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CONTINUOUS ANALOGUE OF THE NEWTON METHOD IN MATHEMATICAL MODELING OF NONLINEAR PARTICLE DYNAMICS PROBLEMS IN THE
ACCELERATOR Polyakova R.V.1, Kovalenko A.D.2, Yudin I.P.3 (Russian Federation)
'Polyakova Rimma Vasilievna - student, 2Kovalenko Alexander Dmitrievich - student, 3Yudin Ivan Pavlovich - student, JOINT INSTITUTE FOR NUCLEAR RESEARCH (JINR), DUBNA
Abstract: an algorithm of the continuous analogue of Newton's method (CANM) is proposed for solving the boundary value problems of beam dynamics in accelerators.
The following results of the numerical simulations of three problems for accelerators are given:
1) Computation of the turn-focusing transport systems at a rapid output of the beam from the accelerator;
2) Optimization for the long "invisible" straight sections (insertions) of the accelerator and coherent analysis of nonlinear aberrations in them;
3) Finding ofperiodic orbits in the accelerator taking into account actual distortions of the magnetic field.
The CANM method proposed in this paper to solve a first problem, allows one to choose in the best way the parameters of the elements of transportation and to arrange as well as to estimate the tolerances on these parameters. The second problem deals with a study of nonlinear aberrations in quadrupole lenses of the "invisible" section of the accelerator and its matching by taking these nonlinearities into account. When solving a third problem, results have been obtained for correction of the orbit of the accelerator for various radii R by using a system of coils with current, located in the straight sections of the accelerator, which create an additional radial component of the magnetic field.
Keywords: particle dynamics equations, Continuous Analogue of the Newton Method, boundary-value problem of charged particles transportation.
НЕПРЕРЫВНЫЙ АНАЛОГ МЕТОДА НЮТОНА В МАТЕМАТИЧЕСКОМ МОДЕЛИРОВАНИИ НЕЛИНЕЙНЫХ ЗАДАЧ ДИНАМИКИ ЧАСТИЦ В
УСКОРИТЕЛЕ
Полякова Р.В.1, Коваленко А.Д.2, Юдин И.П.3 (Российская Федерация)
1'Полякова Римма Васильевна - студент, 2Коваленко Александр Дмитриевич - студент, 3Юдин Иван Павлович - студент, Объединенный институт ядерных исследований (ОИЯИ), г. Дубна
Аннотация: дано описание моделирования краевых задач динамики частиц в ускорителях с использованием численных алгоритмов на базе непрерывного аналога метода Ньютона (НАМН). Приводятся результаты моделирования трех задач:
1) расчет поворотно-фокусирующих систем транспортировки при быстром выводе пучка на ускорителях,
2) оптимизация длинных согласованных промежутков ускорителя и анализ нелинейных аберраций в них,
3) нахождение периодических траекторий в ускорителе с учетом реальных искажений магнитного поля. Метод НАМН, предлагаемый в данной работе для решения первой задачи, позволяет оптимальным образом подобрать параметры элементов систем транспортировки и произвести их расстановку, а также сделать оценку допусков на эти параметры. Во второй задаче было проведено исследование нелинейных аберраций в квадрупольных линзах «невидимого» промежутка ускорителя и согласование его с учетом этих нелинейностей. При решении третьей задачи, получены результаты для коррекции орбит ускорителя на разных радиусах R с помощью системы витков с током, расположенных в прямолинейных промежутках ускорителя, которые создают дополнительную радиальную составляющую магнитного поля.
Ключевые слова: уравнения динамики частиц, непрерывный аналог метода Ньютона, краевая задача транспортировки заряженных частиц.
