By Alexander Wu Chao
Attracts recognition to big theoretical and mathematical points of beam instabilities. Introduces and analyzes a variety of collective instability results for prime strength accelerators. makes use of an important quantity of versions and soluble examples as illustrations.
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Additional resources for Physics of Collective Beam Instabilities in High Energy Accelerators (Wiley Series in Beam Physics and Accelerator Technology)
Example text
60) gives a uniform elliptical distribution with horizontal and vertical extents a and b. The parameters E,,~ in Eq. 58) are the horizontal and vertical beam emittances at the edge of the envelopes. The electric field at position (x, y) produced by the beam charge is obtained by integrating over the beam distribution, dx, dy, (x-4-f + 0 -w (x-xf)2+(y-Yf)2 . (1 61) * The double integral in Eq. 62) The space charge force must also include the magnetic force, which almost cancels the electric force for relativistic beams.
8 Show that this Fourier transform pair satisfies the Parseval theorem, Eq. 26). 4 In the pipe wall (r > 6) and in the region where IzI is of the order of pipe radius b behind the beam, show that B, z=- E, > E, and that each is greater than the next by a factor of the order of x-‘12. The magnetic field penetrates better into the metal than the electric field, and the component of the electric field perpendicular to the metal surface is stopped most effectively by the surface charge on the metal surface.
Wake electric field lines in a resistive wall pipe generated by a point charge q. 35 m m for an aluminum pipe with b = 5 cm). The field line density to the left of the dashed line has been magnified by a factor of 40. ) One can obtain the rate of energy loss of the charge q by equating it to the heat generated in the resistive wall. This gives d&F -=-- 1 1 / dVj? I? 26) = /;m;~i$i)12 for Fourier transform pairs F(z) and F(k). , in the metal wall and Eq. 27) b2 ’ from ir is small. 4. A box for computing the image charge using Gauss’s law.