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A cylindrical mirror analyser with high energy resolution
JOURNAL OF ELECTRON SPECTROSCOPY and Rebated Phenomena
ELSEVIER
Journal of Electron Spectroscopyand Related Phenomena73 (1995) 305-310
A cylindrical mirror analyser with high energy resolution A.A. Trubitsyn Scientific Research Technology Institute, Yablochkov pas. 6, Ryazan 390011, Russia
First received3 May 1994;in final form 8 January 1995
Abstract
The electron-optical properties of a cylindrical-mirror analyser with a non-uniform edge field have been numerically investigated. It is shown that the performance of an instrument with a non-ideal cylindrical field can considerably exceed that of an idealized instrument. Keywords. Cylindrical mirror analyser; Electron energy analyser; Electron optical property
1. Introduction
Among many types of electron energy analyser the cylindrical mirror analyser (CMA) is the one most commonly used. This results from its numerous advantages which include the possibility of obtaining a relatively good resolution at high transmission, simple construction and the possibility of using a built-in electron gun. The finiteness of real instrument size necessitates edge effect elimination. In the common case, complete reduction of edge distortions of electrostatic fields is practically impossible, and therefore real instrument performance differs from idealized instrument performance. Focussing order is the criterion of quality of an energy analyser. The higher it is, the weaker is the conflict between luminosity and resolving power. Here the influence of fringing field perturbation on cylindrical mirror focussing properties are investigated by original numerical methods. The edge effects were reduced by means of several
pairs of fringing rings, insulated from each other and having potentials in the following portions of the outer cylinder potential Vb (inner cylinder potential Va is equal to 0): ui = i V b / ( n + 1), where i is the number of the ring (counting from the inner cylinder) and n is the number of ring pairs. The inner (r~n) and outer (r °ut) radii of the ith correcting ring are calculated by the formulae riin/ra = 0.5[(rb/ra) u' + (rb/ra) ui-~ + d/ra]
rOUt = ri+ in 1 - d where i = 1 , 2 . . . n, u0 = V~ and u, + l = Vb. Here r a and r b are the radii of inner and outer cylinders, and d>~ 0 is the width of insulating gaps between the correcting electrodes. The schematic diagram of such an instrument with two pairs of rings is presented in Fig. 1. In the design of a real device one is to follow the recommendation of Ref. [1]: aperture windows are to be covered by one-dimensional grids consisting of wire pieces stretched along the generators of
0368-2048/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0368-2048(95)02346- 1
306
A.A. TrubitsynlJournal of Electron Spectroscopy and Related Phenomena 73 (1995) 305-310
an inner cylindrical electrode separated by the same angular interval. The merit of a one-dimensional grid is that it does not affect focussing in the axial plane because it scatters a beam only in the azimuthal direction.
numerical integration of an ordinary equation of the first kind dr dz
=* 2. Methods The numerical method of the N order focussing search proposed by Gorelik et al. [2] is based on the solution of the N - 1 non-linear algebraic equation system in terms of the entrance angle oo. For instance, in order to find the second order focussing the equation
is to be solved, where S(E, CY)= rc(E, a) + z,(E,a)t(E,a) and t(E,a) = tan (/I); p and r,, z, are the angle and coordinates of the electron outlet from the region of the electrostatic field gradient and E is the electron energy. The equation F(E, a) = 0 may have several solutions o. for any fixed energy E (as in the case considered below). When the solution a0 of the equation F(E, a) = 0 is found, the focus coordinates are determined by the formulae
zo(E)= ro(E) = r,(E, a01- ko(E) - 44
ao)l@,
ao)
In practice the functions @S(E, a)/dak, @t(E, a)/dd are computed by utilization of the numerical derivative formulae for a discrete set of {oi} and are interpolated for cx # oi. In order to improve the computation accuracy the functions S(E, oi) and t(E, ai) should firstly be smoothed by the method of least squares. The electron trajectories are computed by
differential
(i;-f
~~-~[u(Z,I)-u(Z,r~)+U(Z~,I)-u(Z~,r~)] [W, r) - U(zo,r) + w, ro) - U(ZOlro)l
w
)
suggested in Ref. [3]. Here zo, ro, io, i. are the initial coordinates and the corresponding components of the velocity as regards each step of integration, e and m are the charge and the mass of the particle, U(z, r) is the potential at the point with coordinates (z, Y). Despite the complicated character of the numerical solution of this equation, because it is necessary to apply the rotating coordinate system here, the technique can be recommended for utilization. Its main merit is the absence of the necessity to compute the potential gradient. That is why the method’s limit level of error and the executive computer memory are significantly less than for ordinary techniques. For the sake of validity of the results, the trajectory computations were also supplied by the ordinary Runge-Kutta method. The electrostatic field is calculated by the variable triangular method (VTM) [4], which is one of the quickest. In order to solve Laplace’s equation by VTM we use the following matrix equations with lower and upper matrices (I-_lJ.Ri).W~+, (I-w.R,).
