WO2012150448A1

WO2012150448A1 - Magnets

Info

Publication number: WO2012150448A1
Application number: PCT/GB2012/050953
Authority: WO
Inventors: Ken Peach; Holger Witte; Andrei SERYI
Original assignee: Isis Innovation Limited
Priority date: 2011-05-03
Filing date: 2012-05-01
Publication date: 2012-11-08
Also published as: GB201107318D0

Abstract

A method of designing a magnet for a particle accelerator based on a desired magnetic field to be generated by the magnet comprises the steps of determining a two-dimensional cross-sectional internal space (4) to be enclosed by the magnet, determining from the magnetic field a magnetic vector potential (2) at a plurality of points (P₁, P₂, P₃, P₄, P₅) inside said internal space (4), calculating a current 10 distribution at a plurality of points (I₁, I₂, I₃, I₄, I₅) around the perimeter of the internal space (4) to give the desired magnetic vector potential (2), and determining paths (12, 14) for a plurality of wires (16, 18) to provide the calculated current distribution (10) when current is passed through them.

Description

Magnets

This invention relates to magnets for bending and focussing moving charged particles, particularly in particle accelerators. The invention also relates to methods for designing and manufacturing such magnets.

Particle accelerators have widespread uses in scientific research as probing instruments to investigate the structure of matter, e.g. high energy particle physics, X-ray and neutron scattering, and for a diverse range of industrial applications, e.g. ion implanters for semiconductors, surface hardening, synchrotron radiation sources, and medical applications (radiotherapy, biomedical research and radioisotope production).

Conventionally, charged particles are accelerated in circular accelerators where the particles are kept in the accelerator for multiple revolutions. To keep the particles within the beam pipe of the accelerator, bending magnets are needed to make the particles follow a curved trajectory. Conveniently such accelerators have a circular particle trajectory, but this is not essential and other loop shapes are possible.

The Lorentz force, F=q(E + vxB), describes the force experienced by a charged particle with velocity v and charge q in an electric field E and magnetic field B. Considering when only a magnetic field is present, the force experienced by the particle is perpendicular to both the magnetic field and the component of its velocity perpendicular to the magnetic field. Therefore if the velocity of the particle is only in this direction, i.e. perpendicular to the magnetic field, the trajectory of the particle will be a circle if the magnetic field is homogeneous.

Therefore if a homogeneous magnetic field is created across the area of a circular particle accelerator, perpendicular to the plane of the accelerator and a supply of particles is injected into the magnet, they will circulate within the accelerator as long as the magnetic field is chosen correctly. A magnet able to perform this function is called a dipole magnet which has a homogeneous magnetic field over a certain region and bends particles in a circular path.

Conventional magnets use a solid iron core around which a coil is wound to create the desired field, the core amplifying the magnetic field created by the coil. The maximum field obtainable with such a magnet is approximately 2 T, at which point the iron core saturates and only an insignificant increase in magnetic field is possible as the current in the coil is increased.

The magnetic rigidity of a charged particle describes the relationship between the radius of curvature in a magnetic field with the particles' momentum, which can be derived from balancing the Lorentz force with the centrifugal force to give Bp = p/q where B is the magnetic field, p is the radius of curvature, p is the particle's momentum and q is the particle's charge. For a given application, particles of a certain energy, and therefore momentum, are generally required. For example in radiotherapy using protons, an energy of 250 MeV is required, e.g. to treat tumours at a maximum depth of 25 cm below the skin. This means that once the energy has been set, the magnetic field and radius are inversely proportional to each other, and therefore if an accelerator with a small radius is required, e.g. to fit into a confined hospital space, a large magnetic field is required.

The solution to providing a magnet which can create a magnetic field greater than 2T is to dispense with the iron core and to use electromagnets. In this respect superconducting magnets are very attractive as they require relatively little power, owing to the fact that the electrical resistivity is zero. Superconducting magnets routinely produce magnetic fields in excess of 22 T.

There are a couple of conventional approaches for designing

superconducting electromagnets for particle accelerators. The first method of construction is to arrange a number of coil segments around a circular bore, each of which carries a constant current, with the current running along the axis (z- direction) of the magnet. The net current is varied around the circumference of the bore to give a cosine theta current distribution, i.e. J_z∞ cos(6) for a dipole magnet, where J_z is the z-component of the current and Θ is the azimuth angle. This results in the creation of a dipole magnetic field across the interior of the bore. The current distribution is a higher order sinusoidal term if higher order multipoles are required. However because of the need to provide a cosine theta (or multiple thereof) current distribution, the magnets produced by these designs are necessarily limited to having circular bores.

The second conventional method is from knowledge that a dipole field can be generated in theory by creating two areas of constant current density, the shapes of which correspond to the shapes formed by the non-overlapping portions of two identical but laterally offset ellipses. In practice the field can be

approximated by the use of two sets of coil segments carrying the same constant current, with the current running along the axis (z-direction) of the magnet in opposite directions.

Both of these methods use coil segments to provide the required current distributions, however the finite size of, and need to provide return paths for, each individual coil segment means that there is a limit to how accurate an approximation can be achieved, especially for relatively smaller magnets. Furthermore, both of these conventional methods can only provide multipole magnetic fields.

Another important concern for particle accelerator magnets is the required beam aperture, which needs to be large enough to accommodate the beam. The beam aperture is determined by the optical lattice of the accelerator. For all magnets it is desirable to have as small an aperture as possible to minimise the cost, and for some accelerators this is a particular challenge, e.g. fixed field alternating gradient (FFAG) accelerators, as the aperture is required to be relatively large to accommodate the large radial excursions of the particles in such accelerators. The Applicant has appreciated that when using a magnet design with a circular aperture, e.g. the cosine theta coils, this is wasteful since the excursions are mainly confined to one, generally horizontal, transverse axis and thus a large proportion of the circular aperture is not usefully employed.

Furthermore, in order to create a high quality magnetic field over the entire length of the magnet, the effects of the ends of the coil, which return the current, need to be considered. This is particularly important when a magnet has a large aperture and a short length, giving it a small aspect ratio (ratio of length to radius of aperture). Conventional coil ends make little useful contribution to the magnetic field as they are simply artificial structures which need to be employed to return the current back into the main body of the coil, and thus they reduce the available space for the 'useful' part of the magnet. A further disadvantage of the coil ends is that they introduce field errors.

The present inventors have appreciated that there are shortcomings with conventional electromagnets, as well as their design and manufacture, and the present invention aims to address these.

From a first aspect the invention provides a method of designing a magnet for a particle accelerator based on a desired magnetic field to be generated by the magnet comprising the steps of:

determining a two-dimensional cross-sectional internal space to be enclosed by the magnet, determining from the magnetic field a magnetic vector potential at a plurality of points inside said internal space,

calculating a current distribution at a plurality of points around the perimeter of the internal space to give the desired magnetic vector potential, and

determining paths for a plurality of wires to provide the calculated current distribution when current is passed through them.

From a second aspect the invention provides a computer software product for designing a magnet for a particle accelerator, adapted when run on a suitable data processing means, to perform the steps of:

receiving as inputs a desired magnetic field to be generated by the magnet, and a two-dimensional cross-sectional internal space to be enclosed by the magnet,

determining from the magnetic field a magnetic vector potential at a plurality of points inside said internal space,

returning as an output the paths for a plurality of wires to provide the calculated current distribution when current is passed through them.

The invention also extends to a method for manufacturing a magnet for a particle accelerator comprising the steps of:

designing a magnet according to the method of the first aspect of the invention,

providing a plurality of wires, and

arranging the plurality of wires along the determined paths to provide the calculated current distribution when current is passed through them.

The step of designing the magnet could be performed using the computer software product recited in the second aspect of the invention. The invention also extends to a magnet for a particle accelerator manufactured according to the method of manufacturing the magnet.

The Applicant has appreciated that because a simple linear relationship exists between the current and the magnetic vector potential, which in turn can easily be determined from the desired magnetic field. This provides a

straightforward way of designing and manufacturing a magnet for a particle accelerator, which is applicable for any type of magnetic field, be it multipoles, or non-multipoles, and for any two-dimensional cross-sectional internal space, i.e. bore shape for the magnet. The invention therefore provides a method which is widely applicable for many types of magnets.

The step of determining the paths for a plurality of wires to provide the calculated current distribution in the method of designing a magnet, and the step of arranging the plurality of wires in the method of manufacturing a magnet could comprise any arrangement which delivers the desired current distribution.

However, in a preferred set of embodiments the plurality of wires comprises the turns of at least two concentric tubular coils.

As the wires making up the coils have a finite width, the concentric coils will have slightly different sizes and will not both follow the perimeter of the internal space precisely. Typically, however, the coils will follow the perimeter of the internal space as closely as possible.

By arranging at least two concentric tubular coils in this way, a high quality magnetic field can be provided since they give a continuous current density distribution, thereby avoiding the inevitable discontinuities arising when trying to approximate the current density distribution with separate coil segments as in conventional electromagnets. Furthermore, the arrangement of the coils in accordance with the invention obviates the need for additional, artificial structures at the ends of the coils, as are needed in conventional electromagnets to return the current but which add unwanted components to the magnetic field. The continuous tubular coils can therefore closely reproduce the calculated current density distribution which produces the required fields, e.g. a multipole field or a non- multipole field.

Yet another benefit is that the integral field quality of these concentric coils is very good, as field errors (unwanted harmonics) tend to cancel out. The present invention therefore offers a significant improvement over conventional magnets in mitigating field errors. The whole length of the coils of the present invention can contribute usefully to the desired magnetic field, with the vector addition of the currents in the at least two concentric tubular coils giving the calculated current distribution.

