opalt.asciidoc

@@ -37,25 +37,25 @@ geometry will be implemented.
 Variables in _OPAL-t_
 ~~~~~~~~~~~~~~~~~~~~~
 
-[sec:opalt:canon] _OPAL-t_ uses the following canonical variables to
+_OPAL-t_ uses the following canonical variables to
 describe the motion of particles. The physical units are listed in
 square brackets.
 
 X::
-  Horizontal position latexmath:[$x$] of a particle relative to the axis
+  Horizontal position latexmath:[x] of a particle relative to the axis
   of the element [m].
 PX::
-  latexmath:[$\beta_x\gamma$] Horizontal canonical momentum [1].
+  latexmath:[\beta_x\gamma] Horizontal canonical momentum [1].
 Y::
-  Horizontal position latexmath:[$y$] of a particle relative to the axis
+  Horizontal position latexmath:[y] of a particle relative to the axis
   of the element [m].
 PY::
-  latexmath:[$\beta_y\gamma$] Horizontal canonical momentum [1].
+  latexmath:[\beta_y\gamma] Horizontal canonical momentum [1].
 Z::
-  Longitudinal position latexmath:[$z$] of a particle in floor
+  Longitudinal position latexmath:[z] of a particle in floor
   co-ordinates [m].
 PZ::
-  latexmath:[$\beta_z\gamma$] Longitudinal canonical momentum [1].
+  latexmath:[\beta_z\gamma] Longitudinal canonical momentum [1].
 
 The independent variable is *t* [s].
 
@@ -64,12 +64,15 @@ Integration of the Equation of Motion
 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 
 _OPAL-t_ integrates the relativistic Lorentz equation
-latexmath:[\[\frac{\mathrm{d}\gamma\mathbf{v}}{\mathrm{d}t} =   \frac{q}{m}[\mathbf{E}_{ext} + \mathbf{E}_{sc} + \mathbf{v} \times (\mathbf{B}_{ext} + \mathbf{B}_{sc})]\]]
-where latexmath:[$\gamma$] is the relativistic factor, latexmath:[$q$]
-is the charge, and latexmath:[$m$] is the rest mass of the particle.
-latexmath:[$\mathbf{E}$] and latexmath:[$\mathbf{B}$] are abbreviations
-for the electric field latexmath:[$\mathbf{E}(\mathbf{x},t)$] and
-magnetic field latexmath:[$\mathbf{B}(\mathbf{x},t)$]. To update the
+[latexmath]
+++++
+\frac{\mathrm{d}\gamma\mathbf{v}}{\mathrm{d}t} =   \frac{q}{m}[\mathbf{E}_{ext} + \mathbf{E}_{sc} + \mathbf{v} \times (\mathbf{B}_{ext} + \mathbf{B}_{sc})]
+++++
+where latexmath:[\gamma] is the relativistic factor, latexmath:[q]
+is the charge, and latexmath:[m] is the rest mass of the particle.
+latexmath:[\mathbf{E}] and latexmath:[\mathbf{B}] are abbreviations
+for the electric field latexmath:[\mathbf{E}(\mathbf{x},t)] and
+magnetic field latexmath:[\mathbf{B}(\mathbf{x},t)]. To update the
 positions and momenta _OPAL-t_ uses the Boris-Buneman algorithm
 [langdon].
 
@@ -108,21 +111,24 @@ reference particle, having the central energy and traveling on the
 so-called reference trajectory. Motion of a particle close to this
 fictitious reference particle can be described by linearized equations
 for the displacement of the particle under study, relative to the
-reference particle. In _OPAL-t_, the time latexmath:[$t$] is used as
-independent variable instead of the path length latexmath:[$s$]. The
+reference particle. In _OPAL-t_, the time latexmath:[t] is used as
+independent variable instead of the path length latexmath:[s]. The
 relation between them can be expressed as
-latexmath:[\[\frac{\mathrm{d}}{\mathrm{d} t} = \frac{\mathrm{d}}{\mathrm{d}\mathbf{s}}\frac{\mathrm{d}\mathbf{s}}{\mathrm{d} t} = \mathbf{\beta}c\frac{\mathrm{d}}{\mathrm{d}\mathbf{s}}.\]]
+[latexmath]
+++++
+\frac{\mathrm{d}}{\mathrm{d} t} = \frac{\mathrm{d}}{\mathrm{d}\mathbf{s}}\frac{\mathrm{d}\mathbf{s}}{\mathrm{d} t} = \mathbf{\beta}c\frac{\mathrm{d}}{\mathrm{d}\mathbf{s}}.
+++++
 
 [[global-cartesian-coordinate-system]]
 Global Cartesian Coordinate System
 ++++++++++++++++++++++++++++++++++
 
 We define the global cartesian coordinate system, also known as floor
-coordinate system with latexmath:[$K$], a point in this coordinate
-system is denoted by latexmath:[$(X, Y, Z) \in K$]. In Figure [KS1] of
+coordinate system with latexmath:[K], a point in this coordinate
+system is denoted by latexmath:[(X, Y, Z) \in K]. In Figure [KS1] of
 the accelerator is uniquely defined by the sequence of physical elements
-in latexmath:[$K$]. The beam elements are numbered
-latexmath:[$e_0, \ldots , e_i, \ldots e_n$].
+in latexmath:[K]. The beam elements are numbered
+latexmath:[e_0, \ldots , e_i, \ldots e_n].
 
 image:figures/opalt/coords.png[Illustration of local and global
 coordinates.]
