# Presentation: 613966

## Helical FOFO Snake Simulations

**Helical FOFO Snake Simulations**

**Y. Alexahin**
**(FNAL)**

## Basic Idea

**Basic Idea**

**2**

- • Choose phase advance/period (6-cell period here) > 2integer to obtain p > 0
- • Create rotating B field by tilting (or displacing) solenoids in rotating planes
-                             x*cos(k)+y*sin(k)=0, k=1,2,...
- Example for 6-cell period:
- Solenoid #		1	2	3	4	5	6
- Polarity		+	-	+	-	+	-
- Roll angle	 k 	0	2/3	4 /3	0	2 /3	4 /3

- Channel parameters:
- 200 MHz pillbox RF 2x36cm, Emax=16MV/m
- Solenoids: L=24cm, Rin=60cm, Rout=92cm,
- Absorbers: LH2, total width (on-axis) 6x15cm,
- Total length of 6-cell period 6.12m

## Why double cavities?

**Why double cavities?**

**2’**

- solenoids

- As discussed by D. Neuffer, there is hope not to lose much of gradient due to magnetic field with such configuration

## Periodic Helical Orbits

**Periodic Helical Orbits**

**3**

- 20mrad pitch angle, BLS=25.2 for p=200MeV/c

- Bz/BLS

- By /BLS

- Bx/ BLS

- z

- z

- x

- z-v0*t

- z

- x

- y

- z-v0*t

- z

- Dx

- Dy

- z

- y

- Helical FOFO snake – good for cooling both μ+ and μ-!

- μ-

- μ+

- μ-

## Beta-functions & tunes

**Beta-functions & tunes**

**4**

**The best results with 7mrad pitch angle, no absorber wedge angle:**
- mode		I		II		III
- tune	 	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 = 22/L ImQ = 1/70m
- There is difficulty in equalization of damping rates of the transverse modes
- Bmax=2.3T  j=58A/mm^2, Itot=4.4MA/solenoid

- at the absorber locations
- <  > ~ 70cm
- - compare with MICE’s
- <  > ~ 45cm

## Transmission

**Transmission**

**5**

**Tracking simulations:**
- • “True” action vars  of 1771 particles evenly distributed in tetrahedron
-         (JI + JII)/2.6 + JIII /4 < 1 (cm)
- • Phases chosen at random
- • No decay nor stochastic processes
- Courant-Snyder invariant =2J,
- to compare with normalized emittance multiply by 2:
-  CSImax10cm or 2.2 for N=2cm

- Survival after 25 periods (153m) 97%

## Phase space distributions

**Phase space distributions**

**6**

- z-v0*t

- “Emittances” (cm)		initial 		final
- 6D			 10.3		0.07
- Trans. average		1.99		0.29
- Longitudinal		3.75		1.46

- py

- x

- y

- p

- px

- blue - initial, red - final

**Why momentum acceptance is so large (>60%) in the resonance case?**

## Tune “repulsion” from integer resonance

**Tune “repulsion” from integer resonance**

**7**

**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

- to ensure longitudinal damping.

- Momentum compaction factor:

## Equalization of transverse modes damping rates

**Equalization of transverse modes damping rates**

**8**

- Bob 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 1.6% difference in currents
- mode	I		II		III
- tune	 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)
- Unfortunately this compromised the transmission
- while cooling does not look better :

## Tapering

**Tapering**

**9**

**The damping rates can be equalized at large pitch angles, e.g.**
- increase pitch angle (20mrad here), tunes (via Bz) and introduce absorber wedge angle (0.1 rad conical angle on either side)
- mode		I		II		III
- tune	 	1.330+0.0084i	1.423+0.0095i	0.153+0.0058i
- _eq (mm)		4.7		4.2		2.4
- but the transverse acceptance is worse with larger pitch angle:

- This suggests the following strategy:
-  start with a small pitch angle cooling mostly transversely
-  as the transverse emittance shrinks, increase pitch angle
-  at the final stage the transverse emittance stays ~constant, cooling mostly longitudinal
- By reducing overall dimensions (using 1-cell RF) it is possible to lower equilibrium emittances to ~3mm for 200MHz channel.

## Helical snake for final 6D cooling

**Helical snake for final 6D cooling**

**10**

- By increasing B-field strength it is possible to get phase advance >180/cell and small -function at the solenoid center  much smaller emittance.
- Tune/period > odd_integer for resonant orbit excitation
**Puzzle:**
- 2-cell period (planar snake), Q>1
- 6-cell period, Q>3
- 4-cell period, Q>3
- 6-cell period, Q>5
- Channel parameters (4-cell period):
- 800 MHz pillbox RF 2x8cm, Emax=32MV/m
- Solenoids: L=8cm, Rin=16cm, Rout=26cm,
- Inclination: vertical +3mrad, horizontal +3mrad, vertical -3mrad, horizontal -3mrad
- Absorbers: LiH, total width (on-axis) 4x1.2cm, no wedge angle
- Total length of 4-cell period 1.12m

- p < 0

- p > 0

## Helical snake for final 6D cooling

**Helical snake for final 6D cooling**

**11**

- BLS=154T for p=100MeV/c  Bz_max=18.5T

- Bx/BLS

- By/BLS

- y

- Bz/BLS

- Dx

- Dy

- z

- x

- z

- z

- z

## Helical snake for final 6D cooling

**Helical snake for final 6D cooling**

- min ~1.6cm

- z

- p/100

-  [rad]

- QI

- y

- QII

- orbit lengthening

- p/100

- p

- Longitudinal acceptance limited by nonlinearity, not by insufficient RF bucket height

- x

**12**

## Helical snake for final 6D cooling

**Helical snake for final 6D cooling**

**13**

**3 mrad pitch angle, no absorber wedge angle:**
- mode		I		II		III
- tune	 	3.391+0.0066i	3.280+0.0141i	0.217-0.0047i
- _eq (mm)		0.20		0.08		-
- 3 mrad pitch angle is not enough to obtain longitudinal cooling, larger pitch will reduce separatrix energy width.
- Again, there is difficulty in equalization of damping rates of the transverse modes.
- Difference in + and - solenoid currents helps, but spoils acceptance.
- I tried to superimpose fields of different periodicity and structure, but with no success so far.
-                                                     **Still, the hope is the last to die!**

## Test with G4BL

**Test with G4BL**

**14**

**No absorbers & RF**
- MICCD: magnetic field via vector-potential series expansion in x,y (up to 5th order)
- G4BL: superposition of field-maps

- x [cm]

- Periodic orbit:
- MICCD – red, G4BL – blue
- In the following I used G4BL with field-map computed with MICCD

- z [cm]

- pz [MeV/c]

- z [cm]

- y [cm]

- z [cm]

## Test with G4BL

**Test with G4BL**

**15**

**Absorbers & RF in**
- MICCD: RF phase evenly incremented
- G4BL: RF phase adjusted w.r.t. the reference particle

- x [cm]

- ptot [MeV/c]

- z [cm]

- y [cm]

- pz [MeV/c]

- z [cm]

- something wrong with RF!

- z [cm]

- z [cm]

## Test with G4BL

**Test with G4BL**

**16**

**RF magnetic field **
- MICCD: Bphi and Ez from the same Az
- G4BL: Bphi and Ez computed separately

- x [cm]

- z [cm]

- z [cm]

- y [cm]

- Red: MICCD with Bphi=0
- Blue: G4BL
- The reason of the discrepancy is the practical absence of magnetic RF field in G4BL.
- However small, this field is important: it makes mapping over an RF cavity symplectic.
**Its absence may result in artificial cooling or heating.**