SPuDS Users Guide

SPuDS Users Guide
Written and developed by
Dr. Michael W. Lufaso
e-mail: michael.lufaso@unf.edu
Dr. Patrick Woodward
e-mail: woodward@chemistry.ohio-state.edu
Document update: August 28, 2014
Preface
Disclaimer
By downloading the software programs SPuDS (Structure Prediction Diagnostic
Software) and (TUBERS) (Tilting Using Basic Equation Relationship Software), you
agree to the terms and conditions concerning its use specified in the license agreement
that is provided as part of the distribution. The Software is provided to End User on an as
is basis. No user support is provided or implied. No warranty, express or implied to end
user or to any other person or entity is made.
Acknowledgements
MWL would like to thank Dr. Patrick Woodward in the initialization and assistance
throughout the development of SPuDS. Resources provided by the Ohio State University,
Columbus, OH are acknowledged. Discussions with I.D. Brown (McMaster University)
are appreciated.
Using SPuDS
Publication of results obtained using the program SPuDS or TUBERS should state that
SPuDS and/or TUBERS were used and contain the following reference:
M.W. Lufaso, P.M. Woodward, (2001) "The Prediction of the Crystal Structures of
Perovskites Using the Software Program SPuDS" Acta Cryst., B57, 725-738.
Requests
Requests for additional features (e.g. multiple doping levels on A- and M-site), additional
explanations in the User Manual, tilt angle calculation or any other additional comments
and questions may be addressed to Michael Lufaso (michael.lufaso@unf.edu) and Patrick
Woodward (woodward@chemistry.ohio-state.edu). The software is continually under
development, but research areas that are requested will be addressed first.
SPuDS
2
Contents
1. Introduction
1.1 SPuDS description
1.2 Installation
1.3 Octahedral Tilting Notation
1.4 SPuDS calculation algorithm
2. Crystal Structure Calculation
2.1. Calculation instructions
2.2. Sample calculation of CaTiO3
2.3. Output files
2.3.1. Information contained in output
2.4. Variable temperature crystal structure calculation
2.5. Disordered M-site calculation and Variable Composition
2.6. Jahn-Teller and Octahedral Distortions
2.6.1. Disordered M-site
2.6.2. Ordered M-site
2.7. Pyrochlores
2.8. Anti-perovskites
2.9. Frequently Asked Questions
3. Tilt Angle Calculation using TUBERS
3.1. Overview
3.2. Single M-site perovskites
3.2.1. I4/mcm
3.2.2. Pnma
3.3. Multiple M-site perovskites
3.3.1. SPuDS text version [22 (a0a0c-) and 14 (a-a-a-)]
3.3.2. TUBERS [#10 (a-a-b+) space group P21/n]
3.3.3. TUBERS [#20 (a0b-b-) space group I2/m]
4. Bibliography
5. Appendix
5.1. A
5.2. B
SPuDS
3
Chapter 1
Introduction
1.1
Overview of SPuDS
The software program SPuDS has been developed to predict the crystal structures
of perovskites, including those distorted by tilting of the octahedra. The user inputs the
composition and SPuDS calculates the optimal structure in ten different Glazer tilt
systems for single M-site cations and six for 1:1 rock-salt ordered double perovskites.
This is performed by distorting the structure to minimize the global instability index,
while maintaining rigid octahedra. The location of the A-site cation is chosen so as to
maximize the symmetry of its coordination environment. In its current form SPuDS can
handle up to four different A-site cations in the same structure and two octahedral ions.
Structures predicted by SPuDS are compared with a number of previously determined
structures to illustrate the accuracy of this approach.
Additional details may be found on the following webpage.
http://www.unf.edu/~michael.lufaso/spuds/index.html
1.2
Installation
Click the setup.exe file and install in any folder of your choice. The default
location for installation is C:\Program Files\SPuDS\. Note that you will need to open the
folder containing the spud.exe and tubers.exe files to examine the output files or any log
files. Copy and rename the output.txt file to keep to calculated crystal structure for later
comparison. If you experience error messages regarding "MSVBVM60.DLL file not
found", then unzip the Visual Basic run-time DLL msvbvm60.zip to the Windows\system
(95,98,ME) or Windows\system32 (XP) if it not already installed on your PC. Restart the
computer, and then execute SPuDS again.
1.3
Octahedral Tilting Notation
Octahedral tilting reduces the symmetry of the A-site cation coordination
environment and results in a change in A-X bond lengths. However, there are multiple
ways in which the octahedra can tilt, each leading to a different coordination environment
for the A-site cation(s). A standard notation has been developed to describe octahedral
tilting distortions in perovskites by Glazer (1972). The notation describes a tilt system by
rotations of MX6 octahedra about any of three orthogonal Cartesian axes, which are
coincident with the three axes of the aristotype cubic unit cell. The notation specifies the
magnitude and phase of the octahedral rotations. The letters in Glazer notation indicate
the magnitude of the rotation about a given axis, e.g., the letters a, b, and c imply unequal
tilts about the x, y, and z axes. A superscript is used to denote the phase of the octahedral
tilting in neighboring layers. A positive superscript would denote the neighboring
octahedra tilt in the same direction (in-phase) and a negative superscript implies the tilts
of neighboring octahedra tilt in the opposite direction (out of phase). A superscript of 0
signifies no tilting about that axis. Rotation of one octahedron causes the four adjacent
octahedra in the same layer to rotate in the opposite direction in the same angle. Lattice
connectivity is such that rotations of the octahedra in the layer above and below are not
SPuDS
4
geometrically constrained to the initial rotation and can occur in phase (+ superscript) or
out of phase (- superscript) with respect to the first octahedral layer. The tilting of the
octahedra reduces the symmetry of the undistorted perovskite tilt system a0a0a0. Glazer
derived 23 different tilt systems, which lead to 15 different space groups. There are 15 tilt
systems that can occur in real crystals, each with a different space group for simple
perovskites. Ordered double perovskites, A2MM'O6, have also been described by the
Glazer tilting notation.
