Kinetics effects in multiple intra-beam
scattering.
P. R. Zenkevich, A. E. Bolshakov,
ITEP, Moscow, Russia
Contents.






Introduction.
Invariants of motion and their evolution.
Gaussian IBS model.
Kinetic IBS analysis.
- FPE in momentum space.
- FPE in invariant space.
-Approximate form of FPE.
- Langevin equations map (LEM).
-Longitudinal FPE (“semi-Gaussian” model).
- Binary Collision Map (BCM) and MOCAC code.
Molecular Dynamics.
Conclusions.
CERN, 21.01.09
2
Introduction1

1.
2.
Two kinds of IBS:
Multiple IBS.
Single-event IBS.
Here we consider only the multiple one!
1.
2.
3.

1.
2.
3.
In infinite space due to IBS the charged particles distribution tends to Maxwellian one with
equal energies on all degrees of freedom. This phenomena results in the following effects:
Relaxation of non-Maxwellian initial distribution in Maxwellian one.
Equalization of the temperatures for non-symmetric initial distribution.
In presence of the momentum limitations the diffusion flux results in the particle losses.
In storage rings the beam has the following additional features:
It is limited in momentum /coordinate space.
In different points of the ring the “matched” beam has different temperatures on transverse
degrees of freedom,
Particles with the momentum deviation has different orbits.
The last both effects result in growth of three-dimensional emittance!
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3
Coordinates-momentums vectors.

Let us introduce “coordinate vector” and “dimensionless momentum
vector”:
 1 p 
 p 
 z  zs 




r   x  , P   x 
  
 y 


 y 




Here z – is longitudinal coordinate, zs is longitudinal coordinate of the
equilibrium particle, x is horizontal coordinate, y is the vertical one.
 p is the particle momentum,  p
is its deviation from the equilibrium
momentum.
Px , y
 x, y  px , y / p Here is
are, correspondingly, the horizontal and
vertical components of the particle momentum.

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4
Invariants of motion in circular accelerators.

Linear non-coupled 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
variable s;
 m ,  m ,  m are “Twiss functions” depending on longitudinal
 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;  1  0, 1  0, 1  1
 For coasting beams (CB)
1  0, 1  1

For bunched beams (BB)
 1   s 2 /  2 [(1/  2   ) R]2
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5
Invariants evolution due to multiple IBS.


In scattering event the coordinates does not change; therefore we have:
Here


d ( P12 )


dt


2
2
2

d ( P1 x)  d ( P1 ) 2
dI  d ( x)
  x
 2 D

(D  D2 ) 
dt 
dt
dt
 x dt

2


d ( y)
y


dt


D   x Dx   x Dx
dPi 2
dPi
d (Pi )2
fr
 2 Pi

 2 PF
i i  Di ,i
dt
dt
dt

Friction coefficients
discussed later.
Fi fr
and diffusion coefficients
CERN, 21.01.09
Di ,i
will be
6
Gaussian models.

In Gaussian model it is assumed that distribution function is Gaussian one on all degrees on
freedom, i.e.
i 3
 ( P, r , s )  C N  exp[ 
i 1

Then evolution of each moment of the distribution function is defined by the averaging on the
test and field particles:
d ( PP
i j)
dt


  (u , r , s)( w, r , s)(u , w)dudwdr
Let
for the field particles.
P  u for the “test” particle and
Pw
Using these equations in classical IBS papers (Piwinski-Martini, Bjorken-Mtingwa) it is shown
that an evolution of the average components of the invariant –vector is described by the
following equations:
d I
dt

I i ( P, r , s )
]
Ii
 K ( I , s, t )
K ( I , s) are one dimensional integrals (in B-M model) or two-dimensional
Here functions
integrals (in P-M model), t is “slow” time since usually IBS time is much more than
revolution period.
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7
Differential equations for analysis of
r.m.s. beam parameters

Averaging on the ring we obtain the final result:
1
1

 i (t )
Ii
dI i
 Fi ( I , t )
dt
 
Here F ( I , t )   K ( I , s, t )  , sign
means averaging over the ring
circumference.
 After publication of this theory (beginning of 80-th years) the theory was
developed in the following directions:
1. Creation of numerical codes for IBS simulations (Katayama and
Rao, Mohl and Giannini, BETACOOL).
2. Derivation of analytic expressions for growth rates in different
particular cases.
3. Analysis of so named “Bi-Gaussian distributions.
4. Account of coupling between transverse degrees of freedom.

i
i
i
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8
Why we need in kinetic theory?


