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Unit 3 Field harmonics, Taylor expansions, and pure multipoles Ezio Todesco European Organization for Nuclear Research (CERN) E. Todesco, Milano Bicocca January-February 2016 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 ?41° 49’ 55” N – 88 ° 15’ 07” W 40° 53’ 02” N – 72 ° 52’ 32” W The Biot-Savart law ! What are the components of the magnetic field shape ? What are the beam dynamics requirements ? E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 2 CONTENTS 1. Definition of field harmonics 2. Field harmonics of a current line 3. Validity limits of field harmonics 4. Beam dynamics requirements on field harmonics E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 3 1. FIELD HARMONICS: MAXWELL EQUATIONS Maxwell equations for magnetic field B x B y B z B 0 x y z B 0 J 0 0 E t In absence of charge and magnetized material B y B z B z B x B x B y 0 B , , y x z y x z If James Clerk Maxwell, Scottish (13 June 1831 – 5 November 1879) B z 0 (constant longitudinal field), then z Bx By 0 x y E. Todesco, Milano Bicocca January-February 2016 Bx By 0 y x Unit 3 - 4 1. FIELD HARMONICS: ANALYTIC FUNCTIONS A complex function of complex variables is analytic if it coincides with its power series f ( z ) Cn z n 1 n 1 f x ( x, y ) if y ( x, y ) Cn ( x iy ) n 1 on a domain D ! ( x, y ) D n 1 Note: domains are usually a painful part, we talk about it later A necessary and sufficient condition to be analytic is that f x f y x y 0 f f x y 0 y x called the Cauchy-Riemann conditions E. Todesco, Milano Bicocca January-February 2016 Augustin Louis Cauchy French (August 21, 1789 – May 23, 1857) Unit 3 - 5 1. DEFINITION OF FIELD HARMONICS If B z 0 z f x f y x y 0 f f x y 0 y x By Bx 0 x y Maxwell gives By y Bx 0 x and therefore the function By+iBx is analytic n1 By ( x, y ) iBx ( x, y ) Cn x iy n1 ( x, y ) D Georg Friedrich Bernhard Riemann, German (November 17, 1826 - July 20, 1866) where Cn are complex coefficients Advantage: we reduce the description of a function from R2 to R2 to a (simple) series of complex coefficients Attention !! We lose something (the function outside D) E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 6 1. DEFINITION OF FIELD HARMONICS n 1 By ( x, y ) iBx ( x, y ) Cn x iy n 1 C0 C1 ( x iy ) ... ( x, y ) D Each coefficient corresponds to a “pure” multipolar field A dipole A quadrupole [from P. Schmuser et al, pg. 50] A sextupole 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 superconducting example: JPARC (Japan) Unit 3 - 7 E. Todesco, Milano Bicocca January-February 2016 1. DEFINITION OF FIELD HARMONICS n1 By ( x, y ) iBx ( x, y ) Cn x iy n1 n1 Bn iAn x iy n1 The field harmonics are rewritten as x iy B y iB x 10 B1 (bn ia n ) R n 1 ref 4 n 1 We factorize the main component (B1 for dipoles, B2 for quadrupoles) We introduce a reference radius Rref to have dimensionless coefficients We factorize 10-4 since the deviations from ideal field are 0.01% The coefficients bn, an are called normalized multipoles bn are the normal, an are the skew (adimensional) E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 8 1. DEFINITION OF FIELD HARMONICS x iy B y iB x 10 B1 (bn ia n ) R n 1 ref 4 n 1 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 y 40 - Some wrong ideas about reference radius Wrong statement 1: “the expansion is valid up to the reference radius” + - R ref + 0 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 E. Todesco, Milano Bicocca January-February 2016 -4 0 0 r + - + x -4 0 Unit 3 - 9 40 1. FIELD HARMONICS: LINEARITY 40 40 40 C' n C'' n c' n c ' n +c'' n =C' n +C'' n - c'' n - + - + - + + 0 0 0 -4 0 0 40 -4 0 0 40 + + - - -4 0 0 40 + + - - -4 0 -4 0 -4 0 Linearity of coefficients (very important) Non-normalized coefficients are additive Normalized coefficients are not additive Cn Cn' Cn'' Cn' Cn'' ' '' cn ' c c n n B1 B1 B1'' B1' B1'' Normalization gives handy (and physical) quantities, but some drawbacks – pay attention !! E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 10 CONTENTS 1. Definition of field harmonics 2. Field harmonics of a current line 3. Validity limits of field harmonics 4. Beam dynamics requirements on field harmonics E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 11 2. FIELD OF A CURRENT LINE Field given by a current line (Biot-Savart law) Differential form (international system) r Infinite current line A factor two is given