benchmarks.tex 48.3 KB
 snuverink_j committed Sep 06, 2017 1 2 3 4 \input{header} \chapter{Benchmarks} \label{chp:benchmarks}  snuverink_j committed Sep 07, 2017 5 \section{\textit{OPAL-t} compared with TRANSPORT \& TRACE 3D}  snuverink_j committed Sep 06, 2017 6 7 8 \label{sec:T3D} \subsection{TRACE 3D}  snuverink_j committed Sep 08, 2017 9 TRACE 3D is an interactive beam dynamics program that calculates the envelopes of a bunched beam, including linear space-change forces \ref{Trace_man}. It provides an instantaneous beam profile diagram and delineates the transverse and longitudinal phase plane, where the ellipses are characterized by the Twiss parameters and emittances (total and unnormalized).  snuverink_j committed Sep 06, 2017 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  \subsection{TRACE 3D Units} \label{ssec:T3D_units} TRACE 3D supports the following internal coordinates and units for the three phase planes: \begin{itemize} \item \textbf{horizontal plane:} \\ x [mm] is the displacement from the center of the beam bunch;\\ x' [mrad] is the beam divergence; \item \textbf{vertical plane:}\\ y [mm] is the displacement from the center of the beam bunch; \\ y' [mrad] is the beam divergence; \item \textbf{longitudinal plane:}\\ z [mm] is the displacement from the center of the beam bunch; \\ $\Delta$p/p [mrad] is the difference between the particle's longitudinal momentum and the reference momentum of the beam bunch. \end{itemize} For input and output, however, z and $\Delta$p/p are replaced by $\Delta\phi$ [degree] and $\Delta$W [keV], respectively the displacement in phase and energy. The relationships between these longitudinal coordinates are: z = -\frac{\beta\lambda}{360}\Delta\phi and \frac{\Delta p}{p} = \frac{\gamma}{\gamma +1}\frac{\Delta W}{W}  snuverink_j committed Sep 08, 2017 39 where $\beta$ and $\gamma$ are the relativist parameters, $\lambda$ is the free-space wavelength of the RF and W is the kinetic energy [{MeV}] at the beam center. This internal conversion can be displayed using the \textit{command W} (see \ref{Trace_man} page 42).  snuverink_j committed Sep 06, 2017 40 41 42 43 \subsection{TRACE 3D Input beam} \label{ssec:T3D_input} In TRACE 3D, the input beam is described by the following set of parameters: \begin{itemize}  snuverink_j committed Sep 08, 2017 44 \item \textbf{ER}: particle rest mass [{MeV/\squarec}];  snuverink_j committed Sep 06, 2017 45 \item \textbf{Q}: charge state (+1 for protons);  snuverink_j committed Sep 08, 2017 46 \item \textbf{W}: beam kinetic energy [{MeV}]  snuverink_j committed Sep 11, 2017 47 \item \textbf{XI}: beam current [{mA}]  snuverink_j committed Sep 06, 2017 48 49 50 51 52 53 54 55 56 57 58 59 60 61 \item \textbf{BEAMI}: array with initial Twiss parameters in the three phase planes \begin{center} BEAMI = $\alpha_x , \beta_x, \alpha_y, \beta_y, \alpha_{\phi}, \beta_{\phi}$ \\ \end{center} The alphas are dimensionless, $\beta_x$ and $\beta_y$ are expressed in m/rad (or mm/mrad) and $\beta_{\phi}$ in deg/keV; \item \textbf{EMITI}: initial total and unnormalized emittances in x-x', y-y', and $\Delta\phi$-$\Delta W$ planes. \begin{center} EMITI = $\epsilon_x , \epsilon_y, \epsilon_{\phi}$ \\ \end{center} The transversal emittances are expressed in $\pi$-mm-mrad and in $\pi$-deg-keV the longitudinal emittance. \end{itemize} In this beam dynamics code, the total emittance in each phase plane is five times the RMS emittance in that plane and the displayed beam envelopes are $\sqrt{5}$-times their respective RMS values. \subsubsection{TRACE 3D Graphic Interface} \label{ssec:T3D_graphic}  snuverink_j committed Sep 06, 2017 62 An example of TRACE 3D graphic interface is shown in Figure~\ref{trace}.  snuverink_j committed Sep 06, 2017 63 64 65 66 67 68 69 70 71 72 73 74 \begin{figure}[htbp] \centering \includegraphics[width=\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/Trace.png} \caption{TRACE 3D graphic interface where: (1) input beam in transverse plane (above) and longitudinal plane (below); (2) output beam in transverse plane (above) and longitudinal plane (below); (3) summary of beam parameters such as input and output emittances and desired value for matching function; (4) line lattice with different elements and beam envelope. The color legend is: blue line for horizontal plane, red line for vertical plane, green line for longitudinal plane and yellow line for dispersion.} \label{fig:trace} \end{figure} \clearpage %---------------------------------------------------------------------------------------- % SECTION 2: TRANSPORT %---------------------------------------------------------------------------------------- \subsection{TRANSPORT} \label{sec:TRAN}  snuverink_j committed Sep 08, 2017 75 TRANSPORT is a computer program for first-order and second-order matrix multiplication, intended for the design of beam transport system \ref{bib:transport}. The TRANSPORT version for Windows provides a graphic beam profile diagram, as well as a sigma matrix description of the simulated beam and line \ref{Transport_GUI}.  snuverink_j committed Sep 06, 2017 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 Differently from TRACE 3D, the ellipses are characterized by the sigma-matrix coefficients and the Twiss parameters and emittances (total and unnormalized) are reported as output information. \subsection{TRANSPORT Units} \label{ssec:TRAN_units} At any specified position in the system, an arbitrary charged particle is represented by a vector, whose components are positions, angles and momentum of the particle with respect to the reference trajectory. The standard units and internal coordinates in TRANSPORT are: \begin{itemize} \item \textbf{horizontal plane:} \\ x [cm] is the displacement of the arbitrary ray with respect to the assumed central trajectory;\\ $\theta$ [mrad] is the angle the ray makes with respect to the assumed central trajectory; \item \textbf{vertical plane:}\\ y [cm] is the displacement of the arbitrary ray with respect to the assumed central trajectory;\\ $\phi$ [mrad] is the angle the ray makes with respect to the assumed central trajectory; \item \textbf{longitudinal plane:}\\ l [cm] is the path length difference between the arbitrary ray and the central trajectory;\\ $\delta$ [\%] is the fractional momentum deviation of the ray from the assumed central trajectory. \end{itemize}  snuverink_j committed Sep 08, 2017 91 Even if TRANSPORT supports this standard set of units [cm, mrad and \%]; however using \textbf{card 15}, the users can redefine the units (see page 99 on TRANSPORT documentation \ref{bib:transport} for more details).  