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Focusing of charged particle beams with an energy-angular correlation
Nuclear Instruments and Methods in Physics Research A 427 (1999) 203}208
Focusing of charged particle beams with an energy-angular correlation S.Y. Yavor Iowe Physico-Technical Institute of Russian Academy of Sciences, Politekhnitcheskaya ul. 26, 194021 St. Petersburg, Russia
Abstract Conditions for focusing and the focal line inclination have been calculated for a charged particle beam with an energy-angular correlation as it passes through a system with the lateral dispersion. Formulae are presented for some types of electrostatic and magnetic energy analyzers. The parameters for a planar condenser and a cylindrical de#ector are calculated in detail. 1999 Elsevier Science B.V. All rights reserved. Keywords: Energy analyzer; Energy-angular correlation; Focusing; Focal line; Planar condenser; Cylindrical de#ector
1. Introduction In some electron-optical applications, for example in studies of intersecting beams, #uxes of charged particles arise that are characterized by ordered energy distributions across the beam lateral section rather than by separate energy and angular spreads. In detail, inelastic ion}ion collision processes result in ion #uxes that are products of collisions, whose energy in the laboratory coordinate frame changes in a known way along some direction over its cross section. Therefore, the problem of identifying and recording ions that belong to a speci"c process requires a simultaneous separation of particles with respect to their energies and the angular focusing of the beam with a known energy-angular correlation. Consider qualitatively a focusing of such a beam outgoing a point source. We assume that the energy of particles that emerge with initial angles from
!a to a with respect to the beam axis changes linearly from E !*E to E #*E, where E is the energy of the particle moving along the axial trajectory. When this beam passes through an energy analyzer with the lateral dispersion, it bends. We assume that the angle a is positive for the traject ory with a smaller initial curvature. A particle characterized by the initial angle a with the energy E #*E is de#ected weakly by the analyzer "eld, and consequently the point of its intersection with the axial trajectory is somewhat further away than that of the particle characterized by the same initial angle and the energy E . A particle characterized by the initial angle !a and the energy E !*E is de#ected strongly, and hence will also intersect the beam axis further away than a particle of a monoenergetic beam. Thus, a beam with an energy-angular correlation can exhibit "rst-order focusing which is however weaker than for a monoenergetic beam. Analogous consideration of the beam with
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 2 6 - 5
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S.Y. Yavor / Nuclear Instruments and Methods in Physics Research A 427 (1999) 203}208
a linear energy-angular correlation, where the particle energy decrease with increase of the angle a , shows that focusing in this case is stronger than for a monoenergetic beam. Obviously, the focusing of the beam with an energy-angular correlation is provided by bending of the axial trajectory, that is by the presence of the lateral dispersion in the system. This paper presents a quantitative analysis of the focusing conditions for the charged particle beam with an energy-angular correlation in various electrostatic and magnetic systems possessing the lateral dispersion and the plane of symmetry.
2. Planar condenser Consider the z-axis to be directed along the lower plate of the planar condenser, and the y-axis perpendicularly to it. A charged particle trajectory passes under the angle h with respect to the z-axis (that is with respect to the lower plate of the condenser), as shown in Fig. 1. Then the trajectory
equation reads e< y"! z#z tan h. 4Eg cos h
(1)
Here e is the particle charge, E its initial energy E"0.5mv, < is the potential di!erence between the condenser plates, and g is the gap between these plates. In the trajectory apex its coordinates are Eg y " sin h,
e<
Eg z" sin 2h. e<
(2)
Denote by z the coordinate of the intersection of K the trajectory with the lower condenser plate: 2Eg sin 2h. z " K e<
(3)
Let h be the angle and E the energy correspond ing to the axial trajectory, and h $a the angles and E"E (1$ka ) the energies corresponding to the outer trajectories. Here k is the factor of the energy-angular correlation. With these notations
Fig. 1. Charged particle trajectories in a planar condenser. Solid curves are trajectories for an ordinary beam (k"0), dotted curves are for the beam where the energy increases with the increased angle (k"0.5), dashed lines are for the beam where the energy decreases with the increased angle (k"!0.5). The dimensions are scaled to (2E g)/(e<).
