1 | \\ |
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2 | $x$ is a kinetic variable of the particle : |
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3 | $ x = \log_{10}(\gamma \beta) = \ln(\gamma^{2} \beta^{2})/4.606 $, \linebreak |
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4 | and $\delta(x)$ is defined by |
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5 | \begin{equation} |
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6 | % \label{muion.de1} |
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7 | \begin{array}{rll} |
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8 | \mbox{for} & x < x_0 : & \delta(x) = 0 \\ |
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9 | \mbox{for} & x \in [x_0,\ x_1] : & \delta(x) = 4.606 x - C + a(x_1 - x)^m \\ |
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10 | \mbox{for} & x > x_1 : & \delta(x) = 4.606 x - C |
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11 | \end{array} |
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12 | \end{equation} |
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13 | where the matter-dependent constants are calculated as follows: |
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14 | \begin{equation} |
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15 | % \label{muion.de2} |
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16 | \begin{array}{lcl} |
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17 | h\nu_p & = & \mbox{ plasma energy of the medium } |
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18 | = \sqrt{4\pi n_{el} r_e^3} mc^2/\alpha |
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19 | = \sqrt{4\pi n_{el} r_e} \hbar c \\ |
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20 | C & = & 1 + 2 \ln (I/h\nu_p) \\ |
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21 | x_a & = & C/4.606 \\ |
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22 | a & = & 4.606(x_a - x_0)/(x_1 - x_0)^m \\ |
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23 | m & = & 3 . |
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24 | \end{array} |
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25 | \end{equation} |
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26 | For condensed media |
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27 | $$ |
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28 | \begin{array}{ll} |
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29 | I < 100 \: \mbox{eV} & \left \{ |
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30 | \begin{array}{rll} |
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31 | \mbox{for } C \leq 3.681 & x_0 = 0.2 & x_1 = 2 \\ |
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32 | \mbox{for } C > 3.681 & x_0 = 0.326 C - 1.0 & x_1 = 2 |
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33 | \end{array} \right . \\ |
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34 | I \geq 100 \: \mbox{eV} & \left \{ |
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35 | \begin{array}{rll} |
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36 | \mbox{for } C \leq 5.215 & x_0 = 0.2 & x_1 = 3 \\ |
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37 | \mbox{for } C > 5.215 & x_0 = 0.326 C - 1.5 & x_1 = 3 |
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38 | \end{array} \right . |
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39 | \end{array} |
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40 | $$ |
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41 | and for gaseous media |
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42 | \[ |
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43 | \begin{array}{rlll} |
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44 | \mbox{for} & C < 10. & x_0 = 1.6 & x_1 = 4 \\ |
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45 | \mbox{for} & C \in [10.0,\ 10.5[ & x_0 = 1.7 & x_1 = 4 \\ |
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46 | \mbox{for} & C \in [10.5,\ 11.0[ & x_0 = 1.8 & x_1 = 4 \\ |
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47 | \mbox{for} & C \in [11.0,\ 11.5[ & x_0 = 1.9 & x_1 = 4 \\ |
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48 | \mbox{for} & C \in [11.5,\ 12.25[ & x_0 = 2. & x_1 = 4 \\ |
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49 | \mbox{for} & C \in [12.25,\ 13.804[ & x_0 = 2. & x_1 = 5 \\ |
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50 | \mbox{for} & C \geq 13.804 & x_0 = 0.326 C -2.5 & x_1 = 5 . |
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51 | \end{array} |
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52 | \] |
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