@@ -131,40 +137,46 @@ coordinates.]
 Local Cartesian Coordinate System
 +++++++++++++++++++++++++++++++++
 
-A local coordinate system latexmath:[$K'_i$] is attached to each element
-latexmath:[$e_i$]. This is simply a frame in which latexmath:[$(0,0,0)$]
+A local coordinate system latexmath:[K'_i] is attached to each element
+latexmath:[e_i]. This is simply a frame in which latexmath:[(0,0,0)]
 is at the entrance of each element. For an illustration see
 Figure [KS1]. The local reference system
-latexmath:[$(x, y, z) \in K'_n$] may thus be referred to a global
-Cartesian coordinate system latexmath:[$(X, Y, Z) \in K$].
-
-The local coordinates latexmath:[$(x_i, y_i, z_i)$] at element
-latexmath:[$e_i$] with respect to the global coordinates
-latexmath:[$(X, Y, Z)$] are defined by three displacements
-latexmath:[$(X_i, Y_i, Z_i)$] and three angles
-latexmath:[$(\Theta_i, \Phi_i, \Psi_i)$].
-
-latexmath:[$\Psi$] is the roll angle about the global
-latexmath:[$Z$]-axis. latexmath:[$\Phi$] is the pitch angle about the
-global latexmath:[$Y$]-axis. Lastly, latexmath:[$\Theta$] is the yaw
-angle about the global latexmath:[$X$]-axis. All three angles form
+latexmath:[(x, y, z) \in K'_n] may thus be referred to a global
+Cartesian coordinate system latexmath:[(X, Y, Z) \in K].
+
+The local coordinates latexmath:[(x_i, y_i, z_i)] at element
+latexmath:[e_i] with respect to the global coordinates
+latexmath:[(X, Y, Z)] are defined by three displacements
+latexmath:[(X_i, Y_i, Z_i)] and three angles
+latexmath:[(\Theta_i, \Phi_i, \Psi_i)].
+
+latexmath:[\Psi] is the roll angle about the global
+latexmath:[Z]-axis. latexmath:[\Phi] is the pitch angle about the
+global latexmath:[Y]-axis. Lastly, latexmath:[\Theta] is the yaw
+angle about the global latexmath:[X]-axis. All three angles form
 right-handed screws with their corresponding axes. The angles
-(latexmath:[$\Theta,\Phi,\Psi$]) are the Tait-Bryan angles
+(latexmath:[\Theta,\Phi,\Psi]) are the Tait-Bryan angles
 [bib:tait-bryan].
 
-The displacement is described by a vector latexmath:[$\mathbf{v}$] and
-the orientation by a unitary matrix latexmath:[$\mathcal{W}$]. The
-column vectors of latexmath:[$\mathcal{W}$] are unit vectors spanning
-the local coordinate axes in the order latexmath:[$(x, y, z)$].
-latexmath:[$\mathbf{v}$] and latexmath:[$\mathcal{W}$] have the values:
-latexmath:[\[\mathbf{v} =\left(\begin{array}{c}
+The displacement is described by a vector latexmath:[\mathbf{v}] and
+the orientation by a unitary matrix latexmath:[\mathcal{W}]. The
+column vectors of latexmath:[\mathcal{W}] are unit vectors spanning
+the local coordinate axes in the order latexmath:[(x, y, z)].
+latexmath:[\mathbf{v}] and latexmath:[\mathcal{W}] have the values:
+[latexmath]
+++++
+\mathbf{v} =\left(\begin{array}{c}
     X \\
     Y \\
     Z
   \end{array}\right),
 \qquad
-\mathcal{W}=\mathcal{S}\mathcal{T}\mathcal{U}\]] where
-latexmath:[\[\mathcal{S}=\left(\begin{array}{ccc}
+\mathcal{W}=\mathcal{S}\mathcal{T}\mathcal{U}
+++++
+where
+[latexmath]
+++++
+\mathcal{S}=\left(\begin{array}{ccc}
     \cos\Theta &  0 &  n\Theta \\
     0         &  1 &   0 \\
     -n\Theta &  0 &  \cos\Theta
@@ -174,34 +186,41 @@ latexmath:[\[\mathcal{S}=\left(\begin{array}{ccc}
     1 &  0        &  0 \\
     0 &  \cos\Phi &  n\Phi \\
     0 & -n\Phi &  \cos\Phi
-  \end{array}\right),\]]
-latexmath:[\[\mathcal{U}=\left(\begin{array}{ccc}
+  \end{array}\right),
+\quad
+\mathcal{U}=\left(\begin{array}{ccc}
     \cos\Psi & -n\Psi &  0 \\
     n\Psi &  \cos\Psi &  0 \\
     0        &  0        &  1
-  \end{array}\right).\]]
+  \end{array}\right).
+++++
 
-We take the vector latexmath:[$\mathbf{r}_i$] to be the displacement and
-the matrix latexmath:[$\mathcal{S}_i$] to be the rotation of the local
-reference system at the exit of the element latexmath:[$i$] with respect
+We take the vector latexmath:[\mathbf{r}_i] to be the displacement and
+the matrix latexmath:[\mathcal{S}_i] to be the rotation of the local
+reference system at the exit of the element latexmath:[i] with respect
 to the entrance of that element.
