1 | \section{Hadronic Processes} |
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2 | The purpose of this note is to publish the implementation frameworks used in and |
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3 | provided by {\sc Geant4} for hadronic shower simulation |
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4 | as in the 1.1 release of the program. The |
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5 | implementation frameworks follow the Russian dolls approach to |
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6 | implementation framework design. |
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7 | A top-level, very abstracting implementation framework provides the basic |
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8 | interface to the other {\sc Geant4} categories, and fulfils the most general |
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9 | use-case for hadronic shower simulation. It is refined for more specific |
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10 | use-cases by implementing a hierarchy of implementation frameworks, each level |
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11 | implementing the common logic of a particular use-case package in a concrete implementation of |
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12 | the interface specification of one framework level above, this way refining the |
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13 | granularity of abstraction and delegation. This defines the Russian dolls architectural |
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14 | pattern. |
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15 | Abstract classes are used as the delegation |
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16 | mechanism\footnote{The same can be achieved with template specialisations |
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17 | with slightly improved CPU performance but at the cost of significantly more complex |
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18 | designs and, with present compilers, significantly reduced portability.}. |
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19 | All framework functional requirements were obtained through |
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20 | use-case analysis. In the following we present for each framework level the |
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21 | compressed use-cases, requirements, designs including the flexibility provided, and |
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22 | illustrate the framework functionality with examples. All design patterns cited are to |
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23 | be read as defined in \cite{Patterns}. |
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24 | \section{Level 1 Framework - processes} |
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25 | \begin{figure}[b!] % fig 1 |
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26 | \centerline{\includegraphics[width=14cm]{Level1.eps}} |
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27 | \caption{Level 1 implementation framework of the hadronic category |
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28 | of {\sc Geant4}.} |
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29 | \label{Level1} |
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30 | \end{figure} |
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31 | There are two principal use-cases of the level 1 framework. A user will want to |
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32 | choose the processes used for his particular simulation run, and a physicist |
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33 | will want to write code for processes of his own and use these together with the |
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34 | rest of the system in a seamless manner. |
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35 | \subsection {Requirements} |
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36 | \begin{enumerate} |
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37 | \item \label{L1R1} Provide a standard interface to be used by process implementations. |
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38 | \item \label{L1R2} Provide registration mechanisms for processes. |
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39 | \end{enumerate} |
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40 | \subsection {Design and interfaces} |
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41 | Both requirements are implemented in a sub-set of the tracking-physics interface |
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42 | in {\sc Geant4}. The class diagram is shown in figure \ref{Level1}. |
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43 | All processes have a common base-class {\tt G4VProcess}, from |
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44 | which a set of specialised classes are derived. Three of them are used as base |
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45 | classes for hadronic processes for particles at rest ({\tt G4VRestProcess}), for |
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46 | interactions in flight ({\tt G4VDiscreteProcess}), and for processes like |
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47 | radioactive decay where the same implementation can represent both these extreme cases |
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48 | ({\tt G4VRestDiscreteProcess}). |
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49 | |
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50 | Each of these classes declares two types of methods; one for calculating the |
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51 | time to interaction or the physical interaction length, allowing tracking to request |
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52 | the information necessary to decide on the process responsible for final state |
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53 | production, and one to compute |
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54 | the final state. These are pure virtual methods, and have to be implemented in |
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55 | each individual derived class, as enforced by the compiler. |
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56 | \subsection {Framework functionality} |
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57 | The functionality provided is through the use of process base-class pointers |
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58 | in the tracking-physics interface, and the {\tt G4ProcessManager}. All |
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59 | functionality is implemented in abstract, and registration of derived process classes |
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60 | with the {\tt G4ProcessManager} of an individual particle allows for |
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61 | arbitrary combination of both {\sc Geant4} provided processes, and user-implemented |
