ORIGINAL ARTICLE
Stack of quasi-mosaic thin lamellae as a diffracting
element for Laue lenses
Valerio Bellucci · Riccardo Camattari ·
Vincenzo Guidi · Pierre Bastie
Received: 20 November 2013 / Accepted: 24 April 2014 / Published online: 27 June 2014
© Springer Science+Business Media Dordrecht 2014
Abstract Crystals with curved diffracting planes have been investigated as highefficiency optical components for the realization of a Laue lens for satellite-borne
experiments in astrophysics. Curved crystals implementing the quasi-mosaic effect,
namely an effect of crystalline anisotropy, are able to focus an X-ray beam to a size
far smaller than that of the diffracting element, in turn increasing the focusing power
of a Laue lens. This work provides first results about the feasibility of a self-standing
stack composed of quasi-mosaic crystals. Stacking of crystalline lamellae is a solution to overcome the thickness limitation in existing self-standing quasi-mosaic crystals. Ten thin silicon crystalline lamellae were stacked, and then the planes affected
by the quasi-mosaic effect were tested by polychromatic X-ray diffraction. The multicrystal behaved as one diffracting element, yielding a broad and smooth diffraction
profile. The effective realization of a quasi-mosaic multicrystal opens up the prospective of building a Laue lens with a large integrated reflectivity, which leads to a high
V. Bellucci · R. Camattari · V. Guidi ()
Department of Physics and Earth Sciences, INFN, University of Ferrara, Via Saragat 1/c, 44122
Ferrara, Italy
e-mail: [email protected]
V. Bellucci · R. Camattari · V. Guidi
National Institute of Optics CNR-INO Sensor Lab, Via Branze 45 - 25123 Brescia, Italy
V. Bellucci · V. Guidi
INFN Section of Ferrara, Via Saragat 1/c, 44122, Ferrara, Italy
P. Bastie
Laboratoire de Spectrométrie Physique (CNRS UMR 5588), Université Joseph Fourier, BP 87,
St Martin d’Hères Cedex, 38402, France
P. Bastie
ILL-Institut Laue Langevin, 71 Avenue des Martyrs, 38000 Grenoble, France
2
sensitivity, a necessary condition for the observation of celestial X-ray sources via a
Laue lens.
Keywords Hard X-rays · Laue lens · Diffraction · Stack · Quasi-mosaicity ·
Multicrystal · Lamellae
1 Introduction
The detection of hard X-rays plays an increasingly important role in modern
astronomy. The instruments currently operating in this part of the electromagnetic
spectrum, however, do not use focusing optics, i.e., the measured signal is collected
directly on the sensitive part of the detector itself. Multilayer optics achieve highefficiency focusing up to 80 keV X-rays, though beyond this limit the efficiency of
such optics critically falls off [1]. The construction of high-reflectivity multilayer
mirrors working at energies up to several hundreds of keV was attained very recently
[2]. Nevertheless, these new multimirrors work at very low grazing incidence angles,
below 0.1 ◦ , thus featuring a very low acceptance area for the incident photons.
Moreover, the assembly of a focusing instrument relying on this technology appears
to be rather complex. Up to now, the energy range above 80 keV is left to instruments
detecting photons with no focusing, resulting in low signal-to-noise ratio [3].
The scientific community is searching for a focusing device to overcome the
present limitations [4]. One solution could be the so-called Laue lens, that is a hard
X-ray concentrator built as an ensemble of many crystals oriented in such a way
that the radiation passing through them is diffracted to the lens focus [5, 6]. The
aim is to concentrate as much radiation as possible over a selected energy band [7,
8]. In astrophysics, a Laue lens would allow high sensitivity and high-angular resolution observation of cosmic phenomena producing X-ray emissions [9]. Detailed
mapping of X-ray sources along with characterization of their high-energy emission
spectrum is mandatory to determine their intimate nature. Fine angular scale and
high-sensitivity observations have long been awaited to cast new light on the origin of the positrons that we observe in the galactic center through the 511 keV line
due to annihilation with electrons. The study of supernova dynamics would also be
improved by the observation of the decay lines emitted by radionuclides synthesized
during the explosion, such as the radionuclide 56 Co and the 847 keV line, which is
produced by the decay chain. A detailed analysis of radiation emission by type IA
supernovae and their remnants would provide important information on the explosion
mechanism.
Nuclear medicine would also benefit from a Laue lens. Improving the resolution
and sensitivity of analyses based on tracing radio-nuclei injected into the patient
would allow the detection of physiological processes with sub-millimetric precision,
thereby reducing the dose of the radio-nuclei to inject [10, 11].