=h.U,(z,r) w,,,
=A.
l&+,
and take into consideration that W,, , Ir = 0 and U,lr is fixed on the domain boundary I. Potential U values at the interior points (z, r) of the domain are evaluated by the formula u kil
=
uk+rk+l’
wk,l
in each k + 1 iteration. Here Zis the identity matrix; Ai and A2 are the finite difference two-dimensional Laplace operators with lower and upper matrices, A is the finite difference Laplace operator in the cylindrical coordinates w=2/Jsn
A.A. Trubitsyn/Journalof Electron Spectroscopy and Related Phenomena 73 (1995) 305-310
307
V~ 50"
d
5_8
n ~ ~ V,
¢"//'//--///---//--~~
2.8
~
0.33 Vg
± ,
e.5
f
13.o
,
Z
Fig. 1. Upper part of cross section of the analyser with two pairs o f correcting electrodes and trajectories of electrons with energy E 0
(Eo/Vb = 1.4347) within it.
3. Results and discussion 4
7rhz
~5= ~ sin 2 ~ 4
4 .n27rhr + ~2 Sl 2lr
2 7rhz
a = ~ cos ~
4 h~
2 7rhr 2lr
+ ~ cos - -
hz, h~ are the distances between the nearest grid points and lz, lr are the maximum extensions of the computing region in the z and r directions, respectively. For VTM with the set of Chebyshev parameters % ~-k - 1 + potk ordered in the special way, the number n of iterations required for supplying the computing accuracy e is evaluated by the expression ln(Z/e) n>~no(e ) = ~ 2 V / ~ 4 V / ~ Here
~-0 = 2/(71 + 72), t k = cos((2k -
Po = (1 - ~)/(1 + ~),
, = Ua, -~ = c~7~, v2 = c2~72, "~ = U [ 2 ( 1 + v ~ ) ] , % = ~" ";'I/"Y2,
1)0r/2n)),
3/(4v/-~), C1 and C2 are the minimum and maximum radii of the domain computed. In order to confirm the computational results, the boundary element method (BEM) with the special technique for singular and near-singular integral evaluation [5] was used for the potential problem solution. The numerical technique of the high order focussing search described here was tested by computation of a wide set of electron-optical systems permitting analytical exploration.
The numerical calculations have shown that a CMA with three, four or five pairs of correcting electrodes provides a regime of second order angle focussing corresponding to that of an ideal mirror. The value of the focussing angle, the axis position of the focus and the energy dispersion of the CMA with three pairs of correcting electrodes differ from the corresponding idealized instrument parameters by several percentage units. When the number of correcting electrodes is increased, this difference is further reduced. More important results have been obtained for the analyser with two pairs of correcting rings, i.e. in the case of a significant difference of its field from the idealized one. The corresponding computations allow us to conclude that it is possible to supply a three-times second order focussing in the 36-48 ° angle range. For example, the first angle of the second order focussing for the CMA presented in Fig. 1 is 38.3 °, the second is 42.5 °, and the third is 45.5 °. Fig. 2 shows schematic (a) and real (b) trajectories of electrons leaving the space between the inner and outer cylinders and focussed at points 1, 2 and 3. The increase of focussing multiplicity allows us to improve the consumer characteristics of the analyser. The energy resolution and the luminosity are consumer characteristics of an analyser for electron spectroscopy. Both these parameters are reflected by the instrumental function. Further,
~
(ct)
A.A. Trubitsyn/Journal of Electron Spectroscopy and Related Phenomena 73 (1995) 305-310
308
~~,
(b)
.... 4
\
x
4
Fig. 2. Schematic (a) and real (b) trajectories of the electrons, leaving the space between the inner and the outer cylinders o f the analyser with two pairs of correcting electrodes and focussed at points 1, 2 and 3. Points 4 and 5 are front and back edges of the exit aperture.
