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Third order transfer matrices for real Wien filters with homogeneous main fields
ARTICLE IN PRESS Nuclear Instruments and Methods in Physics Research A 614 (2010) 47–56
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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Third order transfer matrices for real Wien filters with homogeneous main fields Damaschin Ioanoviciu a,b,, Katsushige Tsuno c a
Physics Faculty, Babes Bolyai University Str. M. Kogalniceanu Nr.1 400 084 Cluj-Napoca, Romania National Institute for Research-Development of Isotopic and Molecular Technologies P.O. Box 700, 400 293 Cluj-Napoca, Romania c Electron Optics Solutions Tsuno, 2-10-11, Mihori, Akishima, Tokyo 196-0001, Japan b
a r t i c l e in fo
abstract
Article history: Received 9 July 2009 Received in revised form 17 December 2009 Accepted 22 December 2009 Available online 4 January 2010
The determination of the ion optical aberrations of the Wien filters operated independently or in more complex mass analyzing or charged particle transport systems needs the calculation of the trajectories in a third order approximation. The trajectory calculations for crossed electric and magnetic fields generate unmanageably long formulas for the matrix elements. The reduction of the extent of the matrix elements to usable size can be reached by restricting calculations to less complex configurations as the homogeneous main fields Wien filters, which are probably the most often used crossed field velocity filters. Analytic expressions for transfer matrix elements for homogeneous main fields Wien filters were determined up to third order accuracy level. The elements of three pairs of matrices, those of entry fringing fields, those of the main fields and those of the exit fringing fields were derived for the plane of mass dispersion (radial) and for a direction normal to it (axial). The third order accuracy was kept over all the calculated contributions to the ion trajectory. The first order elements, to be multiplied by first order small quantities were calculated in a second order approximation, those of second order with a first order accuracy, while those of third order were not allowed to contain small quantities in their expressions. The matrix elements describing the ion position, trajectory slope, ion energy and mass were accounted for. In the fringing field matrix elements the effect of the electric and magnetic field boundary curvature radii were included. In a brief application, on one spectrometer geometry, the calculated matrices were used to determine the effect of third order angular aberration generated by the Wien filter alone on the resolution when the second order aberration has been cancelled. & 2009 Elsevier B.V. All rights reserved.
Keywords: Charged particle analyzer Wien filter Homogeneous fields Transfer matrices Third order aberrations
1. Introduction Among the crossed electric and magnetic fields mass analyzers, the velocity filter, called after its inventor Wien filter has the unique property of leaving to pass straight ions of a given velocity. The Wien filter has a wide area of applications, alone or included in more sophisticated ion-optical systems. Alone, the Wien filter produces a mass spectrum from a beam of defined energy ions. A mass spectrum at 4000 resolution was obtained [1], isotopes separated [2] and ions implanted [3] with Wien filters in single focusing arrangements. Double focusing mass spectrometers may incorporate Wien filters as that with oblique incidence [4] or with compact geometry [5]. The linearity
Corresponding author at: Physics Faculty, Babes Bolyai University Str. M. Kogalniceanu Nr.1 400 084 Cluj-Napoca, Romania. E-mail address: [email protected] (D. Ioanoviciu).
0168-9002/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2009.12.054
of the beam axis inside the Wien filters was capitalized in ion microscope [6] and ion beam lithography [7] systems. Wien filters were successfully included and operated in a variety of nuclear instruments as recoil separators and spectrometers. The European Recoil Separator for Nuclear Astrophysics (ERNA) in its configuration reported in Ref. [8], includes four Wien filters. Its ion optics, calculated in Ref. [9] resulted by using differential algebra based code COSY Infinity [10]. The Astrophysics Recoil Separator (ARES) [11] also uses a Wien filter in a configuration simulated by Mathcad [12] and GEANT [13] softwares. The LISE3 double Wien filter [14] has been constructed to improve the separation for exotic nuclei of the Ligne d’Ions Super Epluches (LISE) spectrometer, simulations being ensured by the ZGOUBY computer code [15]. Aberrations calculations were performed also for the VAriable MOde Spectrometer (VAMOS) [16] which operates a Wien filter in some of the spectrometer’s working regimes. The Strong Gradient Electromagnetic Online Recoil separator (St. George) design [17] includes a Wien filter
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with electrodes shaped to ensure main path charged particle trajectory linearity also through the fringing fields. Ion optical calculations for this separator were started at first order level with the TRANSPORT third order code [18], developed to fourth order aberrations level with COSY Infinity, only up to third order coefficients being retained as significant. Most of the mentioned applications of the Wien filters were achieved by homogeneous main field analyzers as those of Ref. [2–5,8,11,14,16,17] as well as most of those delivered by the companies Colutron and Danfysik. Often, the absence of additional mentions besides ‘‘Wien filter’’ means: ‘‘analyzer with homogeneous main fields’’. The conditions to ensure homogeneity of both electric and magnetic fields between parallel plane electrodes and pole faces were defined in [19]. Attempts were made to ensure field homogeneity by shimming and by shaping electrodes as well as pole pieces. The complexity of the description of the ion-optical properties of the crossed field analyzers increases with the complication of electrode and pole piece shape and structure creating the involved fields and with the level of approximation accepted for the trajectory description. The second order particle optical calculations allowed to obtain results which can be displayed in transfer matrix form for crossed fields with circular ion main path [20,21] and Wien filters [22]. These calculations included fringing field effects at the same level of accuracy. Third order calculations were made for very general configurations of crossed electric and magnetic mass analyzers with circular ion main path. In Ref. [23] the third order transfer matrix elements were expressed for both radial coordinate and slope in form of series. Due to the length of the expressions 171 elements of the development were omitted from the radial coordinate development and 172 from that of the slope. In Ref. [24] the general expressions for the crossed fields were disclosed only partially for the same reason, too lengthy formulas, checked by various mathematical methods. Coefficients missing from the paper are available on request. There is a way to obtain so high order matrix elements as third order, still having manageable length expressions by restricting the generality of the formulas to specific analyzer geometries of most frequent use. The ideal way to consider and design ion optical instruments is to calculate the total transfer matrix of the system, from the interaction region to the detector. If analytical expressions of the matrix elements of all instrument constituting ion optical parts are available, then this total transfer matrix can be obtained by successively multiplying the matrices of the components. Cancelling specific elements of the first row of the total matrix, angular and energy focusing can be ensured at some order. Each cancelled matrix element gives a focusing condition. The assembly of the focusing conditions result in a system of equations to be solved simultaneously. The unknowns are some electromagnetic and geometric parameters to be determined. The same parameters can be determined with numerical computer codes as well as by codes performing numerical matrix multiplications as GIOS [25], TRIO [26] or TRANSPORT for instance. Each set of numerical integrations or numerical matrix multiplications results in an individual charged particle trajectory with a numerical coordinate at the detector. Algorithms were developed, including constraint equations, to touch beam focusing, involving groups of trajectories. The analytical and numerical ion optical methods to design systems are complementary and they can be used to check each other. The complete third order ion-optical description of homogeneous main field Wien filters below, includes the calculation of
the transfer matrices, of the entry fringing field region, of the main fields and of the exit fringing field region. The third order treatment implies calculation of the first order matrix elements with a second order accuracy, of those of second order with first order accuracy, while for those of third order neglecting any small terms in their expressions. These transfer matrices allow to determine the ion optical properties of individual homogeneous Wien filters or of the systems including such filters deriving focusing conditions up to third order. They can be used to determine whole instrument aberrations analytic expressions or can be included in computer programs.
2. Entry fringing fields The fringing fields are assumed to be confined to a region of Rd length along the z-axis (Figs. 1 and 2), R being the main path ion trajectory radius if only the homogeneous electric or only the homogeneous magnetic field would act alone. In front of this region, at the entry boundary, plane 1, both the electric and magnetic fields vanish. At the end of the fringing field region, at the plane 2, both x electric field and y magnetic field components reach the main fields values E0 and B0, respectively. On the axis, between the entry point into the fringing fields and the main field region we assume that the two field components increase both according to a function h(z): Eð0; 0; zÞ Bð0; 0; zÞ ¼ ¼ hðzÞ E0 B0
ð1Þ
E0 is directed along Ox, while B0 along Oy. The motion proceeds along z in the coordinate system xzy. In the following derivations the electric and magnetic fringing field distribution shapes are assumed identical, fact that can be obtained by a careful design as reported in [17]. Many earlier Wien filters’ less satisfying performances can be explained by the differences between the two distributions as usually magnetic fringing fields extend farther (reason suggested already in [27]). The usual shape of the fringing fields distributions is given in graphical form as on Fig. 4 of [28] or Fig. 2 of [21]. These distribution shapes are not too much different of that given in Fig. 3, which represent the function h(z) for d= 0.02:
hðzÞ ¼
z3 ðdRÞ3
1015
z z2 þ6 dR ðdRÞ2
! ð2Þ
Here dR is the depth of the fringing field region. The fringing field effects were calculated with the help of the fringing field components of both electric and magnetic fields of [29,30]. The derivatives of the longitudinal distributions of nth order were treated as small quantities of n order. The volume of calculations was reduced accounting for the following reasons: The coefficients of the expressions to be integrated contain the initial quantities, at the level of plane ‘‘1’’, Figs. 1 and 2. At this level the particle initial coordinates are x1 = u1R (radial) and y1 =v1R (axial) while the initial direction is determined by a1 =a tan(a1) (radial) and b1 =a tan(b1) (axial) angles. For the complete definition of the trajectory parameters we need also d and g, the relative energy and mass difference related to the reference particle trajectory. The reference particle possesses a mass m0, a charge e (quantity considered as positive), an energy ej0 in the field-free space. The powers of the trajectory parameters u1, a1, v1, b1, d, g do not change by the integration process. For almost all real world Wien filters the depth of the
ARTICLE IN PRESS D. Ioanoviciu, K. Tsuno / Nuclear Instruments and Methods in Physics Research A 614 (2010) 47–56
49
Fig. 1. Projection of the charged particle trajectory onto the xOz plane, parallel to the main electric field E0. Definition of main trajectory geometric parameters at the reference plane where entry fringing fields begin (plane 1), where main homogeneous fields begin (plane 2), at the end of the main fields (plane 3), at the beginning of the field free region (plane 4).
Fig. 2. Projection of the charged particle trajectory onto the yOz plane, parallel to the main magnetic field B0. Definition of main trajectory geometric parameters at the reference plane where entry fringing fields begin to act (plane 1), at the main homogeneous fields beginning (plane 2), at the end of the homogeneous fields region (plane 3), at the beginning of the field free region (plane 4).
fringing field region is small compared to the radius R of the main path particle trajectory if the magnetic field B0 or the electric field E0 would act alone. If the integrand is small of wk, then the result of an integration over w= z/R will be a small quantity of wk + 1 order. As the accuracy of calculations is of third order, we retain in the final results, only second order powers or less (including negative values) for w in the expressions of trajectory parameters globally of first order, of first order or less in w for those of second order powers in those parameters and containing only w at negative powers for the third order coefficients in u1, a1, v1, b1, d and g. The expression of the electrostatic potential inside the fringing field region originates from that of Ref. [29] simplified for
homogeneous main field: 00 h ðzÞx3 h0 ðzÞxðx2 3y2 Þ h0 ðzÞxzðx2 3y2 Þ jðzÞ ¼ j0 ð1 þ dÞ þ E0 hx þ þ 2 6 6Re 6Re ð3Þ 00 h0 ðzÞy2 ðRe þ zÞ x2 ðR2e h ðzÞRe h0 ðzÞh0 ðzÞzÞ Ex ¼ E0 h 2R2e 2R2e
Ey ¼
E0 h0 ðzÞxyð1þ z=Re Þ Re
Ez ¼ E0 h0 ðzÞx
ð4Þ
ð5Þ
ð6Þ
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Fig. 3. Shape of the fringing field distribution h(w) extending over 0.02R at the entry of the Wien filter.
