On the consideration of interactions in the structures forming the axially symmetric beams of charged particles Текст научной статьи по специальности «Физика»

UDC 517.557 Vestnik of St. Petersburg University. Serie 10. 2014. Issue 4

S. A. Kozynchenko, V. A. Kozynchenko

ON THE CONSIDERATION OF INTERACTIONS IN THE STRUCTURES FORMING THE AXIALLY SYMMETRIC BEAMS OF CHARGED PARTICLES*)

St. Petersburg State University, 7/9, Universitetskaya embankment, St. Petersburg, 199034, Russian Federation

The problem of calculating the Coulomb field of charged particle beam in the injection systems is considered. To calculate the internal field of the beam we use both numerical and analytical methods of solving boundary value problem for the Poisson equation. The first, numerical, method consists in solving the Poisson equation by the finite difference method for the beam field potential with the boundary conditions on the electrodes of the accelerating structure, which depend on the actual configuration of the structure. For the analytical method the axially symmetrical beam of charged particles is represented by a set of annular cylinders. At each cylinder, the transverse beam charge density is assumed to be constant, and the longitudinal density is modeled by a trigonometric polynomial. For each cylinder, the Poisson equation is solved analytically with boundary conditions for the potential in the metal tube of a constant radius. An effective algorithm with parallel computing is proposed for the analytical method of calculation of the internal field. Bibliogr. 11. Il. 4. Table 1.

Keywords: mathematical modeling, accelerators of the charged particles, beams of the charged particles, Coulomb field of the charged beams, calculation of an internal field of beams.

С. А. Козынченко, В. А. Козынченко

ОБ УЧЕТЕ ВЗАИМОДЕЙСТВИЯ В СТРУКТУРАХ ФОРМИРОВАНИЯ АКСИАЛЬНО-СИММЕТРИЧНЫХ ПУЧКОВ ЗАРЯЖЕННЫХ ЧАСТИЦ

Санкт-Петербургский государственный университет, Российская Федерация, 199034, Санкт-Петербург, Университетская наб., 7/9

В работе рассматривается задача расчета кулоновского поля пучка заряженных частиц в системах инжекции. Для расчета внутреннего поля пучка применяются численный и аналитический методы решения краевой задачи для уравнения Пуассона. Численный метод заключается в решении методом сеток уравнения Пуассона для потенциала поля пучка с граничными условиями на электродах ускоряющей структуры, учитывающими реальную геометрию структуры. Для аналитического метода аксиально-симметричный пучок заряженных частиц представляется набором кольцевых цилиндров. В каждом цилиндре поперечная плотность заряда пучка предполагается постоянной, а продольная плотность моделируется тригонометрическим полиномом. Для каждого кольцевого цилиндра аналитически решается уравнение Пуассона с граничными условиями для потенциала в металлической трубе постоянного радиуса. Для аналитического метода расчета внутреннего поля предложен эффективный алгоритм с использованием параллельных вычислений. Библиогр. 11 назв. Ил. 4. Табл. 1.

Ключевые слова: центр коллективного пользования, точные штрафы, необходимые условия оптимальности.

Kozynchenko Sergey Alexo,nd,rovich — candidate of physical and mathematical sciences, senior fellow; e-mail: sergey_kozyntchenko@hotmail.com

Kozynchenko Vladimir Alexo,nd,rovich — candidate of physical and mathematical sciences, associate professor; e-mail: vladkoz@mail.ru

Козынченко Сергей Александрович — кандидат физико-математических наук, старший научный сотрудник; e-mail: sergey_kozyntchenko@hotmail.com

Козынченко Владимир Александрович — кандидат физико-математических наук, доцент; e-mail: vladkoz@mail.ru

*) Work supported by St. Petersburg State University (grant N 9.38.673.2013).

1. Introduction. Currently, linear and circular accelerators, based on different principles of acceleration of charged particles, are widely used in various fields. In this regard, the increasing attention has been paid to the problems of designing and creating the accelerator complexes that provide a formation of precision beams [1-11].

