self-standing bent silicon crystals for very high efﬁciency...

Noname manuscript No.(will be inserted by the editor)

Self-standing bent silicon crystals for very high efficiency

Laue lens

Bellucci Valerio · Camattari Riccardo · Guidi

Vincenzo · Neri Ilaria · Barriere Nicolas

Received: date / Accepted: date

Abstract Silicon mono-crystals have been bent thanks to a series of parallel super-

ficial indentations on one of the largest faces of the crystals. This technique relies

on irreversible compression of the crystal beneath and beside the indentations. This

latter causes deformation with no need for external device, resulting in a uniform self-

standing curvature within the crystal. Indented Si crystals have been characterized at

European Synchrotron Radiation Facility using a monochromatic beam ranging from

150 to 700 keV. Crystals exhibited very high diffraction efficiency over a broad range

of energy, peaking 95% at 150 keV. Measured angular spread of the diffracted beam

was always very close to the morphological curvature of the sample under investiga-

tion, proving that the energy passband of bent crystals can be controlled by simply

imparting a selected curvature to the sample. The method of superficial indentations

was found to offer high reproducibility and easy control of diffraction properties of

the crystals. Moreover the method is cheap and simple, being based on mass produc-

tion tools. A Laue lens made of crystals bent by superficial indentations can provide

high-efficiency concentration of hard x-ray photons, leading significant improvement

in many astrophysical applications.

Keywords Laue lens · Hard x-ray diffraction · bent crystals

1 Introduction

Observation of the sky in the hard x-ray domain would open up a window on a broad

variety of many powerful sources and violent events occurring in the Universe, such

Bellucci Valerio, Camattari Riccardo, Guidi Vincenzo, Neri Ilaria

Department of Physics, University of Ferrara, Via Saragat 1/c, 44122 Ferrara and

CNR - IDASC SENSOR Lab., Italy

Fax: +39 0532 974210

E-mail: [email protected]

Barriere Nicolas

Space Sciences Laboratory, 7 Gauss Way, University of California, Berkeley CA 94720-7450 USA

2 Bellucci Valerio et al.

Fig. 1 X-ray diffraction in Laue geometry in case of an unbent (a) and of a bent crystal (b). Multiple

reflections in case (a) results in maximum 50% diffraction efficiency while in case (b) diffraction efficiency

can reach 100%

as active galactic nuclei, compact object in binary system, pulsar, nova, supernova

and more. However, to take full advantage of this potential, the next generation of

instruments working in this energy domain will have to achieve an improvement in

sensitivity by at least an order of magnitude with respect to existing telescopes [1].

Bragg diffraction in Laue geometry [2] is being proposed to efficiently concentrate

photons at these energies [3,4]. In this configuration, perfect crystals thicker than the

extinction length and mosaic crystals suffer a maximum diffraction efficiency of 50%

because of re-diffraction of the incident beam onto lattice planes. Conversely, curved

crystals prevent this effect due to continuous change of the incidence angle so that

only a single diffraction occurs onto curved crystalline planes (Fig. 1). Unlike for

mosaic crystals, which normally exhibit a Gaussian-like reflectivity passband, a bent

crystal offers a continuum of possible diffraction angles over a finite range, leading

to a rectangular-shape energy passband directly owing to its curvature [2,3].

Thanks to curved crystals it would be possible to build an efficient Laue lens by

concentrating the signal collected over a large area into a small focal spot on the de-

tector, thus significantly increasing the signal to background ratio, and consequently

leading to a sensitivity leap by more than an order of magnitude with respect to ex-

isting telescopes [5]. Such improvement is particularly awaited for the study of the

origin of the positrons annihilating in the Galactic center, visible through the e+ - e-

annihilation line at 511 keV. A study of the distribution of this emission line would

bring new clues concerning the still elusive sources of this antimatter. Despite a 511

keV emission has been observed for more than 30 years towards the Galactic center

[6], the origin of the positrons still remains a mystery. Stellar nucleosynthesis [7–9],

accreting compact objects [10–13], and even the annihilation of exotic dark-matter

particles [14] have all been suggested, thus a deeper investigation has to be done.

For fabrication of a bent crystal, several techniques have been worked out. Bend-

ing can be accomplished by mechanical means, i.e., by deforming a perfect single

crystal [15] through an external device. Mechanically bent crystals have been used

in synchrotrons for many years as high-efficiency monochromators. However, the us-

age of an external device leads to excessive weight, for satellite-borne experiments.

Thus, self-standing bent crystals are mandatory for practical implementation of a fo-

cusing telescope as a Laue lens. Such curved crystal can be produced by applying a

thermal gradient to a perfect single crystal [16] but, this method is energy consum-

ing and not adapted to a space-borne observatory as well. Crystals having curved

Self-standing bent silicon crystals for very high efficiency Laue lens 3

Fig. 2 Grooves were indented on the surface of a Si plate along one direction, either x or y (a). The probe

x-ray beam enters the sample parallel (b) or perpendicular (c) to the grooves

diffracting planes can also be obtained by concentration-gradient technique, i.e., by

growing a two-component crystal with graded composition along the growth axis

[16–19]. However, crystals bent by such a method are not easy to manufacture, this

making the technique hardly applicable for a Laue lens application, for which serial

production of crystals should be envisaged.

It was recently proven that superficial indentations made on a Si crystal is a

method for bending crystals [20]. This technique is based on plastic deformation of

the crystal induced by the grooves. As a result of deformation, a permanent curvature

is produced with no need for external device. Such method is cheap, simple and very

reproducible, thus compatible with mass production.

In this paper we present a systematic study on curved crystals bent by indentations

for the purpose of diffracting high-energy photons for astrophysical observations. In

particular, a Si crystal bent by indentation exhibited record diffraction efficiency of

nearly 95% at 150 keV.