Ez is given at first order only, as the second order term is missing while in the differential equations this component multiplies first order small quantities. Re is the electric field effective boundary radius. The magnetostatic fringing field potential and field components (Ref. [30]) are: 00 h0 ðzÞyðRm þ zÞðy2 3x2 Þ h ðzÞy3 ð7Þ CðzÞ ¼ B0 hy þ 2 6 6Rm Bx ¼ B0
h0 ðzÞxyð1 þz=Rm Þ Rm
ð8Þ
h0 ðzÞx2 ð1 þ z=Rm Þ y2 h0 ðzÞ h0 ðzÞz 00 Bz ¼ B0 h0 ðzÞy By ¼ B0 h 2 h ðzÞ 2 2Rm Rm Rm
ð9Þ Rm being the magnetic field effective boundary radius. The differential equations of the ion trajectories inside the crossed fields of a Wien filter, the electric field directed along Ox, the magnetic field along Oy take the general form [31]:
j½x ð1þ y0 2 Þx0 y0 y þ 00
00
½Ex ð1þ y0 2 Þ þ x0 y0 Ey þ x0 Ez ð1þ x0 2 þ y0 2 Þ 2
rffiffiffiffiffiffiffi ej 2 2 ðy0 Bz þ By Þð1 þx0 þ y0 Þ3=2 ¼ 0 2m
ð10Þ
j½y ð1 þ x0 2 Þy0 x0 x þ 00
00
02
0 0
02
3 3h2 2 uð11d 2dgg2 Þ hh0 ðwÞuvv0 h2 uv0 2 8 2 h h0 ðwÞ w 00 þ u0 2 ð3gdÞ þ u0 v0 v þ v2 d 1þ 4 2q q 00 h0 ðwÞ w h ðwÞ h0 ðwÞ 0 ðd þ gÞ 1 þ ðd þ g Þ vv ðd þ gÞ þ r 4r 4 2 hv0 2 h 3 2 ðg3dÞ ð13Þ ð11d 3d g3dg2 5g3 Þ 4 16 d2 v 1 1 1 1 0 þ w h0 ðwÞuv0 h0 ðwÞu0 v ¼ h ðwÞuv þ þ q r dw2 q2 r2 2 1 2 1 þw 2 þ 2 þ hh0 ðwÞvuu0 þ hu0 v0 þ hh0 ðwÞvu2 q r r q w d w dþg h0 ðwÞ 0 0 þ þ 1þ þ h ðwÞvu 1þ vu ðd þ gÞ r 2r q q 2 0 0 2 2 0 0 0 0 0 0 00 ð14Þ þ 2hh ðwÞv u þ 2h v uu þh ðwÞv ud þ v u u þ hv0 u0 d þ
02
½Ey ð1 þ x Þ þ x y Ex þ y Ez ð1 þ x þ y Þ 2
rffiffiffiffiffiffiffi ej 0 ðx Bz Bx Þð1 þx0 2 þ y0 2 Þ3=2 ¼ 0 2m
0
form appropriate for successive approximation procedure: d2 u h u2 0 1 1 1 1 þh0 ðwÞw 2 þ 2 h ðwÞ ¼ ðgd2huÞ þ þ 2 2 2 q r r dw q 2 h uð5 d g Þ h 00 u0 2 þ 5h3 h ðwÞ h0 ðwÞuu0 þ 2 2 v2 0 1 1 1 1 00 þh0 ðwÞw 2 þ 2 h ðwÞ þ h0 ðwÞvv0 h ðwÞ þ 2 q r r q h 02 h hu3 7 1 2 2 h0 ðwÞ þ v þ ð5d 2dg3g Þ þ þ 2 2 8 3q r 00 7 1 7h ðwÞ þ þ h0 ðwÞw 11h3 þ 2hh0 ðwÞu2 u0 3 3q2 r2 dh0 ðwÞ w h0 ðwÞ w ðd þ gÞ þ u2 1þ 1þ r 2q q 4r 00 dh ðwÞ 3h3 h2 ð11dgÞ uu0 2 þ 4 2 2 2 3 1 3 1 00 0 0 huv 0 þh0 ðwÞw 2 þ 2 h ðwÞ h ðwÞ þ h ðwÞuu d þ 2 q r r q
ð11Þ
These trajectory differential equations are valid for non relativistic charged particles. On the filter axis rffiffiffiffiffiffiffiffiffiffiffiffi 2j0 m0 E0 ¼ ; B0 ¼ E0 ð12Þ R 2ej0 To compensate the effect of E0, oriented along the positive sense of the Ox axis, the applied B0 must be directed in the negative sense of Oy, of the xzy coordinate system. Accounting for these relations, substituting for variables x =uR, y=vR and z = wR, for Re = qR, Rm = rR, as well as for the potential and the field components we obtain the trajectory differential equations in a
Here the following symbols were introduced: dh=dw ¼ h0 ðwÞ;
00
d2 h=dw2 ¼ h ðwÞ
ð15Þ
We start the approximation process with the initial values of the ion parameters at the fringing field entry: u1, a1 = tan(a1), v1, b1 =tan(b1).
ARTICLE IN PRESS D. Ioanoviciu, K. Tsuno / Nuclear Instruments and Methods in Physics Research A 614 (2010) 47–56
For the next approximation we use the above third order equations. The first integration over w gives u0 and v0 , while the second integration gives u and v in the form shown below.
Z
Additionally we define the following symbols: Z w Z w Z w 2 h dw ¼ i1w h2 dw ¼ i2w h3 dw ¼ i3w h0 dw ¼ i4w
w 0
ufirst ¼ u1 þa1 w þ
6 X
si
ð16Þ
1
u0first ¼ a1 þ
13 X
0
Z w Z
0
w
w
0
h3 dw1 dw ¼ j2w
0
ð32Þ
0
ð17Þ Z
7
vfirst ¼ v1 þ b1 wþ s14 þ s15
ð18Þ
v0first ¼ b1 þ s16 þ s17
ð19Þ
w Z w 0
ð31Þ
0
Z w Z
h dw1 dw ¼ j1w 0 0 Z w Z w h2 dw1 dw ¼ j3w 0
si
51
02
h dw1 dw ¼ j4w
Z
0
w
Z
0
Z
w 0
w1
! 02
h dw2
! dw1
dw ¼ k4w
0
ð33Þ
Here the si coefficients have the meanings:
s1 ¼
j1 ðgdÞj2 u1 2
s2 ¼
hh0 2 2 ðv1 u1 Þ 2
ð20Þ
1 1 v21 u21 þu1 a1 v1 b1 þ 2 q r
s3 ¼ ðh0 wi1 Þ
s4 ¼
ð21Þ
u21 v21 1 1 1 2 ðh0 wi1 Þð 2 þ 2 Þ ½j1 ð4a21 4b21 5d þ2dg þ 3g2 Þ 2 8 r q þ 4u1 j2 ðgdÞ20u21 j3
ð22Þ
To easier handle the complex expressions to be integrated, the terms were grouped after the powers of the initial trajectory parameters and those of w, the integrals i1w, i2w, i3w being of first order in w, while j1w, j2w and j3w are of second order. After the substitution of the first solutions into the starting differential equations we arrive to connect the reduced coordinates u and v, as well as their derivatives du/dw and dv/dw at the beginning of the Wien field part with homogeneous fields (index ‘‘2’’, reference plane 2, in Figs. 1 and 2) with the same quantities at the ion entry into the fringing fields (index ‘‘1’’) through relationships of the form: u2 ¼ Du u1 þ Da a1 þ Dd d þ Dg g þ Duu u21 þ Dua u1 a1 þ Dud u1 d 2
þDug u1 g þ Daa a21 þDad a1 d þDag a1 g þ Ddd d þ Ddg dg þDgg g2 hh0 ½3dð3u21 v21 Þ3gv21 þ u1 ðhþ h0 Þð7u21 3v21 Þ 12 7u2 i gd i1 i2 u1 s6 ¼ 4w v21 1 ; s7 ¼ 2 3 2
s5 ¼
2 2 u v 1 1 þ u1 a1 v1 b1 ; s8 ¼ ðhh0 Þ 1 1 þ 2 q r
s10 ¼
þDvv v21 þ Dvb v1 b1 þ Dbb b21 þ Duuu u31 þDuua u21 a1 þ Duud u21 d ð23Þ
þDuug u21 g þDuaa u1 a21 þ Duad u0 a0 d þ Duag u0 a0 g þ Dudd u0 d
þDudg u0 d þDugg u1 þDuvv u1 v21 þDuvb u1 v1 b1 þ Dubb u1 b21 2 þDaaa a30 þ Daad a20 d þ Daag a20 þ Dadd a0 d þ Dadg a1 d þ Dagg a1 2 3 2 þDavv a1 v21 þ Davb a1 v1 b1 þ Dabb a1 b21 þ Dddd d þ Dddg d 2 2 2 3 þDdgg d þ Ddvv dv0 þ Ddvb dv0 b0 þDdbb db0 þ Dggg þ Dgvv v21 þDgvb v1 b1 þ Dgbb b21 ð34Þ
g
ð24Þ
2
g
2
g
g
g
g
h0 ðwÞ 2 2 s9 ¼ ðv1 u1 Þ 2 ð25Þ
u21 v21 1 1 1 2 ðhwi1 Þ 2 þ 2 þ ½i1 ð4a21 4b21 5d þ 2dg þ 3g2 Þ 2 8 r q
þ 4u1 i2 ðg5dÞ20u21 i3
ð26Þ
g g
g
g
g
u02 ¼ D0u u1 þ D0a a1 þ D0d d þ D0g g þ D0uu u21 þ D0ua u1 a1 þ D0ud u1 d þ D0ug u1 g 2
þD0aa a21 þ D0ad a1 d þ D0ag a1 g þ D0dd d þ D0dg dg þ D0gg g2 þ D0vv v21 þD0vb v1 b1 þ D0bb b21 þD0uuu u31 þ D0uua u21 a1 þD0uud u21 d þ D0uug u21 2 þD0uaa u1 a21 þD0uad u1 a1 d þ D0uag u1 a1 þ D0udd u1 d þD0udg u1 d
g
1 ½6dðhh0 Þðu20 v20 Þ þ u0 ðh2 h20 Þð7u21 9v21 Þ 12q v2 u2 hh0 þ 1 1 ½ðd þ gÞðhh0 Þ þu1 ðh2 h20 Þ þ ½2u0 a0 ðd þ u0 ðhþ h0 Þ 4r 2
s11 ¼
v0 b0 ðd þ g þ u0 ðhh0 ÞÞ
i4w u1 ð3v21 7u21 Þ; 6 0 h ðwÞ s13 ¼ ½3dð2u21 v21 Þ3gv21 þ 2hu1 ð7u21 3v21 Þ 12
ð27Þ
g
g
þD0ugg u1 g2 þ D0uvv u1 v21 þD0uvb u1 v1 b1 þ D0ubb u1 b21 þD0aaa a31 2
þD0aad a21 d þD0aag a21 g þ D0add a1 d þD0adg a1 dg þD0agg a1 g2 3 2 þD0avv a1 v21 þ D0avb a1 v1 b1 þ D0abb a1 b21 þ D0ddd d þ D0ddg d þD0dgg d 2 þ D0dvv dv21 þ D0dvb dv1 b1 þD0dbb db21 þ D0ggg 3 þ D0gvv þD0gvb v1 b1 þ D0gbb b21
g
g g
g
g
gv1 ð35Þ
v2 ¼ Dv v1 þDb b1 þ Dvd v1 d þDvg v1 g þ Dvu v1 u1 þ Dva v1 a1 þ Dbd b1 d
s12 ¼
þDbg b1 g þ Dbu b1 u1 þ Dba b1 a1 þ Dvvv v31 þ Dvvb v21 b1 þ Dvdu v1 du1 ð28Þ
þDvda v1 da1 þ Dvgu v1 gu1 þ Dvga v1 ga1 þ Dvuu v1 u21 þ Dvua v1 u1 a1 2
þDvaa v1 a21 þ Dbbb b31 þDbdd b1 d þDbdg b1 dg þ Dbgg b1 g2 1 1 þ v1 a1 þ u1 b1 ; s14 ¼ ðh0 wi1 Þ u1 v1 þ q r 1 1 s15 ¼ u1 v1 ð2j1 i1 wÞ 2 þ 2 a1 b1 j1 r q
þDbdu b1 du1 þ Dbda b1 da1 þ Dbgu b1 gu1 þDbga b1 ga1 þ Dbuu b1 u21 þDbua b1 u1 a1 þ Dbaa b1 a21 ð29Þ
ð36Þ
v02 ¼ D0v v1 þD0b b1 þ D0vd v1 d þD0vg v1 g þ D0vu v1 u1 þ D0va v1 a1 þ D0bd b1 d þD0bg b1 g þ D0bu b1 u1 þ D0ba b1 a1 þ D0vvv v31 þ D0vvb v21 b1 þ D0vdu v1 du1
1 1 s16 ¼ ðh0 hÞ u1 v1 þ a1 v1 þb1 u1 ; q r 1 1 s17 ¼ u1 v1 2 þ 2 ði1 hwÞa1 b1 i1 r q
þD0vda v1 da1 þ D0vgu v1 gu1 þ D0vga v1 ga1 þ D0vuu v1 u21 þ D0vua v1 u1 a1
2
ð30Þ
þD0vaa v1 a21 þ D0bbb b31 þD0bdd b1 d þD0bdg b1 dg þ D0bgg b1 g2 þD0bdu b1 du1 þ D0bda b1 da1 þ D0bgu b1 gu1 þD0bga b1 ga1 þ D0buu b1 u21 þD0bua b1 u1 a1 þ D0baa b1 a21
ð37Þ
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Here Di, Dj0 , Dij, Dij0 , Dijk, Dijk0 are transfer matrix elements, detailed below. First order matrix elements of the u row: Du ¼ 1J2i ;
Da ¼ d;
Dd ¼
J1i ; 2
Dg ¼
J1i 2
ð38Þ
D0bu ¼ ðI1i dÞ
First order matrix elements of the a row: D0a ¼ 1I2i d þ J2i ;
D0d ¼
I1i ; 2
D0g ¼
I1i 2
ð39Þ
Second order matrix elements of the u row: 1 1 1 1 ; Dua ¼ I1i d; Duu ¼ þ I1i 2 qi ri 1 1 1 ; Dvb ¼ dI1i þ 1I1i Dvv ¼ 2 qi ri
"
1 1 1 þ ðI1i dÞ 2 qi ri 1 1 D0bb ¼ ¼ ðI1i dÞ þ qi ri
D0vv ¼
!
