FODO-based Linear Quadrupole PreCooler

M. Berz, D. Errede, C. Johnstone, K. Makino D. Neuffer, and A. Van Ginneken, A. TollestrupWG1, July 3

NuFACT02

Imperial College, London

July 1-6, 2002

Contents

Basics What emittances are required for a neutrino factory? Equilibrium emittances in cooling channels General staging remarks for cooling

What emittances drive the cooling in a neutrino factory

Storage ring design n(rms)

pjk 50-GeV design 1.5 mm rad Fermi 50-GeV design 3.2 mm-rad

RLAs

Study I 1.1 mm-rad Study II <2 mm-rad

Required “cooled” emittance is 1-2 mm-rad for U.S. baseline acceleration scenario

For Japanese FFAG scenario, emittance requirement is 1 cm-rad, almost an order of magnitude larger

To achieve a minimum equilibrium emittance of 1.7 mm-rad @200 MeV/c , has to be ~ 0.4 m.

This is to be compared to the rms emittance after capture of 20 mm-rad, indicating a factor of 10 requirement in transverse cooling for the U.S. Scenario (factor of 2 for Japanese acceleration scenario.

The equilibrium emittance is given by

N,min = (14 MeV)

(mLR.dds)

where is the transverse beta function at the absorber, the relativistic velocity, m the mass of the muon, LR the radiation length of the absorber material, and dE/ds the energy lost in the absorber.

Staging the cooling

Clearly a factor of 10 decrease in transverse emittance is difficult using a single cooling structure large emittances, low betas = large apertures in elements

(max 1/ ) how would you effectively stage an order of magnitude in

cooling

Look at the cooling dynamics

Full simulation: transverse cooling along Quad channel

Observations:

Cooling rates are ~constant until the 1.5 x equilibrium

• lowering does not significantly increase the cooling rate

Conversely, raising the initial uncooled emittance

can be scaled upwards also• very significant for cost and optical design of

cooling at ultra-large emittances after capture.

A staging strategy

Emittance-dependent:

Factor of 2 cooling/plane per stage:

n rms (mm-rad) 20 10 5 2.5**

(m) 1.5 0.75 0.4

Factor 2 2 2

**practical limit for of 0.4 m FFAG scenario

Choice of optical cooling structure:

With the much large one can start thinking

nonsolenoidal structures, i.e. quadrupoles

removing components from apertures of elements, hopefully smaller apertures = lower costs

nonsuperconducting elements

Quadrupole Cooling Channels: General Considerations

Longitudinal and and Transverse Acceptance of Channel Compare acceptances of different optical structures

Insertion of Absorber Location of minimal beam size, both planes Calculate equilibrium emittance limit

Physical Limitations Quadrupole aperture and length constraints Available space between magnets

Match to emittance-exchange channel** Can same optical structure be used for emittance exchange

**not required for FFAG scenario

Optical Structures for Ultra-large Emittance Beams

FODO cell Simple-lens/Alternating Gradient systems have the largest

combined transverse and chromatic acceptance of any optical structure for light/magnetic optics.

Generally, acceptance is limited only by the physical apertures of the components

Before proposing a channel--

What constraints should be imposed on quadrupole design?

Quadrupole Design

Aperture vs. length: role of fringe fields

Maximum poletip field

Design Constraints:

Maximum aperture = Magnet Length

Normal Conducting Quads: Poletip Field < 2T

Separated Components: rf, in particular removed from component apertures; increasing acceptance and

decreasing component cost

Consequences of hoice of Optical Structure

FODO Simplest: alternating focussing and defocussing (in

one transverse plane) lenses A minimum in beta or beam size cannot be achieved

simultaneously in both planes Doublet or Triplet quadrupole system

2 or 3 consecutive, alternating focussing and defocussing quadrupoles

required to form simultaneous low beta points in both planes (interaction regions of colliders, for example)

Acceptance of Quadrupole Channels Transverse Acceptance

Using only linear elements (quadrupoles and/or dipoles), the transverse dynamic aperture is normally larger than the physical aperture (unless a strong resonance is encountered in a long series of cooling cells).

Practically, FODO and doublet/triplet quadrupole channels have transverse acceptances or apertures limited only by poletip strength for a given gradient (8T for superconducting quads and 2T for normal conducting).

Longitudinal Acceptance The FODO cell is a simple lens system and has the largest chromatic acceptance of

any quadrupole-based structure. Lattices based on FODO cells have been designed which transmit up to a factor of 4 change in momentum.

Doublet/triplet-based quadrupole structures are momentum limited to approximately 5% deviation from the central momentum of the channel.

Logistics Because the FODO cell cannot achieve a minimum beta point in both planes, its

valid application is just after capture and phase rotation, where the transverse and longitudinal emittances are very large.

Conversely, the limited momentum acceptance of the triplet/doublet quadrupole channels restrict their implementation to after emittance exchange has occurred.

