ISSN 1061-9348, Journal of Analytical Chemistry, 2016, Vol. 71, No. 14, pp. 1360–1367. © Pleiades Publishing, Ltd., 2016. Published in Russian in Mass-spektrometriya, 2016, Vol. 13, No. 2, pp. 103–110.

ARTICLES

Synthesis of Ion-Optical Devices for the Transportation of Charged Particle Beams in Mass Spectrometers1 V. V. Pavlov* and N. K. Krasnova St. Petersburg Polytechnic University, St. Petersburg, 195251 Russia *e-mail: [email protected] Received January 10, 2016; in final form, March 22, 2016

Abstract—An approach to the creation of corpuscular-optical devices for transforming charged particle beams is discussed. These devices are useful for the optimization and creation of the most convenient configuration of the ion path. The approach relies on an inverse dynamics problem formulated on the basis of the Hamilton–Jacobi equation. Motion in the symmetry plane of a three-dimensional field is considered. The problem is solved by analytical methods. An algorithm for the construction electric fields ensuring particle motion along the desired trajectories is described. It is based on the method of physical analogies. The procedure is illustrated by an example. The calculated two-dimensional potential is extended to the 3D-space by a power series on the transverse coordinate. A device embodiment example is demonstrated on the basis of the calculated field structure. Keywords: inverse dynamics problem, charged particle optics, method of physical analogies, ideal focusing, mass spectrometer DOI: 10.1134/S1061934816140112

INTRODUCTION The effective operation of a state-of-the-art mass spectrometer, including a number of corpuscular and optical elements is determined by the working characteristics of each unit separately, as well as by their optimum adjustment in an integrated ion-optical path. A variety of ion sources and so wide range of mass analyzers require complex schemes for combining these units. For example, the application of beams with wide angular divergence for quadrupole mass spectrometers leads to a decrease in the transmission of stable ions, which adversely affects the accuracy and sensitivity of measurements [1]. The application of diaphragms for the beam collimation leads to an ineffective consumption of assay, and, therefore, the sensitivity of the device decreases. Dynamic mass spectrometers, for example Orbitrap, demand the precise input of the focused ion flux because of their design features [2]. Traditionally electrostatic and magnetic lenses are used in mass spectrometry to form phase volume of an ion beam, which is most suitable for the used type of the mass spectrometer. In some cases when traps or trans-axial systems are used, the lenses provide the monochromation of an ion beam. Almost in all cases, the use of similar devices demands a rectilinear path, which at a great number of functional units generally 1 The article was translated by the authors.

leads to a significant increase in the sizes of the massspectrometric complex. The use of rejecting systems and mirrors is not always efficient, because the configuration of the ion beam is disturbed. In this article, we propose systems that simultaneously accomplish functions of turning devices and lenses, focusing and transporting the beam. In addition, these devices ensure the monochromation of the ion beam because of specific features of the used analytical approaches. A successful search for such systems is done based on the ideology of the inverse dynamics problem, the analytical theory of which was developed by Professor of the St. Petersburg Polytechnic University Yu.K. Golikov and found application in the design of devices with perfect characteristics and unique properties [3]. The essence of the inverse problem is as follows. A system is given together with the constraints, and some unknown forces are applied to it, causing movement with the specified properties. It is required to determine these forces or, otherwise, field structure, which will provide the movement of particles along quite specific trajectories necessary for our purposes. An important point is directly the mathematical representation of the expected properties of the movement, i.e., their mathematical form, and also the formulation of additional conditions ensuring finding of only one solution or, at least, considerably narrowing classes of suitable potential structures. From this point of view,

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inverse problems require a special mathematical theory of the solution, which is based on unique methods and approaches. In our work, the inverse problem formulated on the basis of the Hamilton–Jacobi equation is solved [4]. An approach developed for the synthesis of systems with the specified properties is formulated by the method of physical analogies.