2 Experimental

Indentations manufactured on the surface of a Si plate by a diamond saw is known

to deform the whole crystal, leading to a net and permanent curvature [21]. The ori-

gin of permanent deformation can be intuitively interpreted in terms of irreversible

compression of the crystal beside and beneath the indentations. This region is rich in

dislocations, partly amorphous and its extent is generally limited to some microns,

depending on dicing parameters, e.g., grit and advance speed of the blade [22]. Such

a highly defected region acts as a solid wall for the crystalline material between the

grooves and prevents it from full mechanical relaxation. Thus, the array of elastically

compressed regions behaves as an ”active layer”, imparting internal forces to the

whole crystal, thus bending the remaining crystal below them. It ultimately results

in a net and uniform curvature within the crystal without the usage of any external

device.

Commercially available pure Si wafers were diced to form plates using a high pre-

cision dicing saw (DISCOT M DAD3220), equipped with rotating diamond blades of

various width and grain size. Grooves were manufactured on the surface of the plates

along one direction, i.e., either x or y (Fig. 2a). All plates were 1 mm thick and their

orientation was (111). Fabrication parameters of all the samples are reported in Tab.

1 and an image of one of the samples is shown in Fig. 3. For every crystal, the cur-


vature induced by grooves was measured using an optical profilometer (VEECOT M

NT1100) with 1 µm lateral and 1 nm vertical resolution. The profilometer is equipped

with a stitching system that allows scanning over as wide an area as 10×10 cm2. In

order to account for the initial morphological non-planarity of the samples (wafers are

generally not perfectly flat), subtraction of profile before and after the indentations

was done. Moreover, since the profile of a surface with grooves is altered by their

presence, thus making the analysis more difficult, profilometric characterization was

carried out on the back face of each sample. As a result of the indentation process,

an ellipsoidal surface appeared, with the shortest radius of curvature perpendicular to

the grooves. A typical profilometric pattern of one of the samples is shown in Fig. 4.

Production and optical characterization of all samples have been carried out at Sen-

sor and Semiconductor Laboratory (Ferrara, Italy). All samples were tested through

Table 1 Fabrication parameters of all the samples

Code S24 S31 S71 S72 S81

Size (mm3) 9.8×9.8×1 12.2×12.2×1 25.5×25.5×1 25.5×36.6×1 54.2×30.6×1

Number of15 15 31 25 30

grooves

Direction[110] [110] [110] [211] [110]

of grooves

Pitch of650 780 790 1000 1000

grooves (µm)

Depth of500 500 500 400 400

grooves (µm)

Blade very hard hard hard hard hard

x-ray diffraction during a 6-day run at beamline ID15A of European Synchrotron

Radiation Facility (Grenoble, France). A highly monochromatic and quasi-parallel

beam was tuned to the desired energy, ranging from 150 to 700 keV, thanks to a two-

reflection Laue Si (111) unbent monochromator. Monochromaticity was of the order

of ∆E/E = 2x10−3. The sample holder was set far enough from the detector in order

to allow for sufficient separation of diffracted and transmitted beams even at highest

energy. The characterization of the samples was carried out by performing rocking

curves (RCs), i.e., by recording either the transmitted or diffracted beam intensity

while the crystal was rotated in the beam around the position where the Bragg con-

dition is satisfied. Diffraction and transmission RCs are recorded one after the other,

resulting in two complementary curves as a function of the beam incidence angle.

The FWHM of the RC is a direct measurement of the angular distribution of diffract-

ing planes (hereinafter referred to as angular spread), namely the bending angle of the

crystal. The RCs also exhibit diffraction efficiency of the sample under analysis. For

comparison with previously published measurements, efficiency has been defined as

in Ref. [23], namely the ratio of diffracted beam intensity over the transmitted one.

An advantage of this definition is a directly measurable physical quantity. Transmit-


Fig. 3 Side view of sample S71 with a series of indentations as taken by scanning electron microscope.

The black arrow indicates the pitch of the grooves

Fig. 4 Optical profilometry scanning of the surface without indentations of crystal S71 (a). False-color

representation of deformation is highlighted. Cross sections of the deformation pattern along x (b) and y

(c) directions as taken on the center of the sample with indications of the two main curvature radii

ted beam intensity was recorded by keeping the sample in diffraction condition and

shifting the detector in such a way to measure the beam intensity passing through the

crystal without undergoing diffraction.

All samples were analyzed by diffraction of their (111) planes, the pencil beam

entering the sample at different depths from the grooved surface (coordinate z). Two

different configurations were used, i.e., the beam was set quasi-parallel (hereinafter

referred to as parallel) or perpendicular to the grooves, its size being 50×50 µm2 and

100×50 µm2, respectively. A sketch of the two configurations is shown in Figs. 2b

and 2c.


Table 2 Main performance of all the samples under investigation

Code S24 S31 S71 S71 S72 S81

Photon energy150 150 - 500 150 - 700 150 - 500 150- 600 300

(keV)

Beam configurationPerp. to Perp. to Par. to Perp. to Perp. to Par. to

the grooves the grooves the grooves the grooves the grooves the grooves

Bending angle32.2 35.1 15.7 66.6 55.0 29.5

(arcsec)

Angular spread26.2 23.6 14.1 57.1 49.9 25.7

(arcsec)

Max. diffraction

efficiency at lowest 81.7% 69.1% 94.9% 71.1% 79.4% 86.4%

energy

Averaged diffraction


energy

Finally, experimental data have been analyzed and compared to theoretical expec-

tations through a custom-made software specifically designed for bent crystals and

inspired to the code in Ref. [24]. This latter was developed through Python program-

ming language, which takes advantage of a very large and comprehensive standard

library and is largely used by the scientific community.

