Dispersion of longitudinal momentum distributions induced in fragmentation reactions
A. Bacquias, V. Föhr, D. Henzlova, A. Kelić-Heil, M.V. Ricciardi, K.-H. Schmidt
aa r X i v : . [ nu c l - e x ] N ov Dispersion of longitudinal momentum distributions induced in fragmentation reactions
A. Bacquias a,b,1, ∗ , V. Föhr a,c , D. Henzlova a,2 , A. Kelić-Heil a , M.V. Ricciardi a , K.-H. Schmidt a a GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany b Université de Strasbourg, France c Jyväskylän Yliopisto, Jyväskylä, Suomi/Finland
Abstract
On the basis of systematic measurements of fragmentation reactions, which provide a detailed overview on the velocitydistributions of residual nuclei, an improved description of the kinematical properties of the fragmentation residuesis established. This work is dedicated to the fluctuations of their momentum distributions. In contrast to previousinvestigations, limited to close-to-projectile fragments, we extended our study to the entire production range, downto the lightest observed fragments. In this context, beside the contribution of abrasion and evaporation processes,we considered the effect of the thermal break-up on the width of the momentum distributions. Using approximatedtheoretical descriptions of the different reaction stages, a new analytical formula for the variance of the momentumdistribution is derived, which is well adapted to technical applications.
Keywords: fragmentation, multifragmentation, momentum distribution, analytical model
1. Introduction
Heavy-ions reactions with various target materials atenergies well above the Fermi-energy regime, have gainedincreasing interest for a variety of applications. They areused for the production of rare isotopes in existing [1–4]and future secondary-beam facilities [5–7], they are ex-ploited in the field of nuclear technology for the opera-tion of spallation neutron sources [8, 9], in particular forthe incineration of nuclear waste in an accelerator-drivensystem [10], they are a source of activation due to in-evitable beam losses along high-energy accelerators andin the beam dump [11]. Besides their cross sections, thekinematical characteristics of the fragmentation residuesemerging from these reactions are very important. In in-flight secondary-beam facilities, they determine the emit-tance of the rare-isotope beams; in irradiated constructionmaterial, they are responsible for aging phenomena.In non-central collisions of heavy ions at energies far be-yond the Fermi-energy regime, the geometrical abrasionmodel [12] suggests a clear cut between participant mat-ter, corresponding to the overlap zone of the two nuclei,and spectator matter outside the overlap zone. While theparticipant matter is subject to strong heating and com-pression due to high-energy nucleon-nucleon collisions, thespectator matter is sheared off from the projectile, respec-tively target nucleus, and continues moving on with es-sentially its initial velocity. On a smaller scale, however, ∗ corresponding author Email address: [email protected] (A. Bacquias) Present address: CNRS-IPHC Strasbourg, France Present address: Los Alamos National Laboratory, NM, USA a shift in the mean longitudinal velocity of the spectatormatter is observed. For very peripheral collisions, fric-tion dominates [13], while for less peripheral collisions anopposite effect has been observed [14], which has been at-tributed to the response of the spectator to the participantblast [15]. In addition, the velocity distribution of thespectator matter is characterized by fluctuations aroundits mean value. The magnitude of these fluctuations ismost often larger than the shift of the mean value and,thus, constitutes the most important characteristics of thekinematical properties of the spectators.The basic theoretical understanding of the fluctuationsof the momentum distributions dates back to the pioneer-ing work of Goldhaber in 1974 [16], who described twopossible scenarios, the resulting recoil momentum due toa sequential particle emission, or the random superpositionof the individual Fermi momenta of the nucleons removedin an abrasion process. A quantitative simulation of thevelocity distributions of individual nuclides, also concen-trating on very peripheral collisions, considering the con-tributions of the abrasion and the evaporation stages by afull Monte-Carlo nuclear-reaction code, has been presentedby Hanelt et al. [17]. They have shown that the recoil mo-mentum of evaporated particles modifies the momentumdispersion of final fragments. A comprehensive system-atics on momentum distributions from reactions with rel-ativistic ions, documenting the status of that time, wasestablished by Morrissey in 1989 [18], and an empiricalformula for very peripheral collisions was proposed: σ p k = 150 · ( A p − A f ) (1)where A p and A f represent the mass of the projectile and Preprint submitted to . . . November 12, 2018 he mass of the final fragment, respectively. Although thisempirical formula perfectly reproduces most data near theprojectile, it becomes unrealistic for lighter fragments. In-deed, the predicted dispersion prediction does not decreasetowards low masses and finally gives unrealistically largevalues for very light fragments. One has to realize thatthe width of the momentum has to go down to zero withmass approaching zero! Thus, this formula cannot coverthe whole range of fragments produced and observed infragmentation reactions.Recently, a semi-empirical model describing the momen-tum distribution of the fragmentation residues has been in-troduced by Tarasov [19]. The momentum distributions isrepresented by a convolution of a Gaussian with an expo-nential tail, where the shape of the convoluted distributionand its variation with projectile energy and the masses ofprojectile and fragment are associated with the influenceof abrasion, friction and evaporation on the kinematics ofthe fragments. The free parameters of that distributionare tuned to fit experimental data.The need to build up a model fully based on theoret-ical considerations, taking into account the multiple pro-cesses at play in fragmentation reactions, valid over thewhole mass range of produced residues but compact andfast enough to be used for various applications, is thenobvious.Thanks to a series of extensive experiments on system-atic measurements of fragmentation residues, mostly per-formed at GSI, Darmstadt, the empirical knowledge in thisfield has improved substantially [17, 20–60]. By investigat-ing the projectile-like fragments, a detailed overview oncross sections and velocity distributions of individual nu-clides has been obtained. The high quality of these datacomes not only from the high precision, but also from thefact that, in contrast to experiments on the properties oftarget-like fragments, which suffer from a low-energy cut-off, the kinematical properties of the projectile-like frag-ments can be fully measured. In addition, for a few reac-tions, the final residues were measured over a very broadmass range. The availability of new, high-quality data hasmotivated us to re-visit the understanding of the kinemat-ical properties of the fragmentation residues. The presentwork is dedicated to the fluctuations of their momentumdistributions.It is our aim to study the fluctuations of the momentumdistributions of all fragmentation residues emerging fromthe spectator matter. In contrast to many previous inves-tigations, we consider the whole mass range, from the pro-jectile, respectively target nucleus, down to light chargedparticles.For this purpose, we also treat the influence of the multi-fragmentation stage on the kinematical properties of frag-mentation residues. This break-up of the system due tothermal instabilities becomes important at high beam en-ergies and was not considered in previous works.In the present paper, we develop a model fully basedon theory. We derive analytical approximations from ap- propriate theoretical descriptions of the different stages ofthe collision. This is used to develop a new comprehen-sive analytical formula for the variance of the momentumdistributions, well adapted to technical applications.
2. Experimental data
From the body of data on kinematical properties of frag-mentation residues, we have chosen three experiments per-formed at the FRagment Separator (FRS) in GSI, Darm-stadt. They represent the fragmentation of two light nu-clei ( Fe [54], Kr [24]) and a medium-heavy nucleus(
Xe [56]). These experiments cover a broad mass range:residues have been observed from the vicinity of the pro-jectile down to elements lighter than neon.The FRS is characterized by a high momentum reso-lution and a limited acceptance both in momentum andangle [61]. The momentum resolution (FWHM) amountsto ∆ p/p ≃ · − , the momentum acceptance is ± . ,while the angular acceptance comprises mrad aroundthe beam axis. The limited momentum acceptance is notcrucial, since the full momentum range can be coveredby combining results obtained for different settings of themagnetic fields. We limit the present work to the lon-gitudinal momentum distribution of the residues emittedwithin the angular acceptance of the FRS. Nevertheless,the same formalism can be used for the transverse compo-nent of momentum distributions [20, 62]. Since at the pro-jectile energies considered in the present work, the produc-tion of fragmentation residues is approximately isotropic[20, 62, 63], our description should be valid for the trans-verse momentum dispersion as well.Three measured longitudinal velocity distributions, inthe projectile frame, are given as examples (fig. 1). Theycorrespond to Sn, Ni and C fragments from the re-action
Xe+Pb at 1 A GeV [56]. Over a large mass range,the residues follow a Gaussian distribution. Only the light-est residues show a more complex distribution with anasymmetry or even several humps. Light nuclei can origi-nate from two different processes [49, 53, 64, 65]. In binarymass splits, e.g. very asymmetric fission, Coulomb forceslead to a two-humped longitudinal velocity distribution[34, 49]. In case of multifragmentation, the Gaussian shapeof longitudinal velocity spectra is preserved. Please notethat our model aims at describing fragmentation reactions,and therefore cannot be used to calculate the momentumwidth of fission residues.The data from ref. [24, 54, 56] will be compared withthe predictions of our analytical model in the section 4,but let us first present its fundamental ideas.