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62 | processes. This registration mechanism is a modification on a Chain of Responsibility. |
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63 | It is outside the scope of the current paper, and its |
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64 | description is available from \cite{G4Manual}. |
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65 | \section{Level 2 Framework - Cross-sections and Models} |
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66 | \begin{figure}[b!] % fig 1 |
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67 | \centerline{\includegraphics[width=14cm]{Level2_1.eps}} |
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68 | \caption{Level 2 implementation framework of the hadronic category |
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69 | of {\sc Geant4}; cross-section aspect.} |
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70 | \label{Level2_1} |
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71 | \end{figure} |
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72 | \begin{figure}[b!] % fig 1 |
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73 | \centerline{\includegraphics[width=14cm]{Level2_2.eps}} |
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74 | \caption{Level 2 implementation framework of the hadronic category |
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75 | of {\sc Geant4}; final state production aspect.} |
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76 | \label{Level2_2} |
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77 | \end{figure} |
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78 | \begin{figure}[b!] % fig 1 |
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79 | \centerline{\includegraphics[width=14cm]{Level2_3.eps}} |
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80 | \caption{Level 2 implementation framework of the hadronic category |
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81 | of {\sc Geant4}; isotope production aspect} |
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82 | \label{Level2_3} |
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83 | \end{figure} |
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84 | At the next level of abstraction, only processes that occur for particles |
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85 | in flight are |
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86 | considered. For these, it is easily observed that the sources of cross-sections |
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87 | and final state production are rarely the same. Also, different sources will |
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88 | come with different restrictions. The principal use-cases of the framework are addressing |
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89 | these commonalities. A user might want to combine different cross-sections and |
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90 | final state or isotope production models as provided by {\sc Geant4}, and a physicist might |
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91 | want to implement his own model for particular situation, and add cross-section |
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92 | data sets that are relevant for his particular analysis to the system in a |
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93 | seamless manner. |
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94 | \subsection {Requirements} |
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95 | \begin{enumerate} |
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96 | \item Flexible choice of inclusive scattering cross-sections. |
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97 | \item Ability to use different data-sets for different parts of |
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98 | the simulation, depending on the conditions at the point of interaction. |
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99 | \item Ability to add user-defined data-sets in a seamless manner. |
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100 | \item Flexible, unconstrained choice of final state production models. |
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101 | \item Ability to use different final state production codes |
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102 | for different parts of the simulation, depending on the conditions at the |
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103 | point of interaction. |
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104 | \item Ability to add user-defined final state production models in |
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105 | a seamless manner. |
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106 | \item Flexible choice of isotope production models, to run in |
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107 | parasitic mode to any kind of transport models. |
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108 | \item Ability to use different isotope production codes |
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109 | for different parts of the simulation, depending on the conditions at the |
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110 | point of interaction. |
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111 | \item Ability to add user-defined isotope production models in a |
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112 | seamless manner. |
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113 | \end{enumerate} |
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114 | \subsection {Design and interfaces} |
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115 | The above requirements are implemented in three framework components, one for |
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116 | cross-sections, final state production, and isotope production each. |
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117 | The class diagrams are shown in figure \ref{Level2_1} for the cross-section aspects, |
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118 | figure \ref{Level2_2} for the final state production aspects, and figure \ref{Level2_3} |
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119 | for the isotope production aspects. |
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120 | The three parts are integrated in the {\tt G4HadronicProcess} class, that serves as |
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121 | base-class for all hadronic processes of particles in flight. |
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122 | \subsubsection{Cross-sections} |
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123 | Each hadronic process is derived from {\tt G4HadronicProcess}, which |
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124 | holds a list of ``cross section data sets''. |
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125 | The term ``data set'' is |
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126 | representing an object that encapsulates methods and data for calculating |
