The Touschek Modulein MAD-X

Catia Milardi (LNF-INFN)

Frank Zimmermann (CERN)

Frank Schmidt (CERN)

Topics

•Theoretical approach

•User instructions

•Differences with MAD-8

•Examples

•Limitations

•Further developments

Particles in a relativistic bunch undergo Coulomb scattering involving:

• multiple hits leading to increase of all beam sizes (IBS)

• two particles transforming a small transverse momentum in to a large longitudinal one with consequent loss of both particles (Touschek effect).

The Touschek effect depends on beam energy and on the bunch volume so it is much more concerning for:

• low-energy machines• small emmittance machine• high bunch current

Generalities

Piwinski’s approach

Piwinski, TheTouschek effect in strong focusing storage ring, Desy 98-179, Nov. 1998.

The particle loss rate 1/T due to the Touschek effect is computed:

• considering two arbitrary particles in their center of mass system

• writing the Møller scattering cross section taking into account:

- two-dimensional distribution of transverse momenta (due to both betatron and synchrotron oscillation)

- beam envelopes variation- gaussian distribution in all beam coordinates

€

1

T=

rp2cN p

8πγ 2σ s σ x2σ z

2 −σ p4Dx

2Dz2τ m

F(τ m,B1,B2)

• rp classical particle radius• Np number of particles per bunch• particle energy in rest mass unit• s bunch length• x,z standard deviation of the beam transverse size• p relative momentum spread• Dx,z dispersion function

Particle loss rate due to theTouschek effect

€

B1 =β x

2

2β 2γ 2σ xβ2

1−σ h

2 ˜ D x2

σ xβ2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟+

β z2

2β 2γ 2σ zβ2

1−σ h

2 ˜ D z2

σ zβ2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

B2 = B12 −

β x2β z

2σ h2

β 4γ 4σ xβ4 σ zβ

4 σ p2

σ x2σ z

2 −σ p4Dx

2Dz2

( )

€

1

σ h2

=1

σ xβ2 σ zβ

2 σ p2

˜ σ x2σ zβ

2 + ˜ σ z2σ xβ

2 −σ xβ2 σ zβ

2( )€

˜ σ x,z = σ xβ ,zβ2 + σ p

2 Dx,z2 + ˜ D x,z

2( )€

˜ D x,z = α x,zDx,z + β x,zDx,z'

€

F τ m ,B1,B2( ) = τ m π B12 − B2

2( ) G τ ,B1,B2( )

τ m

∞

∫ dτ

€

τm = β 2 Δpm

p

⎛

⎝ ⎜

⎞

⎠ ⎟

2

Piwinski’s integral

€

G τ ,B1,B2( ) = 2 +1

τ

⎛

⎝ ⎜

⎞

⎠ ⎟2

τ /τ m

1+τ−1

⎛

⎝ ⎜

⎞

⎠ ⎟+1−

1+τ

τ /τ m

−1

2τ4 +

1

τ

⎛

⎝ ⎜

⎞

⎠ ⎟ln

τ /τ m

1+τ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟e−B1τ I0 B2τ( )

τ

1+τ

I0 is the 0th order Bessel function

€

F τ m ,B1,B2( ) = 2τ m π B12 − B2

2( ) G τ ,B1,B2( )

km

π / 2

∫ dk

€

τ =tan2 κ

€

κm = tan−1 τ m

€

G τ ,B1,B2( ) =2τ +1( )

2

τ

τ /τ m

1+τ−1

⎛

⎝ ⎜

⎞

⎠ ⎟+τ − ττ m 1+τ( ) − 2 +

1

2τ

⎛

⎝ ⎜

⎞

⎠ ⎟ln

τ /τ m

1+τ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟e−B1τ I0 B2τ( ) 1+τ

€

dτ

dκ= 2tgκ tg2κ −1( )

… in order to simplify numerical integration

RF momentum acceptance

€

Δpm

p

⎛

⎝ ⎜

⎞

⎠ ⎟

2

=V0 sinϕ s

πηhE2 q2 −1 − arccos

1

q

⎛

⎝ ⎜

⎞

⎠ ⎟

€

=V0

U0

If U0 = 0

€

=αc −1

γ 2 €

Δpm

p

⎛

⎝ ⎜

⎞

⎠ ⎟

2

=2V0

πηhE

computed from the bucket size taking into account the energy loss per turn due to synchrotron radiation U0

V0 RF voltage

s RF synch phase

h harmonic number

q overvoltage factor

The Touschek module can handle machines with more than one RF systems with different frequencies and/or voltages.

