[1208] | 1 | \chapter{Optical Elastic Model} |
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| 2 | |
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| 3 | \section{ The model description } |
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| 4 | |
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| 5 | This model is based on the optical approach in which a nucleus is considered to be a drop of |
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| 6 | an absorptive and refractive medium. Absorption results in the diffraction of a projectile |
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| 7 | hadron, while refraction (scattering) in the surface layer provides some smoothing of the |
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| 8 | diffraction picture. The differential cross section in the center mass system can be written |
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| 9 | \cite{bethe40,ap45,bs98}: |
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| 10 | \[ |
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| 11 | \frac{d\sigma_{el}}{d\Omega}= R^2F_d^2(k\,d\,\theta) |
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| 12 | \left\{\frac{J_1^2(k\,R\,\theta)}{\theta^2}+ |
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| 13 | \left[(\gamma\,k)^2+ |
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| 14 | (\delta\,k^2\,\theta)^2\right]J_0^2(k\,R\,\theta)\right\}, |
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| 15 | \quad d\Omega=2\pi\sin\theta\,d\theta, |
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| 16 | \] |
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| 17 | where $J_k$ is the Bessel function of k-th order ($k = 0,1$) and $\Omega$ and $\theta$ are |
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| 18 | the solid and polar scattering angles, respectively. |
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| 19 | $R$ and $d$ are nuclear geometrical parameters of the Woods-Saxon density form |
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| 20 | \[ |
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| 21 | \rho(r)=\rho_o\left\{1+\displaystyle\exp\left[\frac{(r-R)}{d}\right]\right\}^{-1}, |
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| 22 | \] |
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| 23 | in which $k$ is the projectile wave vector ($k=p/\hbar\,c$, $p$ is the projectile |
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| 24 | momentum multiplied by $c$), and $F_d$ is the dumping factor: |
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| 25 | \[ |
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| 26 | F_d(k\,d\,\theta)=\frac{\pi\,k\,d\,\theta}{\sinh(\pi\,k\,d\,\theta)}. |
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| 27 | \] |
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| 28 | $\gamma$ is the refraction (nuclear scattering) parameter, and $\delta$ is the |
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| 29 | spin-orbital interaction parameter. |
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| 30 | |
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| 31 | The model can be modified for charged particles by taking into account that a simple |
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| 32 | summation of nuclear and Coulomb cross sections does not work. However for small Coulomb |
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| 33 | scattering |
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| 34 | $\delta f_{Coulomb}(\theta)\ll f_{nuclear}(\theta)$ the amplitudes can be summarized: |
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| 35 | \[ |
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| 36 | \frac{d\sigma_{el}}{d\Omega} = | \delta f_{Coulomb}(\theta)+f_{nuclear}(\theta)|^2. |
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| 37 | \] |
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| 38 | This case is assumed in the diffuse optical model considered, where we should correct the |
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| 39 | nuclear refraction (scattering) term: |
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| 40 | \[ |
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| 41 | \gamma\,k \rightarrow \gamma\,k + \frac{n}{2kR}\left[\sin^2(\frac{\theta}{2}) + A_m \right]^{-1}, |
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| 42 | \quad n = \frac{\alpha Z_1 Z_2}{\beta} \ll kR, |
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| 43 | \] |
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| 44 | where $n$ is Zommerfeld parameter for the Coulomb field, $\beta=v/c$ is the particle velocity, |
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| 45 | and $Z_1$ and $Z_2$ are the hadron and nucleus charges, respectively. |
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| 46 | The parameter $A_m$ reflects Coulomb atomic shell screening and can be estimated as: |
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| 47 | \[ |
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| 48 | A_m = \frac{1.13 + 3.76n^2}{(1.77ka_oZ^{-1/3})^2}, |
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| 49 | \] |
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| 50 | where $a_o$ is the Bohr radius, and $Z$ is the atomic number. |
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| 51 | |
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| 52 | This method corresponds to elastic Coulomb-Wentzel scattering recently implemented |
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| 53 | in electromagnetic physics \cite{fer93}: |
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| 54 | \[ |
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| 55 | \frac{d\sigma_{el}^{cw}}{d\Omega}= \frac{n^2}{4k^2} |
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| 56 | \left[\sin^2(\frac{\theta}{2}) + A_m \right]^{-2}, \quad V(r)=\frac{e^2Z_1Z_2}{r^2}\exp(-r/R), |
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| 57 | \quad A_m = (2kR)^{-2}. |
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| 58 | \] |
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| 59 | The Coulomb-Wentzel cross-section reads: |
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| 60 | \[ |
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| 61 | \sigma_{el}^{cw}= \frac{n^2}{k^2}\frac{\pi}{A_m(1+A_m)}, \quad |
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| 62 | \sigma_{el}^{cw}(\theta_1,\theta_2)=2\pi\frac{n^2}{k^2}\frac{\cos\theta_1-\cos\theta_2} |
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| 63 | {(1-\cos\theta_1+2A_m)(1-\cos\theta_2+2A_m)}. |
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| 64 | \] |
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| 65 | This cross-section is much bigger than the integral nuclear cross section. However the |
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| 66 | main contribution comes from very small angles where the differential nuclear elastic |
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| 67 | cross section is approximately constant. In this region the Coulomb correction can be |
