# Presentation: 268657

**                    **
**              ****PIC,REMEX and Final Focus design**

-   Ya. Derbenev

-     **Jefferson Lab**

**      ****Muon Collider Design Workshop**
**                   ****BNL, Dec.3-7, 2007**

## PIC principles

**      ****Resonance dynamics****: amplitudes grow, while phases shrink**
*                             *

**      **
**      ****As result, the beam is focused at absorber plates**
**      ****Absorber effect:**** **** angle  spread at absorber**
**      ****relaxes to the conventional equilibrium**
*:*

**      ****Scattering impact on phases**** is limited by the plates thickness:**
*               *

*** ******Beam size at absorber:***

## Emittance exchange at PIC

**Dispersion and wedges**** are needed in order to maintain **
**    ****the momentum spread at minimum (2.5%)**
**Dispersion ****should be small**** enough at absorbers in order to prevent *****straggling impact***** on horizontal emittance, ****but large in between**** as required for compensating the *****chromatic detunes***
**These constraints are resolved by**** ****snake channel design**** ****and alleviated**** ****by a**** ****small beam size at absorbers**** ****(reduced by 6D cooling and PIC!)**
**Optimal cooling rate partitioning between two planes**
**    ****requires**** beam**** ****rotation or x-y coupling**

- Absorber wedges

- Betatron Function

- Dispersion Function

## Snake channel for PIC

**Snake period**

**Symmetry relatively absorber:****  **

**;**

-     anti-symmetric **;**

- symmetric

**A)**

**B)**

**Normal free oscillations:**
**General free motion:**

- ;

**Tuning options in y-plane:**

**Tuning in x-plane:**

**C)**

## Snake channel on combined bent magnets

**Chose option C:**

**The linear structure parametric resonances avoided!**

**Continuous bend angle: (single dipole)**

**“****Betatron” oscillation periods:**

- The driving parametric resonance can be arranged
- in two planes by modulation of      and

** ****Parametric resonance is introduced in order to damping the symmetric mode to an equilibrium  which is determined by the thickness of absorbers,**** **

**Notes:**
the absobers

## PIC at exact resonance

**Balance equations:**

## PIC tuning  

**Necessary tuning requirement:                                         **
**                             **
**Tuning required for a full PIC effect: **
**Tolerances to linear optics errors:**
**RF voltage/beam energy control?**
**                                 **
**Chromatic and spherical aberrations are known and can be compensated, in principle. The compensation precision requirement is: **
**Coulomb detuning:**** **

## The 3-d order Hamiltonian

**Formulation in terms of a plane reference orbit**
**The third order terms in Hamiltonian result from chromaticity...**
** ****but also from bend:**

**For compensation endeavor, add a ****sextupole field**

**We thus have to fight quite a nasty enemy...**

**Hamiltonian:**

## Compensation for quadratic forces

** **
**The compensation problem is simplified drastically for a symmetric  lattice***** ***
**The compensation conditions are reduced to only two equations:**

**At first, let us formulate the compensation for an asymptotic regime**

**(****e.g. minimum conditions required for approaching the desirable equilibrium)**

**Characteristic behavior of transverse functions:**

**The compensating sextupole field should have the following harmonics:**

## The 4th order Hamiltonian

**The initial 4****th**** order terms:**

**It looks as a dark forest... but even this is not a complete 4th order story.**

- More terms arrive as secondary effects from the 3d order Hamiltonian:

**(Poissons bracket with**

**   ****)**

## Compensation for 5 and 6th order terms

**                          ****Is that really possible?**
**Some idea to simplify the task:    **
**                    ****Introduce coupling resonance,**
**          ****in order to reduce amount of  conditions **

## Use of coupling resonance

**Coupling resonance could replace *****beam rotation***
**   ****as a way to equalize the cooling (and resonance dynamic) rates. Method: ****introduce skew-quads****:**

**Being strong enough compared to detunes, this resonance conserves the sum of actions, only causing the complete x-y exchange (beating)**
**It is critically important to design, namely, two betatron periods ****different by factor of 2 ****: at this condition, the coupling causes beam rotation in space at absorber, ****but not in phase space**** – that agrees with PIC in two planes simultaneously!**

**Beam rotation by coupling resonance also can be used for  reduction of the compensation conditions for high order aberrations, while approaching equilibrium at the end of PIC **

## An approach to compensation control(support instruction to simulation)

**Rely on the Hamilton’s expansion analysis and basic analytical predictions/expectations for dynamics, cooling and equilibrium emittances**
**Introduce necessary compensation knobs, i.e. multipole fields parameters, starting with those related to the lowest order aberrations**
**Observe beam sizes (emittances) at absorber while changing the individual knobs one by one **
**Fix the magnitude of every knob at the moment of minimum emittances**
**Accomplish first compensation for the lowest orders (before introduction of higher orders), then go for the higher orders**
**Note: high order aberrations become important to the end of cooling channel, while approaching the equilibrium**

## PIC scenario

**1. After 6D cooling to a minimum equilibrium, ****accelerate the beam to 300-400 MeV. ****Acceleration will result in**** reduction of spherical aberrations ****(due to decrease of particle angles  **
**2. Start PIC in *****tapered***** *****isochronous****** *****snake channel with ****no RF**** but at ****fast energy loss**** (thick wedge absorbers).  ). Parametric resonance can be strong correspondent to fast energy loss (cooling decrement should exceed detune).  The coupling resonance can be introduced for reduction of compensation conditions.**
**   ****After that momentum drops to about 100 MeV/c, this  stage results in substantial beam focusing at absorbers (hence, cooling)**
**3. After this stage, PIC can be continued in the regular snake channel with RF, thin tapered absorbers, coupling resonance and compensation for aberrations (all order terms from 3 to 6)**

## Compensation for 1st order aberrations of FF

**There are 5 conditions to satisfy in general case:**

**With symmetric CB, the three last conditions are satisfied automatically**

## Second order effects from sexstupoles

**An estimation:**