Particle Accelerator Physics pp 125-174 | Cite as
Electromagnetic Fields
Abstract
Beam dynamics is effected by electromagnetic fields. Generally, magnetic fields are used for relativistic particle guidance and focusing while electric fields are mostly used in the form of electro-static fields or microwaves for acceleration of the particles.
Keywords
Beam Dynamic Curvilinear Coordinate System Fringe Field Hall Probe Beam TransportBeam dynamics is effected by electromagnetic fields. Generally, magnetic fields are used for relativistic particle guidance and focusing while electric fields are mostly used in the form of electro-static fields or microwaves for acceleration of the particles. In this chapter, we will discuss in more detail the magnetic fields and their generation as they are used in beam dynamics. From ( 1.52), ( 1.51) we know how to derive static electric and magnetic fields from a vector or scalar potential by solving their Laplace equations.
6.1 Pure Multipole Field Expansion
Special desired effects on particle trajectories require specific magnetic fields. Dipole fields are used to bend particle beams and quadrupole magnets serve, for example, as beam focusing devices. To obtain an explicit formulation of the equations of motion of charged particles in an arbitrary magnetic field, we derive the general magnetic fields consistent with Maxwells equations.
Although we have identified a curvilinear coordinate system moving together with particles to best fit the needs of beam dynamics, we use in this section first, for simplicity, a fixed, right-handed Cartesian coordinate system (x, y, z). By doing so, we assume straight magnets and neglect the effects of curvature. Later in this chapter, we will derive both the electromagnetic fields and equations of motion in full rigor.
6.1.1 Electromagnetic Potentials and Fields for Beam Dynamics
Earlier we have derived the potentials from the wave equation in a charge and current free static environment. This is the beam environment and we want to formulate fields for beam dynamics there. In the same environment Maxwell’s equations reduce to \(\boldsymbol{\nabla B} = 0\) and \(\boldsymbol{\nabla }\times \boldsymbol{ B} = 0\) and can be used directly. Based on these equations, the magnetic fields can be derived from potentials by ( 1.51) as previously defined. Electrostatic fields are derived from a scalar potential alone according to ( 1.52).
The coefficients λ_{n} are for upright multipoles while the μ_{n} are those of skew multipoles. Upright multipoles are characterized by midplane symmetry which requires that for y = 0 the horizontal fields vanish B_{x}(y = 0) = 0 and only vertical field components exist B_{y}(y = 0) ≠ 0. In beam dynamics we almost exclusively use upright magnets. This ansatz is not the most general solution of the Laplace equation, but includes all main multipole fields used in beam dynamics. Later, we will derive a solution that includes all terms allowed by the Laplace equation in a curvilinear coordinate system. Both, the real and imaginary part, are two independent solutions of the same Laplace equation. All coefficients λ_{n}, μ_{n} are still functions of z although we do not indicate this explicitly.
We distinguish between the electrical potential V_{e} and the magnetic potential V_{m}. Since the Laplace equation is valid for both the electric as well as the magnetic field in a material free region, no real distinction between both fields had to be made. In reality, we rarely design devices which include more than one term of the field expansion. It is therefore appropriate to decompose the general field potential in (6.7) into its independent multipole terms. To keep the discussion simple, we ignore here electric fields.
6.1.2 Fields, Gradients and Multipole Strength Parameter
In (6.7) we used general coefficients which must be related to fields and field gradients. Furthermore, we are looking for energy independent magnet strength parameters which are almost exclusively used in beam dynamics. The particular field patterns for multipole magnets can be derived from the complex potential by differentiation to get the fields (6.6a). Although fields can be derived from both the vector and scalar potential, we will use only the latter to define the fields for beam dynamics.
