Weak Focusing Not to be confused with weak folk cussing. Lawrence originally thought that the cyclotron needed to have a uniform (vertical) field. Actually unstable: protons with p vert 0 would crash into poles. First cyclotron required empirical shimming to eliminate sparking. Radial falloff of field to provide some focusing. Understood after second improved cyclotron did not work. (probably involved some cussing.) Å 1
ÆThe Betatron Coils Iron Yoke Rotational Symmetry Axis R Beam pipe B g in center of pipe Works like a tranformer. Primary winding : coils. Secondary winding : beam. Focusing from beveled gap. Flux inside loop: φ = πr 2 B. (1) ( ) Force: F = qe = ( e) V 2πR = e 2πR dφ dt = 1 2 erd B dt. (2) Cyclotron radius: F = dp dt = erdb g dt. R = p eb g. (3) Wideröe condition: B g = 1 2 B, by comparing Eqs. (1&2). (5) (4) Å 2
Coordinates and Design Particle Y Fixed lab coordinate system θ X s= θ z Z x Local traveling coordinate system Design trajectory y is radius of design orbit or trajectory. Red local system moves with design particle (z = 0). Another particle at (x,y,z) R = +x Å 3
Equations of Motion Assume E = 0 and static B = (B x,b y,0). i. e., no longitudinal magnetic field component. Also assume midplane symmetry B x (x,0,θ) = 0, i. e. only vertical field in midplane at y = 0. F = d p dt = q v B, ( E = 0), F x = d dt (γmẋ) = qvb y, F y = d dt (γmẏ) = qvb y; Å 4
For static B, γ = 0, so d dt (γmẋ) γm (ẍ v2 R ) = γm ( v 2 d 2 ) x R 2 dθ 2 v2 R = qvb y, since θ = s R = vt R, and dt = R v dθ. Recall that for circular motion, the centripital acceleration is v2 R. Rearranging: d 2 x dθ 2 + Similarly in the vertical plane: ( ) qby p R 1 R = 0 d 2 y dθ 2 qb x p R2 = 0. Å 5
Paraxial approximation, i. e. small oscillations about design trajectory x, y,r implies small slopes of the trajectory in the local coordinates dx ds = 1 dx R dθ 1, and dy ds = 1 dx R dθ 1. Define subscript 0 for values of design particle: v 0 = β 0 c for velocity, B 0 for vertical magnetic field on design trajectory. 1 = qb 0 = qb 0 p 0 γ 0 β 0 mc, and p = p 0 +δp. Expand magnetic field in Taylor series: ( ) By B y = B 0 + x ( ) By B x = x 0 0 x+o(2), y +O(2). } From Maxwell: B x y = B y x. Å 6
d 2 x dθ 2 + 1 ( ) 1+ δp p 0 ( 1+ 1 B 0 B y x d 2 x dθ 2 + or by defining the field index ( 1+ B 0 B y x )( x 1+ x ) 1 0 n = B 0 B y x ) 0, 0 x δp p 0. ( 1+ x ) 0, we have to first order: d 2 x δp +(1 n)x =. dθ2 p 0 The inhomogeneous term δp p 0 is called the dispersion term. Å 7
For vertical, recall (from about 4 pages back) d 2 y dθ 2 qb x p R2 = 0, This expands to d 2 y dθ 2 ( 1 δp ) q B y p 0 p 0 x y(+x) 2 = 0, 0 or d 2 y +ny = 0, dθ2 with the dispersion and horizontal effects coming in at higher order. Å 8
Stability Ignoring the dispersion term, for on-momentum (δp = 0) particles, we have d 2 x +(1 n)x = 0, dθ2 (H) d 2 y + ny = 0. (V) dθ2 Clearly for constant n, these can have either sinusoidal or exponential solutions. Stable oscillations will happen in the horizontal for n < 1, and in the vertical when n > 0. For stability in both planes 0 < n < 1, or B 0 < B y x < 0. Å 9
Solutions x(θ) = Acos 1 nθ +Bsin 1 nθ, dx dθ (θ) = A 1 nsin 1 nθ +B 1 ncos 1 nθ. Definition: a prime subscript indicates a derivative with respect to the s coordinate of the design trajectory: x = dx ds = 1 Note that we will generally parameterize trajectories in terms of the arc length variable s of the design orbit rather than in terms of θ. This is because we will have drift spaces and other magnets inserted between the bending magnets. dx dθ. In terms of the initial x 0 and x 0 at θ = 0, our constants A and B are then A = x 0, and B = 1 n x 0. Å 10
