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Dihedral-like electrostatic potential and analysing systems based on it: aberration coefficients and energy resolution
Nuclear Instruments and Methods in Physics Research A 427 (1999) 145}150
Dihedral-like electrostatic potential and analysing systems based on it: aberration coe$cients and energy resolution P.G. Gabdullin, S.N. Davydov*, Yu.K. Golikov Physical Electronics Department, St. Petersburg State Technical University, 29 Polytechnicheskaya st., St. Petersburg 195251, Russia
Abstract The essential feature of the dihedral-like potential U"arctan (y/x), where x and y are Cartesian co-ordinates, is the possibility to compress, i.e. to make narrower in the real space, beams of charged particles. The coe$cient M of the reduction of a ribbon-like beam cross-over can be varied in the wide diapason: 1(M(20. Due to this, the resolution of two- or more-cascade dispersive systems based on U can be more than 10 times higher than the same value of any traditional analyser or monochromator. But aberration coe$cients divided by linear energy dispersion are approximately 10 times bigger in dihedral mirror analyser (DMA), than in the traditional plane mirror analyser (PMA). That is why only 0}23 diverging ribbon-like beams can be treated successfully with the DMA. The experimental model of a two-cascade monochromator, the basic distance between its entrance and exit diaphragms is 150 mm and the entrance slit is 6.4 mm wide, has a resolution of 0.3%. 1999 Elsevier Science B.V. All rights reserved. PACS: 29.30.-h; 29.30.Aj; 29.30Dn; 29.30.Ep Keywords: Analyser; Aberration; Dispersion; Focusing; Monochromator; Resolution
1. Introduction Earlier [1] we considered electron-optical characteristics of some electrostatic potential U"arctan (y/x), the "eld of which was named `the "eld of dihedral anglea. Its equipotential surfaces are halfplanes comings together at the common z-axis,
* Corresponding author. Tel.: #7-812-552-7516; fax: #78122472088. E-mail address: [email protected]. (S.N. Davydov)
where U has a singularity. Let a parallel ribbon-like #ow of thickness G and width *z, having crossed the border U"0, move towards the z-axis. The last only means that the distance between the particles and the z-axis becomes smaller, and does not mean the exact directivity of the particles towards the z-axis (see Fig. 1). In this case, after leaving the "eld through the same equipotential U"0, the beam "nds itself constricted in thickness in the following way: the ratio of the initial section thickness H to the same value at the exit H , both sections being
0168-9002/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 1 5 3 5 - 6
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P.G. Gabdullin et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 145}150
produced by the said zero equipotential, equals the ratio of the entrance's and exit's distances from the z-axis: H /H "x /x . Taking into account the dif ference between the entrance h and exit h angles, one can write down the expression for the compression value: H sin h x sin h G " . M" " H sin h x sin h G
(1)
Due to this feature, which was named telescopic compression of ribbon-like #ow, the potential U possesses a rather high energy dispersion DH, `reduced to the entrancea: DH"DH /H , where D is the usual linear dispersion. Instead of D, DH should be used in the expression of the resolving power of any energy analyser based on U [1]. That is why U-based analysers or monochromators can be of very high energy resolution. Any real #ow has non-zero divergency, and hence it needs to be focused at the exit diaphragm. Aberrations, coexisting together with focusing, have not been calculated yet for the dihedral electron-optical scheme. To know the aberrational blurring m at the exit diaphragm is of special importance in the case of U, because the size of the image is about M times as small as the size of the source.
Fig. 1. Cross-section of some equipotentials (thin solid lines) of the dihedral angle potential U"arctan (y/x) in the region 04U4p. In the picture U"0 at y(0: equipotentials U"0 and U"p are the bordering ones. If a parallel ribbon-like #ow of thickness G and width *z (thick solid arrowed lines) enters the "eld region y50 at an average coordinate x , and leaves this region at x , the #ow cross-section varies in the fashion: H /H "x /x , the width *z remaining invariant.