Magnetic multipole potentials
Dipole | − V_{1} = −B_{x}x − B_{y}y | |
Quadrupole | \(-V _{2} = -\tfrac{1} {2}\)g\(\,(x^{2} - y^{2}) + gxy,\) | |
Sextupole | \(-V _{3} = -\tfrac{1} {6}\underline{s_{3}}\left (x^{3} - 3xy^{2}\right ) + \tfrac{1} {6}s_{3}\left (3x^{2}y - y^{3}\right )\,,\) | |
Octupole | \(-V _{4} = -\tfrac{1} {24}\underline{s_{4}}\left (x^{4} - 6x^{2}y^{2} + y^{4}\right ) + \tfrac{1} {24}s_{4}\left (x^{3}y - xy^{3}\right ),\) | |
Decapole | \(-V _{5} = - \tfrac{1} {120}\underline{s_{5}}\left (x^{5} - 10x^{3}y^{2} + 5xy^{4}\right )\, + \tfrac{1} {120}s_{5}\,\left (5x^{4}y - 10x^{2}y^{3} + y^{5}\right )\) |
Field gradient nomenclature for low order multipoles
Dipole | B_{y} | \(\frac{e} {p_{0}} B_{y} = \frac{1} {\rho }\) | |
Quadrupole | \(\frac{\partial B_{y}} {\partial x} = g\) | \(\frac{e} {p_{0}} \frac{\partial B_{y}} {\partial x} = k\) | |
Sextupole | \(\frac{\partial ^{2}B_{ y}} {\partial x^{2}} = s\) | \(\frac{e} {p_{0}} \frac{\partial B_{y}} {\partial x} = m\) | |
Octupole | \(\frac{\partial ^{3}B_{ y}} {\partial x^{3}} = s_{4}\) | \(\frac{e} {p_{0}} \frac{\partial B_{y}} {\partial x} = r\) | |
Decapole | \(\frac{\partial ^{4}B_{ y}} {\partial x^{4}} = s_{5}\) | \(\frac{e} {p_{0}} \frac{\partial B_{y}} {\partial x} = S_{5}\) |
Upright multipole fields
Dipole | \(\frac{e} {p_{0}} B_{x} = 0\) | \(\frac{e} {p_{0}} B_{y} = \frac{e} {p_{0}} B_{y0}\) | |
Quadrupole | \(\frac{e} {p_{0}} B_{x} = ky\) | \(\frac{e} {p_{0}} B_{y} = kx\) | |
Sextupole | \(\frac{e} {p_{0}} B_{x} = mxy\) | \(\frac{e} {p_{0}} B_{y} = \tfrac{1} {2}m\left (x^{2} - y^{2}\right )\) | |
Octupole | \(\frac{e} {p_{0}} B_{x} = \tfrac{1} {6}r\,\left (3x^{2}y - y^{3}\right )\) | \(\frac{e} {p_{0}} B_{y} = \tfrac{1} {6}s_{4}\,\left (x^{3} - 3xy^{2}\right )\) | |
Decapole | \(\frac{e} {p_{0}} B_{x} = + \tfrac{1} {24}s_{5}\,\left (x^{3}y - xy^{3}\right )\) | \(\frac{e} {p_{0}} B_{y} = + \tfrac{1} {24}s_{5}\,\left (x^{4} - 6x^{2}y^{2} + y^{4}\right )\) |
Rotated or skew multipole fields
Dipole (90^{∘}) | \(\frac{e} {p_{0}} B_{x} = \frac{e} {p_{0}} B_{x0}\) | \(\frac{e} {p_{0}} B_{y} = 0\) | |
\(\text{Quadrupole (}45^{\circ }\text{)}\) | \(\frac{e} {p_{0}} B_{x} = -\underline{k}x\) | \(\frac{e} {p_{0}} B_{y} = +\underline{k}y\) | |
\(\text{Sextupole (}30^{\circ }\text{)}\) | \(\frac{e} {p_{0}} B_{x} = -\frac{1} {2}\underline{m}\left (x^{2} - y^{2}\right )\) | \(\frac{e} {p_{0}} B_{y} = +\underline{m}xy\) | |
\(\text{Octupole (}22.5^{\circ }\text{)}\) | \(\frac{e} {p_{0}} B_{x} = -\frac{1} {6}\underline{r}\left (x^{3} - 3xy^{2}\right )\) | \(\frac{e} {p_{0}} B_{y} = -\frac{1} {6}\underline{r}\left (3x^{2}y - y^{3}\right )\) | |
\(\text{Decapole (}18^{\circ }\text{)}\) | \(\frac{e} {p_{0}} B_{x} = -\frac{1} {24}\underline{s_{5}}\left (x^{4} - 6x^{2}y^{2} + y^{4}\right )\quad\) | \(\frac{e} {p_{0}} B_{y} = + \frac{1} {24}\underline{s_{5}}\left (x^{3}y - xy^{3}\right )\) |
6.1.3 Main Magnets for Beam Dynamics
The feasibility of any accelerator or beam transport line design depends fundamentally on the parameters and diligent fabrication of technical components composing the system. Not only need the magnets be designed such as to minimize undesirable higher order multipole fields but they also must be designed such that the desired parameters are within technical limits. Most magnets constructed for beam transport lines are electromagnets rather than permanent magnets. The magnets are excited by electrical current carrying coils wound around magnet poles or in the case of superconducting magnets by specially shaped and positioned current carrying coils. In this section, we will discuss briefly some fundamental design concepts and limits for most commonly used iron dominated bending and quadrupole magnets as a guide for the accelerator designer towards a realistic design. For more detailed discussions on technical magnet designs we refer to related references, for example [1, 2].
