Kinetics Effects in Multiple
Intra-beam Scattering (IBS).
P.R.Zenkevich*, A.E.Bolshakov*, O. Boine-Frankenheim**
*ITEP, Moscow, Russia
**GSI, Darmstadt, Germany
The work is performed within framework of INTAS/GSI grant
“Advanced Beam Dynamics”
1
Contents.
•
•
•
•
•
Introduction.
One event kinematics.
IBS in infinite medium:
- Focker- Planck (FP) equation in momentum space.
- Langevin map.
-“Binary collision” map.
IBS in circular accelerators:
- FP equation in momentum-coordinate space
- Invariants evolution due to IBS.
- FP equation in invariant space.
- Longitudinal FP equation (“semi-Gaussian” model). .
- MOCAC code and its applications.
Summary.
2
Introduction.
•
•
•
IBS includes:
1) Multiple IBS.
2) Single-event IBS (Touschek effect). The effect can be included in multiparticle codes by straight-forward way. It is out of frame of this review.
Fundamental accelerator papers:
Bjorken, Mtingwa (BM);Piwinski, Martini (PM).
Results: analysis of one-event kinematics and r. m. s. invariants evolution. Main
difference: in BM theory Coulomb logarithm is constant, in PM Coulomb logarithm
is dependent on momentum.
Gaussian model :
theory of rms invariants evolution in PM and BM theories is based on assumption
that the beam has Gaussian distribution on all degrees of freedom.
The reason: IBS results in beam maxwellization!
•
Creation of numerical codes for rms invariants evolution:
- Mohl and Giannini, Katayama and Rao and so on.
- BETACOOL (Meshkov’s group+Zenkevich). The code includes additional
effects: electron cooling, Beam-Target Interaction (BTI) and so on.
3
Kinetic approach.
Why we need kinetic description?
•
•
Solution of kinetic equation is not Gaussian with account of boundary conditions
(particle losses).
Other effects (for example, e-cooling) produce Non-Gaussian tails.
How to investigate these effects?
The simplest way: to solve Focker-Planck (FP) equation.
•
One-dimensional Focker-Planck equations.
-
Spherical symmetry (Globenko, ITEP, 1970).
- One-dimensional longitudinal equation – (Lebedev, Burov, Boine-Frenkenheim).
•
Three dimensional codes.
•
Monte Carlo Code (MOCAC) code (Zenkevich, Bolshakov) (IBS+E-cool+BTI+…).
Six-dmensional codes.
– PTARGET code (Dolinsky) (IBS+E-cool+BTI).
4
One-event kinematics.
• Let us introduce vector of the particle “dimensionless momentum”
• Let for “test” particle
Pu
for “field” particle
• Here
U u w
• Then
Pw
,
U   U 2 2  U 32

1

[U  sin cos   U1 (cos  1)]
2

 1 U U sin   U1U 2 cos 
u   [ 3
sin  U 2 (cos  1)]
2
U


 1 U 2U sin   U1U 3 cos 
sin  U 3 (cos  1)]
 [
2
U


5
Moments evolution 1.
•
Averaging on azimuthal angle we find that for one collision event:
 u   (sin
 (ui )   (u j ) 
•

2
) 2  (u  w)
1

2
(sin ) 2 [ i , j u  w  (ui  wi )(u j  w j )]
2
2
Rutheford cross-section
d  (
•
rp
4 2 sin 2

) 2 sin d d
2
Scattering probability
pscat 
2c     (r )  d
2
6
Moments evolution 2.
•
Average friction force due to multiple IBS
Ffr (r , u ) 
•
dt
 A0 LC ( u  w )
u w
u w
3
Average diffusion coefficient
d (ui )(u j )
dt
•
d u

A0
LC ( u  w )
2
Here Coulomb logarithm
A0 
2 cri 2
 3 4
 i , j u  w  (ui  wi )(u j  w j )
LC ( u  w )  ln(
3
u w
d  u  w
ri
3
)
 (u , w)    (u , w) f ( w, r , t )dw
7
Fokker-Planck equation in momentum space
(infinite medium).
•
Evolution of the distribution function in infinite medium is described by
following FP equation (for constant Coulomb logarithm)
f (u , t )

1
2
  [ Ffr (u , t ) f (u , t )]  
[ Di , j (u , t ) f (u , t )]
t
u
2 i , j ui u j
•
If we neglect weak dependence of logarithm on momentum
H
Ffr (u , t )  A L
u
H (u , t )  2
 2G
D(u , t )  A L
u u
f ( w, t ) 3
d w
uw
G (u , t )   f ( w, t ) u  wd 3 w
•
Here
•
This equation should be solved with initial condition (initial distribution
function and its derivative should be smooth):
f (u ,0)  f 0 (u )
8
Langevin map 1.
• Let at time t distribution in phase space is defined by
1 N
f (u , t )    (u  u j )
N j 1
• Then the friction force and diffusion coefficients are:
2A N ui  u j
i
Ffr (u ) 

