Vertical dynamic aperture with radiation from quadrupoles - - PowerPoint PPT Presentation

Vertical dynamic aperture with radiation from quadrupoles FCCee_z_202_nosol_13.seq at 45.6 GeV

• A. Bogomyagkov, E. Levichev, S. Glukhov, S. Sinyatkin

Budker Institute of Nuclear Physics Novosibirsk

July, 2017

• A. Bogomyagkov (BINP)

FCC-ee DA 1 / 15

6d (SR from BEND, QUAD) and 4d tracking: XY

6d(SR)

x

σ X/ 40 − 30 − 20 − 10 − 10 20 30 40

y

σ Y/ 80 − 60 − 40 − 20 − 20 40 60

=3.8e-004

e

σ =3.1e-008 m,

y

σ =6.3e-006 m,

x

σ

Rx = 35σx Ry = 40σy 4d

x

σ X/ 150 − 100 − 50 − 50 100 150

y

σ Y/ 150 − 100 − 50 − 50 100 150

=3.8e-004

e

σ =3.1e-008 m,

y

σ =6.3e-006 m,

x

σ

Rx = 109σx Ry = 142σy

• A. Bogomyagkov (BINP)

FCC-ee DA 2 / 15

6d (SR from BEND, QUAD) and 6d tracking: XY

6d(SR)

x

σ X/ 40 − 30 − 20 − 10 − 10 20 30 40

y

σ Y/ 80 − 60 − 40 − 20 − 20 40 60

=3.8e-004

e

σ =3.1e-008 m,

y

σ =6.3e-006 m,

x

σ

Rx = 35σx Ry = 40σy 6d

x

σ X/ 100 − 50 − 50 100

y

σ Y/ 150 − 100 − 50 − 50 100 150

=3.8e-004

e

σ =3.1e-008 m,

y

σ =6.3e-006 m,

x

σ

Rx = 109σx Ry = 142σy

• A. Bogomyagkov (BINP)

FCC-ee DA 3 / 15

Introduction

Problem

Why vertical dynamic aperture drops from Ry = 142σy to Ry = 40 ÷ 50σy?

PTC

ptc_create_layout, time=true, model=2, exact=true, method=6, nst=10; ptc_setswitch, fringe=true, debuglevel=1,radiation=true; Yin = y0 + j ∗ dy; Tin = 0.225 ∗ ((Yin/y0)2) ∗ σt; ptc_start, x=0, px=0, y=Yin, py=0,pt=0, T=Tin; ptc_track,icase=6,closed_orbit,dump,maxaper={1,1,1,1,1,1}, turns=5200,ffile=1,element_by_element;

• A. Bogomyagkov (BINP)

FCC-ee DA 4 / 15

6d (SR from BEND, QUAD)

x

σ X/ 2 − 1 − 1 2

px

σ PX/ 20 − 15 − 10 − 5 − 5

PX:X

y

σ Y/ 150 − 100 − 50 − 50 100 150

py

σ PY/ 200 − 150 − 100 − 50 − 50 100 150

PY:Y

t

σ T/ 1 2 3 4 5 6 7

t

σ PT/ 5 − 4 − 3 − 2 − 1 −

PT:T

turn 1000 2000 3000 4000 5000

t

σ PT/ 5 − 4 − 3 − 2 − 1 −

PT:turn

turn 1000 2000 3000 4000 5000

t

σ T/ 0.1 0.2 0.3 0.4 0.5 0.6 0.7

T:turn

x

σ X/ 1 − 0.5 − 0.5 1

y

σ Y/ 50 51 52 53 54 55 56 57 58

=3.8e-004

e

σ =3.1e-008 m,

y

σ =6.3e-006 m,

x

σ

• A. Bogomyagkov (BINP)

