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calculating elastic tensors of monoclinic .pdf



Nom original: calculating elastic tensors of monoclinic.pdf
Titre: A Package for calculating elastic tensors of cubic
Auteur: jam

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IR E L A S T
+

W I E N 2k
A Package for calculating elastic tensors of monoclinic
Phases by using second-order derivative with Wien2k Package
User’s guide, Mono-elastic_13.1 (Release 25.08.2013)
Morteza Jamal
Ghods City-Tehran-Iran

1

MANDATORY CONDITIONS:
In any publication in the scientific literature please reference the program as follows:
M. Jamal, Mono-elastic, http://www.wien2k.at/reg_user/unsupported/cubic-elast/ (2013).

ACKNOWLEDGMENT
I gratefully appreciate B.Z. Yanchitsky and R. Golesorkhtabar for fruitful discussions, P.
Blaha and S. Jalali Asadabadi for suggestions, and Carol Phillips for editing.
I especially appreciate Arash Boochani (Department of Physics, Kermanshah Branch,
Islamic Azad University, Kermanshah, Iran) for providing CPU time.

For suggestions or bug reports please contact the author by email:
m_jamal57@yahoo.com

2

1- Introduction
Mono-elastic is a Package for finding elastic constants of monoclinic symmetries with
Wien2k. This Package calculates elastic constants by second-order derivative ( E”(δ) ) of
Polynomial fit ( E=E(δ) ) of Energy vs. strains (δ) at zero strain (δ =0). This called energy
approach [1].

2- Background theory (energy approach)
Elastic constants are defined by means of a Taylor expansion of the total energy
for the system, with respect to a small strain ( ) of the lattice. If we consider the bravais
lattice vectors of an monoclinic crystal structure as a matrix form
the distortion of
the lattice (
) is expressed by multiplying
with a symmetric (
)
distortion matrix i.e. (

), which is written as,

And in Voigt notation ( It is often convenient to change to the Voigt notation in
order to reduce the number of indices. The Voigt notation replaces
2,

3,

(and

)

4,

(and

)

5,

(and

)

1,

6)

we express the energy of the strained system by means of a Taylor expansion in
the distortion parameters,
3

The linear terms vanish if the strain causes no changes in the volume of the
crystal. Otherwise,
are related to the strain on the crystal and
are elastic
constants and
is the volume of unstrained monoclinic system and we use it to
evaluate the elastic constants.
There are thirteen independent elastic constants for an monoclinic symmetry,
called C11, C22, C33, C44, C55, C66, C12, C13, C23, C16, C26, C36, and C45 . Since we
have thirteen independent elastic constants, we need thirteen different strains to
determine these elastic constants.
For simplicity in writing the equations used in the mono-elastic Package, it is
convenient to rewrite the above equation as :

Which τ represents a linear combination of strain components and C, a linear
combination of elastic constants.
Therefore the thirteen distortions used in the mono-elastic Package are described
as following:

1)

,

2)

,

3)

,

4)

,

5)

,

4

6)

,

7)

,

8)

,

,
,
12)
13)

,
,

5

3- File structure and program flow
The following table describes input and output files for each program of the Mono-elastic
Package.
Program
M_set_elast_lapw

needs
case.struct

generates
init.struct
runcommand1
runcommand2

pwdname
auto_init_lapw
runcommand1
runcommand2

command_init_lapw
M_command_run_lapw
M_setupc11, M_setupc12
M_setupc13, M_setupc22
M_setupc33,
M_setupc44,….
getcalljobM

makestructM
M_modifyjob_lapw
M_calljob_lapw
M_fitdivELC

M_ana_elastc_lapw

M_ana_elast_lapw

M_InverseELC
MassRho
TWS
M_ana_elastorder_lapw
sgroupcheck2_lapw
command_intso_lapw
command_initu_lapw
auto_initso_lapw
auto_initu_lapw

init.struct
pwdname
runcommand1
runcommand2
init.struct
.styp
MONO.job
MONO.job
number.strain
vol.optimize
.styp
VstVene
VstVene
case.outputeos
ELC.fit
ELC.output

ELC-matrix
init.struct
vol.optimize
ELC-matrix
ELCorder.fit
StypX_Y.struct

.infSO
.infLDAU
6

Bold font is OPTIONAL
Italic bold font means it is the user’s choice

.Vper
.styp
MONO.job
number.strain
StypX_Y.struct
vol.optimize
VstVene
ELCorder.fit
ELC.output
ELC.fit
vol.optimize
ELC.ps

ELC-matrix
case.output_elastic
INVELC-matrix
.rho
STDELC-matrix
output-order
case.struct
.infSO
.infLDAU
case.inso
case.inorb
case.indm/c

3-1- Short description for input and output files
case.struct
init.struct
pwdname
runcommand1/2

Is a Wien2k standard struct file.
Is a copy of the case.struct file.
Contains the name of the present work directory.
Contains the run commands for running. It looks similar to:

run_lapw –ec 0.0001 –p –in1new 2

auto_init_lapw

A C-shell program which automatically runs the initialization.
It looks similar to:

#!/bin/csh -f
set RM = not
if ( $RM == 'not' ) then
init_lapw -vxc 13 -ecut -6 -mix 0.2 -numk 3000 -b
else
init_lapw -red 0 -vxc 13 -ecut -6 -mix 0.2 -numk 3000 -b
endif




