Spin pattern for run 15

For the double helicity asymmetries, a set of RHIC fill patterns is used to minimize systematic effects. It was suggested to add a new feature to the fill pattern, namely that the pattern of bunches immediately following to an abort gap is not the same in STAR. (The abort gaps of the two beams are aligned in PHENIX, so this is a feature for STAR only.)

The following requirements were implemented in the existing bunch patterns:
- Balanced number of bunches with same and opposite helicities will reduce the relative luminosity (which is corrected afterwards).
- Polarization directions should alternate in same bunches between fills.
- The PHENIX electronics require a cross check between even and odd bunch crossings, so all combinations have to be present in every fill (balanced is possible).

In the past, four basic patterns were used with alternating polarization directions (or pairs thereof):
(1) +-+--+-+ (2) -+-++-+- (3) ++--++-- (4) --++--++

These were alternated between the two beams, resulting in a total of eight bunch patterns:
P1=B1xY3 P2=B2xY3 P3=B1xY4 P4=B2xY4 P5=B3xY1 P6=B4xY1 P7=B3xY2 P8=B4xY2
STAR collides blue bunch #1 with yellow bunch #81. In order to have a different pattern after the two abort gaps, the eightfold symmetry of the basic pattern has to be changed to twelve bunches. Bunch #5 is now where the flip in the pattern should occur.
Since we also alternate the patterns between the blue and yellow beams, bunch #9 should have the same flip with respect to bunch #1. With a pattern of 12 bunches lenght, it turns out to be very difficult to meet all of the above demands.

We can instead use a basic pattern of length 24 (the patterns are all written by hand for 120 bunches anyway). Then bunches #9 and #17 should be flipped with respect to #1. In this case, we replace the previous patterns by the following:
(1)->(1)(2)(2)  (2)->(2)(1)(1) (3)->(3)(3)(3) (4)->(4)(4)(4)