A new algorithm for the kinetic analysis of intra-beam scattering in storage rings.

P.R.Zenkevich*,O. Boine-Frenkenheim**,A. Ye. Bolshakov*

*ITEP, Moscow, Russia *GSI, Darmstadt, Germany

Acknowledgements.

• This work is performed within framework of INTAS-GSI grant “Advanced Physics Dynamics) Ref. Nr. 03-545584

• I am very grateful for Prof. Hofmann (GSI)

for a possibility to take part in FAIR project

and to Prof. Turketti for the invitation at this very interesting Workshop.

Contents.

• Introduction.• Fokker-Planck equation (FPE) in

momentum space.• Invariants and its evolution. • Simplified model of FPE.• Langevin equations.• Multi-particle code and its applications• Discussion.

Introduction 1.

• IBS includes:1) Multiple IBS.2) Single-event IBS (Touschek effect). This

effect is out of frame of this report.• Multiple IBS results in:

1) Transfer of energy from hot transverse degrees of freedom to cold longitudinal one.

2) Slow growth of 6-dimensional beam emittance due to dispersion function and modulation of Twiss parameters.

Gaussian models and kinetic effects.

• Gaussian model : simulation of rms invariants evolution is based on assumption that the beam

has Gaussian distribution on all degrees of freedom.• Numerical codes for rms invariants evolution:

- Mohl and Giannini, Katayama and Rao. - BETACOOL (IBS+e-cool+Beam-Target Interaction (BTI)).

• Why we need kinetic description? - Solution of kinetic equation is not Gaussian with account of boundary

conditions (particle losses). - Non-linear or stochastic effects (for example, e-cooling or beam-target

interaction) could result in Non-Gausian tails. - These Non-Gaussian tails can be essential; for example particles in

tails can significantly influence on detector noises in colliders.

MOnte-CArlo Code (MOCAC).

• The single known three-dimensional numerical code for IBS study is MOCAC code (Zenkevich, Bolshakov).

• MOCAC program is based on idea to change the real IBS by a set of artificial “scattering” events constructed such a way that the average invariants rates are same as due to real IBS process: so named Binary Collision Model (BCM).

• Main drawback of the code: we need in large number of macro-particles and large computer time.

• Here we proposed some approximate “Approximate Model” (AM) where we calculate a motion of the macro-particles in assumption that the beam is Gaussian one.

Coordinates-momentums.

• Let us introduce “coordinate vector” and “momentum vector”:

• Here z – is longitudinal coordinate, x – is horizontal coordinate,

y – is the vertical one.

• Correspondingly, - is the first component of the momentum,

- is the second component,

- is the third one.

1

,s

p

z z p

r x P x

y y

1

1 pP

p

xy

FPE in coordinate-momentum space 1.

• Evolution of the distribution function in infinite medium is described by following FPE:

• Friction force due to multiple IBS (u corresponds to “test” particle, w to the “field” one)

• Components of the diffusion tensor

2

,,

1[ ] ( )

2fr i ji j i j

fF f D f

t u P P

0 3( ) [ ]fr C F

u wF u A L

u w

2

, 0 3

( )( )[ ]

2i i j jC

i j F

u w u w u wLD A

u w

FPE in coordinate-momentum space 2.

• Here Coulomb logarithm and constant

• Here averaging on the field particles corresponds to the following operator:

20CL

2

0 3 5

2 icrA

[ ( , , )] ( , , ) ( , , )Fu w t u w t f r w t dw

Invariants and its evolution due to IBS. • Linear particle motion is described by conservation of “invariants”:

• Here for m=2,3 are “Twiss functions” depending on longitudinal variable s;

here D and are dispersion function and its derivative;• For coasting beams (CB)

• For bunched beams (BB)

, ,m m m

D

2 22m m m m m m m mI r r P P

1 1

1

,s

p

z z p

r x D P P x D P

y y

1 1 10, 0, 1

1 10, 1

2 2 2 21 / [(1/ ) ]s R

Invariants and its evolution due to IBS.

• In scattering event the coordinates does not change; therefore we have:

• The momentum derivatives

• Here is Kronecker-Kapelli symbol, x x x xD D D

21

22 22 21 1

2

( )

( ) ( )( )2 ( )

( )

xx

y

d P

dt

d Px d PdI d xD D D

dt dt dt dt

d y

dt

2

,03

( ) 3( )( )

2i j i j i i j jC

d PP u w u w u wA L

dt u w

,i j

Approximated form of FPE (1)

The initial assumptions of the model:

• Gaussian beam.

• Coulomb logarithm is constant.

• The components of the friction force with constant coefficients

• The components of the diffusion tensor are constants.

