nonultrarelativistic resistive wall wake fields

Nonultrarelativisticresistive wallwake fields

Diego QuatraroCERN

&Bologna University

Motivations andconcepts

The fields and thetheir calculation

Possible effects onbunch dynamics

Conclusions

Nonultrarelativistic resistive wallwake fields

Diego QuatraroCERN

&Bologna University

April 7, 2008Many thanks to...G. Rumolo, E. Metral, B. Salvant, B. Zotter..


Diego QuatraroCERN

&Bologna University




Conclusions

Outline

Motivations and concepts

The fields and the their calculation

Possible effects on bunch dynamics

Conclusions


Diego QuatraroCERN

&Bologna University




Conclusions

The ideas...1/2

Ring Momentum β γ

PSB @ Injection 0.31 GeV/c ∼ 0.31 ∼ 1.05PSB @ Extraction 1.4 GeV/c ∼ 0.84 ∼ 1.79PS @ Extraction 14 GeV/c ∼ 0.9977 ∼ 14.954


Diego QuatraroCERN

&Bologna University




Conclusions

The ideas...1/2

Ring Momentum β γ


Nonultrarelativistic regime


Diego QuatraroCERN

&Bologna University




Conclusions

The ideas...1/2

Ring Momentum β γ


Nonultrarelativistic regime⇒ different approach should be considered


Diego QuatraroCERN

&Bologna University




Conclusions

The ideas...2/2

i) Electromagnetic field travels with the speed of light...


Diego QuatraroCERN

&Bologna University




Conclusions

The ideas...2/2


ii) Bunches could travel slower than light...PSB injection β ' 0.31.


Diego QuatraroCERN

&Bologna University




Conclusions

The ideas...2/2



iii) The tail can have an effect on the head dynamics.


Diego QuatraroCERN

&Bologna University




Conclusions

The ideas...2/2



iii) The tail can have an effect on the head dynamics.

PSfrag replacements

γ

p+

β ' 1

γ

γ

p+p+

β < 1


Diego QuatraroCERN

&Bologna University




Conclusions

Outline




Conclusions


Diego QuatraroCERN

&Bologna University




Conclusions

Longitudinal wake field 1/7

Definition of the wake field

W ′0(z) =

12π

∫ ∞

−∞Z‖(ω)eiωz/βcdω (1)

non ultrarelativistic beam F. Zimmermann and K. Oide, PRLSTAB 7, 044201 (2004)

Z‖(ω) = iZ0ck2

r

2πω[K0 (kr r) +

I0(kr r)ω2λK1(bkr )K0(bλ) + kr c2

(

λ2 − k2)

K0(bkr )K1(bλ)

ω2λI1(bkr )K0(bλ)− kr c2 (λ2 − k2) I0(bkr )K1(bλ)

]

(2)kr = |ω|/γβc, k = ω/βc, λ2 = −iµ0σω + k2

r andb = beam pipe radius.


Diego QuatraroCERN

&Bologna University




Conclusions


Subtracting the space charge term from Eq. (2) andletting

|λb| 1 → PSB case√

ωµ0σ ' 2 · 10−2[s1/2]√

ω

|kr b| 1 → PSB case ωb/βγc ' 5.3× 10−10[s]ω(3)

we have

Z‖(ω) = iZ0ω

cβ2γ2 I0(r |ω|/γβc)e−2b|ω|/γβc ·

·[

√

|ω|σµ0 (1− i · sgn (ω))√

|ω|σµ0 (1− i · sgn (ω)) +√

2 · sgn (ω) c/βγ (iµ0σ + ω/c2)

]

.

(4)range of validity ω such that |ωb/βγc| 1: shortdistances.


Diego QuatraroCERN

&Bologna University




Conclusions


log− log plot of Z‖(ω) function

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1000 1e+06 1e+09

Z⁄

⁄ (ω

)[Ω

]

ω [Hz]

exact Re(Z) β = 0.31

approximate Re(Z) β = 0.31

exact Im(Z) β = 0.31

approximate Im(Z) β = 0.31

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

1000 1e+06 1e+09 1e+12

Z⁄

⁄ (ω

)[Ω

]

ω [Hz]

exact Re(Z) β = 0.999

approximate Re(Z) β = 0.999

exact Im(Z) β = 0.999

approximate Im(Z) β = 0.999

ωsup as β → 1.


Diego QuatraroCERN

&Bologna University




Conclusions

Longitudinal wake field 4/7FFT of impedance Eq. (2) (Exact) andEq. (4)(Approximate)

1⋅103

1⋅104

1⋅105

1⋅106

1⋅107

1⋅108

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

ln

W0’(z)

[Ω

s-1

]

z[m]

β = 0.1

Exact Wake

Approximate Wake

1⋅103

1⋅104

1⋅105

1⋅106

1⋅107

1⋅108

1⋅109

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

ln

W0’(z)

[Ω

s-1

]z[m]

β = 0.2

Exact Wake

Approximate Wake

1⋅105

1⋅106

1⋅107

1⋅108

1⋅109

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

ln

W0’(z)

[Ω

s-1

]

z[m]

β = 0.3

Exact Wake

Approximate Wake

1⋅104

1⋅105

1⋅106

1⋅107

1⋅108

1⋅109

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

ln

W0’(z)

[Ω

s-1

]

z[m]

β = 0.4

Exact Wake

Approximate Wake


Diego QuatraroCERN

&Bologna University




Conclusions


Field in front of the bunch, FFT of exact formula...

