PHYSICAL REVIEW B 1 MARCH 1999-IVOLUME 59, NUMBER 9

Holography with g rays: Simulations versus experiment fora-57Fe

P. KoreckiInstitute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krako´w, Poland

J. Korecki and W. Karas´Department of Solid State Physics, Faculty of Physics and Nuclear Techniques, University of Mining and Metallurgy,

30-059 Krako´w, Poland~Received 23 July 1998!

The g-ray holography was recently applied@P. Koreckiet al., Phys. Rev. Lett.79, 3518~1997!# to imagethe three-dimensional structure ofa-57Fe with an atomic resolution. A simple theory of the hologram forma-tion based on single-scattering cluster formalism taking into account the polarization effects in the resonantnuclear scattering and absorption processes is presented. The high quality of the real-space reconstruction isdemonstrated, however, problems arising from the cancellation of real twin images are revealed. The realisticsimulations are in a good agreement with the experiment. A method for elimination of the real and twin imagescancellation taking advantage of the strong dependence of the scattering phase on the detuning from theresonance in the nuclear scattering is proposed.@S0163-1829~99!03109-4#

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I. INTRODUCTION

In a previous paper,1 we demonstrated experimentally thpossibility of theg radiation use for the holographic struture imaging on an atomic scale. The idea of so-called innal source holography~ISH! comes from Szo¨ke2 as the de-velopment of Gabor’s3 microscopic principle. A diffractionpattern formed by a radiation emitted from sources localiinside solids may be interpreted as a hologram. Such hgrams can be formed by photoelectrons or fluorescent xemitted from internal sources excited by an external radtion ~e.g., hard x rays!. For a radiation of a sufficiently shorwavelength, the holographic real-space reconstruction gthree-dimensional~3D! structure with an atomic resolutionIn a simple wave picture, the wave directly reaching a ffield detector is taken as the holographic reference wawhereas these ones being additionally scattered onnearby atoms are treated as the object waves. The resuinterference pattern is recorded at once on a screen orlected point by point by moving the detector to cover a psibly high angular range. The holographic reconstructionmade numerically.4

The internal source holography was introduced uselectrons: the photoelectrons, the Auger electrons andlow- and medium-energy electrons elastically and nonelacally scattered.5 Because the interaction of electrons wimatter is strong, only surface layers can be imaged. Moover, the electron-atom scattering as well as the elecemission may be strongly anisotropic and the multiple sctering and self-interference is involved, which causes thatreference and the object waves are ill defined. A considerimprovement of the real-space reconstruction qualitygained if the holograms are acquired for several differenergies and if the multiple-energy reconstruction methare applied.6,7 One of the main advantages of multiplenergy measurements is the elimination of the twin imagwhich can hinder the proper determination of the structu

Recently, also, the x-ray holograms were presented.8–10

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Due to a high transparency of solids for x rays, that kindholography enables the imaging of bulk structures. Onother hand, the weak interaction implies that the holograposcillations in the measured patterns are very weak andsequently the hologram acquisition time is long. The valuethe Thomson scattering amplitude gives the typical hographic oscillations in the range of 0.1–0.3 % of the refence wave as compared to 30–50 % for the electholograms.11

Up to now, two versions of the x-ray holography weperformed. The direct one is called the x-ray fluoresceholography~XFH!.8 In the XFH the sample is excited by thexternal x-ray beam. An angular distribution of the followinfluorescence from individual atoms~internal sources of thehologram! is measured using a far-field detector. The methis limited to a predefined single energy and in additionrequires a sufficiently low wavelength~high Z elements! toaccomplish the atomic resolution.

Gog et al.9 presented the multiple energy x-ray hologrphy ~MEXH! being a time reversed version of the XFH. ThMEXH principle uses the optical reciprocity theorem. Hethe positions of the source and the detector are interchanThe atoms at crystal sites are microscopic holographic detors, and the fluorescence measured in all directions is uonly to probe the electromagnetic field inside a crystal atposition of the detector atom. The holographic patternformed due to the diffraction of the incident beam thcauses a modulation of the total fluorescent yield measuas a function of the sample orientation relative to the progation direction of the incident radiation. Both kinds of hlography, the direct and reversed one, will be called hereISH.

The main advantage of the MEXH is the possibility of thmultiple-energy measurements with the strong tunable schrotron sources. By applying of multiple-energy reconstrtion methods,6,7 the twin image problems can be overcomAs it was predicted theoretically,11–13 for a certain combina-tion of the atomic distances and thek-vector values the over

6139 ©1999 The American Physical Society

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6140 PRB 59P. KORECKI, J. KORECKI, AND W. KARAS

lapping of the real and twin images may lead to a suppsion ~enhancement! of the atomic images in the hologramreconstruction. These periodic effects are due to a diffephase behavior of twin and real images. Another possibto perform the multiple-energy experiments involving tBremsstrahlung was proposed theoretically by Miller aSoerensen14 in terms of QED.

The ISH can be also done with theg radiation. The ideacomes from Tegze and Faigel12 who proposed to use nucleaemitters inside a crystal. In that case, there is no need tothe external excitation source. There are two scattemechanisms responsible for the hologram formation withinternal nuclear emitters. In the most direct way, a nuclradiation could form holograms in similar scattering prcesses as in the x-ray holography. However, when the horaphy is performed with theg radiation, use of the strongesg scattering process, namely the nuclear resonant scattebecomes obvious. The processes of this type can be reaonly by means of the Mo¨ssbauer effect. The resulting holographic pattern may be recorded in the far field using a cventionalg-ray detector, with a moderate resolution. A paallel detection with a position sensitive detector could resin a huge reduction of the acquisition times, making threasonable. The resonant detection using the moving M¨ss-bauer absorber is also possible, which additionally bringssensitivity to a magnetic order in the sample. Theg-ray ho-lography can be also done in the inverse version and thehologram obtained with the resonant scattering of the nucradiation in the reversed geometry was presented recen1

In this paper we will discuss the essential features ofinverse version of theg-ray holography~GRH! based on thesimulated and experimentally obtained holograms andreal-space reconstruction images of thea-57Fe structure. Theprinciples of the discussed version of the GRH are presein Sec. II. Section III contains a simplified theory of thhologram formation taking into account the vector characof the used radiation and the resulting strong polarizateffects. The holographic analysis will be reduced tosingle scattering cluster~SSC! ~Ref. 15! calculations. Thesmall cluster calculations, reflecting the character of ideized holograms are given in Sec. IV. The calculations ofholograms and their reconstruction for the real experimesituation and a comparison with experimental results fothin epitaxial 57Fe film is presented in Sec. V, showing maphysical and methodological limitations of the method.Sec. VI we propose how to eliminate the cancellation ofreal and twin images, being one of the weaknesses ofmethod. In Sec. VII we discuss the perspectives of the Gfor more complex systems.