S.Y. Yavor / Nuclear Instruments and Methods in Physics Research A 427 (1999) 203}208
the coordinates z of the outer trajectories, accuK rate to a, read 2E g z " [sin 2h $a (2 cos 2h #k sin 2h ) K e< #2a(!sin 2h #k cos 2h )].
(4)
Here the "rst term corresponds to the point of the intersection with the lower plate of the axial trajectory, and the focusing condition is the vanishing of the coe$cient with the "rst power of a . Thus we obtain the following value of the initial inclination angle h of the axial trajectory, with which the particle beam with the linear energy-angular correlation focuses at the lower condenser plate: tan 2h "!2/k.
(5)
Eq. (5) shows that the "rst-order focusing is possible and the initial inclination angle for the axial trajectory is de"ned by the correlation coe$cient. Consider an example. Let the initial angle of the beam opening be 2a "0.04, and the relative angu lar deviation across the beam be 2%. Then the value of k is k"$0.5, where the plus sign corresponds to the energy increase with the increased angle, and the minus sign to the energy decrease. Having substituted these values of k in Eq. (5) we obtain that in the "rst case the angle corresponding to the beam focusing at the lower condenser plate is h +523, and in the second case it is h +383. Consequently, the value z decreases as compared K with the case of the monoenergetic beam and in both cases is z +0.97(2E g)/(e<). The value K of y in the "rst case increases: y +
0.62(E g)/(e<), and in the second case decreases: y +0.38(E g)/(e<) (see Fig. 1). In both cases the
focal line coincides with the z-axis. The coe$cient with a in Eq. (4) characterizes the second-order spherical aberration C of the beam with an energy-angular correlation. Taking into account Eq. (5) we have 4E g C"! sin 2h (1#2 cot 2h ). e<
(6)
The "rst term in the brackets corresponds to the spherical second-order aberration of the mono-
205
energetic beam. Besides, in the aberration coe$cient an additional term appeared with the same sign. Thus the spherical aberration of the beam with an energy-angular correlation increases. In our example for both cases we have C" !1.09(4E g)/(e<), that is with the energy increas ing and decreasing with the angle the coe$cient increased by 9%. The dispersion in the planar condenser is *z 2Eg D"E K" sin 2h . D *E e<
(7)
It decreases as compared with the ordinary beam, and in our example reduces by 3%. The normalized dispersion D/Ca also decreases because of the reduced D and increased coe$cient C by 11%. Note that the calculation of the planar condenser parameters can be performed analogously for the linear energy distribution along the beam cross section also in the case of the source and the detector located outside the lower plate.
3. Electrostatic and magnetic de6ectors with circular axial trajectories In the systems under consideration the trajectory coordinate x(z) in the median plane reads in the image space (accurate up to the second order): x(z)"(x"x)x #(x"a)a #(x"d)d#(x"xx)x #(x"xa)x a #(x"xd)x d#(x"aa)a #(x"ad)a d#(x"dd)d.
