- Email: [email protected]
Electron charge density imaging with X-ray holography
15 August 2002
Optics Communications 209 (2002) 273–277 www.elsevier.com/locate/optcom
Electron charge density imaging with X-ray holography F.N. Chukhovskii a, D.V. Novikov b,*, T. Hiort b, G. Materlik b,1 b
a Institute of Crystallography, Leninsky pr. 59, 117333 Moscow, Russia Hamburger Synchrotronstrahlungslabor HASYLAB at Deutsches Elektronen Synchrotron DESY, Notkestr. 85, D-22607 Hamburg, Germany
Received 23 January 2002; received in revised form 21 April 2002; accepted 18 June 2002
Abstract A theory of atom resolving X-ray fluorescence holography is presented. It uses the single-scattering approach to derive from Maxwell’s equations a solution for imaging electron charge density distribution from the observed X-ray holography data. Experimental holograms from a Fe single crystal are used to demonstrate atom position mapping by the new transform versus Barton’s reconstruction algorithm. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.10.)i; 61.10.Dp; 61.10.Yh; 42.40.)i Keywords: X-ray holography; Atom resolving holography; Electron charge density
After many decades of discussions, atomic resolution X-ray holography has recently been realized experimentally as X-ray fluorescence holography (XFH) [1–3] and in its new time-reversed version as multiple energy X-ray holography (MEXH) [4]. In this paper, we will derive in singlescattering approach a rigorous solution to invert the hologram function vðkÞ into the electron charge density (ECD) qðrÞ map of an object. We also present results of atomic image reconstruction of a bcc iron single crystal from XFH holograms measured at the FeKa photon energy to compare
*
Corresponding author. E-mail address: [email protected] (D.V. Novikov). 1 Present address: Diamond Light Source Ltd., Rutherford Appleton Lab., Chilton, Didcot, Oxfordshire, OX11 0QX UK.
and contrast the new transform versus Barton’s multiple energy algorithm [5]. XFH utilizes the coherence properties of a spherical fluorescence wave emitted by a single atom. A fluorescence wave that arrives at the detector without scattering (the reference wave) and the radiation scattered by neighboring atoms (the object wave) interfere and form an interference pattern (hologram) that in principle can be used to obtain three-dimensional images at atomic resolution. The advanced MEXH method is based on the reciprocal geometry, where the positions of emitter and detector in the XFH scheme are inverted. MEXH utilizes a plane X-ray wave to generate an electromagnetic scattering field inside a sample. This interference field is detected by single atoms whose fluorescence yield is monitored as a function of both incidence angle and energy [6].
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 6 8 8 - 7
274
F.N. Chukhovskii et al. / Optics Communications 209 (2002) 273–277
The formation of the XFH hologram can be described in the frame of classical electrodynamics. An X-ray fluorescent wave emitted by an electric dipole moment point source p at the point r ¼ 0 and traveling through a single-scattering atom medium satisfies Maxwell’s equation. At the observation point r far away from the object, the displacement wave field DðrÞ is equivalent to the electric wave field, EðrÞ, and represents a coherent superposition of the direct X-ray fluorescent spherical wave and the object waves. Within the single-scattering object approximation the solution of Maxwell’s equation has the well-known form expðikrÞ DðrÞ ¼ r r p r Z 1 exp½ikjr r0 j
0 dr0 wðr0 Þr0 r 0 4p jr r j 0 exp½ikr
p ; ð1Þ r0 where wðrÞ ¼ ð4pre c2 =x2 Þ~ qðrÞ, re is the classical radius of an electron, q~ðrÞ ¼ qðrÞ q0 ðrÞ being the modified ECD, qðrÞ is the full ECD function of the object and q0 ðrÞ being the ECD of the fluorescent atom at the origin point r ¼ 0. Assuming that the major contribution to the Xray hologram is calculated with an accuracy up to the linear power of the object response (the small cluster approximation), averaging over all the polarization states of the radiant electric dipole moment p [7] and far from the scattering atoms absorption edges, one can find the angular distribution function of the holographic pattern (Z 1 ð1Þ vXFH ðkÞ ¼ Re dr expð ikrÞwðrÞ 4p " 2 f€ ðrÞ 1 þ ðnr=rÞ þ
f_ ðrÞ r
3 ðnr=rÞ
where o f_ ðrÞ ¼ ðexp½ikr =rÞ; or o2 f€ ðrÞ ¼ 2 ðexp½ikr =rÞ; or n ¼ k=k:
2
#)
;
ð2Þ
