A Package for calculating elastic tensors of hexagonal Phases by using second-order derivative with Wien2k Package

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1 IR E L A S T + W I E N 2k A Package for calculating elastic tensors of hexagonal Phases by using second-order derivative with Wien2k Package User s guide, Hex-elastic_13.2 (Release ) Morteza Jamal Ghods City-Tehran-Iran 1

2 MANDATORY CONDITIONS: In any publication in the scientific literature please reference the program as follows: M. Jamal, Hex-elastic, (2012). ACKNOWLEDGMENT I gratefully appreciate B.Z. Yanchitsky for fruitful discussions, P. Blaha and S. Jalali Asadabadi for suggestions, and Carol Phillips for editing. For suggestions or bug reports please contact the author by m_jamal57@yahoo.com 2

3 1- Introduction Hex-elastic is a Package for finding elastic constants of hexagonal 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 2 (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. In this Package we consider the hexagonal crystal structure, which is spanned by three vectors,, and. The Bravais lattice vectors are normally written in a matrix form, i.e.: 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 t o change to the Voigt notation in order to reduce the number of indices. The Voigt notation replaces 1, 2, 3, (and ) 4, (and ) 5, (and ) 6 ) 3

We express the energy of the strained system by means of a Taylor expansion in the distortion parameters, The linear terms vanish if the strain causes no changes in the volume of the crystal.

There are five independent elastic constants for a hexagonal symmetry, called C 11, C 12, C 13, C 33, and C 55 [3].

4 We express the energy of the strained system by means of a Taylor expansion in the distortion parameters, 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 hexagonal system and we use it to evaluate the elastic constants. There are five independent elastic constants for a hexagonal symmetry, called C 11, C 12, C 13, C 33, and C 55 [3]. Since we have five independent elastic constants, we need five different strains to determine these elastic constants. The five distortions used in the hex-elastic Package are described below [2,4]. The first distortion is written as: and it changes the size of the basal plane, while keeping the z axis constant. The symmetry of the strained lattice is therefore still hexagonal and the energy for this distortion can be obtained as: The second type of distortion is a volume conserved distortion and lead to Orthorhombic symmetry and written as: 4

and the energy for this distortion can be obtained as: The third strain we have used is given by This strain changes C lattice parameter and keep the symmetry of the

which produces an object with low symmetry.

5 and the energy for this distortion can be obtained as: The third strain we have used is given by This strain changes C lattice parameter and keep the symmetry of the strained lattice hexagonal and the energy for this distortion can be obtained by The fourth elastic constant, C55, is determined by means of a deformation of the lattice, which produces an object with low symmetry. The deformation is written as: And it leads to triclinic symmetry and the energy for this deformation can be written as Finally, the last strain we have used is volume conserved and keeps the symmetry of the strained lattice hexagonal and can be written as: And the energy for this strain is given by 5

6 And In practice, to calculate elastic constants we fit to a polynomial of degree M and M changes from 2 till N-1, N is number of data, and then elastic constants is computed by using second order derivative ( E (δ) ) of Polynomial fit ( E=E( V, δ) ) of Energy vs. strains (δ) at zero strain (δ =0). In this situation we are able to check the sensitivity of our results (elastic constants) to the order of fit however final results are for a polynomial of degree 2 ( M=2) because we are in the regime of Hooke s law. 6

7 3- File structure and program flow The following table describes input and output files for each program of the Hex-elastic Package. Program needs generates H_set_elast_lapw case.struct init.struct runcommand1 runcommand2 pwdname command_init_lapw H_command_run_lapw H_setup11m12, H_setupc33 H_setupc1112, H_setupczz H_setupc55 getcalljobh makestructh init.struct pwdname runcommand1 runcommand2 init.struct.styp HEX.job auto_init_lapw runcommand1 runcommand2.vper.styp HEX.job number.strain StypX_Y.struct vol.optimize H_modifyjob_lapw H_calljob_lapw HEX.job VstVene H_fitdivELC number.strain ELCorder.fit vol.optimize ELC.output.styp ELC.fit VstVene H_ana_elastc_lapw VstVene case.outputeos ELC.fit vol.optimize ELC.ps H_ana_elast_lapw ELC.output ELC-matrix case.output_elastic H_InverseELC ELC-matrix INVELC-matrix MassRho init.struct.rho vol.optimize H_ana_elastorder_lapw ELCorder.fit output-order sgroupcheck_lapw StypX_Y.struct case.struct command_intso_lapw.infso command_initu_lapw.infldau auto_initso_lapw.infso case.inso auto_initu_lapw.infldau case.inorb case.indm/c Bold font is OPTIONAL Italic bold font means it is the user s choice 7