II. PRINCIPLES OF THE INVERSE g-RAY HOLOGRAPHYBASED ON THE RESONANCE DETECTION:

14.4 keV 57Fe CASE TRANSITION

In the discussed inverse version of the GRH, the photfrom a conventional Mo¨ssbauer source (57Co) are directedon to a sample containing the57Fe isotope, where they arresonantly absorbed by the nuclei. The excited nucleire-emit a g-ray photon or deexcitate to the ground-stathrough the inelastic channel emitting a conversion electrIn the kinematical approximation the incident photon may

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absorbed either directly~the holographic reference wave! orafter being resonantly scattered on nearby nuclei~the objectwaves!. The interference of both processes leads to thelographic oscillations in the total yield of the internal coversion electrons or other products of the decay emittedthe nonelastic channel. The electron yield is measuredfunction of the incident angle of the externalg-ray beam.Obviously, a hologram will be formed only if the local suroundings of the absorbing nuclei are identical and equoriented. It should be also pointed out that the so-cal‘‘atomic resolution’’ in the holographic reconstruction hasbe thought of as the average over the whole crystal.

Properties of the nuclear radiation and nuclear resonscattering16 make them very attractive for holography. Fthe 14.41 keV transition in57Fe the corresponding wavelength of 0.86 Å is sufficiently short for atomic resolutioFor this kind of radiation the nucleus can be treated apoint scatterer, so the form factor is constant. The anganisotropy is caused only by the polarization effects andcause the nuclear transitions in57Fe areM1, the scatteringamplitude is a slow varying function of angles.

The amplitude and phase of the scattered radiation camodified by the Doppler shift of the source energy, whienables also the tuning to only the recoil-free processMoreover, the excited state lifetime is long compared tocharacteristic lattice vibration times of atoms in the samand the Lamb-Mo¨ssbauer factor, describing the thermal efects for the case of slow scattering, is isotropic.17 For mag-netically ordered systems, the levels are split and the scaing and absorption is influenced by the internal magnefields at the nuclei. In consequence, there is a possibilityimage only chosen nuclei, with a particular set of the hypfine interaction parameters, and it is possible to resolve3D magnetic structure of the sample.

In the case of57Fe the conversion coefficienta, respon-sible for the ratio of the nonradiative to radiative decay proability, is 8.17. This value is almost ideal for the holographapplication with the detection through the inelastic channThe detection of conversion electrons is very efficient. Telectrons following the conversion process are easy to deover a full hemisphere above the sample using a simpleportional detector. On the other hand, thea determines thescattering strength given roughly bylGg /G5l/(11a),wherel is the radiation wavelength andGg , G are the ra-diative and total excited level widths, respectively. Tnuclear resonant scattering amplitude value is still very has compared to the x-ray Thomson scattering amplitgiven in the forward direction byZr0 , whereZ is the atomicnumber andr 0 is the classical electron radius. In a sharesonance, the value15 of the nuclear resonant amplitude cabe as high as;440r 0 for 57Fe and may cause holographoscillations of several percent in the measured pattern. Thgradiation scatters on electron shells as well, but those pcesses are much weaker than the nuclear scattering.

The holographic pattern can also reveal strong featucaused by the long-range coherent effect. In the case oternal crystal emitters, they are observed as the Kossel lor standing-wave pattern.18–20The dynamical coherent longrange features may play an important role in the casenearly perfect crystals, where the effect of an anomalousnuclear absorption~similar to the Bormann effect for x

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PRB 59 6141HOLOGRAPHY WITH g RAYS: SIMULATIONS VERSUS . . .

rays21! was predicted.22–24This effect leads to a suppressioof the inelastic channel and it was experimentally confirmby Smirnov and Chumakov.25 In an experimental geometranalogous to the geometry of the inverse holograpexperiment,1 a well collimated incident beam, irradiatingnearly perfect single crystal at the Bragg angles, gave a 5falloff of the conversion electron yield. The observed fetures had a very sharp angular character. In the holograexperiment, a much lower collimation is allowed and, adtionally, the Bragg features may be eliminated when a lopass filter is applied to the hologram,8,9,12 especially whenthe sample is nonperfect. However, for a more complex stem, electronic scattering, as well as the dynamical effeshould be taken into account. They may significantly diststructure images in the holographic reconstruction.

III. HOLOGRAM FORMATION AND THE REAL-SPACERECONSTRUCTION

Kagan and Afanas’ev22,23 explained the effect of an inelastic channel suppression in the nuclear absorption whperfect sample is irradiated at near the Bragg angle.suppression takes place when the total amplitude of thecited state formation is zero due to the cancellation of etromagnetic fields at nuclear sites. In our version of hologphy, the conversion electron emission is also involved,the holographic oscillations are the consequence of the shrange order. The nuclear excitation amplitude at a chocrystal site is affected by the electromagnetic fields produby excited nuclear resonators in the surrounding. In otwords, the scattering of theg-ray photon on nearby nuclemodifies the direct absorption. The modulation of the amtude is observed when the externalg-beam direction isscanned over the angles.

The formalism of the nuclear resonant scattering of thgradiation, needed for the description of the GRH is wknown.22–24,26–30 The scattering approach developedTrammel and Hannon,24 reduced to the kinematical approxmation, is the most useful for our purpose. This singscattering simplification is relevant for nonperfect cryst~small crystallites, mosaic structures, etc.!. For perfect crys-tals, the dynamical approach should be applied. The fiformula for the hologram calculation will be given as a SSequation, which allows to perform an easy test of the hogram and of the real-space reconstruction quality.

In the present calculation, we assume that a photon wiLorenz frequency spectrum is emitted from a conventio57Co source and only one photon is irradiating the sampla given moment. The source is moved with the constantlocity to tune to a particular nuclear transition. We neglethe modification of the photon spectrum in the source andsample due to the absorption~thin source and absorber approximation!. The photon can reach the absorbing nucleidifferent paths, directly or after being single or multiple sctered. Here, only the paths involving single scattering insa cluster around the absorbing nuclei are considered~SSCmodel!. We consider only these absorbing nuclei which dexcitate with the emission of a conversion electron andassume that the conversion electron yield is proportionathe square of the total excitation amplitude. The converselectrons coming from different nuclei are incoherent. T

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nuclear spin-flip scattering is also incoherent and therefornot taken into account, as contributing only to the bacground. The emission from the57Co source is stochasti~time and space! and the individual photons are emitted wia random phase and polarization and the averaging ovephotons should be performed.

