PHYSICAL REVIEW B VOLUME 57, NUMBER 13 1 APRIL 1998-I Atomic holography with x rays B. Adams, D. V. Novikov, T. Hiort, and G. Materlik Hamburger Synchrotronstrahlungslabor HASYLAB am Deutschen Elektronen-Synchrotron DESY, D-22607 Hamburg, Germany E. Kossel II. Institut fušr Experimentalphysik, Universitašt Hamburg, Luruper Chaussee 149, D-22671 Hamburg, Germany Received 25 July 1997 A theoretical treatment of atomic holography with x rays, taking into account the vectorial nature of electromagnetic waves is described. Direct and reciprocal holography are compared and put into the context of x-ray crystallography and the Kossel technique. The theoretical results are compared to experimental data on Cu3Au. S0163-1829 98 00713-9 I. INTRODUCTION only rotational, but no translational order. Far from absorption edges, only Thomson scattering is Atomic holography, a technique very much akin to lens- relevant. Interpretation of the data should be much more less Fourier transform holography,1 was originally proposed straightforward than in electron holography13 because the by Szoške2 for both electrons and x rays. The general problem Thomson scattering phase is isotropic. However, polarization of reconstructing three-dimensional objects from holographic averaging produces a strong anisotropy in the scattering data has already earlier been discussed by Wolf.3 The elec- phase of an equivalent scalar formalism see Sec. II A . Due tron variant of atomic holography is by now a well estab- to the small cross section in the scattering of x rays from lished technique for surface structure analysis.4­10 Tegze and electrons, the interference signal is very weak in most cases, Faigel11 realized the method of direct atomic x-ray hologra- requiring long measuring times and high counting rates. An phy and in parallel, Gog et al.12 developed and proved ex- exception is the phenomenon of Kossel lines which were first perimentally the method of reciprocal atomic holography. observed14 in 1934. Despite the weak scattering contribution Both of these methods use the scattering of x rays to from a single atom, the coherent superposition of scattered obtain information on the neighborhood of atoms of a spe- waves from many atoms in a crystalline sample gives a cific species in a sample. In the direct method, these atoms strong signal in certain crystallographic directions. The are excited to x-ray fluorescence which is scattered from the strength and width of the Kossel lines depends on the crys- neighbors so that in the far field, there is a slight angular talline quality of the sample and the coherence of the illumi- modulation of the fluorescent radiation due to interference of nating wave, i.e., the fluorescence linewidth in the case of the primary fluorescent wave with the scattered ones. In the direct holography and the lateral coherence and monochro- reciprocal method, the direction of an ideally plane wave is maticity in the case of reciprocal holography. The features in scanned relative to the sample. Because this reference wave a holographic data set which change i.e., sharpen with the is scattered from the neighboring atoms and interferes with sample size contain diffractive contributions while the parts the scattered object waves, the local electrical field intensity that are independent of the sample size consist of purely that excites to fluorescence depends on the incidence angle. holographic data. The fluorescence yield is then a measure of the exciting in- Simply put, the holographic information on long range tensity. Therefore, in the reciprocal method, the detector is order in a sample is contained in the Kossel lines while the the fluorescing atom. In both, direct and reciprocal hologra- short range order information can be found in the weak phy, the interference pattern is generated by fully coherent modulation between the Kossel lines-quite in analogy to waves because the differences in optical path lengths are in the technique of crystal truncation rods.15­18 The relation of the order of a few interatomic distances. reciprocal holography to direct holography is the same as the Although an atomic hologram could in principle be ob- relation of the method of standing waves19 to the method of tained from a small cluster, containing just the fluorescent standing waves in reverse.20 atom and its immediate neighbors, real samples always con- The holographic reconstruction procedure is essentially a tain many fluorescent atoms with like and similarly oriented reversal of waves; it is done numerically in the case of x rays neighborhoods. This makes no fundamental difference; the and electrons. In addition to the images of the real atoms, intensity measured at a large sample is simply proportional ghost images may appear in single energy holograms.7 They to what would be obtained from the small cluster. Therefore, may be suppressed by the technique of multiple energy ho- the data set is always treated as if it had been obtained from lography which has been applied for x rays12 and for a cluster with just one fluorescing atom. Atomic holography electrons.8­10 Multiple energy holography is best done with does not attempt to image the sample as a whole. In contrast the reciprocal method because there, the recording wave- to diffraction methods, atomic holography does not rely on length may be chosen quite freely-the x-ray energy only any coherence between the emissions from different atoms. has to lie above the absorption edge of the detecting atomic It is therefore well suited to the study of samples which have species. 