# Presentation: 614019

## Progress with 200MHz FOFO snake (vacuum RF)

- A planar FOFO snake (solenoids tilted and/or displaced in one plane only, e.g. vertical) is an attractive candidate for initial cooling of both μ+ and μ-
- after which they can be separated in HCCs or Guggenheims.
- First considerations were presented in the end of July, next 2 slides are from that presentation.

- Y.Alexahin, MCTF meeting 09/11/2008

- 1

## FOFO snake with 200MHz vacuum RF - first look

- Despite higher gradients achievable with pressurized RF, vacuum RF may be more suitable, especially for the initial cooling where muon beam intensities are expected to be high. Also:
-  no need for thermo-insulation from solenoids (will 3cm gap be enough?)
-  no need in positive momentum compaction*)  may lower the tunes  less beta beat
-  with alternating solenoid polarity the effect of magnetic field on RF may be mild

- Y.Alexahin, MCTF meeting 07/31/2008

- 2

- * This proved not to be true: if c<0 there is strong longitudinal anti-damping due to D’ which has to be compensated with large wedge angle

## Schematic of the channel (particular numbers just a guess):

- 200 MHz pillbox RF 2x20cm
- Solenoids: L=24cm, Rin=40cm, Rout=72cm,
- vertical pitch 20mrad, vertical offset  1.5cm
- Absorbers: LH2, total width 4x8cm

- Y.Alexahin, MCTF meeting 07/31/2008

- 3

- 70cm

- I stuck to 4-solenoids cell, but this was by no means necessary!

## 4

- Positive (and small) momentum compaction is beneficial by a number of reasons :
-  longitudinal cooling can be achieved with a small wedge angle (or no angle at all)
-  smaller (by absolute value) momentum compaction increases longitudinal acceptance
-  no need to invert the wedge  transverse cooling at large amplitudes is not compromised
- c>0 requires that the transverse tune per period Q>1
- How to obtain c>0 while keeping the small phase advance per cell (solenoid + RF)?
- - By increasing the number of cells per period! (determined by periodicity of tilts)
- 6-cell period will require > 60/cell for Q>1
- But first about the optimum length of 200MHz RF cavities for 200MeV/c muons:
- add. length (1/2 solen.+gap)/RF cell (cm)	RF cell length (cm)
- 		0.1				8.1
- 		1				17.1
- 		5				28.0
- 		15				37.8

## 5

- New channel parameters:
- 200 MHz pillbox RF 2x36cm, Emax=16MV/m
- Solenoids: L=24cm, Rin=40cm, Rout=72cm,
- vertical pitch (mrad): +20, -30, +20, -20, +30, -20, no vertical offset
- Absorbers: LH2, total width (on-axis) 4x15cm, horizontal wedge angle tan(/2)=0.05
- Total length of 6-cell period 6.12m

- Bz

- By

- B/BLS

- Bz_max=2.6T (BLS=15) requires j=37.3A/mm^2

- mode		I		II		III
- tune	     1.378 + 0.005 I	     1.308 + 0.014 I	     0.167 + 0.002 I
- _eq (mm)	6.8		2.6		5.9

## 6

- Dispersion (cm)

- There is no way to tap into the 2nd mode: all longitudinal cooling at the expense of the 1st (~horizontal) mode

- periodic orbit (cm)

- dII

- dIII

- dI

- x

- z – v0*t

- Dx

- Dy

- y

- Cooling repartition by 50mrad Be wedge
- dI =ImQI

## 7

- Tracking
-      Lie-transform from “true” invariants JI, II, III  to phase space variables used transverse amplitude-energy correlation taken into account
- 1330 particles distributed in tetrahedron
- 	(JI + JII)/2.34 + JIII /3.6 < 1 (cm)
- Phases chosen at random
- No stochastic force so far
- Survival after 25 periods (153m)  63%

## 8

- JII

- red – lost,
- blue – survived,
- distance along the diagonal reflects JIII

- JI

- first mode is less stable due to weaker damping, JI(max) seems to be small (~1cm), but in physical variables the situation is quite a bit better

## 9

- Evolution of surviving particles distribution

- px

- y

- py

- z

- p

- x

- blue – initial position, red – after 25 periods

## 10

- Some other correlation plots

- Mode II is damped – beams can be separated vertically

- y

- px

- py

- x

## 11

- Emittances (cm)		initial 		final
- 6D			7.95		0.23
- Trans. average		1.92		0.39
- Longitudinal		4.37		3.78

## 12

**Outlook**
- These results should be confirmed with other programs: Adam Lichtl (BNL) is learning how to do this with G4BL.
- There are several possibilities to improve the channel acceptance:
-   changing coil geometry (increase the radius?) can reduce nonlinear field components
-   adjusting relative values of the tilts can reduce nonlinear detuning and RDTs
-   better choice of the tunes! (determined by Bz field)
- Overall ~100 versions to try – requires ~4 months
- Next steps:
-   conjugation of the 200MHz snake with capture/bunch rotation section
-   2-dimensional FOFO snake with tilts and offsets in both transverse directions to form a helix – HCC again!