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Modification of an achromatic mass spectrometer to include transverse focusing
318
Nuclear Instruments and Methods in Physics Research A258 (1987) 318-322 North-Holland, Amsterdam
MODIFICATION OF AN ACHROMATIC MASS SPECTROMETER TO INCLUDE TRANSVERSE FOCUSING Marcel BARIL and Mario NOËL
Centre de Recherche Atomique et Moléculaire, Laval University, Department of Physics, Québec (Québec), Canada GIK 7P4 Modification has been made to a magnetic mass spectrometer, comprising a magnetic prism and a parallel plane mirror, to increase its transmission and to obtain a stigmatic image. This has been done by adding two quadrupole lenses, one between the magnetic prism and the mirror to add some focusing in the transverse direction, the other after the mirror to correct the astigmatism created by the first quadrupole lens . In this paper, we derive all the parameters of the quadrupole lenses needed to ensure this objective. 1. Introduction Theoretical papers by Baril [1,2] have shown the usefulness of the parallel plane electrostatic mirror to eliminate energy dispersion of the ionic beam in a mass spectrometer. Baril and Vallerand [3] verified these ideas experimentally by building an appropriate device . Vallerand [4], for his doctorate thesis, built an achromatic mass spectrometer which combines a parallel plane electrostatic mirror with a uniform field normal incidence magnetic prism. This apparatus lacks any component which has a focusing effect in the transverse direction. Due to this fact, the actual transmission of the apparatus is about twenty percent. This means that the incoming beam at the entrance slit is about five times the outgoing beam after the exit slit. Since we do not want to modify significantly the actual apparatus and since the use of a toroidal electrostatic field [5] or an inhomogeneous magnetic field [6] would imply such a modification, we opted for quadrupole lenses [7] to obtain such transverse focusing and a stigmatic image. A quadrupole lens can be inserted at three places inside the vacuum system without further modification. Baril [8] recently proposed the use of such a modified mass spectrometer to build an ion microscope. In this paper, we present the simplest case study in which we have used two quadrupole lenses . Theoretical results have been obtained using a first order model but can be extended to the second order without difficulty using the model and mathematical tools presented in ref. [2] .
study has shown that it is impossible to obtain a stigmatic image using only one quadrupole lens without greatly reducing the resolving power of the system. Therefore, the problem was divided into two parts. First, introduce the quadrupole lens between the prism and the mirror to increase the transmission of the system between the object slit and the exit slit of the electrostatic mirror. This first lens gives an astigmatic
TRANSVERSAL
LT 1
-4
-HORIZONTAL OH1
!- Ll ---I
0168-9002/87/$03 .50 C Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
i L2
I
PLANE EM (H)
I
000
2.F__-I The new apparatus
LT4
LT3
000
A schematic of the new achromatic mass spectrometer is shown in the central part of fig. 1 . A preliminary
PLANE
I----4 L3
OH2 i i
000
,
o00
i L4
LS
Fig. 1. Schematic of the new achromatic mass spectrometer.