0163-1829/98/57 13 /7526 9 /$15.00 57 7526 © 1998 The American Physical Society 57 ATOMIC HOLOGRAPHY WITH X RAYS 7527 Conventional Gabor holography schemes,21­23 extended scheme for doing this is discussed in Sec. II C. to Ć wavelengths do not reach atomic resolution, one reason Reconstruction of the hologram is done numerically by a at least in the case of x rays being that the interference procedure which is essentially a Fourier transform. The in- pattern is recorded only in a relatively small solid angle, i.e., formation present in one hologram lies on the two- with a small numerical aperture. In atomic holography, either dimensional surface of the Ewald sphere in three- the radiation source in the direct method or the detector in dimensional space. It is therefore incomplete and produces the reciprocal method is inside the sample. Therefore, the ghost images. In the technique of multiple energy x-ray ho- interference pattern is recorded in a solid angle which may lography, the ghost images are suppressed by properly be as large as 4 , providing a resolution in the order of the phased addition of several reconstructions with different wavelength. wave numbers,7 thus extending the Fourier transform to a There are already several theoretical and experimental ar- truly three-dimensional part of reciprocal space. We will dis- ticles which compare electron and x-ray holography and cuss this matter from another viewpoint in Sec. II A. We show the specific strengths of the methods.13,24,25 In this pa- begin our detailed discussion with direct holography of small per, we will discuss direct and reciprocal x-ray holography, clusters, proceed to large crystals in the kinematical approxi- put them into the context of classical crystallographic meth- mation, discuss the connection with the Kossel technique, ods and develop an intuitively simple but formally exact in- look at reciprocal holography, and discuss near-field effects. terpretation of the ghost images and their removal by mul- tiple energy holography. We also show recent experimental A. Direct holography from small clusters results which are in good agreement with the theory. The electric dipole field at point r from a fluorescent atom at the origin with a dipole moment p is given by Jackson,26 II. THEORY Eq. 9.18 : In the following treatment, we always assume the fluo- eik r ik eik r rescing atom-either as emitter in direct holography or as E r k2pn r 3n n *p p 1 r r , detector in reciprocal holography-to be in the coordinate r 2 origin. In reality, the sample contains a large number of such 1 atoms, each of which may be the emitter of fluorescence. We where n r/ r is the normal vector in the direction of r and treat the measured data as an incoherent superposition of the pn n p n . The second term contains a longitudinal holograms produced by them. If all fluorescent atoms are component and is relevant in the near field-which is where located at structurally equivalent sites and their environments the closest neighbor atoms are for typical x-ray wavelengths. all have the same orientation in space, the measured holo- Since multiple scattering is very weak in small clusters, gram is the same as would be obtained from a sample with we consider only single scattering. We restrict ourselves to just one fluorescent atom. In the case of several structurally x-ray energies far from any absorption edges of the scatter- inequivalent sites for the fluorescing atoms, the reconstruc- ing atoms. The Thomson scattering amplitude measured at tion will show a superposition of all neighborhoods. point R due to the field of Eq. 1 from an electron at point r Although the source or the detector is inside the sample, is given in the far field by Jackson,26 Eqs. 14.107 and the intensity of a holograpic recording is calculated in the far 14.99 : field, the justification of which is discussed in Sec. II E. The intensities measured, both in direct and in reciprocal e2 eik R r holography, consist of three contributions: The square of the Er R n n E r mc2 R r . 2 primary wave, an interference term, and the square of the scattered waves. In keeping with holographic terminology, Inserting Eq. 1 and writing k kn leads to the primary wave will henceforth be called the reference wave. For the reconstruction, the interference part is used Er R which preserves the phases of the scattered waves. In con- trast, classical crystallographic methods extract structural in- eik R r eik r formation from the square of the scattered waves which does re R r r not directly reveal absolute phase information. For small clusters, the square of the reference wave is 1 i dominant, the interference term induces a weak angular pn k 3n n *p p k , 3 k r 2 k r modulation of the measured intensity and the square of the scattered waves is negligible. As the crystalline cluster be- where re e2/mc2 is the classical electron radius and comes larger, constructive interference from the many coher- n (R r)/ R r is the normal vector in the direction of ently scattered waves produces strong