1 | \chapter[Hadron-nucleus Elastic Scattering at Medium/High Energy]{Hadron-nucleus Elastic Scattering at Medium and High Energy} |
---|
2 | |
---|
3 | \vspace{2ex} |
---|
4 | \section{Method of Calculation} |
---|
5 | |
---|
6 | The Glauber model \cite{helast.1} is used as an alternative method of |
---|
7 | calculating differential cross sections for elastic and quasi-elastic |
---|
8 | hadron-nucleus scattering at high and intermediate energies. |
---|
9 | |
---|
10 | For high energies this includes corrections for inelastic screening and |
---|
11 | for quasi-elastic scattering the exitation of a discrete level or a state in |
---|
12 | the continuum is considered. |
---|
13 | |
---|
14 | The usual expression for the Glauber model amplitude for multiple scattering |
---|
15 | was used |
---|
16 | |
---|
17 | \begin{equation} |
---|
18 | F(q)=\frac{ik}{2\pi} \int d^2{b}e^{\vec{\mathstrut q} |
---|
19 | \cdot \vec{\mathstrut b}} M(\vec b) . |
---|
20 | \label{helast.eq1} |
---|
21 | \end{equation} |
---|
22 | Here $M(\vec b)$ is the hadron-nucleus amplitude in the impact |
---|
23 | parameter representation |
---|
24 | \begin{equation} |
---|
25 | M(\vec b) = 1-[1-e^{-A\int d^{3}r\Gamma(\vec{\mathstrut b}- |
---|
26 | \vec{\mathstrut s})\rho(\vec {r})}]^A, |
---|
27 | \label{helast.eq2} |
---|
28 | \end{equation} |
---|
29 | $k$ is the incident particle momentum, |
---|
30 | $\vec q = \vec k' - \vec k$ is the momentum transfer, and |
---|
31 | $\vec k'$ is the scattered particle momentum. |
---|
32 | Note that $\left| \vec q \right| ^2 = -t$ - invariant momentum transfer |
---|
33 | squared in the center of mass system. |
---|
34 | $\Gamma(\vec {b})$ is the hadron-nucleon amplitude of |
---|
35 | elastic scattering in the impact-parameter representation |
---|
36 | |
---|
37 | \begin{equation} |
---|
38 | \Gamma(\vec b)=\frac{\displaystyle 1} |
---|
39 | {\displaystyle 2\pi ik^{hN}} |
---|
40 | \int d\vec {q} e^{-\vec {\mathstrut q} \cdot \vec{\mathstrut b}} |
---|
41 | f(\vec {q}). |
---|
42 | \label{helast.eq3} |
---|
43 | \end{equation} |
---|
44 | |
---|
45 | The exponential parameterization of the hadron-nucleon amplitude is usually |
---|
46 | used: |
---|
47 | |
---|
48 | \begin{equation} |
---|
49 | f(\vec {q})=\frac{\displaystyle ik^{hN}\sigma^{hN}}{2\pi} |
---|
50 | e^{-0.5q^2B}. |
---|
51 | \label{helast.eq4} |
---|
52 | \end{equation} |
---|
53 | Here $\sigma^{hN}=\sigma_{tot}^{hN}(1-i\alpha)$, |
---|
54 | $\sigma_{tot}^{hN}$ is the total |
---|
55 | cross section of a hadron-nucleon scattering, $B$ is the slope of the |
---|
56 | diffraction cone and $\alpha$ is the ratio of the real to imaginary parts of |
---|
57 | the amplitude at $q=0$. The value $k^{hN}$ is the hadron momentum in the |
---|
58 | hadron-nucleon coordinate system. |
---|
59 | |
---|
60 | The important difference of these calculations from the usual ones |
---|
61 | is that the two-gaussian form of the nuclear density was used |
---|
62 | |
---|
63 | \begin{equation} |
---|
64 | \rho(r) = C (e^{-(r/R_{1})^2}-pe^{-(r/R_2)^2}), |
---|
65 | \label{helast.eq5} |
---|
66 | \end{equation} |
---|
67 | where $R_1$, $R_2$ and $p$ are the fitting parameters and $C$ is a |
---|
68 | normalization constant. |
---|
69 | |
---|
70 | This density representation allows the expressions for amplitude and |
---|
