Lecture 2 Aspects of Transverse Beam Dynamics
Chandra Bhat
1 Accelerator and Beamline Magnets Dipole Magnet: Dipole magnet is a device used to bend the path of charged particles during beam transport. The radius of curvature of a charged particle in a constant magnetic field perpendicular to its path is, 1 1 eB 0.2998 B[T ] 0.04 I [amp].n Iron Yoke = [m-1] = 0 = ;B = total R ρ p p[GeV / c] 0 h[cm] 1 n=number of turns h=pole gap Quadrupole Magnet: A device used to focus charged particle beam during beam transport.
Particle trajectory Let us see what is the relationship between focal length, f, and the in a magnetic field A quadrupole strength.α Fig. A shows bending of a charged particle in a magnetic field perpendicular to the plane of the paper and “B” l α ρ ρ shows optical analogue of focusing. Thenφ the deflection angle, l r eB eB = − = − = l = − φ l α l 2 = − f p βE ρ Butα the total bending field Bφ is given by, B dB Optics β B φ = φ r = gr dr Then, egrl r 1 eg eg = − = − or = kl; k = = 3 E f f βE p f= focal length Quad strength 2 0.2998 g[Tesla / m] 2µ nI k[m−2 ] = ; g = 0 2 4 Field free region βE[GeV / c] R The quadrupole magnets provide material free aperture and focusing. A conventional quadrupole magnet used in synchrotrons has four iron poles with hyperbolic contours. By = −gx
Bx = −gy Interesting features: The horizontal force component depends only on the horizontal position of the particle trajectory. Similarly Quadrupole Field for the vertical axis. Configuration y Fx = −evgx and Fy = evgy 4 x A single “quad” can provide focusing only in one plane because of the magnetic field configuration. Hence, one needs at least two consecutive quadrupole magnets to Defocusing get overall focusing of all charged particles.
st Focusing 1 1 1 L f1= focal length of 1 quad = + − f nd 5 f f f f f 2= focal length of 2 quad Optics 1 2 1 2 L = separation between two quads Linear Machine: Contains only dipoles and quadrupoles. In these machines the horizontal and vertical motions are “completely decoupled”
f1 3 f2 Sextupole Magnet: These are used for chromatic corrections Iron Yoke during beam transport, storing or beam acceleration. The sextupole magnets generate non-linear magnetic field and introduce horizontal and vertical motion of the beam 1 B = g ' ( x 2 − y 2 ), B = g ' xy y 2 x
6µ nI -1 g ' = 0 [m ] R 3 The momentum-independent sextupole strength is given by, eg' 0.2998 g'[Tesla / m2 ] 6 m = , m[m−3 ] = p p[GeV / c]
1 2 3 B x = gy + sxy + o ( 3 x y − y ) + ... Multipole Field Expansion: General 6 (Vertical bending field) 1 multipole field expansion is given by, B = B + gx + s ( x 2 − y 2 ) + ... y 0 y 2 7 (Horizontal bending field) In an un-coupled case, 1 1 B = 0 and B = B + gx + sx2 + sx3 + ... g, s, o are quadrupole, sextupole and octupole x y 0 y 2 6 strength parameters, respectively
Thus, eBy eB0 y eg es 2 = + x + x + ... 8 p p p 2 p
Dipole Quadrupole Sextupole contributions 4 Beam Transport and FODO Lattice
A beam transport comprises of magnetic elements that form a lattice which guides the charged particle beam from one point to another. Such a beam generally traverses through a vacuum beam pipe. Lattice may be for FODO Period – Beam transfer line – Circular beam storage ring or accelerator – Linear accelerator 1/2QF QD 1/2QF FODO Lattice: Nomenclature A lattice comprising symmetric quadrupole triplets that k(s) Symmetric FODO can focus beam in both X and Y planes or a lattice comprising of Focusing-Drift-Defocusing-Drift with quad strengths. 1/2QF QD 1/2QF
Beamline and Circular Machine: Nomenclature FODO lattice can be repeated as many times as needed k(s) Dipole to form a beamline or a circular machine.