the comparative analysis of the instrumental functions of a CMA with second order focussing and the proposed CMA with three-times second order focussing will be provided. The most correct way of calculating the instrumental function is based on the analysis of the analyser's transmission region in coordinates (E, a). The transmission region is determined in the following way. Firstly, curve E I (a) describing the interconnection of angle a and energy E of the electrons, the trajectories of which "almost touch" the front edge of the ring exit aperture, is calculated. Then an analogous curve Ez(a) for the electrons, the trajectories of which "almost touch"
1.446
L E ~ @V
1.434
~
/ 1.420 34.00
E
~C#-)
d maz
F Em~n
Z m~m 41.50
49.00
~° Fig. 3. Transmission region of the analyser with second order focussing. Potential of the outer cylinder Vb = 1 V.
the opposite edge of the aperture, is calculated. It is obvious that the electrons having energy E, such as El(a)~E(ot)~E2(a), pass through the exit aperture and are recorded by the collector. The work angle range a m i n ~ a ~ a m a x o f the analyser is fixed by the input and output windows of the inner cylindrical electrode, and therefore the transmission region of the analyser with fixed sizes of entrance and exit apertures and slits is described by a system of inequalities O L m i n ~ O l ~ O : m a x , E 1(a) <<.E(a) <~E2(a ), and its basic resolution and luminosity are calculated by the formulae R B = (Ema x - Emin)/Eo × 100% and L = f~0/27r x 100% -- (cos O~min -- COS OLmax)X 100% respectively. Here Emax and Emin are the maximum and the minimum energies of the transmission region, E0 is the analyser's tuning energy and ~0 is the solid angle of collection of the electrons having energy E0. It is necessary to note that the angle range [O~min,O~max]is determined as an optimum for each fixed exit aperture size; namely, the left point Olmin and the right point O~max are found as points of intersection of the straight line E = E0 with curves E2(a ) and El(a), respectively. The range [OLmin,OLmax] is an optimum because when it decreases, the analyser's luminosity also decreases and the instrumental function gets a flat top, and when the range increases, the energy resolution deteriorates without any luminosity increase, i.e. the instrumental function is broadened. According to definition, an instrumental function is the dependence of the intensity of a monoenergetic charge particle flow, emitted from a constant brightness source and passing through
A.A. Trubitsyn/Journalof Electron Spectroscopy and Related Phenomena 73 (1995) 305-310
309
6.5
1.440
2 59_. I007, 2Jr
E,eV
1.435 1.'!339
1.4353
E,eV Fig. 6. Instrumental functions of the analysers having equal resolutions: 1, with second order focussing; 2, with three-times second order focussing.
1.430 34.00
"11.50
"19.00
Fig. 4. Transmission region of the analyser with three-times second order focussing. Potential of the outer cylinder Vb = 1 V.
a receiving slit, on the analyser's tuning energy [6]. In order to plot the instrumental function it is necessary to determine the angle range E ~<~ E O~min~ 6e ~O~ma x falling into the transmission region El(a)<~E(a)<~E2(a ) for each energy Emin~E~Emax, i.e. we must determine the dependence f~ = f~(E), where f~ ~ 27r(cos ~ min E -COS OlEmax).