by the atan integration Félix Savart, French (June 30, 1791-March 16, 1841) r Field in a centre of a circular loop, radius r r E. Todesco, Milano Bicocca January-February 2016 Jean-Baptiste Biot, French (April 21, 1774 – February 3, 1862) Unit 3 - 12 2. FIELD OF A CURRENT LINE: COMPLEX NOTATION Field given by a current line (Biot-Savart law) Infinite current line y Complex notation 40 z 0 =x 0 +iy 0 B=B y +iB x 0 -4 0 0 x 40 Félix Savart, French (June 30, 1791-March 16, 1841) z=x+iy Using the relation -4 0 We obtain the compact very useful notation E. Todesco, Milano Bicocca January-February 2016 Jean-Baptiste Biot, French (April 21, 1774 – February 3, 1862) Unit 3 - 13 2. FIELD HARMONICS OF A CURRENT LINE Field given by a current line (Biot-Savart law) y B( z ) By ( z ) iBx ( z ) I 0 I 1 B( z ) 0 2 ( z z0 ) 2z0 1 z z0 using 40 z 0 =x 0 +iy 0 B=B y +iB x 0 -4 0 x 0 40 z=x+iy -4 0 1 2 3 1 t t t ... t n1 1 t n 1 t 1 !!! Félix Savart, French (June 30, 1791-March 16, 1841) we get I 0 B( z ) 2z0 z n 1 z 0 E. Todesco, Milano Bicocca January-February 2016 n 1 I 0 2z0 Rref n 1 z 0 n 1 x iy R ref n 1 Jean-Baptiste Biot, French (April 21, 1774 – February 3, 1862) Unit 3 - 14 2. FIELD HARMONICS OF A CURRENT LINE Now we can compute the multipoles of a current line at z0 I 0 B( z ) 2z0 z n 1 z 0 n 1 I 0 2z0 Rref n 1 z 0 x iy B y iB x 10 B1 (bn ia n ) R n 1 ref 4 x iy R ref I 010 Rref bn ia n 2z0 B1 z0 4 n 1 x iy z0 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 E. Todesco, Milano Bicocca January-February 2016 n 1 n 1 ¦bn+ian¦ I0 1 B1 Re 2 z0 n 1 5 10 Multipole order n 15 Unit 3 - 15 2. FIELD HARMONICS OF A CURRENT LINE Multipoles given by a current line decay with the order 10 n 1 I 010 Rref ln bn ia n ln n ln 2R B ref 1 z0 4 1 ¦bn+ian¦ I 010 Rref bn ia n 2z0 B1 z0 4 p nq 0.1 0.01 0.001 0 5 10 Multipole order n 15 The slope of the decay is the logarithm of (Rref/|z0|) At each order, the multipole decreases by a factor Rref /|z0| The decay of the multipoles tells you the ratio Rref /|z0|, i.e. where is the coil w.r.t. the reference radius – like a radar … one can detect assembly errors in the magnet through magnetic field shape E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 16 2. FIELD HARMONICS OF A CURRENT LINE Multipoles given by a current line decay with the order 10 n 1 I 010 Rref ln bn ia n ln n ln 2R B ref 1 z0 4 1 ¦bn+ian¦ I 010 Rref bn ia n 2z0 B1 z0 4 p nq 0.1 0.01 0.001 0 5 10 Multipole order n 15 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 E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 17 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 E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 18 3. VALIDITY LIMITS OF FIELD HARMONICS When we expand a function in a power series we lose something 1 1 t t 2 t 3 ... t n1 1 t n 1 y t 1 Example 1. In t=1/2, the function is 1 1 2 1 1/ 2 1/ 2 40 0 -4 0 0 x 40 and using the series one has -4 0 2 3 1 1 1 1 ... 1 0.5 0.25 0.125 1.875 ... 2 2 2 not bad … with 4 terms we compute the function within 7% E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 19 3. VALIDITY LIMITS OF FIELD HARMONICS When we expand a function in a power series we lose something 1 2 3 1 t t t ... t n1 1 t n 1 y 40 t 1 Ex. 2. In t=1, the function is infinite 1 1 11 0 and using the series one has 1 1 1 1 ... 1 1 1 1 ... 2 0 -4 0 0 x 40 -4 0 3 which diverges … this makes sense E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 20 3. VALIDITY LIMITS OF FIELD HARMONICS When we expand a function in a power series we lose something 1 2 3 1 t t t ... t n1 1 t n 1 y 40 t 1 Ex. 3. In t=-1, the function is well defined 1 1 11 2 0 -4 0 0 x 40 BUT using the series one has -4 0 1 (1) 1 1 ... 1 1 1 1 ... 2 3 even if the function is well defined, the series does not work: we are outside the convergence radius E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 21 3. VALIDITY LIMITS OF FIELD HARMONICS When we expand a function in a power series we lose something 1 2 3 1 t t t ... t n1 1 t n 1 y 40 t 1 Ex. 3. In t=-1, the function is well defined 1 1 11 2 0 -4 0 0 x 40 BUT using the series one has -4 0 1 (1) 1 1 ... 1 1 1 1 ... 