8 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 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 b else init_lapw -red 0 -vxc 13 -ecut -6 -mix 0.2 -numk b endif.vper Defines the percent of changes for different strains..styp Defines the type of strain. HEX.job 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= 8

9 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 ( \ ) echo "*******************************" echo $i set name=`echo "$name $i"` echo "*******************************" Styp1_-3.0 \ Styp1_-2.0 \ Styp1_-1.0 \ Styp1 0.0 \ Styp1 1.0 \ Styp1 2.0 \ Styp1 3.0 \ 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, c11-c12, c55, c33, and czz, directories. Otherwise, it sets the optimized volume from the original struct file i.e. case.struct vol.optimize Contains the optimized volume. ELCorder.fit Contains the elastic constants for different values of order of fit. ELC.output Contains the elastic constants for order of fit =2 ELC.fit Contains the data to plot a curve of energy vs value of changes (strains) for each strain type. ELC-matrix Defines the elastic constant matrix for each symmetry. case.output_elastic Contains the final elastic constant values. INVELC-matrix Defines the inverse of elastic constant matrix. 9

10 .rho.infso.infldau Contains density of mass and atomic volume. 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 Flow and short description for programs The Hex-elastic Package consists of several FORTRAN and SHELL SCRIPTS which are described below. A flowchart of the program is shown in the following diagram. H_set_elast_lapw : Makes an elast-constant directory in the present work directory ( PWD ) and c11+c12, c11-c12, c33, c55, and czz directories in the elast-constant directory. The H_set_elast_lapw program also copies information of the "PWD" into the c11+c12, c11- c12, c33, c55, and czz directories and calls "command_init_lapw", H_command_run_lapw, H_setupc1112, H_setupc11m12, H_setupc33, H_setupc55, and H_setupczz programs.. command_init_lapw : Gets information for making "auto_init_lapw". H_command_run_lapw : Gets the run commands for making HEX.job. H_setupcX (X=1112, 11m12, 55,.) : Gets the type of strain and calls the getcalljobh program. getcalljobh: Calls makestructh program and makes the HEX.job file. makestructh : Makes the StypX_Y.struct files where X and Y stand for the type of strain and value of changes, respectively and the vol.optimize file. H_modifyjob_lapw : Edits the job files according to the user s needs. H_calljob_lapw : Calls the HEX.job files for running. H_ana_elast_lapw : 10

11 Calls the H_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 H_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 H_ana_elastorder_lapw program. H_ana_elastc_lapw : Calls the H_fitdivELC program with appropriate libraries for calculating C11-C12, C11+C12, C55, and. H_InverseELC : Makes the Elastic compliance constants generated by inverting the elastic constant matrix. MassRho : Finds density of mass and atomic volume. H_ana_elastorder_lapw : Checks the sensitivity of the elastic constants to the order of fit. sgroupcheck_lapw : Finds the best value of tol in the sgroup [5] program and copies case.struct_sgroup as case.struct. 11

12 H_set_elast_lapw command_init_lapw generates auto_init_lapw H_command_run_lapw generates commandrun1/2 H_setupcX X=1112, 11m12,. getcalljobh generates HEX.job makestructh generates StypX_Y.struct sgroupcheck_lapw HEX.job generates VstVene auto_init_lapw H_ana_elastorder_lapw generates ELCorder.fit H_ana_elast_lapw H_ana_elastc_lapw H_InverseELC MassRho ELASTIC CONSTANTS IS READY generates case.output_elastic and INVELC-matrix Program flow in Tetra-elastic 12 H_fitdivELC calls Libraries generates C 11 -C 12, C 11 +C 12,. Dash arrow means user must run

13 4 Elastic constants calculation 1. Create a struct file and validate it by running "sgroupcheck_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 "H_set_elast_lapw.. 5. Now you must adapt the job files according to your needs (you can run "H_modifyjob_lapw" in Terminal ). It is not necessary to do step 5 if you defined the COMMAND RUN commands in step Now you must run the job files (you can run "H_calljob_lapw" ). It will take some time. 7. Run "H_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 C11-C12 and C55 the symmetry of the hexgonal 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 HEX.job files and a new "savename" (eg. "_use_pbe_rk8"). 13