Perfect crystals diffract within a very narrow energy range, so that they are not useful for building a Laue lens. Mosaic crystals show an enlarged energy passband and
were successfully tested for the purpose [12]. However, they exhibit two drawbacks.
Firstly, diffraction efficiency for mosaic crystals is limited to 50 % owing to the
3
balance between diffracted and transmitted beam inside the crystal [13]. Secondly,
production of mosaic crystals is rather complex and often results in large rejection
rate of the fabricated crystals.
A crystal having curved diffracting planes (CDP crystal) can potentially overcome
these two drawbacks [12, 14, 15]. Its energy passband can be very well controlled,
being proportional to the crystal curvature, and its diffraction efficiency is not limited
because the continuous change in the incidence angle on the bent crystalline planes
prevents re-diffraction of the diffracted beam. Indeed, a scheme relying on Laue
diffraction with bent crystals proved to work for hard X-rays with diffraction efficiency ideally close to unity [16–18]. The energy passband of the photons diffracted
by a CDP crystal is orders of magnitude broader than for a flat crystal, featuring a
uniform transfer function, provided that the crystal curvature is homogeneous.
As the primary curvature of the crystal matches the curvature of the Laue lens
calotte, the crystal itself becomes a focusing element [19]. The incident beam is
concentrated on a spot smaller than the geometrical size of the diffracting element
(Fig. 1), thus allowing the point spread function of the lens to be much more compact than in the case of a lens made of regular flat crystals. For this system to
work, it is necessary that the planes perpendicular to the calotte of the lens exhibit a
curvature too.
Quasi-mosaic (QM) crystal is a particular type of CDP crystal [20, 21]. As a crystal is bent to a primary curvature, e.g. by external forces, anticlastic deformation
takes place, resulting in a secondary curvature in a direction perpendicular to that of
primary curvature. Under specific orientation, another secondary curvature is generated within the crystal through the quasi-mosaic deformation. Quasi-mosaicity is a
mechanical property driven by anisotropy and it is fully explained by the theory of
linear elasticity in an anisotropic medium.
Fig. 1 Cross-section drawing of a Laue lens based on QM crystals. The primary curvature of the lens
is a spherical calotte with radius RP . This structure leads to focalization of diffracted radiation onto the
lens focus located at f = RP /2. The primary curvature also induces a secondary curvature to the planes
affected by quasi-mosaicity, perpendicular to the main surface. The QM curvature increases the integrated
reflectivity and thus the number of photons diffracted onto the focus
4
The planes bent by the QM effect can be used to diffract X-rays. Thus, it is possible
to combine the focusing action due to the primary curvature with the high reflectivity
of CDP built up by the QM effect, therefore increasing the resolution and sensitivity
of an instrument for the observation of X-ray sorces.
Because of technological reasons, it is currently impossible to manufacture selfstanding QM crystals thicker than 2 mm with large enough primary curvature [19].
Since the integrated reflectivity in CDP crystals is proportional to the thickness
traversed by the radiation, this restriction severely limits the integrated reflectivity
of the crystals, and hence the effective area of the lens. A larger thickness would
increase the quantity of photons diffracted and ultimately enhance the sensitivity of
an instrument using this lens.
A possibility to overcome the limit on thickness is the stacking of self-standing
QM crystals one on top of the other. Using the self-focusing geometry, and by
manufacturing composite crystals (multi-crystals) made of a stack of thin lamellae
featuring quasi-mosaicity, the passband and diffraction efficiency of a Laue lens element can be increased. In this paper, we manufactured a stack of ten thin QM lamellae
made of Si. Characterization by X-rays shows that such a configuration can be used
to build a Laue lens.
2 General background
QM was first discussed by Sumbaev [22] in a seminal work about an anomalous
broadening of X-ray diffraction profiles. Crystals with a diamond-like lattice, such
as Si and Ge, exibit QM effect due to their anisotropic elastic properties. For particular crystallographic orientations, the application of a primary curvature to a crystal
plate induces a secondary curvature perpendicular to the first one, in addition to
the anticlastic curvature. The QM curvature is strictly bound to the primary curvature induced to the crystal plate by external bending moments, and on the elastic
properties of the material, i.e. by the components of the compliance tensor Sij .