Introduction
The creation of new accelerators and the reconstruction of the existing ones require solving nonlinear problems, the mathematical modeling of which leads to important and interesting mathematical problems. Many of them are so complex nonlinear problems, including inverse ones, that the only way to study them is to develop numerical algorithms and implement the latter on a PC. These problems include, in particular, the task of determining the optimal (in the sense of some quality criteria) parameters of the systems for transporting charged particles, taking
into account various kinds of nonlinear effects, the task of calculating coordinated "invisible" rectilinear gaps of the accelerator and the problem of pulse recovery (p ) particles recorded by proportional cameras, and that of finding periodic trajectories in the accelerator, considering real distortions of the magnetic field. All of the above-mentioned physical problems are mathematically reduced to the solution of a boundary-value problem for nonlinear ordinary differential equations of the second order. In a number of physical problems [1], the efficiency of a continuous analogue of the Newton method (CANM) [2] was practically shown. The authors of this work found it natural to develop this method for solving nonlinear problems of the dynamics of charged particles, which allows one to choose parameters and arrange elements of physical systems, as well as to evaluate tolerances on them and give specific recommendations for solving the assigned physical problems. Using the proposed method, the following model physical problems were solved:
• calculation of rotary focusing transportation systems for fast beam extraction at accelerators;
• optimization of long coordinated accelerator gaps and analysis of nonlinear aberrations in them;
• finding periodic trajectories in the accelerator, taking into account real distortions of the magnetic field.
Problem 1. Figure 1 shows the diagram of one of the possible options for the accelerator transportation system.
The motion of charged particles in the magnetic field of such a system in the rectangular coordinate system is described by the complete equations:
d2 x A
ds2 BR d2 y _ A
ds2 " BR
s, X y, x's, y 's, Bs , Bx , By , P ) ^ X ^ x' s, y' ^ Bs , Bx , By , Pi )
(1)
where BR is the magnetic hardness of the particle and ^^ = p , p is the momentum of the particle,
A = ^ 1 + (x's)2 + (y's)2 , P are the parameters, whose physical and mathematical meaning is determined in each particular case. The field components B(B, B, B ) in each specific physical case are defined analytically or numerically if the field is specified in the form of a table. Mathematically, the boundary-value problem is formulated as follows: it is necessary to choose parameters of the system P so that the motion trajectory of charged particles satisfies Eq. (1) and the boundary conditions:
x(so, p ) = Xo, X 's(so, p ) = tga0,
y( s0 , P ) = yo , y 's( s0 , P ) = të^
x(sk, p ) = xk = a x ' S(sk, p ) = tg^k = c
y( sk, P ) = yk = b, y ' s(sk, P ) = tg«ki= d-
eee <
fi(sk, P ) = Xk (sk, P ) - a = o,
f2( sk, P ) = yk (sk, P ) - b = o, f3(sk, P) = x'Sk(sk, P)-c = o, f4 ( sk, P ) = yJk (sk, P ) - d = o.
(2)
Fig. 1. Coordinate system adopted in the calculations.
The direction and coordinates of the beam of charged particles at the entrance of the transportation system are set - s0,x0, y0), a0 is the angle in radians between the tangent to the trajectory projection at the point (
s0, x0, y0) onto the "SX" plane and the "S" axis; a01 is the angle in radians between the tangent to the trajectory
projection at the point (s0, x0, y0) onto the "SY" plane and the "S" axis. It is necessary to choose the parameters
( /{, /'-,, /' • • •; in particular, for example, p is the particle momentum), so that for the given initial position and
direction of the particle, its specified final position and direction is (ak, ak]_, sk, xk, yk), ak and akl are the
same as a0 and a01, respectively, at the point (sk, xk, yk). In our case, it was assumed that two rotary magnets M were included in the transportation system, the field gradient of such magnets was large enough f dBR ^
I n = = _ 140 I, therefore, the beam direction at the system output substantially depended not only on the
induction ( B and B ) in the magnets, but also on their position in space (rotations around the axes and
inclinations). A detailed description of the CANM numerical method and of the mathematical modeling algorithm for such a transportation system is given in [2].
In the case when the variation parameter of the system is ounly p , i.e. the particle momentum, instead of the boundary conditions (2), we require the fulfillment of the condition:
F=f2+f2+f2+f2=0.