.Vper
.styp
MONO.job

Defines the percent of changes for different strains.
Defines the type of strain.
A C-shell program which calculates the energy for each strain
by using the Wien2k Package. It looks similar to:

#!/bin/csh -f
#STRAIN TYPE IS
1
#Modify this script according to your needs
unalias rm
set co = 1
set name
set bj
set file
= `pwd`
set file
= $file:t
if (-e VstVene ) then
set i=`/bin/ls VstVene* |wc `
echo " saving pervious VstVene to VstVene_$i[1]"
cp VstVene VstVene_$i[1]
rm VstVene
endif
#
# to reuse previous scf runs (without a new scf run) set answscf=y
# and use the same "savename".
# When you make modifications (RKmax, k-mesh, XC-potentials) choose
# answscf=no and a new savename (eg. "_pbe_rk8_1000k").
set answscf=y
set savename=

7

if (-e cscl.clmsum && \
! -z cscl.clmsum) then
x dstart -super
endif
if (-e cscl.clmup && \
! -z cscl.clmup ) then
x dstart -super -up
x dstart -super -dn
endif
foreach i ( \
Styp1_-3.0
Styp1_-2.0
Styp1_-1.0
Styp1__0.0
Styp1__1.0
Styp1__2.0
Styp1__3.0

\
\
\
\
\
\
\

)
echo "*******************************"
echo $i
set name=`echo "$name $i"`
echo "*******************************"





StypX_Y.struct

VstVene

number.strain
case.outputeos

A Wien2k struct file for each value of changes and for each
strain type where X and Y denote type of strain and value of
changes, respectively.
The main information file, contains values of changes (strains)
and energies for each type of strain, for the calculation of the
elastic constant.
Contains the number of strains.
A Wien2k output of equation of states(EOS).
For finding the best values of elastic constants, find
EOS and then copy the case.outputeos file in the "case"
directory within the c11,c12, c13, c22 , c33, c13, c44, and …
directories. Otherwise, it sets the optimized volume from the
original struct file i.e. case.struct

vol.optimize
ELCorder.fit
ELC.output
ELC.fit
ELC-matrix
STDELC-matrix

Contains the optimized volume.
Contains the elastic constants for different values of order of
fit.
Contains the elastic constants for order of fit =2
Contains the data to plot a curve of energy vs value of changes
(strains) for each strain type.
Defines the elastic constant matrix in WIEN2k Cartesian
coordinates.
Defines the elastic constant matrix, inverse of it and … in
STANDARD Cartesian coordinates.
8

case.output_elastic
INVELC-matrix
.rho

Contains the final elastic constant values.
Defines the inverse of elastic constant matrix.
Contains density of mass and atomic volume.

.infSO

Contains information for making the case.inso file for running
spin-orbit coupling .
Contains information for making the case.inorb and
case.indm/c files for LDA+U calculations.

.infLDAU

3-2- Flow and short description for programs
The Mono-elastic Package consists of several FORTRAN and SHELL SCRIPTS which
are described below. A flowchart of the program is shown in the following diagram.
 M_set_elast_lapw :
Makes an elast-constant directory in the present work directory ( PWD ) and c11, c12,
c13, c22 , c23, c33, c44, and … directories in the elast-constant directory. The
M_set_elast_lapw program also copies information of the "PWD" into the c11, c12, c13,
c22, c23,
c33, c44, and … directories and calls "command_init_lapw",
M_command_run_lapw, M_setupc11, M_setupc12, M_setupc13, M_setupc22,
M_setupc23, M_setupc33, and ….. programs.
.
 command_init_lapw :
Gets information for making "auto_init_lapw".
 M_command_run_lapw :
Gets the run commands for making “MONO.job”.
 M_setupcX (X=11,12, 13,22, 33, ….) :
Gets the type of strain and calls the “getcalljobM” program.
 getcalljobM :
Calls “makestructM” program and makes the “MONO.job” file.
 makestructM :
Makes the “StypX_Y.struct ” files where X and Y stand for the type of strain and
value of changes, respectively and vol.optimize file.
 M_modifyjob_lapw :
Edits the job files according to the user’s needs.
 M_calljob_lapw :
Calls the “MONO.job” files for running.
9

 M_ana_elast_lapw :
Calls the M_ana_elastc_lapw program for calculating elastic constants then calculates
the Voigt, Reuss, and Hill bulk, shear, and the Young modulus as well as the Poisson
ratio. After that it calls the “M_InverseELC” and “MassRho” programs and calculates
sound velocity and Debye temperature then makes two output files in the elastconstant directory with the name case.output_elastic and the INVELC-matrix which
is the Elastic compliance constants generated by inverting the elastic constant matrix.
At the end it calls “M_ana_elastorder_lapw” and TWS programs which TWS
transforms elastic constants from WIEN2k to STANDARD Cartesian coordinates and
makes output file with the name STDELC-matrix.
 M_ana_elastc_lapw :
Calls the “M_fitdivELC” program with appropriate libraries for calculating C11, C12,
C13, C22, C33, and ….
 M_InverseELC :
Makes the Elastic compliance constants generated by inverting the elastic constant
matrix .
 MassRho :
Finds density of mass and atomic volume.
 TWS :
Finds elastic constants in STANDARD Cartesian coordinates.
 M_ana_elastorder_lapw :
Checks the sensitivity of the elastic constants to the order of fit.
 sgroupcheck2_lapw :
Finds the best value of tol in the sgroup[2] program and copies case.struct_sgroup as
case.struct.

