Simulation for experiment on the SPS beam collimation --------------------------------------------------------------

Experimental layout

Beam and crystal parameters

Transverse positions of experimental devices

Roman pot model for simulation

Initial distribution of beam halo particles

Simulation scenario

Results for single crystal in channeling mode, 120 GeV protons

● For two points scheme BC → TAL → BC

● With detectors BC → RP1 → RP2 → TAL → BC

Results for multi crystals in VR mode, 120 GeV and 270 GeV

Model

Beam and crystal parameters--------------------------------------

Energy of protons – 120 GeV and 270 GeV

Normalized emittance at 1 σ – 1.5 mm∙mrad

Beam – unbunched

Tune Qx=26.62 , Qy=26.58

Bend angle α=150 μrad → impact parameters with TAL 6÷8 mm

→ achievable for LHC energy

Crystal length L=1 mm, thickness T=0.5 mm

Compromise between optimal parameters for SPS and required for LHC

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(111) Silicon crystal

BC and RP2 → close to locations with β=βmax

RP1 → small β but phase advance is close to optimal

Transverse positions of experimental devices------------------------------------------------------

BC – distance from closed orbit → Xbc=6σbeam

particles should first hit crystal

RP1, RP2 , TAL → at 6σbeam +Xof shifted by Xof from crystal position

Xof → amplitude increase due to MS in BC with angle 4.25 θms

This practically excludes hits of TAL after BC passage in amorphous mode

Probability to be channeled at following passages increases

TAL → additionally shifted by the value of non-registered area of RPs

6σbeam + Xof + Tnr, where Tnr=800 μm

Roman pot model for simulation---------------------------------------

Detector dead area – 500 μm

Sensitive area distance from RP bottom – 800 μm

Roman pot model for simulation---------------------------------------

Three transverse areas with different content were considered for RPs

1. XRP < X < XRP+150 μm → bottom of RP, Lb=3 cm (Al)

2. XRP+150 μm < X < XRP+300 μm → slot of RP, Ls=400 μm (Fe)

3. X > XRP+300 μm → detector area of RP,

Ld(RP1) = 400 μm (Fe) + 900 μm (Si), Ld(RP2) = 400 μm (Fe) + 1500 μm (Si)

Processes considered for particles crossing RPs

● multiple Coulomb scattering

● ionization losses

● inelastic nuclear interactions

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Initial distribution of halo particles---------------------------------------

Normalized betatron amplitude at BC → xm=xbc+∆xm

Amplitude increase Δxm is a random value

P(∆xm)=exp(-∆xm/λ), λ=0.1 μm, ∆xm=–λ∙ln(ξ1)

Interval betatron phases of particles hit BC

∆φ=arccos(1/(1+∆xm/xbc))

Random phase from this interval

Φ(ξ2)=2∙∆φ(ξ2-0.5)

Horizontal coordinates

x(ξ1,ξ2)=xm(ξ1)∙cos(φ(ξ2))

x΄(ξ1,ξ2)=-xm(ξ1)/βx (sin(φ(ξ2)+α∙cos(φ(ξ2)))

Distributions of vertical coordinates (y, y΄) and momentum deviation δ=∆p/po

P(y)=P(y΄)=P(δ)=δ(0)

Simulation scenario-------------------------

For particle tracking → (1) liner approach was used for SPS

(2) the only aperture restrictions are in the crystal collimation area

We transport particles along SPS through forth azimuths

BC → RP1 → RP2 → TAL using four transfer matrices M(6,6)

Start point → BC azimuth

Final points → (1) absorption in TAL

(2) Inelastic interaction in either BC or RP1, RP2,TAL

Collimation efficiency losses only due to inelastic nuclear interactions

θo=0

θo=20

θo=40

θo=-20

Collimation without detectors-------------------------

Impact parameters Number of passages

Perfect alignment and near

Collimation without detectors-------------------------

Impact parameters Number of passages

θo=75

θo=-75

Far from perfect alignment

Amorphous diffusion

VR drift

Collimation without detectors – orientation dependencies---------------------------------------------------------------