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| 68 | switched off. This avoids double counting of Coulomb scattering in electromagnetic and nuclear |
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| 69 | processes. |
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| 70 | |
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| 71 | \section{Validation} |
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| 72 | |
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| 73 | |
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| 74 | Figs. \ref{pPb1GeV}-\ref{pipC9p92GeVc} show the elastic differential cross sections of protons |
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| 75 | and pions with energies in the range 1 - 301 GeV on different targets: helium, carbon, silicon |
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| 76 | and lead. |
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| 77 | One can see that Coulomb corrections improve the description of differential cross sections |
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| 78 | in the regions of the minima. |
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| 79 | |
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| 80 | \begin{figure*} |
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| 81 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pPbsumT1GeV.eps} |
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| 82 | \vspace{-0.5cm} |
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| 83 | \caption{ Differential elastic scattering cross sections of protons with energy |
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| 84 | 1 GeV on lead. Curves show pure nuclear and Coulomb-corrected models. Points are |
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| 85 | experimental data. The Coulomb cross section dominates at small angles resulting |
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| 86 | in a large integral cross section: $\sigma_{el}^{nucl}=1210$ mb, with $\lambda = 25$ cm, |
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| 87 | $\sigma_{el}^{cw}(2^{0}, 16^{o})=642.65$ mb, with $\lambda = 47$ cm, |
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| 88 | $\sigma_{el}^{cw}(0,1^o)\sim \sigma_{el}^{cw} = 2.36423\cdot10^{9}$ mb, |
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| 89 | with $\lambda = 0.13$ micron.} |
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| 90 | \label{pPb1GeV} |
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| 91 | \end{figure*} |
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| 92 | |
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| 93 | \begin{figure*} |
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| 94 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pSisumT1GeV.eps} |
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| 95 | \vspace{-0.5cm} |
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| 96 | \caption{ Differential elastic scattering cross sections of protons with energy |
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| 97 | 1 GeV on silicon. Curves show pure nuclear and Coulomb-corrected models. Points are |
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| 98 | experimental data. The Coulomb cross section dominates at small angles resulting in |
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| 99 | a large integral cross-section: |
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| 100 | $\sigma_{el}^{cw}(0,1^o)\sim \sigma_{el}^{cw}=5.46779\cdot10^{8}$ mb, |
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| 101 | with $\lambda = 0.366068$ micron.} |
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| 102 | \label{pSi1GeV} |
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| 103 | \end{figure*} |
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| 104 | |
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| 105 | \begin{figure*} |
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| 106 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pPbsum9p92GeVc.eps} |
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| 107 | \vspace{-0.5cm} |
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| 108 | \caption{ Differential elastic scattering cross sections of protons with momentum |
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| 109 | 9.92 GeV/c on lead. Curves show pure nuclear and Coulomb-corrected models. Points are |
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| 110 | experimental data. The Coulomb cross section dominates at small angles resulting in a |
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| 111 | large integral cross section |
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| 112 | $\sigma_{el}^{cw}(0,4 \ mrad)\sim \sigma_{el}^{cw}=2.11854\cdot10^{9}$ mb, |
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| 113 | with $\lambda = 0.143101$ micron. |
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| 114 | $\sigma_{el}^{cw}(4 \ mrad,40 \ mrad)=919.547$ mb, |
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| 115 | with $\lambda = 329689$ micron (33 cm).} |
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| 116 | \label{pPb9p92GeVc} |
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| 117 | \end{figure*} |
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| 118 | |
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| 119 | \begin{figure*} |
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| 120 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pHeT45GeVsum.eps} |
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| 121 | \vspace{-0.5cm} |
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| 122 | \caption{ Differential elastic scattering cross sections of protons with energy |
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| 123 | 45 GeV on helium. Curves show pure nuclear and Coulomb-corrected models. Points are |
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| 124 | experimental data.} |
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| 125 | \label{pHe45GeV} |
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| 126 | \end{figure*} |
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| 127 | |
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| 128 | \begin{figure*} |
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| 129 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pHeT301GeVsum.eps} |
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| 130 | \vspace{-0.5cm} |
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| 131 | \caption{ Differential elastic scattering cross sections of protons with energy |
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| 132 | 301 GeV on helium. Curves show pure nuclear and Coulomb-corrected models. Points are |
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| 133 | experimental data.} |
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| 134 | \label{pHe301GeV} |
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| 135 | \end{figure*} |
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| 136 | |
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| 137 | \begin{figure*} |
