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## m~~:~~~~~Ji~!~~~~ALaboratory Accelerator & Fusion Research<' Division Understanding Modern Magnets through Conformal Mapping Understanding Modern Magnets through Conformal Mapping

### BibTeX

@MISC{Halbach_m~~:~~~~~ji~!~~~~alaboratoryaccelerator,

author = {K Halbach and K Halbach},

title = {m~~:~~~~~Ji~!~~~~ALaboratory Accelerator & Fusion Research<' Division Understanding Modern Magnets through Conformal Mapping Understanding Modern Magnets through Conformal Mapping},

year = {}

}

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### Abstract

A talk given at the Bloch Symposium at Stanford University on Oct.27, 1989. Reminiscences When Prof. Little invited me to present a paper at this symposium, I was very excited not only because it would allow me to talk to a knowledgeable audience about a sUbject about which I have very strong feelings, but even more so because it would give me the opportunity to talk about my interactions with Prof. Bloch during my work with him during my visit at Stanford from 1957 to 1959. While we all know that he was one of the great physicists of this century, it is not nearly as well known that he also was exceptionally generous, gentle and sensitive in his interactions wi th men that were less than his intellectual equals. I will recall here only a few of the many characteristic exchanges that showed his interest in helping a young man become a better physicist, but doing this in such a way that his emotional wellbeing was not harmed. Even though Prof. Bloch knew that I was an experimentalist, he agreed to give me guidance in carrying out some theoretical work that I wanted to do. To this end we met 2-3 times per week in his office, discussing for typically 2 hours both "my" problem as well as the problem he was working on. Characteristic for all these discussions was the fact that he never ever used an argument that I could not fully understand, having calibrated me, and my shortcomings, in a very short time. Our discussions usually became, as most discussions do, a debate in which the participants took opposite views of the topic under discussion. While it is quite clear who would "lose" most of these discussions, it happened on some occasions that my view prevailed. Whenever that happened, he would come to me, pat me on-my shoulder, and would say something like: "Halbach, I am very glad we discussed this to the end, because you were right and I was wrong, and 1 really learned something from this discussion!". While this obviously made me feel very good, he probably never really understood that it is not depressing when one "loses" (1 use this word only because of the lack of a better word, since in this kind of a debate there are clearly no losers) a debate with a giant.-The fact that he, Prof. Bloch, was still learning things from others was a subject that he touched on every now and then, like when he 1 told me that when he started his present work, he did not really understand how to use the density matrix, "Leonard Schiff taught me how to use it".-In his attempts to make me feel less inadequate, he would also occasionally tell me stories that showed an inadequacy of his own, as he did when he professed that he was very lucky that he was not present at the APS meeting in which overhauser was strongly attacked by many prominent people when he first proposed the experiment that would show what is now called the Overhauser Effect, because "I would have said exactly the same fool things that everybody else said". While it is well known that Prof. Bloch was a wonderful teacher, it is less well known how concerned he was with the well being of individuals, and I am very happy to have this opportunity to shed some light on this side of his personality. Introduction When I had to choose, within some narrow range, the topic of this paper, I received great help from a colleague in Berkeley and from Prof. Little when it was suggested that I should pick among the possible sUbj ects of my talk the subj ect that Prof. Bloch would have enjoyed most. Since Prof. Bloch would prefer a scalpel over a sword every time, I hope and think that most people will approve my choice. When one intends to talk about a sUbject that is as old as conformal mapping and one does not want to lose the audience in a very short time, it is advisable to start by explaining both the motivation for the talk as well as the goals one has in mind when giving the talk. This particular talk has been motivated by the increasing frequency with which one hears, from people that ought to know better, statements like: "Conformal mapping is really a thing of the past because of all the marvelous computer programs that we now have". Even though, or more likely because, I have been intimately involved in the development of some large and widely used computer codes, I am deeply disturbed by such statements since they indicate a severe lack of understanding of the purpose of conformal mapping techniques, computers, and computer codes. In my view, conformal mapping can be an extremely powerful computational technique, and the easy availability of computers has made that aspect even more important now than it has been in the past. Additionally, and more importantly, conformal mapping can give very deep and unique insight into problems, giving often solutions to problems that can not be obtained with any other method, in particular not with computers. Wanting to demonstrate in particular the latter part, I set myself two goals for this talk: 1) I want to show with the help of a number of examples that conformal mapping is a unique and enormously powerful tool for thinking about, and solving, problems. Usually one has to write 2 down only a few equations, and sometimes none at all! When I started getting involved in work for which conformal mapping seemed to be a powerful tool, I did not think that I would ever be able to use that technique successfully because it seemed to require a nearly encyclopedic memory, an impression that was strengthened when I saw H.Kober's Dictionary of Conformal Representations (ref. 1). This attitude changed when I started to realize that beyond the basics of the theory of a function of a complex variable, I needed to know only about a handful of conformal maps and procedures. Consequently, my second goal for this talk is to: 2) Show that in most cases conformal mapping functions can be obtained by formulating the underlying physics appropriately. This means particularly that encyclopedic knowledge of conformal maps is not necessary for successful use of conformal mapping techniques. To demonstrate these facts I have chosen examples from an area of physics/engineering in which I am active, namely accelerator physics. In order to do that successfully I start with a brief introduction into high energy charged particle storage ring technology, even though not all examples used in this paper to elucidate my points come directly from this particular field of accelerator technology. This is followed by a brief summary of the most important properties of functions of a complex variable. When reading this introduction into the relevant mathematics, the reader needs to keep in mind that this is not a mathematics essay, but a demonstration how beautiful and powerful, but