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Exercise 22 Find an expression for the echo amplitude if D(δ) is not a constant, but is D(δ) = Dn (δ/σδ )n . 11 that D(δ) is not a constant? Y. R. Baker, Phys. Rev. 188, 326 (1969).  G. Stupakov, SSCL Report 579 (1992).  G. Stupakov and S. Kauﬀmann, SSCL Report 587 (1992). K. -F. L. Colestock, Phys. Rev. Lett. 76, 620 (1996).  O. , Euro. Part. Accel. Conf. 1996.  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.
The sheering mode (d) is unstable if kz < ky . The remaining four modes are with the two particles on the z-axis. 41), we ﬁnd its y-motion is unstable if ky < kz . We thus conclude that when ky > kz , the crystal is oriented along the z-axis. When ky > kz , the crystal is oriented along the yaxis. Ions tend to stay away from the strong focusing dimension in a crystalline beam. 59 There is also a degenerate case when ky = kz . The crystal can be oriented along an arbitrary direction in the y-z plane, but it is barely stable.
We see that the diﬀerence between α and 1 is typically large, thus allowing a good opportunity to measure the diﬀusion strength. In fact, the measurement can in principle be so accurate that it is conceivable that one can explore the diﬀusion coeﬃcient as a function of J using this method. 7 shows the echo amplitude as a function of ξ for various values of α. 4. When α = 1, the location of the echo is not changed, but the magnitude is reduced by a factor of α3 , indicating the sensitive dependence on diﬀusion.