- Email: [email protected]
On the quasi-polynomial 3D potentials of electric and magnetic fields
Accepted Manuscript
On the quasi-polynomial 3D potentials of electric and magnetic fields Nadezhda K. Krasnova , Alexander S. Berdnikov , Konstantin V. Solovyev , Igor A. Averin PII: DOI: Reference:
S2405-7223(17)30012-9 10.1016/j.spjpm.2017.02.004 SPJPM 120
To appear in:
St. Petersburg Polytechnical University Journal: Physics and Mathematics
Received date: Accepted date:
23 February 2017 27 February 2017
Please cite this article as: Nadezhda K. Krasnova , Alexander S. Berdnikov , Konstantin V. Solovyev , Igor A. Averin , On the quasi-polynomial 3D potentials of electric and magnetic fields, St. Petersburg Polytechnical University Journal: Physics and Mathematics (2017), doi: 10.1016/j.spjpm.2017.02.004
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Nadezhda K. Krasnova1, Alexander S. Berdnikov2, Konstantin V. Solovyev1, Igor A. Averin1 1
Peter The Great St. Petersburg Polytechnic University, Politekhnicheskaya, 29, 195251, St. Petersburg, Russian Federation. 2 Institute for Analytical Instrumentation RAS, Rizskiy pr. 26, 190103, St. Petersburg, Russian Federation,
CR IP T
On the quasi-polynomial 3D potentials of electric and magnetic fields
AN US
Spectrographic electron and ion optical structures markedly raise the possibilities of modern energy and mass analysis. Electric and magnetic fields which potentials are expressed by functions homogeneous in Euler’s sense are the effective instrumentation that is used for creating new spectrographic analytical devices with the determined working characteristics. This paper puts forward and discusses some methods for building 3D harmonic and homogeneous in Euler’s sense structures representable as the polynomials of finite degree with respect to one of variables. These strictly mathematical approaches provide a possibility of expanding significantly a class of quasi-polynomial potentials and of enriching modern analytical instrumentation by new spectrographic electrical and magnetic configurations. Functions homogeneous in Euler’s sense; Similarity principle; Thomson formula; Stability of motion
Introduction
M
This article continues the investigations into spectrographic charged-particle
ED
optical structures which can serve as a basis for constructing effective devices with electric as well as magnetic fields. Potential structures of these fields ought to be
PT
homogeneous in Euler’s sense [1, 2]. According to our ideology, this property is very important, and a major condition for designing electric and magnetic
CE
spectrographs with high-performance characteristics such as resolution, sensitivity, transmission, energy dispersion and others, while the overall dimensions of the
AC
field-forming electrodes are rather small, and the device as a whole is compact. The property of homogeneity is expressed in the following analytical form.
A continuous function of three variables, which is the potential U(x, y, z), is homogeneous in Euler’s sense of k-th order, if the identity U x, y, z kU x, y, z ,
(1)
is fulfilled, where k is any real number. If the function U(x, y, z) is differentiable, then it can be described by a differential equation of first order 1
ACCEPTED MANUSCRIPT x
U U U y z kU x, y, z x y z .
(2)
The existence of similar charged particle trajectories in these fields due to the property of homogeneity allows to to create analyzers with multi-channel registration of charged particles with respect to energy (electric spectrographs) or mass number (magnetic spectrographs). In the last 10 years, a number of analytical
CR IP T
methods have been offered [3 – 8] within the framework of the uniform ideology for creating spectrographic devices, and different classes of fields have been built based on them. The properties of some configurations of these families are investigated; the specific schemes of spectrographs are proposed for different applications of energy and mass analysis [9–14].
AN US
Our previous article [15] offered for the first time an analytical theory of 3D harmonic potentials homogeneous in Euler’s sense, which are presented as polynomials of limited power with respect to one of the Cartesian variables. These potentials are called quasi-polynomials. In this study, there are methods and
M
techniques that give new structures for a family of quasi-polynomial potentials
ED
with ideal corpuscular optics characteristics.
(Special case)
CE
PT
Natural degree of homogeneity
In Ref. [15] mentioned above, we have proposed a new algorithm for
AC
synthesis of 3D potentials in order to create effective electric and magnetic spectrographs on their basis. This method can be used for generating potential structures with functions of arbitrary degree of homogeneity k, homogeneous in Euler’s sense. However, it turns out to be impossible to obtain structures with homogeneous functions with the integer degree of homogeneity k by this algorithm. The statement on a particular chosen solution needs to be changed at one of the steps of this synthesis algorithm. That’s why below we present an algorithm to find the 3D potential structures needed. 2
ACCEPTED MANUSCRIPT
First stage. We build the 3D potential as a polynomial of finite power of 2n or (2n – 1) with respect to the coordinate y with coefficients which are homogeneous functions of the corresponding order with respect to two other coordinates x and z. Here it is possible to build the potential in odd form as well as in even one. They are divided into two unmixed families: 1 2 1 2n y U 2,k 2 x, z y U 2 n ,k 2 n x, z 2n ! 2! ,
U x, y, z yU 1,m1 x, z
CR IP T
U x, y, z U 0,k x, z
1 3 1 y U 3,m3 x, z y 2 n1U 2 n1,m2 n1 x, z 2n 1! 3! .
(3) (4)
Second stage. Substitution of decomposition (3) or (4) into 3D Laplace equation U xx U yy U zz 0
(5)
AN US
and combination of the terms with the same powers of variables y gives, instead of a single equation, a chain of equations which are presented as Poisson equations for the corresponding coefficient functions. An exception is the equality corresponding to the coefficient of the highest power of the polynomial. As a result
M
we get a number of equations for the decomposition (3) 2U 0, k
ED
x 2
2U 2, k 2
PT
x 2
2U 0, k z 2
U 2 , k 2
2U 2, k 2 z 2
,
U 4 , k 4
,
AC
CE
..........................,
2U 2 n2, k 2 n2 x 2
2U 2 n2, k 2 n2
2U 2 n , k 2 n x 2
z 2
2U 2 n , k 2 n z 2
U 2 n , k 2 n
(6) ,
0.
,
and also for another decomposition yielding an odd polynomial (4) 2U1, m1 x 2 2U 3, m3 x 2
2U1, m1 z 2 2U 3, m3 z 2
U 3, m3 U 5, m 5
,
(7)
, 3
ACCEPTED MANUSCRIPT
.........................., 2U 2 n 3, m2 n 1 x 2
2U 2 n 3, m2 n 1 z 2
2U 2 n 1, m2 n 1 x 2
U 2 n 1, m2 n 1
2U 2 n 1, m2 n 1 z 2
,
0.
Third stage. We solve the Laplace equation that is the last in the chain given, and choose a harmonic function, homogeneous in Euler’s sense, with the degree of
CR IP T
homogeneity p = k – 2n (in a case of the construction of an even polynomial) or s = m – 2n – 1 (in a case of an odd polynomial) as the generating function U 2 n , p x, z с0 r p
or
AN US
U 2n1, s x, z с0 r s ,
where r x 2 z 2 , arctgz / x ; c0(γ) is an unknown function that is even or odd with respect to the argument (instead of c0 r p cos p or c0 r s sin s , i.e., instead of
M
the functions used in Ref. [15]).