(3)
Equation (3) will be solved by a continuous analogue of the Newton method, based on which, assuming that p = p(t) , we have ^p(t)] = -F[p(t)],
dt
where ^ = 2(f f + f2 f + f3 f + f4 f
dt 1 dt 2 dt 3 dt 4 dt
f JAP, f = dkP, f=APr f JAP,
dt dP " dt dP " dt dP " dt dP "
di±-( \' \' df^-i \' f -( dP ~ (Xk' p' dP ~ p' dP ~ X k) p' dP ~ k) p'
(4)
The problem of fast beam extraction at the Serpukhov accelerator was also of certain mathematical interest, since it was reduced to the inverse boundary-value problem for a second-order differential equation in the polar coordinate system. Mathematically, the problem was set as follows: it is required to solve the equation
p »= + (r +p) -1 (r + p)2F(p) = f (p, p, p'), (6)
rc + p rx
where p = p(p) is the unknown function, p is the argument. Equation (6) is a model for particle motion in the accelerator magnetic field. The polar system is selected as the coordinate system: r = rc + p - polar radius; p -polar angle; r = rc (rc = const) - equilibrium orbit; p(p) - particle deviation from the equilibrium orbit; r -constant, F (p) - given function describing the accelerator magnetic field. Equation (6) is considered at the intervals [p,; p ], [p; p ], • • •, [(f)n ]; pn ], where p0 < p <■■■ <(pn. In the interval [p,; p ], the solution of Eq. (6) is known.
At the points p,,p,---, pn |, the solution p(<p) undergoes a gap, which is defined as follows. Let us consider the intervals [p ,; p ] and [p; p+1 ], (i = 1,2, •••,//-1). Let the solution of Eq. (6) and its derivative in the interval [p;p] at the point p = p have values p = pr(p), p' = p'l(pi), and the solution of the same equation and its derivative in the interval [p; pM ] at the point p = p have values p = p (p ), p' = p'¡(p ),
(l-left, r-right value at the point p p). The left and right values at the point p p are related by:
A(p ) = p(p ) + /p'i(p )
Pr ' ) =ai + p'l' ) (i = 1n -
Here l is the given constant, at for some i are equal to zero, and for other values i they are unknown
quantities, which will be discussed below. Among p(/ = 2,3,-••,//) there are several meanings <plk, k = 1,2,•••,m, (p.j < p.2 < • • • < p m = p), under which the boundary conditions are set for Eq. (6) of one of two types:
pi (p ) = pk oder pi(p k) = p'k, where pk, p'k are the given values. (8)
In Eq. (7) with p = p and p = p k, = 1,2,•••,/?? — 1). <x are the unknown values, and for other values p-^= 0. Let us consider the interval [p;pn ]. It breaks into gaps [p; p ], [p; p ], • • •, [p_j; p ], (p = p , Since the solution of Eq. (6) in the interval [p; p ] is assumed to be known, it follows from (7) that p(p) = p(p) + lp^(p). is known as well. At the point pn , a boundary condition of the type (7) is given. At the points p (z = 2,3, • • •, r -1), the conditions (8) are satisfied for ai = 0 . It is required to find the function p = p(p) and at so that the function p = p(p) satisfies the differential equation (6) in the intervals [p;p], [p; p ], • • •, [p i' P 1 • at 1'1C points (p = p (/ = 2,3, ■ ■ •, r — 1), the relations (7) are satisfied, and at the points p and p , the given boundary conditions (8) are done. At the intervals
[p2;pJ,-",[pm |;pn(]. the task is the same as for the interval [p;pj. To solve the above-described
boundary-value problem, CANM was used; it suggests that instead of Eq. (6), a system of partial differential equations should be considered:
[V" p2 + PV' p + QV = R, 4p
] p , p ,7 where P(p,t,p,pp) = f'p, (p,t,p,pp) = ^-pp, (9)
[ p t = V, p p p p rc +p
Q(p, t, p, p' p) = f' p = 2 -1 +1 [2(rc + p) F (p) + (rc + p)2 F' (p)],
p p (rc +p) rx
/ f \2
R(p, t, p, p f p) = -[p f p2 + f (p, p, p'p)] = -p'p + 2 + rs + p - i(rc + p)2 F (p).
