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Journal of
Applied
Crystallography
Calculation of diffraction efficiency for curved
crystals with arbitrary curvature radius
ISSN 0021-8898
Valerio Bellucci, Vincenzo Guidi,* Riccardo Camattari and Ilaria Neri
Received 24 October 2011
Accepted 2 January 2013
# 2013 International Union of Crystallography
Printed in Singapore – all rights reserved
Department of Physics, University of Ferrara, Via Saragat 1/c, 44122 Ferrara, and CNR-IDASC
SENSOR Laboratory, Brescia, Italy. Correspondence e-mail: [email protected]
A model is proposed to calculate the diffraction efficiency of X-rays in Laue
geometry for curved crystals with an arbitrary value of the curvature radius. The
model generalizes the results based on the dynamical theory of diffraction,
which are valid only for crystals with a radius of curvature lower than the critical
curvature. The model is proposed for any kind of crystal, and its efficiency tends
to one-half in the limit of a thick flat crystal. On the basis of this model, it was
possible to reconsider the results of recently observed diffraction efficiency for
curved crystals. Finally, the model sets an upper limit for diffraction efficiency of
low-curvature curved crystals, this latter being useful in applications such as the
construction of a hard X-ray Laue lens.
1. Introduction
Crystals with curved diffraction planes (CDPs) have recently
become popular in X-ray optics because they allow easy
manipulation of the trajectories of high-energy photons with
efficiency near unity in a broad energy range. A monocrystal
with zero-curvature planes, hereinafter referred to as a ‘flat
crystal’, can diffract photons just within a very narrow energy
range, and its reflectivity is physically limited to 50% at most
(Bellucci, Camattari, Guidi, Neri & Barrière, 2011; Zachariasen, 1945). In fact, a photon has the same probability of
undergoing an even or odd number of diffractions traversing
the crystal thickness. Mosaic crystals, i.e. an ensemble of
microscopic flat crystals slightly misaligned with respect to one
another, can overcome the drawback regarding the energy
range but the limitation to 50% in reflectivity still holds. In
addition, they suffer from poor reproducibility in their fabrication. A method to circumvent these drawbacks is the use of
CDP crystals, whose reflectivity can be close to 100%. In this
case, the continuous curvature of lattice planes changes the
Bragg condition along the crystal thickness, so that a diffracted
photon has a low probability of undergoing re-diffraction
inside the crystal (Fig. 1). The energy passband of the photons
diffracted by these crystals is orders of magnitude broader
Figure 1
X-ray diffraction in Laue geometry in the case of an unbent (a) and of a
bent crystal (b). Multiple reflections in case (a) result in a maximum
diffraction efficiency of 50%, while in case (b) the diffraction efficiency
can be close to 100%.
J. Appl. Cryst. (2013). 46, 415–420
than that for a flat crystal, featuring a uniform transfer function provided that the crystal curvature is homogeneous. The
implementation of CDP crystals in particular geometries can
even result in a focusing effect by the same crystal, making
possible the construction of instruments with very high resolution and sensitivity (Guidi et al., 2011). An interesting
application of crystals with CDPs is the construction of hard
X-ray lenses for astrophysics and nuclear medicine (Frontera
& Ballmoos, 2010).
Up to now, the impossibility of efficiently focusing hard
X-rays left the observation of the sky in this energy range to
direct-view instruments, featuring low sensitivity and modest
angular resolution. In fact, only the spectra of a few of the
strongest sources are known above 70 keV (Frontera et al.,
2005). Focusing optics with this purpose have already been
realized with mosaic crystals, though some limitations were
observed (von Halloin et al., 2004). The introduction of CDP
crystals for the construction of high-efficiency focusing
instruments for photons with energy up to 1 MeV would allow
mapping of the distribution of antimatter and dark matter in
our galaxy. High-efficiency instrumentation would also allow a
detailed analysis of type 1A supernovae, as well as emissions
from compact objects emitting X-rays. Nuclear medicine
would also benefit from this technology. Imaging instruments
in nuclear medicine are based on the detection of hard X-ray
photons emitted by tracing radionuclides injected into the
patient (Roa et al., 2005). Today’s instruments use direct-view
gamma cameras for this purpose, and complex algorithms to
reconstruct the image of the patient’s body, reaching a resolution of a few millimetres at best (Adler et al., 2003). Highefficiency focalization instruments would improve the resolution and sensitivity of these analyses, allowing detection of
physiological processes with submillimetre precision with no
need for reconstruction algorithms, and thereby reducing the
dose of radionuclide that needs to be injected into the patient.
