\documentstyle[prb,aps,english,polish,amsmath]{revtex}
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\begin{document}
\draft
\title{ Holography with $\gamma$-rays - Simulations vs. Experiment on
$\alpha$-$^{57}$Fe}
\author{P.  Korecki}
\address{Institute of Physics, Jagiellonian University,
Reymonta 4, 30-059 Krak\'{o}w, Poland}
\author{J.Korecki and W.Kara\'{s}}
\address{Department of Solid State Physics, Faculty of Physics and Nuclear Techniques,
University of Mining and Metallurgy, 30-059 Krak\'{o}w, Poland}
\maketitle
\begin{abstract}
\baselineskip=24pt
The $\gamma$-ray holography was recently applied [P. Korecki {\em et al.} Phys. Rev. Lett. ${\bf 79}$, 3518 (1997)]
 to image the
three-dimensional structure of $\alpha$-$^{57}$Fe with an atomic resolution. A simple theory
of the hologram formation based on Single Scattering Cluster formalism
taking into account the polarization effects in the resonant nuclear
scattering and absorption processes is presented.  The high quality of the
real-space reconstruction is demonstrated, however problems arising from
the cancellation of real-twin images are revealed.  The realistic
simulations are in a good agreement with the experiment. A method for elimination of the real and twin-images
cancellation taking advantage of the strong dependence of the scattering
phase on the de-tuning from the resonance in the nuclear scattering is proposed.
\end{abstract}
\pacs{76.80.+y, 07.85.-m, 42.40.-i}
\selectlanguage{english}
\baselineskip=24pt
\section{ Introduction}
\label{sec1}
In the previous paper, \cite{ref1} we demonstrated experimentally the
possibility of the $\gamma$ radiation use for the holographic structure
imaging on an atomic scale. The idea of so-called internal source holography
(ISH) comes from Sz\"{o}ke\ \cite{ref2} as the development of Gabor's
\cite{ref3} microscopic principle. A diffraction pattern formed by a
radiation emitted from sources localized inside solids may be interpreted as
a hologram. Such holograms can be formed by photoelectron or fluorescent X-rays
emitted from internal sources excited by an external radiation (e.g. hard
X-rays). For a radiation of a sufficiently short wavelength, the holographic real
space reconstruction gives three-dimensional (3D) structure with an atomic resolution. In a
simple wave picture, the wave directly reaching a far-field detector is
taken as the holographic reference wave, whereas these ones being additionally
scattered on the nearby atoms are treated as the object waves. The resulting
interference pattern is recorded at once on a screen or collected point by
point by moving the detector to cover a possibly high angular range. The
holographic reconstruction is made numerically.\cite{ref4}

The internal source holography was introduced using electrons: the
photoelectrons, the Auger electrons and the low and medium-energy electrons
elastically and non-elastically scattered.\cite{ref5} Because the interaction of
electrons with matter is strong, only surface layers can be
imaged. Moreover, the electron-atom scattering as well as the electron
emission may be strongly anisotropic and the multiple scattering and
self-interference is involved, which causes that the reference and the
object waves are ill defined. A considerable improvement of the real-space
reconstruction quality is gained if the holograms are acquired for several different
energies and if the multiple-energy reconstruction methods are applied.
\cite{ref6,ref7} One of the main advantages of multiple-energy measurements is the
elimination of the twin images, which can hinder the proper determination of
the structure.

Recently, also the X-ray holograms were presented.\cite{ref8,ref9,ref10} Due to a high
transparency of solids for X-rays, that kind of holography enables the
imaging of bulk structures. On the other hand, the weak interaction implies
that the holographic oscillations in the measured patterns are very weak and
consequently the hologram acquisition time is long. The value of the Thomson
scattering amplitude gives the typical holographic oscillations in the range
of 0.1\%-0.3\% of the reference wave as compared to 30\%-50\% for the electron
holograms.\cite{ref11} 

Up to now, two versions of the X-ray holography were
performed. The direct one is called the X-ray Fluorescence Holography (XFH).\cite{ref8} In the
XFH the sample is excited by the external X-ray beam. An angular
distribution of the following fluorescence from individual atoms (internal
sources of the hologram) is measured using a far field detector. The method
is limited to a predefined single energy and in addition, it requires a
sufficiently low wavelength (high Z elements) to accomplish the atomic
resolution.

Gog et al. \cite{ref9} presented the Multiple Energy X-ray Holography (MEXH) being a
time reversed version of the XFH. The MEXH principle uses the optical
reciprocity theorem. Here, the positions of the source and the detector are
interchanged. The atoms at crystal sites are microscopic holographic
detectors, and the fluorescence measured in all directions is used only to
probe the electromagnetic field inside a crystal at the position of the
detector atom. The holographic pattern is formed due to the diffraction of
the incident beam that causes a modulation of the total fluorescent yield
measured as a function of the sample orientation relativ to  the propagation
direction of the incident radiation. Both kinds of holography, the direct
and reversed one, will be called here the ISH.

The main advantage of the MEXH is the possibility of the multiple-energy
measurements with the strong tunable synchrotron sources. By applying of
multiple-energy reconstruction methods,\cite{ref6,ref7} the twin-image problems can be
overcome. As it was predicted theoretically,\cite{ref11,ref12,ref13} for a certain
combination of the atomic distances and the $\vec{k}$-vector values the overlapping
of the real and twin images may lead to a suppression (enhancement) of the
atomic images in the hologram reconstruction. These periodic effects are due
to a different phase behavior of twin and real images. Another possibility
to perform the multiple-energy experiments involving the Bremsstrahlung was
proposed theoretically by Miller and Soerensen \cite{ref14} in terms of QED.

The ISH can be also done with the $\gamma$-radiation. The idea comes from
Tegze and Faigel \cite{ref12} who proposed to use nuclear emitters inside a
crystal. In that case, there is no need to use the external excitation
source. There are two scattering maechanisms responsible for the hologram
formation with the internal nuclear emitters. In the most direct way, a
nuclear radiation could form holograms in similar scattering processes as in
the X-ray holography. However, when the holography is performed with the
$\gamma$-radiation, use of the strongest $\gamma$ scattering process, namely
the nuclear resonant scattering, becomes obvious. The processes of this type
can be realized only by means of the M\"{o}ssbauer effect. The resulting
holographic pattern may be recorded in the far-field using a conventional
$\gamma$-ray detector, with a moderate resolution. A parallel detection with
a position sensitive detector could result in a huge reduction of the
acquisition times, making them reasonable. The resonant detection using the
moving M\"{o}ssbauer absorber is also possible, which additionally brings
the sensitivity to a magnetic order in the sample. The
$\gamma$-ray holography can be also done in the inverse version and the
first hologram obtained with the resonant scattering of the nuclear
radiation in the reversed geometry was presented recently.\cite{ref1}

In this paper we will discuss the essential features of the inverse
version of the $\gamma$-ray holography (GRH) based on the simulated and experimentally
obtained holograms and the real space reconstruction images of the
$\alpha$-$^{57}$Fe structure. The principles of the discussed version of the
GRH are presented in Section \ref{sec2}. Section \ref{sec3} contains a
simplified theory of the hologram formation taking into account the vector
character of the used radiation and the resulting strong polarization
effects. The holographic analysis will be reduced to the Single Scattering
Cluster (SSC) \cite{ref15} calculations. The small cluster calculations,
reflecting the character of idealized holograms are given in Section\
\ref{sec4}. The calculations of the holograms and their reconstruction for
the real experimental situation and a comparison with experimental results
for a thin epitaxial $^{57}$Fe film is presented in Section\ \ref{sec5},
showing main physical and methodological limitations of the method. In
Section\ \ref{sec6} we propose how to eliminate the cancellation of the real
and twin images, being one of the weaknesses of the method. In Section\
\ref{sec7} we discuss the perspectives of the GRH for more complex systems.