319
M. Baril, M. Noël / Modification of an achromatic mass spectrometer
image which is reformed into a stigmatic image by the second quadrupole lens. The choice of the exit slit of the electrostatic mirror as an intermediate imaging plane is purely arbitrary and can be changed without significant loss. Such a modification may be needed to reduce the second order aberrations when these will be calculated. 3. Calculation method The form of the transfer matrix T11 for the horizontal plane of the apparatus is shown in eq. (1) . Elements A13 and A12 of this matrix are zero. They correspond to the achromatic and imaging properties of this system . The last term, A14, indicates that the dispersion of this system is purely in terms of mass. Since we are working only between the entrance slit or object slit and the exit slit of the electrostatic mirror, we will not set the magnification equal to unity since this constraint will correspond to an a focal system for the quadrupole lens. Instead, we will manage so that All = 1/A22 ; 0 0 All A14 A21
A22
0 0
A23
A24
0 1 0 0 0 1 Now, we can consider this transfer matrix as the product of matrices of all components of the spectrometer . This is equivalent to writing the following expression: LH2 ' MH - LHl (2) in which E H equals the transfer matrix of the electrostatic mirror, (1 H equals the transfer matrix of the triplet quadrupole lens in the horizontal plane, M H equals the transfer matrix of the magnetic prism, and LH3, LH2, LHl the transfer matrices of the field-free regions in the horizontal plane . To ease calculation, the lengths L1, L2 and L3 of the field-free regions are expressed as a function of R, the radius of curvature of the main trajectory in the magnetic prism. Using two parameters, this gives eqs. (3) and (4) : Ll = aR, (3) TH = EH - LH3 * OH -
and L2 = AR/2 =
(4) Calculations are also simplified when the length of the field-free region L3 is set equal to L2 . Similarly, we can write the transfer matrix TT for the transversal plane; we obtain eq. (5) where B11 =1/B22 _ B11 TT-I 121
L3 .
0 ~
B22 .
The preceding matrix must equal the following matrix product :
TT = LT2'OT - LTl"
The parameters of the first quadrupole lens triplet must be adjusted so that eqs. (2) and (6) will produce transfer matrices at the form of eqs. (1) and (5) respectively . To do so, we will isolate in eqs. (2) and (6) the transfer matrix corresponding to the electrostatic quadrupole triplet (ESQT) and we will then get a more specific series of equations to solve the problem . From eqs. (2) and (6), we obtain the following expressions for the ESQT transfer matrices, -l ' TH OH - LH3 EH
and
QT =
LT2
LHl
(7)
H1 'LH2,
TT L Ti
(8)
After appropriate calculations, relations between the elements of the ESQT transfer matrices and the parameters of the spectrometer are obtained. For the horizontal plane, we get : QH,ll = 1 - 2xe +y, (1 - x.) = Al1Ll/R - L3 (A21L1/R -1 /A11R ), QH,12=s( 2 +yi) = A,, R - AllLl L2/R - L3 (RA21 - A2l LlL2/R
(10)
+L2/Al1R ), QH,21 -
1
-xe(-2xe +y, (1 -xe)1/S
= A11L1/R -1 /A11R, = Qh,ll = RA21 -A 2l LlL2/R + QH,22 and for the transversal plane QT,11
=1
+2ye-x,(1 +ye)
=
(11) L2/A11R ;
B11 - LT2B21,
(12)
(13) (14)
= s(2 - x i ) = B21LT2LT1 - BuLTI, (15) QT,21 = (1 +Ye)(2Ye - x i (1 +Ye )) = B21, = (16) QWT,22 QT,11 = 1/BI, + B21LT1 In these eight equations, the right hand member corresponds to the transfer matrices of the quadrupole lens triplet as have been developed by Regenstreif [9]. The triplet has been arbitrarily oriented so that the converging-diverging-converging (cdc) plane is parallel to the horizontal plane . Using an iterative process, we have determined the parameter values of the lens which solve this system of equations . In fact, the system has an infinite number of solutions since the number of variables is greater than the number of equations . But, before solving this system, further simplification is considered by noting that terms QH,ll and QH,22 are equal due to the symmetry of the ESQT. This fact enables us to write the following equaQT,1z
I . MASS SPECTROMETERS/SEPARATORS
M. Baril, M. Noël / Modification of an achromatic mass spectrometer
320
tion: A 11 L1 /R - L3(A21L1/R - I/A11R) = RA21 - A21LIL2/R + L2/A11R,
,
(17)
which can be reduced, using relation (4), to the simpler expression: A21 = Alla/R,
(18)
in which only the parameter a appears. From eq. (2), the following expression can be extracted TH,13 - 0 = QH,11(R/2 + L2/2 ) + QH,12/2
L3 (Q H,21(R/2 + L2/2) + QH,22) + K, (19)
which after simplification gives us the condition for chromatism. K has been evaluated by Baril [1]. From this, and after appropriate calculation, the following equation relating All and a is obtained : All = 2/(1 + a) .
