elements.tex 89 KB
 snuverink_j committed Sep 06, 2017 1 2 3 4 5 6 7 8 9 10 \input{header} \chapter{Elements} \label{chp:element} \index{Elements|(} \section{Element Input Format} \label{sec:elm-format} \index{Element!Format} All physical elements are defined by statements of the form  snuverink_j committed Sep 06, 2017 11 \begin{verbatim}  snuverink_j committed Sep 06, 2017 12 label:keyword, attribute,..., attribute  snuverink_j committed Sep 06, 2017 13 \end{verbatim}  snuverink_j committed Sep 06, 2017 14 15 16 17 18 19 where \begin{description} \item[label] \index{Element!Label} \Newline Is the name to be given to the element (in the example QF),  snuverink_j committed Sep 06, 2017 20  it is an {identifier} see~Section~\ref{label}.  snuverink_j committed Sep 06, 2017 21 22 23  \item[keyword] \Newline \index{Element!Keyword}  snuverink_j committed Sep 06, 2017 24  Is a {keyword} see~Section~\ref{label},  snuverink_j committed Sep 06, 2017 25  it is an element type keyword (in the example \texttt{QUADRUPOLE}),  snuverink_j committed Sep 06, 2017 26 27 28 \item[attribute] \Newline \index{Element!Attribute} normally has the form  snuverink_j committed Sep 06, 2017 29 \begin{verbatim}  snuverink_j committed Sep 06, 2017 30 attribute-name=attribute-value  snuverink_j committed Sep 06, 2017 31 \end{verbatim}  snuverink_j committed Sep 06, 2017 32 33 \item[attribute-name] \Newline selects the attribute from the list defined for the element type  snuverink_j committed Sep 06, 2017 34  \texttt{keyword} (in the example \texttt{L} and \texttt{K1}).  snuverink_j committed Sep 06, 2017 35  It must be an {identifier} see~Section~\ref{label}  snuverink_j committed Sep 06, 2017 36 \item[attribute-value] \Newline  snuverink_j committed Sep 06, 2017 37  gives it a {value} see~Section~\ref{attribute}  snuverink_j committed Sep 06, 2017 38 39 40 41 42  (in the example \texttt{1.8} and \texttt{0.015832}). \end{description} Omitted attributes are assigned a default value, normally zero. \noindent Example:  snuverink_j committed Sep 06, 2017 43 \begin{verbatim}  snuverink_j committed Sep 06, 2017 44 QF: QUADRUPOLE, L=1.8, K1=0.015832;  snuverink_j committed Sep 06, 2017 45 \end{verbatim}  snuverink_j committed Sep 06, 2017 46 47 48 49 50 51 52 53 54  \section{Common Attributes for all Elements} \label{sec:Element:common} \index{Element!Common Attributes} The following attributes are allowed on all elements: \begin{kdescription} \item[TYPE]  snuverink_j committed Sep 06, 2017 55  A {string value} see~Section~\ref{astring}.  snuverink_j committed Sep 06, 2017 56 57 58  It specifies an engineering type'' and can be used for element selection. \item[APERTURE]  snuverink_j committed Sep 06, 2017 59  A {string value} see~Section~\ref{astring} which describes  snuverink_j committed Sep 06, 2017 60 61 62 63 64 65 66 67 68 69  the element aperture. All but the last attribute of the aperture have units of meter, the last one is optional and is a positive real number. Possible choices are \begin{itemize} \item \texttt{APERTURE}="\texttt{SQUARE}(\texttt{a,f})" has a square shape of width and height \texttt{a}, \item \texttt{APERTURE}="\texttt{RECTANGLE}(\texttt{a,b,f})" has a rectangular shape of width \texttt{a} and height \texttt{b}, \item \texttt{APERTURE}="\texttt{CIRCLE}(\texttt{d,f})" has a circular shape of diameter \texttt{d}, \item \texttt{APERTURE}="\texttt{ELLIPSE}(\texttt{a,b,f})" has an elliptical shape of major \texttt{a} and minor \texttt{b}. \end{itemize} The option \texttt{SQUARE}(\texttt{a,f}) is equivalent to \texttt{RECTANGLE}(\texttt{a,a,f}) and \texttt{CIRCLE}(\texttt{d,f}) is equivalent to \texttt{ELLIPSE}(\texttt{d,d,f}). The size of the exit aperture is scaled by a factor $f$. For $f < 1$ the exit aperture is smaller than the entrance aperture, for $f = 1$ they are the same and for $f > 1$ the exit aperture is bigger.  snuverink_j committed Sep 06, 2017 70  Dipoles have \texttt{GAP} and \texttt{HGAP} which define an aperture and hence do not recognise \texttt{APERTURE}. The aperture of the dipoles has rectangular shape of height \texttt{GAP} and width \texttt{HGAP}. In longitudinal direction it is bent such that its center coincides with the circular segment of the reference particle when ignoring fringe fields. Between the beginning of the fringe field and the entrance face and between the exit face and the end of the exit fringe field the rectangular shape has width and height that are twice of what they are inside the dipole.  snuverink_j committed Sep 06, 2017 71   snuverink_j committed Sep 08, 2017 72  Default aperture for all other elements is a circle of {1.0}{m}.  snuverink_j committed Sep 06, 2017 73 74  \item[L]  snuverink_j committed Sep 08, 2017 75  The length of the element (default: {0}{m}).  snuverink_j committed Sep 06, 2017 76 \item[WAKEF]  snuverink_j committed Sep 06, 2017 77  Attach wakefield that was defined using the \texttt{WAKE} command.  snuverink_j committed Sep 06, 2017 78 \item[ELEMEDGE]  snuverink_j committed Sep 06, 2017 79  The edge of an element is specified in s coordinates in meters. This edge corresponds to the origin of the local coordinate system and is the physical start of the element. (Note that in general the fields will extend in front of this position.) The physical end of the element is determined by \texttt{ELEMEDGE} and its physical length. (Note again that in general the fields will extend past the physical end of the element.)  snuverink_j committed Sep 06, 2017 80 81 82 83 84 85 86 87 88 \item[PARTICLEMATTERINTERACTION] Attach a handler for particle matter interaction, see \ref{chp:partmatt}. \item[X] X-component of the position of the element in the laboratory coordinate system. \item[Y] Y-component of the position of the element in the laboratory coordinate system. \item[Z] Z-component of the position of the element in the laboratory coordinate system. \item[THETA]  snuverink_j committed Sep 08, 2017 89  Angle of rotation of the element about the y-axis relative to the default orientation, $\mathbf{n} = \transpose{\left(0, 0, 1\right)}$.  snuverink_j committed Sep 06, 2017 90 \item[PHI]  snuverink_j committed Sep 08, 2017 91 92  Angle of rotation of the element about the x-axis relative to the default orientation, $\mathbf{n} = \transpose{\left(0, 0, 1\right)}$ \item[PSI] Angle of rotation of the element about the z-axis relative to the default orientation, $\mathbf{n} = \transpose{\left(0, 0, 1\right)}$ \item[ORIGIN]  snuverink_j committed Sep 06, 2017 93  3D position vector. An alternative to using \texttt{X}, \texttt{Y} and \texttt{Z} to position the element. Can't be combined with \texttt{THETA} and \texttt{PHI}. Use \texttt{ORIENTATION} instead.  snuverink_j committed Sep 06, 2017 94 \item[ORIENTATION]  snuverink_j committed Sep 08, 2017 95  Vector of Tait-Bryan angles \ref{bib:tait-bryan}. An alternative to rotate the element instead of using \texttt{THETA}, \texttt{PHI} and \texttt{PSI}. Can't be combined with \texttt{X}, \texttt{Y} and \texttt{Z}, use \texttt{ORIGIN} instead.  snuverink_j committed Sep 06, 2017 96 97 98 99 100 101 102 \item[DX] Error on x-component of position of element. Doesn't affect the design trajectory. \item[DY] Error on y-component of position of element. Doesn't affect the design trajectory. \item[DZ] Error on z-component of position of element. Doesn't affect the design trajectory. \item[DTHETA]  snuverink_j committed Sep 06, 2017 103  Error on angle \texttt{THETA}. Doesn't affect the design trajectory.  snuverink_j committed Sep 06, 2017 104 \item[DPHI]  snuverink_j committed Sep 06, 2017 105  Error on angle \texttt{PHI}. Doesn't affect the design trajectory.  snuverink_j committed Sep 06, 2017 106 \item[DPSI]  snuverink_j committed Sep 06, 2017 107  Error on angle \texttt{PSI}. Doesn't affect the design trajectory.  snuverink_j committed Sep 06, 2017 108 109 110 111 112 113 114 115 116 117  \end{kdescription} All elements can have arbitrary additional attributes which are defined in the respective section. \clearpage \section{Drift Spaces} \label{sec:drift} \index{DRIFT}  snuverink_j committed Sep 06, 2017 118 \begin{verbatim}  snuverink_j committed Sep 06, 2017 119 label:DRIFT, TYPE=string, APERTURE=string, L=real;  snuverink_j committed Sep 06, 2017 120 \end{verbatim}  snuverink_j committed Sep 06, 2017 121 122 A DRIFT space has no additional attributes. \noindent Examples:  snuverink_j committed Sep 06, 2017 123 \begin{verbatim}  snuverink_j committed Sep 06, 2017 124 125 DR1:DRIFT, L=1.5; DR2:DRIFT, L=DR1->L, TYPE=DRF;  snuverink_j committed Sep 06, 2017 126 \end{verbatim}  snuverink_j committed Sep 06, 2017 127 128 The length of \texttt{DR2} will always be equal to the length of \texttt{DR1}. The reference system for a drift space is a Cartesian coordinate system  snuverink_j committed Sep 06, 2017 129 \ifthenelse{\boolean{ShowMap}}{see~Figure~\ref{straight}}{}.  