N j ( j i ) u i  u j 3
2
D ,  i
A

N
N

j ( j i )
u i  u j  (u i  u j )(u i  u j )
ui  u j
3
• To simulate the particle evolution let us change of the test particle
momentum is
i
u i  Ffr i  t  u dif
• Here the random diffusion kick should satisfy to conditions:
i


u
0
dif


i
i
 (u dif ) (u dif )   D , 
9
Langevin map 2
• We see that Langevin map includes the following steps:
1. Calculation the friction force and diffusion coefficients for the test
particle.
2. Calculation of the momentum change due to friction.
3. Random choice of the the momentum change due to diffusion.
4. Repetition of the process for all particles.
Energy conservation is absent at the second order
on time step. It can result in unphysical growth of
six-dimensional emittance!
• However, we have algorithm which provides energy conservation:
BINARY COLLISION MAP!
10
Collision Map 1
• Let us choose t he 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
u  Ffr  t 
N
i
fr
i
N

j ( j i )
ui  u j
ui  u j
3/ 2
• Azimuthal angle is defined by random choice on interval
[0, 2 ]
.
• We have “invented” this map and included it in code named
“MOCAC (MOnte-Carlo code). However, this idea was suggested
earlier by T. Takizuka and Hirotada Abe (1977).
11
Collision Map 2
• From electrostatic analogy we know, that for code smoothening the
minimal distance between particles is limited.
• Let
u i  w j  0
, then
 scat   max ,  max  2arcsin(
• Computational parameters of the code are
 max , t
At
)
N 03/ 2
.
• Number of “collisions” for each step is equal to N 2 / 2 where N is
number of macro-particles.
• The different algorithm (which is really used for calculations in
MOCAC code and Takizuka-Abe paper) assumes random choice of
“one” partner in each step. For same error we should choose time
step in order to tonepartner  t
.
N
12
FP equation in momentum-coordinate space
• FP equation in indefinite medium can be generalized for the beam
by straight-forward way. Then
f (u , r , t )

1
 
  [ Ffr (u , r , t ) f (u , r , t )]  
[ Di , j (u , r , t ) f (u , r , t )]
t
u
2 i , j ui u j
• Initial condition:
f (u , r , 0)  f 0 (u , r )
f (ubound , rbound , t )  0
• Boundary condition:
• We can use same methods for numerical solution as for infinite
medium (for example, collision map).
• However, particles oscillate in transverse direction; therefore time
step should satisfy to the condition t  1/ f
• To diminish computer time up to reasonable limit we should use
simplified models or different approaches.
13
Longitudinal FP equation 1
•
•
Let us introduce following assumptions:
1)coasting beam;
2) the distributions on transverse degrees of freedom are Gaussian ones
with equal r.m.s values of transverse momentum;
3) dispersion function is equal to zero; such assumption is acceptable if
we are working far below critical energy.
Then
f (u , r , t )  C (t )(u, t ) exp[ 
,
•
Averaging on
(longitudinal) FP
x, y, x, y
equation:
 x x 2   x x2
Ix

 y y 2   y y 2
Iy
]
, we can derive one-dimensional
(u, t ) 
1  2 [ D(u , t )(u, t )]
 [ Ffr (u, t )(u, t )] 
t
u
2
u 2
14
Longitudinal FP equation 2
•
Here the friction force and diffusion coefficients are

Ffr (u, t )   A  f ( w, t ) K fr (u, w)dw

•

A
D(u, t )   f ( w, t ) K dif (u, w)dw
2 
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
15
Dependence of kernels for the friction force and diffusion coefficients on parameter
:
:blue curve  =1, red curve  =2 and green curve  =3.
t  u  w / 2
K
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
2.0
2.5
3.0
t
16
Invariant space 1
•
Linear transverse particle motion is described by conservation of “CourantSnyder invariant”:
•
I     r 2  2  r P    P 2
Here   ,   ,  
are “Twiss functions” depending on longitudinal
variable s; for horizontal motion
r  x  D
p
p
P  x  D
p
p
x 
px
p
here D and D are dispersion function and its derivative;
p
for vertical motion
P  y  y
r  y
•

p
For coasting beams (CB) we can use as longitudinal invariant or momentum
deviation
p
, or its squared value
p
I (
p 2
)
p
17
Invariant space 2.
•
The last form is more symmetrical; In linear approximation for bunched beam (BB)
the invariant can be written as follows:
I   z2  P2
Here
p
P
p
r  z  zs
0, CB


2
 
s
 (1/  2   ) R , BB

z
•
-distance from bunch center,
Let us introduce “invariant vector” with components
I  (I , I x , I y )
and “phase vector”
  x , y , CB
 