(20)
RP = A14/(Alls+d+A),
(21)
Eqs. (18) and (20) will be used to facilitate the search of a solution for the system of equations. One must consider the fact that the resolving power (RP) of the spectrometer is directly related to the magnification All , and thus to the parameter a, by the relation : where s is the width of the object slit, d the width of the image slit and A is a term which groups all aberrations effects . We see that to get high resolving power it is advantageous to keep All as small as possible. Therefore, from eq. (20), the parameter a must be the largest possible. However, the maximum value of a must not be greater than 1 since this would correspond to a negative focal length in the horizontal plane for the quadrupole triplet . This can be easily understood by stating that, as the actual achromatic mass spectrometer works, a =1 corresponds to an afocal lens. Applying the preceding remark, we have limited the range of a to values from 0.9 to 0.99 and therefore we search solutions inside this domain . We still simplify the solution search by setting the value of the parameter Jl, this is equivalent to fixing the total length of the ESQT. Consequently we look for solutions which correspond to the couple (a, X). We give in table 1 a few solutions which seem
satisfactory a priori. We selected, among all possible solutions, those that according to eq. (21) give a high resolving power while simultaneously leading to an acceptable length associated with the first quadrupole lens, which means that Li/2 < Le < L, must be verified and the distance between each element of the quadrupole must be small compared to lengths Le and Li. At each important calculation step, we verified a condition which guarantees that the lens has a convergent effect in both directions (planes). We have evaluated that the transmission of this new mass spectrometer is 4 to 5 times greater than that of the old version . 4. Astigmatism correction Table 1 shows that there is no solution which produces a stigmatic image; we then consider introducing a second quadrupole lens triplet to produce the appropriate stigmatic correction . The geometry of quadrupole lens and voltage applied to it must be adjusted so that the beam be stigmatic at the detector slit. The problem thus is to find the lens magnifications (associated with each direction) that are proportional to the inverse magnifications obtained with the first quadrupole lens. 4.1 . Determination of theparameters for the second ESQT
The condition for stigmatism is obtained by considering the fact that the image distances are identical for both symmetry planes of the quadrupole lens. The mathematical translation is that Qwc equals Qaca and this is expressed by the equation hereafter : 40 = gcdc = gaca = - ( AcacPo + Beac)/( Ccacpo + Dcdo )
= - (A aca Po + Baca)/(Cacd Po + Daca)'
In this equation, p is the distance between the object plane and the entrance plane of the quadrupole lens. The distance between the exit plane of the quadrupole lens and the image plane is equal to q and A, B, C and D are the elements of the transfer matrices of the lens. From eq. (22) we get : Pô (AlC2 - ClA2) + Po (BlC2 - CIB2 + D2A1 - DlA2 ) (23) + D2B1 - D, B2 = 0,
Table 1 Parameters producing imaging and achromatism a
À
AI ,
0.8 0.85 0.9 0.95
0.055 0.038 0.05 0.033
1 .11 1 .08 1 .05 1 .025
B11 0 .862 0 .846 0 .832 0 .815
Le
2 .1 2.1 2.4 2 .5
(22)
L,
s
ke
k,
3 .97 2.91 3 .95 3 .07
3 .62 3 .8 3 .53 3 .65
0 .2196 0 .2006 0 .1968 0.1797
0 .2377 0 .2562 0 .2405 0 .2488
M. Baril, M. Noël / Modification of an achromatic mass spectrometer v,a
ke, ki (cm-1 )
1
321
1 R - 0 7957 0 = 0.1 LE =2
045 ke .
a
curves
kL . o
curves
040
035
030
025 I 0.7
I O9
I 09
I 10
I 11
N
I 12
I 1.3
1 -L 14 15
Fig. 2. Theoretical curves which give the object distance p and the image distance q for a given R as a function of N, the total magnification.
