research papers
The fundamentals of crystal orientation
Danut Dragoi*
ISSN 2053-2733
nComposites, 15527 Spruce Tree Way, Fontana, CA 92336, USA. *Correspondence e-mail: danut.daa@gmail.com
Received 1 October 2017
Accepted 8 December 2017
Edited by A. Altomare, Institute of
Crystallography - CNR, Bari, Italy
Keywords: exact solution for crystal orientation;
parametric crystal orientation; stereographic
projection; extended stereographic projection;
linked spherical triangles.
The method described in this paper improves the old methods of crystal
orientation, applies new parametric equations for crystallography, and increases
the precision and accuracy of measurements. The method applies to inorganic
and organic crystals. A breakthrough in crystal orientation happened about 25
years ago when two equations dependent on the Bragg angle and an arbitrary
direction in the crystal were developed. Unfortunately, they were analytically
insolvable and their unique solution was found numerically. Finding the
numerical solution of crystal orientation is challenging from a mathematical
point of view. In these conditions the numerical solution was found using the
Newton method. The Newton method required a specific programming that
limits the full benefit of the method in the laboratory. In recent years, a new
numerical technique called GRG (generalized reduced gradient), which can be
run on many inexpensive computers, was found to be a good fit for these
equations. The solutions that can be found with the GRG method are now
completed with additional parametric equations; they are easy to use with
computers in many laboratories. In this way, parametrization of nonlinear
equations for X-ray crystal orientation determines the positions of a reference
surface of the single crystal relative to its crystallographic system and to a
goniometer setting with two perpendicular axes of rotation. This approach was
successfully validated and checked for different Si wafers with (111) and (004)
orientation. The paper shows an innovative approach through the parametric
equations in conjunction with exact solutions found with a GRG subroutine. The
results of the method demonstrate the potential for new applications in industry
and research.
1. Introduction
# 2018 International Union of Crystallography
Acta Cryst. (2018). A74
Single-crystal orientation determination is carried out in
industrial as well as research laboratories; however, there are
still some technical problems with regard to the range and
accuracy of the measurements. For example, Kappler et al.
(2011) reported a method with a relatively low accuracy of 0.5
to 3 min (0.8 to 5%). Kim et al. (2013) discussed the limited
accuracy of high-resolution X-ray rocking-curve measurements on a 6 inch (152.4 mm) single-crystal sapphire wafer
with a surface miscut angle less than 3 deviation.
A particular treatment of single-crystal orientation is given
by Kikuchi (1990), resulting in two orientation parameters in
one equation, instead of two equations. This is a drawback to
the derived formulae for orientation of the crystal, the
application of which becomes problematic when the order of
the measurements is reversed. One is supposed to get the
same result when the measurement order is reversed, index 1
swapped with index 2. The explanation of this effect is that the
derivations of the formulae included the approximations for
small-angle deviations and cannot be used for high-precision
orientation measurements. Using !-scan measurements
Hildebrandt & Bradaczek (2004) describe the experiment
https://doi.org/10.1107/S2053273317017594
1 of 7
research papers
Figs. 1 and 2 describe schematically the X-ray diffraction from
an arbitrarily tilted lattice plane, an (hkl) crystallographic
plane, outlined as a green parallelogram in a single-crystal
sample shaped as a cube, when the cube is at a position
marked as = 0 (Fig. 1) and when it is rotated counterclockwise 90 around the axis (Fig. 2). The axis is
perpendicular to the cutting face of the crystal that is facing
the X-ray source. For convenience we select the central
section of the cube, the square plane ()0 in the middle, to be
exposed to X-rays just in the centre of the square outlined in
red. The centre of the cube is the origin of the system of
coordinates, whose Ox, Oy, Oz axes are chosen to be parallel
to three converging edges of the cube (see Figs. 1, 2 and 3).