57J4i ; 6
Duua ¼
D0vb
Duvb ¼ J4i d2K4i ;
1 Davv ¼ ðJ4i d2K4i Þ; 2
1 ; 2
Dvuu ¼
J4i ; 2
Dbuu ¼
J4i d2K4i 2 ð47Þ
J4i I4i d2 ; 2
D0vdu ¼
1 2 1 ; þ 2 qi ri
1 0 1 3 1 I D ¼ ; D0vuu ¼ þ þ 4i ; 2 2ri vga 2 2qi ri I dJ þ4 4i 4i D0bdu ¼ 1; D0buu ¼ 2
Dvju ¼
ð48Þ
0
0
0
ð49Þ
J3i ¼
Duvv ¼
1 Ddvv ¼ ; 4
2K4i J4i d ; 2
ð41Þ J1i ¼
Duud ¼
D0vvb ¼
0
Z d Z
w
0
0
Z d Z
J4i 1 ; 2
1 Dgvv ¼ 4 ð42Þ
J4i ¼
w
0
0
Z d Z
w
0
0
1 I 2 1i
7 ð2K4i J4i dÞ; 2
Dvvb ¼
I4i ; 2
1 ; 2 3 D0vua ¼ ; 2
Third order matrix elements of the u row: Duuu ¼
ð46Þ
Vanishing matrix elements were omitted. The following integrals are involved: Z d Z d Z d Z d 2 I1i ¼ h dw; I2i ¼ h2 dw; I3i ¼ h3 dw; I4i ¼ h0 dw
#
1 1 þ 2 I3i ; ri q2i
I4i ; 2
D0vda ¼
Second order matrix elements of the a row: " ! # 1 1 1 1 1 þ ðI1i dÞ 2 þ 2 þ6I3i ; D0uu ¼ 2 qi ri ri qi 1 1 5 1 ; D0ud ¼ I2i ; D0ug ¼ I2i ; D0ua ¼ ðdI1i Þ þ qi ri 2 2 I1i 5 I1i 3 0 0 0 0 Daa ¼ ; Ddd ¼ I1i ; Ddg ¼ ; Dgg ¼ I1i 2 4 8 8
D0ba ¼ I1i 2d
Third order matrix elements of the b row: D0vvv ¼
ð40Þ
1 1 1; þ qi ri
Third order matrix elements of the v row: Dvvv ¼
D0u ¼ I2i ;
h dw1 dw;
J2i ¼
Z d Z
w
0
0
h2 dw1 dw;
h3 dw1 dw
ð50Þ
2 h0 dw1 dw;
K4i ¼
Z
d
Z
0
w
Z
0
!
w1
2
h0 dw2
! dw1
dw
0
ð51Þ The quantities qi and ri are defined by the relations qi = (Rei/R) and ri = (Rmi/R) where Rei and Rmi are the entry boundary radius of the electric and of the magnetic field respectively.
Third order matrix elements of the a row: 5 1 7 7 1 I ; D0uua ¼ ðJ4i I4i dÞ ; 6qi 2ri 6 4i 2 2 1 2 1 1 1 2 1 0 0 0 ; Duug ¼ þ ; Duvv ¼ þ þ I4i ; Duud ¼ 4 qi ri 4ri 2 qi ri I dJ4i 1 2 1 Duvb ¼ dI4i J4i ; D0avv ¼ 4i ; ; D0dvv ¼ þ 2 4 qi ri 1 D0gvv ¼ ð43Þ 4ri
D0uuu ¼
3. Exit fringing fields The formulas obtained for the entry fringing fields can be used directly to describe the ion trajectories leaving the Wien filter by accounting that there h(0) = 1 and h(d)=0, while the boundary curvature radii must change sign for both electric and magnetic fields, as curvature centre changes position with respect to the coordinate system. The transfer matrix elements become: First order matrix elements of the u row:
First order matrix elements of the v and b rows: Dv ¼ 1;
Db ¼ d;
D0b ¼ 1
ð44Þ
Second order matrix elements of the v row: 1 1 ; Dva ¼ I1i ; Dbu ¼ I1i þ Dvu ¼ I1i qi ri Second order matrix elements of the b row: D0vd ¼
I1i I2i ; "2
D0vg ¼
I2i I1i ; 2
Da ¼ d;
Dd ¼
J1f ; 2
Dg ¼
J1f 2
ð52Þ
First order matrix elements of the a row: ð45Þ D0u ¼ I2f ;
D0a ¼ 1I2f d þ J2f ;
D0d ¼
I1f ; 2
D0g ¼
I1f 2
ð53Þ
Second order matrix elements of the u row: !
#
1 1 1 1 þ þ ðdI1i Þ 2 þ 2 þ I3i I2i ; qi ri ri qi 1 1 D0va ¼ ðI1i dÞ 1; þ qi ri
D0vu ¼
Du ¼ 1j2f ;
" 1 ðdI1f Þ 2 " 1 Dvv ¼ ðI dÞ 2 1f Duu ¼
1 1 þ rf qf
!
# þ1 ;
Dua ¼ I1f ;
! # 1 1 1 ; þ rf qf
Dvb ¼ I1f
ð54Þ
ARTICLE IN PRESS D. Ioanoviciu, K. Tsuno / Nuclear Instruments and Methods in Physics Research A 614 (2010) 47–56
Second order matrix elements of the a row: " ! # 1 1 1 1 1 D0uu ¼ I þ I1f þ þ 6I 2f 3f ; rf 2 qf r2f q2f ! 1 1 5 1 ; D0ud ¼ I2f D0ug ¼ I2f ; þ D0ua ¼ I1f rf qf 2 2
I1f ¼
I4f ¼
J4f ¼
7 1 ð2K4f J4f dÞ; Duud ¼ ; 2 2 J4f 1 ; Duvb ¼ J4f d2K4f ; Davv ¼ ðJ4f d2K4f Þ; Duvv ¼ 2 2 1 1 Ddvv ¼ ; Dgvv ¼ 4 4
ð55Þ
Duua ¼
ð56Þ
D0uud ¼
1 7 I ; 3qf 6 4f
! 1 2 1 ; þ rf 4 qf
D0uvb ¼ dI4f J4f ; D0gvv ¼
D0uua ¼
7 1 ðJ I dÞ þ ; 2 4f 4f 2
D0uug ¼
D0avv ¼
1 1 þI4f ; 2 qf ! 1 2 1 0 ; Ddvv ¼ þ rf 4 qf
1 ; 4rf
I4f dJ4f ; 2
D0uvv ¼
1 4rf
D0vg ¼
I2f ; 2 !
1 1 ; þ rf qf
D0va ¼ 1I1f
J4f Dvvv ¼ ; 2
ð59Þ
1 1 1 1 þ I1f þ 2 I3f ; qf rf rf q2f ! 1 1 D0bu ¼ 1I1f ; D0ba ¼ I1f þ ð60Þ rf qf D0vu ¼
2K4f J4f d Dvvb ¼ ; 2
D0vua ¼
1 ; 2
D0vvb ¼ Dvgu ¼
J4f I4f d1 ; 2
1 ; 2rf
D0bdu ¼ 1;
I3f ¼
0
Z d Z
w
0
0
Z dZ 0
w 0
h21d dw dw;
Z
d
h3 dw
ð63Þ
0
h dw1 dw;
J2f ¼
Z d Z
w
0
0
h2 dw1 dw
K4f ¼
Z
d
Z
0
w
Z
0
w1
! 2
h0 dw2
! dw1
dw
0
4. Main field matrix
Vx2
o
sinot þ
1 a ð Vz2 Þð1cosotÞ
o o
ð66Þ ð67Þ
J4f ; Dvuu ¼ 2
1 D0vga ¼ ; 2
J4f d2K4f Dbuu ¼ 2 ð61Þ
D0buu ¼
1 2 1 ; þ rf 2 qf
Dvuu ¼
I4f dJ4f 1 2
ð1cosotÞ
1 a
I4f 1 ; þ 2 2qi ð62Þ
Not mentioned matrix elements were omitted because they are vanishing. The integrals involved in this transformation are:
a
ð þ Vz2 Þsinot þ t ð68Þ o o o Here a=eE0/m and o =eB0/m, index 2 means quantity at plane 2.
o
The involved velocity components are: V2y ¼ V2 sinb2 ;
V2z ¼ V2 cosa2 cosb2
ð69Þ
while mV22 ¼ e½j0 ð1 þ dÞ þ E0 x ð70Þ 2 The reference particle following the optical axis satisfies the conditions: R¼
!
D0vdu ¼
Vx2
V2x ¼ V2 sina2 cosb2 ; !
Third order matrix elements of the b row: I4f ; 2
h2 dw;
The quantities qf and rf are defined by the expressions qf = (Ref/ R) and rf =(Rmf/R) where Ref and Rmf are the exit boundary radii of the electric and magnetic field, respectively. If into the Eqs. (34)–(37), we substitute the index 2 by 4, index 1 by 3 and the matrix elements (52)–(62) then we connect, at the plane 4 the quantities u4 and v4, a4 = tan a4, b4 = tan b4 to those at plane 3 u3, v3, a3 and b3.
z¼
Third order matrix elements of the v row:
1 D0vda ¼ ; 2
0
J1f ¼
d
ð58Þ
Second order matrix elements of the b row: I2f ; 2
2
h0 dw;
Z
y ¼ y2 þ Vy2 t
D0b ¼ 1
Second order matrix elements of the v row: ! 1 1 ; Dva ¼ dI1f Dbu ¼ dI1f Dvu ¼ ðI1f dÞ þ rf qf
D0vvv ¼
d
x ¼ x2 þ
Db ¼ d;
D0vd ¼
Z
ð57Þ
First order matrix elements of the v and b rows: Dv ¼ 1;
0
I2f ¼
The third order transfer matrices elements of the main field part of a Wien filter can be calculated by integrating the Eqs. (10) and (11) after the substitutions h= 1, h0 =h00 =0. A straightforward way to obtain the third order matrix elements, the only way to fit them into the transfer matrix calculation trend, is to take advantage of the exact equations of motion of the charged particles inside the homogeneous magnetic and electric fields [32]. Accounting for the initial coordinates at the level of the plane where the homogeneous fields begin, index ‘‘2’’, those equations become:
Third order matrix elements of the a row: D0uuu ¼
h dw;
ð65Þ
Third order matrix elements of the u row: 7J4f þ 2 ; 6
d
ð64Þ
I1f I1f 5 D0aa ¼ ; Ddd ¼ I1f ; D0dg ¼ ; 2 4 8 " ! # 3 1 1 1 1 1 0 0 þ I þ I1f þ I Dgg ¼ I1f ; Dvv ¼ 2f 3f ; 8 2 qf rf r2f q2f ! 1 1 1 ; D0bb ¼ I1f þ D0vb ¼ I1f rf qf 2
Duuu ¼
Z
53
m0 V20 2j0 ¼ eB0 E0
ð71Þ
and arrives to the plane 3 at t0 ¼
W
o0
with
o0 ¼
eB0 m0
ð72Þ
V20 is the reference particle velocity along the Oz axis, while ej0 is its energy in the field-free space. In the xyz coordinate system where the Eqs. (66)–(68) were written, the electric and magnetic deviating forces cancel each other if E0 and B0 are both directed in the positive sense of the irrespective axes. We use again dimensionless coordinates defined here by: u=x/ R, v=y/R and w=z/R. To find u and v in a third order approximation we need t in second order approximation as it enters in terms already small of first order. Some charged particle arrives after t=t0(1+ t) at the distance WR, at the plane 3. We put t and W in Eq. (68) and t is calculated in a second order approximation.