Iron dominated magnets are the most commonly used magnets for particle beam transport systems. Only where very high particle energies and magnetic fields are required, superconducting magnets are used with maximum magnetic fields of 6–10 T compared to the maximum field in an iron magnet of about 2 T. Although saturation of ferromagnetic material imposes a definite limit on the strength of iron dominated magnets, most accelerator design needs can be accommodated within this limit.
We are now in a position to determine the fields for any multipole. This will be done in this section for magnetic fields most commonly used in particle transport systems, the bending field and the focusing quadrupole field. Only for very special applications are two or more multipole field components desired in the same magnet like in a gradient bending magnet or synchrotron magnet.
Deflecting Magnets
As mentioned above, vertical bending magnets are rarely used in accelerator physics. Yet, there are special instances, especially in beam transport lines where vertical bending magnets are required. In those cases we would just introduce a vertical curvature κ_{y} in (6.18a) or (6.19a) cover the vertical dispersion function. Outside the bending magnet the dispersion behaves just like a particle trajectory and therefore the quadrupoles do not have to be rotated or modified.
Focusing Device
Synchrotron Magnet
Sometimes a combination of both, the dipole field of a bending magnet and the focusing field of a quadrupole, is desired for compact beam transport lines to form what is called a synchrotron magnet. The name comes from the use of such magnets for early synchrotron accelerators. The fields can be derived just like the dipole and quadrupole fields from the two-term potential (6.7) with n = 1 and n = 2.
Higher Order Multipole Magnets
From this equation it is straight forward to extract an expression for the potential of any multipole field satisfying the Laplace equation. Since both electrical and magnetic fields may be derived from the Laplace equation, we need not make any distinction here and may use (6.32) as an expression for the electrical as well as the magnetic potential.
As mentioned before, it is useful to keep both sets of solutions \(\left (\lambda _{n,}\mu _{n}\right )\) separate because they describe two distinct orientations of multipole fields. For a particular multipole both orientations can be realized by a mere rotation of the element about its axis. Only the solution λ_{n} has what is called midplane symmetry with the property that \(B_{ny}(x,y) = B_{ny}(x,-y)\). In this symmetry, there are no horizontal field components in the midplane, B_{nx}(x, 0) ≡ 0, and a particle travelling in the horizontal mid plane will remain in this plane. We call all magnets in this class upright magnets. The magnets defined by μ_{n} ≠ 0 we call rotated or skew magnets since they differ from the upright magnets only by a rotation about the magnet axis. In real beam transport systems, we use almost exclusively magnetic fields with midplane symmetry.
Vacuum Chamber Material
6.1.4 Multipole Misalignment and “Spill-down”
The original field is still preserved, but now many lower order terms appear. Actually, for a lateral misalignment all lower order magnetic field components appear, a phenomenon that is called “spill-down”. These lower order fields cause orbit distortions, focusing errors and errors in the chromaticity, which all have to be compensated.
6.2 Main Magnet Design Criteria
In this section we will shortly discuss the design criteria for the main beam dynamics magnets like bending magnets and quadrupoles. For more detailed studies on magnets the reader is referred to relevant texts like [2].
6.2.1 Design Characteristics of Dipole Magnets
The expressions for the magnetic potentials give us a guide to design devices that generate the desired fields. Multipole fields are generated mostly in one of two ways: as iron dominated magnets, or by proper placement of electrical current carrying conductors. The latter way is mostly used in high field superconducting magnets, where fields beyond the general saturation level of about 2 T for iron are desired.
