1. USPAS January 2012, Superconducting accelerator magnets Unit 5 Field harmonics Helene Felice, Soren Prestemon Lawrence Berkeley National Laboratory (LBNL) Paolo Ferracin and Ezio Todesco European Organization for Nuclear Research (CERN)

2. USPAS January 2012, 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 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

3. USPAS January 2012, 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. Beam dynamics requirements on field harmonics Random and skew components Specifications (targets) at injection and high field

4. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.4 1. 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)

5. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.5 1. 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)

6. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.6 1. DEFINITION OF FIELD HARMONICS If Maxwell gives 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, 1826 - July 20, 1866)

7. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.7 1. 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) – more common in low energy rings, resistive magnets – one sc example: JPARC (Japan) A dipole A quadrupole [from P. Schmuser et al, pg. 50] A sextupole

8. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.8 1. 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

9. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.9 1. 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 10 -4 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

10. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.10 1. 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

11. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.11 1. 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 !!

12. USPAS January 2012, 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. Beam dynamics requirements on field harmonics Random and skew components Specifications (targets) at injection and high field

13. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.13 2. 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)

14. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.14 2. FIELD HARMONICS OF A CURRENT LINE Now we can compute the multipoles of a current line at z 0

15. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.15 2. 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

16. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.16 2. 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

17. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.17 2. FIELD HARMONICS OF A CURRENT LINE Field given by a current line (Biot-Savart law) – vector potential formalism … … and polar coordinates formalism

18. USPAS January 2012, 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. Beam dynamics requirements on field harmonics Random and skew components Specifications (targets) at injection and high field

19. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.19 3. 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%

20. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.20 3. 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

21. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.21 3. 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

22. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.22 3. 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

23. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.23 3. 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

24. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.24 3. 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

25. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.25 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. Beam dynamics requirements on field harmonics Random and systematic components Specifications (targets) at injection and high field

26. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.26 4. 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

27. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.27 4. 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

28. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.28 4. 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

29. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.29 4. 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

30. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.30 4. 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

31. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.31 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 Biot-Savart: multipoles decay with multipole order as a power law Attention !! Validity limits and convergence domains We outlined some issues about the beam dynamics specifications on harmonics

32. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.32 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 ?

33. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.33 REFERENCES On field harmonics A. Jain, “Basic theory of magnets”, CERN 98-05 (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 2004-003 (2004) pp. 164-168. N. Sammut, et al. “Mathematical formulation to predict the harmonics of the superconducting Large Hadron Collider magnets” Phys. Rev. ST Accel. Beams 9 (2006) 012402.Mathematical formulation to predict the harmonics of the superconducting Large Hadron Collider magnets

34. USPAS January 2012, Superconducting accelerator magnets Unit 5: Field harmonics – 5.34 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 www.wikipedia.orgwww.wikipedia.org for most of the pictures No frogs have been harmed during the preparation of these slides!!