Applying only one bending moment, M, to a rectangular crystal plate results in a
cylindrical curvature as in Fig. 2. In this case, it is possible to demonstrate that, for a
Fig. 2 Rectangular crystal plate with applied one bending moment symbolized by the curved arrows. The
schetched coordinate system is used in the modelization
5
diamond-like lattice, the primary bending of (111) planes generates a QM curvature
of (211) planes [23]. The ratio between the quasi-mosaic and the primary curvature
radii (RQM and RP , respectively) holds
RQM
S22
=−
RP
S42
Where S22 and S42 are the components of the compliance tensor of the material.
Because of the structure of the compliance tensor, the component S22 is never zero.
It follows that, for cylindrical bending, the QM curvature occurs only if S42 is not
zero. Because the (211) reflection is forbidden in a diamond-like lattice, the (422)
reflection was used. Indeed, the QM effect is not a peculiarity of this orientation
as it manifests itself for other lattice planes, and other bending geometries too. The
present geometry was chosen because of the absence of torsion induced by crystalline
anisotropy. This solution makes the analysis of experimental results far clearer.
The QM effect produces a uniform curvature. Thus, for a single crystal, the passband for photon diffraction in a single point exhibits a tipical rectangular distribution
[17, 24]. For a beam with macroscopic dimension, the primary curvature has to
be taken into account, so that the passband is the convolution of two rectangular
functions, one for the primary curvature and another for the secondary one. The convolution is a trapezoidal function. Then, the passband of the stack is the union of
the passbands of all the crystal lamellae. If the lamellae were perfectly aligned, their
passbands would superpose, i.e. the total passband of a stack would be equal to the
passband of an individual lamella. This circumstance is not favorable to build up a
Laue lens because it does not widen the passband and, if the efficiency of each individual crystal is very high, the increase in the total reflectivity of the stack would be
low. The process of crystals stacking one on top of the other naturally induces random misalignements in their relative orientation [25], which would circumvent the
problem.
Figure 3 shows an example of the expected passband due to QM curvature only,
in the case of a stack of five randomly misaligned QM crystals. In this case, the
contribution of the primary curvature has not been included in order to highlight the
contribution of QM passbands. Here, Si (422) QM diffraction planes are bent to a
curvature radius of 140 m, crystal thickness is 2 mm, and the energy of impinging
photons is 150 keV, an optimal condition for a QM stack built for astrophysical applications. In the case shown in the figure, the QM passband of two crystals superpose,
generating only a small increase in reflectivity, and nearly no increase in the total
passband of the stack. The passbands of the other crystals are positioned one near
each other, thereby increasing the total passband width. The passband due to primary
curvature is tipically one order of magnitude larger than that for the QM curvature,
wide enough to smoothen the final passband without leaving gaps. In summary, a balanced tradeoff between diffraction efficiency and passband width is the condition for
which the responses of all the secondary curvatures are not overlapped to each other
and they lie within one individual primary passband. In this case, the arrangement of
the secondary responses is not so crucial and that helps very much in the preparation
of a stack with these characteristics.
6
Fig. 3 Example of the expected response for a QM stack, considering only the contribution due to the
QM curvature. Neglecting the smoothing effect of primary curvature allows to highlight the features of
the QM passband for each crystal composing the stack. In the example the stack is made of 5 crystals
with thickness 2 mm, the QM Si (422) diffraction plane is bent at a radius of 140 m, yielding a passband
of 3 arcsec for each crystal. The energy of impinging photons is 150 keV, a typical energy of operation.
The QM passbands are randomly misaligned with respect to each other, with a mean misalignement of
10 arcsec. Three of the QM passbands are well separated, while two QM passbands superpose one another.
Since diffraction efficiency of each crystal is 61.6 % under this working condition, superposition of the
QM passbands increase diffraction efficiency just up to 85.2 % in the area of superposition
3 Method
Commercially available pure Si wafer was diced to form ten plates by using a high
precision dicing saw (DISCO™DAD3220). The plates were 0.2 mm thick, with lateral dimension 5 mm × 45 mm. Lattice orientation of the largest surface was (111),
while the surface with dimension 0.2 mm × 5 mm had crystalline orientation (211).
By using a low-stress thermal resin, the crystals were bonded one over the other to
form a stack (Fig. 4). This latter was then mounted on a specifically designed rigid
holder realized at Sensors and Semiconductors Laboratory (SSL, Ferrara, Italy). The
holder applied to the stack a controlled primary cylindrical curvature (0.2 m nominally) that resulted in a secondary curvature of the (211) planes due to the QM effect.