The incident X-ray beam that is fixed in the horizontal plane is
shown by the blue line with an arrow pointing towards the
middle of the cube. The diffracted X-ray beam is also coloured
in blue and marked with an arrow that is pointing out. The two
blue lines determine the X-ray incident plane that is not
necessarily parallel to the horizontal plane (H) since their
bisector, the diffraction vector, is pointing either above or
below the horizontal plane. In both figures the incident plane
is designed to be perpendicular to the diffracting plane (hkl)
by just showing the two symmetric Bragg angles. In addition,
the small squares shown in perspective, at the foot of each
projection line, demonstrate the orthogonality of the incident
plane on the diffracting (hkl) plane. The intersection of the
incident plane with the (hkl) plane is marked with a blue
dotted line. The red dotted line is horizontal while the blue
line, the projection of the incident beam on the (hkl) plane, is
not. The two dotted lines that have a common point in O, the
origin of the system of coordinates, form in general a plane
that is not parallel to the horizontal plane (H). We see that the
diffraction selected in the centre of the cube is giving the same
orientation information as the diffraction on the surface of the
cube, for example the square of the cube facing the X-ray
source. By rotating the cube around the h or w axis we get a
maximum of the reflection at angles 1 (Fig. 1) and 2 (Fig. 2).
In Fig. 1 1 is shown as the angle between the incident beam
and the cutting plane ()0 which is the same as the angle
between the direction of the incident beam and the horizontal
red dotted line of the middle square. The other parameter 2
is shown to be defined similarly in Fig. 2 when the cube is
rotated 90 counter-clockwise around the v axis. It is impor-
Figure 1
Figure 2
settings for the orientation of quartz crystals with the precision
limited to three decimal places ( 0.001 ). We notice that the
limitations found in the methods cited are associated with
approximations assumed in the measuring methods that we
expect can be dramatically improved. For this reason we
developed a method of parametric equations for X-ray crystal
orientation that eliminates that inconvenience. The method is
based on exact equations derived for all orientation parameters. For convenience, they are represented on a stereographic projection. In this way the equations can be derived
without approximations or assumptions that can alter the
precision. For X-ray diffraction goniometry it is desirable to
have the highest precision of the goniometer, in order to
maximize the overall precision. It is important to mention that
samples with the same deviation ’, a parameter that will be
described in detail later, cannot be differentiated for orientation by the old method because of a missing parameter. The
missing parameter, v, marked as a vector, was found as an
extra parameter that is required in the rotation operations of
the sample in its own plane. The rotation operator, v,
described later in the text, completes the method and establishes the fundamentals of crystal orientation without the
approximations present in the earlier methods.
2. Development of parametric equations for singlecrystal orientation
Description of the experimental parameter
crystal.
2 of 7
Danut Dragoi
1
for a cube-shaped single
The fundamentals of crystal orientation
Description of the experimental parameter
crystal.
2
for a cube-shaped single
Acta Cryst. (2018). A74
research papers
tant to notice that the angles and do not change in
amplitude while the sample is rotated. In Fig. 1, is defined in
the horizontal plane (H) as the angle between the two horizontal lines obtained as an intersection of ()0 , the middle
vertical plane, and the lattice plane (hkl) with the (H) plane. In
Fig. 1 the angle is similarly defined in the vertical plane (V)
as the angle between the two vertical lines obtained as an
intersection of the middle plane ()0 and crystallographic
plane (hkl) with the vertical plane (V). The situation is
reversed in Fig. 2, when is swapped with . Both and parameters defined in Figs. 1 and 2 are the unknowns in the
system of equations (1) and (2) found earlier by Dragoi
(1992). Beyond the derivation of equations (1) and (2), given
in Dragoi (1992), we mention that for a parallel family of
planes of an object we need two parameters to characterize
the surface orientation. We found that the two parameters to
characterize the surface orientation of single crystals are and
, the two unknowns of equations (1) and (2). The system of
equations (1) and (2) has interesting properties such as: on
swapping the index 1 to 2 and 2 to 1, goes to , and goes to
, making the system invariant as expected. After this
operation one equation takes the form of the other, and the
solution is unique and real. Using equations (1) and (2) and
Figs. 1 and 2, it can be shown that the measurement parameters
cannot be used directly to get the orientation
parameters and . This is simply because the Bragg angle in
most cases is not in the horizontal plane of (H), and cannot be
used in a linear combination with to precisely determine and . This is the net distinction between the old methods and
our method. It is necessary to go through two equations to get
Figure 3
Definition of the variables ’ and .