snuverink_j committed Sep 06, 2017 92 93 94 95 \subsection{TRANSPORT Input beam} \label{ssec:TRAN_input} The input beam is described in \textbf{card 1} in terms of the semi-axes of a six-dimensional erect ellipsoid beam. In terms of diagonal sigma-matrix elements, the input beam in TRANSPORT is expressed by 7 parameters: \begin{itemize}  snuverink_j committed Sep 08, 2017 96 97 98 \item $\sqrt{\sigma_{ii}}$ [cm] represents one-half of the horizontal (i=1), vertical (i=3) and longitudinal extent (i=5); \item $\sqrt{\sigma_{ii}}$ [mrad] represents one-half of the horizontal (i=2), vertical (i=4) beam divergence; \item $\sqrt{\sigma_{66}}$ [\%] represents one-half of the momentum spread;  snuverink_j committed Sep 06, 2017 99 100 101 102 \item p(0) is the momentum of the central trajectory [GeV/c]. \end{itemize} If the input beam is tilted (Twiss alphas not zero), \textbf{ card 12} must be used, inserting the 15 correlations $r_{ij}$ parameters among the 6 beam components. The correlation parameters are defined as following:  snuverink_j committed Sep 08, 2017 103 r_{ij}=\frac{\sigma_{ij}}{\sqrt{\sigma_{ii}gma_{jj}}}  snuverink_j committed Sep 06, 2017 104   snuverink_j committed Sep 06, 2017 105 As explained before, with the \textbf{card 15}, it is possible to transform the TRANSPORT standard units in TRACE-like units. In this way, the TRACE 3D sigma-matrix for the input beam, printed out by \textit{command Z}, can be directly used as input beam in TRANSPORT. An example of TRACE 3D sigma-matrix structure is shown in Figure~\ref{trace_z}.  snuverink_j committed Sep 06, 2017 106 107 108 \begin{figure}[htbp] \centering \includegraphics[width=0.5\textwidth-0.6cm, keepaspectratio=true]{figures/Benchmarks/TRACE_z.png}  snuverink_j committed Sep 08, 2017 109  \caption{Sigma-matrix structure in TRACE 3D \ref{Trace_man}}  snuverink_j committed Sep 06, 2017 110 111 112 113 114  \label{fig:trace_z} \end{figure} From the sigma-matrix coefficients, TRANSPORT reports in output the Twiss parameters and the total, unnormalized emittance. Even in this case, a factor 5 is present between the emittances calculated by TRANSPORT and the corresponding RMS values. \subsubsection{TRANSPORT Graphic Interface} \label{ssec:TRAN_graphic}  snuverink_j committed Sep 08, 2017 115 An improved version of TRANSPORT has been embedded in a new graphic shell written in C++ and is providing GUI type tools, which makes it easier to design new beam lines. A screen shot of a modern GUI Transport interface \ref{Transport_GUI} is shown in Figure~\ref{TRANSPORT}.  snuverink_j committed Sep 06, 2017 116 117 118 \begin{figure}[htbp] \centering \includegraphics[width=\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/TRANSPORT.png}  snuverink_j committed Sep 08, 2017 119  \caption{GUI TRANSPORT graphic interface \ref{Tran_ex}. The continuous lines describe the beam envelope in the vertical plane (above) and horizontal plane (below). The dashed line displays the dispersion. The elements in the beam line are drawn as blue and red rectangles}  snuverink_j committed Sep 06, 2017 120 121 122 123 124 125 126 127 128  \label{fig:TRANSPORT} \end{figure} %\vspace{4cm} %---------------------------------------------------------------------------------------- % SECTION 3: COMPARISON TRACE 3D - TRANSPORT %---------------------------------------------------------------------------------------- \subsection{Comparison TRACE 3D and TRANSPORT} \label{sec:T3D_TRAN}  snuverink_j committed Sep 08, 2017 129 This study has been done following the same trend of the Regression Test in \textit{OPAL} \ref{AMAS}, replacing the electron beam with a same energy proton beam. Due to the different beam rigidity, the bending magnet features have been redefined with a new magnetic field.  snuverink_j committed Sep 06, 2017 130 131 132 133 134 135 136 137 138 139 140 141 142  The simulated beam transport line contains: \begin{itemize} \item drift space (DRIFT 1): 0.250 m length; \item bending magnet (SBEND or RBEND): 0.250 m radius of curvature; \item drift space (DRIFT 2): 0.250 m length. \end{itemize} Keeping fixed the lattice structure, many similar transport lines have been tested adding entrance and exit edge angles to the bending magnet, changing the bending plane (vertical bending magnet) and direction (right or left). In all the cases, the difficulties arise from the non-achromaticity of the system and an increase in the horizontal and longitudinal emittance is expected. In addition, the coupling between these two planes has to be accurately studied. In the following paragraph, an example of Sector Bending magnet (SBEND) simulation with entrance and exit edge angles is discussed. \subsubsection{Input beam}  snuverink_j committed Sep 06, 2017 143 The starting simulation has been performed with TRACE 3D code. According to Section~\ref{T3D_input}, the simulated input beam is described by the following parameters:  snuverink_j committed Sep 06, 2017 144 \begin{verbatim}  snuverink_j committed Sep 06, 2017 145 146 147 148 149 ER = 938.27 W = 7 FREQ = 700 BEAMI = 0.0, 4.0,0.0, 4.0, 0.0, 0.0756 EMITI = 0.730, 0.730, 7.56  snuverink_j committed Sep 06, 2017 150 \end{verbatim}  snuverink_j committed Sep 06, 2017 151   snuverink_j committed Sep 06, 2017 152 Thanks to the TRACE 3D graphic interface, the input beam can immediately be visualized in the three phase plane as shown in Figure~\ref{Input_TRACE}.  snuverink_j committed Sep 06, 2017 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167  \begin{figure}[htbp] \centering \includegraphics[width=0.5\textwidth-0.6cm, keepaspectratio=true]{figures/Benchmarks/Input_Trace.png} \caption{TRACE 3D input beam in the transversal plane (above) and in the longitudinal plane (below)} \label{fig:Input_TRACE} \end{figure} The corresponding sigma-matrix with the relative units is displayed by command Z: \begin{figure}[htbp] \centering \includegraphics[width=0.5\textwidth-0.6cm, keepaspectratio=true]{figures/Benchmarks/TRACE_z_input.png} \caption{TRACE 3D sigma-matrix for the input beam} \label{fig:TRACE_z_Input} \end{figure} Before entering the TRACE 3D sigma-matrix coefficients in TRANSPORT, a changing in the units is required using the \textbf{card 15} in the following way:  snuverink_j committed Sep 06, 2017 168 \begin{verbatim}  snuverink_j committed Sep 06, 2017 169 170 15. 1. 'MM' 0.1 ; //express in mm the horizontal and vertical beam size 15. 5. 'MM' 0.1 ; //express in mm the beam length  snuverink_j committed Sep 06, 2017 171 \end{verbatim}  snuverink_j committed Sep 06, 2017 172 173  At this point, the TRANSPORT input beam is defined by \textbf{card 1}:  snuverink_j committed Sep 06, 2017 174 \begin{verbatim}  snuverink_j committed Sep 06, 2017 175 1.0 1.709 0.427 1.709 0.427 0.11 0.0717 0.1148 /BEAM/ ;  snuverink_j committed Sep 06, 2017 176 \end{verbatim}  snuverink_j committed Sep 06, 2017 177   snuverink_j committed Sep 06, 2017 178 using exactly the same sigma-matrix coefficients of Figure~\ref{TRACE_z_Input}. Other two cards must be added in order to use exactly the TRACE 3D R-matrix formalism:  snuverink_j committed Sep 06, 2017 179 \begin{verbatim}  snuverink_j committed Sep 06, 2017 180 181 16. 