A design of beam formation system largely determines the output characteristics of an accelerating complex. Therefore, when creating accelerators for certain applications, the issues of designing and creating the systems of formation of the charged particle beams play an important role, such as injectors of linear accelerators with spatially uniform quadrupole focusing or cyclotrons. For designing such systems, it is necessary to carry out the simulation and optimization of beam dynamics in electromagnetic fields, which requires developing and improving both mathematical models of charged particle beams and beam dynamics optimization techniques [7-10].

When calculating the beam dynamics, the standard approach is that the beam is modeled by an ensemble of model particles (method of macro particles), a consideration of the Coulomb interaction between the particles being of great importance. The most effective methods are numerical ones for calculating the Coulomb field of a charged particle beam, especially the grid-based numerical solution of boundary value problem for the Poisson equation. However, numerical methods for calculating the beam field are not applicable in optimizing the dynamics of charged beams using methods that require analytical representations of both the internal and the external fields in accelerating structures [10]. Therefore, it seems to be topical to develop models that allow an analytic representation for the Coulomb field of charged particles.

In the present study, we use two methods to calculate the Coulomb field of the systems which form the low-energy axially symmetric beams of charged particles. The first, numerical, method consists in solving the Poisson equation by the finite difference method for the beam field potential with the boundary conditions on the electrodes of the accelerating structure, which depend on the actual configuration of the structure. The second, analytical, method of calculating the Coulomb field of the beam is presented in the paper [11]. The axially symmetrical beam of charged particles is represented by a set of annular cylinders. At each cylinder, the transverse beam charge density is assumed to be constant, and the longitudinal density is modeled by a trigonometric polynomial. For each cylinder, the Poisson equation is solved analytically with boundary conditions for the potential in the metal tube of a constant radius. Such a model provides the possibility of parallel computations of the Coulomb field. Some results of the parallel computation are presented in this paper.

2. Dirichlet problem for the Poisson equation in accelerator injection systems. Let us consider a problem of finding the own field of an ensemble of charged particles in a domain G of a structure consisting of ne round electrodes in the form of thick disks with given potentials U^,...,U^e.

The potential y (x, y, z) of the beam field, defined and continuous in G, is a solution of the boundary value problem for the Poisson equation:

dV (x,y,z) d2f (x,y,z) dV (x,y,z) p (x,y,z)

da? + V + = —' (!)

f (x,y,z) |rG = fo (x,y,z), (2)

where = |Jk rk - boundary of G consisting of piecewise smooth curves rk; f0 - known function. We consider an area G restricted by the dotted line shown in fig. 1. The boundary

rG (u) = (Jr is defined by endpoints of the curves r, such as segments of straight lines, arcs, etc. The part of curvilinear boundary of the i-th electrode between the points A and B, i = 1 , ne, consists of curves Ypi p = rip, rip, 2 < rip, rip < rip — 1 (fig. 1), in which the boundary condition (2) will be:

cp{x,y,z) |r = f/e, j = 1 ,ne.

Fig. 1. Cross-sections of the electrode system in the planes XZ and YZ

Area G within the electrode system is edged by the dotted line (fig. 1). Cross-sections of the electrodes are shown in dark gray, here Uf , ...,Uf - potentials of the electrodes.

In the gaps between the electrodes on the straight sections of the boundary rq, located between points C and D (fig. 1) q = n£, nf,, 3 < nf, n^ < nr — 2, the condition (2) can be written as follows:

v (x,y,z) |r, = U-1 + (Uf - UU)

z — z,

i-1

zf - z,

i-1

On the sections of the boundary (u), other than sections Tp and Tq, the boundary conditions (2) are expressed as follows:

V (x,y,z) |ri = ^q,

ip (x, y, z) |r2 = ^o + (f/f - v0) ■

e

zz

z — z,

e

nr—k—i

-k-i

V (x,y,z) Ir„r_fc+1 = ue

- nr — k + 1 ne 7

here - given value; x, y, Z are fixed.