(8)
Here x(z) is the distance from the axial beam trajectory, the subscript 0 denotes initial values of the parameters, d"*E/E is the relative energy spread, the expressions in parentheses are the aberration coe$cients depending on the coordinate z. Generally, particles with the same values of x and a can have di!erent energies * E; moreover, for particles that leave the same point at di!erent angles there can be additional changes in energy proportional to this angle, that is *E"ka E . So,
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the expression for d can be written as *E * E#ka E "d #ka . d" " E E
(9)
Substituting d from Eq. (9) into the expression of Eq. (8) for the trajectory, we "nd that additional terms appear in it that depend on the initial trajectory angle. The condition of the paraxial focusing is that x(z) is independent of a to "rst order. From this a new focusing equation follows: for z"z , G (x"a) #k(x"d) "0. G G
(10)
For k"0 this becomes the well-known condition for focusing of a monoenergetic beam. Having obtained the coe$cients (x"a) and (x"d) for a speci"c type of the analyzer, and substituting the values of the parameter k as well as the distance l from the point object to the analyzer entrance into Eq. (10), we can calculate the value l that determines the position of the Gaussian plane for a beam with an energy-angular correlation. Obviously, it depends on the magnitude of the energy change over the beam cross section. The magni"cation M and dispersion D are characterized by the coe$cients (x"x) and (x"d) respecG G tively, in which one must substitute the value l found from Eq. (10). The expressions for the coe$cients in Eq. (8) required to calculate the "rst-order parameters in the case of cylindrical and spherical de#ectors, were given in Ref. [1]. For magnetic sectors with uniform and radially nonuniform "elds they were obtained in the matrix form in Ref. [2]. Consider how the coe$cient of the second-order spherical aberration C changes when an energyangular correlation is present in the charged particle beam. Substituting the value of d from Eq. (9) into the expansion of Eq. (8) and grouping the terms with a, we obtain C"(x"aa) #k(x"ad) #k(x"dd) . G G G
(11)
Here the subscript i corresponds to the Gaussian plane. Consider the angle of the inclination of the focal line for the beam with the energy-angular correla-
tion. Let the trajectories emerge from a point at the z-axis. Their coordinates x(z) are determined by Eq. (8) if we set x "0 and substitute d from Eq. (9). The expansion for the angle a(z) accurate to the second order, is a(z)"(a"a)a #(a"d)[d #ka ]#(a"aa)a #(a"ad)a [d #ka ]#(a"dd)[d #ka ]. (12) The expansion coe$cients are determined by the type of the analyzer and depend on the coordinate z. In the image space the expression for the coordinate x(z) in a plane o!set by *z from a certain reference plane takes the following form, taking into account Eqs. (8) and (12): x(z#*z)"[(x"a)#*z(a"a)]a #[(x"d)#*z(a"d)][d #ia ] #[(x"aa)#*z(a"aa)]a #[(x"ad)#*z(a"ad)]a [d #ia ] #[(x"dd)#*z(a"dd)][d #ia ]. (13) The condition that the image be located in this plane is dx/da "0. In the linear approximation with respect to the angle a and parameter d we obtain (x"a)#i(x"d)#(x"ad)d #2i(x"dd)d . (14) *z"! (a"a)#i(a"d)#(a"ad)d #2i(a"dd)d When Eq. (10) holds, that is, when the coe$cients are determined in the Gaussian plane, we have in the linear approximation with respect to d , (x"ad) #2i(x"dd) G G, x"d (x"d) . *z"!d (a"a) #i(a"d) G G G
(15)
The value of *z/x determines the tangent of the inclination angle c of the focal line, for a charged particle beam with an energy-angular correlation, to the Gaussian plane. The angle c depends on the magnitude of the second-order chromatic aberration.
S.Y. Yavor / Nuclear Instruments and Methods in Physics Research A 427 (1999) 203}208
4. Cylindrical de6ector As an example consider the charged particle transport through a cylindrical de#ector for positive and negative values of the parameter i. Fig. 2 shows two identical de#ectors located symmetrically relative to the beam axis. The dashed curves are trajectories in monoenergetic beams. They are also symmetric with respect to this axis, as is the position of the image and the focal lines. The solid curves represent the trajectories in a beam with an energy-angular correlation; from left to right in the plane of the picture the charged particle energy decreases. The "gure clearly demonstrates a strong dependence of trajectories as well as of image positions on the direction of de#ection in the analyzers. The expression for paraxial trajectories in the image space has the form x"x (cos (2u!(2l sin (2u) #a [(l #l ) cos (2u #(1/(2!(2l l ) sin(2u] (16) #(d/2)(1!cos(2u#(2l sin(2u). Here u denotes the angle of de#ection of the analyzer. Eq. (16) is obtained in the sharp-cuto!