3. 1863.153; //proton mass, as ratio of electron mass 22. 0.05 0.0 700 0.0 /SPAC/ ; //space charge card  snuverink_j committed Sep 06, 2017 182 \end{verbatim}  snuverink_j committed Sep 06, 2017 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 \subsubsection{SBEND in TRACE 3D} The bending magnet definition in TRACE 3D requires: \begin{table}[htbp] \centering \caption{Bending magnet description in TRACE 3D and values used in the simulation} \label{tab:Bend_Trace} \begin{tabular}{|l|l|l|} \hline \tabhead{Parameter & Value & Description} \hline NT & 8 & Type code for bending \\ $\alpha$ [deg] & 30 & angle of bend in horizontal plane \\ $\rho$ [mm] & 250 & radius of curvature of central trajectory \\ n & 0 & field-index gradient\\ vf & 0 & flag for vertical bending\\ \hline \end{tabular} \end{table} The edge angles are described with another type code and parameters which include also the fringe field. They must be added before and after the bending magnet if entrance and exit edge angles are present and if the fringe field has to be taken into account. In particular for the entrance edge angle: \begin{table}[htbp] \centering \caption{Edge angle description in TRACE 3D and values used in the simulation} \label{tab:Edge_Trace} \begin{tabular}{|l|l|l|} \hline \tabhead{Parameter & Value & Description} \hline NT & 9 & Type code for edge \\ $\beta$ [deg] & 10 & pole-face rotation \\ $\rho$ [mm] & 250 & radius of curvature of central trajectory \\ g [mm] & 20 & total gap of magnet \\ $K_1$ & 0.36945 & fringe-field factor \\ $K_2$ & 0.36945 & fringe-field factor \\ \hline \end{tabular} \end{table}  snuverink_j committed Sep 08, 2017 219 A same configuration has been used for exit edge angle using $\beta = {5}{^{\circ}}$. The beam envelopes in the three phase planes for this simulation are shown in Figure~\ref{Trace_env}.  snuverink_j committed Sep 06, 2017 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 \begin{figure}[htbp] \centering \includegraphics[width=\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/Trace_SBEND_edge.png} \caption{Beam envelopes in TRACE 3D for a SBEND with entrance and exit edge angles. The blue line describes the beam envelope in the horizontal plane, the red line in the vertical plane, the green line in the longitudinal plane. The yellow line displays the dispersion} \label{fig:Trace_env} \end{figure} \subsubsection{SBEND in TRANSPORT} The bending magnet definition in TRANSPORT requires: \begin{table}[htbp] \centering \caption{Bending magnet description in TRANSPORT and values used in the simulation} \label{tab:Bend_Trans} \begin{tabular}{|l|l|l|} \hline \tabhead{Parameter & Value & Description} \hline Card & 4 & Type code for bending \\ L [m] & 30 & Effective length of the central trajectory \\ $B_0$ [kG] & 250 & Central field strength \\ n & 0 & field-index gradient \\ \hline \end{tabular} \end{table}  snuverink_j committed Sep 06, 2017 243 As for TRACE 3D, the edge angles are described with another card and parameters. In TRANSPORT, however, the fringe field is not automatically included with the edge angle, but it is described by a own card as reported in the Table~\ref{Edge_Trans}.  snuverink_j committed Sep 06, 2017 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 \begin{table}[htbp] \centering \caption{Edge angle and fringe field description in TRANSPORT and values used in the simulation} \label{tab:Edge_Trans} \begin{tabular}{|l|l|l|} \hline \tabhead{Parameter & Value & Description} \hline Card & 2 & Type code for edge \\ $\beta$ [deg] & 10 & pole-face rotation \\ \hline Card & 16 & Type code for fringe field \\ g [mm] & 10 & half-gap of magnet \\ $K_1$ & 0.36945 & fringe-field factor \\ $K_2$ & 0.36945 & fringe-field factor \\ \hline \end{tabular} \end{table}  snuverink_j committed Sep 06, 2017 262 Running the Graphic TRANSPORT version, the beam envelopes in the transverse phase planes for this simulation are shown in Figure~\ref{Tran_env}.  snuverink_j committed Sep 06, 2017 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 \begin{figure}[htbp] \centering \includegraphics[width=0.5\textwidth-0.6cm, keepaspectratio=true]{figures/Benchmarks/TRANS_SBEND_edge.png} \caption{Beam envelopes in TRANSPORT for a SBEND with entrance and exit edge angles. The continuous lines describe the beam envelope in the vertical plane (above) and horizontal plane (below). The dashed line displays the dispersion.} \label{fig:Tran_env} \end{figure} \subsubsection{Beam size and emittance comparison} In the next table, the results of the comparison between TRACE 3D and TRANSPORT in terms of the transversal beam sizes at the end of each element in the line are summarized. \begin{table}[htbp] \centering \caption{Transversal beam size at the end of each element in the line printed out by TRACE 3D and TRANSPORT} \label{tab:Beam_size} \begin{tabular}{|l|l|l|l|l|l|} \hline \multicolumn{2}{|c|}{} & \multicolumn{2}{c|}{\tabheadcell{TRACE 3D}} & \multicolumn{2}{c|}{\tabheadcell{TRANSPORT}} \\ \hline  snuverink_j committed Sep 08, 2017 279  Position & z (m) & $\sigma_x$ (mm) & $\sigma_y$ (mm) & $\sigma_x$ (mm) & $\sigma_y$ (mm) \\  snuverink_j committed Sep 06, 2017 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294  %\hline Input & 0.000 & 1.709 & 1.709 & 1.709 & 1.709 \\ %\hline Drift 1 & 0.250 & 1.712 & 1.712 & 1.712 & 1.712 \\ %\hline Edge & 0.250 & 1.712 & 1.712 & 1.712 & 1.712 \\ %\hline Bend & 0.381 & 1.638 & 1.587 & 1.638 & 1.587 \\ %\hline Edge & 0.381 & 1.638 & 1.587 & 1.638 & 1.587 \\ %\hline Drift 2 & 0.631 & 1.206 & 1.264 & 1.206 & 1.264 \\ \hline \end{tabular} \end{table}  snuverink_j committed Sep 06, 2017 295 The perfect agreement between these two codes arises immediately looking at Figure~\ref{T3D_Tra_env}.  snuverink_j committed Sep 06, 2017 296 297 298 299 300 301 \begin{figure}[htbp] \centering \includegraphics[width=0.5\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/T3D_Tra_SBEND_edge_env.pdf} \caption{Transversal beam size comparison between TRACE 3D and TRANSPORT} \label{fig:T3D_Tra_env} \end{figure}  snuverink_j committed Sep 06, 2017 302 The same comparison has been performed in terms of horizontal and longitudinal emittance, both expressed in $\pi$-mm-mrad. While the vertical emittance remains constant and equal to the initial value ($\epsilon_y =$ 0.730 $\pi$-mm-mrad) , the horizontal and longitudinal emittances are expected growing after the bending magnet. The results are reported in Table~\ref{Emittance} and in Figure~\ref{T3D_Tra_emi}.  