modulations of the R r. measured intensity in directions which lie on cones around The detector at point R is in the far field. Since r R , reciprocal lattice vectors H whose opening angles are we may approximate in Eq. 3 R r 1 R 1 and arcsin( H /2k). This is the origin of the Kossel lines in exp(ik R r ) exp(ik R ik*r) with k kR/ R . The am- the recorded fluorescence. Within a Kossel line, the contri- plitude ED at the detector is obtained by taking the far field bution of the square of the scattered waves to the measured term of Eq. 1 and adding to it the sum over the scattering intensity may not be negligible. It must be subtracted from from all neighbor atoms at positions rj of the emitter of the recorded data to obtain a pure hologram. A possible fluorescence. We use (n p) n p (n *p)n to resolve 7528 ADAMS, NOVIKOV, HIORT, MATERLIK, AND KOSSEL 57 the multiple vector product (pn )k in Eq. 3 , approximate n R/R and write n rj / rj . eik R ei k rj k*rj E D R R pk re j rj f rj ,k 1 i 1 p k2 r k r k j 2 j 3i 3 , 4 k2 r k r j 2 j 1 rj*p rj k rj 2 where f (rj ,k) is the atom form factor for scattering from the direction of rj to k. Since it is defined for the scattering of FIG. 1. Lensless Fourier transform holography a recording, b plane waves, its use implies an approximation see discus- reconstruction of the virtual image Iv and the conjugate virtual im- sion in Sec. II E . The scattering term containing the ratios age I*v with a diverging illuminating beam, c reconstruction of the r real image I e / rj is typically smaller than the first one in Eq. 4 by a r and the conjugate real image Ir* with a converging factor of 10 4­ 10 3. illuminating beam. The intensity that is recorded by an x-ray detector is a sum of three terms i the square of the direct wave, ii an ing from the longitudinal component of the fluorescent wave. interference term, being twice the real part of the product of Furthermore, the near field correction term i/k rj in Ak(rj) the direct wave and the scattering term, and iii the square of and Bk(rj) produces a phase shift in the scattered waves. At the scattered waves. The latter one is negligible for small a recording wavelength of 1.5 Ć, its relative weight to the clusters. We are then left with terms i and ii . With the leading order terms is roughly 10% for the nearest neighbor abbreviations A atoms. Since A k(rj) 1 (k rj ) 2 i(k rj ) 1 and Bk(rj) k(rj) and Bk(rj) are multiplied by factors 2(k r which depend on the angle between k and r j ) 2 2i(k rj ) 1, we obtain j , the apparent scattering phase in the polarization-averaged data is aniso- 1 ei k rj k*rj tropic for the near field. E 2 D R 2 p 2r f r By introduction of effective scattering form factors R2 k eRe j ,k j rj f (rj ,k), given by f(rj ,k) times the content of the paren- theses following f (rj ,k) in Eq. 6 , this equation may be A reduced to the simple form k rj pk 2 2k4 p 2 eik rj ik*rj r ED k 2 1 2re Re A j*p rj k 3R2 j rj f rj ,k k rj Bk rj pk * . rj 2 7 5 which is familiar from derivations of the holographic inten- We assume the fluorescent radiation to be unpolarized and sity with scalar waves,2,24 but with f (rj ,k) instead of therefore take the average E(R) 2 of the detected intensity f (rj ,k). We must note, however, that due to the factors over all orientations of p, i.e., take (4 ) 1 times the integral cos2(k,rj) and sin2(k,rj), the effective scattering factor of k p k 2 and (r f (rj ,k) is generally not physically realizable. Therefore, j*p)(k p k)*(k rj k), respec- tively, over the coordinates and in spherical coordi- there is generally no scalar wave equation for the function nates. They turn out to be 2k4 p 2/3 and that leads to the far field amplitude ED(k). k2 r The second term in Eq. 7 , subsequently called (k), j 2 (k*rj)2 k2 p 2/3, respectively. Since the hologram is an object in reciprocal space see contains the holographic information. This information is in- Sec. III , we write the detected signal in terms of the wave complete because it consists only of the real part of the scat- vector k kR/ R . After collecting some terms and writing tered waves. It can, however, be made complete by combin- cos2(k,r ing measurements made at several wave numbers. This has j) (k*rj)2/(k rj )2, sin2(k,rj) 1 (k*rj)2/ the effect of rotating the phase of the scattering contribution (k rj )2, we arrive at of the atom at rj by the factor exp(ik rj ) contained in it, so 2k4 p 2 ei k r that for different wave numbers, different parts of the scat- j k*rj E tering contribution are projected onto the real axis. D k 2 1 2r 3R2 e Re j rj f rj ,k As stated in the Introduction, direct atomic holography is very similar to lensless Fourier transform holography1 which 1 cos2 k,r sin2 k,r A j j is shown schematically in Fig. 1. Both the real image and the k rj 2 Bk rj 2 . corresponding ghost image in the above reconstruction of 6 atomic holography are real images in the terminology of ho- lography. The factor 1 cos2(k,rj) is an expected result of polarization Reconstruction of the hologram is done numerically. We averaging but the factor sin2(k,rj) is a near field effect, com- develop here a relatively simple scalar reconstruction for- 57 ATOMIC HOLOGRAPHY WITH X RAYS 7529 mula which essentially involves a reversal of waves. As from the artificial sample shows that the