Single-Particle Dynamics

Transcription

1 Single-Particle Dynamics Yannis PAPAPHILIPPOU June 5, 2001

2 Outline The single-particle relativistic Hamiltonian Linear betatron motion and action-angle variables Generalized non-linear Hamiltonian Classical perturbation theory Canonical perturbation method Application to the accelerator Hamiltonian Resonance driving terms and tune-shift The single resonance treatment Secular perturbation theory and resonance overlap criterion Application to the accelerator Hamiltonian - Resonance widths

3 The choice of the working point Outline (cont.) Linear Imperfections and correction Steering error, closed orbit distortion Gradient error Linear coupling Chromaticity Non-linear effects Kinematic effect Fringe-fields Magnet imperfections

4 Outline (cont.) Non-linear correction Sextupole correction Octupole correction Error compensation in magnet design Dynamic aperture Scaling law for magnet fringe-fields Frequency maps

5 Why care about single particle dynamics? Accelerator design focuses on high performance In the case of high-intensity rings, the goal is a low-loss design Losses in the high-intensity are due to the combination of space-charge and magnetic field errors Large transverse emittance in high-intensity low-energy rings amplifies error effects Identification of magnet errors and non-linearities and correction #

6 r p ; e The single-particle relativistic Hamiltonian H(x p t) = c A(x t) 2 + m 2 c 2 + e(x t) c x = (x y z) Cartesian positions p = (p x p y p z ) conjugate momenta A = (A x A y A z ) the magnetic vector potential the electric scalar potential c and e velocity of light and particle charge The ordinary kinetic momentum vector P = mv = p ; e c A with v the particle velocity and = (1 ; v 2 =c 2 ) ;1=2 the relativistic factor.

7 The single-particle relativistic Hamiltonian (cont.) The Hamiltonian is the total energy The total kinetic momentum H E = mc 2 + e H 2 c 2 ; m2 c 2 1=2 P = Use Hamilton s equations ( _x _p) = [(x p) H] to get equations of motion Lorenz equations #

8 H(x 0 p 0 t) = c vut (px ; e A s = (A ^s)(1 + x ) (s) x ) (s) (s) )2 + m2 c 2 + e(x 0 ) The single-particle relativistic Hamiltonian (cont.) Canonical Transformation (x y z p I: p y p z ) 7! (x y s p x p y p s ) x moving the coordinate system on a closed curve, with path length s,ofa particle with reference P momentum 0 in the guiding magnetic field. The new Hamiltonian A x) 2 +(p y ; e c A x) 2 + (p s ; e c As)2 (1 + x c p s = (p ^s)(1 + (s) the local radius of curvature

9 s H 2 c 2 ; m2 c 2 ; (p x ; e c A x) 2 ; (p y ; e c A y) 2 The single-particle relativistic Hamiltonian (cont.) Canonical Transformation (x y s p II: p y p s ) 7! (x y t p x p y ;H) x setting the s path as the independent variable and the ;p momentum s as the new Hamiltonian. = 0 Assuming + x 1 H (;p s ) = ; e c A s ; (s) If A time independent ;! H integral of motion, i.e. constant along the particle trajectories.??? Longitudinal motion neglected (not valid for RSC s)???

10 B y = ; b 0 (s) + b 1 (s)x normalized quadrupole gradient K(s) = b 1 (s) e cp 0 A s (x y s) = ; P 0c 1 2 ; K(s) (s) x 2 + K(s) y2 Linear betatron motion Assume a simple case of linear transverse fields: B x = b 1 (s)y main bending field B 0 b 0 (s) = [T] P 0c e(s) b 1(s) ] [1=m2 B = magnetic rigidity B = P 0c [T m] The vector potential A = (0 0 A s ) with e x + (s) : 2 e 2

11 + K(s) y2 Introduce the momentum = spread P = ( P ;P 0 P ) 0 P 0 Linear betatron motion (cont.) The Hamiltonian can be written as: x 0 H + 1 = 2 ; K(s) x 2 ;P (s) + x 1 q 2 ; p 2 x ; p 2 y P ; 2 2 (s) Equations of motion still non-linear in the canonical momenta! Canonical transformation III: (p x p y ;H) 7! ( p x P p y P ;H P ) rescaling the canonical momenta Expand the square root Throw away terms higher than quadratic (not always a good idea!) new canonical momentum p x = dx ds x0

12 ^H (;p s =P ) = K(s) y2 The new Hamiltonian: Linear betatron motion (cont.) ; 2 + p 2 y + px + 1 x + 1 (s) 1 2 ; K(s) x 2 (s) Hill s equations of betatron motion x ; K(s) x = (s) 1 (s) y K(s)y = 0 1+ Introduce dispersion function 00 x ; 1 D + K x(s)d x = 1 1 (s) where K x (s) 1 (s) 2 ; K(s) and K y(s) K(s).

13 the Courant-Snyder invariant A x y the alpha = ; 0 x y x y and gamma 1+2 x y = x y 0 2 x y the phase advance x y (s) = R s 0 R C 0 Linear betatron motion (cont.) Canonical transformation IV:(x y p x p y ) 7! (x ; x 0 y p x ; p x0 p y ) shifting the coordinate origin to the closed orbit. The new equations of motion x 00 + K x (s)x = 0, y 00 + K y (s)y = 0 : Homogeneous equations with periodic coefficients K x y (s) = K x y (s + C) Floquet Theory p = # x (s) cos( x (s) + 0x ) Ax y = x, q y y (s)cos( y (s) + 0y ) : A the Courant-Snyder amplitude function x y (s) functions d ) x y( the tunes Q x y = 1 2 ds x y(s)

14 Canonical transformation V to action angle variables ( x y J x J y ) p x (s) = ; s x (s) [sin ( x(s) + x (s)) + x (s)cos( x (s) + x (s))] Action-angle variables x(s) = 2 x (s)j x cos ( x (s) + x (s)) p 2J x with The new Hamiltonian x (s)x 0 + x (s)x arctan ; = (s) x x ; x (s) : 0 (J x J y ) = 1 H (Q xj x + Q y J y ) R The equations of motion J x = constant J y = constant and describe a 2-torus in the phase space ( x y J x J y ). x (s) = x0 (s) + Q x(s ; s 0 ) R y (s) = y0 (s) + Q y(s ; s 0 ) R

