1 | \section[Synchrotron Radiation]{Synchrotron Radiation} |
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2 | |
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3 | Synchrotron radiation photons are produced when ultra-relativistic electrons |
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4 | travel along an approximately circular path. In the following treatment, |
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5 | the magnetic field is assumed to be constant and uniform, and the radius of |
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6 | curvature of the electron is assumed to be constant over its trajectory. |
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7 | |
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8 | \subsection{Spectral and Angular Distributions of Synchrotron Radiation} |
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9 | |
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10 | The spectral distribution of the mean number of synchrotron radiation |
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11 | photons, $d\bar{N}/d\omega$, produced by an ultra-relativistic electron |
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12 | along a circular trajectory of length $L$, can be expressed in terms of the |
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13 | mean energy loss spectrum $d\bar{\Delta}/d\omega$ \cite{synch.maier}: |
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14 | |
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15 | \begin{equation} |
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16 | \label{synch.a} |
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17 | \frac{d\bar{N}}{d\omega} = \frac{1}{\omega}\frac{d\bar{\Delta}}{d\omega} = |
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18 | \frac{\sqrt{3}}{2\pi}\alpha\left(\frac{L\gamma}{R}\right)\frac{1}{\omega_{c}} |
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19 | \int_{\omega/\omega_{c}}^{\infty}K_{5/3}(\eta)d\eta . |
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20 | \end{equation} |
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21 | |
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22 | Here, |
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23 | \begin{eqnarray*} |
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24 | \omega & & \mbox{photon energy} \\ |
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25 | \alpha & & \mbox{fine structure constant} \\ |
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26 | R & & \mbox{instantaneous radius of curvature of the trajectory} \\ |
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27 | K & & \mbox{Macdonald function} \\ |
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28 | \omega_c = 1.5\beta(\hbar c/R)\gamma^3 & & \mbox{characteristic energy of synchrotron radiation.} \\ |
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29 | \end{eqnarray*} |
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30 | $\beta$ is the ratio of the electron velocity $v$ to $c$, |
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31 | $\gamma = 1/\sqrt{1 - \beta^2}$, and $\eta$ is an arbitrary integration |
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32 | variable. In the SI system of units: $R(m) = P(GeV/c)/0.3B_{\bot}(T)$ , |
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33 | where $B_{\bot}$ is the component of magnetic flux density perpendicular to |
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34 | the electron velocity, and $P$ is the electron momentum. |
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35 | |
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36 | In order to simulate the energy spectrum of synchrotron radiation using |
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37 | the Monte Carlo method, $\bar{N}_{>\omega}$, the mean number of photons |
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38 | above a given energy $\omega$, must be determined. This is done by |
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39 | integrating Eq.~\ref{synch.a} over energy, after first transforming |
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40 | $d\bar{N}/d\omega$ by using the integral representation of the Macdonald |
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41 | function \cite{synch.abram}: |
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42 | |
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43 | \begin{eqnarray} |
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44 | \label{synch.b} |
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45 | \bar{N}_{>\omega}& = & |
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46 | \int_{\omega}^{\infty}\frac{d\bar{N}}{d\omega'}d\omega' \nonumber\\ |
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47 | & = & |
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48 | \frac{\sqrt{3}}{2\pi}\alpha\left(\frac{L\gamma}{R}\right) |
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49 | \int_0^{\infty}\frac{\cosh\left(\frac{5}{3}t\right)}{\cosh^2(t)} |
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50 | \exp\left[-\frac{\omega}{\omega_c}\cosh(t)\right]dt . |
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51 | \end{eqnarray} |
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52 | Here, $t$ is also an arbitrary integration variable. The latter integral |
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53 | is calculated numerically by the quadrature Laguerre |
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54 | formula \cite{synch.korn}. Calculations indicate that about 50 roots of |
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55 | the Laguerre polynomials are required in order for the accuracy of the |
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56 | integral estimation to be better than $10^{-4}$ \cite{synch.bag}. |
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57 | |
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58 | The Monte Carlo method also requires the mean number of synchrotron |
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59 | radiation photons at all energies, $\bar{N}$ (= $\bar{N}_{>0}$), in order |
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60 | to determine the next occurrence of synchrotron radiation along a |
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61 | trajectory, and to normalize the spectral distribution of the radiation. |
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62 | Setting $\omega = 0$ in Eq.~\ref{synch.b} yields |
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63 | \begin{eqnarray} |
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64 | \bar{N} = \bar{N}_{>0}& = & |
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65 | \frac{\sqrt{3}}{2\pi}\alpha\left(\frac{L\gamma}{R}\right) |
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66 | \int_0^{\infty}\frac{\cosh\left(\frac{5}{3}t\right)}{\cosh^2(t)}dt \nonumber\\ |
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67 | & = & |
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68 | \frac{5}{2\sqrt{3}}\alpha\left(\frac{L\gamma}{R}\right) \approx |
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69 | 10^{-2}\left(\frac{L\gamma}{R}\right) . |
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70 | \end{eqnarray} |
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71 | |
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72 | Qualitatively this result can be manipulated using the fact that the mean |
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73 | number of photons produced along the formation zone length |
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74 | $z \approx R/\gamma$ is proportional to $\alpha$. Then for length $L$, |
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75 | $\bar{N} \approx \alpha L/(R/\gamma)$. Note that when $\gamma\gg1$, and |