We assume that the cluster represents a magneticallydered and uniformly magnetized sample, containing57Fe.The magnetic splitting of the ground and excited stateslarge compared to the natural half-width and only one pticular M1 transition may be considered.

Close to the resonance, the nuclear excitation amplitdue to a photon with the wave vectork and polarizationhp

in a M1 transition is given by:24,26

S~k,hp!}C~ j g1 j e ,mgMme!

E2Eeg1 iG/2Y1M

0 ~ k!* hp

}C~ j g1 j e ,mgMme!

E2Eeg1 iG/2p DpM

1 ~ k!* , ~1!

whereE is the energy of the incident photon,Eeg5Ee2Eg isthe transition energy between the ground and excited nucsublevels,G andGg give the full and the radiative widths o

the excited state,YLMl are the vectorial spherical harmonic

(L51,l50), andDpML and C( . . . ) are therotation matri-

ces forL51 and the Clebsh-Gordan coefficient in notatiof Rose.31 The vectorsh657(eu6ef)/A2 are the leftand right circular polarization vectors,p561 forh6 , j g ,mg , j e ,me are the quantum numbers of the grouand excited levels andM5me2mg .

The total amplitude of the formation of the excited stamay be written as a superposition of the excitation amtudes by a plane wave from the source and by the sphewaves scattered on the nuclei inside the cluster. The scattspherical wave is approximated by a plane wave withamplitude and phase of the spherical one. This approxition, common in holographic applications, is particularly reevant for the nuclear scattering because of the small sizthe nuclei.

For a given photon energyE and polarizationhp the totalexcitation amplitude is given by

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wherer i is the position of thei th nuclei relative to the ab-sorbing nuclei,k0 is the incident photon wave vector, anki5k0r i /r i . The exponential factor is responsible for thfree propagation and it is assumed that the source is in thefield. The amplitudesS0,Si are the excitation amplitudes, thdirect one and the one following the scattering on thei thatom at the positionr i , respectively:

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6142 PRB 59P. KORECKI, J. KORECKI, AND W. KARAS

Sp05S~k0 ,hp!,

~3!Sp8

i5S~ki ,hp8!.

The scattering amplitude of the photon (k0 ,hp) to a pho-ton (ki ,hp8) is the sum of the resonant nuclear scatteramplitudef N and the Thomson scattering amplitude on eltron shellsf E :24

f i~kihp8 ;k0hp!5 f N~kihp8 ;k0hp!1 f E~kihp8 ;k0hp!.~4!

The nuclear coherent resonant scattering amplitudegiven by

f N~kihp8 ;k0hp!

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wheref L-M is the Lamb-Mo¨ssbauer factor and other symbohave been already described. The factor 1/2 comes fromsplitting of the ground state and stands for the internucincoherence, whereas the Clebsh-Gordan coefficient issponsible for the intranuclear incoherence.

The Thomson scattering amplitude is given by

f E~kihp8 ;k0hp!52r 0Fel~uscatt! f DWhp8*•hp , ~6!

wherer 0 is the classical electron radius,Fel(u) is the atomicform factor,uscatt is the scattering angle between vectorsk0

andki , f DW is the Debye-Waller factor, andhp8*•hp is the

polarization factor.To get the formula for the observed conversion elect

yield I (k0) an integration over the energy spectrum and sumation over the polarization of the incident photon shouldperformed. Because the source is stochastic this can be mincoherently:

I ~k0!}E (p561

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represents an energy spectrum of the incident photon,Gs isthe total source line width~for the present calculationGs5G), andE0 is the source energy transition shifted duethe Doppler effect byEs5E0v/c, wherev is the velocity ofthe source.

Equation~7! may be rewritten using a ‘‘holographic’’ notation:

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may be treated as the reference and the object waves~nuclearand electronic! for a given photon energy and polarizatioHere,dp8M

1 (u i) is a real part of the rotation matrixDp8M1 and

u i is the angle between ther i vector, and the quantizationaxis. For compound samples, the summation in theoN isover all nuclei equivalent to the absorbing nucleus and inoE over all the atoms in a cluster. It is seen from Eqs.~10!–~12! that the measured pattern is an incoherent weightedof the individual holograms for a given spectral componeof the incident photon.

The total conversion electron yield given by Eq.~9! maybe written as the sum of a slowly varying ‘‘backgroundI 0(k0), an oscillating term containing holographic informtion H(k0) and higher order terms:

I 0~k0!5C~••• !2FM~u0!

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@~E2E02Es!21Gs

2/4#@~E2Eeg!21G2/4#

D ,

~13!

H~k0!5HN~k0!1HE~k0!. ~14!

The nuclear holographic information is contained in

in the

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PRB 59 6143HOLOGRAPHY WITH g RAYS: SIMULATIONS VERSUS . . .

HN~k0!}3p

8k0C~••• !4Gg f L2MFM~u0!(

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3E dE

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2/4#@~E2Eeg!21G2/4#@Eeg2E2 iG/2#

1c.c., ~15!

whereas theHE(k0), responsible for the scattering on the electronic shell, is much weaker. In Eqs.~13! and ~15!, u0 is theangle between the incident photon and the quantization axis.

The angular polarization factorsFM(u) are

FM~u!5 (p561

dpM1 ~u!25H sin2u, M50

1

2~cos2u11!, M561.

~16!

For clarity, the electronic term as well as the higher-order terms are not written down explicitly, but are includedcalculation.

The most important nuclear holographic term may be written as:

HN~k0!}(i

f T~k0 ,r i ,Es!ei ~k0r i2k0•ri !

r i1c.c., ~17!

where the quantity

f T~k0 ,r i ,Es!5FM~u0!FM~u i ! f ~Es! ~18!

is introduced as the effective amplitude of the scattering and the absorption. It contains the summation over all polarizaspectral components of the incident and scattered photons.

The energy dependent factor in the effective scattering and absorption amplitude is a function of the energy shiftEs :

f ~Es!53p

8k0C~••• !4Gg f L2ME dE

@~E2E02Es!21Gs

2/4#@~E2Eeg!21G2/4#@Eeg2E2 iG/2#

1c.c. ~19!

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To observe the holographic oscillations the propagationrection of the incident photon is to be varied with respectthe crystallographic directions.