snuverink_j committed Sep 07, 2017 130 This is a restricted feature: \texttt{DOPAL-cycl}. In \textit{OPAL-t} drifts are implicitly given, if no field is present.  snuverink_j committed Sep 06, 2017 131 132 133 134 135 136 137  \clearpage \section{Bending Magnets} \label{sec:bend} \index{Bending Magnets|(}  snuverink_j committed Sep 07, 2017 138 139 140 141 Bending magnets refer to dipole fields that bend particle trajectories. Currently \textit{OPAL} supports three different bend elements: \texttt{RBEND}, (valid in \textit{OPAL-t}, see Section~\ref{RBend}), \texttt{SBEND} (valid in \textit{OPAL-t}, see Section~\ref{SBend}), \texttt{RBEND3D}, (valid in \textit{OPAL-t}, see Section~\ref{RBend3D}) and \texttt{SBEND3D} (valid in \textit{OPAL-cycl}, see Section~\ref{SBend3D}).  snuverink_j committed Sep 06, 2017 142 143 144 145 146  Describing a bending magnet can be somewhat complicated as there can be many parameters to consider: bend angle, bend radius, entrance and exit angles etc. Therefore we have divided this section into several parts: \begin{enumerate}  snuverink_j committed Sep 07, 2017 147 \item Section~\ref{RBend,SBend} describe the geometry and attributes of the \textit{OPAL-t} bend  snuverink_j committed Sep 06, 2017 148 149  elements \texttt{RBEND} and \texttt{SBEND}. \item Section~\ref{RBendSBendExamp} describes how to implement an \texttt{RBEND} or \texttt{SBEND} in an  snuverink_j committed Sep 07, 2017 150  \textit{OPAL-t} simulation.  snuverink_j committed Sep 06, 2017 151 \item Section~\ref{SBend3D} is self contained. It describes how to implement an \texttt{SBEND3D} element in  snuverink_j committed Sep 07, 2017 152  an \textit{OPAL-cycl} simulation.  snuverink_j committed Sep 06, 2017 153 154 155 156 \end{enumerate} \input{figures/Elements/RBend}  snuverink_j committed Sep 07, 2017 157 \subsection{RBend (\textit{OPAL-t})}  snuverink_j committed Sep 06, 2017 158 159 \label{ssec:RBend} \index{RBEND}  snuverink_j committed Sep 06, 2017 160 161 An \texttt{RBEND} is a rectangular bending magnet. The key property of an \texttt{RBEND} is that is has parallel pole faces. Figure~\ref{rbend} shows an \texttt{RBEND} with a positive bend angle and a positive entrance edge angle.  snuverink_j committed Sep 06, 2017 162 163 164  \begin{kdescription} \item[L]  snuverink_j committed Sep 06, 2017 165  Physical length of magnet (meters, see Figure~\ref{rbend}).  snuverink_j committed Sep 06, 2017 166 167 168 169 170 171 172 173 174  \item[GAP] Full vertical gap of the magnet (meters). \item[HAPERT] Non-bend plane aperture of the magnet (meters). (Defaults to one half the bend radius.) \item[ANGLE] Bend angle (radians). Field amplitude of bend will be adjusted to achieve this angle. (Note that for  snuverink_j committed Sep 06, 2017 175  an \texttt{RBEND}, the bend angle must be less than $\nicefrac{\pi}{2} + E1$, where \texttt{E1} is the entrance edge angle.)  snuverink_j committed Sep 06, 2017 176 177  \item[K0]  snuverink_j committed Sep 06, 2017 178  Field amplitude in y direction (Tesla). If the \texttt{ANGLE} attribute is set, \texttt{K0} is ignored.  snuverink_j committed Sep 06, 2017 179 180  \item[K0S]  snuverink_j committed Sep 06, 2017 181  Field amplitude in x direction (Tesla). If the \texttt{ANGLE} attribute is set, \texttt{K0S} is ignored.  snuverink_j committed Sep 06, 2017 182 183 184  \item[K1] Field gradient index of the magnet, $K_1=-\frac{R}{B_{y}}\diffp{B_y}{x}$, where  snuverink_j committed Sep 07, 2017 185  $R$ is the bend radius as defined in Figure~\ref{rbend}. Not supported in \texttt{DOPAL-t} any more. Superimpose a \texttt{Quadrupole} instead.  snuverink_j committed Sep 06, 2017 186 187  \item[E1]  snuverink_j committed Sep 06, 2017 188  Entrance edge angle (radians). Figure~\ref{rbend} shows the definition of a positive entrance  snuverink_j committed Sep 06, 2017 189 190  edge angle. (Note that the exit edge angle is fixed in an \texttt{RBEND} element to E2 = ANGLE $\text{\texttt{E2}} = \text{\texttt{ANGLE}} - \text{\texttt{E1}}$).  snuverink_j committed Sep 06, 2017 191 192  \item[DESIGNENERGY]  snuverink_j committed Sep 08, 2017 193  Energy of the reference particle ({MeV}). The reference particle travels approximately the path shown in  snuverink_j committed Sep 06, 2017 194  Figure~\ref{rbend}.  snuverink_j committed Sep 06, 2017 195 196 197  \item[FMAPFN] Name of the field map for the magnet. Currently maps of type \texttt{1DProfile1} can  snuverink_j committed Sep 06, 2017 198 199 200  be used see~Section~\ref{1DProfile1}. The default option for this attribute is \texttt{FMAPN} = \texttt{1DPROFILE1-DEFAULT}'' see~Section~\ref{benddefaultfieldmapopalt}. The field map is used to describe the fringe fields of the magnet see~Section~\ref{1DProfile1}.  snuverink_j committed Sep 06, 2017 201 202 203 204  \end{kdescription} \clearpage  snuverink_j committed Sep 07, 2017 205 \subsection{RBend3D (\textit{OPAL-t})}  snuverink_j committed Sep 06, 2017 206 207 \label{ssec:RBend3D} \index{RBEND3D}  snuverink_j committed Sep 06, 2017 208 209 An \texttt{RBEND3D3D} is a rectangular bending magnet. The key property of an \texttt{RBEND3D} is that is has parallel pole faces. Figure~\ref{rbend} shows an \texttt{RBEND3D} with a positive bend angle and a positive entrance edge angle.  snuverink_j committed Sep 06, 2017 210 211 212  \begin{kdescription} \item[L]  snuverink_j committed Sep 06, 2017 213  Physical length of magnet (meters, see Figure~\ref{rbend}).  snuverink_j committed Sep 06, 2017 214 215 216 217 218 219 220 221 222  \item[GAP] Full vertical gap of the magnet (meters). \item[HAPERT] Non-bend plane aperture of the magnet (meters). (Defaults to one half the bend radius.) \item[ANGLE] Bend angle (radians). Field amplitude of bend will be adjusted to achieve this angle. (Note that for  snuverink_j committed Sep 06, 2017 223  an \texttt{RBEND3D}, the bend angle must be less than $\nicefrac{\pi}{2} + E1$, where \texttt{E1} is the entrance edge angle.)  snuverink_j committed Sep 06, 2017 224 225  \item[K0]  snuverink_j committed Sep 06, 2017 226  Field amplitude in y direction (Tesla). If the \texttt{ANGLE} attribute is set, \texttt{K0} is ignored.  snuverink_j committed Sep 06, 2017 227 228  \item[K0S]  snuverink_j committed Sep 06, 2017 229  Field amplitude in x direction (Tesla). If the \texttt{ANGLE} attribute is set, \texttt{K0S} is ignored.  snuverink_j committed Sep 06, 2017 230 231 232  \item[K1] Field gradient index of the magnet, $K_1=-\frac{R}{B_{y}}\diffp{B_y}{x}$, where  snuverink_j committed Sep 07, 2017 233  $R$ is the bend radius as defined in Figure~\ref{rbend}. Not supported in \texttt{DOPAL-t} any more. Superimpose a \texttt{Quadrupole} instead.  snuverink_j committed Sep 06, 2017 234 235  \item[E1]  snuverink_j committed Sep 06, 2017 236  Entrance edge angle (radians). Figure~\ref{rbend} shows the definition of a positive entrance  snuverink_j committed Sep 06, 2017 237 238  edge angle. (Note that the exit edge angle is fixed in an \texttt{RBEND3D} element to E2 = ANGLE $\text{\texttt{E2}} = \text{\texttt{ANGLE}} - \text{\texttt{E1}}$).  snuverink_j committed Sep 06, 2017 239 240  \item[DESIGNENERGY]  snuverink_j committed Sep 08, 2017 241  Energy of the reference particle ({MeV}). The reference particle travels approximately the path shown in  snuverink_j committed Sep 06, 2017 242  Figure~\ref{rbend}.  snuverink_j committed Sep 06, 2017 243 244 245  \item[FMAPFN] Name of the field map for the magnet. Currently maps of type \texttt{1DProfile1} can  snuverink_j committed Sep 06, 2017 246 247 248  be used see~Section~\ref{1DProfile1}. The default option for this attribute is \texttt{FMAPN} = \texttt{1DPROFILE1-DEFAULT}'' see~Section~\ref{benddefaultfieldmapopalt}. The field map is used to describe the fringe fields of the magnet see~Section~\ref{1DProfile1}.  