Commonly, in the X-ray atomic holography the distances of the neighboring scatterers (atoms) r from the X-ray fluorescence source normalized to the X-ray wavelength k are large, i.e., there is always ðkrÞ 1. Then, by omitting the terms of an order or higher than 1=kr 1 in the integrand of the right-hand side of Eq. (2), and carrying out some simple calculations (in particular, substituting the electric susceptibility wðrÞ via the modified ECD function q~ðrÞ ¼ ðqðrÞ q0 ðrÞÞ, we find the following expression for the hologram function Z expðikrÞ ð1Þ vXFH ðkÞ ¼ re Re dr expð ikrÞ~ qðrÞ r r 2 1þ n : ð3Þ r The obtained equation can be also derived from the results of quantum electrodynamics considerations [8–10]. Expression (3) can be used to invert the holoð1Þ gram functions vXFH ðkÞ collected for the entire wave vector k-space to the electron charge density map of the object. Let us apply an integral transform with the kernel function cosðk R kRÞ extended over the Rfull solid angle XðnÞ and photon R 1 energy range as 0 dk dX cosðk R kRÞ to both the left- and right-hand side of Eq. (3). Omitting the polarization terms and going through direct integration over angles, one obtains Z ð1Þ dX cosðk R kRÞvXFH ðkÞ Z q~ðrÞ ikðRrÞ sinðkjR rjÞ ¼ 2pre Re dr e kjR rj r
sinðkjR þ rjÞ þ eikðRþrÞ : kjR þ rj ð4Þ Integrating Eq. (4) over energies provides then Z q~ðrÞ 2 2p re dr r jRrj Z 1 Z ð1Þ ¼ dk dX cosðk R kRÞvXFH ðkÞ: ð5Þ 0
Finally, applying the Laplace operator DR r2R to both sides of Eq. (5) leads to a solu-
F.N. Chukhovskii et al. / Optics Communications 209 (2002) 273–277
tion for the modified ECD q~ðRÞ , as mapped from the hologram function vXFH ðkÞ : Z q~ðRÞ ’ dk fcosðk R kRÞð1 n R=RÞ re R sinðk R kRÞð1=kRÞ 2 þ 2n R=RgvXFH ðkÞ:
ð6Þ
For centrosymmetric objects, Eq. (6) holds also for polarized radiation. It is interesting to notice that the similar expression for MEXH differs from Eq. (6) only by the sign in front of the terms containing the vector coordinates of the point. This is a direct consequence of the reciprocity of the two methods [4,11]. The rigorous solution Eq. (6) provides the true electron charge density restoration when a complete data set is used. It is worth emphasizing that Eq. (6) allows the direct creation of an ECD map and is free from twin images. To perform the reconstruction procedure according to Eq. (6) data taken in a significantly large energy range are required. This cannot be achieved within the XFH holographic method and hardly – at the current state of experimental technique in general [4,12,13]. However, if the atom positions alone are of interest and since the maxima of an ECD are well separated, even a limited data set appears to be sufficient. As an example, we will use experimental XFH data obtained on the undulator beamline BW1 at the Hamburg Synchrotron Radiation Laboratory
(a)
275
HASYLAB [14]. The radiation after two gold coated focusing mirrors and two symmetrically cut Si (1 1 1) monochromator crystals was incident on a sample placed on a four-circle diffractometer. The beam had the energy of 9 keV and the crosssection of 1 1 mm2 at the sample position. Two energy-dispersive silicon drift detectors with an energy resolution better than 300 eV and angular resolution 1° were rotated around a stationary sample iron single crystal. The sample has the centrosymmetric BCC cubic structure. The XFH holograms were recorded at the FeKa emission line wavelength. The interference patterns display Kossel lines [12,15], the symmetry of which allowed us to extrapolate the experimental holograms over the whole unit sphere in k-space. The interference pattern from a crystalline object encloses at least two strong component. The firstorder scattering component described by Eq. (1) belongs to the hologram. The second-order Bragg scattering component contained in the Kossel lines leads after reconstruction to quasi-Patterson contributions in the ECD restoration [11]. In this work, the Kossel line pattern was removed from the experimental data by filtering. Fig. 1 shows the nomalized amplitude of a reconstructed FeKa hologram along the [1 0 0] and [1 1 1] directions with the origin at the emitting atom at r ¼ 0. As the image is based on holograms recorded at one energy, it contains only one segment of the Fourier expansion in the jkj-space, limited by the natural width of the emission line
(b)
Fig. 1. Reconstructed FeKa hologram image intensity along the (a) [0 0 1] and (b) [1 1 1] directions through the emitting atom at the origin. The circled line corresponds to the new algorithm, the crossed line to the Barton’s reconstruction formula. The images for filtered to remove artifacts. The vertical arrows mark the correct atom positions. The distances are shown in unit cell dimensions.