snuverink_j committed Sep 06, 2017 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 \begin{table}[htbp] \centering \caption{Horizontal and longitudinal emittance comparison between TRACE 3D and TRANSPORT, both expressed in $\pi$-mm-mrad} \label{tab:Emittance} \begin{tabular}{|l|l|l|l|l|l|} \hline \multicolumn{2}{|c|}{} & \multicolumn{2}{c|}{\tabheadcell{TRACE 3D}} & \multicolumn{2}{c|}{\tabheadcell{TRANSPORT}}\\ \hline Position & z (m) & $\epsilon_x$ & $\epsilon_z$ & $\epsilon_x$ & $\epsilon_z$ \\ %\hline Input & 0 & 0.730 & 0.08 & 0.730 & 0.08 \\ %\hline Drift 1 & 0.250 & 0.730 & 0.08 & 0.730 & 0.08 \\ %\hline Edge & 0.250 & 0.730 & 0.08 & 0.730 & 0.08 \\ %\hline Bend & 0.381 & 0.973 & 0.65 & 0.973 & 0.65 \\ %\hline Edge & 0.381 & 0.973 & 0.65 & 0.973 & 0.65 \\ %\hline Drift 2 & 0.631 & 0.973 & 0.65 & 0.973 & 0.65 \\ \hline \end{tabular} \end{table} \begin{figure}[htbp] \centering \includegraphics[width=0.5\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/T3D_Tra_SBEND_edge_emi.pdf} \caption{Emittance comparison between TRACE and TRANSPORT} \label{fig:T3D_Tra_emi} \end{figure} \subsubsection{From TRACE 3D to TRANSPORT} \label{ssec:T3DtoTRAN} \begin{table}[!ht] \centering \caption{Bending magnet features in TRACE 3D and TRANSPORT} \label{tab:Bend_Trace_Tra2} \begin{tabular}{|l|l|l|} \hline \tabhead{Parameter & Trace 3D & Transport} \hline \textbf{Bend card} & 8 & 4 \\ Angle & Input parameter [deg] & Output information [deg] \\ Magn. field & Calculated. [T] & Input parameter [kG] \\ Radius of curv. & Input parameter [mm] & Output information [m] \\ Field-index & Input parameter & Input parameter \\ Effect. length & Calculated [mm] & Input parameter [m] \\ \hline \hline \textbf{Edge card} & 9 & 2 \\ Edge angle & Input parameter [deg] & Input parameter [deg] \\ \hline \hline \textbf{Vertical gap} & 9 & 16.5 \\ Gap & Total [mm] & Half-gap [cm] \\ \hline \hline \textbf{Fringe field card} & 9 & 16.7 / 16.8 \\ $K_1$ & Default: 0.45 & Default: 0.5 \\ $K_2$ & Default: 2.8 & Default: 0 \\ \hline \hline \textbf{Bend direction} & Bend angle sign & Coord. rotation \\ Horiz. right & Angle $>$ 0 & Angle $>$ 0 \\ Horiz. left & Angle $<$ 0 & Card 20 \\ Vertical bend & Card 8, vf $>$ 0 & Card 20 \\ \hline \end{tabular} \end{table} \clearpage %----------------------------------------------------------------------------------------  snuverink_j committed Sep 07, 2017 376 % SECTION: \textit{OPAL}  snuverink_j committed Sep 06, 2017 377 378 %----------------------------------------------------------------------------------------  snuverink_j committed Sep 07, 2017 379 \subsection{Relations to \textit{OPAL-t}}  snuverink_j committed Sep 06, 2017 380 381 \label{sec:OPAL}  snuverink_j committed Sep 07, 2017 382 In \textit{OPAL}, the beam dynamics approach (time integration) is hence completely different from the envelope-like supported by TRACE 3D and TRANSPORT. The three codes support different units and require diverse parameters for the input beam. A summary of their main features is reported in Table~\ref{Features}.  snuverink_j committed Sep 06, 2017 383 384 385  \begin{table}[htbp] \centering  snuverink_j committed Sep 07, 2017 386 \caption{Main features of the three beam dynamics codes: TRACE 3D, TRANSPORT and \textit{OPAL}}  snuverink_j committed Sep 06, 2017 387 388 389 \label{tab:Features} \begin{tabular}{|l|l|l|l|} \hline  snuverink_j committed Sep 07, 2017 390  \tabhead{Code & TRACE 3D & TRANSPORT & \textit{OPAL}}  snuverink_j committed Sep 06, 2017 391 392 393 394 395 396 397 398 399 400  \hline \textbf{Type} & Envelope & Envelope & Time integration \\ %\hline \textbf{Input} & Twiss, Emittance & Sigma, Momentum & Sigma, Energy \\ %\hline \textbf{Units} & mm-mrad, deg-keV & cm-rad, cm-\% & m-$\beta\gamma$ \\ \hline \end{tabular} \end{table}  snuverink_j committed Sep 07, 2017 401 \subsection{\textit{OPAL-t} Units}  snuverink_j committed Sep 06, 2017 402 403 \label{ssec:OPAL_units}  snuverink_j committed Sep 07, 2017 404 \textit{OPAL-t} supports the following internal coordinates and units for the three phase planes:  snuverink_j committed Sep 06, 2017 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419  \begin{itemize} \item \textbf{horizontal plane:} \\ X [m] horizontal position of a particle relative to the axis of the element;\\ PX [$\beta_x\gamma$] horizontal canonical momentum; \item \textbf{vertical plane:}\\ Y [m] vertical position of a particle relative to the axis of the element;\\ PY [$\beta_y\gamma$] horizontal canonical momentum; \item \textbf{longitudinal plane:}\\ Z [m] longitudinal position of a particle in floor-coordinates;\\ PZ [$\beta_z\gamma$] longitudinal canonical momentum; \end{itemize}  snuverink_j committed Sep 07, 2017 420 \subsection{\textit{OPAL-t} Input beam}  snuverink_j committed Sep 06, 2017 421 422 423 424 425 426 427 \label{ssec:OPAL_input} For the input beam, a \texttt{GAUSS} distribution type has been chosen. For transferring the TRANSPORT (or TRACE 3D) input beam in terms of sigma-matrix coefficients, it necessary to: \begin{itemize} \item adjust the units: from mm to m; \item correct for the factor $\sqrt{5}$: from total to RMS distribution;  snuverink_j committed Sep 08, 2017 428 \item multiply for the relativistic factor $\beta\gamma ={0.1224}$ for {7}{MeV} protons;  snuverink_j committed Sep 06, 2017 429 430 \end{itemize}  snuverink_j committed Sep 07, 2017 431  In case of the modified sigma-matrix in Figure~\ref{TRACE_z_Input}, the corresponding \textit{OPAL} parameters for the \texttt{GAUSS} distributions are:  snuverink_j committed Sep 06, 2017 432   snuverink_j committed Sep 06, 2017 433 \begin{verbatim}  snuverink_j committed Sep 06, 2017 434 435 436 437 438 439 440 441  T3D SIGMA OPAL -T ------------------------------------------------------- 1.7088 mm SIGMAX = 1.7088/sqrt(5)e-3 m 0.4272 mrad SIGMAPX = 0.4272/sqrt(5)*0.1224e-3 1.7088 mm SIGMAY = 1.7088/sqrt(5)e-3 m 0.4272 mrad SIGMAPY = 0.4272/sqrt(5)*0.1224e-3 0.1092 mm SIGMAZ = 0.1092/sqrt(5)e-3 m 0.0717 % SIGMAPZ = (0.0717*10)/sqrt(5)*0.1224e-3  snuverink_j committed Sep 06, 2017 442 \end{verbatim}  snuverink_j committed Sep 06, 2017 443   snuverink_j committed Sep 07, 2017 444 At the end of this calculation, the input beam in \textit{OPAL} is:  snuverink_j committed Sep 06, 2017 445   snuverink_j committed Sep 06, 2017 446 \begin{verbatim}  snuverink_j committed Sep 06, 2017 447 448 449 450 451 D1: DISTRIBUTION, TYPE=GAUSS, SIGMAX = 0.7642e-03, SIGMAPX= 0.0234e-03, CORRX= 0.0, SIGMAY = 0.7642e-03, SIGMAPY= 0.0234e-03, CORRY= 0.0, SIGMAZ = 0.0488e-03, SIGMAPZ= 0.0392e-03, CORRZ= 0.0, R61= 0.0, INPUTMOUNITS=NONE;  snuverink_j committed Sep 06, 2017 452 \end{verbatim}  snuverink_j committed Sep 06, 2017 453 454 455 456 457  %---------------------------------------------------------------------------------------- % SECTION: Comparison TRACE 3D and OPAL %----------------------------------------------------------------------------------------  snuverink_j committed Sep 07, 2017 458 \subsection{Comparison TRACE 3D and \textit{OPAL-t}}  snuverink_j committed Sep 06, 2017 459 460 \label{sec:T3D_OPAL}  snuverink_j committed Sep 07, 2017 461 In this section, the comparison between TRACE 3D and \textit{OPAL-t} is discussed starting from \texttt{SBEND} definition in \textit{OPAL-t}. The transport line described in Section~\ref{T3D_TRAN} has been simulated in \textit{OPAL} using 10.000 particles and $10^{-11}$ s time step. The bending magnet features of Table~\ref{Bend_Trace,Edge_Trace} have been transformed in \textit{OPAL} language as:  snuverink_j committed Sep 06, 2017 462   snuverink_j committed Sep 06, 2017 463 \begin{verbatim}  snuverink_j committed Sep 06, 2017 464 465 466 467 468 469 470 471 Bend: SBEND, ANGLE = 30.0 * Pi/180.0, K1=0.0, E1=0, E2=0, FMAPFN = "1DPROFILE1-DEFAULT", ELEMEDGE = 0.250, // end of first drift DESIGNENERGY = 7E+06, // ref energy eV L = 0.1294, GAP = 0.02;  snuverink_j committed Sep 06, 2017 472 \end{verbatim}  snuverink_j committed Sep 06, 2017 473 474 475 476 477  \begin{itemize} \item \textbf{SBEND without edge angles:}  snuverink_j committed Sep 06, 2017 478 \begin{verbatim}  snuverink_j committed Sep 06, 2017 479 480 481 // Bending magnet configuration: K1=0.0, E1=0, E2=0,  snuverink_j committed Sep 06, 2017 482 \end{verbatim}  snuverink_j committed Sep 06, 2017 483 484 485 486 487 488  \begin{figure}[htbp] \begin{center} \subfloat[Transverse beam size]{\includegraphics[width=0.5\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/SBEND_noEdge_Env}} \hspace{1.8cm} \subfloat[Transverse emittance]{\includegraphics[width=0.5\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/SBEND_noEdge_Emi}}  snuverink_j committed Sep 07, 2017 489  \caption{TRACE 13D and \textit{OPAL} comparison: SBEND without edge angles}  snuverink_j committed Sep 06, 2017 490 491 492 493 494 495 496 497  \label{fig:SBEND_noEdge} \end{center} \end{figure} A good overall agreement has been found between the two codes in term of beam size and emittance. The different behavior inside the bending magnet for the horizontal emittance is still undergoing study and it's probably due to a diverse coordinate system in the two codes. \item \textbf{SBEND with edge angles:}  snuverink_j committed Sep 06, 2017 498 \begin{verbatim}  snuverink_j committed Sep 06, 2017 499 500 501 // Bending magnet configuration: K1=0.0, E1=10*Pi/180.0, E2=5* Pi/180.0,  snuverink_j committed Sep 06, 2017 502 \end{verbatim}  snuverink_j committed Sep 06, 2017 503 504 505 506 507 508 509  \begin{figure}[htbp] \begin{center} \subfloat[Transverse beam size]{\includegraphics[width=0.5\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/SBEND_Edges_Env.pdf}} \hspace{1.8cm} \subfloat[Transverse RMS emittance]{\includegraphics[width=0.5\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/SBEND_Edges_Emi.pdf}}  snuverink_j committed Sep 07, 2017 510  \caption{TRACE 3D and \textit{OPAL} comparison: SBEND with edge angles}  snuverink_j committed Sep 06, 2017 511 512 513 514 515 516 517 518 519 520 521  \label{fig:SBEND_Edges} \end{center} \end{figure} Even in this case, a good overall agreement has been found between the two codes in term of beam size and emittance. \item \textbf{SBEND with field index:} The field index parameter K1 is defined as:  snuverink_j committed Sep 11, 2017 522 K1 = \frac{1}{B\rho}\frac{\partial B_y}{\partial x},  snuverink_j committed Sep 06, 2017 523 524   snuverink_j committed Sep 06, 2017 525 Section~\ref{RBend}. Instead, in TRACE 3D the field index parameter n is:  snuverink_j committed Sep 06, 2017 526 527   snuverink_j committed Sep 11, 2017 528 n = -\frac{\rho}{B_y}\frac{\partial B_y}{\partial x}.  snuverink_j committed Sep 06, 2017 529 530   snuverink_j committed Sep 07, 2017 531 In order to have a significant focusing effect on both transverse planes, the transport line has been simulated in TRACE 3D using $n = 1.5$. Since, a different definition exists between \textit{OPAL} and TRACE 3D on the field index, the n-parameter translation in \textit{OPAL} language has been done with the following tests:  snuverink_j committed Sep 06, 2017 532 533 534 535 536 537 538  \begin{itemize}[noitemsep] \item[] TEST 1: K1 $=$ n/$\rho^2$ \item[] TEST 2: K1 $=$ n \item[] TEST 3: K1 $=$ n/$\rho$ \end{itemize}  snuverink_j committed Sep 06, 2017 539 Only the TEST 2 reports a reasonable behavior on the beam size and emittance, as shown in Figure~\ref{SBEND_FI} using:  snuverink_j committed Sep 06, 2017 540   snuverink_j committed Sep 06, 2017 541 \begin{verbatim}  snuverink_j committed Sep 06, 2017 542 543 544 // Bending magnet configuration: K1=1.5 E1=0, E2=0,  snuverink_j committed Sep 06, 2017 545 \end{verbatim}  snuverink_j committed Sep 06, 2017 546 547 548 549 550 551  \begin{figure}[htbp] \begin{center} \subfloat[Transverse beam size]{\includegraphics[width=0.5\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/FI_SBEND_FMDef_Env_T2.pdf}} \hspace{1.8cm} \subfloat[Transverse RMS emittance]{\includegraphics[width=0.5\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/FI_SBEND_FMDef_Emi_T2.pdf}}  snuverink_j committed Sep 07, 2017 552  \caption{TRACE 3D and \textit{OPAL} comparison: SBEND with field index and default field map}  snuverink_j committed Sep 06, 2017 553 554 555 556  \label{fig:SBEND_FI} \end{center} \end{figure}  snuverink_j committed Sep 07, 2017 557 Concerning the emittances and vertical beam size, a perfect agreement has been found, instead a defocusing effect appears in the horizontal plane. These results have been obtained with the default field map provided by \textit{OPAL}. However, a better result, only in the beam size as shown in Figure~\ref{SBEND_FI_test}, is achieved using a test field map in which the fringe field extension has been changed in the thin lens approximation.  