Position Reconstruction in a Dual Phase Xenon Scintillation Detector

Abstract

We studied the application of statistical reconstruction algorithms, namely
maximum likelihood and least squares methods, to the problem of event
reconstruction in a dual phase liquid xenon detector. An iterative method was
developed for in-situ reconstruction of the PMT light response functions from
calibration data taken with an uncollimated gamma-ray source. Using the
techniques described, the performance of the ZEPLIN-III dark matter detector
was studied for 122 keV gamma-rays. For the inner part of the detector (R<100
mm), spatial resolutions of 13 mm and 1.6 mm FWHM were measured in the
horizontal plane for primary and secondary scintillation, respectively. An
energy resolution of 8.1% FWHM was achieved at that energy. The possibility of
using this technique for improving performance and reducing cost of
scintillation cameras for medical applications is currently under study.

Full text

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arXiv:1112.1481v1 [physics.ins-det] 7 Dec 2011
Position Reconstruction in a Dual Phase Xenon
Scintillation Detector
V.N. Solovov, V.A. Belov, D.Yu. Akimov, H.M. Araújo, E.J. Barnes, A.A. Burenkov, V. Chepel, A. Currie,
L. DeViveiros, B. Edwards, C. Ghag, A. Hollingsworth, M. Horn, G.E. Kalmus, A.S. Kobyakin,
A.G. Kovalenko, V.N. Lebedenko, A. Lindote, M.I. Lopes, R. Lüscher, P. Majewski, A.StJ. Murphy, F. Neves,
S.M. Paling, J. Pinto da Cunha, R. Preece, J.J. Quenby, L. Reichhart, P.R. Scovell, C. Silva, N.J.T. Smith,
P.F. Smith, V.N. Stekhanov, T.J. Sumner, C. Thorne and R.J. Walker
Abstract—We studied the application of statistical recon-
struction algorithms, namely maximum likelihood and least
squares methods, to the problem of event reconstruction in
a dual phase liquid xenon detector. An iterative method was
developed for in-situ reconstruction of the PMT light response
functions from calibration data taken with an uncollimated
γ-ray source. Using the techniques described, the performance
of the ZEPLIN-III dark matter detector was studied for
122 keV γ-rays. For the inner part of the detector (R<100 mm),
spatial resolutions of 13 mm and 1.6 mm FWHM were
measured in the horizontal plane for primary and secondary
scintillation, respectively. An energy resolution of 8.1% FWHM
was achieved at that energy. The possibility of using this
technique for improving performance and reducing cost of
scintillation cameras for medical applications is currently under
study.
Index Terms—position reconstruction, scintillation camera,
maximum likelihood, weighted least squares, dark matter,
WIMPs, ZEPLIN-III, liquid xenon, dual phase detectors.
I. INTRODUCTION
A
low energy region <1 MeV, these include medical radio-
nuclide imaging, gamma astronomy and direct dark matter
search experiments. In the latter instance, which motivated
the present work, event localization per se is not relevant for
detection of dark matter particles, but position sensitivity is
important for efficient reduction of the radiation background
and correct identification of the candidate events.
ZEPLIN-III is a dual phase (liquid/gas) xenon detector
built to identify and measure galactic dark matter in the
form of Weakly Interacting Massive Particles (WIMPs).
NUMBER of applications requires measurement of the
interaction coordinates within a particle detector. In the
Corresponding author is V.N. Solovov, solovov@coimbra.lip.pt
E.J. Barnes, C. Ghag, A. Hollingsworth, A.StJ. Murphy, L. Reichhart
and P.R. Scovell are with School of Physics & Astronomy, University of
Edinburgh, UK
H.M. Araújo, A. Currie, M. Horn, V.N. Lebedenko, J.J. Quenby,
T.J. Sumner, C. Thorne and R.J. Walker are with High Energy Physics
group, Blackett Laboratory, Imperial College London, UK
D.Yu. Akimov, V.A. Belov, A.A. Burenkov, A.S. Kobyakin, A.G. Ko-
valenko and V.N. Stekhanov are with Institute for Theoretical and Experi-
mental Physics, Moscow, Russia
V. Chepel, L. DeViveiros, A. Lindote, M.I. Lopes, F. Neves, J. Pinto da
Cunha, C. Silva and V.N. Solovov are with LIP–Coimbra & Department of
Physics of the University of Coimbra, Portugal
B. Edwards, G.E. Kalmus, R. Lüscher, P. Majewski, S.M. Paling,
R. Preece, N.J.T. Smith and P.F. Smith are with Particle Physics Department,
STFC Rutherford Appleton Laboratory, Chilton, UK
The detector measures both the scintillation light (S1) and
the ionisation charge generated in the liquid by interacting
particles and radiation. The ionisation charge drifts upwards
to the liquid surface by means of a strong electric field and
is extracted into a thin layer of gaseous xenon where it
generates UV photons by electroluminescence (S2). Both
the scintillation and electroluminescence light are measured
by a PMT array and the ratio between S1 and S2 allows
to discriminate nuclear recoils (expected to be produced by
elastic scatter of WIMPs off xenon nuclei) from the electron
recoils from β and γ-ray backgrounds. The ZEPLIN-III
experiment operated 1070 m underground at the Boulby
mine (UK) between 2006 and 2011.
