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"Realistic" Ring Cooler Magnetic Fields -- The Next Generation Steve Bracker Workshop on Ring Coolers University of Mississippi March 11-12, 2004

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Don sprang this talk on me without much advance warning. Usually I prepare slides for talks on the plane bus on the way to the conference, so I was relaxed until someone brought to my attention that I was already here. Trouble... Hence I thought it best to rummage around in the archives for a talk I could dust off and revise a bit. Happily I found an old position paper I had put together for Bill Clinton a number of years ago just in case he asked....

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On The Meaning of "Sex" "Realistic" Has come to mean "better than a box field and maybe more or less satisfies Maxwell's Equations". At some point in the design process, we must mean more: 1. The field is that one field of all Maxwellian fields generated by some well-specified apparatus (e.g. coil set). 2. Engineers assure us that such an apparatus can be built to sufficient precision, and operated without causing region-wide power blackouts. 3. Simulations assure us that the apparatus will still perform when constructed and maintained to feasible precision and stability.

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"Whenever I hear 'realistic' and 'field' in the same sentence, I reach for my revolver." Rigor on "Realistic" TIME in Project Vague Speculations Conceptual Design Real Design FabricationInstallationOperation Paraphrasing one not-so-great man... An Idealized Evolution of Rigorous Realism Through Time

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Rigor on "Realistic" TIME in Project Vague Speculations Conceptual Design Real Design FabricationInstallationOperationSuicide An Alternate Trajectory...

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Rigor on "Realistic" TIME in Project Vague Speculations Conceptual Design Real Design FabricationInstallationOperationSuicide Area of Maximum Peril Overwhelming but essential effort to achieve rigorous realism Projects often slowly slide from Conceptual Design to Real Design without much fanfare. Great Peril: that the need for rigorous realism in the design of essential components will be realized very late, so that great strain will be placed on critical manpower late in the design process....

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... or one can just get on with things and hope.

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In building up complex magnetic-field-generating-things (MFGTs), there are only a few computationally interesting sub-things, among them: 1. differential straight-line current elements (Biot-Savart integration, slow) 2. finite-length straight-line current segments (simple analytic expression) 3. cylindrical current sheets (e.g. BSHEET) If magnets for cooling rings can be represented as aggregates of #2 and #3, then there is some hope of generating a truly realistic field map before our sun leaves the main sequence. If magnets for cooling rings can be approximated as aggregates of these sub-things, then there is some hope of generating approximations to a realistic field that are sufficient to test whether cooling rings cool, and how sensitive their performance is to details of the magnetic field map.

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A good magnetic field generator should include the ability to alter MFGTs in ways not too dissimilar to those alterations inevitable due to imperfect construction, operational instability, and inexact field simulation methods. To "demonstrate cooling" (before you build and operate a cooler) you have to show that adequate cooling takes place not in "one ring to rule them all" but a whole ensemble of rings which span the phase space of "rings you may end up with" when all the vicissitudes of design, construction, installation and operation are accounted for.

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P z B at P = (sin - sin ) / z (B points out of page) For /4 = 1 and I = 1, X Y Z P Xp Yp (Xp,Yp,0) (0,0,0) (L,0,0) Xp/Yp = tan (- ) = atan (-Xp/Yp) (L-Xp)/Yp = tan ( ) = atan ((L-Xp)/Yp) Bx = 0 By = 0 Bz = ( /4 I (sin - sin ) / Yp For a straight current segment of length L lying on the X axis from (0,0,0) to (L,0,0), and an observation point P in the XY plane at (Xp,Yp,0), carrying current I in the +X direction: We know how to do finite straight line current carriers...

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X Y Z P Xp D (Xp,D,0) (0,0,0) (L,0,0) Xp/D = tan (- ) = atan (-Xp/D) (L-Xp)/D = tan ( ) = atan ((L-Xp)/D) Bx = 0 Br = 0 Bt = ( /4 I (sin - sin ) / D Changing notation in preparation for generalizing this: Yp -> D By -> Br (radial component) Bz -> Bt (tangential component) Rotating P around the X axis in the Y->Z direction by angle t: D = sqrt(Xp^2 + Zp^2) t = atan (Zp/Xp) Bx = 0 By = -Bt sin (t) Bz = Bt cos (t)...and we can express this in a coordinate system that makes sense to a GEANT simulation.