Fig. 3. Theoretical electrical parameters to be applied to the quadrupole deduced from the preceding figure.
which enables evaluation of po for a given geometry and applied voltage to the ESQT. Subsequently we substitute into eq. (22) the value of po to obtain the corresponding value of qo . By this procedure we obtain families of couples (Po, qo) which produce a conjugated stigmatic image. We retain among these couples those that satisfy the
ratio of the magnifications Gcdc/Gdw = R which is also given by the following relation : (24) R = (Cdcdpo+Adcd)/(CcdcPO+Acdc)* Depending on the geometry of the quadrupole lens triplet that we choose, we can draw curves like those in fig. 2. These curves are drawn for a given R as a
Table 2 Quadrupole lens parameters for stigmatic imaging a
0.8 0.8 0.8 0.85 0.85 0.85 0.90 0.90 0.90 0.95 0.95 0.95
N
1.0 1 .0 1.0 1.0 1.0 1.0 1 .0 1 .0 1 .0 1 .0 1.0 1.0
R 0.776 0.776 0.776 0 .783 0 .783 0 .783 0.792 0.792 0.792 0.796 0.796 0.796
p 13 .07 16 .48 19.95 12.98 16.40 19.84 12.94 16.34 19.74 12.86 16 .21 19 .70
q
11 .36 14.29 17 .33 11 .67 14.80 17 .97 12 .09 15 .35 18 .63 12 .44 15 .79 19 .16
Le
2 .0 2 .0 2.0 2.0 2 .0 2.0 2.0 2.0 2.0 2.0 2 .0 2.0
L,
2.0 3.0 4.0 2.0 3.0 4.0 2.0 3 .0 4 .0 2.0 3.0 4.0
ke
0.3637 0.3276 0 .3007 0.3626 0.3262 0.2994 0.3610 0.3248 0.2982 0.3598 0.3238 0.2973
k,
0.4580 0.3410 0.2736 0.4569 0.3402 0.2730 0.4559 0.3394 0.2723 0.4548 0.3385 0.2780
1. MASS SPECTROMETERS/SEPARATORS
322
M. Baril, M. Noël / Modification of an achromatic mass spectrometer
function of N, the total magnification of the beam when it crosses the second quadrupole lens triplet . We give in fig. 3 the electrical parameters from which the voltage to be applied to the quadrupole lens can be deduced with reference to the curves of the preceding figure . Also, we have grouped in table 2 solutions which correspond to geometries and typical ratios R for which N=1 .
5. Conclusion We have shown that the introduction of two quadrupole lens triplets between elements of an existing achromatic mass spectrometer can increase the performance in terms of transmission and image quality without losing its achromatism. Moreover, the new system ensures a stigmatic image of the entrance slit of the spectrometer at the detector slit, this is a very interesting aspect permitting us to think of using the new apparatus as the principal component of an ion microscope.
However, a second order study must be performed before building such quadrupole lenses (as calculated in this paper) to verify the influence of second order aberrations on the resolving power and image quality of the system . References [1] M. Baril, Can. J. Phys. 47 (1969) 331 . [2] M. Baril, Can. J. Phys. 48 (1970) 2487. [3] M. Baril and P. Vallerand, Can. J. Phys. 52 (1974) 82 . [4] P . Vallerand, Doctorate thesis, Laval University, Québec, Canada (1976) . [5] S. Taya et al ., Int. J. Mass Spectrom . Ion Phys . 26 (1978)
77 . [6] F.G . Ruedenauer, Int. J. Mass Spectrom . Ion Phys. 4 (1970) 195 . [7] C.E.D . Ouwerkerk, A.J.H . Boerboom, A.J .H. Matsuo and T. Sakurai, Int. J. Mass. Spectrom, Ion Phys. 65 (1985) 23 . [8] M. Baril and M. NoO, 1st Int. Conf. on Instrumental Analysis, Beijing (1985) . [9] E. Regenstreif, Focusing of Charged Particles, vol. 1, ed ., Septier (Academic Press, New York, 1967) p . 353 .