snuverink_j committed Sep 06, 2017 249 250 251 252 253 254  \end{kdescription} \clearpage \input{figures/Elements/SBend}  snuverink_j committed Sep 07, 2017 255 \subsection{SBend (\textit{OPAL-t})}  snuverink_j committed Sep 06, 2017 256 257 \label{ssec:SBend} \index{SBEND}  snuverink_j committed Sep 06, 2017 258 259 An \texttt{SBEND} is a sector bending magnet. An \texttt{SBEND} can have independent entrance and exit edge angles. Figure~\ref{sbend} shows an \texttt{SBEND} with a positive bend angle, a positive entrance  snuverink_j committed Sep 06, 2017 260 261 262 263 edge angle, and a positive exit edge angle. \begin{kdescription} \item[L]  snuverink_j committed Sep 06, 2017 264  Chord length of the bend reference arc in meters (see Figure~\ref{sbend}), given by:  snuverink_j committed Sep 06, 2017 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281  \begin{equation*} L = 2 R sin\left(\frac{\alpha}{2}\right) \end{equation*} \item[GAP] Full vertical gap of the magnet (meters). \item[HAPERT] Non-bend plane aperture of the magnet (meters). (Defaults to one half the bend radius.) \item[ANGLE] Bend angle (radians). Field amplitude of the bend will be adjusted to achieve this angle. (Note that practically speaking, bend angles greater than $\frac{3 \pi}{2}$ (270 degrees) can be problematic. Beyond this, the fringe fields from the entrance and exit pole faces could start to interfere, so be careful when setting up bend angles greater than this. An angle greater than or equal to $2 \pi$ (360 degrees) is not allowed.) \item[K0]  snuverink_j committed Sep 06, 2017 282  Field amplitude in y direction (Tesla). If the \texttt{ANGLE} attribute is set, \texttt{K0} is ignored.  snuverink_j committed Sep 06, 2017 283 284  \item[K0S]  snuverink_j committed Sep 06, 2017 285  Field amplitude in x direction (Tesla). If the \texttt{ANGLE} attribute is set, \texttt{K0S} is ignored.  snuverink_j committed Sep 06, 2017 286 287 288  \item[K1] Field gradient index of the magnet, $K_1=-\frac{R}{B_{y}}\diffp{B_y}{x}$, where  snuverink_j committed Sep 07, 2017 289  $R$ is the bend radius as defined in Figure~\ref{sbend}. Not supported in \texttt{DOPAL-t} any more. Superimpose a \texttt{Quadrupole} instead.  snuverink_j committed Sep 06, 2017 290 291  \item[E1]  snuverink_j committed Sep 08, 2017 292  Entrance edge angle ({rad}). Figure~\ref{sbend} shows the definition of a positive entrance  snuverink_j committed Sep 06, 2017 293 294 295  edge angle. \item[E2]  snuverink_j committed Sep 08, 2017 296  Exit edge angle ({rad}). Figure~\ref{sbend} shows the definition of a positive exit edge angle.  snuverink_j committed Sep 06, 2017 297 298  \item[DESIGNENERGY]  snuverink_j committed Sep 08, 2017 299  Energy of the bend reference particle ({MeV}). The reference particle travels approximately the path shown in  snuverink_j committed Sep 06, 2017 300  Figure~\ref{sbend}.  snuverink_j committed Sep 06, 2017 301 302 303  \item[FMAPFN] Name of the field map for the magnet. Currently maps of type \texttt{1DProfile1} can  snuverink_j committed Sep 06, 2017 304 305 306  be used see~Section~\ref{1DProfile1}. The default option for this attribute is \texttt{FMAPN} = \texttt{1DPROFILE1-DEFAULT}'' see~Section~\ref{benddefaultfieldmapopalt}. The field map is used to describe the fringe fields of the magnet see~Section~\ref{1DProfile1}.  snuverink_j committed Sep 06, 2017 307 308 309 310 311  \end{kdescription} \clearpage  snuverink_j committed Sep 07, 2017 312 \subsection{RBend and SBend Examples (\textit{OPAL-t})}  snuverink_j committed Sep 06, 2017 313 \label{ssec:RBendSBendExamp}  snuverink_j committed Sep 07, 2017 314 Describing an \texttt{RBEND} or an \texttt{SBEND} in an \textit{OPAL-t} simulation requires effectively identical commands.  snuverink_j committed Sep 06, 2017 315 There are only slight differences between the two. The \texttt{L} attribute has a different definition for the two  snuverink_j committed Sep 06, 2017 316 types of bends see~Section~\ref{RBend,SBend}, and an \texttt{SBEND} has an additional  snuverink_j committed Sep 06, 2017 317 attribute \texttt{E2} that has no effect on an \texttt{RBEND}, see Section~\ref{SBend}.  snuverink_j committed Sep 06, 2017 318 Therefore, in this section, we will give several examples of how to implement a bend, using the  snuverink_j committed Sep 06, 2017 319 \texttt{RBEND} and \texttt{SBEND} commands interchangeably. The understanding is that the command formats are  snuverink_j committed Sep 06, 2017 320 321 essentially the same.  snuverink_j committed Sep 07, 2017 322 When implementing an \texttt{RBEND} or \texttt{SBEND} in an \textit{OPAL-t} simulation, it is important to note the following:  snuverink_j committed Sep 06, 2017 323 324  \begin{enumerate}  snuverink_j committed Sep 07, 2017 325 \item Internally \textit{OPAL-t} treats all bends as positive, as defined by Figure~\ref{rbend,sbend}.  snuverink_j committed Sep 06, 2017 326 327  Bends in other directions within the x/y plane are accomplished by rotating a positive bend about its z axis.  snuverink_j committed Sep 06, 2017 328 \item If the \texttt{ANGLE} attribute is set to a non-zero value, the \texttt{K0} and \texttt{K0S} attributes  snuverink_j committed Sep 06, 2017 329  will be ignored.  snuverink_j committed Sep 06, 2017 330 331 \item When using the \texttt{ANGLE} attribute to define a bend, the actual beam will be bent through a different angle if its mean kinetic energy doesn't correspond to the \texttt{DESIGNENERGY}.  snuverink_j committed Sep 06, 2017 332 \item Internally the bend geometry is setup based on the ideal reference trajectory, as shown in  snuverink_j committed Sep 06, 2017 333  Figure~\ref{rbend,sbend}.\item If the default field map, \texttt{1DPROFILE-DEFAULT}''  snuverink_j committed Sep 06, 2017 334  see~Section~\ref{benddefaultfieldmapopalt}, is used, the fringe fields will be adjusted  snuverink_j committed Sep 06, 2017 335 336 337 338  so that the effective length of the real, soft edge magnet matches the ideal, hard edge bend that is defined by the reference trajectory. \end{enumerate}  snuverink_j committed Sep 07, 2017 339 For the rest of this section, we will give several examples of how to input bends in an \textit{OPAL-t}  snuverink_j committed Sep 06, 2017 340 simulation. We will start with a simple example using the  snuverink_j committed Sep 06, 2017 341 342 \texttt{ANGLE} attribute to set the bend strength and using the default field map see~Section~\ref{benddefaultfieldmapopalt} for describing the magnet fringe fields see~Section~\ref{1DProfile1}:  snuverink_j committed Sep 06, 2017 343   snuverink_j committed Sep 06, 2017 344 \begin{verbatim}  snuverink_j committed Sep 06, 2017 345 346 347 348 349 350 Bend: RBend, ANGLE = 30.0 * Pi / 180.0, FMAPFN = "1DPROFILE1-DEFAULT", ELEMEDGE = 0.25, DESIGNENERGY = 10.0, L = 0.5, GAP = 0.02;  snuverink_j committed Sep 06, 2017 351 352 \end{verbatim} This is a definition of a simple \texttt{RBEND} that bends the beam in a positive direction 30 degrees (towards  snuverink_j committed Sep 08, 2017 353 the negative x axis as if Figure~\ref{rbend}). It has a design energy of {10}{MeV}, a length of {0.5}{m}, a  snuverink_j committed Sep 08, 2017 354 vertical gap of {2}{\centim} and a {0}{^{\circ}} entrance edge angle. (Therefore the exit edge angle is {30}{^{\circ}}.) We are  snuverink_j committed Sep 06, 2017 355 using the default, internal field map 1DPROFILE1-DEFAULT'' see~Section~\ref{benddefaultfieldmapopalt}  snuverink_j committed Sep 07, 2017 356  which describes the magnet fringe fields see~Section~\ref{1DProfile1}. When \textit{OPAL} is run, you will  snuverink_j committed Sep 06, 2017 357 get the following output (assuming an electron beam) for this \texttt{RBEND} definition:  snuverink_j committed Sep 06, 2017 358   snuverink_j committed Sep 06, 2017 359 \begin{verbatim}  snuverink_j committed Sep 06, 2017 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 RBend > Reference Trajectory Properties RBend > =============================== RBend > RBend > Bend angle magnitude: 0.523599 rad (30 degrees) RBend > Entrance edge angle: 0 rad (0 degrees) RBend > Exit edge angle: 0.523599 rad (30 degrees) RBend > Bend design radius: 1 m RBend > Bend design energy: 1e+07 eV RBend > RBend > Bend Field and Rotation Properties RBend > ================================== RBend > RBend > Field amplitude: -0.0350195 T RBend > Field index (gradient): 0 m^-1 RBend > Rotation about x axis: 0 rad (0 degrees) RBend > Rotation about y axis: 0 rad (0 degrees) RBend > Rotation about z axis: 0 rad (0 degrees) RBend > RBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields RBend > ====================================================================== RBend > RBend > Reference particle is bent: 0.523599 rad (30 degrees) in x plane RBend > Reference particle is bent: 0 rad (0 degrees) in y plane  snuverink_j committed Sep 06, 2017 383 \end{verbatim}  snuverink_j committed Sep 06, 2017 384 The first section of this output gives the properties of the reference trajectory like that described in  snuverink_j committed Sep 07, 2017 385 Figure~\ref{rbend}. From the value of \texttt{ANGLE} and the length, \texttt{L}, of the magnet, \textit{OPAL}  snuverink_j committed Sep 08, 2017 386 calculates the {10}{MeV} reference particle trajectory radius, \texttt{R}. From the bend geometry and the  snuverink_j committed Sep 08, 2017 387 entrance angle ({0}{^{\circ}} in this case), the exit angle is calculated.  