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Z' X' (z'1,0) (z'2, -x'2) (z'3, -x'3) (z'4, -x'4) (z'5, 0) Eight parameters (z'1, x'2, z'2, x'3, z'3, x'4, z'4, z'5) define the shape of the magnet. Corresponding points in the right half by symmetry. 8 straight current segments per magnet. A magnet made up from a small ensemble of finite straight-line current carriers. To avoid unseemly charge buildups, I suppose we should make them all carry the same current, though we might turn individual segments on and off for sensitivity studies. (z'2, x'2) (z'3, x'3) (z'4, x'4)

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Z X Y Z Rcpc Acpc Ycpc Looking down toward -YLooking sideways toward +X Three more parameters Rcpc, Acpc, Ycpc define the entire 12-coil assembly. In total there are 11 parameters defining the field shape. One more (current=I) then defines the field at every point away from a conductor.

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1. A non-Maxwellian "box-field" which has constant B = (0,By,0) between the vertical pairs of coils and (0,0,0) outside it. 2. A Maxwell-compliant single-turn-per-magnet field, computed from the 8 straight-line segments per magnet. Four current "models"

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3. A Maxwell-compliant multiple-turn-per-magnet field stacked in Y, so that each turn has exactly the same shape and size. Requires two more parameters: number of turns in each stack turn-to-turn Y separation 4. A Maxwell-compliant multiple-turn-per-magnet field with layers of coil stacks. Requires two more parameters: layer separation in X-Z plane number of layers in X-Z plane. nStack = 4 dStack nLayer = 3 dLayer

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x'1, x'2, z'2, x'3, z'3, x'4, z'4, x'5 xCoil1, xCoil2, zCoil2, xCoil3, zCoil3, xCoil4, zCoil4, xCoil5 Eight parameters to describe geometry of one (outermost) coil: Four parameters to describe stacking and layering: nStack, dStack, nLayer, dLayer Three parameters to describe distribution of coil assemblies around ring: rCpc, aCpc, yCpc 16-parameter field In array MagnetParam One parameter to set the magnet current: magCurrent

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call Bfield (magnetParam, position, model, field) magnetParam: 16 input reals describing magnet configuration position: 3 input reals specifying (x,y,z) where field is to be found model: 1 - box field 2 - single-coil per magnet 3 - vertical stacks of coils in each magnet 4 - horizontal layers and vertical stacks of coils in each magnet field: 3 output reals returning (Bx, By, Bz) Progressively implemented one model at a time.

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Z' X' Add one more point to the magnet description? A bit of concavity/convexity normal to the particle direction. (z'1,0) (z'2,x'2) (z'3,x'3) (z'4,x'4) (z'5,0) (0,x'6) Questions :

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Add one more parameter to the magnet configuration? nCells: The number of pairs of magnets distributed (uniform angular spacing) around the ring. nCells = 6nCells = 8

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1. This field should explore most of the physically interesting field-shape issues for rings of this type. It is sufficiently general to allow us to test the effects of small perturbations to field shape on cooling performance. True? Discussion Points 2. It is possibly practical to call the field generator directly from GEANT, without the use of a secondary field grid. How many points per second must the generator be capable of to permit this without incurring unacceptable slowdown? How odious is the care and feeding of a secondary field? 3. Ring designs of this kind have appeared so far with 4 and 6 "cells". Are still different numbers of cells contemplated? Should the ring design be generalized (1 more parameter) to allow for N cells? 4. No field map should be believed without a "second opinion". Is there anyone out there who would be willing to undertake a second implementation of exactly this field in a manner that yields Bx,By,Bz directly?

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Particle Positions ExaminedParticle positions with Nonzero Field Ensure that the rotation transforms are working correctly using the box field. Generate a toroid of particle positions (left) and see where the field is non-zero (right).

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Magnets azimuthally centered in 60 degree cells; injection at cell boundary Romulus decided he would prefer to inject into the leading edge of the magnet. One parameter changed (aCpc), and...

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A little Coil Design Tool (an Excel spreadsheet) helps the user compose the coil definitions.

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Next Steps -- 1. Rework parameter definitions as needed; document. 2. Produce the single-coil-per-magnet Maxwellian field. 3. Check 2. by checking with Maxwell, testing simple symmetric cases, etc. 4. Generate grid field (format-compatible with FindFieldAnywhere) if and only if speed dictates. Compare interpolation in grid to primary field. 5. Add stacking and layering if simulation results suggest it's useful.

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