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Low Energy Beam Transport Design Optimization for RIBs R. Baartman TRIUMF May, 2002 1 Optics Scaling One of the characteristic features of heavy ion facilities such as ISAC is that a wide variety of particles can be provided for the experimenters: particle mass, charge and even energy can be customized to the user’s request. How is the optics adjusted when one of these parameters is changed? We can answer this by looking at the scaling of the non-relativistic Hamiltonian, which in Cartesian coordinates is 1 2 H= P − qA + qV (x, y, z). (1) 2m Here, P is not the kinetic momentum p, but the canonical mo- mentum p + q A as appropriate for cases with magnetic ﬁelds given by the vector potential A. All dynamics can be derived from Hamilton’s variational principle: δ (Pidqi − Hdt) = 0 (2) ((P1 , P2 , P3 ) = (Px , Py , Pz ), (q1 , q2 , q3 ) = (x, y, z), and re- peated indices follow summation convention.) We can rescale any variable as long as the variational principle remains true. We want all particles to always follow the same trajectories, so x, y, z are inviolate. To keep track of what we’re doing, let’s do √ it in 2 steps. First scale the momenta by a factor mqV0 . I.e. deﬁne P P∗ = √ (3) mqV0 2 Optics Scaling–cont’d According to the variational principle, we must scale either t or H the same amount. Let’s scale t. Then we get qV0 ∗ 2 ∗ H= P − A + qV, (4) 2 where we’ve deﬁned a new vector potential q A∗ = A. (5) mV0 Now rescale H by qV0 without changing the momenta. This can be done by rescaling t by the reciprocal. We then ﬁnd 1 ∗ 2 ∗ H = ∗ P − A + V ∗, (6) 2 where we’ve introduced the scaled potential V ∗ = V /V0 . (7) Taken together, the 2 steps have scaled time t by the total amount qV0 t∗ = t. (8) m 3 Optics Scaling–cont’d To recap: we’ve transformed the Hamiltonian to one which contains neither charge nor mass, without rescaling spatial coordinates. This proves that trajectories are preserved through the transport system as long as the scaling relations are obeyed. In particular: • Electric ﬁelds are unchanged when changing either mass or charge: in other words, in an entirely electrostatic beamline from source to experiment, all particles have the same trajectory, irrespective of mass or charge. • By eqn. 7, when changing energy, all potentials scale with the energy change. • Magnetic ﬁelds scale as eqn. 5, in particular, the mass separator dipole can go from one mass to the next by scaling the magnetic ﬁeld as the square root of the mass over charge. • Time scales as eqn. 8. If the time scale must remain ﬁxed, for example as in a ﬁxed-frequency linac, the potentials, in particular the voltage on the linac, scale ∝ m/q. 4 RIB LEBTs At beam voltages (energy ÷ charge state) <∼ 1 MV, electrostatic optics is more economical than magnetic. As well, Radioactive Low Energy Ion Beams are of a wide variety of species (masses). Electrostatic optics has the nice scaling feature that element strengths depend only on beam voltage and not on mass (or charge). For these reasons, we concentrate on electrostatic optics. 5 What’s better, a whole lot of little bitty quads, or a few large ones? Size Scale – Economics One would like to transport a beam of given emittance. Say the aperture radius is a, and the distance between focusing elements is Lc. Then maximum angles are a/Lc and acceptance is on the order of ˆ ∼ a2 /Lc. (More precisely, ˆ = a2 /βT , where βT is the Twiss β-function, but βT ∼ Lc). For example, say we want ˆ = 200 mm-mrad. Then the following combinations will work: Length Scale aperture Looks like ... 5m 50 mm CERN-ISOLDE 1m 22 mm TRIUMF-ISAC 2 cm 3 mm an RFQ Also, cost is roughly ∝ a2 (cross-sectional area of vacuum chamber), and ∝ 1/Lc (i.e. the number of optical elements assuming a ﬁxed distance from source to destination). Therefore, cost ∝ a2 /Lc ∼ ˆ, so whether Lc is 5 or 1 metre, the cost is roughly the same. In other words, we have plenty of freedom to choose a scale for the optics. 6 Large scale (like ISOLDE)... 4 x-envelope (cm) 3 y-envelope (cm) 2 1 0 -1 -2 -3 -4 0 50 100 150 200 250 300 350 400 distance (in) Small scale (like ISAC)... 