The Applicant has further appreciated that several advantages may be realised by employing a non-circular bore, i.e. the two-dimensional cross-sectional internal space to be enclosed by the magnet is non-circular, which the present invention facilitates. The first step in the method of designing and manufacturing the magnet is to determine the magnetic field that is to be generated by the magnet, which depends upon the application of the magnet. As already explained, a dipole magnetic field bends charged particles in a circular path, and is the basic type of magnetic field needed to retain particles in a particle accelerator.

However higher order multipole magnetic fields are also needed in particle accelerators because the case of an ideal charged particle circulating in a dipole field does not exist. There are many systematic errors that result in the defocusing of the particle beam: radiation losses in the dipole magnets, gravity, field imperfections, ground motion, alignment of the accelerator, having a limited physical aperture, errors in the power supplies and calibrations, which all result in errors in the magnet strength, and variations in the energy of the particles. As a result of these factors, if purely dipole magnets were employed, particles in the accelerator would tend to spread out transversely and longitudinally and eventually get lost from the accelerator. Other magnets, such as quadrupoles, sextupoles and higher order multipoles are therefore used to compensate for the spread in particles and so reduce the loss of particles from the beam pipe.

There is also an increasing desire to be able to produce magnets which generate magnetic fields which cannot easily be approximated using the ordinary superposition of multipoles (a multipole expansion). Prior to this invention, such non-multipole fields had not been able to be provided in a simple magnet design. Non-multipole magnetic fields have many applications but are particularly important in high intensity particle accelerators, e.g. proton drivers, and allow a large number of particles to be included in each bunch. When a large number of particles, e.g. protons, are included in the bunches, space charge becomes a problem and causes the particles to diverge from the bunch which will eventually be lost if they are not manoeuvred back into position. An appropriate non-multipole magnetic field has the ability to overcome the space charge problem to retain the particles in the bunches, enabling a large number to be included in each bunch. This gives a viable alternative to a high frequency particle accelerator (another way to provide a high intensity accelerator). Such a magnet is considered to be novel and inventive in its own right and thus from a further aspect the invention provides a magnet for a particle accelerator comprising at least two concentric tubular coils, each of said coils comprising a plurality of turns, wherein the turns are arranged to provide a non-multipole magnetic field. From yet a further aspect the invention provides a magnet for a particle accelerator comprising at least two concentric tubular coils, each of said coils comprising a plurality of turns for which the spacing between adjacent turns is not constant around the perimeter of the magnet, wherein the turns are arranged around a non-circular cross-sectional internal space to provide a multipole magnetic field.

The present invention therefore also provides magnets for generating non- multipole magnetic fields and magnets with non-circular bores using concentric tubular coils for generating multipole magnetic fields, which previously has not been possible using conventional techniques.

Once the magnetic field has been determined, the magnetic vector potential is determined from the magnetic field by using the relationship B = V x A . Working dA in the two dimensions of the cross-section of the magnet, this is B = ,

B_x =— - . As will be appreciated, because the magnetic field is derived by taking dy

a spatial derivative of the magnetic vector potential, there is no unique magnetic vector potential which gives the required magnetic field, but an infinite number of possible solutions which vary by a uniform vector potential A₀, because V x A₀ = 0 .

Therefore A₀ can be chosen to be any value which is suitable for the current distribution being provided, i.e. this shift in the magnetic vector potential corresponds to a shift in the current provided in the coils. Therefore A₀ can be conveniently chosen, for example, to minimise the average magnitude of the current.

The magnetic vector potential at a point P (having polar ) due to current / (flowing in the z-direction) is given by: A_z (r, &)

where R is the distance from the point P to the location of the point current /, a is the distance of the point current / to the origin, and μ₀ is the vacuum permeability. Therefore if the magnetic vector potential at some point (determined from the desired magnetic field) is known, the necessary current distribution at another point which will generate this vector potential can be calculated.

The Applicant has devised a particularly simple and straightforward way of calculating the required current distribution for generating a known magnetic vector potential. In one set of embodiments the plurality of points within the two- dimensional cross-sectional internal space, at which the magnetic vector potential is determined, are themselves located on the perimeter of a further two-dimensional cross-sectional internal space. This further two-dimensional cross-sectional internal space could be any shape and size (as long as it falls within the perimeter of the two-dimensional cross-sectional internal space on which the current distribution is calculated), but preferably it is the same shape as the two-dimensional cross- sectional internal space of the bore of the magnet but with a smaller size, e.g. with an area 10% less. A single closed path, e.g. the perimeter of an ellipse, on which the plurality of points at which the magnetic vector potential is determined, is sufficient to define the magnetic vector potential across the whole area of the two- dimensional cross-sectional internal space. In a preferred set of embodiments the number of points used to define the magnetic vector potential is chosen to be between 40 and 80.

In one set of embodiments the plurality of points at which the current distribution is calculated, on the perimeter of the two-dimensional cross-sectional internal space, are chosen at regular steps in theta, the azimuthal angle, e.g. one point every π/20. The number of points chosen at which to determine the magnetic vector potential could be greater than the number of points on the perimeter of the two-dimensional cross-sectional internal space, but in one set of embodiments the number is the same, and preferably is also chosen at the same steps of the azimuthal angle. The plurality of points on the perimeter of the two-dimensional cross-sectional internal space at which the current distribution is calculated correspond to the positions of the turns of the wires.

Once the step of determining the magnetic vector potential at the plurality of points, e.g. on the perimeter of the further two-dimensional cross-sectional internal space, has been performed, in one set of embodiments the following procedure is used to determine the current distribution. If a point P-i is defined, e.g. on the perimeter of the further two-dimensional cross-sectional internal space, at which the magnetic vector potential, A_z is known, then a current at a point on the perimeter of the bore of the magnet can also be defined, which can be related to the magnetic vector potential, A_z using the above relationship. It will appreciated that the relationship between and A_z is linear and therefore can be written in the form A-ifl-i = A_z1, where A_z1 is the magnetic vector potential at the point P-i resulting from the current and An is a constant which absorbs all the geometric terms in the relationship. Similarly, if a further point P₂ is defined, the relationship A₂ li = A_z2 also exists. And generally, (A_n , A_2l , A_nl , A_z2 ..., A_zn ) for all the plurality of points, which in some embodiments will lie on the perimeter of the further two- dimensional cross-sectional internal space.

These relationships between current point sources and the magnetic vector potential can be extended to a plurality of current point sources, l_m, on the perimeter of the two-dimensional cross-sectional internal space of the bore of the magnet and a plurality of points, P_n, at which the magnetic vector potential has been

determined, e.g. on the perimeter of the further two-dimensional cross-sectional

internal space, to give:

This can be seen to be a simple linear equation system of the form Ax = b, from which x, the value of the currents, l_m, needed to generate the required magnetic vector potential A_z, can be calculated, as A_z is known and the values of A_nm can be determined, (e.g. by placing a unit current at each of the plurality of current point sources on the perimeter of the two-dimensional cross-sectional internal space of the bore of the magnet and calculating the values of A_nm).

It will be appreciated that this analytical method of determining the required current distribution, exploiting the relationship between the current sources and the magnetic vector potential, is computationally very straightforward, which therefore reduces both the programming and the processing time required to perform this design step and ensures that the correct solution is obtained. This compares to conventional approaches for designing magnets which, when determining the required current distribution for a desired magnetic field, use a numerical method to solve the problem. Such a numerical method takes the plurality of current sources and varies them randomly using a Monte Carlo input to look for a target function giving a global minimum in order to optimise the values for the current sources. This requires a lot of time, both in terms of implementing the algorithms to run the numerical method and the actual processing time to run them on a computer processor. There is also a possibility that this procedure will return a solution at a local minimum rather the desired solution at the global minimum, thereby not providing the optimum solution. This inferior performance of the conventionally used numerical methods is due to the fact that previously the skilled person has not appreciated the possibility to exploit the simple linear relationship between the current sources and the magnetic vector potential as is done in the present invention.

Numerical methods typically arise from the mindset that to produce a magnetic field, particularly a multipole field for which the beam dynamics in a particle accelerator are well understood, known current distributions need to be used, e.g. cosine-theta or constant current density, which generate known magnetic fields. However these known current distributions have a number of limitations as has been discussed previously, e.g. the cosine-theta distribution is only able to be provide a multipole field with a circular bore magnet, constant current densities are only able to provide multipole fields, and both distributions impose tight restrictions on the location of the current sources needed to produce a high quality magnetic field.

The Applicant has realised that deviating from this conventional wisdom in providing a magnet designed and manufactured according to the present invention affords a lot of design freedom, i.e. the flexibility to provide a magnetic field using a wide variety of cross-sectional shaped bores, and the ability to generate non- multipole magnetic fields. Previously, such approaches have not been

contemplated, because of the prejudice against working with non-multipole fields for which the associated beam dynamics are perceived to be difficult to calculate, and against using non-circular bores, on which it is perceived to be difficult to arrange the coil windings to generate even a simple multipole magnetic field. However, there is now an increasing desire for non-multipole magnetic fields, e.g. to overcome the space charge problem in high intensity particle accelerators, and the present invention provides a method of designing and manufacturing a suitable magnet for this purpose.

