1. Round beams experience at BINP … and other ideas
Workshop on Accelerator R&D for Ultimate Storage Rings
October 30-November 1, 2012, Huairou, Beijing
Round beams experience at BINP … and other ideas
E.Levichev
Budker Institute of Nuclear Physics
Novosibirsk , Russia

2. Round beams concept (RBC)
The Round Beam Concept was proposed more than 20 years ago for the Phi-Factory Project in Novosibirsk *). The RBC requires:
Equal betas at IP
Equal emittances
Equal betatron tunes and no betatron coupling in the arcs
Small fractional tunes
Requirements 1–3 are satisfied by the use of a strong solenoidal beam focusing in the interaction straight. At each passage, the longitudinal field with specific integral along the straight section rotates the transverse oscillation plane over 90° exchanges roles of the two betatron modes, and thereby provides their full symmetry.
*) V.V. Danilov et al., in Proc of the EPAC 1996, Sitges, vol. 2, p

3. RBC advantages Luminosity increases: 
Tune shift from the opposite beam becomes twice smaller than that for the flat beams
Tune shift becomes independent of the longitudinal position in the bunch thereby weakening action of the synchro-betatron resonances
X-Y symmetry of the betatron transfer matrix between collision points results in additional integral of motion: the longitudinal component of the angular momentum is conserved . The transverse motion at IPs becomes equivalent to a one-dimensional motion. For the beam-beam effects, elimination of all betatron coupling resonances is of crucial importance, since they are believed to cause the beam lifetime degradation and blow-up.

4. VEPP-2000 Overview
VEPP-2000 is the first and only e+e- collider operating in the c.m. energy range GeV with the round colliding beams. Main scientific goals of VEPP-2000 include n-nbar and p-pbar study at the production threshold, hadron cross-section measurement, …

5. VEPP-2000 Results
A BB tune shift for the RBC reaches 0.15 and the luminosity increases that obtained at VEPP-2M for the same energy and current. The BB effects relaxed significantly.
Specific luminosity for VEPP-2000 (RB) vs VEPP-2M (flat beams)
Single beam (lattice) resonances

6. VEPP-2000 Conclusion for the USR
RBC works well eliminating coupling BB resonances, increasing the tune shift parameter and luminosity
Sextupoles and other imperfections break the 1D motion
Small fractional tunes contradict emittance minimization requirements
Lattice is rather tricky and problematic for insertion devices for SR generation
It seems that the round beam (in the sense of VEPP-2000!) is hardly can be applied for storage rings to help in reaching of extremely low emittance.
But…

7. Equal Emittances Concept (EEC)
ICFA Beam Dynamics Newsletter, No.57, April 2012, p.48

8. Dynamic aperture and emittance I
We start with sextupole harmonic Hamiltonian (mirror-symmetry cell is supposed):
with 5 types of harmonics (summation over the sextupoles)
corresponding to the following resonances:
A single resonance approximation seems reasonable

9. Dynamic aperture and emittance II
In a compact low emittance cell, where the phase advance between the chromatic sextupoles is small, the main resonant harmonics
can be estimated in a following, very simple, way (details are in (1) and (2)):
Compact TME cell
where
Main harmonics strength:
Estimation vs simulation ,
is the cell natural chromaticity
At some reasonable assumptions
main resonant harmonics strength
E.Levichev and V.Sajaev, Nonlinear phase space study in a low-emittance light source using harmonic approximation, PA, 1997, Vol.56, pp (DBA)
E.Levichev and V.Kvardakov, Nonlinear characteristics of the TME cell, Proc.EPAC06, Edinburgh (TME)

10. Dynamic aperture and emittance III
From known harmonic strength the dynamic aperture can be estimated (fixed point of the resonance separatrix) as
So the dynamic aperture scaling law is:
Single resonance
dynamic aperture
But simulation shows that this scaling low works not only in the vicinity of strong single resonance but in the other tune points as well!
Emittance and DA as a function of the betatron tunes: estimation vs simulation