Fig. 3 shows the transmission region of an analyser having three pairs of correcting electrodes and supplying, as has been mentioned above, second order focussing. The exit aperture is placed at the point of second order focussing and its width is equal to 0.024r a. The optimum angle range is 41.15 ° q: 5.65 °. The analyser lumin6.5"
2~" t00% r
•
1.4~
I..44~
E,eV Fig. 5. Instrumental functions o f the analysers having equal luminosities: 1, with second order focussing; 2, with threetimes second order focussing.
osity L is equal to 6.5%, the basic relative energy resolution RB is equal to 1.14%, and the relative full width at half maximum (FWHM) of the instrumental function RFWHM is 0.55%. Fig. 4 and Fig. 5 (curve 2) present the transmission region and the corresponding instrumental function of the analyser with three-times second order focussing. The exit aperture is placed at the section of the electron flow with minimum diameter (points 4 and 5 in Fig. 2(a)), and its width is equal to 0.0024r a. While plotting the instrumental function we took into consideration the E ,( ~<( E x is fact that the angle range O~min-~O~-~t~ma the sum of several segments for energy E ~ E0, and therefore f~ -- 27r Ei(cos a/Emin -- COSaEmax). By comparing curves 1 and 2 in Fig. 5 we may conclude that the cylindrical field distortion by means of two pairs of correcting rings supplies a ten-fold improvement of the relative energy resolution ( R F w H M z 0 . 0 5 % ) with preserved luminosity. It is necessary to note the possibility of resolution improvement with the help of edge field distortion has also been shown in Ref. [7]. Because the recorded signal level is proportional to the square under the instrumental function, it is useful to compare the instrumental function of the proposed analyser with the classical one, in case both of them supply equal energy resolutions. Fig. 6 shows such instrumental functions, and we can conclude that the three-times second order focussing analyser allows us to increase the signal level approximately three-fold. The stability of the instrument parameters as
310
A.A. Trubitsyn/Journal o f Electron Spectroscopy and Related Phenomena 73 (1995) 305 310
regards various perturbations is very important in practice. The change of the number of the ringed correcting electrodes, arranged near the back edges of the cylinders, does not break the three-times focussing regime, i.e. we can conclude that the main electrostatic field non-uniformity is created by the conical correcting electrodes arranged near the front edges of the cylinders. The three-times focussing regime is preserved if we increase the inclination angle of the section of the conical correcting electrodes to about 70 °. The analyser's transition into the ordinary second order focussing regime (the inclination angle exceeding 70 °) is explained by removal of the non-uniform electrostatic field region out of the region of electron passage. The three-times focussing regime is preserved when the focal distance (the distance between a source and the front edge of the inner cylinder), which is equal to 0.8ra, changes within a 10% range along the z axis; when the reduction of the focal distance exceeds the limit mentioned above, the analyser moves into the ordinary second order focussing regime. This is also explained by removal of the trajectories from the non-uniform electrostatic field region. The numerical exploration has also shown that the conditions of three-times focussing of the second order do not, in practice, depend on the
value of insulating gaps between correcting electrodes in the range d = (0 to 0.05ra) and on correlation of radii of the inner r a and outer r b cylinders.
4. Conclusions
CMA performance can be significantly improved by applying the three-times second order angle focussing regime, which can be obtained by the arrangement of two pairs of correcting rings. Such an instrument can be used in high resolution Auger spectroscopy.
References [1] V.V. Zashkvara, B.U. Ashimbaeva, K.Sh. Chokin and M. Rysavy, J. Electron Spectrosc. Relat. Phenom., 58 (1992) 271. [2] V.A. Gorelik, O.D. Protopopov and A.A. Trubitsyn, Zh. Tekh. Fiz., 8 (1988) 1531. [3] A.A. Trubitsyn, Zh. Vychis. Mat. Mat. Fiz., 7 (1990) 1113. [4] A.A. Samarsky, Introduction to Numerical Methods, Nauka, Moscow, 1987. 15] A.A. Trubitsyn, Zh. Vychis. Mat. Mat. Fiz., 4 (1995) 532. [6] V.P. Afanasiev and S.Ya. Yavor, Electrostatic Energy Analyzers of Charge Particle Flows, Nauka, Moscow, 1978. [7] A.B. Pevzner, Instruments and Methods for X-ray Analysis, Mashinostroenie, Leningrad, Vol. 37, 1988, p. 117.
