,q 

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DEPARTMENT OF PHYSICS 

COLLEGE OF SCIENCES 

OLD DOMINION UNIVERSITY 

NORFOLK, VIRGINIA 23529 

NUCLEAR FRAGMENTATION ENERGY AND HOHE” TRANSFER 

DISTRIBUTIONS IN REJATIVISTIC HEAVY-ION COLT..ISIONS 

BY 

Ferdous Khan, Graduate Research Assistant 

Principal Investigator: Govind S. Khandelwal 

Progress Report 

For the period ended June 30, 1989 

Prepared for 

National Aeronautics and Space Administration 

Langley Research Center 

Hampton, VA 23665 

Under 

Research Grant NCC1-42 

Dr. Lawrence W. Townsend, Technical Monitor 

SSD-High Energy Science Branch 

c 

__ 

(HAS A-CR- I a 550 1) B UCLEM FBAGHEM TA TIG BI 

EOEBGP AND HOtlENTUH TRANSFER DISTRIBUTIONS 

IN BXLATLVSSTIC HEAVY-IO# COLZXSIOBS 

P~oqress Beport, period ended 30 Jon. 1989 (Old Dominion Univ.) 152 p CSCL 20H G3/72 0223916 

I89 - 274 77 

Dnclas 
DEPARTMENT OF PHYSICS 

COLLEGE OF SCIENCES 

OLD DOMINION UNIVERSITY 

NORFOLK, VIRGINIA 23529 

NUCLEAR FRAGMENTATION ENERGY AND NOHE" TRANSFER 

DISTRIBUTIONS IN REIATIVISTIC HEAVY-ION COLLISIONS 

BY 

Ferdous Khan, Graduate Research Assistant 

Principal Investigator: Govind S. Khandelwal 

Progress Report 

For the period ended June 30, 1989 

Prepared for 

National Aeronautics and Space Administration 

Langley Research Center 

Hampton, VA 23665 

Under 

Research Grant NCC1-42 

Dr. Lawrence W. Townsend, Technical Monitor 

SSD-High Energy Science Branch 

Submitted by the 

Old Dominion University Research Foundation 

P.O. Box 6369 

Norfolk, Virginia 23508-0369 

August 1989 

T 
NUCLEAR FRAGMENTATION ENERGY AND no" TRANSFER 

DISTRIBUTIONS IN REIATIVISTIC HEAVY-ION COLLISIONS 

BY 

Ferdous Khan1 

Principal Investigator: Govind S , Khandelwa12 

ABSTRACT 

An optical model description of energy and momentum transfer in 

relativistic heavy-ion collisions, based upon composite particle multiple scattering theory, is presented. Transverse and longitudinal 

momentum transfers to the projectile are shown to arise from the real 

and absorptive part of the optical potential, respectively. Comparisons of fragment momentum distribution observables with experiments are 

made and trends outlined based on our knowledge of the underlying 

nucleon-nucleon interaction. Corrections to the above calculations are 

discussed. Finally, use of the model as a tool for estimating collision impact parameters is indicated. 

lGraduate Research Assistant, Department of Physics, Old Dominion 

University, Norfolk, Virginia 23529. 

2Eminent Professor, Department of Physics, Old Dominion University 

University, Norfolk, Virginia 23529. 

ii 
ACKNO-S 

This is a progress report for the research project "A Study of 

Physical Processes for Space Radiation Protection." 

performance for the research is January 1, 1989 through June 30, 1989. 

The work was supported by the NASA Langley Research Center Cooperative 

Agreement NCC1-42. 

Dr. Lawrence W. Townsend of the Space Systems Division (High Energy 

Science Branch). 

The period of 

The cooperative research agreement was monitored by 

iii 
TABLE OF CONTENTS 

Page 

ABSTRACT .......................................................... ii 

ACKNOWLEDGMENTS ................................................... iii 

LIST OF TABLES .................................................... v 

LIST OF FIGURES ................................................... vi 

CHAPTERS 

I. INTRODUCTION ............................................... 1 

11. MULTIPLE SCATTERING THEORY OF NUCLEUS-NUCLEUS COLLISIONS ... 12 

111. ENERGY AND MOMENTUM TRANSFER IN RELATIVISTIC 

NUCLEUS-NUCLEUS COLLISIONS ............... i.............. 24 

IV. NUMERICAL RESULTS ON ENERGY AND MOMENTUM TRANSFER 

DISTRIBUTIONS ........................................... 38 

A. Momentum Downshifts ...................................... 44 

B. Momentum Widths .......................................... 80 

V. CORRECTIONS ................................................ 88 

VI. SUMMARY... ................................................. 105 

REFERENCES ........................................................ 108 

APPENDICES 

A. Derivations (of Momentum and Energy Transfer Formulae).. 114 

B. V and P m,p,................................. 121 

? mp9 m'p' w 9 

C. Analytical Results Using Gaussian Densities ............. 129 

D. The Equivalent Photon Spectr ............................ 135 

iv 
I 

,1 

Table 

1. 

2. 

3. 

4. 

5. 

6. 

LIST OF TABLES 

&gg 

Nucleon-Nucleon transition amplitude: input parameter 

o(e) for various incident energies ........................... 43 

Nuclear charge distribution parameters from electron 

scattering data .............................................. 45 

Momentum downshifts of projectile fragments in the 

reaction l2C(1.05 A GeV) + Target -+ Projectile 

Fragment + X, where X is unidentified and Targets 

are 9Be, 27Al, 64Cu, Io8Ag and 208Pb .................... 69 

Momentum downshifts of projectile fragments in the 

reaction I2C(2.1 A GeV) + Target -+ Projectile 

Fragment + X, where X is unidentified and Targets 

are 9Be, I2C, 27Al, 64Cu, Io8Ag and *08Pb .................... 70 

Momentum downshifts of projectile fragments in the 

reaction 160(2.1 A GeV) + Target + Projectile 

Fragment + X, where X is unidentified and Targets 

are 9Be, I2C, 27Al, 64Cu, Io8Ag and 208Pb .................... 71 

Transverse momentum widths of projectile fragments in the 

reaction 139~a(1.2 A GeV) + I2C + Projectile Fragment + X, 

where X is unidentified ...................................... 82 
Figure 

vi 

LIST OF FIGURES 

&gg 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

Momentum transfer (MeV/c) to l60 projectile by the 9Be 

target as a function of impact parameter (fm) in the 

reaction 160(2.1 AGeV) + 9Be + Projectile Fragment + X, 

where X is unidentified ...................................... 46 

Momentum transfer (MeV/c) to l60 projectile by the 12C 

target as a function of im act parameter (fm) in the 

reaction 160(2.1 AGeV) + 13C + Projectile Fragment + X, 

where X is unidentified ...................................... 47 

Momentum transfer (MeV/c) to l60 projectile by the 27Al 

target as a function of im act parameter (fm) in the 

reaction 160(2.1 AGeV) + 27A1 + Projectile Fragment + X, 

where X is unidentified ...................................... 48 

Momentum transfer (MeV/c) to l60 projectile by the 64Cu 

target as a function of im act parameter (fm) in the 

reaction 160(2.1 AGeV) + 6'&u + Projectile Fragment + X, 

where X is unidentified ...................................... 49 

Momentum transfer (MeV/c) to l60 projectile by the lo8Ag 

target as a function of im act parameter (fm) in the 

reaction 160(2.1 AGeV) + lg8Ag + Projectile Fragment + X, 

where X is unidentified ...................................... 50 

Momentum transfer (MeV/c) to l6O projectile by the *08Pb 

target as a function of im act parameter (fm) in the 

reaction 160(2.1 AGeV) + 2g8Pb + Projectile Fragment + X, 

where X is unidentified ...................................... 51 

Momentum transfer (MeV/c) to 12C projectile by the 9Be 

target as a function of impact parameter (fm) in the 

reaction 12C (1.05 AGeV) + 9Be + Projectile Fragment + X, 

where X is unidentified ...................................... 52 

Momentum transfer (MeV/c) to 12C projectile by the 12C 

target as a function of impact parameter (fm) in the 

reaction 12C (1.05 AGeV) f 12C + Projectile Fragment + X, 

where X is unidentified ...................................... 53 
vii 

9. Momentum transfer (MeV/c) to 12C projectile by the 27Al 

target as a function of impact parameter (fm) in the 

reaction 12C (1.05 AGeV) + 27Al -+ Projectile Fragment + X, 

where X is unidentified ...................................... 54 

Momentum transfer (MeV/c) to 12C projectile by the 64Cu 

target as a function of impact parameter (fm) in the 

reaction 12C (1.05 AGeV) + 64Cu -+ Projectile Fragment 

+ X, where X is unidentified ................................. 55 

Momentum transfer (MeV/c) to 12C projectile by the lo8Ag 

target as a function of impact parameter (fm) in the 

reaction 12C (1.05 AGeV) + Io8Ag + Projectile Fragment 

+ X, where X is unidentified ................................. 56 

Momentum transfer (MeV/c) to 12C projectile by the 208Pb 

target as a function of impact parameter (fm) in the 

reaction 12C (1.05 AGeV) + 208Pb + Projectile Fragment 

+ X, where X is unidentified ................................. 57 

Momentum transfer (MeV/c) to 12C projectile by the 9Be 

target as a function of impact parameter (fm) in the 

reaction I2C(2.1 AGeV) + 9Be -+ Projectile Fragment + X, 

where X is unidentified ...................................... 58 

10. 

11. 

12. 

13. 

I 

1 

1 

li 

1 

1 

a 

E 

14. Momentum transfer (MeV/c) to 12C projectile by the 12C 

target as a function of im act parameter (fm) in the 

reaction 12C(2.1 AGeV) + 

where X is unidentified ...................................... 59 

-+ Projectile Fragment + X, 

15. Momentum transfer (MeV/c) to 12C projectile by the 27Al 

target as a function of im act parameter (fm) in the 

reaction I2C(2.1 AGeV) + 29A1 + Projectile Fragment + X, 

where X is unidentified ..................................... . 60 

16. Momentum transfer (MeV/c) to projectile by the 64Cu 

target as a function of im act parameter (fm) in the 

reaction 12C(2.1 AGeV) + 6&u + Projectile Fragment + X, 

where X is unidentified ...................................... 61 

17. Momentum transfer (MeV/c) to 12C projectile by the lo8Ag 

target as a function of im act parameter (fm) in the 

reaction 12C(2.1 AGeV) + lg8Ag + Projectile Fragment + X, 

where X is unidentified ...................................... 62 

18. Momentum transfer (MeV/c) to 12C projectile by the 208Pb 

target as a function of im act parameter (fm) in the 

reaction 12C(2.1 AGeV) + 2g8Ag -+ Projectile Fragment + X, 

where X is unidentified ...................................... 63 
19. 

20. 

21. 

22. 

23. 

24. 

25. 

26. 

viii 

Momentum downshifts of projectile fragments in the reaction 

l2C(1.05 AGeV) + Target + Projectile Fra 

X is unidentified and targets are 9Be, Ig, 27Al, 64Cu, 

Io8Ag and 208Pb. Harmonic-well densities were used for 

A < 20, Woods-Saxon for A 2 20 ............................... 

ent + X, where 

Momentum downshifts of projectile fragments in the reaction 

I2C(2.1 AGeV) + Target + Projectile Fra ent + X, 

where X is unidentified and targets are ce, I2C, 27Al, 

64Cu, Io8Ag and 208Pb. 

used for A C 20, Woods-Saxon for A? 20 ...................... Harmonic-well densities were 

Momentum downshifts of projectile fragments in the reaction 

160(2.1 AGeV) + Target + Projectile Fra ent + X 

where X is unidentified and targets are ce, 12C, 27Al, 

64Cu, lo8Ag and 208Pb. Harmonic-well densities were 

used for A C 20, Woods-Saxon for A 2 20 .................... 

Imaginary part of the optical potential Vo t(r) in MeV 

as a function of separation r 

Beryllium at 2.1 A GeV. 

used for both projectile and target and the result 

compared with uniform density caluclations ................... 

in fm for gxygen with 

Harmonic well densities were 

Longitudinal momentum transfer in MeV/c to Oxygen by the 

Beryllium target as a function of impact parameter in fm. 

Harmonic well densities were used for both projectile and 

target and the result compared with uniform density 

calculations ................................................. 

Momentum downshifts of projectile fragments in the 

reaction 160(2.1 A GeV) + Target -* Projectile 

Fragment + X where X is unidentified and targets are 

9Be, I2C, 27Al, 64Cu, Io8Ag and 208Pb. 

(with a modified slope parameter for two-body interaction) 

were used for the projectile and the target. Experimental 

data (ref. 19) have been averaged over isotopes ............. 

Longitudinal momentum transfer (HeV/c) to l6O projectile 

by the 9Be target as a function of impact parameter (fm) 

in the reaction l6O (Einc) + 9Be + Projectile Fragment + X. 

Incident energies Einc = 2.1, 1.05, 0.8, 0.6, 0.4, 0.2 A GeV 

were used .................................................... 

Uniform densities 

Transverse momentum transfer (MeV/c) to l60 projectile 

by the 9Be target as a function of impact parameter (fm) 

in the reaction l6O (Einc) + 9Be -* Projectile Fragment 

+ X. Incident energies Einc - 2.1, 1.05, 0.8, 0.6, 0.4, 

0.2 A GeV were used.. ........................................ 

66 

67 

68 

74 

75 

76 

78 

79 
ix 

27. Transverse momentum widths of projectile fragments 

in the reaction 139~a(2.1 AGeV) + I2C -* Projectile Fragment 

+ X, where X is unidentified. Harmonic-well density was 

used for I2C, Woods-Saxon for 139La .......................... 86 

28. The ratio of phase-shift parameters X(b,z/X(b) as a function 

of b and z (where z is the distance along the beam direction 

and b is the impact parameter) in the reaction l6O (2.1 AGeV) 

+ 9Be + Projectile Fragment + X .............................. 93 

29. First-order correction to calculated longitudinal 

momentum transfer as a function of impact parameter for 

160(2. 1 AGeV) + 9Be.. ........................................ 94 

30. Ratio of change in velocit per nucleon to incident 

velocity per nucleon for 1gO(2.1 AGeV) + 9Be due to 

longitudinal momentum transfer ............................... 95 

31. Transverse momentum transfer (MeV/c) to l6O projectile 

by the 9Be target due to Coulomb interaction compared with 

the nuclear contribution as a function of impact parameter 

(fm) at Einc 9 2.1 AGeV ...................................... 100 

32. Transverse momentum transfer (MeV/c) to l6O projectile 

by the 27Al target due to Coulomb interaction compared with 

the nuclear contribution as a function of impact parameter 

(fm) at Einc 9 2.1 AGeV ...................................... 

Transverse momentum transfer (MeV/c) to l6O projectile 

by the *08Pb target due to Coulomb interaction compared with 

the nuclear contribution as a function of impact parameter 

(fm) at Einc = 2.1 AGeV. ..................................... 

101 

33. 

102 

34. Point-Coulomb potential compared with a uniform charge 

distribution as a function of radial separation r (fm) ....... 104 
I 

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CHAPTER I 

INTRODUCTION 

With the advent of relativistic heavy ion beams at Berkeley, 

Saclay and Dubna, the experimental situation in high-energy nuclear 

interactions has improved dramatically over the previous era of cosmicray heavy-ion physics. Sophisticated experiments have challenged 

theorists to come up with new theoretical tools and insights to understand the new features apparent in relativistic heavy-ion collisions. 

Traditional nuclear physics has been primarily about the nature of 

nuclear matter at or near equilibrium. With heavy-ion beams, the 

possibility of compressing nuclear matter to two or three times the 

normal density and heating to temperatures - 100 MeV has opened up. 

is appropriate at this point then to review the salient features of 

high energy heavy-ion collisions beginning with cosmic-ray heavy ion 

physics. 

It 

Ever since the discovery of Z > 2 (where Z is the nuclear charge) 

components in the primary cosmic radiation by Freier et a1.lv2, which 

fulfilled a prediction by Alfven3, the subject of high energy interactions between nuclei has been of fundamental interest. 

these pioneering studies focussed primarily on interaction mean free 

paths and reaction cross-sections. 

and determination of their isotopic composition was intensely studied 

in order to infer from these data the conditions of their origin, 

possible acceleration mechanisms and subsequent propagation. 

The goal of 

The production of nuclear fragments 

These 

1 
2 

aspects of cosmic-ray heavy ion physics are thoroughly reviewed by 

Shapiro and Silberberg, 

Powell. 6 

Waddington5 and in the classic monograph by 

The qualitative classifications of the nuclear interactions in 

cosmic rays were performed first by Bradt and Peters./ The concepts of 

"Peripheral" (large impact parameter) and "Central" (small impact 

parameter) collisions were introduced by these authors. In a peripheral collision, part of the nucleus overlapping the target is sheared off 

while the remaining fragment proceeds at near the beam velocity. 

projectile and target fragmentation may be described as peripheral 

processes. In a central collision, both the nuclei are destroyed, 

involving high levels of excitation and the emission of large numbers 

of secondary fragments. Nucleons, light fragments and pions are copiously produced in central collisions. 

Both 

Experimental studies of cosmic rays revealed many important features, in spite of low intensities and uncertainties in charge, mass 

and energy determinations. 

cross sections using a semi-empirical "black sphere" expression 

Bradt and Peters7 analyzed the reaction 

with Ap, AT the mass numbers of beam and target, and 6 an overlap 

parameter representing the diffuseness and partial transparency of 

nuclear surfaces. 

tions could be reasonably predicted with equation (1.1)- These crosssections were assumed to be energy-independent for bombarding energies 

from .1 A GeV to 30 A GeV. 

the fragmentation of a certain projectile was also found to be nearly 

With fixed values of ro and 6, reaction cross-secThe average number of fragments produced by 
3 

energy-independent (to within -20%). The Bradt-Peters "black sphere" 

model was later refined and a "grey sphere" 

account for the reduction of geometric cross-section due to the transparency of nuclear surfaces. 

cross-sections were also undertaken successfully by various authors9 lo 

and satisfactory agreement was found with experiment. 

model was proposed8 to 

Optical model calculations of reaction 

For cosmic rays, the reaction products studied were mainly mesons 

and nucleon^^^'^^. 

nucleus interaction as a superposition of independent nucleon-nucleon, 

nucleon-nucleus or alpha-alpha collisions. Although successful in 

achieving a broad understanding of such collisions, precise knowledge 

of high energy interactions of nuclei could not be gained from these 

studies due to low statistics and lack of control over experimental 

conditions. 

