research papers
Calculation of the instrumental function in X-ray
powder diffraction
Journal of
Applied
Crystallography
ISSN 0021-8898
A. D. Zuev
Received 23 June 2005
Accepted 15 February 2006
Adolf-Ehrtmann-Str. 9, Lübeck, D-23564, Germany. Correspondence e-mail: [email protected]
# 2006 International Union of Crystallography
Printed in Great Britain – all rights reserved
A new method for calculating the total instrumental function of a conventional
Bragg–Brentano diffractometer has been developed. The method is based on an
exact analytical solution, derived from diffraction optics, for the contribution of
each incident ray to the intensity registered by a detector of limited size.
Because an incident ray is determined by two points (one is related to the source
of the X-rays and the other to the sample) the effects of the coupling of specific
instrumental functions, for example, equatorial and axial divergence instrumental functions, are treated together automatically. The intensity at any
arbitrary point of the total instrumental profile is calculated by integrating the
intensities over two simple rectangular regions: possible point positions on the
source and possible point positions on the sample. The effects of Soller slits, a
monochromator and sample absorption can also be taken into account. The
main difference between the proposed method and the convolutive approach (in
which the line profile is synthesized by convolving the specific instrumental
functions) lies in the fact that the former provides an exact solution for the total
instrumental function (exact solutions for specific instrumental functions can be
obtained as special cases), whereas the latter is based on the approximations
for the specific instrumental functions, and their coupling effects after the
convolution are unknown. Unlike the ray-tracing method, in the proposed
method the diffracted rays contributing to the registered intensity are
considered as combined (part of the diffracted cone) and, correspondingly,
the contribution to the instrumental line profile is obtained analytically for this
part of the diffracted cone and not for a diffracted unit ray as in ray-tracing
simulations.
1. Introduction
The line profile in X-ray powder diffraction for a monochromatic beam is determined by sample broadening and instrumental aberration. According to Klug & Alexander (1974),
this can be represented as the convolution of a pure diffraction
profile f ðÞ and an instrumental function gð"Þ,
hð"Þ ¼
R1
gðÞ f ð" Þ d:
ð1Þ
1
Here " ¼ 2’ 2, ’ is the diffractometer angle and is the
Bragg angle. The following diffractometer factors affect the
instrumental function: angular non-uniformity of the intensity
distribution, deviation of the flat specimen surface from the
focusing circle, axial divergence, specimen transparency and
the finite width of the receiving slit. Misalignments of the
diffractometer, among them the deviation of the sample plane
from its ideal position, will also cause the instrumental profile
to change. Additional optical elements, such as a crystal
monochromator or analyzer, will also change the instrumental
profile. Alexander (1954) supposed that each of these factors
can be described by separate instrumental functions and that
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doi:10.1107/S0021889806005693
the total instrumental function can be calculated as a convolution of specific instrumental functions. The instrumental
aberration causes three effects: (i) a shift in the peak position,
(ii) a change in profile width and (iii) asymmetry of the profile.
The influence of the different instrumental factors on the
profile, especially for estimating the shift in the peak position,
are considered in detail by Wilson (1963). According to Klug
& Alexander (1974) and Wilson (1963), axial divergence is the
most important contributor to the total instrumental function.
Calculation of the axial instrumental function is considered by
Cheary & Coelho (1998), Masson et al. (2001), Finger et al.
(1994) and Ida (1998).
The most complete approach based on the convolving of
specific instrumental functions is realized in the fundamentalparameter approach (FPA) developed by Cheary & Coelho
(1992). Special attention was given to calculating a specific
instrumental function caused by axial divergence (Cheary &
Coelho, 1998). Representation of the total instrumental
function as a convolution is based on the supposition that
specific instrumental functions are completely independent.
To compensate for lack of knowledge about the influence of
coupling specific instrumental functions in FPA, it is also
J. Appl. Cryst. (2006). 39, 304–314
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necessary to tune the fundamental parameters to allow a best
fit for the experimental data (Cheary et al., 2004). The other
approach to calculating the total instrumental function is ray
tracing, in which the contribution of all possible incident and
diffracted rays to the total intensity is treated numerically. This
approach, for example, was followed by Bergmann et al.
(1998a) in the program BGMN. The approaches based on ray
tracing are time consuming. Kogan & Kupriyanov (1992) have
suggested the calculation of Fourier coefficients for instrumental functions. It was supposed that primary beam intensity,
transmission and absorption in the system can be represented
as a product of ‘n’ instrumental functions. This supposition is
equivalent to assuming that the total instrumental profile can
be given as a convolution of specific instrumental functions.
The dimensions of the focus and the receiving slit in the
equatorial plane were assumed to be so small that they can be
neglected. Honkimäki (1996) suggested a method for calculation of the instrumental function based on the consideration
of all possible paths in the diffractometer, in which the secondorder terms in the equatorial–axial coupling were taken into
account. Recently, Masson et al. (2003) showed that, for highresolution synchrotron powder diffraction, the instrumental
function can be represented as a convolution of four specific
instrumental functions describing the equatorial intensity
distribution, the monochromator and analyzer transfer function, and the axial aberration function.
Here a new comprehensive approach to calculate the total
instrumental function is proposed, in which all aberration
effects are treated simultaneously in the same manner. There
are no limitations on the size of the source, sample or receiving
slit, or the axial or equatorial divergence. The proposed
method, valid over a full range of 2 from 0 to 180 , can be
applied to different diffractometer geometries and can be
implemented in Rietveld refinement programs.
2. Theoretical part
I0 ðÞ ¼ k2 PLG IðA1; A2Þ;
where k2 is a constant, and PLG provides the polarization,
Lorentz and geometry corrections (Kleber et al., 1998). For a
non-polarized incident X-ray beam,
P ¼ ð1 þ cos2 2Þ=2;
the Lorentz factor L is
L ¼ 1=ðsin cos Þ;
and the geometry factor G in the standard form,
G ¼ cos =sin 2;
has two terms: the first, dependent on the fraction of powder
grains contributing to the intensity at the angle , is proportional to cos , and the second is related to the fact that the
linear density of radiation on the film of a Debye–Scherrer
camera or on the receiving slit of a diffractometer with a
radius R decreases proportionally to 1=ðR sin 2Þ with an
increasing diffraction angle . The second term is referred to
as the ‘detection factor’, following van Laar & Yelon (1984).
