THE CODE LISE: new version 4.9
Momentum distributions of fragments.
Universal parametrization

  Contents :

1. Introduction

2. Longitudinal momentum distribution widths

2.1. Systematic of momentum distributions from reactions with relativistic ions of D.J.Morrissey

2.2. Heavy ion projectile fragmentation: A reexamination by W.A.Friedman

2.2.1. "Coulomb correction" of Friedman
2.3. Simple parametrization of fragment reduced widths in heavy ion collisions (R.H.Tripathi et al.)

2.4. Role of intrinsic width in the fragment momentum distributions in heavy ion –collisions (R.H.Tripathi et al.)

2.5. Influence of projectile energy on the reduced width of longitudinal momentum distribution

3. Fragment velocity 3.1. Fragment velocity: removing 8 MeV per ablated nucleon

3.2. Fragment velocity on the basis of surface energy exceed

4. New universal model of momentum distribution on the basis of gaussian and exponent convolution.


1. Introduction

Fragment momentum distributions measured in relativistic heavy ion collisions are typically observed to be Gaussian shaped. Within the framework of the independent particle model, Goldhaber [GOL74] assumed zero net three-momentum in the nucleus and shoed that the parabolic dependence of the momentum width of the gaussian curve can be obtained:

,         /1/ where  is the reduced width and pF is the Fermi momentum. The fragment velocity is supposed to be equal the projectile velocity. The given model long time was unique in program LISE for calculation of width. However, this model is unable to account for the following:
  1. the differences in widths associated with nuclides of the same mass;
  2. the discrepancy between , where pF is obtained from electron scattering experiments;
  3. the observed difference between 
  4. the apparently anomalously small values of s0 observed at lower energies;
  5. the isotope yields from the fragmentation process;
  6. the occurrence an exponential tail in momentum distributions in reactions at low energies.
  7. the reduction of the velocity relation of a fragment to projectile at low energies.
Further it was developed different models for an explanation of these phenomena both theoretical, and empirical parametrizations. A part of their these models already were entered earlier in the LISE code, some are entered in the given version.

Certainly, that each of models has the advantages and lacks depending on energy and masse of projectile and so on. In the new version of program LISE the user can choose depending on a task necessary to him models from 3 models of fragment velocities and from 3 models of momentum distribution widths or take advantage of new universal "convolution" model.

The reviews of models of momentum distribution widths, velocities of fragments, systematization of the reduced width from energy are submitted to attention of readers in the following chapters.

Also for the new version the Universal parametrization which allows to avoid a significant part of those lacks (a,d,f,g) that were inherent in primary statistical model of A.Goldhaber is developed and adapted in the program.

2. Longitudinal momentum distribution widths

2.1. Systematic of momentum distributions from reactions with relativistic ions of D.J.Morrissey

As the practice shows, many physics use for the description given other distributions, in particular take the width from the empirical systematic of Morrissey [MOR89]:

.         /2/ The opportunity of use of this momentum distribution model in the code LISE has appeared in version 4.7. This parametrization is received on a basis of experimental data and represents a linear dependence on the square root of the mass loss as against the parabolic form assumed by Goldhaber’s model.

Fig.1. Dialogue "Production mechanism – Momentum distribution"

2.2 Heavy ion projectile fragmentation: A reexamination by W.A.Friedman

Friedman [FRI83] has shown that a model based on the separation energy of fragments from projectile leads to the same functional form as in Goldhaber’s model and a better representation data (at least for light projectiles/O.T./). This model relates the widths of distributions to the separation energies and an absorptive cutoff radius. In this work it is entered a wave function yF-r(r) which describes the relative separation between the observed fragment (F) and the removed portion of the projectile (R) to calculate only outside of the absorption region:

,        /3/ where mr is the reduced mass and Es is the separation energy. In the calculations he defined after including the lowest order of the Coulomb potential on the bound state wave function tail, that distribution has gaussian form with width: .         /4/ The given model has been incorporated in the new version of the code (see Fig.1).
On the other hand, the fragment will be observed if the process leads to any of its particle-stable excited states. This possibility provides a range and upper value for Es. The user can choose one of three possible separation energy: energy on a basis cluster separation from projectile, excitation energy to the surface exceed, and their sum (see Fig.2).

However the registered nucleus is consequence of prefragment evaporation of neutrons and light charged particles. Prefragment momentum distribution can differ from a registered fragment considerably. To calculate momentum distribution for prefragment the opportunity of prefragment calculation is entered adjusted for momentum distribution in connection with evaporation.

Fig.2. Dialogue “Friedman’s momentum distribution width”


2.2.1 "Coulomb correction" of Friedman

The opportunity of use of the correction for reduced width was added at small energies from distortions due to Coulomb force proposed by W.A.Friedman was incorporated in version 4.7 of the code (see Fig.1):

.        /5/ This correction becomes significant only for energies below 20 MeV/u.