In iron dominated magnets, fields are determined by the shape of the iron surfaces. Just like metallic surfaces are equipotential surfaces for electrical fields, so are surfaces of ferromagnetic material, like iron in the limit of infinite magnetic permeability, equipotential surfaces for magnetic fields. Actually, for practical applications the permeability only has to be large just like the conductivity must be large to make a metallic surface an equipotential surface. This approximate property of iron surfaces can be exploited for the design of unsaturated or only weakly saturated magnets. For preliminary design calculations, we assume infinite permeability. Where effects of finite permeability or magnetic saturation become important, the fields are determined numerically by mathematical relaxation methods. In this text, we will not be able to discuss the details of magnet design and construction but will concentrate only on the main magnet features from a beam dynamics point of view. A wealth of practical experience in the design of iron dominated accelerator magnets, including an extensive list of references, is compiled in a review article by Fischer [1] and a monograph by Tanabe [2].
Excitation Current and Saturation in a Bending Magnet
As a practical example, we consider a magnetic field of 1 T in a dipole magnet with an aperture of 2G = 10 cm. From (6.39), a total electrical excitation current of about 40,000 A is required in each of two excitation coils to generate this field. Since the coil in general is composed of many turns, the actual electrical current is much smaller by a factor equal to the number of turns and the total coil current \(I_{\mathrm{coil}}\) is therefore often measured in units of Ampere\(\cdot \) turns. For example, a coil composed of 40 windings with sufficient cross section to carry an electrical current of 1,000 A would provide the total required current of 40,000 A\(\cdot \) turns.
As a rule of thumb to get a good field quality within an aperture width equal to the full gap height the pole width should be at least 3-times the full gap height. Narrower pole profiles require shimming of the pole profile. There are elaborate way to shape the pole profile for a bending magnet [2] but there are also more simple ways. The drop-off of the field towards the side of the poles can be to some extend extended further out by adding to the pole profile a straight line shim to slightly reduce the pole gap around the edges of the poles. This shim need not be more elaborate than a line segment to reduce the gap followed by a horizontal section to the edge of the pole. Such shims may start around half a full gap size from the center with a gentle slope and rarely a thickness of more than 0. 5–1 mm. We will discuss such shims in more detail in connection with quadrupole design.
Saturation effects are similar to those in a quadrupole magnet which will be discussed in the next section. Like in any magnet the first sign of saturation show up most likely at the pole root where the poles join the return yoke. That is so because much magnetic flux comes into the pole from the sides along the length of the pole thus increasing the magnetic flux density. One way out is to shape the pole pieces like wedges with increasing cross section towards the return yoke. Any saturation in the return yoke is easily avoided by increasing the thick ness of the iron in the return yoke.
6.2.2 Quadrupole Design Concepts
Pole Profile Shimming
While in an ideal quadrupole the field gradient along, say, the x-axis would be constant, we find for a finite pole width a drop off of the field and gradient approaching the corners of poles. Different magnet designer apply a variety of pole shimming methods. In this text we use tangent shimming as described below. The field drop off at the pole edge can be reduced to some extend if the hyperbolic pole profile continues into its tangent close to the pole corner as indicated in Fig. 6.6.
The final design of a magnet pole profile is made with the help of computer codes which allow the calculation of magnet fields from a given pole profile with saturation characteristics determined from a magnetization curve. Widely used computer codes for magnet design are, for example, MAGNET [3] and POISSON [4].
Field errors in iron dominated magnets have two distinct sources, the finite pole width and mechanical manufacturing and assembly tolerances. From symmetry arguments, we can deduce that field errors due to the finite pole width produce only select multipole components. In a quadrupole, for example, only \((2n + 1) \cdot 4\)-pole fields like 12-pole or 20-pole fields are generated. Similarly in a dipole of finite pole width only \((2n + 1) \cdot 2\)-pole fields exist. We call these multipole field components often the allowed multipole errors. Manufacturing and assembly tolerances on the other hand do not exhibit any symmetry and can cause the appearance of any multipole field error.
The particular choice of some geometric design parameters must be checked against technical limitations during the design of a beam transport line. One basic design parameter for a quadrupole is the bore radius R which depends on the aperture requirements of the beam. Addition of some allowance for the vacuum chamber and mechanical tolerance between chamber and magnet finally determines the quadrupole bore radius.
Excitation Current and Saturation
Severe saturation effects at the corners of the magnet pole profile can be avoided if the maximum field gradient, as a rule of thumb, is chosen such that the pole tip field does not exceed a value of B_{p} = 0. 8 − 1 T. This limits the maximum field gradient to g_{max} = B_{p}∕R and the quadrupole length must therefore be long enough to reach the focal length desired in the design of the beam transport line. Saturation of the pole corners introduces higher-order multipoles and must therefore be kept to a minimum.
In addition to pole root saturation, we may also experience return yoke saturation, which is easily avoided by increasing its thickness.