11. Dynamic aperture and emittance IV
Scaling for increasing cells number Nc (cell is the same! no focusing changes):
Hor.emittance
“Natural” DA
Sext.strength
ICFA Beam Dynamics Newsletter, No.57, April 2012, p.43
Horizontal emittance defines the DA reduction and strength of chromatic sextupoles

12. IBS and round beam
Based on: D.Golubenko, S.Nikitin “Touschek effect for 2D collisions”, BINP Preprint , Novosibirsk 1999 (in Russian).
Main idea is that 2D consideration modify the transverse momentum (relative collision velocity) distribution function which, in turn, influence the IBS.
Flat beam distribution function
where
2D distribution function
where
Flat
Round
Round
Flat
2D distribution function
Touschek lifetime for different case of beam “roundness”
IBS should be carefully studied for EEC to envisage if there are advantages for diffraction limited emittance storage ring

13. Space charge effect I
Space charge produces effects similar to those observed in beam-beam collisions: betatron tune spread, beam core blow up, life time reduction, etc. Usually at high energy these effects are suppressed but in the case of extremely low emittance they are not negligible.
E.Levichev, P.Piminov, D.Shatilov, Nonlinear beam dynamics with strong damping and space charge in the CLIC damping ring, Proceedings of PAC09, Vancouver, Canada, TH6PFP093,
For the CLIC DR the space charge tune shift ->:
Particles trapped in high order resonances
Sextupoles
Sextupoles + space charge

14. Space charge effect II Italian SuperB Factory with L = 1036 cm-2s-1
The beam footprint at the tune diagram covers wide area containing many resonances (including those produced by strong chromatic sextupoles) which may cause beam emittance blow up, beam life time degradation, etc.
Italian SuperB Factory with
L = 1036 cm-2s-1
USR example: E = 3 GeV I = 500 mA
x = 50 pm y = 0.5 pm z = 4 mm
L = 360 m
dQy = 0.1
dQy = 0.1

15. Low frequency RF and compact lattice cell seems preferable
Space charge effect II
Some estimation for the flat beam:
Low frequency RF and compact lattice cell seems preferable

16. Main ideas of the EEC for the USR
(1) Bare low emittance compact cells lattice produces usual flat beam with the horizontal emittance much larger and the vertical emittance much lower then the diffraction limit.  A dynamic aperture defined by the horizontal emittance is still quite large and the chromatic sextupoles strength is still technically reasonable.
(2) Damping wigglers with horizontal field reduce the horizontal emittance (due to additional emitted radiation power) but increase the vertical one (wigglers parameters are properly chosen). Both resulting emittances are equal to that defined by the diffraction limit.  A dynamic aperture is not affected much, space charge effect is reduced (just like for the round beams collision), IBS and Touschek lifetime (I hope, but it should be checked) is relaxed.
(3) Additional damping works well against all heating processes (IBS, collective instabilities, high order resonances, etc.).
(4) A long wave RF system seems attractive for IBS, space charge and other collective effects.
Note: A linear betatron coupling should be carefully considered in the above scheme but its using for the beam rounding seems problematic because the linear coupling may cause undesirable effect on nonlinear coupling (see, for instance, G. De Ninno and E. Todesco “Effect of linear coupling on nonlinear resonances in betatron motion”, Phys.Rev. E, v.55, No 2, p.2059, February 1997). Also for a full betatron coupling and for Moebius beam coupling one should be careful with strong chromatic sextupoles, which will influence both transverse oscillation modes.

17. Damping wigglers with horizontal field
See details of damping wigglers design in the talk by Konstantin Zolotarev

18. Summary
Round beam mode in the sense of VEPP-2000 (1D motion at IP) seems has no advantage for USR (my personal opinion)
Equal emittances concept to reach a diffraction limit emittance in both planes seems promissing and should be studied in details
Damping wigglers with horizontal magnetic field look attractive for “rounding” the emittances and to suppress additionally any heating of the beam emittances