6 (3.0 ± ±
1. (3.5 ± ±
5. 170. ± ± ±
1. (3. ± ±
30. 210. ± ± ±
5. 695. ± region. This feature could be attributed to the existenceof at least two different components in the plasma. Todescribe these characteristics, we use a Maxwellian dis-tribution in terms of the energy of the protons in the lab.system ( E ): d NdEd
Ω = c √ π √ E ( kT HH ) / exp( − E/kT HH ) . (8)To this we add a Maxwellian distribution of a movingsource [29]: d NdEd
Ω = c π √ EE ′ ( kT ) exp( − E ′ /kT ) , (9)where E ′ = E − √ EE + E . Both equations are nor-malized to 1/4 π . The latter gives a part of the plasma intranslational motion with respect to the laboratory sys-tem with a collective kinetic energy E . The obtainedfitting curves are shown by thick solid and dashed curvesfor the spectra at 6 and 13 µ m thick targets irradiation,respectively, in Fig. 1. The thin curves show each com-ponent which add into the thick curves. Especially forthe spectra at 13 µ m thick target, we have used a sumof three components, two of them have different extraacceleration energies E . The corresponding fitting pa-rameters are summarized in Tab. II. From the fittingwe can deduce the number of ions in the plasma N i ∼ × and the plasma temperature of kT HH ∼
44 and61 keV for 6 and 13 µ m thick targets, respectively. Thededuced number of plasma ions is in agreement with ouryield estimation from the neutron measurement. More-over one can see clearly that the proton spectra differfrom a Maxwellian but a part of plasma is acceleratedto higher energy with E . Similar features of the pro-ton spectra in laser-produced plasma can be found inSpencer et al. [3], as well. The possible reasons for such a collective acceleration will be discussed in a followingpublication [28].In Fig. 7 and 8 of [8], the authors discuss the ratio ofthe neutron yields at the angle of 23 ◦ to that at the angleof 67 ◦ from the target rear normal. They found that theratio can be larger than 3. In Fig. 2 the ratio of thedifferential cross sections in beam-target experiments[16,17, 18, 19, 20, 21] at laboratory angles 23 ◦ to 67 ◦ is shownby thick curves. There is a discrepancy between [sc72]data [18] and [li73] [19] in the energy region higher than3 MeV. Nevertheless the figure shows clearly that theratio can, indeed, be larger than 3 and less than 3.5 at theincident deuteron energy E L from 1.75 to 1.96 MeV. Thusthe accelerated deuterons which contribute to the nuclearreactions have energies slightly less than 2 MeV. In theCM frame E ∼ E G =900 keV,the neutron yield per solid angle from the HT collision isestimated to be of the order of 10 from Eq. (7), adoptingthe number of accelerated ions given in [8]. This numberis 10 times higher than that in Table I. In other words,to reproduce the experimental data, one needs to assume N i ∼ , as the number of accelerated ions in plasma.In passing we note that the ratio of the neutron yieldtaken at the angle of 20 ◦ to that of 85 ◦ is higher thanthe previous one as shown in Fig. 2 by thin curves withsmaller points. A careful experimental determination ofthis feature would be very useful.We can also derive the plasma temperature corre-sponding to E G =900 keV. Since at this energy we areabove the Coulomb barrier, we can estimate it from theclassical relation: E G = (3 / kT HH , (10)which gives kT HH =600 keV, i.e., higher than the es-timated temperature in Ref. [8]. This contradictionmight be solved by considering that a part of the plasmadeuterons is collectively accelerated at energy E , as wehave shown in the analysis of proton spectra. This ex-tra acceleration energy can reach about 700 keV which isclose to 900 keV. In the presence of an extra acceleration,the plasma temperature cannot be derived simply fromthe relation (10). The difference between the plasma tem-perature deduced from the ratio at two angles and theone estimated by Ref. [8] suggests that the plasma spec-tra is different from a usual Maxwellian distribution. Inother words, the difference justifies the presence of thecollective motion of a part of the plasma. Indeed the col- d i ff. c r . s e c t. r a t i o ( o / o ) neu t r on E L (MeV)th66sc72ja77hu49(neutron)li73 FIG. 2: The ratio of differential cross sections at two angles.The ratio at angles 23 ◦ to 67 ◦ is shown by large points con-nected by curves, while the ratio at angles 20 ◦ to 85 ◦ is shownby smaller points connected by thin curves, for a comparison.The experimental data are retrieved from EXFOR-data sys-tem [30]. hu49-data (pink filled circles) are evaluated fromthe angular distribution of the DD neutron yield [16] andthe others, th66-data (grey open circles) [17], sc72 (red filledsquares) [18], ja77 (green open squares) [20] and li73 (bluecrosses) [19], are converted from the He angular distributionof the DD reaction. lisions between two ions in the plasma which is moving ata collective energy E in the laboratory frame can result in the angular anisotropy of the neutron yield. This isa possible mechanism to explain the angular anisotropyexperimentally observed. IV. CONCLUSIONS
