1. 1 Open Loop Spectrum of Coasting Beams A. Burov RR Talk, March 4 2009

2. Möhl-Schönauer Equation Open loop spectrum can be described by means of the Möhl- Schönauer equation: From here 2 latticewakespace charge Images detuning (Laslett DC tune shift)

3. Tune Shifts 3 For a Gaussian beam, average SC tune shift Dipole portion, for Recycler ≈ 0.5 h=5.08/2 cm, g= 5.31/2 - electric and magnetic half-gaps. Coherent tune shift C – ring circumference, 3.3 km

4. Numbers for the ‘high intensity’ measurements Assuming “high proton intensity” parameters of recent measurements of Jim, Nathan and Martin: it yields: 4 For the lowest harmonic (45 KHz), resistive impedance It is 20 times below the threshold.

5. Beam response at lowest sidebands 5 Re( D )Im( D ) Vertical response functions for 0 current (brown), fast wave (red) and mirrored slow wave (blue). Tune shifts are in units of the rms chromatic tune spread.

6. Same, for chromaticity = -1. 6 Difference between upper and (mirrored) lower sidebands is only due to the Re(Z).

7. Suggestions Since those measurements were so far from the threshold, it is hard to see any traces of Re(Z) there. To see the impedance better, the chromaticity should be reduced to its lowest possible value (where it is still dominated by its linear part). With low chromaticity and good averaging, Re(Z) could be seen at very high harmonics. To measure Re(Z), BTF + Longitudinal Schottky data could be taken in digital forms, and then fitted by 7