71 | differential cross section to be put into analytical form. It was earlier |
---|
72 | used for light \cite{helast.2, helast.3} and medium \cite{helast.4} nuclei. |
---|
73 | Described below is an extension of this method to heavy nuclei. |
---|
74 | The form \ref{helast.eq5} is not physical for a heavy nucleus, but |
---|
75 | nevertheless works rather well (see figures below). The reason is that |
---|
76 | the nucleus absorbs the hadrons very strongly, especially at small impact |
---|
77 | parameters where the absorption is full. As a result only the peripherial part |
---|
78 | of the nucleus participates in elastic scattering. Eq. \ref{helast.eq5} |
---|
79 | therefore describes only the edge of a heavy nucleus. |
---|
80 | |
---|
81 | Substituting Eqs. \ref{helast.eq5} and \ref{helast.eq4} into Eqs. |
---|
82 | \ref{helast.eq1}, \ref{helast.eq2} and \ref{helast.eq3} yields the following |
---|
83 | formula |
---|
84 | |
---|
85 | \begin{eqnarray} |
---|
86 | F(q) = \frac{ik\pi}{2} |
---|
87 | \sum\limits_{k=1}^A (-1)^k {A\choose k} |
---|
88 | [\frac{\sigma^{hN}}{2\pi(R_1^3-pR_2^3)}]^k |
---|
89 | \sum\limits_{m=0}^{k} (-1)^m {k \choose m} |
---|
90 | \left[ \frac{R_1^3}{R_1^2+2B} |
---|
91 | \right]^{k-m} \nonumber |
---|
92 | \end{eqnarray} |
---|
93 | |
---|
94 | \begin{eqnarray} |
---|
95 | \times \qquad {} \left[ |
---|
96 | \frac{pR_2^3}{R_2^2+2B} |
---|
97 | \right]^{m} \left( |
---|
98 | \frac{m}{R_2^2+2B}+\frac{k-m}{R_1^2+2B} \right)^{-1} \nonumber |
---|
99 | \end{eqnarray} |
---|
100 | |
---|
101 | \begin{eqnarray} |
---|
102 | \times \qquad {} \exp\left[ -\frac{-q^2}{4} |
---|
103 | \left( |
---|
104 | \frac{m}{R_2^2+2B}+ |
---|
105 | \frac{k-m}{R_1^2+2B} \right)^{-1} |
---|
106 | \right]. |
---|
107 | \label{helast.eq6} |
---|
108 | \end{eqnarray} |
---|
109 | % \end{eqnarray} |
---|
110 | |
---|
111 | An analogous procedure can be used to get the inelastic screening corrections |
---|
112 | to the hadron-nucleus amplitude $\Delta M(\vec b)$ \cite{helast.5}. |
---|
113 | In this case an intermediate inelastic diffractive state is created which |
---|
114 | rescatters on the nucleons of the nucleus and then returns into the initial |
---|
115 | hadron. Hence it is nessesary to integrate the production cross section over |
---|
116 | the mass distribution of the exited system $\frac{d\sigma^{diff}}{dtdM_x^2}$. |
---|
117 | The expressions for the corresponding amplitude are quite long and so are not |
---|
118 | presented here. The corrections for the total cross-sections can be found in |
---|
119 | \cite{helast.5}. |
---|
120 | |
---|
121 | The full amplitude is the sum $M(\vec b)+ \Delta M(\vec b)$. |
---|
122 | |
---|
123 | The differential cross section is connected with the amplitude in the |
---|
124 | following way |
---|
125 | |
---|
126 | \begin{equation} |
---|
127 | \frac{d\sigma}{d\Omega_{CM}}=\left| F(q) \right|^2, \qquad |
---|
128 | \frac{d\sigma}{|dt|}=\frac{d\sigma}{dq_{CM}^2}= |
---|
129 | \frac{\pi}{k_{CM}^2} \left|F(q) \right|^2. |
---|
130 | \label{helast.eq7} |
---|
131 | \end{equation} |
---|
132 | |
---|
133 | The main energy dependence of the hadron-nucleus elastic scattering |
---|
134 | cross section comes from the energy dependence of the parameters of |
---|
135 | hadron-nucleon scattering |
---|
136 | ($\sigma_{tot}^{hN} \alpha$, $B$ and |
---|
137 | $\frac{d\sigma^{diff}}{dtdM_x^2}$). |