To provide same invariant rates we should average friction force and diffusion coefficients on test and field particles (the notation for this operator is ). Then we obtain:

iK

,i jD

20

3

( )

2C i i

i

A L u wK

u w

2

,0, 3

( )( )

2i j i i j jC

i j

u w u w u wA LD

u w

i i iF K P

Approximated form of FPE (2)

• Averaging on the field and test particles as in Piwinski-Martini and Bjorken-Mtingwa theories we obtain:

• Here the distribution function is defined by

• Here parameters and are:

22

3, 3/ 2( , )

i

i i N

P PD A P s d P

P

23

3/ 22( , )N i

i

i

A PK P s d P

P P

32

1 21

( , ) exp[ ( 2 ] / 4i ii

P s a P PP

ia

2 2 231 2

21 ( ), ,

2 2 2yx

z x x x y

aa aD D

I I I I

2, x x

x

DD D D

I

Approximated form of FPE (3)

• Integrating according to Bjorken-Mtingwa scheme [5] we obtain the following final expressions:

• Here the normalizing constant

• The integrals are:

2, / ( )i N i i iK A INT P

3

, , , ,1

( )i j N i j i i i ji

D A INT INT

2

3 4

,

,4 ,

2

C iN

bx y p

z

NCB

LcL rA

NI I BB

21,1 2 3/ 2 1/ 2

1 2 30

( )4

[( )( ) ] ( )

a dINT

a a a

3,3 2 1/ 2 3/ 21 2 30

4[( )( ) ] ( )

dINT

a a a

3,3 2 1/ 2 3/ 21 2 30

4[( )( ) ] ( )

dINT

a a a

1,2 2 3/ 2 1/ 2

1 2 30

8[( )( ) ] ( )

dINT

a a a

Approximated form of FPE (4)

Using these formulae it was written subroutine, which for set of points with known longitudinal variable s and, correspondingly, known Twiss parameters, dispersion and its derivative) calculates:

1. Average friction coefficients .

2. Average diffusion coefficients .

3. Average rates of moments change

4. Average rates of the invariant change

2 2 2

1 2 3

221 ( )2[ ], , yx

z x x x y

D Da a a

I I I I

2, x x

x

DD D D

I

iK

,i jD( )i jd PP

dtidI

dt

Solution of approximate FPE using Langevin Equations (LE).

• Application of LE to three dimensional FPE with non-diagonal diffusion tensor is not a trivial procedure. At this case the LE can be written in the following generalized form:

• Here are three random numbers with Gaussian distribution and unity dispersion; coefficients take into account correlations between coupled between horizontal and longitudinal degrees of freedom. Averaging on the possible values of the random numbers and on the test and field particles, we obtain the following equations for the coefficients :

3

,1

( ) ( ) ( )i i i i i j jj

P t t P t K P t t t C

j,i jC

1,1 11

1 221 1 2 1 2

1,1

222 22 21

( )1( )

C D

d PPC K K PP

C dt

C D C

3,3 33C D

Macro-particle code using the algorithm.

• The algorithm is included as a possible option (instead of Binary Collision Map) in a multi-particle code MOCAC.

Algorithm of the map consists of the following steps:

• Calculation of supplementary integrals, friction coefficients and components of the diffusion tensor.

• Calculation of the average value of the Coulomb logarithm by averaging on all particles of the beam.

• Calculation of 4 “amplitudes” Ci,j of random jumps in Langevin equations.

• Choice for each particle three random parameters and calculation of values using LE.

• The algorithm was validated by comparison with other methods.

Results of numerical IBS modulation for TWAC

storage ring 1.

0 100 200 300 400 500Time, sec

0.0000

0.0005

0.0010

0.0015

rms p

/p

AM modelBCM model

0 100 200 300 400 500Time, sec

5.0E-6

7.0E-6

9.0E-6

1.1E-5

x ,

y, m

rad

X plane, AM modelY plane, AM modelX plane, BCM modelY plane, BCM model

Results of numerical IBS modulation for TWAC

storage ring 2.

0 100 200 300 400 500Time, sec

0.0000

0.0005

0.0010

0.0015

rms p

/p

Np = 20000Np = 2000Np = 200

Results of numerical IBS modulation for TWAC

storage ring 3.

Validation of the model was made for TWAC storage ring with the following beam parameters:

• kind of ions

• the ion kinetic energy 620 MeV/u,

• number of the ions N= (coasting beam).

The numerical parameters are: • number of macro-particles 20000,

• time step = 0.3 s,

• number of azimuthal points =38,

• number of transverse cells =100 (in BCM model).

1210

1327Al

tazN

CN

Future plans.

• Main purpose is development of numerical codes and its application for GSI project design. Requirements to the codes:

1. Algorithms should satisfy to conservation laws.

2. Codes should be “users friendly”.

3. For each code should be derived “open rules” choice of numerical parameters, applicability and so on.

Our goals: 1. Development of new maps and creation of map library.

2. Investigations of convergence and benchmarking.

3. Comparison with the experiment.