3. Fragmentation mechanism
Let us remind the different descriptions of the processesthat happen during the collision and the de-excitation2 igure 1: Longitudinal velocity distributions in the projectile framefor various fragments (transmitted through the 15 mrad angular ac-ceptance of the FRS) from the reaction
Xe+Pb at 1 GeV pernucleon, taken from [56]. Heavy fragments follow a Gaussian distri-bution. On the contrary, the lightest ones populate an asymmetricdistribution. phase of the fragmentation reaction and study their in-fluence on the width of the longitudinal momentum distri-butions of the fragments.As a typical collision configuration, we will assume aprojectile nucleus hitting a target nucleus (at rest) withan impact parameter in principle different from zero. Inthe overlap region, projectile and target nucleons interactstrongly, while the rest of the projectile and the target nu-cleus, so called spectators, are almost undisturbed by thecollision. This picture is well suited in a wide energy do-main above the Fermi energy, and the model developed inthe present work can thus be used for comparison with var-ious projectile- and target-fragmentation data from avail-able devices (e.g. at GSI, MSU, RIKEN. . . ). In the follow-ing, we will concentrate on the projectile-like fragments,since we will compare the predictions of the present modelwith the projectile-like residues measured at the FRS inGSI. Of course, the model is also applicable to target-likeresidues. We will discuss the contributions of the differ-ent stages of the fragmentation process to the momentumdispersion of the final residues.
Dedicated experiments have long shown that the lon-gitudinal momentum of the heavy fragmentation residuesfollows a Gaussian distribution in the projectile frame. Afew models have been developed to describe the standarddeviation of this distribution.For the standard deviation of momentum in longitudi-nal or transverse directions ( σ p k and σ p T ), the most of-ten used theoretical model is Goldhaber’s prediction forthe dispersion induced in the abrasion process due to theFermi motion of the nucleons [16].A proposition of Goldhaber is to consider the reactionas a sudden cut-off of a part of the projectile, withouttaking into account any further evolution of the remain-ing part of the projectile. Some nucleons are removed in-stantaneously, without inducing momentum transfer. Thisreaction mechanism is referred to as abrasion, and it sug-gests already a removal of matter by friction phenomena,so that its description in [16] is probably too simple, but weshall discuss that after considering the strong implicationsof the assumption made by Goldhaber. An instantaneousremoval of several nucleons from the projectile does in-deed affect the dynamical features of this projectile. Infact, this contribution explains great part of the measuredmomentum width of the fragmentation residues. The re-moved nucleons are well defined by their positions insidethe nucleus at the moment of collision. Yet, the momentaof these nucleons are sampled over a broad distribution.Considering the ensemble of nucleons as a Fermi gas,we know that these constituents of the nucleus have an in-trinsic movement, even at zero temperature. This internalmotion affects the observable features of the projectile-likefragments. In a Fermi gas inside a square-well potentialwith infinitely high borders, there is no correlation be-tween the position and the momentum of a nucleon. Thus,the momentum of an abraded nucleon is a random samplefrom the projectile’s Fermi sphere filled up to the radius p F , in Goldhaber’s view [16]. The authors of the presentpaper are aware of the works of Friedman [66] demon-strating that absorption, in peripheral collisions, preventsfrom sampling over the whole nucleus. Nevertheless, asone can see from ref. [66], the predictions with the absorp-tion taken into account are rather close to the Goldhabermodel. Therefore, and owing to the transparency of thatlatter, it was decided to use Goldhaber model as a basisof the present work.The mean square momentum of a nucleon in a Fermigas is < p > = R p F p d p R p F d p = 35 p F . (2)The momentum variance σ p of the projectile spectatoris given by the sum of the individual contributions fromthe abraded nucleons. Provided the assumption that thisbroadening is equally distributed in all directions, the pro-jection along the beam axis gives σ p k = σ p / (it is also3elevant for transverse momentum studies, see for exam-ple [38]).The variance is then linked with the Fermi momentum p F of the projectile of mass A p and the mass of the frag-ment after the abrasion process, so-called prefragment, A : σ p k = p F · A ( A p − A )( A p − . (3)The Fermi momentum varies only weakly with the massof the nucleus. One can find an expression of the massdependence of the Fermi momentum in reference [67] thatrelies on data from [68]: p F ( A ) = 281 · (1 − A − . ) MeV /c . (4)We can comment that eq. 3 is symmetric regarding themass, so that an abrasion process that removes half of thenucleons in the projectile produces the largest width inthe distribution. In this hypothesis, light fragments andfragments of mass close to that of the projectile shouldhave a similar broadening of their longitudinal momentumdistributions.This model reproduces some data rather well. At thetime of release of Goldhaber’s article, most data availablewere limited to small mass losses, corresponding to theheaviest fragments. Goldhaber’s formula is indeed in quitegood agreement with these data.But Goldhaber’s model is incomplete in the sense thatthe spectator matter after the abrasion process is excited.Therefore, this model does not refer to the measured frag-mentation residues, which result from a subsequent deex-citation stage, but rather to an excited prefragment. Thecomprehension of the deexcitation processes is the key toaddress a model for the final fragment. The removal of nucleons by abrasion is a step that doesnot leave the nucleus cold. It can be seen as a frictionprocess, removing nucleons from the projectile, but alsoinducing excitation energy in the remaining fragment. Itwas shown by different methods [23, 69] that this energyinduction amounts on average to 27 MeV per abraded nu-cleon. Therefore, the spectators may be highly excited,giving rise to the evaporation of a considerable number ofnucleons and light nuclei.The evaporation stage can be described by an appropri-ate code in the frame of the statistical abrasion-ablationmodel [70], using realistic binding energies, Coulomb bar-riers and level densities. Each emitted particle induces arecoil momentum to the corresponding residue accordingto its kinetic energy and momentum conservation. The in-fluence of the evaporation process has been demonstratedby Hanelt et al. [17], stating that the recoil-momentuminduced by sequentially evaporated nucleons should no-ticeably modify the width of momentum distributions.In most cases, the individual contributions of the evap-orated particles are small and just slightly increase the width of the Gaussian distribution resulting from the abra-sion stage. Only in specific cases, like symmetric fission,the recoil momentum is so large that it dominates thekinematic properties of the final fragments, and devia-tions from a Gaussian distribution are observed. Althoughfission plays an important role for very heavy fragments,we will consider only non-fissioning systems in the presentwork.De Jong et al. [71] included the influence of the evapo-ration process on the angular momentum distribution inan analytical formula. The basic idea was to assume acertain proportionality between abraded mass and evapo-rated mass. They obtained satisfactory predictions with asingle parameter. They used ν = 2 as a typical value ofmass loss due to evaporation per abraded nucleon. Thismeans that the abrasion of one nucleon leads on averageto a mass loss of two nucleons by evaporation. Their de-scription was formulated as: σ J = 0 . A / p · ( νA p + A )( A p − A )( ν + 1) ( A p − . (5)We have adapted this idea for expressing the conse-quence of evaporation on the linear momentum by a dedi-cated analytical expression. It will be presented in section4. For the prefragment, the release of nucleons (mostly in-dividually but also as bound clusters) decreases its exci-tation energy. Taking into account the mean separationenergy of one nucleon and the average nuclear tempera-ture inside the prefragment, this “cooling” of the systemallowed by evaporation is considered to be 15 MeV peremitted nucleon on average.Since the angular momentum induced in an abrasion re-action is rather low [71], evaporation from the excited pre-fragment is an isotropic process in the frame of the sourceof emission. Thus, the mean velocity of the prefragment isnot affected by evaporation. The momentum, however, isreduced on average because of the mass loss. In addition,the recoil momenta of the individual evaporated particlestend to increase the width of the momentum distribution.In order to compare explicitly the impact of the recoilprocess with the Fermi contribution discussed before, letus define η , the ratio between the mean individual en-ergy taken from the prefragment by the evaporation ofone nucleon < E evap > and the mean individual energy ofone nucleon inside the nucleus due to the Fermi motion < E fermion > : η = < E evap >< E fermion > . (6)The mean kinetic energy < E fermion > of a nucleon in aFermi gas is: < E fermion > = R p F E d p R p F d p = 35 E F . (7)The average Fermi energy E F of a nucleus amounts toabout 33 MeV; this value leads to < E fermion > ≃ MeV.4he value of < E evap > is not easily estimated, becauseit depends on the charge of the mother nucleus, and ofcourse is different for each type of evaporated particle (neu-tron, proton, alpha, intermediate-mass fragments...). Forsimplicity, we consider only the evaporation of neutronsand protons. Moreover, we assume that they are emit-ted with equal probability, so that the mean recoil energy < E evap > is given by the mean value between neutron andproton kinetic energies E n and E p , respectively. We fix thethermal contribution to E n and E p to 8 MeV; for protons,we estimate the contribution from Coulomb repulsion us-ing the mass A p and nuclear charge Z p of the projectile.In this way, we obtain for η the following expression: η = 12 · " Z p e r ( A / p + 1) ! / < E fermion > , (8)with r =1.4 fm and e the elementary charge.The mean square momenta of a single nucleon, for eitherFermi-gas contribution or for the evaporation recoil, aredirectly proportional to the corresponding mean energy: < p > = 2m n · < E > (9)with m n the mass of a nucleon. The ratio between themean energies is then equivalent to the ratio of momentasquared: η = < E evap >< E fermion > = < p >< p > = 53 · < p >p F , (10)using the expression of < p > from eq. 2. For themean square momentum in the longitudinal direction, onefinally has: < p k evap > = η · p F . (11)We shall use this expression to take the recoil momentuminto account in the formula for σ p k (see details in section“analytical model”). It is commonly accepted that a nucleus starts to showsome thermal instabilities when its excitation energy ex-ceeds a value of about 3 MeV per nucleon. This valuecan easily be reached through the abrasion process. Thekinematics of the fragments is defined when they ceaseto interact, at freeze-out. The momenta of the fragmentsare given by a radial expansion, a random motion of thenascent fragments and by a consecutive Coulomb expan-sion.