The experimental goal was to understand nucleusThe first laboratory acceleration of relativistic heavy-ions was 

accomplished in 1971 at the Princeton Particle Accelerator (PPA), 

shortly followed by the Berkeley Bevatron16. 

particles began in Dubna 1970. 

heavy- ion experiments17. 

BEVALAC became the only high energy heavy ion accelerator in the U.S.A. 

Proposal for a Relativistic Heavy Ion Collider (RHIC) to be built at 

Brookhaven National Laboratory in the 1990's is underway. 

that are also in progress are GANIL in France, Numatron in Japan, 

Nuklotron in the U.S.S.R. and GSI in Darmstadt, West Germany. 

Acceleration of alpha 

Similar proposals were made at CERN for 

With the closing of PPA in 1972, Berkeley 

Projects 

The BEVALAC, proposed by Ghiorso18, employs an 8.5 A MeV heavy ion 

linear accelerator, the SUPERHILAC, to inject the ions into the 

Bevatron which continues the acceleration of these ions to a maximum 
4 

energy of 2.6 A GeV. 

accelerated at the BEVALAC. 

High intensity Uranium beams have recently been 

Experimental techniques at high energy heavy ion accelerators 

combine tools from both traditional low energy nuclear physics as well 

as high energy particle physics because of the wide range of charge, 

mass and energy. 

by AE-E and time of flight over 4s steradians for target fragmentation 

to high energy techniques such as magnetic spectrometers, measurements 

of dE/dx, rigidity, Time of Flight (TOF) for slow projectile particles. For relativistic particles Cerenkov radiation as well as the use 

of streamer chamber or other multiple track detectors are used. Detection by emulsions, plastics and AgCl monocrystals are also used because 

of their wide range of sensitivities, versatility and small demand for 

beam time. 

experiments pose difficulties for charge identification by dE/dx=Z2 

f(P) so that additional capabilities must be incorporated into the 

system. 

the Heavy Ion Study Proceedings published by Lawrence Berkeley Laboratory (and GSI) every two years, where recent information on both experimental and theoretical aspects can be found. 

These techniques range from particle identification 

The availability of wide ranges of charge in heavy ion 

Excellent reviews of these aspects of detector development are 

In this work, we shall examine one aspect of relativistic heavyion collisions in detail. We shall formulate a theoretical framework 

to describe how momentum and energy are transferred to relativistic 

heavy ions. 

fer based on the multiple scattering theory of nucleus-nucleus collisions will be presented. 

An optical model description of momentum and energy trans- 
5 

Aside from the fundamental importance of the topic itself, the 

necessity for understanding these processes (energy-momentum transfer) 

arose due to the increasing sophistication of experiments. Singleparticle inclusive experiments of the type Projectile + Target + Projectile Fragment[+ other products] form the basis of our knowledge on 

heavy ion projectile fragmentation. 

in the fragmentation of 12C and l6O (2.1 A GeV) beams on targets 

ranging from H through Pb that the isotope production cross-sections 

were factorizable into 

It was observed e~perimentallyl~ 

FBT 'E 'T (1.2) 

u 

T where UFBT is the cross-section for producing the fragment; 7; and 7 

are two terms that depend on the beam-fragment and target respectively. 

This suggests that the momentum of fragments in the'projectile rest 

frame should also exhibit independence of target structure and beam 

energy. Indeed, in the rest frame of the projectile, the longitudinal 

momentum distributions of fragments show a statistical Gaussian dependence. Irrespective of projectile, beam energy (2 1.05 A GeV) and 

target nucleus, the longitudinal momentum P distributions for all 

fragments from 12C and l60 projectiles are characterized by Gaussian 

shape with rms width o(P ) = 50 to 200 MeV/c. These longitudinal 

spectra are also downshifted by <PII> = -20 to -130 MeV/c from the 

beam, showing that the mean velocities of these fragments are less than 

that of the beam. The rms widths u(P ) and cr(P1) of the longitudinal 

and transverse momentum spectra are found to be equal to within -lo%, 

consistent with isotropic production of fragments in a frame moving at 

II 

II 

II 
6 

a velocity less than the beam velocity. 

independent of target mass and beam energy, depend on the masses of the 

beam and the fragment. 

of the data 

The widths u(P ) and u(PI), It 

A parabolic shape reproduces the general trend 

I/? AF u(P,,) - 2u0[x(l - x)] , x - - 

A 

P 

(1.3) 

where Ap, AF are the beam and fragment masses respectively, and uo is 

experimentally extracted from the data or predicted by theory. 

The parabolic shape of the widths u(P ) has been the subject of II 

considerable attention of theorists20-22 who explained this dependence 

using conservation of momentum. According to these theories, the 

Gaussian momentum distributions can be understood by treating the 

fragmenting nucleus as a Fermi gas and assuming (i) Momentum conservation, (ii) No correlation among nucleons in the parent nucleus, and 

(iii) Neglecting anti-symmetrization of the single particle states. 

Within these assumptions, one would predict (1.3) where uo is related 

to the Fermi momentum of the projectile (PF) via 

(1.4) 

explaining the isotrop,: production of the fragments. These insigh 

form the basis of much of the analyses of experimental data. 

The simple Fermi gas picture of projectile fragmentation has been 

questioned by many authors. Nuclear structure and binding-energy 
7 

effects in determining the momentum widths has been pointed 

Neglect of anti-symmetrization of single-particle states has been 

questioned by Bert~ch~~ who showed that the momentum widths would be 

reduced from the free Fermi gas value due to correlations. Murphy25 

has investigated the constraints of phase space on treating the projectile as a Fermi gas which emits fragments that are also Fermi gases. 

Recent experimental data26 on the transverse momentum widths of Lanthanum fragments also indicate the inadequacy of the simple Fermi gas 

picture. Indeed, an alternate f~rmulation~~ of projectile fragmentation abandons the Fermi gas picture and attempts to explain widths in 

terms of neutron separation energies. It is clear that one needs to 

bring in sophisticated theoretical insights to address structure and 

binding energy effects, correlations, phase space constraints etc. 

In this work, we attempt to address important but as yet unresolved questions, i.e. how are momentum and energy transferred to heavy 

ions at high energies and how does energy-momentum transfer affect the 

fragment momentum and energy spectra, fragmentation cross-sections and 

their angular distributions. 

To address the above questions, an optical model description of 

momentum and energy transfer between relativistic heavy ion collisions 

within the multiple scattering theory framework will be presented. 

inputs into our calculations will be the well-known nuclear ground 

state densities and NN (nucleon-nucleon) two-body transition amplitudes. 

to address the above questions over & entire energy ranee from low to 

intermediate and extremely high bombarding energies for any proiectileThe 

The energy-dependence of the two-body amplitude will enable us 
8 

target combination. 

will unify a host of data on heavy ion (peripheral) 

New insights gained from this work, it is hoped, 

fragmentation. 

Previously, excitation energy calculations have been undertaken by 

Hufner28 et al. within the Glauber theory framework. 

calculated the excitation energy of "anomalons" detected in heavy ion 

experiments using an impulsive excitation picture. 

and semi-empirical estimates of the excitation energy in heavy ion 

collisions also exist (these are not based on two-body interaction 

parameters). 

Fri~ke~~ has 

Phenomenol~gical~~ 

The new feature of this work is the introduction of a complex 

momentum transfer vector which results from the use of a complex twobody transition amplitude that satisfies unitarity and is used to 

analyze experiments. The real (imaginary) part of the two-body interaction, folded with the appropriate densities for nucleon-nucleus or 

nucleus-nucleus scattering gives rise to the real (imaginary) part of 

the optical potential. The imaginary part accounts for inelastic 

scatterings as well as true absorption, and as will be shown, gives 

rise to longitudinal momentum transfer. The reaction cross-section in 

Glauber theory, for example, is obtained from the imaginary part of the 

phase shift computed from the imaginary part of the optical potential. 

The real part of the complex momentum transfer vector represents transverse momentum transfer due to elastic scattering at high energy. In a 

fragmentation experiment, the projectile (or target) fragments are 

detected. 

the beam velocity i.e. the longitudinal momentum spectra of these 

fragments are "downshifted" by -20 Mev/c to -130 Mev/c depending on the 

fragment. This "momentum downshift" is naturally explained in this 

It has been found that these fragments emerge with less than 
9 

work as due to longitudinal momentum transfer to the projectile by the 

target. This arises due to inelastic scatterings that occur as the 

projectile traverses the target. The transverse momentum transfer due 

to elastic scatterings, is separately obtained from the real part of 

the interaction. Calculation of momentum transfer allows us to calculate the means and widths of transverse (and longitudinal) momentum 

spectra for fragments for any projectile-target combination, using as 

inputs nuclear ground state densities, NN transition amplitude utilizing the currently available theories of projectile fragmentation. 

Since the two-body interaction that is used is energy dependent, 

the validity of these insights can be tested at extremely relativistic 

energies (2 10 A Gev) as well as at lower (5 1 A GeV) bombarding 

energies, assuming the validity of the underlying model at these two 

extremes. Silicon beams have now been accelerated to energies r14.5 

A GeV (at Brookhaven), Oxygen and Sulphur beams to 60 A Gev and 200 A 

GeV at CERN. 

to those at 2.1 A GeV. Lower (< -1 A GeV) 

fragments is harder to come by; the paucity of available data prevents 

a systematic study of the reaction mechanism and its evolution as a 

function of bombarding energy. With the methods described herein, one 

can now perform a theoretical calculation within this model. Such 

calculations have been included in this work for possible future comparisons. 

The momentum spectra of fragments look remarkably similar 

energy data on these same 

Another often studied question in nucleon-nucleus as well as 

nucleus-nucleus collisions is the validity of the constant velocity 

assumption that is frequently made. Within this mode1,we have a 
10 

computationally tractable scheme for addressing this question. 

tudinal momentum transfer to the beam nucleus inherently challenges the 

notion of constant velocity. The magnitude of the momentum transfer is 

responsible for change in velocity. We have developed a scheme that 

takes into account corrections to the constant velocity assumption used 

in our formalism. Another correction we address is the possible Coulomb effects as the electromagnetic fields generated at these energies 

are substantial enough to require such an analysis. 

LongiWe have also derived a theory of energy transfer (based on multiple scattering theory) along the lines of our theory of momentum transfer. 

fragments; it is rather inferred or extracted from observed crosssections (using models such as the "Ablation-Abrasion" model). Our 

calculation of excitation energy for projectiles and projectile fragments introduces a comprehensive framework for performing these energy 

transfer calculations using NN interaction parameters as inputs. No 

comparisons could be made with experimental data because of the absence 

of the latter. However, sophisticated experiments in the future may 

change this situation. 

Experiments are seldom able to measure the excitation energy of 

The remainder of this work is organized as follows: In Chapter 

11, the multiple scattering theory of nucleus-nucleus collisions is 

reviewed and the optical model discussed within this context. 

Chapter 111, the formulation of momentum and energy transfer in 

nucleus-nucleus collisions is made within the multiple scattering 

theory framework. 

transfer (longitudinal and transverse), momentum downshifts and momentum widths, preceded by discussions on the theoretical understanding 

In 

Chapter IV contains numerical results on momentum 
I 

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and experimental facts on these topics. Corrections are discussed in 

Chapter V. The first is the "deceleration correction" arising from 

the transfer modification of the constant velocity assumption at high, 

intermediate and low bombarding energies. 

familiar Coulomb correction. 

insights gained in this work as well as indicating possible directions 

for future research. 

The second is the more 

We conclude by discussing the major new 
I 

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CHAPTER I1 

MULTIPLE SCATTERING THEORY OF NUCLEUS-NUCLEUS COLLISIONS 

Multiple scattering theory provides a reliable and fundamentally 

correct description of hadron-nucleus as well as nucleus-nucleus 

collisions at intermediate (- 100 A MeV to 500 A MeV) and high (>500 

A MeV) incident energies.31-39 In these theories, the complex manybody problem of Ap projectile nucleons interacting with AT target 

constituents is formulated in terms of two-body interactions. The 

success of these theories is well documented and is reviewed quite 

frequently in the literature. 

Our goal in this chapter is to review multiple scattering theory 

in order to formulate the problem of momentum transfer and excitationenergy deposition in nucleus-nucleus collisions. 

main results of the multiple scattering theory of nucleus-nucleus 

collisions. Following Wilson, 39 t4O a set of coupled equations relating 

all entrance channels to all exit channels will be derived. An optical 

potential Vopt will be extracted under certain approximations. 

We shall review the 

1I.a Review of Multiple Scattering Theory 

We shall review multiple scattering theory of nucleus-nucleus 

collisions to find an expression for the multiple scattering series. 

Collision of a composite projectile (mass number Ap) with a composite 

target (mass number AT) will be considered. 

problem and its solution can be found in references 39-42. 

The formulation of this 

The 

12 
8 

I 

I 

8 

8 

I 

1 

t 

1 

8 

8 

8 

1 

I 

I 

t 

I 

t 

a 13 

Hamiltonian for the combined system of N 9 (AP + AT) nucleons can be 

written as 

% 

+C v 

aj 

H-C-T.+C V + C T + C V 

j -1 J i<j ij a-1 a 4 aj 

where Roman indices refer to the projectile constituents and Greek 

indices to targets. The first two terms in (2.1) are, respectively, 

the kinetic energy and potential energy operators for the projectile 

and are written as 

A 

P 

(2.2) ij 

H =C T +C V 

P j-1 j icj 

Similarly, the third and fourth terms are the target kinetic and 

potential energy operators. The last term is the interaction term 

between ath and jth constituents of the target and projectile. One can 

decouple the center of mass motion of the projectile as 

72 

P 

H- +h 

2Mn A P 

P 

where h 

neither on P 

Similar results obtain for the target 

is the internal Hamiltonian of the projectile which depends 

-z 

P 

nor on its canonically conjugate position variable. P 

HT + hT 

2Mn % 

-c 4, , PT- C P a (2.4) 

a-1 

where hT is the internal Hamiltonian of the target. Then the full 

Hamiltonian can be written as 

-c 

P2 (Ap+%) - 

H- + k2 + h- + h, + V, ; fi 9 1 (2.5) 

Y L L 

where the overall center of mass momentum operator is 

-4 --D 

P-P +PT 

P 
8 

8 

8 

8 

8 

I 

I 

8 

I 

I 

t 

I 

8 

8 

I 

1 

8 

I 

1 

14 

and the projectile momentum compared to the overall center of mass is 

-- - --c 

- 'T 

ATp --* AP 

k=P - p- 

(Ap+%) (Ap+%) 

and the interaction is the sum of two body interactions 

% AP 

a=l j-1 aj 

VI- c I: v 

(2.7) 

The first term in (2.5) is the N-body center of mass motion 

energy, decoupled from the other terms. The second is the kinetic 

energy of relative motion of the projectile and the target. The 

projectile relative position variable appears only in the interaction 

term V 

are coupled to the relative motion through the interaction VI. 

separation between the projectile and target becomes larger, V tends 

to zero. We assume that well defined states are prepared in the 

The projectile and target internal Hamiltonians h and hT 

I' P 

As the 

I 

entering state and observed in the final state. We define these (in 

operator notation) to be eigenstates of the free projectile-target 

Hamiltonian 

The full wave function satisfies the Schrodinger equation 

H' - E' (2.10) 

where 3 consists of a superposition of a free state plus a scattered 

state 

3 4 + 'scat 

with 

(2.11) 

- GT 4 'scat 

where the Green's function is defined as 

(E-H -H )-IG-l 

PT 

(2.12) 
1 

8 

I 

I 

8 

1 

8 

8 

8 

1 

8 

8 

I 

8 

i 

11 

1 

I 

8 

and the transition operator is defined as 

T - V+VGT 

15 

(2.13) 

and the wave operator n as 

.9-w (2.14) 

The wave operator satisfies the Lippman-Schwinger equation 

n - i+Gvn (2.15) 

so that the transition operator is formally given by 

T = vn (2.16) 

The goal is to find a series for T. For nucleus-nucleus 

collisions this series was derived by Wilson.40 For a single 

projectile, this series reduces to the familiar Watson3I series. 

the Eikonal approximation, Gla~ber~~ theory is recovered from 

Wilson’s. 

Using 

Wilson39 s40 observed that the transition operator for the 

scattering of a constituent from the jth constituent can be written as 

(2.17) 

which satisfies a Lippman-Schwinger type equation. The wave operator 

which transforms the entering free state up to the collision of a and j 

constituents can be written as 

(2.18) 

The interpretation of the above equation is as follows. The 

propagation to the time just before a and j constituents collide is the 

sum of an operator which brings the initial free state plus the 

scattered part from the scattering of all other p and k constituents. 

The full wave operator then consists of the wave operator whikh 
8 

8 

1 

8 

I 

I 

8 

8 

8 

8 

I 

1 

8 

8 

I 

I 

f 

I 

I 

16 

transforms the system to the a-j collision plus the contribution due to 

scattering of the ath and jth constituents i.e. 

which can be written as 

Wilson then proves 

(2.19) 

(2.20) 

that the series given by equations (2.17)- 

(2.20) 

defined by these equations. 

constitutes an exact representation of the scattering process 

Consider the product 

+v Gt )W ‘vaj aj aj aj 

-t w 

aj aj 

Summing over a and j one obtains 

(2.21) 

(2.22) 

This completes the proof. 39 

The Green’s function G are true N-body operators. One neglects 

binding effects at high energy and replaces G by free N-body operators 

G which satisfy 0 

(E-X T -B TP)Go-l 

jja 

(2.23) 

Watson’s form of the impulse approximation consists of writing t aj as 

t -v +v aj aj aj Go (2.24) 

so that the above operator acts as a two-body transition amplitude. 

iteration of the above the multiple scattering series obtains 

By 

(2.25) 
8 

8 

1 

I 

8 

8 

8 

I 

0 

8 

8 

8 

8 

1 

8 

I 

1 

8 

I 

17 

which constitutes a formal solution to the exact scattering problem. 