Next we consider the solution of the boundary value problem (1), (2) by the finite difference method [2]. Let us associate the computational domain G with a set of discrete points (grid) Qh, formed by intersection of the planes parallel to the coordinate axes: x = xi, y = yj, z = zk, i = 0,nQ, j = 0,nQ, k = 0,nQ. The node (i,j,k) with coordinates (xi,yj,zk) is referred to as an internal node relative to the domain G if (xi,yj,zk) £ G,

x

xi xi— 1 ;

and as an external one otherwise. Let us introduce the notations: h1jk = h

j = hj-i = yj - yj-i; j = hx = xi+i - xi; hj = hVy = yj+i - yj; hj = hk-i =

zk - zk-i; h6jk = hzk = zk+i - zk; h = maxj{suphljk}; - set of internal nodes; Qh - set of external nodes; Qr - set of boundary points of the grid (points of intersection of coordinate grid lines with the boundary r).

On the set of nodes Qh we specify a grid function V = {V (x:i, yj,zk)} = {Vijk} as a set of values, which can be presented in the form of nh-dimensional vector, nh - the total number of nodes considered. The boundary value problem (1), (2) can be written in an operator form [2]

LV = g, (3)

where L - differential operator defined on {v (x, y,z)}.

The differential problem (3) is associated with a difference boundary value problem, which is equivalent to a system of linear equations [2]:

LhV = f,

here V, f - nh-dimensional vectors whose components are defined as values of the grid function at the grid nodes Qh; Lh - finite difference operator defined on the grid functions V.

At the given grid, the following seven-point finite difference approximation to the Poisson equation is considered [2]

where

(^)jk = ((Ax + Ay + Az) V)ijk = 0, (4)

/a ¡\ _ 2ipi-i,j,k 2 V>i,j,fc 2V>i+i, j, k

V x rJijk IX tlx , ,x\ UX UX '

ijk hu(hu + hf) hf_,hf hf(hz_1 + ht)

>2

(dH\ , fef-fef-i^V^ , (fefr - feffef-1 + (fef-i) = [d^)t +-§- +-12- [d^)+0{h)- (5)

Difference operators Ay and Az are defined similarly to Ax. The difference equation (4) approximates the Poisson equation (1) with an error of the first order for a non-uniform grid, and the second order - on a uniform grid [2]. At the nodes near boundary, to construct the difference analogue of the differential operator on a uniform or non-uniform grid the nearest points from Qr are used. In this case, a system of difference equations contains

e

- znr

k

the values of the grid function at all internal nodes, as well as at the boundary nodes of the grid, and in the latter case the boundary condition (2) is approximated exactly [2]

^(xi,yj,Zk) = po(xi,yj,Zk), (Xi,yj,Zk) e (6)

The error of approximation of the difference equation at a node near boundary, as well as the error of the difference problem (4)-(6), is defined by O(h) [2].

A system of linear equations being equivalent to the difference boundary value problem can be written as follows [2]:

A^ = f,

here ip = {ipk}, f = {fk} - vectors, k = 1, nh; A = {a,ki} - nonsingular square matrix of order nh.

General view of a linear iterative method represented in the form [2]

^n+1 = Bn^n + Gn. (7)

In (7) Bn = E — HnA - transition matrix from the n-th to (n + 1)-th iteration (iteration step operator), H0, H1, H2,... - some sequence of the matrices.

For solving the system of difference equations the successive over relaxation (SOR) method can be used. Assume that A - symmetric matrix with diagonal block tridiagonal representation (A = D + M + N, where D - diagonal matrix, M - lower triangular matrix, and N - upper triangular matrix). Then the SOR method is defined as follows [2]:

(D + M) • 4>n+1 = w • f — (N — (1 — w) • A) • 4>n,

here ^n - n-th approximation of the solution (n-th iteration), w - relaxation parameter. The solution is carried out by SOR method when the order of the ^ components is consistent with the view of the tridiagonal matrix A.

3. Mathematical model of a cylindrical beam of charged particles in a metal tube. We assume that the beam is unlimited and periodical in the longitudinal coordinate. Assume also that the beam is in a coaxial circular metal tube of radius a and has azimuthal symmetry. To determine the Coulomb field we consider the beam as a set of annular cylindrical coaxial layers. Each layer is non-uniform in the longitudinal coordinate and in each cross-section the layer density is constant. The intensity vector of the beam Coulomb field is calculated as the sum of the intensity vectors of each layer:

N

E = £ Ei,

i=0

where Ei - Coulomb field intensity vector of the i-th annular layer; N - number of annular layers, zero layer - axial cylindrical layer. This model allows to take into account both longitudinal and transverse beam heterogeneity under calculation of its Coulomb field.

