Quasi-mosaic crystals for high-resolution focusing of hardx-rays through a Laue lens

Camattari R.a, Bellucci V.a, Guidi V. a and Neri I.a

aDepartment of Physics, University of Ferrara, Via Saragat 1/c, 44122 Ferrara and CNR -IDASC SENSOR Lab., Italy

ABSTRACT

Quasi-mosaicity is an effect of secondary bending within a crystal driven by crystalline anisotropy. This effectcan be used to fabricate a series of curved crystals for the realization of a Laue lens. We highlight that crystalsbent by quasi-mosaic effect envisage very high resolution focusing with respect to mosaic crystals. Under sameconditions for energy passband and flux of incident photons, a Laue lens based on quasi-mosaic crystals wouldincrease the signal-to-noise ratio by more than one order of magnitude as for the same lens with mosaic crystals.A proposal for the realization of a self-standing quasi-mosaic crystal is given. Here permanent bending of acrystal is achieved as a result of superficial indentations.

Keywords: curved crystals, Laue lens, x-ray diffraction

1. INTRODUCTION

A Laue lens is an optical component to focus hard x and gamma photons through Bragg diffraction in Lauegeometry within a properly arranged array of crystals disposed as concentric rings with radii spanning over acertain range.1, 2 In astrophysics a Laue lens would allow high-sensitivity observations of many cosmic phe-nomena on a satellite-borne focusing telescope.3 Both hard x-/soft gamma-ray continuum spectra and narrowlines features can be deeply explored with a Laue lens. Important examples are the 511 keV electron-positronannihilation line and the 847 keV produced by the decay chain of the radionuclide 56Co.

High-resolution focusing of diffracted photons within an energy range is an important requirement for allthe applications above. With the aim of wide-passband focusing, a typical component under investigation bythe scientific community is a mosaic crystal, i.e., an aggregation of crystallites whose angular distribution is aGaussian, spread about a nominal direction.4 However, this kind of crystals suffers 50%-maximum in reflectivityat zero absorption (diffraction efficiency) and the spatial resolution of diffracted photons is limited because ofthe so-called mosaic defocusing effect.5, 6 Curved crystals can be used to circumvent these limitations. In fact,curved crystals bent by superficial indentations were recently shown to exhibit a uniform angular distributionwith a total passband proportional to the curvature and a diffraction efficiency close to 90% at 150 keV.7, 8

In this paper we propose to use quasi-mosaicity for crystal bending, i.e., an intriguing effect of anisotropyin crystal deformation. In this configuration, bending results in a primary curvature generating a secondarycurvature within the crystal due to quasi-mosaic effect. Usage of quasi-mosaic (QM ) crystals allows positioningof the crystals in a Laue lens in the same way as for mosaic crystals, i.e., with the diffracting planes perpendicularto the major faces of the crystal (Fig. 1). However, it will be shown that a signal-to-noise ratio more than oneorder of magnitude larger can be attained for QM crystals with respect to the case of mosaic crystals. For aLaue lens made by crystals with diffracting planes perpendicular to the major face of the crystal, focusing canbe fully provided by bending the crystals to a primary curvature equal to that of the whole lens. But, even forsuch a curved crystal, if the diffracting planes were perfectly flat, the reflectivity of the whole lens would be thesame as for an unbent mono-crystal, i.e., a relatively low figure. Indeed, by using QM crystals, it is possibleto encompass the focusing action due to primary curvature with the high reflectivity of bent diffracting planesbuilt up by quasi-mosaicity. Here we also show that since the size of the focal spot of the photons diffractedby a crystal can be controlled by secondary curvature, QM crystals allow focusing with higher resolution withrespect to mosaic crystals.