p p p p r; +p i
In the system (9), the unknown functions V and p depend on p and the additional parameter t: V = V(p, t) , p = p(p, t). The system (9) is solved in the field p<p<p, 0 < t < . The boundary conditions are
defined as follows: V((, t) = 0, V(p, t) = 0 . At the points (, the function V(p, t) for any t satisfies conditions similar to (7). At t = 0, the initial value p(p, 0) = p(0)(p) is set. The accelerator magnetic field (function F (p)) was approximated by the functions:
F(p) = j Cjpn,l(P) f°r [P;PJ ('" =!,n)
P \cfn,2ip) for [p;pj (/ = 2,4,...,n2)
The polynomial coefficients P x (p) and P 2 (p) , as well as the constants Q and C2, were found by the
least squares method based on the table-defined functions F (p) and F (p) . A specific model problem was
solved for the Serpukhov accelerator; Figure 2 shows graphs of the calculated trajectories of the particles output at
different intervals [p; pr ], which satisfy the requirements of the beam extraction system with fairly high accuracy.
The proposed algorithm completely excludes the elements of intuition when solving a boundary-value problem of types (1,2) and (6,9) and allows one to make the selection of variable parameters of transportation systems with sufficiently high accuracy. The equations of particle transportation studied in this paper in Cartesian (1) and polar (6, 9) coordinate systems are model. However, they describe physical processes in accelerators quite well, and the results obtained by mathematical modeling are of some practical interest.
Fig. 2. Graphs of the design trajectories.
Problem 2. In modern hard-focusing accelerators, the organization of I/O (input-output) beam systems, the placement of an accelerating station, etc., require long free gaps (f. g.). In this paper, the calculation of f. g. is carried out, taking into account nonlinear affects in lenses using a continuous analogue of the Newton method. Moreover, we study not only the effect of the mismatch of the gap already agreed upon by linear calculation, but also, which is new, the free gap calculated according to nonlinear aberrations in quadrupole lenses. Figure 3, a shows one of the possible structures of the "invisible" gap for designing a 1.5-GeV superconducting synchrotron, for which studies and the analysis of the influence of nonlinearities in the lenses Fj, F2 and F3 were carried out on the coordination of such a gap.
Fig. 3, a. Structure of the "invisible" gap Fig. 3, b. Structure of the "invisible" gap with
nonlinear corrective elements N.
The physical problem is set as follows. It is required to choose gradients of the lenses, their lengths and drift spaces so that, according to the given coordinates and the direction of the beam at the input of f. g. (
x0, z0, x0, z0), we get the specified coordinates and the beam direction at the exit of f. g. (xk, z^, xk, zk), moreover, xk = x0, zk =— z0, xk = x0, zk = — z0 . This requirement is consistent with obtaining the phase incursion a \yx = 2n for horizontal and A y/z = n -- for vertical betatron vibrations. Mathematical Model of the Problem.