10

M_set_elast_lapw

command_init_lapw
generates
auto_init_lapw

M_command_run_lapw
generates
commandrun1/2

M_setupcX
X=11,12, 13, 22, ….

getcalljobM
generates
MONO.job

makestructM
generates
StypX_Y.struct

Sgroupcheck2_lapw

MONO.job
generates
VstVene

auto_init_lapw

M_ana_elastorder_lapw
generates
ELCorder.fit

M_ana_elast_lapw

M_ana_elastc_lapw

M_InverseELC
MassRho
TWS

ELASTIC CONSTANTS IS READY

M_fitdivELC calls

generates case.output_elastic ,
INVELC-matrix and STDELC-matrix

Libraries

generates C11, C22 ,C33,..
11

Program flow in Mono-elastic

Dash arrow means user must run

4 – Elastic constants calculation
1. Create a struct file and validate it by running "sgroupcheck2_lapw".

2. If Spin-Orbit calculations are required run "command_initso_lapw.
3. If LDA+U calculations are required run "command_initu_lapw" and then
"auto_initu_lapw".
4. Run "M_set_elast_lapw”..
5. Now you must adapt the job files according to your needs (you can run
"T_modifyjob_lapw" in Terminal ).
It is not necessary to do step 5 if you defined the “COMMAND RUN” commands in
step 4.

6. Now you must run the job files (you can run "M_calljob_lapw" ). It will take
some time.
7. Run "M_ana_elast_lapw"..

This package calculates elastic constants by second-order derivative ( E”(δ) ) of
Polynomial fit ( E=E(δ) ) of Energy vs. strains (є) at zero strain (δ =0) so, you must use
values of strain around zero and from the viewpoint of fit convergence, we usually
expect to see a minimum when we plot Energy vs. strain ( this Package plots it ). It is
recommended that the sensitivity of the results is checked to the order of fit. This
program shows them.

4-1– Notes about elastic constants calculation
After using distortions for the calculation of C44, C55, and C66 the symmetry of the
monoclinic compound changes and usually the number of atoms change. So when you
run "command_initso_lapw" or "command_initu_lapw" , in the section name of an
atom, type "all <name of atom>" ( for example: all Mn). With this command, you use SO
or LDA+U calculations for example for all Mn atoms.
When you want to rerun job files with modifications in (RKmax, k-mesh, XC-potentials )
call the “command_init_lapw” and after that choose "answscf=no" in the “MONO.job”
files and a new "savename" (eg. "_use_pbe_rk8").

12

Optionally you can specify more cases by rerunning “M_setupcX” (X=11, 12, 13, 22,
23,… see section 4-3 ). Specify also your ‘‘old’’ cases. The old results will then be
taken automatically into account without recalculation (unless you modify job files i.e:
set answscf=no ).
For the calculation of the best values of elastic constants, please find EOS and then copy
case.outputeos in the "case" directory within the c11, c12, c13, c22, c23, and …
directories. Otherwise, it sets the optimized volume from the original struct file i.e.
case.struct .

4-2– One calculation
To calculate C33 or C44 the following steps should be performed for example for
calculation of C44 :
1. Make a directory for example c44.
2. Make a "case" directory in c44 directory.
3. Make a "case.struct" file in the "case" directory and name it "init.struct".
Create a "pwdname" file and write in it "case." and save it.
4. Run the command_init_lapw
5. chmod +x auto_init_lapw
6. For SO calculations, run the command_initso_lapw.
7. For LDA+U calculations, run the command_initu_lapw and auto_initu_lapw.
7-1. To avoid step 10, you can run “M_command_run_lapw” for setting the
“COMMAND RUN” commands for making “MONO.job” .

8. Run the M_setupc44 program.
9. chmod +x MONO.job file.
10. Modify the MONO.job file.
11. Call MONO.job
12. Call M_ana_elastc_lapw

4-3– Run with more data points
Optionally you can specify more data points, for the calculation of the elastic constants,
by rerunning “M_setupcX” (X=11, 12, 13, 22, 23,… ). Specify also your ‘‘old’’ data
points. The old results will then be taken automatically into account without
recalculation ( unless you modify the job files i.e: set answscf=no ). Please do the
following steps for this goal for example for c44.
1. cd to the “elast-constant” directory.
2. cd to the “c44” directory.
3. cd the “case” directory.
3-1) To avoid step 6, you can run “M_command_run_lapw” for setting the

13

“COMMAND RUN” commands for making the “MONO.job” .

4. Run the M_setupc44 program.
5. If you want to rerun the job files with modifications in (RKmax, k-mesh, XCpotentials ) call “command_init_lapw” and then choose "answscf=no" in
“MONO.job” files and a new "savename" (eg. "_use_pbe_rk8").
6. Modify the MONO.job file.
7. Call MONO.job
8. Call M_ana_elastc_lapw

4-4– Elastic constants calculation for ZrO2
ZrO2 compound is a test case for elastic constants calculation. The ZrO2 structure is
described in detail in the following:
ZrO2
P

