1. Annular Electron Cooling
Y. Derbenev
JLEIC R&D meeting
CASA JLab, April 28, 2016

2. Outline Problem addressed Friction force and cooling decrements
Longitudinal sweep cooling
Fast longitudinal sweep cooling with Annular Beam
Dispersive 6D Sweep Annular Cooling
Making AB
Counter ERAB
Summary

3. Problem or situations addressed
Cooling ion beams of a large transverse temperature
𝑇 ⊥ ≫ 𝑇 ∥
i.e. : 𝛾 𝜃 𝑏 ≫ ∆𝛾 𝛾 , 𝑜𝑟 𝛾 𝜀 ⊥ 𝛽 ≫ ∆𝛾 𝛾 , 𝑜𝑟 𝜎 ⊥ ≫𝛽 ∆𝛾 𝛾 2
Cases in collider ring
Large injected 𝑇 ⊥ due to large 𝜀 ⊥ (Sp.Ch. limitation in SB)
𝑇 ⊥ adiabatic growth at acceleration
Large 𝑇 ⊥ / 𝑇 ∥ at top energy due to IBS

4. Beam transport/cooling agenda
135 MeV (p), 40 MeV/n (Pb) SC linac
4 ( 𝑍 𝐴 )GeV/c/n Small Booster with DC cooler (warm racetrack ring below the Transition Energy). Stripping injection for H- and D- ; stacking by cooling for the rest species
Apply DC longitudinal cooling after stacking in SB
ECR used as 15 GeV/n Large Booster, all species below Tr.En.
GeV(protons) ICR (SF or Cos magnets) (above the Transit. En. for all species ; or, alternatively (if feasible), with imaginary Tr. En.)
Beam coasting at injection energy or somewhat above it
6D cooling while bunching by the HF SC cavities
Final acceleration and cooling
Continuous cooling to maintain luminosity
Note: 1) Stay below or no-crossing Tr.En. as in the presented scenario is highly desirable, especially in ICR; 2) Having imaginary Tr.En. in ICR would be even better – since the negative longitudinal mass instability is eliminated; this allows one to reach a lower initial energy spread in ICR – that is critical in reducing the initial cooling time in ICR. Then we have to check only with the external impedances of a coasting beam in ICR.

5. Where cooling starts: - DC longitudinal Cooling in SB -
Stacking in SB is limited by the transverse Sp.Ch.
Transverse Cooling does not help against Sp.Ch.
But longitudinal cooling could be applied to decrease energy spread
Small energy spread may help one in beam cooling after injection into ICR

6. DC longitudinal El.Cooling in SB
Transverse emittance is limited by the Sp.Ch.
Max. current grows with tr. emittance
Max.tr. emittance is limited by the DA (or mechanical aperture)
Cooling does not help against Sp.Ch.
But cooling could be used to decrease energy spread
Small energy spread may help one in beam cooling after acceleration

7. Cooling rates
Cooling rates Drag (friction) force 𝐹 𝑝 , 𝑟 causes change of particle energy and dynamic invariants (in terms of beam co-moving frame):
< 𝐼 𝛼 > = < 𝜕 𝐼 𝛼 𝜕 𝑝 𝐹 𝑝 , 𝑟 > ;
(averaging over phases)
< 𝑣 ∥ > ∝ < 𝐹 ∥ 𝑝 ,𝑥 > ; < 𝐼 𝑠𝑦𝑛 > ∝ < 𝐹 ∥ 𝑣 ∥ > (𝑠𝑦𝑛)
𝑥= 𝑥 𝑏 + 𝐷 𝑥 𝑣 ∥ ; 𝑥 𝑏 = 𝑎 𝑥 cos Ψ 𝑥 ; 𝑦= 𝑦 𝑏 + 𝐷 𝑦 𝑣 ∥ ; 𝑦 𝑏 = 𝑎 𝑦 cos Ψ 𝑦
𝐼 𝑥 =𝛽( 𝑣 𝑥 𝛾 ) 2 + (𝑥− 𝐷 𝑥 𝑣 ∥ ) 2 𝛽 ; 𝐼 𝑦 =𝛽( 𝑣 𝑦 𝛾 ) 2 + (𝑦− 𝐷 𝑦 𝑣 ∥ ) 2 𝛽 ;
< 𝐼 𝑥 > = 𝛾 −2 < 𝛽𝐹 𝑥 ( 𝑝 ,𝑥)𝑣 𝑥 > − < 𝐷 𝑥 𝛽 𝐹 ∥ 𝑝 ,𝑥 𝑥 𝑏 > ;
< 𝐼 𝑦 > = 𝛾 −2 < 𝛽𝐹 𝑦 ( 𝑝 ,𝑥)𝑣 𝑦 > − < 𝐷 𝑦 𝛽 𝐹 ∥ 𝑝 ,𝑥 𝑦 𝑏 > ;