Finally, the holder was positioned on a hot plate where, at nominal temperature,
the resin solidified, yielding a self-standing curved stack of plates. The production
method is similar to that in Ref. [26]. Production and optical characterizations was
carried out at SSL. To verify the curvature applied to the stack, subtraction of the morphological profiles before and after the mechanical process was done through optical
profilometry (VEECO™NT1100). At the centre of the stack, this analysis was executed for the crystals on the top and on the back faces of the stack (Fig. 5). These two
faces presented a different curvature radius, probably because of the resin was not
able to maintain the crystal plates at the same curvature radii at such high curvature.
7
It was impossible to carry out the same analysis on the edges of the stack, because
the large quantity of glue blur at the clamping points caused significant optical errors
when interferometric analysis was attempted.
Alignement of curved crystals in a self-standing stack was already proved to be
possible with a precision better than 8 arcsec [25]. In that work, the analysis was
performed on crystals with curvature radius of tens of meters, useful for astrophysical applications, but not on planes featuring the QM effect. The self-standing stack
analyzed in this paper was produced with a curvature far larger than in Ref. [25] to
allow a better analysis of misalignements between lamellae. In fact, if the curvature
radius of the lamellae was the value realistically useful for astrophysical applications
(tens of meters), diffraction efficiency of QM planes would be nerly the unity, not
Fig. 4 a Picture of the built stack of ten QM Si lamellae. b Representation of the stack undergoing
diffraction as in the experiment. The beam enters the stack from the concave side, undergoes diffraction
from (422) QM planes, and exits the stack from the opposite side. Realistically, the stacking process
induces a small random misalignement between the bent QM planes belonging to each crystal
8
Fig. 5 Profilometric analysis by white-light interference at the centre of the sample as taken on the convex
(a) and concave (b) side. In both cases it is possible to see a uniform curvature in the Y direction, i.e. the
direction of the primary curvature used for focusing. Curvature on the X direction is produced by anticlastic deformation as a result of the primary curvature. The curvature radii in cases (a) and (b) are different,
probably because of the limited stiffness of the glue used for bonding the crystals together in the stack.
For the case (b), it is possible to see some optical errors caused by the glue blur at the border of the image
9
resulting in a clear distinction of the positions of QM passbands whereas these latter
superpose each other. The optimal condition for a better analysis would be that each
lamella diffracted a very small part of the beam. A decrease in diffraction efficiency
is possible by overbending the diffraction planes beyond a certain limit called the
critical curvature radius RC [14, 27], so that the diffraction efficiency falls off. For
this reason, the QM stack was manufactured with a very large curvature, in order to
have diffraction efficiency of each lamellae to be about 1 %. Thus, the beam interacting with each lamellae would have nearly the same intensity. In this case, each crystal
would produce a diffraction image of nearly equal intensity regardless of passband
superposition between them. The diffraction image of all the crystals would sum up
on the detector, and the superposition of their passbands would be clearly visible in
the total diffraction image as an increase in the counts. As an example, the reflectivity of one lamella under this condition at 100 keV was 1.2 %, while the superposition
of all the photon passband of the lamellae would give a reflectivity of 11 % at the
same energy.
4 Experimental
The stack was tested through X-ray diffraction at Laue Langevin Institute (ILL,
Grenoble, France) by using a hard x-ray diffractometer based on a focusing effect,
occurring when a divergent and policromatic X-ray beam diffracts through a crystal
[28]. This technique is schematically represented in Fig. 6. A high-voltage and finefocus X-ray tube produced a white divergent X-ray beam with energy within 80 and
450 keV. Since Bragg angles were small (0.5 – 1◦ ), diffraction peaks were located
close to the direct beam, thus allowing the observation of peaks from several crystallographic planes. These latter were observed thanks to a high-resolution and sensitive
X-ray image intensifier coupled with a CCD camera, featuring a spatial resolution of
about 0.35 mm (one pixel size). The distance between sample and generator focus,
this latter being 1 × 1 mm2 , was set at 4.45 m, thus determining a lattice tilt maximum sensitivity of 8.1 arcsec. A slit with variable size was positioned just before the
sample in order to delimit the width of the X-ray beam.
The multi-crystal was analyzed with the beam penetrating through its 45 × 5 mm2
surface at different distances from the edge of the sample, quasi-parallel to the (422)
planes. The plate divides the space into a convex and a concave region1 . The beam
entered the sample through the concave side, a sketch of the configuration is shown
in Fig. 4b. We preferred this configuration because it allowed a better resolution
than in the case where the beam entered the sample through the opposite side. Beam
size was chosen to be 0.5 × 5 mm2 . As mentioned in Ref. [28] for a diffractometer based on the method of X-ray focusing in Laue geometry with a divergent and
polychromatic beam, the diffraction spots takes the shape of lines (Fig. 6). Indeed,
the focusing/defocusing effect only occurs in the scattering plane while, in the
1 Here
we adopted the vocabulary definition of convexity.