snuverink_j committed Sep 06, 2017 388 389 390 391 392 393  The second section gives the field amplitude of the bend and its gradient (quadrupole focusing component), given the particle charge ($-e$ in this case so the amplitude is negative to get a positive bend direction). Also listed is the rotation of the magnet about the various axes. Of course, in the actual simulation the particles will not see a hard edge bend magnet, but rather a soft  snuverink_j committed Sep 06, 2017 394 edge magnet with fringe fields described by the \texttt{RBEND} field map file \texttt{FMAPFN} see~Section~\ref{1DProfile1}.  snuverink_j committed Sep 07, 2017 395 So, once the hard edge bend/reference trajectory is determined, \textit{OPAL}  snuverink_j committed Sep 06, 2017 396 then includes the fringe fields in the calculation. When the user chooses to use the default field map,  snuverink_j committed Sep 07, 2017 397 \textit{OPAL} will automatically adjust the position of the fringe fields appropriately so that the soft edge magnet  snuverink_j committed Sep 06, 2017 398 is equivalent to the hard edge magnet described by the reference trajectory. To check that this was done  snuverink_j committed Sep 07, 2017 399 properly, \textit{OPAL} integrates the reference particle through the final magnet description with the fringe fields  snuverink_j committed Sep 06, 2017 400 401 402 403 404 405 included. The result is shown in the final part of the output. In this case we see that the soft edge bend does indeed bend our reference particle through the correct angle. What is important to note from this first example, is that it is this final part of the bend output that tells you the actual bend angle of the reference particle.  snuverink_j committed Sep 06, 2017 406 407 In this next example, we merely rewrite the first example, but use \texttt{K0} to set the field strength of the \texttt{RBEND}, rather than the \texttt{ANGLE} attribute:  snuverink_j committed Sep 06, 2017 408   snuverink_j committed Sep 06, 2017 409 \begin{verbatim}  snuverink_j committed Sep 06, 2017 410 411 412 413 414 415 Bend: RBend, K0 = -0.0350195, FMAPFN = "1DPROFILE1-DEFAULT", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.5, GAP = 0.02;  snuverink_j committed Sep 06, 2017 416 \end{verbatim}  snuverink_j committed Sep 07, 2017 417 The output from \textit{OPAL} now reads as follows:  snuverink_j committed Sep 06, 2017 418   snuverink_j committed Sep 06, 2017 419 \begin{verbatim}  snuverink_j committed Sep 06, 2017 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 RBend > Reference Trajectory Properties RBend > =============================== RBend > RBend > Bend angle magnitude: 0.523599 rad (30 degrees) RBend > Entrance edge angle: 0 rad (0 degrees) RBend > Exit edge angle: 0.523599 rad (30 degrees) RBend > Bend design radius: 0.999999 m RBend > Bend design energy: 1e+07 eV RBend > RBend > Bend Field and Rotation Properties RBend > ================================== RBend > RBend > Field amplitude: -0.0350195 T RBend > Field index (gradient): 0 m^-1 RBend > Rotation about x axis: 0 rad (0 degrees) RBend > Rotation about y axis: 0 rad (0 degrees) RBend > Rotation about z axis: 0 rad (0 degrees) RBend > RBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields RBend > ====================================================================== RBend > RBend > Reference particle is bent: 0.5236 rad (30.0001 degrees) in x plane RBend > Reference particle is bent: 0 rad (0 degrees) in y plane  snuverink_j committed Sep 06, 2017 443 \end{verbatim}  snuverink_j committed Sep 06, 2017 444 445 446 447 448 449 450 The output is effectively identical, to within a small numerical error. Now, let us modify this first example so that we bend instead in the negative x direction. There are several ways to do this: \begin{enumerate} \item  snuverink_j committed Sep 06, 2017 451 \begin{verbatim}  snuverink_j committed Sep 06, 2017 452 453 454 455 456 457 Bend: RBend, ANGLE = -30.0 * Pi / 180.0, FMAPFN = "1DPROFILE1-DEFAULT", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.5, GAP = 0.02;  snuverink_j committed Sep 06, 2017 458 \end{verbatim}  snuverink_j committed Sep 06, 2017 459 \item  snuverink_j committed Sep 06, 2017 460 \begin{verbatim}  snuverink_j committed Sep 06, 2017 461 462 463 464 465 466 467 Bend: RBend, ANGLE = 30.0 * Pi / 180.0, FMAPFN = "1DPROFILE1-DEFAULT", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.5, GAP = 0.02, ROTATION = Pi;  snuverink_j committed Sep 06, 2017 468 \end{verbatim}  snuverink_j committed Sep 06, 2017 469 \item  snuverink_j committed Sep 06, 2017 470 \begin{verbatim}  snuverink_j committed Sep 06, 2017 471 472 473 474 475 476 Bend: RBend, K0 = 0.0350195, FMAPFN = "1DPROFILE1-DEFAULT", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.5, GAP = 0.02;  snuverink_j committed Sep 06, 2017 477 \end{verbatim}  snuverink_j committed Sep 06, 2017 478 \item  snuverink_j committed Sep 06, 2017 479 \begin{verbatim}  snuverink_j committed Sep 06, 2017 480 481 482 483 484 485 486 Bend: RBend, K0 = -0.0350195, FMAPFN = "1DPROFILE1-DEFAULT", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.5, GAP = 0.02, ROTATION = Pi;  snuverink_j committed Sep 06, 2017 487 \end{verbatim}  snuverink_j committed Sep 06, 2017 488 489 490 \end{enumerate} In each of these cases, we get the following output for the bend (to within small numerical errors).  snuverink_j committed Sep 06, 2017 491 \begin{verbatim}  snuverink_j committed Sep 06, 2017 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 RBend > Reference Trajectory Properties RBend > =============================== RBend > RBend > Bend angle magnitude: 0.523599 rad (30 degrees) RBend > Entrance edge angle: 0 rad (0 degrees) RBend > Exit edge angle: 0.523599 rad (30 degrees) RBend > Bend design radius: 1 m RBend > Bend design energy: 1e+07 eV RBend > RBend > Bend Field and Rotation Properties RBend > ================================== RBend > RBend > Field amplitude: -0.0350195 T RBend > Field index (gradient): -0 m^-1 RBend > Rotation about x axis: 0 rad (0 degrees) RBend > Rotation about y axis: 0 rad (0 degrees) RBend > Rotation about z axis: 3.14159 rad (180 degrees) RBend > RBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields RBend > ====================================================================== RBend > RBend > Reference particle is bent: -0.523599 rad (-30 degrees) in x plane RBend > Reference particle is bent: 0 rad (0 degrees) in y plane  snuverink_j committed Sep 06, 2017 515 \end{verbatim}  snuverink_j committed Sep 06, 2017 516 In general, we suggest to always define a bend in the positive x  snuverink_j committed Sep 06, 2017 517 direction (as in Figure~\ref{rbend}) and then use the \texttt{ROTATION} attribute to bend in other  snuverink_j committed Sep 06, 2017 518 519 directions in the x/y plane (as in examples 2 and 4 above).  