Gaussian model often describes the beam behavior with good accuracy; therefore a question appears:
when we should use the general kinetic theory?
Let us consider the one-dimensional Focker-Planck Equation (FPE):
f
D 2 f
 f 
t
2 u 2


Here f (u , t )
is the distribution function on u, friction coefficient
constants. A stationary solution has Gaussian form:
x
2D
f ( x, t )  C exp(  ), x 
x


Thus we have stationary Gaussian solution for variable x in infinite space. The stationary solution is invalid
in the following cases:
1. Friction or diffusion coefficients depend on x.
2. There is a boundary condition (for example, f ( xmax , t )  0).
3. If takes place transient process with initial condition, which is not Gayssian one:
and diffusion coefficient D are
x
)
x0
4. Some particles are born or lost during the process with non-uniform probability.
Thus we see: Gaussian model is valid only in
case of stationary or quasi-stationary solution without boundary.
f ( x, 0)  C exp(

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9
Fokker-Planck (FPE) in momentum space



1.
For pure IBS the beam tends to Gaussian distribution in free space (withoout
boundary) However if we add additional forms of interaction we should use the
general kinetic theory.
Kinetic analysis of the multiple IBS is based on the solution of the Fokker-Planck
equation for the distribution function, which can be written in momentumcoordinate space or in the “invariant space”.
In momentum-coordinate space FPE has the following form:
f (u , r , t )

1
 

[ Fm (u , r , t ) f (u , r , t )]  
[ Dm,m (u , r , t ) f (u , r , t )]
t
um
2 m,m um um



Here f (u , r , t )
is the distribution function; Fm are components of the
friction force; Dm , m are the components of the diffusion tensor (we perform
summation on the repeating indices).
The components of the friction force and the diffusion tensor due to IBS can be
calculated using kinematics of the scattering event, Rutheford cross-section and
the equation for scattering probability.
The friction force is defined by:
F
d u
u w
  A0 [ LC (u , w)
]
3 F
dt
uw
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10
FPE in momentum space 2.

Components of the diffusion tensor
Di , j (u , r , t ) 

d ui
u j
dt
  i , j u  w 2  (ui  wi )(u j  w j ) 


3/ 2
u w


Rate of the second moments evolution
d (ui u j )
dt
 i, j
is Kronecker-Kapelli symbol (we take in mind summation on repeating indices)
,
LC ( u  w )
is Coulomb logarithm, and
 (u , w)  w   (u , w) f ( w, r , t )dw
is the beam distribution function in phase space.

Here
3
 i , j u  w  3(ui  wi )(u j  w j ) 
A0  (r ) 
 LC ( u  w )


3
2 
u

w
 w



Constant
A0 
2 cri 2
 3 4
, where ri is the ion classical radius.
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11
Fokker- Planck equation in invariant space.

FPE equation in invariant space (here scalar
I  I1 , I 2 , I3
) is:
F ( I , t )

1
2

[ Rm ( I , I , t ) F ( I , t )]  
[ Dm,m ( I , I , t ) F ( I , t )]
t

I
2

I

I

m , m
m
m
m


1.
This equation takes place if the distribution on phases is uniform on interval [0, 2. ]
This equation should be solved with corresponding initial and boundary conditions. Initial
condition:
F ( I , 0)  ( I )
Examples of boundary conditions:
Absorbing wall boundary condition
F ( I max , t )  0
I max  I1 , I 2 , I 3
In the simplest case
In general case the boundary conditions are defined by three-dimensional surface in the
invariant space
“Reflecting wall” boundary condition for zero invariant
max
2.
max
max
dF ( I , t )
( I  0)  0
dJ
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12
Evolution of Invariants and their Moments


d I
 R ( I , t )   K ( I , I ) F ( I , t ) dI
dt
0
Then we obtain:
d I  I 
dt

 D ,  ( I , t )   K ,  ( I , I ) F ( I , t )dI
0
Here kernels have form of four-dimensional integrals on phases and longitudinal
variable s. Example (for m=1,3)
1
Km ( I , I ) 
Lper