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127 | total cross sections for a given process in a certain range of validity. |
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128 | The implementations may take any form. It can be a simple equation as well as |
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129 | sophisticated parameterisations, or evaluated data. |
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130 | All cross section data set classes are derived from the abstract class |
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131 | {\tt G4VCrossSectionDataSet}, which declares methods that allow |
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132 | the process inquire, about the applicability of an individual data-set through |
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133 | {\tt IsApplicable(const G4DynamicParticle*, const G4Element*)}, |
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134 | and to de\-le\-gate the calculation of the actual cross-section value through |
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135 | \\ |
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136 | {\tt GetCrossSection(const G4DynamicParticle*, const G4Element*)}. |
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137 | \subsubsection{Final state production} |
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138 | For final state production, we provide the {\tt G4HadronicInteraction} base class. |
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139 | It declares a minimal interface of only one pure virtual method for final state |
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140 | production, |
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141 | \\ |
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142 | {\tt G4VParticleChange* ApplyYourself(const G4Track \&, G4Nucleus \&)}. |
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143 | \\ |
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144 | {\tt G4HadronicProcess} provides a registry for final state |
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145 | production models inheriting from {\tt G4HadronicInteraction}. Again, final state production |
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146 | model is meant in very general terms. This can be an implementation of a |
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147 | quark gluon string model\cite{QGSM}, a sampling code for ENDF/B data formats\cite{ENDFForm}, or a parametrisation |
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148 | describing only neutron elastic scattering off Silicon up to 300~MeV. |
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149 | \subsubsection{Isotope production} |
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150 | For isotope production, a base class ({\tt G4VIsotopeProduction}) is provided. It declares a |
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151 | method ({\tt G4IsoResult * GetIsotope(const G4Track \&, const G4Nucleus \&)}) |
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152 | that calculates and returns the isotope production information. Any concrete isotope |
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153 | production model will inherit from this class, and implement the method. Again, the modelling |
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154 | possibilities are not limited, and the implementation of concrete production models is not |
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155 | restricted in any way. By convention, the {\tt GetIsotope} method returns NULL, if the model |
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156 | is not applicable for the current projectile target combination. |
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157 | \subsection {Framework functionality} |
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158 | \subsubsection{Cross-sections} |
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159 | {\tt G4HadronicProcess} provides registering possibilities for data-sets. |
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160 | A default is provided covering all possible conditions to some approximation. |
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161 | The process stores and retrieves the data-sets through a data-store that acts like a |
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162 | FILO stack (a Chain of Responsibility with a |
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163 | First In Last Out decision strategy). This allows the user to map out |
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164 | the entire parameter space by overlaying |
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165 | cross-section data-sets to optimise the overall result. An example are the cross-sections for |
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166 | low energy neutron transport. If these are registered last by the user, they will be used |
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167 | whenever |
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168 | low energy neutrons are encountered. In all other conditions the system falls back on the |
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169 | default, or data sets with earlier registration dates. |
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170 | The fact that the registration is done through abstract base classes with no side-effects |
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171 | allows the user to implement and use his own cross-sections. An example are special |
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172 | reaction cross-sections of $K^0$-nuclear interactions that might be used for |
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173 | $\epsilon/\epsilon '$ analysis at LHC to control the systematic error. |
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174 | \subsubsection{Final state production} |
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175 | The {\tt G4HadronicProcess} provides a registration service for classes deriving from {\tt |
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176 | G4HadronicInteraction}, and delegates final state production to the applicable model. |
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177 | {\tt G4HadronicInteraction} provides the functionality needed to define and enforce |
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178 | the applicability of a particular model. Models inheriting from {\tt G4HadronicInteraction} |
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179 | can be restricted in applicability in projectile type and energy, and can be |
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180 | activated/deactivated for individual materials and elements. This allows a user to use |
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181 | final state production models in arbitrary combinations, and to write his own models for |
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182 | final state production. The design is a variant of a Chain of Responsability. |