The replacement G4 renders the t essentially two-body operators 

and the series for T above, equation (2.25) becomes a series of twobody operators. 

0 4 

The above series reduces to the Watson3I series when the 

projectile is a single particle, as previously mentioned. Next we 

shall derive an optical potential operator whose Born series is 

equivalent to the multiple scattering series expansion (2.25). 

operator is Vopt, defined from 

opt opt opt opt 

Such an 

T -V +V GT 

as 

From which we obtain 

T-T - C t Gt -... 

Opt aj aj 0.l 

(2.26) 

(2.27) 

(2.28) 

Retaining the first term in (2.26), the optical model is obtained with 

"op t'(Ap'%) (2.29) 

v 

Ap,% are the mass numbers of the 

projectile and the target, respectively. 

The approximate Lippman-Schwinger equation for the effective 

potential operator is given by 

1 where the first order correction to the model is 0 (-1. 

AP+ 

(2.30) 
I 

I 

I 

8 

t 

1 

ia 

1I.b Coupled Channel Equations 

Using the multiple scattering theory of nucleus-nucleus collisions 

following Reference 39-42, a set of coupled equations will be derived. 

An equivalent Schrodinger equation will be extracted from the formalism 

developed so far. We shall focus on the Green’s function 

(2.31) 

k,m,p E - E: - E: - ek + in 

P where Em, ET refer to the projectile and target internal states and 

P 

2M “P A + 21rred 

K 

with the Green‘s function G from (2.31) inserted into (2.11) and 

I 

8 

where 

8 

It 

projecting onto configuration space yields 

(2.32) 

(2.33) 

(2.34a) 
- (rl,r2,. . . rA I 

P 

and 

19 

(2.34b) 

(2.34~) 

(2.35) 

+ -c 

and g 

many-body wavefunctions. 

that the energy transfer is small compared to the incident kinetic 

(6 ) and &r,,(ET) are the projectile and target internal P,m P 

Now, following Foldy and Wale~ka~~, we assume 

energy 

k -k 

mP (2.36) 

Using the closure approximation, (2.33) can be written as 

The equivalent Schrodinger equation follows from the above 

equation 

We express the fact that the projectile and target internal wave 

functions are not eigenstates of the optical potential operator and the 

initial states are mixed into various modes of final excited states as 

follows 

(2.39) 
20 

I 

1 

I I 

I1 

The coupled equations then can be written as42 

which is the desired equation, Here 

(2.41) 

The coupled equations (2.40) for composite particle scattering 

relate all the entrance channels of the system labelled by projectile 

quantum number m and target quantum number p to all the exit channels 

In (2.4a), A 

P 

subscripts m and p label the eigenstates of the projectile and target; 

Mn is the constituent nucleon mass, k is the projectile momentum 

and 4.r are the mass numbers of the projectile and target, 

* 

re la t ive to 

relative to 

-D 

the center of mass, x is the projectile position vector 

the target and 4-L -.c 

The internal coordinates f 

transition amplitude t have been introduced in (2.17). 

and ET have been defined in (2.34) and the 

P 

Qf 

Next we shall write the coupled equations (2.40) in matrix form. 

Introducing the wave vector 

(2.43) 
21 

and the potential matrix 

4 

vll, 00 (2 vll, ol(x) 

- 

-F I 

The coupled equations (2.40) can be written in matrix form as 

(2.45) 

Using the definition of the potential as given by equation (2.42), 

(2.41) can also be written as 

where 

and 

(2.47) 

(2.48) 

This completes our review of multiple scattering theory. We shall 

now discuss the formulation of problems of momentum transfer and 

excitation energy deposition in nucleus-nucleus collisions, guided by 
22 

the multiple scatteri'ng approach. Before we proceed, however, a brief 

review of the optical model is in order, since we shall use the optical 

potential in our work. 

1I.c The Optical Model 

The optical model is defined as the approximation of (2.40) for 

the elastic scattered part as39 p40 

(2.49) 

(2.50) 

(2.51) 

with coupling to various excited internal states neglected. This is 

correct at small momentum transfer or near forward scattering. The 

corresponding approximate wave function is called the coherent 

scattered wave and it dominates the forward scattered component. To 

evaluate the optical potential we calculate the Fourier transform of a 

single term of (2.51) 

(2.52) 

where this term will be recognized as the single scattering term of the 

multiple scattering series; q is the momentum transfer and FplO(q) and 

+ + 

-+ 

F (4) are the Fourier transforms of the single particle density of 

T,O 

the projectile and target, respectively. The nucleon-nucleon 

interaction will be assumed to be constituent averaged as follows40 
I 

I 

I 

I 

I 

I 

I 

D 

1 

I 

I 

I 

I 

I 

d 

I 

I 

E 

/I 

23 

- 

t - (Ap+)-I [NpNT tnn + Z 2 t + (Z N + N Z ) t ] (2.53) P T PP PT PT nP 

where N N are the neutron numbers and Z Z are the proton numbers 

the projectile and target, respectively. The optical potential is 

obtained by evaluating (2.42) (summing over constituents) as40 

W(z - Ap4 I d3;pT (0 I d3F pP (x + < + <'I 

P' T P' T 

-r +--- -c 

(k, e') (2.54) 

are the projectile and target single-particle densities p' 'T where p 

-+ 

t(k,y) is the energy and space dependent two-body transition amplitude. 

We have arrived at the expression for the optical potential used to 

analyze heavy-ion scattering cross sections. The input parameters are 

the nuclear ground state densities and the two-body transition 

amplitudes which will be discussed in Chapter IV. 
E 

:Y 

4 

c 

1 

I I 

'I 

CHAPTER I11 

HOHE" AND ENERGY TRANSFW IN HEAVY-ION COLLISIONS 

In this chapter we shall address the problem of momentum and 

energy transfer in nucleus-nucleus collisions in the framework of 

multiple scattering theory. In the last chapter, we reviewed the 

multiple scattering series, derived the coupled channel equations and 

discussed the optical model. We shall use these tools in the formulation of the problems of momentum and energy transfer in nucleus-nucleus 

collisions. Starting with the reduced Schrodinger equation for the 

combined system, an approximate expression for the wave operator will 

be derived. Momentum transfer to the projectile as well as its excitation energy due to collision will then be evaluated. 

1II.a Momentum Transfer 

Our starting point will be the formulation of the problem of Ap 

projectile constituents colliding with AT target constituents as in 

equation (2.1). Following equations (2.1)-(2.4), the combined 

Hamiltonian has been written in (2.5) as 

H- 7 + k +h +hT+VI 

"P 

(Ap+%) '2 

2M A % P 2Mn(Ap+A+ 

(2.5) 

The N-body center of mass motion energy, the first term in (2.5) has 

been decoupled from the other terms. 

motion kinetic energy, with k as in equation (2.7). The third and 

The second term is the relative 

+ 

fourth terms, hp and hT are the internal Hamiltonians of the projectile 

and target with 

24 
4 

d 

8 

E 

I 

c 

E 

I 

E 

a 

c 

8 

c. 

E 

d 

P 

1 

E 

8 

25 

(3.la) 

(3.lb) 

4 + 

where g (e ) are the internal many-body wave functions of 

the projectile and the target respectively; c 

corresponding eigenvalues. 

((p), + P vm 9P T 

and c T,P are the 

P ,m 

The eigenstates are orthonormal and comIn the overall center of mass frame 

PT = 1 Pa -k 

a 

and P + PT - 0 

P 

Then, the Hamiltonian is 

mn Ap4 H - 2 f2 + h + hT + VI ; pred2Pred P (AP + AT) 

The Schrodinger equation (in coordinate space) 

T2 + h + hT + VI $J (x, E,, ET) - E 1 --4 

We seek solutions of the form4* 

(3.2a) 

(3.2b) 

(3.3a) 

(3.3b) 

(3.3c) 

- reduced mass (3.4) 

can now be written as 
26 

where the bracketed term I 4 (x,( ) I is the desired wave operator. P' T 

Assuming that the source of the beam is along the direction -z, we 

expect 

and 

(3.7) 

Inserting (3.6) into (3.5) yields 

1 ++ 1 '2 --ik*VX+-k+c PO +K P 

+E To 

2c(red 2pred 

(3.9) 

where (3.la), (3.lb) have been used. 

motion (Fermi motion) of the projectile and target respectively. We 

Kp and KT refer to internal 

now note that 

(3.10) 

4-L + 

Solving for the wave operator [ 4 (x,ep,(T)] in (3.9) is in general difficult. 

[ 6 (x,<,,<T)] is assumed to be slowly varying. 

derivate in (3.9) can be dropped.43 

Considerable simplifications occur at high energy when 

-++ + 

Then, the second 

Internal Fermi motion Kp and KT 
27 

can also be neglected. 

approximate wave operator [ 4 (X,~$~,(T)] can be written as 

Using (3.10) and the above assumptions, the 

++ -+ 

r 1 

(3.11) 

.I- 1 

J where 

-e 

9 v - relative velocity 

+ + 

k - pv 

Now (3.6) reads 

(3.12) 

(3.13) 

(3.14) 

Using (3.14), we shall next evaluate the total momentum of the 

projectile. 

the sums of single nucleon momentum operators of the projectile between 

Total momentum can be defined as the expectation value of 

(3.14), i.e. 

(3.15) 
28 

Equation (3.15) is our expression of interest. Note that the 

single nucleon momentum operators act with respect to the internal 

coordinates of the projectile. Note also that we write the interaction 

I+ - 

term V1(x,ep, ET) as the sum of two-body interactions between the ath 

and jth constituent. Since the projectile and target internal 

eigenstates gp,m and gT,u are not eigenstates of the wave operator, the 

above is still a coupled channel problem. 

We shall now make a series of approximations in order to solve 

(3.15). Note that the wave operator in (3.11) was obtained from equation (3.9) by integration. This assumes that the interaction VI 

commutes with itself along different points of a straight line trajectory. 

is appropriate as 

If they do not commute, however, then a power series44 solution 

I- 1 

(3.17) 

+ L 

where bracket [ 

i.e., the argument with earlier z in VI occurs to the left of the VI 

with argument with later z. However, if the interactions VI at dif- 

]+ implies an ordering in analogy with time ordering, 

ferent points along a straightline trajectory commute, then (3.12) 

becomes44 
29 

We will assume that the interactions along a straightline trajectory commute and replace the wave operator (3.17) by (3.18). A plausibility argument will be given to justify the above. 

When the collision time is short compared to the period of orbital 

motion of the constituent nucleons, then the interactions VI at two 

successive positions along a trajectory commute, i.e.45*46 

r 1 

(3.19) 

Ordering in (3.17) becomes inconsequential, and the upper limit in all 

the integrals can be replaced by infinity to yield (3.18). Using 

(3.18), we can rewrite (3.15) as 

L J 

In the light of equations (2.43), (2.44) and (2.45) it may be 

instructive to look at the following decomposition of the potential 

ma tr ix42 

I ’diag ’ ’of f-diag I ’d ’ ’0.d. (3.21) 
30 

where diagonal and off-diagonal elements have been indicated explicitly. It has 

energies, the 

e lement s40 

been shown that for forward scattering at high 

diagonal elements in V dominate over the off-diagonal 

- 

'diag '> 'off-diag 

Then (3.20) can be written as 

(3.22) 

(3.23) 

1 

1 

(3.24) 

+ other terms.............. 

It is now obvious that the other terms in the series will involve 

the off-diagonal elements which are smaller (by (3.22)). 

term in the series involves only diagonal elements. 

The first 

In the spirit of 
31 

the coherent appro~imation~~, we shall retain only the first term in 

(3.24). 

Now, applying the identity45 

[A, [A, B]] + .... i2 eiA B e-iA 9 B + i [A, B] + - 

2 

(3.25) 

leads to 

(3.26) 

Notice that the first term can be thought of as the momentum of the 

projectile before the collision. 

of (3.26) and momentum transfer defined as 

'total - 'before 'transfer 

It may be taken to the left-hand side 

4 4 + 

9 

(3.27) 

(3.28) 

where in the spirit of the optical model discussed in the last chapter, 
8 

1 

1 

I 

I 

I 

c 

t' 

8 

8 

t 

P 

I 

1 

1 

I 

t 

I 

I 

32 

= $oo has been used in going from (3.27) to (3.28). $mP 

Having identified through equation (2.27) the optical potential 

we can substitute the above in equation (3.28) and obtain 

(3.29) 

(3.30) 

Equation (3.30) above is the desired expression for momentum 

transfer in the relativistic heavy ion collisions. 

scattering theory formulation of nucleus nucleus collisions of 

Using the multiple 

Wilson,39940 we have seen that the optical potential can be calculated 

in terms of transition amplitudes rather than two-body potentials. We 

obtained above an expression for momentum transfer in relativistic 

heavy ion collisions using these experimentally well determined transition amplitudes. Equation (3.30) has been explicitly evaluated in 

-e 

I 

*a j 

the Appendix, with the result expressed as Ptran and 'tran 

?j where ;aj is the momentum transfer in the collision 'aj tran tran 

between ath and jth constituents. Explicitly, these are given by 
33 I 

where a and j refer to the ath and jth constituents of the target and 

projectile, respectively; p&(tT) and pj(fp) are their single particle 

densities, and the gradient is with respect to projectile coordinates 

-+ + 

only. The total momentum transfer is 

r 1 

(3.32) 

-* -* 

where p (Ep) and p (ET) are the nuclear densities of the projectile and 

target with mass numbers Ap and AT, respectively, and the two-body 

transition amplitude t has been averaged over constituent type as 4 

(3.33) 1 +NN t +(NZ +NZ)t PTnn PT TP nP 

with Zp, ZT the projectile and target charge numbers; Np, NT the 

neutron numbers and Ap, AT the mass numbers, respectively. (The 

notation for the transition amplitudes is tPP for proton-proton, tnn 

for neutron-neutron, and tnp for neutron-proton.) 

A unique feature of this work is the use of the two-body transition amplitude (in (3.32)) which is complex. The constituent averaged 

two-body transition amplitude is obtained from the first order t-matrix 

used in previous studies of nucleus-nucleus collisions as 
34 

3 - 3/2 

(3.34) 

where e is the two-nucleon kinetic energy in their center of mass 

frame; a(e) is the nucleon-nucleon total cross section; a(e) is the 

ratio of the real to the imaginary part of the forward scattering 

amplitude, and B(e) is the nucleon-nucleon slope parameter. Values for 

these parameters, taken from various compilations, are listed in references 39-42. 

The resulting momentum transfer, equation (3.32), where the transition amplitude (3.34) appears is also complex. The real part of the 

momentum transfer, which comes from the real part of the complex optical potential, is the contribution arising from elastic scattering. It 

is purely transverse. The imaginary component, which comes from the 

absorptive part of the complex optical potential, is the longitudinal 

momentum transfer. To show this, we symbolically rewrite equation 

(3.32) as 

- (QR + i Q,) b t rans fe r (3.35) 

where i ==A and 

direction. If z denotes the beam direction (with -z the source) then 

is the unit vector transverse to the beam 

A h 

we know that 

A A 

ib=-z (3.36) 

since i -4-T is an operator which rotates a unit vector counter 

clockwise through ~/2 radians .47 Therefore (3.35) becomes 
A h 

which can be relabelled as 

--+ A h 

'transfer Ql - QII 

35 

(3.37) 

(3.38) 

The transverse (Ql) and longitudinal (Q ) components can be immediately 

written as 

II 

and 

- 

(3.39) 

- 

(3.40) 

The above has a close analogy in optics where complex refractive 

index is used to account for absorption. 

the incident wave is then along the direction of incidence. Similarly, 

momentum loss due to inelastic scatterings and absorption as the projectile traverses through the target is in the longitudinal direction, 

while the elastic scattering gives rise to transverse momentum transfer 

(specially true at high energy where forward scattering dominates). 

The resulting attenuation of 

IIIb. Energy Transfer 

Having developed the formalism for momentum transfer, we shall 

also find an expression for energy transfer (excitation energy) to the 

projectile. Again, our basis will be the multiple scattering theory 

framework discussed previously. We will assume in light of the discus- 
36 

sion in Appendix B that the sudden approximation is valid, i.e., collision time tcoll is short compared with the period of orbital motion of 

nucleons in the nucleus, 

tion value of the sum of nucleon energy operators (the internal 

Hamiltonian of the projectile HP) with approximate wave functions given 

by (3.14). 

This is best evaluated by taking the expectaExpanding the above yields 
37 

r 1 

(3.42) 

The explicit derivation of the above expressions is detailed in 

Appendix A. The final result for energy transfer is 

with 

-r + 

1 

P 

+ 0 (,I (3.43) 

(3.44) 

Evaluation of the above equation for E* yields an expression for energy 

transfer as 

Q 

LL 

(3.45) 

where the definition of Vopt, equation (3.29) has been used, and we 

neglect terms O(l/Ap). 

previous derivation in section IIIa. 

All the symbols above have been defined in our 
CHAPTEB IV 

NUMERICAL RESULTS ON XoKE" AND ENERGY TRANSFER DISTRIBUTIONS 

In this chapter, we address specifically the momentum and energy 

transfer distributions in nuclear fragmentation reactions. Having 

studied in Chapter I1 the multiple scattering theory of nucleus nucleus 

collisions and in Chapter I11 the formulation of the problem of energy 

and momentum transfer in heavy ion collisions based on the multiple 

scattering theory, we shall discuss a variety of topics in heavy ion 

reactions with these insights. First, we briefly review the experimental situation as it pertains to our topics of interest. We next discuss the currently available theories of heavy ion fragmentation 

reactions. We present calculations on the momentum and energy transfer 

distributions in relativistic heavy ion collisions and indicate how to 

incorporate our results in the existing theoretical framework. Our 

primary focus will be on the momentum spectra of fragments, including 

the effects of momentum transfer on the spectra. Input parameters are 

briefly discussed next. 

widths are presented for a variety of projectile-target combinations 

and compared with experiment. 

