Open midplane designs based on sector coils in superconducting dipole magnets

Department of Physics, Chemistry and Biology

Master’s Thesis

Open midplane designs based on sector coils in

superconducting dipole magnets

Jens Bruér

LITH-IFM-A-EX--08/2008--SE

Department of Physics, Chemistry and Biology Linköpings universitet

iii

Masters’s Thesis

LITH-IFM-A-EX--08/2008--SE

Open midplane designs based on sector coils

in superconducting dipole magnets

Jens Bruér

Supervisor: Ezio Todesco

AT

department,

CERN

Examiner: Peter Münger

v Avdelning, institution

Division, Department

Division of Theoretical Physics

Department of Physics, Chemistry and Biology Linköping University

SE – 581 83 Linköping, Sweden

URL för elektronisk version

http://urn.kb.se/resolve?urn=urn:nbn:s e:liu:diva-12161

ISBN

ISRN: LITH-IFM-A-EX--08/2008--SE _________________________________________________________________ Serietitel och serienummer ISSN

Title of series, numbering ______________________________ Språk Language Svenska/Swedish Engelska/English ________________ Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport _____________

Titel Open midplane designs based on sector coils in superconducting dipole magnets

Title

Författare Jens Bruér

Author

Nyckelord

Keyword

Open midplane designs, superconducting dipoles, magnetic field quality, CERN, LHC

Sammanfattning

Abstract

For some cases of lattice layout in particle accelerators, the major part of the energy deposition coming from the collision is located in the midplane of the magnets. The heat produced might result in a quench in superconducting magnets. One solution for reducing the energy deposition in the coil is to introduce an opening in the midplane, which will lead away most of the particles to a safe zone instead of hitting the superconductors in the magnets.

The aim of this work is to optimize the field quality in dipoles based on the cosθ-design, where an opening in the midplane has been inserted. The equations for finding the solutions for the coil layout for different sizes of the opening are studied, and the solution giving the best field quality for each case is presented. Then, optimization procedures are applied to lay-outs with Rutherford cables. Finally, the resulting field strength from the solutions obtained is presented.

Datum Date 2008-06-02

vii

Abstract

For some cases of lattice layout in particle accelerators, the major part of the energy deposition coming from the collision is located in the midplane of the magnets. The heat produced might result in a quench in superconducting magnets. One solution for reducing the energy deposition in the coil is to introduce an opening in the midplane, which will lead away most of the particles to a safe zone instead of hitting the superconductors in the magnets.

The aim of this work is to optimize the field quality in dipoles based on the cosθ-design, where an opening in the midplane has been inserted. The equations for finding the solutions for the coil layout for different sizes of the opening are studied, and the solution giving the best field quality for each case is presented. Then, optimization procedures are applied to lay-outs with Rutherford cables. Finally, the resulting field strength from the solutions obtained is presented.

ix

Acknowledgement

First and foremost I would like to thank my supervisor at CERN, Ezio Todesco, for the help and guidance during this entire project, providing clear direction of work but also giving me a lot of space for creativity. I would like to thank my examiner Peter Münger at Linköping University, for giving me feedback and taking the time to examine this work. I would also like to thank my college, Franck Borgnolutti, for valuable discussions and for always keeping me in a good mood.

Finally, I would like to thank my flatmates at Rue du Vuache and all my friends in Geneva for making this time a fantastic experience. Thanks also to my family for their never ending support and love. Thank you all!

Jens Bruér

xi

Content

1 Introduction... 1

2 CERN and the LHC project... 3

2.1 Subatomic Physics and the Standard Model... 3

2.2 CERN... 3 2.3 Particle accelerators ... 4 2.4 The LHC project ... 5 2.5 Superconductivity ... 7 2.5.1 Superconducting materials... 7 2.5.2 Superconducting cables ... 8

2.6 The LHC dipole magnets ... 9

2.6.1 The D1 dipole ... 11

2.7 The need of open midplane dipoles in the LHC and beta beams... 11

3 Field quality in superconducting dipoles ... 15

3.1 Different types of magnetic fields... 15

3.2 Mathematical derivation of the multipoles ... 16

3.3 The open midplane... 22

3.3.1 Analytical solution for a one sector layout ... 22

3.3.2 Two sectors ... 24

3.3.3 Three sectors ... 24

3.3.4 The general case... 25

3.4 Evaluation of the short sample field ... 25

4 Theoretical solutions for the coil layout... 31

4.1 Estimation of the multipoles ... 31

4.2 Solutions for different layouts of sector coils ... 32

4.2.1 Different computational methods and computational times ... 32

4.2.2 One sector coil ... 33

4.2.3 Two sector coil... 33

4.2.4 Three sector coil... 38

4.2.5 Four sector coil ... 42

4.3 The short sample field estimation for open midplane layouts ... 49

5 Coil layouts with Rutherford cable ... 51

5.1 One layer, three sector coil ... 52

5.1.1 Five degree opening... 52

5.1.2 Ten degree opening... 57

5.1.3 No opening... 59

5.2 One layer, four sector coil... 61

5.2.1 Five degree opening... 62

5.2.2 Ten degree opening... 69

5.2.3 No opening... 71

5.2.4 The D1 design... 72

5.3 Two layer, five sector coil ... 78

5.4 Short sample field comparison... 81 6 Conclusions... 85 A. Mathematical derivation of the cancellation of the skew and even number

normal multipoles ... 87 B. Calculation of the coil integral ... 88 Bibliography ... 89

xiii

List of Figures

Figure 2.1. Schematic layout of the linear accelerator, showing the drift tubes and the potential gaps between them... 5 Figure 2.2. The LHC with its four detectors, located at each interaction point... 6 Figure 2.3. The two beam paths and the four interaction points... 6 Figure 2.4. Left: The resistance of mercury drops to zero at 4.2 K. Right: The critical surface of a superconducting material. ... 7 Figure 2.5: The critical surfaces of the two alloys niobium-titanium and niobium-tin [6]. 8 Figure 2.6. Left: Cross-section of a strand. Right: Bundles of filaments of

superconducting material in a strand. ... 8 Figure 2.7. Rutherford cable with a keystone angle. The left side is the inner part of the cable; it is slightly thinner than the outer part on the right hand side... 9 Figure 2.8. Cross-section of the LHC main dipole magnet. ... 10 Figure 2.9. Schematic sketch of aiming the two beams together. The interaction point is to the far left of the figure. The magnets denoted with D are dipoles and the magnets denoted with Q are quadrupoles. ... 11 Figure 2.10. A cross-section of a sector coil dipole magnet with an open midplane. ... 12 Figure 2.11. Left: Location of the spray particles (child beams) in a beta beams dipole. Right: Heat deposition in a beta beam dipole, having its peak in the midplane. ... 12 Figure 2.12. Gupta’s block design of an open midplane solution. The figure show the conductors (red blocks) in the first quadrant. ... 13 Figure 3.1. Left: A dipole field. Centre: A quadrupole field. Right: A sextupole field. 15 Figure 3.2. Magnetic field from two line conductors, producing an upward field in the centre... 16 Figure 3.3. Line conductors arranged in a four-fold symmetry... 17 Figure 3.4. The schematic sketch of a dipole coil; r is the apertures radius, w is the width of the sector coil and θ1 is the azimuthal angle... 19