The order of the tilting notation is somewhat arbitrary in crystal structures which
don't have the crystallographic axis parallel to the M-X-M bonds. For example, consider
a perovskite in space group Pnma (tilt system #10, a-b+a-). Looking down the a-axis one
notices, because of the √2ap × 2ap × √2ap cell size relative to the aristotype cubic unit cell,
it does not correspond to any of three orthogonal Cartesian axes which are coincident
with the three aristotype axes. However, looking down the b-axis it is readily apparent
that the octahedral tilting is in-phase (+). The out of phase tilting is in the ac-plane, so ab+a- would be a logical choice.
Space group Pnma perovskite (a-b+a-) #10.
Perovskites crystallizing in space group Pbnm are better described by the tilting
notation a-a-b+. An example is shown below. Similarly, 1:1 ordered double perovskites in
space group P21/n are best described by octahedral tilting notation a-a-b+ (#10).
SPuDS
5
Space group Pbnm perovskite (a-a-b+) #10.
Space group P21/n perovskite (a-a-b+) #10.
1.4
SPuDS Calculation Algorithm
A brief description of the calculation algorithm is given, however the interested
reader is referred to Lufaso & Woodward (2001) for a more detailed description.
Additions to the software have been added that are not part of the version 1 of the
software. SPuDS Calculation Procedure
Optimization of the crystal structure in an unconstrained manner requires
determination of the unit cell dimensions and all free positional parameters. In order to
simplify this process SPuDS restricts the octahedra to remain rigid (six equivalent M-X
distances and all X-M-X angles equal to 90º). This seems to be a reasonable restriction in
light of the fact that most distorted perovskites show very little distortion of the MX6
octahedra, though obviously SPuDS will not work well in systems where octahedral
distortions are expected. Once this restriction is in place the full crystal structure can be
SPuDS
6
generated from two variables: the size of the octahedron and the magnitude of the
octahedral tilting distortion. Additional degrees of freedom must be taken into
consideration in those tilt systems where either the A-cation does not sit on a fixed
position (e.g., a-b+a-) or octahedral distortions cannot be avoided (a+a+c-). SPuDS
calculates the fractional position of each atom at each tilt angle step over a wide range of
octahedral tilt angles. The initial optimization is coarse to allow a wide range of
structures to be calculated quickly and the tilt angle increment is finer for successive (35) optimization routines resulting in a determination of the optimal tilt angle.
The size of the octahedron and the optimum magnitude of the octahedral tilting
distortion are calculated utilizing the bond valence model, which is used to quantitatively
describe inorganic bonding in ionic solids (Brown, 1978). The bond valence, sij,
associated with each cation-anion interaction is calculated using
, where dij is the cation-anion distance. The B parameter is empirically determined, but
can often be treated as a universal constant with a value of 0.37. Rij is empirically
determined for each cation-anion pair based upon a large number of well-determined
bond distances for the cation-anion pair in question. SPuDS uses modern bond valence
parameters from Brown, 2004. The atomic valences, Vi(calc), of the A and M cations and
X anion are calculated by summing the individual bond valences (sij) about each ion.
Six nearest neighbor anions are used for the M cation, six nearest neighbor cations
for the X anion and twelve nearest neighbor anions for the A-site cation are used in the
calculations. No assumption is made about the coordination number of the A-site cation
(valences for twelve A-X interactions are calculated in all cases), but the contribution to
the atomic valence sum becomes smaller as the A-X bond distance increases. The B
cations remain at fixed positions in all space groups generated by simple tilting of the
MX6 octahedra. The M-X bond distance, which determines the size of the octahedron, is
calculated so as to optimize the bond valence sum of the octahedral cation. The X-M-X
bond angles of the MX6 octahedra remain ideal (90°) and the M-X bond distances are
held constant in the calculations. The valence sum of the A-site cation is varied by
changing the size of the octahedral tilting distortion.
The optimized structure is one where the difference between the calculated bond
valence sum and the formal valence (equal to its oxidation state) of each ion is
minimized. This value, which is termed the discrepancy factor (Rao et al, 1998), d, is a
measure of the lattice strains present in the compound. The discrepancy factor is
calculated according to
, where Vi(ox) is the formal valence and Vi(calc) is the calculated bond valence sum for
the ith ion. The overall structure stability is determined by comparing the calculated bond
valence sums with the ideal formal valences. This quantity is referred to as the global
instability index (GII) and is calculated according to (Salinas-Sanchez, 1992):
SPuDS
7
The variables involved in the GII equation are discrepancy factor (di) and N,
which is the number of atoms in the formula unit. During the optimization process the
octahedral tilt angle is stepped incrementally and the individual A-X and M-X bond
distances, discrepancy factors and global instability index are calculated at each step.
After the first optimization process, the procedure is repeated using smaller tilt angle
increments in order to minimize the GII. The stability of perovskite compositions with
different atoms, symmetry, tilt systems and structure can be evaluated by comparing the
GII. The GII value is typically <0.1 valence units (v.u.) for unstrained structures and as
large as 0.2 v.u. in a structure with lattice induced strains. Crystal structures with a GII
greater than 0.2 v.u. are typically found to be unstable, and reports of such structures are
usually found to be incorrect.
SPuDS
8
Chapter 2
Crystal Structure Calculation
2.1
Simple Calculation Instructions
The SPuDS software is composed of several menus. User input is required at
each of these steps described below.
Menu
Menu 1 allows stoichiometry selection for single M-site cations simple
perovskites, 1:1 ordered double perovskites, and disordered perovskites. Select the menu
by clicking the number of the formula of interest (1-23). A box will appear with the
menu number.
Tilt System
Menu 2 selects the tilt system for which the output results will be displayed. A
further description of the Glazer notation may be found in the literature and in the paper
detailing the first version of SPuDS. Click the number of the tilt system of interest, or
choose 23 for the untilted perovskite. A box will appear with the number of the tilt
system. At this point the appropriate boxes for the A-site and M-site cation will appear.
Formula
Input of the ions and oxidation states is performed by clicking the drop down
boxes that appear depending on the Menu selection. Selection the ions and the oxidation
states for each of the cations and anions.
