Elastic properties of copper

This example demonstrates the computation of elastic constants using an embedded atom method (EAM) potential based on user defined functions. The potential form and parameters have been taken from [MisMehPap01]. The example also illustrates the use of XML Inclusions.

In particular this example provides a demonstration of the calculation of both “clamped ion” and “relaxed ion” elastic constants for a hexagonal lattice structure.

Location

examples/elastic_constants_Cu

Input files

  • main.xml: main input file

     1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
    <?xml version="1.0" encoding="iso-8859-1"?>
<job>

  <name>Structural relaxation and elastic constants for copper using an EAM potential</name>
  
  <verbosity>medium</verbosity>
  
  <fitting>
    <BFGS conv-threshold="1e-5" max-iter="20" />
  </fitting>
  
  <atom-types>
    <species>Cu</species>
  </atom-types>
  
  <potentials>
    <xi:include href="potential.xml" xmlns:xi="http://www.w3.org/2003/XInclude" />
  </potentials>
  
  <deformations>
    <deformation id="FCC_lattice_constant">
      <scale>2.55614</scale>
    </deformation>
  </deformations>

  <structures>
    <xi:include href="structures.xml" xmlns:xi="http://www.w3.org/2003/XInclude" />
  </structures>

</job>

  • potential.xml: initial parameter set (included in main input file via XML Inclusions)

      1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
    <eam id='Cu EAM potential' species-a='*' species-b='*'>

  <mapping>
    <pair-interaction species-a='*' species-b='*' function='V' />
    <electron-density species-a='*' species-b='*' function='rho' />
    <embedding-energy species='*' function='F' />			
  </mapping>

  <functions>

    <sum id='V'>
      <user-function id='V_term1'>
	<input-var>r</input-var>
	<expression>
	  (E1*(exp(-2*alpha1*(r-r01)) - 2*exp(-alpha1*(r-r01))) + 
	  E2*(exp(-2*alpha2*(r-r02)) - 2*exp(-alpha2*(r-r02))) + delta) 
	</expression>
	<derivative>
	  E1*(2*alpha1*exp(-alpha1*(r-r01)) - 2*alpha1*exp(-2*alpha1*(r-r01))) 
	  + E2*(2*alpha2*exp(-alpha2*(r-r02)) - 2*alpha2*exp(-2*alpha2*(r-r02)))
	</derivative>
	<param name='E1'>2.01458e2</param>
	<param name='alpha1'>2.97758</param>
	<param name='r01'>0.83591</param>
	<param name='E2'>6.59288e-3</param>
	<param name='alpha2'>1.54927</param>
	<param name='r02'>4.46867</param>
	<param name='delta'>0.86225e-2</param>
	<screening>
	  <user-function id='rho_screening'>
	    <cutoff>5.50679</cutoff>
	    <input-var>r</input-var>
	    <expression>1 - 1/(1 + ((r - cutoff) / h)^4)</expression>
	    <derivative>4 * h^4 * (r-cutoff)^3 / ((h^4 + (r-cutoff)^4)^2)</derivative>
	    <param name='h'>0.50037</param>
	  </user-function>
	</screening>
      </user-function>
      <user-function id='V_term2'>
	<cutoff>5.50679</cutoff>
	<input-var>r</input-var>
	<expression>
	  - (r &lt; rs1 ? S1*(rs1-r)^4 : 0)
	  - (r &lt; rs2 ? S2*(rs2-r)^4 : 0)
	  - (r &lt; rs3 ? S3*(rs3-r)^4 : 0)
	</expression>
	<derivative>
	  - (r &lt; rs1 ? -4*S1*(rs1-r)^3 : 0)
	  - (r &lt; rs2 ? -4*S2*(rs2-r)^3 : 0)
	  - (r &lt; rs3 ? -4*S3*(rs3-r)^3 : 0)
	</derivative>
	<param name='rs1'>2.24000</param>
	<param name='rs2'>1.80000</param>
	<param name='rs3'>1.20000</param>
	<param name='S1'>4.00000</param>
	<param name='S2'>40.00000</param>
	<param name='S3'>1.15000e3</param>
      </user-function>
    </sum>

