1. Kinetics effects in multiple intra-beam scattering.
P. R. Zenkevich, A. E. Bolshakov,
ITEP, Moscow, Russia

2. 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.
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3. Here we consider only the multiple one!
Introduction1
Two kinds of IBS:
Multiple IBS.
Single-event IBS.
Here we consider only the multiple one!
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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4. Coordinates-momentums vectors.
Let us introduce “coordinate vector” and “dimensionless momentum vector”:
Here z – is longitudinal coordinate, is longitudinal coordinate of the equilibrium particle, x is horizontal coordinate, y is the vertical one.
p is the particle momentum, is its deviation from the equilibrium momentum.
Here is are, correspondingly, the horizontal and
vertical components of the particle momentum.
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5. Invariants of motion in circular accelerators.
Linear non-coupled particle motion is described by conservation of “invariants”:
Here for m=2, 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)
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6. Invariants evolution due to multiple IBS.
In scattering event the coordinates does not change; therefore we have:
Here
Friction coefficients and diffusion coefficients will be discussed later.
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7. Gaussian models.
In Gaussian model it is assumed that distribution function is Gaussian one on all degrees on freedom, i.e.
Then evolution of each moment of the distribution function is defined by the averaging on the test and field particles:
Let for the “test” particle and for the field particles.
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:
Here functions are one dimensional integrals (in B-M model) or two-dimensional integrals (in P-M model), t is “slow” time since usually IBS time is much more than revolution period.
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8. Differential equations for analysis of r.m.s. beam parameters
Averaging on the ring we obtain the final result:
Here , 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.
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9. 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):
Here is the distribution function on u, friction coefficient and diffusion coefficient D are constants. A stationary solution has Gaussian form:
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, ).
3. If takes place transient process with initial condition, which is not Gayssian one:
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.
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10. 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 momentum-coordinate space or in the “invariant space”.
In momentum-coordinate space FPE has the following form:
Here is the distribution function; are components of the friction force; 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:
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11. FPE in momentum space 2. Components of the diffusion tensor
Rate of the second moments evolution
Here is Kronecker-Kapelli symbol (we take in mind summation on repeating indices) ,
is Coulomb logarithm, and
is the beam distribution function in phase space.
Constant , where ri is the ion classical radius.
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12. Fokker- Planck equation in invariant space.
FPE equation in invariant space (here scalar ) is:
This equation takes place if the distribution on phases is uniform on interval
This equation should be solved with corresponding initial and boundary conditions. Initial condition:
Examples of boundary conditions:
Absorbing wall boundary condition
In the simplest case
In general case the boundary conditions are defined by three-dimensional surface in the invariant space
2. “Reflecting wall” boundary condition for zero invariant
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13. Evolution of Invariants and their Moments
Then we obtain:
Here kernels have form of four-dimensional integrals on phases and longitudinal variable s. Example (for m=1,3)
Here
Function
For m=1,3
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14. 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 :
, , , ;
2) using “ locality condition” we exclude and find dependence ;
3) then we change averaging on momentum of the field particles by averaging on its invariant-vector , using the expression:
(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:
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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15. 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:
Here are three random numbers with Gaussian distribution and unity dispersion; coefficients 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 :
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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16. Approximate 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 fiction force and diffusion coefficients on test and field particles .Then we obtain:
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P. Zenkevich

17. 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” 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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18. 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.
Then
Averaging on , we can derive one-dimensional (longitudinal) FP
equation:
Let us assume that: 1) dispersion function is equal to zero; 2) horizontal and vertical
r. m. s. emittances are equal. ,
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19. Longitudinal FPE 2
Expressions for longitudinal friction force and diffusion coefficient are:
The kernels are
For high transverse energy ( ) the friction force disappears
( ), and diffusion coefficient tends to constant ( ).
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).

20. Dependence of kernels for the friction force and diffusion coefficients on parameter t=u-w : blue curve =1, red curve =2 and green curve =3.

21. Binary Collision Map 1
Let us choose the scattering angle according to expression
Then work of the friction force and increase of moments because of the diffusion terms coincide with the corresponding exact values:
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).
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22. Structure of program MOCAC

23. Module 1: Transformation from invariant to momentum-coordinate

24. 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
P. Zenkevich

25. 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
, T=620 MeV/u, the booster frequency=1Hz,
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).
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26. Final momentum distribution for HESR calculated by MOCAC code.
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 non-Gaussian beam tails.
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27. Results of numerical IBS simulation for TWAC storage ring 1.
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28. Results of numerical IBS modulation for TWAC storage ring 2.
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29. 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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30. 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
Here are the phase advances per unit length. For string model the space charge potential
Here is the perveance ( ), is a distance between wires i and j.
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31. MOLECULAR DYNAMICS (string model) 2.
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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32. MOLECULAR DYNAMICS (string model) 3.
The results of numerical modeling of Montague resonance by C.Benedetti ( )
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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33. 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
where LC is the ring circumference. For cooled beam and limited number of the stored particles in a ring ( ) 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:
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34. Periodical model, crystalline beam 1.
String (ion < 0.709)
Zigzag (0.709 < ion < 0.964)
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35. Periodical model, crystalline beam 2.
Helix or Tetrahedron (0.964 < ion < 3.10)
Shell + String (3.10 < ion < 5.7)
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36. Conclusions and acknowlengements.
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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