# Solutions to Exercises

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

## Abstract

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
$$\begin{array}{*{20}{c}}{{{ \in }_{x}} = \frac{{\left\langle {{{x}^{2}}\left( s \right)} \right\rangle }}{{{{\beta }_{x}}\left( s \right)}}} \\{ = \int {d{{\phi }_{x}}d{{I}_{x}}2{{I}_{x}}{{{\cos }}^{2}}{{\phi }_{x}}\rho \left( {{{I}_{x}},{{\phi }_{x}}} \right)} } \\{ = \int {d{{\phi }_{x}}d{{I}_{x}}2{{I}_{x}}{{{\cos }}^{2}}{{\phi }_{x}}\rho \left( {{{I}_{x}}} \right)\frac{1}{{2\pi }}} } \\{ = \int {d{{I}_{x}}{{I}_{x}}\rho \left( {{{I}_{x}}} \right) = \left\langle {{{I}_{x}}} \right\rangle .} } \\\end{array}$$
(12.1)

## Keywords

Bunch Length Beam Position Monitor Equilibrium Emittance Beam Lifetime Synchronous Phase
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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