In the case of consideration of scattered intensity for the
entire diffraction cone at the angle , the second term should
be omitted and the geometry factor G becomes cos . Thus we
can write
PLG ¼ ð1 þ cos2 2Þ=sin :
The edges of the receiving slit form a plane rectangle
D1D2D3D4. The detector registers intensity when it intersects
the cone, and this intensity can be given as a curvilinear
integral along the line l,
Pc2
R
IðPc1; Pc2Þ ¼
IðPÞ dl;
ð2Þ
Pc1
where lðPÞ is the unit line intensity at point P lying on l. Pc1
and Pc2 are the points of intersection of the line l and the
detector boundaries.
2.1. General description of the method
Suppose an X-ray coming from surface element dS1 is
scattered by a powder sample at surface element dS2. The
sizes of the elements dS1 and dS2 are small enough to be
considered as points A1 and A2, respectively (Fig. 1). The
number of incident photons emitted from the surface dS1
(containing point A1) to the surface dS2 (containing point A2)
2
is proportional to d12
cos !1 cos !2 dS1 dS2, where d12 is the
distance between A1 and A2, while !1 and !2 are the angles
between the direction of propagation of the X-ray (vector
A1A2) and the corresponding unit vectors normal to the
elementary surfaces dS1 and dS2 (Born & Wolf, 1965). The
incident intensity may be written as
2
dS1 dS2;
IðA1; A2Þ ¼ k1 cos !1 cos !2 d12
with a proportionality coefficient k1 . The scattered photons
form a cone with vertex at A2 and half angle 2. We assume
that the relationship between incident intensity IðA1; A2Þ and
total scattered intensity I0 ðÞ for the Bragg angle is
J. Appl. Cryst. (2006). 39, 304–314
Figure 1
The positions of the incident beam, the diffraction cone and the receiving
slit. dS1 is an element of the source surface and dS2 is an element of the
sample surface. These elements are shown on an enlarged scale above
points A1 and A2.
A. D. Zuev
Instrumental function
305
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If absorption between the scattering point A2 and the
detector is neglected, the integral can easily be calculated,
because the number of photons between rays L1 and L2 of the
cone remains unchanged (e.g. along the curve c1 or c2; Fig. 1),
i.e. (i) the integral does not depend on the path of the integration and (ii) the integrals between the same two rays are
equal. This gives, for the intensity between the rays L1 and L2,
IðL1; L2Þ ¼ ð2Þ1 I0
¼ ð2Þ1 k2 PLG IðA1; A2Þ ;
where I0 is the total scattered intensity for the given reflection
and is the angle between planes 1 and 2. Plane 1 is
determined by line A1A2 (incident ray) and points Pc1, and
plane 2 is determined by the same line A1A2 and point Pc2
(there can be more than two intersection points of line l with
the detector; see xA3). Thus to calculate the integral (2) we
need to know the positions of points Pc1 and Pc2. These
points lie in the detector plane and are the points of intersection between lines D1D2 and/or D2D3 and/or D3D4 and/
or D1D4 and line l. Line l is the intersection of two surfaces:
the cone and the detector plane. The conical section or the
intersection of the diffraction cone and the receiving-slit plane
can be elliptical, parabolic or hyperbolic, depending on the
angle between the receiving-slit plane and the incident ray. In
each case it is possible to find a solution, i.e. the positions of
points Pc1 and Pc2 in space. From the preceding, the main
difference between the proposed method and the ray-tracing
method can be elucidated as follows. In the ray-tracing
approach, scattered rays from the scattered point (here A2)
are considered independently of one another. In the proposed
method, scattered rays from the incident ray (here A1A2) are
considered as a part of the diffracted cone, or in other words,
the integration of all scattered rays contributing to the registered intensity from an arbitrary incident ray can be
performed analytically.
It is easy to show that the product cos is proportional to
the standard geometry correction for the conditions corresponding to the Debye–Scherrer camera.
Table 1
Basic geometric elements and coordinate systems.
Basic
objects
A1
A2
A1A2
d
rd
Ln
Lc
Pv
SD
l
F
Pci
1
2
Description
Coordinate system
Point on source
Point on the sample
Line through A1 and A2
Unit vector parallel to A1A2
Receiving-slit plane
Plane containing A1A2 and perpendicular
to d
Line of intersection of the plane rd and
the receiving-slit plane d
Line of intersection of the plane rd and
the cone (over the sample)
Point of intersection of the line Lc and the
receiving-slit plane d (vertex of the
conic section)
Dandelin sphere, here the inner sphere
inscribed in the diffraction cone and
tangent to the detector plane d at
point F
Conic section
Focus of conic, point of contact between
the detector plane d and the Dandelin
sphere
Points of intersection of the conic section
with the detector boundaries
Plane through A1A2 and Pc1
Plane through A1A2 and Pc2
CS1
CS1
CS1
CS1
CS1, CS2, CS290
CS1
CS1, CS2
CS1
CS1
CS1
CS2
CS1, CS2
CS2, CS1
CS2, CS1
CS2, CS1
2.2. Basic geometric elements and calculation order for a
Bragg–Brentano diffractometer
Figure 2
Fig. 2 schematically shows the scattering geometry of the
Bragg–Brentano diffractometer with additional geometric
objects needed for the consideration.
The diffractometer is adjusted to an angle ’ and has a radius
R. The source dimensions are lf and df along the axial direction and in the equatorial plane, respectively. The sample
dimensions are xs and zs along the x and z axes, respectively.
The receiving slit has the dimensions yd and zd along the axial
direction and in the equatorial plane. We assume there is no
loss in intensity after X-rays pass the receiving slit.
The X-ray from A1 falls onto the sample surface in A2 and
diffracts from this point along a conical surface with a half
angle 2.
The following right-hand orthogonal coordinate systems
related to a diffractometer without misalignments are used.