Once the current distribution at the plurality of points on the perimeter of the two-dimensional cross-sectional internal space has been determined to give the desired magnetic vector potential, the arrangement of the paths of the turns of the coils has to be determined so to give this current distribution when current is passed through the coils of the magnet. In the simplest case this can be m independent current carrying wires extending in the longitudinal direction.

For two tubular coils, the two-dimensional cross-sectional internal space of the coil bore has already been determined, as has the current at each of the plurality of points at positions (x,y) on the perimeter of the two-dimensional internal space. The longitudinal position z_n of the wire along the path of the turns can be n

calculated by evaluating z_n =∑ / for all n. Then, the z-function is normalised and i=

multiplied with an (arbitrary) maximum deflection/excursion per turn: z=z/max(|z|)-d, where d is the maximum deflection. This step of the method results in the determination of a value of the longitudinal position of the wire at each of the plurality of points on the perimeter of the two-dimensional cross-sectional internal space.

This procedure generates a single turn of a coil, with further turns being generated by stepping the subsequent turns by a winding pitch distance, p, in z for each turn. This winding pitch is generally added as a constant function of the azimuthal angle, e.g. z = ρθ/2π. In general, the second coil can be generated by reversing the path in z compared to the first coil (if only two coils are provided), with the radius (or equivalent dimension if an ellipse or other shaped internal space is provided) of the second coil having to increase to allow for the radial thickness of the wire of the turns.

It will be appreciated that if the path of the second coil is the reversal of the path of the first coil in z, and the current is passed through the coils in opposite directions, the transverse components of the magnetic field, which contribute to the useful magnetic field, will add and the solenoidal (axial) components will tend to cancel. Therefore in a preferred set of embodiments, the magnitude of the current passed through each of the at least two coils has the same value. Preferably the current is sent in one direction through the first coil and the opposite direction through the second coil, i.e. the current is arranged to flow in opposite directions through the first and second coils. However locally, the direction of the current distribution is achieved by the direction of the turns in the coils, particularly in the axial direction. Providing concentric coils to generate the magnetic field provides an effective way of generating the required continuous current distribution which, owing to the vector addition of the currents in the at least two concentric coils, is only provided where the current is desired and not where it is unwanted, resulting in a high quality magnetic field, without suffering from any of the problems of conventional electromagnets, e.g. discontinuities between current blocks and end effects from the current return sections. In one set of embodiments, the paths of the turns are arranged such that for the first coil z = ρ(θ) + ΐ(θ), where z is the coordinate along the axis of the coils, Θ is the azimuthal angle, ρ(θ) is a term representing the winding pitch of the turns, and f(6) is a term chosen to provide the required current density distribution, and for the second coil z = ρ(θ) - ΐ(θ). In one set of embodiments f(6) may vary along the length of the coils, but in the set of embodiments in which ΐ(θ) is constant for the length of the coils, the second coil can be thought of as the mirror image of the first coil about a plane through the middle of the coils perpendicular to the axis of the magnet (ignoring the generally negligible term of the winding pitch). In general ρ(θ) is constant along the length of the coils. The inversion of the path of the windings of the first coil (ignoring the term, ρ(θ), for the pitch of the windings) to give the path of the windings of the second coil, i.e. z = ρ(θ) + f(0) to z = ρ(θ) - f(0), acts to minimise the solenoid component of the magnetic field.

The invention has thus far been described in terms of calculating the current distribution at a plurality of points on the perimeter of a two-dimensional internal space and using this to determine the path of the turns of the coils along the length of the magnet. This provides a magnet whose azimuthal magnetic field does not vary along the length of the magnet. However in one set of embodiments it is desired to provide an azimuthal magnetic field which varies along the length of the magnet.

In this set of embodiments the method of designing the magnet comprises repeating the steps of determining the magnetic vector potential at a plurality of points inside said internal space, and calculating the current distribution at a plurality of points around the perimeter of the internal space to give the desired magnetic vector potential, at a plurality of sections along the length of the magnet. Therefore the procedure for calculating the current distribution is repeated a number of times along the length of the magnet to account for the magnetic field varying along the length, so that when the step of determining the paths of the turns of the coils is performed, this takes into account the different current distribution which has been calculated for each section along the length of the magnet. For the previous case in which the magnetic field does not vary along the length of the magnet, the determination of the current distribution is the same at any point along the length of the magnet, and so, as has been described, each subsequent turn of the coils was the same as the previous one, just stepped by the winding pitch. However if the magnetic field varies along the length of the magnet, this will not be the case, and therefore the paths of the turns of the coils needs to vary accordingly.

It will be appreciated that in this set of embodiments the paths of the turns, i.e. the function describing the longitudinal position of the wire, can change turn by turn. This therefore accommodates changes of the vector potential in the longitudinal direction.

Therefore in one set of embodiments this results in a set of coils for which the paths of the turns are arranged such that for the first coil z = ρ(θ) + ΐ(θ), where z is the coordinate along the axis of the coils, Θ is the azimuthal angle, ρ(θ) is a term representing the winding pitch of the turns, and f(6) is a term chosen to provide the required current density distribution, and for the second coil z = ρ(θ) - ΐ(θ), wherein at least one of ρ(θ) and f(6) is varied along the length of the magnet.

The inversion of the path of the windings of the first coil (ignoring the term, ρ(θ), for the pitch of the windings) to give the path of the windings of the second coil, i.e. z = ρ(θ) + ΐ(θ) to z = ρ(θ) - ΐ(θ), acts to minimise the solenoid component of the magnetic field.

It will therefore be appreciated that this aspect of the invention, if ΐ(θ) is varied along the length of the magnet, allows a combined function magnet with different magnetic field components to be provided, with the windings of the coils arranged to vary the coefficients of these components along the length of the magnet. This is advantageous as, for example, it allows two or more different types of magnetic field, e.g. multipoles, a non-multipole magnetic field or a superposition of different types of magnetic fields, to be provided adjacent to each other in a single magnet, instead of needing two separately wound magnets to be provided one after each other along a particle accelerator, or nested within each other, resulting in a significant gap between the magnets. It also allows the strength of a single multipole magnet, e.g. a dipole, to be varied along the length of the magnet. This can be used for field shaping which can be important for some accelerators, e.g. FFAGs, to influence the tunes.

Therefore it can be seen that in one set of embodiments, f(6) describes a superposition of magnetic field components. In another, not necessarily mutually exclusive, set of embodiments, f(6) describes a superposition of magnetic field components whose coefficients vary along the length of the magnet. However, in both of these sets of embodiments, the component could be a non-multipole magnetic field or indeed a superposition of multipole and non-multipole magnetic fields. Therefore in one set of embodiments the current distribution is arranged to provide a multipole magnetic field, a superposition of multipoles, a non-multipole magnetic field, or even a superposition of multipoles and a non-multipole magnetic field.

This set of embodiments therefore allows such gaps between the different magnetic fields (or combination of magnetic fields) to be eliminated, enabling matching of the fringe field extent. Previously with two adjacent magnets, including combined function magnets, it was extremely difficult to match the magnetic field at the ends of each of the magnets resulting in an unstable beam and therefore subsequent loss of particles from the accelerator, whereas the aspect of the invention set out above allows seamless matching between the different field components of the magnet thereby creating a stable tune for the particle accelerator which prevents blow up of the particle beam and the subsequent loss of particles from the accelerator. One example of where this might be required is in a fixed-field alternating gradient particle accelerator.

This seamless matching between the different components of the magnet is difficult to achieve in one layer with previous electromagnets, e.g. conventional cosine-theta magnets with blocks of constant current density, because for two different types of magnet, e.g. multipoles, the longitudinal current has a completely different angular distribution and therefore there is nowhere for the current to go if the two magnets were to be connected. However in the present invention the windings of the coils are arranged so that the components can vary along the magnet and therefore the current distribution is also varied likewise to avoid the discontinuities present in the previous electromagnets.

If ρ(θ) is varied along the length of the magnet, this allows the pitch of the windings of the coils to be varied along the length of the magnet. This allows greater flexibility in the design of a magnet, whether it is a single component or a superposition of components, as the field can be shaped and therefore better performance can be obtained. In one set of embodiments, the density of turns can be increased at the ends of the coils. This is usually the region where the peak field on the wire tends to be lower in comparison to the middle of the coils, so the density of turns can be increased in this area to get more performance.

For the combined function magnet with varying coefficients, the field components of the magnets could be present as a number of zones along the magnet in which the coefficients are constant, e.g. one zone of the magnet could provide a dipole field and the other zone could provide a quadrupole field. In one set of embodiments the change between the different zones is a step change, e.g. the path of the windings of the coils changes immediately from that for a dipole to that for a quadrupole. However in a preferred set of embodiment the varying of the coefficients between each of the different zones is provided in a transition region, i.e. the windings of the coils change gradually between providing one type of magnetic field (or superposition of magnetic fields) to providing a different type of magnetic field. The transition region could comprise only a few turns (in the most extreme case of the transition region being zero turns long, this is the previous case of a step change) or the transition region could comprise the entire coil, i.e. the windings of the coils start at one end of the coil in one magnetic field configuration and gradually change along the whole length of the coil into a different configuration at the other end of the coil. Preferably the transition region comprises ten or fewer turns, preferably five or fewer turns.

It is envisaged that any number of different combinations of magnetic fields could be provided in one set of coils, as well as any number of different

superpositions of fields, e.g. the windings of the coils could change along the length of the magnet from providing a superposition of a dipole and a quadrupole (in any ratio of relative strength) to a superposition of a sextupole and a quadrupole to a superposition of a dipole and an octupole. Thereby different multipoles can be effectively turned on and off at different points along the magnet so that a particle travelling through the magnet experiences the different magnetic fields as it travels through, but with no discernible gap between the different components. It will also be appreciated that in all of these embodiments, the components could include non- multipole magnetic fields.