ELSEVIER
Journal of Electron Spectroscopyand Related Phenomena73 (1995) 305-310
A cylindrical mirror analyser with high energy resolution A.A. Trubitsyn Scientific Research Technology Institute, Yablochkov pas. 6, Ryazan 390011, Russia
First received3 May 1994;in final form 8 January 1995
Abstract
The electron-optical properties of a cylindrical-mirror analyser with a non-uniform edge field have been numerically investigated. It is shown that the performance of an instrument with a non-ideal cylindrical field can considerably exceed that of an idealized instrument. Keywords. Cylindrical mirror analyser; Electron energy analyser; Electron optical property
1. Introduction
Among many types of electron energy analyser the cylindrical mirror analyser (CMA) is the one most commonly used. This results from its numerous advantages which include the possibility of obtaining a relatively good resolution at high transmission, simple construction and the possibility of using a built-in electron gun. The finiteness of real instrument size necessitates edge effect elimination. In the common case, complete reduction of edge distortions of electrostatic fields is practically impossible, and therefore real instrument performance differs from idealized instrument performance. Focussing order is the criterion of quality of an energy analyser. The higher it is, the weaker is the conflict between luminosity and resolving power. Here the influence of fringing field perturbation on cylindrical mirror focussing properties are investigated by original numerical methods. The edge effects were reduced by means of several
pairs of fringing rings, insulated from each other and having potentials in the following portions of the outer cylinder potential Vb (inner cylinder potential Va is equal to 0): ui = i V b / ( n + 1), where i is the number of the ring (counting from the inner cylinder) and n is the number of ring pairs. The inner (r~n) and outer (r °ut) radii of the ith correcting ring are calculated by the formulae riin/ra = 0.5[(rb/ra) u' + (rb/ra) ui-~ + d/ra]
rOUt = ri+ in 1 - d where i = 1 , 2 . . . n, u0 = V~ and u, + l = Vb. Here r a and r b are the radii of inner and outer cylinders, and d>~ 0 is the width of insulating gaps between the correcting electrodes. The schematic diagram of such an instrument with two pairs of rings is presented in Fig. 1. In the design of a real device one is to follow the recommendation of Ref. [1]: aperture windows are to be covered by one-dimensional grids consisting of wire pieces stretched along the generators of
0368-2048/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0368-2048(95)02346- 1
306
A.A. TrubitsynlJournal of Electron Spectroscopy and Related Phenomena 73 (1995) 305-310
an inner cylindrical electrode separated by the same angular interval. The merit of a one-dimensional grid is that it does not affect focussing in the axial plane because it scatters a beam only in the azimuthal direction.
numerical integration of an ordinary equation of the first kind dr dz
=* 2. Methods The numerical method of the N order focussing search proposed by Gorelik et al. [2] is based on the solution of the N - 1 non-linear algebraic equation system in terms of the entrance angle oo. For instance, in order to find the second order focussing the equation
is to be solved, where S(E, CY)= rc(E, a) + z,(E,a)t(E,a) and t(E,a) = tan (/I); p and r,, z, are the angle and coordinates of the electron outlet from the region of the electrostatic field gradient and E is the electron energy. The equation F(E, a) = 0 may have several solutions o. for any fixed energy E (as in the case considered below). When the solution a0 of the equation F(E, a) = 0 is found, the focus coordinates are determined by the formulae
zo(E)= ro(E) = r,(E, a01- ko(E) - 44
ao)l@,
ao)
In practice the functions @S(E, a)/dak, @t(E, a)/dd are computed by utilization of the numerical derivative formulae for a discrete set of {oi} and are interpolated for cx # oi. In order to improve the computation accuracy the functions S(E, oi) and t(E, ai) should firstly be smoothed by the method of least squares. The electron trajectories are computed by
differential
(i;-f
~~-~[u(Z,I)-u(Z,r~)+U(Z~,I)-u(Z~,r~)] [W, r) - U(zo,r) + w, ro) - U(ZOlro)l
w
)
suggested in Ref. [3]. Here zo, ro, io, i. are the initial coordinates and the corresponding components of the velocity as regards each step of integration, e and m are the charge and the mass of the particle, U(z, r) is the potential at the point with coordinates (z, Y). Despite the complicated character of the numerical solution of this equation, because it is necessary to apply the rotating coordinate system here, the technique can be recommended for utilization. Its main merit is the absence of the necessity to compute the potential gradient. That is why the method’s limit level of error and the executive computer memory are significantly less than for ordinary techniques. For the sake of validity of the results, the trajectory computations were also supplied by the ordinary Runge-Kutta method. The electrostatic field is calculated by the variable triangular method (VTM) [4], which is one of the quickest. In order to solve Laplace’s equation by VTM we use the following matrix equations with lower and upper matrices (I-_lJ.Ri).W~+, (I-w.R,).