    <user-function id='rho'>
      <input-var>r</input-var>
      <expression>a * exp(-beta1*(r - r03)^2) + exp(-beta2*(r - r04))</expression>
      <derivative>beta2*(-exp(-beta1*(r-r04))) - 2*a*beta1*(r-r03)*exp(-beta1*(r-r03)^2)</derivative>
      <param name='a'>3.80362</param>
      <param name='r03'>-2.19885</param>
      <param name='r04'>-2.61984e2</param>
      <param name='beta1'>0.17394</param>
      <param name='beta2'>5.35661e2</param>
      <screening>
	<user-function id='rho_screening'>
	  <cutoff>5.50679</cutoff>
	  <input-var>r</input-var>
	  <expression>1 - 1/(1 + ((r - cutoff) / h)^4)</expression>
	  <derivative>4 * h^4 * (r-cutoff)^3 / ((h^4 + (r-cutoff)^4)^2)</derivative>
	  <param name='h'>0.5</param>
	</user-function>
      </screening>
    </user-function>

    <user-function id='F'>
      <input-var>rho</input-var>
      <expression>
	(rho &lt; 1) ?
	(F0 + 0.5*F2*(rho - 1)^2 + q1*(rho-1)^3 + q2*(rho-1)^4 + q3*(rho-1)^5 + q4*(rho-1)^6)
	: 
	(F0 + 0.5*F2*(rho - 1)^2 + q1*(rho-1)^3 + Q1*(rho-1)^4) / (1 + Q2*(rho-1)^3)
      </expression>
      <derivative>
	(rho &lt; 1) ?
	(rho - 1)*(F2 + 3*q1*(rho-1) + 4*q2*(rho-1)^2 + 5*q3*(rho-1)^3 + 6*q4*(rho-1)^4)
	:
	(F2*(rho-1) + 3*q1*(rho-1)^2 + 4*Q1*(rho-1)^3)/(Q2*(rho-1)^3 + 1)
	- (3*Q2*(rho-1)^2 * (F0 + 0.5*F2*(rho-1)^2 + q1*(rho-1)^3 + Q1*(rho-1)^4)) / 
	((Q2*(rho-1)^3 + 1)^2)
      </derivative>
      <param name='F0'>-2.3</param>
      <param name='F2'>1.4</param>
      <param name='q1'>-1.3</param>
      <param name='q2'>-0.9</param>
      <param name='q3'>1.8</param>
      <param name='q4'>3.0</param>
      <param name='Q1'>0.4</param>
      <param name='Q2'>0.3</param>
    </user-function>

  </functions>

</eam>

  • structures.xml: input structures (included in main input file via XML Inclusions)

      1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
    <group>

  <fcc-lattice id="FCC">
    <atom-type>Cu</atom-type>
    <lattice-parameter>3.615</lattice-parameter>
    <properties>
      <atomic-energy target='-3.540'/>
      <lattice-parameter target='3.615'/>
      <bulk-modulus/>
      <C11 target='170.0'/>
      <C12 target='122.5'/>
      <C44 target= '75.8'/>
    </properties>
  </fcc-lattice>

  <hcp-lattice id="HCP (clamped ion elastic constants)">
    <atom-type>Cu</atom-type>
    <lattice-parameter>2.8</lattice-parameter>
    <ca-ratio>1.6</ca-ratio>
    <properties>
      <atomic-energy/>
      <lattice-parameter/> <ca-ratio/>
      <pressure/>
      <pxx/> <pyy/> <pzz/> <pxy/> <pxz/> <pyz/>
      <bulk-modulus/>
      <C11/> <C12/> <C13/> <C33/> <C44/> <C55/> <C66/>
    </properties>
  </hcp-lattice>