snuverink_j committed Sep 06, 2017 558 559 560 561 562 563  \begin{figure}[htbp] \begin{center} \subfloat[Transverse beam size]{\includegraphics[width=0.5\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/FI_SBEND_FMTest_Env_T2.pdf}} \hspace{1.8cm} \subfloat[Transverse RMS emittance]{\includegraphics[width=0.5\textwidth-1cm, keepaspectratio=true]{figures/Benchmarks/FI_SBEND_FMTest_Emit_T2.pdf}}  snuverink_j committed Sep 07, 2017 564  \caption{TRACE 3D and \textit{OPAL} comparison: SBEND with field index and test field map}  snuverink_j committed Sep 06, 2017 565 566 567 568 569  \label{fig:SBEND_FI_test} \end{center} \end{figure} \end{itemize}  snuverink_j committed Sep 07, 2017 570 \subsubsection{From TRACE 3D to \textit{OPAL-t}}  snuverink_j committed Sep 06, 2017 571 572 573 574 \label{ssec:T3DtoOPAL} \begin{table}[!ht] \centering  snuverink_j committed Sep 07, 2017 575 \caption{Bending magnet features in TRACE 3D and \textit{OPAL-t}}  snuverink_j committed Sep 06, 2017 576 577 578 \label{tab:Bend_Trace_OPAL} \begin{tabular}{|l|l|l|} \hline  snuverink_j committed Sep 07, 2017 579  \tabhead{Parameter & Trace 3D & \textit{OPAL-t}}  snuverink_j committed Sep 06, 2017 580  \hline  snuverink_j committed Sep 06, 2017 581  \textbf{Bend card} & 8 & \texttt{SBEND} or \texttt{RBEND} \\  snuverink_j committed Sep 06, 2017 582 583 584 585 586 587 588 589  Angle & Input parameter [deg] & Input/Calc. parameter [rad] \\ Magn. field & Calculated. [T] & Input/Calc parameter [T] \\ Radius of curv. & Input parameter [mm] & Output information [m] \\ Field-index & Input parameter & Input parameter \\ Length & Calculated [mm] & Input/Calc parameter [m] \\ Length type & Effective & Straight \\ \hline \hline  snuverink_j committed Sep 06, 2017 590  \textbf{Edge card} & 9 & \texttt{SBEND} or \texttt{RBEND} \\  snuverink_j committed Sep 06, 2017 591 592 593  Edge angle & Input parameter [deg] & Input parameter [rad] \\ \hline \hline  snuverink_j committed Sep 06, 2017 594  \textbf{Vertical gap} & 9 & \texttt{SBEND} or \texttt{RBEND} \\  snuverink_j committed Sep 06, 2017 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625  Gap & Total [mm] & Total [m] \\ \hline \hline \textbf{Fringe field card} & 9 & FIELD MAP \\ $K_1$ & Default: 0.45 & - \\ $K_2$ & Default: 2.8 & - \\ \hline \hline \textbf{Bend direction} & Bend angle sign & Coord. rotation \\ Horiz. right & Angle $>$ 0 & Angle $>$ 0 \\ Horiz. left & Angle $<$ 0 & Angle $<$ 0 \\ Vertical bend & Card 8, vf $>$ 0 & Coord. rotation \\ \hline \end{tabular} \end{table} \clearpage %---------------------------------------------------------------------------------------- % Conclusion %---------------------------------------------------------------------------------------- \subsection{Conclusion} \label{sec:conclusion} \begin{itemize} \item \textbf{TRACE 3D and TRANSPORT:} \\ - a perfect agreement has been found between these two codes in transversal envelope and emittance; \\ - changing the TRANSPORT units, the input beam parameters, in terms of sigma-matrix coefficients, can directly be imported from TRACE 3D file.  snuverink_j committed Sep 07, 2017 626 \item \textbf{TRACE 3D and \textit{OPAL-t}:} \\  snuverink_j committed Sep 06, 2017 627 628 629 - a good agreement has been found between these two codes in case of sector bending magnet with and without edge angles;\\ % ADA the field index definition should be clarified according to the results of TEST1, TEST2 and TEST3; \\ - the default magnetic field map seems not working properly if the field index is not zero\\  snuverink_j committed Sep 06, 2017 630 - an improvement of the test map used is needed in order to match the TRACE 3D emittance see~Figure~\ref{SBEND_FI_test}.  snuverink_j committed Sep 06, 2017 631 632 633 634 635 \end{itemize} \section{Hard Edge Dipole Comparison with ELEGANT}  snuverink_j committed Sep 07, 2017 636 637 \subsection{\textit{OPAL} Dipole} When defining a dipole (\texttt{SBEND} or \texttt{RBEND}) in \textit{OPAL}, a fringe field map which defines the range of the field and the Enge coefficients is required. If no map is provided, the code uses a default map. Here is a dipole definition using the default map:  snuverink_j committed Sep 06, 2017 638 639 640  \begin{description} \item[Example:]  snuverink_j committed Sep 06, 2017 641 \item \begin{verbatim}  snuverink_j committed Sep 06, 2017 642 643 644 645 646 647 648  bend1: SBEND, ANGLE = bend_angle, E1 = 0, E2 = 0, FMAPFN = "1DPROFILE1-DEFAULT", ELEMEDGE = drift_before_bend, DESIGNENERGY = bend_energy, L = bend_length, WAKEF = FS_CSR_WAKE;  snuverink_j committed Sep 06, 2017 649  \end{verbatim}  snuverink_j committed Sep 06, 2017 650 \end{description}  snuverink_j committed Sep 07, 2017 651 Please refer to Section~\ref{1DProfile1} for the definition of the field map and the default map \texttt{1DPROFILE1-DEFAULT}. It defines a fringe field that extends to 10 cm away from a dipole edge in both directions and it has both $B_y$ and $B_z$ components. This makes the comparison between \textit{OPAL} and other codes which uses a hard edge dipole by default,cumbersome because one needs to carefully integrate thought the fringe field region in \textit{OPAL} in order to come up with the integrated fringe field value (FINT in ELEGANT) that usually used by these codes, e.g. the ELEGANT and the TRACE3D. So we need to find a default map for the hard edge dipole in \textit{OPAL}.  snuverink_j committed Sep 06, 2017 652 653 654 655 656  \subsection{Map for Hard Edge Dipole} The proposed default map for a hard edge dipole can be: \begin{description} \item  snuverink_j committed Sep 06, 2017 657 \begin{verbatim}  snuverink_j committed Sep 06, 2017 658 659 660 661 662 1DProfile1 0 0 2 -0.00000001 0.0 0.00000001 3 -0.00000001 0.0 0.00000001 3 -99.9 -99.9  snuverink_j committed Sep 06, 2017 663 \end{verbatim}  snuverink_j committed Sep 06, 2017 664 665 666 667 668 \end{description} On the first line, the two zeros following \texttt{1DProfile1} are the orders of the Enge coefficient for the entrance and exit edge of the dipole. $2 cm$ is the default dipole gap width. The second line defines the fringe field region of the entrance edge of the dipole which extends from $-0.00000001 cm$ to $0.00000001 cm$. The third line defines the same fringe field region for the exit edge of the dipole. The $3$s on both line don't mean anything, they are just placeholders. On the fourth and fifth line, the zeroth order Enge coefficients for both edges are given. Since they are large negative numbers, the field in the fringe field region has no $B_z$ component and its $B_y$ component is just like the field in the middle of the dipole. \begin{figure}[!htbp] \centering \includegraphics[height=0.5\textwidth-0.6cm, angle = -90, trim = 8mm 10mm 2mm 10mm, clip]{figures/Benchmarks/report-compare-default}  snuverink_j committed Sep 07, 2017 669 \caption{Compare emittances and beam sizes obtained by using the hard edge map (\textit{OPAL}), the default map (\textit{OPAL}), and the ELEGANT}  snuverink_j committed Sep 06, 2017 670 671 \label{fig:plot-compare-default} \end{figure}  snuverink_j committed Sep 06, 2017 672 Figure~\ref{plot-compare-default} compares the emittances and beam sizes obtained by using the hard edge map, the default map and the ELEGANT. One can see that the results produced by the hard edge map match the ELEGANT results when FINT is set to zero.  snuverink_j committed Sep 06, 2017 673 674  \subsection{Integration Time Step}  snuverink_j committed Sep 08, 2017 675 When the hard edge map is used for a dipole, finer integration time step is needed to ensure the accurate of the calculation. Figure~\ref{plot-emit-dt} compares the normalized emittances generated using the hard edge map in \textit{OPAL} with varying time steps to those from the ELEGANT. {0.01}{ps} seems to be a optimal time step for the fringe field region. To speed up the simulations, one can use larger time steps outside the fringe field regions. In Figure~\ref{plot-emit-dt}, one can observe a discontinuity in the horizontal emittance when the hard edge map is used in the calculation. This discontinuity comes from the fact that \textit{OPAL} emittance is calculated at an instant time. Once the beam or part of the beam gets into the dipole, its $P_x$ gets a kick which will result in a sudden emittance change.  