\section{Principles of the inverse $\gamma$-ray holography based on the
resonance detection: 14.4 keV $^{57}$Fe case transition.}
\label{sec2} 
In the discussed, inverse version of the GRH the photons from a conventional
 M\"{o}ssbauer source ($^{57}$Co) are directed on to a sample containing the
$^{57}$Fe isotope, where they are resonantly absorbed by the nuclei. The
excited nuclei can re-emit a $\gamma$-ray photon or deexcitate to the ground
state trough the inelastic channel emitting a conversion electron. In the
kinematical approximation the incident photon may be absorbed either
directly (the holographic reference wave) or after being resonantly
scattered on nearby nuclei (the object waves). The interference of both
processes leads to the holographic oscillations in the total yield of the
internal conversion electrons or other products of the decay emitted in the
non-elastic channel. The electron yield is measured as a function of the
incident angle of the external $\gamma$-ray beam. Obviously, a hologram will
be formed only if the local surroundings of the absorbing nuclei are
identical and equally oriented. It should be also pointed out that the
so-called "atomic resolution" in the holographic reconstruction has to be
thought as the average over the whole crystal.

Properties of the nuclear radiation and nuclear resonant scattering
\cite{ref16} make them very attractive for holography. For the 14.41 keV
transition in
$^{57}$Fe the corresponding wavelength of 0.86\AA\ is sufficiently short
for atomic resolution. For this kind of radiation the nucleus can be treated
as a point scatterer, so the form factor is constant. The angular anisotropy
is caused only by the polarization effects and because the nuclear
transitions in
$^{57}$Fe are M1, the scattering amplitude is a slow varying function of
angles.

The amplitude and phase of the scattered radiation can be modified by the
Doppler shift of the source energy, which enables also the tuning to only
the recoil-free processes. Moreover, the excited state life time is long
compared to the characteristic lattice vibration times of atoms in the
sample and the Lamb-M\"{o}ssbauer factor, describing the thermal effects for
the case of slow scattering, is isotropic.\cite{ref17} For magnetically ordered
systems, the levels are split and the scattering and absorption is
influenced by the internal magnetic fields at the nuclei. In consequence,
there is a possibility to image only chosen nuclei, with a particular set of
the hyperfine interaction parameters, and it is possible to resolve the 3D
magnetic structure of the sample.

In the case of $^{57}$Fe the conversion coefficient $\alpha$, responsible for
the ratio of the non-radiative to radiative decay probability, is 8.17. This
value is almost ideal for the holographic application with the detection
through the inelastic channel. The detection of conversion electrons is very
efficient. The electrons following the conversion process are easy to detect
over a full hemi-sphere above the sample using a simple proportional
detector. On the other hand, the $\alpha$ determines the scattering strength
given roughly by
$\lambda \Gamma_{\gamma}/\Gamma = \lambda/(1+\alpha), $ where $\lambda$  is
the radiation wavelength and $\Gamma_{\gamma}$, $\Gamma$ are the
radiative and total excited level widths, respectively. The nuclear resonant
scattering amplitude value is still very high as compared to the X-ray
Thomson scattering amplitude given in the forward direction by $Zr_0$, where
$Z$ is the atomic number and
$r_0$ is the classical electron radius. In a sharp resonance, the value
\cite{ref15} of
the nuclear resonant amplitude can be as high as
$\sim 440 r_0$ for $^{57}$Fe and may cause holographic oscillations 
of several percents in the measured pattern. The $\gamma$
radiation scatters on electron shells as well, but those processes are much
weaker than the nuclear scattering.

The holographic pattern can also reveal strong features caused by the
long-range coherent effect. In the case of internal crystal emitters, they
are observed as the Kossel lines or standing wave
pattern.\cite{ref18,ref19,ref20} The dynamical coherent long-range features
may play an important role in the case of nearly perfect crystals, where the
effect of an anomalous low nuclear absorption (similar to the Bormann effect
for X-rays \cite{ref21}) was predicted.\cite{ref22,ref23,ref24} This effect
leads to a suppression of the inelastic channel and it was experimentally
confirmed by Smirnov and Chumakov.\cite{ref25} In an experimental geometry
analogous to the geometry of the inverse holographic experiment,\cite{ref1}
a well collimated incident beam, irradiating a nearly perfect single-crystal
at the Bragg angles, gave a 50\% fall-off of the conversion electron yield.
The observed features had a very sharp angular character. In the holographic
experiment, a much lower collimation is allowed and ,additionally, the Bragg
features may be eliminated when a low-pass filter is applied to the
hologram,\cite{ref8,ref9,ref12} especially when the sample is non-perfect.
However, for a more complex system, electronic scattering, as well as the
dynamical effects should be taken into account. They may significantly 
distort structure images in the holographic reconstruction.

\section{Hologram formation and the real space reconstruction}
\label{sec3}

Kagan and Afanas'ev \cite{ref22,ref23} explained the effect of an inelastic
channel suppression in the nuclear absorption when a perfect sample is
irradiated at near the Bragg angle. The suppression takes place when the
total amplitude of the excited state formation is zero due to the
cancellation of electromagnetic fields at nuclear sites. In our version of
holography, the conversion electron emission is also involved, but the
holographic oscillations are the consequence of the short-range order. The
nuclear excitation amplitude at a chosen crystal site is affected by the
electromagnetic fields produced by excited nuclear resonators in the
surrounding. In other words, the scattering of the $\gamma$-ray photon on
nearby nuclei modifies the direct absorption. The modulation of the
amplitude is observed when the external $\gamma$-beam direction is scanned
over the angles.

The formalism of the nuclear resonant scattering of the $\gamma$-radiation, needed
for the description of the GRH is well known.
\cite{ref22,ref23,ref24,ref26,ref27,ref28,ref29,ref30} The scattering
approach developed by Trammel and Hannon, \cite{ref24} reduced to the kinematical
approximation, is the most useful for our purpose. This single-scattering
simplification is relevant for non-perfect crystals (small crystallites,
mosaic structures etc.). For perfect crystals, the dynamical approach should
be applied. The final formula for the hologram calculation will be given as
a SSC equation, which allows to perform an easy test of the hologram and of
the real-space reconstruction quality.

In the present calculation, we assume that a photon with a Lorenz frequency
spectrum is emitted from a conventional $^{57}$Co source and only one photon is
irradiating the sample at a given moment. The source is moved with the
constant velocity to tune to a particular nuclear transition. We neglect the
modification of the photon spectrum in the source and in sample due to the
absorption (thin source and absorber approximation). The photon can reach
the absorbing nuclei on different paths, directly or after being single or
multiple-scattered. Here, only the paths involving single scattering inside
a cluster around the absorbing nuclei are considered (SSC model).  We
consider only these absorbing nuclei which deexcitate with the emission of a
conversion electron and we assume that the conversion electron yield is
proportional to the square of the total excitation amplitude. The conversion
electrons coming from different nuclei are incoherent. The nuclear spin-flip
scattering is also incoherent and therefore is not taken into account, as
contributing only to the background. The emission from the $^{57}$Co source is
stochastic (time and space) and the individual photons are emitted with a
random phase and polarization and the averaging over all photons should be
performed.

We assume that the cluster represents a magnetically ordered and uniformly
magnetized sample, containing $^{57}$Fe. The magnetic splitting of the
ground and excited states are large compared to the natural half-width and
only one particular M1 transition may be considered.

Close to the resonance, the nuclear excitation amplitude due to a photon
with the wave vector $\vec{k}$ and polarization $\vec{\eta}_p$ in a M1 transition
is given by:\cite{ref24,ref26}

%
\begin{equation}
\label{eq:1}
S(\vec{k},\vec{\eta}_p )\propto \frac{C(j_g\, 1\, j_e \, ,m_g\, M \,
m_e)}{E-E_{eg}+i\Gamma /2}\vec{Y}^{0}_{1M}(\hat{\vec{k}})^{*}\vec{\eta}_p  \propto
\frac{C(j_g\, 1\, j_e \, ,m_g\, M \,
m_e)}{E-E_{eg}+i\Gamma /2}p\,D^1_{pM}(\hat{\vec{k}})^{*},
\end{equation}
%
\noindent
where $E$ is the energy of the incident photon, $E_{eg}=E_e-E_g$ is the
	transition energy between the ground and excited nuclear sub-levels,
$\Gamma$  and
	$\Gamma_{\gamma}$ give the full and the radiative widths of the excited state,
	$\vec{Y}^{\lambda}_{LM}$ are the vectorial spherical harmonics
($L=1,\,\lambda = 0$) and $D^L_{pM}$ 
	and $C(..)$ are the rotation matrices for $L=1$ and the Clebsh-Gordan
	coefficient in notation of Rose. \cite{ref31} The vectors $\vec{\eta}_{\pm}=\mp (\vec{e}_{\theta}\pm
\vec{e}_{\phi})/\sqrt{2}$ are the left and right 	circular polarization vectors,
$p=\pm 1$ for $\vec{\eta}_{\pm}$,  $j_g,m_g,j_e,m_e$ are the quantum
	numbers of the ground and excited levels and $M=m_e-m_g$.