14 Optionally you can specify more cases by rerunning H_setupcX (X=1112, 11m12, 55, see section 5-3 ). 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, c11-c12, c55, and directories. Otherwise, it sets the optimized volume from the original struct file i.e. case.struct. 4-2 One calculation To calculate C 33 or C 55 the following steps should be performed for example for calculation of C 55 : 1. Make a directory for example c Make a "case" directory in c55 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 To avoid step 10, you can run H_command_run_lapw for setting the COMMAND RUN commands for making HEX.job. 8. Run the H_setupc55 program. 9. chmod +x HEX.job file. 10. Modify the HEX.job file. 11. Call HEX.job 12. Call H_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 H_setupcX (X=1112, 11m12, 55, ). 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 c Cd to the elast-constant directory. 2. Cd to the c55 directory. 3. cd the case directory. 3-1) To avoid step 6, you can run H_command_run_lapw for setting the 14

15 COMMAND RUN commands for making the HEX.job. 4. Run the H_setupc55 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 HEX.job files and a new "savename" (eg. "_use_pbe_rk8"). 6. Modify the HEX.job file. 7. Call HEX.job 8. Call H_ana_elastc_lapw 4-4 Elastic constants calculation for Ti Ti compound is a test case for elastic constants calculation. The Ti structure is described in detail in the following: Ti H LATTICE,NONEQUIV.ATOMS: P63/mmc MODE OF CALC=RELA unit=bohr ATOM -1: X= Y= Z= MULT= 2 ISPLIT= 4-1: X= Y= Z= Ti1 NPT= 781 R0= RMT= Z: 22.0 LOCAL ROT MATRIX: Select Xc = PBE-GGA, R_Kmax = 7, L_max = 8, and nkpoint = 5000 Two dimensional search of Equation Of State for Ti by using 2Doptimize package. Equation of state: EOS2 (PRB52,8064) info 2 a,b,c,d V0,B(GPa),BP,E Equation of state: Murnaghan info 2 E=E0+[B*V/BP*(1/(BP-1)*(V0/V)**BP +1)-B*V0/(BP-1)]/ Pressure=B/BP*((V0/V)**BP -1) V0,B(GPa),BP,E vol energy de(eos2) de(murnaghan) Pressure(GPa) Sigma:

16 Equation of state: Birch-Murnaghan info 2 E = E0 + 9/16*(B/ )*V0*[(eta**2-1)**3*BP + (eta**2-1)**2*(6-4*eta**2)] --> eta = (V0/V)**(1/3) Pressure = 3/2*B*(eta**7 - eta**5)*(1 + 3/4*(BP-4)*[eta**2-1]) V0,B(GPa),BP,E vol energy de(birch-murnaghan) Pressure(GPa) Sigma: After running job files, you will find the below values for elastic constants of Ti ( do not forget to copy Ti.outputeos in "Ti" directory within c11+c12, c11-c12, c33, c55, and czz directories ). In the following examples you can find the percents that were used for strains. ####################################### # H_ana_elast_lapw analyses Elastic # # constant # # C(2012) by Morteza Jamal # ####################################### ########################################## # H_ana_elastc_lapw analyses Elastic # # constant # # C(2012) by Morteza Jamal # # using case.outputeos # # VstVene # # which have been created by # # HEX.job # ########################################## =============================================================== Order of fit: 2 C11+C12 is: GPa, RMS: E-04 Order of fit: 3 C11+C12 is: GPa, RMS: E-05 Order of fit: 4 C11+C12 is: GPa, RMS: E-12 ****************************************** Polynomial fit for C11+C12 done A RMS of E-04 was achieved using a polynome of degree : 2 At volume= bohr^3 C11+C12 is: a.u or GPa ****************************************** 16

17 Analyze done... Do you want a hardcopy? (y/n) *************************************** You can find data in ELC.output file. *************************************** ########################################## # H_ana_elastc_lapw analyses Elastic # # constant # # C(2012) by Morteza Jamal # # using case.outputeos # # VstVene # # which have been created by # # HEX.job # ########################################## =============================================================== Order of fit: 2 C33 is: GPa, RMS: E-05 Order of fit: 3 C33 is: GPa, RMS: E-05 Order of fit: 4 C33 is: GPa, RMS: E-12 ****************************************** Polynomial fit for C33 done A RMS of E-05 was achieved using a polynome of degree : 2 At volume= bohr^3 C33 is: a.u or GPa ****************************************** Analyze done... Do you want a hardcopy? (y/n) *************************************** You can find data in ELC.output file. *************************************** ########################################## # H_ana_elastc_lapw analyses Elastic # # constant # # C(2012) by Morteza Jamal # # using case.outputeos # # VstVene # # which have been created by # # HEX.job # ########################################## =============================================================== Order of fit: 2 Czz is: GPa, RMS: E-05 Order of fit: 3 Czz is: GPa, RMS: E-05 Order of fit: 4 Czz is: GPa, RMS: E-12 ****************************************** 17