Nuclear Instruments and Methods in Physics Research A258 (1987) 318-322 North-Holland, Amsterdam
MODIFICATION OF AN ACHROMATIC MASS SPECTROMETER TO INCLUDE TRANSVERSE FOCUSING Marcel BARIL and Mario NOËL
Centre de Recherche Atomique et Moléculaire, Laval University, Department of Physics, Québec (Québec), Canada GIK 7P4 Modification has been made to a magnetic mass spectrometer, comprising a magnetic prism and a parallel plane mirror, to increase its transmission and to obtain a stigmatic image. This has been done by adding two quadrupole lenses, one between the magnetic prism and the mirror to add some focusing in the transverse direction, the other after the mirror to correct the astigmatism created by the first quadrupole lens . In this paper, we derive all the parameters of the quadrupole lenses needed to ensure this objective. 1. Introduction Theoretical papers by Baril [1,2] have shown the usefulness of the parallel plane electrostatic mirror to eliminate energy dispersion of the ionic beam in a mass spectrometer. Baril and Vallerand [3] verified these ideas experimentally by building an appropriate device . Vallerand [4], for his doctorate thesis, built an achromatic mass spectrometer which combines a parallel plane electrostatic mirror with a uniform field normal incidence magnetic prism. This apparatus lacks any component which has a focusing effect in the transverse direction. Due to this fact, the actual transmission of the apparatus is about twenty percent. This means that the incoming beam at the entrance slit is about five times the outgoing beam after the exit slit. Since we do not want to modify significantly the actual apparatus and since the use of a toroidal electrostatic field [5] or an inhomogeneous magnetic field [6] would imply such a modification, we opted for quadrupole lenses [7] to obtain such transverse focusing and a stigmatic image. A quadrupole lens can be inserted at three places inside the vacuum system without further modification. Baril [8] recently proposed the use of such a modified mass spectrometer to build an ion microscope. In this paper, we present the simplest case study in which we have used two quadrupole lenses . Theoretical results have been obtained using a first order model but can be extended to the second order without difficulty using the model and mathematical tools presented in ref. [2] .
study has shown that it is impossible to obtain a stigmatic image using only one quadrupole lens without greatly reducing the resolving power of the system. Therefore, the problem was divided into two parts. First, introduce the quadrupole lens between the prism and the mirror to increase the transmission of the system between the object slit and the exit slit of the electrostatic mirror. This first lens gives an astigmatic
TRANSVERSAL
LT 1
-4
-HORIZONTAL OH1
!- Ll ---I
0168-9002/87/$03 .50 C Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
i L2
I
PLANE EM (H)
I
000
2.F__-I The new apparatus
LT4
LT3
000
A schematic of the new achromatic mass spectrometer is shown in the central part of fig. 1 . A preliminary
PLANE
I----4 L3
OH2 i i
000
,
o00
i L4
LS
Fig. 1. Schematic of the new achromatic mass spectrometer.
319
M. Baril, M. Noël / Modification of an achromatic mass spectrometer
image which is reformed into a stigmatic image by the second quadrupole lens. The choice of the exit slit of the electrostatic mirror as an intermediate imaging plane is purely arbitrary and can be changed without significant loss. Such a modification may be needed to reduce the second order aberrations when these will be calculated. 3. Calculation method The form of the transfer matrix T11 for the horizontal plane of the apparatus is shown in eq. (1) . Elements A13 and A12 of this matrix are zero. They correspond to the achromatic and imaging properties of this system . The last term, A14, indicates that the dispersion of this system is purely in terms of mass. Since we are working only between the entrance slit or object slit and the exit slit of the electrostatic mirror, we will not set the magnification equal to unity since this constraint will correspond to an a focal system for the quadrupole lens. Instead, we will manage so that All = 1/A22 ; 0 0 All A14 A21
A22
0 0
A23
A24
0 1 0 0 0 1 Now, we can consider this transfer matrix as the product of matrices of all components of the spectrometer . This is equivalent to writing the following expression: LH2 ' MH - LHl (2) in which E H equals the transfer matrix of the electrostatic mirror, (1 H equals the transfer matrix of the triplet quadrupole lens in the horizontal plane, M H equals the transfer matrix of the magnetic prism, and LH3, LH2, LHl the transfer matrices of the field-free regions in the horizontal plane . To ease calculation, the lengths L1, L2 and L3 of the field-free regions are expressed as a function of R, the radius of curvature of the main trajectory in the magnetic prism. Using two parameters, this gives eqs. (3) and (4) : Ll = aR, (3) TH = EH - LH3 * OH -
and L2 = AR/2 =
(4) Calculations are also simplified when the length of the field-free region L3 is set equal to L2 . Similarly, we can write the transfer matrix TT for the transversal plane; we obtain eq. (5) where B11 =1/B22 _ B11 TT-I 121
L3 .