The first coordinate system, CS1, is fixed. The z axis coincides
with the rotation axis. The origin O of CS1 coincides with the
point of intersection between the rotation axis and the equatorial plane. Without misalignment, point O corresponds to
the center of the sample surface. The x axis is determined by
the line in the equatorial plane that corresponds to the
diffractometer angle ’ ¼ 0. Without misalignment by ’ ¼ 0,
the centers of the source, the sample surface and the receiving
slit lie on the x axis. The y axis is perpendicular to the x and z
axes (Fig. 2).
The second coordinate system, CS2, is floating (shown in
red in Fig. 2). The origin of CS2 coincides with the focus F of
the conic section. The x0 y0 plane lies in the receiving-slit plane
d and the x0 z0 plane lies in the rd plane. The x0 axis coin-
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A. D. Zuev
Instrumental function
The geometry of scattering in a Bragg–Brentano diffractometer (without
Soller slits) with auxiliary elements. Top view. (See Table 1 for symbol
definitions.)
J. Appl. Cryst. (2006). 39, 304–314
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cides with the intersection of the receiving-slit plane d and
the rd plane, and passes through the vertex of the conic. The
direction from F to the vertex of the conic coincides with the
positive direction of the x0 axis. (For details see the figure in
xA2, which represents the scattering in the x0 z0 plane.)
The third coordinate system, CS290 , is fixed to the receiving
slit. It is only used for calculation if 2 ¼ 90. The origin O00 of
the floating coordinate system CS290 coincides with the center
of the receiving slit. The x00 y00 plane lies in the receiving-slit
plane, while the x00 z00 plane lies in the equatorial plane. The
direction of the x00 axis of CS290 coincides with the direction of
the x axis of CS1 by ’ ¼ 0. The position and orientation of the
coordinate system CS290 depends only on the diffractometer
radius R and the angle ’, and does not depend on the line
A1A2.
The principal geometric object in the treatment (if
2 6¼ 90 ) is the Dandelin sphere (Jennings, 1994). Here it is
the inner sphere inscribed in the diffraction cone and tangent
to the plane of the receiving slit (Fig. 2). It is used to find the
position of the point F. It is known that the point at which the
Dandelin sphere touches a plane intersecting a cone is a focus
of the conic section (Jennings, 1994).
The basic geometric objects for defining points Pci are given
in Table 1; the reference coordinate systems are also given. All
these geometric objects can be calculated straightforwardly by
the methods of analytical geometry without any limitations or
assumptions regarding the positions or the sizes of the sources,
the receiving slit, or the values for axial divergence. The
relationships between these objects are given in Appendix A.
The equation describing the conic has the same form for an
ellipse, a hyperbola and a parabola in the polar coordinate
system based on CS2,
% ¼ p=ða þ e cos ’Þ;
where p is the semi-parameter, e is the eccentricity, e < 1 for an
ellipse, e ¼ 1 for a parabola and e>1 for a hyperbola (Bronstein & Semendjajew, 1997).
The intersection points Pc1 and Pc2 (and Pc10 and Pc20 if
needed) of the conic with the straight lines that represent
the receiving-slit boundaries can easily be calculated
(Appendix A).
The next steps are to find coordinates for the points Pci in
the fixed coordinate system SK1 and to calculate the angle
between planes ðPc1; A1; A2Þ and ðPc2; A1; A2Þ passing
through points ðPc1; A1; A2Þ and ðPc2; A1; A2Þ. This gives the
intensity of the diffracted rays produced by the arbitrary
incident ray A1A2 which is registered by the detector. The full
intensity registered by the diffractometer at the angle ’ will be
the sum of the intensities produced by each incident ray A1A2.
The contribution of all factors to the total instrumental
function will automatically be taken into account after integrating over source and sample.
There are two points to be made here. The first is related to
the calculation for 2 ¼ 90 . In this case the diffraction cone
degenerates to a plane. The diffraction plane 90 (degenerated cone) is perpendicular to the line A1A2 and passes
through point A2. The conic section is degenerated to a
J. Appl. Cryst. (2006). 39, 304–314
straight line Lc90 that is the intersection of planes d and 90 .
Points Pci are the points of intersection of the receiving-slit
boundary with the line Lc90 and can easily be calculated. The
second point is related to the switching of the concavity of the
conics near the receiving slit when 2 passes from the range
< 90 to the range > 90. For the diffraction cones with
2 > 90 the center of the Dandelin sphere lies between points
A1 and A2. The focuses of the conics lie above the receiving
slit.
Formulas for calculating rotational and translational parts
of the transformation matrix for coordinate system CS2 as well
as for determinating points Pci are given in Appendix A.
After points Pci are found in the receiving-slit plane, we can
transform them into the initial coordinate system and calculate the angle between two planes determined by points Pc1 or
Pc2 and the common line A1A2.
If we multiply angle by a coefficient, which is the same for
the entire angular range of the detector, we can obtain the
intensities produced by a single ray passing through two
points. To obtain the total intensity, we need to sum all the
possible contributions to the intensities:
R R
Ið’Þ ’
ðA1; A2Þ dS1 dS2:
S1 S2
For a point source of X-rays for samples without tilt this
corresponds to a two-dimensional integral, while for a twodimensional source of X-rays this corresponds to a fourdimensional integral. In both case it can easily be computed.
2.3. Misalignment, absorption, Soller slits, monochromator,
diffractometer geometry
The effects of misalignment, absorption and Soller slits can
be included in the calculation, as well as a different diffractometer geometry. For the case of a diffractometer equipped
with a crystal monochromator it is also possible to provide a
solution in the context of the proposed method. The possible
treatments of these effects are outlined here, without going
into details.1
2.3.1. Misalignment. The positions of the source, the sample
and the receiving slit differ from their ideal positions. Without
loss of generality we can suppose that each misalignment is
only caused by a small rotation about the center of the source,
sample or receiving slit. Each rotation can be described by
three Euler angles. The misalignment leads to a change in the
positions of points A1 and A2 and the corner points of the
receiving slit, which can be easily calculated. The subsequent
treatment is the same as for the case without misalignment.