The self-shielding property of xenon reduces the rate of
background in the interior of the liquid. Using accurate posi-
tion reconstruction to select only events in an inner "fiducial"
volume therefore improves sensitivity to the WIMP signal.
While the depth of the interaction can be inferred very
accurately (few tens of µm FWHM) from the electron drift
time in the liquid (the delay between S1 and S2), the position
in the horizontal plane has to be reconstructed from the light
distribution pattern across the PMT array. Another reason for
analysis of the light distribution is the need to eliminate the
multiple scatter events that can mimic the WIMP interactions
if one of the scatters has occurred in a dead volume of liquid
xenon from where no charge can be extracted.
The active volume of ZEPLIN-III is a flat layer of liquid
xenon (≈40 cm in diameter and 3.5 cm thick) above a
compact hexagonal array of 31 2-inch vacuum ultraviolet-
sensitive PMTs (ETL D730/9829Q) immersed directly in the
liquid [1]. Such a flat geometry makes it (from the point of
view of position reconstruction) rather similar to the well-
studied scintillation camera, which is widely used in areas
as diverse as medical research and experimental astrophysics
[2, 3]. The position of an event in a scintillation camera is
traditionally found by the Anger method which consists in
calculating a centroid of the PMT response [4].
Statistical reconstruction algorithms by maximum likeli-
hood and weighted least squares methods have gained pop-
ularity following the pioneering work of Gray and Macovski
in 1976 [5]. They offer better precision along with the
possibility of checking if the input data correspond to a
valid event. These methods require knowledge of the light

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Figure 1.
showing the PMT array and anode and cathode defining the active volume.
Liquid xenon is shown in blue. The dashed box illustrates the fiducial
volume used for WIMP searches.
Schematic diagram of the ZEPLIN-III WIMP target region,
response functions (LRF) that characterise the response of
a given PMT as a function of position of an isotropic light
source inside the sensitive volume of the detector. Typically,
the LRFs are either measured directly (e.g. by means of
a moving collimated radioactive source) or calculated from
the detector geometry, either analytically or by means of a
Monte Carlo simulation.
In the present work, a method of reconstructing LRFs
in situ from the calibration data obtained by irradiating
the detector by γ-rays from an uncollimated radioactive
source was developed. Based on the set of reconstructed
LRFs, the positions and light yields of scintillation events
in the detector can be readily found using either maximum
likelihood or weighted least squares methods. This procedure
was applied to the WIMP-search data taken with ZEPLIN-III
[6, 7, 8, 9].
II. EXPERIMENTAL SETUP
The target region of the detector is shown in Fig. 1. The
electric field in the active xenon volume (3.9 kV/cm in the
liquid and 7.8 kV/cm in the gas) is defined by a cathode
wire grid 36 mm below the liquid surface and an anode
plate in the gas phase, 4 mm above the liquid. A second
wire grid is located 5 mm below the cathode grid just above
the PMT array. This grid defines a reverse field region which
suppresses the collection of ionisation charge for events just
above the array and helps to isolate the PMT input optics
from the external high electric field. The PMTs are powered
by a common high voltage supply, with the outputs roughly
equalised by means of attenuators (Phillips Scientific 804).
The PMT signals are digitised at 2 ns sampling by 8-bit flash
ADC (ACQIRIS DC265). To expand the dynamic range of
the system, each PMT signal is recorded by two separate
ADC channels: one directly and one after amplification
by a factor of 10 by fast amplifiers (Phillips Scientific
770). The acquired waveforms were analysed by a dedicated
software [10] producing an array of ten parametrised pulses.