18 Polynomial fit for Czz done A RMS of E-05 was achieved using a polynome of degree : 2 At volume= bohr^3 Czz is: a.u or GPa ****************************************** Analyze done... Do you want a hardcopy? (y/n) *************************************** You can find data in ELC.output file. *************************************** ########################################## # H_ana_elastc_lapw analyses Elastic # # constant # # C(2012) by Morteza Jamal # # using case.outputeos # # VstVene # # which have been created by # # HEX.job # ########################################## =============================================================== Order of fit: 2 C11-C12 is: GPa, RMS: E-05 Order of fit: 3 C11-C12 is: GPa, RMS: E-05 Order of fit: 4 C11-C12 is: GPa, RMS: E-12 ****************************************** Polynomial fit for C11-C12 done A RMS of E-05 was achieved using a polynome of degree : 2 At volume= bohr^3 C11-C12 is: a.u or GPa ****************************************** Analyze done... Do you want a hardcopy? (y/n) *************************************** You can find data in ELC.output file. *************************************** ########################################## # H_ana_elastc_lapw analyses Elastic # # constant # # C(2012) by Morteza Jamal # # using case.outputeos # # VstVene # # which have been created by # # HEX.job # ##########################################

19 =============================================================== Order of fit: 2 C55 is: GPa, RMS: E-04 Order of fit: 3 C55 is: GPa, RMS: E-04 Order of fit: 4 C55 is: GPa, RMS: E-12 ****************************************** Polynomial fit for C55 done A RMS of E-04 was achieved using a polynome of degree : 2 At volume= bohr^3 C55 is: a.u or GPa ****************************************** Analyze done... Do you want a hardcopy? (y/n) *************************************** You can find data in ELC.output file. *************************************** Printing final Elastic constant At voulme= bohr^3. ======================================================================= C11+C12 = GPa C11-C12 = GPa C33 = GPa C55 = GPa Czz = C11+C12+2C33-4C13 = GPa ======================================================================= LU decomposition successful Inverse Successful You can find Inverse Matrix in INVELC-matrix file Done ======================================================================= Atom name = Ti Atomic Mass from Periodic table = (gr/mol) Atomic Mass from Periodic table = *10^(-23) (gr) Volume in unit of cm^3 = *10^(-24) (cm^3) Mass of Compound : *10^(-23) (gr) Density of Compound : (gr/cm^3) ======================================================================= C11 = GPa C12 = GPa C13 = GPa C33 = GPa C55 = 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 = (GPa) = (GPa) = (GPa) Prediction VOIGT Shear modulus by using elastic constant values = (GPa) Prediction REUSS Shear modulus by using elastic constant values = (GPa) Prediction HILL Shear modulus by using elastic constant values = (GPa) Prediction VOIGT Young modulus by using elastic constant values = (GPa) 19

20 Prediction REUSS Young modulus by using elastic constant values = (GPa) Prediction HILL Young modulus by using elastic constant values = (GPa) Prediction VOIGT Poisson's coefficient by using elastic constant values =.295 Prediction REUSS Poisson's coefficient by using elastic constant values =.298 Prediction HILL Poisson's coefficient by using elastic constant values =.297 ================================================================================ By using HILL data Transverse elastic wave velocity = (m/s) Longitudinal elastic wave velocity = (m/s) The average wave velocity = (m/s) Debye Temperature = (K) ===================================================================== Press enter key to continue... ############################################## # H_ana_elastorder_lapw analyses Elastic # # constant # # C(2012) by Morteza Jamal # # 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 and 4 We select minimum value for ORDER OF FIT i.e. 4 Press enter key to continue... ######## ORDER OF FIT IS : 2, At volume = (bohr^3) ######## (c11-c12) = (GPa) (c11+c12) = (GPa) (c33) = (GPa) (c55) = (GPa) (czz=c11+c12+2c33-4c13) = (GPa) c11 = (GPa) c12 = (GPa) c13 = (GPa) c33 = (GPa) C55 = (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 = (GPa) = (GPa) = (GPa) Prediction VOIGT Shear modulus by using elastic constant values = (GPa) Prediction REUSS Shear modulus by using elastic constant values = (GPa) Prediction HILL Shear modulus by using elastic constant values = (GPa) 20

