source: trunk/documents/UserDoc/UsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/AbrasionAblation/AbrasionAblation.tex

\chapter{Abrasion-ablation Model}

\section{Introduction}
\label{Intro}
The abrasion model is a simplified macroscopic model for nuclear-nuclear 
interactions based largely on geometric arguments rather than detailed 
consideration of nucleon-nucleon collisions.  As such the speed of the 
simulation is found to be faster than models such as G4BinaryCascade, 
but at the cost of accuracy.  The version of the model implemented is 
interpreted from the so-called abrasion-ablation model described by Wilson 
{\normalsize\it{}et al} \cite{aaWilson},\cite{aaTownsend} together with an 
algorithm from Cucinotta to approximate the secondary nucleon energy 
spectrum \cite{aaCucinotta}.  By default, instead of performing an ablation 
process to simulate the de-excitation of the nuclear pre-fragments, the Geant4 
implementation of the abrasion model makes use of existing and more detailed 
nuclear de-excitation models within Geant4 (G4Evaporation, G4FermiBreakup, 
G4StatMF) to perform this function (see section \ref{deexcitation}).  However, 
in some cases cross sections for the production of fragments with large 
$\Delta$A from the pre-abrasion nucleus are more accurately determined using a 
Geant4 implementation of the ablation model (see section \ref{ablation}).


\noindent The abrasion interaction is the initial fast process in which the 
overlap region between the projectile and target nuclei is sheered-off (see 
figure \ref{fig:1})  The spectator nucleons in the projectile are assumed to 
undergo little change in momentum, and likewise for the spectators in the 
target nucleus.  Some of the nucleons in the overlap region do suffer a change 
in momentum, and are assumed to be part of the original nucleus which then 
undergoes de-excitation.


\noindent Less central impacts give rise to an overlap region in which the 
nucleons can suffer significant momentum change, and zones in the projectile 
and target outside of the overlap where the nucleons are considered as 
spectators to the initial energetic interaction.


\noindent The initial description of the interaction must, however, take into 
consideration changes in the direction of the projectile and target nuclei due 
to Coulomb effects, which can then modify the distance of closest approach 
compared with the initial impact parameter.  Such effects can be important for 
low-energy collisions.


\section{Initial nuclear dynamics and impact parameter}
\label{InitDyn}
For low-energy collisions, we must consider the deflection of the nuclei as a 
result of the Coulomb force (see figure \ref{fig:2}).  Since the dynamics are 
non-relativistic, the motion is governed by the conservation of energy 
equation:

\begin{equation}
E_{tot}  = \frac{1}{2}\mu \dot r^2  + \frac{{l^2 }}{{2\mu r^2 }} + \frac{{Z_P Z_T e^2 }}{r}
\label{eqn1}
\end{equation}
%Equation ??.1

\noindent where:

\(E_{tot}\) $\>$ = total energy in the centre of mass frame;

\(r\),\(\dot r\) $\>$ = distance between nuclei, and rate of change of distance;

\(l\) $\>$ = angular momentum;

\(\mu\) $\>$ = reduced mass of system {\normalsize\it{}i.e.} \(m_1m_2/(m_1+m_2)\);

\(e\) $\>$ = electric charge (units dependent upon the units for \(E_{tot}\) and \(r\));

\(Z_P\), \(Z_T\) $\>$ = charge numbers for the projectile and target nuclei.

\noindent The angular momentum is based on the impact parameter between the 
nuclei when their separation is large, {\normalsize\it{}i.e.}

\begin{equation}
E_{tot}  = \frac{1}{2}\frac{{l^2 }}{{\mu b^2 }}\; \Rightarrow \;l{}^2 = 2E_{tot} \mu b^2 
\label{eqn2}
\end{equation}
%Equation ??.2

\noindent At the point of closest approach, \(\dot r\)=0, therefore:


\begin{equation}
\begin{array}{l}
 E_{tot}  = \frac{{E_{tot} b^2 }}{{r^2 }} + \frac{{Z_P Z_T e^2 }}{r} \\ 
 r^2  = b^2  + \frac{{Z_P Z_T e^2 }}{{E_{tot} }}r \\ 
 \end{array}
\label{eqn3}
\end{equation}
%Equation ??.3

\noindent Rearranging this equation results in the expression:

\begin{equation}
b^2  = r(r - r_m )
\label{eqn4}
\end{equation}
%Equation ??.4

\noindent where:

\begin{equation}
r_m  = \frac{{Z_P Z_T e^2 }}{{E_{tot} }}
\label{eqn5}
\end{equation}
%Equation ??.5

\noindent In the implementation of the abrasion process in Geant4, the 
square of the far-field impact parameter, \(b\), is sampled uniformly subject 
to the distance of closest approach, \(r\), being no greater than 
\(r_P\) + \(r_T\) (the sum of the projectile and target nuclear radii).