Kirchhoff approxi- stated above, this is not perfectly correct because of the lack mation is not necessary-the result, depending only on the of a scalar wave equation that would represent the result of far field approximation exp(ik R rj ) exp(ik R ik*rj) polarization averaging of electromagnetic waves. However, which was used for Eq. 4 , is almost exact. we get a workable approximation and obtain a guideline for The reconstructed wave field, being complex conjugate to application of the same ideas to a fully vectorial formalism. the recording wave field from the artificial sample with the In our scalar formalism, the image that is produced in the virtual counterparts is just the complex conjugate of Eq. 4 reconstruction approximates the field strength that would be with the summation extending over all atoms and the virtual present in a sample with scalar scatterers having the above counterparts compare to Ref. 4 . Specifically, the contribu- effective scattering factors. The field strength is in turn re- tion of an atom at position a to the reconstructed image has lated to the electron density of the sample. a phase exp( ik a ) at a and a phase exp(ik a ) at the posi- First, we construct an artificial sample which will turn out tion a of the virtual counterpart. to be just what the reconstruction of a real sample images. It If the sample contains atoms at the positions a then the is only introduced as a tool for a derivation of the formalism reconstructed amplitude at a will have a contribution with by a simple reversal of waves and not for the reconstruction phase exp( ik a ) from the atom at a and a contribution with procedure itself. Furthermore, it will help to understand the phase exp(ik a ) from the virtual counterpart of the atom at nature of the ghost images, the winking effect, and provides a. These contributions interfere constructively or destruc- a derivation of the reconstruction formalism which does not tively, depending on k a , producing a winking of the sum depend on the Kirchhoff approximation. Since we recon- amplitude in a scan over k. struct within a scalar wave formalism, we take the scattering For a formal calculation of the diffraction pattern of the form factors f (rj ,k) of Eq. 7 . Additionally, for each atom reversed waves, we use Green's formula. The calculation is at position rj , we place another atom which we call ``virtual carried out in the appendix and results in counterpart'' at the position rj . It emits a wave with the same angular distribution given by f (rj ,k)] as the atom at r i j and with a relative phase to the fluorescent atom at the k e ik*rd origin of not exp(ik r k r k . 8 j ) but rather exp( ik rj ). The virtual 2 R S counterpart of the atom at rj is a mathematical construction only. A real atom of the same species which might be sitting at r d k k/R d is now the surface element on the sphere in j would have an atom form factor of f ( rj ,k). We now have the fluorescent spherical wave from the terms of the coordinate k. As mentioned right after Eq. 6 , origin, the scattered waves from all really present atoms and there is a substantial phase shift in the scattering due to near the additional, artificially constructed waves. By construc- field effects. This has the effect of a shift of the apparent tion, the amplitudes from the real atoms and from the virtual atom positions in the reconstructed image. counterparts add up to the real part of the scattering term in In order to remove the ghost images, we take several re- Eq. 7 . If we now multiply the detected intensity of Eq. 7 constructions k(r) with different wave numbers k and cal- by 3Rexp(ikR)/(2k2 p ), we get the amplitude that our artifi- culate the sum: cial arrangement produces on the sphere of radius R. This is the amplitude that we can use in the reversal of waves of the reconstruction process. r k r e ik r . 9 In order to reconstruct the hologram, we calculate the k amplitude k(r) inside a sphere of radius R with an ampli- tude transmission 1 (r Rk/k) on which a converging In this sum, the phase at the positions of the atoms that were spherical wave exp( ik r )/ r is incident. The amplitude actually present in the recording is stationary because it is k(r) 1 (r) exp( ik r )/ r on the inside of this sphere being compensated by the factor exp( ik r ). At ghost image is then the same as would be found if all waves of our arti- positions, a factor exp( 2ik r ) causes partially destructive ficial arrangement were reversed i.e., complex conjugated . interference of the contributions to the sum.7 Therefore, (r) Therefore, by Green's theorem, the intensity everywhere in- now represents the electron density with suppressed ghost side the sphere is the same as the intensity of the waves images. Figure 2 shows this effect in a simulation. Two Cu emitted in the holographic recording. There is a huge inten- atoms were put at the positions 3.75 Ć,0,0 and 3.75 Ć, sity maximum at the origin, corresponding to the emitting 3.75 Ć,0 . Holograms were calculated for the x-ray energies atom and further maxima at the positions rj of the neighbor 15,16, . . . ,30 keV according to Eq. 5 . The top figure shows atoms and at rj , commonly called ghost images, for the a reconstruction with 30 keV according to Eq. 8 for the virtual counterparts. All of them are