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Now we will solve this problem in terms of analytical mechanics. We will use the Hamilton–Jacobi equation, which will be solved by the method of variable separation [5]. Hamilton function H is a function of coordinates and impulses and represents the sum of potential and kinetic energies of the system 2 2 px py (5) + + qx 2 + qy 2, 2m 2m where px and py are impulses in the corresponding coordinates. Knowing that partial derivatives of action functions with respect to coordinates are equal to the corresponding impulses

H =

MATHEMATICAL THEORY Before the description of the implemented approach, we will specify of the terms and equations used. Historically, mechanics was developed in two main trends. One branch, which is conventionally named classical mechanics, proceeded directly from the Newton’s laws of motion. The task consists in the determination of all forces acting on each particle, after which the motion is defined unambiguously if the acting forces are known in each time moment. In the second branch, which is usually named analytical mechanics [4], the treatment of the balance and motion proceeds from two main variables, kinetic energy and force function, and the last variable is often replaced by potential energy. These two fundamental scalars comprise the full dynamics of the most complex material systems. As an example, we will consider the two-dimensional motion of a charged particle of charge q and mass m in a quadratic field.

ϕ ( x, y ) = x 2 + y 2.

(1)

∂S = p , ∂S = p , (6) x y ∂x ∂y we can write down the Hamilton–Jacobi equation in the following form

( )

2q ⎧  x+ x =0 ⎪ m ⎨ 2q ⎪  y+ y =0 ⎩ m is well known and takes the form

(3)

⎧ ⎛ 2q ⎞ ⎛ 2q ⎞ ⎪⎪ x(t ) = C1cos ⎜⎝ m t ⎟⎠ + C 2 sin ⎜⎝ m t ⎟⎠ (4) , ⎨ ⎪ y(t ) = C 3 cos ⎛⎜ 2q t ⎞⎟ + C 4 sin ⎛⎜ 2q t ⎞⎟ ⎪⎩ ⎝ m ⎠ ⎝ m ⎠ where constants C1, C2, C3, C4 are defined by the initial conditions. JOURNAL OF ANALYTICAL CHEMISTRY

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where h is arbitrary constant, and the function S0 is explicitly independent of time. Substituting this expression for S into Eq. (7), we obtain the following expression for the function S0

∂S ∂S H ⎛⎜ x, y, 0 , 0 ⎞⎟ ∂x ∂y ⎠ ⎝ 2 2 ∂S 0 ∂S = 1 + 1 ⎛⎜ 0 ⎞⎟ + qx 2 + qy 2 = h. 2m ∂x 2m ⎝ ∂y ⎠ Note that the Hamiltonian can be written as

( )

(2)

where the point designated differentiation on time. The general solution of the differential equations

2

S = −ht + S 0,

The equations of motion in classical mechanics can be written as

⎧mx = −q ∂ϕ ⎪ ∂x , ⎨ ∂ϕ ⎪my = −q ∂y ⎩

2

1 ∂ S + 1 ⎛ ∂ S ⎞ + qx 2 + qy 2 + ∂ S = 0, (7) ⎜ ⎟ 2m ∂ x 2m ⎝ ∂ y ⎠ ∂t where scalar quantity S is action. The system under consideration is conservative, because the Hamiltonian is explicitly independent of time. Therefore, the solution of Eq. (7) can be found as

(

(9)

)

∂S 0 ∂S (10) + f 2 ⎛⎜ y, 0 ⎞⎟ . ∂x ⎝ ∂y ⎠ In this case, the variables are separated: each function (f1 and f2) includes only one couple of conjugated variables. Let us write H = f1 x,

( )

2

2

1 ∂ S 0 + qx 2 = α , 1 ⎛ ∂ S 0 ⎞ + qy 2 = α , (11) x y ⎜ ⎟ 2m ∂ x 2m ⎝ ∂ y ⎠ where αx, αy are arbitrary constants. Then constant h can be expressed through constants αx and αy according to Eq. (10) as follows: h = α x + α y.