 
-Denoting with latexmath:[$i$] a beam line element, one can compute
-latexmath:[$\mathbf{v}_i$] and latexmath:[$\mathcal{W}_i$] by the
-recurrence relations latexmath:[\[\label{eq:surv}
+Denoting with latexmath:[i] a beam line element, one can compute
+latexmath:[\mathbf{v}_i] and latexmath:[\mathcal{W}_i] by the
+recurrence relations 
+[latexmath]
+++++
+\label{eq:surv}
 \mathbf{v}_i = \mathcal{W}_{i-1}\mathbf{r}_i + \mathbf{v}_{i-1}, \qquad
-\mathcal{W}_i = \mathcal{W}_{i-1}\mathcal{S}_i,\]] where
-latexmath:[$\mathbf{v}_0$] corresponds to the origin of the `LINE` and
-latexmath:[$\mathcal{W}_0$] to its orientation. In _OPAL-t_ they can be
+\mathcal{W}_i = \mathcal{W}_{i-1}\mathcal{S}_i,
+++++
+where
+latexmath:[\mathbf{v}_0] corresponds to the origin of the `LINE` and
+latexmath:[\mathcal{W}_0] to its orientation. In _OPAL-t_ they can be
 defined using either `X`, `Y`, `Z`, `THETA`, `PHI` and `PSI` or `ORIGIN`
-and `ORIENTATION`, see Section [line:simple].
+and `ORIENTATION`, see Section <<line:simple>>.
 
 [[space-charge-coordinate-system]]
 Space Charge Coordinate System
 ++++++++++++++++++++++++++++++
 
 In order to calculate space charge in the electrostatic approximation,
-we introduce a co-moving coordinate system latexmath:[$K_{\text{sc}}$],
+we introduce a co-moving coordinate system latexmath:[K_{\text{sc}}],
 in which the origin coincides with the mean position of the particles
 and the mean momentum is parallel to the z-axis.
 
@@ -210,15 +229,15 @@ Curvilinear Coordinate System
 +++++++++++++++++++++++++++++
 
 In order to compute statistics of the particle ensemble,
-latexmath:[$K_s$] is introduced. The accompanying tripod (Dreibein) of
+latexmath:[K_s] is introduced. The accompanying tripod (Dreibein) of
 the reference orbit spans a local curvilinear right handed system
-latexmath:[$(x,y,s)$]. The local latexmath:[$s$]-axis is the tangent to
+latexmath:[(x,y,s)]. The local latexmath:[s]-axis is the tangent to
 the reference orbit. The two other axes are perpendicular to the
-reference orbit and are labelled latexmath:[$x$] (in the bend plane)
-and latexmath:[$y$] (perpendicular to the bend plane).
+reference orbit and are labelled latexmath:[x] (in the bend plane)
+and latexmath:[y] (perpendicular to the bend plane).
 
 image:figures/opalt/curvcoords.png[Illustration of
-latexmath:[$K_\text{sc}$] and latexmath:[$K_s$]]
+latexmath:[K_\text{sc}] and latexmath:[K_s]]
 
 Both coordinate systems are described in Figure [KS2].
 
@@ -268,9 +287,9 @@ structure speeds up the search for elements that influence the particles
 at a given position in 3D space by minimizing the looping over elements
 when integrating an ensemble of particles. For each time step,
 `IndexMap` returns a set of elements
-latexmath:[$\mathcal{S}_{\text{e}} \subset {e_0 \ldots e_n}$] in case of
+latexmath:[\mathcal{S}_{\text{e}} \subset {e_0 \ldots e_n}] in case of
 the example given in Figure [KS1]. An implicit drift is modelled as an
-empty set latexmath:[$\emptyset$].
+empty set latexmath:[\emptyset].
 
 [[flow-diagram-of-opal-t]]
 Flow Diagram of _OPAL-t_
@@ -281,11 +300,11 @@ method.]
 
 A regular time step in _OPAL-t_ is sketched in
 Figure [OPALTSchemeSimple]. In order to compute the coordinate system
-transformation from the reference coordinate system latexmath:[$K_s$] to
-the local coordinate systems latexmath:[$K'_n$] we join the
-transformation from floor coordinate system latexmath:[$K$] to
-latexmath:[$K'_n$] to the transformation from latexmath:[$K_s$] to
-latexmath:[$K$]. All computations of rotations which are involved in the
+transformation from the reference coordinate system latexmath:[K_s] to
+the local coordinate systems latexmath:[K'_n] we join the
+transformation from floor coordinate system latexmath:[K] to
+latexmath:[K'_n] to the transformation from latexmath:[K_s] to
+latexmath:[K]. All computations of rotations which are involved in the
 computation of coordinate system transformations are performed using
 quaternions. The resulting quaternions are then converted to the
 appropriate matrix representation before applying the rotation operation
@@ -412,7 +431,7 @@ in floor coordinate system
 |41 |dt |ns |Size of time step
 
 |42 |partsOutside |1 |Number of particles outside
-latexmath:[$n \times gma$] of beam, where latexmath:[$n$] is controlled
+latexmath:[n \times gma] of beam, where latexmath:[n] is controlled
 with `BEAMHALOBOUNDARY`
 
 |43 |R0_x |m |X-component of position of particle with ID 0 (only when
@@ -596,7 +615,7 @@ The information that is written can be found in the following table.
 
 |8 |solenoid |1 |Whether the field of a solenoid is present
 
-|9 |rfcavity |latexmath:[$\pm$]1 |Whether the field of a cavity is
+|9 |rfcavity |latexmath:[\pm]1 |Whether the field of a cavity is
 present
 
 |10 |monitor |1 |Whether a monitor is present
@@ -661,83 +680,189 @@ Most accelerator modeling codes use the hard-edge model for magnets -
 constant Hamiltonian. Real magnets always have a smooth transition at
 the edges - fringe fields. To obtain a multipole description of a field
 we can apply the theory of analytic functions.
