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The crystallographic information in localised characteristic loss electron images and diffraction patterns
Ultramicroscopy 7 (1981) 59 64 North-Holland Publishing Company
59
THE CRYSTALLOGRAPHIC INFORMATION IN LOCALISED CHARACTERISTIC LOSS ELECTRON IMAGES AND DIFFRACTION PATTERNS J.C.H. SPENCE Department of Physics, Arizona State University, Tempe, Arizona 85281, USA
The intensity distribution of energy-filtered intermediate and large energy loss images and diffraction patterns has been calculated. It is concluded that satisfactory interpretation of experimentally obtained images and diffraction patterns can only be achieved by matching with the calculated ones and that, in favourable cases, the local projected symmetry of coordination polyhedra may be determined from the observed symmetry of characteristic loss energy-filtered diffraction pattern~
1. Introduction The inelastic scattering of fast electrons in crystals has been studied by many authors (see ref. [1] for a review). The majority of these workers have, however, confined their attention to one of two cases: (i) the development of a theory for delocalised small energy loss processes which are found to preserve diffraction contrast and their effect on elastic diffraction contrast imaging (see for example, refs. [2,3]); and (ii) the theory for the large energy loss localised processes important in X-ray production, electron channeling and cathodoluminescence [4,5]. In addition, there is a large body of work devoted to the inelastic scattering of fast electrons by isolated atoms, based on the Bethe theory, and molecules, such as occurs in electron gas diffraction. This paper is chiefly concerned with predicting the intensity distributions which are expected to occur when energy selected electron diffraction patterns and high resolution images are formed from inelastically scattered electrons at intermediate and large energy losses in crystals. For these single electron excitations the inelastic scattering angle 0E is typically greater than the Bragg angle. The main purpose of this investigation was to determine what, if any, new crystallographic information could be obtained from the directly transmitted characteristic loss electrons themselves rather than from the decay products of these local0304-3991/81/0000-0000/$02.50 © 1981 North-Holland
ized excitations or from their absorptive effect on elastic scattering, as has been studied elsewhere [4,5]. The major uncertainty in this work involves the determination of the localisation L associated with a particular characteristic energy loss AE [4]. In the absence of detailed excited crystal wave functions, L has been taken as a measure of the width of the inelastic scattering potential, obtained as the Fourier Transform of the inelastic scattering amplitude for isolated atoms. However the results of Kuwabara and Cowley [6], in which transitions between delocalised valence electron states of the crystal were observed which correspond to inelastic scattering from a periodic potential clearly indicate the failure of this approach for small losses. The main results from this study are summarized in the last section.
2. Local symmetry information in localised characteristic-loss energy filtered diffraction patterns Since both the localisation L and column width W for characteristic loss excitations may be smaller than the crystal unit cell dimensions, the question arises as to whether the local symmetry about a particular chemical species can be deduced from the observable two-dimensional point group symmetry of the characteristic loss energy filtered electron diffraction pat-
60
J.C.H. Spence / Crystallographic information
tern for this species. Here localisation refers to the volume within which the inelasticalty scattered wave is coherently generated, while local symmetry refers to the projected symmetry of the nearest neighbor coordination polyhedra about the selected atom when this polyhedra is removed from the crystal. This local symmetry should be distinguished from (a) The symmetry of the sublattice of the selected species taken alone, which may be equal to or higher than that of the complete crystal, (b) the site symmetry of particular atoms falling on special positions in the lattice, and (c) the space-group of the crystal as a whole. The ability to determine this local symmetry would then allow the sites of selected light atoms in the lattice to be determined in many cases, both for substitutional and interstitial impurities. In the limit of very large energy losses the inelastically scattered electron wave may be thought of as originating from point sources in the crystal, and so may be calculated by the method developed by von Laue for the X-ray case, using the reciprocity theorem. The resulting energy-filtered electron diffraction pattern is then a Kossel pattern, whose symmetry properties have been discussed elsewhere [7]. Since the inelasticaUy scattered wave at a point B (in the far-field) due to a point source in the crystal at A may be estimated by calculating the wave function at A due to a source at B, and since in the electron case the wave function at A can be found to a good approximation using the column approximation, it might be expected that the inelasticaUy scattered wave at B will depend only on the crystal potential within a thin column a few A in diameter erected about the selected species (column axis parallel to incident beam). Thus the elastic scattering of the inelastically scattered wave within this column acts as a probe which senses the local crystal environment about the selected species. These qualitative ideas are tested by the dynamical calculations which follow. However the crucial parameters on which the success of the method depends are clearly the localisation L, the column width W and the specimen thickness t, as indicated in fig. 1. There are, however, some additional crystallographic considerations. Consider the projected crystal structure shown in fig. 1, and assume that an energy filtered electron diffraction pattern is formed using only those electrons which lose energy to a characteristic excitation localised at atom A. Each coordina-
×tal sublattice site local
I
symmetry
I/---O, O W
I o c a l i s a t i o n L= 2 X E . I ~ E column width w=et<3A
Does t h e e n e r g y selected from A show 2 or
d,p 6 ?