Subsequently, an event filtering tool was used to retain events
Figure 2.
31 photomultiplier envelopes are represented by blue circles. The dashed
circle has 150 mm radius.
The PMT array and the copper grid, viewed from the top. The
with a fast S1 signal preceding a wider S2 one. All multiple
scatter events containing more than one S2 are filtered out.
A57Co radioactive source was used for calibrating the
energy response of the detector. This source emits 122 keV
and 138 keV γ-rays which are rapidly absorbed in liquid
xenon (with attenuation length < 4 mm for these energies)
mostly by photoelectric capture [11]. Consequently, most
of the interactions can be considered point-like with full
energy deposit. The source was positioned at approximately
190 mm above the liquid surface and as close as possible
to the detector axis. The calibration was performed daily
to monitor the detector stability. There were also several
dedicated runs aimed at acquiring sufficient data to train
the positioning algorithms. Before the second science run, a
specially designed rectangular copper grid was placed inside
the chamber, above the sensitive volume (Fig. 2). The grid
structure is 386 mm in diameter, and was manufactured by
diamond wire cutting from a 5.1 mm thick copper plate; the
void pitch is 30 mm and the straight sections are 5 mm
wide. The thickness of the grid was chosen such that it
would attenuate the γ-ray flux from the calibration source
by approximately a factor of 2, creating a shadow image that
can be used to verify and fine-tune position reconstruction.
III. EVENT RECONSTRUCTION METHODS
The problem of event reconstruction consists in finding
the energy (or, rather, the light signal intensityˆ N) and the
position of an event (ˆ x, ˆ y) given a set of the corresponding
PMT output signals Ai. For an event at position r producing
N photons the probability of the i-th PMT detecting ni

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photons is well approximated by the Poisson distribution
[12]:
Pi(ni) =µni
ie−µi
ni!
(1)
where µi = Nηi(r) is the expectation for a number of
photons detected by the i-th PMT out of N initial ones with
ηi(r) being LRFs – the fraction of the photons emitted by a
light source at position r that produce a detectable signal
in the i-th PMT. The corresponding output signal Ai in
the general case is a random variable with an expectation
value proportional to ni. The probability distribution for
Ai depends on the single photoelectron response of the
corresponding PMT and can be quite complex [13, 14].
However, in a few special cases it can be approximated by
simple functions. These special cases include:
• Photon counting. If niis small (say, less than 10) and
the PMT has a narrow single photoelectron distribution
then nican be calculated (almost) unambiguously from
Ai.
• Normal distribution. If ni is large (say, 25 or more)
and the single photoelectron distribution of the PMT is
reasonably symmetric then, following from the central
limit theorem, Aiis approximately normally distributed
with the mean equal to niqsi, where qsiis the average
single photoelectron response of the PMT.
A. Centroid and corrected centroid
The centroid method of position estimation is the oldest
method used by Anger in the first gamma camera in 1957
[4]. It is still widely in use due to its simplicity and
robustness. The position estimate is found as the weighted
average of PMT coordinates with weights determined by the
light distribution across the PMT array:
ˆ x =
?
iXiAifi
?
iAifi
,
ˆ y =
?
?
iYiAifi
iAifi
,
(2)
where (Xi,Yi) are the coordinates of the axis of i-th PMT,
Aiis the measured charge and fiis a flat-fielding coefficient
which compensates for variations in gain and quantum
efficiency across the PMT array. As one can see from
equations (2), no information on LRFs and Ai probability
distribution is necessary for application of this method. On
the other hand, while the centroid method works reasonably
well close to the centre of the detector (up to 100 mm from
the centre in ZEPLIN-III), it becomes increasingly biased
for events in the periphery. Another disadvantage is that it
gives no indication regarding the match of the actual light
distribution to the expected one.
If there exists one-to-one mapping between the true posi-
tion and the one reconstructed by the centroid method then it
is possible to invert this mapping to obtain the unbiased "cor-
rected" estimate from the biased centroid one. In practice,
this is often done by building a look-up table for a number
of known positions on a rectangular grid and interpolating
between these points. Another possibility is to use Monte
Carlo simulation to calculate the forward mapping and then
to use numerical methods to invert it. The latter method was
employed in the ZEPLIN-III event filtering routine. It was
also used to obtain the first approximation in the iterative
LRF reconstruction procedure.
