ELECTROMAGNETIC DESIGN OF ACCELERATOR MAGNETS

ELECTROMAGNETIC DESIGN OF ACCELERATOR MAGNETS S. Russenschuck CERN, 1211 Geneva 23, Switzerland Abstract Analytical and numerical field computation methods for the design of conventional and superconducting accelerator magnets are presented. This report is an extract from a continuously updated ebook that can be downloaded from http://russ.home.cern.ch/russ. 1 Guiding fields for charged particles A charged particle moving with velocity v through an electro-magnetic field is subjected to the Lorentz force according to F = q(v B + E). (1) While the particle moves from the location r 1 to r 2 with v = dr dt, it changes it s energy by r2 r2 E = Fdr = q (v B + E)dr. (2) r 1 r 1 The particle trajectory dr is always parallel to the velocity vector v. Therefore the vector v B is perpendicular to dr, i.e., (v B) dr = 0. The magnetic field cannot contribute to a change in the particle s energy. However, if forces perpendicular to the particle trajectory are needed, magnetic fields can serve for guiding and focusing of particle beams. At relativistic speed, electric and magnetic fields have the same effect on the particle trajectory if E cb. A magnetic field of 1 T is then equivalent to an electric field of strength E = 3 10 8 V/m. A magnetic field of one tesla strength can easily be achieved with conventional magnets (superconducting magnets on an industrial scale can reach up to 10 T), whereas electric field strength in the giga volt / meter range are technically not to be realized. This is the reason why for high energy particle accelerators only magnetic fields are used for guiding the beam. Assuming a constant circular motion on the bending radius r = R (also called the radius of gyration) gives p = mv 0 = qb z R. (3) In charged particle dynamics it is customary to refer to the momentum pc which has the dimension of an energy and to express it in units of GeV. With q expressed in units of the electronic charge, the particle momentum in GeV/c is determined by: {p} GeV/c 0.3{q} e {R} m {B z } T. (4) The term B z R is called the magnetic rigidity and is a measure of the beams stiffness in the bending field. In circular proton machines the maximum energy is basically limited by the strength of the bending magnets. According to Eq. (4) the trajectory radius of the particle increases with the particle momentum. As both the maximum field and the maximum dimensions of the magnets are limited, the magnetic field must be ramped synchronously with the particle energy. Note that the effective radius is between 60% and 70% of the tunnel radius because of the dipole filling factor (space needed for focusing quadrupoles, interconnection regions, cavities etc.) and the straight parts around the collision points. 411

R. RUSSENSCHUCK 2 Conventional and superconducting magnets Fig. 1 left, shows the conventional dipoles for the LEP (Large Electron Positron collider) and the single aperture dipole model used for testing the dipole coil manufacture for the LHC. The field calculations were performed with the CERN field computation program ROXIE. The field representations in the iron yokes are to scale, the size of the field vectors changes with the different field levels. The maximum magnetic induction in the LEP dipoles is about 0.13 T. In order to reduce the effect of remanent iron magnetization, the yoke is laminated with a filling factor of only 0.27. It can be seen that the field is dominated by the shape of the iron yoke. If the excitational current is increased above a density of about 10 A/mm 2, superconducting technology has to be employed. Neglecting the quantum-mechanical nature of the superconducting material, it is sufficient to notice that the maximum achievable current density in the superconducting coil is by the factor of 500 higher than in copper coils. At higher field levels the field quality in the aperture is increasingly affected by the coil layout. Notice the large difference between the field in the aperture (8.3 T) and the field in the iron yoke (max. 2.8 T), which has merely the effect of shielding the fringe field. 3 Field quality in accelerator magnets The quality of the magnetic field is essential to keep the particles on stable trajectories for about 10 8 turns. The magnetic field errors in the aperture of accelerator magnets can be expressed as the coefficients of the Fourier-series expansion of the radial field component at a given reference radius (in the 2-dimensional case). In the 3-dimensional case, the transverse field components are given at a longitudinal position z 0 or integrated over the entire length of the magnet. For beam tracking it is sufficient to consider the transverse field components, since the effect of the z-component of the field (present only in the magnet ends) on the particle motion can be neglected. Assuming that the radial component of the magnetic flux density B r at a given reference radius r = r 0 inside the aperture of a magnet has been measured or calculated as a function of the angular position ϕ, we get for the Fourier-series expansion of the field B r (r 0, ϕ) = (B n (r 0 ) sin nϕ + A n (r 0 ) cos nϕ), (5) n=1 Btot (T) Btot (T) 2.65-2.8 2.50-2.65 2.36-2.50 2.21-2.36 2.06-2.21 1.92-2.06 1.77-1.92 1.62-1.77 1.47-1.62 1.33-1.47 1.18-1.33 1.03-1.18 0.88-1.03 0.73-0.88 0.59-0.73 0.44-0.59 0.29-0.44 0.15-0.29 0.00-0.15 2.65-2.8 2.50-2.65 2.36-2.50 2.21-2.36 2.06-2.21 1.91-2.06 1.77-1.91 1.62-1.77 1.47-1.62 1.32-1.47 1.18-1.32 1.03-1.18 0.88-1.03 0.74-0.88 0.59-0.74 0.44-0.59 0.29-0.44 0.15-0.29 0.00-0.15 Fig. 1: Magnetic field strength in the iron yoke and field vector presentation of accelerator magnets. Left: C-Core dipole (N I = 2 5250 A, B 1 = 0.13 T) with a filling factor of the yoke laminations of 0.27. Right: LHC single aperture coil test facility (N I = 480000 A, B 1 = 8.33 T). Notice that even with increased field in the aperture the field strength in the yoke is reduced in the cos Θ magnet design. 2 412