Fig. 1. Definition of the parameters L and Wwhich control the symmetry of the inelastic scattering from an atom A at a local three-fold axis.
tion polyhedra about A has three-fold symmetry, so that the kinematic filtered pattern from one atom A would show six-fold symmetry. In the more usual case of dynamical scattering where Friedel's law does not apply to the elastic scattering of the inelastically scattered wave, the filtered pattern from either atom A would show three-fold symmetry. The incoherent superposition of two such patterns (related by the crystal two-fold axis) then results in a total faltered pattern showing six-fold symmetry. Note that the three-fold operation is not an element of the crystal plane group, and would not be observed by convergent-beam electron diffraction. Thus, for localised losses in the ZOLZ approximation, the point plane group symmetry of energy selected electron diffraction patterns is equal to the projected local crystal symmetry about the selected atom, with all the crystal plane group operations also applied, and the results summed. The extension to the case where the same species occurs at the centre of different types of coordination polyhedra (e.g. tetrahedral and octohedral) is straightforward. In practice, the problem is to deduce the local symmetry from that of the energy filtered pattern for a particular species. For rotation axes in general the q-fold local projected symmetry of a coordination polyhedra can always be obtained by dividing the n-fold symmetry of the energy filtered pattern by the p-fold projected symmetry of the entire crystal, which can often be obtained by convergent beam electron diffraction, or which may be known. Here p = 2, q = 3 and n = 6 for the example shown in fig. 1. Similar results hold for mirrors, however there are some
£C.H. Spence / Crystallographic information
61
of 0 E difficult to define. At the opposite extreme of approximation for large energy losses with 0E > > 20B, the computational problem is similar to that of STEM microdiffraction using a point source in which a diffraction limited aberrated spherical wave (the electron probe) diverges from a point near the specimen surface. Earlier calculations for this case and an argument based on the integral solution of the Schr6dinger equation [9] also show that these dynamical elastic microdiffraction patterns preserve the projected symmetry of the crystal when taken about the probe centre as origin. The core-loss energy selected patterns reported here were calculated by combining the method of Doyle [10] with that used to compute STEM microdiffraction patterns. Thus a systematics multislice calculation was run using an artificial superlattice whose size exceeded the localisation L and which contained, in one slice, a single inelastic scattering potential at the lattice site of the selected species. All other slices contained many unit cells of the
intractable cases (e.g. a mirror line in the asymmetric unit not parallel to one in the crystal).
3. Calculations Methods for the calculation of the inelastic plasmon and small energy loss single electron scattering in crystals have been described by Doyle, Howie, Rez, Humphreys, Whelan and others (see ref. [8] for a review). These delocalised excitations generate inelastic scattering over a large volume and so preserve diffraction contrast mechanisms. In addition, for a perfect, parallel sided crystal their energy selected diffraction pattern will express the full crystal space group symmetry if 0 E ~ 2AE/E < 0 B, where 0 B is the Bragg angle, AE the energy loss, E the accelerating voltage and 0E is an angle beyond which there is negligible inelastic scattering. The Lorenzian form of the inelastic scattering, however, makes a useful value
selected oxygen
El~'gy
K loss ( . d y n a m i c a l
elastic)
Systematics Znelasti¢ scattering Modified
potential
FWH.M. = 2/~
hydrogeni¢
5 3 3 eV loss
plane wave i l l u m i n a t i o n
©
A
8
© 1 0 0 kV
t , 80,~,
--3A ~ s A
elastic Braggs
i
1
I
i (000)
I Oe/X
i
| 0.25
I 0.5
U (•
~)
Fig. 2. Dynamical systematics calculation of the energy filtered oxygen K-loss intensity for two positions A and B of the oxygen
atom.