Fig. 2. Focusing of a charged particle beam with an energyangular correlation in cylindrical analyzers with di!erent directions of de#ection. For clarity of comparison the entrances of analyzers coincide.
207
approximation for the electrostatic "eld at the edges of the condenser. Substituting into Eq. (16) the expression for d from Eq. (9) and using Eq. (10), we obtain for l : l " sin(2u#(2l cos(2u#(1(2)i(1!cos(2u) . 2l sin (2u!(2 cos (2u!i sin (2u (17) Here and below linear quantities are given in units of radius of curvature r of the axial trajectory. Substituting this value of l into Eq. (16), we can "nd the magni"cation and dispersion of the analyzer. We use now the obtained results to numerically calculate the parameters of the speci"c analyzers shown schematically in Fig. 2. For the analyzer that de#ects the charged particles clockwise (right-hand analyzer), particles with the angle a have the en ergy E #*E and consequently the parameter i in this case is positive. For the analyzer that de#ects the particles counterclockwise (the left-hand analyzer), the angle a corresponds to the energy E !*E and thus i is negative. Assume the initial angular spread of the beam to be 2a "0.04 rad, while the relative energy spread over the beam cross-section is 2*E/E "2%; then i"$0.5. Consider the analyzer with the geometrical characteristics u"p/(2(2)+643 and l "0.7. Calcu lations based in Eq. (17) give for the right-hand analyzer in Fig. 2 a value l>"1.51, and for the left-hand analyzer l\"0.34. For i"0 we obtain l"0.714. The magni"cation and dispersion are M>"!2.14, D>"1.57 and M\"!0.48, D\"0.74, respectively. For the ordinary beam we have M"!1.0, D"1.0. From these results it follows that, although the dispersion for i'0 is larger than for i(0 by roughly a factor of 2, the ratio D/M is larger for i(0. Here we will not give explicitly the lengthy expressions for the second-order chromatic aberration coe$cients (x"ad) and (x"dd) that the present G G in Eqs. (11) and (15). They were calculated using the computer program ISIOS by Yavor and Berdnikov [3]. We calculated these aberrations in the sharpcuto! approximation. Substituting the results of
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S.Y. Yavor / Nuclear Instruments and Methods in Physics Research A 427 (1999) 203}208
the corresponding calculations into Eqs. (11) and (15), we determine the coe$cient of spherical aberration C and the inclination angle c for the examples under consideration. They are, respectively, C>"!2.7, C\"!3.3, C"!2.8, c>"72.53, c\"683, c"703. While the angles of inclination of the focal line are only slightly dependent on i, the coe$cient C shows more signi"cant dependence on this parameter.
6. Conclusion The results of the calculation show that for charged particle beams with an energy-angular correlation a quality of focusing can be achieved in systems with a curvilinear axis, that is similar to the quality of focusing of monochromatic beams. The formulae are obtained that allow one to calculate most important electron-optical characteristics of some electrostatic and magnetic systems that focus such beams.
5. Some other types of de6ectors Using Eqs. (10), (11) and (15), it is not di$cult to "nd the position of the Gaussian plane, the spherical aberration coe$cient C and the focal line inclination for those types of the energy analyzers where the expansions of the trajectories in the vicinity of the optic axis can be represented analytically. For example, for a spherical de#ector we have sin u#l cos u#i(1!cos u) (18) l " (l !i)sin u!cos u Here we use the same notations as for a cylindrical de#ector. In case of a homogeneous magnetic sector "eld Eq. (18) also holds, if i is replaced by 0.5i in this equation.
Acknowledgements The author is grateful to Prof. V.V. Afrosimov for the statement of the problem and valuable discussions. References [1] V.P. Afanasjev, S.Y. Yavor, Electrostatic Energy Analyzers for Charged Particle Beams, Nauka, Moscow, 1978 (in Russian). [2] H. Wollnik, Optics of Charged Particles, Academic Press, Orlando, FL, 1987. [3] M.I. Yavor, A.S. Berdnikov, Nucl. Instr. and Meth. A 363 (1995) 416.
