Simulation of the SPS beam collimation --------------------------------------------

Four points along the ring were considered:

BC - crystal TAL – absorberHD – high dispersion areaRF – accelerating system voltage•

••

•

BC

TAL

HD

RF

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

BC → TALTAL → HDHD → RFRF → BC

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

SPS 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 TAL

(2) Inelastic interactions in BC

HD azimuth → off-momentum halo registration

RF azimuth → change of particle momentumdue to RF voltage V

)2sin(0 C

lhEeV

Beam losses in crystal : experiment ↔ simulation---------------------------------------------

Good agreement in the dependence character and angular width

Observation object → loss dependence on crystal orientation

Disagreement in loss reduction value Rfor channeling

Experiment → R≈6Simulation → R≈30

One of possible reasons of this disagreement → crystal miscut

Crystal with miscut ----------------------

Miscut angle θm is between the crystal plane and its surface

First hit

Beam envelope direction

When crystal plane direction at its entrance isparallel to beam envelope direction

Particles enter BC through its side face in first hits

When impact parameters of particlesb < Δm =0.5 R θm

2 their first passages will be as in

amorphous matter

Beam losses for this perfect orientationwill be increased for crystals with miscut

Δm

Δm – prominent part of crystal surface bend

To estimate dependence of beam losses on crystal miscut -----------------------------------------------

Impact parameters of halo particles should be studied

With taking into account

Betatron oscillationsAmplitude oscillation increase

Synchrotron oscillations

and

Model of crystal with miscut should be developed

Using the surface potential of crystalnear the entrance through its side face

Increase of betatron oscillation amplitude per turn Δxm -------------------------------------------------------

Estimation of halo repopulation time → average increase λ < 1 nm

Assumption → exponential distribution

P(Δxm)=exp(- Δxm / λ)

•

•φ1

φ2

XBC

Case when RF was switched off ---------------------------------------

Distribution of δ is Gaussian with σ = 0.6×10-3 , δ=const

Particles have orbit shift Xε=Dx·δ

Orbit shift determines different values of oscillation amplitudesfor particles, which will touch the crystal

In BC position Dx < 0

For δ > 0 → shift Xε < 0Amplitude of touch is reduced

For δ < 0 → shift Xε > 0Amplitude of touch is increasedXbc

Δ

Amplitude is determined by conditionXbc+Δ=-xm+Dx·δ

Δ – distance between closest trajectory partand BC

Impact parameters in Case when RF was switched off ---------------------------------------

For average amplitude increase λ= 0.5 nm (1) and 0.1 nm (2)

Distribution of impact parameters

Distribution width is about 100 λ50 nm (1) and 10 nm (2)

λ is halo propagation rateParticles begin hit BC when they pass the distance Δ after Nt= Δ/λ turns

Particle with amplitude value at φ=π close to Xbc=-xm+Dx·δ

should be again close to φ=π

Number of turns to be again at the same phase pointNq=100 in our case

when fractional part of tune – fraction(Qh)=0.13

FWHH ≈ 5 μrad

Impact angles in Case when RF was switched off ---------------------------------------

Distribution width is the same for different λ=0.5 nm and 0.1 nm ! Why?

Density of phase points φi of particlesis uniform with Δφq =2π/Nq

Particle can hit BC whenfor its current amplitude value xm

at least one phase point is in the interval (φ1 ,φ2 ) at XBC

Corresponding condition is φ1 = Δφq /2, φ1 = 2 π - Δφq /2

So, the interval of angles should be with present BC parameters

qx

BCq N

XX

2

→ 4.8 μrad

Case of routine operation with RF switched on ---------------------------------------------

Gaussian distributions in longitudinal coordinates (l,δ) were considered

Particles change their momentum due to RF voltage

Trajectory in longitudinal space Horizontal phase ellipse oscillates

For δ=σ period of longitudinal oscillations is about Ts=150 To and orbit shift Xε=Dx·δ=0.47 mm >> Ns·λ (halo propagation value)

