Application: Measurement of the Nonlinear Momentum Compaction

Due to the high precision of the energy measurements the presented techniques can be used to obtain the nonlinear momentum compaction factor $\alpha_2$.

$$\begin{eqnarray*} -\frac{\Delta f_{rf}}{\Delta f_{rf}}=\alpha_1\left(\frac{\Delta E}{E}\right)+\alpha_2{\left(\frac{\Delta E}{E}\right)}^2 \end{eqnarray*}$$

After changing the RF main frequency the shift in energy can be measured. From the first measurement $\alpha_1=5.6\cdot 10^{-4}$ and $\alpha_2=4.4\cdot 10^{-3}$ for the SLS storage ring (uncertainties are missing because only three data points have been obtained). Self-consistency checks with the machine model through dispersion measurements [5] and detuning measurements lead to $\alpha_1=6.0\cdot 10^{-4}$ and $\alpha_2=4.2\cdot 10^{-3}$.