Acta Cryst. (2018). A74
the orientation parameters and first. Exact solutions of a
system of nonlinear equations can be obtained with a precision of more than two or three decimal places in current
mathematical methods. Equations (1) and (2) below are shown
with absolute values in square brackets for convenience:
sinð þ
sinð þ
1Þ
2Þ
¼
¼
sin sinð þ
2
sin ð þ
2Þ
sin sin2 sin sinð þ
2
sin ð þ
1Þ
2Þ
2
2
1Þ
sin sin2 1=2
ð1Þ
1=2 :
ð2Þ
Fig. 3 introduces the definition of the total deviation ’,
which is the angle between the crystallographic plane (hkl),
the parallelogram with coloured green sides, and the plane
()0 , the square with red sides in the middle. Because the ()0
plane is parallel to (), the square of the cube facing the X-ray
source makes the same angle ’ with the lattice plane (hkl). To
show the sides of the angle ’, in Fig. 3 we select the point M on
one corner of the middle square and trace the perpendicular
MP, with P on the intersection line of the planes ()0 and
(hkl). We also trace the line PQ perpendicular on the same
line, with Q on the (hkl) plane, and obtain the angle between
MP and QP as ’, the angle between the planes ()0 and (hkl).
By extending two sides of the parallelogram, the (hkl) plane,
see Fig. 3, we can bring the intersection line, ()0 \ (hkl)
(coloured red), in the cutting plane () as the line c = () \
(hkl). In this way we can define the angle as the angle
between the line c and the horizontal edge of the cube (see
Fig. 3).
Therefore, the scalar is shown as an angle for the slope of
line c that can be associated with the rotation operation
around the vector perpendicular to the cube side facing the
X-ray source. Now we have all elements to show a mathematical relationship between ’ and and . Analysing Figs. 1
and 2 there is not an obvious connection between these three
variables, , and ’.
To find a relationship between these variables we use Fig. 4
which is an extended stereographic projection of the two
planes, the cutting plane (), which is the small circle at the
base of the projection, and the (hkl) plane, the large circle
corresponding with the lattice plane tilted with an angle ’
from the base plane. The angle ’ between circles appears with
its true value ’ conserved due to one of the stereographic
projection properties. In this projection it is important to
mention the spherical triangles that contain the orientation
parameters needed. For each spherical triangle, shown as a
stereographic projection in Fig. 4, we apply the cosine rules for
sides and the law of sines (Wolfram Math World, 2017). We
note that the magnitude and direction of angle , defined in
Fig. 3 as the angle between the horizontal edge of the square
and line c, appear to be similar in the stereographic projection
of Fig. 4. In stereographic projection, Fig. 4, the angle is
shown at about 22 which is similar to that in Figs. 1 and 3. The
horizontal line in Fig. 4 could be any horizontal plane and/or
edge of the cube perpendicular to the h, w axis. Looking at
equations (1) and (2) we recognize that the 1 variable is
Danut Dragoi
The fundamentals of crystal orientation
3 of 7
research papers
adding variable as one argument of the sine function since
they are on the same horizontal plane (H). This formally
means that the 1 variable, the experimental variable, is
measured on the same scale as the variable. In the general
case 1 and 2 are the angles measured between the incident
X-ray beam S0 and the cutting plane () when the maximum
of reflection is reached (Dragoi, 1992). This definition holds
for both oriented and disoriented crystals. In this case ’s are
measured on the same scale as the B scale. Therefore, the
alignment of the X-ray beam direction of I0 intensity can be
done for both B’s and ’s when the surface of the sample at 0
with I0/2 intensity is parallel to the incident X-ray beam.