(12)

Having resolved equalities (11) with respect to S0, we will find the complete integral of the Hamilton– Jacobi equation as a combination of three summands:

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S = +

∫

∫

2m ( α x − qx 2 )dx

2m ( α y − qy )dy − ( α x + α y ) t. 2

(13)

The final equations of motion we will obtain by the differentiation of function S with respect to arbitrary constants αx, αy and setting the derivatives to the new constants βx and βy, respectively, for example:

mdx (14) . 2m ( α x − qx 2 ) After regrouping the constants in Eq. (14) and integrating it, we obtain the following solution: β x = −t +

∫

⎛ qx ⎞ 2q (15) arcsin ⎜ ⎟ = ( (β x − C ) + t ) m , α ⎝ x⎠ where C is an integration constant. Expressing the dependence x(t) in an explicit form, we obtain:

⎛ 2q ⎞ αx (β x − C ) + 2qt ⎟ . (16) sin ⎜ ⎝ m q m ⎠ Expanding the sine of the sum and redesignating constants, we obtain the expression x=

⎛ 2q ⎞ ⎛ 2q ⎞ (17) x(t ) = C1cos ⎜ t ⎟ + C 2 sin ⎜ t ⎟. ⎝ m ⎠ ⎝ m ⎠ Applying the same procedure to the variable y, we obtain: ⎛ 2q ⎞ ⎛ 2q ⎞ (18) y(t ) = C 3 cos ⎜ t ⎟ + C 4 sin ⎜ t ⎟. ⎝ m ⎠ ⎝ m ⎠ Thus, Newton’s and analytical mechanics are two different mathematical descriptions of the same set of natural phenomena. Except for the equivalence of these two theories, there arises a question about mathematical expediency. In the reviewed example, the classical approach appears much less laborious because of the lack of restrictions of particle motion by the specified constraints. However, the analytical approach is more economic and simple for systems with constraints. The specified constraints are considered here naturally, as the movements of the system are considered only along such trajectories that do not contradict the constraints. In the vector approach, it is necessary to consider forces maintaining constraints; therefore it is necessary to suggest various hypotheses concerning these forces. In this study, we work within analytical mechanics, as this ideology is more efficient for the analytical approach creating new corpuscular– optical systems. DIMENSIONLESS MATHEMATICAL MODEL OF MOTION The real motion of a particle in the Euclidean space with Cartesian coordinates X, Y, Z and current time t depends on its charge, mass, acting electromagnetic

fields, and initial conditions. All of this is expressed by dimensional variables and, in total, comprises a physical model of an ion-optical system. Meanwhile, it is easy to simplify calculations and analyze the results by refusing from standard physical measurement units and introducing new special units, unifying all ratios and making them especially dimensionless. Thus, introduction of a system of dimensionless variables [6] replaces a real physical task with a mathematical model of motion, which is simpler and ensures studies with the maximum possible generality. A dimensionless line x(τ), y(τ), z(τ) is assigned to each physical trajectory X(t), Y(t), Z(t). To find a dimensional trajectory by a dimensionless line, one should multiply only functions x(τ), y(τ), z(τ) by the specified linear scale l. Let us show a transition to a dimensionless model of motion for our problem. The Hamilton–Jacobi equation for the motion of a particle with charge q and mass m in an electrostatic field with a potential Φ(x, y) in the plane (x, y) can be written as

( )

2 2 1 ⎡ ∂ S + ⎛ ∂ S ⎞ ⎤ + qΦ ( x, y ) = E , (19) ⎜ ⎟ ⎥ 2m ⎢⎣ ∂ x ⎝ ∂y ⎠ ⎦ where E is the total energy of the particle. If ratio (6) is considered, Eq. (19) can be rewritten as

( ) ( )

2 2 m ⎡ dX + dY ⎤ + qΦ ( x, y ) = E . (20) 2 ⎢⎣ dt dt ⎥⎦ We define dimensional constant l, which is a certain characteristic dimension of the device and use it as a proportion factor between dimensional Cartesian coordinates X, Y and dimensionless quantities x, y

X = lx;Y = ly.

(21)

In addition to l, a characteristic value of potential Φ0 is defined as, for example, the maximum on the module potential on a border of the system Φ = Φ0ϕ(x, y).