6.3 Magnetic Field Measurement
The quality of the magnetic fields translates immediately into the quality and stability of the particle beam. The precision of the magnetic fields determines the predictability of the beam dynamics designs. While we make every effort to construct magnets as precise as possible, we cannot avoid the appearance of higher multipole fields due to finite pole widths or machining and assembly tolerances. Therefore, precise magnetic field measurements are required. While detailed discussions of magnetic field measurement technology exceeds the goals of this book, the issue is too important to ignore completely and we will discuss this topic in an introductory way. For more detailed information, please consult texts like [2].
6.3.1 Hall Probe
By computer controlled precise movement of the Hall probe from point to point within the magnet aperture, the magnetic field can be mapped to high precision. The measurements can then be analysed as to field errors, multipole content and fringe field effects.
6.3.2 Rotating Coil
Practical Considerations
The whole magnetic measurement would record the signals from both coils separately and produce the strength and orientation of the main field for n = N according to (6.49) and (6.50) while the same multipole parameters are derived from the same equations based on the compensated signal and including the calculated sensitivities.
Magnetic field measurements have developed very far and have reached a level of accuracy and precision that fully meets the demands of beam dynamics. Especially, the determination of the multipole content is important to ensure the stability of a beam in, for example, a storage ring. While the effects of multipole fields cannot be analyzed analytically, we may track particles many times around the storage ring in the presence of these multipole fields and thus define beam stability and the dynamic aperture.
6.4 General Transverse Magnetic-Field Expansion*
In the previous section, we discussed solutions to the Laplace equation which included only pure transverse multipole components in a cartesian coordinate system thus neglecting all kinematic effects caused by the curvilinear coordinate system of beam dynamics. These approximations eliminate many higher-order terms which may become of significance in particular circumstances. In preparation for more sophisticated beam transport systems and accelerator designs aiming, for example, at ever smaller beam emittances it becomes imperative to consider higher-order perturbations to preserve desired beam characteristics. To obtain all field components allowed by the Laplace equation, a more general ansatz for the field expansion must be made. Here we restrict the discussion to scalar potentials only which are sufficient to determine all fields [5, 6].
6.4.1 Pure Multipole Magnets
With these definitions of the linear coefficients, we may start exploiting the recursion formula. All terms on the right-hand side of (6.65) are of lower order than the two terms on the left-hand side which are of order n = p + q + 2. The left-hand side is composed of two contributions, one resulting from pure multipole fields of order n and the other from higher-order field terms of lower-order multipoles.
Correspondence between the potential coefficients and multipole strength parameters
A_{00} | |||||||||||
A_{10} | A_{01} | ||||||||||
A_{20} | A_{11} | A_{02} | |||||||||
A_{30} | A_{21} | A_{12} | A_{03} | ||||||||
A_{40} | A_{31} | A_{22} | A_{13} | A_{04} | |||||||
A_{50} | A_{41} | A_{32} | A_{23} | A_{14} | A_{05} | ||||||
⇕ | |||||||||||
0 | |||||||||||
−κ_{y} | κ_{x} | ||||||||||
\(-\underline{k}\) | \(k\) | \(\underline{k}\) | |||||||||
\(-\underline{m}\) | m | \(\underline{m}\) | − m | ||||||||
\(-\underline{r}\) | r | \(\underline{r}\) | r | \(-\underline{r}\) | |||||||
\(-\underline{d}\) | d | \(\underline{d}\) | − d | \(-\underline{d}\) | d |
Magnetic multipole potentials
Dipole | \(-\frac{e} {p_{0}} V _{1} = -\kappa _{y}x +\kappa _{x}y\) | |
Quadrupole | \(-\frac{e} {p_{0}} V _{2} = -\tfrac{1} {2}\underline{k}\,(x^{2} - y^{2}) + kxy,\) | |
Sextupole | \(-\frac{e} {p_{0}} V _{3} = -\frac{1} {6}\underline{m}\left (x^{3} - 3xy^{2}\right ) + \frac{1} {6}m\left (3x^{2}y - y^{3}\right ),\) | |
Octupole | \(-\frac{e} {p_{0}} V _{4} = -\tfrac{1} {24}\underline{r}\left (x^{4} - 6x^{2}y^{2} + y^{4}\right ) + \tfrac{1} {24}r\,\left (x^{3}y - xy^{3}\right ),\) | |
Decapole | \(-\frac{e} {p_{0}} V _{5} = - \tfrac{1} {120}\underline{d}\left (x^{5} - 10x^{3}y^{2} + 5xy^{4}\right )\, + \tfrac{1} {120}d\,\left (5x^{4}y - 10x^{2}y^{3} + y^{5}\right )\) |
Each expression for the magnetic potential is composed of both the real and the imaginary contribution. Since both components differ only by a rotational angle, real magnets are generally aligned such that only one or the other component appears. Only due to alignment errors may the other component appear as a field error which can be treated as a perturbation.