O @ N O ELECTROMAGNETIC DESIGN OF ACCELERATOR MAGNETS with A n (r 0 ) = 1 π π π B r (r 0, ϕ) cos nϕdϕ, (n = 1, 2, 3,...) (6) B n (r 0 ) = 1 π π π B r (r 0, ϕ) sin nϕdϕ. (n = 1, 2, 3,...) (7) If the field components are related to the main field component B N we get with N = 1 for the dipole, N = 2 for the quadrupole, etc.: B r (r 0, ϕ) = B N (r 0 ) (b n (r 0 ) sin nϕ + a n (r 0 ) cos nϕ). (8) n=1 The B n are called the normal and the A n the skew components of the field given in tesla, b n the normal relative, and a n the skew relative field components. They are dimensionless and are usually given in units of 10 4 at a 17 mm reference radius. In practice the B r components are calculated in discrete points ϕ k = kπ P π, k =0,1,2,..,2P -1 in the interval [ π, π) and a discrete Fourier transformation is carried out, i.e., for the normal component: B n (r 0 ) 1 P 2P 1 k=0 B r (r 0, ϕ k ) sin nϕ k. (9) The expression of field quality through the field components is perfectly in line with magnetic measurements using so-called harmonic coils, where the periodic variation of flux in radial or tangential rotating coils is analyzed with a Fast Fourier Transformation (FFT). Consider a so-called tangential coil as sketched in Fig. 2 (left) rotating in the aperture of a magnet. With ϕ = ωt + Θ, where ω is the angular velocity, the flux linkage through the coil at time t is given by Φ(t) = NL = n=1 ϕ+δ/2 ϕ δ/2 2NLr c n B r (r c, ϕ)r c dϕ sin( nδ 2 )[B n(r c ) sin(nωt + nθ) + A n (r c ) cos(nωt + nθ)], (10) H? % + E H & % + E H + E M J + E M J N + E! + A H = E? I D = B J + E! Fig. 2: Left: Cross-section of the long ceramic measuring shaft for the LHC magnets with the three tangential coils centered and aligned with ceramic pins. Right: Radial coil assembly. 3 413

R. RUSSENSCHUCK where N is the number of turns in the rotating coil, r c is the coil radius, L is the length of the rotating coil, δ is the opening angle of the coil and Θ is the positioning angle at t = 0. The voltage signal at time t is then U(t) = dφ dt = 2NLr c ω sin( nδ 2 )[ B n(r c ) cos(nωt + nθ) + A n (r c ) sin(nωt + nθ)]. (11) n=1 With the geometric parameters of the measurement coil resulting in a constant factor (which can be calculated and calibrated) the field harmonics can be obtained by means of the FFT of the voltage signal. 4 Maxwell s equations In this section we will present Maxwell s equations in global and integral form, and in the form of classical vector-analysis. For the solving of Maxwell s equations in various circumstances we further need the constitutive equations as well as the boundary and interface conditions. After a study of the properties of soft and hard magnetic materials we will be able to approximately calculate the main field in conventional accelerator magnets. We first summarize the governing laws of electromagnetism in their global form for all geometrical objects at rest. V m ( a) = I(a) + d Ψ(a), (12) dt U( a) = d Φ(a), dt (13) Φ( V ) = 0, (14) Ψ( V ) = Q(V ). (15) Eq. (12) is Ampère s magnetomotive force law and Eq. (13) is Faraday s law of electromagnetic induction. Eq. (14) is the magnetic flux conservation law and Eq. (15) is Gauss fundamental theorem of electrostatics. In SI units, U denotes the electric voltage [U] = 1 V and V m the magnetomotive force [V m ] = 1 A along the boundary a of a surface a. The electric flux through the boundary surface a = V of a volume V, is denoted Ψ with [Ψ] = 1 C = 1 A s, the magnetic flux is denoted Φ with [Φ] = 1 Wb = 1 V s. I is the electric current [I] = 1 A across the surface a. Q is the electric charge in a volume V, [Q] = 1 C = 1 A s. In integral form, Maxwell s equations read for the stationary case in SI units: H ds = J da + d D da, (16) a a dt a E ds = d B da, (17) a dt a B da = 0, (18) V V D da = V ρ dv. (19) The vector fields E(t, r), H(t, r) are the electric and magnetic field intensities, D(t, r), B(t, r) are the electric and magnetic induction (or flux density), J(t, r) is the electrical current density. These vector fields are assumed to be finite in the entire domain and to be continuous functions of position and time. Discontinuities in the field vectors may occur, however, on surfaces with an abrupt change of the physical properties of the medium. Such discontinuities must therefore be excluded until we have treated the interface conditions in Section 7. The field intensities E and H are integrated along a line, [E] =1 V/m, [H] =1 A/m, whereas the flux and current densities D, B and J are integrated over a surface, [D] =1 A s/m 2, [B] =1 V s/m 2, [J] =1 A/m 2. The electric charge density ρ is integrated on a 4 414

ELECTROMAGNETIC DESIGN OF ACCELERATOR MAGNETS volume, [ρ]= 1 As/m 3. The field intensity vectors and the flux density vectors have different natures. The line integrals of E and H are the electric voltage and magnetomotive force, respectively. The surface integrals of D, B, J are the electric flux, the magnetic flux and the electric current across the surface. As long as the necessary conditions (continuously differentiable vector fields, smooth surfaces with simply connected, closed, piecewise smooth and consistently oriented boundary, volumes with piecewise smooth, closed and consistently oriented surface) hold for the application of the Stokes and Gauss theorem which read curlg da = g ds, (20) a V div g dv = a V g da, (21) and if we assume that the surfaces and volumina are at rest, the field equations can be written as follows: curlh da = (J + D) da, (22) a a t curle da = B da, (23) a a t div B dv = 0, (24) V V div D dv = V ρ dv. (25) The equations can only be true for arbitrary volumes and surfaces if the following equation hold for the integrands: curlh = J + t D, (26) curle = t B, (27) div B = 0, (28) div D = ρ. (29) This is the classical vector analytical form of Maxwell s equations. The use of notation t instead of t is a way to establish t as an operator on the same footing as the differential operators grad, div and curl. Eq. (26) - (29) are Maxwell s equations in classical vector notation which is mainly due to O. Heaviside who eliminated the vector-potential and the scalar potential in Maxwell s original set of equations. Divergence free vector fields such as the magnetic induction are said to be solenoidal. From the first Poincaré lemma, div curlg = 0, it follows directly that div(j + t D) = div J + t ρ = 0 (30) which is called the conservation of charge law. The commutation of the div and t operators is admissible if the fields and charge distributions are smooth. The law can be written in integral form as J da + d ρ dv = 0 (31) dt or in global form as V V I( V ) + d Q(V ) = 0. (32) dt 5 415