3 Results and discussion

The main characteristics of each measured sample are summarized in Tab. 2. As an

example, we extensively describe the features of sample S71. The crystal was initially

measured at 150 keV with the beam penetrating the sample through its 25.5×1 mm2

surface at different depths from the grooved side, parallel and perpendicular to the

grooves. Bending angles of the sample, as measured by optical profilometry, averaged

15.7 and 66.6 arcsec along the [110] and [211] directions, respectively. Fig. 5 shows

both diffracted and transmitted RCs as normalized to transmitted beam intensity (so

that diffraction efficiency is readily displayed).

All RCs exhibited flat-topped and uniform shapes with FWHM of the order of

crystal bending, i.e., it averaged 14.1 and 57.1 arcsec for parallel and perpendicular

cases, respectively. This sample features significantly high efficiency when the beam

is parallel to the grooves, highlighting very homogeneous diffraction pattern with ef-

ficiency about 93.4% over the whole depth. This performance highlights that a bent

crystal can amply break the 50%-efficiency limit. With the beam perpendicular to

the grooves, diffraction efficiency is still a good performance though it varies over

the crystal depth, i.e., it is nearly 50% close to the grooved region, and raises up to

71% deeper into the crystal. Such features are better pointed out in Figs. 6a and 6b,

where diffraction efficiency is shown as a function of coordinates z and y (z and x),

for the parallel (perpendicular) case. Efficiency results constantly close to the unity

in the parallel case while it smoothly varies over the whole depth in the other con-

figuration. However, no dependence on coordinate y (or x) was recorded in any case.

The same dependence was recorded for angular spread for parallel and perpendicular


cases (Fig. 6c and Fig. 6d, respectively). It follows that the curvature is uniform along

coordinate y (or x), though its dependence on coordinate z shows different profile. In-

deed, as the distance from the grooved face increases, the angular distribution slightly

increases for the parallel case. Perpendicularly to the grooves, the variation of the an-

gular spread within the crystal is stronger, i.e., it increases across the groove depth

and decreases outside. This evidence can be ascribed to the fabrication process of

indentations. In fact, generation of mosaicity perpendicularly to the advance speed of

the blade is easier to form than longitudinally because of the stronger action exerted

by the blade on the side walls of the groove. This effect leads to an increase in an-

gular spread, and consequently in energy bandwidth, resulting in efficiency decrease

throughout the whole depth of the grooves. Indeed high efficiency is restored beneath

the grooves, meaning that the curvature of diffracting planes is homogeneous and its

structure is not significantly affected by mosaicity. Finally the sample was measured

at several energies, the beam entering the crystal far from the grooved region, parallel

and perpendicular to the grooves. RCs are shown here for parallel case (Fig. 7). The

sample features significant diffraction efficiency up to 700 keV, ranging from 92%

down to 29%. With the beam perpendicular to the grooves, efficiency keeps lower

than 60% above 200 keV. Next section compares experimental performance to theo-

retical expectations, showing that the decrease in efficiency with energy is completely

in agreement with the dynamical theory of diffraction [25].

4 Simulations

In order to deepen our understanding of diffraction properties of the samples, a simu-

lation code has been developed. The software describes diffraction in both curved and

mosaic crystals and generates the physical quantities of interest. Such quantities, typ-

ically used to qualify the diffraction properties of a crystal, are diffraction efficiency

(defined in section 2) and reflectivity. Reflectivity is defined as the ratio of diffracted

beam intensity over incident beam intensity. For any set of parameters such as crys-

talline material, set of reflection planes and thickness of curved or mosaic crystal,

the code computes reflectivity and diffraction efficiency as a function of photon en-

ergy and angular spread (or mosaicity). Expected performance is then compared to

experimental data.

For curved crystals, reflectivity is given by formula [19,23,25]

Rcurved = [1− e(−π2T0dhkl

ΩΛ20

)]e(−µT0cosθB

)(1)

where the first factor is the diffraction efficiency, and the latter is the attenuation

factor due to linear absorption µ throughout the crystal. T0 is the crystal thickness

traversed by radiation, dhkl the d-spacing of planes (hkl), and Ω the angular spread,

i.e., the bending angle of the crystal. θB is the Bragg angle and Λ0 the extinction

length as defined in Ref. [26] for the Laue symmetric case. In case of a mosaic crystal,

reflectivity is given by formula [2,23]

Rmosaic =1

2[1− e(−2W (∆θ)QT0)]e

(−µT0cosθB

)(2)


Fig. 5 RCs of crystal S71 with the beam parallel to the grooves at several distances from the indented face,

i.e., at (a) z = 0.4 mm, (b) z = 0.6 mm, (c) z = 0.8 mm; all the RCs were recorded at y = 13.9 mm. RCs of

the same crystal with the beam perpendicular to the grooves at (d) z = 0.15 mm, (e) z = 0.55 mm, (f) z =

0.85 mm; all the RCs were recorded at x = 15 mm. The filled red circles plot the intensity of the transmitted

beam, whereas the empty blue circles plot the intensity of the diffracted beam. RCs with rectangular and

homogenous shapes were achieved in all cases, with an energy passband of the order of crystal bending

(about 16 arcsec for the parallel case and 57 arcsec for the perpendicular case) . Efficiency is significantly

high in all cases and close to the unity in the parallel case

where the second factor is the diffraction efficiency, ∆θ the difference between the

angle of incidence and the Bragg angle and W(∆θ ) being the distribution function

of the crystalline orientation. This latter has been calculated at ∆θ=0, i.e., at Bragg

angle, as

W (0) = 2(ln2

π)

12

1

Ω(3)


Fig. 6 Diffraction efficiency of crystal S71 vs. coordinate z (mm) at several positions within the crystal