---|
138 | At interesting energies these parameters were fixed at their well-known |
---|
139 | values. The fitting of the nuclear density parameters was performed |
---|
140 | over a wide range of atomic numbers $(A=4 - 208)$ using experimental data |
---|
141 | on proton-nuclei elastic scattering at a kinetic energy of $T_p=1 GeV$. |
---|
142 | |
---|
143 | The fitting was perfomed both for individual nuclei and for the entire set |
---|
144 | of nuclei at once. |
---|
145 | %%In the last event the following dependensies for nuclei |
---|
146 | %%parameters $(for A=12 \div 208)$ were obtained |
---|
147 | |
---|
148 | %\vspace{2ex} |
---|
149 | %$R_1=4.18A^{0.302}$, $R_2=3.81(A-10)^{0.268}$, $p=0.95$. |
---|
150 | %\hspace{3cm} (8) |
---|
151 | %\vspace{2ex} |
---|
152 | |
---|
153 | %%\vspace{2ex} |
---|
154 | %%$R_1=4.45(A-1)^{0.309}$, $R_2=2.3A^{0.36}$, |
---|
155 | |
---|
156 | %%$p=0.176+0.0017A+8.7.10^{-6}A^2$. |
---|
157 | %%\hspace{3cm} (8) |
---|
158 | %%\vspace{2ex} |
---|
159 | |
---|
160 | It is necessary to note that for every nucleus |
---|
161 | an optimal set of density parameters exists and it |
---|
162 | differs slightly from the one derived for the full set of nuclei. |
---|
163 | |
---|
164 | A comparision of the phenomenological cross sections \cite{helast.6} with |
---|
165 | experiment is presented in Figs. \ref{helast.fig1} - \ref{helast.fig9} |
---|
166 | |
---|
167 | % where the data of the proton-nucleus scattering |
---|
168 | % are imaged. |
---|
169 | |
---|
170 | In this comparison, the individual nuclei parameters were used. |
---|
171 | The experimental data were obtained in Gatchina (Russia) and in |
---|
172 | Saclay (France) \cite{helast.6}. |
---|
173 | The horizontal axis is the scattering angle in the center of mass |
---|
174 | system $\Theta_{CM}$ and the vertical axis is |
---|
175 | $\frac{\displaystyle d\sigma}{\displaystyle d\Omega_{CM}}$ |
---|
176 | in $\frac{\displaystyle mb}{\displaystyle Ster}$. |
---|
177 | %The lower curve in the figures is the diferencial cross-section |
---|
178 | %of a coherent elastic scattering but upper one is the |
---|
179 | %cross-section for noncoherent scattering. |
---|
180 | |
---|
181 | Comparisions were also made for $p ^4He$ elastic scatering at |
---|
182 | $T_ = 1 GeV$[7], $45 GeV$ and $301 GeV$ [3]. The resulting |
---|
183 | cross sections $\frac {d \sigma}{d\left| t \right|}$ are shown |
---|
184 | in the Figs. \ref{helast.fig10} - \ref{helast.fig12}. |
---|
185 | |
---|
186 | In order to generate events the distribution function ${\cal F}$ of a |
---|
187 | corresponding process must be known. |
---|
188 | The differential cross section is proportional to the density distribution. |
---|
189 | Therefore to get the distribution function it is sufficient to integrate the |
---|
190 | differential cross section and normalize it: |
---|
191 | |
---|
192 | \begin{equation} |
---|
193 | {\cal F}(q^2)= |
---|
194 | \frac |
---|
195 | {\displaystyle |
---|
196 | \int\limits_{0}^{q^2}d(q^2) |
---|
197 | \frac {\displaystyle d\sigma}{\displaystyle d(q^2)} |
---|
198 | } |
---|
199 | {\displaystyle |
---|
200 | \int\limits_{0}^{q_{max}^2}d(q^2) |
---|
201 | \frac {\displaystyle d\sigma}{\displaystyle d(q^2)} . |
---|
202 | } |
---|
203 | \label{helast.eq8} |
---|
204 | \end{equation} |
---|
205 | |
---|