2. Slightly diverging beam in dihedral potential Consider electrons emitted with the same energy from a point source O , and moving in the xy-plane (Fig. 2). If the #ow enters the region of grad UO0 and leaves it through the same equipotential U"0 coinciding with the xz-plane, it means that according to the Helmholtz}Lagrange law, the beam linear compressing coe$cient M is inversely proportional to the angular magni"cation: M"G / G "*h /*h . Here *h and *h are beam angular divergences at the entrance and exit, respectively. In the common case of focusing of any order, we can evaluate aberration blurring as m"C (*h /2)I, I where C is the "rst non-zero coe$cient of the I expansion of the image width in power series on the initial half-angular divergence *h admitted in the analyser. If "rst-order focusing takes place, C "C , while for the second-order C "C and I '' I ''' C "C "0. ' '' In the present evaluations of the aberration coe$cients, all the trajectory calculations have been made numerically, by Runge-Kutta method. The x-coordinate of the source O is taken as the length unity, the mass and charge of a particle as the unit mass and charge, respectively. Just in this system of units, potential energy of the particle is numerically equal to the particle's angular coordinate, i.e. to arctan (y/x). In the case of the secondorder focus, the source and image O are in the "eld-free space: the y-coordinates of the source (-S) and image (-Q) are negative. In the particular case of the "rst-order focusing on the border of the "eld, S"Q"0.
Fig. 2. General scheme, showing the beam passage from the source O to the Gaussian image O through the dihedral-like potential U. Thick solid line represents the main (average) trajectory of the beam, while the dashed curves are the outer ones. U"0 at y(0, i.e. the xz-plane is the border of the "eld.
P.G. Gabdullin et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 145}150
3. First-order focusing at the border yⴝ0 At "rst, the situation was examined when both the point source O and the image O were on the plane y"0 (S"Q"0), "rst-order focusing conditions being ful"lled. Thus, m"C (*h /2). In the '' energy diapason 0(=(1.6 (W is the initial kinetic energy of the particle), C turns out to be a little '' bit smaller than the same value of the well-known plane mirror analyser (PMA) under the "rst-order focusing conditions [2] (Fig. 3a). It is convenient to compare these two analysing systems, because the PMA is actually a limiting case of the dihedral-like mirror analyser (DMA) when =P0. As it was shown earlier [1], linear dispersion D of the DMA is smaller than it is in the PMA, especially at high W. This is the reason for a rather steep fall of the dependence D/C "f (=) (Fig. 3b), which '' can be approximated, with an accuracy of 2%, by D/C "0.5 exp (!2.19=), while in the PMA '' D/C "0.5. Comparing the two analysers at '' ="1, we can see that their resolutions are equal to each other, if the angular divergency of DMA is 6.5 times as small. At ="1.6 this ratio equals 23.5.
4. Second-order focusing Varying S50 and Q50, it is possible to achieve the second-order focusing conditions. The results
147
for S"0 and 0.2 are presented in Fig. 4. Aberration broadening can be described in both the cases as m"2C (*h /2). Again, while the ratio D/C " ''' ''' 0.66 for the PMA, similar DMA values for S"0 and 0.2 (Fig. 4b) fall signi"cantly with W as D/C "0.66 exp (!3.22W) and D/C "0.66 ''' ''' exp(!2.33W), respectively. Analysis shows again that though DMA is possible, because of its great resolving power, to treat very wide ribbon-like #ows the angular divergency of them should be as small as possible.
5. Many-cascade DMA systems' synthesis Let us calculate the resolving power of a double-pass DMA, denoting by a superscript (1) or (2) in parentheses the values concerning the "rst or second stage, respectively. Displacement of the image in the image plane of the "rst cascade (i.e. at the exit of the "rst cascade), caused by the small change of energy *= is *x"D*=/=. Because of the second cascade 5 compressing, which can be expressed by the common equation *x"(x/x)*x, this dis placement converts into the smaller value D(*=/=) (x/x) in the image plane of the whole system. But the energy dispersion of the second cascade gives additional contribution to the image displacement: *x"D*=/=. Thus, the 5
Fig. 3. Second-order aberration coe$cient C (a), and linear energy dispersion D divided by C (b) as functions of the energy W. Dashed '' '' lines correspond to the PMA of "rst-order focusing. Solid lines represent the same values for DMA.