Efficiency Average impact parameter Edge fractions

Efficiency at θo=0 → larger 99.9%

For angles ± θc → larger 99%

Edge fractions near θo=0 → smaller 2 and 3%

Collimation with detectors-------------------------

Efficiency Edge fractions

Collimation efficiency Pc for θ≠0 decreases → losses in dead area of detectors

Probability of coordinate registration in RP2 → Pr2 < Pc

Probability of angle registration Pr12 for θ≠0 decreases fast →amplitude increase rate to pass dead area of RP1 is lower

Monitor of beam losses near BC-----------------------------------

Monitor indications ~ particle losses due to inelastic interactions in BC

Beam fraction lost in BC is the compliment to

the other one absorbed in the collimator

Beam losses in BC

Ploss=1-Pc

Losses in VR area (3) are smaller than in random → angular deflections are larger

→ number of BC passages to be absorbed in TAL reduces

Minimum (2) near θo=-α → VR always increases the betatron amplitudes

21

Minimum (1) – due to channeling in BC

3

→ the whole VR area is in one side of beam envelope inclination

Multi reflections by sequence of bent crystals – SVR-----------------------------------------------------------

Optimal bend radius for VR → Ropt=10 Rc, for 120 GeV and (111) Si Rc=0.21 m

VR deflection angle for R=2 m → θvr=21 μrad

Seven subsequent reflections can give a deflection of about 150 μrad

parallel optimized

Angular acceptance for parallel SVR → θac=α-(Nθvr+θc), θc≈20μrad

Our parallel SVR

N=7, (111) Si, L=0.5 mm, R=2m → 250 μrad

θsvr=148.2 μrad

σsvr=27.8 μrad

Collimation by parallel SVR for 120 GeV-----------------------------------------------

Impact parameters Number of passages

θo=-210

θo=-290

θo=0

Middle of MVR area

Edge of MVR area

Channeling in one of crystals

VR in subsequent crystals reduces deflection

Collimation by parallel SVR – orientation dependencies--------------------------------------------------------------

Efficiency Average impact parameter Edge fractions

Efficiency for VR and CH areas – 99%

MVR

Average impact parameter for MVR area → larger than 5 mm

Edge fractions for MVR area → 5% and 9 %

Collimation by parallel SVR – with detectors-----------------------------------------------------

Probability of angle registration Pr12 > 70% for MVR and CH areas

Probability of coordinate registration in RP2 is close to 100% → Pr2 ≈ Pc

Beam losses increase twice – 2%

Collimation efficiency – 98%

Registration efficiencies

Collimation by unparallel SVR for 120 GeV protons-------------------------------------------------------------

Every next crystal is tilted by δθ≈-θvr → δθ=-20 μrad

Acceptance for MVR is limited only by the beam broadening

θac=α-θc-2σmvr, where σmvr – RMS deviation of the multi reflected beam

θo=0

θo=50

θo=120

Particles deflected due to channeling avoid

to be reversely deflected by VR in subsequent BC

They generate max near x=15 mm at TAL

So, positive orientations of SVR are also

appropriate for collimation

Collimation by unparallel SVR for 120 GeV protons-------------------------------------------------------------

Average impact parameter Edge fractions

MVR

CH

MVRCH

MVR area increases twice up to 200 μrad

Edge fractions in MVR area are smaller → 2% and 3.5%

Angular acceptance for channeling increases N times

Collimation by unparallel SVR for 270 GeV protons-------------------------------------------------------------

Optimal bend radius for VR → Ropt=4.6 m for 270 GeV in (110) Si, Rc=0.46 m

VR deflection angle for R=4.6 m → θvr=16 μrad

Ten subsequent reflections can give a deflection of about 160 μrad

Unparallel SVR, δθ=-16 μrad → N=10, (110) Si, L=1 mm, R=4.6 m, α=217 μrad

θmvr=157.8 μrad

σmvr=20.9 μrad

Collimation by unparallel SVR for 270 GeV protons-------------------------------------------------------------

Average impact parameterEfficiency Edge fractions

MVR

CH

MVR CH

Larger acceptance for MVR and increased N times acceptance for CH

Optimized unparallel SVR ←→ parallel SVR

SVR works in CH mode as well as in MVR mode

Collimation in the angular area of 2α width