The rest of this talk will discuss the FODO cooling channel only

Construction of FODO Quad Cooling Cell

1/2 1/2

abs F rf D rf F rf D abs

COOLING CELL PHYSICAL PARAMETERS:

Quad Length 0.6 m

Quad bore 0.6 m

Poletip Field ~1 T

Interquad space 0.4 - 0.5 m

Absorber length 0.35 m *

RF cavity length 0.4 - 0.7 m*

Total cooling cell length 2m (rf extending into magnet)

4m (separated rf)

For applications further upstream at larger emittances, this channel can support a 0.8 m bore, 0.8 m long quadrupole with no intervening drift and without matching to the channel described here.

Acceptance in the Presence of Fringe Fields

Outside the physical 60 cm aperture after applying a known fringe field model

Stable in the presence of fringe fields over a momentum range, dp/p -22% to >+100%

Although it is outside the physical aperture, fringe fields are the limiting effect on the dynamic aperture

Origin of longitudinal Defocussing quad with

field components in beam envelope for the

fringe fields: quatoed acceptance of the channel

Strong only on diagonal BUT Beam envelope quickly

near poletips deviated from the diagonal in a FODO

channel

Fringe-field components

Enhanced physical apertures in a quad:

Star-shaped vacuum chambers can be used in the FODO channel quads effectively increasing their acceptance; this is done in the Fermilab MI for injection/extraction.

However, star chambers have not been assumed in the following simulations.

FODO LATTICE AND INSERTION OF ABSORBER

LATTICE FUNCTIONS OF A FODO CELL FOR A QUAD COOLING CHANNEL, P0=200 MeV/c In a FODO cell, the combined minimum for x and y is at their crossing point, halfway

between quadrupoles. The achievable transverse in the absorber is about 1.6 m, or a factor of 4 larger than the

average transverse in the 2.75 m SFOFO channel (0.4 m). This channel can a full transverse normalized emittance of 2-2.5 for = 20 mm-rad at a p0

of 200 MeV/c.

Longitudinal Acceptance and Lattice Stability at Different Momenta

LATTICE FUNCTIONS for 155, 245, and 300 MeV/c, clockwise.

average at the absorber ranges from 1.57

to 1.90 at 155 and 245 MeV/c, respectively, which represents the momentum range of the 2.75 m SFOFO channel.

max is 3.90 m @155 MeV/c and 3.19 m @245 MeV./c

The acceptance reach of this channel is clearly larger than 300 MeV/c; i.e. avg is still only 2.3 m. and max is 3.5 m

Fodo Cell Properties as a Function of MomentumP

MeV/c

atabsorber

(m)

per

FODOcell

max

min

155 1.57 m .33 3.85 0.35

200 1.63 .23 3.13 0.71

245 1.91 m .18 3.20 1.10

300 2.28 m .15 3.45 1.37

Longitudinal Acceptance and Lattice Stability at Different MomentaBeta Function Dependence on Momentum: Linear Quad

Channel

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

150 200 250 300 350 400

Momentum (MeV/c)

Beta

(m

)

Betamax

Beta (absorber)

Phase Advance vs. Momentum: Linear Quad Channel

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

150 200 250 300 350 400

Momentum (MeV/c)

Ph

ase A

dvan

ce/c

ell

(p

i)

Acceptance of the Linear Quad Channel

With this stability what is the acceptance of the linear quad channel?

More importantly, how does it compare with the upstream SFOFO channels?

Full Normalized Emittance Acceptance of Solenoidal and Quad Channels

20

30

40

50

60

70

80

90

150 200 250 300 350 400

Momentum (MeV/c)

Acc

epta

nce

(p

i-m

m-r

ad)

solenoidal accetance

quad acceptance

Effective Cooling Range

BUT

as we know at the absorber is increasing with momentum, but so is the normalized acceptance

so what is the real cooling range in energy?

The equilibrium emittance is given by

N,min = (14 MeV)

(mLR.dds)

where is the transverse beta function at the absorber, the relativistic

velocity, m the mass of the muon, LR the radiation length of the absorber material, and dE/ds the energy lost in the absorber.

Then, one cools when

N,min/ constant

or

if the unnormalized acceptance is constant vs. momentum

/ constant

Normalized Acceptance and Beta (absorber)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

150 200 250 300 350 400

Momentum (MeV/c)

No

rmal

ized

val

ue

Normalized acceptance

Normalized beta(absorber)

This implies:

Cooling occurs from 150 MeV/c to past 400 MeV/c in this channel

Tracking Studies were performed: with full nonlinear terms

with/without quadrupole fringe fields (different models)

with multiple scattering (windows +absorber)

dE/dx as a function of energy

full energy loss function including straggling and spin

200 MHz sinusoidal rf

Preliminary Tracking Studies of the FODO quad cooling channel

Preliminary Tracking Studies of the FODO quad cooling channel-details

Tracking Studies were performed: with full nonlinear terms with/without quadrupole fringe fields (different models) with multiple scattering (windows +absorber) full straggling implemented (windows + absorber)?