1e+06

1e+09

1e+12

0 0.05 0.1 0.15 0.2 0.25

ln

W0’(z)

[Ω

s-1

]

z[m]

Front field

PS Extraction ⇒ β = 0.998PSB Extraction ⇒ β = 0.83

PSB Injection ⇒ β = 0.31


Diego QuatraroCERN

&Bologna University




Conclusions


Field in front of the bunch, FFT of exact formula...

1e+06

1e+09

1e+12

0 0.05 0.1 0.15 0.2 0.25

ln

W0’(z)

[Ω

s-1

]

z[m]

Front field

PS Extraction ⇒ β = 0.998PSB Extraction ⇒ β = 0.83

PSB Injection ⇒ β = 0.31

..for the Head and the Tail of the bunch...


Diego QuatraroCERN

&Bologna University




Conclusions


10000

1e+06

1e+08

1e+10

1e+12

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

ln

W0’(z)

[Ω

s-1

]

z[m]

LongitudinalField

longitudinal CHAOT. field at β = 0.31F. field at β = 0.31

T. field at β = 0.998F. field at β = 0.998 1e+09

1e+12

-0.005 0.000 0.005

Agreement with well-known Chao’s ultrarelativistic formula


Diego QuatraroCERN

&Bologna University




Conclusions


For the ultrarelativistic case...

1⋅108

1⋅1010

1⋅1012

1⋅1014

-0.1 -0.08 -0.06 -0.04 -0.02 0

ln W

0’(z

) [Ω

s-1

]

z[m]

b = 5 cm

β ≈ 1

Chao’s formula

1⋅1010

1⋅1012

1⋅1014

-0.008 -0.004 0

we have a more complete description of the field close tothe bunch.


Diego QuatraroCERN

&Bologna University




Conclusions

Outline




Conclusions


Diego QuatraroCERN

&Bologna University




Conclusions

Strong head-tail instability 1/3TRANSVERSE plane simple model


Diego QuatraroCERN

&Bologna University




Conclusions

Strong head-tail instability 1/3TRANSVERSE plane simple model → two particles beam

A. Chao, Physics of Collective Instabilities in High Energy Accelerators

i) two particle beam→ charge Ne/2


Diego QuatraroCERN

&Bologna University




Conclusions



i) two particle beam→ charge Ne/2ii) s/c ∈ [0; Ts/2] 1 leads 2 and vice versa for

s/c ∈ [Ts/2; Ts]


Diego QuatraroCERN

&Bologna University




Conclusions




s/c ∈ [Ts/2; Ts]iii) constant wake functions


Diego QuatraroCERN

&Bologna University




Conclusions





a) WT (z) = −W0 → action from the leading to thetrailing

b) WH (z) = W1 → action from the trailing to the leading


Diego QuatraroCERN

&Bologna University




Conclusions






b) WH (z) = W1 → action from the trailing to the leadingiv) betatron oscillation with ωβ frequency


Diego QuatraroCERN

&Bologna University




Conclusions







~x =(

x1, x ′1, x2, x ′2)

⇒ ~x = ~Φ(~x)Equation of motion for half a period TS/2 = π/ωS , then1 → 2


Diego QuatraroCERN

&Bologna University




Conclusions







~x =(

x1, x ′1, x2, x ′2)

⇒ ~x = ~Φ(~x)Equation of motion for half a period TS/2 = π/ωS , then1 → 2

~Φ =

x ′1− (ωβ/c)2 x1 + α1x2

x ′2− (ωβ/c)2 x2 + α2x1

, αj =Nr0Wj

2γCj = 1, 2.

(5)


Diego QuatraroCERN

&Bologna University




Conclusions

Strong head-tail instability 2/3

Basin of stability obtained from 4-th order Runge-Kuttaintegration of Eq. (5).Γ = πNr0W1c2

4γCωsωβ


Diego QuatraroCERN

&Bologna University




Conclusions



4γCωsωβUltrarelativistic beam ⇒


Diego QuatraroCERN

&Bologna University




Conclusions



4γCωsωβUltrarelativistic beam ⇒

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Γ

ωβ / ωs

α1 = 0.0

Dotted region ⇒ unstable system.


Diego QuatraroCERN

&Bologna University




Conclusions


0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Γ

ωβ / ωs

α1 = 0.1α2

Border of stability for u. r. case

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6Γ

ωβ / ωs

α1 = 0.5α2


0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Γ

ωβ / ωs

α1 = -0.1α2


0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Γ

ωβ / ωs

α1 = -0.5α2



Diego QuatraroCERN

&Bologna University




Conclusions

Outline




Conclusions


Diego QuatraroCERN

&Bologna University




Conclusions

I Extent of the wake field concept fornonultrarelativistic case


Diego QuatraroCERN

&Bologna University




Conclusions


I Numerical evidence of front field fornonultrarelativistic bunches


Diego QuatraroCERN

&Bologna University




Conclusions



I For the longitudinal plane the resistive wall front fieldseems to be relevant


Diego QuatraroCERN

&Bologna University




Conclusions



I For the longitudinal plane the resistive wall front fieldseems to be relevant

I Work in the transverse plane still ongoingHowever preliminary simulations based on twoparticle model seem to show a different behaviourconcerning the beam stability

nonultrarelativistic resistive wall wake fields

Documents