D. Rubin
February 7, 2005.

Energy Dependence of Solenoid Compensation and Tune Shift Limit

Introduction

We find that the depressed tuneshift limit that we compute with the beambeam simulation for the low energy (CESR-c) optics is a result of the strong dependence of solenoid compensation on energy. The energy dependence of the coupling compensation is characteristic of the Phase III (superconduting/ permanent magnet hybrid) design and is not specific to the cesr-c incarnation. The simulation indicates that the tune shift parameter would be increased by more than 50% in a solenoid off, compensation off configuration.

Beam beam simulation

The beambeam simulation is consistent with the best specific luminosity we have achieved at 1.89GeV/beam. The calculations and measurements are based on the lattice hibetainj_20040628_v01.lat. There are two sets of luminosity measurements in the plot. The 8X4 data (d265) was collected on September 22, 2004 and the 8X5 data (d303) on October 30, 2004. The data sample represents the highest specific luminosity that we have achieved in 8X4 and 8X5 bunch configurations respectively. The simulation also reproduces luminosity measurements at 5.3GeV/beam in the Phase II lattice bmad_l9a18a000._moverec. The high energy data was collected on March 3, 2001 and is the highest specific luminosity achieved in those conditions. There are no free parameters in the simulation. Radiation damping and excitation are based on tracking through the individual elements. Beam sizes are that equilibrium of beambeam interaction, radiation, nonlinearities etc.

The beambeam tune shift parameter, as determined from the luminosity data for the 1.89 and 5.3GeV conditions are shown for the measurements and the simulation . The beambeam interaction at low energy has a tune shift limit that is only 1/2 of its high energy counterpart. (The beambeam tune shift scales inversely with beam energy.)

The simulation has been used to explore the dependence of limiting tune shift on wiggler nonlinearities, pretzel, and parasitic beambeam effects. No significant dependence is evident. The red points are for the 1.89GeV conditions with 9 trains of 4 bunches. The green and blue are computed for a lattice with linearized wiggler fields with 9X4, and for pretzel off, (single bunch) respectively.

Energy Dependence of Solenoid Compensation

The energy dependence of the compensation is characterized by the chromaticity of the coupling parameters at the IP, by the energy dependence of the dynamic aperture, and most dramatically by the equilibrium vertical phase space at the IP. We compare 4 sets of optics:

(A) 1.89 GeV, hibetainj_20040628_v01.lat
(B) 5.3 GeV, bmad_l9a18a000._moverec (Phase II IR)
(C) 5.3 GeV, bmad_5620mev.lat (Phase III IR)
(D) 1.89 GeV, hibetainj_nosol_v01.lat (solenoid off, quads level)

Coupling Parameters at the Interaction Point

The energy dependence of the IP coupling parameters is shown in the table. We define the coupling chromaticity as d(Cbar)/d(delta_E). We compute Cbar for two different energies. The difference yields d(Cbar)/d(delta_E). There is some small contribution to coupling chromaticity from the vertical separation bump in L3. The tabulated results are computed with vertical separators off.

Optics           A(1.89)    B(phase II)     C(5.3 Phase III)   D(solenoid off)
d(cbar11)/dE      -1.06      -0.75            -0.09             0.0        
d(cbar12)/dE       2.52       0.99             7.40             0.0
d(cbar21)/dE     -16.77       8.95           -27.57             0.0
d(cbar22)/dE       0.32      -0.34             0.21             0.0

The cbar12 matrix element gives the fraction of horizontal emittance that appears in the vertical plane. Energy dependence of coupling is least in the Phase II (B) optics, and greatest in the Phase III high energy optics (C). The cesr-c (A) optics falls in between.