Similarly, the other half of the X-ray intensity having the
direction parallel to the surface can be obtained at 180
rotation of the sample. This classical procedure for alignment
of a diffractometer works well for validation of equations (1)
and (2). We notice the solutions and are expressed in the
same units as the Bragg angle and measurements. In Fig. 4,
the projected spherical triangle whose sides are (, , 90 )
can be solved for and . In this case we apply the cosine rules
for sides as given by Wolfram Math World (2017) and obtain
equation (3):
cos ¼ cos cosð90 Þ:
ð3Þ
For the same triangle we apply the law of sines as given by
Wolfram Math World (2017) and obtain equation (4):
sin ’ sin 90
¼
sin sin ð4Þ
which is equivalent to equation (4a)
sin ¼
sin :
sin ’
ð4aÞ
Since is not a variable of interest we can eliminate it by
adding the square of equation (3) and the square of equation
(4a) and obtain equation (5):
1 ¼ cos2 sin2 þ
sin2 :
sin2 ’
ð5Þ
We obtain similar relations from the other spherical triangle
whose sides are (, , ) and it is linked through the ’ angle
with the previous triangle (, , 90 ). For this triangle we
apply the same procedure using the cosine rules for sides as
given by Wolfram Math World (2017) and obtain equation (6):
cos ¼ cos cos :
ð6Þ
Similarly, we apply the law of sines as given by Wolfram Math
World (2017) and obtain equation (7):
sin ¼
sin :
sin ’
ð7Þ
Again, adding the squares of equations (6) and (7) we eliminate the unnecessary variable as shown in equation (8):
1 ¼ cos2 cos2 þ
sin2 :
sin2 ’
ð8Þ
Separating sin2 and cos2 in equations (5) and (8) we get
equations (9) and (10):
sin2 1
ð9Þ
sin2 ¼ 1 2
sin ’ cos2 sin2 1
:
cos2 ¼ 1 2
sin ’ cos2 ð10Þ
By adding equations (9) and (10) we can eliminate variable and obtain a relationship between ’ and and . After some
algebraic manipulation and changing 1=cos2 for 1 + tan2
and 1=cos2 for 1 + tan2 we obtain equations (11), (11a),
(11b), (12):
1
ð11Þ
1 ¼ 1 þ tan2 þ 1 þ tan2 2 tan2 þ tan2 sin ’
1
tan2 þ tan2 ¼ 0
sin2 ’
1
2
2
¼ 1
tan þ tan 1 2
sin ’
tan2 þ 1 þ tan2 tan2 þ tan2 ¼ tan2 ’:
ð11aÞ
ð11bÞ
ð12Þ
Equation (12) represents the relationship of the variables and and ’, with and determined as the solution of the
system of equations (1) and (2).
Using equations (9) and (10), we can derive equations (13)
and (14):
Figure 4
Extended stereographic projection of two spherical triangles with
orientation parameters.
4 of 7
Danut Dragoi
The fundamentals of crystal orientation
tan ¼ sin tan ’
ð13Þ
tan ¼ cos tan ’:
ð14Þ
As we can see, equations (13) and (14) are solutions of
equation (12). In this case equations (13) and (14) are called
exact parametric equations of crystal orientations. The parametrization of equation (12) was obtained with an independent variable , which is the rotation of the sample in its own
Acta Cryst. (2018). A74
research papers
Table 1
The test of equations (12) and (15) using data from Dragoi (1992).
( )
1
( )
" ( )
n
( )
( )
( )
’ ( )
14.23
14.23
14.36
14.14
17.72
10.74
0.0001
0.0001
89
90
0.100
0.120
3.487
3.493
0.031
0.090
3.493
3.499
12.83
12.83
13.00
12.67
16.19
9.46
0.0001
0.0001
79
80
0.151
0.179
3.363
3.367
0.086
0.054
3.370
3.376
34.6
34.6
35.02
34.16
35.43
33.75
0.0001
0.0001
1
1
0.419
0.441
0.832
0.848
0.671
0.046
0.932
0.956
2
( )
plane. If we wish to determine the amplitude of angle , we
take the ratio of equations (13) and (14) and get equation (15):
tan ¼
tan :
tan ð15Þ
Notice from Fig. 1, when ’ 2:5 equation (15) provides a
slope of about 22 for line c, which is seen at about that angle
in Fig. 3.
