D. Rubin
August 26, 2005

Minimizing energy dependence of compensation

Introduction

As noted in an earlier document , there is not a unique implementation of an anti-solenoid. Using 4-pair compensation, with integrated anti-solenoid opposite integrated CLEO, we find that if the four pairs are Q00 tilt, sc_sk_q01, sc_sk_q02, and sk_q02, that we get significantly better luminosity than if the four pairs are sc_sk_q01, sc_sk_q02, sk_q02, sk_q04 with Q00 tilt fixed at 4.5. We define a new energy dependent coupling constraint so that an IR can be systematically designed to minimize energy dependence

Constraining energy dependence

Consider the 4X4 transport through the compensation region. For on energy trajectories, the matrix is by design block diagonal. But for off energy trajectories, in general, it is not. A normal mode decomposition yields an energy dependent C-matrix. Normalization of the C-matrix requires knowledge of alpha and beta at the endpoints. In order to identify the twiss parameters we assume that the beta functions at start and end are identical, and that alpha at start and end are equal but opposite. We further define the phase advance to be equal to that computed for the same interval of the closed ring.

Examples

In order to take advantage of the constraint, we need additional degrees of freedom. A possibility is Q00 tilt, sc_sk_q01, sc_sk_q02, anti-solenoid Bz (B_anti), and sk_q02. We list Q00-tilt and B_anti as variables, and force 3-pair compensation with sc_sk_q01, sc_sk_q02, and sk_q02. The result is bmad_q0_q1_q2_comp_sk_080605.lat , in which the energy dependence of the C-bar matrix are significantly reduced. 
   dcbar/dE(insert)        -0.0087      0.0486
                            0.0028      0.0054
In this solution B_anti = 0.95 B_anti (nominal) and Q00 tilt = 5.3deg

For the lattice bmad_q0_c_040305.lat , which comes closest to giving solenoid off luminosity the same calculation gives, and where B_anti = B_anti(nominal), and Q00 tilt = 1.9deg, 

 
dcbar/dE(insert)         1.7704   -107.2610
                         0.1627     -9.5477
The summary of the current dependent of the luminosity for the new lattice bmad_q0_q1_q2_comp_sk2_080605.lat along with other configurations shows that minimizing energy dependence is as good as the 1.9 deg solution and no better.

In another configuration we remove one section of the permanent magnet. Then with anti-solenoid fixed at nominal minimize C-bar energy dependence with Q00 tilt as a degree of freedom and force 3-pair compensation with sc_sk_q01, sc_sk_q02, and sk_q02. yielding bmad_short_q0.lat with Q00 tilt 4.86deg, and 


   dcbar/dE(insert)         0.9904    -14.2632
                            0.1031     -1.4399
The luminosity is essentially the same as for the above configurations.

In another example, we fix the angle of the permanent magnet and a 4-pair constraint with sk_sc_q01, sk_sc_q02, asol, sk_q02 as variables. The lattice is bmad_asol_4pair_v7.lat. The luminosity vs current bb_071105_072805_091405_060305_140605_091505 is not terribly definitive. A lumscan_091705 indicates better results at a different tune than was used for the data plotted. The luminosity vs current with opt tunes from scan gives bb_071105_072805_091405_060305_140605_092205