Focusing of charged particle beams with an energy-angular correlation S.Y. Yavor Iowe Physico-Technical Institute of Russian Academy of Sciences, Politekhnitcheskaya ul. 26, 194021 St. Petersburg, Russia
Abstract Conditions for focusing and the focal line inclination have been calculated for a charged particle beam with an energy-angular correlation as it passes through a system with the lateral dispersion. Formulae are presented for some types of electrostatic and magnetic energy analyzers. The parameters for a planar condenser and a cylindrical de#ector are calculated in detail. 1999 Elsevier Science B.V. All rights reserved. Keywords: Energy analyzer; Energy-angular correlation; Focusing; Focal line; Planar condenser; Cylindrical de#ector
1. Introduction In some electron-optical applications, for example in studies of intersecting beams, #uxes of charged particles arise that are characterized by ordered energy distributions across the beam lateral section rather than by separate energy and angular spreads. In detail, inelastic ion}ion collision processes result in ion #uxes that are products of collisions, whose energy in the laboratory coordinate frame changes in a known way along some direction over its cross section. Therefore, the problem of identifying and recording ions that belong to a speci"c process requires a simultaneous separation of particles with respect to their energies and the angular focusing of the beam with a known energy-angular correlation. Consider qualitatively a focusing of such a beam outgoing a point source. We assume that the energy of particles that emerge with initial angles from
!a to a with respect to the beam axis changes linearly from E !*E to E #*E, where E is the energy of the particle moving along the axial trajectory. When this beam passes through an energy analyzer with the lateral dispersion, it bends. We assume that the angle a is positive for the traject ory with a smaller initial curvature. A particle characterized by the initial angle a with the energy E #*E is de#ected weakly by the analyzer "eld, and consequently the point of its intersection with the axial trajectory is somewhat further away than that of the particle characterized by the same initial angle and the energy E . A particle characterized by the initial angle !a and the energy E !*E is de#ected strongly, and hence will also intersect the beam axis further away than a particle of a monoenergetic beam. Thus, a beam with an energy-angular correlation can exhibit "rst-order focusing which is however weaker than for a monoenergetic beam. Analogous consideration of the beam with
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 2 6 - 5
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S.Y. Yavor / Nuclear Instruments and Methods in Physics Research A 427 (1999) 203}208
a linear energy-angular correlation, where the particle energy decrease with increase of the angle a , shows that focusing in this case is stronger than for a monoenergetic beam. Obviously, the focusing of the beam with an energy-angular correlation is provided by bending of the axial trajectory, that is by the presence of the lateral dispersion in the system. This paper presents a quantitative analysis of the focusing conditions for the charged particle beam with an energy-angular correlation in various electrostatic and magnetic systems possessing the lateral dispersion and the plane of symmetry.
2. Planar condenser Consider the z-axis to be directed along the lower plate of the planar condenser, and the y-axis perpendicularly to it. A charged particle trajectory passes under the angle h with respect to the z-axis (that is with respect to the lower plate of the condenser), as shown in Fig. 1. Then the trajectory
equation reads e< y"! z#z tan h. 4Eg cos h
(1)
Here e is the particle charge, E its initial energy E"0.5mv, < is the potential di!erence between the condenser plates, and g is the gap between these plates. In the trajectory apex its coordinates are Eg y " sin h,
e<
Eg z" sin 2h. e<
(2)
Denote by z the coordinate of the intersection of K the trajectory with the lower condenser plate: 2Eg sin 2h. z " K e<
(3)
Let h be the angle and E the energy correspond ing to the axial trajectory, and h $a the angles and E"E (1$ka ) the energies corresponding to the outer trajectories. Here k is the factor of the energy-angular correlation. With these notations
Fig. 1. Charged particle trajectories in a planar condenser. Solid curves are trajectories for an ordinary beam (k"0), dotted curves are for the beam where the energy increases with the increased angle (k"0.5), dashed lines are for the beam where the energy decreases with the increased angle (k"!0.5). The dimensions are scaled to (2E g)/(e<).