doi:10.1107/S0021889813000162
415
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Previous work (Bellucci, Camattari, Guidi, Neri & Barrière,
2011; Barrière et al., 2010) has demonstrated the possibility of
producing good CDP crystals in a reproducible and controlled
manner compatible with mass production. They also showed
high reflectivity and a broad passband. It is the purpose of this
article to provide a model for calculation of the ratio of
diffracted to transmitted intensities in crystals with CDPs
under general conditions and to compare this model with the
experimental results obtained by Bellucci, Camattari, Guidi,
Neri & Barrière (2011) and Barrière et al. (2010).
2. General background
The ratio of diffracted to incident intensities is called reflectivity, and in crystals with CDPs it strongly depends on their
curvature. The reflectivity is composed of two contributions. In
a non-absorbing crystal, the reflectivity would be equal to the
ratio of diffracted to transmitted intensities, called the
diffraction efficiency, . A photon beam traversing a material
undergoes absorption, and the contribution of absorption to
the reflectivity can be expressed by the fraction of photons left
by the absorption process. For a thick flat crystal, the
diffraction efficiency in symmetrical Laue geometry is always
equal to or lower than 50% (Authier, 2001; Zachariasen,
1945). As already mentioned, photons have the same probability of undergoing an even or odd number of diffractions
inside the crystal. If the crystal is thick enough to guarantee
that every photon interacts with the lattice, t0 being the crystal
thickness traversed by the X-ray beam, the beam equally splits
into diffracted and transmitted beams. With CDP crystals, the
probability of a photon undergoing multiple reflections
decreases because of the continuous change in the incidence
angle with respect to the diffraction planes. The theory of
diffraction in curved crystals was widely developed in the past
half century in the context of the dynamical theory of
diffraction, with particular contributions by Malgrange (2002).
The thickness that leads to complete extinction of the
diffracted photon beam in a flat crystal in the case of Laue
symmetric diffraction is defined as extinction length
¼ ðV cos B Þ=ðre CjFh jÞ (Authier, 2001), where V is the
volume of the unit crystalline cell, B the Bragg angle, re the
classical electron radius, the wavelength of the incident
photons, C their polarization factor and Fh the structure factor
of the crystal diffracting planes. The quantity 2W, called the
Darwin width, quantifies the broadening of the intensity
profile while rocking a flat perfect crystal around the Bragg
position. Such an intensity profile is referred to as the rocking
curve (RC) in the literature. In the Laue symmetrical
geometry, W is given by W ¼ d= (Authier, 2001), where d is
the interplanar spacing between the diffraction planes. The
physical quantity RC ¼ 2=ðW Þ is called the critical radius.
If R RC, it is possible to find a simple quantitative expression for diffraction efficiency (Malgrange, 2002).
In a CDP crystal with R RC, it is not possible to obtain
complete extinction of the incident beam as for a flat crystal.
In fact, for a flat crystal, or for a crystal where the variation of
the angle of incidence on the diffracting planes induced by the
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Valerio Bellucci et al.
