Bent solenoids can be used to transport low energy beams as they provide both confinement and dispersion of particle orbits. Solenoids are being considered both as emittance exchange sections and spectrometers in the muon cooling system as part of the study of the muon collider. They present the results of a study of bent solenoids which considers the design of coupling sections between bent solenoids to straight solenoids, drift compensation fields, aberrations, and factors relating to the construction, such as field ripple, stored energy, coil forces and field errors

Norem, J

Norem, J.

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Bent solenoids for spectrometers and emittance exchange sections.

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BENT SOLENOIDS FOR SPECTROMETERSAND EMITTANCE EXCHANGE SECTIONSJ. Norem†, Argonne National Laboratory, Argonne IL, 60439 USA                                                † norem@hep.anl.govAbstractBent solenoids can be used to transport of low energybeams as they provide both confinement and dispersion ofparticle orbits.  Solenoids are being considered both asemittance exchange sections and spectrometers in themuon cooling system as part of the study of the muoncollider.  We present the results of a study of bentsolenoids which considers the design of coupling sectionsbetween bent solenoids to straight solenoids, driftcompensation fields, aberrations, and factors relating tothe construction, such as field ripple, stored energy, coilforces and field errors.1 INTRODUCTIONWhile the optics of beams in straight solenoids is wellunderstood in accelerator physics[1], and the behavior ofcharged particles in toroidal geometries is also wellunderstood in plasma physics[2], the optics of beamswhich traverse both bent and straight sections have nothad significant applications in either field, are not simpleor symmetric, and are not particularly well understood[3].Beams in bent solenoids are subject to drifts perpendicularto the bend plane.  The geometry is shown in Fig. 1.Figure 1, A bend without coupling sections.In a straight solenoid the magnetic field is homogeneous,while in a bend the field is distributed according to 1/R,this means that there is a transition region where the fieldis difficult to parametrize and should be calculatednumerically see Figure 2.Figure 2,  The field in a bent solenoidAs a result this complexity there are a large number ofvariables associated with the design of a solenoidal bend.In addition to the nominal field on axis, bend radius, coilradius and bend length, there is the choice of field tocompensate perpendicular drifts, the bending algorithmused to couple the bend to the straight section, and thealgorithm used to compare the field in the bent andstraight sections.  2 BEND BASICSA particle passing through the magnet system described inFigures 1 and 2 will be mismatched to the bend andexecute an orbit like that shown in figure 3, where thecentrifugal momentum drives the particle to larger radii,produces an upward drift and excites Larmor oscillations atthe end of the bend.Figure 3, particle orbits in bent solenoids, where x and yare perpendicular to the particle motion.The vertical drift is described by the relations          vR BvBv v vR BcB Ldrift c Bdriftdriftm vqB Rv rBBmq B Rv vB B R B BR B B B B R=×= ± × ∇= + =×+= −= =∇ ⊥∇ ⊥⊥γγθ ρρ ρ022 2202 22 22 2120 5|||||| ||( . )/ /( ) / ( ) /∆where Larmor length is L=2 p B r /B, and Br  is themagnetic rigidity.2.1  Compensating FieldsThe vertical drift can be eliminated with a compensatingfield B^  which provides a negative slope of the magneticRr sxyz-0.5 0 0.5 1 1.5 2 2.5x (m)-1012y (m)0246B (T)-0.0100.01x (m) -0.50 0.511.522.533.5s (m)-0.1-0.0500.05y (m)Θ = Bρ/BR∆R∆R0-7803-5573-3/99/$10.00@1999 IEEE. 1417Proceedings of the 1999 Particle Accelerator Conference, New York, 1999field lines so that θ ρdrift B B R B B= − ⊥/ /|| ||  = 0.  Thus asingle momentum can be confined to a plane, while othermomenta will be subject to the same momentumdependence, giving a dispersion          D s B B R ds B Bdrift= = ∫θ ρ ρ/ ~ /|| Θ .The vertical field can be applied either with an externalcoil set or by tipping the coils.  An advantage of tippingthe coils is that the vertical field component can betailored to the local bend radius.  The coils would betipped as shown in figure 4Figure 4, Tipping and turning coils.3 COUPLING SECTIONSIt is possible to minimize the mismatch where straightmeets bend by using a verity of coupling sections.  Themismatch can be caused by two effects: 1) the orbit cannotadjust smoothly to the position of the equilibrium orbit asit goes through the bend, and,  2) the magnetic field linesare displaced from what would be the particle orbit by 1/R(or other) effects.  Both of these effects can minimizedusing a good coil design.3.1 Adiabatic BendsIf the mismatch that causes the oscillations in Figure 3 isspread out over many Larmor lengths and phases, thetransverse oscillations will be canceled, unfortunately atthe cost of a long solenoid.Figure 5, A two dimensional projection of the motion ofa particle in an adiabatic coupler.  