=h.U,(z,r) w,,,
=A.
l&+,
and take into consideration that W,, , Ir = 0 and U,lr is fixed on the domain boundary I. Potential U values at the interior points (z, r) of the domain are evaluated by the formula u kil
=
uk+rk+l’
wk,l
in each k + 1 iteration. Here Zis the identity matrix; Ai and A2 are the finite difference two-dimensional Laplace operators with lower and upper matrices, A is the finite difference Laplace operator in the cylindrical coordinates w=2/Jsn
A.A. Trubitsyn/Journalof Electron Spectroscopy and Related Phenomena 73 (1995) 305-310
307
V~ 50"
d
5_8
n ~ ~ V,
¢"//'//--///---//--~~
2.8
~
0.33 Vg
± ,
e.5
f
13.o
,
Z
Fig. 1. Upper part of cross section of the analyser with two pairs o f correcting electrodes and trajectories of electrons with energy E 0
(Eo/Vb = 1.4347) within it.
3. Results and discussion 4
7rhz
~5= ~ sin 2 ~ 4
4 .n27rhr + ~2 Sl 2lr
2 7rhz
a = ~ cos ~
4 h~
2 7rhr 2lr
+ ~ cos - -
hz, h~ are the distances between the nearest grid points and lz, lr are the maximum extensions of the computing region in the z and r directions, respectively. For VTM with the set of Chebyshev parameters % ~-k - 1 + potk ordered in the special way, the number n of iterations required for supplying the computing accuracy e is evaluated by the expression ln(Z/e) n>~no(e ) = ~ 2 V / ~ 4 V / ~ Here
~-0 = 2/(71 + 72), t k = cos((2k -
Po = (1 - ~)/(1 + ~),
, = Ua, -~ = c~7~, v2 = c2~72, "~ = U [ 2 ( 1 + v ~ ) ] , % = ~" ";'I/"Y2,
1)0r/2n)),
3/(4v/-~), C1 and C2 are the minimum and maximum radii of the domain computed. In order to confirm the computational results, the boundary element method (BEM) with the special technique for singular and near-singular integral evaluation [5] was used for the potential problem solution. The numerical technique of the high order focussing search described here was tested by computation of a wide set of electron-optical systems permitting analytical exploration.
The numerical calculations have shown that a CMA with three, four or five pairs of correcting electrodes provides a regime of second order angle focussing corresponding to that of an ideal mirror. The value of the focussing angle, the axis position of the focus and the energy dispersion of the CMA with three pairs of correcting electrodes differ from the corresponding idealized instrument parameters by several percentage units. When the number of correcting electrodes is increased, this difference is further reduced. More important results have been obtained for the analyser with two pairs of correcting rings, i.e. in the case of a significant difference of its field from the idealized one. The corresponding computations allow us to conclude that it is possible to supply a three-times second order focussing in the 36-48 ° angle range. For example, the first angle of the second order focussing for the CMA presented in Fig. 1 is 38.3 °, the second is 42.5 °, and the third is 45.5 °. Fig. 2 shows schematic (a) and real (b) trajectories of electrons leaving the space between the inner and outer cylinders and focussed at points 1, 2 and 3. The increase of focussing multiplicity allows us to improve the consumer characteristics of the analyser. The energy resolution and the luminosity are consumer characteristics of an analyser for electron spectroscopy. Both these parameters are reflected by the instrumental function. Further,
~
(ct)
A.A. Trubitsyn/Journal of Electron Spectroscopy and Related Phenomena 73 (1995) 305-310
308
~~,
(b)
.... 4
\
x
4
Fig. 2. Schematic (a) and real (b) trajectories of the electrons, leaving the space between the inner and the outer cylinders o f the analyser with two pairs of correcting electrodes and focussed at points 1, 2 and 3. Points 4 and 5 are front and back edges of the exit aperture.