Dynamic Aperture

The dynamic aperture is computed for optics A, B and D . Optics A is the standard 1.89GeV optics and D is the version modified to have the solenoid off. B is the Phase II configuration. Note that the vertical aperture for off energy particles is degraded in optics A. The physical aperture (20 turn aperture) is indicated by the dashed colored lines, and the dynamic aperture (1000 turns) by the solid lines. Attempts to eliminate this energy dependence by carefully tayloring the sextupole distribution have been marginally successful. While it is possible to eliminate the sextupole driven synchro-betatron coupling, the degradation of vertical dynamic aperture with energy is typical of all cesr-c optics. There is plenty of vertical aperture in optics B, (Phase II), even for the relatively large energy offset of 0.008. (Note the different scale). Finally, in the solenoid off optics (D), we see that the dynamic aperture is essentially the same as the physical aperture. We conclude that the marginal vertical dynamic aperture in the cesr-c optics is a result of the energy dependence of the solenoid compensation.

Phase Space

The phase space is computed by tracking a single particle for several damping times (>100,000 turns) through a machine model that includes radiation damping and excitation. The vertical phase space coordinates for the last 1000 turns are plotted for all four sets of optics, A , B , C , and D . The abscissa and ordinate of the plots correspond to vertical displacement and angle respectively. The units are meters and radians. The approximate widths of the distributions as read off the plots is summarized in the table.

Optics                       Delta y (microns) 
 A (1.89 GeV)                      16
 B (Phase II)                       4
 C (Phase III 5620MeV)             14
 D (1.89 GeV, solenoid off)        4.5

We see that the span of the vertical phase space for an equilibrium particle extends to 16 microns in the current CESR-c optics as compared to only 4 microns in the 5.3GeV Phase II configuration. And beta* is indeed smaller in the cesr-c optics at 12mm vs 18mm. The high energy phase III optics (C) also yield a broad distribution, whereas, the low energy solenoid off case (D) is narrow. Evidently, the source of the enlarged vertical beam size is the Phase III IR optics, quite independent of the energy. The vertical enlargement of the beam is presumably due to the energy dependence of the compensation.

We also examine phase space without excitation and damping for direct evidence of the energy dependence of coupling. With all separators off, and synchrotron tune Qz=-0.089, track for 2000 turns, beginning with a 2mm horizontal displacement at detector 8W, (outside the compensation region) and map phase space at the IP. We see that the horizontal amplitude at the IP is 0.5mm, and vertical is 65nm. The energy offset is 1e-5. Then we again track for 2000 turns with initial horizontal displacement of 2mm and energy offset of 8.4e-4 (sig_e/E ~ 8.4e-4). The plot of the phase space at the IP shows beam width reduced by half and the beam height increased to nearly 3microns. For comparison, on and then off energy phase space are shown for the solenoid off optics. There is no increase in beam size with energy.

Beam beam Limit in Solenoid Off Optics

Finally we simulate luminosity in the solenoid off gif , ( ps ) low energy optics (D). The specific luminosity in the solenoid off optics is about 50% greater than in the standard cesr-c optics with 1T solenoid. The solenoid off optics is derived from the hibetainj_20040628_v01.lat lattice. The solenoid and the tilts of the IR quads are set to zero. Quads south of 15 are varied to match the solenoid off IR into the arcs. The sextupole distribution is designed to optimize dynamic aperture in the new lattice. The simulation is for nine trains of 4 bunches. The synchrotron and betatron tunes match the tunes used to compute luminosity in the solenoid on conditions. Having eliminated the energy dependent coupling, the optimal tunes may be different.

Conclusion

The phase III solenoid compensation is the source of significant energy dependent coupling at the IP. Because of the greater energy spread in cesr-c conditions as compared to 5.3GeV configurations, our sensitivity to the chromatic coupling is enhanced at 1.89 GeV. But recall that we never achieved high beam beam tune shift parameter with the Phase III IR at any energy. At 1.89 GeV the results of the coupling are an increase in the vertical beam size at the IP, and a reduction of the beambeam tune shift limit by at least 2/3. In addition there is a substantial reduction of the off energy nonlinear aperture.