6.4.2 Kinematic Terms
Upon closer inspection of (6.84)–(6.86) it becomes apparent that most terms originate from a combination of different multipoles. These equations describe the general fields in any magnet, yet in practice, special care is taken to limit the number of fundamentally different field components present in any one magnet. In fact most magnet are designed as single multipoles like dipoles or quadrupoles or sextupoles etc. A beam transport system utilizing only such magnets is also called a separated-function lattice since bending and focusing is performed in different types of magnets. A combination of bending and focusing, however, is being used for some special applications and a transport system composed of such combined-field magnets is called a combined-function lattice. Sometimes even a sextupole term is incorporated in a magnet together with the dipole and quadrupole fields. Rotated magnets, like rotated sextupoles \(\underline{s_{3}}\) and octupoles \(\underline{s_{4}}\) are either not used or in the case of a rotated quadrupole the chosen strength is generally weak and its effect on the beam dynamics is treated by perturbation methods.
No mention has been made about electric field patterns. However, since the Laplace equation for electrostatic fields in material free areas is the same as for magnetic fields we conclude that the electrical potentials are expressed by (6.82) as well and the electrical multipole field components are also given by (6.84)–(6.86) after replacing the magnetic field (\(B_{x},B_{y},B_{z}\)) by electric-field components (\(E_{x},E_{y},E_{z}\)).
6.5 Third-Order Differential Equation of Motion*
In spite of our attempt to derive a general and accurate equation of motion, we note that some magnet boundaries are not correctly represented. The natural bending magnet is of the sector type and wedge or rectangular magnets require the introduction of additional corrections to the equations of motion which are not included here. This is also true for cases where a beam passes off center through a quadrupole, in which case theory assumes a combined function sector magnet and corrections must be applied to model correctly a quadrupole with parallel pole faces. The magnitude of such corrections is, however, in most cases very small. Equation (6.95) shows an enormous complexity which in real beam transport lines, becomes very much relaxed due to proper design and careful alignment of the magnets. Nonetheless (6.95) and (6.96) for the vertical plane, can be used as a reference to find and study the effects of particular perturbation terms. In a special beam transport line one or the other of these perturbation terms may become significant and can now be dealt with separately. This may be the case where strong multipole effects from magnet fringe fields cannot be avoided or because large beam sizes and divergences are important and necessary. The possible significance of any perturbation term must be evaluated for each beam transport system separately.
In most beam transport lines the magnets are built in such a way that different functions like bending, focusing etc., are not combined thus eliminating all terms that depend on those combinations like \(\kappa _{x}\kappa _{y}\), κ_{x}k or m κ_{x} etc. As long as the terms on the right-hand sides are small we may apply perturbation methods to estimate the effects on the beam caused by these terms. It is interesting, however, to try to identify the perturbations with aberrations known from light optics.
Chromatic terms \(\kappa _{x}(\delta -\delta ^{2} +\delta ^{3})\), for example, are constant perturbations for off momentum particles causing a shift of the equilibrium orbit which ideally is the trivial solution x ≡ 0 of the differential equation \(x^{{\prime\prime}} + (k +\kappa _{ x}^{2})x = 0\). Of course, this is not quite true since κ_{x} is not a constant but the general conclusion is still correct. This shift is equal to \(\varDelta x =\kappa _{x}(\delta -\delta ^{2} +\delta ^{3})/(k +\kappa _{ x}^{2})\) and is related to the dispersion function D by \(D =\varDelta x/\delta\). In light optics this corresponds to the dispersion of colors of a beam of white light (particle beam with finite energy spread) passing through a prism (bending magnet). We may also use a different interpretation for this term. Instead of a particle with an energy deviation \(\delta\) in an ideal magnet κ_{x} we can interpret this term as the perturbation of a particle with the ideal energy by a magnetic field that deviates from the ideal value. In this case, we replace \(\kappa _{x}\,(\delta -\delta ^{2} -\delta ^{3})\) by \(-\,\varDelta \kappa _{x}\) and the shift in the ideal orbit is then called an orbit distortion. Obviously, here and in the following paragraphs the interpretations are not limited to the horizontal plane alone but apply also to the vertical plane caused by similar perturbations. Terms proportional to x^{2} cause geometric aberrations, where the focal length depends on the amplitude x while terms involving x^{′} lead to the well-known phenomenon of astigmatism or a combination of both aberrations. Additional terms depend on the particle parameters in both the vertical and horizontal plane and therefore lead to more complicated aberrations and coupling.