206 | Expressions \ref{helast.eq6} and \ref{helast.eq7} allow analytic integration |
---|
207 | in Eq. \ref{helast.eq8} but the result is too long to be given here. |
---|
208 | |
---|
209 | For light and medium nuclei the analytic expression is more convenient for |
---|
210 | calculations than the numerical integration of Eq. \ref{helast.eq8}, but for |
---|
211 | heavy nuclei the latter is preferred due to the large number of terms in the |
---|
212 | analytic expression. |
---|
213 | |
---|
214 | \section{Status of this document} |
---|
215 | 18.06.04 created by Nikolai Starkov \\ |
---|
216 | 19.06.04 re-written for spelling and grammar by D.H. Wright \\ |
---|
217 | |
---|
218 | |
---|
219 | \begin{latexonly} |
---|
220 | |
---|
221 | \begin{thebibliography}{599} |
---|
222 | |
---|
223 | \bibitem{helast.1} R.J. Glauber, |
---|
224 | in "High Energy Physics and Nuclear Structure", |
---|
225 | edited by S. Devons (Plenum Press, NY 1970). |
---|
226 | |
---|
227 | |
---|
228 | \bibitem{helast.2} R. H. Bassel, W. Wilkin, Phys. Rev., 174, p. 1179, 1968; \\ |
---|
229 | T. T. Chou, Phys. Rev., 168, 1594, 1968; \\ |
---|
230 | M. A. Nasser, M. M. Gazzaly, J. V. Geaga et al., Nucl. Phys., |
---|
231 | A312, pp. 209-216, 1978. |
---|
232 | |
---|
233 | |
---|
234 | \bibitem{helast.3} Bujak, P. Devensky, A. Kuznetsov et al., |
---|
235 | Phys. Rev., D23, N 9, pp. 1895-1910, 1981. |
---|
236 | |
---|
237 | |
---|
238 | \bibitem{helast.4} V. L. Korotkikh, N. I. Starkov, Sov. Journ. of Nucl. Phys., |
---|
239 | v. 37, N 4, pp. 610-613, 1983; \\ |
---|
240 | N. T. Ermekov, V. L. Korotkikh, |
---|
241 | N. I. Starkov, Sov. Journ. of Nucl. Phys., 33, N 6, pp. 775-777, |
---|
242 | 1981. |
---|
243 | |
---|
244 | |
---|
245 | \bibitem{helast.5} R.A. Nam, S. I. Nikol'skii, N. I. Starkov et al., |
---|
246 | Sov. Journ. of Nucl. Phys., v. 26, N 5, pp. 550-555, 1977. |
---|
247 | |
---|
248 | |
---|
249 | \bibitem{helast.6} G.D. Alkhazov et al., Phys. Rep., 1978, C42, N 2, pp. 89-144; |
---|
250 | \bibitem{helast.7} J. V. Geaga, M. M. Gazzaly, G. J. Jgo et al., |
---|
251 | Phys. Rev. Lett. 38, N 22, pp. 1265-1268; \\ |
---|
252 | S. J. Wallace. Y. Alexander, Phys. Rev. Lett. 38, N 22, pp. 1269-1272. |
---|
253 | |
---|
254 | \end{thebibliography} |
---|
255 | |
---|
256 | \end{latexonly} |
---|
257 | |
---|
258 | \begin{htmlonly} |
---|
259 | |
---|
260 | \section{Bibliography} |
---|
261 | |
---|
262 | \begin{enumerate} |
---|
263 | \item R.J. Glauber, |
---|
264 | in "High Energy Physics and Nuclear Structure", |
---|
265 | edited by S. Devons (Plenum Press, NY 1970). |
---|
266 | |
---|
267 | |
---|
268 | \item R. H. Bassel, W. Wilkin, Phys. Rev., 174, p. 1179, 1968; \\ |
---|
269 | T. T. Chou, Phys. Rev., 168, 1594, 1968; \\ |
---|
270 | M. A. Nasser, M. M. Gazzaly, J. V. Geaga et al., Nucl. Phys., |
---|
271 | A312, pp. 209-216, 1978. |
---|
272 | |
---|
273 | |
---|
274 | \item Bujak, P. Devensky, A. Kuznetsov et al., |
---|
275 | Phys. Rev., D23, N 9, pp. 1895-1910, 1981. |
---|
276 | |
---|
277 | |
---|
278 | \item V. L. Korotkikh, N. I. Starkov, Sov. Journ. of Nucl. Phys., |
---|
279 | v. 37, N 4, pp. 610-613, 1983; \\ |
---|
280 | N. T. Ermekov, V. L. Korotkikh, |
---|
281 | N. I. Starkov, Sov. Journ. of Nucl. Phys., 33, N 6, pp. 775-777, |
---|
282 | 1981. |
---|
283 | |
---|
284 | |
---|