15 n;1 = ; n nr n;1 0 b Generalized non-integrable Hamiltonian Transverse vector potential: Basic law of magnetostatics r B = 0 ;! 9 A : B = r A x = B z B, = y Ampére s law in vacuum r B = 0 ;! 9 V : B = " Cauchy-Riemann conditions of analytic functions # A(x + iy) = A z (x y) + iv (x y) = The normal and skew multipole coefficients 1X n=1 ( n + i n )(x + iy) n n;1 0 with r 0 the reference radius, B 0 the main field B 0 and a n;1 = n nr B 0

16 x = D x + x 0 y = D y + y 0 h k x ky (s)xkx y k y Generalized non-integrable Hamiltonian (cont.) The Hamiltonian: The vector potential component A s is: H (;p s =P ) = 1 2 ; 2 + p 2 x y ; p ; e x A s cp (s) A s = (A ^s)(1 + x ) (s) 1X (1 + x = )B 0Re (s) 0 n + ia0 n b + iy) n+1 (x Use transformation x = x IV + D x + x 0 y = y + D y + y 0 The new Hamiltonian is n=0 n X H 0 = H 0 + H where 0 is the integrable Hamiltonian. The h terms x ky (s) k are periodic and depend a on n b0 n and the closed orbit displacements 0 kx ky

17 Classical perturbation theory (Linstead (1882), Poincaré (1892), Von-Zeipel (1916)) Consider a general Hamiltonian with n degrees of freedom H(J ' ) = H 0 (J) +H 1 (J ' ) + O( 2 ) where the non-integrable part H 1 (J ' ) is 2-periodic on and '. If small ;! distorted tori still exist ;! try to streighten up the tori # Canonical Transformation VI: Search a generating function S( J ' ) = J ' + S 1 ( J ' ) + O( 2 ) for transforming old variables to ( J ') so that new H( J) only a function of the new actions.

18 Classical perturbation theory (cont.) By the canonical transformation equations, the old action and new angle are: = J 1( J ' ) + O( 2 ) Inverting the previous equations: = ' 1( J ' ) ' + O( 2 ) = J 1( J ' ) J + O( 2 ) = ' 1( J ' ) ' + O( 2 )??? S 1 is expressed in terms of the new variables???

19 H( J ' ) = H(J( J ') '( J ') ) 1( J ' ) H 0 (J( J ')) = H 0 ( J) 0( J) The new Hamiltonian is: Classical perturbation theory (cont.) O( 2 ) + Expand term by term the old Hamiltonian to leading order 1 ( J ' ) + O( 2 ' Equating the terms of equal order in, we get in zero order H 0 = H 0 ( J) and in first order: H 1 (J( J ') '( J ') ) = H 1 ( J ') +O( 2 ) + H 1 ( J ') 1 1( J ' ) 1( J ' ) + with!( J) 0( J) the unperturbed frequency J

20 Average hh i = 1 H part: n ( J ')d ' H1 2 ; ' 1 Classical perturbation theory (cont.) New Hamiltonian should be a function of J only ;! eliminate ' # Oscillating part: fh 1 g = H 1 ;hh 1 i ' Using the previous equations, H 1 becomes 1 1( J ' ) 1( J ' ) + + hh 1 ( J ')i ' + fh 1 ( J ')g :

21 H 1 ( J) = hh 1 ( J ')i ' 1( J ' ) =!( 1( J ')i ' Classical perturbation theory (cont.) Choose S 1 so that ' dependence is eliminated: = ;fh 1 ( J ')g!( 1( J ' ) New Hamiltonian is a function of the new actions only to leading order!!! J) = H 0 ( J)+ hh 1 ( J ')i ' + O( 2 ) H( with the new frequency vector J) H( J)!( + O( 2 ) J

22 H 1k ( J)e i(k '+p) Classical perturbation theory (cont.) BUT can we find an appropriate generating function S 1? Expand in Fourier series # fh 1 ( J ')g = X k ' = k with ' k n ' n 1. and also k p X The unknown amplitudes S 1k ( J) are: k p S 1 ( J ' ) = S 1k ( J)e i(k '+p) : H 1k ( J) with S 1k ( J) = i k p6= 0 and finally k!( J) + p S( J ') = J ' + i X 1k H J) ei(k ( + 2 '+p) ) O( : p + J)!( k k6=0

23 Classical perturbation theory (cont.) Remarks 1) What about higher orders? In principle, the technique works for arbitrary order, but variables disentagling becomes difficult (even for 2nd order!!!) # Solution ;! Lie transformations (Application in beam dynamics by Dragt and Finn (1976)) 2) What about small denominators? Resonances k!( J) + p = 0 # Solution ;! Superconvergent perturbation techniques - KAM theory (Kolmogorov (1957), Arnold (1962) and Moser (1962))

24 r Jx x (s) H 0 (J x J y x y ) = H 0 (J x J y ) + H 1 (J x J y x y ) h k x ky (s) x (s) k y=2 Application to the accelerator Hamiltonian (Hagedorn (1957), Schoch (1957), Guignard (1976, 1978)) The general acceleretor Hamiltonian: 0 (J x J y x y ) = H 0 (J x J y ) + H 1 (J x J y x y ) H The transverse variable: e i( x(s)+x(s)) + e ;i(x(s)+x(s)) and the equivalent y. The Hamiltonian in action-angle variables: x(s) = 2 The integrable part H 0 (Jx Jy) = 1 R (Q xjx + QyJy ) The perturbation H 1 (Jx Jy x y s) = The coefficients J k x=2 x J k y=2 kxp y j kx ky P kyp g j k l m (s)ei[(j;k) x+(l;m)y ] l (s)e i[(j;k) x(s)+(l;m)y(s)] 2 The indexes 2 k l m: k x = j + k and k y = l + m. j g j k l m (s) = j+k+l+m k x j k y l kx=2 y

25 j+k+l+m 2 2 k x=2 x J k y=2 y J kxx kyx 1X g j k l m p e i[(j;k) x+(l;m)y;p s R ] k x ky (s)k x=2 x (s) k y=2 y (s)e i[(j;k) x(s)+(l;m)y(s)+p s R ] h Resonance driving terms h Coefficients x ky (s) k are periodic ;! in expand in Fourier series H 1 (J x J y x y s) = X j l p=;1 kx ky with the resonance driving terms 1 1 g j k l m p = k x j k y l I 2 For n x = j ; k and n y = l ; m we have the resonance conditions n x Q x + n y Q y = p. Goal of accelerator design and correction systems ;! minimize g j k l m p by Change magnet design so that h k x ky (s) become smaller Introduce magnetic elements capable of creating a cancelling effect