snuverink_j committed Sep 06, 2017 520 As a final \texttt{RBEND} example, here is a suggested format for the four bend definitions if one where implementing  snuverink_j committed Sep 06, 2017 521 522 a four dipole chicane:  snuverink_j committed Sep 06, 2017 523 \begin{verbatim}  snuverink_j committed Sep 06, 2017 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 Bend1: RBend, ANGLE = 20.0 * Pi / 180.0, E1 = 0.0, FMAPFN = "1DPROFILE1-DEFAULT", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.25, GAP = 0.02, ROTATION = Pi; Bend2: RBend, ANGLE = 20.0 * Pi / 180.0, E1 = 20.0 * Pi / 180.0, FMAPFN = "1DPROFILE1-DEFAULT", ELEMEDGE = 1.0, DESIGNENERGY = 10.0E6, L = 0.25, GAP = 0.02, ROTATION = 0.0; Bend3: RBend, ANGLE = 20.0 * Pi / 180.0, E1 = 0.0, FMAPFN = "1DPROFILE1-DEFAULT", ELEMEDGE = 1.5, DESIGNENERGY = 10.0E6, L = 0.25, GAP = 0.02, ROTATION = 0.0; Bend4: RBend, ANGLE = 20.0 * Pi / 180.0, E1 = 20.0 * Pi / 180.0 FMAPFN = "1DPROFILE1-DEFAULT", ELEMEDGE = 2.25, DESIGNENERGY = 10.0E6, L = 0.25, GAP = 0.02, ROTATION = Pi;  snuverink_j committed Sep 06, 2017 559 \end{verbatim}  snuverink_j committed Sep 06, 2017 560   snuverink_j committed Sep 06, 2017 561 Up to now, we have only given examples of \texttt{RBEND} definitions. If we replaced RBend'' in the above  snuverink_j committed Sep 07, 2017 562 examples with SBend'', we would still be defining valid \textit{OPAL-t} bends. In fact, by adjusting the \texttt{L}  snuverink_j committed Sep 06, 2017 563 attribute according to Section~\ref{RBend,SBend}, and by adding the appropriate  snuverink_j committed Sep 06, 2017 564 565 definitions of the \texttt{E2} attribute, we could even get identical results using \texttt{SBEND}s instead of \texttt{RBEND}s. (As we said, the two bends are very similar in command format.)  snuverink_j committed Sep 06, 2017 566 567  Up till now, we have only used the default field map. Custom field maps can also be used. There are two  snuverink_j committed Sep 06, 2017 568 different options in this case see~Section~\ref{1DProfile1}:  snuverink_j committed Sep 06, 2017 569 570 571 572 573  \begin{enumerate} \item Field map defines fringe fields and magnet length. \item Field map defines fringe fields only. \end{enumerate}  snuverink_j committed Sep 07, 2017 574 575 The first case describes how field maps were used in previous versions of \textit{OPAL} (and can still be used in the current version). The second option is new to \textit{OPAL} \textit{OPAL}version{1.2.00} and it has a couple of advantages:  snuverink_j committed Sep 06, 2017 576 577  \begin{enumerate}  snuverink_j committed Sep 06, 2017 578 \item Because only the fringe fields are described, the length of the magnet must be set using the \texttt{L}  snuverink_j committed Sep 06, 2017 579 580 581 582 583  attribute. In turn, this means that the same field map can be used by many bend magnets with different lengths (assuming they have equivalent fringe fields). By contrast, if the magnet length is set by the field map, one must generate a new field map for each dipole of different length even if the fringe fields are the same. \item We can adjust the position of the fringe field origin relative to the entrance and exit points of the  snuverink_j committed Sep 06, 2017 584  magnet see~Section~\ref{1DProfile1}. This gives us another degree  snuverink_j committed Sep 06, 2017 585 586 587 588 589 590 591  of freedom for describing the fringe fields, allowing us to adjust the effective length of the magnet. \end{enumerate} We will now give examples of how to use a custom field map, starting with the first case where the field map describes the fringe fields and the magnet length. Assume we have the following \texttt{1DProfile1} field map:  snuverink_j committed Sep 06, 2017 592 \begin{verbatim}  snuverink_j committed Sep 06, 2017 593 594 595 596 597 598 599 1DProfile1 1 1 2.0 -10.0 0.0 10.0 1 15.0 25.0 35.0 1 0.00000E+00 2.00000E+00 0.00000E+00 2.00000E+00  snuverink_j committed Sep 06, 2017 600 601 \end{verbatim} We can use this field map to define the following bend (note we are now using the \texttt{SBEND} command):  snuverink_j committed Sep 06, 2017 602   snuverink_j committed Sep 06, 2017 603 \begin{verbatim}  snuverink_j committed Sep 06, 2017 604 605 606 607 608 609 610 Bend: SBend, ANGLE = 60.0 * Pi / 180.0, E1 = -10.0 * Pi / 180.0, E2 = 20.0 Pi / 180.0, FMAPFN = "TEST-MAP.T7", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, GAP = 0.02;  snuverink_j committed Sep 06, 2017 611 612 \end{verbatim} \textbf{Notice that we do not set the magnet length using the \texttt{L} attribute.} (In fact, we don't even  snuverink_j committed Sep 06, 2017 613 614 615 include it. If we did and set it to a non-zero value, the exit fringe fields of the magnet would not be correct.) This input gives the following output:  snuverink_j committed Sep 06, 2017 616 \begin{verbatim}  snuverink_j committed Sep 06, 2017 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 SBend > Reference Trajectory Properties SBend > =============================== SBend > SBend > Bend angle magnitude: 1.0472 rad (60 degrees) SBend > Entrance edge angle: -0.174533 rad (-10 degrees) SBend > Exit edge angle: 0.349066 rad (20 degrees) SBend > Bend design radius: 0.25 m SBend > Bend design energy: 1e+07 eV SBend > SBend > Bend Field and Rotation Properties SBend > ================================== SBend > SBend > Field amplitude: -0.140385 T SBend > Field index (gradient): 0 m^-1 SBend > Rotation about x axis: 0 rad (0 degrees) SBend > Rotation about y axis: 0 rad (0 degrees) SBend > Rotation about z axis: 0 rad (0 degrees) SBend > SBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields SBend > ====================================================================== SBend > SBend > Reference particle is bent: 1.0472 rad (60 degrees) in x plane SBend > Reference particle is bent: 0 rad (0 degrees) in y plane  snuverink_j committed Sep 06, 2017 640 641 642 \end{verbatim} Because we set the bend strength using the \texttt{ANGLE} attribute, the magnet field strength is automatically adjusted so that the reference particle is bent exactly \texttt{ANGLE} radians when the fringe fields are included.  snuverink_j committed Sep 06, 2017 643 644 (Lower output.)  snuverink_j committed Sep 06, 2017 645 Now we will illustrate the case where the magnet length is set by the \texttt{L} attribute and only the fringe  snuverink_j committed Sep 08, 2017 646 fields are described by the field map. We change the \textit{TEST-MAP.T7} file to:  snuverink_j committed Sep 06, 2017 647 \begin{verbatim}  snuverink_j committed Sep 06, 2017 648 649 650 651 652 653 654 1DProfile1 1 1 2.0 -10.0 0.0 10.0 1 -10.0 0.0 10.0 1 0.00000E+00 2.00000E+00 0.00000E+00 2.00000E+00  snuverink_j committed Sep 06, 2017 655 \end{verbatim}  snuverink_j committed Sep 06, 2017 656 657 and change the bend input to:  snuverink_j committed Sep 06, 2017 658 \begin{verbatim}  snuverink_j committed Sep 06, 2017 659 660 661 662 663 664 665 666 Bend: SBend, ANGLE = 60.0 * Pi / 180.0, E1 = -10.0 * Pi / 180.0, E2 = 20.0 Pi / 180.0, FMAPFN = "TEST-MAP.T7", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.25, GAP = 0.02;  snuverink_j committed Sep 06, 2017 667 \end{verbatim}  snuverink_j committed Sep 06, 2017 668 669 This results in the same output as the previous example, as we expect.  snuverink_j committed Sep 06, 2017 670 \begin{verbatim}  snuverink_j committed Sep 06, 2017 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 SBend > Reference Trajectory Properties SBend > =============================== SBend > SBend > Bend angle magnitude: 1.0472 rad (60 degrees) SBend > Entrance edge angle: -0.174533 rad (-10 degrees) SBend > Exit edge angle: 0.349066 rad (20 degrees) SBend > Bend design radius: 0.25 m SBend > Bend design energy: 1e+07 eV SBend > SBend > Bend Field and Rotation Properties SBend > ================================== SBend > SBend > Field amplitude: -0.140385 T SBend > Field index (gradient): 0 m^-1 SBend > Rotation about x axis: 0 rad (0 degrees) SBend > Rotation about y axis: 0 rad (0 degrees) SBend > Rotation about z axis: 0 rad (0 degrees) SBend > SBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields SBend > ====================================================================== SBend > SBend > Reference particle is bent: 1.0472 rad (60 degrees) in x plane SBend > Reference particle is bent: 0 rad (0 degrees) in y plane  snuverink_j committed Sep 06, 2017 694 \end{verbatim}  snuverink_j committed Sep 06, 2017 695 696 697  As a final example, let us now use the previous field map with the following input:  snuverink_j committed Sep 06, 2017 698 \begin{verbatim}  snuverink_j committed Sep 06, 2017 699 700 701 