FCC-ee DA 5 / 15

6d (SR from BEND, QUAD) damping

turn 1000 2000 3000 4000 5000

y

σ Y/ 40 − 20 − 20 40 60

}

y

σ Y:turn {part==1, Yin=50

turn 1000 2000 3000 4000 5000

y

σ Y/ 40 − 20 − 20 40 60

}

y

σ Y:turn {part==5, Yin=52

turn 1000 2000 3000 4000 5000

y

σ Y/ 40 − 20 − 20 40 60

}

y

σ Y:turn {part==11, Yin=55

turn 1000 2000 3000 4000 5000

y

σ Y/ 40 − 20 − 20 40 60

}

y

σ Y:turn {part==16, Yin=57.5

turn 200 400 600 800 1000 1200 1400 1600 1800 2000 2200

y

σ Y/ 200 − 100 − 100 200 300 400

}

y

σ Y:turn {part==17, Yin=58

turn 200 400 600 800 1000 1200 1400 1600

y

σ Y/ 400 − 300 − 200 − 100 − 100 200 300

}

y

σ Y:turn {part==18, Yin=58.5

• A. Bogomyagkov (BINP)

FCC-ee DA 6 / 15

History

References

John M. Jowett (SLAC), Introductory Statistical Mechanics for Electron Storage Rings, AIP Conf.Proc. 153 (1987) 864-970 J.M. Jowett (CERN), Nonlinear Dissipative Phenomena In Electron Storage Rings, Lect.Notes Phys. 247 (1986) 343-366 In *Santa Margherita Di Pula 1985, Proceedings, Nonlinear Dynamics Aspects Of Particle Accelerators*, 343-366

• J. Jowett (CERN), Dynamic aperture for LEP: Physics and

calculations, Conf.Proc. C9401174 (1994) 47-71, In *Chamonix 1994, LEP performance* 47-71

Comments

Some tracking plots are similar. There was no mentioning of damping turning into raising.

• A. Bogomyagkov (BINP)

FCC-ee DA 7 / 15

Parameters (radiation ON, no tappering)

Energy: E = 45.6 Gev. Tunes: νs = 0.0413, νy = 0.2217, νs = 0.1366 Damping times [turns]: τs = 1300, τy = 2600, τx = 2600 Energy loss: U0 = 35.96 MeV/turn

Uq(FF, 50σy) = 4 × 0.5 MeV, Uq(FF, 50σx) = 4 × 3 MeV envelope, Ud(B, arc) = 12.4 keV, Uq(QF, 50σx) = 2.8 keV, Uq(QF, 50σy) = 2.5 eV, Uq(QD, 50σx) = 1 keV, Uq(QD, 50σy) = 10 eV

• A. Bogomyagkov (BINP)

FCC-ee DA 8 / 15

Equations of motion: longitudinal

Exact

σ′ = −K0x − p2

x

2 − p2

y

2 p′

t =

• −eV0

p0c

• sin
• φs + 2πσ

λ

• δ(s − s0) − Cγ

2π E4 p0c K 2

0 (1 + 2pt)

− Cγ 2π E4 p0c K 2

1 (x2 + y2)

Average, xβ = 0

σ′ = −αpt − Jy γ 2 p′

t =

• − eV0

p0cΠ

• sin
• φs + 2πσ

λ

• −

U0 p0cΠ(1 + 2pt) − Cγ 2π E4 p0cΠK 2

1 Lqy2 q

• A. Bogomyagkov (BINP)

FCC-ee DA 9 / 15

Synchronous phase

No synchrotron oscillations

σ′ = 0, p′

t = 0,

Jy ∝ exp 2n τy

• sin [φs] =

U0 (−eV0)

• 1 − Jy γ

α

• + Uq(σy)

(−eV0) Jy εy

turn 1000 2000 3000 4000 5000

t

σ T/ 0.1 0.2 0.3 0.4 0.5 0.6 0.7

T:turn {part==1 || part==17}

yin=50σy, τy=-2.6 103 yin=58σy, τy=5 103

1000 2000 3000 4000 5000 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Turn ϕs λRF 2 π σs