\section{Abrasion process}
\label{abrasion}
In the abrasion process, as implemented by Wilson {\normalsize\it{}et al} 
\cite{aaWilson} it is assumed that the nuclear density for the projectile is 
constant up to the radius of the projectile (\(r_P\)) and zero outside.  This 
is also assumed to be the case for the target nucleus.  The amount of nuclear 
material abraded from the projectile is given by the expression:

\begin{equation}
\Delta _{abr}  = FA_P \left[ {1 - \exp \left( { - \frac{{C_T }}{\lambda }} \right)} \right]
\label{eqn6}
\end{equation}
%Equation ??.6

\noindent where F is the fraction of the projectile in the interaction zone, 
\(\lambda\) is the nuclear mean-free-path, assumed to be:

\begin{equation}
\lambda  = \frac{{16.6}}{{E^{0.26} }}
\label{eqn7}
\end{equation}
%Equation ??.7
\noindent \(E\) is the energy of the projectile in MeV/nucleon and \(C_T\) is 
the chord-length at the position in the target nucleus for which the 
interaction probability is maximum.  For cases where the radius of the target 
nucleus is greater than that of the projectile ({\normalsize\it{}i.e.} 
\(r_T > r_P\)):

\begin{equation}
C_T  = \left\{ {\begin{array}{*{20}c}
   {2\sqrt {r_T^2  - x^2 } } & {:x > 0}  \\
   {2\sqrt {r_T^2  - r^2 } } & {:x \le 0}  \\
\end{array}} \right.
\end{equation}

\noindent where:

\begin{equation}
x = \frac{{r_P^2  + r^2  - r_T^2 }}{{2r}}
\label{eqn8}
\end{equation}
%Equation ??.8

\noindent In the event that \(r_P > r_T\) then \(C_T\) is:

\begin{equation}
C_T  = \left\{ {\begin{array}{*{20}c}
   {2\sqrt {r_T^2  - x^2 } } & {:x > 0}  \\
   {2r_T } & {:x \le 0}  \\
\end{array}} \right.
\end{equation}

\noindent where:

\begin{equation}
x = \frac{{r_T^2  + r^2  - r_P^2 }}{{2r}}
\label {eqn9}
\end{equation}
%Equation ??.9

\noindent The projectile and target nuclear radii are given by the expression:

\begin{equation}
\begin{array}{l}
 r_P  \approx 1.29\sqrt {r_{RMS,P}^2  - 0.84^2 }  \\ 
 r_T  \approx 1.29\sqrt {r_{RMS,T}^2  - 0.84^2 }  \\ 
 \end{array}
\label{eqn10}
\end{equation}
%Equation ??.10
\noindent The excitation energy of the nuclear fragment formed by the 
spectators in the projectile is assumed to be determined by the excess surface 
area, given by:

\begin{equation}
\Delta S = 4\pi r_P^2 \left[ {1 + P - \left( {1 - F} \right)^{{\raise0.7ex\hbox{$2$} \!\mathord{\left/
 {\vphantom {2 3}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{$3$}}} } \right]
\label{eqn11}
\end{equation}
%Equation ??.11

\noindent where the functions \(P\) and \(F\) are given in section 
\ref{PandF}.  Wilson {\normalsize\it{}et al} equate this surface area to the 
excitation to:

\begin{equation}
E_S  = 0.95\Delta S
\label{eqn12}
\end{equation}
%Equation ??.12

\noindent if the collision is peripheral and there is no significant 
distortion of the nucleus, or

\begin{equation}
\begin{array}{l}
 E_S  = 0.95\left\{ {1 + 5F + \Omega F^3 } \right\}\Delta S \\ 
 \Omega  = \left\{ {\begin{array}{*{20}c}
   0 & {:A_P  > {\rm 16}}  \\
   {1500} & {:A_P  < 12}  \\
   {1500 - 320\left( {A_P  - 12} \right)} & {:12 \le A_P  \le 16}  \\
\end{array}} \right. \\ 
 \end{array}
\label{eqn13}
\end{equation}
%Equation ??.13
\noindent if the impact separation is such that \(r << r_P\)+\(r_T\).  
\(E_S\) is in MeV provided \(\Delta S\) is in fm$^2$.