'Calculate'
Clicking calculate passes the entered values to the calculation subroutines. Note
that error checking may not determine all errors in the input and crash the software. Email the author if a problem persists.
Options
Define Tilt Angle?
Certain tilt systems (currently # 22, 21, 20, 17, 16, 14, 10, 3) with a single B-site
cation) allow the user to define the tilt angle instead of SPuDS calculating the bond
valence sum optimized tilt angle. In general this box should be unchecked.
Define Temperature?
Click the check box to define the temperature. This option changes the bond
distances calculated from the bond valence parameters. The default temperature is 298 K.
Vary M-site in #23?
Allows M-site expansion in tilt system number 23 (space groups Fm-3m and
Pm-3m).
SPuDS
9
Tilt Sign
Adjusts the phase of the tilt angle (positive or negative). This option is not
implemented in all tilt systems.
Jahn-Teller M-site cation?
Allows for using a Jahn-Teller distorted cation (e.g. Mn3+) with unequal M-X
bond distances.
2.2
Sample Calculation of CaTiO3
The mineral perovskite CaTiO3 crystallizes in space group Pnma in tilt system
a b a (Glazer Number 10). The menu selections should look like the following figure.
- + -
Click Calculate the crystal structure. After the calculation procedure completes,
the software should display the results shown in the next figure.
SPuDS
10
2.3 Output files
Several output files are prepared and described below.
2.3.1 output.txt
This file is shown in the SPuDS main output box. It contains most of the useful
crystal information. The formula, Glazer tilt notation, tolerance factor calculated from
ionic radii and bond valence method, octahedral tilt angle, unit cell volume, Global
Instability Index, temperature of the calculation, bond valence sums, Wyckoff site
occupancies, space group, lattice parameters, fractional coordinates, bond distances and
valences, and M-O-M bond angles.
2.3.2 gii.txt
This file contains the calculated GII for each of the tilt systems. This is displayed
in the rightmost box of the SPuDS program. These values can be used to compare the
relative stability for each tilt system. Note the lowest GII does not necessarily indicate the
most likely tilt system. If the GII is below a certain threshold, effects not considered in
these calculations are the driving for towards crystallization in a particular space group
and tilt system.
2.4 Variable Temperature Crystal Structure Calculation
SPuDS has the capability to perform crystal structure calculations at variable
temperatures. Temperature dependent bond valence parameters are based on ideas
presented in Brown, Dabkowski and McCleary (1997). The dR/dT parameters used in the
calculation of variable temperature bond valence parameters have units of Å K-1. These
parameters modify the bond valence parameter Ro (Brown, 2004) as a function of
temperature using
SPuDS
11
RoT = Ro + (dR/dT)∆T
The values input into SPuDS are given in units of x105, i.e. the user inputs "7.8"
into the text box for the temperature parameter for an A(2+). The default temperature is
298 K, and using this temperature as the input for the text box, results in no requirement
for values in the "T" text boxes. Default temperature parameters values were obtained by
approximation to Fig. 5 in Brown, Dabkowski and McCleary (1997). Values may need to
be changed for each system investigated to accurately reflect the expansion of the bonds
with an increase in temperature.
Oxidation
State
1
2
3
4
5
6
A-site (dR/dT ×105)
M-site (dR/dT)
8
7.8
4.5
-
5
1.9
1.5
1.1
0.9
0.8
A sample calculation is shown for CaTiO3. Check the box "Define Temp. K" and
input the temperature of interest, in this example 1273 K will be used.
SPuDS
12
Note that the octahedral tilt angle decreased from -14.6 degrees to -11.01 degrees.
SPuDS predicted fractional coordinates are given below for a calculation at a temperature
equal to 1273 K.
**********************************************************************
Space Group: Pbnm (#62)
Lattice Parameters (Angstroms)
a = 5.4854
b = 5.5542
c = 7.8051
Atom
Ca
Ti
O
O
Site x
4c 0.5049
4b 0.0000
4c 0.4438
8d 0.7219
y
z
0.5180 0.2500
0.5000 0.0000
-0.0062 0.2500
0.2774 0.0281
**********************************************************************
This may be compared to a high temperature diffraction experimental study of CaTiO3 at
1273 K by Kennedy et al, 1999 shown below. Cell transformations are shown in the
Appendix.
Atom
Ca
Ti
O
O
Site x
4c 0.503(1)
4b 0.0000
4c 0.445(1)
8d 0.7177(5)
y
z
0.519(1) 0.2500
0.5000
0.0000
-0.008(1) 0.2500
0.2802(5) 0.0286(4)
Compare to the SPuDS predicted values, illustrating close agreement between
refined and predicted values, even at 1273 K.
Now consider a phase transition. Calculate the crystal structure of CaTiO3 at 1473
K, but change the space group to Cmcm, which is tilt system 17 (a+b0c-). An out of phase
tilt is no longer present compared to the low temperature crystal structure.
SPuDS
13
SPuDS predicted fractional coordinates for CaTiO3 at 1473 K in space group
Cmcm are:
**********************************************************************
Space group
Cmcm (#63)
Lattice Parameters (Angstroms)
a = 7.7866
b = 7.8492
c = 7.8492
Atom Site x
y
z
Ca
4c 0.0000 0.4894 0.2500
Ca
4c 0.0000 -0.0081 0.2500
Ti
8d 0.2500 0.2500 0.0000
O
8e -0.2187 0.5000 0.0000
O
8f 0.0000 0.2184 0.0316
O
8g 0.2818 0.2520 0.2500
***************************************************
The refined fractional coordinates from Kennedy et al, 1999 are:
****************************************************
Atom Site x
y
z
Ca
4c 0.0000 0.505(5) 0.2500
Ca
4c 0.0000 -0.019(4) 0.2500
Ti
8d 0.2500 0.2500
0.0000
O
8e -0.222(2) 0.5000 0.0000
O
8f 0.0000
0.226(2) 0.032(1)
O
8g 0.280(2) 0.251(2) 0.2500
SPuDS
14
Proceeding to higher temperatures (1523 K) another phase transition occurs to
space group I4/mcm (a0a0c-). The in-phase tilt is no longer present.