-latexmath:[\[\begin{aligned}
+
+[latexmath]
+++++
+\begin{aligned}
         \nabla \cdot \mathbf{B} & = 0 \Rightarrow \exists \quad \mathbf{A} \quad \text{with} \quad \mathbf{B} = \nabla \times \mathbf{A} \\
         \nabla \times \mathbf{B} & = 0 \Rightarrow \exists \quad V \quad \text{with} \quad \mathbf{B} = - \nabla V
-    \end{aligned}\]] Assuming that latexmath:[$A$] has only a non-zero
-component latexmath:[$A_s$] we get latexmath:[\[\begin{aligned}
+    \end{aligned}
+++++
+Assuming that latexmath:[A] has only a non-zero
+component latexmath:[A_s] we get 
+[latexmath]
+++++
+\begin{aligned}
         B_x & = - \frac{\partial V}{\partial x} = \frac{\partial A_s}{\partial y} \\
         B_y & = - \frac{\partial V}{\partial y} = - \frac{\partial A_s}{\partial x} 
-    \end{aligned}\]] These equations are just the Cauchy-Riemann
+    \end{aligned}
+++++
+These equations are just the Cauchy-Riemann
 conditions for an analytic function
-latexmath:[$\tilde{A} (z) = A_s (x, y) + i V(x,y)$]. So the complex
+latexmath:[\tilde{A} (z) = A_s (x, y) + i V(x,y)]. So the complex
 potential is an analytic function and can be expanded as a power series
-latexmath:[\[\tilde{A} (z) = \sum_{n=0}^{\infty} \kappa_n z^n, \quad \kappa_n = \lambda_n + i \mu_n\]]
-with latexmath:[$\lambda_n, \mu_n$] being real constants. It is
+
+[latexmath]
+++++
+\tilde{A} (z) = \sum_{n=0}^{\infty} \kappa_n z^n, \quad \kappa_n = \lambda_n + i \mu_n\]]
+++++
+
+with latexmath:[\lambda_n, \mu_n] being real constants. It is
 practical to express the field in cylindrical coordinates
-latexmath:[$(r, \varphi, s)$] latexmath:[\[\begin{aligned}
+latexmath:[(r, \varphi, s)]
+[latexmath]
+++++
+\begin{aligned}
         x & = r \cos \varphi \quad y = r n \varphi  \\
         z^n & = r^n ( \cos n \varphi + i n n \varphi )
-    \end{aligned}\]] From the real and imaginary parts of equation () we
-obtain latexmath:[\[\begin{aligned}
+    \end{aligned}
+++++
+
+From the real and imaginary parts of equation () we obtain
+
+[latexmath]
+++++
+\begin{aligned}
         V(r, \varphi) & = \sum_{n=0}^{\infty} r^n ( \mu_n \cos n \varphi + \lambda_n n n \varphi ) \\
         A_s (r, \varphi) & = \sum_{n=0}^{\infty} r^n ( \lambda_n \cos n \varphi - \mu_n n n \varphi )
-    \end{aligned}\]] Taking the gradient of latexmath:[$-V(r, \varphi)$]
+    \end{aligned}
+++++
+
+Taking the gradient of latexmath:[-V(r, \varphi)]
 we obtain the multipole expansion of the azimuthal and radial field
-components, respectively latexmath:[\[\begin{aligned}
+components, respectively
+
+[latexmath]
+++++
+\begin{aligned}
         B_{\varphi} & = - \frac{1}{r} \frac{\partial V}{\partial \varphi} = - \sum_{n=0}^{\infty} n r^{n-1} ( \lambda_n \cos n \varphi - \mu_n n n \varphi ) \\
         B_r & = - \frac{\partial V}{\partial r} = - \sum_{n=0}^{\infty} n r^{n-1} ( \mu_n \cos n \varphi + \lambda_n n n \varphi )
-    \end{aligned}\]] Furthermore, we introduce the normal multipole
-coefficient latexmath:[$b_n$] and skew coefficient latexmath:[$a_n$]
-defined with the reference radius latexmath:[$r_0$] and the magnitude of
-the field at this radius latexmath:[$B_0$] (these coefficients can be a
+    \end{aligned}
+++++
+Furthermore, we introduce the normal multipole
+coefficient latexmath:[b_n] and skew coefficient latexmath:[a_n]
+defined with the reference radius latexmath:[r_0] and the magnitude of
+the field at this radius latexmath:[B_0] (these coefficients can be a
 function of s in a more general case as it is presented further on).