Matrices In the horizontal plane this may be written in matrix form as ( ) x x ( ) x0 = M H x 0 = ( cos 1 nθ 1 n sin 1 nθ 1 n sin 1 nθ cos 1 nθ ) (x0 x 0 ). For the vertical matrix equation swap n of Eq. (V) for 1 n of Eq. (H): ( ) ( ) ( y y0 cos nθ n sin ) nθ (y0 ) y = M V y 0 = n sin nθ cos nθ y 0. When n is outside the region of stability for either plane, then we may apply cos(iz) = e z +e z 2 = cosh(z), and sin(iz) = e z e z 2i = sinh(z), to obtain the exponential solutions. Å 11
Betatron tunes Horizontal Betatron Oscillation with tune: Q h = 6.3, i.e., 6.3 oscillations per turn. Vertical Betatron Oscillation with tune: Q v = 7.5, i.e., 7.5 oscillations per turn. Definition: The tune is the number of wavelengths of the oscillation in one complete orbit around the ring. Note that Weak focusing refers to betatron tunes less than 1: Q H < 1, Q V < 1. (Betatron: Q H = 1 2π 1 n2π = 1 n, and QV = n.) Å 12
Dispersion Recall first-order diff. eq. for horiz. motion for betatron (from p. 7): This has an inhomogeneous solution of x δ = 1 n δ, d 2 x δp +(1 n)x =. dθ2 p 0 δp with the definition: δ = p 0. For off-momentum particles, we find the total horiz. position is shifted by x δ : x = x β +x δ = Acos ( 1 nθ ) +Bsin ( 1 nθ ) + 1 n δ. x = dx ds = 1 dx 1 n dθ = ( ) Asin( 1 nθ)+bcos( 1 nθ). Å 13
For the initial condition with θ = 0 we have: x 0 1 0 x 1 n 0 = 0 1 n 0 A B δ 0 0 0 1 δ 0 or after inverting A B = 1 0 1 n 0 1 n 0 x 0 x 0 δ 0 0 0 1 δ 0 whereas at a later θ: x cos 1 nθ sin 1 nθ x = 1 n sin 1 nθ 1 n cos 1 nθ 0 δ 0 0 1 1 n A B δ 0 = M H x 0 x 0 δ 0 where we now have added a third row and column to M H : cos( 1 nθ) 1 n sin( ( ) 1 nθ) 1 n cos( 1 nθ) 1 M H = 1 n sin (1 nθ) cos( 1 1 nθ) 1 n sin( 1 nθ). 0 0 1, Å 14
Dipole bend magnet Å 15
Sector Dipole Bend Magnet Coils B Iron pole faces and flux return θ bend Wedge segment of magnet with n = 0 from θ = 0 to θ bend Uniform vertical field. Horizontal parallel incoming trajectories get focused. Unless θ bend 180. (Think about it.) No vertical focusing for a sector bend. (More discussion later on.) Å 16
Sector bend matrices with no gradient For n = 0 cos( 1 nθ) 1 n sin( ( ) 1 nθ) 1 n cos( 1 nθ) 1 M H = lim n 0 1 n sin( 1 nθ) cos( 1 1 nθ) 1 n sin( 1 nθ) 0 0 1 cosθ sinθ (cosθ 1) = 1 sinθ cosθ sinθ, 0 0 1 Notice the M 21 = 1 sinθ term produces horiz focusing. M V = lim n 0 ( cos nθ n sin nθ n sin nθ cos nθ Here the M 21 is zero. (Looks like a drift element.) ) = ( 1 θ 0 1 ). Å 17
Quadrupole matrix n = B y B 0 x = g, B 0 1 = q B 0 p 0 n = qg 2 = k 2, p 0 1 nθ = 1+k 2 s k = qg p 0. (g =gradient). = 1 2 +ks, Keep s, q, and p 0 constant, let B 0 0, so. 1 nθ kθ. nθ ks. Å 18
1 n = 1 2 +k 1. k 1 n (1 cos 1 nθ) = 1 1 n sin 1 nθ = 1 n k. 1+ 2 k (1 cos ks) 0. 1 1 sin +ks 0. 1+k 2 2 M H = cos( 1 nθ) 1 n sin( ( ) 1 nθ) 1 n cos( 1 nθ) 1 1 n sin (1 nθ) cos( 1 1 nθ) 1 n sin( 1 nθ) 0 0 1 cos ks k 1 sin ks 0 k sin ks cos ks 0. 0 0 1 Å 19
Similarly M V ( cosh ks k 1 sinh ) ks. k sinh ks cosh ks So we find M quad = cos ks k 1 sin ks 0 0 k sin ks cos ks 0 0 0 0 cosh 1 ks k sinh ks, 0 0 k sinh ks cosh ks for a horizontally focusing quadrupole magnet. For a horizontally defocusing quad, just swap the two 2 2 diagonal blocks. Å 20