S.Y. Yavor / Nuclear Instruments and Methods in Physics Research A 427 (1999) 203}208
the coordinates z of the outer trajectories, accuK rate to a, read 2E g z " [sin 2h $a (2 cos 2h #k sin 2h ) K e< #2a(!sin 2h #k cos 2h )].
(4)
Here the "rst term corresponds to the point of the intersection with the lower plate of the axial trajectory, and the focusing condition is the vanishing of the coe$cient with the "rst power of a . Thus we obtain the following value of the initial inclination angle h of the axial trajectory, with which the particle beam with the linear energy-angular correlation focuses at the lower condenser plate: tan 2h "!2/k.
(5)
Eq. (5) shows that the "rst-order focusing is possible and the initial inclination angle for the axial trajectory is de"ned by the correlation coe$cient. Consider an example. Let the initial angle of the beam opening be 2a "0.04, and the relative angu lar deviation across the beam be 2%. Then the value of k is k"$0.5, where the plus sign corresponds to the energy increase with the increased angle, and the minus sign to the energy decrease. Having substituted these values of k in Eq. (5) we obtain that in the "rst case the angle corresponding to the beam focusing at the lower condenser plate is h +523, and in the second case it is h +383. Consequently, the value z decreases as compared K with the case of the monoenergetic beam and in both cases is z +0.97(2E g)/(e<). The value K of y in the "rst case increases: y +
0.62(E g)/(e<), and in the second case decreases: y +0.38(E g)/(e<) (see Fig. 1). In both cases the
focal line coincides with the z-axis. The coe$cient with a in Eq. (4) characterizes the second-order spherical aberration C of the beam with an energy-angular correlation. Taking into account Eq. (5) we have 4E g C"! sin 2h (1#2 cot 2h ). e<
(6)
The "rst term in the brackets corresponds to the spherical second-order aberration of the mono-
205
energetic beam. Besides, in the aberration coe$cient an additional term appeared with the same sign. Thus the spherical aberration of the beam with an energy-angular correlation increases. In our example for both cases we have C" !1.09(4E g)/(e<), that is with the energy increas ing and decreasing with the angle the coe$cient increased by 9%. The dispersion in the planar condenser is *z 2Eg D"E K" sin 2h . D *E e<
(7)
It decreases as compared with the ordinary beam, and in our example reduces by 3%. The normalized dispersion D/Ca also decreases because of the reduced D and increased coe$cient C by 11%. Note that the calculation of the planar condenser parameters can be performed analogously for the linear energy distribution along the beam cross section also in the case of the source and the detector located outside the lower plate.
3. Electrostatic and magnetic de6ectors with circular axial trajectories In the systems under consideration the trajectory coordinate x(z) in the median plane reads in the image space (accurate up to the second order): x(z)"(x"x)x #(x"a)a #(x"d)d#(x"xx)x #(x"xa)x a #(x"xd)x d#(x"aa)a #(x"ad)a d#(x"dd)d.
