V.Balbekov, 12/09/08
HCC
simulation with
wedge
absorbers
V. Balbekov, Fermilab
Muon Collider Design Workshop
December 8-12, 2008
JeffersonLab, Newport News, VA
Is this possible to improve HCC with help of wedge
absorbers?
Primary idea: low kappa channel with small transverse
field, etc.
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V.Balbekov, 12/09/08
B. Palmer:
Low
κ
helix cooling (08/26/08)
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V.Balbekov, 12/09/08
Outline
1. Stability region of “conventional” helical cooling channel (HCC)
2. Peculiarity of helical solenoid channel (HSC)
3. HCC and HSC with small pitch-factor and wedge absorbers
4. Wedge absorber as a tool for diminution of the HCC/HSC field and
beam momentum
5. Cooling degeneration at excessive low momentum
6. HCC with small transverse field
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V.Balbekov, 12/09/08,
p.3
Characteristics of the HCC
• Most characteristics of the HCC:
eigenmodes, eigentunes, beta-function, normalized
dispersion,
normalized transverse decrements
can be found with help of the function F(X) which is normalized
longitudinal magnetic field at
r = XL/2π, θ = 2πz/L:
Bz(X) = Bsol F(X) = Bsol [1+2F’∙I1(X)+F”∙I2(2X)+…]
where z, r, θ are cylindrical coordinates, L – period length, Bsol = Bz(0) –
solenoid field, I1, I2 ...- modified Bessel function, F’, F” … - coefficients.
• LBsol are required additionally to know reference momentum, slippage factor,
and normalized longitudinal decrement.
• Absolute decrements are proportional to the ionization energy loss.
• For Helical Solenoid Channel (HSC, Kashikhin): the function F(X) depends
on relative coil radius Rcoil /L and antisolenoid/solenoid field ratio Bas/Bhs.
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V.Balbekov, 12/09/08
Example 1: HCC (Yonehara design)
Example 2: HSC
F’ = -0.2545, F” = 0.0777, F”’=0,…
Rcoil /L = 0.2733, Bas/Bhs=-0.283
In both cases: homogeneous H2 absorber,
LBsol = 6.97 T-m
to get Prefer = 250 MeV/c at
Xrefer(=κ) = 1
(unite length is L/2π)
Normalization of the decrements:
Dt1+Dt2+Dl = 2β2
!!! No stability at X < ~0.60 -- 0.65 !!!
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V.Balbekov, 12/09/08
Cooling simulation
at ionization energy loss 14.7 MeV/m,
Example 1: HCC
RF 200 MHz, 29.4 MV/m
Example 2: HSC
The results are very similar:
Transverse emittances are about 2 mm ≈ L/500, longitudinal one ~3 mm ≈ L/300.
Transmission 72% - 76 % at 200 MHz,
but falls at higher frequency (reasonable requirement λ >~L ).
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V.Balbekov, 12/09/08
Effect of wedge absorber
In considered cases, either one of transverse modes or longitudinal
mode is unstable at X < ~0.6, if homogeneous absorber is used.
Can a wedge absorber improve the situation?
Plot: HCC normalized cooling decrements in the at:
W = 5 (circles), W = -5 (triangles down), and W = 5
(triangles up).
Wedge strength:
W = d(ln Eion)/dX.
Decrements: orange – trans., magenta – long.
Similar results are observed in HSC.
Wedge absorber does not improve the situation.
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V.Balbekov, 12/09/08
Explanation
Without transverse field and emittance exchange, only one transverse mode can be
really cooled. Other one is neutral (no damping or rising), and longitudinal motion is
unstable at E<~350 MeV.
With additional dipole field and/or emittance exchange, sum of decrements of the
neutral and unstable modes remains negative, if the reference radius is small enough.
Small quadrupole etc. addition does not change the situation.
Quad. dominated HCC has small acceptance.
Therefore low kappa channel with any wedge is unsuitable for 6D cooling.
However, one can use the wedge absorbers to reach a reasonable
longitudinal decrement at pitch-factor ~1, using lower beam momentum
and correspondingly lower magnetic field.