Numerical results on momentum "downshifts" and 

1V.a The Experimental and Theoretical Background 

Most of our information on fragmentation reactions comes from 

single particle inclusive reactions of the type 48-51 

38 
39 

Beam + Target + Fragment + Other Products (4.1) 

Two concepts stand out in describing the projectile (or target) fragmentation data. They are called "limiting fragmentation" and "factorization," respectively. "Limiting fragmentation" implies that 

distribution of fragments approaches a limiting form as the bombarding 

energy is increased. "Factorization" means that the cross section for 

production of a particular fragment (written as UFBT) is a product of a 

F factor TT which depends on the target and VB which depends on the beam 

and fragment, i.e. 

F 

FBT = 'T 'B a (4.2) 

Further analysis52 suggests that another prescription, called "weak 

factorization" could better describe the data 

FT 

FBT 'B 'B a (4.3) 

T where 'B depends both on the beam and the target. 

Experimental measurement^^^ of widths and means of longitudinal 

and transverse momentum spectra of fragments of 12C and l6O on targets 

ranging from H through Pb revealed the following features, In the 

projectile rest frame, the longitudinal momentum distributions for all 

fragments from 12C and l6O (with the exception of protons) show, irrespective of beam energy and target, the following characteristics: 

(a) Gaussian shape, with root mean square widths u(P ) = SO to II 

200 MeV/c and downshifts <P > 9 - 20 to - 130 MeV/c. Thus the mean 

velocities of the fragments are less than that of the beam. 

II 

(b) Rms widths a(P ) and a(P1) are equal to an accuracy of lo%, II 

consistent with isotropic production of fragments. 
8 

I 

8 

8 

8 

I 

'8 

I 

8 

8 

I 

I 

8 

1 

8 

I 

I 

8 

I 

40 

(c) o(P ) and <P > are independent of target mass and beam energy II II 

but depend on the beam and fragment. 

The general trend of the widths o(P ) is reproduced by a parabolic II 

dependence on the fragment mass where 

(4.4) Fragment mass number I 

AF 

O(PJJ) = 200 [x(l-x)l , XI- 1/2 , 

A Projectile mass number P 

(4.4) Fragment mass number I 

AF 

O(PJJ) = 200 [x(l-x)l , XI- 1/2 , 

A Projectile mass number P 

where uo is a constant. 

The parabolic dependence has been explained by a variety of theoretical approaches2°-22 including the conservation of momentum. 

Feshbach and Huang20 and later Goldhaber22 pointed out that the 

Gaussian momentum distributions could be understood by treating the 

fragmenting nucleus as a Fermi gas and assuming (a) momentum conservation (b) no correlation among nucleons and (c) neglect of antisymmetrization of the single particle distributions. 

model, the parabolic dependence of the widths u(P ) on fragment mass AF 

could be reproduced (as in equation (4.4)), as well as the isotropy 

u(PI1 ) = PI) could be understood. More importantly, Goldhaber2* 

also pointed out that the widths and means of momentum distributions 

could be modified according to 

Based on this 

II 

Pi -Px+-\ F 

A 

(4.5a) 

(4.5b) 

2 where ox is the unmodified Gaussian width, <2 is the modified width, 

Qx is the momentum transfer in any direction x; F and A are the frag- 
I 

8 

‘8 

8 

I 

I 

I 

I 

8 

I 

I 

8 

8 

I 

I 

I 

8 

1 

1 

41 

ment and parent mass numbers respectively and unmodified widths are 

related to Fermi momentum PF of the parent as 

2 

F(A-F) ’F 

A 5 

I - 

The longitudinal momentum downshift is given by (4.5b) as 

F 

ApII - - pII - Qll 

(4.6a) 

(4.6b) 

(4.7) 

where Q is the magnitude of the longitudinal momentum transfer 

obtained from equation (3.40). Recalling that Q is a function of 

II 

II 

impact parameter, an appropriate method for choosing it for each fragment is necessary. Before that however, we shall present input parameters required in our calculation. (Note the comprehensive list of 

references 53-88 on the experimental and theoretical overview of the 

subject). 

1V.c. Input Parameters 

Having derived expressions for momentum and energy transfer, we 

shall use them to calculate quantities of interest. 

The NN transition amplitude in coordinate space has been introduced in equation (3.34) and the notation explained. 

Values of u(e), a(e) and B(e) are discussed in references 62 and 

86, including an estimate of experimental uncertainty. Average values 

are given by 
42 

(4.8a) 

Some representative values are listed in Table 1. In addition, 

the "non-diffractive" slope parameters B(e) from these references will 

be replaced by "diffractive" slope parameters 

B(e) - [lo + .5 In (S'/So)] (G~V/C)-~ (4.9) 

with So = 1 (GeV/c)'2 and S' 

gy. This is because diffractive slope parameters are appropriate for 

scattering near forward directions. 

related to the nuclear wave function by 

square of the NN center of mass enerThe single particle densities are 

(4.10a) 

(4. lob) 

for the projectile and target, respectively. The above are understood 

as 

(4.11a) 

(4.11b) 

where Zp, M, are the profectile proton and neutron numbers and ZT, NT 

are the target proton and neutron numbers, respectively. We assume for 
18 

8 

I 

I 

8 

8 

8 

8 

8 

8 

I 

8 

I 

I 

8 

8 

8 

I 

1 

43 

Table 1. 

for various incident energies. 

62,86. 

Values of NN input parame::ters o(e) for NN - np and NN-pp 

Data from compilations in references 

(A GeV) E inc 

.025 

.050 

378.255 

140.38 

130.31 

41.00 

.075 98.35 39.63 

.loo 

.125 

71.73 

59.15 

27.23 

23.83 

.150 52.09 24.13 

,175 

.200 

.225 

.250 

.275 

.300 

.400 

.500 

.600 

.700 

. aoo 

.goo 

1.000 

2.000 

3.000 

46.59 

42.80 

40.09 

37.94 

36.29 

35.15 

34.03 

34.82 

36.03 

36.98 

37.84 

38.76 

39.68 

42.96 

43.19 

23.79 

23.09 

22.45 

22.21 

22.37 

22.78 

25.19 

32.46 

39.28 

43.02 

45.51 

47.01 

47.65 

45.18 

42.50 
44 -L + 

light nuclei Pn (r) = pp (r) since Coulomb repulsion by protons plays 

only a small role in light nuclei. Proton densities are taken from 

compilation of charge radii from electron scattering experiments. The 

effect of finite proton radius is taken into account by extracting the 

matter densities according to reference 62. The nuclear charge density 

is taken as 

(4.12) 

+ 

where pp(r') is the proton charge density. 

In this work, the densities of nuclei for A 2 20 was taken to be 

of the Woods-Saxon type 

(4.13) 

with R the half density radius and a related to the skin thickness 

as t - 4.4a. For A < 20, Harmonic-well densities were used. The 

parameters are listed in Table 2. 

1V.c. Numerical Results 

A. Momentum Downshifts 

Experimental data on momentum downshifts are available19 for the 

fragmentation of l60 and 1% on targets ranging from H through Pb. In 

Figures 1-18, momentum transfers to 160(2.1 A GeV), 12C(2.1 A GeV) and 

12C (1.05 A GeV) projectiles are plotted as a function of impact parameter b (fm). Momentum transfer is in units of MeV/c. 

transverse momentum transfers are shown separately. 

Longitudinal and 
45 

Hw 

Hw 

Hw 

us 

ws 

ws 

us 

~ 

ws 

Table 2: Nuclear charge distribution parameters 

from electron scattering data 

(Hw - Harmonic-well; US - Woods-Saxon) 

Woods-Saxon: pc(r) - po 

Nuc 1 eus 

1 + exp (q) 

9Be 

1% 

160 

27~1 

64cu 

lo8Ag 

139h 

208Pb 

Distribution 

,611 

1.247 

1.544 

2.501 

2.504 

2.354 

2.354 

2.416 

a, fm (HW) or 

R, fm (WS) 

1.791 

1.649 

1.833 

3.05 

4.20 

5.139 

5.71 

6.624 

Harmonic-well: pc(r) - po (l + 7 (y) exp (-$) 
46 

240 I 1 

220 

0 200 

t, 

180 

\ 

c 

- 

- 

- 

Longitudinal 

- 

- 

- 

- 

- 

- 

- 

Transverse - 

201 - - ' - - - - ' - - - - - - - - 

t 40 

' 

Impact parameter, fm 

1:igtirc I: hlotiicnluiit Irnirsfer (McV/c) lo lr,0 yrojcctilc by tlio 9Ue Irrget as a 

Iiiitciioia of intpnct potnmctar (hi) In the tcactlon ''0 (2.1 AGeV) + Ofle -+ 

I'rojcclilc Frngiiieirt + X, wlrate X la unldonllflcd. l~lrrnronlc-wall 

dciisllics wcrc itsctl for hili llrc two/cciilc nnd Ilic? Inrrwt 
47 

- 200 k 

Q) ccr 

1 

k 150- c1 

- 

+J B 

100- 

4) 

0 

a 

E 50- 

Longitudinal 

Transverse 

I I 

4.8 6*2 6.0 6.0 

0' 

Impact parameter, fnr 

I:igtitc 2: Moiirciiiuin tmiufcr (MoV/c) to IQ0 projectile by the I2C target 1111 a 

function of Impact parameter (fm) in thr reacilon l60 (21 AGeV) + 

I'roJecllle Fragmctrl + X, where X Jg unfdaitlfled. Harmonic-well 

densilles were used for k~lr ilia prc$Me and the iatqrt. 

-+ 
300 

0 250 

\ 

a 200 

2 150 

c, 

E 

3 

i, 100 

i 

0 

E 50 

0 

48 

Longltudlnal 

Transverse 

1..1..1..1..1..1.. 

6.6 608 6.0 6.2 6.4 606 6.8 7.0 

Impact parameter, fm 

Figure 3: Momentum transfer (MaV/c) to l60 projeclllc by the A1 target as a 

function of impact parameter (fm) In the rercdon 

Projectile Fragment + X, where X Is unldmllfIad. Humortic-wall density 

was used for the projectile and Woods-Saxon density for the target. 

(21 ACaV) + AI 
49 

350 

k' 

2 200- 

3 

v) 

k * 

u 150: 

8 100- 

a 

0 

E 50- 

300 I 

# 250 *- t 

t r..*a-J 

7.0 7.2 7.4 7.6 7.8 8.0 8.2 0' - a a ' - 

Impact parameter, fm 

Figure 4: Montentum transfer (MeV/c) to '60 projcctlla by the CU target w a 

function of impact parameter (fm) in the reaction *% (2.1 AGcV) + CU 4 

Projectile Fragment + X, where X Is unidentifld. Harmonic-well density 

was used for the projectile and Wds-Saxon dcnslty for the target. 
50 

1 

9 

ff 

a 

300 

< 250 

- 200 k 

Q) 

(w 

id 

c, 

k 150 

E 

3 

E 

0 ' 50 

Fig. 5 

Longitudinal 

8.0 8.2 8.4 8.6 8.8 9.0 9.2 0 

Impact parameter, fin 

Momentum transfer (MeV/c) lo 1- projeclile by the hg tugct as I 

function of Impact parameter (fm) In the rclctlon ''0 (21 ACaV) + Ag 4 

Projectile Fragment + X, where X 1s unldentiflcd. Harmonic-well dcnstty 

was used lot the profcctlle and Woods-Saxon dcnsitv for the target. 
400 

350 

0 

\ 

3 8 300 

250 ccc 

i! 

k 200 

c, 

E 

E 100 

7 150 

d 

c, 

a 

i 

50 

I 51 

- Longitudinal 

0' I I I I 

Impact parameter, fm 

Figure 6: Momentum transfer (MeV/c) to % projccllle by the Pb target 8s a 

function of Impact parameter (fm) In the rerctlon % (21 AGW + Pb 4 

Projectile Fragment + X, where X Is unldenllfled. Humonic-well denslty 

was used for the profectlle and Woods-Smton density for the target. 
52 

140 

0 

\ 

$120 

i k 801 

GlOO 8! 

c, 

60- 7 

E 8 40- 

+Ir 

8 

- 

- 

- 

Tr an aver so 

n I I I 

v 

4.6 4.8 5.0 s.2 6.4 

Impact parameter, fm 

1:igure 7: Momentum transfer (MeV/c) to lyC proJectlle by the ba tnrget as a 

funcllon of Impact parameter (fm) In the rektlon lyC (1.05 AGcV) + 9Be 

-+ Projectile Fragment + X, where X IS unidentified. Hannonlc-well 

densilfe were used for both the projectile and the target. 
53 

160 I 

140 

0 

\ * 

# lZ0: 

L 

~ 100- 

80 

60 

- 

- 

- 

2 

40 

20 

Tr a n sve me 

' 

0' I I 1 I 

- 

4.8 6.0 6.2 8.4 5.6 6.8 

Impact parameter, fm 

I:igiite 8: Moinenlum transfer (MeV/c) lo *e proJecllle by 1110 lYC trrgct IS r 

functloir of Impact perrmcler Urn) In llre rercllon 1v (1.0s AGeV) + 

I2C+ Projectile Frngment + X, where X Is unldcntlflcd. Harmonic-well 

densilies were used for both llre projecille and the target. 
54 

- 60 

40 Transverse 

- 

- - 

20 :

I I I 

6.0 6.2 8.4 0.6 0 

5.8 

Impact parameter, fm 

I:ip,cire 9: Momentum transfer (MeV/c) to prowtlle by the IAl target 89 8 

firnctlon of lmprct parameter (fm) In the reactton 'k (1.05 ACeV) + 

"AI 4 Projectile Fragment + X, where X IS unldentlfled. Harmonlcwell density WIB used for the pro)ectlle, and Waods-slxon was used for 

[lie target. 
55 

180 

160 

0 

\ $ 140- 

L 120 

z 

8! 

k 

H 100- 

8 

k 

c, 

80 

a 

3 * 60- d 

4 40: 

20 

0- 

I 

I 

I 

C 

# 

T 

- Longltudlnal - 

- 

- - 

- 

- - 

- 

Tr a n svo r @e - - 

- - 

1 I I I 

a 

I 

31 

I 

P 

E 

Impact parameter, fm 

I:ip,ute 10: Momentum transfer (MeV/c) to "C projectile by the Cu target as a 

function of impact parameter (fm) In the rcactlon ** (t.oS MeV) + Cu 

4 Projectile Fragment + X, where X le unldmtlftd. Hannonlc-well 

density was used for the projectile, and Wds-Saxon WlS used for the 

t nrge t. 
180 

20 

160 

0 

- - 

I I I I 1 0 

- 120 k 

Y 

80 

56 

Longltudlnal 

Impact parameter, fm 

- .. . . .. ._. . . 

1:igure 11:’ Momentum transfer (MeV/c) to lYC projecllk by the Ag target 16 (I 

function of impact parameter (fm) In the readlon 12C (1.05 ACCV) + Ag 

density was used for the ptofoctlle, and Woads-Saxon was used for the 

I a tgc t. 

. 4 ProJeclile Fragment + X, where X Is unldentlfled. Harmonlc-well 
57 

200 

180 

%60’ 3 

Q) 

c 140 

ccc Lo 

9100 k 

8 

4 40: 

VI 

CI 

E 

3 

CI 

60- 

20 

O+ 

- Longitudinal - 

- 

- - 

- 

- 

- 

- 

Tranrvarso - 

;

- - 

I 1 I 

Impact parameter, fm 

.. 

FIg. 12 Momentum Ircrnsfer (MeV/c) to I2C ProJectIk by the Pb target or a 

function of impact parameter (fm) In the readIan (1.05 ACCV) + Pb 

-+ Projectile Fragment + X, where X Ir unldentlfled. Harmonic-well 

density was used for the proJectlle, and Wdr-Saxon war used for the 

larget. 
160 [ I 

20 I 5 1 I i 

4.6 4.8 5.0 5.2 s.4 

Impact parameter, fm 

Figure 13: Momentum transfer (MeV/c) lo 12C projcclile by the Da target as I 

functlon of Impact parnmetcr (fm) In tho raoctlon (2.1 AGeV) + Otle 

4 ProJectlle Fragment X, where X Is unldentlfld. Hannonlc-well 

densitlm were used for both the profcctlle and the target. 
160 

59 

Impact parameter, fm 

Figtire 14: Momentum transfer (MeV/c) to ** proJecllk by the lk target as a 

fmcllon of Impact parameter (fm) In the nactlon lk (2.1 ACeV) + 

lYC+ Projectile Frngmcnt + X, where X Is unldentlfld. Hnrmonlc-well 

densilles were used for both the projectile and the target. 
I 60 

I 

Longltudlnrl 

Impact parameter, frn 

Fig. 15 Momentum transfer (MeVlc) to proJectlle by the nAI target as a 

function of Impact parameter Urn) In the reactlon 

4 Ptojeclile Fragment + X, where X Ir unldmtlfled. Hamonle-well 

density was used for the proJmtIle, and Wdr-Saxon WIB used for the 

tatEet. 

(2.1 ACeV) + "AI 
61 

Transverse 

60 

40 

20 

7.0 7.2 7.4 7.8 7.8 8.0 0 

- 

- 

- - 

1 I a I 

Impact parameter, fm 

Figure 16: Momentum transfer (MeV/c) lo 

functlon of Impact parameter (fm) In the rarctlon 1% (2.1 AGcV) + Cu 

4 Projectlle Fragment + X, where X IS unldmtlfled. Harmonic-well 

density was usad for tlw projec!lle, and Wds-Sakon was used for the 

I arge t. 

proJedlle by the CU target as I 
62 

60 

40 

160 

- 

- 

k'120 

Q) w 

v) 

k 

9 100 

Y 

80 

Transverse 

20 I I 1 I I 

8.0 8.2 8.4 8.6 8.8 9.0 

Impact parameter, fm 

Fig. 17 Momentum lrrnsfer (MeV/c) to 1v pm~tlle by the Ag target 89 8 

function of impact parameter (fm) In the reaction 'k (2.1 AGcV) + hg 

4 Projectile Fragment + X, where X Is unldentifled. Harmonic-well 

density was used for the ptoJoclfle, and Woods-Saxon was used for the 

target. 
200 

180 

m 

120 

100 

80 

60 

40 

9.6 9.8 10.0 10.2 10.4 

Impact parameter, fkn 

Fig. 18 Momentum transfer (MeV/c) to 12C projectile by the Pb tuget as a 

function of impact parameter (fm) In the reaction 'k (21 AGeV) + Pb 

Projectile Fragment + X, where X Is unfdentifled. Hmnonlc-well 

density was used for he projectile, and Woods-Saxon wag used for the 

target. 
. 