26 Tune-shift and spread First order correction to the tunes ;! derivatives with respect to the action of average part of H 1. For a given term h k x ky (s)xk x y k y: x = Jk x=2;1 x J k y=2 y Q kxx kyx g j k l m I e i[(j;k) x+(l;m)y] j l y = Jk x=2 x J k y=2;1 y Q kxx kyx g j k l m I e i[(j;k) x+(l;m)y] where g j k l m the average of g j k l m (s) around the ring. Remarks j l If Q x y independent of J x y ;! tune-shift If Q x y depends on J x y ;! tune-spread (or amplitude detuning) Q x y = 0 for k x = j + k or k y = l + m odd ;! go to higher order Leading order tune-shift ;! impact of errors and non-linear effects

27 Tune-shift and spread (cont.) Table 1: Tune spread produced by various mechanisms on a 2 MW beam with transverse emittance of 480 mm mrad and momentum spread of 1%. Mechanism Space charge Chromaticity Full tune spread (2 MW beam) 0.08 (1% p=p) Kinematic nonlinearity (480) Fringe field (480) Uncompensated ring magnet error (480) 0.02 Compensated ring magnet error (480) Fixed injection chicane Injection painting bump 0.001

28 ^H(^J ^') = ^H 0 (^J) +" ^H 1 (^J ^') 1 ^') P o k (^J H k1 k (^J)expn = i n [k 1 ^' ^H 1 + (k 1 n 2 + k 2 n 1 ) ^' 1 ] 1 k The single resonance treatment General two dimensional Hamiltonian: ') = H 0 (J) + "H 1 (J ') H(J with the perturbed part periodic in angles: H 1 (J ') = X k 1 k 1 H k1 k 2 (J 1 J 2 )exp[i(k 1 ' 1 + k 2 ' 2 )] Resonance n 1! 1 + n 2! 2 = 0 ;! blow up the solution # Canonical transformation VII: (J ') 7;! (^J ^') eliminate one action: r (^J ') = (n 1 ' 1 ; n 2 ' 2 ) ^J 1 + ' 2 ^J 2 F The transformed Hamiltonian ant the perturbation

29 H(^J ^') = H 0 (^J) + " H 1 (^J ^' 1 ) The single resonance treatment (cont.) Relations between the variables 1 n 1 ^J 1 J 2 = ^J 2 ; n 2 ^ J 1 J = : = n 1 ' 1 ; n 2 ' 2 ^' 2 = ' 2 1 ^' Transformation to a rotating frame where 1 = n 1 _ ' 1 ; n 2 _ ' 2 measures deviation from resonance. _^' Average over the slow ^' angle = ' 2 2 with H 0 (^J) = ^H 0 (^J) and 1 ^' ) = h ^H 1 (^J ^' 1 )i ^'2 = P +1 p=;1 H;pn 1 pn 2 (^J)exp(;ip ^' 1 ) H (^J 1 The averaging eliminated one angle and = J thus + J n 2 1 n ^J invariant of motion. is an

30 ^J 2 ^J 1 = ^J 10 The single resonance treatment (cont.) Assume dominant Fourier harmonics for p = 0 1 H(^J ^ 1 ) = H 0 (^J) +" H 0 0 (^J) + 2" H n1 ;n 2 (^J) cos ^' 1 Introduce ^J 1 = ^J 1 ; ^J 10 moving reference on fixed point and expand H(^J) around it! Hamiltonian describing motion near a resonance: r ( ^J 1 ^ 1 ) H 0 (^J) H ^J 1 2 ^J 1 = ^J 10 ( ^J 1 ) 2 + 2" H n1 ;n 2 (^J)cos ^' 1 Motion near a typical resonance is like that of the pendulum!!! The libration frequency The resonance half width 1 = 2" H n1 ;n 2 H 0 (^J) ^! ^J 1 ^J 1 = ^J 10! 1= =2 ^J 1 max = 2 2" H n1 ;n 2 (^J) B 2 H A C

31 n 3 ^J 1 n n 0 = Resonance overlap criterion (Chirikov (1960, 1979) Contopoulos (1966)) When perturbation grows ;! width of resonance island grows. # Rwo resonant islands ;! overlap orbits diffuse through the resonances. Distance between two resonances 2 1 +n 1 n0 n +n0 ; we get a simple resonance overlap criterion n max + ^J n 0 max ^J n n 0 Considering width of chaotic layer and secondary islands, we have the ^J two thirds rule n max +^J n 0 max 2 3 ^J n n 0 H 0 (^J) ^J 2 = ^J 10 ^J 1 Limitation: geometrical nature ;! difficult to extend in systems with

32 ^ J x ^ J y x ) = (n xq x + n y Q y ; p) ^ J x + ^ J y ^H( ky 2 Single resonance theory for the accelerator Hamiltonian (Hagedorn (1957), Schoch (1957), Guignard (1976, 1978)) The single resonance accelerator Hamiltonian H(J x J y x y s) = 1 R (Q xj x + Q y J y ) + g n x ny 2 kx 2 x J R y cos(n x x + n y y + 0 ; p s R ) J g with x ny ei 0 = g j k l m p n. From the generating function r ( x y ^ J x ^ J y s) = (n x x + n y y ; p s F ) ^ J x + y ^ J y R we get the Hamiltonian R + g n x ny 2 x k x J x ) 2 ^ (n R y J x + ^ ky J y ) 2 cos( ^ x + 0 ) ^ (n

33 kx 2 x ny J x n 2g + ky 2 ky;2 2 Resonance widths The two invariants are the new action and Hamiltonian. In the old variables: 1 J x c ; = J y n y x n c 2 = (Q x ; p )J x +(Q y ; y n + x n p )J y n y + x n y cos(n x x + n y y + 0 ; p s J ) : R n x,n y with opposite sign (difference resonances) ;! bounded motion n x, n y with same sign (sum resonances) ;! unbounded motion??? these are first order perturbation theory considerations??? Distance from the resonance e = n x Q x + n y Q y ; p;! resonance stop band width : kx;2 2 x J = g nx ny e R y (k x n x J x + k y n y J y ) J