Here
L per
 ds  
m
[u ( I ,  , s), w[ I , r ( I ,  , s), s], s][ I , r ( I ,  , s), s) d 
0
u  w  3(ui  wi )2
2
 m (u , w)   m LC ( u  w )
uw
3
3


Function  ( I , r , s )    i ( I , r , s)
i 1
dwm ( I m , rm )
For m=1,3

dI m
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1
( m y)   m ( I m   m y )
2
2
  m ( I m , rm )
Scheme of the kernels calculation (averaging on the
trajectories and longitudinal coordinate s )

.
For transfer from coordinate-momentum to invariant-phase space we should use the following algorithm:
1) express the radius-vector and momentum-vector of the field and test particles through invariants and
phases :
r ( I, ,  , s ) u ( I ,  , ,s )
2) using “ locality condition”
r ( I , , s )
u ( I ,  , s)
,
r ( I ,  , s)  r ( I ,  , s)
;
we exclude

and find dependence ;
u ( I , I ,  , s)

3) then we change averaging on momentum of the field particles by averaging on its invariant-vector , using
the expression:
f (r , u , t )drdu  F ( I , t )W ( I ,  )dId



1.
2.
3.
W (I , )
(here
is Wronskian).

Excluding
from the locality condition and changing of order of integration we find the final expressions
for the kernels.
In general case these kernels are four-dimensional integrals on phases and longitudinal variable, depending
on six parameters:
I1 , I 2 , I 3 , I1 , I 2 , I 3
A number of dimensions can be reduced :
coasting beam : three-dimensional integrals depending on 6 parameters;
coasting beam and the smooth focusing.: two-dimensional integrals depending on 6 parameters;
coasting beam , the smooth focusing and coupled transverse oscillations: two-dimensional integrals
depending on 4 parameters.
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14
Langevin Equations Map (LEM).


The simplest way for numerically solution of the FPE is application of well-known Langevin
equations.
However, 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
Here
are three random numbers with Gaussian distribution and unity dispersion;
C
coefficients
i , j take into account correlations between coupled 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 :
j 1

C 1,1  D11


d ( P1 P2 )
1
 ( K1  K 2 ) P1 P2
C21 
C
dt
1,1


C22   D22  C212


C 3,3 
D33
As I know, Dubna (JINR) guys (A. Sydorin and company) made an attempt to use this
algorithm for the numerical code; I don’t know the result.
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15
Approximate 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 tensorDi , j are constants.
To provide same invariant rates we should average fiction force and
diffusion coefficients on test and field particles .Then we obtain:
AL
Ki  0 C
2
 (u  w ) 2 
i
 i
3 
 u  w T , F
Di , j
AL
 0 C
2
  i , j u  w 2  (ui  wi )(u j  w j ) 


3
uw

T , F
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16
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
their values using LE.
The code has been validated by comparison with other methods.
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17
Longitudinal FPE 1


Let us consider coasting beam and smooth focusing. Moreover, let us introduce following
assumptions:
1) the distributions on transverse degrees of freedom are Gaussian ones with equal r.m.s values
of transverse momentum;
2) dispersion function is equal to zero; such assumption is acceptable if we are working far below
critical energy.
2
2
Then
 x x 2   x x2  y y   y y 
f (u , r , t )  C (t )(u, t ) exp[ 
,

Averaging on
equation:
x, y, x, y
Ix

Iy
, we can derive one-dimensional (longitudinal) FP
(u, t ) 
1  2 [ D(u , t )(u, t )]
 [ Ffr (u, t )(u, t )] 
t
u
2
u 2

Let us assume that: 1) dispersion function is equal to zero; 2) horizontal and vertical
r. m. s. emittances are equal.
CERN, 21.01.09
]
Longitudinal FPE 2

Expressions for longitudinal friction force and diffusion coefficient are:


A
D(u, t )   f ( w, t ) K dif (u, w)dw
2 
Ffr (u, t )   A  f ( w, t ) K fr (u, w)dw