Differential equations describing the trajectory of a charged particle in the magnetic field in the selected Cartesian coordinate system (Fig.3, a) have the form:
z! = e xs Bs—(1+z;2 )Bx+xs zs Bz ]
P (10) x"; =—eji+zj^ [ bs—(i+x;2 )Bz+x'; z; Bx ] ,
where P = mv is the particle momentum, Bx, Bz, Bs are the components of the magnetic field. Equations (10) are valid for any magnetic field, since no simplifying assumptions were made when deriving them. Inside the quadrupole lens of considerable length, the magnetic field is practically independent of ; , i.e., Bs = 0, and the
decomposition of the field components B and B takes the form:
Bz = Gx[1 + d (5z4 — 10x2 z2 + x4) + d10 (9 z8 — 84 z6 x2 +126 x 4 z 4 — 36 z2 x6 + x8) + -] (11)
Bx = Gz[1 + d6 (5x4 —10 x2 z2 + z4) + do (9 x8 — 84 x6 z2 +126 x4 z 4 — 36 x2 z6 + z8) + -]
(12)
where G is the quadrupole gradient of the corresponding lens. This expansion contains only members of the degree 4n +1, n=0, 1,2, • • •. As z2 and x2 < 1, then from Eqs. (10), taking into account the formulas (11), we obtain the equation of motion of the charged particle in the lens in the form: G
x =-x[1 + d (5z4 —10 x2 z2 + x4) + d10 (9 z8 — 84 z6 x2 +126 x4 z4 — 36 z2 x6 + x8)]
Bp G
z = — z[1 + d (5x4 —10 x2 z2 + z4) + d10 (9x8 — 84 x6 z2 + 126x4z4 — 36x2 z6 + z8)] Bp
1 e n-2
where-= —, Bp is the magnetic particle hardness, d = r^ Cn, ^ is the relative magnitude of the field
Bp P
nonlinearity in the lens, r0 is the radius equal to the semi-aperture of the corresponding lens. Mathematically, the
long-gap coherence problem is reduced to the solution of a boundary-value problem for the system of nonlinear differential equations (10) with boundary conditions:
x(;0) = x0' z(;0) = z0' x(;k) = xk> z(;k) = zk>
(13)
x (;0) = x0, z (;0) = z0, x (;k) = z (;k) = moreover, the relation xk = x0, z^ = — z0, xk = x0, z^ = — z0 must be implemented. It is necessary to choose the parameters —, —, — (lens gradients), /j, l2, l3 (lens lengths), L, L, L (drift spaces) and L (length
of the free gap), so that the particle motion trajectory satisfies Eqs. (12) and the boundary conditions (13).
Results of Numerical Calculations. As a zeroth approximation of varied parameters, we took their values obtained using the linear calculation:
—0 = 8645 er/sm, G0 = 10879,79 er/sm, G30 = 10573,6 er/sm,
10 = 5,29 sm, 10 = 17,72 sm, l0 = 20,64 sm, L0 = 180,27 sm, L02 = 3,38 sm, Lg = 700,1 sm, / are the lens lengths, F (i = 1, 2, 3) , L are the drift spaces (see Fig. 3, a).
The dependence of the phase shifts a and a y/2 on the magnitude of the dodecapole nonlinearity in the lenses was calculated for C6 from 0.1 to 1%. The results of this calculation are shown in Fig. 4, from which it can be seen that with nonlinearity in 1% -lenses, phase incursions reach 7.71 +1.2° and 71 + 4 in the horizontal and
vertical planes for the maximum emittance s . The dependence of the phase shifts A \ and A \ on the beam
emittance s for C6 = 0,01 is shown in Fig. 5. In this problem, we performed numerical calculations of the gap
coherence by varying the gradients of the lenses ("linear" coherence) G2, G3, G2, G3 (G2, G3 are the gradients
of the lenses symmetrical to the F , F lenses). In this case, the coherence of the gap is achieved if the symmetric trajectory turns into a slightly asymmetric one (Fig. 6). The phase shift for the "linearly" consistent gap, depending on the beam emittance for C6 = 1%, is shown in Fig. 7. However, the best coherence is given by the so-called
"nonlinear" coherence, which is achieved by introducing nonlinear corrective elements into the gap structure (Fig. 3, b). The calculated optimal parameters of such elements are the following: G =-464,298 er/sm, lN = 13,2 sm, = 10 sm. The phase shifts for the "nonlinearly" consistent gap,
depending on the emittance of the beam for C6 = 1%, is shown in Fig. 8. It can be seen from Figs. 7 and 8 that the
"nonlinear" coherence reduces phase incursions caused by nonlinearities in the lenses in the interval by approximately an order of magnitude. The obtained numerical calculations showed that in coherent gaps, nonlinear aberrations in the lenses caused a noticeable mismatch. To eliminate this effect, the most optimal way is using appropriate nonlinear elements.