\noindent For the abraded region, Wilson {\normalsize\it{}et al} assume that 
fragments with a nucleon number of five are unbounded, 90\% of fragments with 
a nucleon number of eight are unbound, and 50\% of fragments with a nucleon 
number of nine are unbound.  This was not implemented within the Geant4 
version of the abrasion model, and disintegration of the pre-fragment was only 
simulated by the subsequent de-excitation physics models in the 
G4DeexcitationHandler (evaporation, {\normalsize\it{}etc.} or 
G4WilsonAblationModel) since the yields of lighter fragments were already 
underestimated compared with experiment.

\noindent In addition to energy as a result of the distortion of the fragment, 
some energy is assumed to be gained from transfer of kinetic energy across the 
boundaries of the nuclei.  This is approximated to the average energy 
transferred to a nucleon per unit intersection pathlength (assumed to be 13
MeV/fm) and the longest chord-length, \(C_l\), and for half of the 
nucleon-nucleon collisions it is assumed that the excitation energy is:


\begin{equation}
E_X^*  = \left\{ {
\begin{array}{*{20}c}
   {13 \cdot \left[ {1 + \frac{{C_t  - 1.5}}{3}} \right]C_l } & {:C_t  > 1.5{\rm fm}}  \\
   {{\rm 13} \cdot {\rm C}_l } & {:C_t  \le 1.5{\rm fm}}  \\
\end{array}} \right.
\label{eqn14}
\end{equation}
%Equation ??.14

\noindent where:

\begin{equation}
C_l  = \left\{ {
\begin{array}{*{20}c}
   {2\sqrt {r_P^2  + 2rr_T  - r^2  - r_T^2 } } & {r > r_T }  \\
   {2r_P } & {r \le r_T }  \\
\end{array}} \right.
\end{equation}
\begin{equation}
C_t  = 2\sqrt {r_P^2  - \frac{{\left( {r_P^2  + r^2  - r_T^2 } \right)^2 }}{{4r^2 }}} 
\label{eqn15}
\end{equation}
%Equation ??.15

\noindent For the remaining events, the projectile energy is assumed to be 
unchanged.  Wilson {\normalsize\it{}et al} assume that the energy required to 
remove a nucleon is 10MeV, therefore the number of nucleons removed from the 
projectile by ablation is:

\begin{equation}
\Delta _{abl}  = \frac{{E_S  + E_X }}{{10}} + \Delta _{spc} 
\label{eqn16}
\end{equation}
%Equation ??.16

\noindent where \(\Delta _{spc}\) is the number of loosely-bound spectators 
in the interaction region, given by:

\begin{equation}
\Delta _{spc}  = A_P F\exp \left( { - \frac{{C_T }}{\lambda }} \right)
\label{eqn17}
\end{equation}
%Equation ??.17
\noindent Wilson {\normalsize\it{}et al} appear to assume that for half of the 
events the excitation energy is transferred into one of the nuclei (projectile 
or target), otherwise the energy is transferred in to the other (target or 
projectile respectively).

\noindent The abrasion process is assumed to occur without preference for the 
nucleon type, {\normalsize\it{}i.e.} the probability of a proton being abraded 
from the projectile is proportional to the fraction of protons in the original 
projectile, therefore:

\begin{equation}
\Delta Z_{abr}  = \Delta _{abr} \frac{{Z_P }}{{A_P }}
\label{eqn18}
\end{equation}
%Equation ??.18

\noindent In order to calculate the charge distribution of the final fragment, 
Wilson {\normalsize\it{}et al} assume that the products of the interaction lie 
near to nuclear stability and therefore can be sampled according to the 
Rudstam equation (see section \ref{ablation}).  The other obvious condition is 
that the total charge must remain unchanged.