B. Maximum likelihood
The maximum likelihood (ML) technique [2, 3, 15]
consists in finding the set of parameters that maximises
the likelihood of obtaining the experimentally measured
outcome. For the case of photon counting when ni are
known for each PMT, the likelihood function can be easily
calculated from the Poisson distribution (1):
lnL =
?
i
lnP(ni,µi) =
?
i
(nilnµi−µi) −
?
i
ln(ni!).
(3)
Taking into account that µi= Nηi(r), one can write [3]
lnL(r,N) =
?
i
(niln(Nηi(r)) − Nηi(r)) + C ,
(4)
where C does not depend on neither r or N. If the LRFs
ηi(r) are known, the best estimates ˆ r andˆ N can be found in
a straightforward way by maximising function (4). The best
estimate of N at given r,ˆ N(r) can be found analytically:
ˆ N(r) =
?
iηi(r).
ini
?
(5)
By substitutingˆ N for N into (4) one obtains lnLm(r) =
lnL(r,ˆ N(r)), which is a function of the position only.
Thenˆ N and ˆ r are found by maximising lnLm(r) either
analytically or by numerical methods. As a bonus, for the
2D case lnLm(r) can be visualised as a colour map, which
is very useful for either debugging or checking the validity
of a given event.
C. Weighted least squares
If Ai can be considered normally distributed the more
robust weighted least squares (WLS) method can be used
instead of ML [16]. In this case the parameter estimates are
found by minimising the weighted sum of squared residuals
χ2:
χ2=
?
i
wi(Aei− Ai)2,
(6)
where Aei= µiqsi= Nηi(r)qsiis the expected PMT output
charge and wiis the weighting factor which is reciprocal to
the variance of Aei− Ai. The best estimates ˆ r andˆ N are
obtained by finding the global minimum of
χ2(r,N) =
?
i
wi(r,N)(Nηi(r)qsi− Ai)2.
(7)
The N and r minimisations can be separated, as in the
likelihood case, reducing by one the dimensionality of the
problem.

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Figure 3. Iterative reconstruction of the LRFs from57Co calibration data. The top row: the evolution of the distribution of estimated event positions from
S2 pulses. The bottom row: the response of PMT 11 (with centre at (−79.5, −45.9)) versus estimated distance from its centre (dots) and the corresponding
S2 LRFs derived from these distributions (curve). a) Initial position estimates obtained by centroid. b) First iteration. c) Final (5-th) iteration.
D. Method choice
The choice of the WLS method for S2 reconstruction
is straightforward: due to its high light output, the S2
signal statistic is quasi-normal except for the PMTs far
from low energy events. These PMTs may be either ignored
or clustered together so that the photoelectron statistic per
cluster is quasi-normal too.
In the case of S1, the total collected charge (from the
whole PMT array) is equivalent, depending on the event
position, to 1–2 photoelectrons per keV; this means that in
the region of interest for WIMP searches (<50 keV) the S1
distribution is too far from normal to use the WLS method
with confidence. Consequently, the ML method was used.
IV. RECONSTRUCTION OF LIGHT RESPONSE FUNCTIONS
A. Method description
The ML and WLS methods described above rely on the
knowledge of the LRFs ηi(r). For a typical gamma camera,
it is possible to scan the sensitive volume with a collimated
γ-ray source in order to deduce the LRFs. Scanning a dual
phase detector enclosed in a bulky cryostat in this way
would be a rather difficult and error-prone undertaking, so
an alternative method has been developed.
In this method the detector is irradiated by a non-
collimated monoenergetic gamma source and the PMT re-
sponses are recorded event by event. Even if the gamma
source is not collimated, it is still possible to obtain an
estimate for each event position using the centroid or the
corrected centroid method, at least for the central part of the
detector. After a sufficiently large event sample is acquired,
making an additional assumption that the LRF depends
smoothly on r and assuming that all the events produce the
same amount of light, one can get the first approximation for
the LRF η(1)
at different r by a smooth function of r.