For estimating the random motion, two rather diverg-ing ideas were introduced in the literature. In one extreme,it was assumed that the fragments behave like moleculesof an ideal gas [72]. In this case, their individual mo-tion is governed by a Maxwell-Boltzmann distribution and the variance of the momentum distribution of a fragmentof mass A produced in the break-up of a system of mass A bu by multifragmentation is given by the following for-mula [16]: σ p = m n k B T bu · A ( A bu − A ) A bu (12)where T bu is the freeze-out temperature, k B is the Boltz-mann constant and m n denotes the nucleon mass.In the other extreme, the nascent fragments keep theFermi motion of the individual nucleons in the commonsource [35]. In this second case, the distribution followsthe same functional form, but the temperature parametercannot be interpreted as a real temperature.Odeh et al. have shown that the apparent tempera-ture T app extracted from the kinetic-energy spectra of themultifragmentation products is much larger than the tem-perature extracted using isotopic or excited-states popula-tion thermometer [35]. Such large apparent temperatureswould indicate that the process of simultaneous break-upis rather fast and that the system does not have time toreach full thermal equilibrium. In other words, the Fermimotion inside the multifragmenting source is mostly re-sponsible for the broadening of the momentum distribu-tion of the multifragmentation products. Thus, ideas be-hind the Goldhaber formula (eq. 3) should also be validfor describing the momentum dispersion of the multifrag-mentation products. However, one should not forget thatin case of multifragmentation the multifragmenting sourceis not cold as in case of abrasion, but has a finite temper-ature T bu . Thus, the mean velocity of the nucleons insidethe multifragmenting source is larger than the mean veloc-ity of the nucleons inside the cold source. Consequently,the slope parameter (i.e. apparent temperature) of thekinetic-energy spectra of the multifragmentation productsis increased relative to the case of the cold fragmentingsource. This effect has been studied by W. Bauer [73],and he developed an analytical expression for calculatingthe momentum dispersion of a multifragmenting Fermi gasat finite temperature. According to ref. [73], the apparenttemperature T app can be expressed as a function of thereal source temperature T bu through a Taylor expansionby: T app ≈ A bu − AA bu − E F " π (cid:18) T bu E F (cid:19) + O (cid:18) T bu E F (cid:19) ,(13)with A bu the mass of the multifragmenting source, E F itsFermi energy, and A the mass of a produced fragment,which will enter the evaporation stage. This formula isincorporated in our model. The effect of the radial expansion is a lowering of theFermi energy E F and of the Fermi momentum p F down5o values E Fbu and p Fbu according to the following expres-sions: E Fbu = E F · (cid:18) V V bu (cid:19) / (14) p Fbu = p F · (cid:18) V V bu (cid:19) / . (15)In above equations, V is the normal volume, while V bu is the volume at freeze-out. The generally accepted vol-ume increase at freeze-out is still not well established, andvalues between one and six times the normal volume areoften considered [72]. Furthermore, we shall consider the fact that the clus-ters formed in the thermal break-up are charged particles.Thus, Coulomb forces between these fragments have to betaken into account.K. C. Chung et al. developed ideas regarding the effectsof this Coulomb expansion in nuclear fragmentation [74].We summarize here the main ideas and results of theirarticle.Assuming that the expansion is uniform, but that thefragment radii do not change during expansion, it is possi-ble to express the Coulomb contribution to the final kineticenergy: for a fragment of nuclear charge Z and mass A ,at a distance r from the center of a prefragment of mass A pre , nuclear charge Z pre and radius R pre , one has: E Coul = ZZ pre e R r (1 − A/A pre ) . (16)Equation 16 is remarkably simple, especially owing to thefact that the only relevant initial condition is the relativeposition of the fragment ( A , Z ) inside the system at break-up.While that kind of contribution can easily be includedin a Monte-Carlo simulation, its inclusion in an analyticalmodel may be more complex, or require some assumptionsthat we will discuss in the next sections. The impact parameter of the collision determines themass and the excitation energy of the prefragment andhence the mass of the final fragment [75]. Therefore, onecan make the connection between the mass of the finalresidue and the conditions at the beginning of the evapo-ration stage [39]. This is schematically shown in fig. 2.The different stages of the reaction are connected witha reduction of mass. To follow the evolution of a prefrag-ment, one has to read fig. 2 from right to left. The firstprocess of the fragmentation reaction, the abrasion stage,reduces the mass of the projectile, respectively target nu-cleus, inducing substantial excitation energy. The averageexcitation energy follows the line E ∗ , which corresponds E *E * E * *E abrasionbreak−up E* E* A f A GH lim
A A P A evaporation Figure 2: Evolution of the excitation energy of the fragment throughdifferent reaction stages, from a given projectile of mass A p . Theabrasion process is represented by the line E ∗ = ∆ E · ( A p − A ) . Ifmultifragmentation occurs, the system after freeze-out lands on theline E ∗ = A/K · T . In all cases, evaporation goes on average alonglines of the form E ∗ = ε · ( A − A f ) . See text for details. to ∆ E =27 MeV induced by abrasion of one nucleon, asmentioned above.If the spectator after abrasion is left with an excitationenergy of more than about 3 A MeV, it breaks up intoseveral fragments simultaneously. The products of thismultifragmentation process are formed with a freeze-outtemperature T bu of about 5 MeV [39]. Before entering theevaporation process, these fragments fall on the freeze-outline E ∗ . The relation between temperature and excitationenergy is described by the following formula: E ∗ = aT . (17)In this equation, a is the level-density parameter; it isevaluated in a simplified way as A/K
MeV − . With avalue of the inverse level-density parameter K =11 MeVand a freeze-out temperature of 5 MeV, we obtain E ∗ = A MeV.As soon as the system is excited (after abrasion or aftermultifragmentation) above the particle emission threshold,it has the possibility to emit neutrons, light charged par-ticles and light nuclei by evaporation. This deexcitationprocess is represented by the arrows E ∗ . The link betweenthe observed mass A f and the mass of the prefragment be-fore evaporation is given by E ∗ = ε · ( A − A f ) , where ε isthe parameter representing the average energy consumedper evaporated nucleon. We assume a value of ε =15 MeV(see section 3.2).Figure 2 suggests that final fragments lighter than acertain limiting mass A lim are products of multifragmen-tation, while fragments heavier than this limit should beregarded as simple abrasion-evaporation residues.In other words, the present model considers two regionsin mass, corresponding to the two regimes preceeding evap-oration. They are divided by A lim which is obtained by theprojection following an evaporation line E ∗ of the intersec-tion of lines E ∗ and E ∗ . The solution of this geometrical6equirement gives: A lim = K · ε − T K · ε · K · ∆ E ( K · ∆ E + T ) · A p . (18)Again, final residues with masses above A lim are theproducts of a pure abrasion-evaporation process, while fi-nal residues with masses below this limit have experienceda multifragmentation process.