\section{Abraded nucleon spectrum}
\label{spectrum}
Cucinotta has examined different formulae to represent the secondary protons 
spectrum from heavy ion collisions \cite{aaCucinotta}.  One of the models 
(which has been implemented to define the final state of the abrasion process) 
represents the momentum distribution of the secondaries as:

\begin{equation}
\psi (p) \propto \sum\limits_{i = 1}^3 {C_i \exp \left( { - \frac{{p^2 }}{{2p_i^2 }}} \right)}  + d_0 \frac{{\gamma p}}{{\sinh \left( {\gamma p} \right)}}
\label{eqn19}
\end{equation}
%Equation ??.19

\noindent where:

\(\psi (p)\) \indent = number of secondary protons with momentum \(p\) per 
unit of momentum phase space [c$^3$/MeV$^3$];

\(p\) \indent = magnitude of the proton momentum in the rest frame of the 
nucleus from which the particle is projected [MeV/c];

\(C1\), \(C2\), \(C3\) \indent = 1.0, 0.03, and 0.0002;

\(p1\), \(p2\), \(p3\) \indent = \(\sqrt \frac{2} {5} p_F\), 
\(\sqrt \frac{6} {5} p_F\), 500 [MeV/c]

\(p_F\) \indent = Momentum of nucleons in the nuclei at the Fermi surface 
[MeV/c]

\(d_0\) \indent = 0.1

\(\frac 1 \gamma\) \indent = 90 [MeV/c];

\noindent G4WilsonAbrasionModel approximates the momentum distribution for the 
neutrons to that of the protons, and as mentioned above, the nucleon type 
sampled is proportional to the fraction of protons or neutrons in the original 
nucleus.

\noindent The angular distribution of the abraded nucleons is assumed to be 
isotropic in the frame of reference of the nucleus, and therefore those 
particles from the projectile are Lorentz-boosted according to the initial 
projectile momentum.

\section{De-excitation of the projectile and target nuclear pre-fragments 
by standard Geant4 de-excitation physics}
\label{deexcitation}
Unless specified otherwise, G4WilsonAbrasionModel will instantiate the 
following de-excitation models to treat subsequent particle emission of the 
excited nuclear pre-fragments (from both the projectile and the target):

\trivlist
\item 1 G4Evaporation, which will perform nuclear evaporation of 
($\alpha$-particles, $^3$He, $^3$H, $^2$H, protons and neutrons, in 
competition with photo-evaporation and nuclear fission (if the nucleus has 
sufficiently high A).

\item 2 G4FermiBreakUp, for nuclei with \(A \le\) 12 and \(Z \le\) 6.

\item 3 G4StatMF, for multi-fragmentation of the nucleus (minimum energy for 
this process set to 5 MeV).
\endtrivlist

\noindent As an alternative to using this de-excitation scheme, the user may 
provide to the G4WilsonAbrasionModel a pointer to her own de-excitation 
handler, or invoke instantiation of the ablation model (G4WilsonAblationModel).

\section{De-excitation of the projectile and target nuclear pre-fragments by 
nuclear ablation}
\label{ablation}
A nuclear ablation model, based largely on the description provided by 
Wilson {\normalsize\it{}et al} \cite{aaWilson}, has been developed to provide 
a better approximation for the final nuclear fragment from an abrasion 
interaction.  The algorithm implemented in G4WilsonAblationModel uses the same 
approach for selecting the final-state nucleus as NUCFRG2 and determining the 
particles evaporated from the pre-fragment in order to achieve that state.  
However, use is also made of classes in Geant4's evaporation physics to 
determine the energies of the nuclear fragments produced.

\noindent The number of nucleons ablated from the nuclear pre-fragment 
(whether as nucleons or light nuclear fragments) is determined based on the 
average binding energy, assumed by Wilson {\normalsize\it{}et al} to be 
10 MeV, {\normalsize\it{}i.e.}:

\begin{equation}
A_{abl}  = \left\{ {\begin{array}{*{20}c}
   {Int\left( {\frac{{E_x }}{{10MeV}}} \right)} \hfill & {:A_{PF}  > Int\left( {\frac{{E_x }}{{10MeV}}} \right)} \hfill  \\
   {A_{PF} } \hfill & {:otherwise} \hfill  \\
\end{array}} \right.
\label{eqn20}
\end{equation}
%Equation ??.20

\noindent Obviously, the nucleon number of the final fragment, \(A_F\), is 
then determined by the number of remaining nucleons. The proton number of the 
final nuclear fragment (\(Z_F\)) is sampled stochastically using the Rudstam 
equation:

\begin{equation}
\sigma (A_F ,Z_F ) \propto \exp \left( { - R\left| {Z_F  - SA_F  - TA_F^2 } \right|^{{\raise0.7ex\hbox{$3$} \!\mathord{\left/
 {\vphantom {3 2}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{$2$}}} } \right)
\label{eqn21}
\end{equation}
%Equation ??.21

