Yamamura updated the elastic scattering effect on the beam emittance a the LINAC. Since the beam-tail does not contribute the emittance, he re-calculated it within ±7σ as a function of vacuum pressure. Negligibly small effect was seen at the nominal pressure of p=10^{-9} torr, while visible effect would appear at p>10^{-4} torr. Actually, the vertical emittance was calculated to increase with 1% at 10^{-7} torr. With 20mm diameter beam pipe, electrons can be transported within the tail of less than ±400 (4000)σ_{x (y)} at the main LINAC.

This short review motivated by M.Battagolia's talk at the tele-conference, 14 December 2003, Mumbai. One transparency showed "possible luminosity reduction due to collimation wake-fields" with various innermost radii (R_{VTX}) of the vertex detector, where the case of R_{VTX}=1.2cm may cause trouble at E_{CM}< 500GeV, NLC.

There is a well-documented note of "Collimator Wakefield Calculations for ILC-TRC Report", LCC-0101, August 2002 by Peter Tenenbaum (SLAC). Along with this note, the dominant effect of the collimator is a kick (Δy') applied to off-axis beam (y). The kick can be expressed by Δy'=Ky. Assuming that y and y' can be normalized by RMS of incoming jitter and kick angle, respectively, i.e. y=mσ_{y}, &Δy'=nσ_{y'}, the equation becomes to n/m=K σ_{y}/σ_{y'}. Defining an amplification factor as A_{&beta}=K σ_{y}/σ_{y'}, the beam phase space (emittance) increases by 1+A_{&beta}^{2}. Using emittance (&epsilon) and beta function(&beta), A_{&beta}can be expressed by K&beta.

K has been calculated for geometric and resistive wakefields. The geometric one is due to a geometric change of vacuum pipe, which can be estimated by K=1/π^{1/2}·N r_{e}/γ ·&alpha/(&sigma_{z}r) and K= &pi^{1/2}/2·N r_{e}/γ·αh/(&sigma_{z}r^{2}) for circular and rectangular collimators, respectively, in usual case of αr/&sigma_{z}<<1, where α is the taper angle, r and h are half-gap and half-width of the collimator, respectively, γ is a Lorentz factor of electron, N is beam intensity and r_{e} is an electron classical radius.
Also the resistive K can be expressed by K=F_{G} · &Gamma(0.25)/(2 π^{3})^{1/2}·N r_{e}/γ·(c/&sigma_{z}&sigma)^{1/2}·1/r^{2}·[ L/r + 1/α] , where the first term is resistive wake of an untapered vacuum chamber of length L and the second one is that of tapered chamber; &sigma is conductivity which is 5.9 x 10^{7} (1/Ωm) x 9 x 10^{9} for copper. A geometric form factor F_{G} is 1 for circular chamber (cylinder) and &pi^{2}/3 for rectangular one.

Yokaya has also calculated K in case of circular collimator, which is expressed by;
K=(π/2)^{1/2}·N r_{e}/γ·[ (λ/&sigma_{z})^{1/2}/r^{2}( L/r + 1/α ) + α/(2^{1/2}&sigma_{z}r) ] , where first term in [ ] is the resistive and the second is the geometric wake, and resistive depth of collimator is λ=1/(120π&sigma). Compared it with the above ones, Yokoya's geometric one has a larger factor by π/2 and the resistive one has smaller factor by 0.62.

Anyway, the rectangular collimator enhances the wakefield by π/2·h/r compared to the case of circular one, which can be more than 10 for r=200μm and h=3mm. The circular collimators have been assumed in the JLC Design Study(1997).