  <hcp-lattice id="HCP (relaxed ion elastic constants)">
    <atom-type>Cu</atom-type>
    <lattice-parameter>2.8</lattice-parameter>
    <ca-ratio>1.6</ca-ratio>
    <relax-dof>
      <atom-coordinates/>
    </relax-dof>
    <properties>
      <atomic-energy/>
      <lattice-parameter/> <ca-ratio/>
      <pressure/>
      <pxx/> <pyy/> <pzz/> <pxy/> <pxz/> <pyz/>
      <bulk-modulus/>
      <C11/> <C12/> <C13/> <C33/> <C44/> <C55/> <C66/>
    </properties>
  </hcp-lattice>

  <user-structure id="FCC in 111 orientation (relaxed ion elastic constants)">
    <pbc x="true" y="true" z="true"/>
    <cell>
      <a1 x="0.5" y="-0.866025403784" z="0"/>
      <a2 x="0.5" y="+0.866025403784" z="0"/>
      <a3 x="0"   y="0"               z="2.449489742783"/>
      <atoms>
	<atom type="Cu" x="0.0" y="0.0" z="0.0" reduced="true"/>
	<atom type="Cu" x="0.333333333333333" y="0.666666666666667" z="0.333333333333333" reduced="true"/>
	<atom type="Cu" x="0.666666666666667" y="0.333333333333333" z="0.666666666666667" reduced="true"/>
      </atoms>
    </cell>
    <deformation>FCC_lattice_constant</deformation>
    <relax-dof>
      <atom-coordinates/>
    </relax-dof>
    <properties>
      <pressure/>
      <pxx/> <pyy/> <pzz/> <pxy/> <pxz/> <pyz/>
      <bulk-modulus/>
      <C11/> <C12/> <C13/> <C33/> <C44/> <C55/> <C66/>
    </properties>
  </user-structure>

  <bcc-lattice id="BCC">
    <atom-type>Cu</atom-type>
    <lattice-parameter>2.8</lattice-parameter>
    <properties>
      <atomic-energy target='-3.496'/>
      <lattice-parameter/>
      <C11/> <C12/> <C44/>
    </properties>
  </bcc-lattice>

  <diamond-lattice id="DIA">
    <atom-type>Cu</atom-type>
    <lattice-parameter>4.0</lattice-parameter>
    <properties>
      <atomic-energy target='-2.293'/>
      <lattice-parameter/>
      <C11/>
      <C12/>
      <C44/>
    </properties>
  </diamond-lattice>	

  <sc-lattice id="SC">
    <atom-type>Cu</atom-type>
    <lattice-parameter>2.3</lattice-parameter>
    <properties>
      <atomic-energy target='-2.996'/>
      <lattice-parameter/>
      <C11/> <C12/> <C44/>
    </properties>
  </sc-lattice>	

</group>

Output

  • The final properties (as well as parameters) are written to standard output.