276
F.N. Chukhovskii et al. / Optics Communications 209 (2002) 273–277
Dk=k 104 . This is not sufficient to restore the form of the ECD, which varies very fast in the vicinity of the atoms, and to suppress the additional satellite maxima that appear due to the truncation of the series and have no correspondence in the Fe atomic structure. The Fourier-like behavior of the reconstructed image is well seen on one-dimensional plots of the image intensity. The maxima of the single Fourier components still coincide with the real atom positions and can be used for structure mapping. Increasing the range of energies, one can restore not only the atom positions, but also the form of the ECD. For comparison, we applied to the same holograms the widely used Barton’s reconstruction algorithm Z UBARTON ðrÞ ¼ vðkÞ expð iðk r krÞÞ dk ð7Þ which also removes twin images provided that the number and the range of the measured wavelengths is high enough [5,16]. The resulting images (Fig. 1, crossed lines) are very similar to those obtained by new algorithm, but are less resolved. The atom positions given by the maxima of the images obtained by both algorithms coincide well, as an obvious consequence of the similar Fourier based background of both. Maxima observed in the image that have no correspondence in the iron atomic structure might be attributed to artifacts induced under the procedure of background subtraction and Kossel lines removal [3]. The reconstruction formula Eq. (6) can be considered as a rigorous proof of the previously known restoration algorithms [5,15], which are based on the Helmholtz–Kirchoff integral theorem and suggest that the maxima of the wavefield amplitude distribution match the atom positions. Generally, the assertion that the wavefield amplitude peaks at the positions of the scattering atoms is not always valid for crystalline objects, as known for instance from the X-ray standing waves analysis [17]. For non-centrosymmetric objects, the imaginary part of Barton’s algorithm can produce additional false maxima masking the true atom positions. The algorithm Eq. (6) provides not
only the amplitude, but also the sign of the reconstructed image. As the electron charge density can be only positive it allows to discriminate false maxima from the Bragg scattering component and experimental artifacts. It can be also used in iterative structural methods involving constrains in real space. In conclusion, a new theoretical approach, based on classical electrodynamics, shows the potential of atom resolving X-ray holography. The single-scattering solution of Maxwell’s equations is used to derive a relationship which can be interpreted as a rigorous solution of the electron charge density restoration problem using X-ray fluorescence holography. The measured XFH holograms of a Fe single crystal treated by the new threedimensional transform and by Barton’s reconstruction algorithm demonstrate the state of the art of the technique. By improving experimental counting statistics and by taking into account the multiple scattering contribution, the atom resolving X-ray holography can be favorably applied for structure investigations of low-quality single crystals, catalysts, or diluted solid solution. With arrival of new free-electron laser sources, it also has the potential to become a new tool for the electron charge density restoration of non-crystalline materials.
Acknowledgements The authors are indebted to B. Adams, R. Eisenhower and E. Kossel for fruitful discussions and help in conducting the experiments.