Further author information: (Send correspondence to Camattari Riccardo)Camattari Riccardo: E-mail: camattari@fe.infn.it, Telephone: +390532974221

Optics for EUV, X-Ray, and Gamma-Ray Astronomy V, edited by Stephen L. O'Dell, Giovanni Pareschi, Proc. of SPIE Vol. 8147, 81471G · © 2011 SPIE · CCC code: 0277-786X/11/$18 · doi: 10.1117/12.895311

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Figure 1. The primary curvature of a properly oriented crystal leads to a secondary curvature owing to quasi-mosaicity.For a Laue lens, quasi-mosaic curvature of (111) lattice planes resulting from primary bending of (112) crystal plates isproposed for diffraction. In this configuration the (111) diffracting planes are perpendicular to the main surface of theplate, thus positioning of the crystals in the lens would be the same as for mosaic crystals. ΩP and ΩQM are primary andsecondary bending angles, respectively.

2. QUASI-MOSAIC CRYSTALS

Quasi-mosaicity is an anisotropic effect that manifests itself when a properly oriented crystal plate is bent alonga given direction, i.e., quasi-mosaicity depends on the crystallographic direction of the plate under bending.Indeed, a primary curvature imparted to a crystal results in a secondary (quasi-mosaic) curvature of a differentlying of planes due to quasi-mosaic effect (Fig. 1). Historically, quasi-mosaicity was discovered by Sumbaev andpublished in seminal work.9 More recently, this phenomenon was introduced by Ivanov10 to bend Si crystals forsteering high-energy particles via coherent effects in crystals, viz., planar channeling11 and volume reflection.12

Quasi-mosaicity is fully understood in the framework of the theory of linear elasticity. Quasi-mosaicitymanifests itself only along selected crystallographic directions. In particular for a diamond-like lattice, primarydeformation of (112) planes results in quasi-mosaic deformation of (111) planes (Fig. 1). For a Laue lens madeby Si or Ge plates, (111) planes are normally selected because of their high reflectivity. We propose to bend(112) Si or Ge plates as a primary curvature, which results in secondary curvature of (111) diffracting planesaligned with the radial direction of the lens. Thus, QM crystals would be accommodated as tiles on the surfaceof the Laue lens in the same way as mosaic crystals are proposed for usage.3 From the theory of elasticity onecan calculate the ratio between the quasi-mosaic radius of curvature, RQM , and the primary one, RP , this latterbeing considered along the radial direction of the lens. For the case of Ge and previously chosen orientation, itholds

RQM

RP=

S12 + S22

S41 + S42= 2.4 (1)

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Figure 2. Optical profilometry scanning (Veeco (Wyko) - Optical Profiler) of back face (without grooves) of a quasi-mosaiccrystal (a). False-color representation of deformation is highlighted. Cross sections of the deformation pattern along y= [110] (b) and x = [111] (c) directions as taken on the center of the sample. Primary curvature due to indentations isuniform and permanent within the sample

A promising technique for the realization of self-standing QM crystals could be the indentation of a mesh ofgrooves of one of the surfaces of a crystal. This method is based on irreversible compression of the crystal beneathand beside the grooves. This region, being rich in dislocations and partly amorphous, prevents the crystallinematerial between the grooves from full mechanical relaxation. As a result of such a deformation, the remainingcrystal below the grooves is bent to a net and uniform curvature without the usage of any external device. As anexample, a typical profilometric pattern of a QM crystal bent to a primary curvature by indentations is shownin Fig. 2 and an image of a QM crystal is in Fig. 3.

The technique of superficial indentations was proven to be an efficient method for bending crystals, i.e., aSi crystal bent by indentations exhibited record diffraction efficiency of nearly 95% at 150 keV. Such method iseasy and reproducible, simple and cheap, thus compatible with mass production.

3. DISCUSSION AND SIMULATIONS

Here a performance comparison between a mosaic vs. a QM crystal is provided, under optimal condition forpractical implementation in a Laue lens with realistic data.

