Hexagonal (hP)#

Atom arrangement#

Figure 1: Hexagonal lattice structure. X, Y, and Z crystal frame axes are colored red, green, and blue, respectively.

Slip systems#

index	slip direction	plane normal
1	$[2 \bar{1} \bar{1} 0]$	$(0001)$
2	$[\bar{1} 2 \bar{1} 0]$	$(0001)$
3	$[\bar{1} \bar{1} 20]$	$(0001)$

Figure 2: $⟨ 11 \bar{2} 0 ⟩ {0001}$ basal slip system

index	slip direction	plane normal
4	$[2 \bar{1} \bar{1} 0]$	$(0 \bar{1} \bar{1} 0)$
5	$[\bar{1} 2 \bar{1} 0]$	$(\bar{1} 001)$
6	$[\bar{1} \bar{1} 20]$	$(1 \bar{1} 00)$

Figure 3: $⟨ 11 \bar{2} 0 ⟩ {1 \bar{1} 00}$ prismatic slip system

index	slip direction	plane normal
7	$[2 \bar{1} \bar{1} 0]$	$(01 \bar{1} 1)$
8	$[\bar{1} 2 \bar{1} 0]$	$(\bar{1} 011)$
9	$[\bar{1} \bar{1} 20]$	$(1 \bar{1} 01)$
10	$[11 \bar{2} 0]$	$(\bar{1} 101)$
11	$[\bar{2} 110]$	$(0 \bar{1} 11)$
12	$[1 \bar{2} 10]$	$(10 \bar{1} 1)$

Figure 4: $⟨ 11 \bar{2} 0 ⟩ {10 \bar{1} 1}$ pyramidal $⟨ a ⟩$ slip system

index	slip direction	plane normal
13	$[2 \bar{1} \bar{1} 3]$	$(\bar{1} 101)$
14	$[\bar{1} 2 \bar{1} 3]$	$(\bar{1} 101)$
15	$[\bar{1} \bar{1} 23]$	$(10 \bar{1} 1)$
16	$[\bar{2} 113]$	$(10 \bar{1} 1)$
17	$[\bar{1} 2 \bar{1} 3]$	$(01 \bar{1} 1)$
18	$[11 \bar{2} 3]$	$(0 \bar{1} 11)$
19	$[2 \bar{1} \bar{1} 3]$	$(1 \bar{1} 01)$
20	$[\bar{1} 2 \bar{1} 3]$	$(1 \bar{1} 01)$
21	$[11 \bar{2} 3]$	$(\bar{1} 011)$
22	$[2 \bar{1} \bar{1} 3]$	$(\bar{1} 011)$
23	$[1 \bar{2} 13]$	$(01 \bar{1} 1)$
24	$[\bar{1} \bar{1} 23]$	$(01 \bar{1} 1)$

Figure 5: $⟨ 11 \bar{2} 3 ⟩ {10 \bar{1} 1}$ 1st order pyramidal $⟨ c + a ⟩$ slip system

index	slip direction	plane normal
25	$[2 \bar{1} \bar{1} 3]$	$(\bar{2} 112)$
26	$[\bar{1} 2 \bar{1} 3]$	$(1 \bar{2} 12)$
27	$[\bar{1} \bar{1} 23]$	$(11 \bar{2} 2)$
28	$[\bar{2} 113]$	$(2 \bar{1} \bar{1} 2)$
29	$[1 \bar{2} 13]$	$(\bar{1} 2 \bar{1} 2)$
30	$[11 \bar{2} 3]$	$(\bar{1} \bar{1} 22)$

Figure 6: $⟨ 11 \bar{2} 3 ⟩ {11 \bar{2} 2}$ 2nd order pyramidal $⟨ c + a ⟩$ slip system

Twin systems#

$η_{1}$	$K_{1}$	$η_{2}$	$K_{2}$
$⟨ \bar{1} 011 ⟩$	${10 \bar{1} 2}$	$⟨ 10 \bar{1} 1 ⟩$	${10 \bar{1} \bar{2}}$

index	slip direction	plane normal
1	$[1 \bar{1} 01]$	$(\bar{1} 102)$
2	$[\bar{1} 011]$	$(10 \bar{1} 2)$
3	$[01 \bar{1} 1]$	$(0 \bar{1} 12)$
4	$[\bar{1} 101]$	$(1 \bar{1} 02)$
5	$[10 \bar{1} 1]$	$(\bar{1} 012)$
6	$[0 \bar{1} 11]$	$(01 \bar{1} 2)$

Figure 7: $⟨ \bar{1} 011 ⟩ {10 \bar{1} 2}$ T1 tensile twinning in Co, Mg, Zr, Ti, and Be; compressive twinning in Cd and Zn.