In the set of preferred embodiments in which a transition region is provided for changing between the different magnetic field components, the function of the path of the windings in the transition region could be any function which provides the necessary transition between the paths of the windings on either side of the transition region.

Therefore there are many different ways, i.e. functions, in which the windings can change in the transition region between the different magnetic field configurations. The function of the path in the transition region could comprise a linear function of the coefficients of the different magnetic field components in the windings of the coil either side of the transition region, i.e. a linear interpolation across the transition region between the different coefficients of the components of the magnetic field. However different interpolation functions could also be used, e.g. a polynomial interpolation or a spline interpolation.

For the type of combined function coil discussed above the ratio of the different types of magnetic fields is hardwired into the coil (by the values of the different coefficients for the components of the different components) thus eliminating one degree of freedom. In an alternative set of embodiments, when a combined function magnet is required, a plurality of different discrete multipole magnets concentric to each other can be provided. For example, this type of combined function magnet could have two coils creating a dipole field which could be concentric with a magnet comprising two additional coils creating a non- multipole field. The magnetic field produced in such embodiments is a

superposition of the fields created by each coil, thus allowing the different multipole terms or non-multipole fields to be achieved, but with a separate set of coils for each different discrete magnetic field. The path of the turns for each of these coils and the current passing through them can be chosen to give the desired ratio between the strength of the different magnetic fields which are included in the magnet. As each of the components is effectively a separate magnet, the current passing through the coils can be fine-tuned individually to give the desired balance between the different components once the magnet is in operation which is not possible with a "hardwired" combined function magnet, and therefore such an arrangement gives greater flexibility in this regard.

In general, f(6) will not be a simple mathematical formula, such as sin(6), but instead will comprise sections of paths fitted together to give the calculated current density distribution. In a preferred set of embodiments the longitudinal components of the current density generated by the corresponding sections of paths in the first and second coils add to give the required current distribution and therefore the desired transverse magnetic field, and the azimuthal components of the current density cancel to give cancellation of the solenoid field.

In one set of embodiments the common axis of the coils is a straight line. The transverse cross-sectional shape of the coils may take a number of different forms, e.g. a circle but in one set of preferred embodiments the cross-sectional shape of each coil is an ellipse. Other shaped cross-sections are also

contemplated, e.g. a rectangle or a butterfly shape. If a polygonal shape such as a rectangle is provided, it may be provided with rounded corners which make it easier to both wind the turns of the coils and to avoid generating singularities when calculating the current distribution.

In another set of embodiments the axis of the coils is curved. Again the cross-section may also take a number of different forms, e.g. a circle, but again preferably an ellipse, though again other cross-sectional shapes are contemplated, e.g. a rectangle or a butterfly shape. In some of these embodiments therefore the magnet could be part of a torus. As explained previously a dipole magnet deflects a charged particle travelling through the magnetic field, so sometimes coils in the form of a curved tubes will be advantageous. However when using curved tube coils it is more difficult to produce pure multipole fields, e.g. a when trying to create a dipole, the magnet would often include a small component of a quadrupole.

These higher order field components can be compensated by adding a separate multipole coil to create a combined function magnet or by hardwiring extra multipoles into the coil as described above.

As used herein the term 'axis' of a coil should be taken to be a line through the centre of the coil parallel to the sides of the coil.

As has been explained, the present invention is particularly suitable for providing coils with non-circular cross-sections, and such magnets afford a number of advantages. For a magnet based on the cosine-theta principle as discussed above, it is very difficult to arrange the current around the perimeter of an non- circular bore such that a pure multipole magnet is generated. However for a magnet based on the principle of the present invention, a number of different non- circular shapes are possible, particularly ellipses. This stems from the general principle, as previously discussed, that a large variety of magnetic fields can be generated in accordance with the invention by arranging at least two concentric coils, with any shape cross-section, to generate the desired current distribution.

Having a non-circular, and particularly an elliptical, cross-section for the coils can be particularly advantageous for some particle accelerators. This is because the beam aperture, and therefore beam pipe, can be non-circular, e.g. in FFAGs, primarily because the particles are in a circular orbit, so the spread of particles is greatest in the horizontal rather than vertical direction. An elliptical aperture is therefore most suitable to match the shape of the spread of particles. For example, for a particle accelerator with a radius of 6 m, bending protons of energy 250 MeV, the beam aperture is 160 mm x 25 mm. Therefore if a round bore coil was to be used, approximately only 20% of the magnet bore would actually ever be used for the particles. This is a huge waste of magnetic energy as the circular coils are providing a magnetic field in a large volume where it is not needed. Instead, an elliptical aperture is far more suitable as the coils can then match the shape of the particle beam and thereby waste less space. In this arrangement the required magnetic field is only generated in the volume in which it is needed, and therefore results in a more efficient magnet as less magnetic energy is necessary compared to that generated by an equivalent magnet with a circular bore able to accommodate the same beam. As the overall magnetic energy required is reduced, this leads to a more efficient configuration.

Yet another advantage in providing an elliptical coil, or other shaped coil which has one cross-sectional dimension smaller than the other, is that it reduces the size of the magnet in the vertical direction which can aid the extraction of particles from the accelerator. In many particle accelerators the particles are accelerated to a certain energy and then extracted for use elsewhere, e.g. in the many applications described previously. One common method of extracting particles from an accelerator is to use a so-called "kicker" magnet, placed between the other magnets in the accelerator. A kicker magnet is energised as a beam of particles is passing through to "kick" particles from the beam out of the accelerator in a certain direction, generally the smallest dimension to clear the following magnet which is typically horizontally. It will therefore be appreciated that if the magnets are reduced in one dimension owing to having elliptical coils, the kicker magnet will not have to exert as great a force on the particles to deflect them clear of the next magnet in the accelerator in order to extract them.

A further advantageous consequence of providing an elliptical coil, or other shaped coil which is shaped to conform to the region in which the particle beam is located, compared with a circular coil, is that it reduces the surface area of the coils that is needed to be cryogenically cooled during operation. If the coil is cooled using cryocoolers, fewer coolers can be used. This again reduces the energy needed, and therefore the cost of running the magnet.

Other non-circular shapes can also be particularly advantageous in certain circumstances. For example, if a non-multipole magnetic field with singularities in the magnetic vector potential is desired, the singularities cannot be provided by any current distribution as this would require an infinite current. However if the singularities are avoided by providing coils with a bore that does not overlap with the singularities, the required vector potential in the region which falls within the cross-sectional area of the coils can be provided. For this purpose a butterfly shape (this could also be thought of as an hour-glass shape) bore may be useful.

Preferably the coils are arranged to produce a magnetic field in which the solenoid components of the magnetic field generated by the coils are at least partially mutually cancelled. Arranging the coils partially to cancel their solenoid fields enables a high quality magnetic field, e.g. a multipole field to be provided. The arrangement of the present invention is particularly suited to providing such cancellation of the solenoid field. One reason for this is that if currents are passed through each of the coils in different directions and the turns of the coils are suitably arranged, i.e. such that the currents in the transverse directions cancel and add in the axial direction, the effect is that the solenoid (axial) fields cancel, and the desired fields, e.g. the multipole or non-multipole fields, add (e.g. for a dipole this field is perpendicular to the axis), giving a high quality magnetic field. The magnetic field generated by the two (or more) coils of the present invention can therefore have smaller errors than a conventional magnet with coil segments because it is a better approximation to the required current distribution.

Preferably in all aspects of the invention the coils are wound around at least one former. This facilitates production of the coils. In exemplary embodiments the former is as thin as possible whilst retaining sufficient strength to support any pre- stress applied during winding as well as supporting the coil during operation from the Lorentz forces generated when current is passed through the coil.

In one set of embodiments only one former is used, i.e. for the inner coil winding with the outer windings being wrapped around the inner coil. Being able to wind adjacent turns next to each other and dispense with a former for each separate coil allows the magnet, in one set of embodiments, to be manufactured simply by wrapping the coils around a base structure, e.g. a support, a former or a beam pipe, where they may be fixed in place using a resin. Further coils can then be added by winding them around the existing coils and again pressing them into a resin to fix them in place, without needing to provide an intermediate former. This brings a further benefit of increasing the packing factor of the wires making up the coils, i.e. the density of the wires in the magnet, which increases the intensity of the magnetic field generated. This is because the coils can be closer together because there are no intermediate supports which tend to lead to wasted volume, which is having to be magnetised by the coils, but which is not being usefully used by particles that the magnetic field is acting on, i.e. in the bore of the magnet.

In an alternative set of embodiments a separate former is used for each coil. This affords the advantage that each coil is mounted on a former and therefore this helps to improve the alignment of the path the turns of each coil take. The former supporting the innermost coil will of course typically define the maximum aperture of the particle accelerator.

In some embodiments it is envisaged that pins pushed radially into the former at appropriate positions are used to position the coil turns, but preferably the former comprises a groove to receive the coil turns. Having a groove in the former provides a predefined path on the former for the turns and therefore allows the turns to be accurately positioned on the former in order to produce the required magnetic field as well as giving the turns longitudinal support which is particularly important for the higher order multipoles or non-multipole magnetic fields in which the path of the turns is highly curved in the axial direction. The Applicant has found that using a groove produces a better alignment tolerance for the coil turns and therefore a higher quality magnetic field compared to using pins to locate the turns.