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183 | An example would be the likely CMS scenario - the |
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184 | combination of low energy neutron transport with a quantum molecular |
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185 | dynamics\cite{QMD} or chiral |
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186 | invariant phase space decay\cite{CHIPS} model |
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187 | in the case of tracker materials and fast parametrised models for calorimeter materials, |
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188 | with user defined modelling of interactions of spallation nucleons with the most abundant |
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189 | tracker and calorimeter materials. |
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190 | \subsubsection{Isotope production} |
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191 | The {\tt G4HadronicProcess} by default calculates the isotope production information from the |
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192 | final state given by the transport model. In addition, it provides a registering mechanism |
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193 | for isotope production models that run in parasitic mode to the transport models and |
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194 | inherit from {\tt G4VIsotopeProduction}. |
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195 | The registering mechanism behaves like a FILO stack, again based on Chain of |
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196 | Responsibility. |
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197 | The models will be asked for isotope production information |
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198 | in inverse order of registration. The |
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199 | first model that returns a non-NULL value will be applied. |
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200 | In addition, the {\tt G4HadronicProcess} provides the basic infrastructure for accessing and |
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201 | steering of isotope-production information. It allows to enable and disable the calculation |
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202 | of isotope production information |
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203 | globally, or for individual processes, and to retrieve the isotope production information |
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204 | through the |
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205 | \\ |
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206 | {\tt G4IsoParticleChange * GetIsotopeProductionInfo()} |
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207 | method at |
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208 | the end of each step. The {\tt |
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209 | G4HadronicProcess} is a finite state machine that will ensure the {\tt |
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210 | GetIsotopeProductionInfo} returns a non-zero value only at the first call after |
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211 | isotope production occurred. |
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212 | An example of the use of this functionality is the study of activation of a Germanium |
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213 | detector in a high precision, low background experiment. |
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214 | \section{Level 3 Framework - Theoretical Models} |
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215 | \begin{figure}[b!] % fig 1 |
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216 | \centerline{\includegraphics[width=14cm]{Level3_1.eps}} |
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217 | \caption{Level 3 implementation framework of the hadronic category |
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218 | of {\sc Geant4}; theoretical model aspect.} |
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219 | \label{Level3_1} |
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220 | \end{figure} |
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221 | {\sc Geant4} provides at present one implementation framework for theory |
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222 | driven models. The main use-case is that of a user wishing to use theoretical models in |
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223 | general, and to use various intra-nuclear |
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224 | transport or pre-compound models together with models simulating the initial interactions at |
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225 | very high energies. |
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226 | \subsection {Requirements} |
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227 | \begin{enumerate} |
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228 | \item Allow to use or adapt any string-parton or parton transport\cite{VNI} model. |
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229 | \item Allow to adapt event generators, ex. PYTHIA\cite{PYTHIA7} for final state production in shower simulation. |
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230 | \item Allow for combination of the above with any intra-nuclear transport (INT). |
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231 | \item Allow stand-alone use of intra-nuclear transport. |
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232 | \item Allow for combination of the above with any pre-compound model. |
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233 | \item Allow stand-alone use of any pre-compound model. |
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234 | \item Allow for use of any evaporation code. |
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235 | \item Allow for seamless integration of user defined components for any of the above. |
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236 | \end{enumerate} |
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237 | \subsection {Design and interfaces} |
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238 | To provide the above flexibility, the following abstract base classes have been |
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239 | implemented: |
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240 | \begin{itemize} |
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241 | \item {\tt G4VHighEnergyGenerator} |
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242 | \item {\tt G4VIntranuclearTransportModel} |
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243 | \item {\tt G4VPreCompoundModel} |
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244 | \item {\tt G4VExcitationHandler} |
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245 | \end{itemize} |
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246 | In addition, the class {\tt G4TheoFSGenerator} is provided to orchestrate interactions |