2.3.2. Absorption. Consideration of absorption means
firstly that point A2 has a non-zero y component. Secondly, the
contribution to the recorded intensity should be corrected for
each ray by the factor expðl Þ, where is the linear
absorption coefficient and l is the length of the ray in the
sample, which consists of two components; the first is the
length l1 passed by the incident ray in the sample to point A2
from the sample surface, and the second is the distance
1
These effects will be considered in detail in a future paper.
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traversed by the diffracted rays from point A2 to the sample
surface. For the second component the effective length
leff may be introduced. We have l1 ¼ ½ðx20 x2Þ2 þ y20 2 þ
ðz20 z2Þ2 1=2 , and we suppose leff ¼ 1=2ðlL1 þ lL2 Þ, where lL1
and lL2 are the segment lengths from A2 to the sample surface
along lines A2Pc1 and A2Pc2, respectively. A simple example
of the absorption aberration calculated by the proposed
method is given in x3.3.
2.3.3. Soller slits. For the case of Soller slits in the incident
beam, the condition of rays passing through the slits is given
by the inequality
arctan
z20 z2
< slits :
cos ’ðx2 x1Þ sin ’ðy2 y1Þ
For the case of Soller slits in the diffracted rays each point A2
‘sees’ only parts of the receiving slit (Fig. 3). Therefore, instead
of one full receiving slit, several (the number depends on the
positions and geometrical parameters of the Soller slits)
smaller receiving slits should be considered. The corner points
of these new sub-receiving slits for given Soller slits depend on
the position of point A2 and angle ’ and can be easily
calculated. The subsequent calculations of points Pci should
be carried out for each sub-receiving slit. Such consideration
provides an exact solution for the instrumental function for a
diffractometer with Soller slits in the same way as in the case
without Soller slits. Calculations in this case can take more
time (a test calculation with four sub-receiving slits took about
twice as long as the case without Soller slits). Obviously,
related approximations can be devised and tested using the
exact solution.
2.3.4. Monochromator. It should be noted that there is no
satisfactory solution, based on a physically meaningful model,
for incorporating a monochromator during the calculation of
the instrumental function in X-ray powder diffraction (Cheary
et al., 2004). In the context of the proposed method, the case of
the diffractometer with a monochromator can be considered
as follows.
Monochromator in the diffracted beam. According to the
dynamical theory of X-ray scattering, the reflection of the
monochromatic X-rays from the crystal occurs within a certain
region corresponding to the Darwin width of the crystal
(Authier et al., 1996). We assume for simplicity that the
Darwin width represents the rectangular function (see
Appendix B) with the limits BM þ 1 and BM þ 2 , where
BM is the Bragg angle of the monochromator crystal (for an
Si 111 crystal and monochromatic X-rays with the energy
8 keV, BM ¼ 14:31 , 1 ¼ 1:58 105 rad and 2 ¼ 4:77 105 rad). For the point source of the monochromatic X-rays
A2 and the plane crystal monochromator this region represents a ring, with its center at the perpendicular projection A20
of the point A2 on the monochromator plane, and the radii
dM cotðBM þ 1 Þ and dM cotðBM þ 2 Þ, where dM is the
distance between points A2 and A20 . Assuming as before that
the diffracted cone has no width, we can use the same
approach as above. The intersection of the monochromator
plane and the diffracted cone represents a conic section. The
diffracted rays contribute to the recorded intensity only in the
case where the diffraction cone initially intersects the
receiving slit, and then the reflection region. Fig. 4 shows the
reflection region and conic for the following conditions.
Diffractometer radius R ¼ 200:5 mm, ’ ¼ 10 ; the center of
the plane of the Si crystal is 300 mm distant from the rotation
axis. The monochromator is set at the angle BM ¼ þ14:31.
Point A1 lies at the center of the source; the coordinates of
point A2 in the coordinate system CS0 are ð0; 0; 2Þ. The Bragg
angle of the sample is B ¼ 10. Points Di0 ði ¼ 1; 2; 3; 4Þ in
Fig. 4 are the intersection points of the lines from point A2
through the corner points of the receiving slit and the
monochromator plane.
Figure 4
Figure 3
Soller slits in the diffracted cone. A drawing plane passes through the
rotation axis of the diffractometer and the center of the receiving slit. The
z axis is shown on an exaggerated scale. The lamellas of the working
channels of the Soller slits are highlighted in black. Corresponding ‘new’
receiving slits (gray) and the conic section are shown on the right.
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Intersection of the diffracted cone and monochromator plane. A drawing
plane lies on the monochromator surface. The x axis is parallel to the
rotation axis of the diffractometer. The origin coincides with the center of
the crystal surface. The hatched region shows the part of the ring where
the reflection of the monochromatic X-rays is possible. The quadrangle
D10 D20 D30 D40 represents a point projection of the receiving slit from the
point A2 onto the monochromator plane. Note the difference in the axes
scales.
J. Appl. Cryst. (2006). 39, 304–314
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The remainder of the process is similar to that considered
above. The intersection points Pcmi of the conic and reflection
region should be found. Points Pcri and Pcri 0 represent
intersections of the conic with the reflection region boundaries. As in the case with the receiving slit, there are two or
four solutions in the neighborhood of the monochromator
center. Points Pcp1 and Pcp2 represent the intersections of the
conic in the monochromator plane with the quadrangle
D10 D20 D30 D40 . In other words, these are intersections of the
lines A2Pci ði ¼ 1; 2Þ and the monochromator plane.
The reflection region should be calculated for each point
A2. To calculate the total recorded intensity, the integration
should be performed over the wavelength distribution. For an
X-ray tube with a copper anode, as an example, the calculations for two wavelengths k1 and k2 should be performed
for a rough approximation. For a proper approximation, a
wavelength distribution of k1 and k2 lines should be
involved in the calculation. In the fundamental-parameter
approach (Cheary et al., 2004) the k1 and k2 lines are each
represented as a sum of two Lorentzian profiles. The parameters of the Lorentzian profiles are given by Deutsch et al.