J.C.H. Spence / Crystallographic information
62
elastic scattering potential. Then the results of n such multislice calculations were added incoherently, in each one of which the inelastic event was allowed to occur at a successively lower slice. This entire process must then be repeated for each site of the selected species within the unit cell, and so takes full account of the dynamical elastic scattering of the inelastically scattered wave. Wave functions for the inelastic scattering were obtained from the Born approximation, using the modified hydrogenic atomic wave functions for K-shell excitation in isolated atoms given by Egerton [11 ]. Scattering phases are lost in recovering the wave function from the published cross-sections. The use of these wave functions in crystals is expected to be a good approximation for large energy losses for smaller losses a Bloch-wave expansion of the full crystal excited state wave function is needed. For 0E > 20B the calculations reported here predict the usual channelling effects in the presence of an absorption potential. Fig. 2 shows the results of calculations for a onedimensional hypothetical CuO crystal without a centre of symmetry for two positions of the oxygen atom. Here the energy filtered oxygen K-loss intensity is shown along the (h00) systematics line. The positions at which the elastic Bragg peaks would occur are also shown along the abscissa, and the value of 0E
(which exceeds 0 B) is indicated. These calculations are for an 80 A thick specimen, illuminated by a collimated electron beam at an accelerating voltage of 100 kV. Note that the energy selected pattern from the oxygen atom gains a centre of symmetry (not possessed by the crystal as a whole) as this atom is moved away from the copper atom from position B to position A. In this one-dimensional example the "coordination polyhedra" is the single oxygen atom, which possesses a center of symmetry (the only symmetry element possible in one dimension). The important point is that for localised excitations the symmetry of energy f'fltered patterns may be higher than that of the elastic pattern. This and similar calculations have been used to show that for losses greater than about 300 eV in specimens 100 A thick the column width W and localisation L are less than 3 A for 100 kV incident fast electrons. In the absence of true crystal excited state wave functions, however, this is necessarily a rather rough estimate. The effect of increasing the localisation is shown by the results given in fig. 3 for the 110 eV K shell loss of beryllium. Here, with 0E < 0B, peaks are seen to emerge at the sites of the elastic Bragg peaks since the localisation is now larger than the unit cell. Thus, as one increases in energy, the energy filtered diffraction pattern changes continuously from one showing sharp Bragg spots at low energy to a continuous Kossel pattern (containing line features) at larger losses, as observed experimentally [ 12].
4. Imaging at large losses Be A
©e ff
B
®©
K loss
t:so£ 1 0 0 kV Axial
Fig. 3. The energy filtered beryllium K-loss intensity along the systematics line, showing the emergence of peaks at the Bragg positions.
The formation of lattice images from delocalised excitations has been discussed in detail elsewhere [ 13 ]. While formation of a high resolution electron image from large energy-loss energy Filtered electrons presents more severe experimental difficulties (background subtraction, poor statistics, etc.), it is worthwhile considering also some of the fundamental problems which would be involved in the interpretation of these images. The difficulties arise chiefly from the finite localisation L and from the effects of multiple elastic scattering. Fig. 4 shows the image which would be synthesised from the scattering shown in fig. 2 using, for example, a Castaing-Henry imaging energy f'flter tuned to the 533 eV oxygen K-sheU exci-
J.C.H. Spence /Crystallographic information
Oxyoen energy
63
K - shell e x c i t a t i o n filtered
1~ point t=80~,100 Dynamical
TEM
im~g.~
r e s o l u t i o n , a x i a l imaging kV, s y s t e m a t ics
calculation.