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P.G. Gabdullin et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 145}150
Fig. 4. Third-order aberration coe$cient C (a), and linear energy dispersion D divided by C (b) as functions of energy W. Solid lines ''' ''' correspond to the DMA with the point source on the border of the "eld (the xz-plane in Fig. 2; S"0) and with the source placed in the "eld-free space at the distance S"0.2 from the xz-plane.
total image shift at the exit *= *= x #D *x "D 5 = = x *= x D#D . " (2) = x When the beam image moves in the plane of the exit diaphragm, the intensity of the #ow passing through the diaphragm comes to the zero level at the points, the distance between which is
x x (3) *x"2H#m"2H #m. x x Two co-factors (x/x) and (x/x) in the right hand part of Eq. (3) re#ect the diminishing input diaphragm width H in two cascades when it is projected to the exit. Equality of *x and *x , 5 m being neglected, gives the base resolving power of a double-pass device: = R " *=
1 x x x " D # D . (4) 2H x x x The general expression for an n-cascade apparatus can be derived in a similar manner:
1 L x I I\ x G R" D . (5) L 2H x x I G It is evident from Eq. (4) that the contribution to the "nal resolving power from the second cascade is bigger: its energy dispersion is `enhanceda not only by its own compressing (see the factor x/x), but by the "rst-stage compression in addition (the factor x/x). In case of dihedral-like cascades, it is very pro"table to increase their number n. Indeed, if all of them are identical, then the overall dimension of the analyser is approximately proportional to n, while the whole resolving power, according to the expression (5), grows faster than x /x to the power n. The additional extra cascade occurs to be destructive as soon as m becomes comparable with the width of the last output slit. The more is the initial divergence of the beam to be analysed or `monochromizeda, the less is the maximum appropriate number of the cascades. To check experimentally the theoretical results, two-cascade variant of monochromator was chosen.
6. Experimental test of a model electron spectrometer The scheme of the model of a new apparatus designed recently is shown in Fig. 5. The primary
P.G. Gabdullin et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 145}150
149
Fig. 5. The scheme of a model of a high-resolution electron spectrometer based on the double-pass dihedral mirror monochromator (II), and on the analyser (V) with the plane of symmetry. The primary beam is formed by the gun (I). Having passed the monochromator, it is focused by the lens (III) on the target (IV). Re#ected electrons are collected by the channeltron (VI). 1 } shim plates; 2 } suppressing electrodes; 3 } de#ecting and correcting electrodes.
source is the slightly modi"ed electron gun of Erdman and Zipf [3]. It forms a round beam of approximately 3 mm in diameter and 13 angular divergency: *h +13. The "rst stage of mono chromatization is designed in the "rst-order focusing variant: S"Q"0. Such a scheme provides the maximum beam compressing and energy dispersion. To accept 43 diverging #ow passing through the "rst exit slit, the second cascade has second-order focusing, with S"2.1 and Q"0. Both cascades, made of titanium alloy and mounted on ceramic insulators, have the same basic unit of length: x"x"80 mm, and di mensionless pass energy W"1. The whole distance between the wide entrance diaphragm (its width H"6.4 mm) and narrow exit slit (H" 0.1 mm) is L"150 mm. The calculated resolving power of the monochromator is R "200 (resolu tion } 0.5%). Because of the great width of the second cascade entrance window, it is covered with a grid of high transparency. Having passed the both monochromatization steps, the #ow is focused on the target IV by the cone-like lens system III (see Ref. [4]). The target is a metal plate covered with thin luminophor layer to control the beam position and cross-section. A thin molybdenum grid is placed in front of it to screen luminophor charging and possible positive potential applied to the target. The characteristics of the monochromatic #ow re#ected by the target are examined by means of a high-resolution energy analyser, designed and
tested earlier [5]. It has a plane of symmetry, almost ideal focusing in this plane, and its linear energy dispersion equals 12 base dimensions, the last being equal to 50 mm. The whole size of the analyser is approximately 80 mm. At the exit of it, electrons are registered by a channel electron ampli"er (VI). During the experiment, there has been veri"ed the correspondence of the calculated parameters of the monochromator to the experimental ones. All the geometrical parameters were justi"ed within the accuracy of 1.5%, though voltage applied depends not only on the little variations of h , but on the regime of the electron gun operation as well. An experimental test has been carried out at a rather high pass energy E , because there was no magnetic shield around the spectrometer. Full width at half maximum (FWHM) of the peak of elastically re#ected electrons *E was measured as a function of E . Resolving power depends slightly on E and on the tuning of the spectrometer, but in the whole range 100(E (600 eV it can reach the value of 300. It means that at E "100 eV we can observe the peak of elastically re#ected particles of 0.33 eV in FWHM.