SPuDS predicted structural parameters are:
**************************************
Space group I4/mcm (#140)
Lattice Parameters (Angstroms)
a = 5.5098
c = 7.9140
Atom Site x
y
z
Ca
4b 0.0000 0.5000 0.2500
Ti
4c 0.0000 0.0000 0.0000
O
4a 0.0000 0.0000 0.2500
O
8h 0.2056 0.7056 0.0000
**************************************
Experimental refined structural parameters (Kennedy et al, 1999) are:
**************************************
Space group I4/mcm (#140)
Lattice Parameters (Angstroms)
a = 5.4938(2)
c = 7.7815(3)
Atom Site x
y
z
Ca
4b 0.0000
0.5000
0.2500
Ti
4c 0.0000
0.0000
0.0000
O
4a 0.0000
0.0000
0.2500
O
8h 0.2783(5) 0.7783(5) 0.0000
**************************************
SPuDS
15
Note the apparently significant difference in the O position on site 8h. This is
because the octahedral tilting may be arbitrarily positive or negative. In the default case,
the crystal structure has a positive tilt angle. Check the box "Define tilt angle" and
change Tilt Sign from +1 to -1, then recalculate the crystal structure. (Note: this feature
does not work in all tilt systems in the current version of SPuDS, but can be added upon
request.)
Note the tilt angle was +10.07 degrees and is now -10.07 degrees. The crystal
structure parameters are now:
**************************************
Space group I4/mcm (#140)
Lattice Parameters (Angstroms)
a = 5.5098
c = 7.9140
Atom Site x
y
z
Ca
4b 0.0000 0.5000 0.2500
Ti
4c 0.0000 0.0000 0.0000
O
4a 0.0000 0.0000 0.2500
O
8h 0.2944 0.7944 0.0000
**************************************
At a temperature 1623 K, no octahedal tilting is present as detected by diffraction
measurements. SPuDS was used to calculate the crystal structure for the simple primitive
cubic perovskite at 1623 K. A lattice parameter of 3.8933(1) Å was determined by
Kennedy et al, 1999, with the atom positions equivalent. The crystal structure parameters
are shown next.
**********************************************************************
Space group Pm-3m (#221)
SPuDS
16
Lattice Parameters (Angstroms)
a = 3.9592
Atom Site x
y
z
Ca
1b 0.5000 0.5000 0.5000
Ti
1a 0.0000 0.0000 0.0000
O
3d 0.0000 0.0000 0.5000
***************************************************
Comparison of the GII can give some illustration of the phase transitions present in
CaTiO3. The A-site cation increases in size relative to the higher valent octahedral ions.
The effective tolerance factor increases, making the less tilted (and untilted) phases less
unstable compared to tilt system #10 (a-b+a-, Pnma).
CaTiO3 at 298 K
CaTiO3 at 1623 K
Of course the true physics and chemistry is much more complicated than
presented here, but nonetheless it may be useful to understanding phase transitions in
perovskites. The accuracy of the fractional coordinates illustrate how SPuDS may be
used to generate starting structure models in the course of structurally characterizing
materials and also in analyzing known data.
2.5 Disordered Crystal Structure Calculation and Variable Composition
SPuDS has the capability to calculate disordered crystal structures with multiple
A-site and multiple M-site cations for tilt system #10 a-a-b+ (space group Pnma/Pbnm)
and #22 a0a0c- (space group I4/mcm). In its current form, reliable results are only
available for #10 and #22. Use menu 9, 11 or 12 for 1:1 disordered M-site, two A and
two M-site, and two A and three M-site cations, respectively. Simple tilt system #10 has
SPuDS
17
only one crystallographic M-site. An M-X bond distance is calculated according to the
following equations.
b = b(1)*bcomp(1) + b(2)*bcomp(2) + b(3)*bcomp(3)
A new b constant is calculated equal to the weighted average of the three b
constants b(1-3), where bcomp(1-3) is the normalized fractional occupancy.
d = (rb(1)*bcomp(1) + rb(2)*bcomp(2) + rb(3)*bcomp(3)) -b*LOG(
+ [oxb(1)*bcomp(1) + oxb(2)*bcomp(2) + oxb(3)*bcomp(3)]/6.)
The new bond distance is calculated using the equation above. This weighting
scheme has a tendency to result in average bond valence sums of the M-site ions to have
larger than ideal average values.
2.6 Jahn-Teller and Octahedral Distortions
2.6.1 Disordered M-site
Calculation of the crystal structures of compounds containing octahedral
distortions is possible in SPuDS. One only needs to check the appropriate box for JahnTeller M-site. In its current form, only tilt system number 10 (a+b-b-) is supported. Three
boxes should appear (x,y,z).
The description of the Jahn-Teller distortion in combination with the Jahn-Teller
distortion is achieved by applying equations of octahedral tilting to distorted octahedra,
while still retaining nearly 90° O-M-O bond angles. Equations describing the oxygen
fractional positions based on octahedral tilt angle are obtained in a similar method as for
undistorted octahedra, but are more complicated.
Three bond distance factors (x, y, z) are included to modify the magnitude of the
octahedral distortion. Starting values are x = -0.1, y = -0.04, z = 0.18. These values were
chosen to approximate the distortion observed of Mn3+. The resulting Mn-O bond
distances are 1.9765 Å, 1.9194 Å, and 2.1925 Å, which gives a bond valence sum equal
to 3.036 v.u. for Mn3+. The experimental crystal by Rodriguez-Carvajal et al. (1998) has
Mn-O bond distances at room temperature of 1.9680(3) Å, 1.907(1) Å, and 2.178(1) Å.
The bond distance factors influence the bond distances in the following way. A
negative value shortens the M-O bond, whereas as positive value lengthens the M-O
bond. Changing 'x' most strongly influences the M-O2 bond, 'y' the M-O1 bond, and z
the M-O3 bond in the order the M-O bonds are listed in the output file. An example
calculation is shown below for LaMnO3 using the default (-) tilt angle .