QF QD
5 Weak Focusing Accelerators So far we have assumed that the trajectory of the path of any particle in a beam is always perpendicular to the dipole magnetic field in a circular accelerator. Suppose a particle has a vertical N angle ≠ 0. Then the particle follows a spiral path and gets lost. Therefore we need an additional focusing force to keep the particle stable. We know that for an ideal orbit S mv2 mv2 = qvB ⇒ Restoring Force = − qvB = 0 9 R y R y mv2 For any other particle the restoring force is given by, F = − qvB x R y 10 To keep the beam particle focused one needs the gradient component in the magnetic field. dBy R dBy x By = B0 y + x + ... = B0 y (1+ + ...) dx B0 y dx R v x R dBy ≈ B 1− n with n = − ⇐ Field Index r m 0 y +B R B0 y dx 11 R Then one can show that if 0 An optical analogue of weak focusing accelerator is shown here Closed orbit for p >p0 Central Design orbit (Closed orbit for p =p0) ρ ∆p x0 = D(s) p0 Closed orbit for p Combination of focusing and de-focusing B quads shown in Fig. B with f1=-f2 will give 1 1 1 L L = + − = 2 ∴Always focusing f f1 f2 f1 f2 f1 Optical analogue of such an accelerator with the combination of focusing and de-focusing quads isshown in Fig. B . Closed Orbit for the strong focusing lattice Earlier strong focusing accelerators are Accelerators with combined built with combined function magnets. function magnets Recent strong focusing accelerators used ≈± separated function magnets; CERN PS n 288 • Dipoles for bending Fermilab Booster n=165, -207 •Quads for focusing Fermilab Recycler n=620, -598 • Sextupoles for chromatic corrections 8 Coordinate System Reference orbit Y We use an orthogonal right handed coordinate system (Center of the beam) s σ Or Ideal orbit x (y,x,s) that follows the reference orbit particle. Or Design orbit X ρ y ∴This is a moving coordinate system with x, y – deviation of particle trajectory from reference O* orbit at the point of interest “O* ” Trajectory of a particle s – tangential vector at “O* ” σ - individual particle trajectory. This will be our coordinate system in the rest of our analysis Hill’s Equation: σ From this picture, φ 2 ρ Horizontal Plane 2 2 2 2 x dx (d ) = []d 0 ( 0 + x ) + (dx ) ≈ (ds ) 1 + ; 4 d 0 0 = ds x'= 0 dx dσ ds σ Small angle x approximation or d = ds 1 + 13 x ρ 0 ρ σ Part. Traj. The equation of φmotion is going to be with respectφ toρ the x’ s Reference reference orbit. φ orbit Note that in the deflection plane ρ φ 0 φ d 2 x d dφ dφ dx' = d − d ⇒ x'' = = 0 − dφ 0 2 14 0 ρ φ σ ds ds ds ρ Substituting ds d dsρ x d 0 = , d = = 1+ 15 O’ O 0 ρ0 9 We get the equation of motion as, ρ 1 ρ1 x x' ' = − 1 + 16 0 ρ 0 For the monochromatic beam with momentum p0, the curvature of the charged particle in an electromagnetic field is given by, ρ 1 qBy q s 2 2 = = B0 y + gx + (x − y ) + ... ρ p0 p0 2 ρ 1 ≈ + kx 17 ρ 0 ρ Then, 1 1 x ρ1 k 2 very small so we x' ' = − + kx 1 + = − k + 2 x − x 0 0 0 0 ρ 0 can neglect Thus, the equation of motion in horizontal plane becomes 1 x' '+ Kx = 0; with K = k + 2 ρ 0 18a Similarly in vertical plane q dB Notice that this equation a) has not got dipole x''−kx = 0; with k = 0 x p dy strength, b) change in sign 0 18b x The term 2 in the above equation describes the “weak focusing” of a bending ρ0 magnet. In a large accelerator this term can be neglected. 