-latexmath:[\[b_n = - \frac{n \lambda_n}{B_0} r_0^{n-1} \qquad a_n = \frac{n \mu_n}{B_0} r_0^{n-1}\]]
-latexmath:[\[\begin{aligned}
+[latexmath]
+++++
+b_n = - \frac{n \lambda_n}{B_0} r_0^{n-1} \qquad a_n = \frac{n \mu_n}{B_0} r_0^{n-1}
+++++
+
+[latexmath]
+++++
+\begin{aligned}
         B_{\varphi}(r, \varphi) & = B_0 \sum_{n=1}^{\infty} ( b_n \cos n \varphi+ a_n n n \varphi ) \left( \frac{r}{r_0} \right) ^{n-1} \\
         B_r (r, \varphi) & = B_0 \sum_{n=1}^{\infty} ( -a_n \cos n \varphi+ b_n n n \varphi ) \left( \frac{r}{r_0} \right) ^{n-1}
-    \end{aligned}\]] To obtain a model for the fringe field of a
+    \end{aligned}
+++++
+To obtain a model for the fringe field of a
 straight multipole, a proposed starting solution for a non-skew magnetic
-field is latexmath:[\[\begin{aligned}
+field is
+
+[latexmath]
+++++
+\begin{aligned}
         V & = \sum_{n=1}^{\infty} V_n (r,z) n n \varphi \\
         V_n & = \sum_{k=0}^{\infty} C_{n,k}(z) r^{n+2k}
-    \end{aligned}\]] It is straightforward to derive a relation between
+    \end{aligned}
+++++
+
+It is straightforward to derive a relation between
 coefficients
-latexmath:[\[\nabla ^2 V = 0 \Rightarrow \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial V_n}{\partial r} \right) + \frac{\partial^2 V_n}{\partial z^2} = \frac{n^2 V_n}{r^2} = 0\]]
-latexmath:[\[V_n  = \sum_{k=0}^{\infty} C_{n,k}(z) r^{n+2k}\]]
-latexmath:[\[\Rightarrow  \sum_{k=0}^{\infty} \left[ r^{n+2(k-1)} \left[ (n+2k)^2 - n^2 \right] C_{n,k}(z) + r^{n+2k} \frac{\partial^2 C_{n,k}(z)}{\partial z^2} \right] = 0\]]
-By identifying the term in front of the same powers of latexmath:[$r$]
+
+[latexmath]
+++++
+\nabla ^2 V = 0 \Rightarrow \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial V_n}{\partial r} \right) + \frac{\partial^2 V_n}{\partial z^2} = \frac{n^2 V_n}{r^2} = 0
+++++
+
+[latexmath]
+++++
+\[V_n  = \sum_{k=0}^{\infty} C_{n,k}(z) r^{n+2k}
+++++
+
+[latexmath]
+++++
+\Rightarrow  \sum_{k=0}^{\infty} \left[ r^{n+2(k-1)} \left[ (n+2k)^2 - n^2 \right] C_{n,k}(z) + r^{n+2k} \frac{\partial^2 C_{n,k}(z)}{\partial z^2} \right] = 0
+++++
+
+By identifying the term in front of the same powers of latexmath:[r]
 we obtain the recurrence relation
-latexmath:[\[C_{n,k}(z) = - \frac{1}{4k(n+k)}   \frac{d^2 C_{n,k-1}} {dz^2}, k = 1,2, \dots\]]
+
+[latexmath]
+++++
+C_{n,k}(z) = - \frac{1}{4k(n+k)}   \frac{d^2 C_{n,k-1}} {dz^2}, k = 1,2, \dots
+++++
+
 The solution of the recursion relation becomes
-latexmath:[\[C_{n,k} (z) = (-1)^k \frac{n!}{2^{2k} k! (n+k)!} \frac{d^{2k} C_{n,0}(z)}{dz^{2k}}\]]
+
+[latexmath]
+++++
+C_{n,k} (z) = (-1)^k \frac{n!}{2^{2k} k! (n+k)!} \frac{d^{2k} C_{n,0}(z)}{dz^{2k}}
+++++
+
 Therefore
-latexmath:[\[V_n = - \left( \sum_{k=0}^{\infty} (-1)^{k+1} \frac{n!}{2^{2k} k! (n+k)!} C_{n, 0}^{(2k)}(z) r^{2k} \right) r^n\]]
-The transverse components of the field are latexmath:[\[\begin{aligned}
+[latexmath]
+++++
+V_n = - \left( \sum_{k=0}^{\infty} (-1)^{k+1} \frac{n!}{2^{2k} k! (n+k)!} C_{n, 0}^{(2k)}(z) r^{2k} \right) r^n
+
+The transverse components of the field are 
+
+[latexmath]
+++++
+\begin{aligned}
         B_r & = \sum_{n=1}^{\infty} g_{rn} r^{n-1} n n \varphi \\
         B_{\varphi} & = \sum_{n=1}^{\infty} g_{\varphi n} r^{n-1} \cos n \varphi
-    \end{aligned}\]] where the following gradients determine the entire
-potential and can be deduced from the function latexmath:[$C_{n,0}(z)$]
-once the harmonic latexmath:[$n$] is fixed. latexmath:[\[\begin{aligned}
+    \end{aligned}
+++++
+
+where the following gradients determine the entire
+potential and can be deduced from the function latexmath:[C_{n,0}(z)]
+once the harmonic latexmath:[n] is fixed.
