1 | \chapter[Multiple scattering]{Multiple scattering} |
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2 | |
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3 | The G4MultipleScattering class simulates the multiple scattering of |
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4 | charged particles in material. It uses a new multiple scattering (MSC) |
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5 | model which does not use the Moliere formalism \cite{msc.moliere}. |
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6 | This MSC model simulates the scattering of the |
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7 | particle after a given step , computes the mean path length |
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8 | correction and the mean lateral displacement as well. |
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9 | |
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10 | Let us define a few notation first. |
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11 | |
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12 | The true path length ('t' path length) is the total length travelled |
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13 | by the particle. All the physical processes restrict this 't' step. |
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14 | |
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15 | The geometrical ( or 'z') path length is the straight distance between |
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16 | the starting and endpoint of the step , if there is no magnetic field. |
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17 | The geometry gives a constraint for this 'z' step. It should be noted, |
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18 | that the geometrical step length is meaningful in the case of magnetic |
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19 | field, too, but in this case it is a distance along a curved |
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20 | trajectory. |
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21 | |
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22 | The mean properties of the multiple scattering process are determined |
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23 | by the transport mean free path , \(\lambda\) , which is a function of the |
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24 | energy in a given material.Some of the mean properties - the mean lateral |
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25 | displacement and the second moment of cos(theta) - depend on the second |
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26 | transport mean free path, too. (The transport mean free path is called |
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27 | first transport mean free path as well.) |
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28 | |
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29 | The 't'\(\Rightarrow\)'z' (true path length -- geometrical path length) transformation is given by the simple equation |
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30 | |
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31 | \begin{equation} |
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32 | z = \lambda*(1.-exp(-t/\lambda)) \label{msc.a} |
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33 | \end{equation} |
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34 | |
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35 | which is an exact result for the mean values of z , if |
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36 | the differential cross section has an axial symmetry and the energy loss |
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37 | can be neglected . |
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38 | This formula and some other expressions for the first moments of the |
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39 | spatial distribution after a given 'true' path length t have been taken |
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40 | from the excellent paper of Fernandez-Varea et al. \cite{msc.fernandez}, |
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41 | but the expressions have been calculated originally by Goudsmit and |
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42 | Saunderson \cite{msc.goudsmit} and Lewis \cite{msc.lewis}. |
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43 | Inverting eq. \ref{msc.a} the 'z'\(\Rightarrow\)'t' transformation can |
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44 | be written as |
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45 | |
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46 | \begin{equation} |
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47 | t = -\lambda*ln(1.-z/\lambda) \label{msc.b} |
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48 | \end{equation} |
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49 | |
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50 | where \(z < \lambda\) should be required (this condition is fulfilled |
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51 | if z has been computed from eq. \ref{msc.a}). |
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52 | |
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53 | The mean value of \(cos(\theta)\) - \(\theta\) is the scattering angle after a |
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54 | true step length t - is |
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55 | |
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56 | \begin{equation} |
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57 | <cos(\theta)> = exp(-t/\lambda) \label{msc.c} |
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58 | \end{equation} |
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59 | |
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60 | The transport mean free path values have been calculated by Liljequist |
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61 | et al. \cite{msc.liljequist2, msc.liljequist1} for electrons and positrons |
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62 | in the kinetic energy range \(0.1 keV -- 20 MeV\) in 15 materials. The |
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63 | MSC model uses these values with an appropriate interpolation or |
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64 | extrapolation in the atomic number |
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65 | \(Z\) and in the velocity of the particle \(\beta\) , when it is necessary. |
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66 | |
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67 | The quantity \(cos(\theta)\) is sampled in the MSC model according to a model function |
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68 | \(f(cos(\theta))\). The shape of this function has been choosen in such a way, |
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69 | that\(f(cos(\theta))\) reproduces the results of the direct simulation ot the particle |
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70 | transport rather well and eq. \ref{msc.c} is satisfied. |
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71 | The functional form of this model function is |
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72 | |
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73 | \begin{equation} |
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74 | f(x) = p \frac{(a + 1)^2 (a - 1)^2}{2 a} \frac{1}{(a-x)^3} |
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75 | + (1-p) \frac{1}{2} \label{msc.d} |
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76 | \end{equation} |