10 Momentum Dispersion: In reality, the particles in a beam are not monochromatic. Now let us look at a particle that has a small momentum offset, i.e., δ ∆p p → p0 + ∆p = p0 (1+ ); δ = 19 p0 Then the curvatureρ is rewritten as, δ 1 eBy −1 e 2 = (1 + ) = []B0 y + gx + ... []1 − + − ... p0 p0 ρ δ 1 δ 20 ≈ + kx − δ 0 ρ 0 Thus, the equations of motion become δ x' '+ Kx = ρ This has bending term 0 One can combine δ u' '+ K (s)u = ρ(s) and focusing term δ these two into one ρ k(s). K(s) is called spring y ' '+ ky = equation as, 0 ρ y 0 constant 21 These are equations of motion for “strong focusing” charged particle beam accelerators (and for the beam transport lines). This equation is called “Hill’s equation”. Notice that the magnitude of the focusing strength is a free parameter. 11 Solutions for the Equation of Motion (piecewise method) Let us solve the simpler case of Hill’s equations for a monochromatic beam., i.e., δ=0. u' '+ K (s)u = 0 22 The principal solution to this equation (assuming K is a constant) are, 1 For K > 0 : C (s) = cos( K s) & S (s) = sin( K s) K 1 23 For K < 0 : C (s) = cosh( K s) & S (s) = sinh( K s) K which are linearly independent. “C”= cosine like function and “S”= sine like function. They satisfy initial conditions at s= 0 C (0) = 1, C ' (0) = 0, S (0) = 0 S ' (0) = 1 24 Any arbitrary solution can be written as a linear combination of C and S u(s) = C (s)u + S (s)u' u 0 0 with 0 as arbitrary initial coordinates 25 u' (s) = C ' (s)u0 + S ' (s)u'0 u'0 of the particle trajectory In matrix form this is written as, u (s) C (s) S (s) u 0 u 0 26 = = M u ' (s) C ' (s) S ' (s) u '0 u '0 The determinant =CS’-C’S and its derivative, CS’’+C’S’-C’S’-CS’’=CS’’-C”S=0 are independent of s 12 Solutions for the Equation of Motion (cont.) For the initial conditions of s=0 , the 2x2 matrix becomes, C (s) S (s) 1 0 = unit matrix 27 C ' (s) S ' (s) s =0 0 1 Further, with negligible dissipating forces we find that, for any arbitrary beamline, the determinant C (s) S (s) det M = = 1 28 C ' (s) S ' (s) Remark: There are some cases where the above determinant ≠1. This means there might be some accelerating/decelerating or quantum effect to be taken into account. If det M < 1 Damping 29 > 1 Anti - damping 13 Transformation Matrices for the Accelerator Components Here we deal with some commonly used accelerator components 1 A: Pure focusing quadrupole: = 0 , k > 0 , s = l . Set θ = kl = k (s − s0 ) . Then, θ ρ0 sin(θ ) cos( ) 1 0 1 0 1 M QF = θ k ; For a thin lens, lÆ0, M QF = = − kl 1 − 1 − k sin( ) cos( θ ) f 30 1 B: Pure defocusing quadrupole: = 0 , k < 0 , s = l . Set θ = k l = k (s − s0 ). Then, θ ρ0 sinh(θ ) 1 0 cosh( ) 1 0 θ k 1 M DF = ; For a thin lens, lÆ0, M DF = = − kl 1 − 1 − k sinh( ) cosh( θ ) f 31 1 C: Drift space: = 0 , k = 0 , s = L . Set θ = kl = k (s − s0 ) . Then, ρ0 sin 1 L with, lim ϕ with = k L 32 M D = ; 0 1 ϕ ϕ → 0 ϕ D: Quadrupole Doublet: 1 1 0 1 0 1 − L 1 L 1 1 f1 1 1 1 L L M = − 1 − 1 = with = + − = ; f = f = f 0 1 1 1 f * f f f f f 2 1 2 f 2 f1 − * 1 − 1 2 1 2 f f 2 33 14 Transformation Matrices (cont.) E: FODO cell - (Focusing+Drift+Defocusing+Drift) L L2 L2 Nomenclature 1 0 1 0 1− − 2L − k(s) 1 L 1 L f f 2 f M = 1 1 = 1 − 1 L L 0 1 f 0 1 f − 2 1+ f f QF QD 34 FODO cell - symmetric quadrupole triplets Symmetric FODO L2 L k(s) 1 0 1 0 1 0 1 0 1− 2 2L(1− ) 1 L 1 L f f M = 1 1 1 1 = − 1 0 1 1 1 0 1 − 1 L L L2 2 f 2 f 2 f 2 f 1/2QF QD 1/2QF − 2 (1+ ) 1+ 2 2 f f 2 f 35 We make the following point which is critical for beamline or accelerator design. These transformation matrices enable us to follow a charged particle through a transport line/accelerator made of an arbitrary number of drift spaces, quadrupoles and bending magnets. Thus, the final transfer matrix for any system looks like, M = M n M n−1...M1 36 If the sequence of elementary matrices represent components all around a circular accelerator, then we can use this matrix to investigate stability of transverse oscillations. 15 Stability criterion We demand that in a synchrotron or transport line the spacing between lenses and strength of the these lenses should provide stable oscillations for passage of a charged particle beam. This implies the stability criterion is the quantity, n u0 M 37 u'0 must remain finite for an arbitrary value of n. M is the matrix for one turn or repetition period. Then it can be shown that, the stability criterion can be met if, Trace M −1 ≤ ≤ 1 38 2 For FODO lattice (symmetric triplets or otherwise) the stability criterion will be L2 L2 −1 ≤ 1− ≤ 1 ⇒ 0 ≤ 2 − ≤ 2 2 f 2 2 f 2 L 39 or f ≥ Trajectory of an 2 Betatron Oscillations: individual particle Now we can sketch oscillations of a particle traversing through a lattice described above. These oscillations are called betatron oscillations Also, note that the particles do not have QFQD QF any slope and are on axis do not exhibit the oscillations. 16 Solutions for the Equation of Motion (closed form) The equation of motion u ' ' + K ( s ) u = 0 has the following features, 1. The quantity K is a function of s 2. K is periodic for important class of accelerators, e.g., circular accelerators ⇒ K (s + C ) = K (s) C is a repeat distance or “super-period” 40 3. Closely resembles that for simple harmonic motion Then the general solution of such an equation can be given by u(s) = A w(s) cos[ψ (s) + δ )] Quantities independent of s 41 w(s) is a periodic function with periodicity “C” ψ By substituting for u(s) inψ Eq. 22 we get, u' '+ K (s)u = 0 22 u' '+ K (s)u = − A[( 2 w' '+ w ' ' ) sin(ψ + ) + (w' '− w '2 + Kw ) cos( + ) = 0 δ Equating the sine term to zero ⇒ 2 w' '+ w ' ' = 0 = 2 ww ' '+ w 2 ' ' = (w 2 ' )' Cψ1 ψ ; C1ψ is an arbitrary constant of integration or ψ ' = 2 w ψ Thus, the phase of the oscillation of particle advances accordingψ to δ s0 +C ψ C1 Because w(s) is periodic, this integral is ψ (s0 → s0 + C ) = ∆ψ c = 2 ds ψ ∫ independent of choice of s0 s w (s) 0 42 17 Solutions (closed form) (cont.) Now we can express the transformation matrix M in terms of “A” and w(s). To do that, we rewrite the general solution as, ψ u(s) = A1 w(s) cos( ψ (s)) + A2 w(s) sin(ψ (s)) 43 and A2C1 A1C1 u' (s) = A1w'+ cos( ) + A2 w'− sin(ψ ) ψ w w u u’ s=s ψ = 0 A1 Now we set the initial values of andψ at 0 with phase angle and