(2004). By integrating over wavelength, the integration step
(in eV) should be much lower than the Darwin width (in eV)
of the monochromator crystal. For a curved crystal monochromator the reflection region is no longer a ring, but its
contour on the surface of the crystal can be calculated for each
point A2. Accordingly, the intersection of the diffracted cone
with the reflection region can also be obtained.
Monochromator in the incident beam. Placing the crystal
monochromator in the incident beam constrains the possible
positions of points A2 on the sample. For monochromatic
X-rays and a planar crystal monochromator, the reflection
region represents a ring. Correspondingly, the region on the
sample involved in the scattering of X-rays represents an area
limited by two conics. In this case, for simplicity, instead of
points A2 on the sample, points A2m within the reflection
region on the monochromator crystal can be taken. In other
words, the incident rays will be defined by the point A1 on the
source and the point A2m on the monochromator. Clearly,
these rays should fall onto the sample. Because the reflection
region is very narrow, the integration over this region can be
reduced to the integration over the circle within the reflection
region to a good approximation. The limits of the integration
are determined by the condition that reflected rays fall into
the sample. If a curved crystal monochromator is used, the
integration curves no longer form a circle. The shape of the
curves is determined by the shape of the crystal surface. For
each point A1 this curve can be calculated either analytically
(for some simple surfaces) or numerically. For a non-monochromatic X-ray beam, additional integration over the wavelength distribution is required.
2.3.5. Diffractometer geometry. The positions of the source
and the receiving slit should be corrected in the case of a
diffractometer with geometry other than Bragg–Brentano or
for other scanning schemes of the diffractometer. There is no
fundamental difference in the treatment for this case. The ray
A1A2 is determined by the two points A1 and A2. The
J. Appl. Cryst. (2006). 39, 304–314
diffraction cone is determined by the ray A1A2 and point A2.
The intersection of the diffraction cone with the receiving-slit
plane produces a conic section whose parameters can be found
in the same manner as those for the Bragg–Brentano
diffractometer. Intersections of the conic section with the
boundary of the receiving slit determine points Pci.
3. Results
The proposed method for calculating the total instrumental
function can be used to calculate specific instrumental functions. The principal point in the calculations is the finite width
of the receiving slit, i.e. the special instrumental function will
be calculated coupled with the finite width of the receiving slit.
It must be emphasized once again that in the proposed method
the convolution is not used. In fact, there is no need to
calculate specific instrumental functions, but it may be useful
for comparisons with methods based on the convolution
approach or for testing approximations. In the following
sections, comparisons are made between the profiles of some
specific instrumental functions suggested previously by Klug
& Alexander (1974) and later on by others (Ida & Kimura,
1999a,b; Cheary et al., 2004) and the profile calculated by the
proposed method. For the purpose of comparison with the
proposed method, the specific instrumental function used in
the convolution approach was convolved with the instrumental function representing the receiving slit. In the last
section the total instrumental profile calculated by the
proposed method is compared with the profile obtained using
a Monte Carlo ray-tracing simulation (BGMN; Bergmann et
al., 1998a).
3.1. Equatorial aberration
The equatorial aberration is affected by the finite width of
the source and the receiving slit, and the flat-specimen effect.
The first two factors produce rectangular profiles without shift.
The convolution of these two instrumental functions provides
an exact solution as a triangular or trapezoidal profile where
the receiving-slit width equals or does not equal the width of
the source, respectively. This is a simple situation to calculate.
The flat-specimen aberration causes shift and asymmetry of
the diffraction line profile. In the convolution approach, the
exact solution for flat-specimen aberration can be approximated by the expression / "1=2 to obtain a reasonably good
approximation. With the proposed method, integration was
performed by using a grid in the equatorial plane.
3.1.1. Flat-specimen aberration. The approximation for the
specific instrumental function for the flat-specimen aberration
as given by Cheary et al. (2004) and Ida & Kimura (1999a) is
JFS ð"Þ ¼ 1=½2ð""M Þ1=2 ;
"M " 0;
where
2
"M ¼ Lx =ð2RÞ sin 2
and Lx is the specimen length along the equatorial direction.
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The calculation of this type of aberration is simple when
using the proposed method. For the profile in Fig. 5 the grid
used for calculating the flat-specimen aberration coupled with
the receiving-slit function is 1 1 for the source and 20 1 for
the sample; the length of the receiving slit was set to be much
smaller than R sin 2. One can see quite clearly that the results
agree very closely.
3.2. Axial aberration
The specific instrumental function JAX for the axial aberration can be analytically calculated for the special case in
which there is no divergence of the incident rays (Cheary et al.,
2004).
(
j"1 "2 j1 ð"2 ="Þ1=2 ð"1 ="Þ1=2 ; "1 < " < 0
JAX ð"Þ ¼
"2 " "1 ;
j"1 "2 j1 ð"2 ="Þ1=2 1 ;
lines passing through the center of the source or sample in the
axial direction. The results are given in Fig. 6. As can be seen
from Figs. 5 and 6, the axial aberration has a greater impact on
the profile shape than does the equatorial aberration.
3.3. Absorption correction
Absorption correction is important for thick specimens with
a small absorption coefficient for X-rays. In the convolution
approach, the specific instrumental function for a sample with
a thickness T is given by (Cheary et al., 2004; Ida & Kimura,
1999b)
J ð"Þ ¼
"1 ¼ cot 2 Lr Ls
2
2R
2
2
cot 2 Lr þ Ls
; "2 ¼ ;
2
2R
and Ls and Lr are the axial sample and receiving-slit lengths.
The function JAX ð"Þ was convolved with the receiving-slit
function representing the rectangular function. Axial instrumental functions are shown in Fig. 6. The results are consistent
with our model. The analytical approximation for the axial
aberration in the convolution approach is possible only for the
case in which the divergence of incident X-rays is sufficiently
small (less than 1 ) (Cheary et al., 2004). Axial aberration for
the general case can be calculated by using the semi-analytical
approach developed by Cheary & Coelho (1998).