Fig. 4. Dependence of the phase shifts A \ (1) and A \ (2) on the magnitude of the dodecapole nonlinearity C6
Fig. 5. Dependence of the phase shifts A \\ (1) and A \ (2) on the beam emittance for C6 = 1%
Fig. 6. Particle trajectories in the planes x0s and z0s (solid line - linear calculation, dashed - "linear" coherence).
Fig. 7. Dependence of the phase shifts A y/x (1) beam emittance for "linear" coherenc.
Fig. 8. Dependence of the phase shifts A \ (1) and A \ (2) on the A \ (2) on the beam emittance for "nonlinear"
coherence.
Problem 3. The actual characteristics of accelerator magnets always do not coincide to some extent with the calculated ones. When setting up and upgrading the accelerator, it is important to know how much the beam characteristics differ from the designed ones and how this affects the accelerator intensity. When exploring these issues, it is useful to find distorted closed orbits (periodic trajectories) in the accelerator. In this work, we use the numerical method (CANM) for finding spatial trajectories with almost any degree of accuracy, which was applied to the JINR accelerator, having a rather complicated magnetic field distribution in space. Figure 9 shows the diagram of such an accelerator. Further, eight coordinate systems were used in the calculations (see Fig. 9): four cylindrical coordinate systems and four Cartesian ones.
Fig. 9. Accelerator diagram.
The cylindrical coordinate system (p, R, z ) was associated with each accelerator quadrant, starting at the point OiD (i = 1,2,3,4), where i is the accelerator quadrant number. The Cartesian coordinate system (x, y, zD ) was associated with each gap between the quadrants of the accelerator, starting at the point OiD (i = 1,2,3,4), where i
is the accelerator gap number. The axes z of the cylindrical coordinate systems and the Cartesian axis z are directed upward from the plane of the drawing in all cases. For the sake of brevity, we will sometimes talk about some "conditional" coordinate system (7, Q) assuming that if we are talking about a point located in one of the quadrants of the accelerator, then its position is described by the cylindrical coordinate system (p, R, z ) of this quadrant, and if the point is in the gap of the accelerator, then its position is described by the Cartesian coordinate system (x, y, zD ) of this gap. Thus, further ^ denotes p or x ; 7 denotes R or y ; Q denotes z or zD . Mathematical Model of the Problem.
Differential equations of motion of a charged particle in the accelerator magnetic field are given (single-particle model):
7 = $1(£7,Q,7^Q
I , , (14)
Q = $2(Z,7,C,7S,CS)
where $ and $2 are the given functions:
2R,2 Jr+R;+z( , , ,
$1 = R + ' p [-RzpBp -Rpz;BR + (R2 + R/)Bz], (15)
R B0R0R
if the trajectory point is in one of the quadrants of the accelerator (and then in the left part of Eq. (14), 77 2
should be replaced by R^ ); thus, in this case, we write Eq. (14) in the cylindrical coordinate system of one of the
quadrants of the accelerator.
M + y'2 + z'2 , , , ,,
$1 = pp x [ zB + yxzxBy - (1 + y/)Bz ], (16)
B0 R0
if the trajectory point is in one of the gaps of the accelerator (and then in the left part of Eq. (14), 77 2 should be
replaced by y 2 ); thus, in this case, we write Eq. (14) in the Cartesian coordinate system of one of the gaps of the accelerator.
$2 = R21R(» z([RR;Bt-R+zj)BR+(B], (.7) R B0R0R
if the trajectory point is in the quadrant of the accelerator (and then in the left part of Eq. (14), Q 2 should be replaced by z 2).