We introduce a cylindrical coordinate system (z,6,r), where Oz axis coincides with the axis of symmetry of the tube. We assume that an annular layer has an azimuthal symmetry, i.e. coordinates and velocities of the particles do not depend on the polar angle 0. We assume also that the space charge density within an annular cylinder is a periodic function of the longitudinal coordinate Z and is a constant in the cross-section. In this case, the potential u (z, r) satisfies the Poisson equation

1 <9 / <9w(z,r)\ ^ d2u(z,r) r dr \ dr J dz2 ' '

here

( 0,r < Ri,

0 [ 0,r>R2

where t (z) - charge of the beam per unit length, and the boundary conditions

u(z, a) = 0 yz € R,

u(z, r) = u(z + L, r) yz € R, yr € [0, a],

= Vz €*,*<=[<>, a].

We assume that functions u(z,r), du(z,r)/dr, du(z,r)/dz are continuous at r = R1 and r = R2. We introduce the notation:

Ír¡(r), 0 <r < R1, v(r), Ri < r ^ R2, w(r), R2 < r ^ a.

Because of the beam periodicity along the longitudinal coordinate, we consider the function f (z,r) as periodic in z with period L. For modeling of the function f (z,r) we introduce on the axis Oz the grid S = {z¿ = hi, h = L/N, i = 0, jV} with number of nodes N. At the nodes of the grid S we define the function of the beam charge per unit length t (z) based on the location of model particles. By formula (8) we calculate the value of a function f (z, 0) at the nodes of the grid S. We model the function f (z,r) by trigonometric polynomial whose values at the nodes of the grid S coincide with the known ones fi = f (zi, 0):

1 M

r) = -/0C + (/fc(r) cos(wfcz) + fk{r) sin(ujkz)), k=i

0, 0 <r < R1, ( 0, 0 <r < R1,

here cuk=2-f-,M= fck(r) = { f-, R, < r < R2, fsk(r) = I R1<r^ ñ2,

0, R2 <r < a, [ 0, R2 <r < a,

,c 2 -s-^ 2 -Z—1. 2nki 2 ^z—1? . 2nki

fo = w/Efi, fk = jf !2 fiCos—> f* = wz^fiBmifr- (9)

z i=0 z i=0 z z i=0 z

Then the expressions for the longitudinal and transverse components of the intensity vector of the Coulomb field of the cylindrical axially symmetric beam taking into account heterogeneity and periodicity of the charge density in the longitudinal coordinate will look like:

du(z r) _

Ez (z,r) =--^— = (Uck{r)s\nujkz -Usk(r) COS UJkz),

dz

k=i

Er (z,r) = -

du(z,r) 1 du0(r)

dr

2 dr

Y^ f duk(r)

k=l

duk (r) . \

COS UJkZ-\---,- Sin UJkZ

dr dr

where the functions uck(r), usk(r),

duo(r) duk(r) „ duk(r)

dr

dr

dr

calculated by the formulas

(2 (Rj - Rfj hi^ + 2R21hi^ + Rl-Rjj, 0 < r < Ru ]{r) = { _£Sil + ^Jl\nr+îl (2 (Rf _ 2R\\nR2+Rl), Ri < r < R2,

f CC3I0 (fa kr), 0 < r < Ri,

\{r) = <j frk + C\I0far) + C2K0M), i?i < r < i?2 fC (C\Io(ukr) + C6Ko(fakr)), R2 < r < a,

fCIo far), 0 < r < Ri,

[M = <| fsk + CJoM + C2K0far)), R\ f S(C5kIo fa r) + C6Ko(uk r)), R2 < r < a,

< r < R2

duO(r) dr

0, 0 < r < R

, RjfS

2 "T"

ÎL

2 r

2r '