Possible materials out of which to make the former include aluminium, e.g. AI-6063-T6, an epoxy glass fabric laminate, e.g. 10G/40, or stainless steel, e.g. austenitic steel AISI 316L. All of these materials are non-magnetic and strong which therefore provides a good structure onto which the coil windings can be formed to create an accurate path for the turns, without influencing the magnetic field. In embodiments which have a groove on the former, these materials provide a good basis for machining the groove. These materials are also suitable to be used with a cryogenic cooling system which is used to cool the coil windings to cryogenic temperatures, which is necessary if the magnet is superconducting.

The inventors have realised that although the two coil magnet of the present invention offers a significant improvement over a conventional coil in mitigating field errors and the space issue, there is still a residual component of the solenoid field along the axis of the coil which cannot be eliminated. This, they have realised, arises because the two coils which comprise the magnet have slightly different radii owing to them being concentric. Therefore in one set of embodiments the magnet comprises third and fourth concentric tubular coils, i.e. the plurality of wires comprise the turns of third and fourth concentric tubular coils. Preferably the four coils are arranged to produce a magnetic field in which there is substantially no solenoid component of the magnetic field. In a preferred set of embodiments the coils of the four coil magnet are arranged such that apart from their respective radii, the path of the first coil is the same as the path of the fourth coil, and wherein, apart from their respective radii, the paths of the second and third coils are the same as each other. In this embodiment the first coil is the innermost coil, i.e. the one with the smallest radius, the fourth coil is the outermost coil, i.e. the one with the largest radius, and the second and third coils are the middle coils between the first and fourth coils. Except where otherwise discussed, the present invention is not limited to a particular shape of coil. The use of the word radius thus does not imply that the coils have a circular cross-section but rather it is just an indication of the relative size of the coils. This also makes the coil symmetric to the centre plane which means that all point-symmetric field inhomogeneities disappear.

Providing four coils in oppositely-wound pairs, with the inner and outer coils having the same path (at slightly different radii of course) and the intermediate two coils following the same path (again apart from the different radii of the coils), has been found to reduce the problem of the non-cancelling solenoid field in the two-coil magnet significantly since when an equal current is sent through the coils (in one direction for the inner and outer coils and the opposite direction for the two middle coils). In this set of embodiments with four coils, the solenoid field substantially cancels leaving a multipole field remaining. This design of magnet coil is therefore particularly suitable for use when a magnet with a short length and/or large aperture is required, because the cancellation of the solenoid field helps to reduce the field errors significantly at the ends of the magnet, although many other applications are possible.

Although this set of embodiments is defined in terms of two intermediate coils, i.e. the second and the third coils, it should be understood that because these follow the same path, they can be thought of as a single coil with a double layer of turns or a single layer with turns of twice the radial thickness.

Although the invention has thus far been described in terms of two or four coils this should not be considered as limiting the claims to requiring only two or four coils; additional coils could be added, e.g. to provide a current distribution more accurately matched to that calculated, to provide improved cancellation of the solenoid field, to increase the magnetic field, to provide additional multipoles, or for some other reason. Preferably however, if additional coils are provided, these are provided concentrically in groups of two or four, i.e. a magnet with more than two coils would preferably have four, six, eight or ten coils, and a magnet with more than four coils would preferably have eight, twelve or sixteen coils, etc. In the set of embodiments in which groups of four coils are provided, preferably these groups of four coils are each arranged in the same manner as the first-recited group, e.g. with the inner and the outer coils of each group following a first path and the middle coils following a second path, or in whichever arrangement is provided in the first group.

Preferably the number of turns is the same for all the coils. Preferably the winding pitch has the same value for all the coils. Preferably the spacing between adjacent turns is the same for all the coils. Preferably the coils are configured so that the current passing through all the coils has the same absolute value.

Preferably the coils are configured so that current passes through the first coil in the opposite direction to the current passing through the second coil. Where third and fourth coils are provided, preferably the coils are configured so that current passes through the third coil in the same direction as the current passing through the second coil, and the current passes through the fourth coil in the same direction as the current passing through the first coil. These features help to ensure that the magnet in accordance with the invention generates a current distribution, resulting in the desired magnetic field, which is as accurate as possible to the calculated current distribution, and the solenoid field component is reduced as much as possible.

The turns of the coils are preferably arranged to provide a multipole magnet, a superposition of multipole magnets, a non-multipole magnetic field or

superposition of a multipole and a non-multipole field, as has been discussed previously, by arranging the coils to give the required current distribution. In some accelerator designs the magnetic field required from a single magnet is a mixture of the fields produced from different multipoles or non-multipole fields, e.g. mostly a dipole field with a smaller component of a quadrupole field. Such a type of magnet is known as a combined function magnet. This can be achieved in accordance with the present invention, again by arranging the turns of the coils to give the appropriate current distribution for that superposition, as has been discussed previously.

However for the type of combined function coil discussed above the ratio of the different component fields is hardwired into the coil thus eliminating one degree of freedom. In an alternative set of embodiments, when a combined function magnet is required, a plurality of different discrete magnets concentric to each other can be provided. For example, this type of combined function magnet could have two or four coils creating a dipole field which could be concentric with a higher order multiple or a non-multipole magnet comprising two or four additional coils. The magnetic field produced in such embodiments is a superposition of the fields created by each magnet, thus allowing the different component terms to be achieved, but with a separate set of coils for each different discrete magnetic field. The path of the turns for each of these coils and the current passing through them can be chosen to give the desired ratio between the strength of the different magnetic fields which are included in the magnet. As each of the components is effectively a separate magnet, the current passing through the coils can be fine- tuned individually to give the desired balance between the different components once the magnet is in operation which is not possible with a "hardwired" combined function magnet, and therefore such an arrangement gives greater flexibility in this regard.

The different component magnets can be arranged in arbitrary order, but usually it is preferred to position the component magnet which produces the highest magnetic field as the innermost magnet. This helps to minimize the air volume the particular magnet has to magnetise.

As has been explained previously, different types of magnets are used in a particle accelerator to keep the beam of particles within the accelerator. There may be a further requirement that the magnets are arranged for focusing and

defocusing. Focusing and defocusing magnets are typically arranged in a lattice along the beam line of a particle accelerator, alternating between focusing and defocusing magnets. The alternate focusing and defocusing magnets converge and diverge the beam of particles in the accelerator respectively and, like an array of optical lenses, help to collimate the beam of particles in the accelerator and confine the particles within the aperture of the magnets. Without both focusing and defocusing magnets present, the particle beam would quickly be lost from the accelerator.

Focusing and defocusing magnets have different magnetic polarities, so the current direction is reversed to change from one type of magnet to the other. In practice focusing and defocusing magnets may differ in their magnetic field strength, depending on the design of the lattice of the accelerator in order to provide the necessary forces on the particle beam. Therefore the magnet parameters, e.g. number of turns, winding pitch, spacing between adjacent turns, etc, may differ between focusing and defocusing magnets, or it may be possible to use the same magnet design for both types of magnets but with different currents. As magnets focus in one plane and defocus in the other, the magnets in

accordance with the present invention are suitable for use as both focusing and defocusing magnets.

As was mentioned previously, magnets in accordance with the invention could be suitable for any particular use in particle accelerators. However, owing to their suitability in providing a high quality magnetic field in which the field errors at the ends of the magnet are reduced, the magnets of the present invention are particularly suitable for magnets with a short length and/or a large aperture, which gives them a small aspect ratio (ratio of length to radius of aperture), in which the effects of the ends of the coil, which return the current, become important.

Therefore, in one set of embodiments, e.g. suitable for accelerating protons, the coils have a length of between 30 and 80 cm, e.g. between 50 cm and 60 cm, e.g. approximately 55 cm. In another set of embodiments, e.g. suitable for accelerating carbon ions, the coil length is between 80 cm and 150 cm, e.g.

between 90 cm and 140 cm, e.g. between 100 cm and 130 cm, e.g. approximately 1 15 cm. Also, as mentioned previously, the present invention is equally applicable for longer magnets. Indeed the possibility of manufacturing the magnet without intermediate formers is advantageous in the context of longer magnets since it is either too expensive or practically infeasible to produce very long tubular formers

As previously explained, the size of the coils differ from one to another. The mean distance across a coil (e.g. the radius of a circular coil, or the mean of the half axes of an elliptical coil) is in one set of embodiments however between 5 cm and 40 cm, e.g. between 10 cm and 30 cm, e.g. between 10 cm and 25 cm or 15 cm and 30 cm.

Thus it can be seen that the invention can be implemented by a magnet which is short with a large aperture and is therefore suitable for small radius particle accelerators, e.g. of radius approximately 6 m, where a large aperture is required to conserve a large proportion of the injected particles. However it will be appreciated that for the embodiments in which multipole magnets of different orders are nested within each other, e.g. with a dipole in the centre inside a quadrupole which is in turn inside a sextupole, etc., the radii of the coils will be different for the different multipole coils. This contrasts to the length of the coils which will be relatively similar for all the different order multipoles. The invention can be used where the ratio of the coil length to the magnet aperture radius (the "aspect ratio") is less than 15:1 . In one set of embodiments, e.g. suitable for accelerating protons, the ratio is less than 5:1 , e.g. less than 4:1 , and e.g. less than 3:1 . This is small compared to the typical corresponding ratio for a conventional magnet. In another set of the embodiments suitable for accelerating carbon ions the coils are longer and therefore for example the coil aspect ratio is less than 8:1 , e.g. less than 7:1 , e.g. less than 6:1. For magnets with a small aspect ratio, the end effects of the magnetic field become very important, and thus it can be seen that the present invention gives a magnetic field which is particularly suitable for magnets with these dimensions.