Bond Distance Factors
x
0
0
0
-0.05
SPuDS
y
0
0
-0.05
0
z
0
0.18
0
0
Mn-O1
(×2)
2.0165
2.0165
1.9665
2.0166
Mn-O2
(×2)
2.0165
2.0191
2.0165
1.9669
Mn-O3
(×2)
2.0165
2.1926
2.0165
2.0159
BVS
(v.u.)
3.00
2.61
3.14
3.14
18
-0.1
-0.04
Additional examples
-0.05
-0.04
-0.1
-0.08
-0.12
-0.02
-0.1
-0.08
0.18
1.9765
1.9194
2.1925
3.04
0.18
0.22
0.18
0.18
1.9765
1.9365
1.9965
1.9365
1.9692
1.9198
1.8996
1.9191
2.1926
2.232
2.1922
2.193
2.87
3.09
3.05
3.16
The orientation of the observed Jahn-Teller distortion is given using the default
bond distance factors (-0.1, -0.04, 0.18). By changing the value of x from -0.1 to -0.05,
the Mn-O2 bond is lengthened and the Mn bond valence sum decreases to 2.87. The
magnitude of the distortion is adjustable by varying these three parameters.
Consider La0.5Ca0.5Mn3+0.5Mn4+0.5O3, with Mn4+ that often occupies a symmetric
octahedral coordination environment. In the disordered case, the magnitude of the JahnTeller distortion may be expected to be reduced. Using the bond distance factors (-0.07, 0.025, 0.12) the resulting crystal structure would have Mn-O bond distances of 1.9309 Å,
1.8872 Å, and 2.0473 Å and bond valence sums of 3.53 v.u for Mn3+ and 3.47 v.u. for
Mn4+ on the disordered M-site. This approach has the effect of averaging the distortion
around the M-site. The magnitude of the octahedral distortion is difficult to estimate, and
is left up to the user to generate the appropriate octahedral distortion.
When should one use the Jahn-Teller distortion option? There is no simple
answer. For example, La0.5Ca0.5MnO3 (Radaelli et al, 1997) has symmetric octahedra at
T = 300 K, whereas at 1.5 K a short Mn-O bond is formed. Obviously in this case, a
simulation of the room temperature structure would not need a Jahn-Teller distortion,
whereas a low temperature crystal structure is better described by a distorted crystal
structure. A starting point may be (x = 0, y = -0.05, z = 0), which give Mn-O bond
distances of 1.9021 Å, 1.9521 Å, 1.9521 Å in good agreement with the experimental
values of 1.9148(3) Å, 1.951(1) Å, and 1.959(1) Å. The calculated and experimental
crystal structures (mirror image about ab-plane) are shown below in the Pnma setting at
1.5 K with the (-) tilt sign.
La0.5Ca0.5MnO3
Experimental
a (Å)
b (Å)
c (Å)
La/Ca x
La/Ca z
O(4c) x
O(4c) z
O(8d) x
O(8d) y
O(8d) z
5.4466(3)
7.5247(4)
5.4763(3)
0.5195(3)
0.5036(2)
-0.0095(4)
0.4355(2)
0.2720(2)
0.0337(1)
0.7271(2)
SPuDS
calculated
5.4917
7.5256
5.4320
0.5155
0.4999
-0.0053
0.4488
0.2757
0.0270
0.7237
2.6.2 Ordered M-site
Calculation of a charge ordered type structure is also possible. Consider the
calculation the crystal structure of La0.5Ca0.5Mn3+0.5Mn4+0.5O3 with hypothetically rock-
SPuDS
19
salt type ordering Mn3+ and Mn4+. This feature is in testing, but through appropriate
substitution of x,y,z bond distance factors, a crystal structure with one Jahn-Teller
distorted and one octahedrally symmetric ion may be generated. The correct orientations
of the long, medium, and short M-O bonds need to be obtained to match that observed
experimentally. Note that more distortion of the octahedral angles (O-M-O) is expected
with an increased distortion of the M-O bond lengths on a single octahedral site in the
ordered, as opposed to the disordered case where the distortions are cooperative.
Bond Distance Factors
x
0
y
0
z
0
0
0
0.18
0
0.1
0
0.1
0
0
-0.1
-0.04
0.18
Mn-O1
(×2)
2.0212
1.8991
2.1234
1.9016
2.0248
1.8974
2.0677
1.8961
2.0744
1.9081
Mn-O2
(×2)
2.0118
1.9068
2.0896
1.9064
2.12
1.9085
2.0626
1.9113
2.0407
1.9047
Mn-O3
(×2)
2.0174
1.902
2.0209
1.9052
2.1142
1.9018
2.0196
1.904
1.9794
1.9028
BVS M1,2
(JT)(v.u.)
3.00
4.00
2.56
3.99
2.76
4.01
2.75
3.99
2.90
3.98
2.7 Pyrochlores
Structure prediction for simple A2B2O7 pyrochlores is available. The tolerance
factor calculation is based on that of (Isupov, 1958).
2.8 Anti-perovskites
Structure prediction for anti-perovskites is available for formulas of the type
A3OX. The input requires the X and A-sites to be reversed. Sample input is shown
below.
SPuDS
20
2.9 Frequently Asked Questions
Q. Why does the (#10) a-b+a- tilt system, space group Pnma, only have a single tilt
angle?
A. The a-b+a- tilt system has both in phase and out of phase octahedral tilting.
SPuDS uses the assumption that the octahedra do not undergo a distortion if allowed to
remain symmetric (equidistant M-X distances, 90° or 180° X-M-X bond angles), which
requires the in-phase and out-of-phase tilt angles to be coupled to maintain regular
octahedra. More details and the equations that relate the primary and secondary tilt angles
are found in section 3.2.2 of this user manual, section 3 in the SPuDS paper (Lufaso and
Woodward, Acta Cryst. (2001), B57, 725-738) and sequence (iv) on page 3806 in
O’Keeffe and Hyde, Acta Cryst. (1977), B33, 3802-3813.
Q. Why does the (#14, a-a-a-) R-3c always give the lowest GII, but Pnma (#10,
a b a ) are more often observed experimentally?