References [1] A. Sz€ oke, in: D.T. Attwood, J. Bokor (Eds.), Short Wavelength Coherent Radiation: Generation and Applications, AIP Conf. Proc. No. 147, AIP, New York, 1986, p. 361. [2] M. Tegze, G. Faigel, Europhys. Lett. 16 (1991) 41. [3] G. Faigel, M. Tegze, Rep. Prog. Phys. 62 (1999) 355. [4] T. Gog, P.M. Len, G. Materlik, D. Bahr, C.S. Fadley, C. Sanchez-Hanke, Phys. Rev. Lett. 76 (1996) 3132; T. Gog, R.H. Menk, F. Arfelli, P.M. Len, C.S. Fadley, G. Materlik, Synchrotron Radiation News 9 (1996) 30.
F.N. Chukhovskii et al. / Optics Communications 209 (2002) 273–277 [5] J.J. Barton, Phys. Rev. Lett. 67 (1991) 3106. [6] T. Gog, D. Bahr, G. Materlik, Phys. Rev. B 51 (1995) 6761. [7] J.D. Jackson, Clasical Electrodynamics, Wiley, New York, 1975. [8] G.A. Miller, L.B. Soerensen, Phys. Rev. B 56 (1997) 2399. [9] L. Fonda, Phys. Status Solidi B 201 (1997) 329. [10] Y. Nishino, G. Materlik, Phys. Rev. B 60 (1999) 15074. [11] F.N. Chukhovskii, D.V. Novikov, T. Hiort, G. Materlik (in preparation).
277
[12] D.V. Novikov, B. Adams, T. Hiort, E. Kossel, G. Materlik, R. Menk, A. Walenta, J. Synchrotron Radiation 5 (1998) 315. [13] M. Tegze, G. Faigel, S. Marchesini, M. Belakhovsky, A.I. Chumakov, Phys. Rev. Lett. 82 (1999) 4847. [14] T. Hiort, D.V. Novikov, E. Kossel, G. Materlik, Phys. Rev. B 61 (2000) R830. [15] B. Adams, D.V. Novikov, T. Hiort, E. Kossel, G. Materlik, Phys. Rev. B 57 (1998) 7526. [16] J.J. Barton, Phys. Rev. Lett. 61 (1988) 1356. [17] M.J. Bedzyk, G. Materlik, Phys. Rev. B 32 (1985) 6456.
Optics Communications 209 (2002) 273–277 www.elsevier.com/locate/optcom
Electron charge density imaging with X-ray holography F.N. Chukhovskii a, D.V. Novikov b,*, T. Hiort b, G. Materlik b,1 b
a Institute of Crystallography, Leninsky pr. 59, 117333 Moscow, Russia Hamburger Synchrotronstrahlungslabor HASYLAB at Deutsches Elektronen Synchrotron DESY, Notkestr. 85, D-22607 Hamburg, Germany
Received 23 January 2002; received in revised form 21 April 2002; accepted 18 June 2002
Abstract A theory of atom resolving X-ray fluorescence holography is presented. It uses the single-scattering approach to derive from Maxwell’s equations a solution for imaging electron charge density distribution from the observed X-ray holography data. Experimental holograms from a Fe single crystal are used to demonstrate atom position mapping by the new transform versus Barton’s reconstruction algorithm. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 61.10.)i; 61.10.Dp; 61.10.Yh; 42.40.)i Keywords: X-ray holography; Atom resolving holography; Electron charge density
After many decades of discussions, atomic resolution X-ray holography has recently been realized experimentally as X-ray fluorescence holography (XFH) [1–3] and in its new time-reversed version as multiple energy X-ray holography (MEXH) [4]. In this paper, we will derive in singlescattering approach a rigorous solution to invert the hologram function vðkÞ into the electron charge density (ECD) qðrÞ map of an object. We also present results of atomic image reconstruction of a bcc iron single crystal from XFH holograms measured at the FeKa photon energy to compare
*
Corresponding author. E-mail address: [email protected] (D.V. Novikov). 1 Present address: Diamond Light Source Ltd., Rutherford Appleton Lab., Chilton, Didcot, Oxfordshire, OX11 0QX UK.
and contrast the new transform versus Barton’s multiple energy algorithm [5]. XFH utilizes the coherence properties of a spherical fluorescence wave emitted by a single atom. A fluorescence wave that arrives at the detector without scattering (the reference wave) and the radiation scattered by neighboring atoms (the object wave) interfere and form an interference pattern (hologram) that in principle can be used to obtain three-dimensional images at atomic resolution. The advanced MEXH method is based on the reciprocal geometry, where the positions of emitter and detector in the XFH scheme are inverted. MEXH utilizes a plane X-ray wave to generate an electromagnetic scattering field inside a sample. This interference field is detected by single atoms whose fluorescence yield is monitored as a function of both incidence angle and energy [6].