(coordinate y or x) for parallel (a) and perpendicular (b) cases. Same dependence of angular spread is

shown for parallel (c) and perpendicular (d) cases. No dependence on coordinate y (or x) was recorded

in any case. Efficiency near the polished side of the sample is close to the unity while it gently decreases

close to the grooved side in the parallel case and more strongly in the perpendicular case

In this case Ω is the mosaicity of the crystal, i.e., the FWHM of the angular distribu-

tion of the crystallites. Taking into account the kinematical theory approximation, Q

is given by [23,26]

Q =π2dhkl

Λ 20 cosθB

(4)

As in previous section, the results of the simulations will be shown here for sam-

ple S71. In Fig. 8, experimental diffraction efficiency vs. z is compared to the the-

oretical efficiency in case of a perfectly curved crystal and a mosaic crystal. These

latter were calculated taking into account the FWHM of the RCs, thus an experi-

mental uncertainty is included. Due to generation of mosaicity perpendicularly to the

grooves, measured efficiency varies over the crystal depth, being always lower than

the theoretical limit for a perfectly bent crystal, especially within the groove depth.

However, this performance keeps always higher or at worst equal to the theoretical

efficiency for a mosaic crystal, meaning that the indentations allow obtaining a ho-

mogeneous curvature with no significant damage of the crystal. In order to better

understand the behavior of diffraction response, we modeled the crystal as it were

made by two coexisting structures at any coordinate z, i.e., a mosaic crystal and a

perfectly curved crystal. Based on this model, diffraction efficiency was considered

as the superposition of the contributions of the two kinds of crystal. Here, C(z) is

the fraction of perfectly curved crystal-like behavior and [1-C(z)] the contribution of

mosaicity, such that

ε(z) =C(z)εc +[1−C(z)]εm (5)


Fig. 7 RCs of crystal S71 with the beam parallel to the grooves, measured at z = 0.8 mm and y = 13.9 mm.

Beam energy was set at 200 keV (a), 300 keV (b), 400 keV (c), 500 keV (d), 600 keV (e) and 700 keV

(f). The filled red circles plot the intensity of the transmitted beam, whereas the empty blue circles plot the

intensity of the diffracted beam. Efficiency falls off with photon energy according to the dynamical theory

of diffraction though a rectangular shape of the distribution is preserved

where ε(z) is the experimental diffraction efficiency obtained at a given distance from

the top of the crystal, εc the expected diffraction efficiency for a perfectly curved crys-

tal and εm the expected diffraction efficiency in case of a mosaic crystal. As a result,

for the perpendicular case, the fraction of mosaicity [1-C(z)] is close to the unity in

the region of the grooves, and vanishes outside. For the parallel configuration the mo-

saicity fraction keeps about 8% throughout the entire thickness of the sample. Indeed,

in the parallel case, experimental efficiency is constantly close to the theoretical limit

of a perfectly curved crystal.


Fig. 8 Experimental efficiency (blue circles) vs. coordinate z (mm) for crystal S71 with the 150 keV probe

beam perpendicular to the grooves. Red dashed and dotted line represent theoretical efficiencies in case of

a curved and of a mosaic crystal, respectively. An experimental uncertainty is included in both cases. Due

to generation of mosaicity, experimental efficiency varies within the crystal, being lower over the whole

groove depth and increasing outside. However, diffraction efficiency is always higher or at worst equal to

the theoretical contribution given by a mosaic crystal

Diffraction efficiency was studied vs. photon energy. Fig. 9 shows the response

of sample S71, measured with the beam parallel and perpendicular to the grooves.

In the parallel case (Fig. 9a), experimental efficiency is very close to its theoretical

limit over about 15 arcsec angular spread, namely the morphological curvature of the

sample. With the beam perpendicular (Fig. 9b) to the grooves, efficiency is slightly

lower than its theoretical limit but still higher than the theoretical efficiency for a

mosaic crystal.

A behavior similar to that of sample S71 has been seen by all the samples in Tab.

2, this being a representation of the high reproducibility of the method of indentations.

5 Conclusions

The indentation technique has proven to yield self-standing homogeneous and con-

trolled curvature in Si (111) crystals. The crystals have shown significantly high effi-

ciency and broad-band response when subject to x-ray diffraction. The morphological

curvature measured by optical profilometry is in good agreement with the curvature

of the crystalline planes as determined by x-ray diffraction. This feature means that

the energy bandwidth of bent crystals can be very well controlled. Diffraction ef-

ficiency turns out to be different when the beam is parallel or perpendicular to the

indentations and that is ascribed to generation of stronger mosaicity perpendicularly

to the grooves. Indented Si crystal has been shown to efficiently diffract up to 700

keV, proving that optics made of bent crystals can work on a wide energy range in

the hard x / soft gamma ray domain. However, at energy above 300 keV, crystals

with higher atomic number would be more suitable to yield a still higher reflectivity.


Fig. 9 Experimental and theoretical diffraction efficiencies vs. energy, for parallel (a) and perpendicular

(b) cases for crystal S71. RCs were carried out at 0.8 mm and 0.85 mm from the grooved face for parallel

and perpendicular cases, respectively. Theoretical efficiency (red dashed line) was calculated taking into

account the FWHM of the RCs, thus an experimental uncertainty is included. Measured diffraction effi-

ciency is represented by blue circles with their error bars. Red dotted line represents theoretical efficiency

for a mosaic crystal with mosaicity equal to the FWHM of the RCs

Possible candidates for extension of the method of indentations may be Ge or GaAs,

provided that base crystals of these materials possess a mosaicity level comparable to

that of Si.