R - Rl), R2 < r <

Ri < r < R2, a,

dut(r) dr

dr

fCfakCiIi far), 0 < r < Ri,

ffa (Ci Ii fa r) + C2Kifa r)), Ri < r < R2,

fCfak (C 5,Ii(fakr) + C6Ki(fakr)), R2 < r < a,

ffaC3Ii far), 0 < r < Ri,

ffa (CiIifa r) + C2Kifa r)), Ri < r < R2, fSfak (C5Iifar) + C6Ki(fakr)), R2 < r < a,

C2 =

Ii fa Ri)

fai (Ko fa Ri) Ii fa Ri) + Ki fa Ri) Io fa Ri)) '

C\ — C2— —

2

fak aY

Io ( fak R2 )

(Io fa R2) Ko fa a) - Io fa a) Ko fa R2))

_h {ukR2)_

(Ii (fakR2 ) Ko fa a) + Io fa a) Ki fa R2)) '

_K1 (mkR2)_

(Ii fa R2) Ko fa a) + Io fa a) Ki fa R2))

_Kg fai^)_

(Io (fakR2 ) Ko fa a) - Io fa a) Ko fa R2)) '

u

2

u

u

1

a

a

T ( -DMT < \ T ( \TT ( T> \ n Cl II (Uk Ri) - C2 Ki (Uk Ri)

7 = /0 {ukR2) K0 (uka) - /0 {Uka) K0 (ukR2), C3 = -—(--,

Ii (UkRi)

Ko (uka) (CiIi (ukR2) - C2Ki (UkR2)) ~ _ ~ Io (Uka)

— ~T~/--r> \ ts /-—-\~F7~i-—

Ii (UkR2) Ko (Uka) + Io (Uka) Ki (UkR2 V Ko (Uka)'

here fC, fC, f S defined by (9).

4. On the use of an annular cylinder model for considering the interaction in injection systems of accelerators. This section discusses the applicability of the annular cylinder model, considered in section 3 of this paper, for the calculation of a beam field in injection systems performed both with and without use of parallel computing. We consider an injection system of linear accelerator of ions H- consisting of ne = 5 electrodes with potentials Ue = ...= = 0. In the injection system, an ensemble of the charged particles is simulated consisting of five "bunches" (for taking into account a beam periodicity along the longitudinal coordinate) and having both a longitudinal and transverse heterogeneity, as well as an azimuthal symmetry relative to the symmetry axis Oz (see fig. 2). The cross-section in XZ-plane is shown. In the "bunch" cross-section, the charge density has a normal distribution.

Fig. 2. Ensemble of ions H in the injection system of a linear accelerator

To test the applicability of the annular cylinder model the field of considered ensemble of charged particles has been computed. Some results of comparison of the longitudinal components of the field of the particle ensemble acting on the axis of symmetry within the central bunch are shown in fig. 3. The calculations were performed both using an annular cylinder beam model and by solving the boundary value problem for the Poisson equation by the finite difference method.

Performed also were calculations of the longitudinal component of the field of the particle ensemble, which acts on the axis of symmetry of the infinite cylindrical tube of radius a, with a different number of annular cylinders. The results of calculations are shown in fig. 4. In this case, we consider an ensemble of charged particles consisting of one central bunch (see fig. 2) having a normal transverse charge density distribution. It is seen from fig. 4 that, in case of an inhomogeneous radial beam and 6 cylinders, the longitudinal field component on the axis of symmetry of the injection system can increase three-fold as compared to the same value when using one cylinder, which substantiates the usage of the annual cylinder model to calculate the beam field in injection systems.

We investigated a possibility of using parallel computing based on the MPI-1 communication protocol to simulate a field of the ensemble of charged particles using

F(z),H

7.46e-012 6.63e-012 5.81e-012 4.98e-012 4.15e-012 3.32e-012 2.49e-012 1.66e-012 8.29e-013 0

-8.29e-013 -1.66e-012 -2.49e-012 -3.32e-012 -4.15e-012 -4.98e-012 -5.81e-012 -6.63e-012 -7.46e-012

Fig. 3. Graphs of the longitudinal component of the field force of an ensemble of charged particles acting on oZ-axis of system symmetry

the annual cylinder model. Run times of computing the longitudinal component of the field of the particle ensemble along oZ-axis both with and without the use of parallel computing for different numbers of annual cylinders are shown in table. In the case of parallel computations for the cylinders being involved to a beam, both fields and forces are calculated in parallel in separate processes.