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| 138 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pimPbsum9p92GeVc.eps} |
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| 139 | \vspace{-0.5cm} |
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| 140 | \caption{ Differential elastic scattering cross sections of $\pi^{-}$ with momentum |
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| 141 | 9.92 GeV/c on lead. Curves show pure nuclear and Coulomb-corrected models. Points are |
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| 142 | experimental data. } |
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| 143 | \label{pimPb9p92GeVc} |
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| 144 | \end{figure*} |
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| 145 | |
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| 146 | \begin{figure*} |
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| 147 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pipPbsum9p92GeVc.eps} |
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| 148 | \vspace{-0.5cm} |
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| 149 | \caption{ Differential elastic scattering cross sections of $\pi^{+}$ with momentum |
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| 150 | 9.92 GeV/c on lead. Curves show pure nuclear and Coulomb-corrected models. Points are |
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| 151 | experimental data.} |
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| 152 | \label{pipPb9p92GeVc} |
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| 153 | \end{figure*} |
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| 154 | |
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| 155 | \begin{figure*} |
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| 156 | \centering \includegraphics[width=10cm,height=7cm]{hadronic/elastic/pipCsum9p92GeVc.eps} |
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| 157 | \vspace{-0.5cm} |
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| 158 | \caption{ Differential elastic scattering cross sections of $\pi^{+}$ with momentum |
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| 159 | 9.92 GeV/c on carbon. Curves show pure nuclear and Coulomb-corrected models. Points are |
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| 160 | experimental data.} |
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| 161 | \label{pipC9p92GeVc} |
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| 162 | \end{figure*} |
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| 163 | |
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| 164 | \section{Implementation} |
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| 165 | |
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| 166 | This model is implemented in the class {\em G4DiffuseElastic}. It conforms to the general |
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| 167 | interface of hadronic models and performs initialization for different elements during |
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| 168 | tracking. |
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| 169 | |
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| 170 | |
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| 171 | \section{Status of this document} |
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| 172 | 18.11.07 created by V. Grichine \\ |
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| 173 | |
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| 174 | |
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| 175 | |
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| 176 | \begin{latexonly} |
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| 177 | |
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| 178 | \begin{thebibliography}{99} |
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| 179 | |
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| 180 | \bibitem{bethe40} P. Placzec, H.A. Bethe, {\em Phys. Rev.}, {\bf 57} (1940) 1075A. |
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| 181 | |
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| 182 | \bibitem{ap45} A. Akhiezer, I. Pomeranchuk, {\em J. Phys. (USSR)}, {\bf 9} (1945) 471; |
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| 183 | {\em Sov. Phys. USPEKHI}, {\bf 65} (1958) 593. |
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| 184 | |
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| 185 | \bibitem{bs98} Yu.A. Berezhnoy, V.A. Slipko, {\em Int. J. of Mod. Phys.}, {\bf E7} (1998) 723. |
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| 186 | |
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| 187 | \bibitem{fer93} J.M. Fernandez-Varea, R. Moyol, J. Baro, F. Salvat, |
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| 188 | {\em Nucl. Instr. and Math. in Phys. Res.}, {\bf B73} (1993) 447. |
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| 189 | |
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| 190 | |
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| 191 | \end{thebibliography} |
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| 192 | |
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| 193 | \end{latexonly} |
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| 194 | |
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| 195 | \begin{htmlonly} |
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| 196 | |
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| 197 | \section{Bibliography} |
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| 198 | |
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| 199 | \begin{enumerate} |
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| 200 | |
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| 201 | \item P. Placzec, H.A. Bethe, {\em Phys. Rev.}, {\bf 57} (1940) 1075A. |
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| 202 | |
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| 203 | |
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| 204 | \item A. Akhiezer, I. Pomeranchuk, {\em J. Phys. (USSR)}, {\bf 9} (1945) 471; |
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| 205 | {\em Sov. Phys. USPEKHI}, {\bf 65} (1958) 593. |
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| 206 | |
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| 207 | \item Yu.A. Berezhnoy, V.A. Slipko, {\em Int. J. of Mod. Phys.}, {\bf E7} (1998) 723. |
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| 208 | |
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| 209 | \item J.M. Fernandez-Varea, R. Moyol, J. Baro, F. Salvat, |
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| 210 | {\em Nucl. Instr. and Math. in Phys. Res.}, {\bf B73} (1993) 447. |
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| 211 | |
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| 212 | |
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| 213 | \end{enumerate} |
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| 214 | |
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| 215 | \end{htmlonly} |
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| 216 | |
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