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By Alex Chao

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Note that each lattice site is assigned not only three space coordinates Xn , Yn , Zn , but also an energy deviation ∆n . 60) k=n In order for the beam to crystalize, we must require the lattice site coordinates to be periodic with period C. In the smooth approximation, Kx , Ky , ρ as well as the lattice structure are independent of s. The crystal rotates around the accelerator rigidly. 60) vanish. 61) means that a particle with Xn = 0 must be associated with a ∆n in such a way that the extra path length due to Xn is exactly compensated by the extra velocity of the particle due to ∆n .

Consider the case with N = 2. 41) 1/3 ) We next analyze the stability around each of the possible solutions. Take the first case first. 44) 2 2 Note that the y- and z-motions are decoupled from each other. 46) The z-motion is unstable if kz < ky . 6. 6: There are 8 eigenmodes for a 2-particle planar crystal with zfocusing. Four of them with the two particles on the y-axis are shown here. (a) y-motion, 0 mode. (b) y-motion, π-mode. (c) z-motion, 0 mode. (d) zmotion, π-mode. The sheering mode (d) is unstable if kz < ky .

Kauffmann, SSCL Report 587 (1992). K. -F. L. Colestock, Phys. Rev. Lett. 76, 620 (1996). [5] O. , Euro. Part. Accel. Conf. 1996. [6] G. Stupakov and A. Chao, Part. Accel. Conf. 1997. 45 46 3 Crystalline Beams When completed, crystalline beam research should be much more extensive than what will be presented below. Basically we could be re-doing the entire solid state physics – melting point, phonons, specific heat, heat conduction, quantum effects, spin effects – just replacing molecular force by Coulomb force.

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Topics in Accelerator Physics (lecture notes) by Alex Chao

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