Figure 3.5. A two sector dipole coil... 20 Figure 3.6. A three sector dipole coil... 21 Figure 3.7. A one sector coil with an open midplane. The angle θ0 denotes the size of the opening... 23 Figure 3.8. Example of a cross section of the critical surface of Nb-Ti, at 1.9 K. The two lines indicate the current density for the peak field Bp, called the loadline, and the critical

current jc for different values of the magnetic field. ... 26

Figure 3.9. The ratio between the peak field and central field λ as a function of the radial width w... 28 Figure 3.10. Short sample field numerically evaluated for different coil layout, based on Nb-Ti and Nb3Sn... 29 Figure 3.11. The short sample field versus the equivalent width for coils with different angular width Y. Note that the 50 and 60 degrees curves overlap. ... 30 Figure 4.1. Plot over the solutions for a one layer, two sectors coil that set B3 and B5 to

zero. The cross-dotted line denoted G7 is proportional to B7... 34

Figure 4.2. The Total error of G3 G5 and G7 plotted versus θ3, for the two sectors, no

Figure 4.3. Plot over the solutions that set B3 and B5 to zero, for a two sector coil in the

case of a 5 degree opening. ... 36

Figure 4.4. The solutions for the case of a two sector coil with a 10 degree opening... 37

Figure 4.5. The total angular thickness of two sector coils with different values of θ0; from the case of a no opening to the case with a 18 degree opening... 38

Figure 4.6. The solutions that set B3,B5,B7 andB9 to zero, for a three sector layout with no opening. The cross-dotted line is proportional to B11. ... 39

Figure 4.7. A three sector layout with a five degree opening. ... 40

Figure 4.8. Three sectors with a ten degree opening. ... 41

Figure 4.9. The total angular thickness for three sectors, with open midplanes spanning from 0 to 16 degrees. ... 42

Figure 4.10. Parametric study over the four sector case with no opening. The Newton method has been applied to each set of angles, with θ7 and θ0 kept constant... 43

Figure 4.11. Four sectors, no opening with the brute force method. The cross dotted line represent the Total error consisting of G3 to G11... 45

Figure 4.12. Four sectors, five degree opening with the Newton method. ... 46

Figure 4.13. Four sectors with a five degree opening, using the brute force method... 47

Figure 4.14. Four sectors, ten degree opening with the Newton method. ... 48

Figure 4.15. Four sectors, ten degree opening using the brute force method. ... 49

Figure 4.16. The short sample field versus the equivalent width, calculated for openings spanning from zero to ten degrees. ... 50

Figure 5.1. Three sector layout with Rutherford cable, showing the case of the five degree opening. The solid red lines show the theoretical solution. The LHC MB Inner cable is used with an apertures radius of 65 mm. The dot in the second sector indicates where the peak field is located... 53

Figure 5.2. The positioning and inclination angle φ and α, used to specify the position of the sectors when working with Rutherford cable. ... 54

Figure 5.3. Three sector layout with a five degree opening, using the RHIC MB cable, with an aperture radius of 65 mm. ... 56

Figure 5.4. Three sector layout with a ten degree opening and a 65 mm aperture radius. ... 58

Figure 5.5. Starting position of a three sector, no open midplane layout, with LHC MB Inner cable and a 65 mm radius. ... 59

Figure 5.6. Cross-section of a three sector coil that sets all five multipoles to zero. LHC MB Inner cable and a 65 mm aperture radius is used... 61

Figure 5.7. Four sectors with a five degree opening, using LHC MB Inner cable and an aperture radius of 65 mm, case one. ... 62

Figure 5.8. Starting position of the four sectors, five degree opening layout, case 2, with LHC MB Inner cable and a radius of 65 mm... 64

Figure 5.9. The layout of a solution that cancels out the first five multipoles, from the case two configuration. ... 65

Figure 5.10. Layout with LHC MB Inner cable with an aperture radius of 60 mm, for the four sectors, five degree opening case. ... 66

Figure 5.11. Cross section of a coil layout with four sectors and a five degree opening. The first five multipoles are canceled out by this configuration (the exact multipoles and angles are shown on line 4 in table 5.13)... 68

xv Figure 5.12. Four sector, ten degree opening layout with LHC MB Inner cable and 60 mm radius... 70 Figure 5.13. A four sector layout that set all five multipoles to zero for the ten degree opening case. The real solution moved quite far outside the theoretical one. ... 71 Figure 5.14. The second version of the first design of the D1 magnet. The peak field is located on the third sector. ... 73 Figure 5.15. Printout from Roxie, showing the resulting multipoles from the coil

configuration of table 5.20, simulated with an iron yoke. The central field is 5 T... 74 Figure 5.16. D1 1st design layout, third version, adjusted to take the iron contribution into account. All multipoles are set to zero. The peak field is located on the last sector. ... 75 Figure 5.17. Multipoles and nominal field for the 2nd design of D1, calculated with an iron yoke. ... 76 Figure 5.18. The whole coil layout for the 2nd D1 design, with no opening and 65 mm radius... 76 Figure 5.19. The multipoles and the nominal field, calculated with the iron yoke

contribution for the five degree opening design to D1. ... 77 Figure 5.20. Cross section of the coil layout for the open midplane solution to D1. ... 78 Figure 5.21. The starting position for a two layer, five sector coil for the ten degree opening case. The aperture radius is 50 mm and LHC MB Inner cable is used... 79 Figure 5.22. The short sample fields for all solutions found with an aperture radius of 40 mm. The comparison is made with the radial coil width. The line is the analytical

solution of a no open midplane. The two layer solutions all has a ten degree opening. All other solutions are based on one layer, four sector coils. ... 82 Figure 5.23. The short sample fields for solutions with 40 mm aperture radius, here compared to the area of the coil, expressed in the equivalent width. ... 82 Figure 5.24. Short sample field for all solutions with aperture radius of 50 mm compared to the radial coil width. The two layer solutions all has a ten degree opening. All other solutions are based on one layer, four sector coils... 83 Figure 5.25. Short sample field for all solutions with aperture radius of 50 mm,

compared to the coil area, expressed in the equivalent width... 83 Figure 5.26. Short sample fields versus the coil width, with an aperture radius of 60 mm, for solutions based on different sizes of the opening and different types of cables. The two layer solutions all has a ten degree opening. All other solutions are based on one layer, four sector coils... 84 Figure 5.27. Short sample fields versus the equivalent width, with an aperture radius of 60 mm, for solutions with different openings and cables. ... 84

List of Tables

Table 2.1. Main parameters of the LHC MB dipole. ... 10 Table 4.1. Values for the function fn for different aperture radiuses. The aperture radius

of 28 mm is equivalent to the LHC MB dipole. ... 32 Table 5.1. Physical properties for commonly used superconducting cables. The n1 and n2 indicate the number of layer of strands in each direction in the cable, corresponding to the width and thickness respectively, as shown in figure 2.7. The width and thickness are given for insulated cable. ... 51 Table 5.2. Multipoles for different configuration of the layout in figure 4.11, using LHB MB Inner cable. The second line shows the best solution with no restriction, the third line shows the best solution under the restriction that b7, b9 and b11 stay within 1 unit. .. 54