34 The choice of the working point During design, impose periodic structure stronger than 1 Resonance condition n x Q x + n y Q y = p = mn, with m the super-periodicity If p = mn ;! structural or systematic resonances If p 6= mn ;! non-structural or random Major design points for high-intensity rings: Choose the working point far from structural resonances Prevent the break of the lattice supersymmetry

35 The choice of the working point (cont.) (0,1) (0,2) (0,3) (0,4) Horizontal Tune (1,1) (3,0) Vertical Tune (1, 1) (2,1) (2,1) (2, 1) (1,2) (1,2) (4,0) (3, 1) (1,0) (2,0) (3,0) (4,0) (1, 2) (0,3) (3,1) (3,1) (3, 1) (2,2) (2,2) (2, 2) (1,3) (1,3) (1, 3) (1, 3) (0,3) 6.6 (0,4) Horizontal Tune (1,2) (2, 1) Vertical Tune (1,1) (0,3) (0,2) Figure 1: Tune spaces for a lattice with super-period four. The red lines are the structural resonances and the black are the non-structural (up to 4th order). (2, 2) (1,3) (2,2) (1, 2) (3,1) (1,3) (3,0) (1,2) (2,1)

36 Linear Imperfections and Correction Steering error, closed orbit distortion Gradient error Linear coupling Chromaticity

37 Steering error and closed orbit distortion Closed orbit ;! control major concern in high intensity rings. Effect of orbit errors: Take vector potential for a normal multi-pole and x replace y and x + by x y + and y : 1X b n + ia n A nz (x y) = ;B 0 r 0 Re x iy ( ) + n+1 0 r n=0 n Xn=2 X n;2k 2kX m 2k ; B 0 = n 0 r b n n l n n ; 2k (;1) x n;2k;l y 2k;m x l y m k=0 l=0 m=0 2l l # multi-pole feed-down

38 p x y p x y i Z s+c p x y cos(jq x y + x y (s) ; x y ( )j)d Steering error and closed orbit distortion (cont.) Horizontal-vertical orbit distortion (Courant and Snyder 1957) B( ) x y (s) = ; 2sin(Q x y ) B s B( ) with the equivalent magnetic field error s = at. Approximate errors as delta functions n in locations: X i+n x y i = ; x y j p x y j cos(jq x y + x y i ; x y j j) with x y j kick produced by jth element: 2sin(Q x y ) j=i+1 j = B jl j B! dipole field error j = B jl j sin j B! dipole roll j = G jl j x y j B! quadrupole displacement

39 Closed orbit correction Introduce dipole correctors horizontal/vertical located in horizontal/vertical high beta s Introduce random distributions of errors (for simulations) Monitor x y i in BPM s (virtual, for simulations) Minimize x y i globally Use three-bumps method ;! for local correction (or alternative four-bumps)

40 Closed orbit correction (cont.) Figure 2: Horizontal and vertical closed orbit rms displacement for 101 error distributions in the SNS accumulator ring and maximum kicks required for correction (courtesy of C.J. Gardner)

41 Gradient error Key issue for good performance ;! super-periodicity preservation # Only structural resonances are excited Broken super-periodicity (e.g. by random errors) ;! non-structural resonances Causes: # Excessive beam loss Errors in quadrupole strengths (random ans systematic) Injection devices Higher order multi-pole magnets and errors

42 Q x y = 1 Integer Q x y = N and half-integer resonances 2Q x y = N Gradient error (cont.) Observables Distortion of the tune x y (s)k x y (s)ds I Distortion of the optics functions, e.g. beta variation 4 (s) x y (s) = ; 1 x y s+c Z ()Kx y()cos[;2(qx y + x y(s) ; x y())]d x y 2 sin(2qx y) s

43 Monitor x y(s) x y(s) Gradient error correction Introduce TRIM windings on the core of quadrupoles Introduce random distributions of quadrupole errors (for simulations) with BPM s turn-by-turn data (virtual) Monitor Q x y tune-shift in tune-meter (virtual, for simulations) Retune the lattice and minimize beta distortions (especially in areas of maximum beta and dispersion) with TRIM windings Move working point close to integer, half-integer resonances Minimize beta wave (or resonance stop-band width) Remarks: To excite/correct certain resonance harmonic N, strings powered accordingly Individual powering of TRIM windings for flexibility and beam based alignement of BPMs

44 Coupling coefficients: g = 1 R H p x y k s (s)e i[ x y;(qxqy;pn)] d Linear coupling and correction Betatron motion eqs. ;! coupled linear oscillators (xy term in A s ) Effect: Tune-shift Optics function distortion Coupling resonances excitation Q x Q y = N and beam loss Causes: Random rolls in the quadrupoles Skew quadrupole magnet errors (random and systematic) Offsets in sextupoles Observables: Distortion of the tune Q x y Distortion of the optics functions, e.g. beta variation

45 Correction: Linear coupling and correction (cont.) Introduce skew quadrupole correction Introduce random distributions of quadrupole rolls (for simulations) Kick the beam at one plane and monitor tune-shift and beta distortion on the other (virtual, for simulations) Minimize distortions globally with skew quadrupoles Move working point close to coupling resonance Minimize optics distortion and or coupling coefficients (measured by Fourier analysis of turn-by turn BPM data) Remarks: Global coupling corection with two families for each resonance Local coupling correction for each cell (individual powering) Save space with multi-function correctors (dipole + skew quadrupole + multi-pole)

46 Chromaticity Linear equations of motion depend on the energy through p=p Chromaticity: # Tunes and optics functions depend on particle energy x y = ; Q x y : p=p For a linear lattice, the natural chromaticity: x y (s)k(s)ds x y N = ; 1 I 4

47 Chromaticity control Large momentum spread ;! large chromatic tune-shift ;! resonance crossing Instability compensation (Landau dumping) Off momentum orbit on a sextupole ;! quadrupole effect ;! sextupole induced chromaticity: x y (s)b 2 (s) x (s)ds x y S = ; 1 I 2 Correction method: Introduce sextupoles in high beta/dispersion areas Change their strength so as to acheive the desired chromaticity

48 Chromaticity control (cont.) Two families of sextupoles are sufficient to control the horizontal and vertical chromaticity, at first order Problems: Sextupoles introduce a beta/dispersion wave Second order chromaticity is not corrected Solutions: Place them accordingly to cancel the effect (limited flexibility) Use 4 families of sextupoles

49 Chromaticity (cont.) β x δp/p= 0.7% β y δp/p= 0.7% β x δp/p= 0.0% β y δp/p= 0.0% β x δp/p= 0.7% β y δp/p= 0.7% β x δp/p= 0.7% β y δp/p= 0.7% β x δp/p= 0.0% β y δp/p= 0.0% β x δp/p= 0.7% β y δp/p= 0.7% β x,y [m] β x,y [m] η x δp/p= 0.7% η x δp/p= 0.0% η x δp/p= 0.7% S2(F2) S3(F3) η [m] η x δp/p= 0.7% η x δp/p= 0.0% η x δp/p= 0.7% S1(F1) S2(F2) S3(F3) S4(F4) S5(F1) η [m] S [m] S [m] 1.0 Figure 3: Lattice functions of a ring lattice using two (left) and four (right) families of sextupoles.