7
8•
unit cell
15
•16
Fig. 4. High resolution TEM energy filtered image obtained from the scattering shown in fig. 2 at the Scherzer focus (bright field, point resolution 1 A). Note that despite energy filtering for the oxygen loss, the copper atom also appears in the image due to the effects of finite localisation and multiple elastic scattering.
tation. An atomic resolution electron microscope has been assumed (Cs -- 0.01 ram, Scherzer focus) so that all image "artifacts" are due to the scattering process itself. These images are again formed using an incoherent sum of dynamical image calculations for all depths for the inelastic event. Thus the total image intensity Io(x) is given by
lo(x) = ~
I ( x - na),
n
where
l(x) = ~ ~ i
I¢i,j(x)l ~ ,
I
with a the unit cell constant and ~i,j (x) the dynamical image amplitude for an inelastic event in the jth slice occurring at the ith site for the selected species in the unit cell. The simple notion that high resolution characteristic-loss energy selected images reveal only the sites of the selected species in the lattice is thus seen to hold only in the limit of very large losses and very thin specimens. For intermediate cases, such as that shown in fig. 4, the image from each atom will be spread at least over a distance equal to the localisation L and may include images of other atomic spe-
cies which lie within a distance L of the selected species (this is a consequence of the elastic scattering of the inelastically scattered electron). The convenient Rayleigh criterion cannot be used to estimate the spatial resolution of a f'fltered core-loss image in view of the extended tails on the Fourier transform of the inelastic scattering distribution. These filtered images reveal the selected species sublattice symmetry for very large losses, and provide increasing information on the local symmetry about the selected atom as the energy loss is reduced. In addition, all of the normal difficulties associated with the interpretation of elastic high resolution lattice images apply and a similar sensitivity to instrumental parameters can be expected. Since, in the limit of large losses, the inelastic "sources" imaged scatter into a broad range of angles, the images will be less sensitive to crystalline orientation than elastic images, and since the integration through depth is incoherent they may well also be less sensitive to thickness changes. Nevertheless, detailed numerical calculations will be needed for the interpretation of these images at realistic crystal thicknesses for 100 kV electrons. At present these calculations are only practical in two dimensions using array-processor hardware.
64
J.C.H. Spence / Crystallographic information
5. Conclusions The main conclusions which result from the study of many dynamical core loss calculations, some o f which have been reported here, are as follows: (i) That the interpretation o f core loss images will require a detailed comparison with computed images since, for example, atomic resolution images formed in a loss characteristic of a particular species may include atom images o f other chemical species due to elastic scattering o f the inelastically scattered electrons. (ii) That the experimentally observed [12] transition from a sharp Bragg spot pattern to a Kossel pattern with increasing energy loss can be qualitatively accounted for within the multislice algorithm, suitably modified. (iii) That there is some hope that, in favourable cases, the local projected symmetry o f coordination polyhedra may be determined from the observed symmetry o f characteristic loss energy filtered diffraction patterns..
Acknowledgment I am grateful to Drs. P. Goodman, J.M. Cowley and J. Steeds for useful discussions. Work supported
by ARO contract No. DAAG-29-80-C.-0080 and the ASU NSF HREM Facility.
References [ 1 ] J.M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975). [2] A. Howie, Proc. Roy. Soc. (London) A271 (1963) 268. [3] C.J. Humphreys and M.J. Whelan, Phil. Mag. 20 (1969) 165. [4] S.J. Pennycook and A. Howie, Phil. Mag. 41 (1980) 809. [5] G. Lehmpfuhl and J. Tafto, in: Electron Microscopy 1980, Vol. 3, p. 62. [6] S. Kuwabara and J.M. Cowley, J. Phys. Soc. Japan 34 (1973) 1575. [7] J.A. Eades, J. Appl. Cryst. 13 (1980) 368. [8] C.J. Humphreys, Rept. Progr. Phys., in press. [9] J.M. Cowley and J.C.H. Spence, Ultramicroscopy 3 (1979) 433. [10] P.A. Doyle, Acta Cryst. A25 (1969) 569. [11] R.F. Egerton, Ultramicroscopy 4 (1979) 169. [12] J.G. Philip, M.J. Whelan and R.F. Egerton, in: Proc. 8th Intern. Congr. on Electron Microscopy, 1974, Vol. 1, p. 276. [13] A. Craven and C. Colliex, Inst. Phys. Conf. Ser. 36 (1977) 271.