A. The symmetric tilting about the octahedral face results in the crystal structure
obtaining the lowest GII, but does not consider that unfavorably short A-O and O-O
interactions may occur in the hypothetical crystal structure. Chapter 6 and figure 6.4 of
'The Chemical Bond in Inorganic Chemistry; Oxford; 2002 by I.D. Brown give some
useful guidelines for typical O-O distances. Large tilt angles in (#14, a-a-a-) space group
R-3c result in three very short A-O bonds, which are not favorable compared to a more
symmetric coordination environment for the A-site cation. For example, the hypothetical
crystal structure of CaZrO3 in space group R-3c (#14, a-a-a-) has 3 Ca-O bonds of 2.22 Å
with valences of 0.50 v.u. each. That means 75% (1.50 v.u./2 v.u.) of the Ca2+ bonding
requirements is satisfied with only 3 bonds. Contrast that to the three short Ca-O bonds
of CaZrO3 in space group Pnma (#10, a-b+a-), which are 0.366 v.u., 0.324 v.u., and 0.280
v.u. for a total of 0.97 v.u., which is less that 50% of the Ca2+ bonding.
The above example illustrates why space group Pnma perovskites (#10, a-b+a-) are
observed more often than (#14, a-a-a-) space group R-3c perovskites, even though the
GII is lower for R-3c (0.00018 v.u.) compared to Pnma (0.01413 v.u). The GII is just an
indication of the relative structure stability, but does not contain information regarding
the bonding or if unstable bond distances are involved. Below a certain threshold value of
the GII, any tilt system may be stable and additional factors need to be considered, such
as the A-O bonding described above.
- + -
Q. The 40 atom C2/c cell is used to describe the a-b-b- tilting as opposed to the 20
atom I2/a cell. Why?
A. The C2/c setting was the one originally used by Glazer (1972) and also by
Howard and Stokes (1998). The setting chosen in SPuDS was used to give a structure
similar to the C2/c setting of BiMnO3 (ICSD 56842). The structure prediction for the rarely
observed a-b-b- tilt system is still being tested. The octahedral bond distances do not quite
remain ideal. If you have a specific interest in this tilt system, let me know and at some
point I can revisit the code dealing with this subroutine. If the I2/a cell is more convenient,
it might be a possibility to include that as well (like with the Pnma and Pbnm setting).
SPuDS
21
Chapter 3
Tilt Angle Calculation using TUBERS
3.1
Overview
TUBERS may be used to calculate the tilt angles of the octahedra from an
experimental crystal structure. This feature is only available for tilt systems with space
groups P21/n, Pnma, R-3c, I2/m, I4/mcm, and I4/m in the current TUBERS version.
Contact the author (michael.lufaso@unf.edu) for more details or to request a particular
tilt system.
3.2
Single M-site perovskites
Some tilt systems have direct equations to go from fractional coordinate to tilt
angle. In this case, the program will prompt for the fractional coordinate, which must be
near the general position given in the prompt.
#22 (a0a0c-) space group I4/mcm
The fractional coordinate is the only input required for calculating the octahedral
tilt angle in space group I4/mcm perovskites.
3.2.1
SPuDS
22
#10 (a-b+a-) space group Pnma
The calculation of the tilt angle in Pnma requires only the fractional positions of
the O atoms. The fractional positions of the anions must match the setting used for the
SPuDS calculations. It is necessary to convert to Pnma setting from any other setting
(e.g. Pbnm). The tilt angle given is obtained by solving the equations given by O’Keeffe
& Hyde (1977) for the z fractional coordinate of O at Wyckoff position 4c and the x, y,
and z fractional coordinates for the O position at the Wyckoff position 8d. The resulting
tilt angle is given as φ in the TUBERS calculation. A secondary tilt ψ is required. The
relationship between φ and ψ is given below and full details may be found in O’Keeffe &
Hyde (1977). The average φ octahedral tilt angle is also given.
3.2.2
SPuDS
23
6.0
Secondary Rotation psi (degrees)
5.0
4.0
3.0
2.0
1.0
0.0
0
5
10
15
20
25
30
35
Primary Rotation phi (degrees)
3.3
Multiple M-site perovskites
Calculation of the octahedral tilt angle is possible for tilt systems #10 (a-a-b+)
space group P21/n and #20 (a0b-b-) space group I2/m.
3.3.1 SPuDS text version [#22 (a0a0c-) and #14 (a-a-a-)]
(This version of SPuDS is being phase out by TUBERS)
SPuDS
24
The text based version of SPuDS is required for calculation of octahedral tilt
angle for tilt systems 22 (a0a0c-) and 14 (a-a-a-). Equations used to calculate the fractional
coordinate from the tilt angle are complex. Matrix methods are used to obtain the
equation for fractional coordinates based on M-X bond distances and octahedral tilt
angles. The fractional coordinate equations incorporate the lattice parameter into the
equation in terms of the bond distances and tilt angle. Therefore obtaining an equation for
the tilt angle from the fractional coordinate and lattice parameters is not trivial.
Instead, SPuDS takes the average M1-X and M2-X bond distance obtained from
the experimental crystal structure as input. The fractional coordinates are then serially
calculated for a variety of tilt angles into a file named tltangle.txt. Inspection of the file
allows one to obtain the tilt angle by searching for a fractional coordinate near the
experimental crystal structure value. The fractional coordinates must be refined or
converted to approximately the same position.
General
From the experimental crystal structure calculate the mean M1-O and M2-O bond
distances.
Execute text version SPuDS, choose menu 13.
Type tilt system number of interest.
Type number corresponding to simple or ordered.
Enter the mean M1-O bond distance.
Enter the mean M2-O bond distance.
SPuDS will execute and terminate.
Go to the SPuDS folder and open tltangle.txt, search for each fractional coordinate
nearest the experimental value and record the tilt angle. The average for each anion
fractional coordinate is the experimental octahedral tilt angle.
Note, distortions of the octahedra are the cause for discrepancy in the tilt angle. It is up to
the user to weigh each the accuracy of each fractional coordinate and discard any outliers
in order to obtain meaningful results.