This first approximation can now be used to obtain better
estimates for the positions of the events in the sample using
ML or WLS method. Compared to the centroid estimates,
these new estimates are less biased, especially in the case
of peripheral events. Fitting again the PMT response as a
function of coordinates using the updated event positions
gives a second approximation η(2)
The above steps are repeated until some convergence cri-
terion is reached. This can be the fact that the reconstructed
dataset has attained some quality that the physical calibration
events are known to possess, for example monoenergeticity
or some known distribution in the xy plane. Another option
is to iterate until the change in the LRFs on the next step
falls below a pre-defined tolerance.
Some additional regularization may be necessary to force
the iteration to converge. One is the choice of a smoothing
function. Another is the use of some a priori known property
of the LRF; for example in the case of a PMT with a circular
i(r) by fitting the PMT response to the events
i(r).

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photocathode it is reasonable to assume that the LRF has
axial symmetry η(r) = η(r), where r is the distance from
the PMT axis. This type of regularization was used in LRF
reconstruction for ZEPLIN-III.
B. ZEPLIN-III example
In order to collect the data necessary for reconstruction of
the S2 LRFs, the detector was irradiated with γ-rays from
a
distribution of the estimated57Co event positions obtained
with a corrected centroid algorithm. Clearly, the events on
the periphery tend to be misplaced closer to the centre of
the PMT array. The situation deteriorates in the bottom-
right corner where one of the PMTs was not functioning.
However, for the central part of the array, approximately up
to 100 mm from the centre, the centroid performance is good
enough to be used for reconstructing the first approximation
for the LRFs. This is demonstrated in the bottom plot of
Fig. 3(a), where the area of PMT response is plotted versus
the distance from its axis, calculated from the event position
estimated by centroid. The resulting scatter plot was fitted
with a cubic spline (the smooth curve on the plot) which was
used as a first approximation η(1)
PMT. Then the set of LRFs obtained in this way was used
to re-calculate positions of the γ-ray interactions using the
WLS method, producing the position distribution shown on
the top plot of Fig. 3(b), and the cycle was repeated. After
5 iterations, the LRFs converged to the final shape shown
in Fig. 3(c). As one can see, the final distribution of the
estimated event positions clearly shows the projected image
of the copper grid with no significant distortions even in the
region where a PMT response is missing.
57Co source. The top plot in Fig. 3(a) shows the x-y
i(r) for the LRF for a given
V. RESULTS
A. Spatial resolution
In the central part of the chamber, right below the source,
the γ-rays cross the copper grid at normal incidence creating
the sharpest contrast between open and shadow areas. In the
reconstructed event distribution, this transition is smeared
due to finite spatial resolution and, to some extent, by
scattering in the 7-mm anode plate located below the grid. In
other words, the sharpness of the the edges of the projected
image gives an upper limit for the spatial resolution of the
detector for S2 signals. In Fig. 4, the distribution of the y-
positions of the reconstructed events is demonstrated for a
narrow patch in the inner part of the detector (R<100 mm).
The distribution is fitted with a convolutionof a step function
with the Gaussian giving resolution of 1.6 mm FWHM. The
resolution worsens towards the edge of the fiducial volume
due to combination of lower light collection and edge effects,
becoming ~3 mm FWHM at R=150 mm.
The spatial resolution for S1 can be estimated by com-
paring independently reconstructed coordinates for S1 and
S2, Fig. 5(a). The difference between the two, shown in
Fig. 5(b), is approximately normally distributed with FWHM
y, mm
Counts
0
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40
60
80
100
120
140
160
180
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Figure 4. Projection (“shadow”) of the middle bar of the copper grid, used
to estimate the spatial resolution for S2 for the central part of the detector.
of 15.0 mm for the whole fiducial volume and 13.0 mm for
events with R<100 mm. As the contribution of S2 resolution
is obviously negligible, these values correspond to the spatial
resolution for S1.
B. Energy resolution
As demonstrated in [17], there is strong anti-correlation
between scintillation light and extracted charge for electron
recoils in liquid xenon under an applied electric field. The
reason for this is that part of the scintillation light comes
from recombination. For the less dense electron tracks, the
electron extraction efficiency is higher while recombination
(and scintillation output) is lower. Thus, fluctuation of the
electron track density from event to event leads to variations
in light and charge outputs, which in a dual phase detector
leads in turn to anti-correlated variations of S1 and S2 even
for events of the same energy. Consequently, the best energy
estimate for a dual phase detector is a linear combination
of S1 and S2 light outputs. Fig. 6 shows the relationship
between scaled light outputs for S1 and S2 for the events
produced by γ-rays from the
factors were chosen so that the mean of the distribution is
at 125 units for both S1 and S2. One can see that there is
indeed anti-correlation with S1 varying approximately by a
factor of 3 more than S2.