smaller than the maxi- z 0 plane. Ghost images show up at the mirror image po- mum at the origin by a factor of roughly f (r 2 j)re/ rj 2. Be- sitions. For the lower figure, Eq. 9 was used with all six- cause of linear superposition, the maximum at the origin can teen x-ray energies for a reconstruction in the z 0 plane. be suppressed by using (r) instead of 1 (r) in the re- The ghost images are now strongly suppressed. construction. The image that is obtained in a single energy reconstruc- Commonly,4 the diffraction pattern of the reconstruction tion is the complex conjugate of the field strength in the is calculated with the Kirchhoff integral formula, suggesting sample, modified by the near field effects which lead to the use of the Kirchhoff approximation with its mathematical definition of the effective atomic form factors, and is super- inconsistency.26,27 Our argument with the reversal of waves imposed with the ghost images. 7530 ADAMS, NOVIKOV, HIORT, MATERLIK, AND KOSSEL 57 1 r F HeiH*r, FH d3r r e iH*r, 11 H V V where V is the volume of one unit cell and H is the index of the reciprocal lattice vector H. This gives cos k H *r eik r s k F H d3r H C r . 12 The term sin(k H)*r vanishes due to symmetry. For C being a sphere around the origin, the integration is easily done in spherical coordinates (r, , ) with k H along the z axis. In the limit of an infinitely large sphere k has to be replaced by kc k iki with an imaginary part ki 0. The physical reason for this is that we did not take into account the absorption and extinction of the outgoing unscattered wave. The result is then 1 s k 4 FH . 13 H k H 2 k2c In the case of crystalline clusters instead of infinite crystals, the Fourier series is replaced by a Fourier integral, i.e., the sum over H in Eq. 13 is replaced by an integral over H. For H 0, Eq. 13 diverges for ki 0 for all k. This describes the outgoing, unscattered spherical wave of fluo- rescent radiation. FIG. 2. Comparison of single energy top and multiple energy For H 0, the denominator in Eq. 13 becomes as small bottom x-ray holography in a simulation. The horizontal axes are as 2ik 2 2 rki ki if (k H)2 kr , i.e., if the Bragg condition labeled in Ć and the vertical axes, showing k(r) 2 top and arcsin( H /2k) is fulfilled. This is the case on a cone (r) 2 bottom are scaled in arbitrary units. The planes below the around H with an apex angle of 2 , the Kossel cone of grid graphs show contour maps. H. B. Kossel lines The intensity at location R is then As the size of the crystalline cluster grows, Kossel lines E R 2 appear in the hologram. They are quite sharp and localized which reflects the fact that many atoms at different distances E2R2 1 from the origin contribute to the hologram with a wide range 0 0 1 8 re Re FH of spatial frequencies. R2 H k H 2 k2c In the Kossel technique, the shape of the Kossel lines is analyzed to obtain phase information.28­31 To our knowl- F * 2 HFH edge, the theory used is always based on the two-beam ap- 16 2re , 14 k H 2 k2 k H 2 k2 * proximation. Here, we derive a formula in the context of H,H c c kinematical theory that describes the shape of Kossel lines in where E(R) is defined as E(R R k/k). a way that inherently does not require any few-beam ap- proximation. For the sake of simplicity, we do the calcula- C. Isolating the interference term tions with scalar waves but note that the vectorial nature of electromagnetic waves has an influence on the shape of the In order to get a hologram from the detected intensity, the Kossel lines. interference term has to be isolated. Far away from Kossel Far from any absorption edges, there is only Thomson lines, the square of the scattered waves is much smaller than scattering. The sum over the scalar scattering contributions the interference term so that it is sufficient to simply subtract from all atoms in Eq. 4 may then be written in terms of the a constant background. Near the Kossel lines, this is no electron density (r) as longer true. Simply cutting the Kossel lines out of the holo- gram amounts to neglect of holographic information that is contained in the contribution of the interference term to the ei k rj k*rj ei k r k*r s k f r Kossel lines. A way out of this problem is found by analysis j ,k d3r r j rj C r , of the shape more specifically the asymmetry of the Kossel 10 lines. The interference term in Eq. 14 has a pole of first order and the last term has a pole of second order for H H . where C is the volume of the crystal. Therefore, the shape of the Kossel line depends on the ratio The Fourier series of (r) is19 of Re FH to FH 2 which is determined by the position of the 57 ATOMIC HOLOGRAPHY WITH X RAYS 7531 fluorescing atom i.e., by definition the origin relative to the A reconstruction formula which is based on the reversal crystal lattice. It also depends on the absorption of the of waves may be found by application of the concept of the spherical wave of fluorescence which is expressed by ki . virtual counterparts, as described in Sec. II A if the polariza- A fit to the Kossel line belonging to H with the param- tion of the incident radiation is chosen to depend on the eters Re FH and FH 2 allows us to subtract the contribution incident direction in a way which is reciprocal to direct ho- of the last term in Eq. 14 to the detected