R. RUSSENSCHUCK If at every point within a volume V the charge density is constant in time we get div J = 0, (33) J da = 0, (34) V I( V ) = 0, (35) the latter being known as Kirchhoff s node-current law applied in network analysis. 5 Constitutive equations Maxwell s equations constitute two vector equations (2 3 equations) and two scalar equations (alltogether 8 equations) for the unknown E, D, H, B, J and ρ (16 unknowns). If we also consider that Eq. (28) follows from Eq. (27), then Maxwell s equations can only be solved with the additional 9 material relations which are called the constitutive equations B = µh, D = εe, J = κe, (36) where µ, ε, κ are the permeability [µ] = V s A s A m, the permittivity [ε] = V m and the conductivity [κ] = A V m, respectively. (The international IEC standard recommends to use the symbol σ for the conductivity which is, however, also used for the surface charge density. Therefore we use the symbol κ as proposed in DIN 1324). These most simple forms of constitutive equations hold only for linear (field independent), homogeneous (position independent), isotropic (direction independent) and stationary media. The material parameters may, however, depend on the spatial position. If the physical properties in a specimen are the same in all directions, the material is said to be isotropic. In this case it is customary to express the permeability and permittivity as a function of the free space (vacuum) field constants with µ = µ r µ 0 and ε = ε r ε 0 where µ 0 = 4π 10 7 H/m and ε 0 = 8.8542... 10 12 F/m. The permeability and the permittivity of free space are related through the velocity of light in vacuum by c 0 = 1 ε0 µ 0 = 299 792 458 m/s. (37) In a more general case, e.g., with a permanent magnetic or electric polarization (which are volume densities of magnetic and electric dipole moments, respectively) it will prove convenient to introduce new vectors; the electric polarization P el and the magnetic polarization P mag. Often the magnetic polarization is replaced by the magnetization M in units of A/m. The material relations can then be expressed as B = µ 0 H + P mag (H) = µ 0 (H + M(H)), (38) D = ε 0 E + P el (E). (39) Note that the definition of the magnetization is not unique in literature and sometimes M contains µ 0. The polarization vectors are associated with matter and vanish in free space. For linear isotropic material the polarization vectors are parallel to the field vectors and are found to be proportional according to P el = χ e ɛ 0 E and M = χ m H so that for magnetic materials B = µ 0 H + µ 0 χ m H = µ 0 (1 + χ m )H = µ 0 µ r H = µh, (40) where µ r = 1 + χ m is the relative permeability, [µ r ] = 1 E and χ m is called magnetic susceptibility [χ m ] = 1 E. 6 416

B - ) A G, B ELECTROMAGNETIC DESIGN OF ACCELERATOR MAGNETS @ J C H = @ @ E L? K H? K H * 0 C H = @ @ E L @ J Fig. 3: Maxwell s House [3] with scalar and vector-potential as well as the material relations. Left facade: Faraday complex. Right facade: Maxwell s complex. Front facade: Electric fields and charges. Rear facade: Magnetic fields and impressed currents. Note how Ohm s law and the absence of free magnetic charges spoils the otherwise perfect symmetry. 6 Maxwell s House Employing the second Lemma of Poincaré we can express the magnetic field by means of a magnetic vector potential B = curla, (41) as the magnetic flux density is source free. If B in Eq. (27) is replaced by curla one obtains curl(e + t A) = 0 and therefore the electric field can be expressed as E = gradφ t A. (42) In the electrostatic case with t = 0 the electric field is curl free (irrotational), curle = 0, and hence E = gradφ. Since the curl of the magnetic flux density is, in general, non-zero, it cannot always be written as the gradient of a scalar potential function. If, however, a vector-field T is found such that curlt = J, (43) then the vector field H T is curl free, i.e., curl(h T) = 0 and therefore H can be expressed as H = gradφ red m + T. (44) T is called the electric vector potential in the context of the so-called T Ω method for steady-state field problems and φ red m is the reduced magnetic scalar potential. Several options for the choice of T exist, [7]. The most straight forward one is to use the Biot-Savart field H s computed from the impressed current distribution. The structure of the Maxwell equations and the electromagnetic potentials is revealed in a construction called Maxwell s House in [3]. The house is displayed in Fig. 3. For magneto(quasi)static problems with vanishing time derivative ( t = 0) only the back facade of Maxwell s house remains standing erect. Maxwell s equations reduce to with the constitutive equation curlh = J, (45) div B = 0, (46) B = µh. (47) 7 417

R. RUSSENSCHUCK 7 Boundary and interface conditions Subsequently, the closed domain (either 2D or 3D) in which the electromagnetic field is to be calculated will be denoted as Ω. The field quantities B and H satisfy boundary conditions on the piecewise smooth boundary Γ = Ω of the domain Ω, see Fig. 4. Two types of boundary conditions, prescribed on the two disjoint smooth boundaries denoted Γ H and Γ B with Γ = Γ H Γ B, cover all practical cases: On the part Γ B of the boundary the normal component of the magnetic flux density is prescribed. On symmetry planes parallel to the field, on far boundaries, or on outer boundaries of iron yokes surrounded by air (where it can be assumed that no flux leaves the outer boundary) the normal component of the flux density (denoted B n ) is zero. In some special cases the distribution of B n can be estimated along a physical surface, e.g., the flux distribution in the air gap of an electrical machine can be assumed to be sinusoidal. These boundary conditions can be written in the form B n = B n = σ mag on Γ B, (48) where σ mag is the surface density of a fictitious magnetic charge. A surface charge is defined as a charge with infinite density, while the charge per unit surface remains finite: Consider a thin layer of thickness d in which a charge of density ρ mag is present, see Fig. 5 left. In some surface x y of this layer the total charge Q = x ydρ mag is present which is dρ mag per unit surface. If we let ρ mag and d 0 so that dρ mag remains finite, we get the surface charge with the density σ mag = dρ mag, [σ mag ] = 1 V s/m 2. On the part Γ H of the boundary the tangential components of the magnetic field are prescribed. In many cases (as on symmetry planes perpendicular to the field) and on infinitely permeable iron poles, where the field enters at right angle, the tangential components of the field (denoted H t ) are zero. The tangential components of H can also be determined by a real or fictitious surface current density. All these boundary conditions can be written in the form H n = α on Γ H, (49) where α is the density of a real or fictitious electric surface current. A surface current is defined as a current with infinite density on a surface, while the current per unit length remains finite: Consider a thin layer of thickness d in which a current of density J flows, see Fig. 5 right. In some length l of this layer flows the total current I = Jd l which is Jd per unit length. If we let J and d 0 so that Jd remains finite, we get the surface current with the density α = Jd, [α] = 1 A/m. The condition Γ B Γ H n 1 n 2 Γ 12 Ω 1 Ω 2 µ 0 µ 2 Γ H Γ B Fig. 4: Composite material domain with boundary and interface. 8 418