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 6 8 8 - 7
274
F.N. Chukhovskii et al. / Optics Communications 209 (2002) 273–277
The formation of the XFH hologram can be described in the frame of classical electrodynamics. An X-ray fluorescent wave emitted by an electric dipole moment point source p at the point r ¼ 0 and traveling through a single-scattering atom medium satisfies Maxwell’s equation. At the observation point r far away from the object, the displacement wave field DðrÞ is equivalent to the electric wave field, EðrÞ, and represents a coherent superposition of the direct X-ray fluorescent spherical wave and the object waves. Within the single-scattering object approximation the solution of Maxwell’s equation has the well-known form expðikrÞ DðrÞ ¼ r r p r Z 1 exp½ikjr r0 j
0 dr0 wðr0 Þr0 r 0 4p jr r j 0 exp½ikr
p ; ð1Þ r0 where wðrÞ ¼ ð4pre c2 =x2 Þ~ qðrÞ, re is the classical radius of an electron, q~ðrÞ ¼ qðrÞ q0 ðrÞ being the modified ECD, qðrÞ is the full ECD function of the object and q0 ðrÞ being the ECD of the fluorescent atom at the origin point r ¼ 0. Assuming that the major contribution to the Xray hologram is calculated with an accuracy up to the linear power of the object response (the small cluster approximation), averaging over all the polarization states of the radiant electric dipole moment p [7] and far from the scattering atoms absorption edges, one can find the angular distribution function of the holographic pattern (Z 1 ð1Þ vXFH ðkÞ ¼ Re dr expð ikrÞwðrÞ 4p " 2 f€ ðrÞ 1 þ ðnr=rÞ þ
f_ ðrÞ r
3 ðnr=rÞ
where o f_ ðrÞ ¼ ðexp½ikr =rÞ; or o2 f€ ðrÞ ¼ 2 ðexp½ikr =rÞ; or n ¼ k=k:
2
#)
;
ð2Þ
Commonly, in the X-ray atomic holography the distances of the neighboring scatterers (atoms) r from the X-ray fluorescence source normalized to the X-ray wavelength k are large, i.e., there is always ðkrÞ 1. Then, by omitting the terms of an order or higher than 1=kr 1 in the integrand of the right-hand side of Eq. (2), and carrying out some simple calculations (in particular, substituting the electric susceptibility wðrÞ via the modified ECD function q~ðrÞ ¼ ðqðrÞ q0 ðrÞÞ, we find the following expression for the hologram function Z expðikrÞ ð1Þ vXFH ðkÞ ¼ re Re dr expð ikrÞ~ qðrÞ r r 2 1þ n : ð3Þ r The obtained equation can be also derived from the results of quantum electrodynamics considerations [8–10]. Expression (3) can be used to invert the holoð1Þ gram functions vXFH ðkÞ collected for the entire wave vector k-space to the electron charge density map of the object. Let us apply an integral transform with the kernel function cosðk R kRÞ extended over the Rfull solid angle XðnÞ and photon R 1 energy range as 0 dk dX cosðk R kRÞ to both the left- and right-hand side of Eq. (3). Omitting the polarization terms and going through direct integration over angles, one obtains Z ð1Þ dX cosðk R kRÞvXFH ðkÞ Z q~ðrÞ ikðRrÞ sinðkjR rjÞ ¼ 2pre Re dr e kjR rj r
sinðkjR þ rjÞ þ eikðRþrÞ : kjR þ rj ð4Þ Integrating Eq. (4) over energies provides then Z q~ðrÞ 2 2p re dr r jRrj Z 1 Z ð1Þ ¼ dk dX cosðk R kRÞvXFH ðkÞ: ð5Þ 0
Finally, applying the Laplace operator DR r2R to both sides of Eq. (5) leads to a solu-
F.N. Chukhovskii et al. / Optics Communications 209 (2002) 273–277
