VOLUME 76, NUMBER 4 PHYSICAL REVIEW LETTERS 22 JANUARY 1996 Direct Measurement of Diffusion Rates in High Energy Synchrotrons Using Longitudinal Beam Echoes L. K. Spentzouris, J.-F. Ostiguy, and P. L. Colestock Fermi National Accelerator Laboratory, Batavia, Illinois 60510 (Received 6 June 1995) We have made a direct determination of the diffusion rates in a stored, coasting antiproton beam by observing the decay rates associated with beam echoes in the longitudinal plane. The beam echoes, similar to those observed in other fields of physics, are generated by a sequential impulse excitation at harmonics of the beam revolution frequency. The echo envelope follows a characteristic response which, however, can be modified by the presence of even a very weak scattering process, permitting a sensitive determination of the longitudinal diffusion rate in the beam. PACS numbers: 29.20.Lq, 29.27.Bd Of importance to the stored beam lifetime and maxi- mum achievable beam density in a storage ring is the diffusion rate produced by Coulomb interactions between particles or by external noise sources. Typically this is measured by equilibrium emittance measurements or by long-time-scale observations of comparatively slow beam- size growth over a fairly wide range of beam parameters. However, it is possible to obtain an incremental mea- surement of the diffusion rate with the use of a phenom- enon widely observed in a number of areas of physics where scattering processes are sufficiently weak, namely, beam echoes [1–3]. Echo generation is formally a non- linear process, whereby a medium is first given an im- pulse excitation followed by decay of the perturbation through phase mixing of an ensemble of particles; a sec- ond impulse is then applied to reconstruct a portion of the original perturbation after a specific delay. Since the echo reconstruction depends sensitively on the long-time struc- ture of the particle distribution, even weak scattering pro- cesses can cause an observable effect after a relatively short time interval [4,5]. Echoes have been studied extensively in other areas of physics, and it has recently been suggested that similar phenomena might be observed in bunched beams [6–8]. Given the extremely weak scattering processes known to exist in high energy hadron beams, it can well be expected that long-lived echo phenomena can be made to occur. Moreover, the long-time behavior of the echoes depends directly on the diffusion rate and, as such, can be used to give a direct measurement of the same. In this paper we report the first observation of echoes in a coasting beam and use the results to determine the absolute diffusion rates under various operating conditions. A model is developed which is manifestly in agreement with the observations. The echo phenomenon can be viewed as a nonlinear mixing of two waves propagating around the ring. If a short duration rf excitation is applied to the beam at revolution harmonic nv 0 , where v 0 is the revolution frequency of an ideal particle and n is any integer, then a longitudinal wave of the form expinv 0 t  will be readily induced. However, following the excitation pulse, the wave will evolve according to expinv 0 1 k 0 ´t , owing to the energy spread in the beam distribution, where ´ is the energy deviation from the mean energy, and k 0 is the proportionality between energy deviation and frequency deviation. Specifically, k 0  2hv 0 b 2 E 0 , where E 0 is the energy of a particle at the center of the distribution, b  yc is the relativistic b factor, and h is a machine dependent parameter called the “slip factor.” The energy spread results in the decay of the wave amplitude in a so-called Landau damping time. If, however, a second excitation pulse is applied at, say, mv 0 , after a delay Dt , product perturbations can be excited at the difference frequency n 2 mv 0 , by virtue of the amplitude nonlinearity. These second-order currents evolve in time according to expi m 2 nv 0 1 k 0 ´t 2Dt  2 inv 0 1 k 0 ´Dt , and have the property that the energy dependence of the phase can disappear at a time t echo  mm 2 nDt . Thus, at t  t echo , the phase-mixing process has been effectively unwound, resulting in a reconstruction of a portion of the original perturbation. A quantitative model of this phenomenon can be devel- oped for the case where wakefields are negligible, i.e., the free-streaming case, by an elementary construction of the distribution function following each impulse [3]. The ef- fect of collisions has recently been included in this model [4] as an expansion in orders of the kick parameter d, where d is the ratio of the energy gained during each impulse to the beam momentum spread. However, the presence of wakefields modifies the beam response from the simple energy shift associated with free streaming and a quantitative solution requires a perturbation approach, which we follow in this work. We shall return to the free- streaming case when evaluating the echo response in the Fermilab Antiproton Accumulator, since wakefields ap- pear to be negligible in this ring. A perturbation approach including the effects of wake- fields and collisions can be developed using the Vlasov 620 0031-9007 96 76(4) 620(4)$06.00 © 1996 The American Physical Society