Diffraction efficiency for curved crystals
distortion over its thickness is much smaller than the Darwin
width, the entire thickness t0 is available for diffracting the
photons of a particular energy. In the case of a highly curved
crystal, only a fraction of the crystal thickness can diffract the
photons with a particular energy, owing to the large change of
incidence angle between photons and lattice planes (considering a parallel incident beam). As will be explained below,
diffraction efficiency is influenced by two phenomena: the
variation of the fraction of the incident beam that undergoes
extinction (prevalent for R RC), and re-diffraction inside
the traversed crystal thickness (prevalent for R RC). In the
case R RC, the first is the only relevant phenomenon, and
the formula below [elaborated by Barrière et al. (2010)]
provides a good approximation for diffraction efficiency:
¼ 1 expð2 dR=2 Þ:
ð1Þ
Thus, diffraction efficiency strongly depends on the curvature
radius of the CDPs. Of course, equation (1) cannot work for
crystals with low curvature, as it would approach unity as the
curvature radius tends to infinity instead of approaching 0.5 as
it does for a flat crystal. To date, there exists no analytical
theory that quantitatively calculates the diffraction efficiency
for crystals with low curvature. In this case, a complete resolution of the Takagi–Taupin equations would be needed,
which is not simple and is often not possible (Authier, 2001).
Indeed, the applications of CDP crystals sometimes require a
curvature radius in the range where equation (1) is not valid,
and for that reason we developed the model described in the
next section.
3. Modeling
X-ray propagation in a crystal is considered without absorption and the crystal is assumed to be at the Bragg angle for the
first interaction. The basic approach is to divide the crystal
into finite elements, which is similar to the initial assumption
of the lamellar model of White (1950). However, multiple
diffraction is considered in this article, with the consequent rediffraction effects produced by the misalignment of neighboring elements.
One can regard a CDP crystal as a series of very thin flat
crystals (lamellae) slightly misaligned with respect to one
another. The thickness of these crystals is set equal to half the
extinction length (Fig. 2), because within this length, 96% of
the photons in the X-ray beam undergo a single interaction
with the crystal (Authier, 2001). Thus, the total crystal thickness t0 is divided into N lamellae of thickness =2, i.e.
t0 ¼ N=2. The relative misalignment between two subsequent crystals is 0 ¼ =2R and depends on the curvature,
which is considered to be perfectly cylindrical. The beam is
assumed to be monochromatic and arbitrarily narrow, interacting with the lattice every half extinction length, as
explained above. In this case, the totality of the photons in the
beam can be considered to interact with the lamella, but the
interaction is free of re-diffraction processes. In some intuitive
sense, it would be possible to define a lamella as a crystal
thickness within which the totality of the photons interact only
J. Appl. Cryst. (2013). 46, 415–420
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once with the crystal. Naturally, the thickness of a lamella is
dependent on photon energy and the nature of the crystal,
because the extinction length is a function of energy too. In
other words, propagation of an X-ray beam in a curved crystal
can be studied once the relative misalignments between the
lamellae are known.
Diffraction efficiency for a flat and very thin crystal in Laue
geometry is an oscillating function of the crystal thickness. It
approaches unity when the thickness is equal to half of the
extinction length and the angle between the beam and the
diffraction planes is exactly the Bragg angle. If the incoming
beam is slightly misaligned from the Bragg condition,
diffraction efficiency falls off. Following Zachariasen (1945),
the dependence of diffraction efficiency on misalignment can
be described by a Gaussian distribution, whose maximum
attains unity at zero misalignment and whose FWHM equals
the Darwin width, i.e. the standard deviation of the Gaussian is
¼ 2W =2:35.
The effects of misalignment with respect to the Bragg angle
and the variation of the fraction of the incident beam undergoing extinction can be regarded as independent physical
quantities. Thereby, the expression for the diffraction efficiency profile of a single lamella is the product of the two
individual distributions, namely it is equation (1) multiplied by
the misalignment function exp½ðÞ2 =2 2 :
f ð; RÞ ¼ exp ðÞ =ð2 Þ 1 exp dR= :
2
2
2
2
0
pathð3Þ
i;j
1
B 1
B
B 1
B
B 1
¼B
B þ1
B
B þ1
B
@ þ1
þ1
1
1
þ1 C
C
1 C
C
þ1 C
C;
1 C
C
þ1 C
C
1 A
þ1
i ¼ 1; . . . ; 2n ;
j ¼ 1; . . . ; n:
ð3Þ
At the nth order, the fraction of the initial intensity streaming
into the ith branch is
2
2 i;n ¼ ð1 pathðnÞ
Þ=2
exp
dR=
i;1
n1
Q ðnÞ
ðnÞ
pathi;j þ pathi;jþ1 =2
j¼1
#
2
j
P
ðnÞ
exp 0
pathi;k =ð2 2 Þ
k¼1
1 exp 2 dR=2
:
ð4Þ
The diffraction efficiency of the entire crystal after n interactions is equal to the sum of the intensities held by the
branches oriented in the diffraction direction:
n ¼
ð2Þ
Multiple diffraction splits the initial beam into several branches (Fig. 3). The branches continue propagating and interacting with the lattice every half extinction length, so that after
n interactions with the lattice, the initial beam is divided into
2n branches. During this process, the misalignment between
the branches propagating in the crystal and the diffraction
planes varies because of the crystal curvature. The information
about the misalignment between branches and diffraction
planes (Fig. 3) is contained in a matrix. The entries of the
matrix represent misalignment expressed in units of 0, i.e.
