USPAS June 2007, Superconducting accelerator magnets Unit 5 Field harmonics Soren Prestemon and Paolo Ferracin Lawrence Berkeley National Laboratory (LBNL) Ezio Todesco European Organization for Nuclear Research (CERN)

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – 5.2 QUESTIONS How to express the magnetic field shape and uniformity The magnetic field is a continuous, vectorial quantity Can we express it, and its shape, using a finite number of coefficients ? Yes: field harmonics But not everywhere … We are talking about electromagnets: What is the field we can get from a current line ? The mythical Biot-Savart law ! What are the components of the magnetic field shape ? What are the beam dynamics requirements ? 41° 49’ 55” N – 88 ° 15’ 07” W 40° 53’ 02” N – 72 ° 52’ 32” W

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – 5.3 CONTENTS 1. Definition of field harmonics Multipoles 2. Field harmonics of a current line The Biot-Savart law 3. Validity limits of field harmonics 4. Field model and components Geometric, persistent currents, iron saturation 5. Beam dynamics requirements on field harmonics Random and skew components Specifications (targets) at injection and high field

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – FIELD HARMONICS: MAXWELL EQUATIONS Maxwell equations for magnetic field In absence of charge and magnetized material If (constant longitudinal field), then James Clerk Maxwell, Scottish (13 June 1831 – 5 November 1879)

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – FIELD HARMONICS: ANALYTIC FUNCTIONS A complex function of complex variables is analytic if it coincides with its power series on a domain D ! Note: domains are usually a painful part, we talk about it later A necessary and sufficient condition to be analytic is that called the Cauchy-Riemann conditions Augustin Louis Cauchy French (August 21, 1789 – May 23, 1857)

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – DEFINITION OF FIELD HARMONICS If and therefore the function B y +iB x is analytic where C n are complex coefficients Advantage: we reduce the description of a function from R 2 to R 2 to a (simple) series of complex coefficients Attention !! We lose something (the function outside D ) Georg Friedrich Bernhard Riemann, German (November 17, July 20, 1866)

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – DEFINITION OF FIELD HARMONICS Each coefficient corresponds to a “pure” multipolar field Magnets usually aim at generating a single multipole Dipole, quadrupole, sextupole, octupole, decapole, dodecapole … Combined magnets: provide more components at the same time (for instance dipole and quadrupole) – not very common A dipole A quadrupole [from P. Schmuser et al, pg. 50] A sextupole

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – DEFINITION OF FIELD HARMONICS Vector potential Since one can always define a vector potential A such that The vector potential is not unique (gauge invariance): if we add the gradient of any scalar function, it still satisfies Scalar potential In the regions free of charge and magnetic material Therefore in this case one can also define a scalar potential (such as for gravity) One can prove that is an analytic function in a region free of charge and magnetic material

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – DEFINITION OF FIELD HARMONICS The field harmonics are rewritten as (EU notation) We factorize the main component ( B 1 for dipoles, B 2 for quadrupoles) We introduce a reference radius R ref to have dimensionless coefficients We factorize since the deviations from ideal field are  0.01% The coefficients b n, a n are called normalized multipoles b n are the normal, a n are the skew (adimensional) US notation is different from EU notation

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – DEFINITION OF FIELD HARMONICS Reference radius is usually chosen as 2/3 of the aperture radius This is done to have numbers for the multipoles that are not too far from 1 Some wrong ideas about reference radius Wrong statement 1: “the expansion is valid up to the reference radius” The reference radius has no physical meaning, it is as choosing meters of mm Wrong statement 2: “the expansion is done around the reference radius” A power series is around a point, not around a circle. Usually the expansion is around the origin

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – FIELD HARMONICS: LINEARITY Linearity of coefficients (very important) Non-normalized coefficients are additive Normalized coefficients are not additive Normalization gives handy (and physical) quantities, but some drawbacks – pay attention !!

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – 5.12 CONTENTS 1. Definition of field harmonics Multipoles 2. Field harmonics of a current line The Biot-Savart law 3. Validity limits of field harmonics 4. Field model and components Geometric, persistent currents, iron saturation 5. Beam dynamics requirements on field harmonics Random and skew components Specifications (targets) at injection and high field

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – FIELD HARMONICS OF A CURRENT LINE Field given by a current line (Biot-Savart law) using !!! we get Jean-Baptiste Biot, French (April 21, 1774 – February 3, 1862) Félix Savart, French (June 30, 1791-March 16, 1841)

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – FIELD HARMONICS OF A CURRENT LINE Now we can compute the multipoles of a current line

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – FIELD HARMONICS OF A CURRENT LINE Multipoles given by a current line decay with the order The slope of the decay is the logarithm of ( R ref /|z 0 |) At each order, the multipole decreases by a factor R ref /|z 0 | The decay of the multipoles tells you the ratio R ref /|z 0 |, i.e. where is the coil w.r.t. the reference radius – like a radar … we will see an application of this feature to detect assembly errors through magnetic field shape in Unit 21