64 

From the figures, two features are readily apparent. First the 

longitudinal momentum transfer is larger than the transverse, indicating the primarily absorptive nature of nuclear collisions at this 

energy. Second, the predicted momentum transfers decrease rapidly with 

increasing impact parameter. 

Another feature that stands out is that momentum transfers are 

only slightly target dependent. The choice of the impact parameters 

was of consequence in this and will be discussed shortly. The longitudinal momentum transfer to l60 rises from -230 MeV/c for 9Be target to 

-320 MeV/c for 208Pb target at the closest impact parameters (see 

figures 1 and 6). 

-90 MeV/c to -120 MeV/c. The projectile dependence is more noticeable, 

however. 

-155 MeV/c (figure 7) for 9Be and rises to -180 MeV/c (figure 12) for 

208Pb. Transverse momentum transfer is -25 MeV/c - 30 MeV/c only for 

all targets. At incident energy of 2.1 A GeV, however, the energy 

dependence of transverse momentum transfer is noticeable. It ranges 

from -60 MeV/c for the 9Be target (figure 13) to -70 MeV/c for *08Pb 

(figure 18). 

1.05 A GeV and 2.1 A GeV. The increase in transverse momentum transfer 

is related to a(e) which is the energy-dependent ratio 

imaginary part of the NN forward scattering amplitude. 

show that while the longitudinal momentum transfer remains fairly 

constant at high energies, the transverse momentum transfer is highly 

The transverse momentum transfer increase is only 

The longitudinal momentum transfer to 12C (at 1.05 A GeV) is 

Longitudinal momentum transfer hardly changes between 

of real to the 

These trends 

energy-dependent. 
5 

8 

8 

65 

In Figures 19-21 and Tables 3-5, our results on momentum downshifts are reported. Experimental datal9 have been averaged over 

isotopes using 

+ u <P > +... 

ave I uF1 <' (I>F1 F2 llF2 

<p~~' exp UF1 + UF2 +... 

(4.14) 

where OF^, UF2 are the fragmentation cross sections for isotopes 1 and 

2, respectively, and <P >~1, <Pll>~2 

(experimentally observed). 

19-21. To translate the calculated longitudinal momentum 

transfers into "Momentum downshifts," we follow the following prescripare their corresponding downshifts 

These calculations are presented in Figures 

II 

tions. 

(1) The semi-empirical fragmentation code developed at 

NASA Langley is used to assign a range of impact parameters bl - b2, 

b2 - b3, for each fragment AF - 1,2,3 ...( Ap-l) where AF, Ap are the 

fragment and parent mass numbers, respectively. This range is divided 

into - 30 intervals and corresponding longitudinal momentum transfer 

calculated for these values of impact parameters. 

age is done and the average is multiplied by the Goldhaber factor. 

This is the "momentum downshift" for fragment AF. Standard deviation 

of the mean is computed following standard procedures. 

An arithmetic aver- 

(2) An impact parameter b is uniquely assigned by NUCFRAG for 

each fragment AF where AF = 1,2 ...( Ap-l). 

nal momentum transfer is calculated. 

transfer is taken as the same as the above. This average, multiplied 

by the Goldhaber factor (Eq, (4.7)) is the "momentum downshift" for 

that fragment. 

The corresponding longitudiThe average longitudinal momentum 

The results are compared from the two approaches. 
8 

I 

8 

8 

1 

8 

8 

I 

A Theory 

60 

0 

\ 

80 

70 

4 

tn 

- 

- 

A 

Ai A A A 

A 

Fragment maas number 

Figure 19: Momentum downshifts of proJectile frrgmenlr In the reaction I2C 

(1.05 AGcV) + Target ProJectile Prr 

well denshim were used for A < 20, Woodr-Won for A > 20. 

ent + X, where X Is 

unfdcntifled and Iargab ate 'Ue, 1% r AI, Cu, 4 and Pb. Harmonic- 
1 

8 

a 

u 40: 

30- 3 8 20- 

10 

I 

I 

1 

e 

- 

67 

100 I 

0 

0 

0 Theory 

I Experiment 

00 

I 

0 

0 

-2 3 4 5 6 7 8 9 101112 

Fragment maam number 

Fig. 20 Momentum downshifts of projectile fra8ments In the rerctlon I2C 

(2.1 AGeV) + Target 4 ProJcctlle Fragment + X, where X Is unldentlfIed 

and targets are 9Be0 Ik, nA10 Cu, Ag and I%. Harmonlewd1 denrltlm 

were used for A < 20, Woods-Saxon for A > 20. 
I 

I 

8 

8 

8 

8 

I 

I 

I 

8 

8 

8 

8 

I 

8 

1 

& 

'8 

~1 

90 

0 

\ + 80- 

70 

4 

- 

m 9 60- 

50- 

0 

a 40- 

Y 30: 

8 20- 

a 10- 

d 

0 

100 I 

- 

- 

. 

0 

I 

0 

0 

0 

0 Theory 

I Experiment 

0 0 0 

0 

0 

I 

0' - - - - ' - - I-- - ' - - ' - - ' - - 

2 4 6 8 10 12 14 16 

Fragment mass number 

Figure 21: Momentum downshifts of pro)cctlle fragments In the rerctlon l60 

(2.1 ACeV) + target 4 ProJectlle Fragment + X, where X Is unldentified 

and targets are Oh, I*, wAI, Cu, As and Pb. Harmonlawdl densities 

were used for A < 20, WaadeSuon for A > 20. 
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+I 

Y 

8 

d 

m 

- 

d 

rl 

+I 

Q 

rl 

OD 

? 

Y 

- 

d 

rl 

+I 

7 

8: 

d 

OD 

Y 

rl 

+I 

9 

cy 

ra 

8 

* 

+I 

cy 

OD 

- 

9 

x 

In 

+I 

i 

7 

f 

VI 

+I 

2 

3 

f 

- 

m 

In 

d 

+I 

N 

In 

Q 

'I 

? 

a 

8 

f 

+I 

+I 

$1 

c) 

I4 

c 

+ 

4 

c 

r 

$1 

d 

8 

+I 

cy 

OD 

- 

'4 

8 

In 

+I 

- 

? 

8 

In 

+I 

In 

m 

- 

? 

VI 

+I 

cy 

VI 

Q 

? 

- 

3 

d 

'I 

+I 

Q 

rl 

Q 

- 

\o 

9 

3 

+ 

U 

; 7 

C 

.I 

+ i;: . 

'4 

In 

+I 

4 

d 

Q 

4 

- 

OD 

W 

+I 

? 

i 

- 

d 

W 

+I 

W 

rl 

m 

Y 

'I 

- 

W 

VI 

+I 

? 

3 

3 

'I 

8 

- 

In 

+I 

cy 

a0 

- 

a0 

* 

+ 

n 

F 

'3 

s 

- 

a 

.I 

.I 

+ 

z 

F 

- 

c 

.. OD' I 

- 

8 

71 

cy 

W 

+I 

'I 

9 

d 

r- 

- 

c! 

r- 

+I 

3 

0 

OD 

- 

? 

I- 

+I 

'4 

8 

3 

F 

p: 

e 

In 

OD 

- 

I- 

+I 

- 

m 

\o 

4- 

v 

1c 

- 

c 

T 

U 

+ 

+I 

0 

0 

W 

? 

P wd 
72 

Extensive numerical work revealed that the second approach, while being 

more efficient yields results comparable to the first approach, not 

differing by more than 7 MeV/c. When accuracy was desired, the first 

approach was always preferred over the second. 

It may now be worthwhile to discuss the experimental results and 

theoretical calculations. It is clear from figures 19-21 that the 

momentum downshifts are overpredicted in all cases except for a few. 

The source of this discrepancy is the impact parameters obtained from 

the geometrical, semi-empirical code NUCFRAG. The projectile and 

target densities are approximated in the above code as uniform spheres 

with R 2 1.26 All3 (fm ). This is obviously an oversimplification. Electron scattering87 from nuclei reveal that nuclei possess 

diffuse surfaces. Realistic charge distributions (Woods-Saxon for 

example) take the diffuseness into account through skin thickness 

t(fm), which is a measure of the distance where nuclear density falls 

from 90% to 10% of its value. A sharp cut-off of the density thus 

neglects the extended, diffuse nuclear surface. Since realistic nuclear densities were used in the calculations of longitudinal and transverse momentum transfer, it is likely that the impact parameters from 

NUCFRAG are not very realistic, 

make our calculations compatible with uniform density calculations from 

NUCFRAG. 

An alternate procedure would be to 

This was accomplished in two steps. Realistic densities were 

replaced by uniform densities[ R- 1.26 A1/3]in the momentum transfer 

calculations. The zero-range of the two-body interaction in NUCFRAG 

was implemented in our calculation by reducing the range arbitrarily by 
8 

8 

8 

1 

I 

8 

I 

I 

8 

8 

8 

8 

I 

8 

I 

I 

8 

I 

I 

73 

B(e) a factor of five 

equation (4.9) all projectile-target combinations. This amounted to 

B'(e) -.08 fm-2 at 2.1 A GeV, for example. 

i.e. B'(e) 9 - where B(e) was given previously by 

5 

The effect on our calculation can be seen in figures 22-24. In figure 22, the imaginary part of 

the optical potential Vopt(r) for l60 - 9Be collision at 2.1 A GeV is 

plotted using Harmonic-well densities as well as uniform densities with 

variable slope parameter B(e). The first-order optical potential 

follows closely the density distribution of the nuclei (actually extends beyond due to finite range of the interaction). As uniform 

density was substituted and the range of the interaction reduced by a 

factor of 5, the shape of the potential became steeper, thus approximating a sphere with a sharp cutoff. 

momentum transfer to l60 in the collision of the 160-9Be pair (Einc 9 

The effect on the longitudinal 

2.1 A GeV) can be seen in figure 23. Compared with the realistic 

density calculations (see also figure l), the magnitude of the momentum 

transfer is reduced. Setting B(e) = 0 outright involved numerical 

difficulties. 

The momentum downshifts of Oxygen fragments are plotted and 

compared with experimental datal9 in figure 24. The experimental data 

have been averaged according to (4.14). Notice the significant agreement between theory and experiment. 

fragmentation at 2.1 A GeV and 1.05 A GeV. 

Similar results obtain for I2C 

It may be appropriate to point out that the above procedure can be 

reversed and theoretical calculations utilized as an impact parameter 

"gauge". With experimentally observed downshifts as inputs, one can 

calculate with realistic densities the impact parameters where the same 
8 

8 

t 

8 

8 

8 

I 

8 

1 

1 

I 

8 

I 

I 

I 

I 

8 

I 

8 

a - 100- 

3 

3 

- 

74 

450 I 

n 

k 

J 

a I 

501 

0 Harmonic well 

Uniform 

0 Uniform (B(e)/2.) 

Uniform (B(o)/~.) 

0 

0 

b 

Q 

Radial Separation r, fm 

1:ig. 22 Imaginary part of the optlcal potentlal'Vopt(r) In MaV 19 a function of 

separatlon r In fm for Oxygen with Berylllum at 2.1 A GeV. Harmonic 

well densttier were used for both the projectile and the tugct and 

compared with unlform denslty crlculations. Slope parameter was 

. modified to *U*(e)=U(e)/Z and B(e)=U(e)/S respectlvely to slmula!e a 

zero-range Interaction. 
75 

300 

0 

3250 

# 

A Harmonic well 

A Uniform (B(e)/l.) 

0 Uniform (B(e)/2.) fi 

rn Uniform (B(e)/&) 

- 0 

;zoo 

&! VI 

9 150- 

El 

c, 

c, 100 

d 

4 50- 

Q) 

E! 

Impact parameter, fm 

- 8 

=Q 

'4 

b ' 

II 

8 I 

- 

Longitudinal "8 W 

0.- - - -' - - - - - - ' - - ' - - 

FJg. 23 tongttudlnal momentum transfer In MeV/c to Oxygen by the 

Berylllum target as a functlon of impact parameter In fm. Hannonic 

well dcnslttes were used for both the proJ@ctile and the target and 

compared wlth uniform density crlculrtlons. Slope parrmeler WII 

hodifled lo B'(c)=B(c)/Z and D'(e)-B(e)/S rcspcctively to slmulrtc 8 

zero-range Interaction. 
75 

0 0 

0 

0 

0 Theory 

I Experiment 

0 

4 6 8 10 12 14 10 20 

2 

Fragment mas6 number 

Fig. 24 Momentum downshifts ol pmjtCtlla ftrgmenb In tha rerctlon l'o 

(21 ACcV) + Target 

and targets arc%, 1% SAl, Cu, A8 and Pb. Unlf~nn dauitfu were 

used for the proJectifc and the target% Slop pUUnCtCr wU modlfled 

(B(c) = B(c)/S) to rppmximrtc 8 zcr0-m~ Intemctfon. 

haJectih Ftrgmcnt + X, when X b unidenttncd 
77 

downshifts obtain. The practical advantage of these impact parameters 

is that they can be utilized in localizing a reaction event as well as 

used as guides to other calculations such as those of transverse 

momentum transfer, fragmentation cross-sections, and Coulomb dissociation cross-sections. 

The energy dependence of the momentum transfer is a quantity of 

fundamental interest. As noted previously, the NN input parameters are 

energy dependent. 

been plotted for l60 - 9Be pair at incident energies Einc = 0.2, 0.4, 

0.6, 0.8, 1.05 and 2.1 A GeV in figures 25 and 26, respectively. Note 

the gradual increase of longitudinal momentum transfer as the bombarding energy increases, attaining a "limiting" value at Etnc 11 

A GeV. The transverse momentum transfer can be readily obtained from 

the longitudinal momentum transfer by multiplying the latter by a(e), 

where a(e) is the ratio of the real to the imaginary part of the NN 

forward scattering amplitude. As is well-known, a(e) < 0 at 2.1 A GeV 

which implies a repulsive, real part of the optical potential. The 

corresponding transverse momentum transfer is from a repulsive mean 

field and is positive (according to our sign convention). At lower 

bombarding energies, Vopt(r) - - (VReal + iVImag) holds and the corresponding mean field is attractive. 

negative indicating that deflection to negative scattering angles is 

feasible. Coulomb effects have been ignored in our calculations. The 

above predictions should be verifiable in sophisticated experiments in 

the future. 

Longitudinal and transverse momentum transfers have 

Transverse momentum transfer is 
78 

C 

1..#**1*- 

.r( 

M 

! 

Figure 25: 

a 

u 50 

€3 

A 

A 

0 

B 

A 

A 

0 

B 

A 

A 

0 

0 

a 

A 

A 

0 

0 

€3 

A 

A 

0 

0 

0 2.1 A QeV 

0 la05 A QoV 

Am8 A QeV 

A a6 A QeV 

a.4 AaeV 

0.2 A QoV 

Impact parameter, frn 

Longltudlnrl momentum transfer (MeV/c) to 1- profcctlle by the We 

target aj a function of Impact parameter (fm) In the nactlon 

l%(E,,) +9Be-~ProJecdlc Frrgmmt+X. Inddent me@¶ 

E,,,=2.1,1.05,.8,.6,.4 and .2 A CcV were used. 
I 

I 

I 

I 

I 

8 

I 

I 

I 

I 

I 

1 

I 

I 

I 

I 

8 

I 

I 

1.. 1..' I--'-- a 

79 

0 

1uu 

t 

I 

50 t 

L 

Ot 

E d

li t 

0 I E -100 t 

C 

0 

A 

e 

I 

A 

0 

0 

A 

a 

I 

A 

0 

0 

A 

e 

I 

A 

0 

0 

A 

e 

A 

0 

0 

A 

e 

A 

a 

0 

A 

e 

A 

0 

0 

A 

e 

rn 

A 

0 

0 

A 

e 

I 

A 

0 

0 

0 

0 

A 

e 

A 

a 

0 

A 

e 

A 

a 

0 

A 

0 

W 

A 

A .8 A QoV 

Om6 AGoV 

ma4 AQoV 

Am2 AQoV 

Impact parameter, fm 

Figure 26: Transverse momentum transfer (MeV/cj to tQ pto)ccdle by the 9Ue 

target as a function of impact parameter (fm) in the rerctlon l60 (Elne) + 

9Uc 4 Projectile Fragment + X. fnddent energlm E, - 2.1, l.OS, 080 06, .I, 

.2 AGeV were used. 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

‘I 

80 

B. Momentum Widths 

Equation (4.5a) is Goldhaber’s result for the modification of the 

fragment momentum width due to collisional momentum transfer. 

cally, the following equations can be derived from (4.5a) 

Specifin 

2 F(A-F) 2 FL 2 

O2 + - Qll 

A 

!2 F(A-F) 2 F2 2 

Ox - Q 0 +-Qx 

A2 A 

(4.15a) 

(4.15b) 

are the momentum II ’ Q, 2, uL2 are the modified widths, Q where “il 

transfers, and F and A are the fragment and parent mass numbers. 

The first terms in (4.15a) and (4.15b) are the unmodified widths in the 

longitudinal and transverse directions given in terms of Fermi momentum 

as 

2 D 

2 F(A-F) 2 . 2 ‘F 

5 “11 A “o’“-- 0 

2 F(A-F) u2 

UL = 2 

0 A 

(4.16a) 

(4.16b) 

The extra factor of 2 in (4.16b) is due to summing over x and y 

directions. 

Monk et a1,90 measured the Fermi momentum of a range of nuclei 

via electron scattering. 

theory of longitudinal and transverse momentum transfer. 

sions for the longitudinal and transverse momentum transfer in nucleusnucleus collisions are given in equations (3.39) and (3.40). 

We have developed in the last chapter the 

The expres- 
1 

1 

c 

81 

Experimental data on the widths of longitudinal momentum distributions of projectile fragments were measured by Greiner et a1.I9 The 

widths of the transverse momentum distributions, when resolved onto the 

detector plane were found to be similar to the longitudinal widths 

u(P11) = O(PX) = dPy) (4.17) 

within 10% suggesting isotropy of fragment production in the projectile 

frame. 

