The Analysis of Transverse Beam Tail Distributions of Bunches with Non-gaussian Shapes*

The characterization of transverse particle distributions of bunches with non-Gaussian shapes is difficult due to a wide variety of possibilities [l]. Without knowing additional information one can tit a distribution using first-, second-, third-, and higher order moments [2]. These moments can then be used to describe beam shape changes along the accelerator, but with limited knowledge of the physics which caused‘the perturbed shape. However, when the cause of the non-Gaussian distribution is known, a more detailed description of the particle distribution can be constructed. In the Qanford Linear Collider (SLC) non-Gaussian distributions are produced by transverse wakefields in the 3000 m linac. 1 TRANSVERSE TAIL PARAMETERIZATION The effects of transverse wakefields can be calculated given the transverse equation of motion of the particles in the bunch. c’ A!? ds2 ~(2,s) + k2 (2,s) x(z,s) = I” dz’ h(z’)W(z’-z) x(z’,s), where x is the transverse offset from the accelerator axis, s is the distance along the accelerator, z is the longitudinal distance along the bunch, k is the lattice strength, y = E/mc2 , h is the longitudinal particle density, re is the classical radius of the electron, and W is the transverse wakefield which depends on the longitudinal separation of the particle generating the fields and the particle sensing the fields. The solution of this equation shows that oscillating leading particles drive trailing particles to larger amplitudes, generating non-Gaussian distributions 131. A schematic view of such a transverse tail is shown in Figure 1, where the particle centroids move farther and farther off-axis near the end of the bunch. Also, with very strong wakefields or with special conditions used to reduce wakefield emittance growth (BNS damping), the transverse tails can spread out over circles in transverse phase space. * Work supported by US Department of Energy contract AC03?$i$FOO515. # Visitor from the Institute of Nuclear Physics, Novosibirsk, Russia. These longitudinal correlations of the displacement and angle of the various ‘slices’ along the bunch can be seen in Figure 2. The transverse particle distribution pi(x) of a longitudinal slice with label i which is on-axis is given by (2) 1/2KEPi where ox = aPi. The overall distribution f representing the entire bunch is obtained by integrated over the bunch with the appropriate Axi indicating the offsets of the various slices. f(x) = i pi(x + Ax;) h(zi)