59
THE CRYSTALLOGRAPHIC INFORMATION IN LOCALISED CHARACTERISTIC LOSS ELECTRON IMAGES AND DIFFRACTION PATTERNS J.C.H. SPENCE Department of Physics, Arizona State University, Tempe, Arizona 85281, USA
The intensity distribution of energy-filtered intermediate and large energy loss images and diffraction patterns has been calculated. It is concluded that satisfactory interpretation of experimentally obtained images and diffraction patterns can only be achieved by matching with the calculated ones and that, in favourable cases, the local projected symmetry of coordination polyhedra may be determined from the observed symmetry of characteristic loss energy-filtered diffraction pattern~
1. Introduction The inelastic scattering of fast electrons in crystals has been studied by many authors (see ref. [1] for a review). The majority of these workers have, however, confined their attention to one of two cases: (i) the development of a theory for delocalised small energy loss processes which are found to preserve diffraction contrast and their effect on elastic diffraction contrast imaging (see for example, refs. [2,3]); and (ii) the theory for the large energy loss localised processes important in X-ray production, electron channeling and cathodoluminescence [4,5]. In addition, there is a large body of work devoted to the inelastic scattering of fast electrons by isolated atoms, based on the Bethe theory, and molecules, such as occurs in electron gas diffraction. This paper is chiefly concerned with predicting the intensity distributions which are expected to occur when energy selected electron diffraction patterns and high resolution images are formed from inelastically scattered electrons at intermediate and large energy losses in crystals. For these single electron excitations the inelastic scattering angle 0E is typically greater than the Bragg angle. The main purpose of this investigation was to determine what, if any, new crystallographic information could be obtained from the directly transmitted characteristic loss electrons themselves rather than from the decay products of these local0304-3991/81/0000-0000/$02.50 © 1981 North-Holland
ized excitations or from their absorptive effect on elastic scattering, as has been studied elsewhere [4,5]. The major uncertainty in this work involves the determination of the localisation L associated with a particular characteristic energy loss AE [4]. In the absence of detailed excited crystal wave functions, L has been taken as a measure of the width of the inelastic scattering potential, obtained as the Fourier Transform of the inelastic scattering amplitude for isolated atoms. However the results of Kuwabara and Cowley [6], in which transitions between delocalised valence electron states of the crystal were observed which correspond to inelastic scattering from a periodic potential clearly indicate the failure of this approach for small losses. The main results from this study are summarized in the last section.
2. Local symmetry information in localised characteristic-loss energy filtered diffraction patterns Since both the localisation L and column width W for characteristic loss excitations may be smaller than the crystal unit cell dimensions, the question arises as to whether the local symmetry about a particular chemical species can be deduced from the observable two-dimensional point group symmetry of the characteristic loss energy filtered electron diffraction pat-
60
J.C.H. Spence / Crystallographic information
tern for this species. Here localisation refers to the volume within which the inelasticalty scattered wave is coherently generated, while local symmetry refers to the projected symmetry of the nearest neighbor coordination polyhedra about the selected atom when this polyhedra is removed from the crystal. This local symmetry should be distinguished from (a) The symmetry of the sublattice of the selected species taken alone, which may be equal to or higher than that of the complete crystal, (b) the site symmetry of particular atoms falling on special positions in the lattice, and (c) the space-group of the crystal as a whole. The ability to determine this local symmetry would then allow the sites of selected light atoms in the lattice to be determined in many cases, both for substitutional and interstitial impurities. In the limit of very large energy losses the inelastically scattered electron wave may be thought of as originating from point sources in the crystal, and so may be calculated by the method developed by von Laue for the X-ray case, using the reciprocity theorem. The resulting energy-filtered electron diffraction pattern is then a Kossel pattern, whose symmetry properties have been discussed elsewhere [7]. Since the inelasticaUy scattered wave at a point B (in the far-field) due to a point source in the crystal at A may be estimated by calculating the wave function at A due to a source at B, and since in the electron case the wave function at A can be found to a good approximation using the column approximation, it might be expected that the inelasticaUy scattered wave at B will depend only on the crystal potential within a thin column a few A in diameter erected about the selected species (column axis parallel to incident beam). Thus the elastic scattering of the inelastically scattered wave within this column acts as a probe which senses the local crystal environment about the selected species. These qualitative ideas are tested by the dynamical calculations which follow. However the crucial parameters on which the success of the method depends are clearly the localisation L, the column width W and the specimen thickness t, as indicated in fig. 1. There are, however, some additional crystallographic considerations. Consider the projected crystal structure shown in fig. 1, and assume that an energy filtered electron diffraction pattern is formed using only those electrons which lose energy to a characteristic excitation localised at atom A. Each coordina-
×tal sublattice site local
I
symmetry
I/---O, O W
I o c a l i s a t i o n L= 2 X E . I ~ E column width w=et<3A
Does t h e e n e r g y selected from A show 2 or
d,p 6 ?