In the proposed method, the axial aberration function
(coupled with the finite sizes of the receiving slit) for the
general case of a non-parallel incident X-ray beam can be
calculated without difficulty by taking grid points along the
"min " 0;
where
where
expð"=Þ
;
½1 expð"min =Þ
"min ¼ ð2T=RÞ cos ;
¼ sin =ð2RÞ:
Convolution of the aberration function J ð"Þ with the
receiving-slit instrumental function is shown in Fig. 7. Here the
profile calculated according to the proposed method is also
represented. The calculation was carried out with a fixed point
A1 at the center of the source; points A2 lie on the y axis from
the center into the specimen. To eliminate the axial aberration, here due to the length of the receiving slit, the length of
the receiving slit was reduced to 0.1 mm. Comparison of the
axial aberration (Fig. 6) and the absorption effect (Fig. 7)
shows that for samples with a low absorption the influence of
absorption on the line profile may be comparable with the
influence of the axial aberration.
Fig. 7 shows that there is good agreement between these
two approaches in this case as well. It must be emphasized that
the instrumental function J ð"Þ is an approximation. Reefman
Figure 6
Figure 5
Equatorial aberration coupled with the receiving-slit width calculated by
the proposed method (solid line) and as a convolution (open circle) of JFS
with the rectangle function (dashed line) representing the receiving slit.
The vertical line at 2’ ¼ 20 represents the Bragg angle to which the
aberration function is related.
310
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Instrumental function
Axial aberration. Source and sample: axial length 10 mm; receiving slit:
length 10 mm, width 0.25 mm. Solid and dashed lines: calculation by the
proposed method without and with divergence in the incident beam,
respectively. Open circles: calculation as a convolution of JAX with the
rectangle function representing the receiving slit. The vertical line at
2’ ¼ 20 represents the Bragg angle to which the aberration function is
related.
J. Appl. Cryst. (2006). 39, 304–314
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Figure 7
The effect of absorption in the specimen. The absorption coefficient corresponds to that of graphite. Sizes of the receiving-slit: length 0.1 mm,
width 0.25 mm. The calculation according to the proposed method and
the convolution approach are shown as a solid line and open circles,
respectively. The vertical line at 2’ ¼ 20 represents the Bragg angle to
which the aberration function is related.
(1996) has shown, by analyzing the effect of transparency, that
the convolution model is not valid in the general case.
3.4. Total instrumental function
The instrumental function calculated by the proposed
method for two angles was compared with the instrumental
function obtained using the BGMN program (Bergmann et al.,
1998b) (Fig. 8).
In the program geomet of the BGMN package, Monte Carlo
ray tracing is used to calculate the instrumental function. The
advantage of using the ray-tracing simulation is that it
provides a correct instrumental profile. As shown in Fig. 8, the
agreement between these two methods is very good. However,
precise calculations using the ray-tracing method are time
consuming.
4. Discussion and conclusions
An exact analytical solution for the contribution of each
incident X-ray determined by two points (one is related to the
source and the second to the sample) to the recorded intensity
in a conventional Bragg–Brentano diffractometer has been
obtained. On the basis of this solution, the instrumental
function can be calculated as an integral over possible positions of points A1 on the source and points A2 on the sample.
The entire consideration is based on the diffraction optics.
There are no limitations or assumptions about axial or equatorial divergences of the incident or diffracted rays, or about
the dimensions of the source, sample or receiving slit. As a
result of these features, all factors affecting the total instrumental function are taken into account, so that there is no
need to synthesize the total instrumental function by convolving the specific instrumental functions. The method is simple,
and provides many possibilities for analyzing the instrumental
J. Appl. Cryst. (2006). 39, 304–314
Figure 8
Comparison of the calculated instrumental function with the instrumental
function obtained using the BGMN program. Sample sizes: 5 10 mm;
receiving-slit sizes: 0:25 10 mm. Bragg angle: (a) 2 ¼ 20 , (b)
2 ¼ 80 .
function. The effects of misalignments of the diffractometer
and absorption of the sample can also be taken into account.
Implementation of Soller slits in the calculations presents no
special problems because the solution in this case is derived
from the common consideration. A possible treatment for
incorporating the monochromator in the incident as well as in
the diffracted beam is given.
Calculations of the specific and total instrumental profiles
were made for different types of aberrations. Instrumental
functions calculated by the proposed method showed good
agreement with those obtained by other methods in which the
convolution approach or ray tracing were used. The method
considered in this paper is flexible enough to be relatively
easily applied to other scanning schemes of the diffractometer.
The method provides an opportunity to compare the contributions of each factor that impacts on the total instrumental
function. An approximation method (wherever required) for
calculating the instrumental function (total or specific) can be
tested with the approach presented here. Implementation in
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the Rietveld (1969) refinement can be made in the same way
as is realized with Rietveld programs employing a learnt
profile (e.g. see Balzar et al., 2004; Le Bail, 1998). It is known
that the axial aberration provides an important contribution to
the total instrumental function. Thus, as a first approximation,
the axial aberration provides a reasonable instrumental profile
for finding initial values related to the line profile in the
Rietveld refinement. In the following steps all possible aberrations can then be taken into account. To calculate the
instrumental function for the conditions shown in Fig. 8 it is
enough to take 10 10 A1 points for the filament and 10 10
A2 points for the sample. As an example, the difference
P
jyi ð20Þ yi ð10Þj
P
Rð20; 10Þ ¼
yi ð20Þ
results in a value of 2.06%. Here yi ðNÞ denotes the ith point on
the instrumental profile calculated with the grid N N for the
filament and N N for the sample. R(20,5) gives 7.22%. The
calculation time in the latter case is less than 1 s on a computer
with a Pentium 4 3.2 GHz processor.
Convolution of the total instrumental function with the
function representing the sample broadening, and with the
function representing the wavelength distribution, provides
the line profile for the X-ray powder diffraction. In order to fit
to the experimental line profile, only the parameters corresponding to the sample should be varied. This procedure
should result in increased reliability for the structural parameters obtained after fitting because all parameters are
physically meaningful.
APPENDIX A
Intersections of conic and receiving slit
A1. Rotational part
The rotational part of the transformation matrix between
the coordinate systems CS1 and CS2 is determined by the
direction cosines of the three mutually perpendicular vectors.