Acknowledgements The authors gratefully thank N. Abrosimov, P. Bastie, T. Buslaps, G. Roudil and

J. Rousselle for assistance during data taking at ESRF. The authors are also grateful to ASI for partial

financial support through the Laue project.

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Noname manuscript No.(will be inserted by the editor)

Self-standing bent silicon crystals for very high efficiency

Laue lens

Bellucci Valerio · Camattari Riccardo · Guidi

Vincenzo · Neri Ilaria · Barriere Nicolas

Received: date / Accepted: date

Abstract Silicon mono-crystals have been bent thanks to a series of parallel super-

ficial indentations on one of the largest faces of the crystals. This technique relies

on irreversible compression of the crystal beneath and beside the indentations. This

latter causes deformation with no need for external device, resulting in a uniform self-

standing curvature within the crystal. Indented Si crystals have been characterized at

European Synchrotron Radiation Facility using a monochromatic beam ranging from

150 to 700 keV. Crystals exhibited very high diffraction efficiency over a broad range

of energy, peaking 95% at 150 keV. Measured angular spread of the diffracted beam

was always very close to the morphological curvature of the sample under investiga-

tion, proving that the energy passband of bent crystals can be controlled by simply

imparting a selected curvature to the sample. The method of superficial indentations

was found to offer high reproducibility and easy control of diffraction properties of

the crystals. Moreover the method is cheap and simple, being based on mass produc-

tion tools. A Laue lens made of crystals bent by superficial indentations can provide

high-efficiency concentration of hard x-ray photons, leading significant improvement

in many astrophysical applications.

Keywords Laue lens · Hard x-ray diffraction · bent crystals

1 Introduction

Observation of the sky in the hard x-ray domain would open up a window on a broad

variety of many powerful sources and violent events occurring in the Universe, such

Bellucci Valerio, Camattari Riccardo, Guidi Vincenzo, Neri Ilaria

Department of Physics, University of Ferrara, Via Saragat 1/c, 44122 Ferrara and

CNR - IDASC SENSOR Lab., Italy

Fax: +39 0532 974210

E-mail: [email protected]

Barriere Nicolas

Space Sciences Laboratory, 7 Gauss Way, University of California, Berkeley CA 94720-7450 USA


Fig. 1 X-ray diffraction in Laue geometry in case of an unbent (a) and of a bent crystal (b). Multiple

reflections in case (a) results in maximum 50% diffraction efficiency while in case (b) diffraction efficiency

can reach 100%

as active galactic nuclei, compact object in binary system, pulsar, nova, supernova

and more. However, to take full advantage of this potential, the next generation of

instruments working in this energy domain will have to achieve an improvement in

sensitivity by at least an order of magnitude with respect to existing telescopes [1].

Bragg diffraction in Laue geometry [2] is being proposed to efficiently concentrate

photons at these energies [3,4]. In this configuration, perfect crystals thicker than the

extinction length and mosaic crystals suffer a maximum diffraction efficiency of 50%

because of re-diffraction of the incident beam onto lattice planes. Conversely, curved

crystals prevent this effect due to continuous change of the incidence angle so that

only a single diffraction occurs onto curved crystalline planes (Fig. 1). Unlike for

mosaic crystals, which normally exhibit a Gaussian-like reflectivity passband, a bent

crystal offers a continuum of possible diffraction angles over a finite range, leading

to a rectangular-shape energy passband directly owing to its curvature [2,3].

Thanks to curved crystals it would be possible to build an efficient Laue lens by

concentrating the signal collected over a large area into a small focal spot on the de-

tector, thus significantly increasing the signal to background ratio, and consequently

leading to a sensitivity leap by more than an order of magnitude with respect to ex-

isting telescopes [5]. Such improvement is particularly awaited for the study of the

origin of the positrons annihilating in the Galactic center, visible through the e+ - e-

annihilation line at 511 keV. A study of the distribution of this emission line would

bring new clues concerning the still elusive sources of this antimatter. Despite a 511

keV emission has been observed for more than 30 years towards the Galactic center

[6], the origin of the positrons still remains a mystery. Stellar nucleosynthesis [7–9],

accreting compact objects [10–13], and even the annihilation of exotic dark-matter

particles [14] have all been suggested, thus a deeper investigation has to be done.

For fabrication of a bent crystal, several techniques have been worked out. Bend-

ing can be accomplished by mechanical means, i.e., by deforming a perfect single

crystal [15] through an external device. Mechanically bent crystals have been used

in synchrotrons for many years as high-efficiency monochromators. However, the us-

age of an external device leads to excessive weight, for satellite-borne experiments.

Thus, self-standing bent crystals are mandatory for practical implementation of a fo-

cusing telescope as a Laue lens. Such curved crystal can be produced by applying a

thermal gradient to a perfect single crystal [16] but, this method is energy consum-

ing and not adapted to a space-borne observatory as well. Crystals having curved


Fig. 2 Grooves were indented on the surface of a Si plate along one direction, either x or y (a). The probe

x-ray beam enters the sample parallel (b) or perpendicular (c) to the grooves

diffracting planes can also be obtained by concentration-gradient technique, i.e., by

growing a two-component crystal with graded composition along the growth axis

[16–19]. However, crystals bent by such a method are not easy to manufacture, this

making the technique hardly applicable for a Laue lens application, for which serial

production of crystals should be envisaged.

It was recently proven that superficial indentations made on a Si crystal is a

method for bending crystals [20]. This technique is based on plastic deformation of

the crystal induced by the grooves. As a result of deformation, a permanent curvature

is produced with no need for external device. Such method is cheap, simple and very

reproducible, thus compatible with mass production.

In this paper we present a systematic study on curved crystals bent by indentations

for the purpose of diffracting high-energy photons for astrophysical observations. In

particular, a Si crystal bent by indentation exhibited record diffraction efficiency of

nearly 95% at 150 keV.