702 703 704 705 706 Bend: SBend, K0 = -0.1400778, E1 = -10.0 * Pi / 180.0, E2 = 20.0 Pi / 180.0, FMAPFN = "TEST-MAP.T7", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.25, GAP = 0.02;  snuverink_j committed Sep 06, 2017 707 708 \end{verbatim} Instead of setting the bend strength using \texttt{ANGLE}, we use \texttt{K0}. This results in the following output:  snuverink_j committed Sep 06, 2017 709   snuverink_j committed Sep 06, 2017 710 \begin{verbatim}  snuverink_j committed Sep 06, 2017 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 SBend > Reference Trajectory Properties SBend > =============================== SBend > SBend > Bend angle magnitude: 1.0472 rad (60 degrees) SBend > Entrance edge angle: -0.174533 rad (-10 degrees) SBend > Exit edge angle: 0.349066 rad (20 degrees) SBend > Bend design radius: 0.25 m SBend > Bend design energy: 1e+07 eV SBend > SBend > Bend Field and Rotation Properties SBend > ================================== SBend > SBend > Field amplitude: -0.140078 T SBend > Field index (gradient): 0 m^-1 SBend > Rotation about x axis: 0 rad (0 degrees) SBend > Rotation about y axis: 0 rad (0 degrees) SBend > Rotation about z axis: 0 rad (0 degrees) SBend > SBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields SBend > ====================================================================== SBend > SBend > Reference particle is bent: 1.04491 rad (59.8688 degrees) in x plane SBend > Reference particle is bent: 0 rad (0 degrees) in y plane  snuverink_j committed Sep 06, 2017 734 \end{verbatim}  snuverink_j committed Sep 06, 2017 735 736 737 738 739 740 In this case, the bend angle for the reference trajectory in the first section of the output no longer matches the reference trajectory bend angle from the lower section (although the difference is small). The reason is that the path of the reference particle through the real magnet (with fringe fields) no longer matches the ideal trajectory. (The effective length of the real magnet is not quite the same as the hard edged magnet for the reference trajectory.)  snuverink_j committed Sep 08, 2017 741 We can compensate for this by changing the field map file \textit{TEST-MAP.T7} file to:  snuverink_j committed Sep 06, 2017 742 \begin{verbatim}  snuverink_j committed Sep 06, 2017 743 744 745 746 747 748 749 1DProfile1 1 1 2.0 -10.0 -0.03026 10.0 1 -10.0 0.03026 10.0 1 0.00000E+00 2.00000E+00 0.00000E+00 2.00000E+00  snuverink_j committed Sep 06, 2017 750 \end{verbatim}  snuverink_j committed Sep 06, 2017 751 752 We have moved the Enge function origins see~Section~\ref{1DProfile1} outward from the entrance and exit faces of the magnet see~Section~\ref{1DProfile1} by 0.3026 mm. This has the effect of making the  snuverink_j committed Sep 06, 2017 753 754 effective length of the soft edge magnet longer. When we do this, the same input:  snuverink_j committed Sep 06, 2017 755 \begin{verbatim}  snuverink_j committed Sep 06, 2017 756 757 758 759 760 761 762 763 Bend: SBend, K0 = -0.1400778, E1 = -10.0 * Pi / 180.0, E2 = 20.0 Pi / 180.0, FMAPFN = "TEST-MAP.T7", ELEMEDGE = 0.25, DESIGNENERGY = 10.0E6, L = 0.25, GAP = 0.02;  snuverink_j committed Sep 06, 2017 764 \end{verbatim}  snuverink_j committed Sep 06, 2017 765 766 produces  snuverink_j committed Sep 06, 2017 767 \begin{verbatim}  snuverink_j committed Sep 06, 2017 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 SBend > Reference Trajectory Properties SBend > =============================== SBend > SBend > Bend angle magnitude: 1.0472 rad (60 degrees) SBend > Entrance edge angle: -0.174533 rad (-10 degrees) SBend > Exit edge angle: 0.349066 rad (20 degrees) SBend > Bend design radius: 0.25 m SBend > Bend design energy: 1e+07 eV SBend > SBend > Bend Field and Rotation Properties SBend > ================================== SBend > SBend > Field amplitude: -0.140078 T SBend > Field index (gradient): 0 m^-1 SBend > Rotation about x axis: 0 rad (0 degrees) SBend > Rotation about y axis: 0 rad (0 degrees) SBend > Rotation about z axis: 0 rad (0 degrees) SBend > SBend > Reference Trajectory Properties Through Bend Magnet with Fringe Fields SBend > ====================================================================== SBend > SBend > Reference particle is bent: 1.0472 rad (60 degrees) in x plane SBend > Reference particle is bent: 0 rad (0 degrees) in y plane  snuverink_j committed Sep 06, 2017 791 \end{verbatim}  snuverink_j committed Sep 06, 2017 792 793 794 795 Now we see that the bend angle for the ideal, hard edge magnet, matches the bend angle of the reference particle through the soft edge magnet. In other words, the effective length of the soft edge, real magnet is the same as the hard edge magnet described by the reference trajectory.  snuverink_j committed Sep 07, 2017 796 \subsection{Bend Fields from 1D Field Maps (\textit{OPAL-t})}  snuverink_j committed Sep 06, 2017 797 798 799 800 801 802 \label{ssec:opaltrbendsbendfields} \begin{figure}[tbh] \begin{center} \includegraphics[width=\textwidth]{figures/Elements/Enge-func-plot.png} \end{center}  snuverink_j committed Sep 06, 2017 803 \caption{Plot of the entrance fringe field of a dipole magnet along the mid-plane, perpendicular to its entrance face. The field is normalized to 1.0. In this case, the fringe field is described by an Enge function see~Equation~\ref{enge_func} with the parameters from the default \texttt{1DProfile1} field map described in Section~\ref{benddefaultfieldmapopalt}. The exit fringe field of this magnet is the mirror image.}  snuverink_j committed Sep 06, 2017 804 805 806 \label{fig:rbend_enge_fringe} \end{figure}  snuverink_j committed Sep 07, 2017 807 So far we have described how to setup an \texttt{RBEND} or \texttt{SBEND} element, but have not explained how \textit{OPAL-t} uses this information to calculate the magnetic field. The field of both types of magnets is divided into three regions:  snuverink_j committed Sep 06, 2017 808 809 810 811 812 \begin{enumerate} \item Entrance fringe field. \item Central field. \item Exit fringe field. \end{enumerate}  snuverink_j committed Sep 06, 2017 813 This can be seen clearly in Figure~\ref{rbend_field_profile}.  snuverink_j committed Sep 06, 2017 814   snuverink_j committed Sep 06, 2017 815 The purpose of the \texttt{1DProfile1} field map see~Section~\ref{1DProfile1} associated with the element is to define the Enge functions (Equation~\ref{enge_func}) that model the entrance and exit fringe fields. To model a particular bend magnet, one must fit the field profile along the mid-plane of the magnet perpendicular to its face for the entrance and exit fringe fields to the Enge function:  snuverink_j committed Sep 06, 2017 816 817 818 819 820  \label{eq:enge_func} F(z) = \frac{1}{1 + e^{\sum\limits_{n=0}^{N_{order}} c_{n} (z/D)^{n}}} where $D$ is the full gap of the magnet, $N_{order}$ is the Enge function order and $z$ is the distance from the origin  snuverink_j committed Sep 06, 2017 821 of the Enge function perpendicular to the edge of the dipole. The origin of the Enge function, the order of the Enge function, $N_{order}$, and the constants $c_0$ to $c_{N_{order}}$ are free parameters that are chosen so that the function closely approximates the fringe region of the magnet being modeled. An example of the entrance fringe field is shown in Figure~\ref{rbend_enge_fringe}.  snuverink_j committed Sep 06, 2017 822   snuverink_j committed Sep 07, 2017 823 Let us assume we have a correctly defined positive \texttt{RBEND} or \texttt{SBEND} element as illustrated in Figure~\ref{rbend,sbend}. (As already stated, any bend can be described by a rotated positive bend.) \textit{OPAL-t} then has the following information:  snuverink_j committed Sep 06, 2017 824 825 826 827  \begin{align*} B_0 &= \text{Field amplitude (T)} \\ R &= \text{Bend radius (m)} \\  snuverink_j committed Sep 06, 2017 828 n &= -\frac{R}{B_{y}}\diffp{B_y}{x} \text{ (Field index, set using the parameter \texttt{K1})} \\  snuverink_j committed Sep 06, 2017 829 830 831 832 833 834 835 836 F(z) &= \left\{ \begin{array}{lll} & F_{entrance}(z_{entrance}) \\ & F_{center}(z_{center}) = 1 \\ & F_{exit}(z_{exit}) \end{array} \right. \end{align*}  snuverink_j committed Sep 06, 2017 837 Here, we have defined an overall Enge function, $F(z)$, with three parts: entrance, center and exit. The exit and entrance fringe field regions have the form of Equation~\ref{enge_func} with parameters defined by the \texttt{1DProfile1} field map file given by the element parameter \texttt{FMAPFN}. Defining the coordinates:  snuverink_j committed Sep 06, 2017 838 839 840  \begin{align*} y &\equiv \text{Vertical distance from magnet mid-plane} \\  snuverink_j committed Sep 06, 2017 841 \Delta_x &\equiv \text{Perpendicular distance to reference trajectory (see Figure~\ref{rbend,sbend})} \\  snuverink_j committed Sep 06, 2017 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 \Delta_z &\equiv \left\{ \begin{array}{lll} & \text{Distance from entrance Enge function origin perpendicular to magnet entrance face.