7. Conclusions Calculations and the experimental test show that the resolution of a two-cascade dihedral-like
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P.G. Gabdullin et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 145}150
analyser or monochromator can be very high, and that the monochromator designed can hardly be in#uenced by the earth's magnetic "eld. Actually, resolution achieves 0.33% in the device of 150 mm base dimension at 6.4 mm entrance slit width and at pass energy E "100 eV without magnetic pro tection. In any traditional spectrometer of the same size, such a resolution would be achievable, without preliminary deceleration, at the entrance diaphragm width H "0.4}0.5 mm. On the other hand, a two-cascade DMA device is usually able to treat only a beam of less than 13 angular divergency. A dihedral "eld possesses the ability to compress the monoenergetic components of a beam, this ability acting simultaneously with the usual possibilities to focus and separate primary electrons according to their primary energies. Because of this rare property, possible future application of the DMA can be seen in the following "elds. First of all, it can be monochromatization of electron probing beams for di!erent kinds of highresolution electron spectroscopy. Rough numerical estimates show that at low pass energies the principal restriction, imposed by space charge repulsion on the intensity of monoenergetic probing beam [6], may be 2}4 times overcome. To apply successfully several-cascade DMA in this area, additional
investigations are desirable to obtain new low voltage electron guns of wide, intensive, almost parallel primary beams. Another application is energy analysis with high angular resolution of slightly diverging low intensity charged particle beams, when the size of the source is not small, e.g. in the "eld of photoelectron or ion spectroscopy. Possibly, parallel cosmic beta-rays could be a good object for many-cascade DMAs. At last, there can be some technological applications dealing with compressing electron beams in processes like melting.
References [1] S.N. Davydov, Yu.K. Golikov, V.V. Korablev, S.N. Romanov, SPIE Proc. 3345 (January 1998) 136. [2] G.A. Harrower, Rev. Sci. Instrum. 26 (1955) 850. [3] P.W. Erdman, E.C. Zipf, Rev. Sci. Instrum. 53 (1982) 225. [4] S.N. Davydov, V.Yu. Kolomenkov, V.A. Fedotov, Trudy LPI Leningrad, USSR 429 (1989) 73. [5] S.N. Davydov, Yu.A. Kudinov, Yu.K. Golikov, V.V. Korablev, J. Electron Spectrosc. Related Phenomena 72 (1995) 317. [6] M. Mishijima, Y. Kubota, K. Kondo, J. Yoshinobu, M. Onchi, Rev. Sci. Instrum. 58 (1987) 307.
Dihedral-like electrostatic potential and analysing systems based on it: aberration coe$cients and energy resolution P.G. Gabdullin, S.N. Davydov*, Yu.K. Golikov Physical Electronics Department, St. Petersburg State Technical University, 29 Polytechnicheskaya st., St. Petersburg 195251, Russia
Abstract The essential feature of the dihedral-like potential U"arctan (y/x), where x and y are Cartesian co-ordinates, is the possibility to compress, i.e. to make narrower in the real space, beams of charged particles. The coe$cient M of the reduction of a ribbon-like beam cross-over can be varied in the wide diapason: 1(M(20. Due to this, the resolution of two- or more-cascade dispersive systems based on U can be more than 10 times higher than the same value of any traditional analyser or monochromator. But aberration coe$cients divided by linear energy dispersion are approximately 10 times bigger in dihedral mirror analyser (DMA), than in the traditional plane mirror analyser (PMA). That is why only 0}23 diverging ribbon-like beams can be treated successfully with the DMA. The experimental model of a two-cascade monochromator, the basic distance between its entrance and exit diaphragms is 150 mm and the entrance slit is 6.4 mm wide, has a resolution of 0.3%. 1999 Elsevier Science B.V. All rights reserved. PACS: 29.30.-h; 29.30.Aj; 29.30Dn; 29.30.Ep Keywords: Analyser; Aberration; Dispersion; Focusing; Monochromator; Resolution
1. Introduction Earlier [1] we considered electron-optical characteristics of some electrostatic potential U"arctan (y/x), the "eld of which was named `the "eld of dihedral anglea. Its equipotential surfaces are halfplanes comings together at the common z-axis,
* Corresponding author. Tel.: #7-812-552-7516; fax: #78122472088. E-mail address: [email protected]. (S.N. Davydov)
where U has a singularity. Let a parallel ribbon-like #ow of thickness G and width *z, having crossed the border U"0, move towards the z-axis. The last only means that the distance between the particles and the z-axis becomes smaller, and does not mean the exact directivity of the particles towards the z-axis (see Fig. 1). In this case, after leaving the "eld through the same equipotential U"0, the beam "nds itself constricted in thickness in the following way: the ratio of the initial section thickness H to the same value at the exit H , both sections being
0168-9002/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 1 5 3 5 - 6
146
P.G. Gabdullin et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 145}150
produced by the said zero equipotential, equals the ratio of the entrance's and exit's distances from the z-axis: H /H "x /x . Taking into account the dif ference between the entrance h and exit h angles, one can write down the expression for the compression value: H sin h x sin h G " . M" " H sin h x sin h G
(1)
Due to this feature, which was named telescopic compression of ribbon-like #ow, the potential U possesses a rather high energy dispersion DH, `reduced to the entrancea: DH"DH /H , where D is the usual linear dispersion. Instead of D, DH should be used in the expression of the resolving power of any energy analyser based on U [1]. That is why U-based analysers or monochromators can be of very high energy resolution. Any real #ow has non-zero divergency, and hence it needs to be focused at the exit diaphragm. Aberrations, coexisting together with focusing, have not been calculated yet for the dihedral electron-optical scheme. To know the aberrational blurring m at the exit diaphragm is of special importance in the case of U, because the size of the image is about M times as small as the size of the source.