The kernels are
K fr (u  w) 
uw
2
uw
1

(u  w)2
{

exp[
][1


(
]}
u  w 2
4 2
2
uw
1 
(u  w)2
Kdif (u  w,  ) 
exp[
][1


(
]  (u  w) 2 K fr (u, w)
2
 2
4
2


For high transverse energy (   ) the friction force disappears
( Ffr (u, t )  0 ), and diffusion coefficient tends to constant ( D(u, t )  D0 ).
Thus we see that when transverse energy is much more than longitudinal one, IBS adds the
diffusion coefficient in longitudinal FPE. This diffusion coefficient can be taken into account
using Langevin equations.
This method was used in numerical model by O. Boine-Frenkenheim (GSI).
Dependence of kernels for the friction force and diffusion coefficients on parameter
t=u-w : blue curve  =1, red curve

=2 and green curve
K

=3
.
D
1.0
1.5
0.8
t
1.0
0.6
0.4
0.5
0.2
0.0
0.0
0.0
0.0
0.5
1.0
1.5
t
2.0
2.5
3.0
0.5
1.0
1.5
t
2.0
2.5
3.0
Binary Collision Map 1

Let us choose the scattering angle according to expression
sin(

 sc i , j
2
)
A  t
N u u
i
j 3/ 2
Then work of the friction force and increase of moments because
of the diffusion terms coincide with the corresponding exact
values:
A  t N u i  u j
i
i
u fr  Ffr  t 
N

j ( j i )
ui  u j
3/ 2
[0, 2 ]

Azimuthal angle is defined by random choice on interval

We have “invented” this map and included it in code named
“MOCAC (MOnte-Carlo Code). However, later we know that the
idea was suggested earlier by T. Takizuka and H. Abe (1977).
CERN, 21.01.09
.
21
Structure of program MOCAC
MAIN PROGRAM
Intra-beam scattering
Electron cooling
Multiple scattering and energy loss on residual gas
Beam losses because of single interactions
Transformation to momentum-coordinate space
Stochastic cooling
Internal target
Module 1: Transformation from invariant to momentumcoordinate
Module 1
Barrier bucket
Linear synchrotron oscillations
Non-linear synchrotron oscillations
Continuous beam
The list of code parameters
Integration step
Number of
macroparticles
Number of points per
period
Size of transverse
cell
Size of longitudinal
cell
Maximal collision
angle
t  Tibs /100 N per
N mp  100 Nlong Ntr
N per
N mag / 5
tr  min( x , y / 5, D p )
long   s /10
 max  0.5
21.01.09
24
13
Al 27
Dependence of normalized beam intensity and momentum spread on
time for TWAC storage ring calculated by MOCAC code.

The code allows us to make
simulations, which can not be done
using Gaussian model. As an
example, let us consider modeling of
the ion storage in TWAC ring (ITEP,
Moscow) by use of the charge
exchange injection. The initial
particle distribution in the transverse
space is “cutted Maxwellian” one.
The beam parameters: kind of ions
13
Al 27 , T=620 MeV/u, the booster
frequency=1Hz, N part  3 1012
material of the charge-exchange
target: Au, its width 5*10-4 g/cm2.
The simulation results are given at
Fig. 1. We see from the picture that
IBS results in increase of the
momentum spread and significant
beam losses (influence of the
charge-exchange target is small).
CERN, 21.01.09
25
Final momentum distribution for HESR calculated by
MOCAC code.

F
104.0
103.0
102.0
101.0
Real distribution
Gaussian distribution
100.0
10-1.0
-1.0E-4
-5.0E-5
0.0E0
dP/P
5.0E-5
1.0E-4
The plot is calculated
with account of IBS,
electron cooling and
beam target interaction
for ring HESR (FAIR,
Germany) using
MOCAC code. We see
from the picture a
development of nonGaussian beam tails.
CERN, 21.01.09
26
Results of numerical IBS simulation 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
500
0
Time, sec
100
200
300
400
500
Time, sec
CERN, 21.01.09
27
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
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500
28
Code validation.
Dependence of beam invariant on time



Smooth model of TWAC ring
with non-zero dispersion
(D=0.461)
code computational
parameters:
Ngrid = 30*30 (blue curve)
and 5*5 (red curve)
We see regular growth of
invariant deviation for small
number of grid points!
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MOLECULAR DYNAMICS 1.