Equation~17! is similar to the equation derived and usby Barton4 for the formulation of the hologram reconstrution procedure. In our case theH(k0) describes the nonnormalized holographic oscillations of the total electryield following the conversion process, superimposed oslowly varying background described by Eq.~13!.

The 3D structure can be obtained from the normalizhologramx(k0) by a 2D phased Fourier transform:4

U~r!5E ESx~k0!e2 ik0•r ds k0

, ~20!

where

x~k0!5I ~k0!2I 0~k0!

I 0~k0!5

H~k0!

I 0~k0!. ~21!

Usually, the hologram is additionally multiplied by a window function in order to avoid spurious oscillations arisifrom a finite range of integration in thek space. The reastructure is then represented7 by uU(r)u, uU(r)u2 or byurU (r)u2, where the last expression takes into accountradial falloff of the reference or scattered wave.

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To eliminate the anisotropic character of the scatterimodified reconstruction algorithms were proposed and ornally applied for the electron holography. One of them, tscattered wave included Fourier transform~SWIFT!32

method may be easily applied for theg-ray holography andthe modified transform is given by

U~r!5E ES

x~k0!

f ~k0 ,r!*e2 ik0•r ds k0

, ~22!

where thef (k0 ,r) is the scattering amplitude.

IV. BASIC PROPERTIES OF THE g-RAY HOLOGRAM:CALCULATIONS FOR A SMALL CLUSTER

Theg-ray scattering differs considerably from the scatting of electrons and the scattering of x rays above thesorption edge. The polarization effects, a strong energypendence of the scattering amplitude close to the resonaand its different angular character are the most pronoundissimilarities. In this section, we are analyzing the characof the g-ray hologram and the quality of the real-spaceconstruction based on the results obtained in the prevsection. To analyze the basic holographic features the calations are performed for a relatively smalla-57Fe cluster,constructed of only seven coordination spheres~89 nuclei,6.23 Å radius! around the absorbing nucleus. For the simp

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6144 PRB 59P. KORECKI, J. KORECKI, AND W. KARAS

fied case, it is easier to examine the properties of the hgram formed due to a local environment, independentlyexperimental conditions and of the effects being a conquence of the long-range periodicity in the crystal. The hlograms are calculated for the full hemisphere in thek space,and it is assumed that the source is rotated to cover theangular range. The center of the hologram corresponds todirection normal to the surface (kx50, ky50, kz5k). Forthe edges of the hologram the vectork lies in the surfaceplane andk5kuu .

Our calculations should be related directly to the realperimental situation.1 The experimental conditions were optimized to get the best possible hologram~in the sense of thesignal-to-noise ratio!. It implies the choice of one particulanuclear transition out of six, which are possible in the casethe magnetically split levels in57Fe @Fig. 1~a!#. The choiceof the transition should regard not only the intensity of tholographic oscillations but also the fact that certain nucdepending on the position relative to the quantization amay not be illuminated and consequently, not imaged. Tgeometry used for the calculation, depicted in Fig. 1~b!, cor-responds to thea-Fe sample with the normal direction@001#,which is uniformly magnetized in plane in the@010# direc-tion. All patterns are calculated in the sharp resonance.

Figure 1~c! shows the spatial dependence of the convsion electron yield, due to the direct absorption ofg rays~without scattering effects!. The patterns are calculated a

FIG. 1. ~a! Magnetic hyperfine splitting of57Fe levels.~b! Ge-ometry used for hologram calculation. The reference nucleus~ho-logram detector! is located at the origin, the scattering nucleus isposition r i . The anglesu0 andu i are the angles between the manetic field axis and incident and scattered photon propagation dtion, respectively. The R stands for the reference wave, O forobject wave. The factorsFM(u) describe polarization effects inscattering and absorption.~c! Spatial dependence ofFM(u) for M561 and M50 transitions, respectively~dark: low value; light:high value!.

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cording to Eq. ~13! for the transitions61/2→63/2 and61/2→61/2 (M50), respectively. The pattern for thtransition 61/2→61/2 has the same character as for ttransition61/2→63/2 but is weaker because of the depedence on the Clebsh-Gordan coefficient. There is no depdence of the direct absorption on the azimuthal angle in bthe presented patterns.

In the inverse holographic geometry, the direct nucleabsorption plays the role of the reference wave. The anganisotropy of the direct absorption comes from the polarition term determining how strong the absorbing and scating nuclei are illuminated by the reference wave comingrectly from the external source. Obviously, this factor donot influence the normalized hologram intensityx(k0) butfor theM50 transitions may even cancel the total measusignal. More important for the determination of the amptude of the holographic oscillation is to realize@from Fig.1~c!#, how the scattering nucleus at the specific lattice silluminates the absorbing nucleus. This is described byfactor FM(u i).

The patterns shown in Fig. 1~c! represent a dependencetheFM(u) on the angleu5arccos(ky /k) between the photonwave vector~incident or scattered! and the quantization axisIn the pattern for theM50 transition~varying as the sin2 u)there are directions with no emission or absorption presIn consequence, some nuclei would not be seen in the rspace images obtained from the hologram taken forM50transition. For the transitionM561 the dependence is described by cos2 u11.

Figure 2 shows the calculated holograms of seven coonation spheres of thea-57Fe structure. The pure nuclear holograms for the transitionsM561 andM50 are presentedin Figs. 2~a! and 2~b!, respectively. The hologram due to thThomson scattering and subsequent nuclear absorption inM511 transition is shown in Fig. 2~c!. For comparison, anidealized hologram, calculated for the isotropic scatterand absorption and with the square of object wave teneglected, is shown in Fig. 2~d!. Such a ‘‘Gabor-like’’3 ho-logram is calculated for a constant value of the scatteramplitude phasef52p/2. This value corresponds to thsharp nuclear resonant scattering. The intensity of the hgrams in Figs. 2~a!–2~c! is normalized~by the multiplicationfactors given in Fig. 2! to get the same contrast as for thsquare reference wave of Fig. 1~c!. Inspecting the patterns inFig. 1~c!, it is seen that the direct absorption in the normdirection ~which corresponds to the center of the pattermost important from the experimental point of view!, isstronger for theM50 than M51 transition. On the otherhand one can see in Fig. 2 that the holographic signastronger for theM561 transition. This can be understoowhen the angular dependence of theFM(u i) is analyzed. Forthea-Fe structure in our geometry, the nearest neighborsin the off-normal directions relative to the absorbing nucle~illuminated better in theuM u51 transitions!. A signalcaused by the scattering on nuclei lying in the normal dirtion ~next-nearest neighbors—better illuminated in theM50 transitions! is weaker due to the radial falloff of theobject waves. Thus, for this particular structure the hographic oscillations utilizing the61/2→63/2 transition aremore intense and, additionally, are superimposed on a lobackground, which makes the acquisition most efficient. T

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PRB 59 6145HOLOGRAPHY WITH g RAYS: SIMULATIONS VERSUS . . .

amplitude of the holographic oscillation may reach in thcase 3% of the total measured conversion electron yiMoreover, for this transition the nuclei in the neighborhowill be imaged almost isotropically, whereas in the casetheM50 transition the nuclei lying in the directions parallto the quantization axis will not be seen in the real-spimages.