Table 5.3. Sensitivity matrix over the multipoles versus the angles. The starting position is the configuration from the first line in table 5.2. ... 55 Table 5.4. Multipoles for different configuration of three sector coil layouts, made up of LHC MB Inner cable. Ncon represent the number of conductors in sector 1, 2 and 3 respectively. ... 56 Table 5.5. Multipoles from two layouts of a three sector coil, five degree opening, made up of RHIC MB cables. The first line represents the layout shown in figure 5.3... 57 Table 5.6. Multipoles from two different configurations of a three sector coil with a ten degree open midplane, made up of LHC MB Inner cable. ... 58 Table 5.7. Optimization procedure for the three sector, no open midplane layout shown in figure 5.5... 60 Table 5.8. Optimization procedure for a slightly modified layout than the one shown in figure 5.5. One conductor has been added to the first sector and one has been taken away from the second, resulting in a better starting position... 60 Table 5.9. The angle configuration and resulting multipoles from the four sector, five degree opening layout shown in figure 5.7. The starting position is displayed on the first line, while line 2 and 3 show different optimization attempts of this position... 63 Table 5.10. Sensitivity matrix of the starting position of the four sectors, five degree opening layout showed in figure 5.7... 63 Table 5.11. Optimization procedure of the multipoles for the four sector coil with a five degree opening, shown in figure 5.8... 64 Table 5.12. The optimization procedure from the starting point showed in figure 5.10. The Ncon, number of conductors, is 8,4,3,3 for the entire table... 66 Table 5.13. The optimization procedure for a layout with one conductor added to the second sector in figure 5.10. Thus, Ncon is 8,5,3,3 for the entire table... 67 Table 5.14. Solutions of minimizing the five first multipoles, for different types of cables and aperture radiuses... 69 Table 5.15. Optimization procedure for a four sector, ten degree opening layout with LHC MB Inner cable and an aperture radius of 60 mm. ... 70 Table 5.16. Good solutions for different types of cables and aperture radiuses, for the four sectors, ten degree opening case. ... 71 Table 5.17. Different theoretical solutions for the four sectors, no open midplane case. 72

xvii Table 5.18. Different solutions that setting the first five multipoles to zero, for the four sector, no open midplane case. The case column indicates which theoretical solution each configuration is based upon... 72 Table 5.19. The configuration and multipoles of the third version of the D1 design, calculated without an iron yoke. The cable used is LHC MB Inner... 73 Table 5.20. The configuration of the first design of D1, third version, after the fine tuning in Roxie... 74 Table 5.21. Angle configuration and multipoles for the 2nd _{D1 design, without an iron} yoke... 75 Table 5.22. The configuration of the 2nd design of D1, after the fine tuning in Roxie.... 75 Table 5.23. Angle configuration and multipoles for the D1 open midplane design,

without an iron yoke. ... 77 Table 5.24. Optimization procedure of the two layer, five sector layout for the ten degree opening case, with LHC MB Inner cable and an aperture radius of 50 mm... 80 Table 5.25. Sensitivity matrix for the starting position shown in figure 5.21. ... 80 Table 5.26. Good solutions for the two layer, five sector case, with a ten degree opening.

1 Introduction

Mankind has always been striving to get an understanding of the world we live in. Theories of how the universe works have emerged over the years, some being rejected when contradicting new experiments, while some being accurate on some scales but needing complements on other scales. The current theory of how the subatomic world works is called “the Standard Model”. So far experiments have been in accordance with the Standard Model. The model will however be tested at previously unreached energies when the LHC (Large Hadron Collider) start commissioning, CERN’s new particle accelerator. The LHC will be the largest and strongest accelerator ever built, and its collision energies will enable scientist to look deeper into matter than ever before.

The LHC is a circular collider, and to guide the particles into a circular path strong superconducting dipole magnets are needed. Dipole magnets are also needed to aim the beams toward each other before the collision points. The collision produce a lot of spray particles, and a big amount of these will proceed after the collision and fly into the dipolar magnets that aim the two beams toward each other. In the LHC optics, the spray on these magnets is not uniform over the azimuthal angle, but rather concentrated on the horizontal and vertical midplane. This is due to the presence of the final focus quadrupole between the dipole and the interaction point. The impact from the spray particles will produce heat in the magnets, and in superconducting magnets this heat contribution might be enough to disable the superconducting property of the magnet, thus vastly reducing the magnetic field. Therefore, the dipole magnets in the LHC closest to the collision point, called the insertion magnets, are normal conducting and hence unaffected by the heat contribution. Normal magnets are not as powerful as their superconducting counterpart, thus superconducting magnets are the preferred choice. In one of the upgrade plans for the LHC, the normal conducting insertion magnets are replaced by superconducting magnets. To be able to do this, the problem with the heat deposition has to be solved. One way to do this is to avoid the concentration of the heat deposition by open up the midplane of the magnet, producing a gap, and lead the particles into absorbers instead of hitting the magnets. This design is called the open midplane design.

The homogeneity of the magnetic field is called field quality. A homogenous field is necessary to correctly steer the beam. The conventional accelerator dipole magnet, used in all accelerators and also in the upcoming LHC, is a superconducting arc shaped magnet with a closed midplane. For these conventional magnets, the optimization of the field quality is quite thoroughly investigated and designs producing high field quality are implemented in the LHC. For open midplane designs on the other hand, only a limited amount of work has been carried out in the past [10, 11]. This work will investigate the analytical approach to open midplane designs, based on arc shaped sector coils, commonly called the cosθ-design in the accelerator magnet community. The aim is to maintain high field quality while opening up the midplane, and also investigate how the field strength is affected by this action.

This work is made up of four major chapters. In the second chapter, the background of particle physics, CERN, the LHC and superconductivity is given, as well

2 as why the open midplane solution is needed. In the third chapter, the mathematical foundation to find good field quality is derived. In the fourth chapter, the theoretical solutions of these formulas are presented, obtained by numerical methods. These solutions describe how the block of conductors, called sectors, in the coil should be positioned in order to achieve the best field quality for a certain number of sectors. In the fifth chapter, these theoretical solutions are used as starting positions when finding solutions with Rutherford cable. In this chapter, simulation software is used, employing a more exact calculation method than in the previous chapters, making the calculated field quality to be in good accordance to reality. This however causes the optimal solutions to move a bit away from the solutions found in chapter four, why optimizations are needed. This chapter also demonstrates the optimization procedure for several cases, and all good solutions found are presented. The resulting field strengths for these solutions are presented in the end of the chapter. Finally, a short summary and conclusion of the work are given in chapter six.

2 CERN and the LHC project

2.1 Subatomic Physics and the Standard Model

Our understanding of the smallest constituents of matter has dramatically improved over the last century. Theories and experimental results from thousands of physicists have accumulated to a theory today known as “the Standard Model”. According to the theory, matter is made up of twelve fundamental building blocks, divided into two subgroups; quarks and leptons. The quarks build up heavier particles like protons and neutrons, while the lighter electron belongs to the lepton group.

Between the matter particles, i.e. the quarks and the leptons, there are force particles to intermediate the force they exert on each other. There are four fundamental forces that govern the universe; the strong force, the weak force, the electromagnetic force and gravitation. The strong and the weak force are nucleus forces and are only at work at very small distances comparable to the size of an atom, whereas the ranges of the electromagnetic force and gravitation are infinite.