2.2 L = n l LIf the overall bend length is adjusted to be equal to aneven number of Larmor oscillations, the particle can exitfrom the bend at the point of the cusp (see figure  3)where it has no momentum perpendicular to the directionof motion.  This method had two problems: 1) the lengthof the magnet is fixed and, 2) effects due to coil size cancomplicate the effective length of the bend section.Nevertheless, for small dispersion systems this solutioncan be very useful.2.3 L= l L/2 CouplerWhile the mismatch produced at discontinuities isgenerally a problem, it is possible to use the mismatchconstructively to create a half Larmor oscillation of thecorrect size and length to couple into a bend of arbitraryradius.  This can be done by designing a coupling sectionwith half the bend radius of the primary bend and a lengthequal to L = l L/2.  The coupling section then brings theparticle to what would be its equilibrium offset D R  in thebend without residual perpendicular  momentum (fig.6).The use of this technique is complicated by coil radiuseffects which can complicate evaluating the sectionlength, causing mismatch at the ends of the sections.Figure 6. A particle orbit in a L = l L/2 coupler, showingmismatch at the end of the section.2.4 Short Smooth CouplersAn extension of the L = l L/2 coupler is to use a smoothedcurve which has the advantages that discontinuities inboth dQ /ds and its derivatives are avoided.  The expressionused by Fernow and Palmer[4],     ddss s s sΘ=−−−0 51 2. cot( ) cot( )σ σ,where the length over which the transition is made s  =l L/2 seems to be optimum, consistent with the argumentspresented in Sections 3.1 and 3.3, see Figure 7.0.00000.00050.00100.00150.00200.0025r L (m)0.00 0.20 0.40 0.60 0.80s/l LFigure 7, Mismatch produced  as a function of short,smooth coupler length.RTip angle q  = 6.7 p[MeV/c] / R[m] B[T] Yaw angle   h  ~ 0.04 q    r-0.1-0.08-0.06-0.04-0.0200.020.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.0y (m)x (m)∆R-0.1-0.0500.050.10.150.2-0.025 -0.02 -0.015 -0.01 -0.005 0y (m)x (m)1418Proceedings of the 1999 Particle Accelerator Conference, New York, 19994 ABERRATIONSIn addition to the time slewing due to the longer pathtraveled by particles on the outside of a bend, there are anumber of aberrations which complicate the optics.4.1  x’ y’ DependenceParticles with momentum perpendicular to the beam willdrift vertically due to grad B drift.  This can be eliminatedwith equal numbers of bends of opposite direction, but inmost cases with x’<0.3 the drift is small, comparable to ad p/p ~ 2 %.  This is shown in Figure X-0.0200.02x (m) 01234s (m)-0.03-0.02-0.0100.010.020.030.04y (m)Figure X, Grad B drift for orbits with p^  0.4.2 ShearsEven if all translation of the central orbit is canceled usinga good coil design the orbits off center are subject to ashear.  This shear is a result of mismatches between theideal compensation field and that provided by physicalcoils.  The effects of this sheer are shown if Fig Y, andthe shear can be eliminated by equal right and left bendangles.-0.06-0.04-0.02 0 0.020.040.06x (m) 01234s (m)-0.06-0.04-0.0200.020.040.06y (m)Shear DirectionFigure Y, Shears in large beams over a bend.The sum of horizontal and vertical components of theshear should be constant because theperpendicular field components thatcause it are determined by Amperes.law, integrating around the pathshown at right. The relativeamplitude of horizontal and vertical shear can be altered byaltering the geometry of the coils causing thecompensating field.  4.3 Coil Geometry EffectsThe coil geometry interacts strongly with the optics, asshown in figure Z below, which looks at an orbit for twodifferent coil radii.Figure Z, Orbits for coil radii r = 0.3 and 0.03.The sensitivity to coil and geometry parametersemphasizes the importance of using an accuraterepresentation of the problem.  The calculations in thispaper were done using GPT [5], which calculates fieldsfrom coils and tracks the particles through the fields, thusavoiding approximations or simplifications which cancompletely change the nature of the problem.5  CONCLUSIONSThis paper outlines the effects encountered when usingboth bent and straight solenoidal sections.  Emittancegrowth can be produced by mismatches, however it ispossible to minimize the effects of these by optimizingthe coupling section and compensating field geometries.It seems important to track particles from coil geometriessince the effects produced can seem nonintuitive.REFERENCES[1]  S. Humphries, Charged Particle Beams, Wiley, (New York) 1990[2]  F. F. Chen, Introduction to Plasma Physics, Plenum, (New York),1974[3] J. Norem, Submitted to Physicas Review Special Topics/Accelerators and Beams, 1998[4]  R. Fernow, ICOOL: a simuation code for the ionization cooling ofmuon beams, this conference[5]  Pulsar Physics, De Bongerd, NL-3762, X A Soest, The Netherlands(1998)ZRB-0.004-0.003-0.002-0.00100.0010.0020.0030.0040.0050.006-0.005 0 0.005 0.01 0.015 0.02y (m)x (m)1419Proceedings of the 1999 Particle Accelerator Conference, New York, 1999

Bent solenoids for spectrometers and emittance exchange sections

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