3.3.2 TUBERS [#10 (a-a-b+) space group P21/n]
The software program TUBERS allows calculation of octahedral tilt angles for
ordered perovskites in space groups P21/n. The calculation algorithm for determining the
octahedral tilt angle(s) involves using the equations of tilt that SPuDS uses to calculate
fractional coordinates based on M-X bond distances and the octahedral tilt angle. In
addition, the calculation of the tilt angles w [rotation about (001) of the aristotype
perovskite] and j [rotation about (110) of the aristotype perovskite] as described by Groen
et al. (1986) are included for comparison. The formulas for w(B)c and w(B')c appear to be
swapped in the original paper, and have been switched to agree with the tilt angles
obtained in the original paper.
TUBERS utilizes the equations from the SPuDS software to calculate the tilt
angles based on the fractional positions and average M-O bond distances. This method of
SPuDS
25
calculating the octahedral tilt angle has an advantage over the method that separates the
(110) and (001) tilts, because it ensures the octahedra retain approximately corner-sharing
connectivity. The tilt angle calculated by TUBERS links the inherent coupling of the
octahedral tilts about the [001] and [110] axes. For example, the a-a-b+ tilt system is based
on a tilt about the [110] is followed by secondary correction tilt about the [001] axis,
similar to the one made by O’Keeffe, M. & Hyde, B.G. (1977) to maintain corner
connectivity of octahedra in the simple AMX3 perovskite (space group Pbnm/Pnma).
Two different cation sites in the P21/n crystal structure and different size octahedra make
the equations describing the anion fractional positions very complicated. Simplified
equations were solved by matrix algebra methods; however each equation is several
pages in length. Because of the complexity, the equations are not repeated here. The
interested reader can find the complete equations in the appendix of Lufaso, M.W. Ph. D.
(2002) dissertation at http://www.ohiolink.edu/etd/view.cgi?osu1039132858.
In order to calculate the octahedral tilt angles in space group P21/n (a-a-b+), the
required user input are the average M1-O (Wyckoff position 2c) and M2-O (Wyckoff
position 2d) bond distances, the anion (Oxygen) fractional coordinates, and lattice
parameters. The fractional coordinates must approximately match the sample given in
the software. Use the International Tables for Crystallography to generate the additional
positions if the approximate fractional coordinates do not match. A shift of all atoms in
the unit cell will be necessary if the Wyckoff sites of the octahedral cations are 2a and 2b,
rather than 2c and 2d as required by the equations used in TUBERS. Approximate unit
cell dimensions are a ≈ √2ap, b ≈ √2ap and c ≈ 2ap, where ap is the primitive unit cell
length. The cation fractional positions are M(2c) (0, ½,0), M′(2d) (½, 0, 0) and A(4e) (x,
y, z) where x ≈ ½, y ≈ ½ and z ≈ ¼. Anion fractional positions for three cations are X(4e)
(x, y, z) where (x ≈ ¼, y ≈ ¼ and z ≈ 0; x′ ≈ ¼, y′ ≈ ¾ and z′ ≈ 0; x″ ≈ ½, y″ ≈ 0 and z″ ≈
¼).
An example calculation of octahedral tilt angles is performed using an
experimental crystal structure of Nd2TiMgO6 (Groen et al., 1986). Average bond
distances were calculated and input into TUBERS. By convention, M1-O is the average
M-O bond distance for the M-site cation on position 2c, whereas M2-O is for the M-site
cation on position 2d. Given oxygen fractional coordinates, without ESD's, are O1
(0.1960, 0.2100, -0.0494), O2(0.2852, 0.6980, -0.0420), and O3 (0.4130, 0.9746,
0.2559). Note that the y-position of O3 needs shifted by -1 to match the required input.
The resulting fractional coordinates are now O1 (0.1960, 0.2100, -0.0494), O2(0.2852,
0.6980, -0.0420), and O3 (0.4130, -0.0254, 0.2559). These fractional coordinates,
average bond distances, and lattice parameters were input and are shown below.
SPuDS
26
Tilt angles were calculated by pressing the 'Calculate Tilt Angles' button.
The tilt angles of the [TiO6] and [MgO6] are 18.8° and 17.8°, respectively. Note
the tilt angle calculated based on the position of O3z is significantly different. The
position of atom O3z is insensitive to the octahedral tilt angle and is disregarded in the
calculation of the average tilt angle. The average tilt angle is calculated with the other 8
oxygen anions. The spread of the tilt angles is present because of octahedral distortion of
M1-O and M2-O bond lengths and O-M-O bond angles. Note the smaller ion undergoes a
larger tilting in order to maintain corner-sharing connectivity. The tilt angles w(M1)c,
w(M2)c, j(M1)c and j(M2)c are shown, in agreement with the values found in Groen et al.,
1986.
In general, there is close agreement between calculated and theoretical tilt angles
from SPuDS simulations. A plot for multiple space group P21/n perovskite compounds is
shown in the next figure; with open squares SPuDS predicted average tilt angles and
filled square average literature tilt angles.
SPuDS
27
29.0
Tilt Angle (°)
25.0
21.0
17.0
13.0
0.84
0.86
0.88
0.90
0.92
0.94
Tolerance Factor
3.3.3 TUBERS [#20 (a0b-b-) space group I2/m]
The software program TUBERS allows calculation of octahedral tilt angles for
ordered perovskites in space groups I2/m. The calculation algorithm is similar to that
utilized for P21/n perovskites. In this tilt system, only the average M1-O, M2-O bond
distances and O fractional coordinates of O1x and O2z are required. The other fractional
coordinates are not sensitive to the octahedral tilting, so are neglected in the calculation.
The octahedral cations are required to be on Wyckoff sites 2a (0, 0, 0), (1/2, 1/2,
1/2) and 2d (1/2, 1/2, 0) or (0, 0, 1/2). Consider an example Ba2BiBiO6 (Cox & Sleight,
1976) (ICSD CCode 87326). Average Bi3+-O bond lengths are 2.276 Å and Bi5+-O are
2.119. Anion fractional coordinates are O(4i) [0.0604(9), 0, 0.2608(17)] and O(8j)
[0.2613(9), 0.2552(21), -0.0334(4)]. The resulting tilt angles are 9.9° for Bi3+(2a) and
10.6° for Bi5+.