The Standard Model accounts for three of the four fundamental forces. The strong force, the weak force and the electromagnetic force are intermediated by the force particles; gluons, W and Z bosons and photons respectively. But the explanation for gravitation falls short in the Standard Model. The corresponding force particle for gravitation has been named the graviton, although so far it is only hypothetical and has never been observed.

Another unsolved question is why particles have the mass they possess, and why some particles do not have any mass at all. The Standard Model predicts a particle called the Higgs boson, giving mass to all other mass particles. The Higgs boson has never been observed, but is indeed a fundamental part of the Standard Model. The detection of the Higgs boson would be a strong indication that the Standard Model is correct and that we are on the right path. The Higgs boson has such a fundamental importance it has been given various popular nicknames, such as “the holy grail of physics”, and “the God particle”. If the Higgs boson exists, the Standard Model predicts that it has a considerably mass, and so far scientists have not been able to produce the energy required to form this particle. See [1] for more information.

2.2 CERN

CERN is the largest particle physics research centre in the world, situated just outside Geneva, on the border between France and Switzerland. Founded in 1954, CERN is a collaboration between 20 European member states and several observer states. The name CERN is an acronym for the French name Conseil Européen pour la Recherche

Nucléaire, which stands for European Organization for Nuclear Research [2]. The

4 of disposal is large particle accelerators, which make it possible to look deep into matter. With the particle accelerator Super Proton Synchrotron (SPS), CERN was able to discover the W and Z bosons in 1983. Measurements done at the biggest electron-positron collider ever built, called LEP, helped scientist to prove that there was only three generations of particles of matter, an important ingredient of the Standard Model. LEP is now dismantled, to give room for the new, much more powerful accelerator, called LHC, which is expected to start operating in the summer of 2008.

2.3 Particle accelerators

In order to study the world of subatomic particles, high energies are needed to form them. The high energies come mostly from the kinetic energies of particles. To achieve high kinetic energies, charged particles are accelerated with the aid of potential gaps. The energy gained is the product of the voltage of the potential gap and the charge of the particle. For instance, if an electron, with the fundamental charge e, is passed over a potential gap of one volt, the energy has increased by one electron volt (eV), which is the standard unit of energy in particle physics. To convert this unit to the SI unit Joule, one multiplies with the electron’s charge.

The machines which employ this principle to accelerate particles are called particle accelerators. There are two fundamental shapes of these machines; linear accelerators and circular accelerators. In linear accelerators, multiple potential gaps are arranged in a series in order to accelerate the particles to higher and higher energies. Between the gaps are so called drift tubes, where the particles are shielded from the electromagnetic forces and thus no acceleration occurs. The voltage difference is alternating between the drift tubes, see figure 2.1, where a positively charged proton is accelerated in a linear accelerator. When the proton emerges from the first drift tube it is repelled from the positive charge behind it, and attracted to the negative charge in front of it, and is therefore accelerated. When the proton enters the second drift tube, the charges need to switch sign on all places, otherwise the particle would be slowed down instead of accelerated when it emerges from the second tube. By alternating the charges with a sinusoidal wave, the particle can always have a positive charge behind it and a negative in front of it, riding on a ‘charge wave’. Since the particle gain speed for each potential gap it passes, the drift tube must be longer and longer in order for the particle to stay in phase with the wave.

Figure 2.1. Schematic layout of the linear accelerator, showing the drift tubes and the potential gaps between them.

Linear accelerators have an obvious limitation; when the particle comes to the end of the accelerator, the acceleration stops. This problem is avoided with circular accelerators. For more information on particle accelerators, see [14, 15].

2.4 The LHC project

LHC, or the Large Hadron Collider, is a large circular particle accelerator currently in its final phase of completion at CERN. In fact, when finished, it will be the world’s largest accelerator with the highest collision energies ever achieved.

The high energies are needed to look into the finest fabric of matter. Particles are accelerated to almost the speed of light, and are forced to collide with particles in the opposite direction. LHC is mainly designed for accelerating protons. The total energy of the head to head proton collision is 14 TeV. In this energy range, the proton nucleus is ripped apart into its subatomic particles. The higher the energy of the collision, the heavier subatomic particles can be formed. One of the major goals of the LHC project is to find the Higgs boson. The collision energy of 14 TeV exceeds by far the predicted mass of the Higgs boson, meaning that LHC is expected to detect the particle if it exists.

The LHC have a circumference of 27 km and is buried about 100 m underground, in the tunnel that previously housed LEP. Because of the ring shape of the collider, the particle beams must be bended to follow the circular path. This is achieved through superconducting dipole magnets. Since the two particle beams have the same charge but different direction, the beams must be separated and have their own set of dipoles, with opposite direction of the magnetic field. The beams are confined within two vacuum tubes, running parallel with each other along the machine. At four places the two beam paths cross, thus forcing the particles to collide; these are referred to as the interaction points, see figure 2.3. The need of linear stability of the beam requires another type of magnets called quadrupoles. The quadrupoles focus the beam and is the magnetic equivalent to an optical lens.

6

Figure 2.2. The LHC with its four detectors, located at each interaction point.

Figure 2.3. The two beam paths and the four interaction points.

The machine is based on superconductor technology, meaning it has to be cooled down to operate; it has an operating temperature of 1.9 K (-271 C). The superconducting

technology is chosen because of its superior energy efficiency and the strong magnetic field it can generate without heat generation.

2.5 Superconductivity

Superconductivity is a low temperature phenomenon that was discovered by the Dutch physicist H. Kamerlingh in 1911. The main characteristic property is that, below a certain critical temperature, Tc, the electrical resistance suddenly drops to zero. Two other parameters that restrict this remarkable phenomenon is the critical magnetic field,

Bc and the critical current density, Jc. The critical temperature, the critical field and the critical current density defines what is called the critical surface. Below the critical surface is the superconducting phase, where zero resistance (and other extraordinary effects) takes place. Above the critical surface is the normal phase, where the superconductivity vanishes.

Not every material has a superconducting phase. A remarkable fact is that some of the best normal conductors, like copper, silver and gold, are not superconducting materials.

Figure 2.4. Left: The resistance of mercury drops to zero at 4.2 K. Right: The critical surface of a superconducting material.

The critical surface depends on the superconducting material and it defines the properties of the superconductor. In general, the larger the critical surface, the better is the superconductor. The best superconductors tend to be alloys. Usually one has to compromise between factors such as resistance to an external magnetic field, the critical current, the critical temperature, the price of the raw material, how easy it is to manufacture the superconductor, the strength and flexibility of the material, etc. More information on superconductivity can be found in [5].

2.5.1 Superconducting materials

A wide range of materials enter the superconducting phase at the temperature scale of 10 K, however, only two are so far commercially available for large scale magnet production;

8 the alloys niobium-titanium, Nb-Ti, and niobium-tin, Nb3Sn. Below, the critical surfaces of Nb-Ti and Nb3Sn are shown.

Figure 2.5: The critical surfaces of the two alloys niobium-titanium and niobium-tin [6].