50 Chromaticity (cont.) ξ x 2 Sext Families ON ξ y 2 Sext Families ON ξ x 4 Sext Families ON ξ y 4 Sext Families ON ξ x, ξ y 0.0 ξx, ξy δp/p δp/p Figure 4: Plot of the chromaticities x y T as a function of momentum spread, when four families of sextupoles are used. The natural chromaticities are also plotted (distinct red and green lines in the middle of the plot. This plot reflects the successful minimization of the first and second order chromatic terms, accomplished by the four sextupoles families scheme.

51 Non-linear effects Kinematic effect Fringe-fields Magnet imperfections

52 Kinematic effect Kinematic non-linearity! high-order momentum terms in the expansion of the relativistic Hamiltonian Negligible in high energy colliders Noticeable in low-energy high-intensity rings First-order tune-shift: 1X (2k ; 3)!! kx Q x y = 1 2 k 2(k ) ; ;1 J ; k x y Jk; G x y y x G where = H ring x y k; y x ds x y Leading ;! order octupole-type tune-shift k=2 =0 2 2 k (2k)!!

53 By Laplace C equation: = ;C m n+2 ;C [2] m n m+2 n Magnet fringe-field - General field expansion Up to now, considered only transverse! field valid for high-energy colliders but not for low-energy high-intensity rings ;! Fringe-field longitudinal dependence of magnetic field, in magnet edges Consider general 3D-field: where y z) = r(x y z) B(x 2 (x y z) r 2 = 0 Appropriate expansion: (x y z) = 1X 1X C m n (z) xn y m n! m! n=0 m=0

54 1X 1X 1X 1X a n (z) =C 0 n+1 (z) n Bx [1] m n(z) xn y m C The field components: General field expansion (cont.) 1X 1X C m n+1 (z) xn y m B x (x y z) = n! m! n=0 m=0 B y (x y z) = C m+1 n (z) xn y m n! m! m=0 n=0 B z (x y z) = n! m! The usual normal and skew multipole coefficients are: n=0 m=0 B y = b n (0 0 z) (z) 1 n =C (z) (0 0 z) : Note that P k l=0 (;1) k; k [2l] l C m;2k n+2k;2l = C m n

55 Bx(x y z) = By (x y z) = Bz (x y z) = 1X 1X 1X 1X 1X 1X mx m m (;1) (;1) m xn y 2m mx m m (;1) a [2l] b [2l] x n y 2m " mx m +1 x n y 2m [2l] n+2m+1;2l b [2l] n+2m+1;2l a [2l+1] n+2m;2l b 2m +1 2m +1 2m +1 a [2l] n+2m;2l + a [2l+1] n+2m;1;2l + General field expansion (cont.) Consider two cases, for m = 2k (even) or m = 2k +1(odd) kx C 2k n = for n + 2k ; 2l ; 1 0 l=0 (;1) k k l n+2k;2l;1 C 2k+1 n = kx (;1) k k l l=0 n+2k;2l and finally the field components are! y l n=0 m=0 n!(2m)! l=0 [2l] n+2m;2l b l=0 m l m=0 n!(2m)! n=0 y # m+1 X ; l=0 l! y l=0 n=0 l m=0 n!(2m)!

56 Approaches to study fringe-fields 1) Get an accurate magnet model 2a) Integrate equations of motion 2b) Construct a non-linear map i) Hard-edge approximation (fringe-field length to zero) ii) Exact integration of the magnetic field iii) Parameter fit using Enge function 2c) Use your favorite non-linear dynamics tools to analyze the effect

57 m x 2n+1 y 2m+1 m (;1) l (;1) m x 2n y 2m z = y b [1] 0 +O(3) B (;1) m x 2n y 2m+1 [2l] 2n+2m+2;2l b Dipole fringe-field Using the general z-dependent field expansion, for a straight dipole: 1X mx B x = l=0 m n=0 (2n +1)!(2m + 1)! B y = 1X mx m b [2l] 2n+2m;2l l (2n)!(2m)! l=0 m n=0 B z = 1X mx m b [2l+1] 2n+2m;2l l and to leading order: (2n)!(2m + 1)! l=0 m n=0 B x = b 2 xy + O(4) B y b 0 ; 1 = b[2] 0 y b 2(x 2 ; y 2 ) + O(4) 2 Dipole fringe to leading order gives a sextupole-like effect (vertical chromaticity)

58 1X mx (;1) m x 2n y 2m+1 (;1) m x 2n+1 y 2m (;1) m x 2n+1 y 2m+1 z = xyb [1] 1 + O(4) B Quadrupole fringe-field General field expansion for a quadrupole magnet: B x = 1X mx m b [2l] 2n+2m+1;2l l (2n)!(2m +1)! l=0 m n=0 B y = 1X mx m b [2l] 2n+2m+1;2l l : m n=0 (2n +1)!(2m)! l=0 m b [2l+1] 2n+2m+1;2l l B z = m n=0 (2n + 1)!(2m +1)! and to leading order l=0 ; 1 b (3x2 + y 2 )b [2] 1 + O(5) 1 12 B x = y ; 1 b (3y2 + x 2 )b [2] 1 + O(5) 1 12 B y = x The quadrupole fringe to leading order has an octupole-like effect

59 Quadrupole fringe-field (cont.) The hard-edge Hamiltonian (Forest and Milutinovic 1988) Q H f = 12B(1+ p p ) (y 3 p y ; x 3 p x +3x 2 yp y ; 3y 2 xp x ) First order tune spread for an octupole: xa y a 1 a hh a vv hv a where the normalized anharmonicities are A 2J xa y 2J X hh = ;1 a 16B Q i xi xi i X hv = 1 a 16B Q i ( xi yi ; yi xi ) i X vv = 1 a 16B Q i yi yi : i

60 Quadrupole fringe-field (cont.) 5.84 Qy Figure 5: Tune footprints of the SNS ring, based on realistic (blue) and hard-edge (red) quadrupole fringe fields. Qx

61 Normal dipole (n = 0) ;! b 2j Normal quadrupole (n = 1) ;! b 4j+1 Normal sextupole (n = 2);! b 6j+2 Multi-pole errors A perfect 2(n + 1)-pole magnet! (r z) = (r n+1 ; z) which gives n = (2j + 1)(n + 1) ; B 6 /T z /m Figure 6: Dodecapole component in a 21 cm quadrupole with un-shaped ends. The reference radius is 10 cm, and the origin, z = 0, is at the magnet s center.

