First-Principles Study of the Structural Stability and Electronic and Elastic Properties of Helium in α-Zirconium

Table 3

Elastic constants 𝐶_𝑖𝑗 (in GPa), bulk modulus 𝐵 (in GPa), shear modulus 𝐺 (in GPa), Young’s modulus 𝐸 (in GPa), percentage anisotropy in shear 𝐴_𝐺, and universal anisotropic index 𝐴^𝑈 of 𝛼-Zr and Zr-He systems as calculated in the present work compared with experimental values.

	𝐶11	𝐶33	𝐶12	𝐶13	𝐶44	𝐵	𝐺	𝐸	𝐴𝐺	𝐴^𝑈
𝛼-Zr (this work)	138.22	145.17	65.62	70.27	25.14	92.59	31.19	84.13	0.016	0.16
𝛼-Zr (PBE/PAW) [36]	157	158	51	62	15	91	29	80
𝛼-Zr (PW91/UPP) [31]	142	164	64	64	29	92
𝛼-Zr (PW91/UPP) [32]	159.38	180.93	57.96	66.07	17.52	96.16	38.99	103.04
𝛼-Zr (PBE/UPP) [37]	139.4	162.7	71.3	66.3	25.5
𝛼-Zr (Exp.) [38] at 4 K	155.4	172.5	67.2	64.6	36.3
𝛼-Zr (Exp.) [38] at 298 K	143.4	164.8	72.8	65.3	32.0
Zr96He (this work)	139.23	145.14	64.85	69.60	24.43	92.35	31.22	84.17	0.020	0.21
Zr54He (this work)	144.24	147.03	60.91	69.18	23.65	92.61	32.44	87.14	0.034	0.35
Zr36He (this work)	140.99	161.41	65.62	64.74	21.98	92.46	31.47	84.80	0.045	0.47