Brady et al. 26 measured the widths of transverse momentum distributions of projectile fragments in the reaction 139La (1.2 A GeV) + 12C 

+ Projectile Fragment + X. These measurements were made at the Bevalac 

57 

Heavy Ion Super-conducting Spectrometer (HISS) with the MUSIC detector. 

Fragment momentum widths u(Px) were measured in the detector plane and 

fit to the expression 

(4.18) 

with variable uo. 

polated value of Fermi momentum PF = 250 MeV/c (=* ao) underIt was found that Goldhaber theory, based on interpredicted the widths. 

plying an unreasonable value for the Fermi momentum of 377 MeV/c for 

139La. Based on our theory of transverse momentum transfer, we show in 

Values of uo = 169 MeV/c was necessary, imTable 6 and Figure 27 a much improved agreement with the experiment. 

This shows that collisional momentum transfer is substantially responsible for increased transverse widths. 

These calculations were done using the following steps based on 

our expression for transverse momentum transfer. 

used a Woods-Saxon density for 139La and Harmonic-well density for 12C 

In Equation (3.39) we 
c 

c 

P 

E 

E 

1 

T 

I 

8 

1 

E 

c 

f 

.1 

E 

c 

I 

P 

I 

82 

Table 6: Transverse momentum widths ai (Ql) in MeV/c of projectile 

fragments in the reaction 139~a(1.2 AGeV) + 12C -+ Projectile Fragment 

+ X, calculated from equations (4.15b and (4.16b). The last column 

lists the modified widths. When squared, the third column yields the 

modifications due to momentum transfer. 

Pro j ectile 

Fragment 

(mass no.) 

138 

137 

136 

135 

134 

133 

132 

131 

130 

129 

128 

127 

126 

125 

124 

123 

122 

121 

120 

112.0 

157.8 

192.6 

221.5 

246.8 

269 

289.8 

308.6 

326 

342 

357.8 

372 

386.8 

398.8 

411 

423 

434 

445 

455 

15.6 

16.5 

18.0 

19.5 

21.0 

22.6 

24.36 

26.23 

28.2 

30.2 

32.6 

35.05 

37.6 

40.3 

43.15 

46.16 

49.3 

52.6 

56.04 

4p,> 

(MeV/c) 

expt fit 

(ref. 26) 

169 

238 

290 

3 34 

372 

406 

437 

46 5 

492 

517 

540 

561 

582 

602 

620 

638 

655 

671 

687 

159.2 

223.8 

272.9 

313.8 

349.6 

381.0 

410.5 

437.2 

462.0 

484.6 

507 

527.2 

548.3 

565.4 

582.8 

599.9 

615.7 

631.5 

646 
Projectile 

Fragment 

(mass no.) 

119 

118 

117 

116 

115 

114 

113 

112 

111 

110 

109 

io8 

107 

106 

105 

104 

103 

102 

101 

100 

99 

98 

97 

96 

83 

Table 6 (Continued) 

465 

475 

483.7 

492.5 

501 

509 

517 

524.3 

531.5 

538.5 

545 

551.6 

558 

564 

569.6 

575 

580 

586 

590.6 

595.4 

600 

604.3 

608.5 

612.5 

ut (b 1 

ours 

(MeV/c 1 

59.60 

63.34 

67.03 

75.7 

79.78 

84.14 

88.46 

92.9 

97.3 

102 

106.4 

111 

115.6 

125.7 

119 

123.5 

127.9 

132.2 

136.7 

141 

145 

149.2 

153 

157 

702 

716 

7 30 

743 

756 

768 

780 

791 

a02 

812.5 

822 

832.4 

842 

851 

859.6 

868 

876 

884 

891 

898 

905 

912 

918 

924 

660.3 

674.7 

687.3 

700.3 

713 

724.7 

736.5 

747.3 

757.6 

768.3 

778. o 

788 

797.5 

807.5 

814.3 

822.5 

a 30 

a39 

846 

a 54 

861 ~ 

867.5 

a74 

880.3 
Projectile 

Fragment 

(mass no.) 

95 

94 

93 

92 

91 

90 

89 

88 

87 

86 

85 

84 

83 

82 

81 

80 

79 

78 

77 

76 

75 

74 

73 

72 

84 

Table 6 (Continued) 

616 

620 

623.6 

627 

630 

633 

636 

638.7 

641 

643.6 

645.9 

648 

650 

651.8 

653.5 

655 

656.4 

657.6 

658.7 

659.7 

660.5 

661 

661.7 

662.2 

160.5 

164.2 

167.5 

170.9 

173.8 

176.7 

179.5 

182.0 

184.5 

186.7 

188.7 

190.6 

192.2 

193.6 

195 

196 

197 

197.7 

198.3 

198.7 

198.9 

198.9 

198.8 

198.6 

930 

936 

941 

946 

951 

955 

960 

964 

968 

971 

974.6 

978 

980.8 

983.5 

986 

988 

990 

992 

994 

995.5 

996.7 

997.7 

998.6 

999.2 

886 

892 

897.6 

903 

908 

912 

917 

921 

925 

929 

932.7 

936 

939 

942 

944.5 

947 

949 

951 

952 

954 

955 

955 

956.6 

957.3 
Projectile 

Fragment 

(mass no.) 

71 

70 

69 

68 

67 

66 

65 

64 

63 

62 

a5 

Table 6 (Concluded) 

4 (PX) 

(MeV/c) 

Goldhaber 

662.5 

662.6 

662.6 

662.5 

662.2 

661.8 

661.2 

660.5 

659.7 

658.7 

ut (b 1 

ours 

(MeV/c 1 

198.0 

197.6 

196.8 

196 

195 

193.8 

192.5 

191.15 

189.7 

188.0 

999.6 

999.8 

999.8 

999.6 

999.2 

998.6 

997.7 

996.7 

995.5 

994 

957.6 

957.6 

957.5 

957.2 

956.6 

955.8 

954.7 

953.4 

952 

950 
86 

800 

600 

400 

0 li 

0 Theory 

I Experiment 

b 

200 

60 

I I I 

80 100 120 

Fragment ~MS number 

140 

Figure 27: Transverse momentum widths of projectile fragment8 in the reaction 

unidentified. Harmonic-well denslty was used for IF, Woods-Saxon 

for Lanthanum. 

(1.2 AGcV) + 1w 4 Pro)eclile Fra&mmt + X, where X Is 
c 

s 

r 

E 

i 

1 

87 

where the parameters have been listed before in table 2. Root mean 

square momentum transfer QI and squared rms momentum transfer QI were 

calculated, and widths ol(Q1) were calculated using (4.15b). Since the 

experimental widths are in the detector plane and momentum transfer is 

calculated in the reaction plane, information on the azimuth of the 

reaction plane is necessary. Since such information is not available, 

we assumed that the momentum transfer 

tor plane are identical. 

in the reaction plane and detec- 
CHAPTER V 

CORRECTIONS 

Modifications to the above formalism will be addressed in this 

chapter. These are called (i) The Deceleration correction and (ii) The 

Coulomb correction. In theoretical calculations, the assumption of 

constant velocity is frequently made. Momentum transfer introduces an 

asymmetry into the problem; the assumption of constant velocity of beam 

nucleons is then strictly not valid. Our evaluation of momentum transfer in Chapter IV can only be correct if the corrections to our calculations are small. As will become apparent, this correction, related 

to deceleration of the projectile due to momentum transfer is indeed 

small at high energies (but not necessarily so at lower bombarding 

energies). 

made first. Numerical evaluation will be carried out for these corrections, labelled collectively as "The Deceleration corrections'1. 

Modification of the expressions derived previously will be 

Coulomb repulsion of the charges in heavy-ions will be treated 

next. Ever since the beginning of the science of heavy-ions, the 

interplay of nuclear and Coulomb effects has unveiled new insights. 

This is also true for the problem treated here. 

tromagnetic dissociation cross-sections at high energy reveals the 

importance of electromagnetic (EM) fields generated. The WeizsackerWilliams method of virtual quanta will be used to show the importance 

of the EM fields. 

compared with the nuclear contribution. Finally, modifications to the 

An analysis of elecMomentum transfer calculations will be performed and 

88 
89 

point-Coulomb assumption will be made in favor of a more realistic 

charge distribution. 

V.a Deceleration Correction 

The assumption of constant velocity (per nucleon) is frequently 

made in intermediate and high energy nucleus-nucleus as well as hadronnucleus collision calculations. The change in velocity is assumed 

small at these energies. Momentum transfer to the projectile, as 

evaluated in this work will enable us to evaluate the change in velocity (per nucleon) thus testing the assumption of constant velocity. We 

shall develop, in addition a self consistent framework for evaluating 

momentum transfer. 

Consider a beam with kinetic energy per nucleon of T/A GeV. Then 

the velocity per constituent nucleon is 

v - pc = c (1 - 1/72) 1/2 

with 

7-1+- ; Mn = rest energy of nucleon 

Mn 

MeV Momentum transfer AP - to the beam implies a change in velocity 

C 

AV = A@c - per nucleon. 

Mn Ap 

(5.3) 

We parametrize the changed velocity as follows 

v(2) = v + 6(2) (5.4) 

where 6(z) is a path-dependent correction with 6(z) << v. 

Previous expressions for longitudinal and transverse momentum 

QII APL- QL) II transfer read (with AP 
90 

8 

t 

r 1 

and 

L J 

(5.6) 

L J 

The bracketed expressions were z-integrable with a suitable choice 

the two-body amplitude t. 

For 6(z) << v, the following expansion can be used 

Now because of (5.4), this is not so. 

F 1 

(5.7) 

Equation (5.5) now can be written as 

Note that we have obtained a series that takes into account corrections 

to our previous expressions, (5.5) and (5.6). 

is'(5.5), as expected. 

to the asymmetry introduced as a result of momentum transfer. 

The first term in (5.8) 

Successive terms are corrections to (5.5) due 

For 6(z) 
91 

<< v, the higher order corrections should be small at high energies, 

and the series should converge rapidly. 

A reasonable ansatz for 6(z) is 

The above equation (5.9) implies 

with Av as in (5.3) and the phase shift operator is 

Also note that 

6(*) Av 

v Z'* v V 

- + - 9 - , as expected. 

Equation (5.8) can now be written as 

1 

- APiorr +. . . APII - APII 

Av where the correction, O(-) is 

V 

(5.9) 

(5.10) 

(5.11) 

(5.12) 

(5.13) 

(5.14) 

(5.15) 
92 

Numerical evaluation of the ratio appearing in (5.11), X(b,z)/X(b) 

has been carried out for various projectile target combinations for 

impact parameters ranging from 0-15 fm. 

ratio has also been checked. 

9Be (fig. 28) and l6O + 208Pb at incident energy of 2.1 A GeV. 

extreme values of impact parameter (b - 0,7 (fm) for 9Be and b - 0,11 

(fm) for 208Pb) were chosen. 

The energy dependence of this 

This ratio has been compared for l6O + 

Two 

The ratio is almost impact parameter 

independent for the Be target whereas differences of 15-20% are 

observed for the Pb target. Since the ratio - - , the above 

indicates that the z-dependent change in velocity 6(z) is more pronounced for a heavier target (*O*Pb) than a lighter one (9Be) at the 

x(b,z) S(z) 

X(b) Av 

same impact parameter. This is in accord with our physical understanding since 9Be matter density is roughly Gaussian so that the ratio is 

independent of b. For a heavy target such as 208Pb, matter density is 

more appropriately of the Woods-Saxon type so that the ratio depends on 

the impact parameter. Thus the ansatz (5.9) is physically reasonable. 

The resulting asymmetry APcorr was evaluated for l60 (2.1 A GeV) + 9Be 

and is plotted in figure 29 (the correction APcorr has been multiplied 

been multiplied by 10 for display). 

II 

I1 

It can be seen that the correction 

at impact parameters 4.52 - 5.53 fm is merely 3% or less. At smaller 

impact parameters, however, the magnitude of the longitudinal momentum 

transfer as well as the correction to it are slightly greater, indicating the impact parameter dependence of these quantities. 

Another question that we can investigate is the energy dependence 

of the correction APcorr. The energy dependence of longitudinal II 
I 

t 

1 

8 

93 

100 

0 b=I.O fm 

v 

-6 -8 -4 -3 -2 -1 0 1. 2 3 4 6 6 

Distance along beam 2, fm 

Figure 28: The ratio of phastrhift parameter (b, z)/x (b) IS I fundon of b and z 

(where z is the distance along the beam directton md b Is the impact 

parameter) in the reaction (21 AGeV) + 'Be 3 Rojeelile Frrgmgnt + X. 
0 

c, 

n 

k 

4) 

Q 

k 

0 

4 U 

Figure 

Impact parameter, fm 

29: First order correctlon to calculated !onglhrdlnal momentum transfer as 

a fundon of Impact parameter for %(0(21 A C;CV)+% 3 ProJectlle 

Fragment +X. The conectlon Pl1- has been multlplled by 10 for 

comparison purposcs. 
95 

Impact parameter, fm 

Fig. 30 Ratlo of change In velodty per nucleon to Inddent vddty per 

nucleon (as detcnnlned by he lnddcnt kun cncrgy) for 

GeV)+9Be-->Pro~ectlle Frrgment+X due to momentum transfer PI 8 

functlon of Impact parunelet. 

(2.1 A 
96 

momentum transfer has previously been displayed in figure 25. It was 

observed that longitudinal momentum transfer saturates around -1 A GeV. 

At 0.2 A GeV, for example, Longitudinal Momentum Transfer (LMT) is -160 

MeV/c at 4.52 fm for l6O + 9Be. At 3 fm, the correction APcorr is -100 

MeV/c, which is -27% of the longitudinal momentum transfer of 370 

MeV/c. At 2.1 A GeV, there is only a -10% correction at 3 fm. 

II 

Av Finally the ratio - is plotted as a percentage at 2.1 A GeV for 

The ratio is -1.6% 

V 

l60 + 9Be. Note that Av has been defined in (5.3). 

at these impact parameters and higher at 3-4 fm. At 0.2 A GeV, this 

ratio is higher as expected, again indicating that momentum transfer 

introduces relatively greater change of velocity (per nucleon) at .2 

A GeV than at 2.1 A GeV. 

These effects should be experimentally observable at intermediate 

energy heavy ion collisions, Note that the higher order corrections in 

(5.7) have not been evaluated because their magnitude is expected to be 

small at high energies. At intermediate energy, these may not be small 

so that all the higher order terms may need to be taken into account. 

V.b Coulomb Corrections 

A complete treatment of the problem of momentum and energy transfer in relativistic heavy ion collisions must take into account Coulomb 

effects. In low-energy heavy ion collisions, Coulomb effects play a 

significant role. At relativistic energies, the importance of the 

electromagnetic fields generated can be understood by analyzing the 

Coulomb dissociation cross-sections in the fragmentation reactionsg1. 

It was found that these electromagnetic dissociation cross-sections are 
97 

sometimes comparable to the nuclear contribution and may overwhelm the 

latter for high Z (where Z is the charge number) targets such as Lead 

or Uranium. This is specially true of single-nucleon knockout 

reactionsg2 of the type Ap + Target -+ (Ap-1) + X for projectiles such 

as I2C(2.1 A GeV), 160(2.1 A GeV) and l2C(1.05 A GeV) on targets 27Al, 

64Cu, Io8Ag and 208Pb. 

the Giant Dipole Resonance (GDR) 

Quadrupole Resonance (GQR) and their decay contribute primarily to the 

one-nucleon knockout cross-sections, although the contribution of 

magnetic M1 resonance has also been pointed out.93 

dure for the analysis of such Electromagnetic Dissociation crosssections is due to Fermi, Weizsacker and Williams; it is known as the 

Weizsacker-Williams method of virtual quantag4. The EM fields generated by a relativistic projectile are equivalent to two plane wave 

pulses of radiation Pi and P2 impinging on the target, Pi along the 

beam direction and P2 transverse to it. The equivalent photon spectrum 

has been derived in many texts including Jackson; it is included in the 

Appendix for completeness. It can be seen that the photon number 

spectrum scales as Z2 where Z is the charge number of the target; 

hence the overwhelming contribution for high Z targets can be understood. 

Excitation of the giant resonancesg3 such as 

and to some extent the Giant 

The common proceT T 

We shall treat Coulomb corrections at various levels beginning 

with the proton-proton interaction. This will be generalized, via the 

Weizsacker-Williams approach to heavy-ion collisions at relativistic 

energies, 
98 

Consider the two-body transition amplitude of this work for pp 

interaction. This was given by 

with K - 1/2Bpp(e) and t(o) is understood to be 

where upp(e), app(e) and Bpp(e) are the pp cross-section, ratio of the 

real to the imaginary part of the forward scattering amplitude and the 

slope parameter respectively; y is the relative separation between the 

charges. 

transition amplitude at 2.1 A GeV is 

+ 

With the parameters in Table 1 (with app(e) - -.374), the 

+ 

tpp(e,Y) z (35 - 941) exp(-1.19 Y'2) MeV (5.19) 

Momentum transfer to the target proton is semiclassically 

dz t(o) exp(-1.19 (Tf2 + z2)) - 

V 

3 

-(b2 + z2)(1.19) - MeV = (143 - 382 i) b e 

C 

For I b I - 1 fm, z = 1 fm, 

MeV AP -13.2 !! i + 35.4 - i i 

C C A 

A MeV A A 

( -13.2 b - 35.4 Z) - with ib -Z 

C 

(5.20) 

(5.21) 
8 

8 

I 

99 

MeV At a separation of -1.4 fm, an incoming proton imparts -13 - of 

The Coulomb repulsion 

C 

transverse momentum due to strong interactions. 

of the protons contributes only 

(5.22) 2e2 MeV 

bv C 

AP--=3- 

of transverse momentum to the target. However, as the separation grows 

large, the nuclear and Coulomb contributions become comparable. The 

above calculation of the Coulomb effect was performed by evaluating 

the z-integrated force in an impulsive collision. 

electric field contributes, the longitudinal field’s contribution 

Only the transverse 

vanishes due to symmetry. 

For heavy-ions, the above method for treating the Coulomb repulsion can be generalized via the Weizsacker-Williams method. 