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – FIELD HARMONICS OF A CURRENT LINE Multipoles given by a current line decay with the order The semilog scale is the natural way to plot multipoles This is the point of view of Biot-Savart But usually specifications are on a linear scale In general, multipoles must stay below one or a fraction of units – see later This explains why only low order multipoles, in general, are relevant

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – FIELD HARMONICS OF A CURRENT LINE Field given by a current line (Biot-Savart law) – vector potential

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – 5.18 CONTENTS 1. Definition of field harmonics Multipoles 2. Field harmonics of a current line The Biot-Savart law 3. Validity limits of field harmonics 4. Field model and components Geometric, persistent currents, iron saturation 5. Beam dynamics requirements on field harmonics Random and skew components Specifications (targets) at injection and high field

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – VALIDITY LIMITS OF FIELD HARMONICS When we expand a function in a power series we lose something Example 1. In t=1/2, the function is and using the series one has not bad … with 4 terms we compute the function within 7%

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – VALIDITY LIMITS OF FIELD HARMONICS When we expand a function in a power series we lose something Ex. 2. In t=1, the function is infinite and using the series one has which diverges … this makes sense

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – VALIDITY LIMITS OF FIELD HARMONICS When we expand a function in a power series we lose something Ex. 3. In t=-1, the function is well defined BUT using the series one has even if the function is well defined, the series does not work: we are outside the convergence radius

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – VALIDITY LIMITS OF FIELD HARMONICS When we expand a function in a power series we lose something Ex. 3. In t=-1, the function is well defined BUT using the series one has even if the function is well defined, the series does not work: we are outside the convergence radius

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – VALIDITY LIMITS OF FIELD HARMONICS If we have a circular aperture, the field harmonics expansion relative to the center is valid within the aperture For other shapes, the expansion is valid over a circle that touches the closest current line

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – VALIDITY LIMITS OF FIELD HARMONICS If all coefficients are known with infinite accuracy, one can reconstruct the function from the first expansion wherever no singularities are present (analytic continuation) Analytic function: local info gives global values (no need to travel …) f’ has a different convergence domain ! In practice the technique is not used since the measurement accuracy limits this possibility Magnets with non-circular apertures are measured over several circular domains covering the interesting area

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – VALIDITY LIMITS OF FIELD HARMONICS Field harmonics in the heads Harmonic measurements are done with rotating coils of a given length (see unit 21) – they give integral values over that length If the rotating coil extremes are in a region where the field does not vary with z, one can use the 2d harmonic expansion for the integral If the rotating coil extremes are in a region where the field vary with z, one cannot use the 2d harmonic expansion for the integral One has to use a more complicated expansion

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – 5.26 CONTENTS 1. Definition of field harmonics Multipoles 2. Field harmonics of a current line The Biot-Savart law 3. Validity limits of field harmonics 4. Field model and components Geometric, persistent currents, iron saturation 5. Beam dynamics requirements on field harmonics Random and skew components Specifications (targets) at injection and high field

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – OUTLOOK OF THE STATIC FIELD MODEL The final shape of the magnetic field is given by the sum of different contributions The presence of some contributions and their magnitude depends on the design and on the excitation level We will focus on superconducting magnets for the machine in the range of 0.1 to 15 T (and more?) Detector magnets have a different phenomenology Energy swing (from injection to high field) is in the range of a factor 5 to 20 Phenomena at injection and high field are different We will come back on this in Unit 20 We first focus on static effects (no dependence on time) We will follow the assembly stages of the magnet

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – OUTLOOK OF THE STATIC FIELD MODEL 1 - Assembly of the coil: the collared coil Collared coil: the coil surrounded by its retaining structure Room temperature: the current flows in the copper part of the cable Field is given by a sum of Biot-Savart contributions Linear in the current Multipoles independent of the current The assembly is under forces that deform the coil: position is not the nominal one  offset on main field and multipoles Relevant contribution, tentative estimates with FEM (mechanical) In case of collars with permeability different from 1 (stainless steel), a perturbation is added (offset to multipoles and main field) Even with low permeability (  1.005) the contribution is not negligible Can be compute with FEM methods (magnetic) The transverse cross-section of the LHC main dipole collared coil

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – OUTLOOK OF THE STATIC FIELD MODEL 2 - Assembly of the coil: the cold mass (collared coil + iron yoke) Iron yoke is used to enhance the field, to avoid fringe fields, and sometimes as a mechanical support Room temperature: the current flows in the copper part of the cable Iron yoke increases the main field by a factor  (at most a factor two) Iron yoke reduces multipoles by a factor , and adds an offset on multipoles Iron yoke is not saturated  linear, can be computed analytically for simple geometries or with magnetic FEM Iron yoke also changes stress and coil deformations  additional offset, can be computed with mechanical FEM Transverse cross-section of the LHC main dipole cold mass