ARTICLE IN PRESS 54
D. Ioanoviciu, K. Tsuno / Nuclear Instruments and Methods in Physics Research A 614 (2010) 47–56
After substitution in the expressions of u and v, to return to a2 and b2 we use the relations a2 =a2 a32/3 and b2 =b2 b32/3. The expressions for a and b result by derivation of u and v with respect to W. The matrix elements of the transformation from u2, a2, v2, b2 to u3, a3, v3, b3, from plane 2 to plane 3, including the main part of the Wien filter are: First order matrix elements of the u row: c1 1c ; Dg ¼ 2 2 First order matrix elements of the a row:
Du ¼ c;
Da ¼ s;
Dd ¼
s s D0d ¼ ; D0g ¼ 2 2 Second order matrix elements of the u row:
D0u ¼ s;
D0a ¼ c;
ð1cÞð2c þ 3Þ ðc1Þð2c þ 3Þ ; Dua ¼ 2sð1cÞ; Dud ¼ 2 2 2sW þ ðc1Þð2c þ 3Þ ðc1Þð2c1Þ ; Daa ¼ ; Dug ¼ 2 2 ð1cÞð2c þ3Þ Dad ¼ sð1cÞDag ¼ cðsWÞ; Ddd ; 8 2sW þðc1Þð2c þ 3Þ 4sW þ ðc1Þð2c þ 3Þ Ddg ¼ ; Dgg ¼ ; 4 8 1c Dbb ¼ 2
ð73Þ
ð74Þ
Duu ¼
ð75Þ
Second order matrix elements of the a row: sð4c þ1Þ sð4c þ 1Þ ; D0ua ¼ 2ð1cÞð2c þ1Þ; D0ud ¼ ; 2 2 sð14cÞ þ 2cW sð34cÞ ; D0aa ¼ ; D0ad ¼ ð1cÞð2c þ 1Þ; Dug ¼ 2 2 sð4c þ 1Þ D0ag ¼ ðc1Þð2c þ 1Þ þ sW; D0dd ¼ ; 8 sð14cÞ þ2cW sð4c3Þ4cW s D0dg ¼ ; D0gg ¼ ; D0bb ¼ ð76Þ 4 8 2 D0uu ¼
Third order matrix elements of the u row: Duuu ¼ Duud ¼ Duug ¼ Duaa ¼ Duag ¼
ðc1Þð3c2 þ 5c þ 3Þ 3sðc1Þð3c þ 1Þ ; Duua ¼ ; 2 2 3ðc1Þð3c2 þ 5c þ 3Þ ; 4 2 ð1cÞð9c þ11c þ3Þ2Wsð4c þ 1Þ ; 4 2 ð1cÞð9c 5c5Þ 3sðc1Þð3c þ 1Þ ; Duad ¼ ; 2 2 c1 3ðc1Þð3c2 þ5c þ3Þ ½4Wð2c þ1Þ3sð3c þ1Þ; Dudd ¼ ; 2 8
Ddbb ¼
ð1cÞð2c þ 3Þ2Ws 4 Third order matrix elements of the a row:
Dgbb ¼
sð9c2 þ 4c2Þ 3ðc1Þð9c2 þ 5c2Þ ; D0uua ¼ ; 2 2 2 2 3sð9c þ 4c2Þ sð27c 4c10Þ2Wð8c2 þ c4Þ D0uud ¼ ; D0uug ¼ ; 4 4
D0uuu ¼
scð27c28Þ 3ðc1Þð9c2 þ 5c2Þ ; D0uad ¼ 2 2 ð1cÞð27c2 þ 7c10Þ4Wsð4c1Þ 0 ; Duag ¼ 2 2 3sð9c þ 4c2Þ ; D0udd ¼ 8 2 2 sð27c 4c10Þ2Wð8c þc4Þ ; D0udg ¼ 4 2 2 sð27c þ 20c þ6Þ þ4sW þ 4Wð8c2 3c4Þ D0ugg ¼ 8 sð14cÞ 3ðc1Þ2 ð3c þ 2Þ 0 0 ; Daaa ¼ ; Dubb ¼ 2 2 scð27c28Þ scð2827cÞ þ 2Wð8c2 3c4Þ ; D0aag ¼ D0aad ¼ 4 4 3ðc1Þð9c2 þ 5c2Þ ; D0add ¼ 8 ðc1Þð27c2 þ 7c10Þ þ 4sWð4c1Þ D0adg ¼ 4 D0uaa ¼
ðc1Þð9c þ 11c þ 3Þ þ 2Wsð4c þ 1Þ ; 4 2 2 ðc1Þð9c þ7c3Þ4W c þ4Wsð4c1Þ Dugg ¼ 8 ðc1Þð2c þ1Þ 3sðc1Þ2 ; ; Daaa ¼ Dubb ¼ 2 2 ð1cÞð9c2 5c5Þ ðc1Þð9c2 c7Þ þ 2sWð4c3Þ Daad ¼ ; Daag ¼ ; 4 4
Dadd ¼
3sðc1Þð3c þ 1Þ ; 8
Dadg ¼
1c ½3sð3c þ 1Þ4Wð2c þ1Þ 4
ð77Þ
ð78Þ
ðc1Þð27c2 c14Þ4cW 2 þ 16sWð2c1Þ ; 8 2 sð9c þ 4c2Þ ; D0abb ¼ ðc1Þð2c þ1Þ; D0ddd ¼ 16 2 2 sð27c þ 4c10Þ2Wð8c þ c4Þ D0ddg ¼ 16 sð27c2 20c6Þ þ 4sW 2 þ 4Wð8c2 3c4Þ ; D0dgg ¼ 16 2 2 sð14cÞ sð9c 12c þ 2Þ4sW 2Wð8c2 7c4Þ ; D0ggg ¼ ; D0dbb ¼ 4 16
D0agg ¼
sð4c1Þ2cW 4 First order matrix elements of v and b rows:
Dg0 bb ¼
Dv ¼ 1;
Db ¼ W;
D0b ¼ 1
ð79Þ
Second order matrix elements of the v row:
2
Dudg ¼
ðc1Þð2c þ 1Þ ð1cÞð3c2 þ c3Þ þ 2sWð34cÞ þ 4cW 2 ; Dggg ¼ 16 4
Dbd ¼
Ws ; 2
Dbg ¼
sW ; 2
Dbu ¼ Ws;
Dba ¼ c1
ð80Þ
Second order matrix elements of the b row: D0bd ¼
1c ; 2
D0bg ¼
c1 ; 2
D0bu ¼ 1c;
D0ba ¼ s
ð81Þ
Third order matrix elements of the v row: sW sð2c1ÞW ðWsÞð2c1Þ ; Dbdd ¼ ; Dbdg ¼ ; 2 8 4 sð2c1ÞW ð1cÞð2c1Þ Dbdu ¼ ; Dbda ¼ ; 2 2 Wð4c3Þ þ sð12cÞ ðWsÞð2c1Þ Dbgg ¼ ; Dbgu ¼ 8 2 ðc1Þð2c þ 1Þ þ2Ws sð2c1ÞW Dbga ¼ ; Dbuu ¼ ; 2 2 sð2c3Þ þ W Dbua ¼ ð1cÞð2c1Þ; Dbaa ¼ 2 Dbbb ¼
3sð1cÞð3c þ1Þ þ4sW 2 þ 8Wðc1Þð2c þ1Þ ; 8 2 ðc1Þð3c þ 5c þ3Þ ; Dabb ¼ sðc1ÞDddd ¼ 16 2 ð1cÞð9c þ 11c þ 3Þ2Wsð4c þ1Þ ; Dddg ¼ 16 2 2 ðc1Þð9c þ 7c3Þ4cW þ 4sWð4c1Þ ; Ddgg ¼ 16
Dagg ¼
ð82Þ
ARTICLE IN PRESS D. Ioanoviciu, K. Tsuno / Nuclear Instruments and Methods in Physics Research A 614 (2010) 47–56
To operate with a large angular acceptance, second order angular focusing in the mass dispersion plane can be desired. This goal is reached by cancelling the second order aberration coefficient Aaa =0. To cancel this coefficient the Wien filter magnetic boundaries at both its ends should be concavely curved by a radius of 0.49324 m. Then the most significant second and order aberration coefficients are: Aad =33.0796 m Add = 0.06043 m. Accounting for a target of xs = 1.5 mm width, opening angle 11, as = tan(11) and energy spread of d = 0.2% the second order resolution is RII =780.756. If in the calculations we introduce the contribution generated by the matrix elements of third order of the Wien filter only, we obtain a third order angular aberration coefficient Aaaa = 2093.04. By accounting for the absolute value of this coefficient in the resolution formula the resolution drops to RIII = 97.085. By limiting the radial opening angle to 0.51, other conditions kept unchanged the second order resolution RII = 1226.18 is more than twice overestimated with respect to that of third order as RIII = 516.429. The resolution has been calculated by using the following formula:
Third order matrix elements of the b row: c1 D0bbb ¼ ; 2
ðc1Þð4c þ 3Þ D0bdd ¼ ; 8
ð1cÞð4c þ1Þ2sW D0bdg ¼ ; 4
ðc1Þð4c1Þ þ 4sW ðc1Þð4c þ 3Þ ; D0bdu ¼ ; 8 2 sð4c3Þ ð1cÞð4c þ 1Þ2sW D0bda ¼ ; D0bgu ¼ ; 2 2 sð34cÞ þ 2cW ðc1Þð4c þ3Þ D0bga ¼ ; D0buu ¼ ; D0bua ¼ sð4c3Þ; 2 2 D0bgg ¼
D0baa ¼
ð1cÞð4c þ 1Þ 2
55
ð83Þ
Here WR is the length of the homogeneous part of the Wien filter, s= sin(W) and c= cos(W). 5. Application to a specific spectrometer configuration First example: The resolution reduction due the third order angular aberration induced by the Wien filter alone in a second order focusing instrument has been estimated. The main geometric and ion optical parameters are close to those of the spectrometer VAMOS [16] as detailed in [33]. We considered the instrument with the specific parameters defined in Table 1. We assume that the first quadrupole defocuses in the deflexion (horizontal) plane of the instrument while the second exerts a focusing action in that plane. The effect of the elliptic aperture on the focusing action of the second quadrupole was roughly estimated as being proportional to the inverse of the aperture i.e. the arguments of the trigonometric functions in the y directions being O10 times greater that in the horizontal plane. From first order calculations targeting simultaneous x, y and energy focusing at the detector, the following quantities resulted: R inside the Wien filter 1.285794 m, x1 =k1Z1 =1.184464 trigonometric function argument for the first quadrupole, x2 =k2Z2 for the second quadrupole in the x plane. The ki coefficient is defined [34] by the relation: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B e ð84Þ k2i ¼ Ti G0i 2m0 j0
R¼
Dg
ð85Þ
2 jMx jxs þ jAaa ja2s þ 2jAad jas d þjAdd jd þ 2jAaaa ja3s
Second example: The aberration coefficients of a homogeneous main fields Wien filter of length WR, located between two fieldfree spaces, both of 1.92 WR length were computed with the TRIO code combining independent fringing fields of an electrostatic sector and of a magnetic sector field in Ref. [35]. Some of those aberration coefficients, expressed in filter lengths Ajk, Ajkl were reproduced in Table 2 and compared with those calculated with the formulae derived in this paper, expressed in the same length units. Third example: The importance of the fourth order trajectory coefficients can be evaluated for a Wien filter having both source and detector located inside the homogeneous fields (no fringing fields), focusing after W= p, i.e. c= 1 and s= 0. The trajectory coefficients at the image are given below. They resulted from the formulae (66) and (68) processed to account for small quantities up to fourth order.
where BTi, G0i are the magnetic field at the pole tip and half the quadrupole aperture, respectively. Zi is the effective length of the quadrupole. Index ‘‘1’’ and ‘‘2’’ refers to the first and second quadrupole, respectively. The horizontal magnification is then Mx = 0.579717 while the vertical one is 5.26436, with a mass dispersion coefficient of Dg =2.48204 m
uimage ¼ u0 d þ u20 þu0 d þ 3a20 þ
d2 4
3 3 2 u30 u20 d þ9u0 a20 u0 d 2 4
3
þ
9 5 5 15 2 2 15 2 2 15 2 d þ u40 þ u30 d þ a0 d u a þ u d þ u0 a20 d 8 2 4 2 2 0 0 8 0 2
þ
5 15 2 2 5 4 3 97 4 d u0 d a þ a d þ 8 12 0 8 0 64
ð86Þ
Table 1 Some geometric and electro-magnetic parameters of the VAMOS spectrometer. Interaction region first quadrupole distance
Inter-quadrupole distance
Quadrupole Wien filter distance
0.4 m Wien filter length
0.3 m Magnet deflection radius 1.5 m
1m
Wien filter magnet distance
Magnet detector distance
First quadrupole length
Second quadrupole length
0.3 m 0.7 m Magnet deflection angle Beam entry angle
1.5 m Beam exit angle
601
301
0.6 m Entry boundary radius 3m
0.9 m Exit boundary radius 5m
201
Table 2 Aberration coefficients of a symmetrically located Wien filter of length WR between field-free spaces of 1.92 WR. Aberration coefficient
Aaa
Aad
Add
Aaaa
Aaad
Aadd
Addd
Calculated Computed by TRIO+
23.576 25.38
8.148 8.42
0.882 0.92
99.183 623.5
41.158 45.1
6.347 17.8
0.572 0.1
Aberration coefficients in WR units.