3. Testing equations with data from the literature
We can test equations (12) and (15) with data from the
literature. For example, using the data in Table 4 of Dragoi
(1992) gives the values for and ’ shown in the last two
columns of Table 1.
The variables in Table 1 are: ( ) is the Bragg angle, 1 ( ) is
the first reflection reading on the scale when the 2 rotation
is decoupled from (the detector is fixed), 2 ( ) is the second
reflection reading on the scale, similar to 1( ), when the
wafer/sample is rotated counter-clockwise 90 in its own plane,
" ( ) is the chosen precision of the final solutions of equations
(1) and (2) using a numerical method described by Dragoi
(1992), n is the number of iterations, ( ) and ( ) are the
solutions of equations (1) and (2), ( ) and ’ ( ) are the
values obtained with equations (15) and (12), respectively.
As a comment, the values of ( ) tend to be close to the
horizontal line parallel to the flat surface and cutting plane of
two Si wafers, which have different orientations, (111) and
(004), as used by Dragoi (1992). As we can see, the angle ’
tends to be close to 3.5 on (111) Si wafers and about 1 on
(004) Si wafers. The precision for angle ’ in the first two
samples of (111) Si wafer was 0.006 and 0.08 on the third
sample of (004) Si wafer. As we can see from Table 1, only two
measurements are needed to determine the orientation for
one wafer. The values of the first seven columns in Table 1
were obtained by solving the equations using the Newton
method. Now the same values can be obtained using an Excel
spreadsheet that has the Solver capability and the GRG
(generalized reduced gradient) subroutine. Details on how to
apply this new solving technique can be found in the
Appendix. This subroutine is available in almost all computers
and simplifies the work of diffractionists, helping to quickly
find the parameters they need. We think this is an important
breakthrough in applying the GRG subroutine to the parametrization method for crystal orientation.
Acta Cryst. (2018). A74
4. Workflow for single-crystal orientation using X-ray
diffractometers
Fig. 5 shows a workflow of formulae to follow when determining single-crystal orientation using X-ray diffractometers.
In the first step solve the system of equations (1) and (2) using
GRG (see the Appendix). The equations in Fig. 5, (12), (13)
and (14), that were described earlier in the text, are used for
calculating the maximum deviation ’ and the direction of
angle . The symbol ^ is used as the angle between two planes,
(hkl) the lattice plane and () the cutting plane of the sample;
therefore we represent ’ as ’ = () ^ (hkl). The values
obtained in Table 1 were obtained with the GRG subroutine
too. Following the steps in Fig. 5, Table 1 serves as an excellent
template and guide for checking the entire method proposed.
The diffractionist willing to apply this powerful method has
to be careful with the diffractometer slits on the front of the
detector in order to keep the peak shape unaltered.
5. Discussion
A set of orientation parameters describes completely a
method for high-accuracy X-ray crystal orientation. For a
given crystal, we select the crystallographic (hkl) plane closest
to the plane section in the crystal. For the (hkl) plane selected
the Bragg angle B is known and will be treated as a constant
not a variable. As seen from the derivation, two experimental
variables 1 and 2 are needed to find and , the orientation
Figure 5
A flow chart for work on parametric equations for X-ray crystal
orientation.
Danut Dragoi
The fundamentals of crystal orientation
5 of 7
research papers
variables, which are the only unknowns of the system of
equations (1) and (2). Since the two unknowns and satisfy
equation (12) we obtain automatically the maximum deviation
’ of the cut of the crystal. In this situation we find and as
two components of the angle ’. In contrast with the old
methods, and are not the measurement variables. The
precision we get on variables and is much better than that
with current methods described by Hildebrandt & Bradaczek
(2004), Kikuchi (1990), Kim et al. (2013), Kappler et al. (2011)
and Uwe & Armin (2007). Table 3 in Dragoi (1992) shows the
precisions for and to four exact decimal places. The effect
of the goniometer accuracy is seen in Table 4 of Dragoi (1992)
as and are listed to three exact decimal places. In Table 1 of
this work, the calculations were made using an Excel spreadsheet (2010 version) that applies a Solver subroutine called
GRG. We found that the GRG method successfully replaces
the Newton method used previously by Dragoi (1992). The use
of Solver as in Excel 2010 or later versions produces reproducible and excellent results.