Fig. 1. Definition of the parameters L and Wwhich control the symmetry of the inelastic scattering from an atom A at a local three-fold axis.
tion polyhedra about A has three-fold symmetry, so that the kinematic filtered pattern from one atom A would show six-fold symmetry. In the more usual case of dynamical scattering where Friedel's law does not apply to the elastic scattering of the inelastically scattered wave, the filtered pattern from either atom A would show three-fold symmetry. The incoherent superposition of two such patterns (related by the crystal two-fold axis) then results in a total faltered pattern showing six-fold symmetry. Note that the three-fold operation is not an element of the crystal plane group, and would not be observed by convergent-beam electron diffraction. Thus, for localised losses in the ZOLZ approximation, the point plane group symmetry of energy selected electron diffraction patterns is equal to the projected local crystal symmetry about the selected atom, with all the crystal plane group operations also applied, and the results summed. The extension to the case where the same species occurs at the centre of different types of coordination polyhedra (e.g. tetrahedral and octohedral) is straightforward. In practice, the problem is to deduce the local symmetry from that of the energy filtered pattern for a particular species. For rotation axes in general the q-fold local projected symmetry of a coordination polyhedra can always be obtained by dividing the n-fold symmetry of the energy filtered pattern by the p-fold projected symmetry of the entire crystal, which can often be obtained by convergent beam electron diffraction, or which may be known. Here p = 2, q = 3 and n = 6 for the example shown in fig. 1. Similar results hold for mirrors, however there are some
£C.H. Spence / Crystallographic information
61
of 0 E difficult to define. At the opposite extreme of approximation for large energy losses with 0E > > 20B, the computational problem is similar to that of STEM microdiffraction using a point source in which a diffraction limited aberrated spherical wave (the electron probe) diverges from a point near the specimen surface. Earlier calculations for this case and an argument based on the integral solution of the Schr6dinger equation [9] also show that these dynamical elastic microdiffraction patterns preserve the projected symmetry of the crystal when taken about the probe centre as origin. The core-loss energy selected patterns reported here were calculated by combining the method of Doyle [10] with that used to compute STEM microdiffraction patterns. Thus a systematics multislice calculation was run using an artificial superlattice whose size exceeded the localisation L and which contained, in one slice, a single inelastic scattering potential at the lattice site of the selected species. All other slices contained many unit cells of the
intractable cases (e.g. a mirror line in the asymmetric unit not parallel to one in the crystal).
3. Calculations Methods for the calculation of the inelastic plasmon and small energy loss single electron scattering in crystals have been described by Doyle, Howie, Rez, Humphreys, Whelan and others (see ref. [8] for a review). These delocalised excitations generate inelastic scattering over a large volume and so preserve diffraction contrast mechanisms. In addition, for a perfect, parallel sided crystal their energy selected diffraction pattern will express the full crystal space group symmetry if 0 E ~ 2AE/E < 0 B, where 0 B is the Bragg angle, AE the energy loss, E the accelerating voltage and 0E is an angle beyond which there is negligible inelastic scattering. The Lorenzian form of the inelastic scattering, however, makes a useful value
selected oxygen
El~'gy
K loss ( . d y n a m i c a l
elastic)
Systematics Znelasti¢ scattering Modified
potential
FWH.M. = 2/~
hydrogeni¢
5 3 3 eV loss
plane wave i l l u m i n a t i o n
©
A
8
© 1 0 0 kV
t , 80,~,
--3A ~ s A
elastic Braggs
i
1
I
i (000)
I Oe/X
i
| 0.25
I 0.5
U (•
~)
Fig. 2. Dynamical systematics calculation of the energy filtered oxygen K-loss intensity for two positions A and B of the oxygen
atom.