ARTICLE IN PRESS 56
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6. Conclusions The ion trajectories in Wien filters with main homogeneous electric and magnetic fields were determined, in a third order approximation both in the plane of mass dispersion (x direction) and normally to it (y direction). The calculated matrix elements account for the ion position, trajectory slope, ion mass and energy. The analytic expressions of the matrix elements are given with second order accuracy, those of first order, with first order accuracy those of second order and containing no small quantities, those of third order. In this way a third order accuracy is kept over all the contributions to the ion trajectory calculation. The analytic expressions of the transfer matrix elements keep manageable extent being ready to be used in the often encountered class of homogeneous Wien filters. Besides the main fields, transfer matrices were derived for the entry and exit fringing fields with effect of magnetic and electric field boundary curvatures included. As an application, the effect of the third order angular aberration of the Wien filter on the resolution of a spectrometer has been calculated for vanishing second order radial angular aberration. References [1] D. Ioanoviciu, C. Cuna, Int. J. Mass Spectrom. Ion Phys. 25 (1977) 117. [2] I. Wahlin, Nucl. Instr. and Meth. 27 (1964) 55. [3] R.G. Wilson, In: A. Septier (Ed.), Applied Charged Particle Optics, Academic Press, New York, London, Toronto, Sydney, San Francisco, 1980, pp. 45–71. [4] S. Taya, K. Tokiguchi, I. Kanomata, H. Matsuda, Nucl. Instr. and Meth. 150 (1978) 165. [5] D. Ioanoviciu, C. Cuna, Rapid Commun. Mass Spectrom. 9 (1995) 512. [6] Y. Kawanami, T. Ishitami, K. Umemura, Nucl. Instr. and Meth. B 37/38 (1983) 240. [7] T. Shiokawa, P. Hyon Kim, K. Toyoda, S. Namba, J. Vac. Sci. Technol. B 1 (4) (1983) 1117. [8] D. Rogalla, M. Aliotta, C.A. Barnes, L. Campajola, A. D’Onofrio, L. Gialanella, U. Greife, G. Imbriani, A. Ordine, V. Roca, C. Rolfs, M. Romano, C. Sabbarese,
[9]
[10] [11] [12] [13] [14] [15] [16] [17] [18]
[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
D. Schuerrmann, F. Schuemann, F. Strieder, S. Theis, F. Terrasi, H.P. Trautvetter, Nucl. Phys. A 688 (2001) 549c. D. Rogalla, M. Aliotta, C.A. Barnes, L. Campajola, A. D’Onofrio, L. Gialanella, U. Greife, G. Imbriani, A. Ordine, j. Ossmann, V. Roca, C. Rolfs, M. Romano, C. Sabbarese, D. Schuerrmann, F. Schuemann, F. Strieder, S. Theis, F. Terrasi, H.P. Trautvetter, Eur. Phys. J. A 6 (1999) 471. M. Berz, Nucl. Instr. and Meth. A 298 (1990) 473. M. Couder, C. Angulo, W. Galster, J.-S. Graulich, P. Leleux, P. Lipnik, G. Tabacaru, F. Vanderbist, Nucl. Instr. and Meth. A 506 (2003) 26. Mathcad 7 Professional Mathsoft Inc. 1996–97. GEANT Computing Network Division CERN Geneva Switzerland 1993. R. Anne, A.C. Mueller, Nucl. Instr. and Meth. B 70 (1992) 276. F. Meot, Nucl. Instr. and Meth. A 427 (1999) 353. S. Pullanhiotan, M. Rejmund, A. Navin, W. Mittig, S. Bhattacharyya, Nucl. Instr. and Meth. A 593 (2008) 343. M. Couder, G.P.A. Berg, J. Goerres, P.J. LeBlanc, L.O. Lamm, E. Stech, M. Wiescher, J. Hinnefeld, Nucl. Instr. and Meth. A 587 (2008) 35. D. Carey, K. L. Brown, F. Rothacker, Third Order TRANSPORT a computer program for designing charged particle beam transport systems, Stanford Linear Accelerator Center Report Nr. SLAC-R-95-462, 1995, 295 pp. H. Wollnik, G. Muenzenberg, H. Ewald, Nucl. Instr. and Meth. 111 (1973) 355. H. Nakabushi, T. Sakurai, H. Matsuda, Int. J. Mass Spectrom. Ion Phys. 52 (1983) 319. H. Nakabushi, T. Sakurai, H. Matsuda, Int. J. Mass Spectrom. Ion Phys. 55 (1983/1984) 291. D. Ioanoviciu, Int. J. Mass Spectrom. Ion Phys. 11 (1973) 169. X. Ji-ye, C. Ren-shu, Int. J. Mass Spectrom. Ion Phys. 44 (1982) 129. M. Toyoda, H. Hayashibara, T. Sakurai, H. Matsuo, Int. J. Mass Spectrom. Ion Processes 146/147 (1995) 195. H. Wollnik, B. Hartmann, M. Berz, AIP Conf. Proc. 177 (1988) 74. T. Matsuo, Mass Spectrosc. (Japan) 32 (1984) 319. M. Salomaa, H.A. Enge, Nucl. Instr. and Meth. 145 (1977) 279. H. Wollnik, Adv. Electron. Electron. Phys. Suppl. 13B (1980) 133. H. Matsuda, Nucl. Instr. and Meth. 91 (1971) 637. H. Matsuda, H. Wollnik, Nucl. Instr. and Meth. 77 (1970) 40. D. Ioanoviciu, K. Tsuno, G. Rodriguez, Rev. Sci. Instrum. 75 (2004) 4434. G.P. Barnard, Modern Mass Spectrometry, The Institute of Physics, London, 1953, p. 85 (Chapter II, Paragraph 19). /http://www.ganil.fr/vamos/design.htmlS. H. Wollnik, Optics of Charged Particles, Academic Press, Orlando, 1987 p. 51. T. Sakurai, M. Toyoda, H. Hayashibara, T. Matsuo, Int. J. Mass Spectrom. Ion Processes 146/147 (1995) 217.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Third order transfer matrices for real Wien filters with homogeneous main fields Damaschin Ioanoviciu a,b,, Katsushige Tsuno c a
Physics Faculty, Babes Bolyai University Str. M. Kogalniceanu Nr.1 400 084 Cluj-Napoca, Romania National Institute for Research-Development of Isotopic and Molecular Technologies P.O. Box 700, 400 293 Cluj-Napoca, Romania c Electron Optics Solutions Tsuno, 2-10-11, Mihori, Akishima, Tokyo 196-0001, Japan b
a r t i c l e in fo
abstract
Article history: Received 9 July 2009 Received in revised form 17 December 2009 Accepted 22 December 2009 Available online 4 January 2010
The determination of the ion optical aberrations of the Wien filters operated independently or in more complex mass analyzing or charged particle transport systems needs the calculation of the trajectories in a third order approximation. The trajectory calculations for crossed electric and magnetic fields generate unmanageably long formulas for the matrix elements. The reduction of the extent of the matrix elements to usable size can be reached by restricting calculations to less complex configurations as the homogeneous main fields Wien filters, which are probably the most often used crossed field velocity filters. Analytic expressions for transfer matrix elements for homogeneous main fields Wien filters were determined up to third order accuracy level. The elements of three pairs of matrices, those of entry fringing fields, those of the main fields and those of the exit fringing fields were derived for the plane of mass dispersion (radial) and for a direction normal to it (axial). The third order accuracy was kept over all the calculated contributions to the ion trajectory. The first order elements, to be multiplied by first order small quantities were calculated in a second order approximation, those of second order with a first order accuracy, while those of third order were not allowed to contain small quantities in their expressions. The matrix elements describing the ion position, trajectory slope, ion energy and mass were accounted for. In the fringing field matrix elements the effect of the electric and magnetic field boundary curvature radii were included. In a brief application, on one spectrometer geometry, the calculated matrices were used to determine the effect of third order angular aberration generated by the Wien filter alone on the resolution when the second order aberration has been cancelled. & 2009 Elsevier B.V. All rights reserved.
Keywords: Charged particle analyzer Wien filter Homogeneous fields Transfer matrices Third order aberrations
1. Introduction Among the crossed electric and magnetic fields mass analyzers, the velocity filter, called after its inventor Wien filter has the unique property of leaving to pass straight ions of a given velocity. The Wien filter has a wide area of applications, alone or included in more sophisticated ion-optical systems. Alone, the Wien filter produces a mass spectrum from a beam of defined energy ions. A mass spectrum at 4000 resolution was obtained [1], isotopes separated [2] and ions implanted [3] with Wien filters in single focusing arrangements. Double focusing mass spectrometers may incorporate Wien filters as that with oblique incidence [4] or with compact geometry [5]. The linearity
Corresponding author at: Physics Faculty, Babes Bolyai University Str. M. Kogalniceanu Nr.1 400 084 Cluj-Napoca, Romania. E-mail address: [email protected] (D. Ioanoviciu).
0168-9002/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2009.12.054
of the beam axis inside the Wien filters was capitalized in ion microscope [6] and ion beam lithography [7] systems. Wien filters were successfully included and operated in a variety of nuclear instruments as recoil separators and spectrometers. The European Recoil Separator for Nuclear Astrophysics (ERNA) in its configuration reported in Ref. [8], includes four Wien filters. Its ion optics, calculated in Ref. [9] resulted by using differential algebra based code COSY Infinity [10]. The Astrophysics Recoil Separator (ARES) [11] also uses a Wien filter in a configuration simulated by Mathcad [12] and GEANT [13] softwares. The LISE3 double Wien filter [14] has been constructed to improve the separation for exotic nuclei of the Ligne d’Ions Super Epluches (LISE) spectrometer, simulations being ensured by the ZGOUBY computer code [15]. Aberrations calculations were performed also for the VAriable MOde Spectrometer (VAMOS) [16] which operates a Wien filter in some of the spectrometer’s working regimes. The Strong Gradient Electromagnetic Online Recoil separator (St. George) design [17] includes a Wien filter
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with electrodes shaped to ensure main path charged particle trajectory linearity also through the fringing fields. Ion optical calculations for this separator were started at first order level with the TRANSPORT third order code [18], developed to fourth order aberrations level with COSY Infinity, only up to third order coefficients being retained as significant. Most of the mentioned applications of the Wien filters were achieved by homogeneous main field analyzers as those of Ref. [2–5,8,11,14,16,17] as well as most of those delivered by the companies Colutron and Danfysik. Often, the absence of additional mentions besides ‘‘Wien filter’’ means: ‘‘analyzer with homogeneous main fields’’. The conditions to ensure homogeneity of both electric and magnetic fields between parallel plane electrodes and pole faces were defined in [19]. Attempts were made to ensure field homogeneity by shimming and by shaping electrodes as well as pole pieces. The complexity of the description of the ion-optical properties of the crossed field analyzers increases with the complication of electrode and pole piece shape and structure creating the involved fields and with the level of approximation accepted for the trajectory description. The second order particle optical calculations allowed to obtain results which can be displayed in transfer matrix form for crossed fields with circular ion main path [20,21] and Wien filters [22]. These calculations included fringing field effects at the same level of accuracy. Third order calculations were made for very general configurations of crossed electric and magnetic mass analyzers with circular ion main path. In Ref. [23] the third order transfer matrix elements were expressed for both radial coordinate and slope in form of series. Due to the length of the expressions 171 elements of the development were omitted from the radial coordinate development and 172 from that of the slope. In Ref. [24] the general expressions for the crossed fields were disclosed only partially for the same reason, too lengthy formulas, checked by various mathematical methods. Coefficients missing from the paper are available on request. There is a way to obtain so high order matrix elements as third order, still having manageable length expressions by restricting the generality of the formulas to specific analyzer geometries of most frequent use. The ideal way to consider and design ion optical instruments is to calculate the total transfer matrix of the system, from the interaction region to the detector. If analytical expressions of the matrix elements of all instrument constituting ion optical parts are available, then this total transfer matrix can be obtained by successively multiplying the matrices of the components. Cancelling specific elements of the first row of the total matrix, angular and energy focusing can be ensured at some order. Each cancelled matrix element gives a focusing condition. The assembly of the focusing conditions result in a system of equations to be solved simultaneously. The unknowns are some electromagnetic and geometric parameters to be determined. The same parameters can be determined with numerical computer codes as well as by codes performing numerical matrix multiplications as GIOS [25], TRIO [26] or TRANSPORT for instance. Each set of numerical integrations or numerical matrix multiplications results in an individual charged particle trajectory with a numerical coordinate at the detector. Algorithms were developed, including constraint equations, to touch beam focusing, involving groups of trajectories. The analytical and numerical ion optical methods to design systems are complementary and they can be used to check each other. The complete third order ion-optical description of homogeneous main field Wien filters below, includes the calculation of
the transfer matrices, of the entry fringing field region, of the main fields and of the exit fringing field region. The third order treatment implies calculation of the first order matrix elements with a second order accuracy, of those of second order with first order accuracy, while for those of third order neglecting any small terms in their expressions. These transfer matrices allow to determine the ion optical properties of individual homogeneous Wien filters or of the systems including such filters deriving focusing conditions up to third order. They can be used to determine whole instrument aberrations analytic expressions or can be included in computer programs.
2. Entry fringing fields The fringing fields are assumed to be confined to a region of Rd length along the z-axis (Figs. 1 and 2), R being the main path ion trajectory radius if only the homogeneous electric or only the homogeneous magnetic field would act alone. In front of this region, at the entry boundary, plane 1, both the electric and magnetic fields vanish. At the end of the fringing field region, at the plane 2, both x electric field and y magnetic field components reach the main fields values E0 and B0, respectively. On the axis, between the entry point into the fringing fields and the main field region we assume that the two field components increase both according to a function h(z): Eð0; 0; zÞ Bð0; 0; zÞ ¼ ¼ hðzÞ E0 B0
ð1Þ
E0 is directed along Ox, while B0 along Oy. The motion proceeds along z in the coordinate system xzy. In the following derivations the electric and magnetic fringing field distribution shapes are assumed identical, fact that can be obtained by a careful design as reported in [17]. Many earlier Wien filters’ less satisfying performances can be explained by the differences between the two distributions as usually magnetic fringing fields extend farther (reason suggested already in [27]). The usual shape of the fringing fields distributions is given in graphical form as on Fig. 4 of [28] or Fig. 2 of [21]. These distribution shapes are not too much different of that given in Fig. 3, which represent the function h(z) for d= 0.02:
hðzÞ ¼
z3 ðdRÞ3
1015
z z2 þ6 dR ðdRÞ2
! ð2Þ
Here dR is the depth of the fringing field region. The fringing field effects were calculated with the help of the fringing field components of both electric and magnetic fields of [29,30]. The derivatives of the longitudinal distributions of nth order were treated as small quantities of n order. The volume of calculations was reduced accounting for the following reasons: The coefficients of the expressions to be integrated contain the initial quantities, at the level of plane ‘‘1’’, Figs. 1 and 2. At this level the particle initial coordinates are x1 = u1R (radial) and y1 =v1R (axial) while the initial direction is determined by a1 =a tan(a1) (radial) and b1 =a tan(b1) (axial) angles. For the complete definition of the trajectory parameters we need also d and g, the relative energy and mass difference related to the reference particle trajectory. The reference particle possesses a mass m0, a charge e (quantity considered as positive), an energy ej0 in the field-free space. The powers of the trajectory parameters u1, a1, v1, b1, d, g do not change by the integration process. For almost all real world Wien filters the depth of the
ARTICLE IN PRESS D. Ioanoviciu, K. Tsuno / Nuclear Instruments and Methods in Physics Research A 614 (2010) 47–56
49
Fig. 1. Projection of the charged particle trajectory onto the xOz plane, parallel to the main electric field E0. Definition of main trajectory geometric parameters at the reference plane where entry fringing fields begin (plane 1), where main homogeneous fields begin (plane 2), at the end of the main fields (plane 3), at the beginning of the field free region (plane 4).
Fig. 2. Projection of the charged particle trajectory onto the yOz plane, parallel to the main magnetic field B0. Definition of main trajectory geometric parameters at the reference plane where entry fringing fields begin to act (plane 1), at the main homogeneous fields beginning (plane 2), at the end of the homogeneous fields region (plane 3), at the beginning of the field free region (plane 4).
fringing field region is small compared to the radius R of the main path particle trajectory if the magnetic field B0 or the electric field E0 would act alone. If the integrand is small of wk, then the result of an integration over w= z/R will be a small quantity of wk + 1 order. As the accuracy of calculations is of third order, we retain in the final results, only second order powers or less (including negative values) for w in the expressions of trajectory parameters globally of first order, of first order or less in w for those of second order powers in those parameters and containing only w at negative powers for the third order coefficients in u1, a1, v1, b1, d and g. The expression of the electrostatic potential inside the fringing field region originates from that of Ref. [29] simplified for
homogeneous main field: 00 h ðzÞx3 h0 ðzÞxðx2 3y2 Þ h0 ðzÞxzðx2 3y2 Þ jðzÞ ¼ j0 ð1 þ dÞ þ E0 hx þ þ 2 6 6Re 6Re ð3Þ 00 h0 ðzÞy2 ðRe þ zÞ x2 ðR2e h ðzÞRe h0 ðzÞh0 ðzÞzÞ Ex ¼ E0 h 2R2e 2R2e
Ey ¼
E0 h0 ðzÞxyð1þ z=Re Þ Re
Ez ¼ E0 h0 ðzÞx
ð4Þ
ð5Þ
ð6Þ
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Fig. 3. Shape of the fringing field distribution h(w) extending over 0.02R at the entry of the Wien filter.