      1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
    This is program version 0.1.6
Reading job file main.xml
-------------------------------------------------------
Parsing input file(s)
-------------------------------------------------------
-------------------------------------------------------
Potential parameters being optimized (#dof=0):
   NONE
-------------------------------------------------------
-------------------------------------------------------
Number of properties being fitted: 0
-------------------------------------------------------
-------------------------------------------------------
Computing structure properties
Structure 'FCC':
   total-energy: -3.55787 eV
   atomic-energy: -3.55787 eV/atom
   total-volume: 11.8104 A^3
   atomic-volume: 11.8104 A^3/atom
   bulk-modulus: 139.856 GPa
   C11: 171.427 GPa
   C12: 124.138 GPa
   C44: 76.1944 GPa
   lattice-parameter: 3.615 A [0:]
Structure 'HCP (clamped ion elastic constants)':
   total-energy: -6.54577 eV
   atomic-energy: -3.27288 eV/atom
   total-volume: 30.4176 A^3
   atomic-volume: 15.2088 A^3/atom
   pressure: -199122 bar
   bulk-modulus: 25.0959 GPa
   pxx: -205262 bar
   pyy: -205262 bar
   pzz: -186842 bar
   pyz: -1.42541e-10 bar
   pxz: 2.34462e-10 bar
   pxy: 4.31279e-11 bar
   C11: 31.8493 GPa
   C12: 7.33113 GPa
   C13: 14.1413 GPa
   C33: 87.2442 GPa
   C44: 17.2645 GPa
   C55: 17.2645 GPa
   C66: 12.2586 GPa
   lattice-parameter: 2.8 A
   ca-ratio: 1.6
Structure 'HCP (relaxed ion elastic constants)':
   total-energy: -6.54577 eV
   atomic-energy: -3.27288 eV/atom
   total-volume: 30.4176 A^3
   atomic-volume: 15.2088 A^3/atom
   pressure: -199122 bar
   bulk-modulus: 25.0959 GPa
   pxx: -205262 bar
   pyy: -205262 bar
   pzz: -186842 bar
   pyz: -1.42541e-10 bar
   pxz: 2.34462e-10 bar
   pxy: 4.31279e-11 bar
   C11: 29.4757 GPa
   C12: 9.70487 GPa
   C13: 14.1413 GPa
   C33: 87.2442 GPa
   C44: 17.2645 GPa
   C55: 17.2645 GPa
   C66: 9.8872 GPa
   lattice-parameter: 2.8 A
   ca-ratio: 1.6
Structure 'FCC in 111 orientation (relaxed ion elastic constants)':
   total-energy: -10.6736 eV
   atomic-energy: -3.55787 eV/atom
   total-volume: 35.4291 A^3
   atomic-volume: 11.8097 A^3/atom
   pressure: 1.58787 bar
   bulk-modulus: 139.879 GPa
   pxx: 1.58787 bar
   pyy: 1.58787 bar
   pzz: 1.58787 bar
   pyz: -1.25167e-07 bar
   pxz: 1.88275e-12 bar
   pxy: 1.56895e-12 bar
   C11: 224.016 GPa
   C12: 106.64 GPa
   C13: 89.1192 GPa
   C33: 241.533 GPa
   C44: 41.166 GPa
   C55: 41.166 GPa
   C66: 58.6859 GPa
Structure 'BCC':
   total-energy: -3.4849 eV
   atomic-energy: -3.4849 eV/atom
   total-volume: 10.976 A^3
   atomic-volume: 10.976 A^3/atom
   C11: 186.124 GPa
   C12: 191.813 GPa
   C44: 123.432 GPa
   lattice-parameter: 2.8 A [0:]
Structure 'DIA':
   total-energy: 7.19803 eV
   atomic-energy: 3.59901 eV/atom
   total-volume: 16 A^3
   atomic-volume: 8 A^3/atom
   C11: 802.121 GPa
   C12: 1591.55 GPa
   C44: 757.007 GPa
   lattice-parameter: 4 A [0:]
Structure 'SC':
   total-energy: -3.04011 eV
   atomic-energy: -3.04011 eV/atom
   total-volume: 12.167 A^3
   atomic-volume: 12.167 A^3/atom
   C11: 496.813 GPa
   C12: 41.72 GPa
   C44: -43.9108 GPa
   lattice-parameter: 2.3 A [0:]
-------------------------------------------------------
-------------------------------------------------------

Other files

  • lammps_runs/data.lammps: elastic constants computed using Lammps; the original runs can be found in the tgz-archives in the lammps_runs subdirectory.

     1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
    # 1st col: structure
# 2nd col: pressure (GPa)
# 3rd col: lattice constant a (A)
# 4th col: cohesive energy (eV/atom)
# 5th col: elastic constant c11 (GPa)
# 6th col: elastic constant c12 (GPa)
# 7th col: elastic constant c13 (GPa)
# 8th col: elastic constant c33 (GPa)
# 9th col: elastic constant c44 (GPa)
# 10th col: elastic constant c66 (GPa)

FCC     0.00    3.6142   -3.5406  169.2  122.0   -     -       76.0   -         42      96
FCC111  0.00    2.5556   -3.5406  221.6  104.5   87.0  239.1   41.1   58.6     188     144
HCP     0.00    2.5555   -3.5327  205.9  120.2   83.5  214.8   37.0   42.8     957      96
DHCP    0.00    2.5555   -3.5366  206.3  119.8   85.1  226.0   38.8   43.3     188     192