(22)

Dimensionless time τ and dimensionless total energy h we will define by equations as t = T τ; E = H 0h,

(23)

where T is a certain characteristic time (it was not determined by us yet) and H0 is the total energy of a particle. Then Eq. (20) can be rewritten as, ml 2 x + y + qΦ ϕ ( x, y ) = H h, (24) 0 0 2 2 T replacing differentiation with respect to dimensionless time by a point. As the parameter T is an arbitrary constant, we will define the following dependence between the appeared multipliers: 2

2

ml 2 = H = qΦ . 0 0 2 T

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Thus ensuring the isolation of the general multiplier in Eq. (24), which can be neglected. Now we will formulate the problem: let us search for ion-optical systems ensuring the transportation of a particle beam from one point of space to another. We will exclude magnetic fields from our study, paying attention to the search for only among the electrostatic fields. For this purpose, we will use the effective algorithms based on analytical methods. Let a field have plane symmetry z = 0. We require that, in this plane, trajectories starting from point A would come to point B under the action of the field irrespectively of the departure angle. The positions of these points can be specified absolutely randomly as it is required for a certain ion path. Our example illustrates the newly proposed analytical approach, the source and assembly point of particles are symmetrized along one axis relative to the origin of coordinates. METHOD OF PHYSICAL ANALOGIES Let us consider the analytical approach named by the author, Professor of the St. Petersburg Polytechnic University Yu.K. Golikov, “Method of Physical Analogies”. The Hamilton–Jacobi equation can be written in dimensionless variables as

S x2 + S y2 (26) + ϕ ( x, y) = h. 2 Let us suppose that a shortened action S(x, y) (as it defines force function only on one plane rather than in the whole space) is known, then Eq. (26) can be presented as S x2 + S y2 (27) . 2 This equation can be interpreted as a problem for finding of a field structure, the force action of which on a particle is defined by us. It is known that possible mechanical trajectories are lines orthogonal to a family of surfaces S = const for a conservative system [7]. Having two families of orthogonal lines, we can match one family of lines with trajectories, and the other, with action. Therefore, it is possible to determine the potential that is responsible for the motion of a particle along the chosen trajectory, solving the inverse dynamics problem formulated on the basis of the Hamilton–Jacobi equation (27). Also it should be noted that any function (28) S * = F (S ), considered as action has the same constant level lines S* = const, as well as S = const, and, correspondingly, both of them define the same family of trajectories orthogonal to them. Indeed, let us suppose, that P(x, y) = const is a family of trajectories orthogonal to ϕ ( x, y ) = h −

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the family S(x, y) = const, then the identity of orthogonality is true.

∇ S ∇ P = S x Px + S y Py ≡ 0.

(29)

In the case of F(S) = const,

∇ F (S )∇ P = dF S x Px + dF S y Py dS dS (30) dF = (S x Px + S y Py ) = 0 dS because of (29). Substitution of S* into the Hamilton– Jacobi equation provides a new class of potentials containing arbitrary analytical function F(S) in the structure:

ϕ ( x, y ) = h − K (S ) ( S x2 + S y2 ) , 2 (31) dF (S ) K (S ) = 1 . 2 dS It is a difficult task in practice to find two families of orthogonal lines, one of them is for the desirable trajectories. However, to bypass this difficulty, it was possible to borrow the necessary families of orthogonal lines from problems of other branches of physics. Thus, for example, these can be some problems of hydrodynamics or electrostatics. Such use of physical analogies in electron optics allows the researcher to synthesize electric fields with beforehand specified properties. In general, the mentioned algorithm, solving the inverse problem, consists of the following steps: (1) Choice of a physical condition, for which there are two families of mutually transverse lines described by simple analytical expressions. One of families is chosen as shortened action S. (2) Selection of transformation function K(S). (3) Constructing of a solution—potential structure of the field in the symmetry plane ϕ(x, y) by Eq. (31). (4) Finding of a three-dimensional potential ϕ(x, y, z) by the analytical continuation of the found solution from the plane. (5) Analysis of beams behavior in the chosen structure. We will pose the problem of finding a potential structure with the symmetry plane providing an ideal focusing of the charged particle beam in this plane. There are two points M and N in the plane (x, y); it is required to find a potential ϕ(x, y) that provides the focusing of the beam starting from point M and focusing in point N. It would be possible to analytically specify a family of lines connecting the points M and N, then to determine lines orthogonal to them integrating an ordinary differential equation, to take them for contour curves of action S, and then to find ϕ by Eq. (31), having chosen a convenient function K(S). It is desirable to replace these formal computations with more intuitive