tion for the modified ECD q~ðRÞ , as mapped from the hologram function vXFH ðkÞ : Z q~ðRÞ ’ dk fcosðk R kRÞð1 n R=RÞ re R sinðk R kRÞð1=kRÞ 2 þ 2n R=RgvXFH ðkÞ:
ð6Þ
For centrosymmetric objects, Eq. (6) holds also for polarized radiation. It is interesting to notice that the similar expression for MEXH differs from Eq. (6) only by the sign in front of the terms containing the vector coordinates of the point. This is a direct consequence of the reciprocity of the two methods [4,11]. The rigorous solution Eq. (6) provides the true electron charge density restoration when a complete data set is used. It is worth emphasizing that Eq. (6) allows the direct creation of an ECD map and is free from twin images. To perform the reconstruction procedure according to Eq. (6) data taken in a significantly large energy range are required. This cannot be achieved within the XFH holographic method and hardly – at the current state of experimental technique in general [4,12,13]. However, if the atom positions alone are of interest and since the maxima of an ECD are well separated, even a limited data set appears to be sufficient. As an example, we will use experimental XFH data obtained on the undulator beamline BW1 at the Hamburg Synchrotron Radiation Laboratory
(a)
275
HASYLAB [14]. The radiation after two gold coated focusing mirrors and two symmetrically cut Si (1 1 1) monochromator crystals was incident on a sample placed on a four-circle diffractometer. The beam had the energy of 9 keV and the crosssection of 1 1 mm2 at the sample position. Two energy-dispersive silicon drift detectors with an energy resolution better than 300 eV and angular resolution 1° were rotated around a stationary sample iron single crystal. The sample has the centrosymmetric BCC cubic structure. The XFH holograms were recorded at the FeKa emission line wavelength. The interference patterns display Kossel lines [12,15], the symmetry of which allowed us to extrapolate the experimental holograms over the whole unit sphere in k-space. The interference pattern from a crystalline object encloses at least two strong component. The firstorder scattering component described by Eq. (1) belongs to the hologram. The second-order Bragg scattering component contained in the Kossel lines leads after reconstruction to quasi-Patterson contributions in the ECD restoration [11]. In this work, the Kossel line pattern was removed from the experimental data by filtering. Fig. 1 shows the nomalized amplitude of a reconstructed FeKa hologram along the [1 0 0] and [1 1 1] directions with the origin at the emitting atom at r ¼ 0. As the image is based on holograms recorded at one energy, it contains only one segment of the Fourier expansion in the jkj-space, limited by the natural width of the emission line
(b)
Fig. 1. Reconstructed FeKa hologram image intensity along the (a) [0 0 1] and (b) [1 1 1] directions through the emitting atom at the origin. The circled line corresponds to the new algorithm, the crossed line to the Barton’s reconstruction formula. The images for filtered to remove artifacts. The vertical arrows mark the correct atom positions. The distances are shown in unit cell dimensions.