snuverink_j committed Sep 06, 2017 676 677 678 679 680 681 682 683 684 \begin{figure}[!htbp] \centering \includegraphics[width=0.5\textwidth, angle = -90, trim = 0mm 20mm 0mm 8mm, clip]{figures/Benchmarks/report-emit-dt} \caption{Horizontal and vertical normalized emittances for different integration time steps} \label{fig:plot-emit-dt} \end{figure}  snuverink_j committed Sep 06, 2017 685 Figure~\ref{plot-fringe-size,plot-fringe-size-zoom} examine the effects of the fringe field range and the integration time step on the simulation accuracy. Figure~\ref{plot-fringe-size-zoom} is a zoom-in plot of Figure~\ref{plot-fringe-size}. We can conclude that the size of the integration time step has more influence on the accuracy of the simulation.  snuverink_j committed Sep 06, 2017 686 687 688 689 690 691 692 693 694 \begin{figure}[!htbp] \centering \includegraphics[height=0.5\textwidth-0.6cm, angle = -90, trim = 3mm 0mm 2mm 0mm, clip]{figures/Benchmarks/report-fringe-size} \caption{Normalized horizontal emittance for different fringe field ranges and integration time steps} \label{fig:plot-fringe-size} \end{figure} \begin{figure}[!htbp] \centering \includegraphics[height=0.5\textwidth-0.6cm, angle = -90, trim = 3mm 0mm 2mm 0mm, clip]{figures/Benchmarks/report-fringe-size-zoom}  snuverink_j committed Sep 06, 2017 695 \caption{Zoom in on the final emittance in Figure~\ref{plot-fringe-size-zoom}}  snuverink_j committed Sep 06, 2017 696 697 698 699 \label{fig:plot-fringe-size-zoom} \end{figure} \section{1D CSR comparison with ELEGANT}  snuverink_j committed Sep 06, 2017 700 1D-CSR wake function can now be used for the drift element by defining its attribute \texttt{WAKEF = FS\_CSR\_WAKE}. In order to calculate the CSR effect correctly, the drift has to follow a bending magnet whose CSR calculation is also turned on.  snuverink_j committed Sep 06, 2017 701 702 703  \begin{description} \item[Example:]  snuverink_j committed Sep 06, 2017 704 \item \begin{verbatim}  snuverink_j committed Sep 06, 2017 705 706 707 708 709 710 711  bend1: SBEND, ANGLE = bend_angle, E1 = 0, E2 = 0, FMAPFN = 1DPROFILE1-DEFAULT'', ELEMEDGE = drift_before_bend, DESIGNENERGY = bend_energy, L = bend_length, WAKEF = FS_CSR_WAKE;  snuverink_j committed Sep 06, 2017 712 713  \end{verbatim} \item \begin{verbatim}  snuverink_j committed Sep 06, 2017 714 715  drift1: DRIFT, L=0.4, ELEMEDGE = drift_before_bend + bend_length, WAKEF = FS_CSR_WAKE;  snuverink_j committed Sep 06, 2017 716  \end{verbatim}  snuverink_j committed Sep 06, 2017 717 718 719 \end{description} \subsection{Benchmark}  snuverink_j committed Sep 08, 2017 720 The \textit{OPAL} dipoles all have fringe fields. When comparisons are done between \textit{OPAL} and ELEGANT \ref{elegant} for example, one needs to appropriately set the FINT attribute of the bending magnet in ELEGANT in order to represent the field correctly. Although ELEGANT tracks in the ($x, x', y, y', s, \delta$) phase space, where $\delta = \frac{\Delta p}{p_0}$ and $p_0$ is the momentum of the reference particle, the watch point output beam distributions from the ELEGANT are list in ($x, x', y, y', t, \beta\gamma$). If one wants to compare ELEGANT watch point output distribution to \textit{OPAL}, unit conversion needs to be performed, i.e.  snuverink_j committed Sep 06, 2017 721 722 723 724 725 \begin{eqnarray*} P_x &=& x'\beta\gamma, \\ P_y &=& y'\beta\gamma, \\ s &=& (\bar t-t)\beta c . \end{eqnarray*}  snuverink_j committed Sep 07, 2017 726 To benchmark the CSR effect, we set up a simple beamline with 0.1m drift $+$ 30 degree sbend $+$ 0.4m drift. When the CSR effect is turn off, Figure~\ref{plot-emit-csr-off} shows that the normalized emittances calculated using both \textit{OPAL} and ELEGANT agree. The emittance values from \textit{OPAL} are obtained from the {\it .stat} file, while for ELEGANT, the transverse emittances are obtained from the sigma output file (enx, and eny), the longitudinal emittance is calculated using the watch point beam distribution output.  snuverink_j committed Sep 06, 2017 727 728 729 \begin{figure}[!htbp] \centering \includegraphics[height=0.5\textwidth-0.6cm, angle = -90, trim = 3mm 0mm 2mm 0mm, clip]{figures/Benchmarks/emit-csr-off}  snuverink_j committed Sep 07, 2017 730 \caption{Comparison of the trace space using ELEGANT and \textit{OPAL}}  snuverink_j committed Sep 06, 2017 731 732 733 \label{fig:plot-emit-csr-off} \end{figure}  snuverink_j committed Sep 07, 2017 734 When CSR calculations are enabled for both the bending magnet and the following drift, Figure~\ref{plot-dpp-csr-on} shows the average $\delta$ or $\frac{\Delta p}{p}$ change along the beam line, and Figure~\ref{plot-emit-csr-on} compares the normalized transverse and longitudinal emittances obtained by these two codes. The average $\frac{\Delta p}{p}$ can be found in the centroid output file (Cdelta) from ELEGANT, while in \textit{OPAL}, one can calculate it using $\frac{\Delta p}{p} = \frac{1}{\beta^2}\frac{\Delta \overline{E}}{\overline{E}+mc^2}$, where $\Delta \overline{E}$ is the average kinetic energy from the {\it .stat} output file.  snuverink_j committed Sep 06, 2017 735 736 737 \begin{figure}[!htbp] \centering \includegraphics[height=0.5\textwidth-0.6cm, angle = -90, trim = 3mm 0mm 2mm 0mm, clip]{figures/Benchmarks/dpp-csr-on}  snuverink_j committed Sep 07, 2017 738 \caption{$\frac{\Delta p}{p}$ in Elegant and \textit{OPAL}}  snuverink_j committed Sep 06, 2017 739 740 \label{fig:plot-dpp-csr-on} \end{figure}  snuverink_j committed Sep 06, 2017 741 In the drift space following the bending magnet, the CSR effects are calculated using Stupakov's algorithm with the same setting in both codes. The average fractional momentum change $\frac{\Delta p}{p}$ and the longitudinal emittance show good agreements between these codes. However, they produce different horizontal emittances as indicated in Figure~\ref{plot-emit-csr-on}.  