0 ~
B22 .
The preceding matrix must equal the following matrix product :
TT = LT2'OT - LTl"
The parameters of the first quadrupole lens triplet must be adjusted so that eqs. (2) and (6) will produce transfer matrices at the form of eqs. (1) and (5) respectively . To do so, we will isolate in eqs. (2) and (6) the transfer matrix corresponding to the electrostatic quadrupole triplet (ESQT) and we will then get a more specific series of equations to solve the problem . From eqs. (2) and (6), we obtain the following expressions for the ESQT transfer matrices, -l ' TH OH - LH3 EH
and
QT =
LT2
LHl
(7)
H1 'LH2,
TT L Ti
(8)
After appropriate calculations, relations between the elements of the ESQT transfer matrices and the parameters of the spectrometer are obtained. For the horizontal plane, we get : QH,ll = 1 - 2xe +y, (1 - x.) = Al1Ll/R - L3 (A21L1/R -1 /A11R ), QH,12=s( 2 +yi) = A,, R - AllLl L2/R - L3 (RA21 - A2l LlL2/R
(10)
+L2/Al1R ), QH,21 -
1
-xe(-2xe +y, (1 -xe)1/S
= A11L1/R -1 /A11R, = Qh,ll = RA21 -A 2l LlL2/R + QH,22 and for the transversal plane QT,11
=1
+2ye-x,(1 +ye)
=
(11) L2/A11R ;
B11 - LT2B21,
(12)
(13) (14)
= s(2 - x i ) = B21LT2LT1 - BuLTI, (15) QT,21 = (1 +Ye)(2Ye - x i (1 +Ye )) = B21, = (16) QWT,22 QT,11 = 1/BI, + B21LT1 In these eight equations, the right hand member corresponds to the transfer matrices of the quadrupole lens triplet as have been developed by Regenstreif [9]. The triplet has been arbitrarily oriented so that the converging-diverging-converging (cdc) plane is parallel to the horizontal plane . Using an iterative process, we have determined the parameter values of the lens which solve this system of equations . In fact, the system has an infinite number of solutions since the number of variables is greater than the number of equations . But, before solving this system, further simplification is considered by noting that terms QH,ll and QH,22 are equal due to the symmetry of the ESQT. This fact enables us to write the following equaQT,1z
I . MASS SPECTROMETERS/SEPARATORS
M. Baril, M. Noël / Modification of an achromatic mass spectrometer
320
tion: A 11 L1 /R - L3(A21L1/R - I/A11R) = RA21 - A21LIL2/R + L2/A11R,
,
(17)
which can be reduced, using relation (4), to the simpler expression: A21 = Alla/R,
(18)
in which only the parameter a appears. From eq. (2), the following expression can be extracted TH,13 - 0 = QH,11(R/2 + L2/2 ) + QH,12/2
L3 (Q H,21(R/2 + L2/2) + QH,22) + K, (19)
which after simplification gives us the condition for chromatism. K has been evaluated by Baril [1]. From this, and after appropriate calculation, the following equation relating All and a is obtained : All = 2/(1 + a) .