Fig. 1. Cross-section of some equipotentials (thin solid lines) of the dihedral angle potential U"arctan (y/x) in the region 04U4p. In the picture U"0 at y(0: equipotentials U"0 and U"p are the bordering ones. If a parallel ribbon-like #ow of thickness G and width *z (thick solid arrowed lines) enters the "eld region y50 at an average coordinate x , and leaves this region at x , the #ow cross-section varies in the fashion: H /H "x /x , the width *z remaining invariant.
2. Slightly diverging beam in dihedral potential Consider electrons emitted with the same energy from a point source O , and moving in the xy-plane (Fig. 2). If the #ow enters the region of grad UO0 and leaves it through the same equipotential U"0 coinciding with the xz-plane, it means that according to the Helmholtz}Lagrange law, the beam linear compressing coe$cient M is inversely proportional to the angular magni"cation: M"G / G "*h /*h . Here *h and *h are beam angular divergences at the entrance and exit, respectively. In the common case of focusing of any order, we can evaluate aberration blurring as m"C (*h /2)I, I where C is the "rst non-zero coe$cient of the I expansion of the image width in power series on the initial half-angular divergence *h admitted in the analyser. If "rst-order focusing takes place, C "C , while for the second-order C "C and I '' I ''' C "C "0. ' '' In the present evaluations of the aberration coe$cients, all the trajectory calculations have been made numerically, by Runge-Kutta method. The x-coordinate of the source O is taken as the length unity, the mass and charge of a particle as the unit mass and charge, respectively. Just in this system of units, potential energy of the particle is numerically equal to the particle's angular coordinate, i.e. to arctan (y/x). In the case of the secondorder focus, the source and image O are in the "eld-free space: the y-coordinates of the source (-S) and image (-Q) are negative. In the particular case of the "rst-order focusing on the border of the "eld, S"Q"0.
Fig. 2. General scheme, showing the beam passage from the source O to the Gaussian image O through the dihedral-like potential U. Thick solid line represents the main (average) trajectory of the beam, while the dashed curves are the outer ones. U"0 at y(0, i.e. the xz-plane is the border of the "eld.
P.G. Gabdullin et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 145}150
3. First-order focusing at the border yⴝ0 At "rst, the situation was examined when both the point source O and the image O were on the plane y"0 (S"Q"0), "rst-order focusing conditions being ful"lled. Thus, m"C (*h /2). In the '' energy diapason 0(=(1.6 (W is the initial kinetic energy of the particle), C turns out to be a little '' bit smaller than the same value of the well-known plane mirror analyser (PMA) under the "rst-order focusing conditions [2] (Fig. 3a). It is convenient to compare these two analysing systems, because the PMA is actually a limiting case of the dihedral-like mirror analyser (DMA) when =P0. As it was shown earlier [1], linear dispersion D of the DMA is smaller than it is in the PMA, especially at high W. This is the reason for a rather steep fall of the dependence D/C "f (=) (Fig. 3b), which '' can be approximated, with an accuracy of 2%, by D/C "0.5 exp (!2.19=), while in the PMA '' D/C "0.5. Comparing the two analysers at '' ="1, we can see that their resolutions are equal to each other, if the angular divergency of DMA is 6.5 times as small. At ="1.6 this ratio equals 23.5.