The derivation of a parametric representation of nonlinear
equations for X-ray crystal orientation presented here is based
on an extended stereographic projection (Fig. 4). The system
of coordinates chosen in this work is right-handed and its role
is to guide us in finding the right way to calculate the angles of
interest. The origin of the system of coordinates Ox, Oy and
Oz coincides with the centre of the cube. It is also obvious that
the centre of the base circle for stereographic projections is on
the Oy axis of the system of coordinates. In the calculations we
do not have assumptions on space constraints and limitations
on measurement variables that can affect the precision of the
method. The experimental setup is well defined and ready to
be used in many diffractometric systems, like /2, /0 and
90 , /.
From a mathematical point of view, the equations of
orientation are well behaved with no singularities. There are in
total five equations, two for determining the position of the
crystallographic plane in the crystal, equations (1) and (2), one
equation for the total deviation ’, equation (12), and two
equations for parametrization, equations (13) and (14). We
think this approach and description give a better insight and a
complete representation of the determination of all aspects of
single-crystal orientation utilizing X-ray diffraction and a
goniometer. It is worth mentioning that the set of variables
(’, ) can determine the cutting plane of the crystal.
In our work we did not include the analysis of Laue
methods for crystal orientation using EDXRD (energydispersive X-ray diffraction) because that has been treated
elsewhere (Uwe & Armin, 2007). Briefly, according to Uwe &
Armin (2007) a polychromatic X-ray beam can be collected by
a PIN diode (25 mm2) with a resolution of 260 eV. The
physical characteristics of the PIN diode used as an energydispersive detector such as its energy resolution and pixel size
can affect the precision of the measurements. In the Laue
method the ’ angle deviations are limited to 37 . We note
that the size dimension of Laue spots is a limiting precision
factor of the method as it does not work for quartz crystal
orientation (Uwe & Armin, 2007).
6 of 7
Danut Dragoi
The fundamentals of crystal orientation
An interesting discussion on many aspects of technology for
X-ray single-crystal measurement is given by Kappler et al.
(2011), where a high-precision vertical goniometer assures an
angle reproducibility of 0.001 for the XRD-6000 machine
and 0.0002 for the XRD-7000 machine.
Regarding the X-ray beam size cross section, we found that
the thinnest beam possible that can produce a detectable
reflection is sufficient to complete an experiment with the
method described.
6. Conclusion
Exact solutions of crystal orientation by solving the nonlinear
equations for X-ray crystal orientation show on a theoretical
test a precision value in the range of four decimal places,
0.0001 , and measured precision better than three decimal
places (Dragoi, 1992). The equations derived in this work
show a complete description of the orientation of single
crystals. Using an extended stereographic projection, we
derived the exact expression of the maximum deviation ’ of
the orientation and added parameter for in-plane differentiation of samples with constant ’. We also introduced the
parametrization of each component of the maximum deviation
of the orientation. The exact solutions of the equations give
higher precision of the orientation and guide the hardware
goniometry towards the best performance. The symmetry of
equations (1) and (2) in the text shows that these equations
are invariant to the swap variable operation, which removes
the inconvenience found in previous methods (Kikuchi, 1990).
We made the distinction between the measurement variables
of old methods and exact solutions through the equations of
orientation. We introduced the concept of the angle
Figure 6
Solver screenshot in Excel 2010 for solving equations (1) and (2).
Acta Cryst. (2018). A74
research papers
measurement parameter between the incident X-ray beam
direction and any plane cutting surface of a single crystal, the
parameters. The dimensionless nature of the X-ray beam
cross section used in this work suggests the possibility and high
potential to use point-by-point measurements on surfaces of
samples at the microscopic level.