intensity, leaving lography from a polarized i.e., not polarization-averaged the pure hologram. We note, however, that this fit requires source. This work is yet to be done quantitatively. Some the exact theoretical shape of the Kossel line which can be initial work25 has been published on the visibility of atoms in obtained only from a vectorial calculation. different directions with respect to the polarization. The fitting procedure does not require any structural in- formation that is not contained in the holographic recording. The value H of the reciprocal lattice vector around which the E. Near field effects respective Kossel line is centered is contained in the orien- tation and opening angle of the Kossel line and Re F We have used i the far field approximation in the deri- H , F vation of Eq. 4 by replacing the spherical wave that goes H 2 are fit parameters. out to the detector from the scattering atoms by a plane wave. Furthermore, ii the spherical wave exp(ik rj )/ rj in D. Reciprocal holography Eq. 4 and taking the intensity only in the origin in Eq. 15 Most of the discussion of direct holography holds for re- implies the assumption that the fluorescent atom in the origin ciprocal holography as well because the roles of emitter and be pointlike. Finally, iii in the derivation of Eqs. 4 and detector are just reversed. A plane wave E 15 , we assumed pointlike scatterers. 0exp( ik*r) is incident on the sample. In the single scattering approxima- Our assumption i actually depends on the solid angle of tion, the amplitude E(k) at the origin is a sum of the direct the active detector area: The smaller it is, the better justified wave and of contributions that were scattered from neighbor is assumption i . In direct holography, assumption ii is atoms. To calculate it, we take an electron at position r. The justified if the spatial extent of the atomic orbitals that are incident wave induces a dipole moment participating in the x-ray fluorescence is much smaller than p r interatomic distances-a condition which is almost always ek 2E0 exp( ik*r) in it which gives rise to a scat- tered amplitude at the origin according to Eq. 1 with very well satisfied. n r/ r . We decompose r E Assumption iii is valid for the scattering of plane waves 0 r E0 (r*E0)r and form the sum over the scattering contributions from all by introduction of atomic scattering factors. In our case, neighbor atoms by use of the atomic scattering form factors. however, the incoming wave in direct holography or the Neglecting the square of the scattered waves as in Sec. II A, scattered wave in reciprocal holography is spherical. There- we obtain for the intensity at the origin fore, the tabulated values of the scattering factors are some- what inaccurate, depending on how strongly curved the ei k rj k*rj spherical wave is within the electron cloud of the scattering E k 2 Eo 2 2re Re atom. j rj f rj ,k There are some pronounced near field effects which have their origin in the vectorial nature of electromagnetic waves. r A j*E0 2 They lead to the correction terms which decrease as i/k r k rj E0 2 Ak rj Bk rj ... . j rj 2 and 1/k2 rj 2 as compared to the leading order in Eqs. 6 15 and 15 . Furthermore, direct and reciprocal holography dif- fer in these correction terms and are therefore not exactly This equation describes the reciprocal hologram for fully po- equivalent. larized incident radiation. In order to compare this result to Eq. 6 , we take the average E(k) 2 over all possible po- larizations. Contrary to Sec. II A, this average extends not III. ATOMIC HOLOGRAPHY IN RECIPROCAL SPACE over a sphere but only a circle around k, for each k sepa- rately. The averages over E The holographic data is obtained from the surface of the 0 2 are trivial and the average over (E Ewald sphere which is centered at the origin of reciprocal 0*rj)2 gives rj 2 E0 2sin2(k,rj)/2. We have now space and whose radius is given by the wave number. Figure ei k r 3 shows the transition from an infinite lattice to a cluster. j k*rj E k 2 E The infinite lattice has Fourier components located at points 0 2 1 2re Re j rj f rj ,k in reciprocal space. For each reciprocal lattice vector, there is 1 cos2 k,r sin2 k,r a Kossel line which is a circle on the Ewald sphere and is the A j j locus of all wave vectors which fulfil the Bragg condition for k rj 2 Bk rj 2 the respective reciprocal lattice point. 16 As the sampled volume becomes smaller, the Fourier se- ries of Eq. 11 has to be replaced by a Fourier integral, i.e., which-except for a scaling factor-is just the same as Eq. the reciprocal lattice points become spread out. Correspond- 6 . Therefore, the reconstruction procedure that was derived ingly, the Kossel lines are smeared out. As the cluster be- in Sec. II A may be used for reciprocal holography with un- comes very small, the Kossel lines disappear and there is a polarized incident radiation, too. weak modulation of the intensity all over the Ewald sphere. 