2 Experimental

Indentations manufactured on the surface of a Si plate by a diamond saw is known

to deform the whole crystal, leading to a net and permanent curvature [21]. The ori-

gin of permanent deformation can be intuitively interpreted in terms of irreversible

compression of the crystal beside and beneath the indentations. This region is rich in

dislocations, partly amorphous and its extent is generally limited to some microns,

depending on dicing parameters, e.g., grit and advance speed of the blade [22]. Such

a highly defected region acts as a solid wall for the crystalline material between the

grooves and prevents it from full mechanical relaxation. Thus, the array of elastically

compressed regions behaves as an ”active layer”, imparting internal forces to the

whole crystal, thus bending the remaining crystal below them. It ultimately results

in a net and uniform curvature within the crystal without the usage of any external

device.

Commercially available pure Si wafers were diced to form plates using a high pre-

cision dicing saw (DISCOT M DAD3220), equipped with rotating diamond blades of

various width and grain size. Grooves were manufactured on the surface of the plates

along one direction, i.e., either x or y (Fig. 2a). All plates were 1 mm thick and their

orientation was (111). Fabrication parameters of all the samples are reported in Tab.

1 and an image of one of the samples is shown in Fig. 3. For every crystal, the cur-


vature induced by grooves was measured using an optical profilometer (VEECOT M

NT1100) with 1 µm lateral and 1 nm vertical resolution. The profilometer is equipped

with a stitching system that allows scanning over as wide an area as 10×10 cm2. In

order to account for the initial morphological non-planarity of the samples (wafers are

generally not perfectly flat), subtraction of profile before and after the indentations

was done. Moreover, since the profile of a surface with grooves is altered by their

presence, thus making the analysis more difficult, profilometric characterization was

carried out on the back face of each sample. As a result of the indentation process,

an ellipsoidal surface appeared, with the shortest radius of curvature perpendicular to

the grooves. A typical profilometric pattern of one of the samples is shown in Fig. 4.

Production and optical characterization of all samples have been carried out at Sen-

sor and Semiconductor Laboratory (Ferrara, Italy). All samples were tested through

Table 1 Fabrication parameters of all the samples

Code S24 S31 S71 S72 S81

Size (mm3) 9.8×9.8×1 12.2×12.2×1 25.5×25.5×1 25.5×36.6×1 54.2×30.6×1

Number of15 15 31 25 30

grooves

Direction[110] [110] [110] [211] [110]

of grooves

Pitch of650 780 790 1000 1000

grooves (µm)

Depth of500 500 500 400 400

grooves (µm)

Blade very hard hard hard hard hard

x-ray diffraction during a 6-day run at beamline ID15A of European Synchrotron

Radiation Facility (Grenoble, France). A highly monochromatic and quasi-parallel

beam was tuned to the desired energy, ranging from 150 to 700 keV, thanks to a two-

reflection Laue Si (111) unbent monochromator. Monochromaticity was of the order

of ∆E/E = 2x10−3. The sample holder was set far enough from the detector in order

to allow for sufficient separation of diffracted and transmitted beams even at highest

energy. The characterization of the samples was carried out by performing rocking

curves (RCs), i.e., by recording either the transmitted or diffracted beam intensity

while the crystal was rotated in the beam around the position where the Bragg con-

dition is satisfied. Diffraction and transmission RCs are recorded one after the other,

resulting in two complementary curves as a function of the beam incidence angle.

The FWHM of the RC is a direct measurement of the angular distribution of diffract-

ing planes (hereinafter referred to as angular spread), namely the bending angle of the

crystal. The RCs also exhibit diffraction efficiency of the sample under analysis. For

comparison with previously published measurements, efficiency has been defined as

in Ref. [23], namely the ratio of diffracted beam intensity over the transmitted one.

An advantage of this definition is a directly measurable physical quantity. Transmit-


Fig. 3 Side view of sample S71 with a series of indentations as taken by scanning electron microscope.

The black arrow indicates the pitch of the grooves

Fig. 4 Optical profilometry scanning of the surface without indentations of crystal S71 (a). False-color

representation of deformation is highlighted. Cross sections of the deformation pattern along x (b) and y

(c) directions as taken on the center of the sample with indications of the two main curvature radii

ted beam intensity was recorded by keeping the sample in diffraction condition and

shifting the detector in such a way to measure the beam intensity passing through the

crystal without undergoing diffraction.

All samples were analyzed by diffraction of their (111) planes, the pencil beam

entering the sample at different depths from the grooved surface (coordinate z). Two

different configurations were used, i.e., the beam was set quasi-parallel (hereinafter

referred to as parallel) or perpendicular to the grooves, its size being 50×50 µm2 and

100×50 µm2, respectively. A sketch of the two configurations is shown in Figs. 2b

and 2c.


Table 2 Main performance of all the samples under investigation

Code S24 S31 S71 S71 S72 S81

Photon energy150 150 - 500 150 - 700 150 - 500 150- 600 300

(keV)

Beam configurationPerp. to Perp. to Par. to Perp. to Perp. to Par. to

the grooves the grooves the grooves the grooves the grooves the grooves

Bending angle32.2 35.1 15.7 66.6 55.0 29.5

(arcsec)

Angular spread26.2 23.6 14.1 57.1 49.9 25.7

(arcsec)

Max. diffraction


energy

Averaged diffraction


energy

Finally, experimental data have been analyzed and compared to theoretical expec-

tations through a custom-made software specifically designed for bent crystals and

inspired to the code in Ref. [24]. This latter was developed through Python program-

ming language, which takes advantage of a very large and comprehensive standard

library and is largely used by the scientific community.