? ELECTROMAGNETIC DESIGN OF ACCELERATOR MAGNETS y l d Q x I d Fig. 5: Left: Surface charge. Right: Surface current. that the tangential components are zero on the boundary implies H t = 0 n (H n) = 0. (50) In order to establish the interface conditions on a smooth surface (outer oriented by a given crossing direction) between two regions with different magnetic properties, consider two domains Ω 1 with permeability µ 1 and Ω 2 with permeability µ 2 as shown in Fig. 6. Consider a surface element which penetrates the interface and where the vector da lies in the interface plane, as shown in Fig. 6, left. Applying Ampère s law a H ds = a J da to the rectangular loop while letting the height δ 0 yields (H 2 n2 da H 1 n2 da )ds = α da ds. (51) c da da c da Eq. (51) holds for any curve c, if the integrands obey (H 2 H 1 ) n2 da = α da da da, (52) (H 2 H 1 ) n 2 da = α da. (53) Eq. (53) holds for any surface element da in the plane of the interface. It yields α = (H 2 H 1 ) n 2 = (H 1 H 2 ) n, (54) Ω 1 δ µ 1 n B t1 B1 n da n 2 da µ 2 da 1 δ µ 2 da 2 α 1 B n1 Ω 1 B 2 α 2 µ 2 B n2 B t2 n 1 n n 2 Ω 2 Γ 12 Ω 2 Ω 2 Fig. 6: Interface conditions for permeable media. 9 419

R. RUSSENSCHUCK where the surface normal vector n points from Ω 2 to Ω 1 as shown in Fig. 6. If no real or fictitious electric surface currents exist, the tangential components of the magnetic field strength are continuous at the interface H t1 = H t2 (H 1 H 2 ) n = 0. (55) Now consider the volume of the pill-box as shown in Fig. 6 middle. With the flux conservation law V B da = 0 which holds for any closed simply connected surface we get for δ 0, σ mag da = B 1 da 1 + B 2 da 2 = (B 1 B 2 ) n 1 da. (56) a Eq. (56) holds for any surface a if the integrands obey a a σ mag = (B 1 B 2 ) n. (57) If no fictitious magnetic surface charge density exists, the normal component of the magnetic flux density is continuous at the interface B n1 = B n2 (B 1 B 2 ) n = 0. (58) For a boundary of isotropic materials free of surface currents (Fig. 6 right) it follows that tan α 1 tan α 2 = B t1 B n1 B t2 B n2 = µ 1H t1 µ 2 H t2 = µ 1 µ 2, (59) at all points x Γ 12. For µ 2 µ 1 it follows that tan α 2 tan α 1. Therefore for all angles π/2 > α 2 > 0 we get tan α 1 0. The field exits vertically from a highly permeable medium into a medium with low permeability. We will come back to this point when we discuss ideal pole shapes of conventional magnets, see Section 12. Remark: Both B and H are discontinuous at x Γ 12 and therefore curlh and divb cease to make sense there. For any boundary value problem defined on Ω to be well posed, the interface conditions have to be implied. Ways of doing so include weak formulations in the Finite Element technique. 8 Magnetic anisotropy in laminated iron yokes In case of anisotropic magnetic material the permeability has the form of a diagonal rank 2 tensor, so that B = [µ] H with µ x 0 0 [µ] = 0 µ y 0. (60) 0 0 µ z In many materials, such as in rolled metal sheets, the fabrication process produces some regularity in the crystal structure and consequently a dependence of the magnetic properties on the direction. The most well known (and strongest) anisotropy in magnetic materials can be achieved by laminating the iron yokes. Between each of the ferromagnetic laminations of thickness l Fe (magnetically isotropic to first order) there is a non-magnetic (µ = µ 0 ) layer of thickness l 0, as shown schematically in Fig. 7. Consider a lamination in z-direction and the field components B t in the xy-plane. Because of the continuity condition H 0 t = H Fe t = H t we get for the effective macroscopic tangential flux density B t = 1 l Fe + l 0 ( lfe µh t + l 0 µ 0 H t ). (61) 10 420

* O N ELECTROMAGNETIC DESIGN OF ACCELERATOR MAGNETS 0 J. A Fig. 7: On the calculation of the µ tensor for laminated materials. The transversal dimensions are large with respect to l 0 and l Fe. As the normal component of the magnetic flux density is continuous, i.e., Bz 0 = Bz Fe magnetic field intensity can be calculated from With the packing factor 1 H z = l Fe + l 0 λ = ( l Fe B z µ + l 0 = B z, the average ) B z. (62) µ 0 l Fe l Fe + l 0 (63) which is 0.985 for the LHC yokes, we get for the average permeability in the plane of the lamination and normal to the plane of the lamination µ t = λµ + (1 λ)µ 0 (64) µ z = ( λ µ + 1 λ ) 1. (65) µ 0 We have obtained a simple equation for the packing factor scaling of the material characteristic. For laminations in the x and y direction, i.e, with the plane of the laminations normal to the 2D cross-section, the laminations have a strong directional effect and the packing factor scaling is no longer appropriate. A macroscopic model for these circumstances is developed in [5]. 9 Magnetic material Although all materials are either ferro-, dia- or paramagnetic it is customary to talk of magnetic material only in case of ferromagnetic behavior with either a wide hysteresis curve (hard ferromagnetic material and permanent magnets) or soft ferromagnetic material with narrow hysteresis as used for yoke laminations in magnet technology. In diamagnetic substances (e.g. Cu, Zn, Ag, Au, Bi) the orbit and spin magnetic moments cancel in the absence of external magnetic fields. An applied field causes the spin moments to slightly exceed the orbital moments, resulting in a small net magnetic moment which opposes the applied field. The permeability is less than µ 0. In the case of water the magnetic susceptibility χ m is 8.8 10 6. Superconductors in the Meissner phase represent the limiting case of µ = 0, the ideal diamagnet with a 11 421