\noindent Here \(R\)=\(11.8/AF^{0.45}\), \(S\)=\(0.486\), and 
\(T\)=\(3.8 \cdot 10^{-4}\).  Once \(Z_F\) and \(A_F\) have been calculated, 
the species of the ablated (evaporated) particles are determined again using 
Wilson's algorithm.  The number of $\alpha$-particles is determined first, on 
the basis that these have the greatest binding energy:

\begin{equation}
N_\alpha   = \left\{ {\begin{array}{*{20}c}
   {Int\left( {\frac{{Z_{abl} }}{2}} \right)} & {:Int\left( {\frac{{Z_{abl} }}{2}} \right) < Int\left( {\frac{{A_{abl} }}{4}} \right)}  \\
   {Int\left( {\frac{{A_{abl} }}{4}} \right)} & {:Int\left( {\frac{{Z_{abl} }}{2}} \right) \ge Int\left( {\frac{{A_{abl} }}{4}} \right)}  \\
\end{array}} \right.
\label{eqn22}
\end{equation}
%Equation ??.22

\noindent Calculation of the other ablated nuclear/nucleon species is 
determined in a similar fashion in order of decreasing binding energy per 
nucleon of the ablated fragment, and subject to conservation of charge and 
nucleon number.

\noindent Once the ablated particle species are determined, use is made of the 
Geant4 evaporation classes to sample the order in which the particles are 
ejected (from G4AlphaEvaporationProbability, G4He3EvaporationProbability, 
G4TritonEvaporationProbability, G4DeuteronEvaporationProbability, 
G4ProtonEvaporationProbability and G4NeutronEvaporationProbability) and the 
energies and momenta of the evaporated particle and the residual nucleus at 
each two-body decay (using G4AlphaEvaporationChannel, G4He3EvaporationChannel, 
G4TritonEvaporationChannel, G4DeuteronEvaporationChannel, 
G4ProtonEvaporationChannel and G4NeutronEvaporationChannel).  If at any stage 
the probability for evaporation of any of the particles selected by the 
ablation process is zero, the evaporation is forced, but no significant 
momentum is imparted to the particle/nucleus.  Note, however, that any 
particles ejected from the projectile will be Lorentz boosted depending upon 
the initial energy per nucleon of the projectile.

\section{Definition of the functions P and F used in the abrasion model}
\label{PandF}
In the first instance, the form of the functions $P$ and $F$ used in the 
abrasion model are dependent upon the relative radii of the projectile and 
target and the distance of closest approach of the nuclear centres.  Four 
radius condtions are treated.

\noindent
\underline{\bf $r_T > r_P$ and $r_T-r_P \le r \le r_T+r_P$} :

\begin{equation}
P = 0.125\sqrt {\mu \nu } \left( {\frac{1}{\mu } - 2} \right)\left( {\frac{{1 - \beta }}{\nu }} \right)^2  - 0.125\left[ {0.5\sqrt {\mu \nu } \left( {\frac{1}{\mu } - 2} \right) + 1} \right]\left( {\frac{{1 - \beta }}{\nu }} \right)^3 
\end{equation}

\begin{equation}
F = 0.75\sqrt {\mu \nu } \left( {\frac{{1 - \beta }}{\nu }} \right)^2  - 0.125\left[ {3\sqrt {\mu \nu }  - 1} \right]\left( {\frac{{1 - \beta }}{\nu }} \right)^3 
\end{equation} \\

\noindent where:

\begin{eqnarray}
  \nu  = \frac{{r_P }}{{r_P  + r_T }} \\ 
  \beta  = \frac{r}{{r_P  + r_T }} \\ 
  \mu  = \frac{{r_T }}{{r_P }} 
\end{eqnarray}


\noindent
\underline{\bf $r_T > r_P$ and $r < r_T-r_P$} :

\begin{equation}
P =  - 1
\end{equation}
\begin{equation}
F = 1
\end{equation}

\noindent
\underline{\bf $r_P > r_T$ and $r_P-r_T \le r \le r_P+r_T$} :

\begin{eqnarray}
 P = 0.125\sqrt {\mu \nu } \left( {\frac{1}{\mu } - 2} \right)\left( {\frac{{1 - \beta }}{\nu }} \right)^2 
\end{eqnarray}
\begin{eqnarray}
 - 0.125\left\{ {0.5\sqrt {\frac{\nu }{\mu }} \left( {\frac{1}{\mu } - 2} \right) - \left[ {\frac{{\sqrt {1 - \mu ^2 } }}{\nu } - 1} \right]\sqrt {\frac{{2 - \mu }}{{\mu ^5 }}} } \right\}\left( {\frac{{1 - \beta }}{\nu }} \right)^3 \nonumber 
\end{eqnarray}

