Helical FOFO Snake Update

11th Int. Workshop on Neutrino Factories, Superbeams & Beta Beams, Fermilab/IIT, July 20-25, 2009

Helical FOFO Snake Update

Y. Alexahin

(FNAL APC)

Basic Idea 2

absorbers RF cavitiesalternating solenoids

z [cm]

Bx50

B [T]

Bz By 50

x, y [cm]x

z [cm]

y

x

y

• The idea: create rotating B field by periodically tilting solenoids, e.g. with 6-solenoid period.

• Periodic orbits for μ+ and μ- look exactly the same, just shifted by a half period (3 solenoids).

• With tune Q>1 (per period) rD>0 muons with higher momentum make a longer path longitudinal cooling achieved even with planar absorbers

Periodic orbit for p=200MeV/c

HFOFO Update - Y. Alexahin NUFACT09, IIT Chicago July 22, 2009

Why double cavities? 3

solenoids

Double cavity increases packing factor w/o reduction in the transit factor

Hopefully the breakdown RF gradient will not be much reduced due to magnetic field with such configuration


Beta-functions & tunes 4

The best results with 7mrad pitch angle, no absorber wedge angle:

mode I II IIItune 1.239+0.012i 1.279+0.007i 0.181+0.002i_eq (mm) 3.2 4.5 6.9

ImQ=0.007 cooling rate d log / dz = 22/L ImQ = 1/70m

Bmax=2.3T j=58A/mm^2, Itot=4.4MA/solenoid

There is difficulty in equalization of damping rates of the transverse modes

yx

at the absorber locations

< > ~ 70cm

- compare with MICE’s

< > ~ 45cm

Solenoids: L=24cm, Rin=60cm, Rout=92cm, pitch 7mrad, Bzmax=2.35T (p=200MeV/c)RF: 200 MHz pillbox 2x36cm, Emax=16MV/mAbsorbers: 15cm LH2 planar


Tracking with MICCD 5

...2

1

,

20

2

0

020

0

pp

p

p

p

z-v0tx [cm] y [cm]

pPy/p0

blue - initial, red – after 25 periods (153m)

• “True” action variables of 1771 particles evenly distributed in tetrahedron

(JI + JII)/2.6 + JIII /4 < 1 (cm)

• Phases chosen at random

• No decay or stochastic processes

Courant-Snyder invariant =2J,

to compare with normalized emittance multiply by 2:

CSImax10cm or 2.2 for N=2cm

canonical momenta (include vector-potential), p0=200MeV/c

Px/p0


Tune “repulsion” from integer resonance 6

no absorber, no RF

p/200

orbit length/L0

QI, II

Nice surprise:

Large 2nd order chromaticity due to nonlinear field components keeps both tunes from crossing the integer !

p/200

- in contrast to classical HCC with homogeneous absorber where

20/2 p

to ensure longitudinal damping.

Momentum compaction factor:

22.0/11.0 20 p


Excursion - Longitudinal Hamiltonian 7

p

dK p

])(

)(1[)( 0

p

-0 (rad)

Contour plot of the longitudinal Hamiltonian for =0.014 - kinematic nonlinearity limits momentum acceptance, not insufficient RF bucket depth.

where = orbit length/L0

Kinetic energy in quasi-static approximation

=0.014

K

p

=0.007

URF

Longitudinal separatrix reaches maximum at 0.007


Tracking with G4BL 8

blue - initial (the same ensemble), red – after 20 periods (122m)

py/p0 (p-p0)/p0px/p0

x [cm]

y [cm] t [ns]

Stochastic processes on, but no decays

mechanical momenta, p0=200MeV/c


Analysis of G4BL Tracking Data 9

Normal mode emittances (normalized) vs period # (_eq = 0.32, 0.45, 0.69)

G4BL stochs. onMICCD

N/N0

2N

G4BL stochs. off

N [cm]

1N

3N

no decays! With decays transmission over 20 periods (122.4m) is > 85%

Initial/final N [cm]=1.7, 1.8, 2.1 / 0.37, 0.7, 1.1

6D cooling efficiency (following R.Palmer’s definition)

Q6 20

multiply m by 00

Recipe for emittance calculation:

compute normal mode amplitudes am for each muon using lattice eigenvectors

subtract average values

find distribution in action variables Jm =| am |2

fit with exponential function

m

mJf

exp~

Transmission vs period #


Equalization of transverse modes damping rates

R. Palmer proposed to add ~ constant solenoidal field to better mix the transverse modes and equalize their damping rates.