Results for SP2 collimator located at 1358m from IP at 250GeV beam energy are as follows;

**** Yokoya **** circular collimator *** --- energy= 250. N= 7.50000026E+09 &sigmaSince the amplification factor A_{z}= 0.000110000001 --- &beta_{y}= 524. --- &alpha= 0.0199999996 L= 0.00999999978 r= 0.000199999995 --- λ= 4.49818273E-11 σ= 59000000. --- A_{&beta}^{geo}= 0.0174631961 A_{&beta}^{taper}= 0.0222535636 A_{&beta}^{flat}= 0.0222535636 --- A_{&beta}= 0.0619703233 A_{&beta}**2= 0.00384032098 **** NLC **** circular collimator *** --- energy= 250. N= 7.50000026E+09 &sigma_{z}= 0.000110000001 --- &beta_{y}= 524. --- &alpha= 0.0199999996 L= 0.00999999978 r= 0.000199999995 --- h= 0.00322999991 --- σ= 59000000. --- A_{&beta}^{geo}= 0.0114117302 A_{&beta}^{taper}= 0.0357868746 A_{&beta}^{flat}= 0.0357868746 --- A_{&beta}= 0.0829854757 A_{&beta}**2= 0.00688658934 **** NLC **** rectangular collimator *** --- energy= 250. N= 7.50000026E+09 &sigma_{z}= 0.000110000001 --- &beta_{y}= 524. --- &alpha= 0.0199999996 L= 0.00999999978 r= 0.000199999995 --- h= 0.00322999991 --- σ= 59000000. --- A_{&beta}^{geo}= 0.289350182 A_{&beta}^{taper}= 0.0357868746 A_{&beta}^{flat}= 0.0357868746 --- A_{&beta}= 0.360923916 A_{&beta}**2= 0.13026607 Note: PT's paper(LCC-0101) shows A_{&beta}^{geo}=0.2895, A_{&beta}^{taper}=0.0359, A_{&beta}^{flat}=0.0359, A_{&beta}=0.3614

By our request, Akai-san gave a talk on "crab cavity for KEKB". Crab cavity has been developed as a backup scheme for present crossing angle of ±11 mrad at IP of the KEKB, while it shall be adopted as baseline for Super-KEKB with ±15mrad crossing, The crab cavity is used for rotate beams in order to achieve headon collisions effectively. In addition, it could increase the beam-beam tune shift several times to Δν=0.2 at KEKB, which has been estimated by K.Ohmi.

One major issue in designing the crab cavity is to damp parasitic modes, since generally an operating (crabbing and TM110) mode is not the lowest frequency mode contrary to accelerating cavities (TM010 mode). Therefore, an appropriate design has been invented for damping parasitic modes which become higher frequencies than the crabbing mode (501.7MHz). Wave guides or beam pipe with cut-off frequency higher than the operating mode can damp all Higher Order Mode (HOM).

The superconductive crab cavity has a squashed cell with larger horizontal size about twice than vertical size and a relatively short cell length. Another feature is a coaxial coupler with notch filter in order to maintain Q-value for the crabbing mode. The design has been performed by MAFIA calculations. Experimental verification ( cold test) was also conducted with a full scale cavity at KEK by Hosoyama et al. . The cold test achieved a surface peak field of about 27MV/m with a Q_{o} value of 10^{9}, which exceeded the design values of 21MV/m@and Q_{o} of 10^{9}.

Phase error is the most relevant in tolerances of the crab cavity, which is also the most concerned one at linear colliders. The phase error would cause horizontal shift of bunches at IP. The displacement should be much smaller than the beam size. Therefore, the phase error should be largely less than 4.3 degree. At the KEKB, the error comes from different timing of RF reference pulse between HER and LER, an intensity dependence (2.4 degree) and a phase change due to bunch gap (2.7 or 4.9 degree depending on the cavity type). These errors can be compensated. The experimental verification of crab crossing will be scheduled at the KEKB in 2005, where two crab cavities will be installed each for two rings, i.e. without second cavity in the same ring, at Nikko experimental hall.

At Super-KEKB the crab cavity must be improved for the 10A beam, while the present cavity is designed for 2A beam at KEKB. Reducing the HOM power together with heavier damping, the design is almost completed. The performance looks very promising.

At linear colliders, the phase tolerance is 0.2 degree at x-band (11.4GHz) crab cavity. Assuming the tolerance is inversely proportional to the frequency of crab cavity, the translated one would be 0.01 degree at the KEKB-crab cavity of 501.7MHz. So, the requirement is looser 10 times than ours at least.