Consider the collision of heavy-ions with charge numbers +,ZT 

respectively, with relative velocity v - @c per nucleon. The EM 

fields generated at the projectile by the target (and vice-versa) can 

be found from equation (D.l) of Appendix D with q = ZTe. The momentum 

transfer in an impulsive collision is 

(5.23) 

with the longitudinal contribution vanishing due to symmetry. 

The above equation is the basis of our calculations. For I2C and 

l6O projectiles (at 2.1 AGeV) on various targets ranging from Be 

through Pb, we have previously calculated the transverse and longitudinal momentum transfer in Chapter IV. 

in Figures 31-33 and compared with the nuclear contribution. 

Coulomb contribution is now shown 

As can be 
I 

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8 

~8 

8 

8 

8 

I 

E 

I 

I 

1 

I 

I 

I 

I 

8 

8 

100 

c 

120 

id 

k 

c, 

E 15 =r 

d 

c, 

4) 

E 10 i 

4) 

m 

4) 

k5 

b B I N N 

6.2 6.4 6.6 6.8 

Impact parameter, fm 

Figure 31: Transverse momentum transfer IMeV/c) to *' 0 ProJectlle by the 9Dc 

target due to Coulomb Intcractlon compared with the nuclear 

contributfon as I functlon of Impact parmeter (fm) at E,,,-2.1 A CcV. 
I 

8 

8 

8 

8 

I 

8 

I 

1 

8 

I 

I 

8 

I 

I 

8 

8 

8 

8 

50 0 

\ 

3 

- 40 k 

E 

i k 

* 30 

3 

d 

E 

E 2o 

* 

4) 

8 

Q) 

v) 

b 

g 10 

m 

101 

J 

I I I I 

i 

G 

7.2 7.4 7.6 7.8 8.0 0 

7*0 

Impact parameter. f’ 

Figure 32: Transverse momentum transfer (MeV/c) lo % I”#Clllu by llrv 

target due to Coulomb Intcractlon compared with the nuclear 

contrlbutlon as I function of Impact parameter (fm) at I3,,=2.1 A CeV. 
200 

180 

0 

\ 

E 

8 

-160 k 

P40 

5 120 

EI 

3100 

a 8 80 

0 

60 

+Ir 

Q) 

Q1 

k 40 

4) 

> 

- - n 

u C 

c Coulomb 1 

c: 

Figure 33: 

102 

I 

0' I I I I I Y 

10.66 10.86 11.06 11.28 11.48 11.66 

Impact parameter, frm 

Transverse momentum trrnsfn (MeV/c) to '"0 Projdle by the Pb 

target due to Coulomb lntcrrcdon mmprred with the nudew 

contributlon 19 a functlon of Impact prruncter (fm) 8t E,,=2.1 A GcV. 

I 
I 

~8 

l8 

8 

I 

8 

I 

8 

I 

I 

8 

I 

I 

8 

I 

8 

I 

8 

i' 103 

readily seen for heavier targets, Coulomb effects are non-negligible 

and may modify the transverse momentum spectra of fragments substantially. Note that an impact parameter cutoff bmin was included in the 

calculations. 

from the Coulomb. 

This was done to separate the strong interaction effects 

Inherent in the Weizsacker-Williams approach is the point-Coulomb 

assumption for the heavy-ions. For impact parameters b > bmin this is 

a reasonable assumption. However, for collisions with b < bmin where 

bmin - k.1 (Projectile) + R0.1 (Target) (i.e. the sum of 10% charge 

radii of the projectile and target), the point Coulomb assumption 

becomes questionable. This can be seen in the departure of the potential for a unit charge in a uniform charge distribution from that of 

the point-Coulomb potentialg5 i. e. 

, r>R (5.24) 

r 

with R the uniform charge radius of the distribution. This departure 

in the potential has been plotted in figure 34. The corresponding 

momentum transfer is reduced in magnitude as the overlap increases. 

For heavy ions described as two uniform charge distributions, extension 

of the above argument throws into doubt, for b < bmin, the WeizskkerWilliams approach based on the point-Coulomb field. 
I 

8 

I 

8 

I 

I 

8 

8 

8 

I 

I 

I 

I 

I 

I 

I 

I 

I 

8 

P 

3 0.30- 

0 

0 

0.25 

c, 

d 

0 

PI 0.20 

E 

.r( 

n 

r-4 

- 

- 

104 

II 6 0.401 

4 t t - N 0.36 

" * 0.15 

0 

cw " 0.10 

t Q) 

2 0.0s 5 

Point Coulomb 

-dial distance r, fin 

Figure 34: Departure of the Coulomb potential felt by an Incident cliarge witltln a 

uniform charge distribution (mPb, 2-82) of rrdlur R-1.26(A0*(1 /3)) 

from the point Coulomb fleld. The constant C In Veeu,(r)=C/r has been 

set to unity. 
CHAPTER VI 

SUlMARY 

In this work, an optical model description of momentum (longitudinal and transverse) transfer and energy deposition in relativistic 

heavy-ion collisions has been presented within the multiple scattering 

theory framework. 

been evaluated and compared with experiment for various projectiletarget combinations. Momentum "downshifts" of projectile fragments in 

the collision of I2C (2.1 AGeV) , l6O (2.1 AGeV) 

with targets ranging from 9Be through 208Pb have been evaluated using 

the above theory and compared with target-averaged data from experiments. Transverse momentum widths of Lanthanum fragments in the reaction of 139La (1.2 AGeV) with I2C have also been calculated. 

Quantitative as well as qualitative agreement has been found with 

experiment. Thus the theory of this work can account for many features 

of heavy ion momentum spectra in a comprehensive fashion. 

Longitudinal and transverse momentum transfer have 

and 12C (1.05 AGeV) 

Energy transfer calculations can also be undertaken using the 

optical model description along the above lines. 

significant because energy transfer determines the various channels 

available for de-excitation (in the two stage Abrasion-Ablation model 

of relativistic heavy ion collisions). 

the future will hopefully address this question. 

tions are already in progress in this respect and await comparisons 

with future experimental measurements. 

Such calculations are 

Sophisticated experiments in 

Theoretical calcula105 
106 

We have provided, through "The Deceleration correction," a method 

for investigating and correcting the oft-used constant velocity assumption. Momentum transfer to the projectile could invalidate such an 

assumption. Based on our calculations, we find at relativistic speeds, 

that this change is minor. Specifically, we found that the first-order 

correction to our calculations were small at high energies (v = c). 

These corrections, however, were found to be substantial at lower 

bombarding energies, provided the impulse approximation remains valid 

at these energies. 

The very important question of Coulomb effects has also been 

addressed. This has been done within the Weizsacker-Williams method of 

virtual quanta. We find a substantial momentum transfer (transverse) 

due to repulsion of the charges. 

butions in place of the point-Coulomb assumption made in the 

Weizsacker-Williams approach resulted in reduction of the magnitude of 

momentum transfer as the collision impact parameters grew smaller. 

Use of more realistic charge distriThe future directions for research in this area remain open. Only 

the single scattering term of the multiple scattering series has been 

utilized so far. The importance of the double-scattering term and its 

physical meaning within the context of our work remains to be explored 

in detail. 

the framework of energy-momentum conservation requires the relativistic 

theory of nucleus-nucleus interactions. Although a satisfactory theory 

of proton-nucleus multiple scattering formalism (Dirac phenomenology) 

already exists, similar approaches have had limited success for heavyions. An alternate approach may be the theory currently known as 

"Quantum Hadrodynami~s"~~ which is a relativistic field theory of 

A comprehensive theory of energy-momentum transfer within 
I 

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I 

107 

strongly interacting mesons and baryons. Semi-classical solutions for 

energy-momentum transfer can be obtained from the above theory for the 

collision of heavy-ions. Such approaches already show substantial 

promise. Future work should definitely proceed along these lines. 
REFWWCES 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

17. 

P. Freier, E.J. Lofgren, E.P. Ney, F. Oppenheimer, H.L Bradt and 

G. Peters, Phys. Rev. 74, 213 (1948). 

P. Freier, E.J. Lofgren, E.P. Ney, and F. Oppenheimer, Phys. Rev. 

74, 1818 (1948). 

H. Alfven, Nature 143, 435 (1939). 

M.M. Shapiro and R. Silberberg, Ann. Rev. Nucl. Sci. a, 323 

(1970). 

C.J. Waddington, Prog. Nucl. Phys. 8, 1 (1960). 

C.F. Powell, P.H. Fowler, D.H. Perkins, "The Study of Elementary 

Particles by the Photographic Method," (Pergamon, London, 1959). 

H.L. Bradt and B. Peters, Phys. Rev. 74, 1828.(1948); H.L. Bradt 

and B. Peters, Phys. Rev. z, 1779 (1949); H.L. Bradt and B. 

Peters, Phys. Rev. 80, 943 (1950). 

Y. Eisenberg, Phys. Rev. 96, 1378 (1954). 

G. Alexander and G. Yekutieli, Nuovo Cimento 19, 103 (1961). 

G.Alexander, J. Avidan, A. Anvi, G. Yekutieli, Nuovo Cimento 20, 

648 (1961). 

P.L. Jain, E. Lohrmann, M. Teucher, Phys. Rev. 115, 643 (1959). 

Y. Tsuzuki, J. Phys SOC. Jpn., 16, 2131 (1961). 

K. Rybicki, Nuovo Cimento a, 1437 (1963). 

F. Abraham, J. Gierula, R. Levi-Setti, K. Rybicki and C.H. Tsao, 

Phys. Rev. m, 1110 (1967). 

B. Andersson, I. Otterlund and K. Kristiansson, Ark. Phys. 2 527 

(1966). 

M.G. White, M. Isaila, K. Prelec and H.L Allen, Science 174, 1121 

(1971); H.A. Grunder, W.D. Hartsough and E.J. Lofgren, Science 

174, 1128 (1971). 

F.J.M. Farley, "Speculations on Nucleus-Nucleus Collisions with 

the ISR," CERN NP Intern. Rep. 70-26 (1970), unpublished. 

108 
109 

18. A. Ghiorso, H. Grunder, W. Hartsough, G. Lambertson, E. Lofgren, 

K. Lou, R. Main, R. Mobley, R. Morgado, W. Salsig and F. Selph, 

IEE Trans. Nucl. Sci. NS-20 (3), 155 (1973). 

19. D.E. Greiner, P.J. Lindstrom, H.H. Heckman, B. Cork and F.S. 

Bieser, Phys. Rev. Lett. x, 152 (1975). 

20. H. Feshbach and K. Huang, Phys. Lett. B u, 300 (1973). 

21. 3.V. Lepore and J.R. Riddell Jr., Lawrence Berkeley Laboratory 

Report No. 3086 (1974), unpublished. 

22. A.S. Goldhaber, Phys. Lett. B 52, 306 (1974). 

23. N. Masuda and F. Uchiyama, LBL Report 4263 (1975), unpublished. 

24. G.F. Bertsch, Phys. Rev. Lett. G, 472 (1981). 

25. M.J. Murphy, Phys. Lett. B 56, 23 (1975). 

26. F.P. Brady, W.B. Christie, J.L. Romero, C.E. Tull, b. McEachern, 

M.L. Webb, J.C. Young, H.J. Crawford, D.E. Greiner, P.J. 

Lindstrom and H. Sann, Phys Rev. Lett. 60, 1699 (1988). 

27. W. A. Friedman, Phys. Rev. C, 27(#2), 569 (1983). 

28. J. Hufner, K. Schafer and B. Schurman, Phys. Rev. C. u, 1888 

(1975). 

29. S.H. Fricke, Ph.D. Thesis, University of Minnesota (1984), 

unpublished; B.J. Bayman, P.J. Ellis and Y.C. Tang, Phys Rev. 

Lett. 53, 1322 (1984). 

30. J.D. Bowman, W.J. Swiatecki and C.F. Tsang, LBL Report No. LBL2908 (1973), unpublished. 

31. K.M. Watson, Phys. Rev. 89, 575 (1953). 

32. M.L. Goldberger and K.M. Watson, "Collision Theory," (John Wiley 

and Sons, N.Y.), pp. 749 (1964). 

33. R.J. Glauber, Phys. Rev. 100, 242 (1955). 

34. V. Franco and R.J. Glauber, Phys. Rev. 142, 1195 (1966). 

35. L. Zamick, Ann. Phys, 21, 550 (1963). 

36. V. Franco, Phys. Rev. 175, 1376 (1968). 

37. 0. Kofoed-Hansen, Nuovo Cimento 60A, 621 (1969). 
I 110 

38. W. Czyz and L.C. Maximon, Ann. Phys. 52, 59 (1969). 

I 

39. J.W. Wilson, Phys. Lett. 52B, 149 (1974). 

40. J.W. Wilson, Ph.D. Thesis, College of William and Mary (1975), 

unpublished. 

41. J.W. Wilson and L.W. Townsend, Can. Journal of Physics 59, 1569 

(1981). 

42. J.W. Wilson and C.M. Costner, NASA Technical Note D-8107 (1975). 

43. L.L. Foldy and J.D. Walecka, Ann. Phys. (NY), 54, 447 (1969). 

44. R.J. Glauber, "Lectures in Theoretical Phys.," Vol. I, Ed. W.E. 

Brittin and L.G. Dunham, (Interscience Publishers, Inc., N.Y.), 

pp. 364 (1959). 

45. E. Merzbacher, "Quantum Mechanics," (John Wiley & Sons, Inc., New 

York), pp. 167 (1970). 

46. K. Gottfried, "Quantum Mechanics," Vol. I, (W.A. Benjamin, Mass.), 

pp. 240 (1966). 

47. P.M. Morse and H. Feshbach, "Methods of Theoretical Physics," 

(McGraw-Hill, New York) Part I, pp. 349 (1953). 

48. H.H. Heckman, D.E. Greiner, P.J. Lindstrom and F.S. Bieser, Phys. 

Rev. Lett. 2, 926 (1972). 

49. P.J. Lindstrom, D.E. Greiner, H.H. Heckman, B. Cork and F.S. 

Bieser, LBL Report No. 3650 (1975). 

50. D.E. Greiner, P.J. Lindstrom, F.S. Beiser and H.H. Heckman, Nucl. 

Inst. Methods u, 21 (1974). 

51. H.H. Heckman, Heavy Ion Fragmentation Experiments at the Bevatron, 

NASA CR-142589 (1975). 

52. D.L. Olson, B.L. Berman, D.E. Greiner, H.H. Heckman, P.J. 

Lindstrom and H.J. Crawford, Phys. Rev. C. 2, 1602 (1983). 

53. J. Jaros, Ph.D. Thesis, Univ. of Calif., Berkeley (1975), 

unpublished; LBL Report No. 3849 (1975), unpublished. 

54. H.H. Heckman, D.E. Greiner, P.J. Lindstrom, H. Shwe, Phys. Rev. 

C JJ 1735 (1978). 

55. G.M. Chernov, K.G. Gulamov, U.G. Gulyamov, S.Z. Nasyrov, L.N. 

Svechnikova, Nucl. Phys. A a, 478 (1977). 

56. B. Jakobsson and R. Kullberg, Phys. Ser. u, 327 (1976). 
111 

57. B. Judek, Proc. Int, Conf. Cosmic Rays, 14th Munchen, 2342 

(1975). 

58. R. Kullberg, K. Kristiansson, 8. Lindkvist and I. Otterland, Nucl. 

Phys. A 280, 431 (1977). 

59. H.L. Bradt and B. Peters, Phys. Rev. n, 54 (1950). 

60. S. Barshay, C.B. Dover, J.P. Vary, Phys. Rev. C u, 360 (1975). 

61. P.J. Karol, Phys. Rev. C JJ, 1203 (1975). 

62. J.W. Wilson and C.M. Costner, NASA Technical Note No. D-8107 

(1975), unpublished. 

63. J.W. Wilson and L.W. Townsend, Can. Journal of Physics 59, 1569 

(1981). 

64. 1L. Anderson, Ph.D. Thesis, Univ. of Calif., Berkeley (1977), 

unpublished. 

65. J. Papp, J. Jaros, L. Schroeder, J. Staples, H. Steiner, A. Wagner 

and J. Wiss, Phys. Rev. Lett. 10, 601 (1975). 

66. R.K. Bhaduri, Phys. Lett. 508, 211 (1974). 

67. M. Buenerd, C.K. Gelbke, B.G. Harvey, D.L. Hendrie, J. Mahoney, A. 

Menchaca-Roca, C. Olmer, D.K. Scott, Phys. Rev. Lett. 37, 1191 

(1976). 

68. C.K. Gelbke, M. Buenerd, D.L. Hendrie, J. Mahoney, M.C. Mermaz, C. 

Olmer, D.K. Scott, LBL Report No. 5826 (1977). 

69. C.K. Gelbke, D.K. Scott, M. Bini, D.L. Hendrie, J.L. Laville, J 

Mahoney, M.C. Mermaz, C. Olmer, Phys. Lett. 8 70, 415 (1977). 

70. V. Franco and G.K. Varma, Phys. Rev. C 15, 1375 (1977); V. Franco 

and A. Tekou, Phys. Rev. C u, 658 (1977); V. Franco and W.T. 

Nutt, Phys. Rev, C u, 1347 (1978). 

71. T.F. Cleghorn, MS. Thesis, Univ. of Minnesota (1967); Report No. 

CR- 104. 

72. T.F. Cleghorn, P.S. Freier and C.J. Waddington, Can. J. Phys. 46, 

572 (168). 

73. R. Serber, Phys. Rev. 18, 304 (1949). 

74. J. Gosset, H.H. Gutbrod, W.G. Meyer, A.M. Poskanzer, A. Sandoval, 

R. Stock and G.D. Westfall, Phys. Rev. C u, 629 (1977). 

75. L.F. Oliveira, R. Donangelo and J.O. Rasmussen, Phys. Rev. C 19 

826 (1979). 
112 

76. M. Bleszynski and C. Sander, Nucl. Phys. A m, 525 (1979). 

77. L.W. Townsend, Can. J. Phys. fi, 93 (1983). 

78. L.W. Townsend, J.W. Wilson and J.W. Norbury, Can. J. Phys. 63, 135 

(1985). 

79. L.W. Townsend, J.W. Wilson, F.A. Cucinotta and J.W. Norbury, Phys. 

Rev. C 2, 1491 (1986). 