Preliminary estimates of cooling were obtained by: 1. determining the invariant phase ellipse of the quad channel

2. tracking in 1 cm steps along the x axis to determine the dynamic aperture of the channel

3. particles were then launched on the outer stable invariant ellipse at various x,x’ coordinates and on one inner phase ellipse near the calculated equilibrium emittance

4. particle positions were plotted for 10 cells, but at increasing distance down the length of the cooling channel (cells 21-30, 31-40, 101-110, for example) until the cooling converged.

5. the rough cooling factor was obtained by comparing the outermost stable ellipse with the final ellipse which clearly contained the majority of the particles.

Quad Cooling Channel Simulationby COSY Infinity

Muons (180MeV/c to 245MeV/c) Magnetic Quadrupoles (k=2.88) Liquid H Absorber: -dE/dx = -12MeV/35cm Cavities: Energy gain +12MeV/Cell to compensate the

loss in the absorber

K. Makino Emittance Exchange Workshop at LBNL, October 3-19, 2001

4m Cell

Tracking the Quad Cooling CellsMomentum: 220 MeV/c, Starting from x=2cm,4cm,…,30cm, for 100 Cells

(a) Without Cooling (b) With Cooling (no scattering)(c) With Cooling and Scattering (d) Pseudo-Invariant Ellipses with Cooling (damping factor corrected)

K. Makino Emittance Exchange Workshop at LBNL, October 3-19, 2001

(a) (b)

(c) (d)

Tracking the Quad Cooling Cells with ScatteringMomentum: 220 MeV/c, Starting from x=10cm, 15cm Pseudo-Invariant Ellipses

(a) Initial Ellipses (b) for 11-20 Cells (c) for 31-40 Cells (d) for 101-110 Cells

(a) (b)

(c) (d)

Full Simulation includes:

dE/dx as a function of energydE/dx curves have now been loaded as a function of energy into COSY;

i.e. energy lost in each absorber depends on the particle’s energy. Straggling

A. Van Ginneken’s energy loss function has been interfaced to COSY.

Low energy tail of the straggling function is believed to be an important

loss mechanism in the early channels AND the spin of the particle

affects the average energy of the distribution. In this simulation the

energy of the reference particle is calculated with full straggling and spin.

The energy loss of other particles are calculated relative to this

reference particle following the dE/dx curve. rf bucket

A sinusoidal 200-MHz rf waveform has been implemented assuming a

gradient of 10MV/m.

Straggling: Energy distribution after hydrogen absorber

Straggling: Energy distribution after Al window

Generation of particle energy distribution after absorber

Quantifying the Cooling: Description of a Merit Factor

TRANSVERSE

Load an elliptical distribution in x,x’ which corresponds to the stable phase ellipses

of a quad channel without cooling. The exact shape will be a Gaussian whos rms

width is 1/2 - 1/2.5 the half aperture of the quadrupoles, which is 30cm (max =3.1 m,

@p=200MeV/c). An initial to final rms ratio can be calculated for the transverse

cooling factor.

LONGITUDINAL

Two cases will be loaded: one with little longitudinal loss, and one “filling” the

bucket for comparison.

Minimal bucket loss

E = 12 MeV; rms bunch length = 7.5 cm

s = 60°; = ±54° for a 3 distribution

Maximul bucket loss

E = 24 MeV; rms bunch length = 15 cm

s = 60°; = ±108° for a 3 distribution

Longitudinal and transverse losses will form the overall transmission factor.

The product of the transmission and cooling factors combined with the decay

losses will provide the merit factor for this emittance range.

Full simulation: transverse cooling along channel

Tracking of particles launched along the diagonal in x,y

Goals Achieved with a the FODO-based Quad Cooling Channel

Cool transversely by a factor of 2 in the emittance of each plane: from an full normalized emittance of about 80-125 mm-rad to 30-40 mm-rad in each plane.

Inject cleanly (without matching) into an emittance exchange channel using the same FODO-based optical cell.

Recool transversely with the same channel. After this channel, inject into more sophisticated cooling channels

such as solenoidal ones to cool to the final required emittances This is the only cooling needed for an FFAG scenario (emittance

exchange is not needed)

Future Design Studies:

Move the central energy of the channel to the minimum in the total strength of the reheating terms (~300-400 MeV/c) by increasing the poletip strength of the quadrupoles (they are still normal conducting; I.e. <2T).

Although this, in principle, halves the momentum spread, this quad channel will still accept more the -22%-100% total momentum bite for effective cooling due to its naturally high longitudinal acceptance.

This implies a momentum bile of 270 MeV/c - 700 MeV/c

Compare cooling rates, final emittances, and losses at the higher momenta Load “exact” particle distributions from target/capture/buncher string

Investigate superconducting rf and increased absorber lengths. (The quadrupole end fields fall much more quickly than solenoidal ones

and if quads are normal conducting, mirror plates can be used)

Evaluate other absorbers: He?, LiH (rotating drum target)