(8)
Here x(z) is the distance from the axial beam trajectory, the subscript 0 denotes initial values of the parameters, d"*E/E is the relative energy spread, the expressions in parentheses are the aberration coe$cients depending on the coordinate z. Generally, particles with the same values of x and a can have di!erent energies * E; moreover, for particles that leave the same point at di!erent angles there can be additional changes in energy proportional to this angle, that is *E"ka E . So,
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the expression for d can be written as *E * E#ka E "d #ka . d" " E E
(9)
Substituting d from Eq. (9) into the expression of Eq. (8) for the trajectory, we "nd that additional terms appear in it that depend on the initial trajectory angle. The condition of the paraxial focusing is that x(z) is independent of a to "rst order. From this a new focusing equation follows: for z"z , G (x"a) #k(x"d) "0. G G
(10)
For k"0 this becomes the well-known condition for focusing of a monoenergetic beam. Having obtained the coe$cients (x"a) and (x"d) for a speci"c type of the analyzer, and substituting the values of the parameter k as well as the distance l from the point object to the analyzer entrance into Eq. (10), we can calculate the value l that determines the position of the Gaussian plane for a beam with an energy-angular correlation. Obviously, it depends on the magnitude of the energy change over the beam cross section. The magni"cation M and dispersion D are characterized by the coe$cients (x"x) and (x"d) respecG G tively, in which one must substitute the value l found from Eq. (10). The expressions for the coe$cients in Eq. (8) required to calculate the "rst-order parameters in the case of cylindrical and spherical de#ectors, were given in Ref. [1]. For magnetic sectors with uniform and radially nonuniform "elds they were obtained in the matrix form in Ref. [2]. Consider how the coe$cient of the second-order spherical aberration C changes when an energyangular correlation is present in the charged particle beam. Substituting the value of d from Eq. (9) into the expansion of Eq. (8) and grouping the terms with a, we obtain C"(x"aa) #k(x"ad) #k(x"dd) . G G G
(11)
Here the subscript i corresponds to the Gaussian plane. Consider the angle of the inclination of the focal line for the beam with the energy-angular correla-
tion. Let the trajectories emerge from a point at the z-axis. Their coordinates x(z) are determined by Eq. (8) if we set x "0 and substitute d from Eq. (9). The expansion for the angle a(z) accurate to the second order, is a(z)"(a"a)a #(a"d)[d #ka ]#(a"aa)a #(a"ad)a [d #ka ]#(a"dd)[d #ka ]. (12) The expansion coe$cients are determined by the type of the analyzer and depend on the coordinate z. In the image space the expression for the coordinate x(z) in a plane o!set by *z from a certain reference plane takes the following form, taking into account Eqs. (8) and (12): x(z#*z)"[(x"a)#*z(a"a)]a #[(x"d)#*z(a"d)][d #ia ] #[(x"aa)#*z(a"aa)]a #[(x"ad)#*z(a"ad)]a [d #ia ] #[(x"dd)#*z(a"dd)][d #ia ]. (13) The condition that the image be located in this plane is dx/da "0. In the linear approximation with respect to the angle a and parameter d we obtain (x"a)#i(x"d)#(x"ad)d #2i(x"dd)d . (14) *z"! (a"a)#i(a"d)#(a"ad)d #2i(a"dd)d When Eq. (10) holds, that is, when the coe$cients are determined in the Gaussian plane, we have in the linear approximation with respect to d , (x"ad) #2i(x"dd) G G, x"d (x"d) . *z"!d (a"a) #i(a"d) G G G
(15)
The value of *z/x determines the tangent of the inclination angle c of the focal line, for a charged particle beam with an energy-angular correlation, to the Gaussian plane. The angle c depends on the magnitude of the second-order chromatic aberration.
S.Y. Yavor / Nuclear Instruments and Methods in Physics Research A 427 (1999) 203}208
4. Cylindrical de6ector As an example consider the charged particle transport through a cylindrical de#ector for positive and negative values of the parameter i. Fig. 2 shows two identical de#ectors located symmetrically relative to the beam axis. The dashed curves are trajectories in monoenergetic beams. They are also symmetric with respect to this axis, as is the position of the image and the focal lines. The solid curves represent the trajectories in a beam with an energy-angular correlation; from left to right in the plane of the picture the charged particle energy decreases. The "gure clearly demonstrates a strong dependence of trajectories as well as of image positions on the direction of de#ection in the analyzers. The expression for paraxial trajectories in the image space has the form x"x (cos (2u!(2l sin (2u) #a [(l #l ) cos (2u #(1/(2!(2l l ) sin(2u] (16) #(d/2)(1!cos(2u#(2l sin(2u). Here u denotes the angle of de#ection of the analyzer. Eq. (16) is obtained in the sharp-cuto!