A direct comparison of the pure nuclear holograms wthe idealized one from Fig. 2~d! proves their nearly opticaproperties. As there is no form factor and due to the fact tthe scattering phase shift is angle independent, the nucholograms are almost identical with the Gabor hologramThey should provide high fidelity real-space images, whthe images obtained from the single-energy electron hgrams are of rather poor quality. The nuclear holograposcillation is stronger than that of x-ray holograms whimakes the acquisition times reasonable. Their quality is acertainly better than in the single-energy XFH.

The Thomson hologram shown in Fig. 2~c! is muchweaker than the pure nuclear hologram in the sharp renance. It is even much weaker than it could be expected fthe direct comparison of the maximum values of the amtudes derived from Eqs.~5! and~6!. Additionally, the effec-tive scattering and absorption amplitude includes an intetion over the photon Lorenz energy spectrum and this shoagain enhance the Thomson electronic scattering, whicenergy independent. To explain the relative suppressiothe electronic processes, the anisotropy of the atomic f

FIG. 2. Holograms calculated for small cluster~radius 6.23 Å!around the reference nucleus.~a! and ~b! Pure nuclear hologramobtained for different transitions:M51 andM50. ~c! Hologramresulting from the Thomson atomic scattering and subsequenM51 nuclear absorption.~d! Gabor-like hologram obtained for aisotropic scattering with a constant scattering phase shift2p/2.The multiplication factors refer to the maximum values from F1~c!. The patterns are presented in thek-vector scale in the fullhemisphere.~dark: low intensity; light: high intensity!.

d.

f

e

atar

s.e-

ic

o

o-mi-

a-ldisofm

factor has to be considered. It has a maximum in the forwdirection. In other directions, the amplitude may be an orof magnitude weaker. In the holographic geometry, thesorbing nucleus is illuminated from all directions. Fnuclear holograms, the scattering on several nuclei can ctribute constructively or destructively due to the comparastrength of the scattering in all directions. For electronscattering, in the most favorable case only one atom is inpreferential forward scattering direction.

It is interesting to note that the Thomson hologra~shown forM511) has a lower symmetry than the nucleone. Asymmetries of a similar character were experimentobserved for the Braggg-ray scattering on a57Fe foil.33 Theasymmetries observed in our Thomson hologram result frcomplicated polarization effects present inE1 atomic scat-tering and subsequentM1 nuclear absorption process, anthey are a direct consequence of the interference betweeobject and reference waves. Is should be pointed out thathologram is quite different from the XFH one. The diffeences arise from the resonant detection by a polarinuclear detector. However, the sum of the Thomson hograms obtained forM511 and M521 as well as thesingleM50 hologram have the same symmetry as the pnuclear one. It corresponds to the symmetry of the cryreduced by the magnetic field in they direction.

The real-space images of the cluster structure, obtaineapplying the holographic transform~20! to the M561 ho-logram from Fig. 2~a!, are presented in Fig. 3. The figurshows the 2D slices of the 3D real-space structure. Theaged area is always 14314 Å 2 and the absorbing nucleu~the reference nucleus! occupies the 000 site. The nodesthe square grid indicate the lattice sites in the bcc structThe plots are presented in linear inverted grayscale andthreshold for suppression of smaller intensities is used. Fures 3~a! and 3~b! show the~001! cuts parallel to the surfaceat z50 Å ~containing the reference nucleus! and at z51.43 Å ~the distance corresponding to the interplanspacing in thea-Fe structure above the reference nucleu!.The~100! vertical cut of the real-space in the distance 1.43from the emitter is shown in Fig. 3~c!. The sample is mag-netized in the@010# direction. There are distinct maxima athe crystal lattice sites but different nuclei are imaged wdifferent intensities. The intensity depends on the nuclpositions relative to the quantization axis. This is connecwith polarization effects in the scattering and absorption.expected, the nuclei at positions 100 and 100 are imawith lower intensities as their counterparts in the@010# di-rection. The correction for the anisotropy of the object wavcan be done in the SWIFT method, which is very simplethis case. Because the factorFM(u i) does not depend onintegration variables in Eq.~22! the reconstructed imagshould be just divided by it. Figure 4 shows a~001! real-space cut@as in Fig. 3~a!# after anisotropy correction. Allnuclei equidistant from the absorbing nuclei are imaged nwith the same intensity.

The real-space images in Figs. 3 and 4 show also mfrustrating features. The intensities of the imaged nuclei vindependently on the position relative to the quantizataxis in a way, which cannot be explained directly bysimple radial falloff of the scattered waves. In the~001!planeof the reference nucleus, all nuclei inside the cluster are

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6146 PRB 59P. KORECKI, J. KORECKI, AND W. KARAS

ible but the images of the more distant 110~and symmetryequivalent! nuclei are stronger than those of 010~even afterSWIFT correction!. In the planes above and below, effecthat are even more spurious are observed. In Fig. 3~b!, thenearest neighbor at position12

12

12 and the nucleus at positio

32

12

12 as well as the nuclei in the symmetry related positio

are well imaged. However, the nuclei at the crystal site32 0 1

2

and the symmetry related ones are not visible. In addit

FIG. 3. The real-space images of the cluster structure obtaby applying the holographic transform to theM561 hologramfrom Fig. 2~a!. ~a! The ~001! cut in the plane of the referencnucleus.~b! The ~001! cut 1.43 Å~interplanar spacing in thea-Festructure! above the reference nucleus.~c! The ~100! cut in distance1.43 Å from reference nucleus. The area of the imaged neighhood is 14314 Å 2. Inverted gray scale~light: low intensity; dark:high intensity! is used for intensity, and no threshold value felimination of weaker intensities is applied. The nodes of the squgrid depict the lattice sites in the bcc structure.

s

,

four strong features are artificially imaged near the unoc

pied 1112 -type sites. They come from the strong images

the nuclei in 110-type positions, which show through tplane below. In the vertical~100! cut, the situation concerning visibility of different nuclei is similar but, due to thedifferent vertical and in-plane spatial resolutions, the shaof imaged nuclei is elongated along the normal axis.