+
+[latexmath]
+++++
+\begin{aligned}
         g_{rn} (r,z) & = \sum_{k=0}^{\infty} (-1)^{k+1} \frac{n! (n+2k)}{2^{2k} k! (n+k)!} C_{n,0}^{(2k)}(z)r^{2k} \\   
         g_{ \varphi n} (r,z) & = \sum_{k=0}^{\infty} (-1)^{k+1} \frac{n!n}{2^{2k} k! (n+k)!} C_{n,0}^{(2k)}(z)r^{2k}
-    \end{aligned}\]] The z-directed component of the filed can be
-expressed in a similar form latexmath:[\[\begin{aligned}
+    \end{aligned}
+++++
+
+The z-directed component of the filed can be
+expressed in a similar form
+
+[latexmath]
+++++
+\begin{aligned}
         B_z & = - \frac{\partial V}{\partial z} =  \sum_{n=1}^{\infty} g_{zn} r^n n n \varphi \\
         g_{zn} & = \sum_{k=0}^{\infty} (-1)^{k+1} \frac{n!}{2^{2k} k! (n+k)!} C_{n,0}^{2k+1} r^{2k}
-    \end{aligned}\]] The gradient functions
-latexmath:[$g_{rn}, g_{\varphi n}, g_{zn}$] are obtained from
-latexmath:[\[\begin{aligned}
+    \end{aligned}
+++++
+
+The gradient functions
+latexmath:[g_{rn}, g_{\varphi n}, g_{zn}] are obtained from
+
+[latexmath]
+++++
+\begin{aligned}
         B_{r,n} & = - \frac{\partial V_n}{\partial r} n n \varphi = g_{rn} r^{n-1} n n \varphi \\
         B_{\varphi,n} & = - \frac{n}{r} V_n \cos n \varphi = g_{\varphi n} r^{n-1} \cos n \varphi \\
         B_{z,n} & = - \frac{\partial V_n}{\partial z} n n \varphi = g_{zn} r^{n} n n \varphi
-    \end{aligned}\]] One preferred model to approximate the gradient
+    \end{aligned}
+++++
+
+One preferred model to approximate the gradient
 profile on the central axis is the k-parameter Enge function
-latexmath:[\[\begin{aligned}
+
+[latexmath]
+++++
+\begin{aligned}
         C_{n,0}(z) & = \frac{G_0}{1+exp[P(d(z))]}, \quad G_0 = \frac{B_0}{r_0^{n-1}} \\
         P(d) & = C_0 + C_1 \left( \frac{d}{\lambda} \right) + C_2 \left( \frac{d}{\lambda} \right)^2 + \dots + C_{k-1} \left( \frac{d}{\lambda} \right)^{k-1} 
-    \end{aligned}\]] where latexmath:[$d(z)$] is the distance to the
-field boundary and latexmath:[$\lambda$] characterizes the fringe field
+    \end{aligned}
+
+where latexmath:[d(z)] is the distance to the
+field boundary and latexmath:[\lambda] characterizes the fringe field
 length.
 
 [[fringe-field-of-a-curved-multipole]]
@@ -747,9 +872,9 @@ Fringe field of a curved multipole
 _(fixed radius)_
 
 We consider the Frenet-Serret coordinate system
-latexmath:[$ ( \hat{\mathbf{x}}, \hat{\mathbf{s}}, \hat{\mathbf{z}} )$]
-with the radius of curvature latexmath:[$ \rho $] constant and the scale
-factor latexmath:[$ h_s = 1 + x/ \rho$]. A conversion to these
+latexmath:[ ( \hat{\mathbf{x}}, \hat{\mathbf{s}}, \hat{\mathbf{z}} )]
+with the radius of curvature latexmath:[ \rho ] constant and the scale
+factor latexmath:[ h_s = 1 + x/ \rho]. A conversion to these
 coordinates implies that latexmath:[\[\begin{aligned}
         \nabla \cdot \mathbf{B} & = \frac{1}{h_s} \left[ \frac{\partial (h_s B_x )}{\partial x} + \frac{\partial B_s}{\partial s} + \frac{\partial (h_s B_z )}{\partial z} \right] \\
         \nabla \times \mathbf{B} & = \frac{1}{h_s} \left[ \frac{\partial B_z}{\partial s} -  \frac{\partial (h_s B_s )}{\partial z} \right] \hat{\mathbf{x}} + \left[ \frac{\partial B_x}{\partial z} - \frac{\partial B_z}{\partial x} \right] \hat{\mathbf{s}} + \frac{1}{h_s} \left[ \frac{\partial (h_s B_s)}{\partial x} - \frac{\partial B_x}{\partial s} \right] \hat{\mathbf{z}} \nonumber
@@ -761,8 +886,8 @@ The most general form of the expansion is latexmath:[\[\begin{aligned}
          B_x & = z \sum_{i,k=0}^{\infty} a_{i,k} x^i z^{2k} \label{eq:02}\\