Ez is given at first order only, as the second order term is missing while in the differential equations this component multiplies first order small quantities. Re is the electric field effective boundary radius. The magnetostatic fringing field potential and field components (Ref. [30]) are: 00 h0 ðzÞyðRm þ zÞðy2 3x2 Þ h ðzÞy3 ð7Þ CðzÞ ¼ B0 hy þ 2 6 6Rm Bx ¼ B0
h0 ðzÞxyð1 þz=Rm Þ Rm
ð8Þ
h0 ðzÞx2 ð1 þ z=Rm Þ y2 h0 ðzÞ h0 ðzÞz 00 Bz ¼ B0 h0 ðzÞy By ¼ B0 h 2 h ðzÞ 2 2Rm Rm Rm
ð9Þ Rm being the magnetic field effective boundary radius. The differential equations of the ion trajectories inside the crossed fields of a Wien filter, the electric field directed along Ox, the magnetic field along Oy take the general form [31]:
j½x ð1þ y0 2 Þx0 y0 y þ 00
00
½Ex ð1þ y0 2 Þ þ x0 y0 Ey þ x0 Ez ð1þ x0 2 þ y0 2 Þ 2
rffiffiffiffiffiffiffi ej 2 2 ðy0 Bz þ By Þð1 þx0 þ y0 Þ3=2 ¼ 0 2m
ð10Þ
j½y ð1 þ x0 2 Þy0 x0 x þ 00
00
02
0 0
02
3 3h2 2 uð11d 2dgg2 Þ hh0 ðwÞuvv0 h2 uv0 2 8 2 h h0 ðwÞ w 00 þ u0 2 ð3gdÞ þ u0 v0 v þ v2 d 1þ 4 2q q 00 h0 ðwÞ w h ðwÞ h0 ðwÞ 0 ðd þ gÞ 1 þ ðd þ g Þ vv ðd þ gÞ þ r 4r 4 2 hv0 2 h 3 2 ðg3dÞ ð13Þ ð11d 3d g3dg2 5g3 Þ 4 16 d2 v 1 1 1 1 0 þ w h0 ðwÞuv0 h0 ðwÞu0 v ¼ h ðwÞuv þ þ q r dw2 q2 r2 2 1 2 1 þw 2 þ 2 þ hh0 ðwÞvuu0 þ hu0 v0 þ hh0 ðwÞvu2 q r r q w d w dþg h0 ðwÞ 0 0 þ þ 1þ þ h ðwÞvu 1þ vu ðd þ gÞ r 2r q q 2 0 0 2 2 0 0 0 0 0 0 00 ð14Þ þ 2hh ðwÞv u þ 2h v uu þh ðwÞv ud þ v u u þ hv0 u0 d þ
02
½Ey ð1 þ x Þ þ x y Ex þ y Ez ð1 þ x þ y Þ 2
rffiffiffiffiffiffiffi ej 0 ðx Bz Bx Þð1 þx0 2 þ y0 2 Þ3=2 ¼ 0 2m
0
form appropriate for successive approximation procedure: d2 u h u2 0 1 1 1 1 þh0 ðwÞw 2 þ 2 h ðwÞ ¼ ðgd2huÞ þ þ 2 2 2 q r r dw q 2 h uð5 d g Þ h 00 u0 2 þ 5h3 h ðwÞ h0 ðwÞuu0 þ 2 2 v2 0 1 1 1 1 00 þh0 ðwÞw 2 þ 2 h ðwÞ þ h0 ðwÞvv0 h ðwÞ þ 2 q r r q h 02 h hu3 7 1 2 2 h0 ðwÞ þ v þ ð5d 2dg3g Þ þ þ 2 2 8 3q r 00 7 1 7h ðwÞ þ þ h0 ðwÞw 11h3 þ 2hh0 ðwÞu2 u0 3 3q2 r2 dh0 ðwÞ w h0 ðwÞ w ðd þ gÞ þ u2 1þ 1þ r 2q q 4r 00 dh ðwÞ 3h3 h2 ð11dgÞ uu0 2 þ 4 2 2 2 3 1 3 1 00 0 0 huv 0 þh0 ðwÞw 2 þ 2 h ðwÞ h ðwÞ þ h ðwÞuu d þ 2 q r r q
ð11Þ
These trajectory differential equations are valid for non relativistic charged particles. On the filter axis rffiffiffiffiffiffiffiffiffiffiffiffi 2j0 m0 E0 ¼ ; B0 ¼ E0 ð12Þ R 2ej0 To compensate the effect of E0, oriented along the positive sense of the Ox axis, the applied B0 must be directed in the negative sense of Oy, of the xzy coordinate system. Accounting for these relations, substituting for variables x =uR, y=vR and z = wR, for Re = qR, Rm = rR, as well as for the potential and the field components we obtain the trajectory differential equations in a
Here the following symbols were introduced: dh=dw ¼ h0 ðwÞ;
00
d2 h=dw2 ¼ h ðwÞ
ð15Þ
We start the approximation process with the initial values of the ion parameters at the fringing field entry: u1, a1 = tan(a1), v1, b1 =tan(b1).
ARTICLE IN PRESS D. Ioanoviciu, K. Tsuno / Nuclear Instruments and Methods in Physics Research A 614 (2010) 47–56
For the next approximation we use the above third order equations. The first integration over w gives u0 and v0 , while the second integration gives u and v in the form shown below.
Z
Additionally we define the following symbols: Z w Z w Z w 2 h dw ¼ i1w h2 dw ¼ i2w h3 dw ¼ i3w h0 dw ¼ i4w
w 0
ufirst ¼ u1 þa1 w þ
6 X
si
ð16Þ
1
u0first ¼ a1 þ
13 X
0
Z w Z
0
w
w
0
h3 dw1 dw ¼ j2w
0
ð32Þ
0
ð17Þ Z
7
vfirst ¼ v1 þ b1 wþ s14 þ s15
ð18Þ
v0first ¼ b1 þ s16 þ s17
ð19Þ
w Z w 0
ð31Þ
0
Z w Z
h dw1 dw ¼ j1w 0 0 Z w Z w h2 dw1 dw ¼ j3w 0
si
51
02
h dw1 dw ¼ j4w
Z
0
w
Z
0
Z
w 0
w1
! 02
h dw2
! dw1
dw ¼ k4w
0
ð33Þ
Here the si coefficients have the meanings:
s1 ¼
j1 ðgdÞj2 u1 2
s2 ¼
hh0 2 2 ðv1 u1 Þ 2
ð20Þ
1 1 v21 u21 þu1 a1 v1 b1 þ 2 q r
s3 ¼ ðh0 wi1 Þ
s4 ¼
ð21Þ
u21 v21 1 1 1 2 ðh0 wi1 Þð 2 þ 2 Þ ½j1 ð4a21 4b21 5d þ2dg þ 3g2 Þ 2 8 r q þ 4u1 j2 ðgdÞ20u21 j3
ð22Þ
To easier handle the complex expressions to be integrated, the terms were grouped after the powers of the initial trajectory parameters and those of w, the integrals i1w, i2w, i3w being of first order in w, while j1w, j2w and j3w are of second order. After the substitution of the first solutions into the starting differential equations we arrive to connect the reduced coordinates u and v, as well as their derivatives du/dw and dv/dw at the beginning of the Wien field part with homogeneous fields (index ‘‘2’’, reference plane 2, in Figs. 1 and 2) with the same quantities at the ion entry into the fringing fields (index ‘‘1’’) through relationships of the form: u2 ¼ Du u1 þ Da a1 þ Dd d þ Dg g þ Duu u21 þ Dua u1 a1 þ Dud u1 d 2
þDug u1 g þ Daa a21 þDad a1 d þDag a1 g þ Ddd d þ Ddg dg þDgg g2 hh0 ½3dð3u21 v21 Þ3gv21 þ u1 ðhþ h0 Þð7u21 3v21 Þ 12 7u2 i gd i1 i2 u1 s6 ¼ 4w v21 1 ; s7 ¼ 2 3 2
s5 ¼
2 2 u v 1 1 þ u1 a1 v1 b1 ; s8 ¼ ðhh0 Þ 1 1 þ 2 q r
s10 ¼
þDvv v21 þ Dvb v1 b1 þ Dbb b21 þ Duuu u31 þDuua u21 a1 þ Duud u21 d ð23Þ
þDuug u21 g þDuaa u1 a21 þ Duad u0 a0 d þ Duag u0 a0 g þ Dudd u0 d
þDudg u0 d þDugg u1 þDuvv u1 v21 þDuvb u1 v1 b1 þ Dubb u1 b21 2 þDaaa a30 þ Daad a20 d þ Daag a20 þ Dadd a0 d þ Dadg a1 d þ Dagg a1 2 3 2 þDavv a1 v21 þ Davb a1 v1 b1 þ Dabb a1 b21 þ Dddd d þ Dddg d 2 2 2 3 þDdgg d þ Ddvv dv0 þ Ddvb dv0 b0 þDdbb db0 þ Dggg þ Dgvv v21 þDgvb v1 b1 þ Dgbb b21 ð34Þ
g
ð24Þ
2
g
2
g
g
g
g
h0 ðwÞ 2 2 s9 ¼ ðv1 u1 Þ 2 ð25Þ
u21 v21 1 1 1 2 ðhwi1 Þ 2 þ 2 þ ½i1 ð4a21 4b21 5d þ 2dg þ 3g2 Þ 2 8 r q
þ 4u1 i2 ðg5dÞ20u21 i3
ð26Þ
g g
g
g
g
u02 ¼ D0u u1 þ D0a a1 þ D0d d þ D0g g þ D0uu u21 þ D0ua u1 a1 þ D0ud u1 d þ D0ug u1 g 2
þD0aa a21 þ D0ad a1 d þ D0ag a1 g þ D0dd d þ D0dg dg þ D0gg g2 þ D0vv v21 þD0vb v1 b1 þ D0bb b21 þD0uuu u31 þ D0uua u21 a1 þD0uud u21 d þ D0uug u21 2 þD0uaa u1 a21 þD0uad u1 a1 d þ D0uag u1 a1 þ D0udd u1 d þD0udg u1 d
g
1 ½6dðhh0 Þðu20 v20 Þ þ u0 ðh2 h20 Þð7u21 9v21 Þ 12q v2 u2 hh0 þ 1 1 ½ðd þ gÞðhh0 Þ þu1 ðh2 h20 Þ þ ½2u0 a0 ðd þ u0 ðhþ h0 Þ 4r 2
s11 ¼
v0 b0 ðd þ g þ u0 ðhh0 ÞÞ
i4w u1 ð3v21 7u21 Þ; 6 0 h ðwÞ s13 ¼ ½3dð2u21 v21 Þ3gv21 þ 2hu1 ð7u21 3v21 Þ 12
ð27Þ
g
g
þD0ugg u1 g2 þ D0uvv u1 v21 þD0uvb u1 v1 b1 þ D0ubb u1 b21 þD0aaa a31 2
þD0aad a21 d þD0aag a21 g þ D0add a1 d þD0adg a1 dg þD0agg a1 g2 3 2 þD0avv a1 v21 þ D0avb a1 v1 b1 þ D0abb a1 b21 þ D0ddd d þ D0ddg d þD0dgg d 2 þ D0dvv dv21 þ D0dvb dv1 b1 þD0dbb db21 þ D0ggg 3 þ D0gvv þD0gvb v1 b1 þ D0gbb b21
g
g g
g
g
gv1 ð35Þ
v2 ¼ Dv v1 þDb b1 þ Dvd v1 d þDvg v1 g þ Dvu v1 u1 þ Dva v1 a1 þ Dbd b1 d
s12 ¼
þDbg b1 g þ Dbu b1 u1 þ Dba b1 a1 þ Dvvv v31 þ Dvvb v21 b1 þ Dvdu v1 du1 ð28Þ
þDvda v1 da1 þ Dvgu v1 gu1 þ Dvga v1 ga1 þ Dvuu v1 u21 þ Dvua v1 u1 a1 2
þDvaa v1 a21 þ Dbbb b31 þDbdd b1 d þDbdg b1 dg þ Dbgg b1 g2 1 1 þ v1 a1 þ u1 b1 ; s14 ¼ ðh0 wi1 Þ u1 v1 þ q r 1 1 s15 ¼ u1 v1 ð2j1 i1 wÞ 2 þ 2 a1 b1 j1 r q
þDbdu b1 du1 þ Dbda b1 da1 þ Dbgu b1 gu1 þDbga b1 ga1 þ Dbuu b1 u21 þDbua b1 u1 a1 þ Dbaa b1 a21 ð29Þ
ð36Þ
v02 ¼ D0v v1 þD0b b1 þ D0vd v1 d þD0vg v1 g þ D0vu v1 u1 þ D0va v1 a1 þ D0bd b1 d þD0bg b1 g þ D0bu b1 u1 þ D0ba b1 a1 þ D0vvv v31 þ D0vvb v21 b1 þ D0vdu v1 du1
1 1 s16 ¼ ðh0 hÞ u1 v1 þ a1 v1 þb1 u1 ; q r 1 1 s17 ¼ u1 v1 2 þ 2 ði1 hwÞa1 b1 i1 r q
þD0vda v1 da1 þ D0vgu v1 gu1 þ D0vga v1 ga1 þ D0vuu v1 u21 þ D0vua v1 u1 a1
2
ð30Þ
þD0vaa v1 a21 þ D0bbb b31 þD0bdd b1 d þD0bdg b1 dg þ D0bgg b1 g2 þD0bdu b1 du1 þ D0bda b1 da1 þ D0bgu b1 gu1 þD0bga b1 ga1 þ D0buu b1 u21 þD0bua b1 u1 a1 þ D0baa b1 a21
ð37Þ
ARTICLE IN PRESS 52
D. Ioanoviciu, K. Tsuno / Nuclear Instruments and Methods in Physics Research A 614 (2010) 47–56
Here Di, Dj0 , Dij, Dij0 , Dijk, Dijk0 are transfer matrix elements, detailed below. First order matrix elements of the u row: Du ¼ 1J2i ;
Da ¼ d;
Dd ¼
J1i ; 2
Dg ¼
J1i 2
ð38Þ
D0bu ¼ ðI1i dÞ
First order matrix elements of the a row: D0a ¼ 1I2i d þ J2i ;
D0d ¼
I1i ; 2
D0g ¼
I1i 2
ð39Þ
Second order matrix elements of the u row: 1 1 1 1 ; Dua ¼ I1i d; Duu ¼ þ I1i 2 qi ri 1 1 1 ; Dvb ¼ dI1i þ 1I1i Dvv ¼ 2 qi ri
"
1 1 1 þ ðI1i dÞ 2 qi ri 1 1 D0bb ¼ ¼ ðI1i dÞ þ qi ri
D0vv ¼
!