10
Fig. 6 Schematic in-plane representation of the polychromatic diffractometer used in the experiment.
Diffraction by a flat perfect crystal in Laue geometry results in focusing the divergent polychromatic Xray beam to a line, at a distance from the sample equal to the distance between sample and source. If the
crystal presents mosaicity or bending, the line broadens to a more complex diffraction pattern, according
to the features of the specimen
perpendicular direction, the radiation propagates straight. The width of the diffraction lines is directly related to lattice distortion of the crystal under analysis. Figure
7a shows a diffraction image of the centre of the stack, while Fig. 7b shows a diffraction image taken near the edge on the stack. The pattern for (422) diffraction spot
in the centre of the stack is broad and uniform, even if some asymmetries are visible. The asymmetry is probably the result of some torsion in the crystal, which
however does not appreciably influence the performance of the stack. At the opposite, the same diffraction pattern taken near the edge of the stack presents different
well-defined peaks and valleys. This is a result of the manufacturing process used to
obtain the stack. In fact, the holder used for bending the stack induced nearly no curvature at the clamping points of the lamellae, thus producing a stack of nearly flat
crystals at the edges. Then, the bending contribution due to both primary and QM
curvatures is smaller than the misalignment between the plates, showing the structure of the multi-crystal in the diffraction pattern. In Fig. 7a and b the diffraction spot
for (202) asymmetric reflection is also visible, rotated by + 30◦ relative to the (422)
spot. The (422) plane is not the only one affected by crystalline anisotropy. In fact,
the (202) asymmetric plane also has a curvature. Moreover, because the (202) plane
has a higher electron density than for the (422) plane, hence the integrated reflectivity
is about four times larger and its diffraction spot results more intense. Nevertheless,
an evaluation of the alignment of lamellae by the analysis of this reflection would be
difficult. In fact, asymmetric lattice planes present a strong torsion driven by crystalline anisotropy. Even if the final effect of the torsion on the performance of a
Laue lens may be small, and asymmetric planes may certainly be used for their high
11
Fig. 7 Unprocessed experimental 2D diffraction image for the centre (a) and for the edge of the QM stack
(b). QM (422) diffraction distribution is highlighted in the bottom-right of the 2D image, while another
diffraction spot corresponding to (202) asymmetric reflection is clearly visible. (202) reflection is always
more intense than the (422), because of the larger electron density of this lattice plane. The
shape of the
diffraction spot produced by an unbent perfect crystal would be a line with width δ = a 2 + (2tθB )2 .
Because the second term is far smaller than the first term, the line would be as wide as the source dimension
a = 1 mm, i.e. (23 pm 8) arcsec. (a) Both distributions have an oblique-shape, which highlights some
torsion in the stack. A smooth shape is evident, which means that the crystals composing the stack are
far less misaligned than the contribution of the primary curvature. (b) From the 2D image is evident the
misalignement between the crystals composing the stack with some torsion. In fact, both the QM (422) and
the (202) diffraction distributions are divided into multiple spots, each one with different shape. Because
of the smaller curvature of the crystals, there is no complete overlap. A misalignement between each
crystal larger than in case (a) is also evident
reflectivity, the analysis of the alignment of lamellae in a single stack would be
difficult with asymmetric planes. Hence, we took into consideration just the (422)
symmetric reflection in spite of its moderate strenght.
12
5 Results and discussion
Figure 8a and b show the cross section of the diffracted spot at the centre and near
the border of the stack, with their errors. The standard deviation was calculated from
the counting rate far from the diffraction spot, in order to consider the errors due to
electronic noise and diffused photons, which are present in an experimental setup
relying on an uncollimated detector.
For a crystal with flat diffraction planes in the configuration sketched in Fig. 6,
it comes out that the width δ of the X-ray beam at the detector, under small-angle
approximation, is given by [25]
δ=
a2
l
+ 2tθB ± 2f
RP
2
Fig. 8 a, b experimental data with their uncertainty in units of the standard deviation at the centre and at
the edge of the QM stack. The black-, gray-, white-shaded areas correspond to an experimental uncertainty
of less than one, two, and three standard deviations respectively, c and d compare the experimental data
with the theoretical calculation for the mean distribution of the diffraction cross-section at the centre
and at the edge of the QM stack. In each case, the noise baseline was subtracted to experimental data.