The total amplitude of the formation of the excited state may be written as
	a superposition of the excitation amplitudes by a plane wave from the
	source and by the spherical waves scattered on the nuclei inside the
	cluster. The scattered spherical wave is approximated by a plane wave with
	an amplitude and phase of the spherical one. This approximation, common in
	holographic applications, is particularly relevant for the nuclear scattering
	because of the small size of the nuclei.

For a given photon energy $E$ and polarization $\vec{\eta}_{p}$ the total excitation
	amplitude is given by:

%
\begin{equation}
S^{tot}_p(E)~=~S^{0}_{p}(E)~+~\sum_i \sum_{p^{'}=\pm
1}\,S^i_{p^{'}}(E)\frac{f_i(\vec{k}_i\,\vec{\eta}_{p^{'}}\,
;\vec{k}_0\,\vec{\eta}_p )}{r_i}\,e^{\textstyle{i(k_0r_i-\vec{k}_0\cdot
\vec{r}_i)}}
\label{eq:2}
\end{equation}
%
\noindent
where $\vec{r}_i$ is the position of the i-th nuclei relative to the absorbing
nuclei, $\vec{k}_0$ is the incident photon wave vector and
$\vec{k}_i=k_0\vec{r}_i/r_i$, The
exponential factor is responsible for the free propagation and it is assumed
that the source is in the far field. The amplitudes $S^0$, $S^i$ are the
excitation amplitudes, the direct one and the one following the scattering
on the i-th atom at the position $\vec{r}_i$, respectively:

%
\begin{align}
\label{eq:3}
\nonumber
S^0_{p}&=S(\vec{k}_0\, ,\vec{\eta}_p)\\
&\\
\nonumber
S^i_{p^{'}}&=S(\vec{k}_i\, ,\vec{\eta}_{p^{'}})
\end{align}
%

The scattering amplitude of the photon $(\vec{k}_0,\vec{\eta}_p)$ to a photon
$(\vec{k}_i,\vec{\eta}_{p^{'}})$ is the
sum of the resonant nuclear scattering amplitude $f_N$ and the Thomson
scattering amplitude on electron shells $f_E$:\cite{ref24}

%
\begin{equation}
f_i(\vec{k}_i\,\vec{\eta}_{p^{'}}\,
;\vec{k}_0\,\vec{\eta}_p )~=~f_N(\vec{k}_i\,\vec{\eta}_{p^{'}}\,
;\vec{k}_0\,\vec{\eta}_p )~+~f_E(\vec{k}_i\,\vec{\eta}_{p^{'}}\,
;\vec{k}_0\,\vec{\eta}_p )
\label{eq:4}
\end{equation}
%

The nuclear coherent resonant scattering amplitude is given by:

\begin{equation}
f_N (\vec{k}_i\, \vec{\eta}_{p^{'}}\,;\vec{k}_0\,\vec{\eta}_p )
~=~\frac{1}{2}\frac{3}{4k_0}f_{L-M} C^2(j_g\, 1\, j_e \, ,m_g\, M \,m_e)
\Gamma_{\gamma}
\frac{pp^{'}\, D^1_{p^{'}M}(\hat{\vec{k}}_i) D^1_{pM}(\hat{\vec{k}}_0)^{*}}{E_{eg}-E-i\Gamma/2}\, ,
\label{eq:5}
\end{equation}
\noindent
where $f_{L-M}$ is the Lamb-M\"{o}ssbauer factor and other symbols have been
already described. The factor $1/2$ comes from the splitting of the ground
state and stands for the inter-nuclear incoherence, whereas the
Clebsh-Gordan coefficient is responsible for the intra-nuclear incoherence.

The Thomson scattering amplitude is given by:

%
\begin{equation}
f_E (\vec{k}_i\, \vec{\eta}_{p^{'}}\,;\vec{k}_0\,\vec{\eta}_p
)~=~-r_0F^{el}(\theta_{scatt})f_{D-W} \vec{\eta}_{p^{'}}^{*}\cdot
\vec{\eta}_p \, ,
\label{eq:6}
\end{equation}
\noindent
where $r_0$ is the classical electron radius, $F^{el}(\theta)$ is the atomic
	form factor, $\theta_{scatt}$ is the scattering angle between vectors
$\vec{k}_0 $ and $\vec{k}_i $ , $f_{D-W}$ is the Debye-Waller
	factor, and $\vec{\eta}_{p^{'}}^{*}\cdot
\vec{\eta}_p$ is the polarization factor.

To get the formula for the observed conversion electron yield $I(\vec{k}_0)$ an
	integration over the energy spectrum and summation over the polarization
	of the incident photon should be performed. Because the source is
	stochastic this can be made incoherently:


%
\begin{equation}
I(\vec{k}_0) \propto \int \, \sum_{p=\pm 1}
\Bigl | S^{0}_{p}(E)+\sum_{i} \sum_{p^{'}=\pm 1} \, 
S^i_{p^{'}}(E)\frac{f_i}{r_i}~e^{\textstyle{i(k_0r_i-\vec{k}_0\cdot
\vec{r}_i)}}  \Bigr | ^2 |A^0(E)|^2\, dE
\label{eq:7}
\end{equation}
%
\noindent
where:

%
\begin{equation}
A^0 \propto \frac{1}{E-(E_0+E_s)+i\Gamma_s /2}
\label{eq:8}
\end{equation}
%
\noindent
represents an energy spectrum of the incident photon, $\Gamma_s$ is the
	total source line width (for the present calculation $\Gamma_s=\Gamma$ )
	and $E_0$ is the source energy transition shifted due to the Doppler
	effect by $E_s=E_0v/c$, where $v$ is the velocity of the source.

Equation\ \eqref{eq:7} may be rewritten using a "holographic" notation:


%
\begin{equation}
\label{eq:9}
I(\vec{k}_0)\propto \int \, \sum_{p=\pm 1}
\left | r_p+o^N_p(E)+o^E_p\right|^2 \, \frac{C(\cdots
)^2}{[(E-E_0-E_s)^2+\Gamma^2_s /4][(E-E_{eg})^2+\Gamma^2 /4]}
%
\end{equation}
\noindent
where:

\begin{align}
\label{eq:10}
r_p&=pD^1_{pM}(\vec{k}_0)^{*}\, ,\\
o_N&=\frac{3}{8k_0}\frac{pC(\cdots)\Gamma_{\gamma}f_{L-M}}{(E_{eg}-E-i\Gamma
/2)}\,\sum_i\sum_{p^{'}=\pm1}\frac{e^{\textstyle{i(k_0r_i-\vec{k}_0\cdot
\vec{r}_i)}}}{r_i}\,d^1_{p^{'}M}(\theta_i)^2D^1_{pM}(\hat{\vec{k}}_0)^{*}\\
o_E&=-\frac{e^2}{mc^2}\sum_i\sum_{p^{'}=\pm1}\, F^{el}(\theta_{scatt})
f_{D-W}\frac{e^{\textstyle{i(k_0r_i-\vec{k}_0\cdot
\vec{r}_i)}}}{r_i}p^{'}D^1_{p^{'}M}(\hat{\vec{k}}_i)^{*}\,
\vec{\eta}^{*}_{p^{'}}\cdot \vec{\eta}_{p}
\end{align}
\noindent
may be treated as the reference and the object waves (nuclear and
	electronic) for a given photon energy and polarization. Here, 
$d^1_{p^{'}M}(\theta_i) $ is 	a real part of the rotation matrix
$D^1_{p^{'}M} $ and $\theta_i $ is the angle between the $\vec{r}_i $
vector, and the quantization axis. For compound
	samples, the summation in the $o_N$ is over all nuclei equivalent to the
	absorbing nucleus and in the $o_E$ over the all atoms in a cluster. It is
	seen from Eqns\ (10-12) that the measured pattern is an incoherent weighted
	sum of the individual holograms for a given spectral component of the
	incident photon.