3 Results and discussion

The main characteristics of each measured sample are summarized in Tab. 2. As an

example, we extensively describe the features of sample S71. The crystal was initially

measured at 150 keV with the beam penetrating the sample through its 25.5×1 mm2

surface at different depths from the grooved side, parallel and perpendicular to the

grooves. Bending angles of the sample, as measured by optical profilometry, averaged

15.7 and 66.6 arcsec along the [110] and [211] directions, respectively. Fig. 5 shows

both diffracted and transmitted RCs as normalized to transmitted beam intensity (so

that diffraction efficiency is readily displayed).

All RCs exhibited flat-topped and uniform shapes with FWHM of the order of

crystal bending, i.e., it averaged 14.1 and 57.1 arcsec for parallel and perpendicular

cases, respectively. This sample features significantly high efficiency when the beam

is parallel to the grooves, highlighting very homogeneous diffraction pattern with ef-

ficiency about 93.4% over the whole depth. This performance highlights that a bent

crystal can amply break the 50%-efficiency limit. With the beam perpendicular to

the grooves, diffraction efficiency is still a good performance though it varies over

the crystal depth, i.e., it is nearly 50% close to the grooved region, and raises up to

71% deeper into the crystal. Such features are better pointed out in Figs. 6a and 6b,

where diffraction efficiency is shown as a function of coordinates z and y (z and x),

for the parallel (perpendicular) case. Efficiency results constantly close to the unity

in the parallel case while it smoothly varies over the whole depth in the other con-

figuration. However, no dependence on coordinate y (or x) was recorded in any case.

The same dependence was recorded for angular spread for parallel and perpendicular


cases (Fig. 6c and Fig. 6d, respectively). It follows that the curvature is uniform along

coordinate y (or x), though its dependence on coordinate z shows different profile. In-

deed, as the distance from the grooved face increases, the angular distribution slightly

increases for the parallel case. Perpendicularly to the grooves, the variation of the an-

gular spread within the crystal is stronger, i.e., it increases across the groove depth

and decreases outside. This evidence can be ascribed to the fabrication process of

indentations. In fact, generation of mosaicity perpendicularly to the advance speed of

the blade is easier to form than longitudinally because of the stronger action exerted

by the blade on the side walls of the groove. This effect leads to an increase in an-

gular spread, and consequently in energy bandwidth, resulting in efficiency decrease

throughout the whole depth of the grooves. Indeed high efficiency is restored beneath

the grooves, meaning that the curvature of diffracting planes is homogeneous and its

structure is not significantly affected by mosaicity. Finally the sample was measured

at several energies, the beam entering the crystal far from the grooved region, parallel

and perpendicular to the grooves. RCs are shown here for parallel case (Fig. 7). The

sample features significant diffraction efficiency up to 700 keV, ranging from 92%

down to 29%. With the beam perpendicular to the grooves, efficiency keeps lower

than 60% above 200 keV. Next section compares experimental performance to theo-

retical expectations, showing that the decrease in efficiency with energy is completely

in agreement with the dynamical theory of diffraction [25].

4 Simulations

In order to deepen our understanding of diffraction properties of the samples, a simu-

lation code has been developed. The software describes diffraction in both curved and

mosaic crystals and generates the physical quantities of interest. Such quantities, typ-

ically used to qualify the diffraction properties of a crystal, are diffraction efficiency

(defined in section 2) and reflectivity. Reflectivity is defined as the ratio of diffracted

beam intensity over incident beam intensity. For any set of parameters such as crys-

talline material, set of reflection planes and thickness of curved or mosaic crystal,

the code computes reflectivity and diffraction efficiency as a function of photon en-

ergy and angular spread (or mosaicity). Expected performance is then compared to

experimental data.

For curved crystals, reflectivity is given by formula [19,23,25]

Rcurved = [1− e(−π2T0dhkl

ΩΛ20

)]e(−µT0cosθB

)(1)

where the first factor is the diffraction efficiency, and the latter is the attenuation

factor due to linear absorption µ throughout the crystal. T0 is the crystal thickness

traversed by radiation, dhkl the d-spacing of planes (hkl), and Ω the angular spread,

i.e., the bending angle of the crystal. θB is the Bragg angle and Λ0 the extinction

length as defined in Ref. [26] for the Laue symmetric case. In case of a mosaic crystal,

reflectivity is given by formula [2,23]

Rmosaic =1

2[1− e(−2W (∆θ)QT0)]e

(−µT0cosθB

)(2)


Fig. 5 RCs of crystal S71 with the beam parallel to the grooves at several distances from the indented face,

i.e., at (a) z = 0.4 mm, (b) z = 0.6 mm, (c) z = 0.8 mm; all the RCs were recorded at y = 13.9 mm. RCs of

the same crystal with the beam perpendicular to the grooves at (d) z = 0.15 mm, (e) z = 0.55 mm, (f) z =

0.85 mm; all the RCs were recorded at x = 15 mm. The filled red circles plot the intensity of the transmitted

beam, whereas the empty blue circles plot the intensity of the diffracted beam. RCs with rectangular and

homogenous shapes were achieved in all cases, with an energy passband of the order of crystal bending

(about 16 arcsec for the parallel case and 57 arcsec for the perpendicular case) . Efficiency is significantly

high in all cases and close to the unity in the parallel case

where the second factor is the diffraction efficiency, ∆θ the difference between the

angle of incidence and the Bragg angle and W(∆θ ) being the distribution function

of the crystalline orientation. This latter has been calculated at ∆θ=0, i.e., at Bragg

angle, as

W (0) = 2(ln2

π)

12

1

Ω(3)


Fig. 6 Diffraction efficiency of crystal S71 vs. coordinate z (mm) at several positions within the crystal