(20)
RP = A14/(Alls+d+A),
(21)
Eqs. (18) and (20) will be used to facilitate the search of a solution for the system of equations. One must consider the fact that the resolving power (RP) of the spectrometer is directly related to the magnification All , and thus to the parameter a, by the relation : where s is the width of the object slit, d the width of the image slit and A is a term which groups all aberrations effects . We see that to get high resolving power it is advantageous to keep All as small as possible. Therefore, from eq. (20), the parameter a must be the largest possible. However, the maximum value of a must not be greater than 1 since this would correspond to a negative focal length in the horizontal plane for the quadrupole triplet . This can be easily understood by stating that, as the actual achromatic mass spectrometer works, a =1 corresponds to an afocal lens. Applying the preceding remark, we have limited the range of a to values from 0.9 to 0.99 and therefore we search solutions inside this domain . We still simplify the solution search by setting the value of the parameter Jl, this is equivalent to fixing the total length of the ESQT. Consequently we look for solutions which correspond to the couple (a, X). We give in table 1 a few solutions which seem
satisfactory a priori. We selected, among all possible solutions, those that according to eq. (21) give a high resolving power while simultaneously leading to an acceptable length associated with the first quadrupole lens, which means that Li/2 < Le < L, must be verified and the distance between each element of the quadrupole must be small compared to lengths Le and Li. At each important calculation step, we verified a condition which guarantees that the lens has a convergent effect in both directions (planes). We have evaluated that the transmission of this new mass spectrometer is 4 to 5 times greater than that of the old version . 4. Astigmatism correction Table 1 shows that there is no solution which produces a stigmatic image; we then consider introducing a second quadrupole lens triplet to produce the appropriate stigmatic correction . The geometry of quadrupole lens and voltage applied to it must be adjusted so that the beam be stigmatic at the detector slit. The problem thus is to find the lens magnifications (associated with each direction) that are proportional to the inverse magnifications obtained with the first quadrupole lens. 4.1 . Determination of theparameters for the second ESQT
The condition for stigmatism is obtained by considering the fact that the image distances are identical for both symmetry planes of the quadrupole lens. The mathematical translation is that Qwc equals Qaca and this is expressed by the equation hereafter : 40 = gcdc = gaca = - ( AcacPo + Beac)/( Ccacpo + Dcdo )
= - (A aca Po + Baca)/(Cacd Po + Daca)'
In this equation, p is the distance between the object plane and the entrance plane of the quadrupole lens. The distance between the exit plane of the quadrupole lens and the image plane is equal to q and A, B, C and D are the elements of the transfer matrices of the lens. From eq. (22) we get : Pô (AlC2 - ClA2) + Po (BlC2 - CIB2 + D2A1 - DlA2 ) (23) + D2B1 - D, B2 = 0,
Table 1 Parameters producing imaging and achromatism a
À
AI ,
0.8 0.85 0.9 0.95
0.055 0.038 0.05 0.033
1 .11 1 .08 1 .05 1 .025
B11 0 .862 0 .846 0 .832 0 .815
Le
2 .1 2.1 2.4 2 .5
(22)
L,
s
ke
k,
3 .97 2.91 3 .95 3 .07
3 .62 3 .8 3 .53 3 .65
0 .2196 0 .2006 0 .1968 0.1797
0 .2377 0 .2562 0 .2405 0 .2488
M. Baril, M. Noël / Modification of an achromatic mass spectrometer v,a
ke, ki (cm-1 )
1
321
1 R - 0 7957 0 = 0.1 LE =2
045 ke .
a
curves
kL . o
curves
040
035
030
025 I 0.7
I O9
I 09
I 10
I 11
N
I 12
I 1.3
1 -L 14 15
Fig. 2. Theoretical curves which give the object distance p and the image distance q for a given R as a function of N, the total magnification.