The total conversion electron yield given by\ \eqref{eq:9} may be written as
	the sum of a slowly varying "background" $I_0(\vec{k}_0)$, an
oscillating term containing
	holographic information $H(\vec{k}_0)$ and higher order terms:

\begin{equation}
I_0(\vec{k}_0)~=~C(\cdots )^2 F_M(\theta_0 )
\left (
\int \frac{dE}{[(E-E_0-E_s)^2+\Gamma^2_s /4][(E-E_{eg})^2+\Gamma^2 /4]}
\right ) 
\label{eq:13}
\end{equation}

%
\begin{equation}
H(\vec{k}_0)~=~H_N(\vec{k}_0)+H_E(\vec{k}_0)
\label{eq:14}
\end{equation}
%
The nuclear holographic information is contained in:

%
\begin{equation}
\label{eq:15}
\begin{split}
H_N(\vec{k}_0) \propto &\frac{3\pi}{8k_0}C(\cdots )^4\Gamma_{\gamma} f_{L-M}
F_M(\theta_0)\sum_i \frac{e^{\textstyle{i(k_0r_i-\vec{k}_0\cdot
\vec{r}_i)}}}{r_i}\, F_M(\theta_i) \times \\ 
&\int \frac{dE}{[(E-E_0-E_s)^2+\Gamma^2_s /4][(E-E_{eg})^2+\Gamma^2
/4][E_{eg}-E-i\Gamma /2] } ~+~c.c\\
&
\end{split}
\end{equation}
%
\noindent

whereas the $H_E(\vec{k}_0)$ , responsible for the scattering on the electronic shell,
is much weaker. In Eqns\ [13,15], $\theta_0 $ is the angle between the incident photon
and the quantization axis. 

The angular polarization factors $F_M(\theta) $are:

%
\begin{equation}
F_M(\theta)~=~\sum_{p=\pm 1} 
d^1_{pM}(\theta)^2~=~
\left \{ 
\begin{array}{ll}
\sin^2\theta,&M=0 \\
\frac{1}{2}(\cos^2\theta + 1), & M=\pm 1 
\end{array}
\right .
\label{eq:16}
\end{equation}
%

\noindent
For clarity, the electronic term as well as the higher order terms are not
written down explicitly, but are included in the calculation.

The most important nuclear holographic term may be written as:

%
\begin{equation}
H_N(\vec{k}_0) \propto \sum_i f_T(\vec{k}_0 ,\vec{r}_i
,E_s) \frac{e^{\textstyle{i(k_0r_i-\vec{k}_0\cdot \vec{r}_i)}}}{r_i} + c.c
\label{eq:17}
\end{equation}
%

\noindent
where the quantity:

%
\begin{equation}
\label{eq:18}
f_T(\vec{k}_0 ,\vec{r}_i
,E_s) ~=~F_M(\theta_{0})F_M(\theta_{i})f(E_s)
\end{equation}
%
is introduced as the effective amplitude of the scattering and the
absorption. It contains the summation over all polarization and spectral
components of the incident and scattered photons.

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

%
\begin{equation}
\label{eq:19}
\begin{split}
f(E_s)&=\frac{3\pi}{8k_0}C(\cdots )^4\Gamma_{\gamma} f_{L-M}\times \\
&\int \frac{dE}{[(E-E_0-E_s)^2+\Gamma^2_s /4][(E-E_{eg})^2+\Gamma^2
/4][E_{eg}-E-i\Gamma /2] } ~+~ c.c\\
&
\end{split}
\end{equation}
%

\noindent
To observe the holographic oscillations the propagation direction of the
incident photon is to be varied with respect to the crystallographic
directions.

Equation \eqref{eq:17} is similar to the equation derived and used by Barton
\cite{ref4} for
the formulation of the hologram reconstruction procedure. In our case the
$H(\vec{k}_0)$ describes the non normalized holographic oscillations of the total
electron yield following the conversion process, superimposed on a slowly
varying background described by Eq.\ \eqref{eq:13}.

The 3D structure can be obtained from the normalized hologram
$\chi(\vec{k}_0)$ by a
	2D-phased-Fourier transform:\cite{ref4}

%
\begin{equation}
\label{eq:20}
U(\vec{r})~=~\underset{\!\!S}{\, \iint}\chi(\hat{\vec{k}}_0)\textstyle{e}^{\textstyle{-i\vec{k}_0\cdot
\vec{r}}}\, d\sigma_{\hat{\vec{k}}_0}
\end{equation}
%

\noindent
where:


%
\begin{equation}
\label{eq:21}
\chi(\vec{k}_0)~=~\frac{I(\vec{k}_0)
-I_0(\vec{k}_0)}{I_0(\vec{k}_0)}~=~\frac{H(\vec{k}_0)}{I_0(\vec{k}_0)}
\end{equation}
%

\noindent
Usually, the hologram is additionally multiplied by a window function in
order to avoid spurious oscillations arising from a finite range of
integration in the k-space. The real structure is then represented\ \cite{ref7} by
$|U(\vec{r})|$, $|U(\vec{r})|^2$ or by $|rU(\vec{r})|^2$, 
where the last expression takes into
account the radial fall-off of the reference or scattered wave.

To eliminate the anisotropic character of the scattering, modified
reconstruction algorithms were proposed and originally applied for the
electron holography. One of them, the Scattered Wave Included Fourier
Transform (SWIFT) \cite{ref32} method may be easily applied for the
$\gamma$-ray holography
and the modified transform is given by:

\begin{equation}
\label{eq:22}
U(\vec{r})~=~\underset{\!\!S}{\,
\iint}\frac{\chi(\hat{\vec{k}}_0)}{f(\vec{k}_0,\vec{r})^{*}}\textstyle{e}^{\textstyle{-i\vec{k}_0\cdot
\vec{r}}}\, d\sigma_{\hat{\vec{k}}_0}
\end{equation}

\noindent
where the $f(\vec{k}_0,\vec{r})$ is the scattering amplitude.

\section{Basic properties of the $\gamma$-ray hologram. Calculations for a small
	cluster}
\label{sec4}
 The $\gamma$-ray scattering differs considerably from the scattering of
	electrons and the scattering of X-rays above the absorption edge. The
	polarization effects, a strong energy dependence of the scattering
	amplitude close to the resonance, and its different angular character are
	the most pronounced dissimilarities. In this section, we are analyzing the
	character of the
	$\gamma$-ray hologram and the quality of the real space reconstruction based on
	the results obtained in the previous section. To analyze the basic
	holographic features the calculations are performed for a relatively small
	$\alpha$-$^{57}$Fe cluster, constructed of only seven coordination spheres (89 nuclei,
	 6.23\AA\ radius) around the absorbing nucleus. For the simplified case, it is
	easier to examine the properties of the hologram formed due to a local
	environment, independently of experimental conditions and of the effects
	being a consequence of the long-range periodicity in the crystal. The
	holograms are calculated for the full hemisphere in the $\vec{k}$-space, and it is
	assumed that the source is rotated to cover the full angular range. The
	center of the hologram corresponds to the direction normal to the surface
	($k_x=0,~ k_y=0,~k_z=k$). For the edges of the hologram the vector
$\vec{k} $ lies in the 	surface plane and $\vec{k}=\vec{k}_{||}$.

Our calculations should be related directly to the real experimental
	situation.\cite{ref1} The experimental conditions were optimized to get the best
	possible hologram (in the sense of the signal-to-noise ratio). It implies
	the choice of one particular nuclear transition out of six, which are
	possible in the case of the magnetically split levels in $^{57}$Fe 
(Fig.\ \ref{fig:1}(a)). 
	The choice of the transition should regard not only the intensity of the
	holographic oscillations but also the fact that certain nuclei, depending
	on the position relative to the quantization axis, may be not illuminated
	and consequently, not imaged.  The geometry used for the calculation,
	depicted in Fig.\ \ref{fig:1}(b), corresponds to the $\alpha$-Fe sample with the normal
	direction [001], which is uniformly magnetized in-plane in the [010]
	direction. All patterns are calculated in the sharp resonance.

