Yamamura has simulated the capture threshold in two cases of sine-wave RF pulse and one in the accelerating structure (iris), where initial phases of electrons were set to be &pi at t=0.

In case of the sine-wave, final phases (&Phi_{f}), longitudinal distance (z_{f}) and longitudinal momenta(p_{z,f}) were plotted as a function of accelerating gradient of E_{o}=59 through 66MV/m at t = 33usec and z=10km. Since z_{f} reaches 10km above E_{o}=61MV/m, the capture threshold was estimated to be 61-62MV/m.

He just started to calculate the case of the accelerating structure by using MAFIA. First calculation was &Phi_{f} as a function of initial phase (&Phi_{i}) with E_{o}=10MV/m and t=10nsec. There were two bumps at &Phi_{i}=0.2&pi and 1.2&pi . A case of the sine-wave was also calculated by the same method, whose result shows a symmetrical distribution around &Phi_{i}=&pi.

The capture threshold should be analytically calculated as pointed out at previous meeting.

Kuroda estimated position errors (dx, dy) of sextupole magnets (SD0, SF1, SD4, SF5 and SF6) in the FF system, which blow up beam sizes by 50% in both directions at IP. i.e. 1.5&sigma^{*}_{x}=300nm and 1.5&sigma^{*}_{y}=4nm. These position errors mostly affect on the beam sizes at IP. The smallest horizontal position error is 2um at SD0 and SD4, and the smallest vertical one is 0.6-0.7um at SD0 and SD4, too.

Yamaoka has conducted frequency response analysis for support tube, whose central part is thin CFRP-tube, by ANSYS-FEM.
The analysis could calculate the frequency response with an input force of F_{o} sin(ωt) in the frequency range from 0 to 1000Hz.
He estimated the input force from ground motions which have been measured at ATF-floor, 5pm, 10 February 2004, since force (F) is product of mass (m) and acceleration (a); F=m a . The acceleration was about constant to be 2 x 10^{-7} m/s^{2} for f (=&omega/2&pi)>3Hz, and the mass is estimated to be 90 tons as its own weight. The support tube has 80cm diameter and 16m total length, whose central part of ±2m length is a CFRP-tube of 3-10 mm thickness (t) and the other part is 10cm thick tungsten tube. So, it consists of one CFRP tube and two tungsten-tube (QC-L and QC-R) where the final quarupoles and tungsten-mask are installed. The most relevant parameter is relative displacement (v_{12}) at ±2m in vertical direction for the L*=2 final quadrupole. The bearings are located at ±3.85m, ±7m and ±8m from the center. The QC-L and QC-R are assumed to have different first resonant vibration-mode, too, which is Δf=0, 1, 3 and 5Hz.

Major results are listed below;

v_{12} : Relative Displacement (nm) at ±2m | ||||||||
---|---|---|---|---|---|---|---|---|

First resonant mode | Second mode | |||||||

Tungsten tube | Central tube | Central tube | ||||||

QC-L - QC-R | CFRP | tungsten | CFRP | tungsten | ||||

Δf (Hz) | 3mm | 5mm | 10mm | 100mm | 3mm | 5mm | 10mm | 100mm |

0 | 0.098 | 0.055 | 0.016 | 9.1E-5 | 0.21 | 0.21 | 0.19 | 0.08 |

1 | 0.13 | 0.080 | 0.035 | 1.1E-3 | 0.17 | 0.18 | 0.18 | 0.081 |

3 | 0.17 | 0.13 | 0.076 | 3.3E-3 | 0.094 | 0.12 | 0.14 | 0.082 |

5 | 0.20 | 0.17 | 0.11 | 5.6E-3 | 0.056 | 0.078 | 0.11 | 0.084 |

In order to "absorb" the displacement, the CFRP tube is not rigid enough compared to the tungsten tube, although it could correlate the displacements as vibrational modes. Generally flexural rigidity is a product of Young's modulus (E) and moment of inertia(I). The CFRP tube of 5mm thickness has 1/39 flexural rigidity of tungsten tube of 10cm thickness. In previous prototype measurements with the relative flexural rigidity of 1/512, two "different" first modes of the cantilevers could merge into a single mode. So he estimated the thinnest thickness to recover the correlation, where the second mode appear. He found that 3mm is the thinnest for the CFRP tube. The correlation may help in an active vibration-free control of the support tube. In future, he will investigate a bearing structure with high resonant mode frequency as much as possible for less vibrational amplitude, then re-analyze it.

Kubo investigated effects of malfunction of quad-movers and BPMs which are used in orbit correction at the main LINAC. The LINAC has the final beam energy of 250GeV and the "GLC roadmap" configuration with 962 quadrupole magnets equally spaced with 6.56m. Every Q-magnet has a mover with a 50nm stepping motor and a BPM with 200nm position resolution. Quadrupoles with malfunctioned movers ("missing quad-movers") were removed from the fitting and correction. Beam positions at the malfunctioned ("missing") BPMs were interpolated by an equation written in the PDF file from adjacent BPM-positions. The missing objects were randomly selected. In case of the missing BPMs, results have been obtained as a function of the total number (N_{t}) and the consecutive number (n_{c}).

First, all the quadrupoles and accelerating structures are displaced by the ATL rule (σ^{2}=ATL) with AT= 1 x 10^{-12}m, where L is a distance between two components. For an example, T is calculated to 1x10^{5}s ( about 1 day) with A=1x10^{-17}m/s. Reading all the BPMs with 200nm Gaussian error, positions of the quadrupoles are corrected. Then, all the accelerating structures are aligned with 5um Gaussian error. These corrections are iterated in 8 times. After the iteration, the emittance is calculated at end of the LINAC. Average emittance is estimated by 100 times simulations, where the statistical error is calculated by RMS/SQRT(100-1) .

The missing quad-movers have no significant emittance growth since the additional growth is 20% at most even with 700 missing movers.

In the case of the missing BPMs, the additional growth is less than 20% if n_{c}=1,2 and N_{t}=300. When N_{t} runs over 60 with n_{c}=3, the emittance suddenly jumps up. With n_{c}=4, the growth rate becomes slow again and it is less than 20% even with N_{t}=100. He found very large growth periodically at n_{c}=3+4m, where m=0,1,2... (N_{t}=100), which is probably due to some resonant effect.
In order to avoid this periodic growth, the simplest way is just neglect one of adjacent BPMs as missing one.

As a conclusion, the malfunction of quad-movers and BPMs will not be a serious problem if they can be identified.