SPuDS
28
Bibliography
M.W. Lufaso, P.M. Woodward, (2001). "The Prediction of the Crystal Structures of
Perovskites Using the Software Program SPuDS" Acta Cryst., B57, 725-738.
S.-H. Byeon, M. W. Lufaso, J. B. Parise, P. M. Woodward, T.Hansen (2003). "HighPressure Synthesis and Characterization of Perovskites with Simultaneous Ordering of
Both the A- and M-site Cations, CaCu3Ga2M2O12 (M = Sb, Ta)" Chem. Mater., 15, 37983804.
O’Keeffe, M. & Hyde, B.G. (1977). "Some Structures Topologically Related to Cubic
Perovskite (E21), ReO3 (D09) and Cu3Au (L12)" Acta Cryst. B33, 3802-3813.
Groen, W.A., van Berkel, F.P.F., & IJdo, D.J.W. (1986). "Dineodymium Magnesium
Titanate(IV). A Rietveld refinement of Neutron Powder Diffraction Data." Acta Cryst.
C42, 1472-1475.
Brown, Dabkowski and McCleary (1997). "Thermal Expansion of Chemical Bonds" Acta
Cryst. B53, 740-761.
Brown, I.D. (2004). Accessed June 2004
http://www.ccp14.ac.uk/ccp/web-mirrors/i_d_brown/bond_valence_param/bvparm.cif
Rao, G.H., Barner, K., Brown, I.D. (1998) "Bond-valence analysis on the structural
effects in magnetoresistive manganese perovskites" J. Phys.: Condens. Matter, 10, L757.
Rodriguez-Carvajal et. al. Phys. Rev. B 57(6), R3189-R3192 (1998).
Salinas-Sanchez,A ., Garcia-Munoz,J.L., Rodriguez-Carvajal,J., Saez-Puche,R.;
Martinez, J.L. (1992). "Structural Characterization of R2BaCuO5 (R = Y, Lu, Yb, Tm,
Er, Ho, Dy, Gd, Eu and Sm) Oxides by X-Ray and Neutron-Diffraction" J. Solid State
Chem. 100, 201.
Kennedy, B.J., Howard, C.J., Chakoumakos, B.C. (1999). "Phase Transitions in
perovskite at elevated temperatures - a powder neutron diffraction study." J. Phys.:
Condens, Matter 11, 1479-1488.
Glazer, A.M. (1972). "The Classification of Tilted Octahedra in Perovskites" Acta Cryst.
B28, 3384-3392.
Cox, D.E.;Sleight, A.W. (1976)"Crystal structure of Ba2Bi(3+)Bi(5+)O6" Solid State
Communications, 19, 969-973.
SPuDS
29
Appendix A
Conversion of Pbnm
**********************************************************************
Space Group: Pbnm (#62)
Lattice Parameters (Angstroms)
a = 5.4671(2)
b = 4.4823(2)
c = 7.7461(3)
Atom
Ca
Ti
O
O
Site x
y
z
4c 0.003(1)
0.519(1) 0.2500
4a 0.0000
0.0000 0.0000
4c -0.055(1) -0.008(1) 0.2500
8d 0.2177(5) 0.2802(5) 0.0286(4)
**********************************************************************
Note that the Ti is located on site 4b in the SPuDS predicted crystal structure,
whereas it is on the 4a site in the Kennedy et al, 1999 crystal structure. The site
symmetry is -1 in both the 4a and 4b site, so shift the unit cell by (0.5, 0, 0). The resulting
fractional coordinates are now:
Atom
Ca
Ti
O
O
Site x
y
z
4c 0.503(1)
0.519(1) 0.2500
4b 0.5000
0.0000 0.0000
4c 0.445(1) -0.008(1) 0.2500
8d 0.7177(5) 0.2802(5) 0.0286(4)
The equivalent positions for the 4b site in space group Pnma are (0,0,0.5),
(0.5,0,0), (0,0.5,0.5) and (0.5,0.5,0), therefore the equivalent positions in Pbnm are
(0.5,0,0), (0, 0.5,0), (0.5,0,0.5) and (0,0.5,0.5). We can rewrite the positions as:
Atom
Ca
Ti
O
O
Site x
4c 0.503(1)
4b 0.0000
4c 0.445(1)
8d 0.7177(5)
y
z
0.519(1) 0.2500
0.5000
0.0000
-0.008(1) 0.2500
0.2802(5) 0.0286(4)
*****************************************************
Space group
Cmcm (#63)
Lattice Parameters (Angstroms)
a = 7.7642(6)
b = 7.7674(4)
c = 7.7667(4)
SPuDS
30
Atom Site x
y
z
Ca
4c 0.0000 0.505(5) 0.2500
Ca
4c 0.0000 -0.019(4) 0.2500
Ti
8d 0.2500 0.2500
0.0000
O
8e 0.278(2) 0.0000
0.0000
O
8f 0.0000 0.226(2) 0.032(1)
O
8g 0.280(2) 0.251(2) 0.2500
***************************************************
Note that O on site 8e are not easily comparable between the SPuDS and
experimental crystal structure, so find equivalent positions. Site positions are (x, 0,0), (-x,
0, 0.5), (-x,0,0), and (x, 0, 1/2) combined with (0.5, 0.5, 0) to generate all 8 positions for
the site. Adding (0.5, 0.5, 0) to the O at site 8e, generating the equivalent position, gives
the following fractional coordinates after a shift of minus one unit cell in the x direction.
SPuDS
31
Appendix B
Neutron Powder diffraction simulations (λ = 1.5403 Å)
SrFeO3 (Sp. grp. Pm-3m)
Sr(Ti0.65Fe0.35)O3-δ (Sp. grp. I4/mcm)
NaTaO3 (Sp. grp. Pbnm)
SPuDS
32
SPuDS
33
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