Fig.\ \ref{fig:1}(c) shows the spatial dependence of the conversion electron yield, due
to the direct absorption of $\gamma$-rays (without scattering effects). The
patterns are calculated according to Eq. (13) for the transitions
$\pm 1/2\rightarrow \pm 3/2$ and $\pm 1/2\rightarrow \pm 1/2$ ($M=0$),
respectively. The  pattern for the transition $\pm
1/2\rightarrow \pm 1/2$ has the same character as for the transition $\pm
1/2\rightarrow \pm 3/2$
but is weaker because of the dependence on the Clebsh-Gordan coefficient.
There is no dependence of the direct absorption on the azimuthal angle in
the both presented patterns.

In the inverse holographic geometry, the direct nuclear absorption plays the
role of the reference wave.  The angular anisotropy of the direct absorption
comes from the polarization term determining how strong the absorbing and
scattering nuclei are illuminated by the reference wave coming directly from
the external source. Obviously, this factor does not influence the
normalized hologram intensity $\chi (\vec{k}_0) $ but for the $M=0$ transitions may even cancel the total measured signal.
More important for the determination of the amplitude of the holographic
oscillation is to realize (from Fig.\ \ref{fig:1}(c )), how 
the scattering nucleus at the specific lattice site illuminates the absorbing
nucleus . This is described by the factor $F_M(\theta_i)$.

The patterns shown in Fig.\ \ref{fig:1}(c ) represent a dependence of the
$F_M(\theta)$ on the
angle $\theta=\arccos(k_y/k)$ between the photon wave vector (incident or scattered)
and the quantization axis. In the pattern for the $M=0$ transition (varying as
the $\sin^2\theta$ ) there are directions with no emission or absorption present. In
consequence, some nuclei would not be seen in the real space images obtained
from the hologram taken for $M=0$ transition. For the transition $M=\pm 1$ the
dependence is described by $\cos^2\theta+1$ .

Fig.\ \ref{fig:2}  shows the calculated holograms of seven coordination spheres of the
$\alpha$-$^{57}$Fe structure. The pure nuclear holograms for the transitions
$M=\pm 1$ and
	$M=0$ are presented in Figs\ \ref{fig:2}(a) and \ \ref{fig:2}(b), respectively. The hologram due
	to the Thomson scattering and subsequent nuclear absorption in the $M=+1$
	transition is shown in Fig.\ \ref{fig:2}(c). For comparison, an idealized hologram,
	calculated for the isotropic scattering and absorption and with the square
	of object wave terms neglected, is shown in Fig.\ \ref{fig:2}(d). Such a "Gabor-like"
	\cite{ref3} hologram is calculated for a constant value of the scattering
	amplitude phase $\phi=-\pi /2$. This value corresponds to the 
sharp nuclear resonant
	scattering. The intensity of the holograms in Figs\ \ref{fig:2}(a-c) is normalized
	(by the multiplication factors given in Fig.\ \ref{fig:2}) to get the same contrast
	as for the square reference wave of Fig.\ \ref{fig:1}(c). Inspecting the patterns in
	Fig.\ \ref{fig:1}(c), it is seen that the direct absorption in the normal direction
	(which corresponds to the center of the patterns, most important from the
	experimental point of view), is stronger for the $M=0$ than $M=1$ transition. 
	On the other hand one can see in Fig.\ \ref{fig:2} that the holographic signal is
	stronger for the $M=\pm 1$ transition. This can be understood when the angular
	dependence of the $F_M(\theta_i)$ is analyzed . For the $\alpha$-Fe structure in our
	geometry, the nearest neighbors are in the off-normal directions relative
	to the absorbing nucleus (illuminated better in the $|M|=1$ transitions). A
	signal caused by  the scattering on nuclei lying in the normal
	direction (next nearest neighbors - better illuminated in the $M=0$
	transitions) is weaker due to the radial fall-off of the object waves. 
	Thus, for this particular structure the holographic oscillations utilizing
	the $\pm 1/2\rightarrow \pm 3/2$ transition are more intense and,
additionally, are
	superimposed on a lower background, which makes the acquisition most
	efficient. The amplitude of the holographic oscillation may reach in this
	case 3\% of the total measured conversion electron yield. Moreover, for
	this transition the nuclei in the neighborhood will be imaged almost
	isotropically, whereas in the case of the $M=0$ transition the nuclei lying
	in the directions parallel to the quantization axis will not be seen in
	the real space images.

A direct comparison of the pure nuclear holograms with the idealized one
	from Fig.\ \ref{fig:2}(d) proves their nearly optical properties. As there is no the
	form factor and due to the fact that the scattering phase shift is angle
	independent, the nuclear holograms are almost identical with the Gabor
	holograms. They should provide high-fidelity real-space images, while the
	images obtained from the single-energy electron holograms are of rather
	poor quality. The nuclear holographic oscillation is stronger than that of
	X-ray holograms which makes the acquisition times reasonable. Their
	quality is also certainly better than in the single-energy XFH.

The Thomson hologram shown in Fig.\ \ref{fig:2}(c) is much weaker than the pure nuclear
	hologram in the sharp resonance.  It is even much weaker than it could be
	expected from the direct comparison of the maximum values of the
	amplitudes derived from Eqs (5) and (6). Additionally, the effective
	scattering and absorption amplitude includes an integration over the
	photon Lorenz energy spectrum and this should again enhance the Thomson
	electronic scattering, which is energy independent. To explain the
	relative suppression of the electronic processes, the anisotropy of the
	atomic form factor has to be considered.  It has a maximum in the forward
	direction. In other directions, the amplitude may be an order of magnitude
	weaker. In the holographic geometry, the absorbing nucleus is illuminated
	from all directions. For nuclear holograms, the scattering on several
	nuclei can contribute constructively or destructively due to the
	comparable strength of the scattering in all directions. For electronic
	scattering, in the most favorable case only one atom is in the
	preferential forward scattering direction.

It is interesting to note that the Thomson hologram (shown for $M=+1$) has a
lower symmetry than the nuclear one. Asymmetries of a similar character
were experimentally observed for the Bragg $\gamma$-ray scattering on a
$^{57}$Fe foil.\cite{ref33} The asymmetries observed in our Thomson hologram result from
complicated polarization effects present in E1 atomic scattering and
subsequent M1 nuclear absorption process, and they are a direct consequence
of the interference between the object and reference waves. Is should be
pointed out that this hologram is quite different from the XFH one. The
differences arise from the resonant detection by a polarized nuclear
detector. However, the sum of the Thomson holograms obtained for $M=+1$ and
$M=-1$ as well as the single $M=0$ hologram have the same symmetry as the pure
nuclear one. It corresponds to the symmetry of the crystal reduced by the
magnetic field in the $\hat{\vec{y}}$ direction.

	The real-space images of the cluster structure, obtained by applying the
holographic transform (19) to the $M=\pm 1$ hologram from Fig.\
\ref{fig:2}(a), are presented in Fig.\ \ref{fig:3}. The figure shows the 2D
slices of the 3D real-space structure. The imaged area is always
14x14\AA$^2$ and the absorbing nucleus (the reference nucleus) occupies the
$0 0 0$ site. The nodes of the square grid indicate the lattice sites in the
bcc structure. The plots are presented in linear inverted grayscale and no
threshold for suppression of smaller intensities is used. Figs\
\ref{fig:3}(a) and \ \ref{fig:3}(b) show the (001) cuts parallel to the
surface: at z=0 \AA\ (containing the reference nucleus) and at z=1.43 \AA\
(the distance corresponding to the inter-planar spacing in the $\alpha$-Fe
structure above the reference nucleus). The (100) vertical cut of the
real-space in the distance 1.43\AA\ from the emitter is shown in Fig.\
\ref{fig:3}(c). The sample is magnetized in the [010] direction. There are
distinct maxima at the crystal lattice sites but different nuclei are imaged
with different intensities. The intensity depends on the nucleus positions
relative to the quantization axis. This is connected with polarization
effects in the scattering and absorption. As expected, the nuclei at
positions $100$ and
$100$ are imaged with lower intensities as their counterparts in the [010]
direction. The correction for the anisotropy of the object waves can be done
in the SWIFT method, which is very simple in this case. Because the factor
$F_M(\theta_i)$ does not depend on integration variables in Eq. (22) the
reconstructed image should be just divided by it.  Fig.\ \ref{fig:4} shows a
(001) real-space cut (as in Fig.\ \ref{fig:3}(a)) after anisotropy
correction. All nuclei equidistant from the absorbing nuclei are imaged now
with the same intensity.

