Homework Wednesday 1 062415 In the alternating gradient

Homework, Wednesday 1. 06/24/15 In the alternating gradient synchrotron (AGS) at BNL protons are injected at a total energy of 2. 355 Ge. V and extracted at an energy of 23. 812 Ge. V. a) What is the stable spin direction? (no snakes) b) Suppose a longitudinally polarized proton is injected into the AGS (no snakes). How many times does the polarization precess about the vertical direction (“up”) per turn (i) at injection energy, (ii) at extraction energy? c) Suppose a vertically polarized proton is injected into the AGS. Should the ramp rate d /dt be much larger or much smaller than the resonance strength e in order to ensure complete spin flip when accelerating the beam through that resonance? 2. The spinor rotation matrix for precession by an angle about the j-axis (j=x, y, s) is given by exp(i j/2). Assuming an initial polarization that is purely a) horizontal b) vertical c) longitudinal, determine the final polarization after precession about the longitudinal (s) axis by an angle and sketch the final orientation in each of these cases. Note: exp(i j) = cos( )+i jsin( ).

3. Consider a hexagonally-shaped accelerator with a type-I Siberian snake. a) With a proton kinetic energy of 108. 4 Me. V, what is G ? (G=1. 7928) b) Show that at the location of the snake the stable spin direction is in the longitudinal direction (i. e. parallel to the nominal particle velocity). Hint: solve for directional cosines using one turn map, Eq. 10. 25 c) Draw the orientation of the stable spin direction in each of the 6 straight sections of the ring d) Assuming that the beam is fully polarized, what are the amplitudes of the components of the transverse beam polarization measured at the location P? snake P beam direction 4. . Loss factor and vertical head-tail instability a) The FCC-ee may have a (single-bunch) cavity loss factor of 1000 V/p. C. When running on the Z pole (45. 5 Ge. V/beam) a beam consists of about 15000 bunches, each with about Nb=2 x 1011 e±. What is the energy lost per electron per turn due to impedance? [The energy loss due to synchrotron radiation is 30 Me. V/turn. ] What is the total power lost by the beam at the RF cavities? Consider a circumference of 100 km. b) In the CERN SPS at a chromaticity of Qy’=-0. 5 with Nb=2 x 1010 protons per bunch the growth time of the vertical head-tail instability is about 5000 turns, Under these conditions what would be the growth time for a bunch with Nb=1011 protons? What would be the growth times for Qy’=-1 and for Qy’=0. ?