Stability, Mechanical Properties and Anisotropic Elastic Properties of Ga<sub>x</sub>Mg<sub>y</sub> Compounds

Liu, LinJing; Lian, Liangchong; Yu, Jie

doi:10.1590/1980-5373-MR-2018-0624

LinJing Liu

Department of Mechanical and Electrical Engineering, Hunan Biological and Electromechanical Polytechnics, Changsha, 400126, China

http://orcid.org/0000-0002-6697-361X

Liangchong Lian

Department of Mechanical and Electrical Engineering, Hunan Biological and Electromechanical Polytechnics, Changsha, 400126, China

Jie Yu ^**e-mail: yujieone@163.com

Faculty of Material Science and Engineering, Kunming University of Science and Technology, Kunming, 650093, China

*e-mail: yujieone@163.com

Species	Space group	Composition at.% Ga	Lattice constants (Å)			E_coh ( eV/atom)	ΔH_r(eV/atom)
Species	Space group	Composition at.% Ga	a	b	c	E_coh ( eV/atom)	ΔH_r(eV/atom)
α-Mg	P63/mmc	0	3.209	3.209	5.211	-1.858	0
Ga₂Mg₅	Ibam	28.57	7.004^b b This work	13.663^b b This work	5.987^b b This work	-2.504^b b This work	-0.120^b b This work
			7.017^g g Exp. In Ref.43	13.708^g g Exp. In Ref.43	6.020^g g Exp. In Ref.43		-0.113ⁱ i Exp. In Ref.37
			7.036^c c Cal. In Ref.28	13.747^c c Cal. In Ref.28	6.028^c c Cal. In Ref.28		-0.119^c c Cal. In Ref.28
GaMg₂	P-62^c c Cal. In Ref.28	33.33	7.730^b b This work	7.730^b b This work	6.976^b b This work	-2.605^b b This work	-0.133^b b This work
			7.794^f f Exp. In Ref.42	7.794^f f Exp. In Ref.42	6.893^f f Exp. In Ref.42		-0.121ⁱ i Exp. In Ref.37
			7.805^c c Cal. In Ref.28		6.941^c c Cal. In Ref.28		-0.130^c c Cal. In Ref.28
GaMg	I41/aZ	50.00	10.523^b b This work	10.523^b b This work	5.539^b b This work	-2.941^b b This work	-0.162^b b This work
			10.530^e e Exp. In Ref.41		5.530^e e Exp. In Ref.41		-0.135ⁱ i Exp. In Ref.37
			10.690^c i Exp. In Ref.37		5.555^c i Exp. In Ref.37		-0.142^c i Exp. In Ref.37
H-Ga₂Mg	P63/mmc	66.67	4.370^b b This work	4.370^b b This work	6.692^b b This work	-3.229^b b This work	-0.144^b b This work
			4.343^h h Exp. In Ref.44	4.343^h h Exp. In Ref.44	6.982^h h Exp. In Ref.44
O-Ga₂Mg	Pbam	66.67	6.813^b b This work	16.259^b b This work	4.040^b b This work	-3.234^b b This work	-0.148^b b This work
			6.802^a a Exp. In Ref.39	16.346^a a Exp. In Ref.39	4.111^a a Exp. In Ref.39		-0.118ⁱ i Exp. In Ref.37
			6.868^c c Cal. In Ref.28	16.457^c c Cal. In Ref.28	4.139^c c Cal. In Ref.28		-0.120^c c Cal. In Ref.28
Ga₅Mg₂	I4/mmm	71.43	8.568^b b This work	8.568^b b This work	7.125^b b This work	-3.313^b b This work	-0.140^b b This work
			8.627^d d Exp. In Ref.40		7.111^d d Exp. In Ref.40		-0.103ⁱ i Exp. In Ref.37
			8.725^c c Cal. In Ref.28	8.725^c c Cal. In Ref.28	7.178^c c Cal. In Ref.28		-0.112^c c Cal. In Ref.28
α-Ga	Cmca	100	3.706 4	4.8891	7.971	-3.701	0