The real-space images in Figs\ \ref{fig:3} and \ \ref{fig:4} show also more frustrating features. 
  The intensities of the imaged nuclei vary independently on the position
  relative to the quantization axis in a way, which can not be explained
  directly by a simple radial fall-off of the scattered waves. In the
  (001)plane of the reference nucleus, all nuclei inside the cluster are
  visible but the images of the more distant $110$ (and symmetry equivalent)
  nuclei are stronger than those of $010$ (even after SWIFT correction). In
  the planes above and below, effects that are even more spurious are
  observed. In Fig.\ \ref{fig:3}(b), the nearest neighbor at position 
$\frac{1}{2}\,\frac{1}{2}\,\frac{1}{2} $ and
  the nucleus at position $\frac{3}{2}\,\frac{1}{2}\,\frac{1}{2} $ as well 
as  the nuclei in the symmetry related positions are well imaged. 
However, the nuclei at the crystal site
  $\frac{3}{2}\,0\,\frac{1}{2} $ and the symmetry related ones are not visible.  
In addition,
  four strong features are artificially imaged near the unoccupied 
$11\frac{1}{2}$-type sites.  They come from the strong images of the nuclei in
$110$-type  positions,
  which show through the plane below. In the vertical (100) cut, the
  situation concerning visibility of different nuclei is similar but, due to
  the different vertical and in-plane spatial resolutions, the shape of
  imaged nuclei is elongated along the normal axis.

The effects described above are mostly due to the overlapping of the twin
  and real-images and they were discussed extensively in 
Refs\ 11,13.
  Because of a different phase behavior of the real and twin images, the
  intensity of a given atom/nucleus in the real-space image may oscillate as
  a function of its position relative to the reference atom/nucleus and the
  wavelength used for the hologram acquisition. For some values of
$r_i/\lambda$, the
  reconstructed images can be enhanced, suppressed or even vanish
  completely. The oscillations of the  image intensity for a nucleus  at
the  position $+\vec{a}$ having a counterpart nucleus in $-\vec{a}$ are shown in 
Fig.\ \ref{fig:5}.

 The holographic pattern is very sensitive to the combination of the
  structural parameters and the radiation wavelength used.
For the real  situation the interplay of the real and twin images 
can hinder the proper structure
  determination as it was discussed above. Another combination could be much 
more favourable as it is
shown on the example of the hologram calculated for a different hypothetical
lattice constant a=3.02\AA\ .  The hologram (Fig.\ \ref{fig:6}(a)) differs
very
  much from that obtained for $a_{\alpha -Fe}$=2.87 \AA\ (compare Fig.\
  \ref{fig:2}(a)). The (001) cut of the real-space is presented in Fig.\
  \ref{fig:6}(b).  The reconstructed image is noticeably different as
  compared to the case for the true lattice constant (compare Fig.\
  \ref{fig:3}(b)).  All nuclei in the plane above the reference nucleus are
  well imaged. This example shows that the problem of twin-images can be
  very troublesome in the $\gamma$-ray holography and the structure
  determination process can be ambiguous in the single-energy holography.

\section{Real sample - experiment and simulations.}
\label{sec5}
	In this section, we will present a comparison between simulations and
experimentally obtained data. Details of the experimental procedure were
described previously.\cite{ref1} The total yield of the electrons following the
conversion processes was measured as a function of the sample orientation
relative to  the $\gamma$-rays propagation direction. The sample was a
2000\AA\  $^{57}$Fe film grown by the Molecular Beam Epitaxy on a polished 
10x10x1mm$^3$ MgO(001)
single crystal substrate. The film consisted of small equally oriented $^{57}$Fe(001)
crystallites. The average crystallite diameter was smaller than 100\AA\ . The
kinematical approximation in the calculation is for this sample appropriate
and the dynamical effects should be considerably suppressed. The sample was
magnetized along the in-plane [010] direction. The conventional 100 mCi
$^{57}$Co(Rh) M\"{o}ssbauer source of 8mm diameter was placed 15cm from the
sample. The hologram was acquired in the sharp resonance for the source Doppler
tuned to the $M=\pm 1$ nuclear transitions. The data were normalized to an
off-resonant signal. The angular pattern was recorded on a $d\theta_1 \times
d\theta_2=1.80^{\circ}\times 1.80^{\circ}$
mesh in the range from $-43.2^{\circ}$ to $43.2^{\circ}$ for two angles of 
rotation around the [100]
and [010] directions.

For comparison with the simulation, the normalized experimental pattern was
transformed to the $\vec{k}$-vector mesh and then it was two-fold symmetrized. 
A slowly
varying background intensity coming mainly from the reference wave was
subtracted from the total signal $I(\vec{k}_0)$. In the background elimination
process, the low-frequency gaussian convolution $I(\vec{k}_0)\otimes
G(\vec{k}_0)$ was subtracted
from the measured pattern giving the normalized hologram:\cite{ref34}

%
\begin{equation}
\label{eq:23}
\chi(\vec{k}_0)~=~I(\vec{k}_0)~-~I(\vec{k}_0)\otimes G(\vec{k}_0)
\end{equation}
%

\noindent
presented in Fig.\ \ref{fig:7}(a).

The simulation of a real hologram was performed for a cluster of 300
	coordination spheres ($\sim 10^4$ nuclei,$\sim 30$\AA\  radius) of the
	$\alpha$-$^{57}$Fe structure on a coarse angular grid analogous to that one used in the
	experiment. In the simulation, the integration over the source and sample
	areas as well as the real source-to-sample distance were included. Due to
	this procedure the hologram becomes more blurred. The background
	(reference wave) removal was the same as in the case of the experimental
	data to take into account possible distortions of the real-space
	introduced by this procedure. The calculated hologram is presented in
Fig.\ \ref{fig:7}(a). The intensity distributions in the experimental and calculated
	holograms are very similar, especially when one keeps in mind a high
	sensitivity to the lattice parameter.

Differences in the relative intensities between the simulated and
	experimental holograms are probably due to the imperfect procedure of the
	background removal. When single profiles of the holograms are compared,
	the agreement between the experiment and the calculations is especially
	well emphasized, as can be seen from Fig.\ \ref{fig:7}(b).  The real-space images of
	the structure obtained by applying the
holographic transform to the calculated and experimental holograms are shown
in Figs\ \ref{fig:8}(a-c) and \ \ref{fig:8}(d-f), respectively. In both cases, the nearest
neighbors in the $\alpha$-Fe structure are well imaged. In cuts obtained from the
experimental hologram, the images of the nearest neighbors are shifted by
about 0.2\AA\  from the true positions. It is caused by  experimental errors
in the sample orientation. The limited angular range used for the hologram
recording results in a moderate resolution. The theoretical resolution is
calculated according to Ref.\ 35 and it is shown as the vertical and
horizontal bars on the plots. The poor vertical resolution makes the images
of the $\frac{1}{2}\,\frac{1}{2}\,\frac{1}{2}$-type nuclei elongated along [001] axis in 
Figs\ \ref{fig:8}(c ) and
8(f). The overlapping of the real and twin images enhances this effect and
in the (001) plane of the reference nucleus, the four maxima corresponding
to "wings" of those nuclei are dominant. The images of all other nuclei
beyond the first coordination shell are strongly suppressed. This is a
consequence of a poor angular resolution, resulting from the relatively
short source-sample distance and the finite size of the source and the
sample, but also due to the coarse angular grid. It is however interesting
to note, that the nuclei at $\frac{3}{2}\,\frac{1}{2}\,\frac{1}{2} $ and the symmetry equivalent sites
are not imaged at all. This effect can be also observed in Fig.\ \ref{fig:3}, for the
idealized experimental situation, and it is clearly due to the cancellation
of the real and twin images. Some differences of the relative intensity
ratios in the experimental and theoretical real-space cuts may be also
caused by the experimental error in tuning to the resonance energy. The
phase of the scattering amplitude is a rapid function of the small shifts
from the resonance energy, and the scattering phase determines the
overlapping of real and twin images.