Despite the fact that Nb3Sn is clearly better than Nb-Ti for every one of the three critical parameters, Nb-Ti is still more commonly used because of its outstanding ductility, a very important property for cable fabrication. Nb3Sn, on the other hand, is a brittle material which makes cable production more difficult.

2.5.2 Superconducting cables

Superconducting magnets are electromagnets, thus the magnetic field is produced by leading strong currents through a superconducting cable. The LHC superconducting magnets are based on Nb-Ti cables. The cable is not made of pure superconducting material, but of a mixture of copper and Nb-Ti. The Nb-Ti is embedded in copper matrixes, in very thin filaments, with diameters in the μm scale. The copper matrixes are called strands and are around 1 mm in diameter, see figure 2.6.

Figure 2.6. Left: Cross-section of a strand. Right: Bundles of filaments of superconducting material in a strand. 20 15 10 15 10 20 5 103 5 104 105 106 107 temperature (K) current density (A/cm )2 Nb Sn3 Nb-Ti magnetic field (T) critical J-H-T surface

Superconducting magnets are usually based on Rutherford cable, which typically has 20-40 strands wired together, in a rectangular shape with a small keystone angle. The keystone angle is defined by the inner and outer thickness of the cable, which differs slightly; meaning that the cable is not perfectly rectangular, see figure 2.7. The keystone angle is present in order to make the coil shape circular, when multiple cables are stacked upon each other. For more information on the Rutherford cable used in the LHC, see [4].

Figure 2.7. Rutherford cable with a keystone angle. The left side is the inner part of the cable; it is slightly thinner than the outer part on the right hand side.

The embedding of the superconductor in copper is done to avoid instabilities called flux jumps and to provide both mechanical stability and a heat sink. If the superconducting cable experiences a heat pulse, which heats it up above the critical surface, it will leave the superconducting phase and enter the normal phase. Nb-Ti has a fairly high resistance in the normal phase, and the Joule effect will add more and more heat to the process, heating up the strand even more. The accumulated heat will cause the neighboring strands to heat up and leave the superconducting phase as well. This will cause a chain reaction and soon the whole cable will be out of the superconducting phase. This event is called a quench, and limits the performance on the magnet.

When one strand goes onto the normal phase, the copper matrixes, with low resistance at any temperature, takes over the current from the Nb-Ti, relieving it from the heat development. This enables the superconducting material to cool down and go back to the superconducting phase. If the initial heat pulse is not to large, this mechanism can prevent a quench.

2.6 The LHC dipole magnets

The majority of the magnets in the LHC are the main bending (MB) dipole magnets, with a total number of 1232. Their purpose is to bend the beam into a circular path. The LHC MB is based on a twin-aperture design, meaning the two beam pipes are confined within the same iron yoke and cryostat, see figure 2.2. The iron yoke will enhance the magnetic field in the beam tube and at the same time isolate the field from the outside of the magnet. Data for the MB are given in table 2.1. See [3] for more information.

10

Parameter Value Unit

Injection field 0.54 T

Current at injection field 763 A Nominal field (7 TeV beam energy) 8.33 T Current at nominal field 11850 A

Ultimate field 9 T

Operating temperature 1.9 K

Cold mass length at 293 K 15.18 m

Total mass ~27.5 t

Inner coil diameter at 293 K 56.00 mm Number of conductor blocks / pole 6

Number of turns / pole, inner layer 15 Number of turns / pole, outer, layer 25

Table 2.1. Main parameters of the LHC MB dipole.

The dipole magnet coils have the shape of two symmetric circular arcs, and the particle beam is located in the very centre of these two arcs, where the beam tube is indicated in figure 2.8. The strength of this central field is one of the major parameters of the performance of the dipole. Further investigations of the central field will be done in chapter 3.4.

2.6.1 The D1 dipole

Before each interaction point, there are straight sections where the aiming of the two beams is fine tuned in order to make the cross section of the interaction point as small as possible, causing more particles to collide. The fine tuning of the aiming is done with quadrupoles. Just before the interaction points, the two beams are in principle parallel, with a very small crossing angle. In the bending sections, the beams are separated by about 20 cm, see figure 2.8. To force the two beams together before each interaction point, two dipole magnets are used, one to bend the beams towards each other (D2), and one to straighten out their path when the beams are close enough to each other (D1), see figure 2.9. The figure shows the right side of the interaction point; the left side of the interaction point is the mirror image of this figure. This means that the D1 is the first dipole before and after each interaction point. The D2 is a twin aperture magnet while the D1 is a single aperture magnet, meaning the two beams share the same beam pipe. This is possible due to the different direction of the two beams, resulting in that they are separated in different directions after the interaction point, despite having the same electrical charge. The magnets in the straight sections before the interaction points are called insertion magnets [3].

Figure 2.9. Schematic sketch of aiming the two beams together. The interaction point is to the far left of the figure. The magnets denoted with D are dipoles and the magnets denoted with Q are quadrupoles.

2.7 The need of open midplane dipoles in the LHC and beta beams

The particle beam will be located at the very centre of the dipole, which has a circular shape. At the collision point, the proton nucleus are scattered into a wide range of subatomic particles. In addition to the decays of the subatomic particles of interest, there are also a lot of non interesting waste products after a collision. A substantial amount of spray particles will continue after the collision point and smash into the first dipole magnet, that is, the D1. This gives a heat contribution to the magnet, which could result in a quench. This is the reason why the D1 magnets currently installed in the LHC are normal conducting magnets. Superconducting magnets are preferable to normal conducting magnets due to higher fields; hence one upgrade for the LHC would be to replace the normal conducting D1 dipoles with superconducting ones. To do this the heat deposition imposing on the magnets has to be avoided in some way. See [8] for more information.

12 The heat deposition will mainly be concentrated to the midplane of the dipole, which is the crossing plane of the two beams. By introducing a gap in the magnet, located in the midplane (see figure 2.10), the majority of the waste will pass through the gap instead of hitting the dipole. In this way the quench can be avoided. This solution is also desired in the new beta beams decay ring, which is one of the future CERN projects. In this ring, the accelerated particles are unstable nuclei, which produce a continuous spray of decaying particles, concentrated in the midplane, see figure 2.11 [9]. Also in this case an open midplane can avoid the heat deposition in the coil.

-50 0 50 -50 0 50 + + -- _Midplane

Figure 2.10. A cross-section of a sector coil dipole magnet with an open midplane.

Figure 2.11. Left: Location of the spray particles (child beams) in a beta beams dipole. Right: Heat deposition in a beta beam dipole, having its peak in the midplane.

Earlier works on open midplane solutions exists, but are usually based on block design [10, 11], see figure 2.12. The conventional layout is the sector coil layout, which is the

arc shaped design used in currently operating accelerators (RHIC, Tevatron etc), but all these designs have closed midplanes.

The focus of this work is to investigate how the magnetic field quality and field strength are effected by opening up the midplane of dipoles based on sector coils, and try to optimize these quantities.

Figure 2.12. Gupta’s block design of an open midplane solution. The figure show the conductors (red blocks) in the first quadrant.