Terms depending also on the energy deviation \(\delta\), on the other hand, give rise to chromatic aberrations which are well known from light optics. Specifically, the term \((k + 2\kappa _{x}^{2})x\delta\) is the source for the dependence of the focal length on the particle momentum. Some additional terms can be interpreted as combinations of aberrations described above.
6.6 Longitudinal Field Devices
The linear dependence of the integrated radial fields on the distance r from the axis constitutes linear focusing capabilities of solenoidal fringe fields. Such solenoid focusing is used, for example, around a conversion target to catch a highly divergent positron beam. The positron source is generally a small piece of a heavy metal like tungsten placed in the path of a high energy electron beam. Through an electromagnetic cascade, positrons are generated and emerge from a point like source into a large solid angle. If the target is placed in the center of a solenoid the radial positron motion couples with the longitudinal field to transfer the radial particle momentum into azimuthal momentum. At the end of the solenoid, the azimuthal motion couples with the radial field components of the fringe field to transfer azimuthal momentum into longitudinal momentum. In this idealized picture a divergent positron beam emerging from a small source area is transformed or focused into a quasi-parallel beam of larger cross section. Such a focusing device is called a \(\lambda /4\)-lens, since the particles follow one quarter of a helical trajectory in the solenoid.
In other applications large volume solenoids are used as part of elementary particles detectors in high energy physics experiments performed at colliding-beam facilities. The strong influence of these solenoidal detector fields on beam dynamics in a storage ring must be compensated in most cases. In still other applications solenoid fields are used just to contain a particle beam within a small circular aperture like that along the axis of a linear accelerator.
6.7 Periodic Wiggler Magnets
Particular arrays or combinations of magnets can produce desirable results for a variety of applications. A specially useful device of this sort is a wiggler magnet [7] which is composed of a series of short bending magnets with alternating field excitation. Most wiggler magnets are used as sources of high brightness photon beams in synchrotron radiation facilities and are often also called undulators. There is no fundamental difference between both. We differentiate between a strong field wiggler magnet and an undulator, which is merely a wiggler magnet at low fields, because of the different synchrotron radiation characteristics. As long as we talk about magnet characteristics in this text, we make no distinction between both types of magnets. Wiggler magnets are used for a variety of applications to either produce coherent or incoherent photon beams in electron accelerators, or to manipulate electron beam properties like beam emittance and energy spread. To compensate anti-damping in a combined function synchrotron a wiggler magnet including a field gradient has been used for the first time to modify the damping partition numbers [8]. In colliding-beam storage rings wiggler magnets are used to increase the beam emittance for maximum luminosity [9]. In other applications, a very small beam emittance is desired as is the case in damping rings for linear colliders or synchrotron radiation sources which can be achieved by employing damping wiggler magnets in a different way [10].
6.7.1 Wiggler Field Configuration
Upon closer inspection of the precise beam trajectory we observe a lateral displacement of the beam within a wiggler magnet. To compensate this lateral beam displacement, the wiggler magnet should begin and end with only a half pole of length λ_{p}∕4 to allow the beams to enter and exit the wiggler magnet parallel with the unperturbed beam path.
The parameter K is a characteristic wiggler constant defining the wiggler strength and is not to be confused with the general focusing strength \(K =\kappa ^{2} + k\). Coming back to the distinction between wiggler and undulator magnet, we speak of a wiggler magnet if \(K \gg 1\) and of an undulator if \(K \ll 1.\) Of course, many applications happen in a gray zone of terminology when \(K \approx 1.\)
6.8 Electrostatic Quadrupole
The potential of the four electrodes is alternately \(V = \pm \frac{1} {2}gR^{2}.\) This design can be somewhat simplified by replacing the hyperbolic metal surfaces by equivalently sized metallic tubes as shown in Fig. 6.22 (right). Numerical computer simulation programs can be used to determine the degradation of the quadrupole field due to this simplification.