2 3 even if the function is well defined, the series does not work: we are outside the convergence radius E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 22 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 40 60 - - + + + + 0 0 -4 0 0 40 + + -4 0 0 + + - - 40 - - -4 0 -6 0 For other shapes, the expansion is valid over a circle that touches the closest current line E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 23 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 2d field harmonics B (T) B (T) 2d field harmonics -1 -1 0 length (m) 15 0 length (m) 15 B (T) 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 no 2d field harmonics One has to use a more complicated expansion -1 0 E. Todesco, Milano Bicocca January-February 2016 length (m) 15 Unit 3 - 24 CONTENTS 1. Definition of field harmonics 2. Field harmonics of a current line 3. Validity limits of field harmonics 4. Beam dynamics requirements on field harmonics E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 25 4. BEAM DYNAMICS REQUIREMENTS Main component of the dipoles (field) This is ensuring the orbit along the ring Typically a absolute knowledge of the magnetic field within 0.1% A spread between magnets of the order of 0.1% A reproducibility of the order of 0.01% We will see that the tolerances needed for building superconducting magnets naturally guarantee these levels Cables are positioned within 0.05 mm, apertures of the order of 25 mm, this gives 0.2% error in field, and similar values for spread For reproducibility it is critical to cycle the magnets E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 26 4. BEAM DYNAMICS REQUIREMENTS Main component of the quadrupoles (gradient) The tune has to be controlled within 0.001 Tune is proportional to quadrupole main component If the total tune is large (for instance 60 in the LHC) even 0.01% variation of quadrupole force is visible Corrector elements and feedback solve the problem – magnets alone cannot reach this level In general a absolute knowledge of quadrupole main component within 0.2% is achievable, a spread of 0.1%, and a reproducibility of 0.01% Note that accelerators with larger number of cells (larger tune) become more difficult E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 27 4. BEAM DYNAMICS REQUIREMENTS Sextupolar components Sextupole gives chromaticity – to be controlled on a edge of a cliff (positive but smaller than 10, negative values make the beam unstable) Dipoles have sextupolar components that needs to be controlled within 0.1 units (normalized multipoles) Example: in the LHC 1 units of b3 in the dipoles give 40 chromaticity units – so one need to know and control b3 within 0.05 units, This is within reach of measurement systems, and with proper precycling reproducibility can be guaranteed E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 28 4. BEAM DYNAMICS REQUIREMENTS High order mutipoles Rule of thumb (just to give a zero order idea): field harmonics have to be of the order of 0.1 to 1 unit ¦bn+ian¦ 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 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 E. Todesco, Milano Bicocca January-February 2016 5 10 Multipole order n 15 Unit 3 - 29 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 E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 30 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 ? E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 31 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. E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 32 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.org for most of the pictures E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 33 1. DEFINITION OF FIELD HARMONICS Vector potential Since B 0 one can always define a vector potential A such that A B The vector potential is not unique (gauge invariance): if we add the gradient of any scalar function, A' A f it still satisfies A' A f A B Scalar potential In the regions free of charge and magnetic material B 0 Therefore in this case one can also define a scalar potential (such as for gravity) g B One can prove that A ig is an analytic function in a region free of charge and magnetic material E. Todesco, Milano Bicocca January-February 2016 Unit 3 - 34 2. FIELD HARMONICS OF A CURRENT LINE Field given by a current line (Biot-Savart law) – vector potential formalism … y B Az r 1 Az Br r 40 z 0 =x 0 +iy 0 B=B y +iB x 0 -4 0 2 2 0 I R 0 I 0 2 0 cos( 0 ) Az ( , ) ln ln 2 0 2 0 0 x 40 z=x+iy -4 0 y z 0 = 0 exp(i 0 ) 40 … and polar coordinates formalism n 1 I Br 0 20 n 1 0 I B 0 20 n 1 0 n 1 sin n 0 B 0 -4 0 0 x 40 z = exp(i ) cosn 0 E. Todesco, Milano Bicocca January-February 2016 -4 0 Unit 3 - 35