The effects described above are mostly due to the olapping of the twin and real images and they were discusextensively in Refs. 11 and 13. Because of a different phbehavior of the real and twin images, the intensity of a givatom/nucleus in the real-space image may oscillate as a ftion of its position relative to the reference atom/nucleus athe wavelength used for the hologram acquisition. For sovalues ofr i /l, the reconstructed images can be enhancsuppressed or even vanish completely. The oscillations ofimage intensity for a nucleus at the position1a having acounterpart nucleus in2a are shown in Fig. 5.

The holographic pattern is very sensitive to the combition of the structural parameters and the radiation walength used. For the real situation the interplay of the rand twin images can hinder the proper structure determtion as it was discussed above. Another combination co

ed

r-

re

FIG. 4. Same as Fig. 3~a! but after a correction for polarizationeffects.

FIG. 5. Intensity oscillation of the nuclear images at the potions 6a due to the overlapping of the real and twin images. Tdistances of the nuclei, in the neighborhood of the reference nucfor the a-Fe structure, are depicted by arrows.

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PRB 59 6147HOLOGRAPHY WITH g RAYS: SIMULATIONS VERSUS . . .

be much more favorable as it is shown on the example ofhologram calculated for a different hypothetical lattice costant a53.02 Å. The hologram@Fig. 6~a!# differs verymuch from that obtained foraa2Fe52.87 Å @compare Fig.2~a!#. The ~001! cut of the real space is presented in F6~b!. The reconstructed image is noticeably different as copared to the case for the true lattice constant@compare Fig.3~b!#. All nuclei in the plane above the reference nucleuswell imaged. This example shows that the problem of twimages can be very troublesome in theg-ray holography andthe structure determination process can be ambiguous insingle-energy holography.

V. REAL SAMPLE—EXPERIMENT AND SIMULATIONS

In this section, we will present a comparison betwesimulations and experimentally obtained data. Details ofexperimental procedure were described previously.1 The to-tal yield of the electrons following the conversion proceswas measured as a function of the sample orientation relato the g rays propagation direction. The sample was2000 Å 57Fe film grown by the molecular-beam epitaxy oa polished 1031031 mm3 MgO(001) single-crystal substrate. The film consisted of small equally orient57Fe(001) crystallites. The average crystallite diameter wsmaller than 100 Å . The kinematical approximation in tcalculation is for this sample appropriate and the dynameffects should be considerably suppressed. The samplemagnetized along the in-plane@010# direction. The conven-tional 100 mCi57Co(Rh) Mossbauer source of 8 mm diameter was placed 15 cm from the sample. The hologramacquired in the sharp resonance for the source Doppler tu

FIG. 6. ~a! Hologram (M51 transition! for a small bcc clusterwith a lattice constanta53.02 Å @to compare with Fig. 2~a!#. ~b!The resulting ~001! real-space cut 1.43 Å above the referennucleus@to compare with Fig. 2~b!#.

e-

.-

e

he

ne

svea

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alas

ased

to theM561 nuclear transitions. The data were normalizto an off resonant signal. The angular pattern was recoron a du13du251.80°31.80° mesh in the range from243.2° to 43.2° for two angles of rotation around the@100#and @010# directions.

For comparison with the simulation, the normalized eperimental pattern was transformed to thek-vector mesh andthen it was twofold symmetrized. A slowly varying background intensity coming mainly from the reference wawas subtracted from the total signalI (k0). In the backgroundelimination process, the low-frequency Gaussian convolutI (k0) ^ G(k0) was subtracted from the measured pattern ging the normalized hologram34

x~k0!5I ~k0!2I ~k0! ^ G~k0! ~23!

presented in Fig. 7~a!.

FIG. 7. ~a! Calculated and measuredg-ray holograms for thea-Fe structure. The patterns are presented in thek-vector scale.~b!Profiles of the holographic oscillation for constantkx values of0 Å 21 and 3.6 Å21 ~points: experiment; lines: simulation!.

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erictn-ialc-.

6148 PRB 59P. KORECKI, J. KORECKI, AND W. KARAS

FIG. 8. Real-space images of thea-Fe struc-ture obtained from holograms in Fig. 7~a!. Thereal-space cuts calculated from simulation ashown in Figs. 8~a!–8~c!, the cuts obtained fromthe measured hologram are depicted in Fig8~d!–8~f!. The area of the imaged neighborhoois 14314 Å 2. Inverted gray scale~light: low in-tensity; dark: high intensity! is used for intensity,and no threshold value for elimination of weakintensities is applied. The nodes of the grid depthe lattice sites in the bcc structure. The horizotal and vertical bars correspond to the spatresolution resulting from the experimentally acessiblek-vector range for hologram acquisition

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The simulation of a real hologram was performed forcluster of 300 coordination spheres (;104 nuclei, ;30 Åradius! of the a-57Fe structure on a coarse angular ganalogous to that one used in the experiment. In the simtion, the integration over the source and sample areas asas the real source-to-sample distance were included. Duthis procedure the hologram becomes more blurred.background~reference wave! removal was the same as in thcase of the experimental data to take into account possdistortions of the real space introduced by this proceduThe calculated hologram is presented in Fig. 7~a!. The inten-sity distributions in the experimental and calculated hograms are very similar, especially when one keeps in minhigh sensitivity to the lattice parameter.

Differences in the relative intensities between the simlated and experimental holograms are probably due toimperfect procedure of the background removal. Whsingle profiles of the holograms are compared, the agreem

a-elltoe

lee.

-a

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between the experiment and the calculations is especwell emphasized, as can be seen from Fig. 7~b!. The real-space images of the structure obtained by applying the hgraphic transform to the calculated and experimental hograms are shown in Figs. 8~a!–8~c! and 8~d!–8~f!,respectively. In both cases, the nearest neighbors in thea-Festructure are well imaged. In cuts obtained from the expmental hologram, the images of the nearest neighborsshifted by about 0.2 Å from the true positions. It is causedexperimental errors in the sample orientation. The limitangular range used for the hologram recording resultsmoderate resolution. The theoretical resolution is calculaaccording to Ref. 35 and it is shown as the vertical ahorizontal bars on the plots. The poor vertical resolutimakes the images of the12

12

12 -type nuclei elongated along

@001# axis in Figs. 8~c! and 8~f!. The overlapping of the reaand twin images enhances this effect and in the~001! planeof the reference nucleus, the four maxima corresponding

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PRB 59 6149HOLOGRAPHY WITH g RAYS: SIMULATIONS VERSUS . . .