solve for and A2, which gives, A1=u0/w and A2=(wu0’-u0w’)/C1. Then the transfer matrix in terms of phase advance ψ (s0+C)= ∆ψ looks like, ψ ww' ψ w2 ψ cos(∆ c )α− sin(∆ c ) sin(∆ c ) u C1 ψ C1 u β = γ 2 ψ u' 1+ (ww'/ C1ψ) ww' u' s0 +C − sin(∆ ) cos(∆ ) + sin(∆ ) s0 2 c β c ψ c w / C1 ψ C1 cos(∆ ) + sin(∆ β) ψ sin(∆ ) u 44 c c α c = − sin(∆ c ) cos(∆ c ) − sin(∆ψ c ) u' s0 with γ w2 (s) 1 d (s) ww' 1 +α 2 (s) = , α(s) = − = − and (s) = 45 C1 2 ds C1 β The functions α, β and γ are called “twiss” parameters or “Courant –Snyder” parameters 18 Tune of an accelerator- Solutions (closed form) (cont.) The phase advance can be written as, (after setting C1= 1) ψ s0 +C 1 ψ∆ = ds 46 c ∫ s β (s) 0 β s2 1 ψ 1 And can be shown that ∆ s →s = ds and '= is the local phase advance 1 2 ∫ (s) β (s) s1 In particular for a circular machine,ν the number of betatron oscillations per turn 1 1 = π ds 2 ∫ β (s) 47 is called the “tune” of the machine.β The general solution to the equation of motionψ becomes γ u(s) = A (s) cos[ (s) + δ ] 48 In this case one can showα that 2 2 2 u + 2 uu '+ βu' = A ⇐ Courant –Snyder invariant 49 is invariant at any point in the lattice. Notice that this is an equation of ellipse in (u,u’)-space with A2 as its area. Hence, we conclude that the phase-space enclosed by the particle is constant (if it is not accelerted). 19 Solutions (closed form) (cont.) In a circular machine, each time a particle passes a particular point in the ring its coordinates will appear as a point on the ellipse given by its amplitude and its slope at that point. Such a phase-space ellipse looks like, u’ u’ u′ u slope=−γ /α x′ = γ ⋅A2 max u’ u x′ = A2 / β s3 int θ u s2 a u Notice that the orientation of the x = β⋅A2 max ellipse changes from location to b 2 s1 xint = A /γ location through the lattice because the twiss parameters change from point to point. But the area remains constant. Admittance: This is the phase space area associated with the largest ellipse that the accelerator will accept. Emittance: The minimum phase space area which embeds all particles in a beam. ⇐ ε Or, the area of the phase space ellipse spanned by the largest amplitude particle in the beam. We have two emittances in our study of transverse dynamics: horizontal and vertical emittances. 20 A Few Remarks on Emittance and RMS Emittance In reality, the transverse emittance of a beam as defined earlier, is of limited merit because it may become very large if one includes all beam particles. For example for a Gaussian beam the emittance becomes infinite. This problem can be reduced by quoting the emittance of a certain fraction of the beam, e.g. 90%, yielding ε90%. One may use the rms emittance that averages over all particles with a weight given by the distance of the particles from the “center”: 2 ∑′x⋅c x,x 2 ∑(x− x) ⋅c x,x′ all () x 2 2 2 all () e.g.: ε = x ⋅ x′ − x⋅x′ , with x = , x= rms ∑c()x,x′ ∑c x,x′ all all () ∑(x′− x′)2 ⋅c x,x′ ∑x′⋅c x,x′ ∑(x−x)⋅(x′− x′)⋅c()x,x′ 2 all () () all x′ = , x′= all , and x⋅x′ = , ∑′c()x,x ∑c()x,x′ all ∑c()x,x′ all all 50 This is the semi-axis-product of an ellipse. For a Gaussian distribution, the ellipse contains 39% of the beam. (ε90% = 4.6⋅εrms) At Fermilab, we