The run time of computing the longitudinal component of the field of ensemble of charged particles at Oz-axis for various numbers of annular cylinders

The number cylinders Time without MPI, s Time with MPI, s

2 3.272 4.197

4 4.025 1.973

8 5.576 1.037

16 8.687 0.798

32 15.368 0.751

64 28.205 0.690

Table shows that with increasing the number of cylinders the run time of calculating the field with parallel computing is reduced compared to the run time

F(z), H

число колец 6

- число колец 5

- число колец 4 число колец 3 число колец 2

- число колец 1

6.34е-012 5.64е-012 4.93е-012 4.23е-012 3.52е-012 2.82е-012 2.11е-012 1.41е-012 7.05е-013

-7.05е-013 -141е-012 —2.11е-012 -2.82е-012 -3.52е-012 ^.23е-012 ^.93е-012 -5.64е-012 -6.34е-012

Fig. 4. Graphs of the longitudinal component of the field of the ensemble of charged particles acting on the oZ-axis for various numbers of annular cylinders

without parallelization. This demonstrates the applicability and efficiency of the parallel computation based on the MPI-1 to calculate the fields of injection systems using annular model.

The studies carried out show the suitability of the annular model for calculation of the field of charged particle ensembles in the injection systems of accelerators.

5. Conclusion. The paper presents the calculations of the internal Coulomb field of the ensemble of charged particles in the injection system using both analytical and numerical methods for modeling the internal field. Comparison of the results of calculations using numerical and analytical modeling techniques shows the effectiveness of the analytical method of calculation in injection systems, including the effectiveness of parallelization.

References

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7. Kozynchenko S. A., Ovsyannikov D. A. Optimization mathematical models of beam dynamics in the injection systems with real geometry. 4th Intern. Scientific Conference on Physics and Control, PhysCon-2009, 1-4 September 2009. Catania, Italy (www.physcon2009.diees.unit.it).

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Литература

1. Bondarev B, Durkin A., Ivanov Y., Shumakov I., Vinogradov S., Ovsyannikov A., Ovsyannikov D. The LIDOS.RFQ.Designer development // Proc. of Particle Accelerator Conference. 2001. Vol. 4. P. 2947-2949.

2. Ovsyannikov D. A. Mathematical modeling and optimization of beam dynamics in accelerators // RuPAC Contributions to the Proceedings - 23rd Russian Particle Accelerator Conference. 2012. P. 68-72.

3. Drivotin O. I., Ovsyannikov D. A. Self-consistent distributions for charged particle beam in magnetic field // Intern. J. of Modern Physics. 2009. Vol. A24. P. 816-842.

4. Drivotin O. I., Ovsyannikov D. A. Modeling of self-consistent distributions for longitudinally non-uniform beams // Nuclear Instruments and Methods in Physics Research, section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 2006. Vol. 558, issue 1. P. 112-118.

5. Drivotin O. I., Ovsyannikov D. A. Particle distributions for beam in electric field // Proc. of the IEEE Particle Accelerator Conference. 1999. Vol. 3. P. 1857-1859.

6. Ovsyannikov D. A., Ovsyannikov A. D., Antropov I. V., Kozynchenko V. A. BDO-RFQ code and optimization models // Proc. of Intern. Conference Physics and Control. 2005. P. 282-288.

7. Kozynchenko S. A., Ovsyannikov D. A. Optimization mathematical models of beam dynamics in the injection systems with real geometry // 4th Intern. Scientific Conference on Physics and Control. PhysCon-2009. 1-4 September 2009. Catania, Italy (www.physcon2009.diees.unit.it).

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The article is recommended for publication by prof. D. A. Ovsyannikov. The article is received by the editorial office on June 26, 2014. Статья рекомендована к печати проф. Д. А. Овсянниковым. Статья поступила в редакцию 26 июня 2014 г.