314_P21/c
RELA
10.12752
9.81229
9.93072 90.00000 90.00000
ATOM -1: X=0.21044544 Y=0.77565959 Z=0.54339126
MULT= 4
ISPLIT= 8
-1: X=0.28955456 Y=0.22434041 Z=0.04339126
-1: X=0.78955456 Y=0.22434041 Z=0.45660874
-1: X=0.71044544 Y=0.77565959 Z=0.95660874
Zr1
NPT= 781 R0=0.00001000 RMT=
1.9500
LOCAL ROT MATRIX:
1.0000000 0.0000000 0.0000000
0.0000000 1.0000000 0.0000000
0.0000000 0.0000000 1.0000000
ATOM -2: X=0.34539854 Y=0.56806354 Z=0.83192841
MULT= 4
ISPLIT= 8
-2: X=0.15460146 Y=0.43193646 Z=0.33192841
-2: X=0.65460146 Y=0.43193646 Z=0.16807159
-2: X=0.84539854 Y=0.56806354 Z=0.66807159
O 1
NPT= 781 R0=0.00010000 RMT=
1.7500
LOCAL ROT MATRIX:
1.0000000 0.0000000 0.0000000
0.0000000 1.0000000 0.0000000
0.0000000 0.0000000 1.0000000
ATOM -3: X=0.47603875 Y=0.95042689 Z=0.25750770
MULT= 4
ISPLIT= 8
-3: X=0.02396125 Y=0.04957311 Z=0.75750770
-3: X=0.52396125 Y=0.04957311 Z=0.74249230
-3: X=0.97603875 Y=0.95042689 Z=0.24249230
O 2
NPT= 781 R0=0.00010000 RMT=
1.7500
LOCAL ROT MATRIX:
1.0000000 0.0000000 0.0000000
0.0000000 1.0000000 0.0000000
0.0000000 0.0000000 1.0000000
4
NUMBER OF SYMMETRY OPERATIONS

99.63400

Select Xc = PBE-GGA, R_Kmax = 8, L_max = 10, and nkpoint = 500

14

Z:

40.00000

Z:

8.00000

Z:

8.00000

In the following examples you can find the percents that were used for strains.

#######################################
# M_ana_elast_lapw analyses Elastic #
#
constant
#
#
C(2013) by Morteza Jamal
#
#######################################
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using case.outputeos
#
#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.709920
-0.010000
-29999.714460
0.000000
-29999.716424
0.010000
-29999.715971
0.020000
-29999.713226
===============================================================
Order of fit: 2 C11 is:
366.8981 GPa, RMS: 0.403159E-04
Order of fit: 3 C11 is:
366.8981 GPa, RMS: 0.183876E-05
Order of fit: 4 C11 is:
364.9787 GPa, RMS: 0.162695E-11
******************************************
Polynomial fit for C11 done
A RMS of 0.403159E-04 was achieved using a polynome of degree : 2
At volume= 972.9393 bohr^3
C11 is:
0.024941 a.u or
366.8981 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.
***************************************
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using case.outputeos
#
#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.709564
-0.010000
-29999.714566
0.000000
-29999.716424

15

0.010000
-29999.715180
0.020000
-29999.710931
===============================================================
Order of fit: 2 C11+C22-2C12 is:
466.6136 GPa, RMS: 0.196989E-04
Order of fit: 3 C11+C22-2C12 is:
466.6136 GPa, RMS: 0.301631E-05
Order of fit: 4 C11+C22-2C12 is:
469.7624 GPa, RMS: 0.539599E-11
******************************************
Polynomial fit for C11+C22-2C12 done
A RMS of 0.196989E-04 was achieved using a polynome of degree : 2
At volume= 972.9393 bohr^3
C11+C22-2C12 is:
0.031720 a.u or
466.6136 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.
***************************************
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using case.outputeos
#
#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.709593
-0.010000
-29999.714625
0.000000
-29999.716424
0.010000
-29999.715045
0.020000
-29999.710509
===============================================================
Order of fit: 2 C11+C33-2C13 is:
481.9992 GPa, RMS: 0.108952E-04
Order of fit: 3 C11+C33-2C13 is:
481.9992 GPa, RMS: 0.193176E-05
Order of fit: 4 C11+C33-2C13 is:
479.9826 GPa, RMS: 0.281797E-11
******************************************
Polynomial fit for C11+C33-2C13 done
A RMS of 0.108952E-04 was achieved using a polynome of degree : 2
At volume= 972.9393 bohr^3
C11+C33-2C13 is:
0.032766 a.u or
481.9992 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.
***************************************
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using case.outputeos
#

16

#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.710244
-0.010000
-29999.714677
0.000000
-29999.716424
0.010000
-29999.715595
0.020000
-29999.712325
===============================================================
Order of fit: 2 C22 is:
388.4121 GPa, RMS: 0.345214E-04
Order of fit: 3 C22 is:
388.4121 GPa, RMS: 0.122352E-05
Order of fit: 4 C22 is:
389.6894 GPa, RMS: 0.460172E-11
******************************************
Polynomial fit for C22 done
A RMS of 0.345214E-04 was achieved using a polynome of degree : 2
At volume= 972.9393 bohr^3
C22 is:
0.026404 a.u or
388.4121 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.
***************************************
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using case.outputeos
#
#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.710546
-0.010000
-29999.715003
0.000000
-29999.716424
0.010000
-29999.714828
0.020000
-29999.710232
===============================================================
Order of fit: 2 C22+C33-2C23 is:
456.2655 GPa, RMS: 0.510319E-05
Order of fit: 3 C22+C33-2C23 is:
456.2655 GPa, RMS: 0.281694E-06
Order of fit: 4 C22+C33-2C23 is:
455.9715 GPa, RMS: 0.363798E-11
******************************************
Polynomial fit for C22+C33-2C23 done
A RMS of 0.510319E-05 was achieved using a polynome of degree : 2
At volume= 972.9393 bohr^3
C22+C33-2C23 is:
0.031016 a.u or
456.2655 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.