Problems
6.1 (S). Show that the electrical power in the excitation coil is independent of the number of turns. Show also that the total electrical power in a copper coil depends only on the total weight of the copper used and the current density.
6.2 (S). Design an electrostatic quadrupole which provides a focal length of 10 m in the horizontal plane for particles with a kinetic energy of 10 MeV. The device shall have an aperture with a diameter of 10 cm and an effective length of 0.1 m. What is the form of the electrodes, their orientation and potential?
6.3 (S). In the text, we have derived the fields from a scalar potential. We could also derive the magnetic fields from a vector potential \(\boldsymbol{A}\) through the differentiation \(\boldsymbol{B} =\boldsymbol{ \nabla }\times \boldsymbol{ A}\). For purely transverse magnetic fields, show that only the longitudinal component A_{z} ≠ 0 must be non zero. Derive the vector potential for a dipole and quadrupole field and compare with the scalar potential. What is the difference between the scalar potential and the vector potential?
6.4 (S). Derive the pole profile (aperture radius r = 1 cm) for a combined function magnet including a dipole field to produce for a particle beam of energy E = 50 GeV a bending radius of ρ = 300 m, a focusing strength k = 0. 45 m^{−2} and a sextupole strength of m = 23. 0 m^{−3}.
6.5 (S). Strong mechanical forces exist between the magnetic poles when a magnet is energized. Are these forces attracting or repelling the poles? Why? Consider a dipole magnet ℓ = 1 m long, a pole width w = 0. 2 m and a field of B = 1. 5 T. Estimate the total force between the two magnet poles?
6.6 (S). Following the derivation of ( 5.7) for a bending magnet, derive a similar expression for the electrical excitation current in A⋅ turns of a quadrupole with an aperture radius R and a desired field gradient g. What is the total excitation current necessary in a quadrupole with an effective length of \(\ell=\) 1 m and R = 3 cm to produce a focal length of f = 50 m for particles with an energy of cp = 500 GeV?
6.7 (S). Consider a coil in the aperture of a magnet as shown in Fig. 6.14. All n windings are made of very thin wires and are located exactly on the radius R. We rotate now the coil about its axis at a rotation frequency ν. Such rotating coils are used to measure the multipole field components in a magnet. Show analytically that the recorded signal is composed of harmonics of the rotation frequency ν. What is the origin of the harmonics?
6.8 (S). Explain why a quadrupole with finite pole width does not produce a pure quadrupole field. What are the other allowed multipole field components ignore mechanical tolerances and why?
6.9 (S). Through magnetic measurements the following vertical magnetic multipole field components in a quadrupole are determined. At x = 1. 79 cm and y = 0 cm: B_{2} = 0. 3729 T, \(B_{3} = 1.25 \times 10^{-4}\) T, \(B_{4} = 0.23 \times 10^{-4}\) T, \(B_{5} = 0.36 \times 10^{-4}\) T, \(B_{6} = 0.726 \times 10^{-4}\) T, \(B_{7} = 0.020 \times 10^{-4}\) T, \(B_{8} = 0.023 \times 10^{-4}\) T, \(B_{9} = 0.0051 \times 10^{-4}\) T, \(B_{10} = 0.0071 \times 10^{-4}\) T. Calculate the relative multipole strengths at x = 1 cm normalized to the quadrupole field at 1 cm. Why do the 12-pole and 20-pole components stand out with respect to the other multipole components?
6.10 (S). Derive the equation for the pole profile of an iron dominated upright octupole with a bore radius R. Ignore longitudinal variations. To produce a field of 0.2 T at the pole tip \(\left (R = 3\text{cm}\right )\) what total current per coil is required?
6.11 (S). Calculate and design the current distribution for a pure air coil, superconducting dipole magnet to produce a field of B_{0} = 5 T in an aperture of radius R = 3 cm without exceeding an average current density of \(\ \hat{\jmath} = 1,000\) A/mm^{2}.
6.12. Derive an expression for the current distribution in air coils to produce a combination of a dipole, quadrupole and sextupole field. Express the currents in terms of fields and field gradients.
Footnotes
- 1.
The author thanks Jyh-Chyuan Jan, Cheng-Ying Kuo and Ping J. Chou from NSRRC, Taiwan for the pictures showing the effect of a magnetized vacuum chamber based on simulations.
- 2.
Consistent with the definitions of magnet strengths, the underlined quantities represent the magnet strengths of rotated multipole magnets.
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