The resolution and the size of the imaged area can be improved by recording
the hologram in a wider angular range, with a better angular resolution. The
polarization effects are easy to be eliminated with the SWIFT method. So,
the main limitation in the single-energy $\gamma$-ray holography remains the effect
of the cancellation of the real and twin images.

\section{Elimination of real and twin images cancellation in $\gamma$-ray
holography}
\label{sec6}
	In the case of the  $\gamma$-ray holography utilizing the M\"{o}ssbauer effect, it
	is possible to vary the energy of the incident beam only in a close
	vicinity of the resonance. The possible energy change is then of the order
	$\Gamma $ . The multi-energy methods require a remarkable change in the phase of
	the propagation factor $\exp(ikr)$ due to a change of the $\vec{k}$ value. This is
	inaccessible by such a small energy variation as can be realized in the
	M\"{o}ssbauer experiment. In this section, we propose a method, exploiting
	the fact, that the scattering phase itself is a rapidly varying function
	of the energy near the resonance. We will also discuss the experimental
	feasibility of this method.

The effect of the cancellation of the nuclear images for a system consisting
	of a reference nucleus and two scattering nuclei in positions $+\vec{a}$ and
$-\vec{a}$ can
	be explained simply using arguments from Ref.\ 11. The function
$U(\vec{r})$,
	representing the real-space image, is calculated for this system by
	substituting the value of the holographic oscillations due to the nuclei
	in positions $+\vec{a}$ and -$\vec{a}$, given by Eq. (17) into the Fourier holographic
	transform from Eq. (20). Then, the reconstructed real-space in the
	position $\vec{r}$ is given by:

%
\begin{equation}
\label{eq:24}
\begin{split}
U(\vec{r})&=\underset{\!\!S}{\, \iint} \Bigl ( \frac{f_T(\vec{k},\,
+\vec{a},\, E)^{*}}{a}\textstyle{e}^{\textstyle{-ika}}\,
\textstyle{e}^{\textstyle{i\vec{k}\cdot \vec{a}}}\\
&+\frac{f_T(\vec{k},\,-\vec{a},\, E)^{*}}{a}
\textstyle{e}^{\textstyle{-ika}}\,\textstyle{e}^{\textstyle{-i\vec{k}\cdot
\vec{a}}} \Bigr )   \textstyle{e}^{\textstyle{-i\vec{k}\cdot \vec{r}}}\,
d\sigma_{\hat{\vec{k}}}+ c.c\\
&
\end{split}
\end{equation}
%

\noindent
The exponential factors oscillate, except for $\vec{r}=\pm \vec{a}$, and hence, the image
function has maxima for $\vec{r}=\pm \vec{a}$. Consequently, the reconstructed image in
the position $+\vec{a}$ is dominated by two terms:

%
\begin{equation}
\label{eq:25}
U(+\vec{a})~\simeq ~\frac{1}{a}\textstyle{e}^{-ika}\underset{\!\!S}{\, \iint
}f(\vec{k},\,
+\vec{a},\, E)^{*}\, d\sigma_{\hat{\vec{k}}}~+
~\frac{1}{a}\textstyle{e}^{ika}\underset{\!\!S}{\, \iint}f(\vec{k},\,
-\vec{a},\, E)\, d\sigma_{\hat{\vec{k}}}
\end{equation}
%

\noindent
where the first one corresponds to the real image while the second one leads
to the non-physical twin image. From the properties of the effective
scattering and absorption amplitude given by Eq. (18) and from the fact that
the $F_M(\theta)$ is real and $F_M(\theta)=F_M(\pi - \theta)$ it follows that:

%
\begin{equation}
\label{eq:26}
U(+\vec{a})~\propto ~ |f(E)|\left( \textstyle{e}^{-ika}\textstyle{e}^{-i\phi(E)}~+~
\textstyle{e}^{ika}\textstyle{e}^{i\phi(E)}\right)
\end{equation}
%

\noindent
 where $\phi(E) $ is the phase of the effective amplitude given by Eq. (18)
. For a given energy ($\vec{k}$ vector value) the image function
$U(+\vec{a})$ oscillates between real and imaginary values due to the
propagation phase factors $\exp(\pm ika)$ and for some values of a equals zero.
>From Eq. (26) it is obvious that the condition for cancellation changes and
depends on the value of the scattering amplitude phase factors $\exp(\pm
i\phi(E))$.
So, for example, the real space image obtained from a $\gamma$-ray hologram
recorded in the sharp resonance and the image obtained from a single XFH
hologram for the same energy will differ, because different pairs of atoms
at $\pm\vec{a}$ could be affected by the cancellation. The Thomson scattering 
amplitude is almost pure real while the nuclear resonant scattering
amplitude is pure imaginary.

When the M\"{o}ssbauer effect is used for holography, the scattering phase
can be changed directly by the de-tuning from the resonance. Of course, it is
inevitably accompanied by a rapid fall-off of the resonant signal. If, however, two
holograms would be taken for the two particular energy values $E=E_1$ and
$E=E_2$,
not too far from the resonance, the holographic transformation:

%
\begin{equation}
\label{eq:27}
U^{'}(\vec{r})~=~\sum_{E=E_{1,2}}\frac{U_E(\vec{r})}{f(E)^{*}}\textstyle{e}^{ikr}
\end{equation}
%

\noindent
could remove the problem of the real and twin images cancellation. In Eq.
(27) the $U_E(\vec{r})$ is a usual single holographic image obtained by the
applying the holographic transform from Eq.\eqref{eq:20} to the hologram
recorded for a particular energy $E=E_1$ or $E=E_2$. The phase $\phi(E)$ of
the scattering amplitude
$f(E)$ depends strongly on the energy shift from the resonance. It is
reasonable to take the two holograms symmetrically, on both sides of the
resonance and denote $\phi(E_1)=\phi_1$ and $\phi(E_2)=\phi_2$. So, if for
each of them the phase differs by $\pm \Delta 2$ from the phase at the sharp
resonance
$\phi(E_0)=\pi /2$, we can write that $\phi_1=-\pi /2-\Delta /2$ and
$\phi_2=-\pi /2+\Delta /2$. Inserting the image functions $U_E(+\vec{a})$ recorded
for the phase $\phi_1 $ and $\phi_2 $ to the proposed holographic transform
(Eq.\ \eqref{eq:27}) we
get :

%
\begin{equation}
\label{eq:28}
U^{'}(+a)~=~2-2\textstyle{e}^{2ika}\,\cos\, \Delta
\end{equation}
%

\noindent
and the reconstructed image intensity $|U(+a)|$  is proportional to:

%
\begin{equation}
\label{eq:29}
|U^{'}(+a)|~\propto~\sqrt{1+\cos^2\Delta -2\cos(2ka)\cos\Delta} 
\end{equation}
%

\noindent
If $\Delta=\pi /2 $ then the oscillation of the reconstructed image
intensity is completely removed for an arbitrary pair of nuclei at the positions
$\pm \vec{a} $ relative to the reference nucleus. For realistic experiments the $\Delta
= \pi/2 $
condition is impossible to achieve and for holograms recorded for a smaller
phase difference the oscillation can not be perfectly removed. This
situation illustrates Fig.\ \ref{fig:9}. The intensity oscillations of images 
for nucleus pairs at $\pm \vec{a} $ are plotted as a function of the
$\vec{a} $ length for
$\Delta = 0 $ (single
hologram, as in Fig.\ \ref{fig:5}), $\Delta = \pi/2 $ (idealized situation) and $\Delta
= \pi/3 $ (a realistic
value used for the experiment simulation, corresponding to the de-tuning
from resonance by $\pm 1.75\Gamma_{exp}=\pm 3.5\Gamma_{s} $. Apparently, the situation, which is
experimentally possible, brings certain improvement. The oscillations are
yet present but real and twin images never cancel. This fact has direct
consequences for the reconstruction quality. Fig.\ \ref{fig:10} shows the real-space
cuts obtained for  $\Delta = \pi/2 $ and  $\Delta   
= \pi/3 $. In the idealized case of $\Delta = \pi/2$ all nuclei
are visible with almost equal intensity. For the experimentally realistic
hologram recorded   off resonance, the atomic scattering becomes also
important, and because it is not taken into account in the reconstruction
procedure, the real-space is distorted. In the realistic case, all nuclei
are still visible but certain artifacts are present. Images of nuclei at the
positions $\frac{3}{2}\,\frac{1}{2}\,\frac{1}{2} $  and the symmetry
equivalent ones can be recognized, whereas
in the image obtained from the single in-resonance hologram they were not
present. In addition, the nearest neighbors are imaged with a stronger
intensity as compared to the next nearest neighbors, as it could be expected
from the spherical character of the scattered waves.