57J4i ; 6
Duua ¼
D0vb
Duvb ¼ J4i d2K4i ;
1 Davv ¼ ðJ4i d2K4i Þ; 2
1 ; 2
Dvuu ¼
J4i ; 2
Dbuu ¼
J4i d2K4i 2 ð47Þ
J4i I4i d2 ; 2
D0vdu ¼
1 2 1 ; þ 2 qi ri
1 0 1 3 1 I D ¼ ; D0vuu ¼ þ þ 4i ; 2 2ri vga 2 2qi ri I dJ þ4 4i 4i D0bdu ¼ 1; D0buu ¼ 2
Dvju ¼
ð48Þ
0
0
0
ð49Þ
J3i ¼
Duvv ¼
1 Ddvv ¼ ; 4
2K4i J4i d ; 2
ð41Þ J1i ¼
Duud ¼
D0vvb ¼
0
Z d Z
w
0
0
Z d Z
J4i 1 ; 2
1 Dgvv ¼ 4 ð42Þ
J4i ¼
w
0
0
Z d Z
w
0
0
1 I 2 1i
7 ð2K4i J4i dÞ; 2
Dvvb ¼
I4i ; 2
1 ; 2 3 D0vua ¼ ; 2
Third order matrix elements of the u row: Duuu ¼
ð46Þ
Vanishing matrix elements were omitted. The following integrals are involved: Z d Z d Z d Z d 2 I1i ¼ h dw; I2i ¼ h2 dw; I3i ¼ h3 dw; I4i ¼ h0 dw
#
1 1 þ 2 I3i ; ri q2i
I4i ; 2
D0vda ¼
Second order matrix elements of the a row: " ! # 1 1 1 1 1 þ ðI1i dÞ 2 þ 2 þ6I3i ; D0uu ¼ 2 qi ri ri qi 1 1 5 1 ; D0ud ¼ I2i ; D0ug ¼ I2i ; D0ua ¼ ðdI1i Þ þ qi ri 2 2 I1i 5 I1i 3 0 0 0 0 Daa ¼ ; Ddd ¼ I1i ; Ddg ¼ ; Dgg ¼ I1i 2 4 8 8
D0ba ¼ I1i 2d
Third order matrix elements of the b row: D0vvv ¼
ð40Þ
1 1 1; þ qi ri
Third order matrix elements of the v row: Dvvv ¼
D0u ¼ I2i ;
h dw1 dw;
J2i ¼
Z d Z
w
0
0
h2 dw1 dw;
h3 dw1 dw
ð50Þ
2 h0 dw1 dw;
K4i ¼
Z
d
Z
0
w
Z
0
!
w1
2
h0 dw2
! dw1
dw
0
ð51Þ The quantities qi and ri are defined by the relations qi = (Rei/R) and ri = (Rmi/R) where Rei and Rmi are the entry boundary radius of the electric and of the magnetic field respectively.
Third order matrix elements of the a row: 5 1 7 7 1 I ; D0uua ¼ ðJ4i I4i dÞ ; 6qi 2ri 6 4i 2 2 1 2 1 1 1 2 1 0 0 0 ; Duug ¼ þ ; Duvv ¼ þ þ I4i ; Duud ¼ 4 qi ri 4ri 2 qi ri I dJ4i 1 2 1 Duvb ¼ dI4i J4i ; D0avv ¼ 4i ; ; D0dvv ¼ þ 2 4 qi ri 1 D0gvv ¼ ð43Þ 4ri
D0uuu ¼
3. Exit fringing fields The formulas obtained for the entry fringing fields can be used directly to describe the ion trajectories leaving the Wien filter by accounting that there h(0) = 1 and h(d)=0, while the boundary curvature radii must change sign for both electric and magnetic fields, as curvature centre changes position with respect to the coordinate system. The transfer matrix elements become: First order matrix elements of the u row:
First order matrix elements of the v and b rows: Dv ¼ 1;
Db ¼ d;
D0b ¼ 1
ð44Þ
Second order matrix elements of the v row: 1 1 ; Dva ¼ I1i ; Dbu ¼ I1i þ Dvu ¼ I1i qi ri Second order matrix elements of the b row: D0vd ¼
I1i I2i ; "2
D0vg ¼
I2i I1i ; 2
Da ¼ d;
Dd ¼
J1f ; 2
Dg ¼
J1f 2
ð52Þ
First order matrix elements of the a row: ð45Þ D0u ¼ I2f ;
D0a ¼ 1I2f d þ J2f ;
D0d ¼
I1f ; 2
D0g ¼
I1f 2
ð53Þ
Second order matrix elements of the u row: !
#
1 1 1 1 þ þ ðdI1i Þ 2 þ 2 þ I3i I2i ; qi ri ri qi 1 1 D0va ¼ ðI1i dÞ 1; þ qi ri
D0vu ¼
Du ¼ 1j2f ;
" 1 ðdI1f Þ 2 " 1 Dvv ¼ ðI dÞ 2 1f Duu ¼
1 1 þ rf qf
!
# þ1 ;
Dua ¼ I1f ;
! # 1 1 1 ; þ rf qf
Dvb ¼ I1f
ð54Þ
ARTICLE IN PRESS D. Ioanoviciu, K. Tsuno / Nuclear Instruments and Methods in Physics Research A 614 (2010) 47–56
Second order matrix elements of the a row: " ! # 1 1 1 1 1 D0uu ¼ I þ I1f þ þ 6I 2f 3f ; rf 2 qf r2f q2f ! 1 1 5 1 ; D0ud ¼ I2f D0ug ¼ I2f ; þ D0ua ¼ I1f rf qf 2 2
I1f ¼
I4f ¼
J4f ¼
7 1 ð2K4f J4f dÞ; Duud ¼ ; 2 2 J4f 1 ; Duvb ¼ J4f d2K4f ; Davv ¼ ðJ4f d2K4f Þ; Duvv ¼ 2 2 1 1 Ddvv ¼ ; Dgvv ¼ 4 4
ð55Þ
Duua ¼
ð56Þ
D0uud ¼
1 7 I ; 3qf 6 4f
! 1 2 1 ; þ rf 4 qf
D0uvb ¼ dI4f J4f ; D0gvv ¼
D0uua ¼
7 1 ðJ I dÞ þ ; 2 4f 4f 2
D0uug ¼
D0avv ¼
1 1 þI4f ; 2 qf ! 1 2 1 0 ; Ddvv ¼ þ rf 4 qf
1 ; 4rf
I4f dJ4f ; 2
D0uvv ¼
1 4rf
D0vg ¼
I2f ; 2 !
1 1 ; þ rf qf
D0va ¼ 1I1f
J4f Dvvv ¼ ; 2
ð59Þ
1 1 1 1 þ I1f þ 2 I3f ; qf rf rf q2f ! 1 1 D0bu ¼ 1I1f ; D0ba ¼ I1f þ ð60Þ rf qf D0vu ¼
2K4f J4f d Dvvb ¼ ; 2
D0vua ¼
1 ; 2
D0vvb ¼ Dvgu ¼
J4f I4f d1 ; 2
1 ; 2rf
D0bdu ¼ 1;
I3f ¼
0
Z d Z
w
0
0
Z dZ 0
w 0
h21d dw dw;
Z
d
h3 dw
ð63Þ
0
h dw1 dw;
J2f ¼
Z d Z
w
0
0
h2 dw1 dw
K4f ¼
Z
d
Z
0
w
Z
0
w1
! 2
h0 dw2
! dw1
dw
0
4. Main field matrix
Vx2
o
sinot þ
1 a ð Vz2 Þð1cosotÞ
o o
ð66Þ ð67Þ
J4f ; Dvuu ¼ 2
1 D0vga ¼ ; 2
J4f d2K4f Dbuu ¼ 2 ð61Þ
D0buu ¼
1 2 1 ; þ rf 2 qf
Dvuu ¼
I4f dJ4f 1 2
ð1cosotÞ
1 a
I4f 1 ; þ 2 2qi ð62Þ
Not mentioned matrix elements were omitted because they are vanishing. The integrals involved in this transformation are:
a
ð þ Vz2 Þsinot þ t ð68Þ o o o Here a=eE0/m and o =eB0/m, index 2 means quantity at plane 2.
o
The involved velocity components are: V2y ¼ V2 sinb2 ;
V2z ¼ V2 cosa2 cosb2
ð69Þ
while mV22 ¼ e½j0 ð1 þ dÞ þ E0 x ð70Þ 2 The reference particle following the optical axis satisfies the conditions: R¼
!
D0vdu ¼
Vx2
V2x ¼ V2 sina2 cosb2 ; !
Third order matrix elements of the b row: I4f ; 2
h2 dw;
The quantities qf and rf are defined by the expressions qf = (Ref/ R) and rf =(Rmf/R) where Ref and Rmf are the exit boundary radii of the electric and magnetic field, respectively. If into the Eqs. (34)–(37), we substitute the index 2 by 4, index 1 by 3 and the matrix elements (52)–(62) then we connect, at the plane 4 the quantities u4 and v4, a4 = tan a4, b4 = tan b4 to those at plane 3 u3, v3, a3 and b3.
z¼
Third order matrix elements of the v row:
1 D0vda ¼ ; 2
0
J1f ¼
d
ð58Þ
Second order matrix elements of the b row: I2f ; 2
2
h0 dw;
Z
y ¼ y2 þ Vy2 t
D0b ¼ 1
Second order matrix elements of the v row: ! 1 1 ; Dva ¼ dI1f Dbu ¼ dI1f Dvu ¼ ðI1f dÞ þ rf qf
D0vvv ¼
d
x ¼ x2 þ
Db ¼ d;
D0vd ¼
Z
ð57Þ
First order matrix elements of the v and b rows: Dv ¼ 1;
0
I2f ¼
The third order transfer matrices elements of the main field part of a Wien filter can be calculated by integrating the Eqs. (10) and (11) after the substitutions h= 1, h0 =h00 =0. A straightforward way to obtain the third order matrix elements, the only way to fit them into the transfer matrix calculation trend, is to take advantage of the exact equations of motion of the charged particles inside the homogeneous magnetic and electric fields [32]. Accounting for the initial coordinates at the level of the plane where the homogeneous fields begin, index ‘‘2’’, those equations become:
Third order matrix elements of the a row: D0uuu ¼
h dw;
ð65Þ
Third order matrix elements of the u row: 7J4f þ 2 ; 6
d
ð64Þ
I1f I1f 5 D0aa ¼ ; Ddd ¼ I1f ; D0dg ¼ ; 2 4 8 " ! # 3 1 1 1 1 1 0 0 þ I þ I1f þ I Dgg ¼ I1f ; Dvv ¼ 2f 3f ; 8 2 qf rf r2f q2f ! 1 1 1 ; D0bb ¼ I1f þ D0vb ¼ I1f rf qf 2
Duuu ¼
Z
53
m0 V20 2j0 ¼ eB0 E0
ð71Þ
and arrives to the plane 3 at t0 ¼
W
o0
with
o0 ¼
eB0 m0
ð72Þ
V20 is the reference particle velocity along the Oz axis, while ej0 is its energy in the field-free space. In the xyz coordinate system where the Eqs. (66)–(68) were written, the electric and magnetic deviating forces cancel each other if E0 and B0 are both directed in the positive sense of the irrespective axes. We use again dimensionless coordinates defined here by: u=x/ R, v=y/R and w=z/R. To find u and v in a third order approximation we need t in second order approximation as it enters in terms already small of first order. Some charged particle arrives after t=t0(1+ t) at the distance WR, at the plane 3. We put t and W in Eq. (68) and t is calculated in a second order approximation.