285 | \item R.A. Nam, S. I. Nikol'skii, N. I. Starkov et al., |
---|
286 | Sov. Journ. of Nucl. Phys., v. 26, N 5, pp. 550-555, 1977. |
---|
287 | |
---|
288 | |
---|
289 | \item G.D. Alkhazov et al., Phys. Rep., 1978, C42, N 2, pp. 89-144; |
---|
290 | \bibitem{helast.7} J. V. Geaga, M. M. Gazzaly, G. J. Jgo et al., |
---|
291 | Phys. Rev. Lett. 38, N 22, pp. 1265-1268; \\ |
---|
292 | S. J. Wallace. Y. Alexander, Phys. Rev. Lett. 38, N 22, pp. 1269-1272. |
---|
293 | |
---|
294 | \end{enumerate} |
---|
295 | |
---|
296 | \end{htmlonly} |
---|
297 | |
---|
298 | |
---|
299 | \begin{figure} |
---|
300 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_Be9_1.eps} |
---|
301 | \caption{Elastic proton scattering on $^9$Be at 1 GeV} |
---|
302 | \label{helast.fig1} |
---|
303 | \end{figure} |
---|
304 | \begin{figure} |
---|
305 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_B11_1.eps} |
---|
306 | \caption{Elastic proton scattering on $^{11}$B at 1 GeV} |
---|
307 | \label{helast.fig2} |
---|
308 | \end{figure} |
---|
309 | \begin{figure} |
---|
310 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_C12_1.eps} |
---|
311 | \caption{Elastic proton scattering on $^{12}$C at 1 GeV} |
---|
312 | \label{helast.fig3} |
---|
313 | \end{figure} |
---|
314 | \begin{figure} |
---|
315 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_016_1.eps} |
---|
316 | \caption{Elastic proton scattering on $^{16}$O at 1 GeV} |
---|
317 | \label{helast.fig4} |
---|
318 | \end{figure} |
---|
319 | \begin{figure} |
---|
320 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_Si28_1.eps} |
---|
321 | \caption{Elastic proton scattering on $^{28}$Si at 1 GeV} |
---|
322 | \label{helast.fig5} |
---|
323 | \end{figure} |
---|
324 | \begin{figure} |
---|
325 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_Ca40_1.eps} |
---|
326 | \caption{Elastic proton scattering on $^{40}$Ca at 1 GeV} |
---|
327 | \label{helast.fig6} |
---|
328 | \end{figure} |
---|
329 | \begin{figure} |
---|
330 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_Ni58_1.eps} |
---|
331 | \caption{Elastic proton scattering on $^{58}$Ni at 1 GeV} |
---|
332 | \label{helast.fig7} |
---|
333 | \end{figure} |
---|
334 | \begin{figure} |
---|
335 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_Zr90_1.eps} |
---|
336 | \caption{Elastic proton scattering on $^{90}$Zr at 1 GeV} |
---|
337 | \label{helast.fig8} |
---|
338 | \end{figure} |
---|
339 | \begin{figure} |
---|
340 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_Pb208_1.eps} |
---|
341 | \caption{Elastic proton scattering on $^{208}$Pb at 1 GeV} |
---|
342 | \label{helast.fig9} |
---|
343 | \end{figure} |
---|
344 | |
---|
345 | \begin{figure} |
---|
346 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_He4_1.eps} |
---|
347 | \caption{Elastic proton scattering on $^4$He at 1 GeV} |
---|
348 | \label{helast.fig10} |
---|
349 | \end{figure} |
---|
350 | \begin{figure} |
---|
351 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_He4_45.eps} |
---|
352 | \caption{Elastic proton scattering on $^4$He at 45 GeV} |
---|
353 | \label{helast.fig11} |
---|
354 | \end{figure} |
---|
355 | \begin{figure} |
---|
356 | \includegraphics[scale=0.7]{hadronic/he_elastic/p_He4_301.eps} |
---|
357 | \caption{Elastic proton scattering on $^4$He at 301 GeV} |
---|
358 | \label{helast.fig12} |
---|
359 | \end{figure} |
---|