' 2
' 2 >
J1 + y '2 + z'
$2 = go X [-yxBx + (1 + z'xZ)By - yxzxB2 ], (18)
B0 R0
if the trajectory point is in the gap of the accelerator (and then in the left part of Eq. (14), Q 2 should be replaced by z 2). In Eqs. (15) - (18), B0 and R are the constants, B0, BR, Bz are the given functions of p, R, z ; Bx, By, B are the given functions of x, y, z . The periodic solution 7 = 7(£), Q=Q(£) of Eqs. (14) in the interval a b is found, which satisfies the boundary conditions:
J 7(a) = 7(b) J Q(a) = Q(b)
7a) = 7,(b) jO) = Qe(b) (19)
The numbers a and b are such that the solution found on [ a, b ] corresponds to one full turn of the particle in the accelerator. A continuous analogue of the Newton method for this problem leads to the solution of the system of equations:
- (^ )] - (O^ U - (O J- (O^ v = [] -0! (4, ], C, ], £ )],
- (O2)!' U4 - (02)r u - (O2)' v4 - (O2)' v = C -02(4,r,^,rt,4t)],
](4,0) = ro(4),
C(4,0) = Co(4),
f u(a, t) = u(b, t), K(a, t) = u^b, t),
|C(a) = Co(b),
I Co (a) = Co(b),
(21)
]=u£ =v
The unknown functions ] = T(4, t), C = C(4, t), u = u(4, t), v = v(4, t) are subject to the following initial and boundary conditions:
\ To(a) = To(bX
[ r(a) = r<o(b),
[ v(a, t) = v(b, t), [v_4 (a, t) = v4(b, t),
For the functions ] = ](4, t), C = C(4, t), which are the solution to Eq. (20) according to the general theory of a continuous analogue of the Newton method, generally speaking, the following conditions must be satisfied:
: lim r(4,t)=r(4), lim C(4,t)=C(4),
t ^ +» t ^ +» where ](4) and C(4) are the solution to the original problem (14) - (19). The functions T0 (4), Co (4) , which are initial approximations of the unknown functions ](4) and C(4) , in the
conditions (21) are set before solving the problem. The more accurately the functions ] and Co describe the
desired periodic trajectory, the faster the process converges. Calculation of the Magnetic Field Components. Accounting for Distortions of the Magnetic Field.
The components of the accelerator leading magnetic field in space can be calculated based on the measured distribution of the field exponent n(R) in the magnetic median plane z = o using Maxwell equations and the
representations Bz (0, R, z) and B (0, R, z) in the vicinity z = o of Taylor series:
(22)
Bz (0, R, z) = Bz (0, R, o) - — [
2 R
1 SBZ (0, r, o), a2Bz (0, R,o) 1 a2bz (0, R,o)
dR
■ +
dR2
+ ■
R2
aO2
(23)
Br (0, R z) =
dBz (0, R, o) dR
z ~ [
1 d2Bz (0, R, o) 1 dBz (0, R, o) 2 d2Bz (0, R,o)
6 R
aR2
R2
aR
R3
a02
] +... (24)
Table 1. z^ (0, R) is the function defined by Table.
2
3
0 z (0; R = 272o) z (0; R = 288o)
0o z1(0o; R1) z1(0o; r2)
01 z1(01; R) z1(0; R2)
z:(0; R) z1(0,; R)
0 z,(0„; R) z:(0„; R)
Table 2. The radial orbits under these conditions.
№ of the trajectory Ro Bo
1 2770 (sm) -164,9753
2 2800 -163,2077
3 2830 -161,4776
4 2840 -160,9090
5 2850 -160,3444
Fig. 10. Graphs of the periodic trajectories.
For several reasons, the geometric median plane of the accelerator did not coincide with the plane z = 0. It was
deformed to some surface z = z (0, R) . In the considered case, this surface was set (according to the measurement
results) for each quadrant of the accelerator by two-dimensional Table 1. The following assumptions were made: 1) the distorted median geometric and magnetic surfaces of the accelerator coincide; the equation of this surface is
z = z (0, R), where z7 (0, R) is the function defined by Table 1; 2) the element of the surface z = z7 (0, R) in some vicinity of the point N(0, R, z1 (0, R)) lies in the plane P , passing through the point N and forming the angle a with the plane z = 0 in the radial direction and the angle fi in the azimuthal direction, where a and fi are determined from the conditions: tga = (z7 )R, tgf = (z7 )0 . Further, in order to find the components of the magnetic field at the quadrant point of the accelerator M(0, R, z), we acted as follows. First, the components of the magnetic field B- , B- were found using the formulas (23), (24) in the auxiliary coordinate system (0, R, z) ,
the beginning of which O lies in the plane P , and z is the distance from the point M to the plane P. Then, the values B- , B- were decomposed in the directions (0, R, z) of the cylindrical quadrant coordinate system, and
the values obtained in this case were taken as B^, BR, B field components included in Eqs. (14). Between the
quadrants, it was considered that Bx = By = Bz = 0.