Although the high quality nature of the magnetic fields make the magnets of the present invention suitable for short magnets, they are equally applicable for longer magnets.

Magnets in accordance with all aspects of the invention could be

conventional electromagnets, superconducting or hybrid magnets. Where it is a conventional electromagnet, if cooling is required, the coils may be e.g. cooled by water. In one set of embodiments the wires are made from NbTi superconductor, though other materials such as Nb₃Sn and high temperature superconductor materials are also envisaged. The copper to superconductor ratio may be as high as 20:1 , but preferably the copper to superconductor ratio is between 1.2:1 and

2.1 :1 , preferably approximately 1 .35:1 . Superconducting wires allow high magnetic fields to be reached which are not possible using a conventional electromagnet.

In one set of embodiments the wire used to wind the coils is a single filament wire, e.g. a superconducting NbTi wire with 54 NbTi filaments embedded in a Cu matrix, for example a single filament rectangular wire. In another set of embodiments the wire used to wind the coils is a Rutherford cable (a multi filament wire), e.g. with 5 strands each with a diameter of about 1 mm giving outer dimension of about 3 mm ^χ 2 mm.

In the embodiments in which superconducting wires are used, the coils can be cooled below the critical temperature of the superconductor using a bath cryostat, but many other methods known to those skilled in the art for realising the cryostat could alternatively be used. For a superconductor made of NbTi the critical temperature is approximately 9.2 K.

Preferably the plurality of turns for each coil comprises between 100 and 400 turns, preferably between 200 and 300 turns, preferably between 240 and 260 turns. Providing a large number of turns for a coil helps to provide a high, uniform magnetic field. The number of turns will vary depending on a number of factors including the desired magnetic field, the type of wire used and what sort of particles the accelerator including the magnet is designed to accommodate.

Preferably a combined function magnet in accordance with the present invention is arranged to deliver a peak magnetic field between 1 T and 8 T, e.g. between 2 T and 6 T, e.g. between 4 T and 5 T, e.g. about 4.5 T. As described previously a combined function magnet is a superposition of a number of different magnetic fields which each have a different value for their peak magnetic field. Examples of such values for the different multipoles for a lattice of magnetic length 314.4 mm, with the peak fields calculated at a radius of 0.14 m are: 1 .95 T for a dipole, 1.65 T for a quadrupole, 0.71 T for a sextupole, and 0.19 T for an octupole. The combined field of all these multipoles in such a combined function magnet varies across the horizontal direction, i.e. x, from 0.8 T to 4.5 T.

A magnet capable of delivering high strength magnetic fields set out above is suitable for inclusion in a small radius particle accelerator, e.g. of radius 6 m where a large magnetic field is needed to bend protons with an energy of approximately 250 MeV. As with many of the other dimensions and values which describe the magnet coil, the peak magnetic field differs between different multipoles and whether the magnet is being used for proton or carbon acceleration.

The magnet provided by the present invention is suitable to be used in a number of different particle accelerators, e.g. fixed field alternating gradient (FFAG) accelerators which have a number of applications, such as in hospitals for radiotherapy treatment; in scientific research for neutrino factories, muon sources and proton drivers; and in industry for accelerator driven subcritical reactors (ADSR). Also, as has been explained previously, the magnet provided by the present invention is particularly suitable to be used in a particle accelerator for providing a high intensity particle, e.g. proton, beam, which includes some of the above examples.

Certain preferred embodiments of the invention will now be described, by way of example only, with reference to the accompanying drawings in which:

Fig. 1 shows a schematic view of a step in the method of designing a magnet;

Fig. 2 shows a plot of the current density distribution required to generate a magnetic vector potential for a dipole magnet; Fig. 3 shows an isometric view of the structure of the current paths to give the calculated current distribution shown in Fig. 2;

Fig. 4a shows a side view of the physical implementation of the coils arranged along the current paths shown in Fig. 3;

Fig. 4b shows a cross-sectional view of the coils of Fig. 4a along the line A-

A;

Fig. 5 shows a plot of the multipole components generated by the coils shown in Figs. 4a and 4b;

Fig. 6 shows a plot of the current density distribution required to generate a magnetic vector potential for a quadrupole magnet;

Fig. 7a shows a cross-sectional view of two coils arranged to the current distribution shown in Fig. 6;

Fig. 7b shows a side cross-sectional view of the coils of Fig. 7a;

Fig. 8 shows a plot of the multipole components generated by the coils shown in Fig. 7;

Fig. 9 shows a plot of the two-dimensional cross-sectional distribution for a magnetic vector potential for generating a non-multipole magnetic field;

Fig. 10 shows a plot of the current distribution on required to give the magnetic vector potential shown in Fig. 9;

Fig. 1 1 shows an isometric view of the structure of the current paths to give the calculated current distribution shown in Fig. 10;

Fig. 12a shows a cross-sectional view of the current paths shown in Fig. 1 1 ; Fig. 12b shows a side view of the current paths shown in Fig. 1 1 ;

Fig. 13 shows a cross-sectional view of a set of points at which the current distribution has been calculated to give the current distribution shown in Fig. 10, and a plot of the magnetic vector potential generated by the current distribution; and

Fig. 14 shows a plot of the multipole components generated by the current paths shown in Fig. 1 1.

Fig. 1 shows a schematic view of a magnetic vector potential 2 that is desired to be generated by a magnet. The magnetic vector potential could be chosen to be one to generate any type of magnetic field, e.g. multipole or non- multipole, but in this case it has been chosen to be a dipole magnetic field. Also in this case, the shape of the bore of the magnet is desired to be an ellipse and therefore in accordance with the invention, an ellipse 4 is drawn so to cover the cross-sectional area which will be enclosed by the magnet. Within this ellipse 4, a further ellipse 6 is drawn, and a set of points, Pi, P₂, P₃, P₄,Ps, etc, are set out at regular points along the perimeter of the inner ellipse 6, e.g. at steps of π/20 in the azimuthal angle around the ellipse 6.

At these set of points, P_n, the magnetic vector potential, A, is determined from the magnetic field by using the relationship B = V x A , where B is the magnetic field desired to be generated by the magnet. Working in the two

dA. „ dA, dimensions of the cross-section of the magnet, this is B As dx dy the magnetic field is derived by taking a spatial derivative of the magnetic vector potential, there is no unique magnetic vector potential which corresponds to the magnetic field, but an infinite number of possible solutions which vary by a uniform vector potential A₀, because V x A₀ = 0 . Therefore A₀ can be chosen to be any value which is suitable for the current distribution being provided. A fixed magnetic vector potential, e.g. A₀, is generated by a constant current in z (for circular structures), which allows the current to be shifted, e.g. to make the integral of the current around the bore of the magnet zero.

Once the magnetic vector potential, A_z, has been determined at the set of points, P_n, a current distribution at a set of points l , , U, , etc, on the perimeter of the oute n be calculated, using the relationship

, Θ) , which gives the magnetic vector potential at a point P

(having coordinates (/-,0)) due to current / (flowing in the z-direction), where R is the distance from the point P to the location of the point current / and a is the distance to the origin of the point current /. This results in a linear equation system being set up for the set of current point sources, l_m, on the perimeter of the outer ellipse 4, and the set of points P_n, at which the magnetic vector potential has been determined on the erimeter of the inner ellipse 6, giving

This can be seen to be a simple linear

equation system of the form Ax = b, from which x, the value of the currents, l_m, needed to generate the required magnetic vector potential, A_z, can be calculated, as A_z is known and the values eft A_nm can be determined, e.g. by placing a unit current at each of the set of current point sources on the perimeter of the outer ellipse 4 and calculating the values of A_nm).

Fig. 2 shows the calculated current density distribution 10 at the set of points, l_m, on the perimeter of the outer ellipse 4, which are required to give the desired magnetic vector potential, A_z. The current distribution 10 can be seen to deviate from a sinusoidal curve, which it would have followed had the bore of the magnet been circular, i.e. a cosine-theta distribution to give a dipole magnetic field.

Once the current has been calculated at the set of points, l_m, on the perimeter of the outer ellipse 4, the paths of the turns of the concentric tubular coils, which will provide the calculated current density distribution 10 when current is passed through them, need to be determined. The longitudinal position z_n of the n

wire can be calculated by evaluating z_n =∑ / for all n. Then, the z-function is i=

normalised and multiplied with an (arbitrary) maximum deflection/excursion per turn: z=z/max(|z|)-d, where d is the maximum deflection. This results in the determination of a value of the longitudinal position of the wire at each of the set of points on the perimeter of the outer ellipse 4.

Fig. 3 shows the paths 12, 14 which have been determined for two coils having a cross-section corresponding to the outer ellipse 4. The step of

determining the paths of the turns of the coils is performed in two dimensions, i.e. the longitudinal position of the wire on the outer ellipse 4 is determined for a single turn of the coils. However, to provide a magnetic field for a certain length along a section of a particle accelerator requires the coil to have multiple turns, as is shown in Fig. 3, which each follow the same path around the axis of the coil but are stepped from each other by a winding pitch. The first coil 12 is a mirror image of the second coil 14 about a plane through the middle of the coils perpendicular to the axis of the coils, i.e. the plane z = 0. It can be seen that the turns 12, 14 of the two coils are at a generally constant gradient (this being the same for both coils) to the axis of the coils, but angled in opposite directions so that the turns cross each other at a generally constant angle at the points on the ellipse where y = 0, i.e. Θ = 0 and π on the plot of Fig. 2. At the points where x = 0, i.e. θ = π/2 and 3π/2, the turns 12, 14 of the coils are generally parallel to each other.