17

***************************************
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using case.outputeos
#
#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.709983
-0.010000
-29999.714572
0.000000
-29999.716424
0.010000
-29999.715663
0.020000
-29999.712381
===============================================================
Order of fit: 2 C33 is:
396.4612 GPa, RMS: 0.304318E-04
Order of fit: 3 C33 is:
396.4612 GPa, RMS: 0.174842E-05
Order of fit: 4 C33 is:
394.6360 GPa, RMS: 0.162695E-11
******************************************
Polynomial fit for C33 done
A RMS of 0.304318E-04 was achieved using a polynome of degree : 2
At volume= 972.9393 bohr^3
C33 is:
0.026951 a.u or
396.4612 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.
***************************************
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using case.outputeos
#
#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.708260
-0.010000
-29999.714376
0.000000
-29999.716424
0.010000
-29999.714376
0.020000
-29999.708260
===============================================================
Order of fit: 2 C44 is:
154.2082 GPa, RMS: 0.319110E-05
Order of fit: 3 C44 is:
154.2082 GPa, RMS: 0.319110E-05
Order of fit: 4 C44 is:
155.0410 GPa, RMS: 0.000000E
******************************************
Polynomial fit for C44 done
A RMS of 0.319110E-05 was achieved using a polynome of degree : 2
At volume=

972.9393 bohr^3

18

C44 is:
0.010483 a.u or
154.2082 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.
***************************************
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using case.outputeos
#
#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.704706
-0.010000
-29999.713498
0.000000
-29999.716424
0.010000
-29999.713498
0.020000
-29999.704706
===============================================================
Order of fit: 2 C44+C55+2C45 is:
221.5061 GPa, RMS: 0.147956E-05
Order of fit: 3 C44+C55+2C45 is:
221.5061 GPa, RMS: 0.147956E-05
Order of fit: 4 C44+C55+2C45 is:
221.1199 GPa, RMS: 0.162695E-11
******************************************
Polynomial fit for C44+C55+2C45 done
A RMS of 0.147956E-05 was achieved using a polynome of degree : 2
At volume= 972.9393 bohr^3
C44+C55+2C45 is:
0.015058 a.u or
221.5061 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.
***************************************
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using case.outputeos
#
#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.711405
-0.010000
-29999.715167
0.000000
-29999.716424
0.020000
-29999.711405
===============================================================
Order of fit: 2 C55 is:
94.8424 GPa, RMS: 0.751807E-06
Order of fit: 3 C55 is:
94.8560 GPa, RMS: 0.181899E-11

19

******************************************
Polynomial fit for C55 done
A RMS of 0.751807E-06 was achieved using a polynome of degree :

2

At volume= 972.9393 bohr^3
C55 is:
0.006447 a.u or
94.8424 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.
***************************************
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using case.outputeos
#
#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.703825
-0.010000
-29999.713056
0.000000
-29999.716424
0.010000
-29999.714762
0.020000
-29999.708778
===============================================================
Order of fit: 2 C11+4C66+4C16 is:
765.8798 GPa, RMS: 0.217945E-03
Order of fit: 3 C11+4C66+4C16 is:
765.8798 GPa, RMS: 0.658425E-05
Order of fit: 4 C11+4C66+4C16 is:
759.0065 GPa, RMS: 0.230086E-11
******************************************
Polynomial fit for C11+4C66+4C16 done
A RMS of 0.217945E-03 was achieved using a polynome of degree : 2
At volume= 972.9393 bohr^3
C11+4C66+4C16 is:
0.052063 a.u or
765.8798 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.
***************************************
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using case.outputeos
#
#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.701645
-0.010000
-29999.712636

20

0.000000
-29999.716424
0.010000
-29999.713715
0.020000
-29999.705171
===============================================================
Order of fit: 2 C22+4C66+4C26 is:
984.2220 GPa, RMS: 0.193666E-03
Order of fit: 3 C22+4C66+4C26 is:
984.2220 GPa, RMS: 0.238504E-05
Order of fit: 4 C22+4C66+4C26 is:
981.7323 GPa, RMS: 0.363798E-11
******************************************
Polynomial fit for C22+4C66+4C26 done
A RMS of 0.193666E-03 was achieved using a polynome of degree : 2
At volume= 972.9393 bohr^3
C22+4C66+4C26 is:
0.066906 a.u or
984.2220 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.
***************************************
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using case.outputeos
#
#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.704487
-0.010000
-29999.713226
0.000000
-29999.716424
0.010000
-29999.714362
0.020000
-29999.707270
===============================================================
Order of fit: 2 C33+4C66+4C36 is:
797.4801 GPa, RMS: 0.725372E-04
Order of fit: 3 C33+4C66+4C36 is:
797.4801 GPa, RMS: 0.266139E-05
Order of fit: 4 C33+4C66+4C36 is:
794.7019 GPa, RMS: 0.430451E-11
******************************************
Polynomial fit for C33+4C66+4C36 done
A RMS of 0.725372E-04 was achieved using a polynome of degree : 2
At volume= 972.9393 bohr^3
C33+4C66+4C36 is:
0.054212 a.u or
797.4801 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.
***************************************
##########################################
# M_ana_elastc_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#