ARTICLE IN PRESS 54
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After substitution in the expressions of u and v, to return to a2 and b2 we use the relations a2 =a2 a32/3 and b2 =b2 b32/3. The expressions for a and b result by derivation of u and v with respect to W. The matrix elements of the transformation from u2, a2, v2, b2 to u3, a3, v3, b3, from plane 2 to plane 3, including the main part of the Wien filter are: First order matrix elements of the u row: c1 1c ; Dg ¼ 2 2 First order matrix elements of the a row:
Du ¼ c;
Da ¼ s;
Dd ¼
s s D0d ¼ ; D0g ¼ 2 2 Second order matrix elements of the u row:
D0u ¼ s;
D0a ¼ c;
ð1cÞð2c þ 3Þ ðc1Þð2c þ 3Þ ; Dua ¼ 2sð1cÞ; Dud ¼ 2 2 2sW þ ðc1Þð2c þ 3Þ ðc1Þð2c1Þ ; Daa ¼ ; Dug ¼ 2 2 ð1cÞð2c þ3Þ Dad ¼ sð1cÞDag ¼ cðsWÞ; Ddd ; 8 2sW þðc1Þð2c þ 3Þ 4sW þ ðc1Þð2c þ 3Þ Ddg ¼ ; Dgg ¼ ; 4 8 1c Dbb ¼ 2
ð73Þ
ð74Þ
Duu ¼
ð75Þ
Second order matrix elements of the a row: sð4c þ1Þ sð4c þ 1Þ ; D0ua ¼ 2ð1cÞð2c þ1Þ; D0ud ¼ ; 2 2 sð14cÞ þ 2cW sð34cÞ ; D0aa ¼ ; D0ad ¼ ð1cÞð2c þ 1Þ; Dug ¼ 2 2 sð4c þ 1Þ D0ag ¼ ðc1Þð2c þ 1Þ þ sW; D0dd ¼ ; 8 sð14cÞ þ2cW sð4c3Þ4cW s D0dg ¼ ; D0gg ¼ ; D0bb ¼ ð76Þ 4 8 2 D0uu ¼
Third order matrix elements of the u row: Duuu ¼ Duud ¼ Duug ¼ Duaa ¼ Duag ¼
ðc1Þð3c2 þ 5c þ 3Þ 3sðc1Þð3c þ 1Þ ; Duua ¼ ; 2 2 3ðc1Þð3c2 þ 5c þ 3Þ ; 4 2 ð1cÞð9c þ11c þ3Þ2Wsð4c þ 1Þ ; 4 2 ð1cÞð9c 5c5Þ 3sðc1Þð3c þ 1Þ ; Duad ¼ ; 2 2 c1 3ðc1Þð3c2 þ5c þ3Þ ½4Wð2c þ1Þ3sð3c þ1Þ; Dudd ¼ ; 2 8
Ddbb ¼
ð1cÞð2c þ 3Þ2Ws 4 Third order matrix elements of the a row:
Dgbb ¼
sð9c2 þ 4c2Þ 3ðc1Þð9c2 þ 5c2Þ ; D0uua ¼ ; 2 2 2 2 3sð9c þ 4c2Þ sð27c 4c10Þ2Wð8c2 þ c4Þ D0uud ¼ ; D0uug ¼ ; 4 4
D0uuu ¼
scð27c28Þ 3ðc1Þð9c2 þ 5c2Þ ; D0uad ¼ 2 2 ð1cÞð27c2 þ 7c10Þ4Wsð4c1Þ 0 ; Duag ¼ 2 2 3sð9c þ 4c2Þ ; D0udd ¼ 8 2 2 sð27c 4c10Þ2Wð8c þc4Þ ; D0udg ¼ 4 2 2 sð27c þ 20c þ6Þ þ4sW þ 4Wð8c2 3c4Þ D0ugg ¼ 8 sð14cÞ 3ðc1Þ2 ð3c þ 2Þ 0 0 ; Daaa ¼ ; Dubb ¼ 2 2 scð27c28Þ scð2827cÞ þ 2Wð8c2 3c4Þ ; D0aag ¼ D0aad ¼ 4 4 3ðc1Þð9c2 þ 5c2Þ ; D0add ¼ 8 ðc1Þð27c2 þ 7c10Þ þ 4sWð4c1Þ D0adg ¼ 4 D0uaa ¼
ðc1Þð9c þ 11c þ 3Þ þ 2Wsð4c þ 1Þ ; 4 2 2 ðc1Þð9c þ7c3Þ4W c þ4Wsð4c1Þ Dugg ¼ 8 ðc1Þð2c þ1Þ 3sðc1Þ2 ; ; Daaa ¼ Dubb ¼ 2 2 ð1cÞð9c2 5c5Þ ðc1Þð9c2 c7Þ þ 2sWð4c3Þ Daad ¼ ; Daag ¼ ; 4 4
Dadd ¼
3sðc1Þð3c þ 1Þ ; 8
Dadg ¼
1c ½3sð3c þ 1Þ4Wð2c þ1Þ 4
ð77Þ
ð78Þ
ðc1Þð27c2 c14Þ4cW 2 þ 16sWð2c1Þ ; 8 2 sð9c þ 4c2Þ ; D0abb ¼ ðc1Þð2c þ1Þ; D0ddd ¼ 16 2 2 sð27c þ 4c10Þ2Wð8c þ c4Þ D0ddg ¼ 16 sð27c2 20c6Þ þ 4sW 2 þ 4Wð8c2 3c4Þ ; D0dgg ¼ 16 2 2 sð14cÞ sð9c 12c þ 2Þ4sW 2Wð8c2 7c4Þ ; D0ggg ¼ ; D0dbb ¼ 4 16
D0agg ¼
sð4c1Þ2cW 4 First order matrix elements of v and b rows:
Dg0 bb ¼
Dv ¼ 1;
Db ¼ W;
D0b ¼ 1
ð79Þ
Second order matrix elements of the v row:
2
Dudg ¼
ðc1Þð2c þ 1Þ ð1cÞð3c2 þ c3Þ þ 2sWð34cÞ þ 4cW 2 ; Dggg ¼ 16 4
Dbd ¼
Ws ; 2
Dbg ¼
sW ; 2
Dbu ¼ Ws;
Dba ¼ c1
ð80Þ
Second order matrix elements of the b row: D0bd ¼
1c ; 2
D0bg ¼
c1 ; 2
D0bu ¼ 1c;
D0ba ¼ s
ð81Þ
Third order matrix elements of the v row: sW sð2c1ÞW ðWsÞð2c1Þ ; Dbdd ¼ ; Dbdg ¼ ; 2 8 4 sð2c1ÞW ð1cÞð2c1Þ Dbdu ¼ ; Dbda ¼ ; 2 2 Wð4c3Þ þ sð12cÞ ðWsÞð2c1Þ Dbgg ¼ ; Dbgu ¼ 8 2 ðc1Þð2c þ 1Þ þ2Ws sð2c1ÞW Dbga ¼ ; Dbuu ¼ ; 2 2 sð2c3Þ þ W Dbua ¼ ð1cÞð2c1Þ; Dbaa ¼ 2 Dbbb ¼
3sð1cÞð3c þ1Þ þ4sW 2 þ 8Wðc1Þð2c þ1Þ ; 8 2 ðc1Þð3c þ 5c þ3Þ ; Dabb ¼ sðc1ÞDddd ¼ 16 2 ð1cÞð9c þ 11c þ 3Þ2Wsð4c þ1Þ ; Dddg ¼ 16 2 2 ðc1Þð9c þ 7c3Þ4cW þ 4sWð4c1Þ ; Ddgg ¼ 16
Dagg ¼
ð82Þ
ARTICLE IN PRESS D. Ioanoviciu, K. Tsuno / Nuclear Instruments and Methods in Physics Research A 614 (2010) 47–56
To operate with a large angular acceptance, second order angular focusing in the mass dispersion plane can be desired. This goal is reached by cancelling the second order aberration coefficient Aaa =0. To cancel this coefficient the Wien filter magnetic boundaries at both its ends should be concavely curved by a radius of 0.49324 m. Then the most significant second and order aberration coefficients are: Aad =33.0796 m Add = 0.06043 m. Accounting for a target of xs = 1.5 mm width, opening angle 11, as = tan(11) and energy spread of d = 0.2% the second order resolution is RII =780.756. If in the calculations we introduce the contribution generated by the matrix elements of third order of the Wien filter only, we obtain a third order angular aberration coefficient Aaaa = 2093.04. By accounting for the absolute value of this coefficient in the resolution formula the resolution drops to RIII = 97.085. By limiting the radial opening angle to 0.51, other conditions kept unchanged the second order resolution RII = 1226.18 is more than twice overestimated with respect to that of third order as RIII = 516.429. The resolution has been calculated by using the following formula:
Third order matrix elements of the b row: c1 D0bbb ¼ ; 2
ðc1Þð4c þ 3Þ D0bdd ¼ ; 8
ð1cÞð4c þ1Þ2sW D0bdg ¼ ; 4
ðc1Þð4c1Þ þ 4sW ðc1Þð4c þ 3Þ ; D0bdu ¼ ; 8 2 sð4c3Þ ð1cÞð4c þ 1Þ2sW D0bda ¼ ; D0bgu ¼ ; 2 2 sð34cÞ þ 2cW ðc1Þð4c þ3Þ D0bga ¼ ; D0buu ¼ ; D0bua ¼ sð4c3Þ; 2 2 D0bgg ¼
D0baa ¼
ð1cÞð4c þ 1Þ 2
55
ð83Þ
Here WR is the length of the homogeneous part of the Wien filter, s= sin(W) and c= cos(W). 5. Application to a specific spectrometer configuration First example: The resolution reduction due the third order angular aberration induced by the Wien filter alone in a second order focusing instrument has been estimated. The main geometric and ion optical parameters are close to those of the spectrometer VAMOS [16] as detailed in [33]. We considered the instrument with the specific parameters defined in Table 1. We assume that the first quadrupole defocuses in the deflexion (horizontal) plane of the instrument while the second exerts a focusing action in that plane. The effect of the elliptic aperture on the focusing action of the second quadrupole was roughly estimated as being proportional to the inverse of the aperture i.e. the arguments of the trigonometric functions in the y directions being O10 times greater that in the horizontal plane. From first order calculations targeting simultaneous x, y and energy focusing at the detector, the following quantities resulted: R inside the Wien filter 1.285794 m, x1 =k1Z1 =1.184464 trigonometric function argument for the first quadrupole, x2 =k2Z2 for the second quadrupole in the x plane. The ki coefficient is defined [34] by the relation: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B e ð84Þ k2i ¼ Ti G0i 2m0 j0
R¼
Dg
ð85Þ
2 jMx jxs þ jAaa ja2s þ 2jAad jas d þjAdd jd þ 2jAaaa ja3s
Second example: The aberration coefficients of a homogeneous main fields Wien filter of length WR, located between two fieldfree spaces, both of 1.92 WR length were computed with the TRIO code combining independent fringing fields of an electrostatic sector and of a magnetic sector field in Ref. [35]. Some of those aberration coefficients, expressed in filter lengths Ajk, Ajkl were reproduced in Table 2 and compared with those calculated with the formulae derived in this paper, expressed in the same length units. Third example: The importance of the fourth order trajectory coefficients can be evaluated for a Wien filter having both source and detector located inside the homogeneous fields (no fringing fields), focusing after W= p, i.e. c= 1 and s= 0. The trajectory coefficients at the image are given below. They resulted from the formulae (66) and (68) processed to account for small quantities up to fourth order.
where BTi, G0i are the magnetic field at the pole tip and half the quadrupole aperture, respectively. Zi is the effective length of the quadrupole. Index ‘‘1’’ and ‘‘2’’ refers to the first and second quadrupole, respectively. The horizontal magnification is then Mx = 0.579717 while the vertical one is 5.26436, with a mass dispersion coefficient of Dg =2.48204 m
uimage ¼ u0 d þ u20 þu0 d þ 3a20 þ
d2 4
3 3 2 u30 u20 d þ9u0 a20 u0 d 2 4
3
þ
9 5 5 15 2 2 15 2 2 15 2 d þ u40 þ u30 d þ a0 d u a þ u d þ u0 a20 d 8 2 4 2 2 0 0 8 0 2
þ
5 15 2 2 5 4 3 97 4 d u0 d a þ a d þ 8 12 0 8 0 64
ð86Þ
Table 1 Some geometric and electro-magnetic parameters of the VAMOS spectrometer. Interaction region first quadrupole distance
Inter-quadrupole distance
Quadrupole Wien filter distance
0.4 m Wien filter length
0.3 m Magnet deflection radius 1.5 m
1m
Wien filter magnet distance
Magnet detector distance
First quadrupole length
Second quadrupole length
0.3 m 0.7 m Magnet deflection angle Beam entry angle
1.5 m Beam exit angle
601
301
0.6 m Entry boundary radius 3m
0.9 m Exit boundary radius 5m
201
Table 2 Aberration coefficients of a symmetrically located Wien filter of length WR between field-free spaces of 1.92 WR. Aberration coefficient
Aaa
Aad
Add
Aaaa
Aaad
Aadd
Addd
Calculated Computed by TRIO+
23.576 25.38
8.148 8.42
0.882 0.92
99.183 623.5
41.158 45.1
6.347 17.8
0.572 0.1
Aberration coefficients in WR units.
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6. Conclusions The ion trajectories in Wien filters with main homogeneous electric and magnetic fields were determined, in a third order approximation both in the plane of mass dispersion (x direction) and normally to it (y direction). The calculated matrix elements account for the ion position, trajectory slope, ion mass and energy. The analytic expressions of the matrix elements are given with second order accuracy, those of first order, with first order accuracy those of second order and containing no small quantities, those of third order. In this way a third order accuracy is kept over all the contributions to the ion trajectory calculation. The analytic expressions of the transfer matrix elements keep manageable extent being ready to be used in the often encountered class of homogeneous Wien filters. Besides the main fields, transfer matrices were derived for the entry and exit fringing fields with effect of magnetic and electric field boundary curvatures included. As an application, the effect of the third order angular aberration of the Wien filter on the resolution of a spectrometer has been calculated for vanishing second order radial angular aberration. References [1] D. Ioanoviciu, C. Cuna, Int. J. Mass Spectrom. Ion Phys. 25 (1977) 117. [2] I. Wahlin, Nucl. Instr. and Meth. 27 (1964) 55. [3] R.G. Wilson, In: A. Septier (Ed.), Applied Charged Particle Optics, Academic Press, New York, London, Toronto, Sydney, San Francisco, 1980, pp. 45–71. [4] S. Taya, K. Tokiguchi, I. Kanomata, H. Matsuda, Nucl. Instr. and Meth. 150 (1978) 165. [5] D. Ioanoviciu, C. Cuna, Rapid Commun. Mass Spectrom. 9 (1995) 512. [6] Y. Kawanami, T. Ishitami, K. Umemura, Nucl. Instr. and Meth. B 37/38 (1983) 240. [7] T. Shiokawa, P. Hyon Kim, K. Toyoda, S. Namba, J. Vac. Sci. Technol. B 1 (4) (1983) 1117. [8] D. Rogalla, M. Aliotta, C.A. Barnes, L. Campajola, A. D’Onofrio, L. Gialanella, U. Greife, G. Imbriani, A. Ordine, V. Roca, C. Rolfs, M. Romano, C. Sabbarese,
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