3 Field quality in superconducting dipoles

3.1 Different types of magnetic fields

The bending force of the dipole magnets is ruled by the famous equation

) (v B q

F= × , (3.1)

where F is the force, q is the charge of the particle, v is the velocity of the particle, and B is the magnetic field. The force is perpendicular to the velocity of the particle and the magnetic field. The particle path is of course always parallel to the circular arc of the collider. To produce an inward force, pointing straight to the centre of the circle, the magnetic field should be either pointed straight upward or straight downward, depending on the direction of the beam. A proton beam circulating clockwise requires a magnetic field pointing upward and for an anti clockwise proton beam a field pointing downward.

To get the best precision in driving particles, a purely homogenous dipole field is the ideal. A pure dipole field is created between a north pole and a south pole on a magnet, whereas two north poles and two south poles arranged in a symmetric manner so the same polarity are opposing each other creates a quadrupole, see figure 3.1.

Figure 3.1. Left: A dipole field. Centre: A quadrupole field. Right: A sextupole field.

As seen from the figure, these magnets produce fields with a certain symmetry. However, in general, magnetic fields are more complicated than those shown above and not necessary symmetric, with field lines pointing in arbitrary directions. But as we shall see, an arbitrary magnetic field can be built up by these pure magnetic fields, that is, have a dipole component, a quadrupole component and so forth. The pure magnetic fields are called multipoles.

In superconducting magnets, the magnetic field is induced through currents. The field lines generated by a straight conductor follow the Bio-Savart law:

3 2 r r dl I dB= ⋅ × π μ _(3.2)

16

From this equation we see that by placing two conductors on each side of a reference point with currents in opposite directions, we produce a magnetic field pointing upward in the centre, see figure 3.2. However, these two conductors will not produce a pure dipole field, as we soon will see.

Figure 3.2. Magnetic field from two line conductors, producing an upward field in the centre.

We will show in section 3.2 that a single line current produces a magnetic field consisting of components from all multipoles. As mentioned, a multipole is the generalization of a pure magnetic field, such as the dipole field, of a certain order n. In figure 3.1 we show examples of dipole, quadrupole and sextupole fields, corresponding to multipole order 1, 2 and 3 respectively. Since we want a homogenous field for the bending magnets, a pure dipole field is desired. To better handle the multipoles, also called the field harmonics, a mathematical description is needed.

3.2 Mathematical derivation of the multipoles

Following complex formalism, a line current at position z0 = x0 + iy0 produces a magnetic

field B(z) at position z = x + iy, which, according to Bio-Savart law, reads as:

) ( 2 ) ( 0 0 z z I z B − = π μ _(3.3)

Since B is analytical, we can expand the function in a Taylor series

0 1 1 1 0 0 0 1 1 0 0 0 _, 2 2 ) ( z z R z z R z I z z z I z B n n n n n < ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − =

∑

∑

∞ = − − ∞ = − π μ π μ _(3.4)

where R is a reference radius, typically chosen to 2/3 of the apertures radius; defined in figure 3.4. We denote the constants with:

+

1 0 0 0 2 − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = n n z R z I C π μ _{. (3.5)}

Since z0 is complex, Cn is also complex, and the series now reads

∑

∑

∞ = − ∞ = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 1 1 1 1 ) ( ) ( n n n n n n n R z iA B R z C z B . (3.6)

Magnetic fields given by current lines are additive; therefore, if we place the conductors in a four-fold symmetry according to figure 3.3, all An, called the skew multipoles, vanish,

and also all Bn, called the normal multipoles, with even n disappear.

+ + _ _ +I +I -I -I θ0 ρ0

Figure 3.3. Line conductors arranged in a four-fold symmetry.

For a mathematical derivation of this, see Appendix A. Hence, after this arrangement, what remains of our series is

∑

∞ = − − ⎟ + ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 1 4 5 2 3 1 ) 1 ( 2 1 2 ) ( n n n R z B R z B B R z B z B _K (3.7)

where the first coefficient reads

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = 0 0 1 1 Re 2 4 z I B π μ _{, (3.8)}

18 and in general ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = − n n n z R I B 0 1 0 _Re 1 2 4 π μ _{. (3.9)}

We see that the first component in the series (3.7) is constant while the other ones have a

z-dependence. If we had B(z) = B1, we would have a homogenous field. From this we

draw the conclusion that B1 corresponds to a pure dipolar field. In a similar way, B2

corresponds to a pure quadrupole field, B3 to a sextupole field, and so on.

Since we want a homogenous field, B(z) = B1 would be the ideal. We saw that all

even term cancel out from the symmetry alone. We would now like to put all the other odd terms, B3, B5 and so forth, to zero as well. Can we arrange the conductors so that the

resulting field cancels out certain multipoles? The answer is yes, as can be seen easily if we write the multipole coefficients in polar coordinates:

) cos( 2 4 0 1_{ρ0 θ0} π μ _{R n} I B n n n − − − = . (3.10)

In order to cancel out Bn, one has to place the conductor in a way so that the angle of the

polar coordinate satisfies the condition

K , 2 , 1 , 0 , 2 0 = +k k= nθ π π (3.11) or in degrees K , 2 , 1 , 0 , 180 90 0 + ° = ° = k n k n θ (3.12)

For example, to cancel out B3, we choose θ0 = 30º. However, as seen before,

superconductors have a limit on the maximum current density. In magnets for particle accelerators, stronger field is required than just four line conductors can provide. It is therefore necessary to add more and more conductors to get a higher resulting field and at the same time lower the current density. The conductors are the strands in the Rutherford cable, normally 20-40 in one cable as mentioned in section 2.5.2. By winding the coil with multiple turns of superconducting cable, usually in the range of 10 to 30, the number of conductors is increased even further. The resulting magnetic field is the sum of all conductors. However, sums over a large number of components can be cumbersome; the continuous formalism is more handy and appropriate for analytical estimates.

Imagine a large number of conductors, homogenously placed over an area A. They are placed in such a way that four-fold symmetry still applies, with currents I on the right hand side and currents –I on the left hand side. Instead of calculating the sum of all contributions, we can approximate the sum with an integral over the area A, the sum of currents being approximated with a homogenous current density, spread out over the same area, see figure 3.4. The area A represents the magnet coil. The parameters

defining the coil are the apertures radius r, the radial coil width w, and the azimuthal angle θ1.

Note that in a cable with a keystone angle, as shown in figure 2.7, the current density is not homogenous, due to the smaller area in the inner part of the cable and at the same time having the same currents passing in its conductors. However, this difference in current density is small and is neglected with this approximation. The homogenous approximation is good enough for our goals, i.e. determining coil layouts that produce a pure dipolar field.

+ +

-r w θ₁ A/2 A/2

Figure 3.4. The schematic sketch of a dipole coil; r is the apertures radius, w is the width of the sector coil and θ1 is the azimuthal angle.

The resulting integral becomes

θ ρ ρ θ ρ π μ d d n j R B A n n n

∫∫

− − − = 2 / 1 0 _{cos( )} 2 2 , (3.13) which simplifies to

(

( )

)

sin( ), 2 ) 2 ( 2 2 2 ₁ 1 0 _{+ − ≠} − − = − _{r w} − _r − _{n n} n n jR B n n n n

_π

θ

μ

(3.14)

20 See appendix B for the mathematical derivation. For further convenience, we split the double integral (3.13) into the single integrals corresponding to the angular and the non-angular dependence, and write the multipoles as

) sin( ) , ( ) cos( ) , ( ) cos( 2 2 1 ' 1 1 0 1 1 1 1 θ θ θ θ θ ρ ρ π μ θ θ θ θ n w r F d n w r F d n d j R B n n w r r n n n = = = − =

∫

∫

∫

− + − − − . (3.15)

The magnetic field depends on the three parameters r,w and θ1, defining the sectors of the

coil. The sectors are built up by superconducting cables and the spaces in between are made of passive elements, usually copper wedges. In any case, there are no currents outside the sector coils.