Fig. 2. Focusing of a charged particle beam with an energyangular correlation in cylindrical analyzers with di!erent directions of de#ection. For clarity of comparison the entrances of analyzers coincide.
207
approximation for the electrostatic "eld at the edges of the condenser. Substituting into Eq. (16) the expression for d from Eq. (9) and using Eq. (10), we obtain for l : l " sin(2u#(2l cos(2u#(1(2)i(1!cos(2u) . 2l sin (2u!(2 cos (2u!i sin (2u (17) Here and below linear quantities are given in units of radius of curvature r of the axial trajectory. Substituting this value of l into Eq. (16), we can "nd the magni"cation and dispersion of the analyzer. We use now the obtained results to numerically calculate the parameters of the speci"c analyzers shown schematically in Fig. 2. For the analyzer that de#ects the charged particles clockwise (right-hand analyzer), particles with the angle a have the en ergy E #*E and consequently the parameter i in this case is positive. For the analyzer that de#ects the particles counterclockwise (the left-hand analyzer), the angle a corresponds to the energy E !*E and thus i is negative. Assume the initial angular spread of the beam to be 2a "0.04 rad, while the relative energy spread over the beam cross-section is 2*E/E "2%; then i"$0.5. Consider the analyzer with the geometrical characteristics u"p/(2(2)+643 and l "0.7. Calcu lations based in Eq. (17) give for the right-hand analyzer in Fig. 2 a value l>"1.51, and for the left-hand analyzer l\"0.34. For i"0 we obtain l"0.714. The magni"cation and dispersion are M>"!2.14, D>"1.57 and M\"!0.48, D\"0.74, respectively. For the ordinary beam we have M"!1.0, D"1.0. From these results it follows that, although the dispersion for i'0 is larger than for i(0 by roughly a factor of 2, the ratio D/M is larger for i(0. Here we will not give explicitly the lengthy expressions for the second-order chromatic aberration coe$cients (x"ad) and (x"dd) that the present G G in Eqs. (11) and (15). They were calculated using the computer program ISIOS by Yavor and Berdnikov [3]. We calculated these aberrations in the sharpcuto! approximation. Substituting the results of
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the corresponding calculations into Eqs. (11) and (15), we determine the coe$cient of spherical aberration C and the inclination angle c for the examples under consideration. They are, respectively, C>"!2.7, C\"!3.3, C"!2.8, c>"72.53, c\"683, c"703. While the angles of inclination of the focal line are only slightly dependent on i, the coe$cient C shows more signi"cant dependence on this parameter.
6. Conclusion The results of the calculation show that for charged particle beams with an energy-angular correlation a quality of focusing can be achieved in systems with a curvilinear axis, that is similar to the quality of focusing of monochromatic beams. The formulae are obtained that allow one to calculate most important electron-optical characteristics of some electrostatic and magnetic systems that focus such beams.
5. Some other types of de6ectors Using Eqs. (10), (11) and (15), it is not di$cult to "nd the position of the Gaussian plane, the spherical aberration coe$cient C and the focal line inclination for those types of the energy analyzers where the expansions of the trajectories in the vicinity of the optic axis can be represented analytically. For example, for a spherical de#ector we have sin u#l cos u#i(1!cos u) (18) l " (l !i)sin u!cos u Here we use the same notations as for a cylindrical de#ector. In case of a homogeneous magnetic sector "eld Eq. (18) also holds, if i is replaced by 0.5i in this equation.
Acknowledgements The author is grateful to Prof. V.V. Afrosimov for the statement of the problem and valuable discussions. References [1] V.P. Afanasjev, S.Y. Yavor, Electrostatic Energy Analyzers for Charged Particle Beams, Nauka, Moscow, 1978 (in Russian). [2] H. Wollnik, Optics of Charged Particles, Academic Press, Orlando, FL, 1987. [3] M.I. Yavor, A.S. Berdnikov, Nucl. Instr. and Meth. A 363 (1995) 416.