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76 | $R \sim \gamma$, $\bar{N}$ does not depend on the electron energy but is |
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77 | defined by the values of $L$ and $B_{\bot}$ only. Instead, it is the mean |
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78 | energy loss due to synchrotron radiation $\bar\Delta$, corresponding to a |
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79 | trajectory of length $L$, that displays the characteristic |
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80 | relativistic rise: |
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81 | |
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82 | \begin{equation} |
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83 | \bar{\Delta} = \int_0^{\infty}\omega\frac{d\bar{N}}{d\omega}d\omega = |
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84 | \frac{2}{3}\alpha\hbar c\left(\frac{L\gamma^2}{R^2}\right)\beta\gamma^2 = |
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85 | \frac{8\bar{N}}{15\sqrt{3}}\omega_{c} |
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86 | \approx 0.31\bar{N}\omega_{c} \sim \gamma^2 . |
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87 | \end{equation} |
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88 | |
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89 | The angular distribution of synchrotron radiation produced by |
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90 | ultra-relativistic electrons shows a clear 'searchlight' effect. Most of |
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91 | the photons are radiated within an angular range of order $1/\gamma$ |
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92 | centered on the electron trajectory direction. In the interesting region |
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93 | of $\gamma > 10^3$ the angular resolution of X-ray and gamma detectors |
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94 | usually does not allow the details of the angular distribution to be |
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95 | measured. Therefore, the angular distribution is set to be flat in the |
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96 | range $0 - 1/\gamma$. |
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97 | |
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98 | \subsection{Simulating Synchrotron Radiation} |
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99 | |
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100 | The distance $x$ along the electron/positron trajectory to the next |
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101 | occurrence of a synchrotron radiation photon is simulated according to the |
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102 | exponential distribution, $exp(-x\bar{N}/L)$. The energy $\omega$ of the |
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103 | photon is simulated according to the distribution |
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104 | $\bar{N}_{>\omega}/\bar{N}$. The direction of the photon |
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105 | ($\theta$, $\varphi$) is generated relative to the local $z$-axis which is |
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106 | taken to be along the instantaneous direction of the electron. $\theta$ |
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107 | and $\varphi$ are distributed randomly in the ranges [0, $1/\gamma$] and |
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108 | [0, $2\pi$], respectively. |
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109 | |
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110 | \subsection{Status of this document} |
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111 | 15.10.98 created by V.Grichine \\ |
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112 | 02.12.02 re-written by D.H. Wright \\ |
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113 | |
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114 | \begin{latexonly} |
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115 | |
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116 | \begin{thebibliography}{99} |
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117 | |
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118 | \bibitem{synch.maier} R. Maier, |
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119 | {\em Synchrotron Radiation, CERN Report 91-04, 97-115} (1991). |
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120 | |
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121 | \bibitem{synch.abram} Edited by M. Abramovwitz and I.A. Stegan, |
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122 | {\em Handbook of Mathemaatical Functions, |
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123 | NBS Applied Mathematics Series 55} (1964). |
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124 | |
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125 | \bibitem{synch.korn}G.A. Korn and T.M. Korn, |
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126 | {\em Mathematical Handbook for scientists and |
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127 | engineers, McGRAW-HILL BOOK COMPANY, INC} (1961). |
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128 | |
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129 | \bibitem{synch.bag} A.V. Bagulya and V.M. Grichine, |
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130 | {Bulletin of Lebedev Institute, no.9-10, 7 } (1998). |
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131 | |
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132 | \end{thebibliography} |
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133 | |
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134 | \end{latexonly} |
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135 | |
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136 | \begin{htmlonly} |
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137 | |
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138 | \subsection{Bibliography} |
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139 | |
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140 | \begin{enumerate} |
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141 | \item R. Maier, |
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142 | {\em Synchrotron Radiation, CERN Report 91-04, 97-115} (1991). |
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143 | |
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144 | \item Edited by M. Abramovwitz and I.A. Stegan, |
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145 | {\em Handbook of Mathemaatical Functions, |
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146 | NBS Applied Mathematics Series 55} (1964). |
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147 | |
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148 | \item G.A. Korn and T.M. Korn, |
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149 | {\em Mathematical Handbook for scientists and |
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150 | engineers, McGRAW-HILL BOOK COMPANY, INC} (1961). |
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151 | |
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152 | \item A.V. Bagulya and V.M. Grichine, |
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153 | {Bulletin of Lebedev Institute, no.9-10, 7 } (1998). |
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154 | |
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155 | \end{enumerate} |
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156 | |
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157 | \end{htmlonly} |
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