2.3. Simple parametrization of fragment reduced widths in heavy ion collisions (R.H.Tripathi et al.)

A systematic analysis of the observed reduced widths obtained in relativistic heavy ion fragmentation reactions was used to develop a phenomenological parametrization [TRI94a] of these data:

.         /6/ In new version 4.9 of program LISE this parametrization is entered as the amendment (see Fig.1) to reduced width of other models as. .    /7/ 2.4. Role of intrinsic width in the fragment momentum distributions in heavy ion –collisions (R.H.Tripathi et al.)

In work [TRI94b] has been demonstrated comparison of intrinsic widths incorporating correlations in conjunction with dynamical effect with intrinsic widths alone without correlations [GOL74]. They suggested the momentum distribution of the observed fragments depends upon the initial (intrinsic) momentum distribution of the projectile nucleus and upon the momentum transferred by the collision (dynamics):

,      /8/ where sint is the intrinsic width due to internal Fermi motion of the nucleons, and sdyn is the dynamical distribution. This model is not used in the code.

2.5. Influence of projectile energy on the reduced width of longitudinal momentum distribution

Systematics of reduced widths for projectile like fragments [RAM85,BOR86] have shown abnormal behavior at small energies. From energy 100 AMeV sharp recession of width (see Fig.3) is observed. Different hypotheses were offered, but any of them can not describe a course of a curve at energy of some tens MeV per nucleon. It spoke as the contribution of other mechanisms of reactions, or influence of Coulomb forces. The concept of the contribution of other mechanisms was entered first of all for an explanation low-energetic exponential tail, accompanying gaussian distribution. Coulomb interaction starts to play a role in area below 20 AMeV.

Fig.3. Systematic of reduced widths of the linear momentum distributions of fragments measured in various reactions and at different bombarding energies [RAM85]. The points (triangles) from ref. [GRE75] have been corrected. See details in the text.

With input of new universal parametrization on the basis of convolution (it will be in more detail submitted in the following chapters) it is possible to explain experimental results. As a result of work on parametrization it was shown, that the width is proportional to a root of velocity. In the figure the given dependence is shown by a red line. The greater consent is achieved for low energies, when to this dependence the Coulomb amendment is added: magenta dash-dot curve. However, how to explain a strong divergence with the data at energies 60-100 AMeV? The answer consists in a method of measurement of width of momentum distribution. At energy 100 AMeV the momentum distribution under the form is very close to gaussian and accordingly experimenters measure full width on maximal height to extract reduced width: a dark blue line received from new universal parametrization. At lower energies however the contribution exponential tail is more appreciable, that is why physics describe distribution by several others. Roughly it can be presented if to consider the right part of distribution as gaussian and to take right halfwidth for calculations of the given width: in figure 3 it is shown by green dotted curve. Two last curves was calculated for reaction 40Ar + Al-> 25Mg.

Momentum distributions of 25Mg in reaction 40Ar(50AMev)+Al calculated by two methods are presented on Fig.4, where it is well seen an exponential tail in distribution on the basis of new parametrization. At initial corrected sigma0 is equal to 43 () as a result of convolution it turns out 82 of full width and 63 of right halfwidth.

Fig.4. Momentum distributions of  25Mg in reaction 40Ar(50AMev)+Al calculated by two methods: new universal convolution method (red curve), Morrissey parametrization for the momentum width and Rami model for relation of velocities (blue curve).

3. Fragment velocity
As it was already mentioned, velocity of fragmentation reaction products is equal to projectile velocity. In initial versions of the program velocity was considered independent of energy, masses of fragments and an ion of beam. The user in manual entered the relation of velocity of a fragment to a beam. At small energies the relation of velocities is experimentally shown decreases (see Fig.5). Many works show change of this relation due to the contribution of other mechanisms of reactions to an output of products, that accordingly changes the form of momentum distribution.
Fig.5. Ration of the ejectile to projectile velocities versus the mass of the fragment in the 40Ar(26.5AMev)+ 68Zn reaction [RAM85]. The solid and dotted curves correspond to the two types of velocity calculation: suggestion that for each ablated nucleon requires 8 MeV[BOR83] and surface energy exceeds [RAM85].


3.1. Fragment velocity: removing 8 MeV per ablated nucleon

First estimation [BOR83] of velocity issuing from a fragmentation reaction: if one conjectures that in the fragmentation process the nucleons are removed from the projectile to another and that an average of 8 MeV is required for each (this corresponds to the solid curve in fig.5):

.    /9/ However experimental data show, that the estimation underestimates velocity, in consequence of that began possible to change in the program a quality of energy necessary to ablate one nucleon. The estimation of the velocity relation is inserted into the program (see Fig.6).

3.2. Fragment velocity on the basis of surface energy exceed

If one assumes that the projectile is sheared in two, then several nucleons bonds have to be broken simultaneously. The number broken can be treated as being proportional to the surface geometry. In the this case [RAM85]:

,         /10/ where S (MeV) is the surface energy of contact and equal to 2g s (this corresponds to the dotted curve in fig.5). g is the nuclear surface tension coefficient (0.95 MeV/fm2) and s the area of the interface between the abraded zone and the remaining fragment. s is calculated using clean-cut abrasion model of Gosset et al.[GOS77]. The given estimation of the velocity relation is added in the new version of the program (see fig.6). The option calculation of the prefragment mass for use of this value instead of the fragment mass further for velocity calculation is entered.