misalignment divided by 0. The rows of such a matrix
represent the 2n branches and the n columns represent the
interactions undergone by the beam. As an example, for n = 3,
this matrix holds:
1
1
þ1
þ1
1
1
þ1
þ1
2n
P
i;n ð1 pathðnÞ
i;n ½i; nÞ=2:
ð5Þ
i¼1
According to the model, in the case of a flat crystal, the
intensity of the initial beam is a single branch, which is either
the diffracted beam if n is an odd number or the transmitted
beam if n is even. Hence, the diffraction efficiency is either 1 or
0, respectively.
The elaboration of the model has called for the discretization at ‘deterministic’ locations of the interaction between
X-rays and the lattice, but for real situations the interaction
Figure 3
Figure 2
Schematic representation of the diffraction process in a Laue crystal.
Dotted lines represent the diffraction planes. The X-ray beam interacts
with the crystal by traversing the entire crystal thickness t0. The crystal
can be divided into elements with thickness =2.
J. Appl. Cryst. (2013). 46, 415–420
Subdivision of the initial beam into branches inside the crystal for
subsequent steps n (Borrmann triangle). The number beside each branch
represents the misalignment with respect to the Bragg angle at the next
interaction point of the beam with the CDPs. Misalignment is expressed
in units of 0, i.e. misalignment divided by 0. The angles between the lines
of different layers are physically the same, but in the figure they are
plotted differently to avoid line overlap.
Valerio Bellucci et al.
Diffraction efficiency for curved crystals
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occurs continuously. For a thick crystal, the effect of continuous interaction, ruled by the laws of probability, changes the
effective thickness within which the beam is completely
diffracted. This stochastic effect is equivalent to a crystal with
deterministic diffraction locations but with random thickness.
Thus, for a given thickness, the diffraction efficiency of the
outgoing diffracted beam is the average of the two extreme
values 0 and 1, which is 0.5 for a flat crystal. The same holds for
a curved crystal, the real diffraction efficiency being the
average between the extreme values as calculated by the
model. Hence, the diffraction efficiency of the entire crystal at
the nth order is the average between the intensities held by the
branches in the diffraction directions at the order n and at the
order (n + 1):
ntot ¼ ðn þ nþ1 Þ=2:
ð6Þ
Since it is necessary to consider the contribution to diffraction
efficiency up to the order n + 1, the corresponding number of
lamellae is N = n + 1.
Figure 4
Diffraction efficiency versus curvature radius in a Ge crystal. Photons
with energy E = 150 keV are diffracted by (111) CDPs. The dashed curve
represents the prediction of dynamical diffraction theory through
equation (1). The vertical line indicates the critical radius RC = 48.5 m
for Ge(111). The dotted horizontal line is the theoretical limit to
diffraction efficiency for a flat crystal ( = 0.5). The set of curves are
diffraction efficiency ntot at n = 1, 3, 5, 7, 9, 12 and 124, corresponding to a
number of lamellae N = n + 1 = 2, 4, 6, 8, 10, 13 and 125. With =2 =
78.8 mm, the corresponding crystal thicknesses are t0 = 0.158, 315, 0.473,
0.630, 0.788, 1.024 and 9.85 mm. The latter thickness is for the Ge crystal
that was experimentally tested in Fig. 6(b). An increase in the number of
lamellae tends to be even less effective.