8. Non-dispersive cooling at 𝑻 ∥ << 𝑻 ⟘
𝑣 ∥ ∝ < 𝐹 ∥ > ∝ −< 3 ( 𝑣 ⊥ 𝑢 𝑑 ) 2 ln 𝜌 𝑚𝑎𝑥 𝑟 𝑐 𝑢 ∥ 𝑢 𝑑 3 >
𝑢 𝑑 3 =[( 𝑣 𝑥 ) 2 +( 𝑣 𝑦 ) 2 +( 𝑢 ∥ ) 2 ] 3/2
𝑣 𝑥 = 𝑣 𝑥0 𝑠𝑖𝑛 Ψ 𝑥 ; 𝑣 𝑦 = 𝑣 𝑦0 𝑠𝑖𝑛 Ψ 𝑦 ;
Dominating contribution over betatron phases belong to a small interval:
Δ Ψ 𝑥,𝑦 ≤ 𝑢 ∥ 𝑣 𝑥,𝑦0 ≪1
< 𝐹 ∥ > ∝ 1 𝑣 𝑥0 𝑣 𝑦0
Dominating contribution over betatron phases belong to a small interval
< 𝑑 𝑑𝑡 ( 𝑣 𝑥 ) 2 > ∝ < 𝐹 𝑥 𝑣 𝑥 > ∝ − < 𝑛 𝑒 ( 𝑣 ⊥ ) 2 −2( 𝑢 ∥ ) 𝑢 𝑑 5 ( 𝑣 𝑥 ) 2 𝑙𝑜𝑔 𝜌 𝑚𝑎𝑥 𝑟 𝑐 >

9. Dispersive rates 𝐷 𝑥 𝑣 ∥ ≪ 𝑥 𝑏
𝐷 𝑥 𝑣 ∥ ≪ 𝑥 𝑏
𝐹 ∥ 𝑝 ,𝑥 = 𝐹 ∥ 𝑝 , 𝑥 𝑏 + 𝐷 𝑥 𝑣 ∥ 𝜕 𝜕 𝑥 𝑏 𝐹 ∥ 𝑝 , 𝑥 𝑏 , 𝑦 𝑏 + 𝐷 𝑦 𝑣 ∥ 𝜕 𝜕 𝑦 𝑏 𝐹 ∥ 𝑝 , 𝑥 𝑏 , 𝑦 𝑏
( 𝐼 𝑥 ) 𝐷 =− < 𝐷 𝑥 𝛽 𝐹 ∥ 𝑝 , 𝑥 𝑏 , 𝑦 𝑏 𝑥 𝑏 > ; ( 𝐼 𝑦 ) 𝐷 =− < 𝐷 𝑦 𝛽 𝐹 ∥ 𝑝 , 𝑥 𝑏 , 𝑦 𝑏 𝑦 𝑏 >
𝑣 ∥ ∝ < 𝐷 𝑥 𝜕 𝜕 𝑥 𝑏 + 𝐷 𝑦 𝜕 𝜕 𝑦 𝑏 )𝐹 ∥ 𝑝 , 𝑥 𝑏 , 𝑦 𝑏 > 𝑣 ∥
𝐼 𝑠𝑦𝑛 ∝ < 𝑣 ∥ ∥ 𝑣 ∥ > (𝑠𝑦𝑛)
𝐹 ∥ 𝑥 𝑏 = 𝐹 + 𝑥 𝑏 + 𝐹 − 𝑥 𝑏
𝐹 + 𝑥 𝑏 = 𝐹 + −𝑥 𝑏 ; 𝐹 − 𝑥 𝑏 =− 𝐹 − −𝑥 𝑏
( 𝐼 𝑥 ) 𝐷 = − < 𝐷 𝑥 𝛽 𝐹 − 𝑥 𝑏 𝑥 𝑏 >
( 𝑣 ∥ ) 𝐷 ∝ + < 𝐷 𝑥 𝜕 𝜕 𝑥 𝑏 𝐹 − 𝑝 , 𝑥 𝑏 > 𝑣 ∥ ⟶ ( Λ ∥ ) 𝐷 = −< 𝐷 𝑥 𝜕 𝜕 𝑥 𝑏 𝐹 − 𝑝 , 𝑥 𝑏 >
( 𝐼 𝑥 ) 𝐷 = − 𝐼 𝑥 < 𝐷 𝑥 𝑎 𝑥 𝐹 − ( 𝑥 𝑏 )cos Ψ 𝑥 > ⟶ ( Λ 𝑥 ) 𝐷 = < 𝐷 𝑥 𝑎 𝑥 𝐹 − ( 𝑥 𝑏 )cos Ψ 𝑥 >