	α-Mg	Ga₂Mg₅	GaMg₂	GaMg	H-Ga₂Mg	O-Ga₂Mg	Ga₅Mg₂	α-Ga
C₁₁	80.9	93.3	88.2	90.1	136.9	69.1	114.6	125.6
C₂₂	-	80.7	-	-	-	102.8	-	93.8
C₃₃	101.7	72.9	71.9	96.2	133.0	78.0	137.1	127.7
C₄₄	47.2	37.9	25.6	36.5	29.5	23.9	40.1	12.2
C₅₅	-	30.2	-	-	-	22.1	-	21.22
C₆₆	-	23.9	-	21.7	-	29.3	34.6	16.8
C₁₂	20.1	26.6	31.2	30.0	34.0	37.2	47.6	115.8
C₁₃	1.1	27.2	31.1	35.7	16.8	56.6	20.4	26.1
C₂₃	-	31.2	-	-	-	34.8	-	43.3
C₁₆				4.3			0
C₁₆				4.3			0

Species	α-Mg	Ga₂Mg₅	GaMg₂	GaMg	H-Ga₂Mg	O-Ga₂Mg	Ga₅Mg₂	α-Ga
B_V	34.2	46.3	48.3	53.3	60.2	56.3	60.3	59.5
B_R	34.2	46.1	47.9	53.1	59.9	56.1	60.3	58.6
B	34.2	46.2	48.1	53.2	60.1	56.2	60.3	59.1
G_V	41.0	29.2	26.3	30.6	44.7	23.1	41.5	26.9
G_R	39.0	28.2	26.1	29.1	40.7	18.3	39.8	20.9
G	40.0	28.7	26.2	29.8	42.7	20.7	40.6	23.9
E	86.3	71.3	66.5	75.3	103.6	55.3	99.5	63.2
B/G	0.86	1.61	1.84	1.78	1.41	2.71	1.49	2.47
σ	0.08	0.24	0.27	0.26	0.21	0.34	0.23	0.32
λ	7.53	27.1	30.6	33.3	31.6	42.4	33.2	43.2
µ	40	28.7	26.2	29.8	42.7	20.7	40.6	23.9
H_v	17.8	5.16	3.64	4.39	9.05	0.66	7.99	1.44

Species	A₁	A₂	A₃	A_B	A_G	A^U
α-Mg	1.05	1.05	1	0. 06%	2.62%	0.2704
Ga₂Mg₅	1.35	1.33	0.79	0. 26%	1.82%	0.191251
GaMg₂	1.04	1.04	1.00	0. 41%	0.38%	0.046665
GaMg	1.27	1.27	0.72	0. 19%	2.51%	0.261498
H- Ga₂Mg	0.50	0.50	1.00	0. 25%	4.68%	0.496409
O-Ga₂Mg	2.81	0.79	1.20	0. 18%	11.59%	1.31504
Ga₅Mg₂	0.76	0.76	1.03	0	2.09%	0.214562
α-Ga	0.24	0.63	0.40	0.76%	12.6%	1.450765

Species	ρ	ν_l	ν_t	ν_m
α-Mg	1.74	7092.7	4794.6	5890.5
Ga₂Mg₅	3.03	5279.8	3077.6	3271.5
GaMg₂	3.27	5039.1	2830.6	3689.7
GaMg	4.07	4778.5	2705.9	3519.7
H-Ga₂Mg	4.92	4877.2	2946.0	3761.9
O-Ga₂Mg	6.41	4152.0	2063.5	2766.2
Ga₅Mg₂	5.05	4760.3	2835.4	3636.0
α-Ga	6.41	3767.1	1930.9	2572.1

Direction	[100]			[001]			[110]
	[100]ν_l	[001] ν_t1	[010] ν_t2	[001]ν_l	[100] ν_t1	[010] ν_t2	[110]ν_l	[001] ν_t1	[110] ν_t2
Ga₅Mg₂	4.764	2.818	2.618	5.210	2.618	2.618	4.788	2.818	2.576
GaMg	4.705	2.995	2.309	4.862	2.309	2.309	4.482	2.995	2.717

Direction	[100]			[010]			[001]
	[100]ν_l	[010] ν_t1	[001] ν_t2	[010]ν_l	[100] ν_t1	[001] ν_t2	[001]ν_l	[100] ν_t1	[010] ν_t2
O-Ga₂Mg	3.770	2.455	2.132	4.596	2.455	2.216	4.005	2.132	2.216
Ga₂Mg₅	5.549	2.809	3.159	5.162	2.809	3.535	4.908	3.159	3.535
GaMg₂	2.951	5.193	2.798	- -	- -	- -	4.689	2.798	2.798
H-Ga₂Mg	3.233	5.275	2.449	- -	- -	- -	5.199	2.449	2.449