62 Non-linear correction Sextupole Skew-sextupole Octupole Error compensation in magnet design Dynamic aperture

63 Sextupole correction Cause: Chromaticity sextupoles Sextupole errors in dipoles Dipole fringe-fields Effect: Zero first order tune-spread, octupole-like second order Excitation of sextupole resonances 3Q x = N or Q x 2Q y = N Method: Introduce sextupole correctors (non-dispersive areas) Minimize globally sextupole driving terms

64 Sextupole correction (cont.) Sextupole Res. Driving Terms' Norm after correction before correction ξ x,y Figure 7: Norm of sextupole resonance driving terms before correction (dashed line) and after correction (solid line) with dedicated sextupole correctors. The resonance norms are reduced after the correction by as much as a factor of four.

65 Excitation of skew sextupole resonances 3Q y = N or 2Q x Q y = N Skew sextupole correction Cause: Skew sextupole errors (dipole roll) Effect: Zero first order tune-spread, octupole-like second order Method: Introduce skew sextupole correctors Tune their strength to globally minimize skew sextupole driving terms

66 Octupole correction Cause: Quadrupole fringe-fields Kinematic effect Octupole errors in magnets Sextupole, skew sextupole tune-spread Effect: Tune-spread linear in actions Excitation of normal octupole resonances 4Q x = N, 2Q x 2Q y = N 4Q and = N y

67 Octupole correction (cont.) Method: Introduce octupole correctors (in zero dispersion areas) Tune their strength to globally minimize tune-spread and/or driving terms The anharmonicities become 3 A hh = a hh + O j 2 xj X 16B j 6 A hv = a hv ; O j xj yj X 16B j 3 A vv = a vv + O j 2 yj : X 16B j

68 Octupole correction (cont.) β (m) β x β y Path (m) K1 K2 K Path (m) Figure 8: Top: the functions in the first half of the SNS straight section. x Bottom: integrated strengths of three families of octupoles versus location of the third family.

69 J 2 x Error compensation in magnet design Example: dodecapole in quadrupoles Tune-spread: xa X = y 5i Q i b D i 8B x J y A J i B where D i denotes the 3 2 matrix J 2 y xi ;6 2 xi yi 3 xi 2 2 xi yi 6 xi 2 yi ; 3 yi ;3 A : i.e. quadratic in the actions. Method of correction ;! Shape ends of the quadrupoles (local correction)

70 Error compensation in magnet design (cont.) 0.84 Qy Qx Figure 9: Tune footprints of a ring lattice with a dodecapole error in the quadrupoles of b 5 = 60 units; results are from tracking data (blue) or the analytic estimate (red).

71 Error compensation in magnet design (cont.) 0.81 Qy Qx Figure 10: Comparison of tune-shift plots using body (red) and local (blue) compensation of the dodecapole component in the SNS ring quadrupoles.

72 Dynamic aperture) 1000 DA without sextupoles 1000 DA with sextupoles Emittance (π mm mrad) δp/p=0 δp/p=-0.02 δp/p=0.02 Physical Aperture δp/p=+/-0.02) ( Physical Aperture δp/p=0) ( Emittance (p mm mrad) dp/p=0 dp/p=-0.02 dp/p=0.02 Physical Aperture dp/p=+/-0.02) ( Physical Aperture dp/p=0) ( ε x /(ε x +ε y ) e x /(e x +e y ) Figure 11: Dynamic aperture for the working point (6.3,5.8), without (left) and with (right) sextupoles.

73 z = y b [1] 0 +O(3) B Scaling law for magnet fringe-field Field of a streight dipole magnet at leading order: B x = b 2 xy + O(4) B y b 0 ; 1 = b[2] 0 y b 2(x 2 ; y 2 ) + O(4) 2 The change of transverse momentum imparted by the dipole field is: p b = ;e Z body v z b 0 dz ;ev z b 0 L eff with L eff = R body b 0dz=b 0 the effective length and b 0 the main dipole field.

74 Scaling law for magnet fringe-field (cont.) The momentum kick imparted by the fringe field: where p f x y = pf x y (k) +pf x y (?) f x(k) = e R p fringe v z y 0 B z (x y z)dz f y (k) = ; e R p fringe v z x 0 B z (x y z)dz the momentum increments caused by the longitudinal component of the magnetic field and f x(?) = ; e R p v z B y (x y z)dz fringe f y (?) = e R p v z B x (x y z)dz fringe the momentum increments by the transverse components of the magnetic field.

75 p f x (k) = e Z p f x (?) = ;e Z p f y (k) = ;e Z p f y (?) = e Z Scaling law for magnet fringe-field (cont.) The momentum increments of the particle caused by the longitudinal component: fringe fringe v z y 0 B z dz ev z b 0 yy 0 v z x 0 B z dz ;ev z b 0 yx 0 : The momentum increments caused by the transverse component of the dipole fringe fields are: v z B y dz ev z b 0 yy 0 v z B x dz 0 : The total momentum increments: p f x 2ev z b 0 yy 0 p f y ;ev z b 0 yx 0 :

76 f 2 i + h(p f y ) 2 q i = ev z b 0 4hy 2 y 02 i + hy 2 x 02 i x) qh(p Scaling law for magnet fringe-field (cont.) The total rms transverse momentum kick by the fringe field: where h:i denotes the average over the angle variables and (p f? ) rms = The rms transverse momentum kick becomes 2 y 02 i = (1 +3 y) 2 y hy i = hy 2 ihx 02 i = (1 + x) y x y hy : x x 4 f ) s + 3 y ) 2 (1 y (p 0 b z ev = rms? 8 (1 + x) y x y + x 4