4. Analytical model
Our goal is to obtain an analytical description of themomentum dispersion of fragmentation residues. Such anapproach requires some approximations.We have seen that formula 3 represents only the abrasionstage of the fragmentation reaction and does not includethe evaporation nor the multifragmentation stages.Nevertheless, formula 3 serves as a basis for the presentwork, since the Fermi motion is the dominant effect.Most of our results come from geometrical deductionsof the reaction scheme depicted in fig. 2. We already ex-pressed the mass limit under which an observed fragmenthas presumably undergone break-up (eq. 18).We denote the mass of the prefragment just beforeevaporation by A GH (GH stands for Goldhaber), sincewe will use it as a variable in Goldhaber’s formula. Inthe abrasion-evaporation scenario, the mass before evap-oration is obtained at the intersection between lines E ∗ and E ∗ (see fig. 2). In the abrasion-multifragmentation-evaporation scenario, the mass before evaporation is thesolution of the intersection of lines E ∗ and E ∗ . For final-fragment masses close to the projectile ( A f ≥ A lim ), weobtain: A GH = εA f + ∆ EA p (∆ E + ε ) , (19)and if A f < A lim , A GH = K · εK · ε − T · A f . (20)In the abrasion stage, the influence of the Fermi mo-tion is described by the Goldhaber formula (eq. 3), andthe standard deviation of the longitudinal momentum dis-tribution of abrasion-evaporation products including thechange in mass due to evaporation is given by the follow-ing formula: σ p k Fermi = (cid:18) A f A GH (cid:19) · p F · A GH ( A p − A GH )( A p − . (21)The Fermi momentum p F in the above equation is di-rectly obtained from eq 4. On the other hand, in the caseof multifragmentation, one must consider that the Gold-haber formula (eq. 3) does not account for the volumeexpansion during the break-up nor for the finite tempera-ture of the multifragmenting source. The volume expan-sion during the break-up is described via equations 14 and 15, where we assume a freeze-out volume equal 1.5 timesthe normal volume. The influence of finite temperatureof the multifragmenting source is described according toBauer’s analytical model presented by eq. 13, where weassume a value of 5 MeV for the temperature at freeze-out T bu [39].Since the descriptions of abrasion and break-up are bothbased on the Fermi momentum, we can adapt the expres-sion of p Fbu , into an apparent Fermi momentum: e p Fbu = p F · (cid:18) V V bu (cid:19) / · " π (cid:18) T bu E Fbu (cid:19) . (22)The fact that we use an apparent Fermi momentum incase of multifragmentation requires that we write the cor-responding contributions to the momentum dispersion intwo terms: one contribution from the abrasion process(formation of a prefragment of mass A bu out of a pro-jectile A p ), and another contribution from the multifrag-mentation (formation of a product of mass A GH out of anensemble of A bu nucleons).In case of multifragmentation, A GH is expressed byeq. 20, but we do not know a priori the mass A bu of thesource of multifragmentation residues. As a reasonnableestimate, one can take the heaviest fragment that can en-ter multifragmentation. Let us recall that the abrasionstage is inducing an increase of excitation energy in theprefragment, by the amount of about ∆ E = 27 MeV perabraded nucleon. Once the critical point leading to multi-fragmentation is reached, further abrasion of a few nucle-ons should lead to a break-up into smaller pieces, tendingrapidly to vaporization. Final multifragmentation residueslarger than alpha particles are then most probably pro-duced by a prefragment of mass close to the theoreticalmaximum.We calculated the mass A bu of the heaviest system un-dergoing multifragmentation. It corresponds to the colli-sions in which the abrasion has introduced enough energyto excite the nucleus to the critical temperature of 5 MeV,and is obtained from the intersection of the lines E ∗ and E ∗ shown in figure 2: A bu = K · ∆ EK · ∆ E + T A p . (23)Thus, the total contribution from the Fermi motion inthe abrasion and the break-up on the kinematics of multi-fragmentation products is written: σ p k Fermi = (cid:18) A f A bu (cid:19) · p F · A bu ( A p − A bu )( A p − (cid:18) A f A GH (cid:19) · e p · A GH ( A bu − A GH )( A bu − (24)with e p Fbu the apparent Fermi momentum at break-up,given by eq. 22.Apart from the effects of the Fermi motion, we men-tioned that in case of multifragmentation, the expansion7ue to Coulomb forces should be implemented in themodel. Let us reconsider eq. 16. Many variables aresubject to fluctuations from one event to another, butthrough simple assumptions, one can simplify this equa-tion. Isospin thermometer experiments [23, 39] have shownthat the nuclear composition of prefragments just beforethe sequential evaporation steps is compatible with a sce-nario in which the prefragment keeps “in memory” the ratio Z p /A p of the projectile. Both processes, abrasion and mul-tifragmentation, preserve this value on average. It meansthat both the mother nucleus and its considered daugh-ter have on average the same proportion of neutrons andprotons. We express then the atomic number of these twonuclei in the following form: Z = A · Z p A p (25)where A, Z design the features of each considered part ofthe nuclear ensemble. The effect of neutron distillation [76,77] on the mean
N/Z ratio of the fragments, which is smallcompared to the effect of evaporation, was neglected. Letus stress that this ratio is not preserved by the evaporationprocess, so that this formula does not apply to the finalfragment.The system entering multifragmentation was taken tobe the heaviest, with its mass A bu given by eq. 23; itscharge Z bu is determined by the fixed ratio Z/A . Thisheaviest mother nucleus will generate the biggest electricfield. This assumption should give us an upper estimateof the Coulomb repulsion, and at least allow us to checkwhether the magnitude of such an effect is compatible withthe observed velocity distributions.The distance r between the fragments at their forma-tion is also crucial. We assume that the production of afragment has a uniform probability inside the volume ofthe system. The variance of such a distribution of proba-bility in one direction (in our case, along the longitudinalaxis) is then the one of a uniform distribution betweenan interval [- r max ; + r max ]; such variance takes the form (2 r max ) / . The integration of the electrical field in thespherical charged source up to the bounds of this intervalleads to a Coulomb contribution to the longitudinal mo-mentum that we name P Coul . The Coulomb contributionto the variance of the spectra is then written: σ p k Coul = 13 · P (26)with the contribution to the momentum (see eq. 16) takenas: P = 2 A GH · u · A GH A bu Z p e A p R · r · (1 − A GH A bu ) (27)with u = 931 . MeV/c and e = 1 . MeV · fm. r max isthe maximum distance between the centre of mass of thefragment and the one of the whole system, so it is theradius of the mother nucleus, reduced by the radius of theproduced piece: r max = R bu − R GH . The further refinement in the description of the pro-cesses undergone by the observed fragment is related to thedynamics acting in the evaporation step. The sequentialaspect of the evaporation gives rise to recoil momentum,broadening the momentum distributions.For this approach, we look first at the variance of thelongitudinal velocity distributions. If we start from theprefragment, we may assume that the variance after evap-oration of one nucleon is modified by the induced recoilmomentum; the additional contribution to the variance ofthe velocity distribution is: σ v = p ( A GH − . (28)We suppose that a single particle is evaporated with themean momentum p evap . Then for a number n of emittedparticles, we can write: σ v n = p · n X i =0 A GH − i ) . (29)For a large number A GH (compared to the number of evap-orated particles n ), the difference between two steps of thesum is rather small. The sum can then be approximatedby an integral. With the upper bound of the integral being n = A GH − A f , we obtain this integrated relation for thefinal fragment: σ v evap = p · (cid:18) A f − A GH (cid:19) . (30)Since for the momentum variance, we have: σ p evap = A f σ v evap , (31)we can deduce the following, expressing p along thelongitudinal axis as in eq. 11 (with η evaluated in eq. 8): σ p k evap = A f · p F η · (cid:18) A f − A GH (cid:19) . (32) σ p k evap represents the contribution from the recoil of evap-orated particles to the variance of the longitudinal momen-tum; A GH has to be taken as in equations 19 and 20.Our final analytical formula is the quadratic sum ofthe different contributions, the Fermi motion (from eq. 21for masses beyond A lim or eq. 24 for masses below A lim ),Coulomb repulsion between the multifragmentation prod-ucts (eq. 26, applied below A lim ), and the recoil induced byevaporation (formula 32 for the whole mass range). Thelongitudinal momentum dispersion of final fragments is ex-pressed by: σ p k = σ p k Fermi + σ p k evap beyond A lim σ p k Fermi + σ p k evap + σ p k Coul below A lim (33)In fig. 3, we confront our predictions with three sets ofdata, measured in the reactions Xe+Pb at 1 A GeV [56],8 igure 3: Model in its full form as the square-root of eq. 33 (plainline) and Morrissey’s systematics (dashed line) compared to differentdata sets [24, 54, 56]. Kr+Be at 0.5 A GeV [24] and Fe+p at 1 A GeV [54]respectively. These data present longitudinal momentumdispersion of fragmentation residues over a broad massrange. Other published data concentrating only on veryperipheral collisions (i.e. masses very close to the mass ofthe projectile) are easily reproduced by our model as wellas by Morrissey’s systematics. Since the present modeldoes not consider the fission process, data containing fis-sion fragments were disregarded.The shape of the global prediction for σ p k of spectatorfragments is in good accordance with the overall trend ofthe data available for this observable (see the confrontationwith some data sets in fig. 3).It is interesting to note that, although the abrasion pic-ture, which is a basis of the present model, is not expectedto be valid for targets as light as protons, our model isable to reproduce such data (see fig.. 3) with a remarkablygood agreement.The degree of sophistication seems sufficient to repro-duce the data. Our analytical formula is in general accor-dance with the data, even for light fragments and no sys- tematic deviation is observed. In this mass region, wheresmall residues can originate from more violent collisions,Coulomb interaction is reduced by the expansion of thesystem at break-up and our prediction could have beentoo large; this is not the case.The good agreement of our model with rather compre-hensive sets of experimental data give some confidence thatthe ingredients of our model may correctly represent thephysics of the fragmentation reaction. A final conclusion,however, would require to investigate the sensitivity of theresults to the ingredients of the model and to the valuesof the parameters. Such study is beyond the scope of thepresent work.