Unit vector nd ¼ ðcos ’; sin ’; 0Þ is perpendicular to the
receiving-slit plane and parallel to the z0 axis of the coordinate
system CS2. Unit vector nrd is perpendicular to the plane rd
and defines the direction of the y0 axis of CS2.
nrd ¼
A2. Translational part
The next step is to find the translation part, or the vector
from point O(0,0,0) to the point of focus of the conic section.
Firstly, we should find the line Lc of intersection of plane
rd with the conical surface. We are interested only in the line
from A2 to the detector with a positive y component. The
direction of the line Lc is given by the unit vector
k ¼ k cos 2 þ g sin 2;
where
g ¼ nrd k:
(Vector k lies in the plane rd and forms an angle 2 with the
ray A1A2.) The scattered ray from A2 parallel to k intersects
the receiving-slit plane d at point Pv , which is the vertex of
the conic section. The coordinates of Pv may be obtained from
(Bronstein & Semendjajew, 1997)
xPv ¼ xA2 kx ;
yPv ¼ yA2 ky ;
zPv ¼ zA2 kz ;
with
¼
ndx xA2 þ ndy yA2 þ ndz zA2 R
:
ndx x þ ndy y þ ndz z
It is known that the position of the focus of the conic section
coincides with the point of contact of the sphere inscribed in
the cone with the secant plane (Jennings, 1994). For further
calculations it is useful to consider the geometrical relationship between the rays and the geometrical elements in the
plane rd (Fig. 9).
Circle c represents the intersection of the inscribed sphere
with the plane rd. Point F represents the focus of the conic
section. The distance d from point A2 to point Pv is given by
the formula
d ¼ ½ðxPv xA2 Þ2 þ ðyPv yA2 Þ2 þ ðzPv zA2 Þ2 1=2 :
k nd
k nd
¼
;
sin½arccosðk nd Þ ½1 ðk nd Þ2 1=2
where k is the unit vector parallel to the incident ray A1A2.
Unit vector ni defines the direction of the x0 axis of CS2.
ni ¼ nrd nd :
The sought-for rotation matrix is given by
0 sin ’ð sin ’ þ cos ’Þ
x
y
2 1=2
ðx cos ’ þ y sin ’Þ B ½1cos
B ’ð sin ’ þ cos ’Þ
M ¼ B ½1 ð cosx ’ þ siny ’Þ2 1=2
x
y
@
z
½1 ðx cos ’ þ y sin ’Þ2 1=2
z sin ’
½1 ðx cos ’ þ y sin ’Þ2 1=2
z cos ’
½1 ðx cos ’ þ y sin ’Þ2
x sin ’ þ y cos ’
½1 ðx cos ’ þ y sin ’Þ2 1=2
cos ’
1
C
sin ’ C
C:
A
0
Figure 9
Scattering in the plane rd .
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We now need to know the angle 0 between unit vectors k and
ni ; the latter is parallel to the line of intersection of planes d
and rd . We have
cos 0 ¼ ni k:
Angle ffA2Pv C ¼ ¼ 0 =2. The distance d1 from point A2 to
point C (the center of the inscribed sphere) may be obtained
by considering the triangle A2Pv C:
d1 ¼ d sin =sin ffA2CPv :
The position of the center of the inscribed sphere may be
calculated now as
xC ¼ xA2 þ x d1 ;
yC ¼ yA2 þ y d1 ;
Now we are in a position to calculate the conic equation in the
receiving-slit plane. Points D1, D2, D3 and D4 can be obtained
by using the common transformation (2). Fig. 10 shows the
results of calculations for one of the cases.
To illustrate more clearly the mutual positions of points D1,
D2, D3 and D4, the x-axis range was scaled out by a factor of
20. With such a transformation, the right angles between the
pairs of lines D1D2 and D3D4, and D2D3 and D1D4, are not
conserved. In the case shown in Fig. 10 there are four points of
intersection of the detector boundaries with the conic. There
are actually two common possibilities: two or four intersections.
To find the point of intersection of the straight line representing the detector boundary with the conic, we use two
equations:
% ¼ p=ð1 þ e cos ’Þ;
zC ¼ zA2 þ z d1 :
ðy yD1 Þ=ðyD2 yD1 Þ ¼ ðx xD1 Þ=ðxD2 xD1 Þ:
The radius r of the inscribed sphere may be given as
r ¼ d1 sin 2:
Point F is the intersection of the perpendicular to plane d
passing through point C with plane d. Its coordinates are
xF ¼ xC þ
If we assume that y ¼ kx þ c and put y ¼ % sin ’, x ¼ % cos ’,
the first equation can be transformed into
p sin ’
kp cos ’
¼
þc
1 þ e cos ’ 1 þ e cos ’
ndx r;
or
yF ¼ yC þ ndy r;
zF ¼ zC þ ndz r:
The vector Vtransl ¼ ðxF ; yF ; zF Þ represents the translation
vector from the initial coordinate system CS1 to the coordinate system with the origin at point F.
The relationship between coordinates x0, y0 , z0 of point X in
the new coordinate system CS2 and coordinates x, y, z in the
coordinate system CS1 is
!
!
x xF
x0
y0 ¼ M y yF :
z0
z zF
sinð’ Þ ¼ ;
ð3Þ
where
cos ¼ p=½ p2 þ ðkp þ ecÞ2 1=2 ;
sin ¼ ðkp þ ecÞ=½ p2 þ ðkp þ ecÞ2 1=2
and
¼ c=½ p2 þ ðkp þ ecÞ2 1=2 :
Equation (3) has no solution if jj > 1, one solution if jj ¼ 1,
and two solutions if jj < 1:
A3. Parameters of conics and positions of points L1 and L2
To find the parameters p and e of the conic section we may
use two equations:
rj’¼0 ¼ p=ð1 þ eÞ;
rj’¼=2 ¼ p:
Segment Pv F ¼ r cot =2 gives the value rj¼0, and we can
easily obtain for the semi-parameter p
p ¼ ½ðBDÞ2 ðFDÞ2 1=2 ;
where
FD ¼ r sinð2’ þ 2 =2Þ;
BD ¼ ðd1 þ CDÞ tan 2’:
J. Appl. Cryst. (2006). 39, 304–314
Figure 10
Conic section and position of the receiving slit in the receiving-slit plane.