In conclusion, we have discussed different possible ion-collision mechanisms, which can result in the observedanisotropic neutron spectra. When analyzing the protonspectra, we have found that there are at least two plasmacomponents: one is approximated by a Maxwellian dis-tribution and the other has a collective motion relativeto the former component. Comparing the ratios in theneutron counts at two angles in the experiment to the dif-ferential cross sections measured in conventional beam-target experiments, the most effective energy is estimatedto be 0.9 MeV with corresponding plasma temperatureof 600 keV, which is higher than the estimated plasmatemperature in Ref. [8]. We have discussed a possible so-lution of this contradiction, in connection with the collec-tive motion of a part of the plasma, which can in princi-ple explain the observed neutron angular anisotropy. Wesuggest that an experimental determination of the neu-tron angular anisotropy in coincidence with the plasmadistribution would be very interesting and give useful in-formation on the mechanisms at play. [1] K. Ledingham, P. McKenna, and R. Singhal, Science , 1107 (2003).[2] A. J. Mackinnon, Y. Sentoku, P. K. Patel, D. W. Price,S. Hatchett, M. H. Key, C. Andersen, R. Snavely, andR. R. Freeman, Phys. Rev. Lett. , 215006 (2002).[3] I. Spencer, K. W. D. Ledingham, P. McKenna, T. Mc-Canny, R. P. Singhal, P. S. Foster, D. Neely, A. J. Lan-gley, E. J. Divall, C. J. Hooker, et al., Phys. Rev. E ,046402 (2003).[4] K. Lee, S. H. Park, Y.-H. Cha, J. Y. Lee, Y. W. Lee,K.-H. Yea, and Y. U. Jeong, Phys. Rev. E , 056403(2008).[5] S. Karsch, S. D¨usterer, H. Schwoerer, F. Ewald, D. Habs,M. Hegelich, G. Pretzler, A. Pukhov, K. Witte, andR. Sauerbrey, Phys. Rev. Lett. , 015001 (2003).[6] P. A. Norreys, A. P. Fews, F. N. Beg, A. R. Bell, A. E.Dangor, P. Lee, M. B. Nelson, H. Schmidt, M. Tatarakis,and M. D. Cable, Plasma Phys. Control. Fusion , 175(1998).[7] N. Izumi, Y. Sentoku, H. Habara, K. Takahashi,F. Ohtani, T. Sonomoto, R. Kodama, T. Norimatsu,H. Fujita, Y. Kitagawa, et al., Phys. Rev. E , 036413(2002).[8] H. Habara, R. Kodama, Y. Sentoku, N. Izumi, Y. Kita-gawa, K. A. Tanaka, K. Mima, and T. Yamanaka, Phys.Rev. E , 036407 (2004).[9] D. Hilscher, O. Berndt, M. Enke, U. Jahnke, P. V. Nick-les, H. Ruhl, and W. Sandner, Phys. Rev. E , 016414 (2001).[10] S. Fritzler, Z. Najmudin, V. Malka, K. Krushelnick,C. Marle, B. Walton, M. S. Wei, R. J. Clarke, and A. E.Dangor, Phys. Rev. Lett. , 165004 (2002).[11] T. Ditmire, J. Zweiback, V. Yanovsky, T. Cowan,G. Hays, and K. Wharton, Nature , 489 (1999).[12] F. Buersgens, K. W. Madison, D. R. Symes, R. Hartke,J. Osterhoff, W. Grigsby, G. Dyer, and T. Ditmire, Phys.Rev. E , 016403 (2006).[13] V. S. Belyaev, A. P. Matafonov, V. I. Vinogradov, V. P.Krainov, V. S. Lisitsa, A. S. Roussetski, G. N. Ignatyev,and V. P. Andrianov, Phys. Rev. E , 026406 (2005).[14] S. Kimura, A. Anzalone, and A. Bonasera, Phys. Rev. E , 038401 (2009).[15] A. Bonasera, A. Caruso, C. Strangio, M. Aglione, A. An-zalone, S. Kimura, D. Leanza, A. Spitaleri, G. Imme,D. Morelli, et al., Fission and properties of neutron-richnuclei - Proceedings of the Fourth International Confer-ence p. 503 (2008).[16] G. Hunter and H. Richards, Phys. Rev. , 1445 (1949).[17] R.B.Theus, W.I.McGarry, and L.A.Beach, Nucl. Phys. , 273 (1966).[18] R. Schulte, M. Cosack, A. Obst, and J. Weil, Nucl. Phys.A , 609 (1972).[19] H. Liskien and A.Paulsen, Nuclear Data Tables (NuclearData Sect.A) , 569 (1973).[20] J. N.Jarmie, Phys. Rev. C , 15 (1977).[21] R. E. Brown and N. Jarmie, Phys. Rev. C , 1391 (1990).[22] J. Ziegler, URL .[23] J. Fuchs, Y. Sentoku, E. d’Humieres, T. E. Cowan,J. Cobble, P. Audebert, A. Kemp, A. Nikroo, P. An-tici, E. Brambrink, et al., Physics of plasmas , 053105(2007).[24] D. D. Clayton, Principles of Stellar Evolution and Nu-cleosynthesis (University of Chicago Press, 1983).[25] D. W. Forslund, J. M. Kindel, and K. Lee, Phys. Rev.Lett. , 284 (1977).[26] F. Begay and D. W. Forslund, Phys. Fluids , 1675 (1982).[27] C. Angulo, M. Arnould, M. Rayet, P. Descouvemont,D. Baye, C. Leclercq-Willain, A. Coc, S. Barhoumi,P. Aguer, C. Rolfs, et al., Nucl. Phys. A , 3 (1999).[28] S. Kimura and A. Bonasera, to be submitted.[29] T. C. Awes, G. Poggi, C. K. Gelbke, B. B. Back, B. G.Glagola, H. Breuer, and V. E. Viola, Phys. Rev. C ,89 (1981).[30] URL