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physical reasonings, making use of electrostatic fields as analogy. It is known that force lines connect two electric charges equal in magnitude but opposite in sign and the force lines and equipotentials are described by simple expressions. Therefore, it is favorable to interpret force lines as particle trajectories in a certain field, which is a subject of definition, and to take the corresponding potential of a combination of point charges for the function S. All versions of models of fluxes representing ideal focusing, including the one under consideration, possess common basic drawback. It consists in the model field strength infinitely growing in the points of start and focusing; therefore, these points will be singular for the required potential only if the function K(S) does not annihilate these singularities. The choice of K(S) does not affect the geometry of trajectories in the symmetry plane (XOY plane), but changes the time parameters of motion and also the trajectories and distribution of the potential outside the plane z = 0. The mentioned arbitrariness in the choice of K(S) can be used, first, for the submission of potential ϕ(x, y) to additional physical conditions, secondly, for the selection of fields providing the cross focusing of the beam (along the z axis). Let us chose a pair of linear charges with different charge signs located in points (–1,0), (1,0) parallel to the z axis as a physical system [8]. At such configuration, the potential in the XOY plane can be written as

which is one of combinations described in [9]. Calculations of the two-dimensional potential providing ideal focusing in the symmetry plane at a total energy of h = 1 by Eq. (31) has led to the result

( x + 1) + y (32) ϕ* = 1 ln . 2 ( 2 x − 1) + y 2 According to the method of physical analogies, which was discussed above, particle trajectories are matched to field lines, and equipotentials of such structure are associated with lines of constant action. At the same time, the action function has the form

Δ xy = ∂ 2 + ∂ 2 . ∂x ∂y The study of the accuracy grade of the threedimensional potentials computed by Eq. (36) and their applicability to the calculation charged particle trajectories near the symmetry plane at different numbers of expansion terms has been carried out earlier, and the results were published in the article [9]. The error due to the substitution of several numbers of expansion terms (36) instead of their infinite numbers in a performance of the three-dimensional potential has the accuracy required and implemented in the production of complex shaped electrodes of the system. Here we carry out calculations taking into account only members up to of the sixth degree on z inclusively. The trajectories of charged particles in the field with the potential (36) were calculated by the Mathematica software [10], and marks of trajectories in the vicinity of the finish point (x, y, z) = (1, 0, 0) are shown in Fig. 1 at different initial angles α ∈ [10°; 80°] in the plane XOY and angle γ = –1° to the same plane. It can be seen similar structures do not provide the cross focusing of the ion beam. The analysis of the first derivative on the z coordinate from the three-dimensional potential (36) led to the same result. To be fair, it should be noted that the potentials synthesized within the symmetry plane had the problem of the sta-

2

2

( x + 1) + y 2 (33) S = 1 ln . 2 ( x − 1) 2 + y 2 Thus, substituting Eq. (33) into Eq. (31) gives a chance to determine the potential ϕ, which provides ideal focusing point–point in the XOY plane. To provide a physical consistency of the model, function K(S) has to compensate the singularities appearing in the points of the start and finish of particles, that is, to be zero in those points, in which action tends to infinity. As action represents a logarithmic function, the selection of K(S) should be carried out among the functions containing exponential dependence in their structure. As an example, we choose function of regularization K(S) as 2

K (S ) =

2 8 = , 2 cosh ( S ) ( e −S + e S ) 2

(34)

ϕ ( x, y ) = 1 −

4

(1 + x

)

2 2

.