Results of Calculations. The above-stated method was used to calculate periodic trajectories for the following values of the constants R and B0, included in Eqs. (14) - (18) (see Table 2). As a result of the calculation, for
each periodic trajectory, tables of the function values J] = J](4) and C = C(4) are obtained. Table 3 shows the
values of the function R = R (0), (i = 1,2,3,4,5) , i.e. the trajectory number (at the ends of each quadrant).
Figure 10 shows the function graphs C = C (4), (i = 1,2,3,4,5 , i.e. the trajectory numbers, in the figure they are indicated at the right end of the trajectories). Each of the trajectories was considered so that 48 steps were taken in the quadrant O. The step along t was taken equal to unity (r = 1). At one step along t, the program consumes about one minute of the computer time. To calculate the first and second trajectories, two steps for each one were required along t. For the rest of the trajectories there were four steps. The orbits of the synchrophasotron calculated by the aforesaid method show that the field heterogeneity with respect to R and the deformation of the magnet quadrants lead to a significant distortion of the vertical orbits (especially on R more than 2800 cm) and a decrease
in the working aperture of the vacuum chamber. Moreover, as can be seen from Table 2, the radial orbits under these conditions are slightly distorted. The orbit distortion can probably be compensated with the help of additional
corrections of the accelerator field. This correction can be carried out by a system of turns with a current located in the rectilinear gaps of the accelerator, which create an additional radial component of the field.
Conclusions
When creating new charged particle accelerators and reconstructing the existing ones, the ability to solve the above problems, with the help of mathematical modeling, to ensure the required properties of the beam extracted from the accelerator is important, which significantly reduces both the time and cost of their creation. This problem falls into a number of complex mathematical problems, some of which are described above. The results of solving each of such problems determine some parameters of the accelerator and, therefore, the cost of its equipment. A general approach to solving the problems of transporting charged particles, matching straight-line gaps of accelerators, and finding periodic trajectories is proposed; it is based on a continuous analogue of the Newton method (CANM). The developed method is effective for solving the described model boundary-value problems associated with the design of accelerator systems. A further development of the proposed numerical algorithm as applied to the calculation of the described transportation systems may become the optimal choice of system parameters taking into account not only the beam direction, but also the size of the beam "spot".
The work is dedicated to the memory of Professor E.P. Zhidkov and Professor I.A. Shelaev.
Table 3. The values of the function R — R (p), (i — 1,2,3,4, 5) .
№ of the trajectory 1 2 3 4 5
Quadrant <P R = Ri(p) R = R» R = R(v) R = R» R = R(<P)
0 2769,951 2799,946 2829,861 2839,733 2849,527
I 1 — ж 2 ......,971 ......,965 ......,885 ......,757 ......,544
1 — ж ......,979 ......,972 ......,895 ......,769 ......,553
II 2 ж ......,011 2800,005 ......,954 ......,852 ......,652
ж ......,013 ......,007 ......,959 ......,860 ......,663
III 3 — ж 2 2769,995 2799,987 ......,924 ......,807 ......,597
IV 3 — ж 2 2ж ......,987 ......,952 ......,979 ......,947 ......,909 ......,862 ......,787 ......,732 ......,572 ......,522
References / Список литературы
1. Zhidkov E.P., Puzynin I. V. et al. Continuous Analogue of the Newton Method in Nonlinear Problems of Physics // Phys. Part. Nucl. 1973. V. 4, No. 1 (in Russian).
2. Polyakova R. V., Yudin I.P. Continuous Analogue of the Newton Method in Beam Dynamics Problems. RUDN Univ. Bull. 2011. No. 2. P. 71-84.