use ε95% = 6⋅εrms. 21 β Transverse Beam Dynamicsψ in Terms of Betatron Functions In term of twissβ parameters the trajectory of a particleβ can be described by, u(s) = A1 (s) cos( (s)) + A2 (s) sin(ψ (s)) β ψ α Now setting β = β , at ψ = 0, s = 0 and u(0) = u , u' (0) = u' ββ α 0 ψ 0 0 α1 and with ψ ' = theψ transformation in terms of twiss parameters will be, β αα ββ ψ ψ β [cos( ) + sin( )] 0 sin( ) u u = 0 β ψ 0 α u' 1 0 u0 ' [( 0 − )cos( ) − (1+ 0 )sin( )] [cos( ) − sin(ψ )] 0 51 The ψ is the phase advance from s=0 to s=s. In general, this can be for any s1 Æ s2. 22 Envelope and Envelope-Equations The quantity, E(s) = u = εβ (s) max 52 gives the maximum transverse size of the beam at any point in an accelerator or beam transport. This is called “beam envelope”. It is important to note that ± εβ ( s ) represents an envelope embedding all particles at a point in the lattice. ε The beam divergenceα is given by, 1 +β 2 (s) u' = = εγ (s) 53 max (s) (s) Wherever α=0 the beam envelope E(s) has a local waist minimum, “waist”. At this point εβ α ε s E ' (s) = − = 0. εβ β All particles in the beam with emittance ε followψ the trajectory given by, u(s) = (s) cos[ ( s) + δ i )] Here the quantity δi is an arbitrary phase constant for the particle i. Substituting this in Hill’s equation, and equatingββ the cos(ψ+δ ) term to zero, we get, β i 2 ' '− '2 +4 Kβ 2 = 4 54 23 Envelope-Equations (Cont.) With some mathematical rearrangements, one can write 2 ε called the “envelope equation”. E ' ' (s) − 3 + KE (s) = 0 E (s) 55 This plays very significant role in designing beamlines, accelerators and beam injection and extraction regions. If we take into account the Coulomb mean field due to all particles in the beam the above equation will takes the following form, ε 2 G E ' ' (s) + KE (s) − − = 0 E 3 (s) 4 E (s) 56 2 I peak with G = 3 ; I peak = peak current, I 0 = 17000Amp [ I 0 /( βγ ) ] ; electron characteristic current βγ is relativistic factor . The above equation is called KV-envelope equation. This plays very important role in understanding beam transport of space-charge dominated beam in accelerators such as high intensity Photo-injector. 24 Transformationγ of Twiss Parameters Let us take the initial conditionsα as u=u0, u’=u’0 and A=ε. Then the Courant –Snyder invariant becomes, β 2 2 0u0 + 2 0u0u0 '+ 0u0 ' = ε A 57 This represents the particle beam with emittance ε at s=s0=0. Then the transformation matrix as s≠ 0 is given by, u(s) C(s) S(s) u0 = β u'(s) C'(s) S'(sβ)u'0 α Solvingγ these equations for u and u’ and inserting themα in Eq. A we get, 0 0 β γ α γ 2 2 β = C 0 − 2SC 0 + S 0 2 2 β 58 u + 2 uuβ'+ u' = ε with = −CC' + (S'C + SC'α) − SS' 0 α 0 0 α 2 2 γ = C' 0 − 2S'C' 0 + S' γ 0 Or in matrix form theγ transformation of twiss parameters can be written as 2 2 β C − 2SC S 0 = − CC' (S'C + SC') − SS α 0 59 2 2 C' − 2S'C' S' γ 0 The initial set of twiss parameters will be established from the parameters of injection region and the rest can be evaluated using the above transformation matrix. 25 Schematic of the beam in FODO lattice Positively charged Beam QF QF QD Y Y Y QF QD QF Mid Mid Mid point point point FODO lattice, Twiss parameters, Envelope s 27