(35)

+y The potential found by solving the inverse problem formulated on the basis of the Hamilton–Jacobi equation by the choice of action in the form (33), provides the ideal focusing of a charged particle beam in the XOY plane. Particles focus in the point (1, 0), starting from the point (–1, 0). It should be noted that the field structure (35) belongs to a class of transaxial ideal focusing systems, described in the monograph [6]. Their outstanding features consist in a possibility of the analytical extension of the field into the 3D-space in the coordinates r, z, but requires the application of powerful methods of the theory of functions of a complex variable. In the present research, ion beams in the small vicinity of the symmetry plane are analyzed, which allowed us to apply approximate calculations of the distribution of the 3D field structure, using the wellknown decomposition in a power series of a small value z: ϕ ( x, y, z ) =

∞

∑ k =0

2

( − 1) k k Δ ϕ ( x, y ) z 2k ; ( 2k ) ! xy 2

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Let the base L of our device be 10 cm. We will determine a linear scale factor l, knowing that distance on the x coordinate between the points of start and finish was 2 units in the dimensionless model:

–0.02

–0.04

(37) l = L = 0.05 m . x Let us choose Φ0 = 1000 V and according to Eqs. (21) and (22), we will express the potential in volts

z –0.06

4000 (38) V, (1 + 400 X 2 + 400Y 2 ) where X and Y are measured in meters. Let us put q = 1.6 × 10–19 C and m = 1.66 × 10–27 kg = 1 Da, then we will be able to determine the time scaling factor T and the value H0 by Eq. (25) Φ = 1000 −

–0.08 0.996

0.998 x

Fig. 1. Finish points of particle trajectories in the detector plane.

bility of trajectories in their shift from the symmetry plane. It was necessary to investigate this problem individually for each calculated field structure. One of possible solutions was to change the calculated potential with a potential similar in the structure but ensuring the stability of the beam along the cross coordinate with a loss of perfect quality of focusing in the symmetry plane. This approach represents a new problem, and its solution can be considered as a particular research. Figure 2 presents the basic characteristics in the detector plane for beams with the average angle αav = 5°, 15°, 25°, …, 85°, deviation of angle Δα = 10°, starting at the angle γ = –1°. The following notations are used: Δx is beam width on the x coordinate; Δz is beam width on the z coordinate; x is the minimum x coordinate; z is the maximum z coordinate; Sspot is the area of beam spot, Sspot = ΔxΔz; ΔSslit is the area of the output slit providing the transmission of ion beams with ~ ~ γ ∈ [–1°; 1°]. Sslit = (x0 – x)|2z |, x0 is the x coordinate of the ideal focusing point. In our case, x0 = 1. Analyzing the graphs, one can draw a conclusion that the choice of an exit angle of particles αav = 85° provides the optimum operating conditions in this system. To realize the created potential structure, it was necessary to shape the field-forming electrodes to equipotential surfaces. The two surfaces between which the analyzed beam passed were chosen proceeding from the computed modes. Further the entry and exit of the system were designed according to calculations, and, by applying corresponding voltages to the electrodes and guiding the beam to the field area, the necessary charged particle trajectories were provided. At the end, in order to design a device, the calculated dimensionless model had to be transformed to the real one with a dimensional system of units, for example SI, scaling all parameters by the corresponding factors. JOURNAL OF ANALYTICAL CHEMISTRY

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T =

ml 2 = 1.61 × 10 −7 s, qΦ 0

H 0 = q Φ 0 = 1.6 × 10 − 16 J.

(39) (40)