 , x , y , BB
18
Evolution of Invariants and their Moments 1
•
In scattering event the coordinates does not change; we have:
Rm 
Dm,m
•
•
d (Im )
 A0  LC (u , w) Lm (u , w) f ( w, r )dw
dt
d (I m )(I m )

 0.5 A0  LC (u , w) Lm,m (u , w) f ( w, r )dw
dt
Here Lm ,m (u , w) and Lm (u , w)
Let us define operator
•
For m=1,3
•
For m=2
are defined by:
u  w  4um (um  wm )  (um  wm ) 2
2
Lm (u, v) 
uw
3
Lm   m Lm
L2  H x 2 L1  K x L1,2   x L2
H x   x Dx 2  2 x Dx Dx   x ( Dx ) 2
K x   x Dx   x Dx
19
Evolution of Invariants and their Moments 2
•
For the second order moments of the invariants with m=1,3 we find:
Lm,m  4m2um2 (um )2
•
For the second order of invariant with m=2
L2,2  12 L1,1 (u , w)  212 L1,2 (u , w)  22 L2,2 (u , w)
1  H x 2u1  K x u2
 2   2u2  K x u1
 i , j u  w  (ui  wi )(u j  w j )
2
•
Here the coefficients
•
For the invariant with m=1,2
Li , j 
1  H x 2u1  K x u2
u w
3
L1,2 (u, w)  2u11L1,1 (u, w)  2u22 L1,2 (u, w)
 2   2u2  K x u1
20
Evolution of Invariants and their Moments 3
•
Calculation of evolution of invariants moments:
-Expression of momentae with account of (“locality condition”) for “field”
particle through its invariant and coordinates:
w  w(r , I , s)
-Change of variables in integrals over local distribution function:
dw 
dw(r , I , s )
dI
dI
- Transfer for test particle from variables
using the expressions:
u, r
to variables I ,
u  u ( I , , s), r  r ( I , , s)
- Averaging on invariants and phases of test particle and on variable s.
21
Evolution of Invariants and their Moments 4
•
d I
Then we obtain:

 R( I , t )   K ( I , I ) F ( I , t )dI
dt
0

d I m  I m
 Dm,m ( I , t )   K m ,m ( I , I ) F ( I , t )dI
dt
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
•
Lper
 ds  
m
[u ( J ,  , s), w[ J , r ( J ,  , s), s], s][ J , r ( J ,  , s), s)d 
0
u  w  3(ui  wi )2
2
Here
 m (u , w)   m LC ( u  w )
uw
3
3
•
•
Function  ( J , r , s)    i ( J , r , s)
i 1
For m=1,3
dwm ( I m , rm )
1

  m ( I m , rm )
2
2
dJ m
( m y)   m ( I m   m y )
22
Focker- Planck equation in invariant space 1.
• FP equation is:
F ( I , t )

1
2
 
[ Rm ( J , t ) F ( J , t )]  
[ Dm,m (J , t ) F ( J , t )]
t
2 m,m J mJ m
 J m
•
•
[0, 2 ]
The distribution on phases is uniform on interval
This equation should be solved with following initial and boundary conditions:
F ( J ,0)  F0 ( J )
•
•
Initial distribution function should be smooth with its first derivative.
Absorbing wall boundary condition
F (J , t)  0
max
•
m ( J , t)
( J m  0)  0
dJ
“Zero flux” boundary condition
Here flux
( J , t )   Rm ( J , t ) F ( J , t )  
m

[ Dm,m ( J , t ) F ( J , t )]
J m
23
24
Application of Langevin method for solution of FP
equation in invariant space 1.
•
If we know the kernels we can use for numerical solution “Langevin method”. Let
1
F (J , t) 
N
•
•
Then
1
R 
N
n
n  N

n
n
K (J , J )
N
 (J  J
n
)
n 1
Dm,m
n ( n   n )
n
1 n  N

Dm,m ( J n , J n )

N n ( n   n )
Change of invariant:
J n  J fr n   J dif n
•
Here change of the invariant due to “friction” is defined by:
J fr n  R n  t
25
Application of Langevin method for solution of FP equation in
invariant space 2.
•
Statistical correlation there is only for longitudinal and horizontal motion; therefore
all diagonal elements are equal to zero, besides m=1,2. Then diffusion vector can
be found by random choice from the distribution:
f (I1 , I 2 , I 3 ) 
here
•
D2,2 I12  2 D1,2 I1I 2  D1,1I 2 2 I 32
exp[