Figure 6
Figure 5
The same physical quantities as in Fig. 4 as a function of normalized
radius. In addition, the simulation was repeated within the interval of
photon energy 150–5000 keV with steps of 150 keV in the cases of both Si
and Ge. For each number of lamellae all the curves overlap each other
irrespective of the energy and of the material.
418
Valerio Bellucci et al.
Diffraction efficiency for curved crystals
(a) Rocking curve of an Si(111) curved crystal with size 25.5 25.5 1.0 mm analyzed through one of its longest sides by 150 keV photons. At
this energy, the crystal thickness traversed by X-rays is t0 = 25.5 mm,
corresponding to N = 139 lamellae. (1) Open circles: the intensity of the
diffracted beam divided by the intensity of the transmitted beam, i.e.
diffraction efficiency. (2) Filled circles: the intensity of the transmitted
beam over the intensity of the transmitted beam when no diffraction
occurs, i.e. transmission efficiency. The experimental diffraction efficiency
is 94.0 (30)%, while the FWHM of the distribution is 15.4 (11)00 . (3)
Dashed lines: prediction of the model about the theoretical diffraction
efficiency [93.1 (42)%]. The simulation accounts for the uncertainties of
experimental parameters; dashed lines define the tolerance range of
theoretical predictions. (b) Rocking curve of a Ge(111) curved crystal
with size 9.8 9.8 1.0 mm analyzed through one of its longest sides by
150 keV photons. At this energy, the traversed crystal thickness t0 =
9.8 mm, corresponding to N = 125 lamellae. (4), (5), (6) Same meanings as
for curves (1), (2), (3), respectively. The experimental diffraction
efficiency is 58.1 (19)%, while the FWHM of the distribution is
22.0 (17)00 . The theoretical diffraction efficiency is 67.6 (44)%, a value
slightly larger than the experimental diffraction efficiency owing to partial
mosaicity in the crystalline structure (Bellucci, Camattari, Guidi, Neri &
Barrière, 2011).
J. Appl. Cryst. (2013). 46, 415–420
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4. Discussion and results
Fig. 4 shows the calculated diffraction efficiency ntot for an Si
crystal as a function of the curvature radius for several values
of n. The diffraction efficiency tends to 0.5 as R ! 1, i.e. for a
flat crystal. Indeed, the diffraction efficiency perfectly overlaps
with the expectation of the dynamical theory [equation (1)] in
the range R RC. As the crystal becomes thicker owing to an
increase of n, the change in the efficiency curve becomes even
less important. The critical radius is the condition at which the
change in the orientation of lattice planes over half the
extinction length, 0 , is equal to the Darwin width. Since 0
depends on curvature, as the curvature radius exceeds the
critical radius, the effect of misalignment between near points
of the lattice dominates with respect to the effect of variation
of the extinction length, producing an increase in diffraction
efficiency with respect to the case of a flat crystal. As the
curvature radius becomes smaller than the critical radius, the
effect of variation of the extinction length dominates, thus
reducing the diffraction efficiency.
The analysis has been repeated for Ge, which has also been
used to fabricate curved crystals for X-ray diffraction and
presents a similar crystalline structure but higher atomic
number and electronic density than Si. These characteristics
lead to a smaller extinction length for Ge than for Si. As an
example, for diffraction of photons with energy E = 150 keV
by a Ge(111) crystal RC = 48.5 m, while for Si(111) RC = 267 m.
For a given n, the efficiency can be expressed in terms of
normalized curvature radius R/RC as in Fig. 5. Here, the efficiency is the same for both materials, reflecting the fact that
the information regarding X-ray diffraction in a bent crystal is
conveyed by the critical radius.