Let us consider a mosaic crystal with the shape of a square tile of size LM and with a certain mosaicity ΩM .If we assume a parallel incoming beam impinging onto the crystal, the spatial distribution of the photons that arediffracted on the focal plane is given by the convolution of the tile size of the crystal with the normal distributionof the crystallites.5 Thus, the square image of each mosaic crystal suffers from blurring due to mosaicity, thecontribution of all the crystals of a whole ring being the overlap of the blurred images of all the crystal tiles.

The final response is dominated by one of the two effects in the convolution, depending on the proportionbetween mosaicity and the ratio between the tile size with the focusing length. If mosaicity is negligible, the focal

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Figure 3. A picture of quasi-mosaic sample with a mesh of grooves onto its surface

spot only depends on the tile size, if instead it dominates, the intensity of the focal spot is Gaussian-distributed.In practical cases, as those of,5 mosaicity is not negligible though it is not the major effect.

With a QM crystal with tile size LQM , it is possible to focalize the photons along radial direction by simplybending the crystal to a primary curvature equal to that of the whole lens. In this case the focal spot is arectangle with base LQM in the direction where focusing does not occur and height being determined by thequasi-mosaic curvature within the crystal. Under same conditions for the incoming beam and by assuming thatthe passband for photon energy for a mosaic crystal equals that for a QM crystal, it comes clear that the size ofthe spot on the focal plane is certainly smaller for a QM crystal than for a mosaic one.

To deepen this argument, simulations of focusing capability of QM vs. mosaic crystals have been done.Firstly, for both kinds of crystals we computed the reflectivity, i.e., the ratio between diffracted beam intensityand incident beam intensity. For mosaic case, reflectivity is given by formula4

rM =1

2[1− e−2W (Δθ)QT0 ]e

−μT0cosθB (2)

where T0 is the crystal thickness traversed by radiation, Δθ is the difference between the angle of incidence andBragg angle θB, μ the linear absorption coefficient within the crystal and W (Δθ) the distribution function ofcrystallite orientations, namely

W (Δθ) = 2(ln2

π)

12

1

ΩMe−ln2( Δθ

ΩM/2)2

(3)

where ΩM is the mosaicity of the crystal, i.e., the angular distribution of the crystallites. Finally, consideringthe kinematical theory approximation13 , Q is given by

Q =π2dhklΛ20cosθB

(4)

where dhkl is the d -spacing of planes (hkl) and Λ0 the extinction length as defined in14 for the Laue symmetriccase.

For a QM crystal, reflectivity is given by formula13

rQM = [1− e(−π2T0dhkl

ΩQMΛ20

)]e

−μT0cosθB (5)

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Table 1. Main features of the simulationmosaic case quasi-mosaic case

Material Cu GeAtomic number 29 32ΔE (keV) 1.26 1.26Number of rings 1 1Tile size (mm) 15 9.6Tile thickness (mm) 1.85 0.6Mosaicity (arcsec) 77.5 -

where, in this case, ΩQM represents the bending angle of the curved diffracting planes. With QM crystals, theselatter are (111) planes, thus ΩQM can be calculated as a function of primary curvature through Eq. (1).

For the simulation, we considered the crystal tiles to be disposed as concentric rings onto the spherical capof a Laue lens with a certain focal distance f. The energy passband of a full ring is given by formula15 , whichholds for both mosaic and QM configurations,

ΔE ≈ 2ΩEf

r(6)

r being the radius of the ring and E the mean energy diffracted by a ring, depending on crystal reflection planesas given by formula15

E =hc

2dhkl sin(12 tan

−1( rf ))≈ 12.4

f

dhklr(7)

where the approximation of small angles has been considered in last two formulae.

In order to provide a numerical comparison, an incoming photon beam with uniform energy distributioncentered at E = 100 keV and with energy spread ΔE has been considered to impinge parallel to the axis of thelens, this latter taken with a focal distance of 20 m and made of a single ring. Cu and Ge have been chosenas materials for mosaic and QM cases, respectively. The former is normally proposed as a mosaic crystal todiffract photons at those energies, the latter, having an atomic number comparable to that of Cu and beingcommercially available as high perfection mono-crystals. The choice of a single ring has been taken for a moredirect comparison between the two configurations.