4 x-envelope (cm) 3 y-envelope (cm) 2 1 0 -1 -2 -3 -4 0 50 100 150 200 250 300 350 400 distance (in) 7 Size Scale – Stray Fields Are there other considerations? Yes! The equation of motion for regularly-focused particles is 2 x + x/βT = 1/ρ (9) where ρ = (Bρ)/Bstray is the radius of curvature of the beam n the stray ﬁeld. So on average, the displacement ∆x due to the 2 stray ﬁeld is βT /ρ ∼ L2 /ρ. This effect is nasty because it is c mass-dependent through Bρ. To avoid light masses being steered off-axis compared with heavy masses at the same voltage, we would like ∆x to be small compared with the beam √ √ size βT ∼ LC. In other words, LC (ρ2 )1/3 (10) This clearly favours small optics. Example: In ISAC, the smallest Bρ is 400 gauss-metres. This is for the lightest mass (A = 6) at the energy 2 keV/u required for the RFQ. In the earth’s magnetic ﬁeld, we ﬁnd ρ=1 km. For acceptance =100 mm-mrad, we would like Lc 5 m. In fact, we have Lc = 1 m. This makes us fairly tolerant, but CERN-ISOLDE at Lc ∼ 3 m is much more susceptible. 8 Size Scale – Space Charge RIBs are generally of such low intensity that space charge is not an issue. However, in setting up stable beams for commissioning purposes, it is useful to know how much current is allowed before space charge becomes a non-negligible effect. To ﬁnd this, we can compare the space charge and emittance terms in the Kapchinsky-Vladimirsky envelope equation. a + k2 (z)a + 2 /a3 + 2A/(a + b) = 0 (11) b + k2 (z)b + 2 3 /b + 2A/(a + b) = 0, (12) where a and b are x- and y-envelopes, A is the space charge parameter. space charge potential well depth I × 30 Ω A= = (13) beam voltage βVB This gives the following condition for the negligibility of space charge. 2 A /a2 , 2 /b2 ∼ (beam divergence)2 ∼ /LC (14) This means that in spite of smaller beam sizes and stronger space charge forces, stronger focusing allows higher current. Example: In ISAC, LC ∼ 1 m, and a good beam has emittance 10 mm-mrad. This leads to A 10−5 . As VB ∼ 30 kV, and β = 0.002, we ﬁnd that to avoid space charge effects, we need I 20 µA. 9 One often wonders: Are large quad apertures better because they make the nonlinear ﬁelds weaker, or worse because they make the nonlinear ﬁelds longer? In fact the two effects cancel. Size Scale – Aperture (What about nonlinearity?) The quadrupole potential ﬁeld k 2 V (x, y) = (x − y 2 ) (15) 2 is a solution to Laplace’s equation, but only if the quadrupole is inﬁnitely long (k=constant). For ﬁnite quads, we use the expansion k 2 k k V (x, y, z) = (x − y 2 ) − (x4 − y 4 ) + (x6 − y 6 ) − ... (16) 2 24 720 (Note that this assumes no quad ‘shape’ errors.) The quartic term gives a cubic force term which leads to the following focusing error, −1 7 3 1 2 ∆x = x − xy , (17) f 2 LQ 6 2 where LQ is the quad length and f the focal length. It is important to note that this is independent of aperture size or fringe ﬁeld hardness: indeed, the aberration is not affected by changing the fringe ﬁeld shape. In a given context of ﬁxed optics and quad locations, the only way to reduce quad aberration is to lenghten the quads. BTW, similar formulas can be found for other types of optical elements. 10 Nonlinearity – cont’d It is clear that for the higher order errors to have a negligible effect on the beam quality, the distortion in the emittance ellipse, ∆x , should be small compared with the local divergence /x. 11 Nonlinearity – cont’d Roughly speaking, ∆ ∆x x4 = ∼ 2 (18) /x f LQ In a fairly smooth-focusing transport channel, x2 = βT ∼ LC and f ∼ LC, so somewhat surprisingly the optics length scale cancels and ∆ = (19) LQ Even if the emittance is as large as 100 mm-mrad, this gives the easily-satisﬁed constraint LQ 0.1 mm. However, this is for a regular FODO transport case. Matching sections often have much tighter constraints. An example is when matching to an RFQ. Since the RFQ requires a very small beam, the matching system must transform a relatively large beam with small divergence to a small beam with large divergence. The magniﬁcation in the case of the ISAC RFQ is xf /xi = 1/5. The focal length of the last doublet is 1/5 of the typical FODO focal length of 1 m. Apply eqn. 18, with same x, but 5 times smaller f . This makes the aberration 25 times worse, now requiring LQ 2.5 mm. 12 Catch-22 