Beta Function Measurement

An important ingredient for the successful compensation of the linear optical distortions is the precise measurement of the average beta functions at the location of the quadrupoles. The presented procedure makes use of the fact that a betatron tune change $\delta\nu$ is related to the perturbation $\delta k(s)$ of the focussing strength and the beta function $\beta(s)$ around the ring:

$$\delta\nu=-\frac{1}{4\pi}\oint{\beta(s)\delta k(s)ds}.$$

Thus a change in the strength $\delta k(s_q)$ of quadrupole $q$ allows the average beta function $\bar\beta(s_q)$ at position $s_q$ to be measured, by observing $\delta\nu$ as a function of $\delta k(s_q)$. The error of this measurement defines the limit for the correction of the linear optics perturbations. Thus a sophisticated procedure has been implemented which takes into account known magnetic hysteresis effects and restores the betatron tunes rather than the original quadrupole currents. In addition, the quadrupoles are measured ``magnet family'' wise in order to minimize the residual optical perturbations. In order to increase the precision further, the tune is observed for 5 different currents. The beta functions are then derived from a least square fit. As a result average measurement errors of $\approx\pm$ 1 % have been achieved.