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – OUTLOOK OF THE STATIC FIELD MODEL 3 - Cool-down to 2-4 K: the “geometric” Additional deformations in the coil  offset in the multipoles Contraction of the structure – rescaling, little effect Change of pre-stress – larger effect – estimated with mechanical FEM This gives the “geometric” Field is linear in the current, multipoles independent of the current Cannot be directly measured, but can be estimated from measurements 4 – At injection energy: persistent currents Magnetization of the superconductor – large effect at low fields – can be computed knowing the cable magnetization (see Unit 13) 5 – At collision: iron saturation and Lorentz forces If the field in the iron is larger than 2 T, the iron saturates and its permeability changes, going from  10 3 to 1 – this affects the main field and multipoles – can be computed with magnetic FEM At high field the induced Lorentz forces can deform the coil – can be estimated with mechanical FEM

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – OUTLOOK OF THE STATIC FIELD MODEL 6 – Beam screen and cold bore The cold bore and the beam screen inside affect the multipoles, depending on their permeability This gives an additional offset on multipoles Summary of static field model Relevance estimated for the case of LHC dipoles

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – 5.32 CONTENTS 1. Definition of field harmonics Multipoles 2. Field harmonics of a current line The Biot-Savart law 3. Validity limits of field harmonics 4. Field model and components Geometric, persistent currents, iron saturation 5. Beam dynamics requirements on field harmonics Random and systematic components Specifications (targets) at injection and high field

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – – BEAM DYNAMICS REQUIREMENTS Integral values for a magnet For each magnet the integral of the main component and multipoles are measured Main component: average over the straight part (head excluded) Magnetic length: length of the magnet as If there were no heads, and the integrated strength is the same as the real magnet

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – – BEAM DYNAMICS REQUIREMENTS Integral values for a magnet Average multipoles: weighted average with the main component Systematic and random components over a set of magnets Systematic: mean of the average multipoles Random: standard deviation of the average multipoles

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – – BEAM DYNAMICS REQUIREMENTS Beam dynamics requirements Rule of thumb (just to give a zero order idea): systematic and random field harmonics have to be of the order of 0.1 to 1 unit (with R ref one third of magnet aperture radius) Higher order are ignored in beam dynamics codes (in LHC up to order 11 only) Note that spec is rather flat, but multipoles are decaying !! – Therefore in principle higher orders cannot be a problem

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – – BEAM DYNAMICS REQUIREMENTS Beam dynamics requirements: one has target values for range for the systematic values (usually around zero, but not always) maximum spread of multipoles and main component The setting of targets is a complicated problem, with many parameters, without a unique solution Example: can we accept a larger random b 3 if we have a smaller random b 5 ? Usually it is an iterative (and painful) process: first estimate of randoms by magnet builders, check by accelerator physicists, requirement of tighter control on some components or of adding corrector magnets … Random normal multipoles in LHC main quadrupole: measured vs target Systematic skew multipoles in LHC main quadrupole: measured vs target Systematic multipoles in LHC main dipole: measured vs target

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – – BEAM DYNAMICS REQUIREMENTS Beam dynamics requirements (tentative) Dynamic aperture requirement: that the beam circulates in a region of pure magnetic field, so that trajectories do not become unstable Since the beam at injection is larger of a factor  (E c /E i ), the requirement at injection is much more stringent  no requirement at high field Exception: the low beta magnets, where at collision the beta functions are large, i.e., the beam is large Chromaticity, linear coupling, orbit correction This conditions on the beam stability put requirements both at injection and at high field Mechanical aperture requirement Can be limited by an excessive spread of the main component – or bad alignment

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – 5.38 SUMMARY We outlined the Maxwell equations for the magnetic field We showed how to express the magnetic field in terms of field harmonics Compact way of representing the field Attention !! Validity limits and convergence domains We outlined the main contributions to field harmonics given by the successive assembly stages of a magnet We outlined some issues about the beam dynamics specifications on harmonics

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – 5.39 COMING SOON Coming soon … It is useful to have magnets that provide pure field harmonics How to build a pure field harmonic (dipole, quadrupole …) [pure enough for the beam …] with a superconducting cable ? Which field/gradient can be obtained ?

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – 5.40 REFERENCES On field harmonics A. Jain, “Basic theory of magnets”, CERN (1998) 1-26 Classes given by A. Jain at USPAS P. Schmuser, Ch. 4 On convergence domains of analytic functions Hardy, “Divergent series”, first chapter (don’t go further) On the field model A.A.V.V. “LHC Design Report”, CERN (2004) pp N. Sammut, et al. “Mathematical formulation to predict the harmonics of the superconducting Large Hadron Collider magnets” Phys. Rev. ST Accel. Beams 9 (2006) Mathematical formulation to predict the harmonics of the superconducting Large Hadron Collider magnets

USPAS June 2007, Superconducting accelerator magnets Unit 5: Field harmonics – 5.41 ACKOWLEDGEMENTS A. Jain for discussions about reference radius, multipoles in heads, vector potential S. Russenschuck for discussions about vector potential G. Turchetti for teaching me analytic functions and divergent series, and other complicated subjects in a simple way for most of the pictures