Fig.6. Dialogue "Production mechanism – Fragment velocity"

Velocities of fragments calculated by different methods in the program LISE for reactions
40Ar+Al->*Mg (setting nucleus is 25Mg) are shown in Fig.7.

Fig.7. Velocities of fragments calculated by different methods in the program LISE for reactions 40Ar+Al->*Mg.

4. New universal model of momentum distribution on the basis of gaussian and exponent convolution.

Developed in the new version of the program the model of momentum distribution is universal: definition width of distribution depending on beam energy and energy of prefragment excitation, an estimation of the most probable fragment velocity, occurrence a low-energetic exponential tail. Attempt to describe experimental distributions of products of a fragmentation with strongly pronounced gaussian form of beam velocity and with a low-energetic tail at lower energies was undertaken entering convolution between gaussian and exponent. In a basis of model the postulate lays, that energy necessary on division of a projectile on prefragment and participants, and also on prefragment excitation acts ONLY from kinetic energy of projectile. The given assumption lays also in a basis of estimations of speed of a fragment that was described in the previous chapters (equations /9,10/). Obtaining of final distribution needs to be divided into two stages:

  1. obtaining prefragment momentum distribution in view of kinetic energy loss,
  2. evaporation process for prefragment.
Assuming, that one of convolution components is the initial momentum distribution of prefragment according to the equation /1/, it is possible to write both these stages for prefragment and fragment momentum distributions in a general view:          /11/

.             /12/

where t* is the temperature of prefragment. Solution of the first equation for momentum distribution in the beam direction, assuming t0 tending to infinity, it is possible to write then as follows: ,             /13/ where p0pf is the more probable momentum of prefragment, which corresponds of the velocity of projectile. Assuming, that the second stage does not carry the big contribution to the form of distribution, and only shifts the distribution from p0pf to p0 (p0 is the more probable momentum of prefragment), it is possible then to write in final view the equation /12/ as result of both stages as follows: ,             /14/ where addends from the exponent were moved in normalizing coefficient.

For determination of t the evident form was offered:

,                /15/ where ES is the energy spent on break of a particle in prefragment and participants, and on prefragment excitation. As mass the following possible values were used: the mass of prefragment, the reduced mass of prefragment and residual, the fragment mass, a difference between mass of the projectile and prefragment. The best approximation was received as mass in the equation /15/ use of prefragment mass. However in result of fit it was found out, that t must be inversely proportional to fragment velocity to explain of an amplitude of exponential tail. Whence at once follows, that spf used in the equation /14/ should be proportional to a root of velocity. Thus their final kind with the account Coulomb amendments can be submitted as: ,                 /16/

.             /17/

In process of the data fit it was noticed, that shift on velocity much more than it was expected, in this connection it was necessary to insert additional the amendment into error function to compensate this shift which is proportional to t : .             /18/

Fig.8. Energy spectra and results of their fit by the formula /18/.

35 spectra from the following works [BOR86,BLU86,DAR86,GRE75,MER86,MOU81,VIY79] were used for fit. In works where distributions from energy were presented, spectra were transformed in momentum distribution in view of the appropriate amendments (see the description to the version 4.5 "Transformation of distributions"). From fit it was received, that value of s0 in expression /17/ is equal to 91,5. Energy spectra and results of their fit by the formula /18/ shown in figure 8.

The experimental data was fitted for three different values of separation energy: energy separation prefragment from projectile, excitation energy of prefragment, and their sum. The best consent was received in a case of "tau" for the second case, for definition of velocity shift the best result was for the third case. All these three opportunities are given in the program (see Fig.9).

Fig.9. Dialogue "Convolution of Gaussian and exponent" called from dialogue"Production mechanism".


[BOR83] V.Borrel et al., Z.Phys.A 314 (1983) 191.

[BOR86] V.Borrel et al., Z.Phys.A 324 (1986) 205.

[BLU86] Y.Blumenfeld et al., Nucl.Phys A455 (1986) 357.

[DAR86] R.Dayras et al., Nucl.Phys. A460 (1986) 299.

[GRE75] D.E.Greiner et al., Prys.Rev.lett. 35 (1975) 152.

[GOL74] A.S.Goldhaber, Phys.Lett. B53 (1974) 306.

[GOS77] J.Gosset et al., Phys.Rev. C16 (1977) 629.

[FRI83] W.A.Friedman, Phys.Rev. C27(1983) 569.

[MOR89] D.J.Morrissey, Phys.Rev. C39 (1989) 460.

[MOU81] J.Mougey et al., Phys.Lett. B105 (1981) 25.

[MER86] M.C.Mermaz et al., Z.Phys. A324 (1986) 217.

[RAM85] F.Rami et al., Nucl.Phys. A444 (1985) 325.

[TRI94a] R.K.Tripathi, L.W.Towsend, Phys.Rev. C49(1994) 2237.

[TRI94b] R.K.Tripathi, L.W.Towsend, F.Khan, Phys.Rev. C49(1994) R1775.

[VIY79] Y.P.Viyogi et al., Phys.Rev.Lett. C42 (1979) 33.