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247 | between these classes. |
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248 | The class diagram is shown in figure \ref{Level3_1}. |
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249 | |
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250 | {\tt G4VHighEnergyGenerator} serves as base class for parton transport or parton |
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251 | string models, and for Adapters to event generators. This class declares two methods, |
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252 | {\tt Scatter}, and {\tt GetWoundedNucleus}. |
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253 | |
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254 | The base class for INT inherits from |
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255 | {\tt G4HadronicInteraction}, making any concrete implementation usable as |
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256 | stand-alone model. In doing so, it |
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257 | re-declares the {\tt ApplyYourself} interface of {\tt |
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258 | G4HadronicInteraction}, and adds a second interface, {\tt Propagate}, for further propagation |
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259 | after high energy interactions. {\tt Propagate} takes as arguments |
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260 | a three-dimensions model of a wounded nucleus, and a set of secondaries with energies |
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261 | and positions. |
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262 | |
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263 | The base class for pre-equilibrium decay models, {\tt G4VPreCompoundModel}, |
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264 | inherits from {\tt G4HadronicInteraction}, again |
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265 | making any concrete implementation usable as stand-alone model. It adds a pure virtual |
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266 | {\tt DeExcite} method |
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267 | for further evolution of the system when intra-nuclear transport assumptions |
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268 | break down. {\tt DeExcite} takes a {\tt G4Fragment}, the {\sc Geant4} representation of an |
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269 | excited nucleus, as argument. |
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270 | |
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271 | The base class for evaporation phases, {\tt G4VExcitationHandler}, declares an |
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272 | abstract method, {\tt BreakItUP()}, for compound decay. |
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273 | \subsection {Framework functionality} |
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274 | {\tt G4TheoFSGenerator} inherits from {\tt G4HadronicInteraction}, and hence can be |
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275 | registered as model for final state production with a hadronic process. It allows to |
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276 | register a concrete implementation of {\tt G4VIntranuclearTransportModel} and |
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277 | {\tt G4VHighEnergyGenerator}, and delegates initial interactions, and intra-nuclear transport |
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278 | of the corresponding secondaries to the respective classes. The design is a complex |
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279 | variant of a Strategy. The most spectacular application |
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280 | of this pattern is the use of parton-string models for string excitation, quark molecular |
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281 | dynamics for correlated string decay, and quantum molecular dynamics for transport, a |
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282 | combination which |
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283 | promises to result in a coherent description of quark gluon plasma in high energy nucleus |
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284 | nucleus interactions. |
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285 | |
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286 | {\tt G4VIntranuclearTransportModel} provides registering mechanisms for concrete |
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287 | implementations of {\tt G4VPreCompoundModel}, and provides concrete intra-nuclear transports |
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288 | with the possibility to delegate pre-compound decay to these models. |
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289 | |
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290 | {\tt G4VPreCompoundModel} provides a registering mechanism for compound decay through the |
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291 | {\tt G4VExcitationHandler} interface, and provides concrete implementations with the |
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292 | possibility to delegate the decay of a compound nucleus. |
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293 | |
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294 | The concrete scenario of {\tt G4TheoFSGenerator} using a dual parton model and a classical |
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295 | cascade, which in turn uses an exciton pre-compound model that delegates to an evaporation |
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296 | phase, would be the following: {\tt G4TheoFSGenerator} receives the conditions of the |
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297 | interaction; a primary particle and a nucleus. It asks the dual parton model to perform |
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298 | the initial scatterings, and return the final state, along with the by then damaged nucleus. |
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299 | The nucleus records all information on the damage sustained. |
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300 | {\tt G4TheoFSGenerator} forwards all information to the classical cascade, that propagates the particles in the |
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301 | already damaged nucleus, keeping track of interactions, further damage to the nucleus, |
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302 | etc.. Once the cascade assumptions |
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303 | break down, the cascade will collect the information of the current state of the hadronic |
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304 | system, like excitation energy and number of excited particles, |
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305 | and interpret it as a pre-compound system. It delegates the decay of this to the |
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306 | exciton model. The exciton model will take the information provided, and calculate |