\section{Conclusions}
\label{sec7}
We have presented the basic theory of the inverse $\gamma$-holography. The atomic
resolution hologram is formed due the resonant nuclear scattering of the
electromagnetic radiation. The acquisition is made by the resonance
M\"{o}ssbauer detection of the de-excitation products via the inelastic
channel (conversion electrons). The absorbing and re emitting nuclei are
microscopic holographic detectors. Due to a small size of nuclei, the
character of the nuclear scattering process is almost ideal for holographic
applications. The simulations performed in the frame of the SSC model proved
however, that even for the idealized hologram spurious images and effects
are observed in the real space reconstruction. They result from the
overlapping of the real and twin images, typical for all kinds of
single-energy holography. The polarization effects in the scattering and
absorption are easy to remove by a modified holographic transform. We showed
also explicitly the small role of the Thomson scattering in the hologram
formation for the $\alpha$-$^{57}$Fe structure.  Despite the obvious limitations of the single energy
$\gamma$-holography, the real space of the $\alpha$-Fe structure could be reconstructed
based on the experimental hologram. The hologram and the real-space image
agree well with these resulting from the simulations. We proposed also a way how
to eliminate the cancellation of twin and real images exploiting the strong
dependence of the scattering phase on the energy, close to the resonance. 

It
becomes a challenge to use the most attractive property  of the $\gamma$-holography
with the resonance
detection , namely its ability to image the magnetic structure
with the hyperfine field resolution. The structure of magnetite (Fe$_3$O$_4 $), in
which Fe occupies two different crystallographic positions, is a well suited
candidate. The magnetic hyperfine field for both sites differs enough to
allow the hologram to be acquired separately for different M\"{o}ssbauer
transitions. However, it has to be  noticed that for such a complicated system
the hologram quality (statistics, the angular resolution and range) should
to be considerably improved. In addition, it is expected that the hologram
will be much more complicated due the increased role of the Thomson
scattering and strong dynamical effects.

\section{Acknowledgments}
Financial  support by the Polish Science Research Council under Grants No. 2
P03B 06913 and No. 2 P03B 08010 is gratefully acknowledged.

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%
\newpage
%fig
%
\section*{ Figure Captions}
\baselineskip=24pt
\begin{figure}[h] 
\baselineskip=24pt
\caption{(a) Magnetic hyperfine splitting of $^{57}$Fe levels.
(b) Geometry used for hologram calculation. The reference nucleus (hologram
detector) is located at the origin, the scattering nucleus is at position
$\vec{r}_i$. The angles $\theta_0$ and $\theta_i$ are the angles between the magnetic field axis and
incident and scattered photon propagation direction, respectively. The R
stands for the reference wave, O for the object wave. The factors
$F_M(\theta) $ describe polarization effects in scattering and absorption. (c) Spatial
dependence of $F_M(\theta) $ for $M=\pm 1$ and $M=0$ transitions, respectively (dark - low
value, light - high value) } 
\label{fig:1} 
\end{figure}
%
%
\begin{figure}[h] 
\caption{ Holograms calculated for small cluster (radius
6.23\AA\ ) around the reference nucleus. (a,b) Pure nuclear holograms obtained
for different transitions: $M=1$ and $M=0$. (c) Hologram resulting from the
Thomson atomic scattering and subsequent $M=1$ nuclear absorption. (d)
Gabor-like hologram obtained for an isotropic scattering with a constant
scattering phase shift $-\pi /2 $. The multiplication factors refer to the
maximum values from Fig.\ \ref{fig:1}(c). The patterns are presented in the
$\vec{k} $-vector
scale in the full hemi-sphere. (dark - low intensity, light - high
intensity).} 
\label{fig:2} 
\end{figure}
%
%
\begin{figure}[h] 
\caption{The real-space images of the cluster structure
obtained by applying the holographic transform to the $M=\pm 1$ hologram from
Fig. \ref{fig:2}(a). (a) The (001) cut in the plane of the reference nucleus. (b) The
(001) cut 1.43\AA\  (inter-planar spacing in the $\alpha$-Fe structure) above the
reference nucleus. (c) The (100) cut in distance 1.43\AA\  from reference
nucleus. The area of the imaged neighborhood is 14x14\AA$^2$. Inverted gray scale
(light - low intensity, dark - high intensity) is used for intensity, and
no threshold value for elimination of weaker intensities is applied. The
nodes of the square grid depict the lattice sites in the bcc structure.}
\label{fig:3} 
\end{figure}
%
%
\begin{figure}[h]
\caption{Same as Fig.\ \ref{fig:3}(a) but after a correction for polarization effects.}
\label{fig:4}
\end{figure}
%
%
\begin{figure}[h]
\caption{Intensity oscillation of the nuclear images at the positions $\pm
\vec{a}$ due
to the overlapping of the real and twin images. The distances of the nuclei,
in the neighborhood of the reference nucleus for the $\alpha$-Fe structure, are
depicted by arrows.}
\label{fig:5}
\end{figure}
%
%
\begin{figure}[h]
\caption{ (a) Hologram ($M=1$ transition) for a small bcc cluster with a lattice
constant a=3.02\AA\  (to compare with Fig.\ \ref{fig:2}(a)). (b) The resulting (001)
real-space cut 1.43\AA\  above the reference nucleus (to compare with Fig.\
\ref{fig:2}(b)).
}
\label{fig:6}
\end{figure}
%
%
\begin{figure}[h]
\caption{ (a) Calculated and measured $\gamma$-ray holograms for the
$\alpha$-Fe structure.
The patterns are presented in the $\vec{k}$-vector scale. (b) Profiles of the
holographic oscillation for constant $k_x$ values of 0\AA$^{-1}$ and
3.6\AA$^{-1}$ (points -
experiment, lines - simulation).
}
\label{fig:7}
\end{figure}
%
%
\begin{figure}[h]
\caption{Real-space images of the $\alpha$-Fe structure obtained from holograms in
Fig.\ \ref{fig:7}(a). The real-space cuts calculated from simulation are shown in 
Figs\ \ref{fig:8}(a)-(c), the cuts obtained from the measured hologram are depicted in 
Figs\ \ref{fig:8}(d)-(f). The area of the imaged neighborhood is 14x14\AA$^2$. Inverted gray
scale (light - low intensity, dark - high intensity) is used for intensity,
and no threshold value for elimination of weaker intensities is applied. The
nodes of the grid depict the lattice sites in the bcc structure. The
horizontal and vertical bars correspond to the spatial resolution resulting
from the experimentally accessible $\vec{k}$-vector range for hologram acquisition.
}
\label{fig:8}
\end{figure}
%
%
\begin{figure}[h]
\caption{ Intensity oscillation of the nuclear images at position $\pm
\vec{a}$ on
applying the holographic transform from Eq.\ \eqref{eq:25} to two holograms which
differ in phase of the scattering amplitude by $\Delta $. Dashed line corresponds to
$\Delta = \pi/2$ (idealized situation), dotted line to $\Delta =\pi / 3$ (simulation of real
experiment) and solid  line corresponds to $\Delta =0$ (only one hologram at
resonance and $\phi =- \pi /2$, as in Fig.\ \ref{fig:6}). The distances of the nuclei, in the
neighborhood of the reference nucleus for the $\alpha$-Fe structure, are depicted by
arrows.}
\label{fig:9}
\end{figure}
%
\begin{figure}[h]
\caption{ The (001) cut of real space 1.43\AA\  above the reference nucleus
obtained by applying the modified holographic transform from 
Eq.\ \eqref{eq:25} to two
simulated holograms for $\Delta = \pi/2$ (a) and $\Delta =\pi / 3$ (b)
}
\label{fig:10}
\end{figure}
%
\end{document}