The achievements of this paper are useful for revisiting
some results of a measurement campaign run at ESRF in 2010
(Bellucci, Camattari, Guidi, Neri & Barrière, 2011). The aim of
the experiment was the realization of high-efficiency optics for
hard X-rays by diffraction in crystals curved by the method of
grooving (Bellucci, Camattari, Guidi & Mazzolari, 2011). A
series of Si and Ge crystals was analyzed by X-ray diffraction
of their CDPs. A typical diffraction plot for an Si crystal,
Figure 7
Theoretical rocking curves (the intensity of the diffracted beam divided by the intensity of the transmitted beam, i.e. diffraction efficiency th) of an
Si(111) curved crystal with size 25.5 25.5 1.0 mm analyzed through one of its longest sides by 150 keV photons, at four curvature radii, R. The
FWHMs of the distributions are always 15.400 . The theoretical prediction fits the experimental data (open circles) well for (c); a curve joining the
experimental data points has been drawn to guide the eye. The curvature radii are R = RC / 4 = 68.75 m (a), R = RC = 275 m (b), R = 1.28RC = 341 m (c), R
= 1.89RC = 519.75 m (d). The diffraction efficiency predicted by dynamical theory would, respectively, be 0.793, 1, 1 and 1; the diffraction efficiency
predicted by the model th = 0.793, 0.998, 0.931 and 0.676.
J. Appl. Cryst. (2013). 46, 415–420
Valerio Bellucci et al.
Diffraction efficiency for curved crystals
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5. Conclusions
Table 1
Comparison of experimental results with theory for Si and Ge crystal
plates.
An Si crystal plate with dimensions 25.5 25.5 1.0 mm was analyzed
through one of its longest sides; the error on geometrical dimensions is
25 mm. Photons with energy E = 150 keV were diffracted by (111) CDPs. In
these conditions, the critical radius was RC = 275 m, while the curvature radius
of the diffraction planes was R = 341 (25) m. The analysis was repeated with a
Ge crystal of dimensions 9.8 9.8 1.0 mm analyzed through one of its
longest sides with the same photon energy and diffraction planes. In these
conditions the critical radius was RC = 48.5 m while the curvature radius of the
diffraction planes was R = 92 (7) m. Values in parentheses are standard
uncertainties on the least significant digits. exp and dyn stand for the
experimental diffraction efficiency and the diffraction efficiency calculated by
the dynamical theory, respectively. The errors on the theoretical diffraction
efficiency th are calculated by propagating the error bounds of the
experimentally measured curvature radius and thickness, as half the difference
between the extreme values of efficiency.
Si
Ge
E (keV)
R/RC
exp (%)
dyn (%)
th (%)
150
150
1.28 (9)
1.89 (15)
94.0 (30)
58.1 (19)
100
100
93.1 (42)
67.6 (44)
A model of Laue diffraction in a curved crystal has been
developed, whose results agree very well with those of the
dynamical theory (Barrière et al., 2010) for R RC. The
model also produces the same results for a flat crystal when
the curvature radius is infinitely large (R ! +1) and provides
a quantitative description of diffraction efficiency when
neither of these cases are applicable. The model allows a
refinement in the interpretation of previously achieved
experimental data for curved crystals in the range R > RC. The
model sets an upper limit of efficiency for CDP crystals, even
in the region R > RC. This knowledge is important when
designing new schemes for a Laue lens in next-generation
satellite-borne experiments in astrophysics.
We recognize financial support by ASI through the LAUE
project.
References
obtained by rocking the crystal with respect to the incident
X-ray beam, is shown in Fig. 6(a). The energy passband of the
photons that a curved crystal can diffract is determined by its
curvature. Indeed, most of the samples characterized at ESRF
showed a diffraction efficiency consistent with equation (1),
but for the samples with moderate curvature the experiment
highlighted a discrepancy between measured efficiency and
the theoretical expectations relying on the dynamical theory
of diffraction (Table 1). Indeed, under some conditions of
application, the curvature radius of crystals exceeded the
critical radius.
The measured diffraction efficiency for Si (Fig. 6a) is
consistent with the prediction of the model developed in this
paper in all cases. In fact, the predicted efficiency is
93.1 (42)%, while a diffraction efficiency of 94.0 (30)% was
recorded experimentally. For Ge, the diffraction efficiency was
somewhat lower than the theoretical prediction, inasmuch as
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420
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Diffraction efficiency for curved crystals
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