From formula (3.14) we have

1 , ) sin( 2 1 0 1=− w n= j B θ π μ _(3.16) 3 , ) 3 sin( 1 1 3 2 1 2 0 3 ⎟ = ⎠ ⎞ ⎜ ⎝ ⎛ ₋ + = n r w r jR B θ π μ _(3.17) 5 , ) 5 sin( 1 ) ( 1 15 2 3 3 1 4 0 5 ⎟⎟ = ⎠ ⎞ ⎜⎜ ⎝ ⎛ − + = n r w r jR B θ π μ _(3.18)

The multipoles decrease with n, so the first priority should be to cancel out the lowest index multipoles.

We see that in order to cancel out B3, we shall now choose θ1 = 60º. To cancel out

B5, we can choose either θ1 = 36º or θ1 = 72º. Since the angles do not coincide, we

cannot terminate both B3 and B5 with only one sector. We now look at the case with two

sectors; defined by the three angles θ1, θ2 and θ3 according to figure 3.5.

Figure 3.5. A two sector dipole coil.

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 Θ1 Θ2 Θ3

To calculate the integral over these sectors, we use the formula (3.15) to add and subtract the integrals of the corresponding sectors:

(

θ θ θ

)

( )

θ θ θ θ θ θ θ θ θ θ θ θ θ n n n n n G w r F n n n w r F d n d n d n w r F B ) , ( ) sin( ) sin( ) sin( ) , ( ) cos( ) cos( ) cos( ) , ( 1 2 3 ' 3 3 1 1 2 2 = + − = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − =

∫

∫

∫

− − − (3.19)

Here a new function is defined, Gn, which is the angular dependence of Bn. In the

following part of this work, Gn will always denote the angular dependency, but the

number of sine functions the function consists of will differ, depending on how many sectors the coil is made up from.

Given a certain apertures radius r with a certain width w, Fn(r,w) is constant. The

only parameters that affect the multipoles are the angles. To cancel out both B3 and B5

we want to solve the system with the two non-linear equations

0 ) 5 sin( ) 5 sin( ) 5 sin( ) , , ( 0 ) 3 sin( ) 3 sin( ) 3 sin( ) , , ( 1 2 3 1 2 3 5 5 1 2 3 1 2 3 3 3 = + − = ∝ = + − = ∝ θ θ θ θ θ θ θ θ θ θ θ θ G B G B (3.20) Setting Gn to zero is equivalent of setting Bn to zero, because of the proportionality

between the two functions. We see that by adding one sector we now have three parameters to solve two equations. With the freedom of the extra parameter, we might be able to also set B7 to zero. This will be investigated in chapter 4.

By adding one more sector, that is, three sectors in total, we gain two additional parameters, see figure 3.6,

Sector 1 Sector 2 Sector 3 θ1 θ2 θ3 θ4 θ5

22 and the multipoles for a three sector coil become:

(

)

) ( ) , ( ) sin( ) sin( ) sin( ) sin( ) sin( ) , ( ... ... ... ... ... ) , ( 1 2 3 4 5 ' 3 3 1 1 2 2 4 4 5 5 θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ n n n n n G w r F n n n n n w r F w r F B = = + − + − = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + − =

∫

∫

∫

∫

∫

− − − − − (3.21)

With five parameters, we might set five multipoles to zero. Choosing the lowest orders, this will be: B3, B5, B7, B9 and B11. In general, k sectors will give 2k-1 parameters to play

with. This applies to a non-open midplane.

It is worth mentioning that it is also possible to create a pure dipolar field with only one sector, 0º-90º. This is achieved if the current density in the sector is proportional to the cosine of the azimuthal angle of the position. This model is known as the cosθ-model [7]. However, this arrangement is hard to achieve in reality, which is the reason why uniform current densities are the common choice in applications. By making the sector sizes decreasing with order, as shown in figure 3.6, the cosθ-model is roughly approximated. The arc-circle sector coil dipole design as presented in this work is therefore sometimes referred to as the cosθ approach.

3.3 The open midplane

In section 2.7 we saw that one way to avoid waste particles from the collision is to open up the midplane of the dipole. In this section we will see how this action affects the equations for the multipoles, for the one, two, three and four sector case.

3.3.1 Analytical solution for a one sector layout

By open up the midplane, the equations are slightly modified. The midplane is defined as the plane located at y = 0 in the Cartesian coordinate system. Starting with the one sector case, we have two angles defining the sector, see figure 3.7,

θ0

θ1

Figure 3.7. A one sector coil with an open midplane. The angle θ0 denotes the size of the opening.

where θ0 denotes the azimuthal width of the opening gap from the midplane, and θ1 the

end of the sector coil. Note that when looking at the whole coil, including all four quadrants, the real width of the opening gap is 2θ0 . The resulting multipoles become

(

sin( ) sin( )

)

) , ( ... ... ) , ( _{1 0} ' 0 0 1 1 θ θ θ θ θ θ n n w r f w r f B_{n n ⎟}⎟= _n − ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − =

∫

∫

− − (3.22) To put the multipole to zero, we set

0 ) sin( ) sin( ) , ( _{1 0} = ₁ − ₀ = ∝G θ θ nθ nθ B_{n n} (3.23)

The analytical solution is

K , 2 , 1 , 0 , ) 1 2 ( 2 2 _{1 0 0} 0 1 = + = − + =− + k+ k = n n k n or n k π θ π θ π θ π θ θ (3.24) or in degrees K , 2 , 1 , 0 , ) 1 2 ( 180 360 0 1 0 1 + = ° + − = ° + = k k n or k n θ θ θ θ (3.25)

24 We see that the first solution is not physically applicable when n=3 and θ1≠θ0. To cancel

out B3 with a one sector open midplane, the only physically viable solution is 0

1 60 θ

θ = °− (3.26)

However, this case is not of particular interest since we want to terminate more coefficients than just B3.

3.3.2 Two sectors

With two sectors, we get a four parameter space instead of two. Working under the condition that the gap is fixed, we have three free parameters. The multipoles become:

(

)

) ( ) , ( ) sin( ) sin( ) sin( ) sin( ) , ( ... ... ... ... ) , ( 0 1 2 3 ' 3 3 0 0 1 1 2 2 θ θ θ θ θ θ θ θ θ θ θ θ θ n n n n n G w r F n n n n w r F w r F B = = − + − = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − =

∫

∫

∫

∫

− − − − (3.27)

In order to cancel out B3 and B5, we have the following system of equations:

0 ) 5 sin( ) 5 sin( ) 5 sin( ) 5 sin( 0 ) 3 sin( ) 3 sin( ) 3 sin( ) 3 sin( 0 1 2 3 5 5 0 1 2 3 3 3 = − + − = ∝ = − + − = ∝ θ θ θ θ θ θ θ θ G B G B . (3.28) For convenience, from here on the angular function Gn will be written without the

angular dependence notation. It is understood that it is dependent of the angles θi anyway.

