Solutions to Exercises

  • Michiko G. Minty
  • Frank Zimmermann
Open Access
Part of the Particle Acceleration and Detection book series (PARTICLE)


From (1.12) at a fixed location s we can write \(x = \sqrt {2{I_x}{\beta _x}} \cos {\phi _x},\) where φx includes the initial phase φ0 We then have
$$\begin{array}{*{20}{c}}{{{ \in }_{x}} = \frac{{\left\langle {{{x}^{2}}\left( s \right)} \right\rangle }}{{{{\beta }_{x}}\left( s \right)}}} \\{ = \int {d{{\phi }_{x}}d{{I}_{x}}2{{I}_{x}}{{{\cos }}^{2}}{{\phi }_{x}}\rho \left( {{{I}_{x}},{{\phi }_{x}}} \right)} } \\{ = \int {d{{\phi }_{x}}d{{I}_{x}}2{{I}_{x}}{{{\cos }}^{2}}{{\phi }_{x}}\rho \left( {{{I}_{x}}} \right)\frac{1}{{2\pi }}} } \\{ = \int {d{{I}_{x}}{{I}_{x}}\rho \left( {{{I}_{x}}} \right) = \left\langle {{{I}_{x}}} \right\rangle .} } \\\end{array}$$


Bunch Length Beam Position Monitor Equilibrium Emittance Beam Lifetime Synchronous Phase 
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  1. 1.
    R. Assmann, A. Chao: Dispersion in the Presence of Strong Transverse Wake- fields. In: Proc. PAC 97, Vancouver, 1997 (IEEE, Piscataway 1998) P. 1523Google Scholar

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Authors and Affiliations

  • Michiko G. Minty
    • 1
  • Frank Zimmermann
    • 2
  1. 1.DESY - MDEHamburgGermany
  2. 2.AB Division, ABP GroupCERNGeneva 23Switzerland

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