          B_s & = z \sum_{i,k=0}^{\infty} c_{i,k} x^i z^{2k} \label{eq:03}
     \end{aligned}\]] Maxwell’s equations
-latexmath:[$ \nabla \cdot \mathbf{B} = 0 $] and
-latexmath:[$ \nabla \times \mathbf{B} = 0 $] in the above coordinates
+latexmath:[ \nabla \cdot \mathbf{B} = 0 ] and
+latexmath:[ \nabla \times \mathbf{B} = 0 ] in the above coordinates
 yield
 latexmath:[\[\frac{\partial}{\partial x} \left( (1+x/ \rho) B_x \right) + \frac{\partial B_s}{\partial s} + (1+x/ \rho) \frac{\partial B_z}{\partial z} = 0 \label{eq:21}\]]
 latexmath:[\[\begin{aligned}
@@ -773,13 +898,13 @@ latexmath:[\[\begin{aligned}
 ([eq:03]) into Maxwell’s equations allows for the derivation of
 recursion relations. ([eq:23]) gives
 latexmath:[\[\sum_{i,k=0}^{\infty} a_{i,k} (2k+1) x^i z^{2k} = \sum_{i,k=0}^{\infty} b_{i,k} i x^{i-1} z^{2k}\]]
-Equating the powers in latexmath:[$x^i z^{2k}$]
+Equating the powers in latexmath:[x^i z^{2k}]
 latexmath:[\[a_{i,k} = \frac{i+1}{2k+1} b_{i+1,k} \label{eq:11}\]] A
 similar result is obtained from ([eq:24]) latexmath:[\[\begin{aligned}
         \sum_{i,k=0}^{\infty} \partial_s b_{i,k} x^i z^{2k} & = \left( 1+ \frac{x}{\rho} \right) \sum_{i,k=0}^{\infty} c_{i,k} (2k+1) x^i z^{2k} \\
         \Rightarrow c_{i,k}  & + \frac{1}{\rho} c_{i-1,k} = \frac{1}{2k+1} \partial_s b_{i,k} \label{eq:12} 
     \end{aligned}\]] The last equation from
-latexmath:[$\nabla \times \mathbf{B} = 0$] should be consistent with the
+latexmath:[\nabla \times \mathbf{B} = 0] should be consistent with the
 two recursion relations obtained. ([eq:22]) implies
 latexmath:[\[\sum_{i,k=0}^{\infty} \left[ \frac{i+1}{\rho} c_{i,k} x^i + c_{i,k} i x^{i-1} \right] z^{k+1} = \sum_{i,k=0}^{\infty} \partial_s a_{i,k} x^i z^{2k}\]]
 latexmath:[\[\Rightarrow \frac{\partial_s a_{i,k}}{i+1} = \frac{1}{\rho} c_{i,k} + c_{i+1,k}\]]
@@ -788,16 +913,16 @@ the relations are consistent. Furthermore, the last required relations
 is obtained from the divergence of *B*
 latexmath:[\[\sum_{i,k=0}^{\infty} \left[ \frac{a_{i,k} x^i z^{2k+1}}{\rho}  + i a_{i,k} x^{i-1} z^{2k+1} + \frac{i a_{i,k} x^i z^{2k+1}}{\rho}  + \partial_s c_{i,k} x^i z^{2k+1} + 2kb_{i,k}x^i z^{2k-1} \right] = 0 \nonumber\]]
 latexmath:[\[\Rightarrow \partial_s c_{i,k} + \frac{2(k+1)}{\rho} b_{i-1,k+1} + 2(k+1) b_{i,k+1} + \frac{1}{\rho} a_{i,k} + (i+1) a_{i+1,k} + \frac{1}{\rho} a_{i,k} = 0 \nonumber\]]
-Using the relation ([eq:11]) to replace the latexmath:[$a$] coefficients
-with latexmath:[$b$]’s we arrive at
+Using the relation ([eq:11]) to replace the latexmath:[a] coefficients
+with latexmath:[b]’s we arrive at
 latexmath:[\[\partial_s c_{i,k} + \frac{(i+1)^2}{\rho (2k+1)} b_{i+1,k} + \frac{(i+1)(i+2)}{2k+1} b_{i+2,k} + \frac{2(k+1)}{\rho} b_{i-1,k+1} + 2(k+1) b_{i,k+1} = 0\]]
 All the coefficients above can be determined recursively provided the
-field latexmath:[$B_z$] can be measured at the mid-plane in the form
+field latexmath:[B_z] can be measured at the mid-plane in the form
 latexmath:[\[B_z(z=0) = B_{0,0} + B_{1,0}x + B_{2,0} x^2 + B_{3,0} x^3 + \dots\]]
-where latexmath:[$B_{i,0}$] are functions of latexmath:[$s$] and they
-can model the fringe field for each multipole term latexmath:[$x^n$]. As
-an example, for a dipole magnet, the latexmath:[$B_{1,0}$] function can
-be model as an Enge function or latexmath:[$tanh$].
+where latexmath:[B_{i,0}] are functions of latexmath:[s] and they
+can model the fringe field for each multipole term latexmath:[x^n]. As
+an example, for a dipole magnet, the latexmath:[B_{1,0}] function can
+be model as an Enge function or latexmath:[tanh].