80. M.P. Guthrie, Oak Ridge National Laboratory Technical Memorandum 

#3119 (1970). 

81. H.W. Bertini, R.T. Santoro and O.W. Hermann, Phys. Rev. C 14, 590 

(1976). 

82. R.K. Smith and M. Danos, Proc. Meeting Heavy-Ion Collisions, 363 

(1977). 

83. C.Y. Wong, Proc. 5th High Energy Heavy Ion Study, p. 61 (Lawrence 

Berkeley Laboratory Report No. LBL-12652), 1981. 

84. F. Reif, "Statistical and Thermal Physics," p. 39 (McGraw-Hill, 

N.Y. 1965). 

85. A.I. Khinchin, "Mathematical Foundations of Statistical 

Mechanics," p. 166 (Dover, NY 1949). 

86. L.W. Townsend and J.W. Wilson, NASA Reference Publication No. RP1134 (1985). 

87. R. Hofstadter and H.R. Collard, "Landolt-Bornstein Numerical Data 

and Functional Relationships in Science and Technology," Group I, 

Vol. 2, 21 (ed. H. Schopper, Springer-Verlag, 1967). 

88. C.W. DeJaeger, H. DeVries and C. DeVries, "Atomic Data & Nucl. 

Data," Vol. u, No. 5/6, 479 (1974). 

89. J.W. Wilson, L.W. Townsend and F.F. Badavi, Nucl. Inst. Methods in 

Phys. Res., B18, 225 (1987). 

90. E.J. Moniz, I. Sick, R.R. Whitney, J.R. Ficence, R.D. Kephart and 

W.P. Trower, Phys. Rev. Lett. 26, 445 (1971). 

91. J.W. Norbury and L.W. Townsend, NASA TP-2527, 1986. 

92. H.H. Heckman and P.J. Lindstrom, Phys. Rev. Lett. 36, 193 (1976) 

93. C.A. Bertulani and G. Baur, Phys, Rep. 163 (5 & 6), 1988. 

94. A.D. Jackson, "Classical Electrodynamics," p. 719 (John Wiley & 

Sons, New York, 1975). 
113 

95. M.A. Preston and R.K. Bhaduri, "Structure of the Nucleus," p. 89 

(Addison-Wesley, Mass., 1975). 

96. B.D. Serot and J.D. Walecka, "The Relativistic Nuclear Many-Body 

Problem," Vol. 16, Advances in Nuclear Physics (Plenum, N.Y. 

1986). 
APPENDIX A 

DERIVATIONS (OF HOHE" AND ENERGY TRANSFER FoRH[TLAE) 

In this Appendix, we shall go through the derivation of the final 

result for momentum transfer pdj +-ran between the ath target constituent 

and j& projectile constituent (equation (3.31)), 

and the total momentum transfer (equation (3.32)), 

--5 AT AP pj 

'tot 1 1 tran a-1 j-1 

(A. 2) 

where the symbols have been explained in the text in Chapter 3. 

Equation (A.l) was derived from the expression 

AT I AP 

where the projecti-e and target many-body wave functions 

(A. 3) 

Bp,o> and 

IgT,o> are written in terms of single particle states in a Slater 

determinant as follows 

114 
I 

i 115 

and where the single particle states have been orthonormalized. 

ground state single-particle densities are defined as 

The 

X 

AP =c 

j -1 

(A. 5) 

where c means all permutations of <pj have to be taken. For a = 

1,2 ... AT target and j = 1,2 ... Ap projectile nucleons, momentum transfer 

due to collision of aj pair is -=j P tran and the total momentum is obtained 

by summing over all such pairs. 

Explicitly, we can write 

Perm 
116 

r 1 

11 pj I - 

tran Ap! AT! 

AT pa 

1 

(A. 7) 

as claimed. 

The derivation of the energy transfer in relativistic heavy-ion 

collisions is analogous to the above derivation for momentum transfer. 
117 

We assume that the energy of the projectile can be evaluated by taking 

the expectation value of the sum of projectile energy operators between 

the full state vector (i.e. equation (3.14)) 

where Hp - Internal Hamiltonian of the projectile 

+ c "ij 

1 H I-- 

2mn j-1 fPj i<j P 

(A. 9) 

Explicitly, the above (A.8) can be written as 

Using the identity 

i2 eiABe-iA - B + i [A,B] + - [A,[A,B]] +... 

2! 

(A. 10) 

(A. 11) 
Using 

[A,BC] = [A,B]C + B[A,C] 

s, [ s, jI 92 Ep-! 

118 

can be evaluated using (A.15). 

One obtains the operator expression 

Similarly, 

r 

I 

- 

j -1 EPj 

(A. 13) 

(A. 14) 

(A. 15) +C 

L J 
119 

Combining all the terms, one obtains 

The first term in (A.16) will be recognized as the initial energy 

of the projectile Eo,proj. 

energy transfer defined as 

This can be taken to the left hand side and 

-E- Eo (A. 17) 

Etran 

The second and the fourth terms can be combined. The first term 

in the resulting expression can be converted into a surface integral by 

use of the divergence theorem. This will vanish if the single particle 

states vanish sufficiently fast at infinity. 

can be shown to contribute nothing to the excitation energy for eveneven nuclei such as 12C, l6O considered in this work. 

The remaining expression 

For odd-even 

1 nuclei, this contribution is O(-), therefore this has been ignored. 

The final result for energy transfer is then 

AP 
120 

The rest of the calculation is analogous to the momentum transfer 

calculations i.e. one obtains 

tot A~ A~ aj 

1 1 Etran Etran j-1 a-l 

(A. 19) 

where 

Eaj - Energy transfer in the collision of the a-j pair tran 2 - _-- I 

d3ip Pj(ip) I d3i, Pa(iT) VtP I-Eaj(X..C --I 2mn [- 

(A. 20) 

and 

- - 

where pj((p) and pa((T) are the single particle density of the jth 

projectile and ath target nucleon and pp(ifp), p~(if~) are the nuclear 

densities of the projectile and the target, as before. 
~ 

I 

I 

I 

I 

I 

We want to discuss how our expression for momentum, equation 

(3.20) 

We expect an analogy on physical grounds because momentum transfer in a 

collision is directly related to the gradient of the interaction in a 

one to one fashion. 

relates to Wilson's expression for potential, equation (2.41). 

The coupled equations in (2.40) 

define a potential matrix 9 with matrix elements 

with 

so that the matrix looks like (2.44) with 

po ,OO(X) 

+ <x'> voo,lo(x)' - 4 

voo, 01 

(Z). . vlo , ol(x) %o, 10 

vol , 01 (5 vol,lo(x)' - 1 3 

121 

(B. 4) 
122 

We introduced in (3.20) similar expressions for the momentum of 

the projectile and its matrix elements were defined in (3.20). 

Equation (3.20) 

operator 

defines a coupled problem because the approximate wave 

induces transition in the internal eigenstates g 

projectile and target, respectively. 

and g~,~ of the 

P ,m 

Equation (3.20) defines a matrix 

F(x). 

This expression for momentum 

L 

) 

can be compared to Wilson's expression, (B.2) above. 
I 

I 

E 

I 

123 

One can recognize that there is a close connection between the two 

expressions. This connection should be one to one, since for each 

channel defined by the eigenstates m,m' of the projectile and p,p' of 

the target, an interaction V induces a momentum transfer 

P 

Wlm'P' 

mp,m'p' * 

The Sudden Approximation 

The connection between our derivation and the more familiar 

formalism of time-dependent Schrodinger theory will be discussed 

briefly. 

Consider 

where 

Ho and Hi are the unperturbed Hamiltonian and the interaction, 

respectively. 

$,(t)> is the time dependent Schrodinger state vector and where A h 

In the "Interaction Representation" the above 

reads45146,47 

A h 

i Hot -5. Hot 

i 2 1 $,(t>> - e (-Ho+HO+H1) e 

at 

with 

(B. loa) 
A A 

A iHot A -iHot 

HI(t) - e 1 He 

The time evolution operator is defined as 

Writing the above as an integral equation, one obtains 

h 

u (t, to) - 1 - - It (t') i (t', to) dt' + ... fi to I 

On iteration, the above yields 

124 

(B.lOb) 

(B.ll) 

(B.12) 

(B.13) 

(B. 14) 

The above is a formal solution to the time-development of the 

state vector from initial time to to final time t. 

series in the strength of the interaction HI(t) and it is time-ordered, 

i.e., earlier times occur to the left of later times. 

It is a power 

A 

Explicitly, 
125 

Consider adiabatic switching of the interaction. Define46 

A 

-t t A HI(t) = e I IH1 (B. 17) 

which implies that instead of turning on the interaction at T = (t-to), 

turn it on and off slowly. As t + 2 - A AA 

and the solutions to the Schrodinger equation take the form 

A 

with Ho I$ > = E I$ > for stationary states. 

0 

Then in this limit 

which is independent of time. Therefore, 

Next, the sudden approximation will be discussed to justify 

(3.19). We can write46 

(B.18) 

(B. 19) 

(B.20) 

(B.21) 
I 

1 

'P 

I; 

I 

z 

I 

1 

I 

f 

I 

8 

1 

I 

I 

I 

I 

E 

a 

126 

rA A 

(B.22) 

It has been shown in reference [47] that a necessary condition for 

the sudden approximation to be valid is that the interactions at two 

different times commute 

rA A 1 

(B. 23) 

L J 

so that time-ordering is unimportant in (B.22). 

[471 

Following reference 

(B.24) 

which is the desired result. 

Having derived (B.24) it is necessary to relate these equations 

(B.22)-(B.24) to our previous discussion in equations in Chapter 111. 

Essential to the derivation of equation (3.19) was the commutativity of 

interaction matrices at different points along a straight line 

trajectory 

(B.25) 
I 

E 

E 

E 

E 

I 

# 

1 

I 

I 

I 

I 

E 

I 

8 

E 

1 

I 

a 

127 

In the Interaction Representation, this is entirely equivalent 

to4' 

A 

HI(tf), HI(tf') - 0 (B. 26) 

L 2 

provided one identifies dz' - vdt' and dz" - vdt" assuming constant 

velocity per nucleon. The latter can be justified on the basis of t 

high energy assumption. 

The physical basis for the sudden approximation in the high energy 

context is as follows; the collision time tcoll << tnuc where tnuc is 

the period associated with orbital motion of the nucleons in the 

nucleus. This condition 

tcoll << tnuc (B.27) 

is met in relativistic heavy ion collisions. We shall argue this by 

tcoll 

tnuc 

calculating the ratio - . For incident beam energies above 1 

GeV/N the nucleon velocity = ,9c = c. For the distance we shall 

take typical nuclear diameter = 6 fm. 

tnUc we will consider the period of nucleons in a Fermi gas i.e. 

Then tcoll = -. fm For 

C 

. For a typical nucleus with Fermi energy = 1 

tnuc - (E/h)-l - - - 

v 

tnuc 

40 MeV, the ratio is 

1 tic = 197.3 MeV.fm (B.28) 1 

t 2n (tic) 5 

=- Lcoll_ 6 fm x 40 MeV 

nuc 
128 

which is small enough to justify the Sudden Approximation. Although 

details of the nuclear diameter and Fermi energy may vary, at 

relativistic energies the ratio will be small. 

Lorentz factor in the above argument, but the conclusions remain 

unchanged because we have overestimated the Fermi energy (not all 

nucleons are at the surface) and have considered the diameter of the 

nucleus and not of the nucleons themselves. 

We have omitted the 
1 

3 

E 

I 

z 

'I 

a 

I 

s 

I 

i 

t 

t 

I 

& 

I 

II 

I 

APPENDIX c 

ANALYTICAL RESULTS USING GAUSSIAN DENSITIES 

Analytical results are useful in getting physical insights as well 

as establishing the validity of numerical calculations. We present in 

this section analytical calculations of momentum and energy transfer 

using Gaussian densities. 

transfer between ath constituent of the projectile and jth constituent 

We know from Chapter I11 that momentum 

of the target is 

where the notation was explained following equation (3.32). The total 

momentum transfer is obtained by summing over a and j as 

+ L) 

- c' c PaJ 

a1 

'trans 

L =p J-w 

A A hn AAA -P 

Define a vector G = Gb b + GI i with (X, Y, Z) = (b, I, Z) (C.3) 

with 

129 
130 

so that equation (C.l) be rewritten as 

and with 

-2 - 

t 9 - t(o) e -IC(;' + $ - FT) 

where t(o) is assumed to be real. 

* 

The components of G can be written with Ep,b (Ep,J as the 

+ b F. 

components of Fp 

the components of ET along b (and perpendicular to b). 

along b (and perpendicular to b) and ET,^ (tTI ) 

+ A h 

and 

Define the thickness function 

where 

(C. lo) 
1 

P 

i 

t Similarly 

131 

(C. 11) 

(C.12) 

where ap, aT are the root mean square radii of projectile and target, 

respectively. Therefore, 

or 

where J 

or 

where we have used (C.7), and 

(C. 14) 

(C.15) 

(C. 16) 
e 1’ 

2 On evaluation this yields (with a - 3/2aT), 

substituting (C.18) into (C.16) yields 

exp [ - &{P + fP’b)’ + L}] 

132 

(C.17) 

(C. 18) 

(C.19) 

Similarly, using the same method as in (C.16)-(C.19), we obtain 

which yields the two components in (C.15). 

(C. 20) 
8 Now we need to 

--* Ap AT + 

'trans = c c pa, 

1 

133 

evaluate (C. 14) 

2 

On evaluation this yields (a' 9 3/2ap), 

3 3 

V 2 2nap 2na~ 

Ah 

Pb b 9 b ApA~ (2K) - 

r 

n2aa'b 

(aa' + K(a+a')) 

- 

I aa' kb2 

(aa' + K(a+a')) exp 1- c - 

aa'k 

aa' + k(a+a') 

where x = [ ] 

Thus, finally, 

A A 

I -P -.) 

= c Paj -%b+Pl 

4 'trans 

A 

- Pb b and PI 9 0 due to symmetry 

(C.21) 

(C. 22) 

(C.23) 

(C.27) 

which is the analytical result for momentum transfer. 
;I 134 

The above result holds for only real two-body amplitudes t(o). For 

complex t(o), there is a longitudinal component of momentum transfer as 

we have shown in Chapter 111. 

the latter case. 

The above result can be generalized to 
APPENDIX D 

THE EQUIVALENT PHOTON SPECTRUn 

In this Appendix, we shall derive via the Weizshker-Williams 

method of virtual quanta the equivalent photon spectrum generated by 

the target at the projectile. 

For an incident particle of charge q, velocity V = p(c) passing 

a system S at an impact parameter b, the spectrum of equivalent 

radiation is obtained from the electromagnetic fieldsg4 

- 

where El(t), E2(t) are the electric fields along the beam direction and 

transverse to it and B3(t) is the accompanying magnetic field. For V 

= c, the fields El(t), E2(t) and B3(t) are completely equivalent to 

plane wave pulses of radiation Pi and P2 incident on S, Pi along beam 

direction, P2 transverse to it. The equivalency is not exact for P2, 

since there is no magnetic field accompanying El(t). If the 

motion is nonrelativistic in the frame S, then the particles in this 

frame respond to electric forces only so that one can add an extra 

magnetic term to the fields without affecting the results. This field, 

Bl(t) = p El(t) will be shown to be of minor importance in calculations. 

135 
136 

The equivalent frequency spectrum (energy per unit area per unit 

frequency interval) of the pulses Pi, P2 are given by 

with 

W 

E2(w) - 2 I E2(t)eiwt dt -W 

and similarly for El(w). Explicitly, these Fourier integrals are 

7Vt 

-6 

i wbx/yV dx 

; xe 

(1 + x2) 3/2 

r 1 

r 1 

(D.5) 

where KO(K1) are the modified Bessel functions of zeroth (first) order. 

The frequency spectra are 
137 

and 

These have been plotted in reference 94 (Figure 15.7). The 

1 

- and is of minor 

2 

To obtain the energy incident per unit freintensity of pulse P2 involves a factor of 

7 

importance for V = c. 

quency interval, one sums the frequency spectra over all impact parameters. This is 

(D. 10) 

dw J dw 

where b,in is a minimum impact parameter beyond which other interactions take over (strong interactions in heavy ion collisions, for 

example). The result is 

For low frequencies 7v 

bin 

w << - , equation (D.11) reduces to 
138 

I 

i 

8 

I 

1 

1 

I 

I 

I 

I 

I 

8 

I 

I 

(D.12) 

For high frequencies (w >> TV/bmin), the result is an exponential falloff 

The number spectrum of virtual quanta is 

dI (w) 

dw 

- dw 9 N(fiw) d (fiw) (fiw) - E N(E)dE 

(D.13) 

(D.14) 

so that 

(D.15) 

Wbmin 

,XIe2 For q - Z e, a -- (D.17) 

From (D.9), the equivalent frequency spectrum for pulse P2 is 

shown to contain a factor - . At relativistic speeds, this pulse p2 1 

2 

7 

is , therefore, of negligible importance. 

for adding a magnetic field Bl(t) 9 B El(t). 

This was the justification 

The time integral of the 
139 

field El(t) 

contribute to momentum transfer. The frequency spectrum confirms the 

above and justifies our retaining the transverse component in (5.21). 

was shown to yield zero by symmetry, so that this did not 
1 

8 

1 

I 

AUTOBIOGRAPHICAL STATEMENT 

Ferdous Khan was born on the 20th of June, 1957 in Chittagong, 

Bangladesh, the son of Yunus Khan and Khaleda Khan. 

He was graduated from Favjdarhat Cadet College in Chittagong, 

Bangladesh in 1975. 

Connecticut from 1977-1981 and earned a Bachelor of Arts degree in 

Physics. 

University of Rhode Island at Kingston, R.I. 

studies at Old Dominion University in the Fall of 1984 under NASA 

research grant NCCI-42. Ferdous has coauthored -10 publications in 

heavy-ion physics, radiation physics as well as atomic theory. 

work as a postdoctoral research associate at Old Dominion University 

and the High Energy Science Branch at NASA Langley Research Center from 

the Fall of 1989. 

He then attended Yale College in New Haven, 

He earned his Master of Science degree in Physics from the 

He started his graduate 

He will 

140