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For the case with solutions we should select only the points
that lie on the detector boundaries.
A4. Angle between planes P1 and P2
After the points are found in the receiving-slit plane, we can
transform them into the initial coordinate system and calculate the angle between the two planes determined by points L1
or L2 and the common line A1A2. Plane i (i ¼ 1; 2) passing
through point Li and containing ray A1A2 may be represented
as (Bronstein & Semendjajew, 1997)
y yA1
z zA1 x xA1
xA2 xA1 yA2 yA1 zA2 zA1 ¼ 0;
xLi xA1 yLi yA1 zLi zA1
or in the standard form Ai x þ Bi y þ Ci z þ Di ¼ 0, where
y y
zA2 zA1 A2
A1
Ai ¼ yLi yA1 zLi zA1 ¼ ðyA2 yA1 ÞðzLi zA1 Þ ðyLi yA1 ÞðzLi zA1 Þ;
x x
A2
A1
Bi ¼ xLi xA1
zA2 zA1 zLi zA1 ¼ ðxA2 xA1 ÞðzLi zA1 Þ ðxLi xA1 ÞðzLi zA1 Þ;
x x
A2
A1
Ci ¼ xLi xA1
yA2 yA1 yLi yA1 ¼ ðxA2 xA1 ÞðyLi yA1 Þ ðxLi xA1 ÞðyLi yA1 Þ:
The angle
between two planes is given by (Bronstein &
Semendjajew, 1997)
¼ arccos
½ðA21
A1 A2 þ B1 B2 þ C1 C2
:
þ B21 þ C12 ÞðA22 þ B22 þ C22 Þ1=2
Figure 11
The reflectivity profile of an Si 111 crystal. The rectangular profile (shown
dashed) was used to calculate the reflection region in Fig. 4.
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APPENDIX B
Darwin width of the Si 111 crystal
The Bragg rocking curve for the Si 111 reflection at 8 keV was
calculated with the program XCRYSTAL (Sanchez del Rio et
al., 1997). Fig. 11 shows the results of the calculations.
The author thanks Dipl.-Ing. N. Endlich for his technical
support and especially Professor Dr A. Ertel for useful and
stimulating discussions during the author’s scientific work in
the X-ray laboratory at the University of Advanced Science,
Lübeck.
References
Alexander, L. (1954). J. Appl. Phys. 25, 155–161.
Authier, A., Lagomarsino, S. & Tanner, B. K. (1996). Editors. X-ray
and Neutron Dynamical Diffraction: Theory and Applications,
NATO Science Series B, Vol. 357. New York: Plenum Press.
Balzar, D., Audebrand, N., Daymond, M., Fitch, A., Hewat, A.,
Langford, J., Bail, A. L., Louër, D., Masson, O., McCowan, C., Popa,
N., Stephens, P. & Toby, B. (2004). J. Appl. Cryst. 37, 911–924.
Bergmann, J., Friedel, P. & Kleeberg, R. (1998a). CPD Newsl. 20, 5–8.
Bergmann, J., Friedel, P. & Kleeberg, R. (1998b). http://www.bgmn.de.
Born, M. & Wolf, E. (1965). Principles of Optics. London: Pergamon
Press.
Bronstein, I. N. & Semendjajew, (1997). Taschenbuch der Mathematik. Frankfurt Am Main: Thun.
Cheary, R. W. & Coelho, A. (1992). J. Appl. Cryst. 25, 109–121.
Cheary, R. W. & Coelho, A. (1998). J. Appl. Cryst. 31, 851–861.
Cheary, R. W., Coelho, A. A. & Cline, J. (2004). J. Res. Natl Inst.
Stand. Technol. 109, 1–25.
Deutsch, M., Förster, E., Hölzer, G., Härtwig, J., Hämäläinen, K.,
Kao, C.-C., Huotari, S. & Diamant, R. (2004). J. Res. Natl Inst.
Stand. Technol. 109, 75–98.
Finger, L. W., Cox, D. & Jephcoat, A. (1994). J. Appl. Cryst. 27, 892–
900.
Honkimäki, V. (1996). J. Appl. Cryst. 29, 617–624.
Ida, T. (1998). Rev. Sci. Instrum. 69, 2268–2272.
Ida, T. & Kimura, K. (1999a). J. Appl. Cryst. 32, 634–640.
Ida, T. & Kimura, K. (1999b). J. Appl. Cryst. 32, 982–991.
Jennings, G. A. (1994). Modern Geometry with Applications. New
York: Springer-Verlag.
Kleber, W., Bautsch, H.-J. & Bohm, J. (1998). Einführung in die
Kristallographie. Munich: Oldenbourg.
Klug, H. & Alexander, L. (1974). X-ray Diffraction Procedure for
Polycrystalline and Amorphous Materials. New York: John Wiley.
Kogan, V. A. & Kupriyanov, M. F. (1992). J. Appl. Cryst. 25, 16–25.
Laar, B. van & Yelon, W. B. (1984). J. Appl. Cryst. 17, 47–54.
Le Bail, A. (1998). Advances in X-ray Analysis, Vol. 42, pp. 191–203.
Denver: JCPDS – International Centre for Diffraction Data.
Masson, O., Dooryhee, E. & Fitch, A. (2003). J. Appl. Cryst. 36, 286–
294.
Masson, O., Guinebretière, R. & Dauger, A. (2001). J. Appl. Cryst. 34,
436–441.
Reefman, D. (1996). Powder Diffr. 11, 107.
Rietveld, H. M. (1969). J. Appl. Cryst. 2, 65–71.
Sanchez del Rio, M., Ferrero, C. & Mocella, V. (1997). Proc. SPIE,
3152, 136–146.
Wilson, A. E. C. (1963). Mathematical Theory of X-ray Powder
Diffractometry. Eindhoven: Philips Technical Library.
J. Appl. Cryst. (2006). 39, 304–314