An idea of the method consists of the direct calculation of the particles trajectories with
account of the external electromagnetic fields and “particle-particle” Coulomb interactions. The
main technical problem of the molecular dynamics is too large computational volume because
of the big number of particles and small integration step, which is necessary to resolve close
collisions between particles (typically, a particle needs 102 steps in order to cover the average
particle distance). Let us consider two options of this method:
“String” model developed by Bologna group for IBS simulations [13].
Three-dimensional model of “periodical cells” used in BETACOOL code [14] for a simulation of
the crystalline ion beams.
Let us denote number of particles in the beam Q=eNp, Np is a number of particles per beam,
e is the particle electric charge, N is a number of macro-particles per beam. For constant
focusing lattice with non-equal tunes the single particle Hamiltonian is
1 2  2 2 
2
H  P  0 x x  0 y y  i
2
2
2
2

Here 0 x / 0 y
potential
are the phase advances per unit length. For string model the space charge
1 2  2 2 
2
H  P  0 x x  0 y y  i
2
2
2
2

Here
 is the perveance (

q Q
m  02
),
ri , j
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is a distance between wires i and j.
30
MOLECULAR DYNAMICS (string model) 2.

1.
2.
3.
Let us mark that this 2-D model has evident
drawbacks:
the diffusion and friction coefficients are quite
different from diffusion and friction coefficients in true
3-D IBS with point-like Coulomb potentials;
2-D model does not describe the longitudinal heating
of the “cold” longitudinal degree of freedom due to
energy exchange with “hot” transverse degrees (this
effect probably is the most important phenomena).
Nevertheless there are some effects, which can be
modeled using this theory, for example, relaxation
initial distribution with different transverse
temperatures or crossing of the Coulomb coupling
resonance (so named “Montague” resonance”).
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31
MOLECULAR DYNAMICS (string model) 3.



The results of numerical modeling of
Montague resonance by C.Benedetti
(  x  y )
Emittances evolution during the
dynamical crossing of the Montague
resonance. Tune ramp over 30 (red),
800 (blue) and 2500 (cyan) turns.
We see a generation of the transition
asymmetry due to IBS.
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32
MOLECULAR DYNAMICS (periodical model).



In “periodical model” we assume that the beam consists of periodical cells with
length about 2az (here az is the vertical beam radius). The particle charge and
mass correspond to the real particle; Coulomb potential of each particle is defined
by standard Green function of the point-like electrical charge . If a particle goes
outside the cell boundary, then a new particle with same value of the momentum
enters in a cell from the opposite boundary.
Number of particles in cell
Nc  2 Naz / LC
where LC is the ring circumference. For cooled beam and limited number of the
stored particles in a ring (105 -106) a number of particles in cell is small (typically
NC<10) and simulations can be made without serious difficulties. These
simulations have shown that such cooled beam transfers in “crystalline” form
where IBS is suppressed.
A shape of the crystals depends on the dimensionless linear density of particles
lion defined as follows:
1/ 3
ion 
N  3 rion 


C  2k 05  02 
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33
Periodical model, crystalline beam 1.

String (ion < 0.709)

Zigzag (0.709 < ion
< 0.964)
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34
Periodical model, crystalline beam 2.

Helix or Tetrahedron
(0.964 < ion < 3.10)

Shell + String (3.10 <
ion < 5.7)
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35
Conclusions and
acknowlengements.

1.
2.
The multiple IBS is very important in storage rings with high phase density of the
accumulated beam. We see that the last main advances are connected with new
perspective methods of numerical IBS modeling:
“Collective maps” in momentum space;
Molecular dynamics methods.

Both methods were successfully applied to new physical problems such as
calculation of the beam losses and non-Gaussian tails, analysis of the IBS effects
during crossing of Montague resonance and simulation of crystalline beams.
These methods are continuously developed and in near future we can expect
their further progress.

I am grateful to O. Boine-Frenkenheim for the useful collaboration, as well
as to C. Benedetti and A. Smirnov for interesting discussions on the
molecular dynamics.
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