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 macroparticles 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”:
•
 1 p 
 p 
 z  zs 




r   x  , P   x 
  
 y 


 y 




Here z – is longitudinal coordinate, x – is horizontal coordinate,
y – is the vertical one.
•
Correspondingly, P  1 pp - is the first component of the momentum,
x
- is the second component,

y
- is the third one.
1
FPE in coordinate-momentum space 1.
•
Evolution of the distribution function in infinite medium is described by
following FPE:
f

1
2
  [ Ffr f ]  
( Di , j f )
t
u
2 i , j Pi Pj
•
Friction force due to multiple IBS (u corresponds to “test” particle, w to the
“field” one)
Ffr (u )  A0 LC [
•
u w
u w
]
3 F
Components of the diffusion tensor
L u  w  (ui  wi )(u j  w j )
Di , j  A0 C [
]F
3
2
u w
2
FPE in coordinate-momentum space 2.
•
Here Coulomb logarithm
A0 
LC  20
and constant
2 cri 2
 3 5
• Here averaging on the field particles corresponds to the
following operator:
[(u , w, t )]F   (u , w, t ) f (r , w, t )dw
Invariants and its evolution due to IBS.
•
Linear particle motion is described by conservation of “invariants”:
I m   m rm 2  2 m rm Pm   m Pm 2
•
Here for m=2,3
 m , m ,  m
on longitudinal variable s;
are “Twiss functions” depending
 1 p 
  p 
 z  zs 




r   x  D P1  , P   x  D P1 




y
y








•
here D and D are dispersion function and its derivative;
For coasting beams (CB)  1  0,1  0, 1  1
•
For bunched beams (BB)
1  0, 1  1
 1   s 2 /  2 [(1/  2   ) R]2
Invariants and its evolution due to IBS.
•
In scattering event the coordinates does not change; therefore we have:


d ( P12 )


dt




2
2
2
d ( P1 x)
dI
d ( x)
 d ( P1 )
2
2 

 x
 2 D

(D  D )
dt
dt
dt
x
dt




d ( y) 2


y


dt


•
The momentum derivatives
d ( PP
i j)
dt
•
AL
 0 C
2
u  w  i , j  3(ui  wi )(u j  w j )
2
Here  i , j is Kronecker-Kapelli symbol, D   x Dx   x Dx
uw
3
Approximated form of FPE (1)
The initial assumptions of the model:
•
•
•
•
Gaussian beam.
Coulomb logarithm is constant.
The components of the friction force Fi   Ki Pi with constant coefficients K i
The components of the diffusion tensor Di , j 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:
A0 LC (ui  wi )2
Ki 
3
2
u w
A L  i , j u  w  (ui  wi )(u j  w j )
 0 C
3
2
u w
2
Di , j
Approximated form of FPE (2)
• Averaging on the field and test particles as in Piwinski-Martini and
Bjorken-Mtingwa theories we obtain:
2
Di ,i  AN 



  ( P, s )
P  Pi 2
3/ 2
Ki 
d 3P
P
AN
Pi 2



  ( P, s )
Pi 2
3/ 2
d 3P
P
• Here the distribution function is defined by
3
( P, s)  exp[( ai Pi 2  2 P1 P2 ] / 4
i 1
• Here parameters
ai
and
a1
1
 2 ( D 2  D 2 ) a2  x a3 2 y


, 
, 
2
Iz
x Ix
2
Ix 2
Iy

are:

2 D
, D  D x  D x
Ix
Approximated form of FPE (3)
•
Integrating according to Bjorken-Mtingwa scheme [5] we obtain the
following final expressions:
3
Di , j  AN ( i , j  INTi ,i  INTi , j )
K i  AN INTi ,i / ( Pi )
2
•
i 1
Here the normalizing constant
AN 
•
cLC ri
2
4   3 4 I x



,  
Iy  p

 2
N
, CB
L
Nb
, BB
 z
The integrals are:

 (a2   )d 
INT1,1  4 
[(a1   )(a2   )   2 ]3/ 2 (a3   )1/ 2
0

d
INT3,3  4 
[(a1   )(a2   )   2 ]1/ 2 (a3   )3/ 2
0

d
[(a1   )(a2   )   2 ]1/ 2 (a3   )3/ 2
0
INT3,3  4 

d
[(a1   )(a2   )   2 ]3/ 2 (a3   )1/ 2
0
INT1,2  8 
Approximated form of FPE (4)

Using these formulae it was written
subroutine, which for set of points
with
s and, correspondingly,
known Twiss
2 y
2 variable
1 known
 2 ( D 2 longitudinal
D2 )
2 D
a1  2[

], a2  x , a3 


, D  D  x  D x
Iz
 x I x dispersion
I x andIits
parameters,
derivative)
calculates:
Ix
y
1. Average friction coefficients
.
2. Average diffusion coefficients
K
D
d ( PP )
3. Average rates of moments change
dt
dI
4. Average rates of the invariant
change
dt
i
i, j
i
i
j
.
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:
3
Pi (t  t )  Pi (t )  Ki Pi (t )t  t  Ci , j j
j 1
•
j
Here
are three random numbers with Gaussian distribution and
unity dispersion; coefficients Ci , j 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 :

C 1,1  D11


d ( P1 P2 )
1
 ( K1  K 2 ) P1 P2
C21 
C1,1
dt


C22   D22  C212

C 3,3 
D33
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.
1.1E-5
0.0010
AM model
BCM model
x ,y, m rad
rms p/p
0.0015
9.0E-6
X plane, AM model
Y plane, AM model
X plane, BCM model
Y plane, BCM model
7.0E-6
0.0005
5.0E-6
0.0000
0
100
200
300
400
Time, sec
500
0
100
200
300
400
Time, sec
500
Results of numerical IBS modulation for TWAC
storage ring 2.
rms p/p
0.0015
0.0010
Np = 20000
Np = 2000
Np = 200
0.0005
0.0000
0
100
200
300
Time, sec
400
500
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 Al27 13
the ion kinetic energy 620 MeV/u,
number of the ions N= 1012 (coasting beam).
The numerical parameters are:
•
•
•
•
number of macro-particles 20000,
t = 0.3 s,
time step
number of azimuthal points N az =38,
N C =100 (in BCM model).
number of transverse cells
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.