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High-energy polarized proton beams : a modern view by Georg Heinz Hoffstaetter

By Georg Heinz Hoffstaetter

"This monograph starts off with a evaluate of the fundamental equations of spin movement in particle accelerators. It then reports how polarized protons will be sped up to numerous tens of GeV, utilizing as examples the preaccelerators of HERA, a 6.3 km lengthy cyclic accelerator at DESY/Hamburg. numerous equipment for computing the invariant spin box, the adiabatic spin invariant, and the amplitude-dependent spin track are presented.  Read more...

1 advent; References; 2 Spin Dynamics; 2.1 TheEquationofSpinMotion; 2.2 Spin movement in round Accelerators; References; three First-Order Spin movement; 3.1 LinearizedSpin-OrbitMotion; 3.2 First-OrderResonanceSpectrum; 3.3 OptimalChoicesofSiberianSnakes; References; four Higher-Order Spin movement; 4.1 Higher-OrderResonancesandSnakeSchemes; 4.2 acquiring n(z)byStroboscopicAveraging; 4.3 acquiring n(z)byAnti-damping; References; five Polarized Beams in different Very-High-Energy Proton Accelerators; 5.1 The Relativistic Heavy Ion Collider (RHIC); 5.2 TEVATRON; 5.3 LHC; References; Index.

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1 . For N Siberian Snakes in a ring with spin rotations otherwise only around the vertical, the following three conditions are required: • ∆Ψ = 0, to make ν0 independent of energy. • N is even, to make n0 vertical in the arcs of the ring. • ∆ϕ = π2 , to make ν0 = 12 . As long as the amplitude-dependent spin tune ν does not change too strongly over the orbital amplitudes, all imperfection resonances and, because the orbital tunes cannot be 1/2, also all first-order intrinsic resonances are avoided by the insertion of such Siberian Snakes [43], and polarized beam acceleration to very high energy could become possible.

For this, n0 should be vertical in the arcs, which requires an even number N of Siberian Snakes. 78) with ∆ϕ = ϕN − ϕN −1 ± . . − ϕ1 . For N Siberian Snakes in a ring with spin rotations otherwise only around the vertical, the following three conditions are required: • ∆Ψ = 0, to make ν0 independent of energy. • N is even, to make n0 vertical in the arcs of the ring. • ∆ϕ = π2 , to make ν0 = 12 . As long as the amplitude-dependent spin tune ν does not change too strongly over the orbital amplitudes, all imperfection resonances and, because the orbital tunes cannot be 1/2, also all first-order intrinsic resonances are avoided by the insertion of such Siberian Snakes [43], and polarized beam acceleration to very high energy could become possible.

This distinction will not be made here. To analyze whether s3 = S · n0 is an adiabatic invariant, I consider a precession vector Ω 0 (θ, τ ) that depends on a slowly changing parameter τ . For fixed τ , the rotation matrix for one turn has a unit rotation vector n0 (θ, τ ) and one can again define the 2π-periodic coordinate system using this vector together with the unit vectors m(θ, τ ) and l(θ, τ ). Because these vectors have unit length and constitute a right-handed orthogonal coordinate system for all values of τ , their variation with τ can only be a rotation around some vector η(θ, τ ), ∂τ n0 = η(θ, τ ) × n0 , ∂τ (m + il) = η(θ, τ ) × (m + il) .

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