77 Scaling law for magnet fringe-field (cont.) Ratio between momentum increment produced by dipole fringe field to the dipole body (p f? ) rms 1 s (1 + 3 y ) 2 y L eff (1 x) y x y + : + x 4 8 If small: (p b? ) rms (p f? ) rms (p b? ) rms? L eff? where the rms beam transverse emittance. When or y large: (p f? ) rms where the maximum of x or y.? L eff (p b? ) rms

78 Scaling law for magnet fringe-field (cont.) The quadrupole magnetic field in leading order: B x = y ; 1 b (3x2 + y 2 )b [2] 1 + O(5) 1 12 B y = x ; 1 b (3y2 + x 2 )b [2] 1 + O(5) 1 12 z = xyb [1] 1 + O(4) B b where (z) 1 the transverse field gradient. The momentum increments for body part b x = ;ev zb 1 xl p eff p b y = ev zb 1 yl eff L where eff R body b 1dz=b 1 is the effective length. =

79 p f x (?) ;ev zb 1 2xyy 0 + (x 2 + y 2 )x 0 p f y (?) ev zb 1 ;2xx 0 y + (x 2 + y 2 )y 0 : Scaling law for magnet fringe-field (cont.) The momentum increments from the longitudinal field component p f x (k) ev zxyy 0 b 1 p f y (k) ;ev zxyx 0 b 1 and the momentum increment by the transverse component of the fringe fields 2xx 0 y + (x 2 + y 2 )y 0 : 4 4 Combining the contributions 2xyy 0 ; (x 2 + y 2 )x 0 f x ev zb 1 p 4 f y ev zb 1 p 4

80 (p f? ) rms ev zb 1 + (1+5 y) x 2 y 3 y + 3 y (1 +5 x ) x 3 x + 3 +(1 +5 y ) y 3 y + 3 y (1 + y ) 2 x ; 8 x y x y +2(1+ 3 x ) 2 y 2 x x (1 + x ) 2 y ; 8 x y x y + 2(1 + 3 y ) 2 x x (1 + x ) 2 y ; 8 x y x y + 2(1 + 3 y ) 2 x x 2 y ) 1=2 Scaling law for magnet fringe-field (cont.) The total rms transverse momentum kick 16 Ratio between momentum increment from fringe field to that of the body f ) ( + (1 5 2 x y 3 1 x x (1 + x y ) ) 2 x ; 8 x y x y + 2(1 +3 x ) 2 y 2 x? (p rms 8L eff x y ( x x + y y ) 2 x y ( x x + y y ) Crude estimate, for small: (p f? ) rms? L eff (p b? ) rms

81 and when s are large: (p f? ) rms where is the maximum of x or y. (p b? ) rms? L eff

82 ( + iy) n+1 b [1] n (z) (x (x + iy)n+1 [(n + 3)x ; i(n + 1)y] b [2] n (z) ; (x + iy)n+1 [(n +1)x ; i(n +3)y] b [2] n (z) ; + O(n + 3) ) Re f(x + iy) g z b ev L n eff n n! Im f(x + iy) g z b ev L n eff n n! Scaling law for magnet fringe-field (cont.) General multipole expansion to leading order: + O(n + 4) ) ( (x + iy) n b n (z) B x (x y z) =Im 4(n + 2)! n! + O(n +4) ) ( + iy) n b n (z) (x B y (x y z) =Re 4(n +2)! n! B z (x y z) =Im (n +1)! The body part R body e v z B y (x y z)dz ; p b x = ; R body e where L eff = R body b n(z)dz=b n is the effective length. p b y = v z B x (x y z)dz

83 The fringe part: Scaling law for magnet fringe-field (cont.) p f x(k) z b n ev + 1)! Im (x + iy) n+1 y 0 (n : and p f y (k) ; z b n ev + 1)! Im (x + iy) n+1 x 0 (n p f x(?) ;ev zb n +1)! Re (x + iy) n (n +1)xx 0 +(n + 3)yy 0 + i(n ; 1)xy 0 ; i(n + 1)yx 0 4(n f y (?) ev zb n p +1)! Im (x + iy) n (n + 3)xx 0 + (n +1)yy 0 + i(n +1)xy 0 ; i(n ; 1)yx 0 4(n

84 The total momentum increments: p f x ; z b n ev + 1)! Re (x + iy) n (n + 1)(x ; iy)(x 0 + iy 0 ) +2iy 0 (x + iy) 4(n p f y : b n ev z 1)! Im (x + iy) n (n + 1)(x ; iy)(x 0 + iy 0 ) ; 2x 0 (x + iy) + 4(n

85 " nx l=0 n;l x Scaling law for magnet fringe-field (cont.) ev z b n 2(n ; l) ; l n 2l n;l x 2X (p f? ) rms " nx l=0 l y n;l x l y n l m ( x y x y ) m x 2;m y g m=0 2 n+3 (n + 1)! l (p b? ) rms ev zb n L eff n 2(n ; l) n ; l l 2l l n;l l # 1=2 y x y l 2 n n!

86 2 + 1)(2l + 1) ; n l (n 2 + 4n +5)(2l + 1) ; n l (n 2 +4n + 5) ; n l (n ; 8(n + 1) h (2l + 1) ; n 2l(2n + 1)(;1) l ; 2ni ; 2(5n + 2ln + 2)(;1) l ; 2ni + 2(n + 2)(;1) l ; 2ni ; ; l)(;1) l ; 2ni +(n The coefficients [1 + (2l + 3) y ] h 2l g n l 0 ( x y x y ) = (l + 1)(l +2) 8(n + 1)(n ; l)(2l + 3)(;1)l; 2n 2l x y y ; x (l + 1)(l + 2) [1 + (2n ; h 2l g n l 1 ( x y x y ) = x (n ; l + 1)(l +1) (2n ; 2l + 1)[1 +(2l + 1 h 2l + y (n ; l + 1)(l + 1) (2n ; 2l + 1) x y l 2l (n ; l + 1)(l + 1) h; + 1 ; n l 2 2n(;1) l ; 2ni 2l + n (2n ; 2l + 1)[1 +(2n ; 2l +3) x ] g n l 2 ( x y x y ) = (n ; l + 1)(n ; l +2)

