Table 2 Lattice parameters, energies (E), cohesive energies (E^coh), energy differences (ΔE), and elastic constants of HCP, BCC and FCC Ti, relaxed surface energies (σ), vacancy formation energy (E_v) of HCP Ti and the unstable stacking fault energy (γ_usf) of BCC Ti from DFT, experiment (Expt), DP, EAM⁴, and MEAM⁶.

HCP

a (Å)

2.936

2.947^a

2.934

2.951

2.930

c/a

1.583

1.586^a

1.586

1.589

1.596

E (eV/atom)

−7.834

–

−7.833

−5.401

−4.831

E^coh (eV/atom)

5.34

4.85^b

4.85

5.40

4.83

C₁₁ (GPa)

171.6

176.1^c

157.7

164.8

174.3

C₁₂ (GPa)

86.0

86.9^c

85.6

87.8

94.7

C₁₃ (GPa)

74.4

68.3^c

78.8

82.6

72.3

C₃₃ (GPa)

189.0

190.5^c

188.2

165.2

187.9

C₄₄ (GPa)

42.7

50.8^c

39.2

57.7

57.7

σ_basal (J/m²)

1.95

–

1.94

2.26

1.47

σ_prism (J/m²)

2.00

–

1.99

2.29

1.55

σ_pyr.I (J/m²)

1.91

–

1.90

2.31

1.52

σ_pyr.II (J/m²)

2.09

–

2.07

2.43

1.70

E_v (eV)

2.06

–

2.41

1.73

2.19

BCC

a (Å)

3.252

–

3.253

3.242

3.272

E (eV/atom)

−7.724

–

−7.725

−5.313

−4.720

ΔE_BCC (eV/atom)^d

0.110

–

0.108

0.088

0.111

E^coh (eV/atom)

5.23

–

4.74

5.31

4.72

C₁₁ (GPa)

90.2

–

92.9

276.8

94.9

C₁₂ (GPa)

113.8

–

108.8

283.6

111.0

C₄₄ (GPa)

39.9

–

40.3

34.3

52.9

\({\gamma }_{{{{\rm{usf}}}}}^{\{110\}}\) (J/m²)^e

0.19

–

0.28

–

0.21

\({\gamma }_{{{{\rm{usf}}}}}^{\{112\}}\) (J/m²)

0.24

–

0.36

–

0.25

\({\gamma }_{{{{\rm{usf}}}}}^{\{123\}}\) (J/m²)

0.23

–

0.34

–

0.24

FCC

a (Å)

4.108

–

4.108

4.131

4.147

E (eV/atom)

−7.777

–

−7.778

−5.349

−4.792

ΔE_FCC (eV/atom)

0.057

–

0.055

0.052

0.039

E^coh (eV/atom)

5.28

–

4.79

5.35

4.79

C₁₁ (GPa)

133.0

–

145.1

–^f

125.8

C₁₂ (GPa)

94.2

–

95.3

–

83.5

C₄₄ (GPa)

58.6

–

59.4

–

58.7