Energy Scaling of Spin Tune due to RHIC Snakes

For a ring like RHIC with two full Siberian snakes on opposite sides of the ring, the spin tune for a flat orbit will be 1/2 if the snake rotation axes are perpendicular, {Delta}{phi} = {phi}{sub 9}-{phi}{sub 3} = {pi}/2. Here {phi}{sup 9} and {phi}{sub 3} are respectively the direction of the rotation axes of the 9 o'clock and 3 o'clock snakes relative to the design trajectory as shown in Figure 1. If the two snakes are slightly detuned by the same amount such that the rotation axes are no longer perpendicular, then the deviation of the closed-orbit spin tune {nu}{sub 0} from 1/2 is given by {Delta}{nu}{sub 0} {approx_equal} ({Delta}{mu}){sup 2}/4{pi} cosG{gamma}{pi} - 2{Delta}{phi}/{pi} {approx_equal} 2{Delta}{phi}/180{sup o} with G{gamma} at a half integer, and where {Delta}{mu} is the deviation of snake rotation angle from 180{sup o}. It should be noted that there is a sign ambiguity in {Delta}{mu}{sub 0} since a spin tune of 0.495 is also a spin tune of 0.505, depending on the direction taken along the stable spin axis. In order to understand the effect of energy scaling on the snake axis direction, I have integrated the trajectory and spin rotation through a model of a RHIC snake (bi9-snk7) and found the energy (U) dependence of the snake axis angle {phi}{sub 9} and rotation angle {mu} as shown. A {approx_equal} p{sup -2} scaling of errors is typical in helical snakes. To first order, the orbit excursion drops as p{sup -1} and the spin precessions about transverse fields increase as {gamma} giving an approximate cancellation with energy, so we do not expect much change during the field ramp. The next order term which comes in is primarily proportional to p{sup -2}; although naively one might expect a slight effect inversely proportional to the velocity since {gamma}/p {proportional_to} c/{nu} {approx_equal} 1 + 1/2{gamma}{sup 2}