• A. Bogomyagkov (BINP)

FCC-ee DA 10 / 15

Equations of motion: longitudinal

Average, xβ = 0

σ′ = −αpt − Jy γ 2 p′

t =

• − eV0

p0cΠ

• sin
• φs + 2πσ

λ

• −

U0 p0cΠ(1 + 2pt) − Uq(σy) p0cΠ y2

q

σ2

q,y

yq = 2 |A|

• βq,y cos(ψq + kys) = Afq + A∗f ∗

q

Solution

pt = Be−αts cos (kss) − Jy γ 2α − Uq(σy) p0cΠ Jy k2

s σ2 q,y

β′

q,y

− Fy 4k2

y

ei2kys − F ∗

y

4k2

y

e−i2kys Fy = −Uq(σy) p0cΠ Jyfqf ′

q

σ2

y

, ky = 2π{νy} Π

• A. Bogomyagkov (BINP)

FCC-ee DA 11 / 15

Equations of motion: vertical

Exact

y′ = py(1 − pt) p′

y = K1y − py

Cγ 2π E4 p0c K 2

0 − pyy2 Cγ

2π E4 p0c K 2

1

Map of quadrupole radiation

∆py = −py,0y2 Cγ 2π E4 p0c K 2

1 Lq ,

Cγ 2π E4 p0c K 2

1 Lq = Uq(σy)

E0σ2

y

≈ 0.7m−2

Map of quadrupole fringe

∆py = −py,0y2 K1 4 , K1/4 ≈ 0.15m−2

• A. Bogomyagkov (BINP)

FCC-ee DA 12 / 15

Vertical dynamic aperture limit

Solving: parameter variation and averaging

y(s) = A(s)fq(s) + A(s)∗fq(s)∗, py(s) = A(s)f ′

q(s) + A(s)∗f ′ q(s)∗

Solution

J′

y = −2αyJy + Uq(σy)

4p0c cos(2ψq){νy} νy

• −1 + (β′

y/2)2

βy

• J2

y

Πσ2

q,yk2 y

DA limit

J′

y = 0

Jy = U0 Uq(σy) 4σ2

q,yk2 y

cos(2ψq) {νy}

νy

−1+(β′

y/2)2

βy

∝ K 2 K 2

1 Lq

Jy ≈ 50σy

• A. Bogomyagkov (BINP)

FCC-ee DA 13 / 15

Parametric resonance and Van der Pol oscillator

Exact: p′′

y = K1py(1 − pt) − p′ y

Cγ 2π E4 p0c K 2

0 − (pyy2)′ Cγ

2π E4 p0c K 2

1

Illustration: y′′ + k2

y

• 1 − F1y2 cos(2kys)
• y + 2αy′ = 0

y[0]=1.5 103

1000 2000 3000 4000 5000

• 1500
• 1000
• 500

500 1000 1500 Turn y

y[0]=1.73 103

1000 2000 3000 4000 5000

• 6000
• 4000
• 2000

2000 4000 6000 Turn y

Van der Pol oscillator: y′′ + k2

y y + 2αy′

1 − F1y2 = 0

• A. Bogomyagkov (BINP)

FCC-ee DA 14 / 15

Conclusion for vertical plane at 45.6 GeV

1

Observed and studied a new effect limiting dynamic aperture.

2

Radiation from FF quadrupoles modulates pt at double betatron frequency.

3

Map of radiation from FF quadrupole is similar to quadrupole fringe and of the same value.

4

Parametric resonance in vertical motion changes damping. It is

• bserved in tracking and obtained by equations.

5

Vertical dynamic aperture is limited by this effect.

6

Chromaticity of vertical beta function in the quadrupole will change estimations, because particle amplitude will depend on its

• energy. Minimization of β and β′ chromaticity in the quadrupole is

beneficial.

• A. Bogomyagkov (BINP)

FCC-ee DA 15 / 15