21 Prediction VOIGT Young modulus by using elastic constant values = (GPa) Prediction REUSS Young modulus by using elastic constant values = (GPa) Prediction HILL Young modulus by using elastic constant values = (GPa) Prediction VOIGT Poisson's coefficient by using elastic constant values =.295 Prediction REUSS Poisson's coefficient by using elastic constant values =.298 Prediction HILL Poisson's coefficient by using elastic constant values =.297 ======================================================================= ######## ORDER OF FIT IS : 3, At volume = (bohr^3) ######## (c11-c12) = (GPa) (c11+c12) = (GPa) (c33) = (GPa) (c55) = (GPa) (czz=c11+c12+2c33-4c13) = (GPa) c11 = (GPa) c12 = (GPa) c13 = (GPa) c33 = (GPa) C55 = (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 = (GPa) = (GPa) = (GPa) Prediction VOIGT Shear modulus by using elastic constant values = (GPa) Prediction REUSS Shear modulus by using elastic constant values = (GPa) Prediction HILL Shear modulus by using elastic constant values = (GPa) Prediction VOIGT Young modulus by using elastic constant values = (GPa) Prediction REUSS Young modulus by using elastic constant values = (GPa) Prediction HILL Young modulus by using elastic constant values = (GPa) Prediction VOIGT Poisson's coefficient by using elastic constant values =.295 Prediction REUSS Poisson's coefficient by using elastic constant values =.298 Prediction HILL Poisson's coefficient by using elastic constant values =.297 ======================================================================= ######## ORDER OF FIT IS : 4, At volume = (bohr^3) ######## (c11-c12) = (GPa) (c11+c12) = (GPa) (c33) = (GPa) (c55) = (GPa) (czz=c11+c12+2c33-4c13) = (GPa) c11 = (GPa) c12 = (GPa) c13 = (GPa) c33 = (GPa) C55 = (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 = (GPa) = (GPa) = (GPa) 21

22 Prediction VOIGT Shear modulus by using elastic constant values = (GPa) Prediction REUSS Shear modulus by using elastic constant values = (GPa) Prediction HILL Shear modulus by using elastic constant values = (GPa) Prediction VOIGT Young modulus by using elastic constant values = (GPa) Prediction REUSS Young modulus by using elastic constant values = (GPa) Prediction HILL Young modulus by using elastic constant values = (GPa) Prediction VOIGT Poisson's coefficient by using elastic constant values =.368 Prediction REUSS Poisson's coefficient by using elastic constant values =.378 Prediction HILL Poisson's coefficient by using elastic constant values =.373 ======================================================================= You can find these data in the output-order file. C11 = 182 GPa C12 = 73 GPa C13 = 69 GPa C33 = 188 GPa C55 = 44 GPa Our calculation Other(TB) 6 Exp 6 C C C C C

5 Installation of the Hex-elastic package The

namely hex-elastic-deriativemethod-2012-p.tar.

firstly copy the file to a directory of your choice.

23 5 Installation of the Hex-elastic package The Hex-elastic package comes as a compressed tar file namely hex-elastic-deriativemethod-2012-p.tar.gz or hex-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 hex-elastic.tar.gz cd hex-elastic 23

24 Run buildhirelast_lapw This program helps you to create the "Makefile" and then compile hex-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. 24

25 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. 25

26 The Environment Variable ELASTH_PATH is then defined and added to the end of the.bashrc file. Thus you will be able to call hex-elastic s programs for any location. 26

27 If you view the.bashrc file you can see Now, logout from your Linux system and then login. 27

28 6 References [1] R. Stadler, W. Wolf, R. Podloucky, G. Kresse, J. Furthmller, J. Hafner, Phys. Rev. B 54 (1996) [2] L. Fast, J. M. Wills, B. Johansson and O. Eriksson, Phys. Rev. B 51 (1995). [3] D.C. Wallace, Solid State Phys. 25 (1970) 301 [4] [5] B. Z. Yanchitsky, A. N. Timoshevskii, Determination of the space group and unit cell for a periodic solid, Comp. Phys. Comm. 139 (2001) [6] Michael J. Mehl and Dimitrios A. Papaconstantopoulos, Phys. Rev. B 54 (1996). 28

A Package for calculating elastic tensors of rhombohedral Phases by using second-order derivative with Wien2k Package

IR ELAST + WIEN2k A Package for calculating elastic tensors of rhombohedral Phases by using second-order derivative with Wien2k Package User s guide, Rhom-elastic_13.1 (Release 16.09.2013) Morteza Jamal