(coordinate y or x) for parallel (a) and perpendicular (b) cases. Same dependence of angular spread is

shown for parallel (c) and perpendicular (d) cases. No dependence on coordinate y (or x) was recorded

in any case. Efficiency near the polished side of the sample is close to the unity while it gently decreases

close to the grooved side in the parallel case and more strongly in the perpendicular case

In this case Ω is the mosaicity of the crystal, i.e., the FWHM of the angular distribu-

tion of the crystallites. Taking into account the kinematical theory approximation, Q

is given by [23,26]

Q =π2dhkl

Λ 20 cosθB

(4)

As in previous section, the results of the simulations will be shown here for sam-

ple S71. In Fig. 8, experimental diffraction efficiency vs. z is compared to the the-

oretical efficiency in case of a perfectly curved crystal and a mosaic crystal. These

latter were calculated taking into account the FWHM of the RCs, thus an experi-

mental uncertainty is included. Due to generation of mosaicity perpendicularly to the

grooves, measured efficiency varies over the crystal depth, being always lower than

the theoretical limit for a perfectly bent crystal, especially within the groove depth.

However, this performance keeps always higher or at worst equal to the theoretical

efficiency for a mosaic crystal, meaning that the indentations allow obtaining a ho-

mogeneous curvature with no significant damage of the crystal. In order to better

understand the behavior of diffraction response, we modeled the crystal as it were

made by two coexisting structures at any coordinate z, i.e., a mosaic crystal and a

perfectly curved crystal. Based on this model, diffraction efficiency was considered

as the superposition of the contributions of the two kinds of crystal. Here, C(z) is

the fraction of perfectly curved crystal-like behavior and [1-C(z)] the contribution of

mosaicity, such that

ε(z) =C(z)εc +[1−C(z)]εm (5)


Fig. 7 RCs of crystal S71 with the beam parallel to the grooves, measured at z = 0.8 mm and y = 13.9 mm.

Beam energy was set at 200 keV (a), 300 keV (b), 400 keV (c), 500 keV (d), 600 keV (e) and 700 keV

(f). The filled red circles plot the intensity of the transmitted beam, whereas the empty blue circles plot the

intensity of the diffracted beam. Efficiency falls off with photon energy according to the dynamical theory

of diffraction though a rectangular shape of the distribution is preserved

where ε(z) is the experimental diffraction efficiency obtained at a given distance from

the top of the crystal, εc the expected diffraction efficiency for a perfectly curved crys-

tal and εm the expected diffraction efficiency in case of a mosaic crystal. As a result,

for the perpendicular case, the fraction of mosaicity [1-C(z)] is close to the unity in

the region of the grooves, and vanishes outside. For the parallel configuration the mo-

saicity fraction keeps about 8% throughout the entire thickness of the sample. Indeed,

in the parallel case, experimental efficiency is constantly close to the theoretical limit

of a perfectly curved crystal.


Fig. 8 Experimental efficiency (blue circles) vs. coordinate z (mm) for crystal S71 with the 150 keV probe

beam perpendicular to the grooves. Red dashed and dotted line represent theoretical efficiencies in case of

a curved and of a mosaic crystal, respectively. An experimental uncertainty is included in both cases. Due

to generation of mosaicity, experimental efficiency varies within the crystal, being lower over the whole

groove depth and increasing outside. However, diffraction efficiency is always higher or at worst equal to

the theoretical contribution given by a mosaic crystal

Diffraction efficiency was studied vs. photon energy. Fig. 9 shows the response

of sample S71, measured with the beam parallel and perpendicular to the grooves.

In the parallel case (Fig. 9a), experimental efficiency is very close to its theoretical

limit over about 15 arcsec angular spread, namely the morphological curvature of the

sample. With the beam perpendicular (Fig. 9b) to the grooves, efficiency is slightly

lower than its theoretical limit but still higher than the theoretical efficiency for a

mosaic crystal.

A behavior similar to that of sample S71 has been seen by all the samples in Tab.

2, this being a representation of the high reproducibility of the method of indentations.

5 Conclusions

The indentation technique has proven to yield self-standing homogeneous and con-

trolled curvature in Si (111) crystals. The crystals have shown significantly high effi-

ciency and broad-band response when subject to x-ray diffraction. The morphological

curvature measured by optical profilometry is in good agreement with the curvature

of the crystalline planes as determined by x-ray diffraction. This feature means that

the energy bandwidth of bent crystals can be very well controlled. Diffraction ef-

ficiency turns out to be different when the beam is parallel or perpendicular to the

indentations and that is ascribed to generation of stronger mosaicity perpendicularly

to the grooves. Indented Si crystal has been shown to efficiently diffract up to 700

keV, proving that optics made of bent crystals can work on a wide energy range in

the hard x / soft gamma ray domain. However, at energy above 300 keV, crystals

with higher atomic number would be more suitable to yield a still higher reflectivity.


Fig. 9 Experimental and theoretical diffraction efficiencies vs. energy, for parallel (a) and perpendicular

(b) cases for crystal S71. RCs were carried out at 0.8 mm and 0.85 mm from the grooved face for parallel

and perpendicular cases, respectively. Theoretical efficiency (red dashed line) was calculated taking into

account the FWHM of the RCs, thus an experimental uncertainty is included. Measured diffraction effi-

ciency is represented by blue circles with their error bars. Red dotted line represents theoretical efficiency

for a mosaic crystal with mosaicity equal to the FWHM of the RCs

Possible candidates for extension of the method of indentations may be Ge or GaAs,

provided that base crystals of these materials possess a mosaicity level comparable to

that of Si.

Acknowledgements The authors gratefully thank N. Abrosimov, P. Bastie, T. Buslaps, G. Roudil and

J. Rousselle for assistance during data taking at ESRF. The authors are also grateful to ASI for partial

financial support through the Laue project.

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