21

#
using case.outputeos
#
#
VstVene
#
#
which have been created by
#
#
MONO.job
#
##########################################
-0.020000
-29999.711182
-0.010000
-29999.715126
0.000000
-29999.716424
0.010000
-29999.715118
0.020000
-29999.711248
===============================================================
Order of fit: 2 C66 is:
98.4538 GPa, RMS: 0.118798E-04
Order of fit: 3 C66 is:
98.4538 GPa, RMS: 0.983513E-07
Order of fit: 4 C66 is:
98.4281 GPa, RMS: 0.325391E-11
******************************************
Polynomial fit for C66 done
A RMS of 0.118798E-04 was achieved using a polynome of degree : 2
At volume= 972.9393 bohr^3
C66 is:
0.006693 a.u or
98.4538 GPa
******************************************
Analyze done.....
Do you want a hardcopy? (y/N)
***************************************
You can find data in ELC.output file.
***************************************
Printing final Elastic constant At voulme= 972.9393 bohr^3 .
=======================================================================
C11 = 366.8981 GPa
C22 = 388.4121 GPa
C33= 396.4612 GPa
C44 = 154.2082 GPa
C55 = 94.8424 GPa
C66= 98.4538 GPa
C11+C22-2C12 = 466.6136 GPa
C11+C33-2C13 = 481.9992 GPa
C33+C22-2C23 = 456.2655 GPa
C44+C55+2C45 = 221.5061 GPa
C11+4C66+4C16 = 765.8798 GPa
C22+4C66+4C26 = 984.2220 GPa
C33+4C66+4C36 = 797.4801GPa
=======================================================================
LU decomposition successful
Inverse Successful
You can find Inverse Matrix in INVELC-matrix file
Done
=======================================================================
Atom name = Zr
Atomic Mass from Periodic table =
91.2240 (gr/mol)
Atomic Mass from Periodic table =
15.1483*10^(-23) (gr)
Atom name = O
Atomic Mass from Periodic table =
15.9994 (gr/mol)
Atomic Mass from Periodic table =
2.6568*10^(-23) (gr)
Atom name = O
Atomic Mass from Periodic table =
15.9994 (gr/mol)
Atomic Mass from Periodic table =
2.6568*10^(-23) (gr)

22

Volume in unit of cm^3 =
144.1746*10^(-24) (cm^3)
Mass of Compound :
81.8478*10^(-23) (gr)
Density of Compound :
5.6770 (gr/cm^3)
=======================================================================
C11 = 366.8981 GPa
C22 = 388.4121 GPa
C33 = 396.4612 GPa
C44 = 154.2082 GPa
C55 = 94.8424 GPa
C66 = 98.4538 GPa
C12 = 144.3483 GPa
C13 = 140.6800 GPa
C16 = 1.2916 GPa
C23 = 164.3039 GPa
C26 = 50.4986 GPa
C36 = 1.8009 GPa
C45 = -13.7722 GPa
Prediction VOIGT Bulk modulus by using elastic constant values
Prediction REUSS Bulk modulus by using elastic constant values
Prediction HILL Bulk modulus by using elastic constant values

= 227.826 (GPa)
= 224.010 (GPa)
= 225.918 (GPa)

Prediction VOIGT Shear modulus by using elastic constant values = 116.329 (GPa)
Prediction REUSS Shear modulus by using elastic constant values = 108.362 (GPa)
Prediction HILL Shear modulus by using elastic constant values = 112.345 (GPa)
Prediction VOIGT Young modulus by using elastic constant values = 298.228 (GPa)
Prediction REUSS Young modulus by using elastic constant values = 279.945 (GPa)
Prediction HILL Young modulus by using elastic constant values = 289.111 (GPa)
Prediction VOIGT Poisson's coefficient by using elastic constant values = .281
Prediction REUSS Poisson's coefficient by using elastic constant values = .291
Prediction HILL Poisson's coefficient by using elastic constant values = .286
=======================================================================
By using HILL data
Transverse elastic wave velocity = 4448.54 (m/s)
Longitudinal elastic wave velocity = 8135.19 (m/s)
The average wave velocity = 4960.65 (m/s)
Debye Temperature = 644.834 (K)
=======================================================================
Press enter key to continue....
##############################################
# M_ana_elastorder_lapw analyses Elastic
#
#
constant
#
#
C(2013) by Morteza Jamal
#
#
using ELCorder.fit files
#
##############################################
CHECK THE SENSITIVITY
OF YOUR RESULT TO THE ORDER OF FIT
Press enter key to continue....
Order of fit for calculations were 4,4,4,4,4,4
4,4,4,4,4,3, and 4
We select minimum value for ORDER OF FIT i.e. 3
Press enter key to continue....
######## ORDER OF FIT IS : 2 , At volume = 972.93925 (bohr^3) ########
C11 = 366.898 GPa
C22 = 388.412 GPa
C33= 396.461 GPa
C44 = 154.208 GPa
C55 = 94.842 GPa
C66= 98.454 GPa
C11+C22-2C12 = 466.614 GPa

23

C11+C33-2C13 = 481.999 GPa
C33+C22-2C23 = 456.266 GPa
C44+C55+2C45 = 221.506 GPa
C11+4C66+4C16 = 765.880 GPa
C22+4C66+4C26 = 984.222 GPa
C33+4C66+4C36 = 797.480 GPa
_______________________________________________________________________
C11
C44
C12
C23