7532 ADAMS, NOVIKOV, HIORT, MATERLIK, AND KOSSEL 57 FIG. 3. The Ewald sphere in reciprocal space. Left: An infinite lattice, two reciprocal lattice vectors H1, H2, their Kossel lines and one Bragg reflex from k to H1 k. Right: A cluster, the reciprocal lattice points and the Kossel lines become smeared out. IV. EXPERIMENT The theory was tested by measurements in the multiple FIG. 5. A hologram of Cu3Au with lower resolution than in Fig. energy mode because it has a good angular resolution and 4. Horizontal axis: 0°,1°, . . . ,360°; vertical axis provides a possibility of wide parameter variation. The ex- 45°,50°, . . . ,85°. The data was symmetrized to four-fold sym- periments were carried out with synchrotron radiation at HA- metry. SYLAB on beamlines BW1 and CEMO. A Si 111 double crystal monochromator provided a monochromatic colli- Kossel lines as described in Sec. II C. However, far from the mated incident beam with energies from 9 to 30 keV and Kossel lines the contrast may be attributed to the linear in- E/E 10 4. A flat polished single crystal of Cu3Au was terference term in Eq. 4 alone and is consequently evalu- mounted on a four-circle goniometer to provide rotation by ated in the holographic approach. the incidence angles inclination, measured from the sur- The hologram of Fig. 5 was measured at 9.35 keV with an face and azimuth over a wide angular range. Energy angular step size which was five times larger in the direc- dispersive silicon drift detectors32,33 with an energy resolu- tion than in Fig. 4. The large step size served as a low pass tion of 300 eV at 10 keV and counting rates of 2 105 s 1 filter and excluded the high frequency Bragg scattering com- were used to register separately fluorescent radiation in the ponents. A reconstruction of the hologram in the base plane Cu K and Au L lines. The technique of multiple energy of the Cu x-ray holography was applied in a modified way, as the de- 3Au face centered cubic lattice cell is shown in Fig. 6. The central detecting copper atom is excluded from the tectors kept a constant angle to the sample surface, but did reconstructed image by the reconstruction procedure accord- not follow the azimuthal rotation of the sample. ing to Eq. 8 , integrating only over that part of the sphere on Figure 4 shows an experimental hologram, recorded with which holographic data was taken. Cu K radiation at an incident energy of 24 keV. The back- ground from the reference wave was removed and correc- tions for absorption were applied. Kossel lines, belonging to Bragg diffraction of the incident radiation are clearly visible. Due to beamtime limitations, the density of data points was too low to allow a separation of the contributions to the FIG. 4. A hologram of Cu3Au with Kossel lines. Horizontal FIG. 6. Reconstruction from the hologram of Fig. 5. The smaller axis: 0°,1°, . . . ,360°; vertical axis: 19°,20°, . . . ,90°. and larger spots are at the positions of Cu and Au, respectively. 57 ATOMIC HOLOGRAPHY WITH X RAYS 7533 The nearest neighbors of the detecting Cu atoms are Au tion and near-field effects, have shown the relation of x-ray and the next nearest neighbors are Cu. Au atoms, having holography to crystallography using Bragg reflections. We more than twice as many electrons than Cu, produce much have also derived a compact formula for Kossel lines in the more pronounced maxima in the reconstructed image. kinematical approximation. This allows a separation of the contributions to the Kossel lines which belong into the holo- gram. Our holograms from a crystal of Cu V. DISCUSSION 3Au show clear evidence of Kossel lines. X-ray holography with atomic resolution provides an x- The hologram obtained experimentally is in good agree- ray optical image of the atomic short range order. The for- ment with the theory that was developed in this text. The malism developed here also includes long range order by evaluation of experimental results from single crystals will way of the Kossel lines. Thus, the holographic interpretation be discussed in a forthcoming paper.34 of scattering data avoids the phase problem of crystallogra- phy. The emphasis of atomic holography as presented here is ACKNOWLEDGMENTS on the immediate atomic neighborhood of the fluorescing atoms. This application requires interference information The authors are indebted to P. M. Len and V. Kaganer for from a large solid angle with moderate angular resolution. valuable discussions and especially acknowledge the inspira- As evident from the exponent in Eq. 7 , the spatial fre- tion from discussions with F.N. Chukhovskii and his com- quency of the contribution of a scattering atom to the inter- ments on the manuscript. ference pattern scales with its distance from the origin. Therefore, the angular resolution required depends on the APPENDIX: HOLOGRAPHIC RECONSTRUCTION spatial range of the reconstruction. Since all atoms of a large crystal contribute to the interference pattern with their re- We insert a modulated converging spherical wave spective spatial frequencies, the Kossel lines are sharp and 1 (r ) exp( ikR)/R with (r ) defined as k kr / localized. In the case of a large crystal, the square of the R)] and the Green's function (r r ) exp(ik r r )/ scattered waves which was neglected in Eq. 5 , becomes 4 r r see Jackson,26 Eq. 9.122 into Green's formula noticeable. It then becomes possible to determine the posi- to obtain an expression for the amplitude k(r) at a point r tion of the fluorescing atom relative to the crystal lattice by inside the sphere. Since (k) depends only on the direction analysis of the shape of the Kossel lines as given by Eq. 14 . and not the