the comparative analysis of the instrumental functions of a CMA with second order focussing and the proposed CMA with three-times second order focussing will be provided. The most correct way of calculating the instrumental function is based on the analysis of the analyser's transmission region in coordinates (E, a). The transmission region is determined in the following way. Firstly, curve E I (a) describing the interconnection of angle a and energy E of the electrons, the trajectories of which "almost touch" the front edge of the ring exit aperture, is calculated. Then an analogous curve Ez(a) for the electrons, the trajectories of which "almost touch"
1.446
L E ~ @V
1.434
~
/ 1.420 34.00
E
~C#-)
d maz
F Em~n
Z m~m 41.50
49.00
~° Fig. 3. Transmission region of the analyser with second order focussing. Potential of the outer cylinder Vb = 1 V.
the opposite edge of the aperture, is calculated. It is obvious that the electrons having energy E, such as El(a)~E(ot)~E2(a), pass through the exit aperture and are recorded by the collector. The work angle range a m i n ~ a ~ a m a x o f the analyser is fixed by the input and output windows of the inner cylindrical electrode, and therefore the transmission region of the analyser with fixed sizes of entrance and exit apertures and slits is described by a system of inequalities O L m i n ~ O l ~ O : m a x , E 1(a) <<.E(a) <~E2(a ), and its basic resolution and luminosity are calculated by the formulae R B = (Ema x - Emin)/Eo × 100% and L = f~0/27r x 100% -- (cos O~min -- COS OLmax)X 100% respectively. Here Emax and Emin are the maximum and the minimum energies of the transmission region, E0 is the analyser's tuning energy and ~0 is the solid angle of collection of the electrons having energy E0. It is necessary to note that the angle range [O~min,O~max]is determined as an optimum for each fixed exit aperture size; namely, the left point Olmin and the right point O~max are found as points of intersection of the straight line E = E0 with curves E2(a ) and El(a), respectively. The range [OLmin,OLmax] is an optimum because when it decreases, the analyser's luminosity also decreases and the instrumental function gets a flat top, and when the range increases, the energy resolution deteriorates without any luminosity increase, i.e. the instrumental function is broadened. According to definition, an instrumental function is the dependence of the intensity of a monoenergetic charge particle flow, emitted from a constant brightness source and passing through
A.A. Trubitsyn/Journalof Electron Spectroscopy and Related Phenomena 73 (1995) 305-310
309
6.5
1.440
2 59_. I007, 2Jr
E,eV
1.435 1.'!339
1.4353
E,eV Fig. 6. Instrumental functions of the analysers having equal resolutions: 1, with second order focussing; 2, with three-times second order focussing.
1.430 34.00
"11.50
"19.00
Fig. 4. Transmission region of the analyser with three-times second order focussing. Potential of the outer cylinder Vb = 1 V.
a receiving slit, on the analyser's tuning energy [6]. In order to plot the instrumental function it is necessary to determine the angle range E ~<~ E O~min~ 6e ~O~ma x falling into the transmission region El(a)<~E(a)<~E2(a ) for each energy Emin~E~Emax, i.e. we must determine the dependence f~ = f~(E), where f~ ~ 27r(cos ~ min E -COS OlEmax).