To compare QM and mosaic crystals, the energy passband and flux of incident photons have been chosen tobe the same for the two cases. According to the requirements given by the scientific community about mosaicityand crystal tile size for the mosaic case, e.g.,3 we fixed LM to be 1.5 cm. Thus, the optimal mosaicity ΩM ,at those energy and crystal dimension, turned out to be 77.5 arcsec. Therefore, these parameters determinean energy passband ΔE = 1.26 keV. Finally, a crystal thickness of T0 = 1.85 mm has been chosen in order tooptimize the reflectivity rM (Eq.2) throughout the whole ring of the lens.

For the QM case, the primary radius of curvature of the crystal must be twice the focal length of the lens, i.e.,40 m. As the energy passband is given by the convolution of primary and quasi-mosaic curvature, ΔE strictlydepends on LQM . In order to diffract photons over the same range as for the mosaic crystals, LQM turned out tobe 0.96 cm. T0 is a free parameter that determines both focusing capability and reflectivity of the lens. Indeed,large values of T0 lead to higher reflectivity and lower spatial resolution on the focal plane, while smaller valuesof T0 smoothly decrease reflectivity but significantly increase the focusing capability. An optimum value of T0

is determined by perfect matching of the focal spot with the detector size. In this simulation, we chose T0 = 0.6mm.

Main features used for simulation are reported in Tab. 3 and results are shown in Fig. 4. Here a cross-sectionof the photon distribution in the focal plane, in the cases of mosaic and QM crystals is shown. As visible, undersame conditions of lens geometry, ΔE and flux of photons hitting the crystals, the focal spot for the QM caseis significantly smaller and a higher signal intensity can be attained. This latter fact means that focusing withhigher resolution can be accomplished if the lens is formed by QM crystals.

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Figure 4. Cross-section of photon distribution in the focal plane for the cases of mosaic and quasi-mosaic crystals. For themosaic case the crosses and the continuum line represent Monte Carlo simulation and analytical modelling, respectively.For quasi-mosaic crystal the circles and the dashed line represent Monte Carlo simulation and analytical modelling,respectively. Total counts are normalized to mosaic case. FWHM turns out to be 17.2 mm for mosaic case while 0.4 mmfor quasi-mosaic one.

To compare the two focal spots, we resorted to the signal-to-noise ratio as follows

StN =S

N∝ S

Vdet∝ S

rfocus2(8)

where S is the signal count within a specified encircled area with radius rfocus, N is the background, Vdet is theuseful volume of the detector and rfocus is half of the FWHM of photon distribution in the focal plane. Thebackground has been taken proportional to the volume of detector, this latter facing the incoming x-rays with asurface as large as the encircled area. Due to different focalization properties, the obtained FWHMs of photondistribution in the focal plane for QM and mosaic cases is significantly different.

As a result of simulation, the quotient between the signal-to-noise ratios for the cases of QM and mosaiccrystals turned out to be

StNQM

StNM> 10 (9)

4. CONCLUSIONS

The effect of quasi-mosaicity in Ge crystals can be exploited in efficient focusing of high-energy photons in aLaue lens. The size of the focal spot determined by a Laue lens made by QM crystals is significantly smallerthan for mosaic crystals under same experimental conditions. Thereby, the focusing capability of a Laue lens can

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be largely improved through QM crystals. The choice of 100 keV as nominal energy made photoelectric effectto dominate over Compton scattering in interaction with the detector. At larger energies, where Compton effecttakes over, the method can still be pursued, provided that the detector is an imager, i.e., it identifies the firstCompton interaction.

Indentations manufactured onto one of the surfaces of a crystal can be used as an efficient method forrealization of self-standing QM crystals. The primary curvature, resulting in the quasi-mosaic curvature, due toindentations would be permanent and uniform with no need for external bending devices.

5. ACKNOWLEDGMENTS

This work has been partially supported by the LAUE project of ASI.

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