Calculations show that, for a beam of particles with αav = 85°, Δα = 10° and initial energy E = 1.6 × 10–16 J = 1 keV, the equipotential surfaces corresponding to voltage 50 and –500 V can be chosen as constructive elements of the device. It is known that, in systems with an electrostatic field, trajectories of charged particles depend only on their total energy. This fact means that the device transports ions along the same trajectories regardless of their mass, and only transit time will change. At the same time, the device provides the monochromatization of the beam, which makes desirable its use with mass analyzers without focusing on energy. The estimates carried out for the specified operation conditions show energy dispersion in relation to the base of the device of about 0.8 and resolution of about 0.1%. The structure found has relatively complex equipotential surfaces. However, because of the geometrical features of these surfaces, the device can be constructed using milling machines with program control. Nowadays, because of technical evolution, this problem can be solved successfully. Figure 3 shows a possible design of an ion-optical system on the basis of the calculated potential structure. The device is only an example, showing a possibility of the application of the described approach. Before recommending this system for realization and introduction into the practice of designing of mass spectrometers, it is necessary to make additional calculations of optical properties, an assessment of the influence of small variations of initial conditions: angular divergence of the beam from the central plane, energy dispersion, and the value of charge, replacement of a point source and the detector by finite-size slits, and inaccuracies of fabrication. The system and the above questions set fundamentals and open a big sphere for work in this direction.

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×10–3

×10–3

(a)

(b)

999

–0.6 998

Δx –0.4

x˜ 997

–0.2 996

20

40

Δz

60

20

80

αav°

×10–3 14

40

×10–3 0 (c)

10

αav°

60

80

(d)

2 z˜ 4

6 6 2 20 ×10

40

–6

αav°

60

20

80

40

×10–3

8

αav°

60

80

(f)

(e) 0.6

6 Sspot 4

Sslit 0.4

2

0.2

20

40

αav°

60

80

20

40

αav°

60

80

Fig. 2. The basic characteristics of an ion beam in the detector plane.

Another approach to the practical embodiment of the proposed system is also possible. The surfaces of the field forming electrodes are replaced by surfaces simpler in shape, such as spheres, cubes, tori, etc. In this case, the conditions of ideal focusing in the symmetry plane are disturbed, but there is a possibility to simplify the design from the viewpoint of its fabrication. Such an approach in itself is a complex problem, and its solution demands additional time and the development of heuristic methods and approaches mostly analytical. Shaping the field forming electrodes of the system by constructively simpler surfaces

with the retention of properties characteristic for an ideal system, is a separate problem, and it is beyond the scopes of the present article. CONCLUSIONS An approach to the creation of ion-optical systems by the required properties of particle motion in them is formulated. Considering electrostatic fields with a plane of symmetry and formulating the inverse problem on the basis of the Hamilton–Jacobi equation

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x, m 0.0

ion path for the mass spectrometer offers a number of advantages. The replacement of several electrostatic or magnetic lenses by a device similar to the one proposed in the article allowed us to simplify and reduce the price of the construction of a mass spectrometer, and also, to reduce its overall dimensions, taking into account the transformation and transportation of the ion beam. Noteworthy is the easier adjustment of our device and the ion path in total. There is one more advantage, the reduction of requirements to power supply. The aforesaid offers prospects for the development of similar devices and their application in the mass analyzing systems.

0.05

–0.05

0.04 y, m 0.02

0.0 –0.04

–0.02

0.0 z, m

1367

0.02

0.04

ACKNOWLEDGMENTS This study was supported by the Committee on Science and the Higher School of the Government of St. Petersburg.

Fig. 3. Device embodiment example. Equipotential surfaces are {–500, 50 V}.

provides a basis for this approach. At the same time, solution in an analytical form can be obtained. The algorithm associated with the theorem of analytical mechanics about the orthogonality of the trajectory of motion of a material point and constant action lines was used to find electric fields ensuring the motion of charged particle along the required trajectories. Two families of orthogonal curves were matched to trajectories and action, respectively. The application of a method of physical analogies at this stage allowed us to avoid computational difficulties associated with the construction of curves orthogonal to the desirable trajectories. The necessary families are borrowed from the problems of electrostatics. The approach presented was illustrated by specific calculations. The application of the method of physical analogies to the system of two linear charges allowed us to construct a potential field providing ideal focusing in the symmetry plane. The calculated twodimensional potential, which is a distribution of the three-dimensional harmonious field in the symmetry plane, was extended in space by a power series on the transverse coordinate. Embodiments of the ion-optical device were proposed on the basis of the calculated field structures. The use of the system described instead of the traditional conjugating elements in the construction of an

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2016