5. Conclusion
A new description of the width of the momentum distri-butions of fragmentation residues at relativistic energies isproposed. It describes the experimental data over a verybroad mass range and includes most features of the knownphysical processes at play, constituting a great progresscompared to previous formulae.The present new description has several advantagescompared to predictions traditionally used. Morrissey pro-vided an analytical description of the data but only in thelimit of small mass losses. Furthermore, his formula is em-pirical and gives no insight into the physics of the reaction.Goldhaber introduced the idea that the Fermi motion pro-vides the major contribution to the shape of the momen-tum distribution of the surviving fragments. The problemis that the quantity referred to in his model is not the finalmass of the fragment which is observed.Going deeper into the description of the processes oc-curring in the collisions and using results provided by ex-periments, we suggested a new formula. It considers allrelevant processes the system is undergoing in an analyti-cal formulation. This description reproduces the data verywell.Our formula gives a good estimate of the kinematics ofprojectile-like fragments in the relativistic-energy domain,which is the energy range concerning the applications men-tioned in the introduction. Moreover, the physics basisof our description provides a realistic prediction of thelongitudinal momentum width of fragmentation products,in the whole energy regime where the abrasion is a validmodel for the collision phase, i.e. for energies above theFermi energy.9 ppendix
To ease the use of the present model for practical applications, we propose in the present appendix the summary ofneeded equations, to calculate the momentum width of the fragmentation residues. We also list the needed parameterswith their recommended values.The present model is valid over a large energy range (from about 50 A MeV to at least 10 A GeV), where fragmen-tation (and multifragmentation) processes occur. It applies to all kinds of projectiles (from light nuclei to uranium),independently from the target nucleus (therefore suited for any target, from protons to uranium).
Input parameters: A p : projectile mass Z p : projectile atomic number A f : final fragment massThe predictions are calculated for projectile-like fragments in the projectile frame, but if A p and Z p are replaced by thetarget mass A t and nuclear charge Z t , the predictions apply to target-like fragments in the laboratory frame. Calculated output: σ p k : dispersion of the linear momentum along the longitudinal axis for the final fragment (product of fragmentationreactions, excluding fission and fission-like processes). Model parameters: r : radius = 1.4 fm e : squared elementary charge = 1.44 MeV · fm u : atomic mass unit = 931.5 MeV/ c ∆ E : average induced excitation energy per abraded nucleon = 27 MeV < E fermion > : mean energy of one nucleon from Fermi motion = 20 MeV K : inverse level-density parameter K = A/a = 11
MeV ε : average consumed energy per evaporated particle = 15 MeV E thermkin : average contribution of the thermal motion to the kinetic energy of evaporated neutrons or protons = 8 MeV T bu : break-up temperature = 5 MeV V exp = V bu /V : volume expansion factor = 1.5 Summary of equations:
For a given projectile ( A p , Z p ), we calculate: p F = 281 · (1 − A − . p ) MeV /c , E F = p F
2u MeV , η = 12 · " E thermkin + E thermkin + Z p e r ( A / p + 1) ! / < E fermion > .The mass limit for multifragmentation (see text) can also be defined: A lim = K · ε − T ε · ∆ EK · ∆ E + T · A p . For A f ≥ A lim (final-fragment masses close to the projectile), multifragmentation did not occur. We need the followingequations: A GH = εA f + ∆ E · A p ∆ E + ε , σ p k Fermi = (cid:18) A f A GH (cid:19) · p F · A GH ( A p − A GH ) A p − , σ p k evap = A f · p F η · (cid:18) A f − A GH (cid:19) . 10he total momentum dispersion for this mass range is given by: σ p k = σ p k Fermi + σ p k evap . For A f < A lim , multifragmentation occurs. The following ingredients are required: E Fbu = E F V / , e p Fbu = p F V / · " π (cid:18) T bu E Fbu (cid:19) , A bu = K · ∆ EK · ∆ E + T · A p , A GH = K · εK · ε − T · A f .Then we can calculate: σ p k Fermi = (cid:18) A f A bu (cid:19) · p F · A bu ( A p − A bu ) A p − (cid:18) A f A GH (cid:19) · e p · A GH ( A bu − A GH ) A bu − ,And also: σ p k evap = A f · p F η · (cid:18) A f − A GH (cid:19) .Needed to calculate the contribution from Coulomb forces, the distance between the centre of mass of the producedfragment and the one of the whole system (i.e. the multifragmentation source) is taken to be r max = R bu − R GH ,with R bu = r ( A bu /V exp ) / and R GH = r ( A GH /V exp ) / .We define P = 2 A GH · u · A GH A bu Z p e A p R · r · (cid:18) − A GH A bu (cid:19) ,entering the expression of the Coulomb contribution to the momentum dispersion: σ p k Coul = 13 · P .The total momentum dispersion for this mass range is given by: σ p k = σ p k Fermi + σ p k evap + σ p k Coul . 11 eferences
Xe and
Auprojectiles, Z. Phys. A 346 (1993) 43.[18] D. Morrissey, Systematics of momentum distributions from re-actions with relativistic ions, Phys. Rev. C 39 (1989) 460.[19] O. Tarasov, Analysis of momentum distributions of projectilefragmentation products, Nucl. Phys. A 734 (2004) 536 – 540.[20] D. E. Greiner, P. J. Lindstrom, H. H. Heckman, B. Cork, F. S.Bieser, Momentum distributions of isotopes produced by frag-mentation of relativistic C and O projectiles, Phys. Rev.Lett. 35 (1975) 152.[21] G. D. Westfall, R. G. Sextro, A. M. Poskanzer, A. M. Zebelman,G. W. Butler, E. K. Hyde, Energy spectra of nuclear fragmentsproduced by high energy protons, Phys. Rev. C 17 (1978) 1368–1381.[22] J. Hubele, P. Kreutz, V. Lindenstruth, J. C. Adloff,M. Begemann-Blaich, P. Bouissou, G. Imme, I. Iori, G. J.Kunde, S. Leray, Z. Liu, U. Lynen, R. J. Meijer, U. Milkau,A. Moroni, W. F. J. Müller, C. Ngô, C. A. Ogilvie, J. Pochodza-lla, G. Raciti, G. Rudolf, H. Sann, A. Schüttauf, W. Seidel,L. Stuttge, W. Trautmann, A. Tucholski, Statistical fragmen-tation of Au projectiles at E/A=600 MeV, Phys. Rev. C 46(1992) R1577.[23] K.-H. Schmidt, T. Brohm, H.-G. Clerc, M. Dornik, M. Fauer-bach, H. Geissel, A. Grewe, E. Hanelt, A. Junghans, A. Magel,W. Morawek, G. Münzenberg, F. Nickel, M. Pfützner, C. Schei-denberger, K. Sümmerer, D. Vieira, B. Voss, C. Ziegler, Distri-bution of Ir and Pt isotopes produced as fragments of AGeV