‘‘wings’’ of those nuclei are dominant. The images of aother nuclei beyond the first coordination shell are stronsuppressed. This is a consequence of a poor angular retion, resulting from the relatively short source-sample dtance and the finite size of the source and the sample,also due to the coarse angular grid. It is, however, interesto note that the nuclei at32

12

12 and the symmetry equivalen

sites are not imaged at all. This effect can also be observeFig. 3, for the idealized experimental situation, and itclearly due to the cancellation of the real and twin imagSome differences of the relative intensity ratios in the expmental and theoretical real-space cuts may also be causethe experimental error in tuning to the resonance energy.phase of the scattering amplitude is a rapid function ofsmall shifts from the resonance energy, and the scattephase determines the overlapping of real and twin image

The resolution and the size of the imaged area canimproved by recording the hologram in a wider angurange, with a better angular resolution. The polarizationfects are easy to be eliminated with the SWIFT method.the main limitation in the single-energyg-ray holographyremains the effect of the cancellation of the real and twimages.

VI. ELIMINATION OF REAL AND TWIN IMAGESCANCELLATION IN g-RAY HOLOGRAPHY

In the case of theg-ray holography utilizing the Mo¨ss-bauer effect, it is possible to vary the energy of the incidbeam only in a close vicinity of the resonance. The possenergy change is then of the orderG. The multienergy meth-ods require a remarkable change in the phase of the prgation factor exp(ikr) due to a change of thek value. This isinaccessible by such a small energy variation as can beized in the Mossbauer experiment. In this section, we ppose a method, exploiting the fact, that the scattering phitself is a rapidly varying function of the energy near tresonance. We will also discuss the experimental feasibof this method.

The effect of the cancellation of the nuclear images fosystem consisting of a reference nucleus and two scattenuclei in positions1a and2a can be explained simply using arguments from Ref. 11. The functionU(r), representingthe real-space image, is calculated for this system by sututing the value of the holographic oscillations due to tnuclei in positions1a and 2a, given by Eq.~17! into theFourier holographic transform from Eq.~20!. Then, the re-constructed real space in the positionr is given by

U~r!5E ESS f T~k,1a,E!*

ae2 ikaeik•a

1f T~k,2a,E!*

ae2 ikae2 ik•aDe2 ik•r ds k1c.c.

~24!

The exponential factors oscillate, except forr56a, andhence, the image function has maxima forr56a. Conse-quently, the reconstructed image in the position1a is domi-nated by two terms:

ylu--utg

in

.i-by

heeng.e

rf-o,

n

tle

a-

al--se

ty

ang

ti-

U~1a!.1

ae2 ikaE E

Sf ~k,1a,E!* ds k

11

aeikaE E

Sf ~k,2a,E!ds k , ~25!

where the first one corresponds to the real image whilesecond one leads to the nonphysical twin image. Fromproperties of the effective scattering and absorption amtude given by Eq.~18! and from the fact that theFM(u) isreal andFM(u)5FM(p2u) it follows that

U~1a!}u f ~E!u~e2 ikae2 if~E!1eikaeif~E!!, ~26!

wheref(E) is the phase of the effective amplitude givenEq. ~18!. For a given energy (k-vector value! the image func-tion U(1a) oscillates between real and imaginary valudue to the propagation phase factors exp(6ika) and for somevalues ofa equals zero. From Eq.~26! it is obvious that thecondition for cancellation changes and depends on the vof the scattering amplitude phase factors exp@6if(E)#. So,for example, the real-space image obtained from ag-ray ho-logram recorded in the sharp resonance and the imagetained from a single XFH hologram for the same energy wdiffer, because different pairs of atoms at6a could be af-fected by the cancellation. The Thomson scattering amtude is almost pure real while the nuclear resonant scatteamplitude is pure imaginary.

When the Mo¨ssbauer effect is used for holography, tscattering phase can be changed directly by the detufrom the resonance. Of course, it is inevitably accompanby a rapid falloff of the resonant signal. If, however, twholograms would be taken for the two particular energy vuesE5E1 and E5E2 , not too far from the resonance, thholographic transformation

U8~r!5 (E5E1,2

UE~r!

f ~E!*eikr ~27!

could remove the problem of the real and twin images ccellation. In Eq.~27! theUE(r) is a usual single holographiimage obtained by applying the holographic transform froEq. ~20! to the hologram recorded for a particular energyE5E1 or E5E2 . The phasef(E) of the scattering amplitudef (E) depends strongly on the energy shift from the renance. It is reasonable to take the two holograms symmcally, on both sides of the resonance and denotef(E1)5f1 and f(E2)5f2 . So, if for each of them the phasdiffers by 6D/2 from the phase at the sharp resonanf(E0)5p/2, we can write thatf152p/22D/2 andf252p/21D/2. Inserting the image functionsUE(1a) recordedfor the phasef1 andf2 to the proposed holographic tranform @Eq. ~27!# we get

U8~1a!5222e2ika cosD, ~28!

and the reconstructed image intensityuU(1a)u is propor-tional to

uU8~1a!u}A11cos2 D22 cos~2ka!cosD. ~29!

If D5p/2 then the oscillation of the reconstructed imaintensity is completely removed for an arbitrary pair of n

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6150 PRB 59P. KORECKI, J. KORECKI, AND W. KARAS

clei at the positions6a relative to the reference nucleus. Frealistic experiments theD5p/2 condition is impossible toachieve and for holograms recorded for a smaller phaseference the oscillation cannot be perfectly removed. Tsituation illustrates Fig. 9. The intensity oscillations of images for nucleus pairs at6a are plotted as a function of tha length for D50 ~single hologram, as in Fig. 5!, D5p/2~idealized situation! and D5p/3 ~a realistic value used fothe experiment simulation, corresponding to the detunfrom resonance by61.75Gexp563.5Gs . Apparently, thesituation, which is experimentally possible, brings certimprovement. The oscillations are yet present but realtwin images never cancel. This fact has direct consequefor the reconstruction quality. Figure 10 shows the real-spcuts obtained forD5p/2 andD5p/3. In the idealized caseof D5p/2 all nuclei are visible with almost equal intensitFor the experimentally realistic hologram recorded off renance, the atomic scattering becomes also important,because it is not taken into account in the reconstrucprocedure, the real-space is distorted. In the realistic casenuclei are still visible but certain artifacts are present. Imaof nuclei at the positions32

12

12 and the symmetry equivalen

ones can be recognized, whereas in the image obtainedthe single in-resonance hologram they were not presenaddition, the nearest neighbors are imaged with a stronintensity as compared to the next-nearest neighbors, acould be expected from the spherical character of the stered waves.

