# Presentation: 768652

## Pulsed ‘Dogbone’ RLA for Muon Collider 

- Pulsed ‘Dogbone’ RLA for Muon Collider_** **_

## RLA for Muons

**‘****Dogbone’** (Single Linac) RLA has advantages over the Double-Linac RLA ‘Racetrack’
-  **FODO** Optics is superior to Triplet focusing  more passes are possible with the FODO scheme
**Pulsed **linac Optics.... proposed by Rol Johnson ....even larger number of passes is possible if the quadrupole focusing can be increased as the beam energy increases.....subject to a new SBIR with Muons Inc.

## ‘Racetrack’ vs ‘Dogbone’  RLA (both + and - species )

- ‘Racetrack’ vs ‘Dogbone’  RLA (both + and - species )

- E

- better orbit separation at linac’s end ~ energy difference between consecutive passes (2E)
- allows both charges to traverse the Linac in the same direction (more uniform focusing profile
- the droplets can be reduced in size according to the required energy
- both charge signs can be made to follow a Figure-8 path (suppression of depolarization effects) Chuck Ankenbrandt

## FODO vs Triplet focusing  ‘flat focusing' linac profile*

- 1-pass, 3-5 GeV

- phase adv. drops much faster in the horizontal plane

- phase adv. diminish uniformly in both planes

- FODO

- Triplet

- FODO vs Triplet focusing  ‘flat focusing' linac profile*

- Bob Palmer*

## FODO vs Triplet focusing  ‘flat focusing' linac profile

- 2-pass, 5-7 GeV

- phase adv. drops much faster in the horizontal plane

- FODO vs Triplet focusing  ‘flat focusing' linac profile

- phase adv. diminish uniformly in both planes

- FODO

- Triplet

## FODO vs Triplet focusing  ‘flat focusing' linac profile

- 3-pass, 7-9 GeV

- no phase adv. in the horizontal plane

- FODO vs Triplet focusing  ‘flat focusing' linac profile

- FODO

- Triplet

## FODO vs Triplet focusing  ‘flat focusing' linac profile

- 4-pass, 9-11 GeV

- phase adv. diminish uniformly in both planes

- FODO

- Triplet

- FODO vs Triplet focusing  ‘flat focusing' linac profile

- no phase adv. across linac- beam envelopes not confined

## FODO vs Triplet focusing  ‘flat focusing' linac profile

- 5-pass, 11-13 GeV

- phase adv. diminish uniformly in both planes

- FODO

- Triplet

- FODO vs Triplet focusing  ‘flat focusing' linac profile

- no phase adv. Across the  linac- beam envelopes not confined

## FODO vs Triplet focusing  ‘flat focusing' linac profile

- FODO

- Triplet

- FODO vs Triplet focusing  ‘flat focusing' linac profile

- 6-pass, 13-15 GeV

- phase adv. diminish uniformly in both planes

- no phase adv. Across the linac- beam envelopes not confined

##  RLA requirements 

-  Simultaneous acceleration of both + - species
-  Manageable orbit separation at recirculation arcs
-  Beam dynamics challenges  RLA Optics solutions_** **_
-  Phase slippage in the linacs
-  Multi-pass linac optics
-  ‘Droplet’ arc lattice
-  Orbit separation - switchyard
-  7.5-pass Dogbone RLA - Linear Optics/Lattices designed

- ‘Dogbone’ RLA with FODO Focusing

## Possible RLA based Acceleration Scenario for MC

**7.5 pass**

- 4 GeV/pass

- 32.5 GeV

**Dogbone RLA I**

## Possible RLA based Acceleration Scenario for MC

**Low Energy** **RLA** (**2.5 GeV to 32.5 GeV**)
- 7.5-pass ‘Dogbone’ RLA: Linac (**4 GeV/pass** ) + 7 droplet Arcs
- SRF: 200-400 MHz
- Top-to-injected energy ratio = 13
**High Energy** **RLA** (**32.5 GeV to 753.5 GeV**)
- 7-pass ‘Dogbone’ RLA: Linac (ILC 4000 m, **103 GeV/pass** ) + 7 droplet Arcs
- Tesla SRF: 1300 MHz, 25 MeV/m
- Top-to-injected energy ratio =23
- No injection at the middle of linac necessary    Harold Kirk