\begin{eqnarray}
 F = 0.75\sqrt {\mu \nu } \left( {\frac{{1 - \beta }}{\nu }} \right)^2 
\end{eqnarray}
\begin{eqnarray}
 - 0.125\left[ {3\sqrt {\frac{\nu }{\mu }}  - \frac{{\left[ {1 - \left( {1 - \mu ^2 } \right)^{{\raise0.7ex\hbox{$3$} \!\mathord{\left/
  {\vphantom {3 2}}\right.\kern-\nulldelimiterspace}
 \!\lower0.7ex\hbox{$2$}}} } \right]\sqrt {1 - \left( {1 - \mu } \right)^2 } }}{{\mu ^3 }}} \right]\left( {\frac{{1 - \beta }}{\nu }} \right)^3 \nonumber
\end{eqnarray}


\noindent
\underline{\bf $r_P > r_T$ and $r < r_T-r_P$} :

\begin{equation}
P = \left[ {\frac{{\sqrt {1 - \mu ^2 } }}{\nu } - 1} \right]\sqrt {1 - \left( {\frac{\beta }{\nu }} \right)^2 } 
\end{equation}
\begin{equation}
F = \left[ {1 - \left( {1 - \mu ^2 } \right)^{{\raise0.7ex\hbox{$3$} \!\mathord{\left/
 {\vphantom {3 2}}\right.\kern-\nulldelimiterspace}
\!\lower0.7ex\hbox{$2$}}} } \right]\sqrt {1 - \left( {\frac{\beta }{\nu }} \right)^2 } 
\end{equation}

\begin{figure}
  \includegraphics[scale=0.8]{hadronic/theory_driven/AbrasionAblation/abrasion.eps}
\caption{In the abrasion process, a fraction of the nucleons in the projectile 
and target nucleons interact to form a fireball region with a velocity between 
that of the projectile and the target.  The remaining spectator nucleons in 
the projectile and target are not initially affected (although they do suffer 
change as a result of longer-term de-excitation).}
\label{fig:1}       % Give a unique label
\end{figure}

\begin{figure}
  \includegraphics[scale=0.55, angle=0]{hadronic/theory_driven/AbrasionAblation/deflection.eps}
\caption{Illustration clarifying impact parameter in the far-field (\(b\)) and 
 actual impact parameter (\(r\)).}
\label{fig:2}       % Give a unique label
\end{figure}

\section{Status of this document}
 18.06.04 created by Peter Truscott \\

\begin{latexonly}

\begin{thebibliography}{}

\bibitem{aaWilson}
J W Wilson, R K Tripathi, F A Cucinotta, J K Shinn, F F Badavi, S Y Chun, 
J W Norbury, C J Zeitlin, L Heilbronn, and J Miller, "NUCFRG2: An evaluation 
of the semiempirical nuclear fragmentation database," NASA Technical Paper 
3533, 1995.

\bibitem{aaTownsend}
Lawrence W Townsend, John W Wilson, Ram K Tripathi, John W Norbury, 
Francis F Badavi, and Ferdou Khan, "HZEFRG1, An energy-dependent semiempirical 
nuclear fragmentation model," NASA Technical Paper 3310, 1993.

\bibitem{aaCucinotta}
Francis A Cucinotta, "Multiple-scattering model for inclusive proton 
production in heavy ion collisions," NASA Technical Paper 3470, 1994.

\end{thebibliography}

\end{latexonly}

\begin{htmlonly}

\section{Bibliography}

\begin{enumerate}
\item
J W Wilson, R K Tripathi, F A Cucinotta, J K Shinn, F F Badavi, S Y Chun, 
J W Norbury, C J Zeitlin, L Heilbronn, and J Miller, "NUCFRG2: An evaluation 
of the semiempirical nuclear fragmentation database," NASA Technical Paper 
3533, 1995.

\item
Lawrence W Townsend, John W Wilson, Ram K Tripathi, John W Norbury, 
Francis F Badavi, and Ferdou Khan, "HZEFRG1, An energy-dependent semiempirical 
nuclear fragmentation model," NASA Technical Paper 3310, 1993.

\item
Francis A Cucinotta, "Multiple-scattering model for inclusive proton 
production in heavy ion collisions," NASA Technical Paper 3470, 1994.

\end{enumerate}

\end{htmlonly}