The method proposed above reveals similarities betwtheg-ray holography and other methods suggested earlieovercome the phase problem in the resonant scattering onuclear radiation. For example, the phase could be demined directly, as proposed theoretically,26 by the analysis ofthe Kossel line pattern, which results from the interferenbetween the direct emitted wave and the wave diffractedBragg angles. The multiphase holography has also cer

FIG. 9. Intensity oscillation of the nuclear images at positi6a on applying the holographic transform from Eq. 25 to twholograms which differ in phase of the scattering amplitude byD.Dashed line corresponds toD5p/2 ~idealized situation!, dotted lineto D5p/3 ~simulation of real experiment! and solid line corre-sponds toD50 ~only one hologram at resonance andf52p/2, asin Fig. 6!. The distances of the nuclei, in the neighborhood ofreference nucleus for thea-Fe structure, are depicted by arrows.

if-is

g

dese

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mIneritt-

ntoher-

eatin

correspondence to preliminary studies,36,37 in which thephase is derived from the Bragg intensity, modulated asfunction of the detuning from the resonance by the interfence between the nuclear resonant and Rayleigh electrscattering. In contrast to the holography, the methods invoing the Bragg diffraction require perfect single-crystallinsamples and a high collimation.

VII. OUTLOOK AND CONCLUSIONS

Finally, we would like to discuss some questions relato practical applications of theg-ray holography for the di-rect visualization of the real three-dimentional structure wthe atomic resolution. The main limitation of the methoresults from the fact that only particular samples can be alyzed. First, in order to exploit fully the properties of thnuclear resonant scattering, the analyzed sample shouldcompound containing exclusively~or at least in several tenpercent! a Mossbauer isotope, which usually has a smabundance. Additionally, in the direct version of the holoraphy, the radioactive nuclei should be introduced intosample in the amount that is sufficient for recording the hlogram in a reasonable time. Second, like in the other Imethods, the local surroundings of the reference nuclei hto be identical and equivalently oriented. Obviously, tsample fulfilling the above requirements has to be madetificially and, for practical reasons, the group of suitable stems is limited to epitaxial layers. In the inverse variantholography, the sample should contain about 100 ML

e

FIG. 10. The~001! cut of real space 1.43 Å above the referennucleus obtained by applying the modified holographic transfofrom Eq. ~25! to two simulated holograms forD5p/2 ~a! and D5p/3 ~b!.

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PRB 59 6151HOLOGRAPHY WITH g RAYS: SIMULATIONS VERSUS . . .

57Fe in order to achieve reasonable acquisition times. Seral types of the57Fe samples, obtainable with the molecubeam epitaxy, like metallic1 or oxide38 films, and mono-atomic superlattices,39 are already nowadays good accessicandidates. In the direct version of the holography, the pallel detection with a position sensitive detector can conserably shorten the acquisition time, setting the number ofradioactive atoms at 1014 in case of57Co. The nonresonanvariant of the acquisition scheme excludes however intering features ofg-ray holography not present in other ISmethods. Only the resonant detection allows to tune tospecific Mossbauer transition between the sublevels splithe internal magnetic and to visualize the 3D magnetic strture of the sample. For example, in the structure of magne(Fe3O4) Fe occupies two different crystallographic positioand the magnetic hyperfine fields for both sites differ enouto allow the hologram to be acquired separately for theferent Mossbauer transitions. However, it has to be noticthat for such a complicated system the hologram qua~number of counts, the angular resolution, and range! shouldbe considerably improved. In addition, it is expected thathologram will be much more complicated due to thecreased role of the Thomson scattering and strong dynameffects. The resonant detection allows also for eliminationreal twin images by the multiple phase method proposeddescribed in Sec. VI.

The g-ray holography can be performed in the most ecient way for 57Fe. Its advantages come from the mostvorable nuclear parameters among all Mo¨ssbauer isotopeand from the frequent use of iron in epitaxial technologThe nuclear properties predestinate for holography a151Eu and119Sn. However, due to the limitation in the souractivity (119Sn) and the low resonance cross-section va(151Eu), the holographic detection limit is for them muchigher than for iron.

At present, several synchrotron radiation facilities asuitable for theg holography. The measurement through tinelastic channel could be performed by the conversion x-photons instead of the conversion electrons. The presendiation intensity, which for typical x-ray applications is vehigh (3.531012 photons/s/eV for 14.4 keV radiation i

on,

IP

m

v-r

er--e

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e-alfd

--

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yra-

ESRF Grenoble!, does not bring expected advantages forholography by the nuclear resonant scattering. The beamtensity corresponds only to 17 000 photons/s per the57Fenuclear level natural widthG0 , to be compared with abou53105 photons/s from a 200 mCi57Co irradiating thesample in a reasonable experimental geometry. One ofstrongest points of the synchrotron radiation, its perfect climation, becomes in some sense a disadvantage due tostrong dynamical diffraction effects~similar to Kossel linepattern! that have a very sharp angular character.

Concluding, we have presented the basic theory ofinverse g holography. The atomic resolution hologramformed due the resonant nuclear scattering of the electromnetic radiation. The acquisition is made by the resonaMossbauer detection of the de-excitation products viainelastic channel~conversion electrons!. The absorbing andre-emitting nuclei are microscopic holographic detectoDue to a small size of nuclei, the character of the nuclscattering process is almost ideal for holographic applitions. The simulations performed in the frame of the Smodel proved however, that even for the idealized hologrspurious images and effects are observed in the real-sreconstruction. They result from the overlapping of the rand twin images, typical for all kinds of single-energy hlography. The polarization effects in the scattering andsorption are easy to remove by a modified holographic traform. We showed also explicitly the small role of thThomson scattering in the hologram formation for thea-57Festructure. Despite the obvious limitations of the singenergyg holography, the real space of thea-Fe structurecould be reconstructed based on the experimental hologThe hologram and the real-space image agree well with thresulting from the simulations. We proposed also a way hto eliminate the cancellation of twin and real images exploing the strong dependence of the scattering phase onenergy, close to the resonance.

ACKNOWLEDGMENT

Financial support by the Polish Science Research Couunder Grant No. 2 P03B 06913 and No. 2 P03B 08010gratefully acknowledged.

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