=
=
=
=

366.898 GPa
C22 = 388.412 GPa
C33 = 396.461 GPa
154.208 GPa
C55 = 94.842 GPa
C66 = 98.454 GPa
144.3480 GPa
C13 = 140.6800 GPa
C16 = 1.2915 GPa
164.3035 GPa
C26 = 50.4985 GPa
C36 = 1.8007 GPa
C45 = -13.7720 GPa
=======================================================================
Prediction VOIGT Bulk modulus by using elastic constant values
Prediction REUSS Bulk modulus by using elastic constant values
Prediction HILL Bulk modulus by using elastic constant values

= 227.826 (GPa)
= 849.725 (GPa)
= 538.775 (GPa)

Prediction VOIGT Shear modulus by using elastic constant values = 113.402 (GPa)
Prediction REUSS Shear modulus by using elastic constant values = 127.718 (GPa)
Prediction HILL Shear modulus by using elastic constant values = 120.560 (GPa)
Prediction VOIGT Young modulus by using elastic constant values = 291.792 (GPa)
Prediction REUSS Young modulus by using elastic constant values = 364.873 (GPa)
Prediction HILL Young modulus by using elastic constant values = 336.575 (GPa)
Prediction VOIGT Poisson's coefficient by using elastic constant values = .286
Prediction REUSS Poisson's coefficient by using elastic constant values = .428
Prediction HILL Poisson's coefficient by using elastic constant values = .395
=======================================================================
######## ORDER OF FIT IS : 3 , At volume = 972.93925 (bohr^3) ########
C11 = 366.898 GPa
C22 = 388.412 GPa
C33= 396.461 GPa
C44 = 154.208 GPa
C55 = 94.856 GPa
C66= 98.454 GPa
C11+C22-2C12 = 466.614 GPa
C11+C33-2C13 = 481.999 GPa
C33+C22-2C23 = 456.266 GPa
C44+C55+2C45 = 221.506 GPa
C11+4C66+4C16 = 765.880 GPa
C22+4C66+4C26 = 984.222 GPa
C33+4C66+4C36 = 797.480 GPa
_______________________________________________________________________
C11
C44
C12
C23

=
=
=
=

366.898 GPa
C22 = 388.412 GPa
C33 = 396.461 GPa
154.208 GPa
C55 = 94.856 GPa
C66 = 98.454 GPa
144.3480 GPa
C13 = 140.6800 GPa
C16 = 1.2915 GPa
164.3035 GPa
C26 = 50.4985 GPa
C36 = 1.8007 GPa
C45 = -13.7790 GPa
=======================================================================
Prediction VOIGT Bulk modulus by using elastic constant values
Prediction REUSS Bulk modulus by using elastic constant values
Prediction HILL Bulk modulus by using elastic constant values

= 227.826 (GPa)
= 849.725 (GPa)
= 538.775 (GPa)

Prediction VOIGT Shear modulus by using elastic constant values = 113.405 (GPa)
Prediction REUSS Shear modulus by using elastic constant values = 127.722 (GPa)
Prediction HILL Shear modulus by using elastic constant values = 120.563 (GPa)

24

Prediction VOIGT Young modulus by using elastic constant values = 291.798 (GPa)
Prediction REUSS Young modulus by using elastic constant values = 364.884 (GPa)
Prediction HILL Young modulus by using elastic constant values = 336.582 (GPa)
Prediction VOIGT Poisson's coefficient by using elastic constant values = .286
Prediction REUSS Poisson's coefficient by using elastic constant values = .428
Prediction HILL Poisson's coefficient by using elastic constant values = .395
=======================================================================
You can find these data in the output-order file.

We did all of these calculations in WIEN2k cartesian coordinate.
WIEN2k cartesian coordinate is different with
STANDARD cartesian coordinate for monoclinic compound.
So, we calculate elastic constants and ... in STANDARD cartesian
coordinate
gamma is:

99.634000

Elastic constants matrix in unit of (GPa) in WIEN2k cartesian
coordinate
366.898
144.348
140.680
0.000
0.000
1.292

144.348
388.412
164.304
0.000
0.000
50.499

140.680
164.304
396.461
0.000
0.000
1.801

0.000
0.000
0.000
154.208
-13.772
0.000

0.000
0.000
0.000
-13.772
94.842
0.000

1.292
50.499
1.801
0.000
0.000
98.454

Elastic constants matrix in unit of (GPa) in STANDARD cartesian
coordinate
418.246
164.237
131.002
164.237
396.461
140.747
131.002
140.747
363.757
0.000
0.000
0.000
36.246
-2.198
9.094
0.000
0.000
0.000
LU decomposition successful

0.000
0.000
0.000
101.050
0.000
-22.796

Inverse Successful
Done
_________________________________
The output file is STDELC-matrix.
---------------------------------

25

36.246
-2.198
9.094
0.000
85.107
0.000

0.000
0.000
0.000
-22.796
0.000
148.001

If you view the STDELC-matrix file you can see
Elastic constants matrix in unit of (GPa) in WIEN2k cartesian
coordinate
366.898
144.348
140.680
0.000
0.000
1.292

144.348
388.412
164.304
0.000
0.000
50.499

140.680
164.304
396.461
0.000
0.000
1.801

0.000
0.000
0.000
154.208
-13.772
0.000

0.000
0.000
0.000
-13.772
94.842
0.000

1.292
50.499
1.801
0.000
0.000
98.454

Elastic constants matrix in unit of (GPa) in STANDARD cartesian
coordinate
418.246
164.237
131.002
0.000
36.246
0.000