The momentum acceptance is computed assuming:• The maximum phase on the separatrix is defined by the reference RF having the lowest frequency and voltage V ≠ 0.• All systems are synchronized with the reference RF.

Different RF arrangements are not supported and would require a modification of the module !!

Momentum acceptancein presence of several RF systems

€

U0 =2

3

rp

(mc 2)3β 3E0

4I2

€

U0 =4

3π

rp

(mc 2)3β 3E0

4R1

ρ 2

€

1

ρ 2=

1

2πR

1

ρ 20

2πR

∫ ds =1

2πRI2

Energy loss per turn due to SR

U0 = 0 when I2 is:vanishing,undefined or not computed

is computed from I2 returned by the TWISS_summ table when the TWISS chrom option is invoked

The BEAM command computes the synchrotron tune qs from the RF cavity parameters.Then s and p for leptons come from:

€

s =LηE t

2πqs

€

p =2πqsE t

Lη

About s and p

Et longitudinal emittanceL machine circumference

Defining Et < 0 in the BEAM command allows the user to assign the s and p values

Using a more accurate routine (CJYDBB) for I0 calculation produces negligible differences in the evaluation of the Piwinski G function, but removing the discontinuity results in a reduction by a factor 2 in the overall execution time.

For a generic element in the DANE lattice

0

1000

2000

3000

4000

5000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

G(τ,B1,B

2) CJYDBB

(G τ,B1,B

2) DBESIO from CERNLib

κ

1100

1200

1300

1400

1500

1600

0.07 0.075 0.08 0.085 0.09 0.095

G(τ,B1,B

2) CJYDBB

G(τ,B1,B

2) DBESIO from CERNLib

κ

Touschek lifetime evaluation in MAD-8

T is computed when the BMPM TOUSCH option is invoked.

The BMPM module could not handle:• combined function dipoles• machine elements with TILT ≠ 0

It was based on some simplifying assumptions about:

• beam emittance• damping partition numbers

Computation formalism see Gygi, Keil, Jowett, BeamParam, CERN/LEP-TH/88-2

………RFCAV: rfcavity, l = lcav, volt = 0.09, lag = 0.49, harmon = 120, tfill = 1., shunt = 0.2, pg = 0.1;

RESBEAM;

BEAM, PARTICLE = positron, energy = .51,sequence = wholekf , npart = 2.E+10, ex = .4E-6 , ey = 1.1964E-9 , et = -6.E-6 , sige = .0003, sigT = .02, radiate = true;……… use, period = wholekf;

twiss, chrom, sequence = wholekf, centre, table, file =wholekf0.txt, save;

touschek, tolerance = 1.e-07, file = name;

STOP;

Input Instructions

http://mad.home.cern.ch/mad/touschek/touschek.html

Output File

name beamline element

s azimuth of the element center

tli element contribution to 1/T

tliw tli*Le/L Le element length, L ring circumference

tlitot sum of loss rates tliw through the beamline elements

The Touschek lifetime T is:

T = 1/tlitot @ the end of the beamline

0

0.0005

0.001

0.0015

0.002

0 20 40 60 80 100

azimuth [m]

Contributions tliw to the Loss Ratefrom the elements of the DA NE lattice

Touschek lifetime, emittances etc. vs. RF voltagefor CLIC (DR)

beam parameters were computedby Maxim and include IBS effect

Machine E [GeV] ESR [MeV] T [s] T [h]

DANE 0.51 9.68 10-3 3661.59 1.017

CLIC DR 2.424 2.19 183.45 0.051

LHC (inj.) 450. 1.14 10-7 192.64 105 5351.

LHC (top) 7000. .00668 552.85 105 15357.

Touschek lifetime evaluation

http://frs.home.cern.ch/frs/mad-X_examples/touschek/

DANE experiences limitations of the transverse aperture

due to the presence of:• COLLIMATORS affecting physical aperture• WIGGLERS and SEXTUPOLES having non-linear

terms affecting dynamic aperture

As a consequence the maximum stable relative momentum

deviation evaluated from RF acceptance is larger than the

real one leading to an optimistic evaluation of the Touschek

lifetime by a factor ≈2.

Further developments

Introduce the dynamic momentum Acceptance Ap

in the evaluation of the Touschek Lifetime (if smaller than RF acceptance)

Since Ap depends on:•geometrical aperture• tune shift on amplitude from non linear terms• tune proximity to resonances

it can be evaluated by tracking processes only and then passed as an input parameter to the Touschek module.

Introduce the exact evaluation of the equilibrium energy spread, in order to take into account rings with SRFF

Add Uo to the input parameters in order to deal with high order modes