10. Dispersive cooling at 𝑻 ∥ << 𝑻 ⟘
< 𝐼 𝑥 > ∝ − 𝛾 −2 < 𝑓 𝑒 𝑑 3 𝑣 𝑒 𝛽 ( 𝑣 ⊥ ) 2 −2( 𝑢 ∥ ) 2 𝑢 𝑑 5 ( 𝑣 𝑥 ) 2 ln 𝜌 𝑚𝑎𝑥 𝑟 𝑐 >+ + < 𝐷 𝛽 𝑥 𝑏 𝑓 𝑒 𝑑 3 𝑣 𝑒 [3 ( 𝑣 ⊥ 𝑢 𝑑 ) 2 ln 𝜌 𝑚𝑎𝑥 𝑟 𝑐 +1] 𝑢 ∥ 𝑢 𝑑 3 >

11. Non-dispersive Annular Cooling
< 𝑣 ∥ > ∝ < 𝐹 ∥ > ∝ −3< 𝑛 𝑒 ( 𝑣 ⊥ ) 𝑢 𝑑 𝑢 ∥ 𝑙𝑜𝑔 𝜌 𝑚𝑎𝑥 𝑟 𝑐 >
< 𝑑 𝑑𝑡 ( 𝑣 𝑥 ) 2 > ∝ < 𝐹 𝑥 𝑣 𝑥 > ∝ − < 𝑛 𝑒 ( 𝑣 ⊥ ) 2 −2( 𝑢 ∥ ) 𝑢 𝑑 5 ( 𝑣 𝑥 ) 2 𝑙𝑜𝑔 𝜌 𝑚𝑎𝑥 𝑟 𝑐 >
Gain in cooling rate: ~ 𝑎 𝑤 ;

12. Dispersive Annular Cooling
𝐹 ∥ =𝑛(𝑥,𝑦) 𝑓 ∥ ( 𝑣 ∥ , 𝑣 ⊥ )⟶ 𝑛 0 (𝑥− ∆ 𝑥 , 𝑦− ∆ 𝑦 ) 𝑓 0 ( 𝑣 ∥ − 𝑣 0 , 𝑣 ⊥ )
𝑛 0 𝑥−∆ ≈ 𝑛 0 𝑥 − ∆ 𝑥 𝜕 𝑛 0 𝜕𝑥 − ∆ 𝑦 𝜕 𝑛 0 𝜕𝑦 ;
𝑓 0 𝑣 ∥ − 𝑣 0 , 𝑣 ⊥ ≈ 𝑓 0 𝑣 0 , 𝑣 ⊥ − 𝑣 ∥ 𝜕 𝑓 0 𝜕 𝑣 ∥
( 𝑣 ∥ ) 𝐷 ∝ + < 𝐷 𝑥 𝜕 𝜕 𝑥 𝑏 𝐹 − 𝑝 , 𝑥 𝑏 > 𝑣 ∥ = −𝐷 𝑥 ∆ 𝑥 < 𝑓 0 𝜕 𝜕𝑥 𝜕 𝑛 0 𝜕𝑥 > 𝑣 ∥
( 𝐼 𝑥 ) 𝐷 =−< 𝐷 𝑥 𝑎 𝑥 𝐹 − ( 𝑥 𝑏 )cos Ψ 𝑥 > 𝐼 𝑥 ⟶ 𝐷 𝑥 𝑎 𝑥 ∆ 𝑥 < 𝑓 0 𝜕 𝑛 0 𝜕𝑥 cos Ψ 𝑥 > 𝐼 𝑥
Gain in cooling rate: ~ 𝐷 𝑤

13. Longitudinal drag force of EC
||

14. Dispersive Annular Coolingw y  x
n(x)

15. Making AB

16. Counter AB ERL

17. Summary
It sounds promising - to be explored carefully.