Fig. 3 shows the transmission region of an analyser having three pairs of correcting electrodes and supplying, as has been mentioned above, second order focussing. The exit aperture is placed at the point of second order focussing and its width is equal to 0.024r a. The optimum angle range is 41.15 ° q: 5.65 °. The analyser lumin6.5"
2~" t00% r
•
1.4~
I..44~
E,eV Fig. 5. Instrumental functions o f the analysers having equal luminosities: 1, with second order focussing; 2, with threetimes second order focussing.
osity L is equal to 6.5%, the basic relative energy resolution RB is equal to 1.14%, and the relative full width at half maximum (FWHM) of the instrumental function RFWHM is 0.55%. Fig. 4 and Fig. 5 (curve 2) present the transmission region and the corresponding instrumental function of the analyser with three-times second order focussing. The exit aperture is placed at the section of the electron flow with minimum diameter (points 4 and 5 in Fig. 2(a)), and its width is equal to 0.0024r a. While plotting the instrumental function we took into consideration the E ,( ~<( E x is fact that the angle range O~min-~O~-~t~ma the sum of several segments for energy E ~ E0, and therefore f~ -- 27r Ei(cos a/Emin -- COSaEmax). By comparing curves 1 and 2 in Fig. 5 we may conclude that the cylindrical field distortion by means of two pairs of correcting rings supplies a ten-fold improvement of the relative energy resolution ( R F w H M z 0 . 0 5 % ) with preserved luminosity. It is necessary to note the possibility of resolution improvement with the help of edge field distortion has also been shown in Ref. [7]. Because the recorded signal level is proportional to the square under the instrumental function, it is useful to compare the instrumental function of the proposed analyser with the classical one, in case both of them supply equal energy resolutions. Fig. 6 shows such instrumental functions, and we can conclude that the three-times second order focussing analyser allows us to increase the signal level approximately three-fold. The stability of the instrument parameters as
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regards various perturbations is very important in practice. The change of the number of the ringed correcting electrodes, arranged near the back edges of the cylinders, does not break the three-times focussing regime, i.e. we can conclude that the main electrostatic field non-uniformity is created by the conical correcting electrodes arranged near the front edges of the cylinders. The three-times focussing regime is preserved if we increase the inclination angle of the section of the conical correcting electrodes to about 70 °. The analyser's transition into the ordinary second order focussing regime (the inclination angle exceeding 70 °) is explained by removal of the non-uniform electrostatic field region out of the region of electron passage. The three-times focussing regime is preserved when the focal distance (the distance between a source and the front edge of the inner cylinder), which is equal to 0.8ra, changes within a 10% range along the z axis; when the reduction of the focal distance exceeds the limit mentioned above, the analyser moves into the ordinary second order focussing regime. This is also explained by removal of the trajectories from the non-uniform electrostatic field region. The numerical exploration has also shown that the conditions of three-times focussing of the second order do not, in practice, depend on the
value of insulating gaps between correcting electrodes in the range d = (0 to 0.05ra) and on correlation of radii of the inner r a and outer r b cylinders.
4. Conclusions
CMA performance can be significantly improved by applying the three-times second order angle focussing regime, which can be obtained by the arrangement of two pairs of correcting rings. Such an instrument can be used in high resolution Auger spectroscopy.
References [1] V.V. Zashkvara, B.U. Ashimbaeva, K.Sh. Chokin and M. Rysavy, J. Electron Spectrosc. Relat. Phenom., 58 (1992) 271. [2] V.A. Gorelik, O.D. Protopopov and A.A. Trubitsyn, Zh. Tekh. Fiz., 8 (1988) 1531. [3] A.A. Trubitsyn, Zh. Vychis. Mat. Mat. Fiz., 7 (1990) 1113. [4] A.A. Samarsky, Introduction to Numerical Methods, Nauka, Moscow, 1987. 15] A.A. Trubitsyn, Zh. Vychis. Mat. Mat. Fiz., 4 (1995) 532. [6] V.P. Afanasiev and S.Ya. Yavor, Electrostatic Energy Analyzers of Charge Particle Flows, Nauka, Moscow, 1978. [7] A.B. Pevzner, Instruments and Methods for X-ray Analysis, Mashinostroenie, Leningrad, Vol. 37, 1988, p. 117.