J.C.H. Spence / Crystallographic information
62
elastic scattering potential. Then the results of n such multislice calculations were added incoherently, in each one of which the inelastic event was allowed to occur at a successively lower slice. This entire process must then be repeated for each site of the selected species within the unit cell, and so takes full account of the dynamical elastic scattering of the inelastically scattered wave. Wave functions for the inelastic scattering were obtained from the Born approximation, using the modified hydrogenic atomic wave functions for K-shell excitation in isolated atoms given by Egerton [11 ]. Scattering phases are lost in recovering the wave function from the published cross-sections. The use of these wave functions in crystals is expected to be a good approximation for large energy losses for smaller losses a Bloch-wave expansion of the full crystal excited state wave function is needed. For 0E > 20B the calculations reported here predict the usual channelling effects in the presence of an absorption potential. Fig. 2 shows the results of calculations for a onedimensional hypothetical CuO crystal without a centre of symmetry for two positions of the oxygen atom. Here the energy filtered oxygen K-loss intensity is shown along the (h00) systematics line. The positions at which the elastic Bragg peaks would occur are also shown along the abscissa, and the value of 0E
(which exceeds 0 B) is indicated. These calculations are for an 80 A thick specimen, illuminated by a collimated electron beam at an accelerating voltage of 100 kV. Note that the energy selected pattern from the oxygen atom gains a centre of symmetry (not possessed by the crystal as a whole) as this atom is moved away from the copper atom from position B to position A. In this one-dimensional example the "coordination polyhedra" is the single oxygen atom, which possesses a center of symmetry (the only symmetry element possible in one dimension). The important point is that for localised excitations the symmetry of energy f'fltered patterns may be higher than that of the elastic pattern. This and similar calculations have been used to show that for losses greater than about 300 eV in specimens 100 A thick the column width W and localisation L are less than 3 A for 100 kV incident fast electrons. In the absence of true crystal excited state wave functions, however, this is necessarily a rather rough estimate. The effect of increasing the localisation is shown by the results given in fig. 3 for the 110 eV K shell loss of beryllium. Here, with 0E < 0B, peaks are seen to emerge at the sites of the elastic Bragg peaks since the localisation is now larger than the unit cell. Thus, as one increases in energy, the energy filtered diffraction pattern changes continuously from one showing sharp Bragg spots at low energy to a continuous Kossel pattern (containing line features) at larger losses, as observed experimentally [ 12].
4. Imaging at large losses Be A
©e ff
B
®©
K loss
t:so£ 1 0 0 kV Axial
Fig. 3. The energy filtered beryllium K-loss intensity along the systematics line, showing the emergence of peaks at the Bragg positions.
The formation of lattice images from delocalised excitations has been discussed in detail elsewhere [ 13 ]. While formation of a high resolution electron image from large energy-loss energy Filtered electrons presents more severe experimental difficulties (background subtraction, poor statistics, etc.), it is worthwhile considering also some of the fundamental problems which would be involved in the interpretation of these images. The difficulties arise chiefly from the finite localisation L and from the effects of multiple elastic scattering. Fig. 4 shows the image which would be synthesised from the scattering shown in fig. 2 using, for example, a Castaing-Henry imaging energy f'flter tuned to the 533 eV oxygen K-sheU exci-
J.C.H. Spence /Crystallographic information
Oxyoen energy
63
K - shell e x c i t a t i o n filtered
1~ point t=80~,100 Dynamical
TEM
im~g.~
r e s o l u t i o n , a x i a l imaging kV, s y s t e m a t ics
calculation.