Fourth and subsequent stages. Then we search the rest of the multipliers
ED
U 2 n j , p j x, z ( U 2 n j 1, s j x, z )
of the lower powers of y, solving the corresponding Poisson equations with the
PT
right parts that were found at the previous stage, and satisfying the condition of being symmetrical or anti-symmetrical with respect to the coordinate z.
CE
We should note that we shall search for uncertain coefficient functions as a
AC
particular solution presented in the following form c j r p j .
This is a common form for a function of two variables which is homogeneous in Euler’s sense with the corresponding degree of homogeneity. The procedure described is repeated until a chain of recurrent equations is stopped at the first term of decomposition (3) or (4).
4
ACCEPTED MANUSCRIPT
Besides, there are variants if the coefficient multiplying the higher power of y is a harmonic and homogeneous function with zero degree of homogeneity, and therefore it will be defined as U x, z U 0 const
or
instead of formulas
x
2
U x, z U 0
x
2
k z z 2 cos k arctg U 0 r k cos k x
z2
sin k arctg xz U r sin k k
k
0
AN US
U x, z U 0
CR IP T
z U x, z U 0 arctg U 0 x
in the case of polynomials of arbitrary degree of homogeneity k or m [15]. The final results are given below; there are configurations that differ greatly
M
compared with the common formulas presented in Ref. [15].
ED
Potentials which are symmetrical with respect to z and with even power of y k 0 : U 0 x, y, z 1
y
PT
k 1 : U 2 x, y, z
;
2
r 2 cos r sin r ;
CE
k 2 : U 2 x, y, z y 2
1 2 r 2 ;
AC
y4 3 3 k 2 : U x, y, z cos 2 2 3 y 2 r 2 r 2 sin 2 4 2 r ; 4
(8)
y4 9 3 k 3 : U 4 x, y, z cos 6 y 2 r r 3 r 3 6 y 2 r sin r 3 cos 3 8 2 r ;
y 6 15 y 4 45 y 2 r 5 3 15 3 k 3 : U 6 x, y, z cos 3 3 r r sin 3 4r 8 8 8 r ;
5
ACCEPTED MANUSCRIPT 3 k 4 : U 4 x, y, z y 4 3 y 2 r 2 r 4 8 ; y 6 15 4 45 2 2 5 4 15 4 45 2 2 k 4 : U 6 x, y , z cos 2 2 y y r r r y r sin 2 2 4 8 4 2 r 45 4 r cos 4 32
CR IP T
y 8 14 y 6 35 4 35 2 2 35 4 35 4 k 4 : U 8 x, y, z cos 4 4 y y r r r sin 4 3 r2 4 4 64 16 r .
Potentials which are symmetrical ith respect to z and with odd power of y k 1 : U1 x, y, z y ; y 3 3 yr 2 cos 3 yr sin r ;
k 3 : U 3 x, y, z y 3
3 2 yr 2 ;
AN US
k 2 : U 3 x, y, z
y5 15 15 k 3 : U 5 x, y, z 2 5 y 3 yr 2 cos 2 yr 2 sin 2 4 2 r ;
M
(9)
ED
y5 45 3 15 k 4 : U 5 x, y, z 10 y 3 r yr cos yr 3 10 y 3 r sin 5 yr 3 cos 3 8 2 r
PT
;
y 7 21 y 5 105 3 35 3 105 3 k 4 : U x, y, z 3 y r yr cos 3 yr sin 3 r 4 r 8 8 8 ;
CE
7
AC
Potentials which are anti-symmetrical with respect to z and with even power
of y
k 0 : U 0 x, y, z
k 1 : U 2 x, y, z
; (10)
y 2 sin r cos r ;
6
ACCEPTED MANUSCRIPT 1 k 2 : U 2 x, y, z y 2 r 2 2 ; y4 3r 2 k 2 : U 4 x, y, z sin 2 2 3 y 2 cos 2 r 2 ; y4 3 3 2 3 k 3 : U 4 x, y, z sin r 6 y r r 3 cos 2 r 8 ;
x, y, z sin 3 y3
15 y 4 45 y 2 r 15 3 r cos 3 r 4 r 8 8 ; 6
3 k 4 : U 4 x, y, z y 4 3 y 2 r 2 r 4 8 ;
CR IP T
k 3:U
6
AN US
y 6 15 4 45 2 2 5 4 45 2 2 15 4 k 4 : U 6 x, y, z sin 2 2 y y r r y r r cos 2 2 4 8 2 4 r 45 4 r sin 4 32
;
y 8 14 y 6 35 4 35 2 2 35 4 k 4 : U x, y, z sin 4 4 y y r r cos 4 2 r 3 r 4 4 16 .
ED
M
8
Potentials which are symmetrical with respect to z and with odd power of y
PT
k 1 : U1 x, y, z y ;
CE
k 2 : U 3 x, y, z
y3 sin 3 yr cos r ;
AC
3 k 3 : U 3 x, y, z y 3 yr 2 2 ; y5 15 k 3 : U 5 x, y, z 2 5 y 3 sin 2 yr 2 cos 2 2 r ;
(11)
y 5 15 15 k 4 : U 5 x, y, z yr 3 sin 10 y 3 r yr 3 cos r 8 2 ; y 7 21 y 5 105 3 105 3 k 4 : U 7 x, y, z 3 y r sin 3 yr cos 3 4 r 8 8 r .
7
ACCEPTED MANUSCRIPT
This number of potentials which are homogeneous in Euler’s sense with natural orders of homogeneity k are an addition to a family of quasi-polynomial 3D potentials homogeneous with any other orders of homogeneity [15]. But this class can be expanded if we explore the transformations of such type when functions remain harmonic and homogeneous in Euler’s sense, but the degree of
CR IP T
homogeneity can differ from the initial one.
Rotation, scaling and parallel shifting of coordinate system
It is well known that the Laplace equation keeps its form under transformations such as scaling, rotation and shifting of the Cartesian coordinate system. Besides, if scaling and rotation are used, a function remains homogeneous
AN US
in Euler’s sense. That’s why if we substitute the variables such as the 3D rotation of the common type [16], it is possible to get new analytical expressions for homogeneous harmonic potentials based on existing analytical formulae. However, as it is easy to see, the rotation in the plane xz yields a linear combination of
M
symmetric and anti-symmetric quasi-polynomials that have already been obtained.
ED
When the degree of homogeneity is an integer and a quasi-polynomial is an ordinary harmonic polynomial, it proves impossible to build fundamentally new
PT
analytical expressions for 3D potentials in this manner, as all expressions obtained will be linear combinations with constant coefficients of existing polynomials,
CE
except the case of the permutation of coordinate axes. However, for the case where the degree of homogeneity is not an integer but real or where the quasi-polynomial
AC
is not a harmonic polynomial, it can be possible to get new analytical expressions for 3D potentials fot electric and magnetic fields using this method. It also seems promising to use linear combinations with constant coefficients
of harmonic functions homogeneous in Euler’s sense, generated by means of independent rotations of the coordinate system. Notably, for pure quasipolynomials this similar linear combination is again a known quasi-polynomial.
Thomson (Kelvin) formula 8
ACCEPTED MANUSCRIPT
Treatise [16] established that if U x, y, z is an arbitrary harmonic function, then the function 1 x y z U * x, y, z U 2 , 2 , 2 ,
(12)
where x 2 y 2 z 2 , will be harmonic as well. Substitution of variables x , x y2 z2 2
y
y z , z 2 2 2 x y z x y2 z2 2
CR IP T
x
(13)
is an inversion in a sphere, and the function remains harmonic and homogeneous. The given transformation can be used for synthesizing electron and ion optical systems, Refs. [3, 4] are examples of this application.
AN US
The fact that the 3D function U * x, y, z remains harmonic can be verified by substituting formula (12) to 3D Laplace equation (5). Besides, it should be noted that formula (12) is not the only one of this type. For example, if we employ inversion and then implement transformation of 3D rotation with respect to the
M
origin of coordinates, allowing for parameterization using three independent [16], we can obtain formula (12) where the
ED
parameters in the general form
numerators of the arguments of the U function will be linear combinations of
PT
variables x, y, z with constant coefficients. The unique property of the transformation expressed by Thomson formula
CE
(12) is that if U x, y, z is a function homogeneous in Euler’s sense with the degree of homogeneity k, then the function U * x, y, z will be homogeneous in Euler’s
AC
sense as well, but with the degree of homogeneity k 1 [3]. If we apply the transformation (12) again, then a reverse transformation will make the transition from the function U * x, y, z to the function U x, y, z , and the degree of homogeneity restores its previous value k k 1 k 1 1 k .
Let us illustrate this statement.
9
ACCEPTED MANUSCRIPT
Applying the transformation (12) we get a new harmonic function homogeneous in Euler’s sense, U * x, y, z . As the function U * x, y, z is homogeneous in Euler’s sense with the degree of homogeneity k, and identity (1) is satisfied, this function can be presented in the following form
CR IP T
x y z 1 U 2 , 2 , 2 2 k U x, y, z .
Then the new function U * x, y, z will be written as U * x, y , z
1
2 k 1
U x, y , z ,
which guarantees that this function is homogeneous in Euler’s sense with the order
AN US
(–k – 1). Here we take into account the fact that the function U(x, y, z) itself is homogeneous in Euler’s sense with order k (the details can be found in the first volume of treatise [17] in Appendix of chapter 1 dedicated to spherical harmonic functions).
M
It is easy to verify that transformation (12) retains the properties of an even or odd potential function with respect to the z as well as y coordinate. So, if we
ED
choose quasi-polynomial potentials (8)–(11) or (9)–(12) from Ref. [15] as a base with the degree of homogeneity k*= –k – 1, then, by applying Thomson formula
PT
(12) to them, we can construct new potentials presented in analytical form with the
CE
symmetry we need, expressed by functions homogeneous in Euler’s sense with the necessary degree of homogeneity k.
AC
Figs. 1–3 show equal potential surfaces of the fields with potentials
presented by formulae (8)–(11) in analytical form and also the fields with transformed potentials, Thomson formula (12) is applied to these fields. If we take a potential structure homogeneous in Euler’s sense with the order k = 3 U5+ (x, y, z) (9), then by applying transformation (12), we get a configuration in the following form
10
ACCEPTED MANUSCRIPT V5 x, y, z
x
1
2
y2 z2
7
y 5 15 2 15 2 3 2 5 y yr cos 2 yr sin 2 , 4 2 r
(14)
with the degree of homogeneity k = -4. By means of Thomson transformation (12) applied to the potential U2– (x, y, z) that is anti-symmetric with respect to the z coordinate and expressed by a function homogeneous in Euler’s sense with the order k = 2 (10), we get a structure
V x, y , z 2
y2 r2 / 2
x
2
y2 z2
5
CR IP T
V2–(x, y, z), which is anti-symmetric and homogeneous with the order k = -3 .
(15)
The potential expressed by a function homogeneous in Euler’s sense (k =-2)
x
y 2
y z 2
2 3
,
AN US
V1 x, y, z
(16)
is the result obtained through transformation (12) from the potential structure U1– (x, y, z), which is a quasi-polynomial with odd powers of y and anti-symmetric
M
with respect to the z coordinate (11).
ED
Fig. 1. 3D views of equal potential surfaces of the fields with potentials U5+ (x, y, z)
PT
(a) and V5+(x, y, z) (b) (see formulae (9) and (14), respectively). Fig. 2. 3D views of equal potential surfaces of the fields with potentials U2– (x, y, z)
CE
(а) and V2–(x, y, z) (b) (see formulae (10) and (15), respectively ).
AC
Fig. 3. 3D views of equal potential surfaces of the fields with potentials U1– (x, y, z) (а) and V1–(x, y, z) (b) (see formulae (11) and (16), respectively ).
Control of motion stability in electrostatic quasi-polynomial fields
Quasi-polynomials of the given degree presented in lists (8)–(11) and (9)– (12) from Ref. [15] are not unique. It is easy to understand that if we append an 11
ACCEPTED MANUSCRIPT
arbitrary linear combination with constant coefficients consisting of polynomials of lower degree to a quasi-polynomial of n-th degree and with the degree of homogeneity k, then as a result we get a quasi-polynomial of the same type again. Some variations in such combinations allow to optimize the properties of a charged particle system. One of the important criteria of optimization is the stability of charged particle trajectories with respect to small deviations from the median plane
CR IP T
[18]. If we use the sufficient criterion of stability formulated in Ref. [18], then we can determine what coefficients of quasi-polynomials of lower degree (or additional homogeneous functions of other type) should be chosen in order to correct the local instability of trajectories. Let us consider some examples. This criterion is expressed in the following form. If the charged particles
AN US
move in a field with a symmetry plane z = 0, then their motion will be stable near this plane when the following condition is satisfied U xx U yy
z 0
0.
(17)
Example 1. Let us build a field U (x, y, z) as a combination of two other
M
potentials of the same order which are homogeneous in Euler’s sense, one of them
ED
is U2+(x, y, z) and the other U4+(x, y, z) (see formula (8)). Thus,
PT
U x, y, z U 2 x, y, z hU 4 x, y, z y 2
y4 1 2 3 3 r h cos 2 2 3 y 2 r 2 r 2 sin 2 , 2 4 2 r
(18)
CE
where h is a mixing coefficient. In order to satisfy to the criterion of stability, the parameter h should be
AC
negative, h < 0, and this holds true for the entire space. Fig. 4 shows equal potential surfaces of this field. The surfaces have a rather complex cone-like form but the axis y is the place of clamping of equal potential surfaces, and as a result forces acting on charged particles near the plane z = 0 appear, making particles return to the initial plane.
12
ACCEPTED MANUSCRIPT
Fig. 4. 3D view of equal potential surfaces of the field with potential U (x, y, z) expressed by formula (18). Example 2. Let us take a combination of potentials U2+(x, y, z) and U4+(x, y, z) (see formulae (8)) with different orders of homogeneity (19)
CR IP T
1 3 U x, y, z U 2 x, y, z hU 4 x, y, z y 2 r 2 h y 4 3 y 2 r 2 r 4 . 2 8
Some equal potentials are presented in Fig. 5. For h < 0 the criterion of stability is fulfilled, but not in the whole space, limited by either y > x/2 or y < –x/2.
Fig. 5. 3D view of equal potential surfaces of the field with potential U (x, y, z)
AN US
expressed by formula (19).
Example 3. Let us build a field U (x, y, z) as a combination of two other fields with homogeneous potentials of different orders of homogeneity U3+(x, y, z) (k = 3) and
M
U3+(x, y, z) (k = 2) according to formula (9)
ED
U x, y, z U 3 x, y, z hU 3 x, y, z y 3
y 3 3 yr 2 3 2 yr h cos 3 yr sin . (20) 2 r
Fig. 6 shows equal potentials of field (20). Charged particle trajectories in the field
CE
PT
are stable, but in a limited space.
AC
Fig. 6. 3D view of equal potential surfaces of the field with potential U (x, y, z) expressed by formula (20).
Conclusion
We considered quasi-polynomial 3D potential structures for electric and
magnetic fields which can be expressed in analytical form. Potentials are built on the base of functions that are homogeneous in Euler’s sense and established as polynomials of finite degree. The algorithm for synthesizing such potentials has been expanded for the case of functions homogeneous in Euler’s sense with integer 13
ACCEPTED MANUSCRIPT
orders of homogeneity. Here the procedure of recurrent calculation of polynomial coefficients has an additional feature absent for homogeneous quasi-polynomials of the general form, this subject was analyzed in our previous study [15]. Additionally, the class of quasi-polynomial 3D potentials can be expanded by new structures constructed by transformations of the coordinate system, such as scaling, rotating and parallel shifting. The function remains harmonic and
CR IP T
homogeneous in Euler’s sense through the application of all these transformations. All methods and transformations generate homogeneous potentials with the degree of homogeneity k > 0. Applying the transformation based on the Thomson formula we obtain a new potential configuration of the given class that are homogeneous in
AN US
Euler’s sense with the degree of homogeneity k < 0.
REFERENCES
[1] G.M. Fichtenholz, Course of differential and integral calculus: Vol. 1,
M
Moscow, Fizmatlit, 2001.
[2] V.I. Smirnov, Course of higher mathematics, Vol. 1, Moscow, Nauka,
ED
1974.
[3] N.K. Krasnova, Theory and synthesis of dispersion and focusing electron
PT
optics sphere, doctoral thesis, St. Petersburg, 2013. [4] Yu.K. Golikov, N.K. Krasnova, Application of electric fields uniform
170.
CE
in the Euler sense in electron spectrography, Technical Physics. 57 (2) (2011) 164–
AC
[5] N.K. Krasnova, Two-dimensional power-type electronic spectrographs
with a symmetry plane, Technical Physics. 57 (6) (2011) 843–849. [6] N.K. Krasnova, O.A. Abramenok, A family of field structures with a
plane of symmetry for electron spectrography, St. Petersburg State Polytechnical University Journal. Physics and Mathematics. (2) (2011) 85–92. [7]
N.K. Krasnova, Ideal focusing in the theory of electrostatic
spectrographs, Technical Physics. 58 (8) (2012) 1143–1147. 14
ACCEPTED MANUSCRIPT
[8] Yu.K. Golikov, N.K. Krasnova, Theory of synthesis of electrostatic energy analyzers, St. Petersburg, Izdatelstvo Politekhnicheskogo Universiteta, 2010. [9] A.S. Berdnikov, I.A. Averin, Yu.K. Golikov, Static mass spectrographs of new type used electric and magnetic fields homogeneous in Euler’s sense. I, Mass-spektrometriya. 12 (4) (2015) 272–281.
CR IP T
[10] A.S. Berdnikov, I.A. Averin, Yu.K. Golikov, Static mass spectrographs of new type used electric and magnetic fields homogeneous in Euler’s sense. II, Mass-spektrometriya. 13 (1) (2016) 11–20.
[11] I.A. Averin, Electrostatic and magnetostatic electron spectrographs based on Euler’ homogeneous potentials with non-integer orders, Nauchnoye
AN US
priborostroyeniye. 25 (3) (2015) 35–44.
[12] Yu.K. Golikov, N.K. Krasnova, Analytical structures of electrical generalized homogeneous spectrographic sphere, Nauchnoye priborostroyeniye. 24 (1) (2014) 50–58.
P.G. Gabdullin, Yu.K. Golikov, N.K. Krasnova, S.N. Davydov,
M
[13]
ED
Application of Donkin’s formula in the theory of energy analyzers: Part II, Technical Physics 45 (3) (2000) 330–333. [14]
Yu.K. Golikov, N.K. Krasnova, O.A. Abramyonok, Electric
PT
spectrographs of charged particle flows with potentials of Euler’s type, Prikladnaya
CE
fizika. (5) (2011) 69–73. [15] A.S. Berdnikov, I.A. Averin, N.K. Krasnova, K.V. Solovyev, Quasi-
AC
polynomial 3D electric and magnetic potentials homogeneous in Euler’s sense, St. Petersburg State Polytechnical University Journal. Physics and Mathematics. (1) (2017) (in press). [16] G. Korn, T. Korn, A handbook of mathematics for research workers and engineeers. Moscow, Nauka, 1973. [17] W. Thomson, P.G. Tet, Treatise on natural philosophy, Part II, URL: http://name.umdl.umich.edu/ABR1665.0001.001. 15
ACCEPTED MANUSCRIPT
[18] A.S. Berdnikov, N.K. Krasnova, A sufficient criterion of stability and compactness of the plane ionic beams in the 3D electrical and magnetic fields with plane symmetry, Nauchnoye priborostroyeniye. 25 (2) (2015) 69–90.
THE AUTHORS
CR IP T
Nadezhda K. Krasnova
Peter the Great St. Petersburg Polytechnic University
29 Politekhnicheskaya St., St. Petersburg, 195251, Russian Federation
AN US
[email protected]
Alexander S. Berdnikov
Institute for Analytical Instrumentation RAS
26 Rizskiy Ave., 190103, St. Petersburg, Russian Federation
M
[email protected]
ED
Konstantin V. Solovyev
Peter the Great St. Petersburg Polytechnic University
PT
29 Politekhnicheskaya St., St. Petersburg, 195251, Russian Federation
CE
[email protected]
AC
Igor A. Averin
Peter the Great St. Petersburg Polytechnic University 29 Politekhnicheskaya St., St. Petersburg, 195251, Russian Federation [email protected]
16
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 1а
17
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 1b
18
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 2а.
19
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 2b
20
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 3а
21
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 3b
22
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 4.
23
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 5.
24
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 6.
25
On the quasi-polynomial 3D potentials of electric and magnetic fields Nadezhda K. Krasnova , Alexander S. Berdnikov , Konstantin V. Solovyev , Igor A. Averin PII: DOI: Reference:
S2405-7223(17)30012-9 10.1016/j.spjpm.2017.02.004 SPJPM 120
To appear in:
St. Petersburg Polytechnical University Journal: Physics and Mathematics
Received date: Accepted date:
23 February 2017 27 February 2017
Please cite this article as: Nadezhda K. Krasnova , Alexander S. Berdnikov , Konstantin V. Solovyev , Igor A. Averin , On the quasi-polynomial 3D potentials of electric and magnetic fields, St. Petersburg Polytechnical University Journal: Physics and Mathematics (2017), doi: 10.1016/j.spjpm.2017.02.004
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Nadezhda K. Krasnova1, Alexander S. Berdnikov2, Konstantin V. Solovyev1, Igor A. Averin1 1
Peter The Great St. Petersburg Polytechnic University, Politekhnicheskaya, 29, 195251, St. Petersburg, Russian Federation. 2 Institute for Analytical Instrumentation RAS, Rizskiy pr. 26, 190103, St. Petersburg, Russian Federation,
CR IP T
On the quasi-polynomial 3D potentials of electric and magnetic fields
AN US
Spectrographic electron and ion optical structures markedly raise the possibilities of modern energy and mass analysis. Electric and magnetic fields which potentials are expressed by functions homogeneous in Euler’s sense are the effective instrumentation that is used for creating new spectrographic analytical devices with the determined working characteristics. This paper puts forward and discusses some methods for building 3D harmonic and homogeneous in Euler’s sense structures representable as the polynomials of finite degree with respect to one of variables. These strictly mathematical approaches provide a possibility of expanding significantly a class of quasi-polynomial potentials and of enriching modern analytical instrumentation by new spectrographic electrical and magnetic configurations. Functions homogeneous in Euler’s sense; Similarity principle; Thomson formula; Stability of motion
Introduction
M
This article continues the investigations into spectrographic charged-particle
ED
optical structures which can serve as a basis for constructing effective devices with electric as well as magnetic fields. Potential structures of these fields ought to be
PT
homogeneous in Euler’s sense [1, 2]. According to our ideology, this property is very important, and a major condition for designing electric and magnetic
CE
spectrographs with high-performance characteristics such as resolution, sensitivity, transmission, energy dispersion and others, while the overall dimensions of the
AC
field-forming electrodes are rather small, and the device as a whole is compact. The property of homogeneity is expressed in the following analytical form.
A continuous function of three variables, which is the potential U(x, y, z), is homogeneous in Euler’s sense of k-th order, if the identity U x, y, z kU x, y, z ,
(1)
is fulfilled, where k is any real number. If the function U(x, y, z) is differentiable, then it can be described by a differential equation of first order 1
ACCEPTED MANUSCRIPT x
U U U y z kU x, y, z x y z .
(2)
The existence of similar charged particle trajectories in these fields due to the property of homogeneity allows to to create analyzers with multi-channel registration of charged particles with respect to energy (electric spectrographs) or mass number (magnetic spectrographs). In the last 10 years, a number of analytical
CR IP T
methods have been offered [3 – 8] within the framework of the uniform ideology for creating spectrographic devices, and different classes of fields have been built based on them. The properties of some configurations of these families are investigated; the specific schemes of spectrographs are proposed for different applications of energy and mass analysis [9–14].
AN US
Our previous article [15] offered for the first time an analytical theory of 3D harmonic potentials homogeneous in Euler’s sense, which are presented as polynomials of limited power with respect to one of the Cartesian variables. These potentials are called quasi-polynomials. In this study, there are methods and
M
techniques that give new structures for a family of quasi-polynomial potentials
ED
with ideal corpuscular optics characteristics.
(Special case)
CE
PT
Natural degree of homogeneity
In Ref. [15] mentioned above, we have proposed a new algorithm for
AC
synthesis of 3D potentials in order to create effective electric and magnetic spectrographs on their basis. This method can be used for generating potential structures with functions of arbitrary degree of homogeneity k, homogeneous in Euler’s sense. However, it turns out to be impossible to obtain structures with homogeneous functions with the integer degree of homogeneity k by this algorithm. The statement on a particular chosen solution needs to be changed at one of the steps of this synthesis algorithm. That’s why below we present an algorithm to find the 3D potential structures needed. 2
ACCEPTED MANUSCRIPT
First stage. We build the 3D potential as a polynomial of finite power of 2n or (2n – 1) with respect to the coordinate y with coefficients which are homogeneous functions of the corresponding order with respect to two other coordinates x and z. Here it is possible to build the potential in odd form as well as in even one. They are divided into two unmixed families: 1 2 1 2n y U 2,k 2 x, z y U 2 n ,k 2 n x, z 2n ! 2! ,
U x, y, z yU 1,m1 x, z
CR IP T
U x, y, z U 0,k x, z
1 3 1 y U 3,m3 x, z y 2 n1U 2 n1,m2 n1 x, z 2n 1! 3! .
(3) (4)
Second stage. Substitution of decomposition (3) or (4) into 3D Laplace equation U xx U yy U zz 0
(5)
AN US
and combination of the terms with the same powers of variables y gives, instead of a single equation, a chain of equations which are presented as Poisson equations for the corresponding coefficient functions. An exception is the equality corresponding to the coefficient of the highest power of the polynomial. As a result
M
we get a number of equations for the decomposition (3) 2U 0, k
ED
x 2
2U 2, k 2
PT
x 2
2U 0, k z 2
U 2 , k 2
2U 2, k 2 z 2
,
U 4 , k 4
,
AC
CE
..........................,
2U 2 n2, k 2 n2 x 2
2U 2 n2, k 2 n2
2U 2 n , k 2 n x 2
z 2
2U 2 n , k 2 n z 2
U 2 n , k 2 n
(6) ,
0.
,
and also for another decomposition yielding an odd polynomial (4) 2U1, m1 x 2 2U 3, m3 x 2
2U1, m1 z 2 2U 3, m3 z 2
U 3, m3 U 5, m 5
,
(7)
, 3
ACCEPTED MANUSCRIPT
.........................., 2U 2 n 3, m2 n 1 x 2
2U 2 n 3, m2 n 1 z 2
2U 2 n 1, m2 n 1 x 2
U 2 n 1, m2 n 1
2U 2 n 1, m2 n 1 z 2
,
0.
Third stage. We solve the Laplace equation that is the last in the chain given, and choose a harmonic function, homogeneous in Euler’s sense, with the degree of
CR IP T
homogeneity p = k – 2n (in a case of the construction of an even polynomial) or s = m – 2n – 1 (in a case of an odd polynomial) as the generating function U 2 n , p x, z с0 r p
or
AN US
U 2n1, s x, z с0 r s ,
where r x 2 z 2 , arctgz / x ; c0(γ) is an unknown function that is even or odd with respect to the argument (instead of c0 r p cos p or c0 r s sin s , i.e., instead of
M
the functions used in Ref. [15]).
Fourth and subsequent stages. Then we search the rest of the multipliers
ED
U 2 n j , p j x, z ( U 2 n j 1, s j x, z )
of the lower powers of y, solving the corresponding Poisson equations with the
PT
right parts that were found at the previous stage, and satisfying the condition of being symmetrical or anti-symmetrical with respect to the coordinate z.
CE
We should note that we shall search for uncertain coefficient functions as a
AC
particular solution presented in the following form c j r p j .
This is a common form for a function of two variables which is homogeneous in Euler’s sense with the corresponding degree of homogeneity. The procedure described is repeated until a chain of recurrent equations is stopped at the first term of decomposition (3) or (4).
4
ACCEPTED MANUSCRIPT
Besides, there are variants if the coefficient multiplying the higher power of y is a harmonic and homogeneous function with zero degree of homogeneity, and therefore it will be defined as U x, z U 0 const
or
instead of formulas
x
2
U x, z U 0
x
2
k z z 2 cos k arctg U 0 r k cos k x
z2
sin k arctg xz U r sin k k
k
0
AN US
U x, z U 0
CR IP T
z U x, z U 0 arctg U 0 x
in the case of polynomials of arbitrary degree of homogeneity k or m [15]. The final results are given below; there are configurations that differ greatly
M
compared with the common formulas presented in Ref. [15].
ED
Potentials which are symmetrical with respect to z and with even power of y k 0 : U 0 x, y, z 1
y
PT
k 1 : U 2 x, y, z
;
2
r 2 cos r sin r ;
CE
k 2 : U 2 x, y, z y 2
1 2 r 2 ;
AC
y4 3 3 k 2 : U x, y, z cos 2 2 3 y 2 r 2 r 2 sin 2 4 2 r ; 4
(8)
y4 9 3 k 3 : U 4 x, y, z cos 6 y 2 r r 3 r 3 6 y 2 r sin r 3 cos 3 8 2 r ;
y 6 15 y 4 45 y 2 r 5 3 15 3 k 3 : U 6 x, y, z cos 3 3 r r sin 3 4r 8 8 8 r ;
5
ACCEPTED MANUSCRIPT 3 k 4 : U 4 x, y, z y 4 3 y 2 r 2 r 4 8 ; y 6 15 4 45 2 2 5 4 15 4 45 2 2 k 4 : U 6 x, y , z cos 2 2 y y r r r y r sin 2 2 4 8 4 2 r 45 4 r cos 4 32
CR IP T
y 8 14 y 6 35 4 35 2 2 35 4 35 4 k 4 : U 8 x, y, z cos 4 4 y y r r r sin 4 3 r2 4 4 64 16 r .
Potentials which are symmetrical ith respect to z and with odd power of y k 1 : U1 x, y, z y ; y 3 3 yr 2 cos 3 yr sin r ;
k 3 : U 3 x, y, z y 3
3 2 yr 2 ;
AN US
k 2 : U 3 x, y, z
y5 15 15 k 3 : U 5 x, y, z 2 5 y 3 yr 2 cos 2 yr 2 sin 2 4 2 r ;
M
(9)
ED
y5 45 3 15 k 4 : U 5 x, y, z 10 y 3 r yr cos yr 3 10 y 3 r sin 5 yr 3 cos 3 8 2 r
PT
;
y 7 21 y 5 105 3 35 3 105 3 k 4 : U x, y, z 3 y r yr cos 3 yr sin 3 r 4 r 8 8 8 ;
CE
7
AC
Potentials which are anti-symmetrical with respect to z and with even power
of y
k 0 : U 0 x, y, z
k 1 : U 2 x, y, z
; (10)
y 2 sin r cos r ;
6
ACCEPTED MANUSCRIPT 1 k 2 : U 2 x, y, z y 2 r 2 2 ; y4 3r 2 k 2 : U 4 x, y, z sin 2 2 3 y 2 cos 2 r 2 ; y4 3 3 2 3 k 3 : U 4 x, y, z sin r 6 y r r 3 cos 2 r 8 ;
x, y, z sin 3 y3
15 y 4 45 y 2 r 15 3 r cos 3 r 4 r 8 8 ; 6
3 k 4 : U 4 x, y, z y 4 3 y 2 r 2 r 4 8 ;
CR IP T
k 3:U
6
AN US
y 6 15 4 45 2 2 5 4 45 2 2 15 4 k 4 : U 6 x, y, z sin 2 2 y y r r y r r cos 2 2 4 8 2 4 r 45 4 r sin 4 32
;
y 8 14 y 6 35 4 35 2 2 35 4 k 4 : U x, y, z sin 4 4 y y r r cos 4 2 r 3 r 4 4 16 .
ED
M
8
Potentials which are symmetrical with respect to z and with odd power of y
PT
k 1 : U1 x, y, z y ;
CE
k 2 : U 3 x, y, z
y3 sin 3 yr cos r ;
AC
3 k 3 : U 3 x, y, z y 3 yr 2 2 ; y5 15 k 3 : U 5 x, y, z 2 5 y 3 sin 2 yr 2 cos 2 2 r ;
(11)
y 5 15 15 k 4 : U 5 x, y, z yr 3 sin 10 y 3 r yr 3 cos r 8 2 ; y 7 21 y 5 105 3 105 3 k 4 : U 7 x, y, z 3 y r sin 3 yr cos 3 4 r 8 8 r .
7
ACCEPTED MANUSCRIPT
This number of potentials which are homogeneous in Euler’s sense with natural orders of homogeneity k are an addition to a family of quasi-polynomial 3D potentials homogeneous with any other orders of homogeneity [15]. But this class can be expanded if we explore the transformations of such type when functions remain harmonic and homogeneous in Euler’s sense, but the degree of
CR IP T
homogeneity can differ from the initial one.
Rotation, scaling and parallel shifting of coordinate system
It is well known that the Laplace equation keeps its form under transformations such as scaling, rotation and shifting of the Cartesian coordinate system. Besides, if scaling and rotation are used, a function remains homogeneous
AN US
in Euler’s sense. That’s why if we substitute the variables such as the 3D rotation of the common type [16], it is possible to get new analytical expressions for homogeneous harmonic potentials based on existing analytical formulae. However, as it is easy to see, the rotation in the plane xz yields a linear combination of
M
symmetric and anti-symmetric quasi-polynomials that have already been obtained.
ED
When the degree of homogeneity is an integer and a quasi-polynomial is an ordinary harmonic polynomial, it proves impossible to build fundamentally new
PT
analytical expressions for 3D potentials in this manner, as all expressions obtained will be linear combinations with constant coefficients of existing polynomials,
CE
except the case of the permutation of coordinate axes. However, for the case where the degree of homogeneity is not an integer but real or where the quasi-polynomial
AC
is not a harmonic polynomial, it can be possible to get new analytical expressions for 3D potentials fot electric and magnetic fields using this method. It also seems promising to use linear combinations with constant coefficients
of harmonic functions homogeneous in Euler’s sense, generated by means of independent rotations of the coordinate system. Notably, for pure quasipolynomials this similar linear combination is again a known quasi-polynomial.
Thomson (Kelvin) formula 8
ACCEPTED MANUSCRIPT
Treatise [16] established that if U x, y, z is an arbitrary harmonic function, then the function 1 x y z U * x, y, z U 2 , 2 , 2 ,
(12)
where x 2 y 2 z 2 , will be harmonic as well. Substitution of variables x , x y2 z2 2
y
y z , z 2 2 2 x y z x y2 z2 2
CR IP T
x
(13)
is an inversion in a sphere, and the function remains harmonic and homogeneous. The given transformation can be used for synthesizing electron and ion optical systems, Refs. [3, 4] are examples of this application.
AN US
The fact that the 3D function U * x, y, z remains harmonic can be verified by substituting formula (12) to 3D Laplace equation (5). Besides, it should be noted that formula (12) is not the only one of this type. For example, if we employ inversion and then implement transformation of 3D rotation with respect to the
M
origin of coordinates, allowing for parameterization using three independent [16], we can obtain formula (12) where the
ED
parameters in the general form
numerators of the arguments of the U function will be linear combinations of
PT
variables x, y, z with constant coefficients. The unique property of the transformation expressed by Thomson formula
CE
(12) is that if U x, y, z is a function homogeneous in Euler’s sense with the degree of homogeneity k, then the function U * x, y, z will be homogeneous in Euler’s
AC
sense as well, but with the degree of homogeneity k 1 [3]. If we apply the transformation (12) again, then a reverse transformation will make the transition from the function U * x, y, z to the function U x, y, z , and the degree of homogeneity restores its previous value k k 1 k 1 1 k .
Let us illustrate this statement.
9
ACCEPTED MANUSCRIPT
Applying the transformation (12) we get a new harmonic function homogeneous in Euler’s sense, U * x, y, z . As the function U * x, y, z is homogeneous in Euler’s sense with the degree of homogeneity k, and identity (1) is satisfied, this function can be presented in the following form
CR IP T
x y z 1 U 2 , 2 , 2 2 k U x, y, z .
Then the new function U * x, y, z will be written as U * x, y , z
1
2 k 1
U x, y , z ,
which guarantees that this function is homogeneous in Euler’s sense with the order
AN US
(–k – 1). Here we take into account the fact that the function U(x, y, z) itself is homogeneous in Euler’s sense with order k (the details can be found in the first volume of treatise [17] in Appendix of chapter 1 dedicated to spherical harmonic functions).
M
It is easy to verify that transformation (12) retains the properties of an even or odd potential function with respect to the z as well as y coordinate. So, if we
ED
choose quasi-polynomial potentials (8)–(11) or (9)–(12) from Ref. [15] as a base with the degree of homogeneity k*= –k – 1, then, by applying Thomson formula
PT
(12) to them, we can construct new potentials presented in analytical form with the
CE
symmetry we need, expressed by functions homogeneous in Euler’s sense with the necessary degree of homogeneity k.
AC
Figs. 1–3 show equal potential surfaces of the fields with potentials
presented by formulae (8)–(11) in analytical form and also the fields with transformed potentials, Thomson formula (12) is applied to these fields. If we take a potential structure homogeneous in Euler’s sense with the order k = 3 U5+ (x, y, z) (9), then by applying transformation (12), we get a configuration in the following form
10
ACCEPTED MANUSCRIPT V5 x, y, z
x
1
2
y2 z2
7
y 5 15 2 15 2 3 2 5 y yr cos 2 yr sin 2 , 4 2 r
(14)
with the degree of homogeneity k = -4. By means of Thomson transformation (12) applied to the potential U2– (x, y, z) that is anti-symmetric with respect to the z coordinate and expressed by a function homogeneous in Euler’s sense with the order k = 2 (10), we get a structure
V x, y , z 2
y2 r2 / 2
x
2
y2 z2
5
CR IP T
V2–(x, y, z), which is anti-symmetric and homogeneous with the order k = -3 .
(15)
The potential expressed by a function homogeneous in Euler’s sense (k =-2)
x
y 2
y z 2
2 3
,
AN US
V1 x, y, z
(16)
is the result obtained through transformation (12) from the potential structure U1– (x, y, z), which is a quasi-polynomial with odd powers of y and anti-symmetric
M
with respect to the z coordinate (11).
ED
Fig. 1. 3D views of equal potential surfaces of the fields with potentials U5+ (x, y, z)
PT
(a) and V5+(x, y, z) (b) (see formulae (9) and (14), respectively). Fig. 2. 3D views of equal potential surfaces of the fields with potentials U2– (x, y, z)
CE
(а) and V2–(x, y, z) (b) (see formulae (10) and (15), respectively ).
AC
Fig. 3. 3D views of equal potential surfaces of the fields with potentials U1– (x, y, z) (а) and V1–(x, y, z) (b) (see formulae (11) and (16), respectively ).
Control of motion stability in electrostatic quasi-polynomial fields
Quasi-polynomials of the given degree presented in lists (8)–(11) and (9)– (12) from Ref. [15] are not unique. It is easy to understand that if we append an 11
ACCEPTED MANUSCRIPT
arbitrary linear combination with constant coefficients consisting of polynomials of lower degree to a quasi-polynomial of n-th degree and with the degree of homogeneity k, then as a result we get a quasi-polynomial of the same type again. Some variations in such combinations allow to optimize the properties of a charged particle system. One of the important criteria of optimization is the stability of charged particle trajectories with respect to small deviations from the median plane
CR IP T
[18]. If we use the sufficient criterion of stability formulated in Ref. [18], then we can determine what coefficients of quasi-polynomials of lower degree (or additional homogeneous functions of other type) should be chosen in order to correct the local instability of trajectories. Let us consider some examples. This criterion is expressed in the following form. If the charged particles
AN US
move in a field with a symmetry plane z = 0, then their motion will be stable near this plane when the following condition is satisfied U xx U yy
z 0
0.
(17)
Example 1. Let us build a field U (x, y, z) as a combination of two other
M
potentials of the same order which are homogeneous in Euler’s sense, one of them
ED
is U2+(x, y, z) and the other U4+(x, y, z) (see formula (8)). Thus,
PT
U x, y, z U 2 x, y, z hU 4 x, y, z y 2
y4 1 2 3 3 r h cos 2 2 3 y 2 r 2 r 2 sin 2 , 2 4 2 r
(18)
CE
where h is a mixing coefficient. In order to satisfy to the criterion of stability, the parameter h should be
AC
negative, h < 0, and this holds true for the entire space. Fig. 4 shows equal potential surfaces of this field. The surfaces have a rather complex cone-like form but the axis y is the place of clamping of equal potential surfaces, and as a result forces acting on charged particles near the plane z = 0 appear, making particles return to the initial plane.
12
ACCEPTED MANUSCRIPT
Fig. 4. 3D view of equal potential surfaces of the field with potential U (x, y, z) expressed by formula (18). Example 2. Let us take a combination of potentials U2+(x, y, z) and U4+(x, y, z) (see formulae (8)) with different orders of homogeneity (19)
CR IP T
1 3 U x, y, z U 2 x, y, z hU 4 x, y, z y 2 r 2 h y 4 3 y 2 r 2 r 4 . 2 8
Some equal potentials are presented in Fig. 5. For h < 0 the criterion of stability is fulfilled, but not in the whole space, limited by either y > x/2 or y < –x/2.
Fig. 5. 3D view of equal potential surfaces of the field with potential U (x, y, z)
AN US
expressed by formula (19).
Example 3. Let us build a field U (x, y, z) as a combination of two other fields with homogeneous potentials of different orders of homogeneity U3+(x, y, z) (k = 3) and
M
U3+(x, y, z) (k = 2) according to formula (9)
ED
U x, y, z U 3 x, y, z hU 3 x, y, z y 3
y 3 3 yr 2 3 2 yr h cos 3 yr sin . (20) 2 r
Fig. 6 shows equal potentials of field (20). Charged particle trajectories in the field
CE
PT
are stable, but in a limited space.
AC
Fig. 6. 3D view of equal potential surfaces of the field with potential U (x, y, z) expressed by formula (20).
Conclusion
We considered quasi-polynomial 3D potential structures for electric and
magnetic fields which can be expressed in analytical form. Potentials are built on the base of functions that are homogeneous in Euler’s sense and established as polynomials of finite degree. The algorithm for synthesizing such potentials has been expanded for the case of functions homogeneous in Euler’s sense with integer 13
ACCEPTED MANUSCRIPT
orders of homogeneity. Here the procedure of recurrent calculation of polynomial coefficients has an additional feature absent for homogeneous quasi-polynomials of the general form, this subject was analyzed in our previous study [15]. Additionally, the class of quasi-polynomial 3D potentials can be expanded by new structures constructed by transformations of the coordinate system, such as scaling, rotating and parallel shifting. The function remains harmonic and
CR IP T
homogeneous in Euler’s sense through the application of all these transformations. All methods and transformations generate homogeneous potentials with the degree of homogeneity k > 0. Applying the transformation based on the Thomson formula we obtain a new potential configuration of the given class that are homogeneous in
AN US
Euler’s sense with the degree of homogeneity k < 0.
REFERENCES
[1] G.M. Fichtenholz, Course of differential and integral calculus: Vol. 1,
M
Moscow, Fizmatlit, 2001.
[2] V.I. Smirnov, Course of higher mathematics, Vol. 1, Moscow, Nauka,
ED
1974.
[3] N.K. Krasnova, Theory and synthesis of dispersion and focusing electron
PT
optics sphere, doctoral thesis, St. Petersburg, 2013. [4] Yu.K. Golikov, N.K. Krasnova, Application of electric fields uniform
170.
CE
in the Euler sense in electron spectrography, Technical Physics. 57 (2) (2011) 164–
AC
[5] N.K. Krasnova, Two-dimensional power-type electronic spectrographs
with a symmetry plane, Technical Physics. 57 (6) (2011) 843–849. [6] N.K. Krasnova, O.A. Abramenok, A family of field structures with a
plane of symmetry for electron spectrography, St. Petersburg State Polytechnical University Journal. Physics and Mathematics. (2) (2011) 85–92. [7]
N.K. Krasnova, Ideal focusing in the theory of electrostatic
spectrographs, Technical Physics. 58 (8) (2012) 1143–1147. 14
ACCEPTED MANUSCRIPT
[8] Yu.K. Golikov, N.K. Krasnova, Theory of synthesis of electrostatic energy analyzers, St. Petersburg, Izdatelstvo Politekhnicheskogo Universiteta, 2010. [9] A.S. Berdnikov, I.A. Averin, Yu.K. Golikov, Static mass spectrographs of new type used electric and magnetic fields homogeneous in Euler’s sense. I, Mass-spektrometriya. 12 (4) (2015) 272–281.
CR IP T
[10] A.S. Berdnikov, I.A. Averin, Yu.K. Golikov, Static mass spectrographs of new type used electric and magnetic fields homogeneous in Euler’s sense. II, Mass-spektrometriya. 13 (1) (2016) 11–20.
[11] I.A. Averin, Electrostatic and magnetostatic electron spectrographs based on Euler’ homogeneous potentials with non-integer orders, Nauchnoye
AN US
priborostroyeniye. 25 (3) (2015) 35–44.
[12] Yu.K. Golikov, N.K. Krasnova, Analytical structures of electrical generalized homogeneous spectrographic sphere, Nauchnoye priborostroyeniye. 24 (1) (2014) 50–58.
P.G. Gabdullin, Yu.K. Golikov, N.K. Krasnova, S.N. Davydov,
M
[13]
ED
Application of Donkin’s formula in the theory of energy analyzers: Part II, Technical Physics 45 (3) (2000) 330–333. [14]
Yu.K. Golikov, N.K. Krasnova, O.A. Abramyonok, Electric
PT
spectrographs of charged particle flows with potentials of Euler’s type, Prikladnaya
CE
fizika. (5) (2011) 69–73. [15] A.S. Berdnikov, I.A. Averin, N.K. Krasnova, K.V. Solovyev, Quasi-
AC
polynomial 3D electric and magnetic potentials homogeneous in Euler’s sense, St. Petersburg State Polytechnical University Journal. Physics and Mathematics. (1) (2017) (in press). [16] G. Korn, T. Korn, A handbook of mathematics for research workers and engineeers. Moscow, Nauka, 1973. [17] W. Thomson, P.G. Tet, Treatise on natural philosophy, Part II, URL: http://name.umdl.umich.edu/ABR1665.0001.001. 15
ACCEPTED MANUSCRIPT
[18] A.S. Berdnikov, N.K. Krasnova, A sufficient criterion of stability and compactness of the plane ionic beams in the 3D electrical and magnetic fields with plane symmetry, Nauchnoye priborostroyeniye. 25 (2) (2015) 69–90.
THE AUTHORS
CR IP T
Nadezhda K. Krasnova
Peter the Great St. Petersburg Polytechnic University
29 Politekhnicheskaya St., St. Petersburg, 195251, Russian Federation
AN US
[email protected]
Alexander S. Berdnikov
Institute for Analytical Instrumentation RAS
26 Rizskiy Ave., 190103, St. Petersburg, Russian Federation
M
[email protected]
ED
Konstantin V. Solovyev
Peter the Great St. Petersburg Polytechnic University
PT
29 Politekhnicheskaya St., St. Petersburg, 195251, Russian Federation
CE
[email protected]
AC
Igor A. Averin
Peter the Great St. Petersburg Polytechnic University 29 Politekhnicheskaya St., St. Petersburg, 195251, Russian Federation [email protected]
16
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 1а
17
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 1b
18
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 2а.
19
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 2b
20
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 3а
21
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 3b
22
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 4.
23
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 5.
24
ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
AN US
CR IP T
Fig. 6.
25