snuverink_j committed Sep 06, 2017 742 743 744 \begin{figure}[!htbp] \centering \includegraphics[height=0.5\textwidth-0.6cm, angle = -90, trim = 3mm 0mm 2mm 0mm, clip]{figures/Benchmarks/emit-csr-on}  snuverink_j committed Sep 07, 2017 745 \caption{Transverse emittances in ELEGANT and \textit{OPAL}}  snuverink_j committed Sep 06, 2017 746 747 748 \label{fig:plot-emit-csr-on} \end{figure}  snuverink_j committed Sep 08, 2017 749 One important effect to notice is that in the drift space following the bending magnet, the normalized emittance $\epsilon_x(x, P_x)$ output by \textit{OPAL} keeps increasing while the trace-like emittance $\epsilon_x(x, x')$ calculated by ELEGANT does not. This can be explained by the fact that with a relatively large energy spread (about $3\%$ at the end of the dipole due to CSR), {\bf an correlation} between transverse position and energy can build up in a drift thereby induce emittance growth. However, this effect can only be observed in the normalized emittance calculated with $\epsilon_x(x, P_x) = \sqrt{\langle x^2 \rangle \langle P_x^2\rangle - \langle xP_x \rangle^2}$ where $P_x = \beta\gamma x'$, not the trace-like emittance which is calculated as $\epsilon_x(x, x') = \beta\gamma\sqrt{\langle x^2 \rangle \langle x'^2 \rangle - \langle xx' \rangle^2}$ \ref{prstab2003}. In Figure~\ref{plot-emit-csr-on}, a trace-like horizontal emittance is also calcualted for the \textit{OPAL} output beam distributions. Like the ELEGANT result, this trace-like emittance doesn't grow in the drift. However, their differences come from the ELEGANT's lack of CSR effect in the fringe field region.  snuverink_j committed Sep 06, 2017 750   snuverink_j committed Sep 07, 2017 751 \section{\textit{OPAL} \& \texttt{Impact-t}}  snuverink_j committed Sep 11, 2017 752 This benchmark compares rms quantities such as beam size and emittance of \textit{OPAL} and \texttt{Impact-t} \ref{qiang2005, qiang2006-1, qiang2006-2}. A {\bf cold} {10}{mA} H+ bunch is expanding in a {1}{m} drift space. A Gaussian distribution, with a cut at 4 $\sigma$ is used. The charge is computed by assuming a {1}{MHz} structure i.e. $Q_{\text{tot}}=\frac{I}{\nu_{\text{rf}}}$. For the simulation we use a grid with $16^{3}$ grid point and open boundary condition. The number of macro  snuverink_j committed Sep 06, 2017 753 754 particles is $N_{\text{p}} = 10^{5}$.  snuverink_j committed Sep 07, 2017 755 \subsection{\textit{OPAL} Input}  snuverink_j committed Sep 06, 2017 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 \begin{longexample} OPTION, ECHO = FALSE, PSDUMPFREQ = 10, STATDUMPFREQ = 10, REPARTFREQ = 1000, PSDUMPLOCALFRAME = FALSE, VERSION=10600; TITLE, string="Gaussian bunch drift test"; REAL Edes = 0.001; // GeV REAL CURRENT = 0.01; // A REAL gamma=(Edes+PMASS)/PMASS; REAL beta=sqrt(1-(1/gamma^2)); REAL gambet=gamma*beta; REAL P0 = gamma*beta*PMASS; D1: DRIFT, ELEMEDGE = 0.0, L = 1.0; L1: LINE = (D1); Fs1: FIELDSOLVER, FSTYPE = FFT, MX = 16, MY = 16, MT = 16, BBOXINCR=0.1; Dist1: DISTRIBUTION, TYPE = GAUSS, OFFSETX = 0.0, OFFSETY = 0.0, OFFSETZ = 15.0e-3, SIGMAX = 5.0e-3, SIGMAY = 5.0e-3, SIGMAZ = 5.0e-3, OFFSETPX = 0.0, OFFSETPY = 0.0, OFFSETPZ = 0.0, SIGMAPX = 0.0 , SIGMAPY = 0.0 , SIGMAPZ = 0.0 , CORRX = 0.0, CORRY = 0.0, CORRZ = 0.0, CUTOFFX = 4.0, CUTOFFY = 4.0, CUTOFFLONG = 4.0; Beam1: BEAM, PARTICLE = PROTON, CHARGE = 1.0, BFREQ = 1.0, PC = P0, NPART = 1E5, BCURRENT = CURRENT, FIELDSOLVER = Fs1; SELECT, LINE = L1; TRACK, LINE = L1, BEAM = Beam1, MAXSTEPS = 1000, ZSTOP = 1.0, DT = 1.0e-10; RUN, METHOD = "PARALLEL-T", BEAM = Beam1, FIELDSOLVER = Fs1, DISTRIBUTION = Dist1; ENDTRACK; STOP; \end{longexample}  snuverink_j committed Sep 06, 2017 796 \subsection{\texttt{Impact-t} Input}  snuverink_j committed Sep 06, 2017 797 \begin{longexample}  snuverink_j committed Sep 06, 2017 798 !Welcome to \texttt{Impact-t} input file.  snuverink_j committed Sep 06, 2017 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 !All comment lines start with "!" as the first character of the line. ! col row 1 1 ! ! information needed by the integrator: ! step-size, number of steps, and number of bunches/bins (??) ! ! dt Ntstep Nbunch 1.0e-10 700 1 ! ! phase-space dimension, number of particles, a series of flags ! that set the type of integrator, error study, diagnostics, and ! image charge, and the cutoff distance for the image charge ! ! PSdim Nptcl integF errF diagF imchgF imgCutOff (m) 6 100000 1 0 1 0 0.016 ! ! information about mesh: number of points in x, y, and z, type ! of boundary conditions, transverse aperture size (m), ! and longitudinal domain size (m) ! ! Nx Ny Nz bcF Rx Ry Lz 16 16 16 1 0.15 0.15 1.0e5 ! ! ! distribution type number (2 == Gauss), restart flag, space-charge substep ! flag, number of emission steps, and max emission time ! ! distType restartF substepF Nemission Temission 2 0 0 -1 0.0 ! ! sig* sigp* mu*p* *scale p*scale xmu* xmu* ! 0.005 0.0 0.0 1. 1. 0.0 0.0 0.005 0.0 0.0 1. 1. 0.0 0.0 0.005 0.0 0.0 1. 1. 0.0 0.0462 ! ! information about the beam: current, kinetic energy, particle ! rest energy, particle charge, scale frequency, and initial cavity phase ! ! I/A Ek/eV Mc2/eV Q/e freq/Hz phs/rad 0.010 1.0e6 938.271998e+06 1.0 1.0e6 0.0 ! ! ! ======= machine description starts here ======= ! the following lines, which must each be terminated with a '/', ! describe one beam-line element per line; the basic structure is ! element length, ???, ???, element type, and then a sequence of ! at most 24 numbers describing the element properties ! 0 drift tube 2 zedge radius ! 1 quadrupole 9 zedge, quad grad, fileID, ! radius, alignment error x, y ! rotation error x, y, z ! L/m N/A N/A type location of starting edge v1 v23 / 1.0 0 0 0 0.0 0.5 / \end{longexample} \subsection{Results}  snuverink_j committed Sep 06, 2017 857 A good agreement is shown in the Figure~\ref{plot-opal-impact1,plot-opal-impact2}. This proves to some extend the compatibility of the  snuverink_j committed Sep 07, 2017 858 space charge solvers of \textit{OPAL} and \texttt{Impact-t}.  snuverink_j committed Sep 06, 2017 859 860 861 862 863 864  \begin{figure}[!htbp] \centering \includegraphics[width=0.5\textwidth-0.6cm, angle = 0, trim = 20mm 0mm 15mm 0mm, clip]{figures/Benchmarks/opal-impact-1MHz-x} \hspace{1cm} \includegraphics[width=0.5\textwidth-0.6cm, angle = 0, trim = 20mm 0mm 15mm 0mm, clip]{figures/Benchmarks/opal-impact-1MHz-y}  snuverink_j committed Sep 07, 2017 865 \caption{Transverse beam sizes and emittances in \texttt{Impact-t} and \textit{OPAL}}  snuverink_j committed Sep 06, 2017 866 867 868 869 870 871 \label{fig:plot-opal-impact1} \end{figure} \begin{figure}[!htbp] \centering \includegraphics[width=0.5\textwidth-0.6cm, angle = 0, trim = 20mm 0mm 15mm 0mm, clip]{figures/Benchmarks/opal-impact-1MHz-z}  snuverink_j committed Sep 07, 2017 872 \caption{Longitudinal beam size and emittance in \texttt{Impact-t} and \textit{OPAL}}  snuverink_j committed Sep 06, 2017 873 874 875 876 \label{fig:plot-opal-impact2} \end{figure} \input{footer}