Varying the Calibration Constant and the Tune

As already mentioned, the amplitudes of theoretical prediction and of data do not match perfectly. Aside from the modulation of their difference due to beta-beat, there seems to be a threshold in difference. This again leads to the assumption that under circumstances the calibration factor of the corrector magnets was not evaluated correctly. Figure 20 shows minimization of the difference between theoretical predictions and measured values after changing the horizontal calibration constant:

$$calconst_{horizontal}: 8.5\cdot10^{-3} \longrightarrow 16.6\cdot10^{-3}$$

Figure 20: Comparison of the amplitudes of measured data (with adapted calibration constant) and theoretical prediction. Their difference has a mean of $3\cdot 10^{-6}$ and a RMS of $1.5\cdot 10^{-4}$.

This leads to a minimized difference of the two plots (their difference has a mean of $3\cdot 10^{-6}$ and a RMS of $1.5\cdot 10^{-4}$), but it doesn't give us any proof considering the properties of our corrector magnets. Once again the calibration constant would have to be changed by a great factor which at the moment cannot be legitimated. The fact that the difference between the two calibration constants is almost exactly a factor two is even more bothering.

There is however another possible approach: Perhaps the $\beta$-function used by the Tracy code is not identical to the actual $\beta$-function when we took experimental data. Recall that in equation 6 the phase and the $\beta$-function depend on each other as a result of solving Hill's equation [4]:

$$\phi(s)=\int_{0}^{s}\frac{d\overline{s}}{\beta(\overline{s})}$$ (10)

Therefore a beta-beat will have influence on the phase. This influence may be only small because of the inverse relationship but nevertheless it should be taken into account. Of course if the phase is shifted by only a small deviation this will have great consequences for the BPM readings because their position (in reference to phase advance) is shifted. Even if this shift is small, the position readings may be completely off because of the strong oscillation of the beam position around reference orbit.

Thus the Tracy code was modified to include the possibility of tune adjustment (by quadrupole magnets). The vertical beam positions were in good agreement with theoretically calculated values, but horizontal positions were way off. Therefore the horizontal tune was adjusted in the Tracy evaluation until good agreement between experimental data and theoretical values was achieved. The result is shown in figure 21 (the difference between measured data and theoretical values has a mean of $1.5\cdot 10^{-5}$ and a RMS of $2\cdot 10^{-4}$). The corresponding tune values were (tune in vertical direction left unchanged):

$$Q_{x}=15.1$$

$$Q_{y}=9.6$$

Figure 21: Nicely matching horizontal beam positions after adapting the Tracy tune value. The difference between measured data and theoretical values has a mean of $1.5\cdot 10^{-5}$ and a RMS of $2\cdot 10^{-4}$.

So this method basically illustrates a way of tune measurement. We can now be assured that the tune of the storage ring (while we took experimental data) was at the values $Q_{x,y}$ stated above.