4. Second-order focusing Varying S50 and Q50, it is possible to achieve the second-order focusing conditions. The results
147
for S"0 and 0.2 are presented in Fig. 4. Aberration broadening can be described in both the cases as m"2C (*h /2). Again, while the ratio D/C " ''' ''' 0.66 for the PMA, similar DMA values for S"0 and 0.2 (Fig. 4b) fall signi"cantly with W as D/C "0.66 exp (!3.22W) and D/C "0.66 ''' ''' exp(!2.33W), respectively. Analysis shows again that though DMA is possible, because of its great resolving power, to treat very wide ribbon-like #ows the angular divergency of them should be as small as possible.
5. Many-cascade DMA systems' synthesis Let us calculate the resolving power of a double-pass DMA, denoting by a superscript (1) or (2) in parentheses the values concerning the "rst or second stage, respectively. Displacement of the image in the image plane of the "rst cascade (i.e. at the exit of the "rst cascade), caused by the small change of energy *= is *x"D*=/=. Because of the second cascade 5 compressing, which can be expressed by the common equation *x"(x/x)*x, this dis placement converts into the smaller value D(*=/=) (x/x) in the image plane of the whole system. But the energy dispersion of the second cascade gives additional contribution to the image displacement: *x"D*=/=. Thus, the 5
Fig. 3. Second-order aberration coe$cient C (a), and linear energy dispersion D divided by C (b) as functions of the energy W. Dashed '' '' lines correspond to the PMA of "rst-order focusing. Solid lines represent the same values for DMA.
148
P.G. Gabdullin et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 145}150
Fig. 4. Third-order aberration coe$cient C (a), and linear energy dispersion D divided by C (b) as functions of energy W. Solid lines ''' ''' correspond to the DMA with the point source on the border of the "eld (the xz-plane in Fig. 2; S"0) and with the source placed in the "eld-free space at the distance S"0.2 from the xz-plane.
total image shift at the exit *= *= x #D *x "D 5 = = x *= x D#D . " (2) = x When the beam image moves in the plane of the exit diaphragm, the intensity of the #ow passing through the diaphragm comes to the zero level at the points, the distance between which is
x x (3) *x"2H#m"2H #m. x x Two co-factors (x/x) and (x/x) in the right hand part of Eq. (3) re#ect the diminishing input diaphragm width H in two cascades when it is projected to the exit. Equality of *x and *x , 5 m being neglected, gives the base resolving power of a double-pass device: = R " *=
1 x x x " D # D . (4) 2H x x x The general expression for an n-cascade apparatus can be derived in a similar manner:
1 L x I I\ x G R" D . (5) L 2H x x I G It is evident from Eq. (4) that the contribution to the "nal resolving power from the second cascade is bigger: its energy dispersion is `enhanceda not only by its own compressing (see the factor x/x), but by the "rst-stage compression in addition (the factor x/x). In case of dihedral-like cascades, it is very pro"table to increase their number n. Indeed, if all of them are identical, then the overall dimension of the analyser is approximately proportional to n, while the whole resolving power, according to the expression (5), grows faster than x /x to the power n. The additional extra cascade occurs to be destructive as soon as m becomes comparable with the width of the last output slit. The more is the initial divergence of the beam to be analysed or `monochromizeda, the less is the maximum appropriate number of the cascades. To check experimentally the theoretical results, two-cascade variant of monochromator was chosen.
6. Experimental test of a model electron spectrometer The scheme of the model of a new apparatus designed recently is shown in Fig. 5. The primary
P.G. Gabdullin et al. / Nuclear Instruments and Methods in Physics Research A 427 (1999) 145}150
149
Fig. 5. The scheme of a model of a high-resolution electron spectrometer based on the double-pass dihedral mirror monochromator (II), and on the analyser (V) with the plane of symmetry. The primary beam is formed by the gun (I). Having passed the monochromator, it is focused by the lens (III) on the target (IV). Re#ected electrons are collected by the channeltron (VI). 1 } shim plates; 2 } suppressing electrodes; 3 } de#ecting and correcting electrodes.
