# Presentation: 704217

## 
Fringe focusing ring cooler for PIC 

V. Balbekov,  MCTF meeting 01/15/09

            FFR has very large aperture and minimal aberrations.
            Is it possible to use it for PIC ? 

_**Fringe focusing ring cooler for PIC**__** **_
- V. Balbekov,  MCTF meeting 01/15/09
-             FFR has very large aperture and minimal aberrations.
-             Is it possible to use it for PIC ?

- V.Balbekov, MCTF 01/15/09

- 01

## V.Balbekov, MCTF 01/15/09

- _Aberrations_

- 2

_**Possibility of using a ring accelerator for ionization cooling of muons**__  __(V.Balbekov, PAC-1999, p.315)_

- _Layout_

- _Emittances_

- _How does it work_

- 02

##         My conclusion was:  It is very hard (impossible) to suppress   
         aberrations of a strong focusing cooling channel (ring or linear)
         as needed for resonance cooling (PIC).

 The publication: Y. Derbenev and R.Johnson,  “Parametric-resonance 
   Ionization Cooling” has renewed interest to the problem       (2005). 

 However, investigation of snake-like channels with quads or 
   solenoids does not bring desirable results (D. Newsham,  R.Sah, 
   A.Bogacz, T.Roberts,  K.Yonehara, and Yu-Chiu Chao  (2007). 

 Properties (aberrations) of Epicycling Helical Solenoid are unclear yet
  (Y. Derbenev,  2008,  A. Afanasev).

-        Therefore I decides to consider weak focusing ring which can be 
          achromatic even without correction (talk at PIC workshop, 2007).   

-          aberrations of a strong focusing cooling channel (ring or linear)
-          as needed for resonance cooling (PIC).
-  The publication: Y. Derbenev and R.Johnson,  “Parametric-resonance
-    Ionization Cooling” has renewed interest to the problem       (2005).
-  However, investigation of snake-like channels with quads or
-    solenoids does not bring desirable results (_D. Newsham_,  _R.Sah_,
-    A.Bogacz, T.Roberts,  K.Yonehara, and Yu-Chiu Chao  (2007).
-  Properties (aberrations) of Epicycling Helical Solenoid are unclear yet
-   (Y. Derbenev,  2008,  A. Afanasev).
- -        Therefore I decides to consider weak focusing ring which can be
-           achromatic even without correction (talk at PIC workshop, 2007).

- V.Balbekov, MCTF 01/15/09

- 03

## V.Balbekov, MCTF 01/15/09

- - Symmetric ring with Bz~1/R**n**  is achromatic system:
- Q**x****2**** **= 1-n,      Q**y****2** = n,      0  < n < 1
-  However, only one of the tunes can be half-integer
-   as is needed for PIC.    E.g. at n = 0.75:
- Q**x** = 0.5,  Q**y**** **= 0.866
- - Therefore,  only one-directed PIC is possible in this case.
** **
**                 **

_**Weak focusing ring cooler with combine function magnet**_

- - Weak focusing ring accelerator with
-    drift spaces solves this part of the
-     problem.     E.g.,  in the left picture
-                    n = 0.5,    Q**x** = Q**y** = 1
- - However, chromaticities are rather large:
-                      ξ**x** = ξ**y** =-0.565
-  - And they are incorretible by sextupoles
-     because  β**x** = β**y**  in this case.
**                **

- 04

## V.Balbekov, MCTF 01/15/09

- - Scaling fringe focusing ring is achromatic owing to full
-    geometric and dynamic similarity of closed orbits at different
-    momentum.   See example of a tetra-ring (1 quadrant).
- - However, horizontal tune is wrong in this case being between  1
-    and ~1.5.  E.g.,
-                      Q**x** = 1.103,    Q**y** = 1     at     α = 46.24**o**.
- - Non-scaling addition to the drift is needed to decrease
-    horizontal tune.
- - The geometry is presented to provide
-  Q**x0** = Q**y0** = 1      at     S = R**0 **(0.780 P/P**0 **- 0.238),  α = 42.68**o**
- - Because of non-scaling addition, chromaticity ≠ 0 now,
-     however, it is 5-6 times less of previous one:
-                            ξ**x** = 0.122,   ξ**y** = 0.087
- - The chromaticity is correctable by sextupoles
-    because      β**x** ≠  β**y**     (see next page).
**                 **

_**Ring cooler with flat, fringe focusing magnet (n=0)**__** **_
** **
**                 **

