Simulation of the LHC beam collimation --------------------------------------------

Four points along the ring were considered:

BC - crystal SCOL – secondary collimatorTCOL – tertiary collimatorRF – accelerating system voltage•

••

•

BC

SCOL

TCOL

RF

Four linear 6-D transfer matrices M(6,6) were used to transport particles between

BC → SCOLSCOL → TCOL

TCOL → RFRF → BC

Particle coordinates – (x, x′, y, y′, l, δ)

Simulation of the LHC beam collimation (horizontal) --------------------------------------------

BC positions: TCP.A6L7.B1, s= 19795.1844 xbc=6σx=1.557 mm (7 TeV/c) → 5.949 mm (0.45 TeV/c)

SCOL positions: TCSG.6R7.B1, s=20140.5234, xscol=7σx=2.88 mm (7 TeV/c) → 10.01 mm (0.45 TeV/c)

TCOL position: TCLA.B6R7.B1, s=20178.4634

RF position: in the middle of the cavities, s=9996.79with lattice parameters interpolated

with voltage summed

LHC beam emittance ε=0.5×10-9 m·rad for 7 TeV/cε=7.3×10-9 m·rad for 0.45 TeV/c

LHC azimuths characterization---------------------------------

Start point → BC azimuthHalo generation

Halo particles begin hit BC after some turn numbersDue to increase of particle oscillation amplitudes

Final points → (1) absorption in SCOL

(2) Inelastic interactions in BC

TCOL azimuth → halo registration

RF azimuth → change of particle momentumdue to RF voltage V

)2sin(0 C

lhEeV

Peculiarities of the LHC beam collimation--------------------------------------------

Different distances from the orbit for collimators For crystal collimator Δbc=xbc(0.45 TeV/c) – xbc(7 TeV/c) = 4.392 mm

Corresponding change of beam envelope direction ΔXP = (αx/βx)·Δbc = 62.38 µrad

Critical channeling angle for (110) Si bent with R=60 mθcb=9.89 μrad (0.45 TeV/c) → 1.96 μrad (7 TeV/c)

Multiple Coulomb scattering in 3 mm Siθms=5.91 μrad (0.45 TeV/c) → 0.38 μrad (7 TeV/c)

Ratio of coherent to incoherent scatteringθcb/θms = 1.67 (0.45 TeV/c) → 5.17 (7 TeV/c)

--------------------------

Inelastic nuclear cross-section σin = 507 mb (0.45 TeV/c) → 567.5 mb (7 TeV/c)

--------------------------

Impact parameters and angles for the first hits-------------------------------------

0.45 TeV/c 7 TeV/c

QH=64.28ΔXP ≈ 10 μrad

QH=64.31 ΔXP ≈ 1 μrad

With betatron amplitude increase per turn as in the SPS

Different phase point density

Impact parameters with SCOL for perfect alignment-------------------------------------

0.45 TeV/c 7 TeV/c

Impact parameters with BC for amorphous orientation-------------------------------------

0.45 TeV/c 7 TeV/c

Before extraction (blue) and inelastic interactions in crystal (red)

The whole crystal works Only crystal surface works

Impact parameters with BC for VR orientation-------------------------------------

0.45 TeV/c 7 TeV/c

In both cases the whole crystal works

Channeling efficiency and beam losses-------------------------------------

0.45 TeV/c 7 TeV/c

Losses in AM 18% and 75%, respectivelyEfficiency is larger than 80% in the range of 25 µrad and 5 µrad, respectively

Crystal imperfections: miscut and torsion-------------------------------------

Miscut angle: 60 µrad

Loss increase for θo=0, ΔL= 180% Channeling reduction for θo=0, ΔPch=0.7%

Crystal torsion:1 µrad/mm → ΔL=24% , ΔPch=0.27%

2 µrad/mm → ΔL=39% , ΔPch=0.6%

Optimal crystal parameters - ?-------------------------------

L=3 mm, α=50 µrad, R=60 m → L=4 mm, α=50 µrad, R=80 m → ΔL=0 , ΔPch=0.28%

L=3 mm, α=40 µrad, R=75 m → ΔL=-9.4% , ΔPch=0.8%

Impact parameterswith SCOL for α=40 µrad

Conclusions----------

1. Very large beam losses 75% occur in AM crystal (mainly in its surface)

Avoid AM-orientations for 7 TeV/cKeep crystal in CH or VR modes

2. Range of 80% channeling: 25 µrad (0.45 TeV/c) and 5-6 µrad (7 TeV/c) 90% channeling: 20 µrad and 3 µrad

3. Effects of crystal imperfections will be sufficiently smallwhen miscut angle ≤ 50 µrad and torsion ≤ 1 µrad/mm

4. Optimal crystal parameters : L=3-4 mm at α=50 µrad

5. Main problem is goniometerfast and accurate and reproducible crystal orientations