distance from the origin, the radial component of This leads to the methods of standing waves and standing vanishes and we are left with waves in reverse which are the limiting cases of reciprocal and direct holography, respectively. The standing wave method is usually done in the context 1 eik r r e ik r 1 r n of dynamical scattering theory, as is appropriate to large per- k r 4 S r r r fect crystals. If the crystal is large but not quite so perfect-or slightly off the Kossel lines-the result 14 1 r 1 r r which was developed in the kinematical approximation is ik ik d . r r r r r r valid. The phase angle arctan(Re FH / FH ) which can be obtained by a fit to the shape of the Kossel line belonging to A1 the reciprocal lattice vector H as described in Sec. II C is the direct analog to the coherent position of the standing wave r is on the sphere, n is the inward surface normal vector of method. the spherical integration surface, i.e., n *r r R , It is important to realize that the reconstruction formula and d is the surface element on the sphere in terms of the that was derived in Sec. II A does not give an image of the coordinate r . Since we are interested in the values of k electron density. It rather returns an image that is related to near the origin, we can make use of some simplifications the electrical wave field in the sample at recording time, which are similar to those made before Eq. 4 and being the complex conjugate of the wave field amplitude in the auxiliary sample that was constructed in Sec. II A as a tool for the derivations. A full reconstruction of the electron eik r r eikRe ir*k density will require further steps: first corrections for the r r R , n * r r R, influence of the near field terms, such as the apparent shift of the atom positions, described in Sec. II A and then a proce- dure to obtain the electron density from the amplitude of the 1 1 scattered waves inside the sample. This will certainly require ik ik, ik ik. A2 multiple energy x-ray holography to probe the atomic scat- r r r tering form factors at different momentum transfers for con- stant scattering angle. In order to suppress the huge maximum at the origin which is just the focus of the converging spherical wave, we use (k) instead of 1 (k) and obtain now VI. CONCLUSIONS We have discussed both direct and reciprocal x-ray holog- i k e ik*rd raphy in an intuitively appealing way and including polariza- k r 2 R k . A3 S 7534 ADAMS, NOVIKOV, HIORT, MATERLIK, AND KOSSEL 57 1 H. A. Smith, Principles of Holography Wiley, New York, 1969 . 19 M. Bedzyk and G. Materlik, Phys. Rev. B 32, 6456 1985 . 2 A. Szoške, in Short Wavelength Coherent Radiation: Generation 20 T. Gog, D. Bahr, and G. Materlik, Phys. Rev. B 51, 6761 1995 . and Applications, edited by T. Atwood and J. Boker, AIP Conf. 21 D. Gabor, Nature London 161, 777 1948 . Proc. No. 147 AIP, New York, 1986 , p. 361. 22 A. Tonomura, Adv. Phys. 41, 59 1992 . 3 E. Wolf, Opt. Commun. 1, 153 1969 . 23 M. Howells, C. Jacobsen, J. Kirz, R. Feder, K. McQuaid, and S. 4 J. J. Barton, Phys. Rev. Lett. 61, 1356 1988 . Rothman, Science 238, 514 1987 . 5 D. K. Saldin, G. R. Harp, B. L. Chen, and B. P. Tonner, Phys. 24 P. M. Len, T. Gog, D. Novikov, G. Materlik, and C. S. Fadley, Rev. B 44, 2480 1991 . Phys. Rev. B 55, R3323 1997 . 6 S. Thevuthasan, G. S. Herman, A. P. Kaduwela, R. S. Saiki, Y. J. 25 P. M. Len, C. S. Fadley, and G. Materlik, in X-Ray and Inner- Kim, W. Niemczura, M. Burger, and C. S. Fadley, Phys. Rev. Shell Proceses, Proceedings of the 17th International Confer- Lett. 67, 469 1991 . ence, edited by R. L. Johnson, B. F. Sonntag, and H. Schmidt- 7 J. J. Barton, Phys. Rev. Lett. 67, 3106 1991 . Bocking, AIP Conf. Proc. No. 389 AIP, New York, 1997 , p. 8 S. Y. Tong, H. Li, and H. Huang, Phys. Rev. Lett. 67, 3102 295. 1991 . 26 J. D. Jackson, Classical Electrodynamics Wiley, New York, 9 L. J. Terminello, J. J. Barton, and D. A. Lapiano-Smith, Phys. 1975 . Rev. Lett. 70, 599 1993 . 27 A. Sommerfeld, Optics Academic, New York, 1964 . 10 M. Zharnikov, M. Weinelt, P. Zebisch, M. Stichler, and H.-P. 28 J. M. Cowley, Acta Crystallogr. 17, 33 1964 . Steinrušck, Phys. Rev. Lett. 73, 3548 1994 . 29 J. P. Hannon and G. T. Trammell, in Mošssbauer Effect Method- 11 M. Tegze and G. Faigel, Europhys. Lett. 16, 41 1991 . ology, edited by I. J. Gruverman, C. W. Seidel, and D. K. Diet- 12 T. Gog, P. M. Len, G. Materlik, D. Bahr, C. S. Fadley, and C. erly Plenum, New York, 1974 , Vol. 9. Sanchez-Hanke, Phys. Rev. Lett. 76, 3132 1996 . 30 D. Stephan, and V. Geist, Exp. Tech. Phys. Berlin 34, 153 13 P. M. Len, S. Thevutasan, C. S. Fadley, A. P. Kaduwela, and M. 1986 . A. Van Hove, Phys. Rev. B 50, 11 275 1994 . 31 J. T. Hutton, G. T. Trammell, and J. P. Hannon, Phys. Rev. B 31, 14 W. Kossel, V. Loeck, and H. Voges, Z. Phys. 94, 139 1935 . 743 1985 . 15 M. v. Laue, Ann. Phys. Leipzig 26, 55 1936 . 32 ROšNTEC, Rudower Chaussee 6, D-12489 Berlin, Germany. 16 A. M. Afanas'ev et al., Acta Crystallogr., Sect. A: Found. Crys- 33 P. Lechner et al., Nucl. Instrum. Methods Phys. Res. A 377, 346 tallogr. A41, 227 1985 . 1996 . 17 S. R. Andrews and R. A. Cowley, J. Phys. C 18, 6427 1985 . 34 D. V. Novikov, B. Adams, T. Hiort, E. Kossel, and G. Materlik, 18 I. K. Robinson, Phys. Rev. B 33, 3830 1986 . J. Synchrotron Radiat. to be published .