→

Thus, particles touch BC when their momentum is about its maximum, δm

Impact parameters and angles for Case with RF switched on ---------------------------------------------------------

To hit BC a particle should have betatron phase near φ=π and longitudinal position near (0,δm )

When particles have already Xm > XBC they will miss the crystal many times

Their impact parameters are increased by about Ns times, Xim ~ 1μm

Simulation for crystal with miscut -----------------------------------

Conditions for entrance and exit through the side face

should be correctly taken into account

Potential of the crystal surface was used near the entrance

through the side face

Loss reduction change for crystal with miscut -----------------------------------------

Crystal 4 has miscut angle θm=200 μradIts length 2 mm and bend angle 176 μrad → R=11.36 m

Prominent part of the surface bend for crystal 4 → Δm=0.23 μmIt is smaller but comparable with average impact parameters

Our simulation results showlosses of halo particles for perfect orientation of the crystal with miscut

are really increased

These additional bean losses are largerwhen halo propagation rate is smaller

For amplitude increase λ=0.5 nm per turn → 42%

For λ=0.1 nm per turn → 120%

Beam losses in crystal for its amorphous orientation --------------------------------------------------

Multiple Coulomb scattering in crystal is consideredin Gaussian approach

This gives good description of dechanneling due to scattering by crystal nuclei in short crystals

Particles of this tail possessing large angular kickscan more quickly reach TAL aperture This should reduce beam losses value

Real angular distribution behind the crystal in amorphous orientationbesides Gaussian part has longer tails about 1%

due to strong single Coulomb scattering by crystal nuclei

Passage number

Simulation for our case showsAmorphous level of beam losses

may be lower by about 10%

Off-momentum halo registration in TAL2 - determination of collimation leakage--------------------------------------------------X-coordinate of absorber xTAL → boundary of betatron motion = envelope

Additional X-displacement for off-energy particle with δ≠0 in TAL2 azimuth

Xδ(sTAL2) = Dx(sTAL2)·δ

Xβm(s) = xTAL (β(s)/βTAL)1/2

Maximum of Xδ is determined by bucket half-height δh=1.539×10-3

For TAL2 azimuth s=sTAL2 → Dx=3.4015

For run with Pb ions with pz=120 GeV/c → xTAL=-8.361 mm

Xβm(sTAL2) = -8.4915 mm

Maximum of total displacement for bunched particle in TAL2

min Xβ,δ = Xβm(sTAL2) + Xδ(STAL2,δh) = -13.7264 mm

Xδ(STAL2,-δh)=-5.2349 mm

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

Off-momentum halo for 120 GeV/c protons in TAL2 azimuth----------------------------------------------------------------------

Ionization losses of protons in bent crystal, ∆δ < 0, lead to their debunchingHalo tail appears with X < - max Xβ,δ

Particles will be lost somewhere in the ring if their coordinate at TAL2 X < Xleak – leakage boundary coordinate

Xleak ? ↓

Large part of debunched particles have possibility to be collimated at subsequent turns

Xleak is determined by value of δl for which at HD azimuth SAL with smallest aperture XAL

Xβm(sAL) + |Xδ(SAL,δl)| = XAL

Off-momentum halo for 120∙Z GeV/c Pb ions in TAL2 azimuth----------------------------------------------------------------------

Xleak ? ↓

Ionization losses increases up to 6.6 GeV for one passageThree passages through non-aligned crystal are sufficient for debunching

Strong energy losses lead to betatron amplitude increase They more quickly reach TAL aperture

To increase channeling efficiency due to contribution of multiple passagesIt is necessary to increase Xoff distance between BC and TAL

Conclusion --------------------------------------------------

Simulation show

(1) Miscut really increases beam losses in channeling orientation

This increase may be larger than 100%

(2) Calculated level of beam losses for amorphous orientationshould be corrected because of contribution from strong single Coulomb scattering

It decreases by about 10%

(1) and (2) decreasing the calculated loss reductionconsiderably improve agreement of simulation with experiment

Halo propagation rate λ should be measured to use its correct value in simulations