Au projectiles: a thermometer for peripheral nuclear colli-sions, Phys. Lett. B 300 (1993) 313.[24] M. Weber, C. Donzaud, J. Dufour, H. Geissel, A. Grewe,D. Guillemaud-Mueller, H. Keller, M. Lewitowicz, A. Magel,A. Mueller, G. Munzenberg, F. Nickel, M. Pfutzner,A. Piechaczek, M. Pravikoff, E. Roeckl, K. Rykaczewski, M. Saint-Laurent, I. Schall, C. Stephan, K. Summerer,L. Tassan-Got, D. Vieira, B. Voss, Longitudinal momenta andproduction cross-sections of isotopes formed by fragmentationof a 500 A MeV Kr beam, Nucl. Phys. A 578 (1994) 659–672.[25] A. Magel, H. Geissel, B. Voss, P. Armbruster, T. Aumann,M. Bernas, B. Blank, T. Brohm, H.-G. Clerc, S. Czajkowski,H. Folger, A. Grewe, E. Hanelt, A. Heinz, H. Irnich, M. de Jong,A. Junghans, F. Nickel, M. Pfutzner, A. Piechaczek, C. Rohl,C. Scheidenberger, K.-H. Schmidt, W. Schwab, S. Steinhauser,K. Summerer, W. Trinder, H. Wollnik, G. Munzenberg, Firstspatial isotopic separation of relativistic uranium projectile frag-ments, Nucl. Instr. Meth. B 94 (1994) 548.[26] A. Schüttauf, W. D. Kunze, A. Worner, M. Begemann-Blaich,T. Blaich, D. R. Bowman, R. J. Charity, A. Cosmo, A. Fer-rero, C. K. Gelbke, C. Gro[ss], W. C. Hsi, J. Hubele, G. Imme,I. Iori, J. Kempter, P. Kreutz, G. J. Kunde, V. Lindenstruth,M. A. Lisa, W. G. Lynch, U. Lynen, M. Mang, T. Mohlenkamp,A. Moroni, W. F. J. Muller, M. Neumann, B. Ocker, C. A.Ogilvie, G. F. Peaslee, J. Pochodzalla, G. Raciti, F. Rosen-berger, T. Rubehn, H. Sann, C. Schwarz, W. Seidel, V. Serfling,L. G. Sobotka, J. Stroth, L. Stuttge, S. Tomasevic, W. Traut-mann, A. Trzcinski, M. B. Tsang, A. Tucholski, G. Verde, C. W.Williams, E. Zude, B. Zwieglinski, Universality of spectatorfragmentation at relativistic bombarding energies, Nucl. Phys.A 607 (1996) 457–486.[27] M. de Jong, K. H. Schmidt, B. Blank, C. Bockstiegel,T. Brohm, H. G. Clerc, S. Czajkowski, M. Dornik, H. Geis-sel, A. Grewe, E. Hanelt, A. Heinz, H. Irnich, A. R. Jung-hans, A. Magel, G. Munzenberg, F. Nickel, M. Pfutzner,A. Piechaczek, C. Scheidenberger, W. Schwab, S. Steinhauser,K. Summerer, W. Trinder, B. Voss, C. Ziegler, Fragmentationcross sections of relativistic
Pb projectiles, Nucl. Phys. A628 (1998) 479–492.[28] A. R. Junghans, M. de Jong, H.-G. Clerc, A. V. Ignatyuk, G. A.Kudyaev, K.-H. Schmidt, Projectile-fragment yields as a probefor the collective enhancement in the nuclear level density, Nucl.Phys. A 629 (1998) 635.[29] J. Benlliure, P. Armbruster, M. Bernas, C. Böckstiegel, S. Cza-jkowski, C. Donzaud, H. Geissel, A. Heinz, C. Kozhuharov,P. Dessagne, G. Münzenberg, M. Pfützner, C. Stéphan, K.-H.Schmidt, K. Sümmerer, W. Schwab, L. Tassan-Got, B. Voss,Production of medium-weight isotopes by fragmentation in 750 A MeV
U on
Pb collisions, Eur. Phys. J. A 2 (1998)193–198.[30] J. Reinhold, J. Friese, H.-J. Körner, R. Schneider, K. Zeitelhack,H. Geissel, A. Magel, G. Münzenberg, K. Sümmerer, Projectilefragmentation of
Xe at
Elab = 790 A MeV, Phys. Rev. C 58(1998) 247–255.[31] S. Fritz, C. Schwarz, R. Bassini, M. Begemann-Blaich, S. J.Gaff-Ejakov, D. Gourio, C. Gro, G. Imme, I. Iori, U. Kleinevo,G. J. Kunde, W. D. Kunze, U. Lynen, V. Maddalena, M. Mahi,T. Mohlenkamp, A. Moroni, W. F. J. Muller, C. Nociforo,B. Ocker, T. Odeh, F. Petruzzelli, J. Pochodzalla, G. Rac-iti, G. Riccobene, F. P. Romano, A. Saija, M. Schnittker,A. Schuttauf, W. Seidel, V. Serfling, C. Sfienti, W. Traut-mann, A. Trzcinski, G. Verde, A. Worner, H. Xi, B. Zwieglinski,Breakup density in spectator fragmentation, Phys. Lett. B 461(1999) 315–321.[32] J. Benlliure, K. H. Schmidt, D. Cortina-Gil, T. Enqvist, F. Far-get, A. Heinz, A. R. Junghans, J. Pereira, J. Taieb, Productionof neutron-rich isotopes by cold fragmentation in the reaction
Au + Be at 950 A MeV, Nucl. Phys. A 660 (1999) 87 – 100.[33] J. Benlliure, K. H. Schmidt, D. Cortina-Gil, T. Enqvist, F. Far-get, A. Heinz, A. R. Junghans, J. Pereira, J. Taieb, Erratumto "Production of neutron-rich isotopes by cold fragmentationin the reaction
Au + Be at 950 A MeV"[nucl. phys. a 660(1999) 87-100], Nucl. Phys. A 674 (2000) 578 – 578.[34] T. Enqvist, J. Benlliure, F. Farget, K.-H. Schmidt, P. Arm-bruster, M. Bernas, L. Tassan-Got, A. Boudard, R. Legrain,C. Volant, C. Böckstiegel, M. de Jong, J. P. Dufour, System-atic experimental survey on projectile fragmentation and fission nduced in collisions of U at 1 A GeV with lead, Nucl. Phys.A 658 (1999) 47–66.[35] T. Odeh, R. Bassini, M. Begemann-Blaich, S. Fritz, S. J. Gaff-Ejakov, D. Gourio, C. Groß, G. Immé, I. Iori, U. Kleinevoß,G. J. Kunde, W. D. Kunze, U. Lynen, V. Maddalena, M. Mahi,T. Möhlenkamp, A. Moroni, W. F. J. Müller, C. Nociforo,B. Ocker, F. Petruzzelli, J. Pochodzalla, G. Raciti, G. Ric-cobene, F. P. Romano, A. Saija, M. Schnittker, Fragment ki-netic energies and modes of fragment formation, Phys. Rev.Lett. 84 (2000) 4557–4560.[36] T. Enqvist, W. Wlazlo, P. Armbruster, J. Benlliure, M. Bernas,A. Boudard, S. Czajkowski, R. Legrain, S. Leray, B. Mustapha,M. Pravikoff, F. Rejmund, K.-H. Schmidt, C. Stéphan, J. Taieb,L. Tassan-Got, C. Volant, Isotopic yields and kinetic energies ofprimary residues in 1 A GeV
Pb + p reactions, Nucl. Phys.A 686 (2001) 481–524.[37] F. Rejmund, B. Mustapha, P. Armbruster, J. Benlliure,M. Bernas, A. Boudard, J. P. Dufour, T. Enqvist, R. Legrain,S. Leray, K.-H. Schmidt, C. Stéphan, J. Taieb, L. Tassan-got,C. Volant, Measurement of isotopic cross sections of spallationresidues in 800 A MeV
Au + p collisions, Nucl. Phys. A 638(2001) 540–565.[38] G. Hüntrup, T. Streibel, W. Heinrich, Investigation of trans-verse momentum distributions for fragments produced in reac-tions of
Au and
Pb projectiles with different targets inthe energy range from 1.0 to 158 GeV/nucleon, Phys. Rev. C65 (2001) 014605.[39] K. Schmidt, M. Ricciardi, A. Botvina, T. Enqvist, Productionof neutron-rich heavy residues and the freeze-out temperaturein the fragmentation of relativistic
U projectiles determinedby the isospin thermometer, Nucl. Phys. A 710 (2002) 157.[40] T. Enqvist, P. Armbruster, J. Benlliure, M. Bernas, A. Boudard,S. Czajkowski, R. Legrain, S. Leray, B. Mustapha, M. Pravikoff,F. Rejmund, K.-H. Schmidt, C. Stéphan, J. Taieb, L. Tassan-Got, F. Vivès, C. Volant, W. Wlazlo, Primary-residue produc-tion cross sections and kinetic energies in 1 A GeV
Pb ondeuteron reactions, Nucl. Phys. A 703 (2002) 435–465.[41] A. Stolz, T. Faestermann, J. Friese, P. Kienle, H.-J. Körner,M. Münch, R. Schneider, E. Wefers, K. Zeitelhack, K. Süm-merer, H. Geissel, J. Gerl, G. Münzenberg, C. Schlegel, R. S.Simon, H. Weick, M. Hellström, M. N. Mineva, P. Thirolf, Pro-jectile fragmentation of
Sn at