Fig. 3. Theoretical electrical parameters to be applied to the quadrupole deduced from the preceding figure.
which enables evaluation of po for a given geometry and applied voltage to the ESQT. Subsequently we substitute into eq. (22) the value of po to obtain the corresponding value of qo . By this procedure we obtain families of couples (Po, qo) which produce a conjugated stigmatic image. We retain among these couples those that satisfy the
ratio of the magnifications Gcdc/Gdw = R which is also given by the following relation : (24) R = (Cdcdpo+Adcd)/(CcdcPO+Acdc)* Depending on the geometry of the quadrupole lens triplet that we choose, we can draw curves like those in fig. 2. These curves are drawn for a given R as a
Table 2 Quadrupole lens parameters for stigmatic imaging a
0.8 0.8 0.8 0.85 0.85 0.85 0.90 0.90 0.90 0.95 0.95 0.95
N
1.0 1 .0 1.0 1.0 1.0 1.0 1 .0 1 .0 1 .0 1 .0 1.0 1.0
R 0.776 0.776 0.776 0 .783 0 .783 0 .783 0.792 0.792 0.792 0.796 0.796 0.796
p 13 .07 16 .48 19.95 12.98 16.40 19.84 12.94 16.34 19.74 12.86 16 .21 19 .70
q
11 .36 14.29 17 .33 11 .67 14.80 17 .97 12 .09 15 .35 18 .63 12 .44 15 .79 19 .16
Le
2 .0 2 .0 2.0 2.0 2 .0 2.0 2.0 2.0 2.0 2.0 2 .0 2.0
L,
2.0 3.0 4.0 2.0 3.0 4.0 2.0 3 .0 4 .0 2.0 3.0 4.0
ke
0.3637 0.3276 0 .3007 0.3626 0.3262 0.2994 0.3610 0.3248 0.2982 0.3598 0.3238 0.2973
k,
0.4580 0.3410 0.2736 0.4569 0.3402 0.2730 0.4559 0.3394 0.2723 0.4548 0.3385 0.2780
1. MASS SPECTROMETERS/SEPARATORS
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M. Baril, M. Noël / Modification of an achromatic mass spectrometer
function of N, the total magnification of the beam when it crosses the second quadrupole lens triplet . We give in fig. 3 the electrical parameters from which the voltage to be applied to the quadrupole lens can be deduced with reference to the curves of the preceding figure . Also, we have grouped in table 2 solutions which correspond to geometries and typical ratios R for which N=1 .
5. Conclusion We have shown that the introduction of two quadrupole lens triplets between elements of an existing achromatic mass spectrometer can increase the performance in terms of transmission and image quality without losing its achromatism. Moreover, the new system ensures a stigmatic image of the entrance slit of the spectrometer at the detector slit, this is a very interesting aspect permitting us to think of using the new apparatus as the principal component of an ion microscope.
However, a second order study must be performed before building such quadrupole lenses (as calculated in this paper) to verify the influence of second order aberrations on the resolving power and image quality of the system . References [1] M. Baril, Can. J. Phys. 47 (1969) 331 . [2] M. Baril, Can. J. Phys. 48 (1970) 2487. [3] M. Baril and P. Vallerand, Can. J. Phys. 52 (1974) 82 . [4] P . Vallerand, Doctorate thesis, Laval University, Québec, Canada (1976) . [5] S. Taya et al ., Int. J. Mass Spectrom . Ion Phys . 26 (1978)
77 . [6] F.G . Ruedenauer, Int. J. Mass Spectrom . Ion Phys. 4 (1970) 195 . [7] C.E.D . Ouwerkerk, A.J.H . Boerboom, A.J .H. Matsuo and T. Sakurai, Int. J. Mass. Spectrom, Ion Phys. 65 (1985) 23 . [8] M. Baril and M. NoO, 1st Int. Conf. on Instrumental Analysis, Beijing (1985) . [9] E. Regenstreif, Focusing of Charged Particles, vol. 1, ed ., Septier (Academic Press, New York, 1967) p . 353 .