Given a specified width of the opening, θ0 is fixed and hence we have a three

parameter space again. Two parameters will be used to set B3 and B5 to zero, but as in

the case of no open midplane, we expect to be able to put also B7 to zero with the use of

the last free parameter. 3.3.3 Three sectors

In analogy with the no open midplane case, three sectors will give a five parameters space, considering θ0 is constant. The multipoles are:

(

)

) ( ) , ( ) sin( ) sin( ) sin( ) sin( ) sin( ) sin( ) , ( ... ... ... ... ... ... ) , ( 0 1 2 3 4 5 ' 0 0 3 3 1 1 2 2 4 4 5 5 θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ n n n n n G w r F n n n n n n w r F w r F B = = − + − + − = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + − + − =

∫

∫

∫

∫

∫

∫

− − − − − − . (3.29)

With a five parameter space, we might be able to cancel out the first five multipoles; B3,

B5, B7, B9 and B11. To cancel out these multipoles, we have the following system of

equations: 0 ) 11 sin( ) 11 sin( ) 11 sin( ) 11 sin( ) 11 sin( ) 11 sin( 0 ) 9 sin( ) 9 sin( ) 9 sin( ) 9 sin( ) 9 sin( ) 9 sin( 0 ) 7 sin( ) 7 sin( ) 7 sin( ) 7 sin( ) 7 sin( ) 7 sin( 0 ) 5 sin( ) 5 sin( ) 5 sin( ) 5 sin( ) 5 sin( ) 5 sin( 0 ) 3 sin( ) 3 sin( ) 3 sin( ) 3 sin( ) 3 sin( ) 3 sin( 0 1 2 3 4 5 11 0 1 2 3 4 5 9 0 1 2 3 4 5 7 0 1 2 3 4 5 5 0 1 2 3 4 5 3 = − + − + − = = − + − + − = = − + − + − = = − + − + − = = − + − + − = θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ G G G G G (3.30)

keeping in mind that Bn is proportional to Gn.

3.3.4 The general case

In general, to cancel out k multipoles, we are given k equations set to zero. In order to solve this system we need at least k parameters. By adding additional sectors, we get access to more parameters. With θ0 fixed, m sectors give 2m-1 free parameters. If we

allow ourselves to adjust also θ0, m sectors give 2m free parameters. In general, m

sectors give the following multipoles:

(

)

) ( ) , ( ) sin( ) sin( ) sin( ) sin( ) , ( _{2 1 2 2 1 0} θ θ θ θ θ n n m m n n G w r F n n n n w r F B = = − + + − = _{− − K} (3.31) To cancel out k multipoles, we need 2m> k. To cancel out the k first multipoles, with the aid of m sectors, we are given the following set of equations:

0 ) ) 1 2 sin(( ) ) 1 2 sin(( ) ) 1 2 sin(( 0 ) ) 1 2 sin(( ) ) 1 2 sin(( ) ) 1 2 sin(( 0 ) 5 sin( ) 5 sin( ) 5 sin( ) 5 sin( 0 ) 3 sin( ) 3 sin( ) 3 sin( ) 3 sin( 0 2 2 1 2 1 2 1 2 0 2 2 1 2 1 2 1 2 0 1 2 2 1 2 5 5 0 1 2 2 1 2 3 3 = + − + + − + = ∝ = − − + − − − = ∝ = − + + − = ∝ = − + + − = ∝ − − + + − − − − − − − − θ θ θ θ θ θ θ θ θ θ θ θ θ θ k k k G B k k k G B G B G B m m k k m m k k m m m m K K M K K (3.32)

These are non-linear equations that cannot be solved analytically. To solve these, numerical methods will be employed.

3.4 Evaluation of the short sample field

Since the particle beam is located in the centre of the dipole cross-section, the magnetic field at this location (the central field) is of the highest interest. The maximum obtainable field in the centre of the dipole is denoted by short sample field, and it is the central field resulted when the current density in the coil is equal to the critical current density. The

26 short sample field is with other words the limit of the performance of the magnet. In this section we will make a mathematical derivation of this field.

Neglecting the contribution of the iron yoke, the magnetic field is entirely given by the magnetic coils. We have the symmetry as shown in figure 3.3 and assume that the current density j is uniform over the sector coil. We define

• The field B[T] at the centre of the dipole

• The peak field Bp[T], which means the highest value of the magnetic field in

the coil. For uniform j, the peak field will be located on the contour of the coil.

From Eq (3.3) we have that the magnetic field is proportional to the current, and hence also proportional to the current density:

, γ j B= (3.33) , λγ j B_p = (3.34)

where γ[T m2/A] is the central field (in T) per unit of current density (in A/m2) and λ is the ratio between the peak field the central field. The cosθ-design has the optimal λ=1, meaning the central field has the same value as the peak field. For Nb-Ti, the critical current density is approximately given by:

, , ) (b B B b c j_c =κ − < (3.35)

where κ is the filling factor between the area of the superconducting material versus the total area of the conductor, c is a constant ~ 6.00×108 A/(Tm2) and b is the critical magnetic field, in literature usually denoted as B*c2. For the Nb-Ti this value is around 13 T.

The current density can be expressed in terms of the peak field by rearranging Eq. (3.34). This function is called the loadline. The peak field is limited by the critical current density. 0 1000 2000 3000 0 5 10 15 B (T) j( A /m m 2 ) jc= κ c(b-B) Bp=λγ j [Bp,c,jp,c]

Figure 3.8. Example of a cross section of the critical surface of Nb-Ti, at 1.9 K. The two lines indicate the current density for the peak field Bp, called the loadline, and the critical current jc for

The maximum peak field is reached when the current density is equal to the critical current density of the superconductor. This is the critical peak field and it is the point where the loadline hits the critical current density jc in figure 3.8. To get the critical peak

field, we substitute Eq. (3.35) into Eq. (3.34):

λγ κ λγ ( , ) ,c c pc p j c b B B = = − (3.36)

This is the highest field that can possible be reached in the magnets before leaving the superconducting state. By rearranging this equation, we can solve out the critical peak field explicitly: λγ λγ κ κ c cb B_pc + = 1 , (3.37)

By comparing Eq. (3.37) and (3.34) we conclude that the critical current density that corresponds to the critical peak field can be written:

λγ κ κ c cb j_pc + = 1 , . (3.38)

This current gives the maximum field in the centre as well. By substituting Eq. (3.38) into Eq. (3.33) we obtain the short sample field:

γ λγ κ κ c cb B_ss + = 1 . (3.39)

The constants κ, c and b are known, so in order to calculate the short sample field the functions γ and λ have to be evaluated.

Consider a sector with an angular width θ, with no open midplane, from Eq. (3.16) we had that the dipolar component is

) sin(

1 w θ

B ∝ , (3.40)

and since the other multipole components have a z-dependence, the magnetic field is equal to the dipolar component in the centre, giving:

0 , ) sin( 2 0 1 =− = =B jw z B θ π μ . (3.41) The central field per unit of current density γ can therefore be written

w j B 0 γ γ = = , (3.42)