"Realistic" Ring Cooler Magnetic Fields --The Next Generation

Steve BrackerWorkshop on Ring Coolers

University of MississippiMarch 11-12, 2004

Don sprang this talk on me without much advance warning.

Usually I prepare slides for talks on the plane bus on the way tothe conference, so I was relaxed until someone brought to myattention that I was already here. Trouble...

Hence I thought it best to rummage around in the archives fora talk I could dust off and revise a bit. Happily I found an oldposition paper I had put together for Bill Clinton a number ofyears ago just in case he asked . . . .

On The Meaning of "Sex" "Realistic"

Has come to mean "better than a box field andmaybe 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 fieldsgenerated by some well-specified apparatus (e.g. coil set).

2. Engineers assure us that such an apparatus can bebuilt 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.

"Whenever I hear 'realistic' and 'field' in the samesentence, I reach for my revolver."

Rigor on"Realistic"

TIME in Project

VagueSpeculations

ConceptualDesign

RealDesign

Fabrication Installation Operation

Paraphrasing one not-so-great man . . .

An Idealized Evolution of Rigorous Realism Through Time

Rigor on"Realistic"

TIME in Project

VagueSpeculations

ConceptualDesign

RealDesign

Fabrication Installation Operation Suicide

An Alternate Trajectory . . .

Rigor on"Realistic"

TIME in Project

VagueSpeculations

ConceptualDesign

RealDesign

Fabrication Installation Operation Suicide

Area of Maximum Peril

Overwhelming butessential effort toachieve rigorous

realism

Projects often slowly slide from Conceptual Design to Real Designwithout much fanfare. Great Peril: that the need for rigorousrealism in the design of essential components will be realized verylate, so that great strain will be placed on critical manpower latein the design process....

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

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 aresufficient to test whether cooling rings cool, and how sensitive their performanceis to details of the magnetic field map.

A good magnetic field generator should include the ability to alter MFGTs inways 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 toshow 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 mayend up with" when all the vicissitudes of design, construction, installation andoperation are accounted for.

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 = 0By = 0Bz = (/4 I (sin - sin ) / Yp

For a straight current segment oflength L lying on the X axis from(0,0,0) to (L,0,0), and an observationpoint P in the XY plane at (Xp,Yp,0),carrying current I in the +X direction:

We know how to dofinite straight linecurrent carriers...

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 = 0Br = 0Bt = (/4 I (sin - sin ) / D

Changing notation in preparationfor 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 senseto a GEANT simulation.

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 themagnet. Correspondingpoints in the right half bysymmetry. 8 straightcurrent segments per magnet.

A magnet made up froma small ensemble of finitestraight-line current carriers.To avoid unseemly chargebuildups, I suppose we shouldmake them all carry the samecurrent, though we mightturn individual segments on andoff for sensitivity studies.

(z'2, x'2)

(z'3, x'3)

(z'4, x'4)

Z

X

Y

Z

Rcpc

AcpcYcpc

Looking down toward -Y Looking sideways toward +X

Three more parameters Rcpc, Acpc, Ycpcdefine the entire 12-coil assembly. Intotal there are 11 parameters definingthe field shape. One more (current=I)then defines the field at every pointaway from a conductor.

1. A non-Maxwellian "box-field" which has constant B = (0,By,0) between the verticalpairs of coils and (0,0,0) outside it.

2. A Maxwell-compliant single-turn-per-magnet field, computed from the 8 straight-linesegments per magnet.

Four current "models"

3. A Maxwell-compliant multiple-turn-per-magnet field stacked in Y, so that eachturn 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 = 4dStack

nLayer = 3

dLayer

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 fieldIn array MagnetParam

One parameter to set the magnet current:

magCurrent

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.

Z'

X'

Add one more point to themagnet description? A bitof concavity/convexity normalto 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:

Add one more parameter to the magnet configuration?

nCells: The number of pairs of magnets distributed (uniform angular spacing)around the ring.

nCells = 6 nCells = 8

1. This field should explore most of the physically interesting field-shape issuesfor 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 mustthe generator be capable of to permit this without incurring unacceptableslowdown? 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". Arestill different numbers of cells contemplated? Should the ring design begeneralized (1 more parameter) to allow for N cells?

4. No field map should be believed without a "second opinion". Is thereanyone out there who would be willing to undertake a second implementationof exactly this field in a manner that yields Bx,By,Bz directly?

BfieldTest1 X vs Z

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Z

X

BfieldTest1 X vs Z

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Z

X

Particle Positions Examined Particle 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).

BfieldTest1 X vs Z

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BfieldTest1 X vs Z

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Z

XMagnets azimuthally centered in 60 degree cells; injection at cellboundary

Romulus decided he would preferto inject into the leading edge ofthe magnet. One parameter changed(aCpc), and . . .

BfieldTest1 X vs Z

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Z

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BfieldTest1 X vs Z

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Cell Geometry

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3040506070

8090100

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X

P1

P2

P3

P4

P5

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

Cell Geometry

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P1

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P4

P5

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 primaryfield.

5. Add stacking and layering if simulation results suggest it's useful.

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