It will be appreciated that when a current / is passed through the turns 12, 14 of the coils, the current being of the same magnitude for both coils and being passed in opposite directions through the coils, the current density distribution as shown in Fig. 2 will be generated. This is because in the sections of the coils at the x = 0 where the turns are parallel to each other, the currents flow in opposite directions in the separate coils. The vector addition of these currents, i.e. / + (-/), gives a net resultant current of zero, and therefore the current density distribution is zero in this region, corresponding to θ = π/2 and 3π/2 on the plot in Fig. 2.

Furthermore, with the turns of the coils running perpendicular to the axis of the coils, there is no component of the current flowing in the direction parallel to the axis.

In the sections of the coils where the turns cross each other at an angle, i.e. at y = 0, the vector addition of the currents in the two coils 12, 14 results in the components in the azimuthal direction cancelling, and the components in the direction parallel to the axis of the coils adding. Therefore in these regions the resultant current density distribution is in the direction parallel to the axis of the coils, and zero in the azimuthal direction. As the current travels in the opposite direction on the opposite side of the coils, this results in the current density distributions on opposite sides of the coils being in opposite directions.

The final step comprises actually winding the turns of the physical coils, which generates the coils 16, 18 shown in Figs. 4a and 4b. The physical coils 16, 18 follow the current paths 12, 14 shown in Fig. 3 as closely as possible, but owing to the physical thickness of the wires, both coils cannot be wound at exactly the position of the ellipse of the magnet bore, i.e. the outer ellipse 4 on which the current paths 12, 14 are determined, and so are wound concentric to each other, as can be clearly seen in Fig. 4b.

The magnet, when assembled, therefore comprises an inner coil 16 arranged concentrically within an outer coil 18. Each of the coils 16, 18 has an elliptical cross-section, giving the magnet an elliptical bore. Both the inner coil 16 and the outer coil 18 comprise a plurality of turns which are wound in a generally circumferential manner around the common axis of the coils. Although not shown, the inner coil 16 could be formed on a support to provide support for the windings. It is not necessary to provide an intermediate support between the inner coil 16 and the outer coil 18, the other coil 18 can simply be wound around the inner coil 16. As the wire of the inner coil 16 is being wound round the support (if one is used), and then when the outer coil 18 is wound around the inner coil 16, a tension is applied to the wire in order to keep the wire on the correct path. After winding the coil it is usually impregnated with epoxy resin to keep the wire in place and to aid electrical insulation. Alternatively a pre-impregnated fibre cloth can be used.

The wire can either be a single filament wire such as a rectangular filament wire with a copper to superconductor ratio of 1.3:1 (obtained from Oxford

Instruments Superconducting Technology, 600 Milik Street, PO Box 429, Carteret, NJ, 07008, USA), or a Rutherford cable which comprises a number of individual strands. With either the single filament wire or the Rutherford cable, the wire is usually insulated before being wound.

In operation, a current / is passed through the turns of the coils 16, 18 which, as already explained, generates the current density distribution shown in Fig. 2 and therefore the desired magnetic field. Fig. 5 shows a plot of the coefficients of the multipole components of the dipole magnet as shown in Figs. 4a and 4b, when currents of equal magnitude are passed through the coils from opposite ends of the magnet. The vertical axis 20 denotes the value of the multipole coefficients at a reference radius r₀ and the horizontal axis 22 denotes the distance along the coil. Starting from the left hand side of the coil (as viewed in Fig. 4a) and from the left hand side of the plot in Fig. 5, the magnitude of the dipole coefficient 24 increases rapidly to a relatively constant value. Travelling rightwards through the magnet, the dipole magnetic field 24 remains at a relatively constant value throughout the length of the magnet, and then falls back to zero in a symmetric manner when the end of the magnet is reached. Fig. 5 also shows that the magnetic field generated by the dipole magnet is almost purely composed of the dipole component. Only the sextupole component 26 can be seen to be non-zero at the coil entrance and exit, but its magnitude is negligible compared to that of the dipole component 24.

Fig. 6 shows a similar plot of a current density distribution 1 10 to that shown in Fig. 2, but instead that has been calculated for an elliptical bore coil for generating a quadrupole magnetic field. The same procedure that was described for calculating the current density distribution shown in Fig. 2 for a dipole magnetic field was used to generate the current density distribution 1 10 shown in Fig. 6, i.e. decide on a desired magnetic field - in this case a quadrupole field, determine the cross-sectional shape of the magnet - in this case an ellipse, determine the magnetic vector potential at a set of points on an ellipse within the bore of the magnet, and then calculate the current distribution at a set of points on the outer ellipse to give the current density distribution 1 10. As with the case of the dipole field and magnet in Figs. 1 to 5, two concentric coils 1 16, 1 18 are used to generate the calculated current distribution 1 10 in order to provide the desired magnetic field, and these coils are shown in Figs. 7a and 7b, Fig. 7b being a side view cross-section along the line B-B in Fig. 7a. Again, each of the coils 1 16, 1 18 has an elliptical cross-section as shown in Fig. 7a, giving the magnet an elliptical bore. The paths taken by the turns of the coils 1 16, 1 18 can be better seen from the side view in Fig. 7b, which show that the turns of the coils 1 16, 1 18 take a different path in order to produce the required current distribution for the quadrupole magnet compared to the dipole magnet. Again, apart from its slightly different radius, the inner coil 1 16 is a mirror image of the outer coil 1 18 about a plane through the middle of the coils perpendicular to the axis of the coils.

The operation of the two coil quadrupole magnet shown in Figs. 7a and 7b is very similar to the operation of the dipole magnet of Figs. 4a and 4b. A current / is passed through the turns of the coils 1 16, 1 18, the current being of equal magnitude for both coils. The current is passed in one direction (e.g. left to right in Fig. 7b) for the inner coil 1 16 and in the opposite direction (right to left in Fig. 7b) for the outer coil 1 18. In the sections of the coils where the turns are parallel to each other, the currents flow in opposite directions in the separate coils. The vector addition of these currents, i.e. / + (-/), gives a net resultant current of zero, and therefore the current distribution is zero in this region, i.e. there is no component of the current flowing in the direction parallel to the axis. This therefore generates the calculated current distribution shown in Fig. 6.

Fig. 8 shows a plot of the coefficients of the multipole components of the quadrupole magnet as shown in Figs. 7a and 7b, when currents of equal magnitude are passed through the coils from opposite ends of the magnet. The vertical axis 120 denotes the value of the multipole coefficients at a reference radius r₀ and the horizontal axis 122 denotes the distance along the coil. Starting from the left hand side of the coil (as viewed in Fig. 7b) and from the left hand side of the plot in Fig. 8, the magnitude of the quadrupole coefficient 124 increases rapidly to a relatively constant value. Travelling rightwards through the magnet, the quadrupole magnetic field 124 remains at a relatively constant value throughout the length of the magnet, and then falls back to zero in a symmetric manner when the end of the magnet is reached. Fig. 8 also shows that the magnetic field generated by the quadrupole magnet is almost purely composed of the quadrupole component. Only the dipole component 126 and the octupole component 128 can be seen to be non-zero at the coil entrance and exit, but their magnitude is negligible compared to that of the quadrupole component 124. If a four coil arrangement were to be used, the dipole component has been shown to be eliminated.

As has been explained previously, the present invention is also suitable for providing magnets with a non-multipole magnetic field. Fig. 9 shows a plot of a two- dimensional cross-sectional distribution for a normalised magnetic vector potential 203 for generating a non-multipole magnetic field. At the points (±0.01 ,0) m there are singularities 205, 207 in the magnetic vector potential which would require infinite currents to generate, which is clearly unphysical. Therefore these points need to lie outside of the two-dimensional cross-sectional space chosen on which to calculate the current distribution, e.g. a butterfly or hour-glass shaped space could be used, but in this embodiment an ellipse is chosen (see Figs. 1 1 , 12a and 13).

The current density distribution 210 calculated for generating this non- multipole magnetic vector potential is shown on Fig. 10, which is similar to those shown for the dipole and quadrupole in Figs. 2 and 6 respectively. The large magnitude current at Θ = 0, π and 2π (0 and 2π corresponding to the same point) corresponds to the points on the ellipse which are closest to the singularities 205, 207 in the magnetic vector potential 203. The same procedure, as for the dipole and quadrupole, was used to generate the current distribution 210 shown in Fig. 10, i.e. decide on a desired magnetic vector potential, determine the cross-sectional shape of the magnet - in this case an ellipse, , and then calculate the current distribution at a set of points on the outer ellipse to give the current density distribution 210.

Figs. 1 1 , 12a and 12b show the paths 212, 214 which have been

determined for two coils having a cross-section corresponding to the ellipse on which the current distribution was calculated. As for the dipole and the quadrupole, the step of determining the paths of the turns of the coils 212, 214 is performed in the two dimensions of the ellipse cross-section, with the longitudinal position of the wire on the ellipse being determined for a single turn of the coils. The complete path of the turns, i.e. for the whole length of the magnet, is then simply obtained by repeating this path for all the turns of the coils 212, 214. The first coil 212 is a mirror image of the second coil 214 about a plane through the middle of the coils perpendicular to the axis of the coils, i.e. the plane z = 0. As can be seen from the paths of the turns of the coils 212, 214, and from the calculated current distribution 210, the paths do not follow a simple pattern as for the dipole and the quadrupole, particularly in the sections near to the singularities in the magnetic vector potential.

