Retrieving nuclear information from protons propagating through a thick target

The multiple scattering of high-energy particles in a thick target is formulated in an impact parameter representation. A formalism similar but not identical to that of Molière is obtained. We show that calculations of particle beam broadening due to multiple Coulomb scattering alone can be given in closed form. The focus of this study is on determining whether the broadening of the Coulomb angular distribution prevents the retrieval of nuclear-interaction information from the measurement of the angular distributions of charged particles scattered from a thick target. For this purpose, we study multiple scattering with both the nuclear and Coulomb interactions included, and we do not make a small-angle expansion. Conditions for retrieving nuclear information from high-energy protons propagating through a block of material are obtained.

Figure 1

Crosses: ${\varphi }_{1c}(B)$ for ${\sigma }_{1c}(Q)=(Q^2+1){}^{-5/2}$. Solid curve: corresponding multistep ${\Phi }_{\mathit{Mc}}(B)$ if $\nu =4$. Circles and dashed curve: ${\varphi }_{1c}(B)$ and ${\Phi }_{\mathit{Mc}}(B)$ (with $\nu =4)$ for ${\sigma }_{1c}=(Q^2+1){}^{-4}$. All profiles normalized to 1 at $B=0$.

Figure 2

Crosses: ${\sigma }_{1c}(Q)=(Q^2+1){}^{-5/2}$. Solid curve: corresponding ${\sigma }_{\mathit{Mc}}(Q)$ for $\nu =4$. Circles: ${\sigma }_{1c}(Q)=(Q^2+1){}^{-4}$. Dashed curve: corresponding ${\sigma }_{\mathit{Mc}}(Q)$ for $\nu =4$. All cross sections normalized to 1 at $Q=0$.

Figure 3

Plots of $\log {}_{10}{\sigma }_1$ (solid curve) and $\log {}_{10}{\sigma }_{1c}$ (dashed curve) as functions of $Q$.

Figure 4

Plots of $\log {}_{10}{\varphi }_1,$ $\log {}_{10}{\varphi }_{1n}$, and $\log {}_{10}{\varphi }_{1c}$ as functions of $B$.

Figure 5

Dependences on $Q$ of $\log {}_{10}{\sigma }_1$ of single scattering and $\log {}_{10}{P}_M$ of multiple scattering when $\nu =4$, 9, and 16.

Figure 6

Plots of $\log {}_{10}{\sigma }_{1c}(Q)$ and $\log {}_{10}{P}_{\mathit{Mc}}(Q)$ for $\nu =4,$ 9 and 16.

Figure 7

Plots of $\log {}_{10}{P}_{\mathit{Mn}}(Q)$ for $\nu =4$, 9, and 16.

Figure 8

Plots of $\log {}_{10}|{\sigma }_{1n}(Q)|$ for a nuclear signal, and probability distribution $\log {}_{10}{P}_{\mathit{Mn}}^{\text{'}}(Q)$ when $\nu =4$, 9 and 16.

Figure 9

Dependences on $\log {}_{10}\nu$ of $\log {}_{10}\langle Q^2\rangle$ for $P_M(Q)$ (solid curve), $P_{\mathit{Mc}}(Q)$ (dashed curve), and $P_{\mathit{Mn}}(Q)$ (dotted curve). Note that the slopes $\simeq 1$ when $\nu \gg 10$.

Figure 10

Differential cross sections of $p{\text{−}}^{208}$Pb elastic scattering at 20 GeV.

Figure 11

Dependences on $Q$ of $\log {}_{10}{\sigma }_1$ and of $\log {}_{10}{\sigma }_M$ for $\nu =2000$ and $\nu =10000$. All with $r={10}^{-2}$.

Figure 12

Dependences on $Q$ of $\log {}_{10}{\sigma }_1$ and of $Y=\log {}_{10}{\sigma }_M$ for $\nu =2000$ and $\nu =10000$. All with $r=0.5\times {10}^{-2}$.

Figure 13

Dependences on $Q$ of $\log {}_{10}{\sigma }_1$ and of $\log {}_{10}{\sigma }_M$ for $\nu ={10}^5$ and $\nu ={10}^6$. All with $r={10}^{-3}$.