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307 | transitions and emissions, until the number of excitons in the system equals the mean number of |
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308 | excitons expected in equilibrium for the current excitation energy. Then it hands over to the |
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309 | evaporation phase. The evaporation phase decays the residual nucleus, and the Chain of |
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310 | Command rolls back to {\tt G4TheoFSGenerator}, accumulating the information produced in |
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311 | the various levels of delegation. |
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312 | \section{Level 4 Frameworks - String Parton Models and Intra-Nuclear Cascade} |
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313 | \begin{figure}[b!] % fig 1 |
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314 | \centerline{\includegraphics[width=14cm]{Level4_1.eps}} |
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315 | \caption{Level 4 implementation framework of the hadronic category |
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316 | of {\sc Geant4}; parton string aspect.} |
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317 | \label{Level4_1} |
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318 | \end{figure} |
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319 | \begin{figure}[b!] % fig 1 |
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320 | \centerline{\includegraphics[width=14cm]{Level4_2.eps}} |
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321 | \caption{Level 4 implementation framework of the hadronic category |
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322 | of {\sc Geant4}; intra-nuclear transport aspect.} |
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323 | \label{Level4_2} |
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324 | \end{figure} |
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325 | The use-cases of this level are related to commonalities and |
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326 | detailed choices in string-parton models and cascade models. They are use-cases of an expert |
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327 | user wishing to alter details of a model, or a theoretical physicist, wishing to study |
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328 | details of a particular model. |
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329 | \subsection {Requirements} |
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330 | \begin{enumerate} |
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331 | \item Allow to select string decay algorithm |
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332 | \item Allow to select string excitation. |
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333 | \item Allow to select concrete implementation of three-dimensional model of the nucleus |
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334 | \item Allow to select concrete implementation of final state and cross-sections in |
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335 | intra-nuclear scattering. |
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336 | \end{enumerate} |
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337 | \subsection {Design and interfaces} |
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338 | To fulfil the requirements on string models, two abstract classes are provided, |
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339 | the {\tt G4VPartonStringModel} and the {\tt G4VStringFragmentation}. |
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340 | The base class for parton string models, {\tt G4VPartonStringModel}, |
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341 | declares the {\tt GetStrings()} |
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342 | pure virtual method. {\tt G4VStringFragmentation}, the pure abstract |
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343 | base class for string fragmentation, declares the interface for string fragmentation. |
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344 | |
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345 | To fulfil the requirements on intra-nuclear transport, two abstract classes are provided, |
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346 | {\tt G4V3DNucleus}, and {\tt G4VScatterer}. |
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347 | At this point in time, the usage of these intra-nuclear transport related classes |
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348 | by concrete codes is not enforced by designs, as the details of the |
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349 | cascade loop are still model dependent, and more experience has to be gathered to achieve |
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350 | standardisation. It is within the responsibility of the implementers of concrete |
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351 | intra-nuclear transport |
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352 | codes to use the abstract interfaces as defined in these classes. |
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353 | |
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354 | The class diagram is shown in figure \ref{Level4_1} for the string parton model aspects, and |
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355 | in figure \ref{Level4_2} for the intra-nuclear transport. |
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356 | |
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357 | \subsection{Framework functionality} |
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358 | Again variants of Strategy, Bridge and Chain of Responsibility are used. |
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359 | {\tt G4VPartonStringModel} implements the initialisation of a three-dimensional model of a |
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360 | nucleus, and |
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361 | the logic of scattering. It delegates secondary production to string |
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362 | fragmentation through a {\tt G4VStringFragmentation} pointer. It provides a registering |
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363 | service for the concrete string fragmentation, and delegates the string excitation to derived |
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364 | classes. Selection of string excitation is through selection of derived class. Selection of |
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365 | string fragmentation is through registration. |
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366 | |
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367 | \section{Level 5 Framework - String De-excitation} |
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368 | |
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369 | \begin{figure}[b!] % fig 1 |