164.237
396.461
140.747
0.000
-2.198
0.000

131.002
140.747
363.757
0.000
9.094
0.000

0.000
0.000
0.000
101.050
0.000
-22.796

36.246
-2.198
9.094
0.000
85.107
0.000

0.000
0.000
0.000
-22.796
0.000
148.001

Elastic compliance matrix in unit of (1/GPa) in STANDARD cartesian
coordinate
0.00313
-0.00106
-0.00069
0.00000
-0.00129
0.00000

-0.00106
0.00329
-0.00091
0.00000
0.00063
0.00000

-0.00069
-0.00091
0.00335
0.00000
-0.00009
0.00000

0.00000
0.00000
0.00000
0.01025
0.00000
0.00158

Prediction VOIGT Bulk modulus =
227.826
Prediction REUSS Bulk modulus =
224.010
Prediction HILL Bulk modulus
=
225.918
Prediction VOIGT Shear modulus =
116.330
Prediction REUSS Shear modulus =
108.362
Prediction HILL Shear modulus =
112.346
Prediction VOIGT Young modulus =
298.231
Prediction REUSS Young modulus =
279.947
Prediction HILL Young modulus =
289.114
Prediction VOIGT Poisson's coefficient =
Prediction REUSS Poisson's coefficient =
Prediction HILL Poisson's coefficient =
***************************************
By using HILL data
Transverse elastic wave velocity
Longitudinal elastic wave velocity
The average wave velocity
Debye Temperature

=
=
=
=

26

-0.00129
0.00063
-0.00009
0.00000
0.01232
0.00000
(GPa)
(GPa)
(GPa)
(GPa)
(GPa)
(GPa)
(GPa)
(GPa)
(GPa)
0.282 (GPa)
0.292 (GPa)
0.287 (GPa)

4448.567
8135.213
4960.675
644.839

(m/s)
(m/s)
(m/s)
(K)

0.00000
0.00000
0.00000
0.00158
0.00000
0.00700

In WIEN2k(unrelax)
366.9
388.4
396.5
154.2
94.8
98.4
144.4
140.7
164.3
1.3
50.5
1.8
-13.8

In WIEN2k(relax)

C11
C22
C33
C44
C55
C66
C12
C13
C23
C16
C26
C36
C45

C11
C22
C33
C44
C55
C66
C12
C13
C23
C15
C25
C35
C46

In STD(unrelax)
418.3
396.5
363.8
101.1
85.1
148.0
164.2
131.0
140.8
36.3
-2.2
9.1
-22.8

Exp3
361.0
408.0
258.0
99.9
81.2
126.0
142.0
55.0
196.0
-21.3
31.2
-18.2
-22.7

117.7
82.9

27

5– Installation of the Mono-elastic package
The Mono-elastic package comes as a compressed tar file namely “mono-elastic.tar.gz”.
To install the package firstly copy the file to a directory of your choice.
Now, uncompress and expand it as:
 tar –zxvf mono-elastic.tar.gz
 cd mono-elastic
 Run buildMIRelast_lapw
This program helps you to create the "Makefile" and then compile mono-elastic. By
default, the Makefile expects the lapack_lapw and blas_lapw to be in the location
../SRC_lib. This should be changed to the correct location by modifying the FOPT
parameter as shown below.

This Program helps you to define Fortran compiler, Fortran options, and Library
options if you have installed WIEN2k. As you can see here this program defines
Fortran compiler, Fortran options, and Library options as automatically. Otherwise
you can define compiler and linker options as well as the path of mkl library
depending on the your selected system.
/home/MyLib/mkl/lib/em64t is the path of my mkl library.
To make “Makefile” by the lapack_lapw and blas_lapw libraries in the location
../SRC_lib and gfortran use the following options:
Fortran compiler: gfortran
Fortran options: -ffree-form
Library options (Lapack and BLAS): $(FOPT) –L/home/physicsprogram/SRC_lib –lpthread –static –llapack_lapw –lblas_lapw

the location ../SRC_lib should be changed to the correct location by modifying the
FOPT parameter as shown above.
PS: To install with -ffree-form, you should compile the lapack_lapw and
blas_lapw libraries with -ffree-form options. Otherwise it might was caused error.

28

If you view the OPTIONS file of the WIEN2k package you can use the FOPT,
LDFLAGS, and R_LIBS of it for compiling.

After defining the Fortran compiler, Fortran options, and Library options press Enter key.

29

The Environment Variable ELASTM_PATH is then defined and added to the end of the
.bashrc file. Thus you will be able to call mono-elastic’s programs for any location.

If you view the .bashrc file you can see

Now, logout from your Linux system and then login.

30

6– References
[1] R. Stadler, W. Wolf, R. Podloucky, G. Kresse, J. Furthmller, J. Hafner, Phys. Rev. B
54 (1996) 1729.
[2] B. Z. Yanchitsky, A. N. Timoshevskii, Determination of the space group and unit cell
for a periodic solid, Comp. Phys. Comm. 139 (2001) 235–242.
[3] S. Chan, Y. Fang, M. H. Grinsditch, Z. Li, M. V. Nevitt, W. M. Robertson, and E. S.
Zouboulis, J. Am. Ceram. Soc. 74 (1991) 1742 .

31


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