In (c), (d), the continuous black curve represents experimental data. The green dashed curve represents
the theoretical expected diffraction distribution, taking into account the position of QM passbands of the
crystals composing the stack, the smoothing contribution of primary curvature, the torsion and the finite
size of the source. The red solid line curve represents the best-fit positions of the QM passbands and their
superposition, in units of the number of passbands superposing in each point. At the stack centre (c) the
misalignement between passbands is lower than 16 arcsec, the stack responds as one diffracting element,
resulting in a broad and smooth diffraction profile. At the stack edge (d) the passbands shrink down
because of the smaller curvature, and the misalignements rise, so that they do not appreciably superpose
13
The sign (+) holds for the beam entering the sample through the concave side, the
sign (-) holds in the opposite case. a is the size of the X-ray source, t the crystal
thickness traversed by radiation, f the sample-to-detector distance, l the length of
the crystal seen by the beam, bent to a radius of curvature RP . The term 2tθB is
l
the contribution of the broadening due to the thickness of the sample, while 2f
Rp
represents the contribution of primary bending. In the case of a QM stack, an effect
of image enlargement due to QM passband has also to be considered. This term is
2f t
RQM , with RQM the radius of curvature of QM diffraction planes. The expected cross
section of the diffracted spot is obtained
by convolving
the term of image enlargement
2f l
due to QM passband with the term 2tθB ± r due to the crystal geometry, and
then convolving the result with the spread due the finite size of the souce, a.
Figure 8c and d compare experimental data with theoretical expectations. In
these cases, the noise baseline was subtracted, too. The red step-like distribution
represents the best fit for the superposition of the QM passbands only, without
the contribution due to the primary curvature. The green dotted curve represents
the theoretical diffraction profile, without considering the random noise. The QM
passbands positions were derived through computer simulation, by changing the
positions of the passbands until the diffraction profile best fits the experimental
data.
At the centre of the stack (Fig. 8c), a progressive primary bending was assumed
along the stack thickness, from 0.20 m at the top face of the stack, to 0.28 m at
the back face, the instrumental uncertainty being 5 %. Then, the primary curvature
contributed to the diffraction distribution with a broadening from 368 arcsec to 516
arcsec. The QM passband width of lamellae was know through the theory of elasticity
to range from 42 arsec to 59 arcsec. The expected diffraction profile was calculated
as the summation of the contribution of each lamella in the stack. The torsion of
the stack results in further broadening of the cross section by (148 ± 16) arcsec.
Through best fit of the QM passbands and taking into account the primary curvature
and the torsion, it resulted in a misalignement between the lamellae less than 16 arcsec (2 pixels), consistently with the misalignement already achieved in Ref. [25]. A
typical lamella built with a curvature radius useful for straight application in astrophysics would have a QM passband width in the order of few arcsecs. In this case, a
misalignement between passbands in the order of 16 arcsecs or lower would be ideal.
At the opposite, the single contributions of the lamellae are well visible at the edge
of the stack (Fig. 8d). The maximum misalignements between lamellae is (694 ± 8)
arcsec, while the mean misalignement is (172 ± 8) arcsec. In fact, the manufacturing method is likely to induce stronger stress and extra random misalignements at
the clamping points with the holder, i.e. at the edges of the stack. In this case, the
smaller curvature also lowers the passband due to primary curvature to less than the
misalignement between crystals. The curvature radius at the border can be calculated
by changing the primary curvature in order to match the width of the peaks in the
diffraction data. The primary curvature radius results to be (3.94 ± 0.60) m, so that
the spread due to the primary curvature is (26 ± 4) arcsec, and the expected QM
passband is (3 ± 0.5) arcsec. The uncertainty is mainly due to the resolution of the
detector.
14
The experimental measurements allow one to conclude that a stack of quasimosaic crystals can be built up with suitable relative misalignement to achieve a
broad and smooth passband. Despite the crystals were far more bent than in the case
of Ref. [25], the relative alignement remained good. The relative alignement of the
QM crystals allows one to stack a multitude of thin lamellae to obtain thick diffracting elements. Clearly, for the preparation method we used in this paper, it is necessary
to cut out the parts near the edges at the clamp positions. The quality of the primary
curvature may be improved by testing different bonding methods. For example, a
holder with curved surfaces may be implemented during the manufacturing of the
stack, in order to induce a proper curvature at the clamping points, i.e. at the edges of
the stack. Another possibility may be to stack self-standing bent lamellae [29], thus
avoiding the use of a holder and restraining the role of a glue in keeping the lamellae
together.
