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