276
F.N. Chukhovskii et al. / Optics Communications 209 (2002) 273–277
Dk=k 104 . This is not sufficient to restore the form of the ECD, which varies very fast in the vicinity of the atoms, and to suppress the additional satellite maxima that appear due to the truncation of the series and have no correspondence in the Fe atomic structure. The Fourier-like behavior of the reconstructed image is well seen on one-dimensional plots of the image intensity. The maxima of the single Fourier components still coincide with the real atom positions and can be used for structure mapping. Increasing the range of energies, one can restore not only the atom positions, but also the form of the ECD. For comparison, we applied to the same holograms the widely used Barton’s reconstruction algorithm Z UBARTON ðrÞ ¼ vðkÞ expð iðk r krÞÞ dk ð7Þ which also removes twin images provided that the number and the range of the measured wavelengths is high enough [5,16]. The resulting images (Fig. 1, crossed lines) are very similar to those obtained by new algorithm, but are less resolved. The atom positions given by the maxima of the images obtained by both algorithms coincide well, as an obvious consequence of the similar Fourier based background of both. Maxima observed in the image that have no correspondence in the iron atomic structure might be attributed to artifacts induced under the procedure of background subtraction and Kossel lines removal [3]. The reconstruction formula Eq. (6) can be considered as a rigorous proof of the previously known restoration algorithms [5,15], which are based on the Helmholtz–Kirchoff integral theorem and suggest that the maxima of the wavefield amplitude distribution match the atom positions. Generally, the assertion that the wavefield amplitude peaks at the positions of the scattering atoms is not always valid for crystalline objects, as known for instance from the X-ray standing waves analysis [17]. For non-centrosymmetric objects, the imaginary part of Barton’s algorithm can produce additional false maxima masking the true atom positions. The algorithm Eq. (6) provides not
only the amplitude, but also the sign of the reconstructed image. As the electron charge density can be only positive it allows to discriminate false maxima from the Bragg scattering component and experimental artifacts. It can be also used in iterative structural methods involving constrains in real space. In conclusion, a new theoretical approach, based on classical electrodynamics, shows the potential of atom resolving X-ray holography. The single-scattering solution of Maxwell’s equations is used to derive a relationship which can be interpreted as a rigorous solution of the electron charge density restoration problem using X-ray fluorescence holography. The measured XFH holograms of a Fe single crystal treated by the new threedimensional transform and by Barton’s reconstruction algorithm demonstrate the state of the art of the technique. By improving experimental counting statistics and by taking into account the multiple scattering contribution, the atom resolving X-ray holography can be favorably applied for structure investigations of low-quality single crystals, catalysts, or diluted solid solution. With arrival of new free-electron laser sources, it also has the potential to become a new tool for the electron charge density restoration of non-crystalline materials.
Acknowledgements The authors are indebted to B. Adams, R. Eisenhower and E. Kossel for fruitful discussions and help in conducting the experiments.
References [1] A. Sz€ oke, in: D.T. Attwood, J. Bokor (Eds.), Short Wavelength Coherent Radiation: Generation and Applications, AIP Conf. Proc. No. 147, AIP, New York, 1986, p. 361. [2] M. Tegze, G. Faigel, Europhys. Lett. 16 (1991) 41. [3] G. Faigel, M. Tegze, Rep. Prog. Phys. 62 (1999) 355. [4] T. Gog, P.M. Len, G. Materlik, D. Bahr, C.S. Fadley, C. Sanchez-Hanke, Phys. Rev. Lett. 76 (1996) 3132; T. Gog, R.H. Menk, F. Arfelli, P.M. Len, C.S. Fadley, G. Materlik, Synchrotron Radiation News 9 (1996) 30.
F.N. Chukhovskii et al. / Optics Communications 209 (2002) 273–277 [5] J.J. Barton, Phys. Rev. Lett. 67 (1991) 3106. [6] T. Gog, D. Bahr, G. Materlik, Phys. Rev. B 51 (1995) 6761. [7] J.D. Jackson, Clasical Electrodynamics, Wiley, New York, 1975. [8] G.A. Miller, L.B. Soerensen, Phys. Rev. B 56 (1997) 2399. [9] L. Fonda, Phys. Status Solidi B 201 (1997) 329. [10] Y. Nishino, G. Materlik, Phys. Rev. B 60 (1999) 15074. [11] F.N. Chukhovskii, D.V. Novikov, T. Hiort, G. Materlik (in preparation).
277
[12] D.V. Novikov, B. Adams, T. Hiort, E. Kossel, G. Materlik, R. Menk, A. Walenta, J. Synchrotron Radiation 5 (1998) 315. [13] M. Tegze, G. Faigel, S. Marchesini, M. Belakhovsky, A.I. Chumakov, Phys. Rev. Lett. 82 (1999) 4847. [14] T. Hiort, D.V. Novikov, E. Kossel, G. Materlik, Phys. Rev. B 61 (2000) R830. [15] B. Adams, D.V. Novikov, T. Hiort, E. Kossel, G. Materlik, Phys. Rev. B 57 (1998) 7526. [16] J.J. Barton, Phys. Rev. Lett. 61 (1988) 1356. [17] M.J. Bedzyk, G. Materlik, Phys. Rev. B 32 (1985) 6456.