The derivation of formulae combined with experimental
verification establishes a significant method free of approximations and assumptions for orienting single crystals.
The computerized capability of any laboratory today allows
full application of this method. The results obtained in this
work were successfully checked, guaranteeing that the method
has great potential in many new applications of crystallography, like microelectronics, crystal growth characterization, protein single crystals, X-ray goniometry and materials
science.
APPENDIX A
Equations (1) and (2) were previously solved numerically
by Dragoi (1992), using the Newton method. A new method
called GRG that can be found in later Excel versions, for
example the 2010 version, can be successfully applied. Fig. 6
shows a screenshot of the Solver subroutine in Excel (2010
version). We give a short description of the method that the
user can easily apply. Transform equations (1) and (2) as F1
and F2 equations as suggested by Dragoi (1992):
F1 ¼ sinð þ
F2 ¼ sinð þ
1Þ
2Þ
sin sinð þ
2
sin ð þ
2Þ
sin sin2 sin sinð þ
2
sin ð þ
1Þ
2Þ
2
2
1Þ
sin sin2 1=2
ð16Þ
1=2 : ð17Þ
Equations (16) and (17) are easy to calculate in an Excel
spreadsheet. For this calculation we need a table similar to
Table 1 in which we add step by step the following numerical
values:
(i) Add the Bragg angle in the first column of the table in a
new Excel spreadsheet. This value is not a variable; it is a
constant that can be found in tables or can be calculated.
(ii) Add the two measurements 1 and 2, and skip the
columns of " and n.
Acta Cryst. (2018). A74
(iii) Initialize the solutions and with some arbitrary
numbers. In some situations it is better to select values close to
the difference between 1 and 2.
(iv) In two separate cells outside the spreadsheet table,
evaluate F1 and F2, which are given by equations (16) and (17).
Next to these cells calculate the sum F1 + F2.
(v) Call Solver in Excel, which is in the Data menu, and
select as target, Set Objective, the cell with sum F1 + F2. Select
the radio button Value Of: as zero. In the field By Changing
Variable Cells: select both cells of and that will change
automatically their values when Solver is run. In the Subject to
the Constraints area add one by one the cells where F1 and F2
are calculated. Leave the checkbox Make Unconstraint Variables Non-Negative unchecked. In the field of Select a Solving
Method chose GRG Nonlinear. Select Options, and get a new
window in which you type the value of 0.0001 in the Constraint
Precision field for All methods. Then check the box Use
Automatic Scaling, then select next GRG Nonlinear tab and
enter the value 0.0001, select radio button Forward on Derivatives field and select the checkbox Require Bounds on
Variables. In this work we did not try the Evolutionary
method.
(vi) Run Solver by clicking on Solve. After a short period of
time a new window will open with the information that the
solution was found. The new numerical values under and are the solutions of equations (1) and (2).
References
Dragoi, D. (1992). J. Appl. Cryst. 25, 6–10.
Hildebrandt, G. & Bradaczek, H. (2004). J. Optoelectron. Adv. Mater.
6, 5–21.
Kappler, R., Lydon, D. & Turnquist, T. (2011). Use of X-ray
Diffraction Analysis to Determine the Orientation of Single-Crystal
Materials. American Laboratory, published online 8 April 2011.
https://www.americanlaboratory.com/914-Application-Notes/1593Use-of-X-ray-Diffraction-Analysis-to-Determine-the-Orientationof-Single-Crystal-Materials/.
Kikuchi, T. (1990). Rigaku J. 7, 27–35.
Kim, C. S., Bin, S. M., Jeon, H.-G., O, B. & Choi, Y. D. (2013). J. Appl.
Cryst. 46, 1298–1305.
Uwe, P. & Armin, G. (2007). EDXRD for Out-of-the-Box Crystal
Orientation. Bruker Corporation.
Wolfram Math World (2017). Spherical Trigonometry, http://
mathworld.wolfram.com/SphericalTrigonometry.html.
Danut Dragoi
The fundamentals of crystal orientation
7 of 7