]
0  t
D3,3t
(2t )3/ 2 ( 0 D3,3 )1/ 2
0  D1,1D2,2  D1,2 2
The last step - comparison of new values of the invariants with boundary
conditions.
If the particle is outside the absorbing wall it is considered as lost
If one of the positive components becomes negative (transfer through reflecting
wall) its value is changed on opposite one.
26
MOCAC code and its applications1.
• Idea of code is change of kernel calculation by successive
application of the “binary collision” map in momentum space.
Algorithm steps:
1. Random choice of macro-particle phases for given set of macroparticles invariants.
2. Calculation of momentae and coordinates of macro-particles.
3. Computation of macro-particles distribution on the space cells.
4. Application of collision map and each macro-particle using “local
ensemble” for each cell.
5. Calculation of new set of macro-particles invariants.
27
MOCAC code and its applications2.
• This operation can be considered as “collective map” in invariant
space; each particle is “test” and “field” particle simultaneously.
• The map is repeated through the time interval t
• If the magnetic lattice is non-uniform, it is presented as set of
discrete points corresponding different longitudinal coordinates.
Each point is characterized by its set of Twiss parameters and
dispersion function. The points are distributed uniformly on the
lattice period.
• The map is made through each time interval in new point of period
(averaging on the lattice).
28
The list of code parameters
Integration step
t  Tibs /100 N per
Number of
macroparticles
Number of points per
period
Size of transverse cell
N mp  100 Nlong Ntr
N per
N mag / 5
tr  min( x , y / 5, D p )
Size of longitudinal cell
long   s /10
Maximal collision angle
max  1
29
Code validation.
Dependence of r. m. s. momentum spread on time
• Smooth model of TWAC ring
with zero dispersion.
• Beam and ring parameters:
kind of ions:
27
13=1.66,
T=620 MeV/u,
Al
ring = 251.0 m , Q=9.3.
• Code computational
parameters: Npart = 100000, ,
Ngrid = 30*30, t = 0.01 sec, ,
max=1.0.
30
Code validation.
Dependence of beam invariant oscillations on time.
• Smooth model of TWAC ring
with zero dispersion.
• Beam invariant
I  p
2
1 Ix
1 Iy
( 2 ) 


2 x 2  y
1
31
Code validation.
Dependence of r. m. s. momentum spread on time
• Smooth model of TWAC ring
with non-zero dispersion
(D=0.461)
• Beam and ring parameters:
kind of ions: 27 Al 13
T=620 MeV/u, =1.66,
=0.0116203, ring = 251.0
m, Q=9.3.
• code computational
parameters:
Ngrid = 30*30 (blue curve)
and 5*5(red curve), Npart =
100000, t = 0.01 sec, ,
max=1.0
32
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!
33
Code validation.
Dependence of r. m. s. momentum spread on time
•
Smooth model of TWAC ring with nonzero dispersion (D=0.461) and high ion
energy (not very far from critical one).
•
Beam and ring parameters:
27
13
Al
kind of ions:
T=3800 MeV/u, =5.05
=0.0116203, ring = 251.0m, Q=9.3.
•
code computational parameters:
Ngrid = 30*30 (blue curve)
and 5*5(red curve),
Npart = 100000, t = 10 sec, ,
max=1.0
•
We see regular growth for red curve
(small number of points in the grid)!
34
Code validation.
Dependence of beam invariant on time
•
Smooth model of TWAC ring with nonzero dispersion (D=0.461) and high ion
energy (not very far from critical one).
•
Beam and ring parameters:
kind of ions:
T=3800 MeV/u, =5.05
=0.0116203, ring = 251.0m, Q=9.3.
•
code computational parameters:
Ngrid = 30*30 (blue curve)
and 5*5(red curve),
Npart = 100000, t = 10 sec, ,
max=1.0
•
We see regular growth for red curve
(small number of points in the grid)!
35
Comparison with ESR experiments
(
238
U 92
)
36
Numerical modeling of multi-turn injection in TWAC 1.
•
Fig. 7. Dependence of r. m. s.
momentum spread on time for real
option of TWAC lattice and multiturn charge-exchange injection.
•
Beam parameters: kind of ions,
T=620 MeV/u, booster
frequency=1Hz,
number of
13
27
Al
injected particles
Npart=
per booster cycle,
charge-exchange target: Au
3 1010
g/cm2.
•
4
We
increase of longitudinal
5 10see
temperature due to IBS
maxwellization.
37
Numerical modeling of multi-turn injection in TWAC 2.
38
Summary
1. Using semi-Gaussian model it is derived longitudinal integro-differential FP
equation, which can be solved using grid or macro-particle method.
2. For matched beam IBS is described by three-dimensional integrodifferential FP equation for invariant-vector.
3. It is developed multi-particle code MOCAC, which allows us to solve FP
equation using “binary collisions” method.
4. Validation of the code has shown its long term stability for appropriate
choice of numerical parameters.
Future plans:
•
Development of new IBS “collective maps” and creation of map library.
•
Investigations of convergence and benchmarking.
•
Comparison with the experiment.
•
Application to GSI project.
39