6 Conclusions
A self-standing stack of QM crystals was produced at SSL. The stack was made of
ten Si (111) crystalline lamellae, in order to test the feasibility of stacking a multitude
of QM crystals. The QM multicrystal was then analyzed at ILL by polychromatic
X-ray diffraction. The diffraction response was broad and smooth at the centre of
the crystal, the contribution of each crystal composing the stack was not separable,
and the stack behaved as one diffracting element. Misalignement between crystals
resulted to be lower than 16 arcsecs, a misalignement of interest for building QM
stacks for astrophysical applications. The quality of primary curvature near the edges
of the stack may be improved by implementing a different bonding technique in the
manufacturing process.
The applicability, and the validity of this experiment is not limited by the material, the geometrical dimension, the number of lamellae, or the crystalline reflection
implemented for this particular stack. In fact, other materials exhibit quasi-mosaicity
(e.g. Ge) as well as other crystalline reflections. Asymmetric diffraction planes
may also be used for their high reflectivity. The manufacturing parameters of the
stack have to be choosen in order to match the characteristics of the Laue lens to
be built.
The work provides first results about the feasibility of a self-standing stack composed by QM crystals. The effective realization of a QM multicrystal opens up the
prospective of solving the main problem concerning QM crystals, i.e. the limited
thickness they can be manufactured with currently available technology. Then, a QM
diffracting element concentrating a large quantity of photons can be envisaged, with
a proportional increase in sensitivity of an instrument implementing a Laue lens composed by stacks of QM crystals. A high sensitivity concentration instrument for hard
X-rays would lead to a better understanding of astrophysical phenomena producing
X and gamma rays, and of their intimate nature.
Acknowledgments
The authors are thankful to INFN for financial support through the LOGOS project.
15
References
1. Madsen, K.K., Harrison, F.A., Mao, P.H., Christensen, F.E., Jensen, C.P., Brejnholt, N., Koglin, J.,
Pivovaroff, M.J.: Optimizations of Pt/SiC and W/Si multilayers for the nuclear spectroscopic telescope
array. Proc, SPIE 7437, 743716 (2009)
2. Della Monica Ferreira, D., et al.: Hard x-ray/soft gamma-ray telescope designs for future astrophysics
missions. Proc. SPIE 8861, 886116 (2013)
3. Harrison, F.A., et al.: The nuclear spectroscopic telescope array (NuSTAR) high-energy x-ray mission.
A.P.J. 770, 103 (2013)
4. Lindquist, T.R., Webber, W.R.: A focusing X-ray telescope for use in the study of extraterrestrial
X-ray sources in the energy range 20-140 keV. Canadian Journal of Physics. Proceedings of the
10th International Conference on Cosmic Rays, Calgary, Alberto, 19–30 June 1967, vol. 46, p. 1103
46 1103
5. Von Ballmoos, P., Halloin, H., Evrad, J., Skinner, G., Abrosimov, N., Alvarez, J., Bastie, P., Hamelin,
B., Hernanz, M., Jean, P., Knodlseder, J., Smither, R.K.: CLAIRE: first light for a gamma-ray lens.
Exp. Astron. 20, 253–267 (2005)
6. Knodlseder, J., von Ballmoos, P., Frontera, F., Bazzano, A., Christensen, F., Hernanz, M., Wunderer,
C.: GRI: focusing on the evolving violent universe. Exp. Astron. 23, 121–138 (2009)
7. Pellicciotta, D., Frontera, F., Loffredo, G., Pisa, A., Andersen, K., Courtois, P., Hamelin, B., Carassiti,
V., Melchiorri, M., Squerzanti, S.: IEEE Trans. Nucl. Sci. 53, 253–258 (2006)
8. Pisa, A., Frontera, F., Loffredo, G., Pellicciotta, D., Auricchio, N.: Laue lens development for hard
X-rays (>60 keV). Exp. Astron. 20, 219–228 (2005)
9. Frontera, F., Von Ballmoos, P.: Laue gamma-ray lenses for space astrophysics: status and prospects.
X-Ray Optic. Instrument. 215375, 18 (2010)
10. Roa, D., Smither, R., Zhang, X., Nie, K., Shieh, Y., Ramsinghani, N., Milne, N., Kuo, J., Redpath, J.,
Al-Ghazi, M., Caligiuri, P.: Development of a new photon diffraction imaging system for diagnostic
nuclear medicine. Exp. Astron. 20, 229–239 (2005)
11. Adler, L.P., Weinberg, I.N., Bradbury, M.S., Levine, E.A., Lesko, N.M., Geisinger, K.R., Berg, W.A.,
Freimanis, R.I.: Method for combined FDG-PET and radiographic imaging of primary breast cancers.