source is the slightly modi"ed electron gun of Erdman and Zipf [3]. It forms a round beam of approximately 3 mm in diameter and 13 angular divergency: *h +13. The "rst stage of mono chromatization is designed in the "rst-order focusing variant: S"Q"0. Such a scheme provides the maximum beam compressing and energy dispersion. To accept 43 diverging #ow passing through the "rst exit slit, the second cascade has second-order focusing, with S"2.1 and Q"0. Both cascades, made of titanium alloy and mounted on ceramic insulators, have the same basic unit of length: x"x"80 mm, and di mensionless pass energy W"1. The whole distance between the wide entrance diaphragm (its width H"6.4 mm) and narrow exit slit (H" 0.1 mm) is L"150 mm. The calculated resolving power of the monochromator is R "200 (resolu tion } 0.5%). Because of the great width of the second cascade entrance window, it is covered with a grid of high transparency. Having passed the both monochromatization steps, the #ow is focused on the target IV by the cone-like lens system III (see Ref. [4]). The target is a metal plate covered with thin luminophor layer to control the beam position and cross-section. A thin molybdenum grid is placed in front of it to screen luminophor charging and possible positive potential applied to the target. The characteristics of the monochromatic #ow re#ected by the target are examined by means of a high-resolution energy analyser, designed and
tested earlier [5]. It has a plane of symmetry, almost ideal focusing in this plane, and its linear energy dispersion equals 12 base dimensions, the last being equal to 50 mm. The whole size of the analyser is approximately 80 mm. At the exit of it, electrons are registered by a channel electron ampli"er (VI). During the experiment, there has been veri"ed the correspondence of the calculated parameters of the monochromator to the experimental ones. All the geometrical parameters were justi"ed within the accuracy of 1.5%, though voltage applied depends not only on the little variations of h , but on the regime of the electron gun operation as well. An experimental test has been carried out at a rather high pass energy E , because there was no magnetic shield around the spectrometer. Full width at half maximum (FWHM) of the peak of elastically re#ected electrons *E was measured as a function of E . Resolving power depends slightly on E and on the tuning of the spectrometer, but in the whole range 100(E (600 eV it can reach the value of 300. It means that at E "100 eV we can observe the peak of elastically re#ected particles of 0.33 eV in FWHM.
7. Conclusions Calculations and the experimental test show that the resolution of a two-cascade dihedral-like
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analyser or monochromator can be very high, and that the monochromator designed can hardly be in#uenced by the earth's magnetic "eld. Actually, resolution achieves 0.33% in the device of 150 mm base dimension at 6.4 mm entrance slit width and at pass energy E "100 eV without magnetic pro tection. In any traditional spectrometer of the same size, such a resolution would be achievable, without preliminary deceleration, at the entrance diaphragm width H "0.4}0.5 mm. On the other hand, a two-cascade DMA device is usually able to treat only a beam of less than 13 angular divergency. A dihedral "eld possesses the ability to compress the monoenergetic components of a beam, this ability acting simultaneously with the usual possibilities to focus and separate primary electrons according to their primary energies. Because of this rare property, possible future application of the DMA can be seen in the following "elds. First of all, it can be monochromatization of electron probing beams for di!erent kinds of highresolution electron spectroscopy. Rough numerical estimates show that at low pass energies the principal restriction, imposed by space charge repulsion on the intensity of monoenergetic probing beam [6], may be 2}4 times overcome. To apply successfully several-cascade DMA in this area, additional
investigations are desirable to obtain new low voltage electron guns of wide, intensive, almost parallel primary beams. Another application is energy analysis with high angular resolution of slightly diverging low intensity charged particle beams, when the size of the source is not small, e.g. in the "eld of photoelectron or ion spectroscopy. Possibly, parallel cosmic beta-rays could be a good object for many-cascade DMAs. At last, there can be some technological applications dealing with compressing electron beams in processes like melting.
References [1] S.N. Davydov, Yu.K. Golikov, V.V. Korablev, S.N. Romanov, SPIE Proc. 3345 (January 1998) 136. [2] G.A. Harrower, Rev. Sci. Instrum. 26 (1955) 850. [3] P.W. Erdman, E.C. Zipf, Rev. Sci. Instrum. 53 (1982) 225. [4] S.N. Davydov, V.Yu. Kolomenkov, V.A. Fedotov, Trudy LPI Leningrad, USSR 429 (1989) 73. [5] S.N. Davydov, Yu.A. Kudinov, Yu.K. Golikov, V.V. Korablev, J. Electron Spectrosc. Related Phenomena 72 (1995) 317. [6] M. Mishijima, Y. Kubota, K. Kondo, J. Yoshinobu, M. Onchi, Rev. Sci. Instrum. 58 (1987) 307.