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V.Balbekov, 12/09/08
Example 1: HCC without/with wedge absorber
W/o wedge
F’ = -0.2545 => -0.2535
F” = 0.0777 => 0.0780
W= 0
=> 0.05
With wedge
L = 1m
Bsol = 6.97 => 5.24 T
Xrefer (=κ) = 1
Prefer = 250 => 189 MeV/c
E’refer= 14.7 MeV/m
(wedge absorber – 5%)
RF 200 MHz, 29.4 MV/m
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V.Balbekov, 12/09/08
Example 2: HSC without/with wedge absorber
W/o wedge
Antisol/Sol -0.2733=>-0.283
Rcoil/L = 0.2733
2πRhelix/L = 1
With wedge
W = 0
=> 0.07
L =1m
Bsol = 6.98 => 5.24 T
Xrefer (=κ) = 1
Prefer = 250 => 189 MeV/c,
E’refer = 14.7 MeV/m
(wedge absorber – 7%)
RF 200 MHz, 29.4 MV/m
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V.Balbekov, 12/09/08
RESULTS
• About 25% reduction of operational beam momentum and required
magnetic field of the HCC is possible with wedge absorbers.
• Achievable beam emittance nearly does not change, and
transmission a bit falls in the case (the fall is more in the helical
solenoid).
• However, an attempt of more momentum/field reduction leads to a
strong degradation resulting a huge particles loss.
Important question: what is a reason of the degradation?
• The main reason is a deterioration of longitudinal phase space.
• Stronger dependence dE/dx on velocity at lower energy has but a
minor effect .
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V.Balbekov, 12/09/08
Lower momentum HCC with wedge absorber
Parameters
F’
= -0.196
FXrefer (=κ) = 0.62,
F”
= 0...
Prefer = 159 MeV/c,
W
= 0.57
F = 50 MHz
L
=1m
V’ = 29.4 MV/m
Bsol = 5.23 T
Longitudinal ph. space
Cooling simulation
E’refer = 14.7 MeV/m
Parameters vs momentum
Cooling is impossible at these parameters
because of huge particles loss.
The reason is “bad” longitudinal phase
space because of small slippage factor .
Blue – longitudinal
phase trajectory at
betatron oscillations
Betatron motion results fast phase
oscillations and bring about an additional
deterioration of the space.
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V.Balbekov, 12/09/08
Cooling at artificial media: dE/dx does not depend on E
HSC with artificial cooling media is considered. Longitudinal emittance is less than “normal”
one because of more longitudinal decrement. Transmission slightly increases also. However,
the transmission drops as before when beam momentum decreases from 250 MeV/c to
189 MeV/c:
76% / 62% “normal”
79% / 68% “artificial”
It means that dE/dx(E) dependence produces a minor effect.
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V.Balbekov, 12/09/08
Lower transverse field / pitch-factor
Parameters
(V.B., MCnote 146, 2000)
L = 1.8 m
Bsol = 5 T
Xrefer (κ) = 0.14
Prefer = 240 MeV/c
LiH, 260 mrad, 3.9 cm
E’ = 7 MeV/m
201 MHz, 14 MV/m
F’ =-0.06, F”...= 0, W = 1.9
No equilibrium
transverse emittance
Old simulation
Both simulation demonstrate that the cooling is unachievable in this regime.
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V.Balbekov, 12/09/08
Conclusion
1.
HCC or HSC with small transverse field and small pitch-factor (kappa) are
unsuitable for 6D cooling, both with and without wedge absorbers.
2.
Wedge absorber can be used to decrease momentum and field of a “conventional”
HCC or HSC. 25% reduction is achievable without unreasonable degradation of
the cooling. The wedge absorbers should provide 5-7% of ionization energy loss,
95-93% -- high pressure hydrogen.
3.
More momentum/field reduction leads to unacceptable particles loss. The main
reason is a deterioration of longitudinal phase space because of an approach of
transition energy to the working area. My attempts to remedy this flaw were
unsuccessful.
4.
After optimization, production LBsol can be decreased from ~7 T-m (no wedge) to
~5 T-m (with wedge). Achievable transverse emittance is ~L/500 in both cases.
5.
RF separatrix should be long enough to neutralize dependence of time of flight on
betatron oscillations. The wavelength λ ≈ L (or more) looks reasonably providing
achievable longitudinal emittance ~L/300.
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