Å 21
Drift matrices atan( x 0 ) x x 0 In a field free space our trajectories will be straight lines: l ( x x ( y y ) ) = M H,drift ( x0 x 0 = M V,drift ( y0 y 0 ) ) = = ( 1 l 0 1 ( 1 l 0 1 )( x0 x 0 )( y0 y 0 ), ), Å 22
Forshadowing of Liouville s Theorem Note that the determinants of all these matrices is identically 1. ( ) M drift = 1 l = 1, 0 1 ( cos 1 nθ M H = 1 n sin ) 1 nθ 1 n sin 1 nθ cos 1 nθ = 1, ( cos nθ n sin ) nθ M V = n sin nθ cos nθ = 1. M quad = 1. These transformations preserve area in (x,x ) and (y,y ) spaces. x = dx/ds = p x pz p x p for small angles. This small-angle approximation is known as the paraxial approximation. Much more about Liouville s theorem to come later. Å 23
Gradient Fields and Forces For a dipole bend: B y larger for smaller gap between poles in an iron magnet. Tilt poles B y x 0 Polefaces opening outward (a&c) less horiz focusing. (Perhaps even net defocusing.) Polefaces closing outward (b&d) more horiz focusing. Å 24
Example Weak-focusing Synchrotron Ring made of four 90 sector bends. Put drift spaces between magnets. Beam injected at one drift space with pulsed injection kicker. Another straight section may have an rf cavity for acceleration. Bend field must increase as beam accelerates. rf frequency must increase with velocity for constant cirumference. Å 25
Three possible periodic-cell configurations: beam from right to left. Ignore kicker and rf cavity for present. Periodic cell: 1/4 of ring: Book uses left periodic cell. Let s do the middle one. ), or M cell = ( 1 l0 0 1 M cell = M H,drift (l 0 )M H,bend ( π 2 ) ( cos 1 nπ 2 1 n 1 n sin 1 nπ 2 sin 1 nπ 2 cos 1 nπ 2 ). Å 26
M cell = ( cos 1 nπ 2 l 0 1 n 1 n sin 1 nπ 2 1 n sin 1 nπ 2 +l 0 cos 1 nπ 2 sin 1 nπ 2 cos 1 nπ 2 ). For now skip rest of 2.6 in book. Probably easier to understand after we do Chapter 5. Å 27
Momentum compaction factor For a flat ring, the closed design orbit must have ds (s) = 2π. In straight sections with no field =. 1 From the cyclotron radius, we have = qb, so substituting, we find p p = q B ds. 2π The design circumference is L = ds. Momentum compaction is ratio: fractional change in circumference, i. e. fractional change in momentum α p = dl L / dp p = p L dl dp. Å 28
Hubble Telescope example What s the momentum compaction of the HST? Nearly circular orbit at r = R +559km (from Wikipedia). F = GMm r 2 p 2 r = GMm 2 2ln(p)+ln(r) = ln(gmm 2 ) 2 dp p + dr r = 0, dl L = dr α p = dl L r = 2dp p = ma = m v2 r = p2 mr, / dp p = 2. Å 29
Mom. comp. of our WF synchrotron Circumference: L = 2π+4l 0. L+dL = 2π(+d)+4l 0. dl L = 2πd 1 = 2π+4l 0 1+2l 0 /(π) d dp p = d + db B = α p = p = qb ( 1+ B B 1 ( ). 1+ 2l 0 π (1 n) ) d = (1 n)d. Å 30
Beam position monitors RHIC stripline beam monitor. PEP beam button monitor. from S. R. Smith, SLAC-PUB- 7244 (1966). Å 31
BPM data from RHIC transfer line Left: Raw signal of single bunch passing a stripline. (Unfiltered) Right: Electrode voltages measured with 20 MHz filters. Position V 1 V 2 V 1 +V 2. Å 32
BPM turn-by-turn data RHIC turn-by-turn meas. Left col. is horizontal. Right col. is vertical. Decoherence from chromaticity ξ = 1 Q dq dδ. Chromaticity adj by sextupoles. Moreaboutξ laterinweek. Å 33
Tune measurement from turn-by-turn data Top: data from horiz. BPM. Middle: Raw FFT. Tune from FFT(128 turns). Bottom: Fit to single peak. Double peak caused by HV coupling. Note beating in top graph. Å 34