Elab = 1 A GeV, Phys. Rev.C 65 (2002) 064603.[42] INDRA and ALADIN collaborations, Fragment in peripheralheavy-ion collisions: from neck emission to spectator decays,Phys. Lett. B 566 (2003) 76.[43] J. Taieb, K.-H. Schmidt, E. Casarejos, L. Tassan-Got, P. Arm-bruster, J. Benlliure, M. Bernas, A. Boudard, E. Casarejos,S. Czajkowski, T. Enqvist, R. Legrain, S. Leray, B. Mustapha,M. Pravikoff, F. Rejmund, C. Stephan, C. Volant, W. Wlazlo,Evaporation residues produced in the spallation reaction
U+ p at 1 A GeV, Nucl. Phys. A 724 (2003) 413–430.[44] K. A. Gladnishki, Z. Podolyák, P. H. Regan, J. Gerl, M. Hell-ström, Y. Kopatch, S. Mandal, M. Górska, R. D. Page, H. J.Wollersheim, A. Banu, G. Benzoni, H. Boardman, M. La Com-mara, J. Ekman, C. Fahlander, H. Geissel, H. Grawe, E. Kaza,A. Korgul, M. Matos, M. N. Mineva, C. J. Pearson, C. Plettner,D. Rudolph, C. Scheidenberger, K.-H. Schmidt, V. Shishkin,D. Sohler, K. Sümmerer, J. J. Valiente-Dobón, P. M. Walker,H. Weick, M. Winkler, O. Yordanov, Angular momentumpopulation in the projectile fragmentation of
U at 750MeV/nucleon, Phys. Rev. C 69 (2004) 024617.[45] P. Armbruster, J. Benlliure, M. Bernas, A. Boudard, E. Casare-jos, S. Czajkowski, T. Enqvist, S. Leray, P. Napolitani,J. Pereira, F. Rejmund, M.-V. Ricciardi, K.-H. Schmidt,C. Stéphan, J. Taieb, L. Tassan-Got, C. Volant, Measurementof a complete set of nuclides, cross sections, and kinetic energiesin spallation of
U 1 A GeV with protons, Phys. Rev. Lett. 93(2004) 212701.[46] P. Napolitani, K.-H. Schmidt, A. S. Botvina, F. Rejmund,L. Tassan-Got, C. Villagrasa, High-resolution velocity mea- surements on fully identified light nuclides produced in Fe +hydrogen and Fe + titanium, Phys. Rev. C 70 (2004) 054607.[47] M. V. Ricciardi, A. V. Ignatyuk, A. Kelic, P. Napolitani, F. Rej-mund, K. H. Schmidt, O. Yordanov, Complex nuclear-structurephenomena revealed from the nuclide production in fragmenta-tion reactions, Nucl. Phys. A 733 (2004) 299–318.[48] D. Henzlova, L. Audouin, J. Benlliure, A. Botvina, A. Boudard,E. Casarejos, J. Ducret, T. Enqvist, L. Got, A. Heinz, V. Henzl,A. Junghans, B. Jurado, A. Kelic, A. Krasa, T. Kurtukian,S. Leray, P. Napolitani, M. Ordonez, J. Pereira, R. Pleskac,F. Rejmund, M. Ricciardi, K.-H. Schmidt, C. Schmitt,C. Stephan, C. Villagrasa, C. Volant, A. Wagner, O. Yordanov,Isotopically resolved residues from the fragmentation of pro-jectiles with largely different
N/Z - the isospin-thermometermethod, Nucl. Phys. A 749 (2005) 110–113.[49] M. V. Ricciardi, P. Armbruster, J. Benlliure, M. Bernas,A. Boudard, S. Czajkowski, T. Enqvist, A. Kelic, S. Leray,R. Legrain, B. Mustapha, J. Pereira, F. Rejmund, K.-H.Schmidt, C. Stephan, L. Tassan-Got, C. Volant, O. Yordanov,Light nuclides produced in the proton-induced spallation of
U at 1 GeV, Phys. Rev. C 73 (2006) 014607.[50] L. Audouin, L. Tassan-Got, P. Armbruster, J. Benlliure,M. Bernas, A. Boudard, E. Casarejos, S. Czajkowski, T. En-qvist, B. Fernandez-Dominguez, B. Jurado, R. Legrain,S. Leray, B. Mustapha, J. Pereira, M. Pravikoff, F. Rej-mund, M.-V. Ricciardi, K.-H. Schmidt, C. Stephan, J. Taieb,C. Volant, W. Wlazlo, Evaporation residues produced in spal-lation of
Pb by protons at 500 A MeV, Nucl. Phys. A 768(2006) 1–21.[51] E. Casarejos, J. Benlliure, J. Pereira, P. Armbruster, M. Bernas,A. Boudard, S. Czajkowski, T. Enqvist, R. Legrain, S. Leray,B. Mustapha, M. Pravikoff, F. Rejmund, K.-H. Schmidt,C. Stephan, J. Taieb, L. Tassan-Got, C. Volant, W. Wla-zlo, Isotopic production cross sections of spallation-evaporationresidues from reactions of
U (1 A GeV) with deuterium,Phys. Rev. C 74 (2006) 044609.[52] J. Kurcewicz, Z. Liu, M. Pfützner, P. Woods, C. Mazzocchi,K.-H. Schmidt, A. Kelic, F. Attallah, E. Badura, C. Davids,T. Davinson, J. Döring, H. Geissel, M. Górska, R. Grzywacz,M. Hellström, Z. Janas, M. Karny, A. Korgul, I. Mukha, C. Plet-tner, A. Robinson, E. Roeckl, K. Rykaczewski, K. Schmidt,D. Seweryniak, K. Sümmerer, H. Weick, Production cross-sections of protactinium and thorium isotopes produced in frag-mentation of
U at 1 A GeV, Nucl. Phys. A 767 (2006) 1 –12.[53] P. Napolitani, K.-H. Schmidt, L. Tassan-Got, P. Armbruster,T. Enqvist, A. Heinz, V. Henzl, D. Henzlova, A. Kelic,R. PleskaC, M. V. Ricciardi, C. Schmitt, O. Yordanov, L. Au-douin, M. Bernas, A. Lafriaskh, F. Rejmund, C. Stephan,J. Benlliure, E. Casarejos, M. F. Ordonez, J. Pereira,A. Boudard, B. Fernandez, S. Leray, C. Villagrasa, C. Volant,Measurement of the complete nuclide production and kineticenergies of the system
Xe+hydrogen at 1 GeV per nucleon,Phys. Rev. C 76 (2007) 064609.[54] C. Villagrasa-Canton, A. Boudard, J.-E. Ducret, B. Frnandez,S. Leray, C. Volant, P. Armbruster, T. Enqvist, F. Hammache,K. Helariutta, B. Jurado, M. V. RIcciardi, K.-H. Schmidt,K. Suemmerer, F. Vives, O. Yordanov, L. Audouin, C.-O.Bacri, L. Ferrant, P. Napolitani, F. Rejmund, C. Stephan,L. Tassan-Got, J. Benliure, E. Casarejos, M. Fernandez-Ordonez, J. Pereira, S. Czajkowski, D. Karamanis, M. Pravikoff,J. S. George, R. A. Mewaldt, N. Yanasak, M. Wiedenbeck, J. J.Connell, T. Faestermann, A. Heinz, A. Junghans, Spallationresidues in the reaction Fe + p at 0.3 A , 0.5 A , 0.75 A , 1.0 A ,and 1.5 A GeV, Phys. Rev. C 75 (2007) 044603.[55] W. Trautmann, R. Bassini, M. Begemann-Blaich, A. Fer-rero, S. Fritz, S. J. Gaff-Ejakov, C. Gross, G. Imme, I. Iori,U. Kleinevoss, G. J. Kunde, W. D. Kunze, A. L. Fevre, V. Lin-denstruth, J. L. ukasik, U. Lynen, V. Maddalena, M. Mahi,T. Mohlenkamp, A. Moroni, W. F. J. Muller, C. Nociforo,B. Ocker, T. Odeh, H. Orth, F. Petruzzelli, J. Pochodzalla, . Raciti, G. Riccobene, F. P. Romano, T. Rubehn, A. Saija,H. Sann, M. Schnittker, A. Schüttauf, C. Schwarz, W. Seidel,V. Serfling, C. Sfienti, A. Trzcinski, A. Tucholski, G. Verde,A. Worner, H. Xi, B. Z. A. Collaboration, Thermal and chem-ical freeze-out in spectator fragmentation, Phys. Rev. C 76(2007) 064606.[56] D. Henzlova, K.-H. Schmidt, M. V. Ricciardi, A. Kelić,V. Henzl, P. Napolitani, L. Audouin, J. Benlliure, A. Boudard,E. Casarejos, J. E. Ducret, T. Enqvist, A. Heinz, A. Junghans,B. Jurado, A. Krása, T. Kurtukian, S. Leray, M. F. Ordóñez,J. Pereira, R. Pleskač, F. Rejmund, C. Schmitt, C. Stéphan,L. Tassan-Got, C. Villagrasa, C. Volant, A. Wagner, O. Yor-danov, Experimental investigation of the residues produced inthe Xe+Pb and
Xe+Pb fragmentation reactions at A GeV, Phys. Rev. C 78 (2008) 044616.[57] J. Benlliure, M. Fernández-Ordóñez, L. Audouin, A. Boudard,E. Casarejos, J. E. Ducret, T. Enqvist, A. Heinz, D. Henzlova,V. Henzl, A. Kelic, S. Leray, P. Napolitani, J. Pereira, F. Rej-mund, M. V. Ricciardi, K.-H. Schmidt, C. Schmitt, C. Stéphan,L. Tassan-Got, C. Volant, C. Villagrasa, O. Yordanov, Produc-tion of medium-mass neutron-rich nuclei in reactions inducedby