R. RUSSENSCHUCK B µ max 4 J = J E 1 H H A L A H I E > A * K @ = H O, E I F =? A A J B r µ 0 M s 4 A L A H I E > A * K @ = H O, E I F =? A A J H B c µ 0 H H µ dif Fig. 8: Hysteresis curve for a ferromagnetic material. Coercive field H B c, remanence B r. Saturation magnetization M s. Red: Normal magnetization curve. complete shielding of the external field. Diamagnetic samples brought to either pole of a magnet will be repelled. The diamagnetic effect in materials is so low that it is easily overwhelmed in materials where the spin and orbit magnetic moments are unequal. In ideal paramagnets the individual magnetic moments do not interact with each other and take random orientation in space due to thermal agitation. When an external field is applied, the magnetic moments line up in the field direction resulting in positive susceptibilities χ m in the order of 10 5 to 10 3 basically independent of field strength and without hysteresis behavior. Paramagnetic substances include the rare earth elements, platinum, sodium and oxygen. 9.1 Ferromagnetic material Ferromagnetic substances (which include iron, nickel and cobalt as well as alloys of these elements) cannot be characterized by simple, single valued constitutive laws, as different B(H) relations (called magnetization curves) can be measured depending on the history of the excitation. The magnetization curves, see Fig. 8, can be measured by means of so-called permeameters consisting of an annulus of ferromagnetic material with N toroidal windings that excite the field of modulus H = NI 2πr. (66) The induced voltage in the pick-up coil (which is wound directly onto the specimen) is proportional to the rate of change of the flux U = d dt Φ = d Ba, (67) dt where a is the cross-section of the ring. Time integration ( Udt = Ba) yields the corresponding values of I and U and consequently the hysteresis curve for H and B. For an easy exchange of specimen these permeameters are made with split coils for B(H) measurements at ambient temperatures, see Fig. 9 left. 12 422

ELECTROMAGNETIC DESIGN OF ACCELERATOR MAGNETS Fig. 9: Left: Split coil permeameter used for the warm measurements of LHC yoke iron samples (180 turns for the excitation coil, 90 turns for the pick-up coil. Right: Superconducting excitation coil (about 3000 turns) and pickup coil wound directly onto a glass-epoxy box containing the ferromagnetic specimen for cyrogenic measurements. However, the total resistance of the excitation coil is inherently high due to the large number of contacts (two per turn) resulting in high power dissipations at high currents. Thus for low temperature measurements involving superconducting excitation coils, the coil (consisting of more than 3000 turns) is wound directly onto a toroidal glass-epoxy case with an automatic winding machine. A specimen prepared for measurements is shown in Fig. 9 right. If the applied field to a specimen is increased to saturation and then decreased, the decrease in flux density is not as rapid as the increase along the initial (virgin) magnetization curve. When H reaches zero there remains a residual flux density or remanence B r. In order to reduce B to zero, a negative field Hc B has to be applied which is called the coercive field. The phenomenon that causes B to lag behind H, so that the magnetization curves for increasing and decreasing fields are not the same, is called hysteresis. Hysteresis curves for soft and hard ferromagnets are shown in Fig. 10. A hysteresis loop can be represented in terms B(H) or M(H). In a soft ferromagnet, the fields involved in the hysteresis loop are much smaller than the corresponding magnetization (Fig. 10 left) and plotting B(H) instead of M(H) makes only a tiny difference. However, in permanent magnet material H and M have the same order of magnitude and the B(H) loops differs considerably from the M(H) curve, see Fig. 12 left 1. The coercive field Hc B is in the order of 50-100 A/m in non-oriented Si-Fe alloys and low-carbon steels used in electrical motors. The low-carbon steel used for the LHC yoke laminations is specified to have a coercivity of less than 60 A/m. Low-carbon steels are good choices for yoke lamination because they are easy to handle (draw, bend, and punch) and are fairly inexpensive. Coercive fields are decreased to about 10 A/m in grain-oriented Si-Fe alloys used in transformer cores. Extremely soft materials can be obtained from nickel alloys (usually called permalloys) with about 80% of nickel and 20% of iron. Fig. 11 shows the measured virgin B(H) curve of the low carbon steel laminations used for the yoke of the LHC main dipoles, the corresponding M(B) curve and the relative permeability as a function of the magnetic induction. The measurements were performed at 4.2 K with a ring specimen, a toroidal superconducting excitation coil and a copper search coil, in magnetic flux densities of up to 7.4 tesla. Table 1 gives the measured temperature and stress dependence on the coercive field, remanence and maximum permeability of yoke laminations used in a LHC model magnet. The properties of the magnetization curves M(H) are governed by two mechanisms known as exchange coupling and anisotropy. Exchange coupling between electron orbitals in the crystal lattice favors long-range spin ordering over macroscopic distances and is isotropic in space. At temperatures above a critical value (for iron about 770 o C) which is called the Curie temperature, the exchange cou- 1 Thus the need to distinguish between the two coercive fields H B c and H M c. 13 423