} \\ & \text{Not defined, Enge function is always 1 in this region.} \\ & \text{Distance from exit Enge function origin perpendicular to magnet exit face.} \end{array} \right. \end{align*} using the conditions \begin{align*} \nabla \cdot \overrightarrow{B} &= 0 \\ \nabla \times \overrightarrow{B} &= 0 \end{align*} and making the definitions: \begin{align*} F'(z) &\equiv \diff{F(z)}{z} \\ F''(z) &\equiv \diff[2]{F(z)}{z} \\ F'''(z) &\equiv \diff[3]{F(z)}{z} \end{align*} we can expand the field off axis, with the result: \begin{align*}  snuverink_j committed Sep 08, 2017 866 B_x(\Delta_x, y, \Delta_z) &= -\frac{B_0 \frac{n}{R}}{\sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z}}} e^{-\frac{n}{R} \Delta_x} n \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right] F(\Delta_z) \\  snuverink_j committed Sep 06, 2017 867 B_y(\Delta_x, y, \Delta_z) &= B_0 e^{-\frac{n}{R} \Delta_x} \cos \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right] F(\Delta_z) \\  snuverink_j committed Sep 08, 2017 868 869 B_z(\Delta_x, y, \Delta_z) &= B_0 e^{-\frac{n}{R} \Delta_x} \left\{\frac{F'(\Delta_z)}{\sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}}} n \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right] \right. \\ &- \frac{1}{2 \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}}} \left(F'''(\Delta_z) - \frac{F'(\Delta_z) F''(\Delta_z)}{F(\Delta_z)} \right) \left[ \frac{n \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right]}{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right. \\  snuverink_j committed Sep 06, 2017 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 &- \left. \left. y \frac{\cos \left[ \left( \sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}} \right) y \right]}{\sqrt{\frac{n^2}{R^2} + \frac{F''(\Delta_z)}{F(\Delta_z)}}} \right] \right\} \end{align*} These expression are not well suited for numerical calculation, so, we expand them about $y$ to $O(y^2)$ to obtain: \begin{itemize} \item In fringe field regions: \begin{align*} B_x(\Delta_x, y, \Delta_z) &\approx -B_0 \frac{n}{R} e^{-\frac{n}{R} \Delta_x} y \\ B_y(\Delta_x, y, \Delta_z) &\approx B_0 e^{-\frac{n}{R} \Delta_x} \left[ F(\Delta_z) - \left( \frac{n^2}{R^2} F(\Delta_z) + F''(\Delta_z) \right) \frac{y^2}{2} \right] \\ B_z(\Delta_x, y, \Delta_z) &\approx B_0 e^{-\frac{n}{R} \Delta_x} y F'(\Delta_z) \end{align*} \item In central region: \begin{align*} B_x(\Delta_x, y, \Delta_z) &\approx -B_0 \frac{n}{R} e^{-\frac{n}{R} \Delta_x} y \\ B_y(\Delta_x, y, \Delta_z) &\approx B_0 e^{-\frac{n}{R} \Delta_x} \left[ 1 - \frac{n^2}{R^2} \frac{y^2}{2} \right] \\ B_z(\Delta_x, y, \Delta_z) &\approx 0 \end{align*} \end{itemize}  snuverink_j committed Sep 07, 2017 888 These are the expressions \textit{OPAL-t} uses to calculate the field inside an \texttt{RBEND} or \texttt{SBEND}. First, a particle's position inside the bend is determined (entrance region, center region, or exit region). Depending on the region, \textit{OPAL-t} then determines the values of $\Delta_x$, $y$ and $\Delta_z$, and then calculates the field values using the above expressions.  snuverink_j committed Sep 06, 2017 889   snuverink_j committed Sep 07, 2017 890 \subsection{Default Field Map (\textit{OPAL-t})}  snuverink_j committed Sep 06, 2017 891 892 893 894 895 896 \label{ssec:benddefaultfieldmapopalt} \index{RBEND!Default Field Map} \index{SBEND!Default Field Map} \index{Default Field Map} \index{1DPROFILE1-DEFAULT} Rather than force users to calculate the field of a dipole and then fit that field to find Enge coefficients  snuverink_j committed Sep 08, 2017 897 for the dipoles in their simulation, we have a default set of values we use from \ref{enge} that are set  snuverink_j committed Sep 06, 2017 898 when the default field map, \texttt{1DPROFILE1-DEFAULT}'' is used:  snuverink_j committed Sep 06, 2017 899 900 901 902 903 904 905 906 907 908 909 910 911  \begin{align*} c_{0} &= 0.478959 \\ c_{1} &= 1.911289 \\ c_{2} &= -1.185953 \\ c_{3} &= 1.630554 \\ c_{4} &= -1.082657 \\ c_{5} &= 0.318111 \end{align*} The same values are used for both the entrance and exit regions of the magnet. In general they will give good results. (Of course, at some point as a beam line design becomes more advanced, one will want to find Enge coefficients that fit the actual magnets that will be used in a given design.)  snuverink_j committed Sep 06, 2017 912 The default field map is the equivalent of the following custom \texttt{1DProfile1} (see Section~\ref{1DProfile1} for an explanation of the field map format) map:  snuverink_j committed Sep 06, 2017 913   snuverink_j committed Sep 06, 2017 914 \begin{verbatim}  snuverink_j committed Sep 06, 2017 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 1DProfile1 5 5 2.0 -10.0 0.0 10.0 1 -10.0 0.0 10.0 1 0.478959 1.911289 -1.185953 1.630554 -1.082657 0.318111 0.478959 1.911289 -1.185953 1.630554 -1.082657 0.318111  snuverink_j committed Sep 06, 2017 930 \end{verbatim}  snuverink_j committed Sep 08, 2017 931 As one can see, the default magnet gap for \texttt{1DPROFILE1-DEFAULT'}'' is set to {2.0}{\centim}. This value  snuverink_j committed Sep 06, 2017 932 can be overridden by the \texttt{GAP} attribute of the magnet (see Section~\ref{RBend,SBend}).  snuverink_j committed Sep 06, 2017 933 934 935 936 937 938 939 940 941  \clearpage \subsection{SBend3D (OPAL-CYCL)} \label{ssec:SBend3D} \index{SBEND3D} % NOTE: SBEND3D, RINGDEFINITION in elements.tex and \ubsection {3D fieldmap} in % opalcycl.tex all refer to each other - if updating one check for update on % others to keep them consistent. The SBend3D element enables definition of a bend from 3D field maps. This can be  snuverink_j committed Sep 06, 2017 942 used in conjunction with the \texttt{RINGDEFINITION} element to make a ring for  snuverink_j committed Sep 07, 2017 943 tracking through \textit{OPAL-cycl}.  snuverink_j committed Sep 06, 2017 944   snuverink_j committed Sep 06, 2017 945 \begin{verbatim}  snuverink_j committed Sep 06, 2017 946 label: SBEND3D, FMAPFN=string, LENGTH_UNITS=real, FIELD_UNITS=real;  snuverink_j committed Sep 06, 2017 947 \end{verbatim}  snuverink_j committed Sep 06, 2017 948 949 950 951 952 953 954 955 956 957 958 959  \begin{kdescription} \item[FMAPFN] The field map file name. \item[LENGTH\_UNITS] Units for length (set to 1.0 for units in mm, 10.0 for units in cm, etc). \item[FIELD\_UNITS] Units for field (set to 1.0 for units in T, 0.001 for units in mT, etc). \end{kdescription} Field maps are defined using Cartesian coordinates but in a polar geometry with the following restrictions/conventions: \begin{enumerate}  snuverink_j committed Sep 07, 2017 960 \item 3D Field maps have to be generated in the vertical direction (z coordinate in \textit{OPAL-cycl}) from z = 0 upwards. It cannot be generated symmetrically about z = 0 towards negative z values.  snuverink_j committed Sep 06, 2017 961 962 963 964 \item Field map file must be in the form with columns ordered as follows: [$x, z, y, B_{x}, B_{z}, B_{y}$]. \item Grid points of the position and field strength have to be written on a grid in ($r, z, \theta$) with the primary direction corresponding to the azimuthal direction, secondary to the vertical direction and tertiary to the radial direction. \end{enumerate}  snuverink_j committed Sep 06, 2017 965 Below two examples of a \texttt{SBEND3D} which loads a field maps with different units. The \texttt{triplet} example has units of cm and fields units  snuverink_j committed Sep 06, 2017 966 of Gauss, where the \texttt{Dipole} example (Figure~\ref{sbend3d1}) uses meter and Tesla. The first 8 lines in the field map are ignored.  