Breast J. 9(3), 163 (2003)
12. Barriére, N., Rousselle, J., von Ballmoos, P., Abrosimov, N.V., Courtois, P., Bastie, P., Camus, T.,
Jentschel, M., Kurlov, V.N., Natalucci, L., Roudil, G., Frisch Brejnholt, N., Serre, D.: Experimental and theoretical study of the diffraction properties of various crystals for the realization of a soft
gamma-ray Laue lens. J. Appl. Cryst. 42, 834–845 (2009)
13. Zachariasen, W.H.: Theory of X-rays Diffraction in Crystals. Wiley, New York (1945)
14. Malgrange, C.: X-ray Propagation in distorted crystals: from dynamical to kinematical theory. Cryst.
Res. Tech. 37, 654–662 (2002)
15. Keitel, S., Malgrange, C., Niemoller, T., Schneider, J.R.: Diffraction of 100 to 200 keV X-rays from an
Si1−x Gex gradient crystal: comparison with results from dynamical theory. Acta Cryst. A55, 855–863
(1999)
16. Barriére, N., Guidi, V., Bellucci, V., Camattari, R., Buslaps, T., Rousselle, J., Roudil, G., Arnaud,
F.-X., Bastie, P., Natalucci, L.: High diffraction efficiency at hard X-ray energy in a silicon crystal
bent by indentation. J. App. Cryst. 43, 1519–1521 (2010)
17. Bellucci, V., Camattari, R., Guidi, V., Neri, I., Barriére, N.: Self-standing bent silicon crystals for very
high efficiency Laue lens. Exp. Astron. 31, 45–58 (2011b)
18. Guidi, V., Barrière, N., Bellucci, V., Camattari, R., Neri, I.: Bent crystals by surface grooving method
for high-efficiency concentration of hard x-ray photons by a Laue lens. Proc. SPIE, 81471E-81471E-7
(2011)
19. Camattari, R., Battelli, A., Bellucci, V., Guidi, V.: Highly reproducible quasi-mosaic crystals as optical
components for a Laue lens. Exp. Astron. 37, 1–10 (2014)
20. Guidi, V., Bellucci, V., Camattari, R., Neri, I.: Proposal for a Laue lens with quasi-mosaic crystalline
tiles. J. Appl. Cryst. 44, 1255–1258 (2011b)
21. Camattari, R., Bellucci, V., Guidi, V., Neri, I.: Quasi-mosaic crystals for high-resolution focusing of
hard x-rays through a Laue lens. Proc. SPIE, 81471G-81471G-8 (2011)
22. Sumbaev, O.: Reflection of gamma-rays from bent quartz plates. Sov. Phys. JETP 5, 1042–1044
(1957)
16
23. Bellucci, V., Camattari, R., Guidi, V.: Proposal for a Laue lens relying on hybrid quasi-mosaic curved
crystals. Astron. Astrophys. 560, 58 (2013)
24. Camattari, R., Guidi, V., Bellucci, V., Neri, I., Frontera, F., Jentschel, M.: Self-standing quasi-mosaic
crystals for focusing hard X-rays. Rev. Sci. Instrum. 84, 053110 (2013)
25. Neri, I., Camattari, R., Bellucci, V., Guidi, V., Bastie, P.: Ordered stacking of crystals with adjustable
curvatures for hard X- and [gamma]-ray broadband focusing. J. Appl. Cryst. 46, 953–959 (2013)
26. Smither, R.K.: High resolution x-ray and gamma ray imaging using diffraction lenses with mechanically bent crystals. US 7468516 B2 (2008)
27. Bellucci, V., Guidi, V., Camattari, R., Neri, I.: Calculation of diffraction efficiency for curved crystals
with arbitrary curvature radius. J. Appl. Cryst. 46, 415–420 (2013)
28. Stockmeier, M., Magerl, A.: A focusing Laue diffractometer for the investigation of bulk crystals.
J. Appl. Cryst. 41, 754–760 (2008)
29. Guidi, V., Bellucci, V., Camattari, R., Neri, I.: Curved crystals for high-resolution focusing of X and
gamma rays through a Laue lens. Nucl. Instrum. Meth. B 309, 249–253 (2013)