6 6 6 6 R. RUSSENSCHUCK pling disappears. Anisotropy favors spin orientation along certain symmetry axes of the lattice. The study of the quantum origin of these mechanisms is not needed in our phenomenological treatment of the material properties in field computation. Many of the phenomena of the magnetization curve, i.e., the three sections between the toe, the instep and the knee can be described by means of the domain theory by Weiss, see for example [4]. In an unmagnetized (and unstrained) piece of iron the directions in which the domains are magnetized are either distributed at random (in parallel to one of the six crystal axes) or in such a way that the resultant magnetization of the specimen is zero. Application of a magnetic field only changes the direction of the magnetization in a given volume and not the magnitude. This is attained by a reversible and later irreversible boundary displacement of the domains. Saturation in high field is attained by a reversible process of rotation within the domains. 9.2 Magnetostriction A ferromagnetic specimen changes its dimensions by some parts per million when it is magnetized. This effect is referred to as magnetostriction (positive for materials showing expansion and negative for contraction). The effect is due to magneto-crystalline anisotropy which gives rise to energy variations when the relative positions of magnetic ions in the lattice are modified. It is usually distinguished between Joule magnetostriction (the change of dimension transversely to the field) and volume magnetostriction. In case of inverse magnetostriction or stress anisotropy, the deformation caused by an externally applied stress favors certain magnetization directions. Table 1 gives the measured temperature and stress dependence on the coercive field, remanence and maximum µ r of yoke laminations for a LHC model magnet. An aluminum ring around the ring specimen provided for mechanical stress in the order of 20 MPa. The main dipole magnets for LEP were built with a small packing factor of 0.27, realized by regularly spaced magnetic steel laminations and spaces filled with cement mortar. This solution provided for mechanical rigidity at low price. Mortar shrinkage at hydration had an effect on the longitudinal magnet geometry which was well controlled by means of four tie rods. In the transverse plane, however, the steel laminations opposed the shrinkage of the mortar layers so that tensile stresses built up in the mortar (about 10 MPa, near the upper limit of mortar yield strength) and compressive stresses built up in the iron laminations (at about 30 MPa due to a different elastic modulus and thickness of the layers). This resulted in an unacceptable B/B in the bending field at low excitation caused by the reduction of the! 5 E. A µ 0 M T. A @ * % % # & # µ 0 M # # ) # # # # # H #! µ 0 H # µ 0 H Fig. 10: Left: M(B) hysteresis curve for soft (3%Si-Fe) grain oriented laminations used in transformer cores. Right: M(B) hysteresis curve for a sintered Fe 77Nd 15B 8 permanent magnet. Loop width differ by a factor of 10 5. The low-carbon steel used for the LHC yoke laminations is specified to have a coercive field Hc B of less than 80±10 A/m at room temperature. 14 424

* 6 0 6 6 H! 6 ELECTROMAGNETIC DESIGN OF ACCELERATOR MAGNETS K 6 K "!! " # $ )! "! " * * Fig. 11: Measured virgin B(H) curve of the iron yoke laminations for the LHC main dipole (stressed at 20 MPa and measured at cryogenic temperature of 4.2 K), the corresponding µ 0M(B) curve and the relative permeability as a function of the magnetic induction [?]. maximum µ r due to magnetostriction. By means of a hydraulic system, transverse forces were exerted all along the poles in order to relieve the mortar induced compressive stresses in the yoke laminations. 9.3 Permanent magnets In dealing with permanent magnets, the section of the hysteresis loop in the second quadrant of the B(H) and M(H) diagrams are of interest. If the loop is the major hysteresis loop, it is called the demagnetization curve. It is desirable that the material has a high remanence (as it determines the maximum possible flux density in a circuit) and a high coercive field Hc M so that the magnet will not easily be de-magnetized. Therefore the maximum product (BH) max is a good figure of merit. A discussion of the operation point of permanent magnets in a magnetic iron circuit is given in Section 10.3. Rare earth materials like SmCo 5 and NdFeB are sintered from a powder with grain sizes of about 5µm. These grains are magnetically highly anisotropic along one crystaline direction. The powder is then exposed to a strong magnetic field (so that the grains rotate until their magnetically preferred axis is aligned to the magnetic field), subjected to a high pressure and sintered. Finally the sintered and machined material is exposed to a very strong magnetization field in parallel to the previously established direction. A typical B(H) relationship for B and H parallel to the magnetically preferred axis of the grains is shown in Fig. 12 left. The demagnetization curve is for rare earth materials basically a straight line with a differential permeability of db/dh 1.04µ 0 1.08µ 0, so that the coercive field µ 0 Hc B is about 4-8% less than B r. In the perpendicular direction, the typical values for the relative differential permeability are in the range of 1.02 to 1.08. Because the permeabilities are so close to µ 0 we will treat the material as vacuum with an impressed (field dependent) magnetization. Temperature T Stress Coercive field Hc B Remanence B r max µ r K MPa A/m T 300 0 68.4 1.07 5900 77 0 79.6 1.12 5600 4.2 0 85.1 1.06 4800 4.2 20 110. 0.67 2460 Table 1: Measured temperature and stress dependence on the coercive field H B c, remanence B r and maximum µ r of the LHC yoke laminations. 15 425

6! 6 6 R. RUSSENSCHUCK B(H) M(H) B A @ O E K 1 H * H 5 = = H E K + > = J ) E + " & $ " B. A H H E J A! 6! µ 0H! H µ 0H ) " & $ & $ " " Fig. 12: Left: Typical B(H) and M(H) curves for hard ferromagnetic substances (permanent magnets). M and H are both pointing into the direction of the easy axis of the grains in the sinter material. Right: Demagnetization curves for different permanent magnet materials at room temperature. 9.4 Magnetization currents and fictitious magnetic charges In the presence of ferromagnets the magnetic field can be calculated as in vacuum, if all currents (including the magnetization currents) are explicitly considered Hence curlb = µ 0 (J free + J mag ) = µ 0 J free + µ 0 curl M(H). (68) ( ) B µ0 M(H) curl = J free. (69) µ 0 The magnetic induction B is always source free but the magnetic field H is not: ( ) B µ0 M(H) div H = div = div M(H) (70) which gives rise to the definition of a fictitious magnetic charge of density µ 0 ρ mag = div µ 0 M(H). (71) A fictitious magnetic surface charge density, c.f. Fig. 13, is then defined as σ mag = µ 0 M(H) da da = µ 0M(H) n. (72) H B M(H) Fig. 13: Field, magnetic induction and magnetization in a permanent magnet. 16 426

