1 | The final state of elastic scattering is described by sampling the differential |
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2 | scattering cross-sections |
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3 | ${{\rm d} \sigma \over {\rm d} \Omega}$. |
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4 | Two representations |
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5 | are supported |
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6 | for the normalised differential cross-section for |
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7 | elastic scattering. |
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8 | The first is a tabulation of the differential cross-section, |
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9 | as a |
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10 | function of the cosine of the scattering angle $\theta$ and the kinetic energy |
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11 | $E$ of the incoming |
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12 | neutron. |
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13 | $${{\rm d} \sigma \over {\rm d} \Omega}~=~ |
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14 | {{\rm d} \sigma \over {\rm d} \Omega}\left(\cos{\theta,~E}\right)$$ |
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15 | The tabulations used are normalised by $\sigma/(2\pi)$ so the integral of the |
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16 | differential cross-sections over the scattering angle yields unity. |
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17 | |
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18 | In the second representation, the normalised cross-section are represented |
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19 | as a series of legendre polynomials $P_l(\cos{\theta})$, and the |
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20 | legendre coefficients $a_l$ are |
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21 | tabulated as a function of the incoming energy of the neutron. |
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22 | $${2\pi\over\sigma (E)}{{\rm d} \sigma \over {\rm d} \Omega}\left(\cos{\theta,~E}\right)~=~ |
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23 | \sum_{l=0}^{n_l} {2l+1\over 2}a_l(E)P_l(\cos{\theta})$$ |
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24 | |
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25 | Describing the details of the sampling procedures is outside the scope |
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26 | of this paper. |
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27 | |
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28 | An example of the result we show in figure \ref{elastic} for the elastic |
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29 | scattering of 15~MeV neutrons off Uranium |
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30 | a comparison of the simulated angular distribution of the scattered neutrons |
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31 | with evaluated data. |
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32 | The points are the evaluated data, |
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33 | the histogram is the Monte Carlo prediction. |
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34 | |
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35 | In order to provide full |
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36 | test-coverage for the algorithms, similar tests have been performed for |
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37 | ${\rm^{72}Ge}$, |
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38 | ${\rm^{126}Sn}$, |
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39 | ${\rm^{238}U}$, |
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40 | ${\rm^{4}He}$, and |
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41 | ${\rm^{27}Al}$ for a set of neutron kinetic energies. |
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42 | The agreement is very good for all values of scattering angle and neutron |
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43 | energy investigated. |
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44 | \begin{figure}[b!] % fig 1 |
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45 | % \centerline{\epsfig{file=hadronic/lowEnergyNeutron/neutrons/plots/elastic.u238.14mev.costh.epsi,height=5.5in,width=3.5in}} |
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46 | \includegraphics[angle=0,scale=0.6]{hadronic/lowEnergyNeutron/neutrons/plots/elastic.u238.14mev.costh.epsi} |
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47 | \vspace{10pt} |
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48 | \caption{Comparison of data and Monte Carlo for the angular distribution of |
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49 | 15~MeV neutrons scattered |
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50 | elastically off Uranium ($^{238}U$). The points are evaluated data, and the histogram is |
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51 | the Monte Carlo prediction. The lower plot excludes the forward peak, to better show |
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52 | the Frenel structure of the angular distribution of the scattered neutron.} |
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53 | \label{elastic} |
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54 | \end{figure} |
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