(\begin{matrix} σ_{1} \\ σ_{2} \\ σ_{3} \\ τ_{4} \\ τ_{5} \\ τ_{6} \end{matrix}) = (\begin{matrix} C_{11} C_{12} C_{13} C_{14} C_{15} C_{16} \\ C_{22} C_{23} C_{24} C_{25} C_{26} \\ C_{33} C_{34} C_{35} C_{36} \\ C_{44} C_{45} C_{46} \\ C_{55} C_{56} \\ C_{66} \end{matrix}) (\begin{matrix} ε_{1} \\ ε_{2} \\ ε_{3} \\ γ_{4} \\ γ_{5} \\ γ_{6} \end{matrix})

\begin{matrix} C_{11} > 0, C_{33} > 0, C_{44} > 0, C_{66} > 0, C_{11} - C_{12} > 0, \\ C_{11} + C_{33} - C_{13} > 0, 2 (C_{11} + C_{12}) + C_{33} + 4 C_{13} > 0, \end{matrix}

\begin{matrix} C_{11} + C_{12} + C_{33} + 2 C_{12} + 2 C_{23} > 0, \\ C_{11} + C_{22} > 2 C_{12}, C_{22} + C_{33} > 2 C_{33}, \\ C_{11} + C_{33} > 2 C_{13}, C_{ii} > 0 (i = 1 - 6) \end{matrix}

\begin{matrix} \frac{1}{B} = (S_{11} + S_{12} + S_{13}) l_{1}^{2} + (S_{12} + S_{22} + S_{23}) l_{2}^{2} + \\ (S_{13} + S_{23} + S_{33}) l_{3}^{2} \end{matrix}

\begin{matrix} \frac{1}{E} = S_{11} l_{1}^{4} + S_{22} l_{2}^{4} + S_{33} l_{3}^{4} + (2 S_{12} + S_{66}) l_{1}^{2} l_{2}^{2} + \\ (2 S_{13} + S_{55}) l_{1}^{2} l_{3}^{2} + (2 S_{23} + S_{44}) l_{2}^{2} l_{3}^{2} \end{matrix}

\begin{matrix} [100] = [010] \\ [100] υ_{l} = \sqrt{\frac{C_{11}}{ρ}}; [001] υ_{t 1} = \sqrt{\frac{C_{44}}{ρ}}; \\ [010] υ_{t 2} = \sqrt{\frac{C_{66}}{ρ}} \end{matrix}

\begin{matrix} [110] \\ [110] υ_{l} = \sqrt{\frac{(C_{11} + C_{12} + 2 C_{66})}{2 ρ}}; \\ [001] υ_{t 1} = \sqrt{\frac{C_{44}}{ρ}}; [1 \overset{─}{1} 0] v_{t 2} = \sqrt{\frac{(C_{11} - C_{12})}{2 ρ}} \end{matrix}

\begin{matrix} [100] \\ [100] υ_{l} = \sqrt{\frac{C_{11}}{ρ}}; [010] υ_{t 1} = \sqrt{\frac{C_{66}}{ρ}}; \\ [001] v_{t 2} = \sqrt{\frac{C_{55}}{2 ρ}} \end{matrix}

\begin{matrix} [010] \\ [010] υ_{l} = \sqrt{\frac{C_{22}}{ρ}}; [100] υ_{t 1} = \sqrt{\frac{C_{66}}{ρ}}; \\ [001] v_{t 2} = \sqrt{\frac{C_{44}}{ρ}} \end{matrix}

\begin{matrix} [001] \\ [001] υ_{l} = \sqrt{\frac{C_{33}}{ρ}}; [010] υ_{t 1} = \sqrt{\frac{C_{55}}{ρ}}; \\ [010] v_{t 2} = \sqrt{\frac{C_{44}}{ρ}} \end{matrix}

\begin{matrix} [100] \\ [100] υ_{l} = \sqrt{\frac{(C_{11} - C_{12})}{ρ}}; [010] υ_{t 1} = \sqrt{\frac{C_{11}}{ρ}}; [001] v_{t 2} = \sqrt{\frac{C_{44}}{ρ}} \end{matrix}

\begin{matrix} [001] \\ [001] υ_{l} = \sqrt{\frac{C_{33}}{ρ}}; [100] υ_{t 1} = \sqrt{\frac{C_{44}}{ρ}}; [010] v_{t 2} = \sqrt{\frac{C_{44}}{ρ}} \end{matrix}

[1] *e-mail: yujieone@163.com

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