not always appreciated, mathematics can be if used by a physicist or engineer to solve some real life problems. High Energy Charged Particle Storage Rings High energy in this context means that the particles move with a velocity that is very close to the velocity of light. Storing particles at that velocity for something like ten hours means that they travel a distance of the order of the diameter of the orbit of the planet farthest from the sun, Pluto. This means that the trajectories of the particles must be bent so that they follow a closed path, and that the beam needs to be focused and refocused all the time in order to assure a long life time of the beam in the storage ring. If one exposes a charged particle that moves with the veloci ty v through a magnetic field B in the direction perpendicular to its traj ectory, it experiences a force that is equal in strength to that caused by an electric field of strength E=vB. This means that in order to generate the same force on a high energy particle as that produced by a magnetic field of the order of one Tesla g one would need to apply an electric field of the order of 3*10 Vim. Since such large DC fields can not be 3 generated without electrical breakdown, one uses exclusively magnetic fields to focus and bend (into a closed trajectory) high energy charged particle beams. In order to focus a high energy charged particle, it is desirable for a number of reasons to provide a restoring (i. e. focusing) force that is proportional to the distance of the particle from the desired orbit, the same as in a harmonic oscillator. If a particle with charge e travels in the z-direction through a magnet that provides a magnetic field By (x, y, z), that field component causes the x-component of the momentum to change by l:.pr=eJ By(X,y,Z)"Vdt=eJ By (x,y,z)dz (1) From this equation it is clear that one is mostly interested in the integral of fields over the length of magnets, and that one wants those integrals to be linear functions of the cartesian coordinates for the magnets that are used to focus the particles. For that reason we study now the mathematical properties of the integrals over three-dimensional vacuum fields taken over the whole length of a magnet, i.e. from the field-free region on one side of the magnet to the field-free region on the other side of the magnet. Applying this integration to the two magnetostatic equations leads (because integration and differentiation commute) to with the integral over the fields indicated by underlining the letter B. These are clearly the same differential equations that are valid for two-dimensional fields, and I will be dealing exclusively with such fields from now on, making the use of the underlining unnecessary. An additional change in notation is convenient because of the two-dimensionality of all subsequent equations: from now on, z is defined by z=x+iy, with i 2 =-1. If one writes down the equation for the change in the y component of the momentum due to B x ' one finds that if the field is focusing the particles in one direction, it always de focuses them in the other direction, seemingly making focusing in both directions impossible. However It is clear that under these conditions all the standard rules of differentiation are valid. Therefore lines 3 and 4 (and the specific example in line 8) are straight forward, as are the consequences, line 5 (the Cauchy-Riemann-condition) and line 6, the latter expressing that both the real and the imaginary part of a function of a complex variable satisfy the Laplace equation. While we often use functions whose real and imaginary parts are non-geometric quanti ties, like the x and y components of fields, or vector and scalar potentials, a geometric interpretation is also possible and useful. The function w(z)=u+iv can be used to map the x-y plane onto the u-v plane. If w(z) is an analytical function as defined above, line 7 shows that the angle between two intersecting lines in the x-y plane is the same as the angle between their maps in the u-v plane (provided neither of the derivatives WI or Zl are zero), i.e. the map is conformal. Line 9 states, without proof, how one can calculate from the knowledge of the real or imaginary part of an analytical function of z on the circumference of a circle the value of the complete function in the interior of the circle. This solution to the Dirichlet problem in a circle is valid only if the function has no singUlarity inside the circle. switching now from mathematics to physics, line 10 shows the relationships between the x-and y-components of the magnetic field in vacuum on one hand, and the component A of the vector potential that is perpendicular to the x-y plane, and the scalar potential V on the other hand. Having fortui tously chosen the appropriate notation in the mathematics part, comparison with line 5 shows that A and V can be considered as real and imaginary part of the analytical function F (z), customarily called the complex potential. From line 10 follow directly lines 11 and 12. Since the result that it is Hx-iH y and not Hx+iH y that is an analytical function of z has been derived here ln a rather abstract way, lines 13 and 14 show how one can come to the same resul t in a more direct and elementary way. Line 15 shows the complex potential and its constituent parts for the case of the Ubiquitous quadrupole, the magnet used for linear focusing in accelerators. Also shown are the scalar potential surfaces that 5 will produce such a field distribution. Design of non-dipole magnets in dipole geometry If one were to apply an arbitrary coordinate transformation u(x,y), v(x,y) to the geometry of the field-producing entities (potential surfaces and/or current filament locations), one would get a new geometry of the field-producing entities, but one would also have to solve transformed magnetostatic differential equations that, in that new geometry, would look very different from the equations shown in In order to make proper design decisions in dipole geometry, 6 one needs to know the relationship between (non-ideal) fields, and field errors, in the w-and z-planes. From the equations in It should be map and has follows for ideal field, two planes: noted that w'=dw/dz comes from the chosen conformal nothing to do with the actual fields. From equ. 6 the field errors I i . e. actual deviations from the that the relative field errors are identical in the 2) Using w(z), map, only in the above established region of the xy-plane, both the boundary of the allowed region and the good field region(s) from the z-plane into the w-plane. The result of those mappings are given for the sextupole in 3) Design the polefaces of the dipole in w-geometry. These poles are I for our specific example, also shown in One of the most interesting insights obtained from the design of the example-sextupole in w geometry is the great difference in the size of the two good field regions that are not that much different in size in the z-plane, thus showing that a much greater effort is required to get a good field in the larger good field region than it is to achieve the same field quality in the smaller good field region. For some applications it is advantageous to use sextupoles that have a field that increases less strongly than an ideal sextupole on the x-axis, and