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77 | |
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78 | where \( x= cos(\theta)\) , \( 0 \leq p \leq 1\) and \( a > 1\) . The model |
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79 | parameters \(p\) and \(a\) depend on the path length t , the energy of the |
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80 | particle and the material.They are not independent parameters , they should |
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81 | satisfy the constraint |
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82 | |
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83 | \begin{equation} |
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84 | \frac{p}{a} = exp(-\frac{t}{\lambda}) \label{msc.e} |
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85 | \end{equation} |
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86 | |
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87 | which follows from eq. \ref{msc.c} . |
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88 | |
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89 | The mean lateral displacement is given by a more complicated formula |
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90 | (see the paper \cite{msc.fernandez} ), but this quantity also can be calculated |
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91 | relatively easily and accurately. |
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92 | |
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93 | It is worth to note that in this MSC model there is no step limitation |
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94 | originated from the multiple scattering process. Another important feature |
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95 | of this model |
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96 | that the total 'true' path length of the particle does not depend the |
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97 | length of the steps . Most of the algorithms used in simulations do not have |
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98 | these properties. |
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99 | |
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100 | In the case of heavy charged particles ( \(\mu,\pi,proton,etc.\) ) the |
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101 | mean transport free path is calculated from the \(e+/e-\) \(\lambda\) values |
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102 | with a 'scaling'. |
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103 | |
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104 | In its present form the model computes and uses {\em mean} path length |
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105 | corrections and lateral displacements, the only {\em random} quantity is |
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106 | the scattering angle \(\theta\) which is sampled according to the model |
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107 | function \( f \). |
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108 | |
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109 | The G4MultipleScattering process has 'AlongStep' and 'PostStep' |
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110 | parts. |
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111 | |
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112 | The AlongStepGetPhysicalInteractionLength function performs the\linebreak |
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113 | \mbox{'t' step \(\Rightarrow\) 'z' step} transformation . It should be called after the |
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114 | other physics GetPhysicalInteractionLength functions but before |
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115 | the GetPhysicalInteractionLength of the transportation process.The |
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116 | reason for this restriction is the following: The physics processes |
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117 | 'feel' the true path length travelled by the particle , the geometry |
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118 | (transport) uses the 'z' step length.If we want to compare the minimum |
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119 | step size coming from the physics with the constraint of the geometry, |
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120 | we have make the transformation. |
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121 | |
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122 | The AlongStepDoIt function of the process performs the inverse, |
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123 | 'z'\(\Rightarrow\)'t' transformation.This function should be called after the |
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124 | AlongStepDoIt of the transportation process , i.e. after the particle |
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125 | relocation determined by the geometrical step length, but before applying |
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126 | any other (physics) AlongStepDoIt. |
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127 | |
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128 | The PostStepGetPhysicalInteractionLength part of the multiple |
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129 | scattering process is very simple , it sets the force flag to 'Forced' |
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130 | in order to ensure the call of the PostStepDoIt in every step and |
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131 | returns a big value as interaction length (that means that the multiple |
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132 | scattering process does not restrict the step size). |
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133 | |
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134 | \section{Status of this document} |
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135 | 9.10.98 created by L. Urb\'an. |
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136 | \\5.12.98 editing by J.P. Wellisch. |
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137 | |
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138 | |
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139 | \begin{thebibliography}{99} |
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140 | \bibitem{msc.moliere} |
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141 | {\em Z. Naturforsch. 3a (1948) 78. } |
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142 | \bibitem{msc.fernandez}J. M. Fernandez-Varea et al. |
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143 | {\em NIM B73 (1993) 447.} |
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144 | \bibitem{msc.goudsmit}S. Goudsmit and J. L. Saunderson. |
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145 | {\em Phys. Rev. 57 (1940) 24. } |
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146 | \bibitem{msc.lewis} H. W. Lewis. |
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147 | {\em Phys. Rev. 78 (1950) 526. } |
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148 | \bibitem{msc.liljequist1} D. Liljequist and M. Ismail. |
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149 | {\em J.Appl.Phys. 62 (1987) 342. } |
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150 | \bibitem{msc.liljequist2} D. Liljequist et al. |
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151 | {\em J.Appl.Phys. 68 (1990) 3061. } |
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152 | \end{thebibliography} |
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