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370 | \centerline{\includegraphics[width=14cm]{Level5_1.eps}} |
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371 | \caption{Level 5 implementation framework of the hadronic category |
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372 | of {\sc Geant4}; string fragmentation aspect.} |
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373 | \label{Level5_1} |
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374 | \end{figure} |
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375 | The use-case of this level is that of a user or theoretical physicist wishing to understand |
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376 | the systematic effects involved in combining various fragmentation functions with |
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377 | individual types of string fragmentation. Note that this framework level is meeting the |
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378 | current state of the art, making extensions and changes of interfaces in subsequent |
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379 | releases likely. |
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380 | \subsection {Requirements} |
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381 | \begin{enumerate} |
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382 | \item Allow the selection of fragmentation function. |
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383 | \end{enumerate} |
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384 | \subsection {Design and interfaces} |
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385 | A base class for fragmentation functions, {\tt G4VFragmentationFunction}, is provided. |
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386 | It declaring the {\tt GetLightConeZ()} interface. |
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387 | \subsection {Framework functionality} |
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388 | The design is a basic Strategy. The class diagram is shown in figure \ref{Level5_1}. |
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389 | At this point in time, the usage of the {\tt G4VFragmentationFunction} is not enforced |
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390 | by design, but made available from the {\tt G4VStringFragmentation} to an implementer of a |
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391 | concrete string decay. {\tt G4VStringFragmentation} provides a registering mechanism for the |
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392 | concrete fragmentation function. It delegates the calculation of $z_f$ of the hadron to |
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393 | split of the string to the concrete implementation. Standardisation in this area is expected. |
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394 | \section{Summary} |
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395 | A set of implementation frameworks has been provided for hadronic shower simulation. Finding |
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396 | framework functional requirements through use-case analysis has proven to be a highly |
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397 | effective tool. Framework components were found through bundling use-cases. Framework |
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398 | interfaces were defined by the need for delegation and flexibility; framework functionality |
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399 | was found from detailed requirements analysis. |
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400 | |
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401 | The Russian dolls approach, as defined in this paper, to framework design has proven very effective. Having layered |
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402 | implementation frameworks, and keeping abstractions in the upper levels of the framework |
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403 | hierarchy simple and general has |
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404 | proven to result in a structured and well suited solution to a complex problem. |
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405 | Addressing more specific use-cases in lower |
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406 | level frameworks that implement the interfaces of the more general framework has kept the |
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407 | system surprisingly extendible. It allows for distributed, and largely decoupled contributions |
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408 | of many scientists. |
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409 | \begin{thebibliography}{999} |
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410 | \bibitem{Geant4} The GEANT~4 Collaboration, \\ |
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411 | CERN/DRDC/94--29, DRDC/P58 1994. |
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412 | \bibitem{Patterns} E. Gamm et al., Design Patterns, Addison-Wesley Professional Computing |
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413 | Series, 1995. |
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414 | \bibitem{G4Manual} |
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415 | {\scriptsize http://cern.ch/geant4/G4UsersDocuments/Overview/html/index.html} |
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416 | \bibitem{QGSM} Kaidalov A. B., Ter-Martirosyan K. A., Phys. Lett. {\bf B117} |
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417 | 247 (1982); |
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418 | \bibitem{ENDFForm} |
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419 | Data Formats and Procedures for the Evaluated Nuclear Data |
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420 | File, National Nuclear Data Center, Brookhave National Laboratory, Upton, NY, USA. |
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421 | \bibitem{QMD} For example: VUU and (R)QMD model of high-energy heavy ion collisions. |
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422 | H. Stocker et al., Nucl. Phys. A538, 53c-64c (1992) |
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423 | \bibitem{CHIPS} P.V. Degtyarenko, M.V. Kossov, H.P. Wellisch, Eur. Phys J. A 8, 217-222 |
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424 | (2000) |
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425 | \bibitem{VNI} VNI 3.1, Klaus Geiger (Brookhaven), BNL-63762, |
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426 | Comput. Phys. Commun. 104, 70-160 (1997) |
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427 | \bibitem{PYTHIA7} M. Bertini, L. L\"onnblad, T. Sj\"orstrand, Pythia version 7-0.0 -- a |
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428 | proof-of-concept version, LU-TP 00-23, hep-ph/0006152, May 2000 |
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429 | |
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430 | \end{thebibliography} |
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431 | |
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