- 05

## V.Balbekov, MCTF 01/15/09

-    - Left:  schematic and sizes of the cooler. At equilibrium
-       bending radius  1 m, required magnetic field is
-       B(T) = P(MeV/c)/300    (1.27 T  at  P = 380 MeV/c).
-    - Down:  tracking in hard edges approximation.
-       Left:   closed orbits at  ΔP/P**0** = -0.5 (0.1) 0.5
-       Right:  trajectory at amplitude of betatron oscillations
-       ~80 cm,  ΔP/P**0** = 0.5,
**                 **

_**Ring cooler with flat, fringe focusing magnet (tracking)**_
** **
**                 **

** **
**                 **

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## V.Balbekov, MCTF 01/15/09

- - Beta-fulction of of the cooler at R = 1 m
-   (one quadrant, β ~ R).
- - Owing to modulation of β-functions, chromaticity
-   correction and parametric resonance excitation
-   are achievable in both  directions.
- - Suitable places for the correctors are marked.

_**Beta-functions and chromaticity**_
**                 **

- - Thin-lens approximation is used for sextupoles:
-                        **∫ **B**y**ds = S(x**2**-y**2**),
-    SA = 0.085 T/m,  SB =-0.129 T/m,  P=380 MeV/c
- - Tune addition is plotted vs  ΔP/P**0**.
- - Tune w/o correction are plotted for comparison
-     (scale 1:10)
- - Tune spread is suppressed by factor about 10.
- - More suppression requires higher nonlinearities

- 07

## V.Balbekov, MCTF 01/15/09

- - In this example,  gradients for the parametric resonance are:
**               ∫ **Gds  =          A:   0.052 T,       B: -0.055T
- - Drift space is decreased on 3 mm to compensate tune perturbation
- - Below:   red – horizontal phase space,  blue - vertical
- -  Observation points (absorbers) marked

_**Parametric  resonance**_

- ∆P/P =-0.1
**                 **

- ∆P/P = 0.1
**                 **

- ∆P/P = 0
**                 **

- 08

## V.Balbekov, MCTF 01/15/09

- - Transformation of horizontal phase ellipse due to parametric
-    resonance and cooling is pesented at different initial
-    conditions and  ∆P/P.  15 periods between the scans.
- - There are significant spherical aberrations at P**trans**/mc >~0.3
-    Needed correction ?
**                 **

_**Parametric  resonance and absorbers**_
**                 **

- ∆P/P =-0.1
**                 **

- ∆P/P = 0.1
**                 **

- ∆P/P = 0
**                 **

- 09

## V.Balbekov, MCTF 01/15/09

- - The cooler with absorbers 2 x 8.9 MeV is considered
- - Stochastic effects neglected
- - Incorporated acceleration
- - Horizontal (top) and vertical (bottom) phase space are
-   shown at injection (blue), after 15 turns (red),
-   and after 50 turns (dark).
- - Evolution of the emittances is plotted below.
-   They are restricted by spherical aberrations.
- - Dashed lines – the same at absorbers 13.3 MeV
-   and proportionally stronger quads. Not only cooling time,
-   but achievable emittances are less.

_**Cooling without stochastic effects**_
** **
**                 **

- 10

## V.Balbekov, MCTF 01/15/09

- - The same as above cooler is considered, but the beam
- -  Has 2 times more initial angle spread
- - The spread about conserves at the cooling and produces
-    more spherical aberrations.
- - As a result, minimal emittance about 10 times more then
-    in previous case.
- - Octupole correction is needed ?

_**Simulation at bigger angle spread**_
** **
**                 **

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## Conclusion 

 Fringe Focusing Ring cooler is easy adjusted for PIC at Q = 1. 
 Its basic chromaticity is rather small and easy correctible. 
 Next step should be: simulation with realistic absorbers, 
  investigation of spherical aberrations and their suppression 
  (if needed)
 Also important: momentum optimization, because the FFR has 
  large dispersion resulting large contribution of straggling to 
  betatron oscillations. 
 Further: simulation with realistic accelerating system and   
  magnetic field
- Injection and extraction problems – next page 

-  Fringe Focusing Ring cooler is easy adjusted for PIC at Q = 1.
-  Its basic chromaticity is rather small and easy correctible.
-  Next step should be: simulation with realistic absorbers,
-   investigation of spherical aberrations and their suppression
-   (if needed)
-  Also important: momentum optimization, because the FFR has
-   large dispersion resulting large contribution of straggling to
-   betatron oscillations.
-  Further: simulation with realistic accelerating system and
-   magnetic field
- - Injection and extraction problems – next page

- V.Balbekov, MCTF 01/15/09

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## V.Balbekov, MCTF 01/15/09

_**Addition:  injection-extraction problems**_
**Inverse Cyclotron**
- _D. Summers_,  S. Bracker,  L. Cremaldy,  R. Godarg,  R. Palmer  (2006)
- _T. Hart_, K. Paul, K. Beard (2008)
** **
**                 **

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