A more detailed analysis of the plot on Fig. 6 yields the
coefficients of the linear combination with the best energy
resolution: E = S2∗0.715+S1∗0.285. Using this formula,
an energy resolution of 10.6% FWHM was obtained at
122 keV for the whole fiducial volume – see Fig. 7(a).
For the central spot with R<50 mm, where the effects
from Compton scattering of incoming γ-rays in copper are
57Co source. The scaling

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a)
S2 y position, mm
S1 y position, mm
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Figure 5.
S2 demonstrate, as expected, very strong correlation, (b) the S1 spatial
resolution for the whole fiducial volume (top) and for the events with
R<100 mm where R is the distance from the axis of the chamber.
(a) The independently reconstructed y-coordinates for S1 and
minimal, the resolution is 8.1% FWHM and the two lines of
the57Co source are clearly resolved as shown in Fig. 7(b).
VI. CONCLUSIONS
Position sensitivity is crucially important for a modern
dark matter detector as it allows one to drastically reduce
the background by considering only events inside an inner
fiducial volume away from any detector surfaces. A position-
sensitive detector also offers better energy resolution as it
becomes possible to apply a position-dependentcorrection to
the energy. In the case of a scintillation detector, the optimal
performance of the position estimation algorithm depends on
S2 energy estimate, a.u.
S1 energy estimate a.u.
60
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7080 90100 110 120 130 140 150 160 170
Figure 6.Anti-correlation between S1 and S2 signals.
how well the set of the PMT LRFs describes the detector
response to scintillation events.
In the present work, a novel method for iterative recon-
struction of the light response functions from the calibration
data acquired with uncollimated γ-ray source was developed
and its suitability has been proven for the real detector.
Using the reconstructed LRFs and applying the weighted
least squares and maximum likelihood methods to position
and energy reconstruction, the performance of the ZEPLIN-
III detector was studied for 122 keV γ-rays. The measured
performance for the inner part of the detector (R<100 mm)
is as follows:
• spatial resolution of 13 mm FWHM in the horizontal
plane for scintillation signal (S1);
• spatial resolution of 1.6 mm FWHM for electro-
luminescence signal (S2);
• energy resolution of 8.1% FWHM for the combined
(S1 and S2) signal.
A more detailed description of the implementation of the
position reconstruction algorithms and their impact on the
WIMP search with ZEPLIN-III will be published as a
separate paper. The developed method can also be applied
in scintillation cameras for medical imaging for correction
of non-uniformities and improving non-linearity associated
with both the scintillation crystal and the PMT array as well
as those due to the position reconstruction algorithm. The
success of the new method in mitigating significant perform-
ance irregularities suggests that hardware components may
be subject to less stringent requirements, thereby reducing
the cost of scintillation cameras. The method can also be of
advantage for regular quality control of gamma cameras.

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a)
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P1
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Figure 7.
combination of S1 and S2 light yields for the whole fiducial volume (a)
and for the central spot with R<50 mm (b).
The spectrum of57Co γ-ray energy estimated from a linear
VII. ACKNOWLEDGEMENTS
The UK groups acknowledge the support of the Sci-
ence & Technology Facilities Council (STFC) for the
ZEPLIN–III project and for maintenance and operation
of the underground Palmer laboratory which is hosted
by Cleveland Potash Ltd (CPL) at Boulby Mine, near
Whitby on the North-East coast of England. The pro-
ject would not be possible without the co-operation of
the management and staff of CPL. We also acknow-
ledge support from a Joint International Project award,
held at ITEP and Imperial College, from the Russian
Foundation of Basic Research (08-02-91851 KO a) and
the Royal Society. LIP–Coimbra acknowledges financial
support from Fundação para a Ciência e Tecnologia
(FCT) through the project-grants CERN/FP/109320/2009,
CERN/FP/116374/2010and
well as the postdoctoral grants SFRH/BPD/27054/2006,
SFRH/BPD/47320/2008 and SFRH/BPD/63096/2009. This
work was supported in part by SC Rosatom, contract
#H.4e.45.90.11.1059 from 10.03.2011.
PTDC/FIS/67002/2006,as
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