Xe projectiles at 1 A GeV on a beryllium target, Phys.Rev. C 78 (2008) 054605.[58] H. Alvarez-Pol, J. Benlliure, E. Casarejos, L. Audouin,D. Cortina-Gil, T. Enqvist, B. Fernández, A. Junghans, B. Ju-rado, P. Napolitani, J. Pereira, F. Rejmund, K. Schmidt,O. Yordanov, Production cross-sections of neutron-rich Pb andBi isotopes in the fragmentation of
U, Eur. Phys. J. A 42(2009) 485–488.[59] H. Alvarez-Pol, J. Benlliure, E. Casarejos, L. Audouin,D. Cortina-Gil, T. Enqvist, B. Fernández-Domínguez, A. R.Junghans, B. Jurado, P. Napolitani, J. Pereira, F. Rejmund,K.-H. Schmidt, O. Yordanov, Production of new neutron-richisotopes of heavy elements in fragmentation reactions of
Uprojectiles at 1 A GeV, Phys. Rev. C 82 (2010) 041602.[60] D. Henzlova, A. S. Botvina, K.-H. Schmidt, V. Henzl, P. Napoli-tani, M. V. Ricciardi, Symmetry energy of fragments producedin multifragmentation, Jour. Phys. G 37 (2010) 085010.[61] H. Geissel, P. Armbruster, K. H. Behr, A. Brunle, K. Burkard,M. Chen, H. Folger, B. Franczak, H. Keller, O. Klepper,B. Langenbeck, F. Nickel, E. Pfeng, M. Pfutzner, E. Roeckl,K. Rykaczewski, I. Schall, D. Schardt, C. Scheidenberger, K. H.Schmidt, A. Schroter, T. Schwab, K. Summerer, M. Weber,G. Munzenberg, T. Brohm, H. G. Clerc, M. Fauerbach, J. J.Gaimard, A. Grewe, E. Hanelt, B. Knodler, M. Steiner, B. Voss,J. Weckenmann, C. Ziegler, A. Magel, H. Wollnik, J. P. Du-four, Y. Fujita, D. J. Vieira, B. Sherrill, The GSI projectilefragment separator (FRS): a versatile magnetic system for rel-ativistic heavy ions, Nucl. Instr. Meth. B 70 (1992) 286–297.[62] H. Feshbach, K. Huang, Fragmentation of relativistic heavyions, Phys. Lett. B 47 (1973) 300–302.[63] A. S. Goldhaber, H. H. Heckman, High energy interactions ofnuclei, Ann. Rev. Nucl. Part. Sci. 28 (1978) 161–205.[64] A. Bacquias, Kinematical properties of spectator fragments inheavy-ion collisions at relativistic energies, Ph.D. thesis, ULPStrasbourg, 2008.[65] P. Napolitani, K.-H. Schmidt, L. Tassan-Got, Inclusive selectionof intermediate-mass-fragment formation modes in the spalla-tion of
Xe, J. Phys. G: Nucl. Part. Phys. 38 (2011) 115006.[66] W. A. Friedman, Heavy ion projectile fragmentation: A reex-amination, Phys. Rev. C 27 (1983) 569–577.[67] M. Giacomelli, L. Sihver, J. Skvarc, N. Yasuda, R. Ilic, Pro-jectilelike fragment emission angles in fragmentation reactionsof light heavy ions in the energy region
200 MeV/nucleon:Modeling and simulations, Phys. Rev. C 69 (2004) 064601.[68] E. J. Moniz, I. Sick, R. R. Whitney, J. R. Ficenec, R. D.Kephart, W. P. Trower, Nuclear fermi momenta from quasielas-tic electron scattering, Phys. Rev. Lett. 26 (1971) 445.[69] B. G. Harvey, Beam-velocity fragment yields and momentain nucleus-nucleus collisions from 20 MeV/nucleon to 200GeV/nucleon, Phys. Rev. C 45 (1992) 1748. [70] J.-J. Gaimard, K.-H. Schmidt, A reexamination of the abrasion-ablation model for the description of the nuclear fragmentationreaction, Nucl. Phys. A 531 (1991) 709.[71] M. de Jong, A. Ignatyuk, K.-H. Schmidt, Angular momentumin peripheral fragmentation reactions, Nucl. Phys. A 613 (1997)435.[72] J. Bondorf, A. Botvina, A. Iljinov, I. Mishustin, K. Sneppen,Statistical multifragmentation of nuclei, Physics Reports 257(1995) 133.[73] W. Bauer, Temperatures of fragment kinetic energy spectra,Phys. Rev. C 51 (1995) 803–805.[74] K. C. Chung, R. Donangelo, H. Schechter, Dynamical effects inthe Coulomb expansion following nuclear fragmentation, Phys.Rev. C 36 (1987) 986.[75] J. Hubele, P. Kreutz, J. C. Adloff, M. Begemann-Blaich,P. Bouissou, G. Imme, I. Iori, G. J. Kunde, S. Leray, V. Linden-struth, Z. Liu, U. Lynen, R. J. Meijer, U. Milkau, A. Moroni,W. F. J. Müller, C. Ngô, C. A. Ogilvie, J. Pochodzalla, G. Rac-iti, G. Rudolf, H. Sann, A. Schüttauf, W. Seidel, L. Stuttge,W. Trautmann, A. Tucholski, Fragmentation of gold projec-tiles: From evaporation to total disassembly, Z. Phys. A 340(1991) 263–270.[76] H. S. Xu, M. B. Tsang, T. X. Liu, X. D. Liu, W. G. Lynch,W. P. Tan, A. Vander Molen, G. Verde, A. Wagner, H. F. Xi,C. K. Gelbke, L. Beaulieu, B. Davin, Y. Larochelle, T. Lefort,R. T. de Souza, R. Yanez, V. E. Viola, R. J. Charity, L. G.Sobotka, Isospin fractionation in nuclear multifragmentation,Phys. Rev. Lett. 85 (2000) 716–719.[77] V. Baran, M. Colonna, M. Di Toro, V. Greco, Nuclear frag-mentation: Sampling the instabilities of binary systems, Phys.Rev. Lett. 86 (2001) 4492–4495.200 MeV/nucleon:Modeling and simulations, Phys. Rev. C 69 (2004) 064601.[68] E. J. Moniz, I. Sick, R. R. Whitney, J. R. Ficenec, R. D.Kephart, W. P. Trower, Nuclear fermi momenta from quasielas-tic electron scattering, Phys. Rev. Lett. 26 (1971) 445.[69] B. G. Harvey, Beam-velocity fragment yields and momentain nucleus-nucleus collisions from 20 MeV/nucleon to 200GeV/nucleon, Phys. Rev. C 45 (1992) 1748. [70] J.-J. Gaimard, K.-H. Schmidt, A reexamination of the abrasion-ablation model for the description of the nuclear fragmentationreaction, Nucl. Phys. A 531 (1991) 709.[71] M. de Jong, A. Ignatyuk, K.-H. Schmidt, Angular momentumin peripheral fragmentation reactions, Nucl. Phys. A 613 (1997)435.[72] J. Bondorf, A. Botvina, A. Iljinov, I. Mishustin, K. Sneppen,Statistical multifragmentation of nuclei, Physics Reports 257(1995) 133.[73] W. Bauer, Temperatures of fragment kinetic energy spectra,Phys. Rev. C 51 (1995) 803–805.[74] K. C. Chung, R. Donangelo, H. Schechter, Dynamical effects inthe Coulomb expansion following nuclear fragmentation, Phys.Rev. C 36 (1987) 986.[75] J. Hubele, P. Kreutz, J. C. Adloff, M. Begemann-Blaich,P. Bouissou, G. Imme, I. Iori, G. J. Kunde, S. Leray, V. Linden-struth, Z. Liu, U. Lynen, R. J. Meijer, U. Milkau, A. Moroni,W. F. J. Müller, C. Ngô, C. A. Ogilvie, J. Pochodzalla, G. Rac-iti, G. Rudolf, H. Sann, A. Schüttauf, W. Seidel, L. Stuttge,W. Trautmann, A. Tucholski, Fragmentation of gold projec-tiles: From evaporation to total disassembly, Z. Phys. A 340(1991) 263–270.[76] H. S. Xu, M. B. Tsang, T. X. Liu, X. D. Liu, W. G. Lynch,W. P. Tan, A. Vander Molen, G. Verde, A. Wagner, H. F. Xi,C. K. Gelbke, L. Beaulieu, B. Davin, Y. Larochelle, T. Lefort,R. T. de Souza, R. Yanez, V. E. Viola, R. J. Charity, L. G.Sobotka, Isospin fractionation in nuclear multifragmentation,Phys. Rev. Lett. 85 (2000) 716–719.[77] V. Baran, M. Colonna, M. Di Toro, V. Greco, Nuclear frag-mentation: Sampling the instabilities of binary systems, Phys.Rev. Lett. 86 (2001) 4492–4495.