87

88 s 2n + 2 Scaling law for magnet fringe-field (cont.) For flat beam (vertical y y 0 vanishes) and the total transverse rms momentum increment for the body (p b? ) rms q b x )2 i ev zb n L eff h(p r 2n n n? where n the average of n and? the rms beam transverse emittance. For the fringe: 2 n n! n (p f? ) rms q f x) 2 i ev zb n h(p [1+(2n 2 ] n +3) n+2? 2) + 2(n 2 n+3 n! n + 1

89 Scaling law for magnet fringe-field (cont.) The ratio of the rms momentum transverse kicks: (p f? ) rms (p b? ) rms s? + 1) [1 + (2n (2n +3) 2 ] 8L eff n n +2) 1)(n + (n Considering beta functions are not varying rapidly: (p f? ) rms When anomalously large: (p b? ) rms? L eff (p f? ) rms? L eff (p b? ) rms

90 n=2 n=2? (2n ; 1)!! ;n 1 1=2 ; n 1) ;n (1=2 2 3F " nx l=0 1=2 Scaling law for magnet fringe-field (cont.) For a round beam, transverse emittances are equal x = y =?. For simplicity, x y = and x y = and n n and the total transverse rms momentum increment for the body: (p b? ) rms ev zb n L eff n! 2 n=2 n! The rms momentum kick given by the fringe field is: f? ) rms ev zb n n=2 n=2+1? (p 2l g n l ()# 1=2 2(n ; l) ; l n l 2 n+3 (n + 1)!

91 n=2 C n() 3F 2 (1=2 ;n ;n 1 1=2 ; n 1)(2n ; 1)!! Scaling law for magnet fringe-field (cont.) The ratio of the rms momentum transverse kicks is: (p f? ) rms? n=2 L eff where the coefficient C n is: (p b? ) rms 1=2 2 4 P ; 2(n;l) ; 2l n l gn l () n;l l=0 n! C n () = 8(n + 1) For small functions: (p f? ) rms? L eff and for large: (p b? ) rms (p f? ) rms? L eff (p b? ) rms

92 Frequency Maps Perturbation theory ;! insight about non-linear Problem: Construction of normal forms or action variables cannot be applied for large perturbations Need a method to represent the systems global dynamics # Frequency map analysis (Laskar 1988, 1990) Algorithm to precisely compute frequencies associated with KAM tori of tracked orbits

93 Frequency Maps (cont.) NAFF algorithm ;! quasi-periodic approximation, truncated to order N, f 0 j (t) = N X k=1 a j k e i! jkt with f 0 j (t) a j k 2 C and j = 1 ::: n, of a complex function f j (t) = q j (t) + ip j (t), formed by a pair of conjugate variables determined by usual numerical integration, for a finite time span t =. Recover the frequency vector = ( 1 2 ::: n ) ;! parameterizes KAM tori.

94 F : R n ;! R n Frequency Maps (cont.) Construct frequency map by repeating procedure for set of initial conditions. Example: Keep all q the variables constant, and explore the p momenta to produce the F map : pj q=q0 ;! : Dynamics of the system analyzed by studying the regularity of the frequency map.

95 F.M.A is applied to tracking data N, Frequency Maps (cont.) f 0 j (t) = N X k=1 a j k e i! jkt with f 0 j (t) a j k 2 C and j = 1 ::: n, of a complex function f j (t) = q j (t) + ip j (t), formed by a pair of conjugate variables determined by usual numerical integration, for a finite time span t =. Recover the frequency vector = ( 1 2 ::: n ) ;! parameterizes KAM tori.

96 F : R n ;! R n Frequency Maps (cont.) Construct frequency map by repeating procedure for set of initial conditions. Example: Keep all q the variables constant, and explore the p momenta to produce the F map : pj q=q0 ;! : Dynamics of the system analyzed by studying the regularity of the frequency map.

97 R 2 ;! R 2 Frequency Maps (cont.) F.M.A is applied to tracking data ( = 500 turns), for large number of initial conditions ( 10 4 ). Particle coordinates distributed uniformly on Courant-Snyder invariants A x0 and A y0, at different ratios A x0 =A y0. The map is F : (I x I y )j p x py=0 ;! ( x y )

98 Frequency Maps (cont.) Example: Frequency maps for working point (6.3,5.8) of SNS accumulator ring. Inject 1000 particles with different amplitudes up to a Maximum emittance of 480 mm mrad Five different momentum spreads (p=p = 0 0:1 0:15) Small magnet field errors (10 ; ;4 level) Quadrupole fringe fields in the hard-edge approximation

99 Frequency Maps (cont.) (Q x,q y )=(6.3,5.8) without sextupoles (Q x,q y )=(6.3,5.8) with sextupoles 5.95 (1,3) δp/p=(1.5%,1%,0, 1%, 480 π mm mrad (3,1) 5.95 δp/p=(1.5%,1%,0, 1%, 480 π mm mrad (1,3) (1,2) (1,2) (3,1) Vertical Tune (1,1) (2,2) (0,4) Vertical Tune (1,1) (2,2) (0,4) (2, 1) 5.65 (2,1) (1, 3) (2, 2) (0,3) (3, 1) (4,0) (1, 2) Horizontal Tune (3,0) (2, 1) (2,1) (1, 3) (2, 2) (0,3) (3, 1) (4,0) Horizontal Tune (3,0) (1, 2) Figure 12: Frequency maps for the working point (6.3,5.8), without (left) and with (right) sextupoles.

100 Frequency Maps (cont.)

101 e Horizontal Tune ( 4,7) ( 3,6) ( 9,5) ( 8,4) Vertical Tune (3,7) ( 15,4) 0 6 σ 6 10 σ σ σ o lost orbit W.P (3, 6) W.P. (2, 5) (1, 4) (4,3) ( 5,1) (5,2) ( 6,2) (6,1) ( 7,3) ( 8,4) ( 10,6) (6,1) (4,6) ( 2,5) (7,0) (13,1)

102 jdj 2 x0 + I2 y0 )1=2 R : (I Diffusion Maps Calculate tune for two equal and successive time spans and compute diffusion vector: Dj t= = j t2(0 =2] ; j t2(=2 ] Plot the points in (A x0 A y0 ) space with a different colors grey for stable (jdj 10 ;7 )to black for strongly chaotic particles (jdj > 10 ;2 ). Diffusion quality factor: D QF =

103 Diffusion Maps (cont.)

104 D < D < D < D < D < D < D 16 Vertical Position (σ) ion (σ) Horizontal Position (σ) D < D < D < D < D < D < D