This can be achieved by powering e.g. negative solenoids with slightly lower current.

With just (I+ - I-)/ I+ = 1.6% :

mode I II IIItune 1.211+0.0100i 1.301+0.0108i 0.196+0.0003i_eq (mm) 3.8 3.2 36.5

(fast transverse cooling w/o longitudinal blowup was the intent):

10

y

z

x

Unfortunately, there is no appreciable gain in equilibrium emittance due to large -wave excited with unequal field in + and - solenoids.

Transmission also suffers (red vs. blue)

- computed without stochastic effects and decay

N/N0

period #


Damping Rate Equalization with Quadrupole Field 11


Most efficient “equalizer” is constant quadrupole field superimposed on the main solenidal field.

With a gradient of just 0.057 T/m:

mode I II IIItune 1.240+0.0088i 1.286+0.0087i 0.178+0.0036i_eq (mm) 3.9 3.8 4.7(w/o quad was 3.2 4.5 6.9)

z

y (cm)Dy (cm)

z

Dx (cm)

x (cm)

The sum of betas smaller - no increase in transverse emittance (despite smaller long. emittance)

50% stronger long. damping due to larger dispersion (smaller solenoid pitch required)

The effect does not depend on quad tilt (and polarity) as well as on the muon sign - the channel is still good for both!

G4BL Simulations with Quad “Equalizer” 12


1.71, 1.79, 2.20 / 0.554, 0.584, 1.006

1.66, 1.80, 2.05 / 0.367, 0.701, 1.096

N/N0

period #

period #

period #

2N

3N

1N

1N

2N

3N

with quad

There is not so much reduction in final emittances due to the damping rate equalization as increase in the initial values due to changes in nonlinear transformation from “true” normal forms (Here emittances in cm)

w/o quadwith quad

w/o quad:

Initial / final emittances (cm) with quad: w/o quad

Absorber Container and RF Cavity Walls 13


A MODERN AIRCRAFT ALUMINUM ALLOY (thanks to Don Summers)

Aluminum Alloy Yield Tensile Radiation Name Composition Density Strength Strength Length % by weight g/cc ksi ksi cm 300K 300K 20K 6061-T6 1.0Mg .6Si .3Cu .2Cr 2.70 40 45 68 8.86 2090-T81 2.7Cu 2.2Li .12Zr 2.59 74 82 120 9.18

Supposedly 2090-T81 walls can be made 0.1mm thick (as compared to 0.18mm of 6061-T6 in MICE).

Three cases considered below:No container nor cavity walls (original lattice)Two 0.1mm 2090-T81 container walls / absorber, no safety walls nor cavity walls Two 0.1mm 2090-T81 container walls / absorber + three 0.38mm Be walls / two RF cavities

Q1 Q2 Q3 1N (mm) 2N (mm) 3N (mm) ERF (MV/m)No walls 1.239+0.012i 1.279+0.007i 0.181+0.0025i 3.2 4.5 6.9 16.0Abs. walls 1.241+0.012i 1.281+0.007i 0.185+0.0026i 3.7 5.1 7.5 16.3+ RF walls 1.243+0.013i 1.286+0.007i 0.191+0.0033i 4.0 5.8 7.3 17.5

Though the effect of the walls is tolerable (<30% increase in the equilibrium emittance) it would be worth while to look for a lighter material for the absorber walls (Be?)

G4BL Simulations with Walls 14


Final emittances for the cases 1 and 3:1N (mm) 2N (mm) 3N (mm) 6D (mm^3)

No walls 3.67 7.01 10.96 218.2With all walls 3.63 7.77 10.22 212.9

- Strange result, may be due to a faster cooling with 9% increase in losses and the RF gradient

period # period #

1N

2N

3N

N/N0

with wallsw/o walls

Summary & Outlook 15

Tilting solenoids of ASOL channel (Dave’s nomenclature) increases momentum acceptance and provides some longitudinal cooling with only mild deterioration of transverse acceptance and cooling (look for the optimum!)

Transverse damping rates can be equalized by adding quadrupole field without compromising transverse or momentum acceptance

HFOFO with 200MHz double RF gives T < 6mm, L ~10mm

Another 200MHz section with single RF can give T ~ 3mm, L ~10mm

Combine HFOFO cooling with Daves bunching / rotation

Higher phase advance per cell small betas @ absorbers

Better options for the absorber and RF cavity windows?


Helical FOFO Snake Update

Documents

Transcript of Helical FOFO Snake Update