situation: Making the quad longer helps, but increases the beam size, which hinders. Nonlinearity – cont’d The situation is actually worse than this because of using quadrupoles rather than stigmatic lenses. The beam in the next-to-last quadrupole is much larger in the defocusing plane of the last quadrupole than in the focusing plane. This forces the last two quads close together, and requires a very short last lens. 2 50 pi mm-mrad x-envelope (cm) 1.5 y-envelope (cm) x-focal power (arb.) y-focal power (arb.) 1 0.5 0 -0.5 -1 -1.5 0 10 20 30 40 50 60 70 80 distance (in) 13 Optimization technique There are many optimization techniques for designing and tuning beam transport lines. Some are built into the transport codes themselves. Almost all of these work on the basis of reducing an error to zero by ﬁnding local derivatives of the error with respect to the parameters. This ﬁnds only local minima and is not useful for more than about 3 variable parameters. A technique I use is simulated annealing. The book Numerical Recipes by Press, Flannery, et al. gives the following description. Offered a succession of options, a simulated thermodynamic system was assumed to change its conﬁguration from energy E1 to energy E2 with probability p = exp[(E2 − E1 )/(kT )]. Notice that if E2 < E1 , this probability is greater than unity; in such cases the change is arbitrarily assigned a probability p = 1, i.e., the system always took such an option. This general scheme, of always taking a downhill step while sometimes taking an uphill step, has come to be known as the Metropolis algorithm. To make use of the Metropolis algorithm for other than thermodynamic systems, one must provide the following elements: 1. A description of possible system conﬁgurations. 2. A generator of random changes in the conﬁguration; these changes are the options presented to the system. 3. An objective function E (analog of energy) whose minimization is the goal of the procedure. 4. A control parameter T (analog of temperature) and an annealing schedule which tells how it is lowered from high to low values, e.g., after how many random changes in conﬁguration is each downward step in T taken, and how large is that step. 14 Simulated Annealing applied to Beam Optics Design Applied to the design of a transport channel, we can identify these 4 points as 1. A set of parameters such as quad strengths, locations, and their allowed ranges. 2. We use the vertices of a “simplex” in the space of all possible parameters, and change these randomly, independently according to T . For T = 0, the technique reduces to the standard “simplex” method of minimization. 3. E = the sum of all possible effects to be minimized, such as mismatch, emittance growth, etc. with their appropriate weights. 4. Use T such that T = 1 implies that parameters are varied through their entire ranges, specify N the number of random changes at each temperature, and α(< 1), the factor with which T is multiplied at each step. For as few as 3 parameters, α can be 0.8, with N = 5, resulting in only about 100 evaluations of the beamline to achieve 10−4 accuracy. But this number of evaluations increases exponentially with number of parameters. For 9 parameters, need α = 0.98, N = 50, and this results in roughly 104 − 105 evaluations before convergence is obtained. 15 Conclusions • Optics of any scale can be made to work: elements can be many metres apart, down to cm apart. • Smaller scale means more focusing per unit length. This makes the beam less sensitive to perturbations (stray ﬁelds, misalignments, space charge, ...) • Smaller scale also means more customizablility for adding new experiments. • Beam transport design can be accomplished with simple tools like ﬁrst order matrix codes, augmented by “back of the envelope” formulas for higher orders. (This is not true however of aberration-limited designs like spectrometers.) 16 17

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low energy, Energy Beam, beam transport, energy electron, ion beam, cross section, ion source, electron beam, TRANSPORT SYSTEM, beam diagnostics

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posted: | 8/19/2011 |

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