It will be appreciated that when a current / is passed through the turns 212, 214 of the coils, the current being of the same magnitude for both coils and being passed in opposite directions through the coils, the current density distribution as shown in Fig. 10 will be generated.

Again, as with the dipole and the quadrupole, two concentric coils are provided to generate the calculated current distribution 210 in order to generate the desired magnetic field, the paths taken by the turns of the coils having been determined and shown in Figs. 1 1 , 12a and 12b. Again, the coils have an elliptical cross-section giving the magnet an elliptical bore, and, apart from its slightly different radius, the inner coil is a mirror image of the outer coil about a plane through the middle of the coils perpendicular to the axis of the coils.

The operation of the two coil non-multipole field magnet is very similar to the operation of the dipole and quadrupole magnets. A current / is passed through the turns of the coils, the current being of equal magnitude for both coils but passed in opposite directions through the coils. The vector addition of the currents around the coils generates the net calculated current distribution.

Fig. 13 shows a simulation of the magnetic vector potential 21 1 that is generated by the non-multipole magnet, with the current distribution having been evaluated at the points around the ellipse 204 which defines the two-dimensional cross-sectional internal space of the magnet. As can be seen, this is an accurate representation of the desired magnetic vector potential 203 as shown in Fig. 9.

Fig. 14 shows a plot of the representation of the coefficients of the multipole components of the non-multipole magnetic field (with many high order multipole component also being present), when currents of equal magnitude are passed through the coils from opposite ends of the magnet. The vertical axis 220 denotes the value of the multipole coefficients at a reference radius r₀ and the horizontal axis 222 denotes the distance along the coil. Starting from the bottom of the coil (as viewed in Figs. 1 1 and 12b) and from the left hand side of the plot in Fig. 14, the magnitude of the quadrupole 224, octupole 226 and duodecapole 228 components increase rapidly to a relatively constant value. Travelling upwards through the magnet, the quadrupole 224, octupole 226 and duodecapole 228 magnetic fields remain at a relatively constant value throughout the length of the magnet, before rising slightly and then falls back to zero when the end of the magnet is reached. Also starting from the bottom of the coil, the magnitude of the dipole 223, sextupole 225 and decapole 227 components increase in the opposite (negative) direction and quickly fall back to zero, and then rise to a positive value at the other end of the magnet. The vector potential is difficult to approximate using a conventional multipole expansion as there are too many higher order multipole components present. There are contributions from these higher order multipole components, but these are negligible compared to the other terms.

It will be appreciated by those skilled in the art that many variations and modifications to the embodiments described above may be made within the scope of the various aspects of the invention set out herein. It will be appreciated that the present invention gives a general method of designing and then manufacturing a wide range of different types of magnets, it is not limited to any particular magnetic field, be it a multipole, a non-multipole field, or any superposition of magnetic fields. Furthermore it is not limited to any particular shape of magnet, any size or shape of bore could be used.

More than two coils could be employed even to produce a single magnet. Also it is not essential for the oppositely-directed coils to have similar paths, their pitch and currents could be manipulated instead to give a similar result. This is particularly the case when the magnetic field changes along the length of the magnet, and the current distribution is calculated at a number cross-sections along the length in order to generate the required magnetic vector potential.

Claims

1 . A method of designing a magnet for a particle accelerator based on a desired magnetic field to be generated by the magnet comprising the steps of: determining a two-dimensional cross-sectional internal space to be enclosed by the magnet,

2. A method as claimed in claim 1 , wherein the plurality of wires comprise turns of at least two concentric tubular coils.

3. A method as claimed in claim 2, wherein each of said coils comprises a plurality of turns for which the spacing between adjacent turns is not constant around the perimeter of the magnet.

4. A method as claimed in claim 2 or 3, wherein the path of the second coil is generated by reversing the path in z compared to the first coil, where z is the coordinate along the axis of the coils.

5. A method as claimed in claim 2, 3 or 4, wherein the magnitude of the current passed through each of the at least two coils has the same value.

6. A method as claimed in any of claims 2 to 5, wherein the current is sent in one direction through the first coil and the opposite direction through the second coil.

7. A method as claimed in any of claims 2 to 6, wherein the paths of the turns are arranged such that for the first coil z = ρ(θ) + ΐ(θ), where z is the coordinate along the axis of the coils, Θ is the azimuthal angle, ρ(θ) is a term representing the winding pitch of the turns, and f(6) is a term chosen to provide the required current density distribution, and for the second coil z = ρ(θ) - ΐ(θ).

8. A method as claimed in claim 7, wherein ρ(θ) is constant along the length of the coils.

9. A method as claimed in claim 7, wherein at least one of ρ(θ) and f(6) is varied along the length of the magnet.

10. A method as claimed in claim 7, 8 or 9, wherein f( Θ) describes a

superposition of magnetic field components.

1 1 . A method as claimed in claim 10, wherein the coefficients of the

superposition of magnetic field components vary along the length of the magnet.

12. A method as claimed in claim 1 1 , wherein the magnetic field components provide a plurality of zones along the magnet in which the coefficients are constant.

13. A method as claimed in claim 12, wherein the varying of the coefficients between each of the different zones is provided in a transition region.

14. A method as claimed in claim 13, wherein the transition region comprises ten or fewer turns, preferably five or fewer turns.

15. A method as claimed in any of claims 2 to 14, wherein the coils are arranged to produce a magnetic field in which the solenoid components of the magnetic field generated by the coils are at least partially mutually cancelled.

16. A method as claimed in any of claims 2 to 15, wherein the plurality of wires comprise the turns of third and fourth concentric tubular coils.

17. A method as claimed in claim 16, wherein the four coils are arranged to produce a magnetic field in which there is substantially no solenoid component of the magnetic field.

18. A method as claimed in claim 16 or 17, wherein the coils of the four coil magnet are arranged such that apart from their respective radii, the path of the first coil is the same as the path of the fourth coil, and wherein, apart from their respective radii, the paths of the second and third coils are the same as each other.

19. A method as claimed in claim 16, 17 or 18, wherein the coils are configured so that current passes through the third coil in the same direction as the current passing through the second coil, and the current passes through the fourth coil in the same direction as the current passing through the first coil.

20. A method as claimed in any of claims 2 to 19, wherein the number of turns is the same for all the coils.

21 . A method as claimed in any preceding claim, wherein the two-dimensional cross-sectional internal space is non-circular.

22. A method as claimed in any preceding claim, wherein the desired magnetic field is a non-multipole magnetic field.

23. A method as claimed in any preceding claim, wherein the plurality of points within the two-dimensional cross-sectional internal space, at which the magnetic vector potential is determined, are themselves located on the perimeter of a further two-dimensional cross-sectional internal space.

24. A method as claimed in claim 23, wherein the further two-dimensional cross- sectional internal space is the same shape as the two-dimensional cross-sectional internal space of the bore of the magnet but with a smaller size.

25. A method as claimed in any preceding claim, wherein the number of points used to define the magnetic vector potential is between 40 and 80.

26. A method as claimed in any preceding claim, wherein the plurality of points at which the current distribution is calculated, on the perimeter of the two- dimensional cross-sectional internal space, are at regular steps in the azimuthal angle.

27. A method as claimed in claim 26, wherein the plurality of points at which the magnetic vector potential is determined is chosen at the same steps of the azimuthal angle as the plurality of points at which the current distribution is calculated.

28. A method as claimed in any preceding claim, wherein the number of points chosen at which to determine the magnetic vector potential is the same as the number of points at which the current distribution is calculated.

29. A method as claimed in any preceding claim, comprising repeating the steps of determining the magnetic vector potential at a plurality of points inside said internal space, and calculating the current distribution at a plurality of points around the perimeter of the internal space to give the desired magnetic vector potential, at a plurality of sections along the length of the magnet.

30. A method for manufacturing a magnet for a particle accelerator comprising the steps of:

designing a magnet according to the method as claimed in any preceding claim,

providing a plurality of wires, and

31 . A magnet for a particle accelerator manufactured according to the method as claimed in claim 30.

32. A magnet for a particle accelerator comprising at least two concentric tubular coils, each of said coils comprising a plurality of turns, wherein the turns are arranged to provide a non-multipole magnetic field.

33. A magnet for a particle accelerator comprising at least two concentric tubular coils, each of said coils comprising a plurality of turns for which the spacing between adjacent turns is not constant around the perimeter of the magnet, wherein the turns are arranged around a non-circular cross-sectional internal space to provide a multipole magnetic field.

34. A magnet as claimed in claim 32 or 33, wherein the coils are wound around at least one former.

35. A magnet as claimed in claim 34, wherein the former comprises a groove to receive the coil turns.

36. An accelerator comprising a magnet as claimed in any of claims 31 to 35.

37. An accelerator as claimed in claim 36 suitable for accelerating protons.

38. An accelerator as claimed in claim 36 suitable for accelerating carbon ions.

39. An accelerator as claimed in claim 36, 37 or 38 wherein the accelerator is a fixed field alternating gradient accelerator.

40. A computer software product for designing a magnet for a particle accelerator, adapted when run on a suitable data processing means, to perform the method as claimed in any of claims 1 to 30.

41 . A computer software product for designing a magnet for a particle accelerator, adapted when run on a suitable data processing means, to perform the steps of: