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Calculation of a mass-spectrometer with a sector magnet, an electrostatic prism and a transaxial lens
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Physics PhysicsProcedia Procedia 100(2008) (2008)425–433 000–000 www.elsevier.com/locate/procedia www.elsevier.com/locate/XXX
Proceedings of the Seventh International Conference on Charged Particle Optics
Calculation of a mass-spectrometer with a sector magnet, an electrostatic prism and a transaxial lens . . Baisanova , G. . Doskeevb, I.F. Spivak-Lavrovb ∗ b
a Military Institute of Air Defense Forces, Kazakhstan Aktobe State University after name K. Zhubanov, Kazakhstan Br. Zhubanov St. 263, 030000 Aktobe, Kazakhstan
Elsevier only:received Receivedindate here; form revised date here; date here 2008 Received 9 Julyuse 2008; revised 9 July 2008;accepted accepted 9 July
Abstract We have considered a static mass-spectrometer with a double focusing beam, which is used in combination with a uniform sector magnetic field, a three electrode electrostatic prism and a transaxial lens. The chosen electrostatic system not only ensures focusing of the beam on energy, but also vastly increases the beam before entering in the magnetic field. This, in combination with the theorem on the "quality" parameter, allows greater resolution of the mass-spectrometer. The electric and magnetic fields of the mass-spectrometer are given analytically. By numerical integration of the differential equations the trajectories of the charged particles in curvilinear coordinates are calculated and the characteristics of the mass-spectrometer are deduced. This enables the parameters for the mass-spectrometer to be chosen. © 2008 Elsevier B.V. All rights reserved. PACS: 07.75.+h; 41.85.Lc; 41.85.-p; 41.20.Cv; 41.20.Gz Keywords: Mass spectrometers; Sector magnets; Electrostatic prism; Analytic field model
1. Introduction Earlier the sector mass-spectrometer with the electrostatic prism was considered in [1]. In the present work the theory advanced in [2,3], is used for account of the corpuscular-optical characteristics of the new schema of static mass-spectrometer with the double-focusing type. Is considered of mass-spectrometer, in which in the combination with by sector homogeneous magnetic field also is used three-electrode electrostatic prism and transaxial lens. Such electrostatic system not only ensures focusing of beam on energy, but also vastly increases the beam before entering in the magnetic field. That in the consent with the theorem about parameter "quality" [4] allows to improve efficiency of mass-spectrometer. Proposed mass-spectrometer is schematically represented on Fig. 1, where the motion of monochromatic ion beam of equal mass in mass-spectrometer is also shown. In this picture the slit of ions source is marked as number 1; numbers 2, 3, 4 are marked electrodes of electrostatic three electrode prism with potentials V0, V1, V2 accordingly; 4
∗ Coresponding author: Tel: + (3132) 561893; Fax: + (3132) 567843 E-mail address: [email protected]
doi:10.1016/j.phpro.2008.07.123
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and 5 are electrodes of transaxial lens with potentials V2 and V0 accordingly; 6 is poles of sector magnet with the angle of beam deflection, equal ΦH; 7 is a slit of ions receiver. Coming out of point A of ion source divergence ions beam first speed up, being refracted on the border of electrodes 2, 3, but then slow on the border of electrodes 3, 4 leaving this border perpendicularly, and is then increased by transaxial lens, which, working as diffusing lens, enlarges angular divergence in the beam. Here with rumpled representation of the source transfers to point A′. Then extended beam of ions is rejected by sector magnet 6 and is focused in the slit of ion receiver 7.
Fig. 1. Principle scheme of the mass-spectrometer.
Fig. 2. Schematic sketch of the mass-spectrometer and Cartesian coordinates.
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2. Theory The offered mass-spectrometer is schematically represented in a Fig. 2 together with accompanying Cartesian coordinates. A stroke the dashed line represents an axial trajectory of a beam. The axes Oy and Oy1 coincide with borders of electrodes of a prism, on which the potentials V0, V1, V2 specified on figure move. The corner between borders of electrodes of a prism is equal γe. The electrodes of transaxial lens with radius is equal Re have the potentials V2 and V0, its center coincides with the beginning of the Cartesian system of coordinates x2 O2 y2. The rule of a beginning of coordinates O2 is set in distances Lex and Ley. The borders of magnetic poles of the sector magnet coincide with axes O3 y3 and O3 y4. The corner between borders of magnetic poles is equal γH. The rule of a beginning of coordinates O3 is set in distances LHx and LHy. In Fig. 2 the corners σ1 and σ2, formed by an axial trajectory with borders of magnetic poles are shown also. The letters A and A′ designate cracks of a source and receiver of ions accordingly. This work uses the curvilinear coordinates q, z, s [2,3]. The curvilinear axis s lays in mean plane and coincides with the axial trajectory of beam; the axis z has constant direction, perpendicular to the mean plane z = 0; the axis q lays in the mean plane and is directed by the normal to the axial trajectory and in each point of axial trajectory the units form the right system. The electric and magnetic fields are described by dimensionless potentials and , which are connected with electrostatic ϕ and magnetic ω potentials by expressions [2-4]: eω eϕ , (1) =− =− m c2 m c2 where e − an electric charge of the particles in the concerned beam, m – mass of particles driven on an axial trajectory, − the velocity of light in vacuum. Potential (q, z, s ) = F (q, z, s ) + ε f 0 also is normalized in such a 2 manner that it is equal to zero there, where the speed of particles with mass m and energy mc f (s ) driven on an axial trajectory is equal to zero too. Here f (s ) = F (0, 0, s ) – distribution of electrostatic potential on an axial trajectory; h ≡ h (s ) = z (0, 0, s ) – distribution of size of intensity of a magnetic field on an axial trajectory laying in a mead plane z = 0. The index "0" means, that the appropriate size undertakes in area outside of a field in the plane s = 0 , where the crack of a source of ions is located, and the indexes at and designate private derivative by the appropriate coordinates. In this work γ and ε characterize relative variations of mass and energy in the beam as 2 respects to the average mass m and average energy mc f 0 correspondingly. Using the law of refraction for refracting sides of an electrostatic prism and lens, and also the approximation of sharp border for magnetic field, is possible to receive the approached formulas for the characteristics of offered mass-spectrometer in a horizontal direction [5]. However this does not allow to calculate focusing properties in a vertical direction, and also the aberration of system. For realization of detailed account of the mass-spectrometer we used the analytical expressions for electrical and magnetic fields received with the help of methods of the theory of functions for complex variable [6]. The electrical field of the mass-spectrometer was defined by function: F (x, y , z ) = F1 (u , v |V0 , V1 )+ F1 (u1 , v1 |V1 , V2 ) + F1 (u 2 , v 2 |V2 , V0 ) − V1 − V2 .
(2)
Here F1 (u, v |V1 , V2 ) = V2 +
u ≡ u ( x, z ) = − exp v ≡ v ( x, z ) = exp
π d
V1 − V2
π d
π
arctg
x sin
π
π
z
x cos
d
d
z
1− u 1+ u + arctg v v
(3)
(4)
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u1 ≡ u1 ( x, y, z ) = − exp
u2 = v2 =
−
πR
−
ρ
sin
π
cos
π
d
Re πR d
Re
d
π
v1 ≡ v1 ( x, y, z ) = exp
ρ
π
d
π
x1 sin
x1 cos
d
π d
z
(5)
z
z
d
(6)
z
d
In last formulas d – distance between electrodes in vertical direction, x1 = x cos γ e − y sin γ e y1 = x sin γ e + y cos γ e
(7)
ρ = x22 + y22
(8)
x 2 = x cos γ e − y sin γ e − Lex . y 2 = x sin γ e + y cos γ e − Ley
(9)
The expression (2) well describes a field of electrostatic system of the mass-spectrometer with indefinitely narrow backlashes between electrodes, when areas of a field of separate cracks, where there is a difference of potentials, enough far are removed from each other. In the same approximation there was a magnetic field of the mass-spectrometer. Thus was considered that magnetic screens with equal to zero magnetic potential adjoin to magnetic poles. The magnetic field of a sector magnet was defined by function:
( x, y , z ) =
W1 (u 3 , v3 | − C ), W1 (u 4 , v 4 | C ),
x3 cos x3 cos
γH 2
γH 2
− y 3 sin
− y 3 sin
γH 2
γH 2
≤ 0;
.
(10)
≥ 0.
Here C
W1 (u , v | C ) =
arctg
1+ u
π v π π u 3 = − exp x3 sin z a a π π v3 = exp x3 cos z a a u 4 = exp − v 4 = exp −
π a
π
x 4 sin
π a
− arctg
1− u , v
z
(11)
(12)
(13)
π
x 4 cos z a a In last formulas a – distance between poles of a magnet in vertical direction,
x3 = (x1 − LH x )cos σ 1 − (y1 − LH y )sin σ 1
(14)
x4 = x3 cos γ H − y3 sin γ H
(15)
y3 = (x1 − LH x )sin σ 1 + (y1 − LH y )cos σ 1
y 4 = x3 sin γ H + y3 cos γ H
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and C = a , were r0 – radius of axial trajectory at the center of a magnet, where a magnetic field 2 r0 homogeneous. The axial trajectory was defined by numerical integration of system of the differential equations: Fy F h + y′ x′′ = 1 − x′ 2 x − x′y ′ 2F 2F 2F
(
)
(
) 2FF − x′y′ 2FF − x′
y ′′ = 1 − y ′ 2
y
(16)
h
. (17) 2F Here strokes designate differentiation on s. The entry conditions for these equations were set by the following parities: x0 = − l s cos i , x0′ = cos i , y 0 = l s sin i + re , y 0′ = − sin i , where l s defines the distance up to crack of source of ions, re – sizes of the prism, and i – the asymptotic corner x
of fall of an axial trajectory on the first side of a prism (see Fig. 2). The connections between Cartesian x, y, z and the curvilinear coordinates q, z, s is set by parities: x = xc (s ) − q y c′ (s ) , y = y c (s ) + q xc′ (s ) ,
(18)
z=z. ( ) ( ) x s y s Here c , c define an axial trajectory in mean plane z = 0. The calculation of trajectories of charged particles was carried out by numerical integration of the differential equations of a trajectory in the curvilinear coordinates. These equations can be written down in the following kind: ′ r ′2 q z′ s r′ (19) q ′′ + (1 − k q ) k + q ′ ln − (1 − k q ) z , = + 2 r′ 2 (1 + γ ) 1 − k q ′ z ′′ + z ′ ln
where
r′
′ ln
r′
=
=
r′2 2
z
+
r′ 2
(1 + γ )
(1 − k q )
r ′2 s r′ 1 q′ + 1 − k q 2 Φ (1 − k q ) 2 (1 + γ )
(
q′ s , 1− k q
q
−
z
− z′
q
)+ q k ′ + 2 q′k
(20)
.
(21)
In the formulas (19)-(21) strokes designate differentiation on s – length of an arch of an axial trajectory; k – curvature of an axial trajectory determined by a parity fq h ; (22) − k= 2f 2f r ′ = q ′ 2 + z ′ 2 + (1− k q ) . 2
(23)
Each particle of a beam is described by two the curvilinear coordinates q and z, which characterize a deviation of a particle from an axial trajectory. Thus the six-parametrical family of trajectories leaving from a ion source is considered which differ in two initial coordinates q0 and z 0 , their by derivative q′0 and z′0 , and also the deviation on mass γ and energy ε. Such approach allows to characterize behavior of all beam as a whole, expecting simultaneously plenty of trajectories with the various entry conditions and different meanings of parameters ε and γ , and without the assumption of them to small size.
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The numerical integration of the equations (16), (17) and (19), (20) the Adams method with an automatic choice of a step was carried out and the beginning points were found by the Krilov method. The program for the calculation is written on Visual Basic for Applications in environment Excel. As unit of length was chosen d = a , it is the distance between electrodes and magnetic poles. The relative accuracy at a presence of an axial trajectory varied and achieved 10-10. Were selected the relation of potentials V1/V0 and the value of corner γ e , at which the axial trajectory perpendicularly the second refracting side of prism. The received results will well be coordinated to the approached estimations which have been carried out with use of results of works [4, 5].
3. Computed Results We carried calculation of the mass-spectrometer with the following parameters, wish are presented in Table 1. In a Fig. 3 is represented the calculated axial trajectory of beam in the Cartesian system of coordinates x y . The thick lines in a Fig. 3 mark the projections of borders of electrodes and magnetic poles on the mean plane. Table 1 The parameters of mass-spectrometer and the beam´s emittance. Element
Prism
Lens
Magnet
Beam´s emittance
i
75°
ls
40
γe
12.4732°
re
25
V1/V0
20
V2/V0
3.27665
Lex
28.9
Ley
28.96731
Re
25
Lhx − Lex
3.9
Lhy
12.05002
r0
25
γH
58°
σ1 = σ2 = σ
–15°
ΦH
28°
ε
0.001
qs
0.007
qs′
0.001
zs
0.050
zs′
0.001
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Fig. 3. The calculated axial trajectory of beam in mass-spectrometer.
The beam profile in mass-spectrometer in horizontal and vertical directions is submitted in a Fig. 4 and Fig. 5 accordingly. On a Fig. 4 the beam profile in horizontal directions corresponds to small values z 0 and z′0 , shaped lines represent also extreme trajectories of a beam of particles with a relative deviation on mass γ = 0,001.
Fig. 4. The beam profile in mean plane of the mass-spectrometers.
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Fig. 5. The beam profile in vertical dimension.
The angle magnification of mass-spectrometer is = 5.2911. The highest resolution is reached at s = sc r = 127.5, where is located the crossover of beam. The Gauss plane corresponds the value s = s g = 127.58. In the mean plane at s = sc r = 127.5 it is possible to write down the following approached expression: q c r = q (s cr ) = M q 0 + D m γ + A2 q 0′ + A5ε + A11 q 02 + A12 q 0 q 0′ + A2 2 q 0′ 2 + A15 q 0 ε + A2 5 q 0′ ε + A55 ε 2 .
(24)
Here the image magnification M = 1/Γ = 0.18899, the mass dispersion Dm = 10.981= 0.43924 r0 , the relation Dm/M = 2.3241, the coefficients A2 = 0,15774 and A5 = – 0, 014625. The values of second order aberration coefficients are given in the Table 2. Table 2 The value of second order aberration coefficients mass-spectrometer.
A11
A12
A2 2
A15
A25
A55
0.63155
38.355
582.54
10.994
374.64
– 18.126
By the opportunities this mass-spectrometer is comparable with [7], but has more simple design. So if to put as in [7] d = a = 6 mm, then the value q s = 0,007 corresponds the width of slit of ions source of equal 84 m . Thus depending on values z s and z′s can be received the mass resolution R and the aberrations of mass-spectrometer cr for q s = 0,007, q′s = 0,001, ε = 0,001 at s = s c r is presented in the Table 3. It was supposed, that in beam − z s ≤ z 0 ≤ z s and − z ′s ≤ z 0′ ≤ z ′s ; q1 ≤ qc r ≤ q 2 and − d z ≤ z c r ≤ d z . Table 3 Dependence of the mass resolution from the vertical sizes of beam. Entrance
zs
Aberrations and resolution
Crossover
z ′s
q1
q2
qma x
dz
r
R
0
0
–0.001618
0.002221
0.003839
0
0.001193
2866
0.0125
0.00025
–0.002589
0.002157
0.004810
0.23
0.002164
2287
0.025
0.0005
–0.005354
0.002025
0.007575
0.43
0.004929
1452
0.05
0.001
–0.009434
0.001148
0.011660
0.60
0.009010
944
Let's note that the method, used in this work, of account allows to analyze passage even of very wide beams of the charged particles, when the application of the aberration theory is not so possible. So in Fig. 5 is given the beam
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profile in vertical dimension received at the greatest possible the value z s and z′s , at which particles yet do not touch of the field surfaces.
References [1] [2] [3] [4] [5] [6] [7]
D. Ioanoviciu, C. Cuna, G. Kalmutchi and C. Toma, Rev. Roum. Phys. 31 (1986) 693. I.F. Spivak-Lavrov, Nucl. Instr. & Meth. A 363 (1995) 485. I.F. Spivak-Lavrov, Zh. Tekh. Fiz. 64 (1994) 140. L.G. Glikman, I.F. Spivak-Lavrov,. Letters to Zh.Tekh. Fiz. 16 (1990) 26. O.A. Baisanov, G.A. Doskeev, I.F. Spivak-Lavrov. Izv. NAN RK. Ser. phys.-mat. 2 (2006) 41. G.A. Doskeev, I.F. Spivak-Lavrov, Zh. Tekh. Fiz. 59 (1989) 144. M. Ishihara, Y. Kammei, H. Matsuda, Nucl. Instr. & Meth. A 363 (1995) 440.
Physics PhysicsProcedia Procedia 100(2008) (2008)425–433 000–000 www.elsevier.com/locate/procedia www.elsevier.com/locate/XXX
Proceedings of the Seventh International Conference on Charged Particle Optics
Calculation of a mass-spectrometer with a sector magnet, an electrostatic prism and a transaxial lens . . Baisanova , G. . Doskeevb, I.F. Spivak-Lavrovb ∗ b
a Military Institute of Air Defense Forces, Kazakhstan Aktobe State University after name K. Zhubanov, Kazakhstan Br. Zhubanov St. 263, 030000 Aktobe, Kazakhstan
Elsevier only:received Receivedindate here; form revised date here; date here 2008 Received 9 Julyuse 2008; revised 9 July 2008;accepted accepted 9 July
Abstract We have considered a static mass-spectrometer with a double focusing beam, which is used in combination with a uniform sector magnetic field, a three electrode electrostatic prism and a transaxial lens. The chosen electrostatic system not only ensures focusing of the beam on energy, but also vastly increases the beam before entering in the magnetic field. This, in combination with the theorem on the "quality" parameter, allows greater resolution of the mass-spectrometer. The electric and magnetic fields of the mass-spectrometer are given analytically. By numerical integration of the differential equations the trajectories of the charged particles in curvilinear coordinates are calculated and the characteristics of the mass-spectrometer are deduced. This enables the parameters for the mass-spectrometer to be chosen. © 2008 Elsevier B.V. All rights reserved. PACS: 07.75.+h; 41.85.Lc; 41.85.-p; 41.20.Cv; 41.20.Gz Keywords: Mass spectrometers; Sector magnets; Electrostatic prism; Analytic field model
1. Introduction Earlier the sector mass-spectrometer with the electrostatic prism was considered in [1]. In the present work the theory advanced in [2,3], is used for account of the corpuscular-optical characteristics of the new schema of static mass-spectrometer with the double-focusing type. Is considered of mass-spectrometer, in which in the combination with by sector homogeneous magnetic field also is used three-electrode electrostatic prism and transaxial lens. Such electrostatic system not only ensures focusing of beam on energy, but also vastly increases the beam before entering in the magnetic field. That in the consent with the theorem about parameter "quality" [4] allows to improve efficiency of mass-spectrometer. Proposed mass-spectrometer is schematically represented on Fig. 1, where the motion of monochromatic ion beam of equal mass in mass-spectrometer is also shown. In this picture the slit of ions source is marked as number 1; numbers 2, 3, 4 are marked electrodes of electrostatic three electrode prism with potentials V0, V1, V2 accordingly; 4
∗ Coresponding author: Tel: + (3132) 561893; Fax: + (3132) 567843 E-mail address: [email protected]
doi:10.1016/j.phpro.2008.07.123
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and 5 are electrodes of transaxial lens with potentials V2 and V0 accordingly; 6 is poles of sector magnet with the angle of beam deflection, equal ΦH; 7 is a slit of ions receiver. Coming out of point A of ion source divergence ions beam first speed up, being refracted on the border of electrodes 2, 3, but then slow on the border of electrodes 3, 4 leaving this border perpendicularly, and is then increased by transaxial lens, which, working as diffusing lens, enlarges angular divergence in the beam. Here with rumpled representation of the source transfers to point A′. Then extended beam of ions is rejected by sector magnet 6 and is focused in the slit of ion receiver 7.
Fig. 1. Principle scheme of the mass-spectrometer.
Fig. 2. Schematic sketch of the mass-spectrometer and Cartesian coordinates.
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2. Theory The offered mass-spectrometer is schematically represented in a Fig. 2 together with accompanying Cartesian coordinates. A stroke the dashed line represents an axial trajectory of a beam. The axes Oy and Oy1 coincide with borders of electrodes of a prism, on which the potentials V0, V1, V2 specified on figure move. The corner between borders of electrodes of a prism is equal γe. The electrodes of transaxial lens with radius is equal Re have the potentials V2 and V0, its center coincides with the beginning of the Cartesian system of coordinates x2 O2 y2. The rule of a beginning of coordinates O2 is set in distances Lex and Ley. The borders of magnetic poles of the sector magnet coincide with axes O3 y3 and O3 y4. The corner between borders of magnetic poles is equal γH. The rule of a beginning of coordinates O3 is set in distances LHx and LHy. In Fig. 2 the corners σ1 and σ2, formed by an axial trajectory with borders of magnetic poles are shown also. The letters A and A′ designate cracks of a source and receiver of ions accordingly. This work uses the curvilinear coordinates q, z, s [2,3]. The curvilinear axis s lays in mean plane and coincides with the axial trajectory of beam; the axis z has constant direction, perpendicular to the mean plane z = 0; the axis q lays in the mean plane and is directed by the normal to the axial trajectory and in each point of axial trajectory the units form the right system. The electric and magnetic fields are described by dimensionless potentials and , which are connected with electrostatic ϕ and magnetic ω potentials by expressions [2-4]: eω eϕ , (1) =− =− m c2 m c2 where e − an electric charge of the particles in the concerned beam, m – mass of particles driven on an axial trajectory, − the velocity of light in vacuum. Potential (q, z, s ) = F (q, z, s ) + ε f 0 also is normalized in such a 2 manner that it is equal to zero there, where the speed of particles with mass m and energy mc f (s ) driven on an axial trajectory is equal to zero too. Here f (s ) = F (0, 0, s ) – distribution of electrostatic potential on an axial trajectory; h ≡ h (s ) = z (0, 0, s ) – distribution of size of intensity of a magnetic field on an axial trajectory laying in a mead plane z = 0. The index "0" means, that the appropriate size undertakes in area outside of a field in the plane s = 0 , where the crack of a source of ions is located, and the indexes at and designate private derivative by the appropriate coordinates. In this work γ and ε characterize relative variations of mass and energy in the beam as 2 respects to the average mass m and average energy mc f 0 correspondingly. Using the law of refraction for refracting sides of an electrostatic prism and lens, and also the approximation of sharp border for magnetic field, is possible to receive the approached formulas for the characteristics of offered mass-spectrometer in a horizontal direction [5]. However this does not allow to calculate focusing properties in a vertical direction, and also the aberration of system. For realization of detailed account of the mass-spectrometer we used the analytical expressions for electrical and magnetic fields received with the help of methods of the theory of functions for complex variable [6]. The electrical field of the mass-spectrometer was defined by function: F (x, y , z ) = F1 (u , v |V0 , V1 )+ F1 (u1 , v1 |V1 , V2 ) + F1 (u 2 , v 2 |V2 , V0 ) − V1 − V2 .
(2)
Here F1 (u, v |V1 , V2 ) = V2 +
u ≡ u ( x, z ) = − exp v ≡ v ( x, z ) = exp
π d
V1 − V2
π d
π
arctg
x sin
π
π
z
x cos
d
d
z
1− u 1+ u + arctg v v
(3)
(4)
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u1 ≡ u1 ( x, y, z ) = − exp
u2 = v2 =
−
πR
−
ρ
sin
π
cos
π
d
Re πR d
Re
d
π
v1 ≡ v1 ( x, y, z ) = exp
ρ
π
d
π
x1 sin
x1 cos
d
π d
z
(5)
z
z
d
(6)
z
d
In last formulas d – distance between electrodes in vertical direction, x1 = x cos γ e − y sin γ e y1 = x sin γ e + y cos γ e
(7)
ρ = x22 + y22
(8)
x 2 = x cos γ e − y sin γ e − Lex . y 2 = x sin γ e + y cos γ e − Ley
(9)
The expression (2) well describes a field of electrostatic system of the mass-spectrometer with indefinitely narrow backlashes between electrodes, when areas of a field of separate cracks, where there is a difference of potentials, enough far are removed from each other. In the same approximation there was a magnetic field of the mass-spectrometer. Thus was considered that magnetic screens with equal to zero magnetic potential adjoin to magnetic poles. The magnetic field of a sector magnet was defined by function:
( x, y , z ) =
W1 (u 3 , v3 | − C ), W1 (u 4 , v 4 | C ),
x3 cos x3 cos
γH 2
γH 2
− y 3 sin
− y 3 sin
γH 2
γH 2
≤ 0;
.
(10)
≥ 0.
Here C
W1 (u , v | C ) =
arctg
1+ u
π v π π u 3 = − exp x3 sin z a a π π v3 = exp x3 cos z a a u 4 = exp − v 4 = exp −
π a
π
x 4 sin
π a
− arctg
1− u , v
z
(11)
(12)
(13)
π
x 4 cos z a a In last formulas a – distance between poles of a magnet in vertical direction,
x3 = (x1 − LH x )cos σ 1 − (y1 − LH y )sin σ 1
(14)
x4 = x3 cos γ H − y3 sin γ H
(15)
y3 = (x1 − LH x )sin σ 1 + (y1 − LH y )cos σ 1
y 4 = x3 sin γ H + y3 cos γ H
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and C = a , were r0 – radius of axial trajectory at the center of a magnet, where a magnetic field 2 r0 homogeneous. The axial trajectory was defined by numerical integration of system of the differential equations: Fy F h + y′ x′′ = 1 − x′ 2 x − x′y ′ 2F 2F 2F
(
)
(
) 2FF − x′y′ 2FF − x′
y ′′ = 1 − y ′ 2
y
(16)
h
. (17) 2F Here strokes designate differentiation on s. The entry conditions for these equations were set by the following parities: x0 = − l s cos i , x0′ = cos i , y 0 = l s sin i + re , y 0′ = − sin i , where l s defines the distance up to crack of source of ions, re – sizes of the prism, and i – the asymptotic corner x
of fall of an axial trajectory on the first side of a prism (see Fig. 2). The connections between Cartesian x, y, z and the curvilinear coordinates q, z, s is set by parities: x = xc (s ) − q y c′ (s ) , y = y c (s ) + q xc′ (s ) ,
(18)
z=z. ( ) ( ) x s y s Here c , c define an axial trajectory in mean plane z = 0. The calculation of trajectories of charged particles was carried out by numerical integration of the differential equations of a trajectory in the curvilinear coordinates. These equations can be written down in the following kind: ′ r ′2 q z′ s r′ (19) q ′′ + (1 − k q ) k + q ′ ln − (1 − k q ) z , = + 2 r′ 2 (1 + γ ) 1 − k q ′ z ′′ + z ′ ln
where
r′
′ ln
r′
=
=
r′2 2
z
+
r′ 2
(1 + γ )
(1 − k q )
r ′2 s r′ 1 q′ + 1 − k q 2 Φ (1 − k q ) 2 (1 + γ )
(
q′ s , 1− k q
q
−
z
− z′
q
)+ q k ′ + 2 q′k
(20)
.
(21)
In the formulas (19)-(21) strokes designate differentiation on s – length of an arch of an axial trajectory; k – curvature of an axial trajectory determined by a parity fq h ; (22) − k= 2f 2f r ′ = q ′ 2 + z ′ 2 + (1− k q ) . 2
(23)
Each particle of a beam is described by two the curvilinear coordinates q and z, which characterize a deviation of a particle from an axial trajectory. Thus the six-parametrical family of trajectories leaving from a ion source is considered which differ in two initial coordinates q0 and z 0 , their by derivative q′0 and z′0 , and also the deviation on mass γ and energy ε. Such approach allows to characterize behavior of all beam as a whole, expecting simultaneously plenty of trajectories with the various entry conditions and different meanings of parameters ε and γ , and without the assumption of them to small size.
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The numerical integration of the equations (16), (17) and (19), (20) the Adams method with an automatic choice of a step was carried out and the beginning points were found by the Krilov method. The program for the calculation is written on Visual Basic for Applications in environment Excel. As unit of length was chosen d = a , it is the distance between electrodes and magnetic poles. The relative accuracy at a presence of an axial trajectory varied and achieved 10-10. Were selected the relation of potentials V1/V0 and the value of corner γ e , at which the axial trajectory perpendicularly the second refracting side of prism. The received results will well be coordinated to the approached estimations which have been carried out with use of results of works [4, 5].
3. Computed Results We carried calculation of the mass-spectrometer with the following parameters, wish are presented in Table 1. In a Fig. 3 is represented the calculated axial trajectory of beam in the Cartesian system of coordinates x y . The thick lines in a Fig. 3 mark the projections of borders of electrodes and magnetic poles on the mean plane. Table 1 The parameters of mass-spectrometer and the beam´s emittance. Element
Prism
Lens
Magnet
Beam´s emittance
i
75°
ls
40
γe
12.4732°
re
25
V1/V0
20
V2/V0
3.27665
Lex
28.9
Ley
28.96731
Re
25
Lhx − Lex
3.9
Lhy
12.05002
r0
25
γH
58°
σ1 = σ2 = σ
–15°
ΦH
28°
ε
0.001
qs
0.007
qs′
0.001
zs
0.050
zs′
0.001
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Fig. 3. The calculated axial trajectory of beam in mass-spectrometer.
The beam profile in mass-spectrometer in horizontal and vertical directions is submitted in a Fig. 4 and Fig. 5 accordingly. On a Fig. 4 the beam profile in horizontal directions corresponds to small values z 0 and z′0 , shaped lines represent also extreme trajectories of a beam of particles with a relative deviation on mass γ = 0,001.
Fig. 4. The beam profile in mean plane of the mass-spectrometers.
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Fig. 5. The beam profile in vertical dimension.
The angle magnification of mass-spectrometer is = 5.2911. The highest resolution is reached at s = sc r = 127.5, where is located the crossover of beam. The Gauss plane corresponds the value s = s g = 127.58. In the mean plane at s = sc r = 127.5 it is possible to write down the following approached expression: q c r = q (s cr ) = M q 0 + D m γ + A2 q 0′ + A5ε + A11 q 02 + A12 q 0 q 0′ + A2 2 q 0′ 2 + A15 q 0 ε + A2 5 q 0′ ε + A55 ε 2 .
(24)
Here the image magnification M = 1/Γ = 0.18899, the mass dispersion Dm = 10.981= 0.43924 r0 , the relation Dm/M = 2.3241, the coefficients A2 = 0,15774 and A5 = – 0, 014625. The values of second order aberration coefficients are given in the Table 2. Table 2 The value of second order aberration coefficients mass-spectrometer.
A11
A12
A2 2
A15
A25
A55
0.63155
38.355
582.54
10.994
374.64
– 18.126
By the opportunities this mass-spectrometer is comparable with [7], but has more simple design. So if to put as in [7] d = a = 6 mm, then the value q s = 0,007 corresponds the width of slit of ions source of equal 84 m . Thus depending on values z s and z′s can be received the mass resolution R and the aberrations of mass-spectrometer cr for q s = 0,007, q′s = 0,001, ε = 0,001 at s = s c r is presented in the Table 3. It was supposed, that in beam − z s ≤ z 0 ≤ z s and − z ′s ≤ z 0′ ≤ z ′s ; q1 ≤ qc r ≤ q 2 and − d z ≤ z c r ≤ d z . Table 3 Dependence of the mass resolution from the vertical sizes of beam. Entrance
zs
Aberrations and resolution
Crossover
z ′s
q1
q2
qma x
dz
r
R
0
0
–0.001618
0.002221
0.003839
0
0.001193
2866
0.0125
0.00025
–0.002589
0.002157
0.004810
0.23
0.002164
2287
0.025
0.0005
–0.005354
0.002025
0.007575
0.43
0.004929
1452
0.05
0.001
–0.009434
0.001148
0.011660
0.60
0.009010
944
Let's note that the method, used in this work, of account allows to analyze passage even of very wide beams of the charged particles, when the application of the aberration theory is not so possible. So in Fig. 5 is given the beam
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profile in vertical dimension received at the greatest possible the value z s and z′s , at which particles yet do not touch of the field surfaces.
References [1] [2] [3] [4] [5] [6] [7]
D. Ioanoviciu, C. Cuna, G. Kalmutchi and C. Toma, Rev. Roum. Phys. 31 (1986) 693. I.F. Spivak-Lavrov, Nucl. Instr. & Meth. A 363 (1995) 485. I.F. Spivak-Lavrov, Zh. Tekh. Fiz. 64 (1994) 140. L.G. Glikman, I.F. Spivak-Lavrov,. Letters to Zh.Tekh. Fiz. 16 (1990) 26. O.A. Baisanov, G.A. Doskeev, I.F. Spivak-Lavrov. Izv. NAN RK. Ser. phys.-mat. 2 (2006) 41. G.A. Doskeev, I.F. Spivak-Lavrov, Zh. Tekh. Fiz. 59 (1989) 144. M. Ishihara, Y. Kammei, H. Matsuda, Nucl. Instr. & Meth. A 363 (1995) 440.