6! ELECTROMAGNETIC DESIGN OF ACCELERATOR MAGNETS This allows formally a treatment of magnetostatic problems in the same way as electrostatic problems using a magnetic scalar potential. 10 One-dimensional field calculation for conventional magnets 10.1 C-core dipole Consider the magnetic circuit shown in Fig. 14 (left), a c-shaped iron yoke with two coils of all-together N turns around it. If the gap thickness is small compared to all other dimensions, the fringe field around the gap will be small and we assume that the flux of B through any cross section of the yoke (and across the air-gap) will be constant. With Ampère s law a H ds = a J da, we can write With µ r 1 we get the easy relation H i s i + H 0 s 0 = V m, (73) 1 µ 0 µ r B i s i + 1 µ 0 B 0 s 0 = N I. (74) B 0 = µ 0N I s 0. (75) 10.2 Quadrupole For the quadrupole we can split up the integration path as shown in Fig. 14 right, from the origin to the pole (s 1 ) along an arbitrary path through the iron yoke (s 2 ) and back inside the aperture along the x-axis (s 3 ). Neglecting the magnetic resistance of the yoke we get H ds = H 1 ds + H 3 ds = N I. (76) s 1 s 3 As we will see later, in a quadrupole the field is defined by it s gradient g with B x = gy and B y = gx. Therefore the modulus of the field along the integration path s 1 is H = g µ 0 x 2 + y 2 = g µ 0 r. (77) Along the x-axis (s 3 ) the field integral is zero because H s. Therefore r0 0 Hdr = g µ 0 r0 0 rdr = g µ 0 r 2 0 2 = N I, (78) I H = Fig. 14: Magnetic circuit of a conventional dipole magnet (left) a quadrupole magnet (right). Neglecting the magnetic resistance of the iron yoke, an easy relation between the air gap field and the required excitational current can be derived. 17 427

R. RUSSENSCHUCK = I I = M Fig. 15: Left: C-core dipole with excitation coil and varying yoke surface. Notice the orientation of the surface for the integration of the magnetic field strength and the surface of the integration of the magnetic flux density. Right: C-core dipole with permanent magnet excitation. Disregarding the magnetic resistance of the iron yoke and all fringe fields, an easy relation between the air-gap field and the size of the permanent magnet can be derived. or g = 2µ 0NI r0 2. (79) Notice that for a given NI the field decreases linearly with the gap size of the dipole, whereas the gradient in a quadrupole magnet is inverse proportional to the square of the aperture radius r 0. 10.3 Permanent magnet excitation For a magnet with permanent magnet excitation as shown in Fig. 15 (right), we write the equation for the magnetic circuit as H i s i + H 0 s 0 + H m s m = 0. (80) In the absence of fringe fields we get with the pole surface a 0 and the magnet surface a m : B m a m = B 0 a 0 = µ 0 H 0 a 0. (81) For µ r 1 we can again neglect the magnetic resistance of the yoke and from Eq. (80) it follows that H 0 s 0 = H m s m, (82) 1 a m B m s 0 = H m s m, µ 0 a 0 (83) B m µ 0 H m = s m s 0 a 0 a m = P, (84) where P is called the permeance coefficient which becomes zero for s 0 s m (open circuit) and becomes for s m s 0 (short circuit). The case of a m > a 0 is usually referred to as the flux concentration mode. The permeance coefficient defines the point on the demagnetization curve which is the branch of the permanent magnet hysteresis curve in the second quadrant. From Eq. (81) and (82) we derive Therefore H 0 = B m a m s m = µ 0 H 0 a 0 H 0 s 0 H m. (85) (a m s m )( B m H m ) V m ( B m H m ) =. (86) µ 0 (a 0 s 0 ) µ 0 V 0 18 428

# # ELECTROMAGNETIC DESIGN OF ACCELERATOR MAGNETS 5 + 5 + Fig. 16: C-Core magnet with permanent magnet excitation. Top: With permanent magnet brought to the air gap in order to reduce the fringe field. Display of vector potential and arrows representing the magnetic flux density are to scale. For a given magnet volume, the maximum air gap field can be obtained by dimensioning the magnetic circuit in such a way that B m H m is maximum. Neglecting the leakage flux may be a rough treatment of the field problem, in particular for magnetic flux densities exceeding 1 T in the yoke, or for large air gaps. Flux leakage is proportional to the magnetic potential difference, i.e., the m.m.f. V m = H ds between the poles. The design shown in Fig. 16 (bottom) is therefore a poor one, as there are large areas at high potential differences. The structure in Fig. 16 (top) with the permanent magnets brought to the air gap shows considerably less leakage flux. For the use of permanent magnet material in accelerator magnets, demagnetization due to irradiation and thermal fluctuations has to be considered. As no permanent damage to the crystalline structure of the magnet occurs, it is always possible to re-magnetize the magnet after irradiation to the nominal level. 11 Potential formulations for magnetostatic field problems We will now show that in the aperture of a magnet (two-dimensional, current free region) both the magnetic scalar-potential as well as the vector-potential can be used to solve Maxwell s equations: H = gradφ = Φ x e x Φ y e y, (87) B = curl(e z A z ) = A z y e x A z x e y, (88) and that both formulations yield the Laplace equation. These fields are called harmonic and the field quality can be expressed by the fundamental solutions of the Laplace equation. Lines of constant vectorpotential give the direction of the magnetic field, whereas lines of constant scalar potential define the ideal pole shapes of conventional magnets. Every vector field can be split into a source free and a curl free part. In case of the magnetic field with H = H s + H m, (89) the curl free part H m arises from the induced magnetism in ferromagnetic materials and the source free part H s is the field generated by the prescribed sources (can be calculated directly by means of Biot Savart s law). With curlh m = 0 it follows that H = gradφ m + H s (90) 19 429