snuverink_j committed Sep 06, 2017 967   snuverink_j committed Sep 06, 2017 968 \begin{verbatim}  snuverink_j committed Sep 06, 2017 969 triplet: SBEND3D, FMAPFN="fdf-tosca-field-map.table", LENGTH_UNITS=10., FIELD_UNITS=-1e-4;  snuverink_j committed Sep 06, 2017 970 \end{verbatim}  snuverink_j committed Sep 06, 2017 971   snuverink_j committed Sep 08, 2017 972 The first few links of the field map \textit{fdf-tosca-field-map.table}:  snuverink_j committed Sep 06, 2017 973   snuverink_j committed Sep 06, 2017 974 \begin{verbatim}  snuverink_j committed Sep 06, 2017 975 976 977 978 979 980 981 982 983 984 985 986  422280 422280 422280 1 1 X [LENGU] 2 Y [LENGU] 3 Z [LENGU] 4 BX [FLUXU] 5 BY [FLUXU] 6 BZ [FLUXU] 0 194.01470 0.0000000 80.363520 0.68275932346E-07 -5.3752492577 0.28280706805E-07 194.36351 0.0000000 79.516210 0.42525693524E-07 -5.3827955117 0.17681348191E-07 194.70861 0.0000000 78.667380 0.19766168358E-07 -5.4350026348 0.82540823165E-08 .....  snuverink_j committed Sep 06, 2017 987 \end{verbatim}  snuverink_j committed Sep 06, 2017 988   snuverink_j committed Sep 06, 2017 989 \begin{verbatim}  snuverink_j committed Sep 06, 2017 990 Dipole:SBEND3D,FMAPFN="90degree_Dipole_Magnet.out",LENGTH_UNITS=1000.0, FIELD_UNITS=-10.0;  snuverink_j committed Sep 06, 2017 991 \end{verbatim}  snuverink_j committed Sep 08, 2017 992 The first few links of the field map \textit{90degree\_Dipole\_Magnet.out}:  snuverink_j committed Sep 06, 2017 993 \begin{verbatim}  snuverink_j committed Sep 06, 2017 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005  4550000 4550000 4550000 1 X [LENGTH_UNITS] Z [LENGTH_UNITS] Y [LENGTH_UNITS] BX [FIELD_UNITS] BZ [FIELD_UNITS] BY [FIELD_UNITS] 0 4.3586435e-01 5.0000000e-02 1.2803431e+00 0.0000000e+00 1.6214000e+00 0.0000000e+00 4.2691532e-01 5.0000000e-02 1.2833548e+00 0.0000000e+00 1.6214000e+00 0.0000000e+00 4.1794548e-01 5.0000000e-02 1.2863039e+00 0.0000000e+00 1.6214000e+00 0.0000000e+00 ...  snuverink_j committed Sep 06, 2017 1006 \end{verbatim}  snuverink_j committed Sep 06, 2017 1007 1008   snuverink_j committed Sep 07, 2017 1009 This is a restricted feature: \textit{OPAL-cycl}.  snuverink_j committed Sep 06, 2017 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027  \begin{figure}[tb] \begin{center} \includegraphics[width=0.58\textwidth]{figures/Elements/sbend3d-1} \includegraphics[width=0.4\textwidth]{figures/Elements/sbend3d-2} \end{center} \caption{A hard edge model of $90$ degree dipole magnet with homogeneous magnetic field. The right figure is showing the horizontal cross section of the 3D magnetic field map when $z = 0$} \label{fig:sbend3d1} \end{figure} \index{Bending Magnets|)} \clearpage \section{Quadrupole} \label{sec:quadrupole} \index{QUADRUPOLE}  snuverink_j committed Sep 06, 2017 1028 \begin{verbatim}  snuverink_j committed Sep 06, 2017 1029 1030 label:QUADRUPOLE, TYPE=string, APERTURE=real-vector, L=real, K1=real, K1S=real;  snuverink_j committed Sep 06, 2017 1031 \end{verbatim}  snuverink_j committed Sep 06, 2017 1032 1033  The reference system for a quadrupole is a Cartesian coordinate system  snuverink_j committed Sep 07, 2017 1034 \ifthenelse{\boolean{ShowMap}}{see~Figure~\ref{straight}}{}. This is a restricted feature: \texttt{DOPAL-cycl}.  snuverink_j committed Sep 06, 2017 1035   snuverink_j committed Sep 06, 2017 1036 A \texttt{QUADRUPOLE} has three real attributes:  snuverink_j committed Sep 06, 2017 1037 1038 1039 \begin{kdescription} \item[K1] The normal quadrupole component $K_1=\diffp{B_y}{x}$.  snuverink_j committed Sep 08, 2017 1040  The default is ${0}{T\perm}$.  snuverink_j committed Sep 06, 2017 1041 1042 1043 1044 1045 1046  The component is positive, if $B_y$ is positive on the positive $x$-axis. This implies horizontal focusing of positively charged particles which travel in positive $s$-direction. \item[K1S] The skew quadrupole component. $K_{1s}=-\diffp{B_x}{x}$.  snuverink_j committed Sep 08, 2017 1047  The default is ${0}{T\perm}$.  snuverink_j committed Sep 06, 2017 1048 1049 1050 1051  The component is negative, if $B_x$ is positive on the positive $x$-axis. \end{kdescription} \noindent Example:  snuverink_j committed Sep 06, 2017 1052 \begin{verbatim}  snuverink_j committed Sep 06, 2017 1053 1054 QP1: Quadrupole, L=1.20, ELEMEDGE=-0.5265, FMAPFN="1T1.T7", K1=0.11;  snuverink_j committed Sep 06, 2017 1055 \end{verbatim}  snuverink_j committed Sep 06, 2017 1056 1057 1058 1059 1060 1061 1062 1063  \clearpage \section{Sextupole} \label{sec:sextupole} \index{SEXTUPOLE}  snuverink_j committed Sep 06, 2017 1064 \begin{verbatim}  snuverink_j committed Sep 06, 2017 1065 1066 label: SEXTUPOLE, TYPE=string, APERTURE=real-vector, L=real, K2=real, K2S=real;  snuverink_j committed Sep 06, 2017 1067 1068 \end{verbatim} A \texttt{SEXTUPOLE} has three real attributes:  snuverink_j committed Sep 06, 2017 1069 1070 1071 1072 \begin{kdescription} \item[K2] The normal sextupole component $K_2=\diffp[2]{B_y}{x}$.  snuverink_j committed Sep 08, 2017 1073  The default is ${0}{T\per\squarem}$.  snuverink_j committed Sep 06, 2017 1074 1075 1076 1077  The component is positive, if $B_y$ is positive on the $x$-axis. \item[K2S] The skew sextupole component $K_{2s}=-\diffp[2]{B_x}{x}$.  snuverink_j committed Sep 08, 2017 1078  The default is ${0}{T\per\squarem}$.  snuverink_j committed Sep 06, 2017 1079 1080 1081  The component is negative, if $B_x$ is positive on the $x$-axis. \end{kdescription} \noindent Example:  snuverink_j committed Sep 06, 2017 1082 \begin{verbatim}  snuverink_j committed Sep 06, 2017 1083 S:SEXTUPOLE, L=0.4, K2=0.00134;  snuverink_j committed Sep 06, 2017 1084 \end{verbatim}  snuverink_j committed Sep 06, 2017 1085 The reference system for a sextupole is a Cartesian coordinate system  snuverink_j committed Sep 06, 2017 1086 \ifthenelse{\boolean{ShowMap}}{see~Figure~\ref{straight}}{}.  snuverink_j committed Sep 06, 2017 1087 1088 1089 1090 1091 1092  \clearpage \section{Octupole} \label{sec:octupole} \index{OCTUPOLE}  snuverink_j committed Sep 06, 2017 1093 \begin{verbatim}  snuverink_j committed Sep 06, 2017 1094 1095 label:OCTUPOLE, TYPE=string, APERTURE=real-vector, L=real, K3=real, K3S=real;  snuverink_j committed Sep 06, 2017 1096 1097 \end{verbatim} An \texttt{OCTUPOLE} has three real attributes:  snuverink_j committed Sep 06, 2017 1098 1099 1100 1101 \begin{kdescription} \item[K3] The normal sextupole component $K_3=\diffp[3]{B_y}{x}$.  snuverink_j committed Sep 08, 2017 1102  The default is ${0}{Tm^{-3}}$.  snuverink_j committed Sep 06, 2017 1103 1104 1105 1106 1107  The component is positive, if $B_y$ is positive on the positive $x$-axis. \item[K3S] The skew sextupole component $K_{3s}=-\diffp[3]{B_x}{x}$. % $K_{3s}=\frac{1}{B \rho}\diffp[3]{B_x}{x}$.  snuverink_j committed Sep 08, 2017 1108  The default is ${0}{Tm^{-3}}$.  snuverink_j committed Sep 06, 2017 1109 1110 1111  The component is negative, if $B_x$ is positive on the positive $x$-axis. \end{kdescription} \noindent Example:  snuverink_j committed Sep 06, 2017 1112 \begin{verbatim}  snuverink_j committed Sep 06, 2017 1113 O3:OCTUPOLE, L=0.3, K3=0.543;  snuverink_j committed Sep 06, 2017 1114 \end{verbatim}  snuverink_j committed Sep 06, 2017 1115 The reference system for an octupole is a Cartesian coordinate system  snuverink_j committed Sep 06, 2017 1116 \ifthenelse{\boolean{ShowMap}}{see~Figure~\ref{straight}}{}.  snuverink_j committed Sep 06, 2017 1117 1118 1119 1120 1121  \clearpage \section{General Multipole} \label{sec:multipole} \index{MULTIPOLE}  snuverink_j committed Sep 07, 2017 1122 A \texttt{MULTIPOLE} is in \textit{OPAL-t} is of arbitrary order.  snuverink_j committed Sep 06, 2017 1123 \begin{verbatim}  snuverink_j committed Sep 06, 2017 1124 1125 label:MULTIPOLE, TYPE=string, APERTURE=real-vector, L=real, KN=real-vector, KS=real-vector;  snuverink_j committed Sep 06, 2017 1126 \end{verbatim}  snuverink_j committed Sep 06, 2017 1127 1128 \begin{kdescription} \item[KN]  snuverink_j committed Sep 06, 2017 1129  A real vector see~Section~\ref{anarray},  snuverink_j committed Sep 06, 2017 1130 1131  containing the normal multipole coefficients, $K_n=\diffp[n]{B_y}{x}$  snuverink_j committed Sep 08, 2017 1132  (default is ${0}{Tm^{-n}}$).  snuverink_j committed Sep 06, 2017 1133 1134  A component is positive, if $B_y$ is positive on the positive $x$-axis. \item[KS]  snuverink_j committed Sep 06, 2017 1135  A real vector see~Section~\ref{anarray},