## Phase Slippage in the Linacs 

- Phase Slippage in the Linacs** **

- Phase slippage of a semi-relativistic muon beam injected with the initial energy **E****0** and accelerated by ******E** in a linac of length, **L  - **assuming uniformly spaced RF cavities phased for a speed-of-light particle

- RLA I (2.5 GeV to 32.5 GeV)

- RLA II (32.5 GeV to 753.5 GeV)

## ‘Pulsed’ vs ‘Fixed’ Dogbone RLA (8-pass) 

- Quad pulse would assume 500 Hz cycle ramp with the top pole field of 1 Tesla.
- Equivalent to: maximum quad gradient of Gmax =2 kGauss/cm (5 cm bore radius) ramped over  = 10-3 sec from the initial gradient of G0 =0.1 kGauss/cm (required by 900 phase advance/cell FODO structure at 3 GeV). G8 =13 G0 = 1.3 kGauss/cm
- These parameters  are based on similar applications for ramping corrector magnets such as the new ones for the Fermilab Booster Synchrotron that have 1 kHz capability

## ‘Pulsed’ vs ‘Fixed’ Dogbone RLA (8-pass) 

- For simplicity, we consider a linear ramp according to the following formula:
- A single bunch traveling with a speed of light along the Linac with quads ramped as above   ‘sees’ the following quad gradient passing through i-th cell along the Linac (i = 1,...20)
- 	where                      is the cell length and i defines the bunch position along the Linac.
- For multiple passes through the Linac (the index n defines the pass number) the above formula can be generalized as follows:
- 	where         is the full Linac length and         is the length of the lowest energy droplet arc. Here we also assume that the energy gain per linac is much larger than the injection energy.

## ‘Pulsed’ vs ‘Fixed’ Dogbone RLA (8-pass) 

- 1-pass, 3-7 GeV

- ‘Pulsed’ vs ‘Fixed’ Dogbone RLA (8-pass)** **

- Pulsed

- Fixed

## ‘Pulsed’ vs ‘Fixed’ Dogbone RLA (8-pass) 

- 8-pass, 28-32 GeV

- ‘Pulsed’ vs ‘Fixed’ Dogbone RLA (8-pass)** **

- Pulsed

- Fixed

- phase adv. diminishes down to 1800

## ‘Pulsed’ vs ‘Fixed’ Dogbone RLA (8-pass) 

- 12-pass, 47-51 GeV

- ‘Pulsed’ vs ‘Fixed’ Dogbone RLA (8-pass)** **

- Pulsed

- Fixed

- no phase adv. across the linac- beam envelopes not confined

- phase adv. diminishes down to 1800

## ‘Pulsed’ Dogbone ILC - example  

- number of passes

## Mirror-symmetric ‘Droplet’ Arc  –  Optics  

- 16 cells

- 2 cells

- 2 cells

- (out = in ,and out = -in , matched to the linacs)

## Conclusions

**‘****Dogbone’** RLA – preferred configuration
- better orbit separation for higher passes
- offers symmetric solution for simultaneous acceleration of + and -
**FODO** lattice (900 deg. phase adv/cell) allows to accommodate large number of passes  in the linac (8-pass RLA example)
- uniform phase advance decrease in both planes
- smaller variation of Twiss function – easier match to the Arcs
**Pulsed linac Optics **Dogbone RLA looks very encouraging...needs further studies...
- Increase from 8-pass (Fixed Optics) to 12-pass (Pulsed Optics) for 500 m long 4 GeV pass example
**Muon Collider Acceleration **Scheme with two Dogbone RLAs
**RLA I:****	****2.5-32 GeV**
**RLA II:****	****32-753 GeV**