7
8•
unit cell
15
•16
Fig. 4. High resolution TEM energy filtered image obtained from the scattering shown in fig. 2 at the Scherzer focus (bright field, point resolution 1 A). Note that despite energy filtering for the oxygen loss, the copper atom also appears in the image due to the effects of finite localisation and multiple elastic scattering.
tation. An atomic resolution electron microscope has been assumed (Cs -- 0.01 ram, Scherzer focus) so that all image "artifacts" are due to the scattering process itself. These images are again formed using an incoherent sum of dynamical image calculations for all depths for the inelastic event. Thus the total image intensity Io(x) is given by
lo(x) = ~
I ( x - na),
n
where
l(x) = ~ ~ i
I¢i,j(x)l ~ ,
I
with a the unit cell constant and ~i,j (x) the dynamical image amplitude for an inelastic event in the jth slice occurring at the ith site for the selected species in the unit cell. The simple notion that high resolution characteristic-loss energy selected images reveal only the sites of the selected species in the lattice is thus seen to hold only in the limit of very large losses and very thin specimens. For intermediate cases, such as that shown in fig. 4, the image from each atom will be spread at least over a distance equal to the localisation L and may include images of other atomic spe-
cies which lie within a distance L of the selected species (this is a consequence of the elastic scattering of the inelastically scattered electron). The convenient Rayleigh criterion cannot be used to estimate the spatial resolution of a f'fltered core-loss image in view of the extended tails on the Fourier transform of the inelastic scattering distribution. These filtered images reveal the selected species sublattice symmetry for very large losses, and provide increasing information on the local symmetry about the selected atom as the energy loss is reduced. In addition, all of the normal difficulties associated with the interpretation of elastic high resolution lattice images apply and a similar sensitivity to instrumental parameters can be expected. Since, in the limit of large losses, the inelastic "sources" imaged scatter into a broad range of angles, the images will be less sensitive to crystalline orientation than elastic images, and since the integration through depth is incoherent they may well also be less sensitive to thickness changes. Nevertheless, detailed numerical calculations will be needed for the interpretation of these images at realistic crystal thicknesses for 100 kV electrons. At present these calculations are only practical in two dimensions using array-processor hardware.
64
J.C.H. Spence / Crystallographic information
5. Conclusions The main conclusions which result from the study of many dynamical core loss calculations, some o f which have been reported here, are as follows: (i) That the interpretation o f core loss images will require a detailed comparison with computed images since, for example, atomic resolution images formed in a loss characteristic of a particular species may include atom images o f other chemical species due to elastic scattering o f the inelastically scattered electrons. (ii) That the experimentally observed [12] transition from a sharp Bragg spot pattern to a Kossel pattern with increasing energy loss can be qualitatively accounted for within the multislice algorithm, suitably modified. (iii) That there is some hope that, in favourable cases, the local projected symmetry o f coordination polyhedra may be determined from the observed symmetry o f characteristic loss energy filtered diffraction patterns..
Acknowledgment I am grateful to Drs. P. Goodman, J.M. Cowley and J. Steeds for useful discussions. Work supported
by ARO contract No. DAAG-29-80-C.-0080 and the ASU NSF HREM Facility.
References [ 1 ] J.M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975). [2] A. Howie, Proc. Roy. Soc. (London) A271 (1963) 268. [3] C.J. Humphreys and M.J. Whelan, Phil. Mag. 20 (1969) 165. [4] S.J. Pennycook and A. Howie, Phil. Mag. 41 (1980) 809. [5] G. Lehmpfuhl and J. Tafto, in: Electron Microscopy 1980, Vol. 3, p. 62. [6] S. Kuwabara and J.M. Cowley, J. Phys. Soc. Japan 34 (1973) 1575. [7] J.A. Eades, J. Appl. Cryst. 13 (1980) 368. [8] C.J. Humphreys, Rept. Progr. Phys., in press. [9] J.M. Cowley and J.C.H. Spence, Ultramicroscopy 3 (1979) 433. [10] P.A. Doyle, Acta Cryst. A25 (1969) 569. [11] R.F. Egerton, Ultramicroscopy 4 (1979) 169. [12] J.G. Philip, M.J. Whelan and R.F. Egerton, in: Proc. 8th Intern. Congr. on Electron Microscopy, 1974, Vol. 1, p. 276. [13] A. Craven and C. Colliex, Inst. Phys. Conf. Ser. 36 (1977) 271.