 
 [[fringe-field-of-a-curved-multipole-1]]
 Fringe field of a curved multipole
@@ -806,12 +931,12 @@ Fringe field of a curved multipole
 _(variable radius of curvature)_
 
 The difference between this case and the above is that
-latexmath:[$\rho$] is now a function of latexmath:[$s$],
-latexmath:[$\rho(s)$]. We can obtain the same result starting with the
+latexmath:[\rho] is now a function of latexmath:[s],
+latexmath:[\rho(s)]. We can obtain the same result starting with the
 same functional forms for the field ([eq:01]), ([eq:02]), ([eq:03]). The
 result of the previous section also holds in this case since no
-derivative latexmath:[$\frac{\partial}{\partial s}$] is applied to the
-scale factor latexmath:[$h_s$]. If the radius of curvature is set to be
+derivative latexmath:[\frac{\partial}{\partial s}] is applied to the
+scale factor latexmath:[h_s]. If the radius of curvature is set to be
 proportional to the dipole filed observed by some reference particle
 that stays in the centre of the dipole
 latexmath:[\[\rho (s) \propto B(z=0, x=0, s) = B_x (z=0,x=0) = b_{0,0}(s)\]]
@@ -824,29 +949,29 @@ _This is a different, more compact treatment_ The derivation is more
 clear if we gather the variables together in functions. We assume again
 mid-plane symmetry and that the z-component of the field in the
 mid-plane has the form latexmath:[\[B_z (x, z=0, s) = T(x) S(s)\]] where
-latexmath:[$T(s)$] is the transverse field profile and
-latexmath:[$S(s)$] is the fringe field. One of the requirements of the
-symmetry is that latexmath:[$B_z(z) = B_z(-z)$] which using a scalar
-potential latexmath:[$\psi$] requires
-latexmath:[$\frac{\partial \psi}{\partial z}$] to be an even function in
-z. Therefore, latexmath:[$\psi$] is an odd function in z and can be
+latexmath:[T(s)] is the transverse field profile and
+latexmath:[S(s)] is the fringe field. One of the requirements of the
+symmetry is that latexmath:[B_z(z) = B_z(-z)] which using a scalar
+potential latexmath:[\psi] requires
+latexmath:[\frac{\partial \psi}{\partial z}] to be an even function in
+z. Therefore, latexmath:[\psi] is an odd function in z and can be
 written as
 latexmath:[\[\psi = z f_0(x,s) + \frac{z^3}{3!} f_1(x,s) + \frac{z^5}{5!} f_3(x,s) + \dots\]]
 The given transverse profile requires that
-latexmath:[$f_0(x,s) = T(x)S(s)$], while latexmath:[$\nabla^2 \psi = 0$]
+latexmath:[f_0(x,s) = T(x)S(s)], while latexmath:[\nabla^2 \psi = 0]
 follows from Maxwell’s equations as usual, more explicitly
 latexmath:[\[\frac{\partial}{\partial x} \left( h_s \frac{\partial \psi}{\partial x} \right) + \frac{\partial}{\partial s} \left( \frac{1}{h_s} \frac{\partial \psi}{\partial s} \right) + \frac{\partial}{\partial z} \left( h_s \frac{\partial \psi}{\partial z} \right) = 0\]]
-For a straight multipole latexmath:[$h_s = 1$]. Laplace’s equation
+For a straight multipole latexmath:[h_s = 1]. Laplace’s equation
 becomes
 latexmath:[\[\sum_{n=0} \frac{z^{2n+1}}{(2n+1)!} \left[ \partial_x^2 f_n(x,s) + \partial_s^2 f_n(x,s) \right] + \sum_{n=1} f_n(x,s) \frac{z^{n-1}}{(n-1)!} = 0\]]
-By equating powers of latexmath:[$z$] we obtain the recursion relation
+By equating powers of latexmath:[z] we obtain the recursion relation
 latexmath:[\[f_{n+1}(x,s) = - \left( \partial_x^2 + \partial_s^2 \right) f_n(x,s)\]]
-The general expression for any latexmath:[$f_n(x,s)$] is then obtained
+The general expression for any latexmath:[f_n(x,s)] is then obtained
 from the mid-plane field by latexmath:[\[\begin{aligned}
         f_n(x,s) & = (-1)^n \left( \partial_x^2 + \partial_s^2 \right)^n f_0(x,s) \\
         f_n(x,s) & = (-1)^n \sum_{i=0}^n \binom{n}{i}T^{(2i)}(x) S^{(2n-2i)}(s)
     \end{aligned}\]] For a curved multipole of constant radius
-latexmath:[$h_s = 1 + \frac{x}{\rho} \quad \text{with} \quad \rho = const.$]
+latexmath:[h_s = 1 + \frac{x}{\rho} \quad \text{with} \quad \rho = const.]
 The corresponding Laplace’s equation is
 latexmath:[\[\left( \frac{1}{\rho h_s} \partial_x + \partial_x^2 + \partial_z^2 + \frac{\partial_s^2}{h_s^2} \right) \psi = 0\]]
 Again we substitute with the functional form of the potential to get the
@@ -854,7 +979,7 @@ recursion latexmath:[\[\begin{aligned}
         f_{n+1}(x,s) & = - \left[ \frac{1}{\rho + x} \partial_x + \partial_x^2 + \frac{\partial_s^2}{(1+x/ \rho)^2} \right] f_n(x,s) \\
         f_{n+1}(x,s) & = (-1)^n \left[ \frac{1}{\rho + x} \partial_x + \partial_x^2 + \frac{\partial_s^2}{(1+x/ \rho)^2} \right]^n f_0(x,s)
     \end{aligned}\]] Finally consider what changes for
-latexmath:[$\rho = \rho (s)$]. Laplace’s equation is
+latexmath:[\rho = \rho (s)]. Laplace’s equation is
 latexmath:[\[\left[ \frac{1}{\rho h_s} \partial_x + \partial_x^2 + \partial_z^2 + \frac{\partial_s^2}{h_s^2} + \frac{x}{\rho^2 h_s^3} (\partial_s \rho) \partial_s \right] \psi = 0\]]
 The last step is again the substitution to get
 latexmath:[\[\begin{aligned}
@@ -862,6 +987,6 @@ latexmath:[\[\begin{aligned}
         f_{n}(x,s) & = (-1)^n \left[ \frac{\partial_x}{\rho h_s} + \partial_x^2 + \partial_z^2 + \frac{\partial_s^2}{h_s^2} + \frac{x}{\rho^2 h_s^3} (\partial_s \rho) \partial_s \right]^n f_0(x,s) \label{eq:40}
     \end{aligned}\]] If the radius of curvature is proportional to the
 dipole field in the centre of the multipole (the dipole component of the
-transverse field is a constant latexmath:[$T_{dipole}(x) = B_0$] and
+transverse field is a constant latexmath:[T_{dipole}(x) = B_0] and
 latexmath:[\[\rho(s) = B_0 \times S(s)\]] This expression can be
 replaced in ([eq:40]) to obtain a more explicit equation.