Beam traversing the structure on the symmetry plane vertical coordinate: we can observe that the maximum value of the longitudinal loss factor occurs at the geometric center of the structure as it has to be expected because the distance between the beam and the holes is the smallest one.

About the transverse loss factor we note that it is directly proportional to the derivative of the longitudinal one as the Panofsky-Wenzel theorem states for coupling impedances.

Coupling impedance study of the SPS cold to warm transitions

B. Spataro, D. Alesini, LNF-INFN, Frascati, Italy;

M. Migliorati, A. Mostacci, L. Palumbo, University of Rome "La Sapienza'', Italy; F. Ruggiero, CERN, Switzerland.

In many papers, the interaction between a relativistic particle beam and a vacuum chamber with holes is usually described in terms of coupling impedance and loss factor. The interference among the holes is the main source of wake fields and losses. This note focuses on the impedances evaluation of SPS cold to warm transitions. A comparison between an analytical model and the numerical results is presented.

NUMERICAL APPROACH

In order to set up the SPS machine as final injector for the LHC, the substitution of some components was required. Measurements made with the beam showed that a reduction of the machine impedance was mandatory. In particular in the cold to warm region it has been measured an unacceptable heating (up to 4 W/m dissipated power). For this reason, a detailed study of the coupling impedance budget for both the cold and warm sections was carried out.

Quantitative results related to the energy losses, parasitic resonances, longitudinal and transverse coupling impedance have been obtained by MAFIA simulations in time domain.

ANALITYCAL APPROACH

The coupling impedance of a circular coaxial beam pipe with N pumping holes has been studied extensively by means of a modified Bethe theory [1,2].

It has been considered a single Gaussian bunch with σ=12cm to evaluate the short range wake potential over the bunch length. The long range wake potential has been calculated by assuming a smaller bunch length (σ=1cm) over a distance of 6 m behind the bunch. The impedance of the structure has been estimated by the Fourier transform of this long range wake potential.

MAFIA simulated structures: The cold and warm transitions have been treated separately.

b=inner radius of the coaxial beam pipe; d=outer radius;

αe,m,=elect. and magn. Polarizabilities;

D=hole spacing;

α=attenuation constant (in the case of ohmic losses at room temperature α depends on the and, in practical cases, the ohmic dissipation is very small).

RESULTS

Both real and imaginary parts depend on the interference among the holes leading to resonance peaks in the impedance at n=nc /D.

Since the structure needs a very large number of mesh points, we have considered four different cases:

a) 38 holes and 74 holes with 2 mm mesh sizes;

b) 146 holes with 4 mm longitudinal and 2 mm transverse mesh sizes.

Far from resonance and with the approximations discussed in [1,2] the loss facor is:

LOSS PARAMETERS

COUPLING IMPEDANCE

Loss parameters as a function of the vertical and horizontal coordinates.

Real part of coupling impedance at low frequencies.

The parabolic behavior is clear from the plots and can be highlighted by a polynomial fit.

The ratio ai/aj is very close to (Ni/Nj)2 as it should be from theory.

74 holes

38 and 74 holes

To completely characterize the structure, we have calculated the 6 m long range wake potential considering a Gaussian bunch of σ=1 cm traveling trough the structure 4 mm away from the holes. The Fourier transform of this wake potential gives the impedance of the structure at that beam position.

At low frequency (up to 4 GHz) the impedance is purely inductive. At low frequency the imaginary part scales with ω, as expected from theory. The real part is proportional to ω2 and exhibits resonant peaks. The maximum amplitude of the resonances depends on the proximity of the beam to the slotted wall, but they are always present. In the simulations, their amplitude scales with N. The parasitic resonances in the 4-12 GHz frequency range are very far from the bunch spectrum cut-off. It is worth noticing that the strongest resonance is peaked at about f=9.27 GHz. This frequency corresponds to a wavelength equal to the holes distance.

Even though, in the present study, the chamber has a beam pipe with elliptical cross-section, the analytical methods (valid for a circular bem pipe) remains extremely useful to check the order of magnitude of the discussed numerical results. As a result for the cold transition with a number of holes N=146, by assuming an internal radius equal to b=4.3 cm we obtain Z/n=12.3 μΩ and P=0.55mW. Instead asuuming an equivalent chamber radius [3] beq=3.74 cm, we get Z/n=16.5 μΩand P=0.43 mW. Both results are in good agreement with the simulation ones.

WARM TRANSITION

The detailed study of the warm transition did not give specific problem and we will present the final results only. The longitudinal impedance of the global structure is estimated to be Z/n=0.31 mΩ. The vertical transverse impedance is Zty=707 Ω/m while the horizontal one is Ztx=382 Ω/m.

146 holes

CONCLUSIONS

We presented the study of the cold to warm transitions of the SPS machine. The numerical estimations of the coupling impedance have been compared to a theoretical model showing a good agreement. Large beam off-set deposits higher power that could affect the machine operation. The obtained ohmic losses are much less than those measured in the SPS cold transitions. However even losses due to the pumping holes (of the order of few tens of mW per meter) lead to heat overload. Therefore it is advisable an optimization of the beam pipe shape.

Calculated with an average current of 0.7mA/bunch, a revolution time of T=23 s and a number of bunches Nb=288.

The horizontal transverse loss factor is three time higher than the vertical one. For this reason we present results related only to the horizontal case.

To investigate the behavior of the power losses as a function of the beam pipe shape, different simulations were done changing the smaller axis of the elliptical beampipe. The figure shows the power losses as function of the horizontal displacement for different values of the smaller beam pipe axis. The case with a=36 mm corresponds to the actual one while that one with a=42 mm to a circular cross section.

REFERENCES

[1] A.Mostacci, L.Palumbo and F.Ruggiero, “Impedance and loss factor of a coaxial liner with many holes: effect of the attenuation'', Phys.~Rev.~ST-AB,{\bf 2},124401, 1999.

[2] A. Mostacci, “Energy lost by a particle beam in a lossy coaxial liner with many holes”, LHC Project Report 199, February 2002.

[3 L. Palumbo, et al., “Coupling Impedance in a Circular Particle Accelerator, a Particular Case: Circular Beam, Elliptical Chamber”, IEEE Trans. on Nuclear Scienze, Vol. NS-31, n. 4, August 1984.