The Cockcroft Institute
Post-graduate
Lecture Courses: Academic Year 2005-6
Specialist
Course
Spin Dynamics and
Polarisation
Tutors: Prof DP Barber (DESY and Liverpool), Dr G Moortgat-Pick
(Durham), Dr G Court (Liverpool)
Location and timing : The Cockcroft
Institute (CCLRC Daresbury Laboratory) over 5th June - 9th June 2006
Prerequisites : A solid
knowledge of undergraduate-level classical mechanics including
Hamiltonians, a solid understanding of undergraduate-level quantum
mechanics, a basic understanding of single particle dynamics in
accelerators and storage rings including the phenomenology of
synchrotron radiation.
Syllabus
After an introduction including some history and an update on the
current status of the field, the course will include a selection from
the following topics :
1. The meaning of "Spin" in accelerator physics
Spin in the rest frame of the particle.
Spin motion at rest in a
magnetic field: classical and quantum descriptions. The
Thomas-Bargmann-Michel-Telegdi (T-BMT) equation of spin motion (and
Thomas precession). The simple consequences of the T-BMT equation: the
gyromagnetic anomaly and Thomas precession. Linear acceleration.
Electrons, muons, protons, deuterons, He^3. Measurement of (g-2)/2 for
muons. The concepts: "polarisation", "local polarisation" and "beam
polarisation".
2. Basic phenomenology for rings-protons
The vector n_0(theta). Spin tune on the
design orbit and the closed
orbit. Perturbation to the spin motion by synchro-betatron motion.
Fourier analysis and spin-orbit resonances. Resonance strength and its
dependence on emittance and energy. The problem with deuterons. The
"single resonance model" (SRM) on the closed orbit. Spin tune gaps.
Classification of spin-orbit resonances.
3. Polarisation preservation for accelerating protons
The Froissart-Stora formula, NMR.
"Imperfection resonances","Intrinsic
resonances". Tune jumping, Examples. Siberian Snakes, Examples. Partial
Siberian Snakes, Examples. Adiabatic control of "resonance strengths".
Spin rotators. The status at RHIC.
4. Practical spin-orbit tracking algorithms
Ignore Stern-Gerlach effects. 3x3
matrices (SO(3)), quarternion algebra
(SU(2)). Linearised spin motion: the "SLIM" formalism. The first step
beyond linearised spin motion: sideband resonances. Differential
algebra.
5. Modern approaches
The invariant spin field (ISF):the
vector n(z;theta), local coordinate
systems. The amplitude dependent spin tune. Spectral analysis of spin
motion. The maximum equilibrium polarisation and the maximum
time-averaged polarisation. Beam history and the actual beam:
factorisation of the time-averaged polarisation. The ISF in the SRM.
Common misconceptions (e.g."spin closed orbit", "perturbed spin tune").
Generalised resonance strength and generalised F-S effects. The
adiabatic invariant for spin motion.
6. Calculating the ISF
Perturbative methods. (SMILE,
Successive diagonalisation).
Non-perturbative methods:-Analytical:SODOM, MILES
-Numerical:Stroboscopic averaging, adiabative antidamping. -On orbital
resonance (analytical or numerical). The uniqueness of the ISF.
7. Electrons (are not protons)
Overview of the effects of synchrotron
radiation on spins.
Irreversibility (electrons) vs reversibility (protons). The
Sokolov-Ternov effect. The Baier-Katkovformula. The dependence on g.
There is no such thing as a free lunch: spin diffusion =>
depolarisation. The equilibrium polarisation. The SLIM formalism for
depolarisation: spin diffusion wrt the vector n^0.
8. Maximising electron polarisation
The 2x6 spin-orbit coupling matrix of
the SLIM formalism. The influence
of spin rotators and Siberian Snakes. Strong spin matching and spin
transparency. Harmonic synchro-betatron spin matching. Harmonic closed
orbit spin matching.
9. The Derbenev-Kondratenko (D-K) formula
The D-K formula: spin diffusion wrt
thevector n(z;theta). A common
misconception. dn/ddelta with linearised spin motion. Kinetic
polarisation. The MIT-Bates ring and the AmPs ring. Getting it
together: the quantum mechanical approach. More on the classification
of spin-orbit resonances: synchrotron
sideband resonances.
10. Really getting it together: the Fokker-Planck equation for spin
The Fokker-Planck equation for orbital
motion. The corresponding
Fokker-Planck equation for the polarisation density.
Inclusion of spin flip and K-P terms. An educational model:horizontal
electron polarisation (c.f.muons). Equation of motion of the spin
density matrix of fermions.
11. Pragmatism:spin-orbit Monte-Carlo simulation of spin diffusion
Approximating photon emission. Examples
with a model ring. Beam-beam
effects. Insights. ILC damping rings:e.g.the effect of wigglers.
12. Electron rings: examples
HERA:27.5GeV. LEP:46-105GeV, energy
measurement: gravitation and the
TGV. MIT-Bates:900MeV. CEBAF:6GeV. eRHIC5-10GeV
13. Sources and Polarimetry
Protons. Electrons
Further information may be found at http://www.desy.de/~mpybar.
Return to education page