R. RUSSENSCHUCK and we get: div B = 0, (91) div µ( gradφ m + H s ) = 0, (92) div µgradφ m = div µh s. (93) While a solution of Eq. 93 is possible, the two parts of the magnetic field H m and H s tend to be of similar magnitude (but opposite direction) in non-saturated magnetic materials, so that cancellation errors occur in the computation. For regions where the current density is zero, however, curlh = 0 and the field can be represented by a total scalar potential and therefore we get H = gradφ (94) µ 0 div gradφ = 0, (95) 2 Φ = 0, (96) which is the Laplace equation for the scalar potential. The vector-operator Nabla is defined in Cartesian coordinates as and the Laplace operator = ( x, y, z ) (97) = 2 = 2 x 2 + 2 y 2 + 2 z 2. (98) The Laplace operator itself is essentially scalar. When it acts on a scalar function the result is a scalar, when it acts on a vector function, the result is a vector. Because of div B = 0 a vector potential A can be introduced: B = curl A. We then get curla = µ 0 (H + M), (99) H = 1 µ 0 curla M, (100) 1 curl curla = J + curlm, µ 0 (101) 1 ( 2 A + grad diva) = J + curlm, µ 0 (102) Since the curl (rotation) of a gradient field is zero, the vector-potential is not unique. The gradient of any (differentiable) scalar field ψ can be added without changing the curl of A: A 0 = A + gradψ. (103) Eq. (103) is called a gauge-transformation between A 0 and A. B is gauge-invariant as the transformation from A to A 0 does not change B. The freedom given by the gauge-transformation can be used to set the divergence of A to zero diva = 0, (104) 20 430

ELECTROMAGNETIC DESIGN OF ACCELERATOR MAGNETS which (together with additional boundary conditions) makes the vector-potential unique. Eq. (104) is called the Coulomb gauge, as it leads to a Poisson type equation for the magnetic vector-potential. Therefore, from Eq. (102) we get after incorporating the Coulomb gauge: 2 A = µ 0 (J + curlm). (105) In the two-dimensional case with no dependence on z, z = 0 and J = J z, A has only a z-component and the Coulomb gauge is automatically fulfilled. Then we get the scalar Poisson differential equation 2 A z = µ 0 J z. (106) For current-free regions (e.g. in the aperture of a magnet) Eq. (106) reduces to the Laplace equation, which reads in in cylindrical coordinates r 2 2 A z r 2 + r A z r + 2 A z = 0. (107) ϕ2 11.1 Harmonic fields A solution of the homogeneous differential equation (107) reads A z (r, ϕ) = (C1 n r n + C2 n r n )(D1 n sin nϕ + D2 n cos nϕ). (108) n=1 Considering that the field is finite at r = 0, the C2 n have to be zero for the vector-potential inside the aperture of the magnet while for the solution in the area outside the coil all C1 n vanish. Rearranging Eq. (108) yields the vector-potential in the aperture: A z (r, ϕ) = r n (C n sin nϕ D n cos nϕ), (109) n=1 and the field components can be expressed as B r (r, ϕ) = 1 A z r ϕ = nr n 1 (C n cos nϕ + D n sin nϕ), (110) B ϕ (r, ϕ) = A z r n=1 = nr n 1 (C n sin nϕ D n cos nϕ). (111) n=1 Each value of the integer n in the solution of the Laplace equation corresponds to a different flux distribution generated by different magnet geometries. The three lowest values, n=1,2, and 3 correspond to a dipole, quadrupole and sextupole flux density distribution. The solution in Cartesian coordinates can be obtained from the simple transformations For the dipole field (n=1) we get B x = B r cos ϕ B ϕ sin ϕ, (112) B y = B r sin ϕ + B ϕ cos ϕ. (113) B r = C 1 cos ϕ + D 1 sin ϕ, (114) B ϕ = C 1 sin ϕ + D 1 cos ϕ, (115) B x = C 1, (116) B y = D 1. (117) 21 431

R. RUSSENSCHUCK This is a simple, constant field distribution according to the values of C 1 and D 1. Notice that we have not yet addressed the conditions necessary to obtain such a field distribution. For the pure quadrupole (n=2) we get from Eq. (110) and (111): B r = 2 r C 2 cos 2ϕ + 2 r D 2 sin 2ϕ, (118) B ϕ = 2 r C 2 sin 2ϕ + 2 r D 2 cos 2ϕ, (119) B x = 2(C 2 x + D 2 y), (120) B y = 2( C 2 y + D 2 x). (121) The amplitudes of the horizontal and vertical components vary linearly with the displacements from the origin, i.e., the gradient is constant. With a zero induction in the origin, the distribution provides linear focusing of the particles. It is interesting to notice that the components of the magnetic fields are coupled, i.e., the distribution in both planes cannot be made independent of each other. Consequently a quadrupole focusing in one plane will defocus in the other. Repeating the exercise for the case of the pure sextupole (n=3) yields: B r = 3 r C 3 cos 3ϕ + 3 r D 3 sin 3ϕ, (122) B ϕ = 3 r C 3 sin 3ϕ + 3 r D 3 cos 3ϕ, (123) B x = 3C 3 (x 2 y 2 ) + 6D 3 xy, (124) B y = 6C 3 xy + 3D 3 (x 2 y 2 ). (125) Along the x-axis (y=0) we then get the expression for the y-component of the field: B y = D 1 + 2D 2 x + 3D 3 x 2 + 4D 4 x 3 +... (126) If only the two lowest order elements are used for steering the beam, forces on the particles are either constant or vary linear with the distance from the origin. This is called a linear beam optic. It has to be noted that the treatment of each harmonic separately is a mathematical abstraction. In practical situations many harmonics will be present and many of the coefficients C n and D n will be non-vanishing. A successful magnet design will, however, minimize the unwanted terms to small values. It has to be stressed that the coefficients are not known at this stage. They are defined through the (given) boundary conditions on some reference radius or can be calculated from the Fourier series expansion of the (numerically) calculated field, ref. Eq. (5), in the aperture using the relations A n = nr n 1 0 C n and B n = nr n 1 0 D n. (127) 12 Ideal pole shapes of conventional magnets Equipotentials are surfaces where Φ is constant. For a path ds along the equipotential it therefore results dφ = gradφ ds = 0 (128) i.e., the gradient is perpendicular to the equipotential. With the field lines (lines of constant vector potential) leaving highly permeable materials perpendicular to the surface (ref. Section 7), the lines of total magnetic scalar potential define the pole shapes of conventional magnets. As in 2D (with absence of magnetization and free currents) the z-component of the vector potential and the magnetic scalar potential both satisfy the Laplace equation, we already have the solution for a dipole field: Φ = C 1 x + D 1 y. (129) So C 1 = 0, D 1 0 gives a vertical (normal) dipole field, C 1 0, D 1 = 0 yields a horizontal (skew) dipole field. The equipotential surfaces are parallel to the x-axis or y-axis depending on the values of C 22 432