$η_{1}$	$K_{1}$	$η_{2}$	$K_{2}$
$⟨ \bar{1} \bar{1} 26 ⟩$	${11 \bar{2} 1}$	$⟨ 1120 ⟩$	${0002}$

index	slip direction	plane normal
7	$[2 \bar{1} \bar{1} 6]$	$(\bar{2} 111)$
8	$[\bar{1} 2 \bar{1} 6]$	$(11 \bar{2} 1)$
9	$[\bar{1} \bar{1} 26]$	$(2 \bar{1} \bar{1} 1)$
10	$[\bar{2} 116]$	$(\bar{1} 2 \bar{1} 1)$
11	$[1 \bar{2} 16]$	$(\bar{1} 012)$
12	$[11 \bar{2} 6]$	$(\bar{1} \bar{1} 21)$

Figure 8: $⟨ \bar{1} \bar{1} 26 ⟩ {11 \bar{2} 1}$ T2 tensile twinning in Co, Re, and Zr.

$η_{1}$	$K_{1}$	$η_{2}$	$K_{2}$
$⟨ 10 \bar{1} \bar{2} ⟩$	${10 \bar{1} 1}$	$⟨ 30 \bar{3} 2 ⟩$	${10 \bar{1} \bar{3}}$

index	slip direction	plane normal
13	$[\bar{1} 10 \bar{2}]$	$(\bar{1} 101)$
14	$[10 \bar{1} \bar{2}]$	$(10 \bar{1} 1)$
15	$[0 \bar{1} 1 \bar{2}]$	$(0 \bar{1} 11)$
16	$[1 \bar{1} 0 \bar{2}]$	$(1 \bar{1} 01)$
17	$[\bar{1} 01 \bar{2}]$	$(\bar{1} 011)$
18	$[01 \bar{1} \bar{2}]$	$(01 \bar{1} 1)$

Figure 9: $⟨ 10 \bar{1} \bar{2} ⟩ {10 \bar{1} 1}$ C1 compressive twinning in Mg.

$η_{1}$	$K_{1}$	$η_{2}$	$K_{2}$
$⟨ 11 \bar{2} \bar{3} ⟩$	${11 \bar{2} 2}$	$⟨ 22 \bar{4} 3 ⟩$	${11 \bar{2} \bar{4}}$

index	slip direction	plane normal
19	$[2 \bar{1} \bar{1} \bar{3}]$	$(2 \bar{1} \bar{1} 2)$
20	$[\bar{1} 2 \bar{1} \bar{3}]$	$(\bar{1} 2 \bar{1} 2)$
21	$[\bar{1} \bar{1} 2 \bar{3}]$	$(\bar{1} \bar{1} 22)$
22	$[\bar{2} 11 \bar{3}]$	$(\bar{2} 112)$
23	$[1 \bar{2} 1 \bar{3}]$	$(1 \bar{2} 12)$
24	$[11 \bar{2} \bar{3}]$	$(11 \bar{2} 2)$

Figure 10: $⟨ 11 \bar{2} \bar{3} ⟩ {11 \bar{2} 2}$ C2 compressive twinning in Ti and Zr.

Interaction Matrices#

Slip-Slip#

index	label	description
1	S1	basal self-interaction
2	1	basal/basal coplanar
3	3	basal/prismatic collinear
4	4	basal/prismatic non-collinear
5	S2	prismatic self-interaction
6	2	prismatic/prismatic
7	5	prismatic/basal collinear
8	6	prismatic/basal non-collinear
9	$-$	basal/pyramidal $⟨ a ⟩$ non-collinear
10	$-$	basal/pyramidal $⟨ a ⟩$ collinear
11	$-$	prismatic/pyramidal $⟨ a ⟩$ non-collinear
12	$-$	prismatic/pyramidal $⟨ a ⟩$ collinear
13	$-$	pyramidal $⟨ a ⟩$ self-interaction
14	$-$	pyramidal $⟨ a ⟩$ non-collinear
15	$-$	pyramidal $⟨ a ⟩$ collinear
16	$-$	pyramidal $⟨ a ⟩$ /prismatic non-collinear
17	$-$	pyramidal $⟨ a ⟩$ /prismatic collinear
18	$-$	pyramidal $⟨ a ⟩$ /basal non-collinear
19	$-$	pyramidal $⟨ a ⟩$ /basal collinear
20	$-$	basal/1. order pyramidal $⟨ c + a ⟩$ semi-collinear
21	$-$	basal/1. order pyramidal $⟨ c + a ⟩$
22	$-$	basal/1. order pyramidal $⟨ c + a ⟩$
23	$-$	prismatic/1. order pyramidal $⟨ c + a ⟩$ semi-collinear
24	$-$	prismatic/1. order pyramidal $⟨ c + a ⟩$
25	$-$	prismatic/1. order pyramidal $⟨ c + a ⟩$ semi-coplanar?
26	$-$	pyramidal $⟨ a ⟩$ /1. order pyramidal $⟨ c + a ⟩$ coplanar
27	$-$	pyramidal $⟨ a ⟩$ /1. order pyramidal $⟨ c + a ⟩$
28	$-$	pyramidal $⟨ a ⟩$ /1. order pyramidal $⟨ c + a ⟩$ semi-collinear
29	$-$	pyramidal $⟨ a ⟩$ /1. order pyramidal $⟨ c + a ⟩$ semi-coplanar
30	$-$	1. order pyramidal $⟨ c + a ⟩$ self-interaction
31	$-$	1. order pyramidal $⟨ c + a ⟩$ coplanar
32	$-$	1. order pyramidal $⟨ c + a ⟩$
33	$-$	1. order pyramidal $⟨ c + a ⟩$
34	$-$	1. order pyramidal $⟨ c + a ⟩$ semi-coplanar
35	$-$	1. order pyramidal $⟨ c + a ⟩$ semi-coplanar
36	$-$	1. order pyramidal $⟨ c + a ⟩$ collinear
37	$-$	1. order pyramidal $⟨ c + a ⟩$ /pyramidal $⟨ a ⟩$ coplanar
38	$-$	1. order pyramidal $⟨ c + a ⟩$ /pyramidal $⟨ a ⟩$ semi-collinear
39	$-$	1. order pyramidal $⟨ c + a ⟩$ /pyramidal $⟨ a ⟩$
40	$-$	1. order pyramidal $⟨ c + a ⟩$ /pyramidal $⟨ a ⟩$ semi-coplanar
41	$-$	1. order pyramidal $⟨ c + a ⟩$ /prismatic semi-collinear
42	$-$	1. order pyramidal $⟨ c + a ⟩$ /prismatic semi-coplanar
43	$-$	1. order pyramidal $⟨ c + a ⟩$ /prismatic
44	$-$	1. order pyramidal $⟨ c + a ⟩$ /basal semi-collinear
45	$-$	1. order pyramidal $⟨ c + a ⟩$ /basal
46	$-$	1. order pyramidal $⟨ c + a ⟩$ /basal
47	8	basal/2. order pyramidal $⟨ c + a ⟩$ non-collinear
48	7	basal/2. order pyramidal $⟨ c + a ⟩$ semi-collinear
49	10	prismatic/2. order pyramidal $⟨ c + a ⟩$
50	9	prismatic/2. order pyramidal $⟨ c + a ⟩$ semi-collinear
51	$-$	pyramidal $⟨ a ⟩$ /2. order pyramidal $⟨ c + a ⟩$
52	$-$	pyramidal $⟨ a ⟩$ /2. order pyramidal $⟨ c + a ⟩$ semi collinear
53	$-$	1. order pyramidal $⟨ c + a ⟩$ /2. order pyramidal $⟨ c + a ⟩$
54	$-$	1. order pyramidal $⟨ c + a ⟩$ /2. order pyramidal $⟨ c + a ⟩$
55	$-$	1. order pyramidal $⟨ c + a ⟩$ /2. order pyramidal $⟨ c + a ⟩$
56	$-$	1. order pyramidal $⟨ c + a ⟩$ /2. order pyramidal $⟨ c + a ⟩$ collinear
57	S3	2. order pyramidal $⟨ c + a ⟩$ self-interaction
58	16	2. order pyramidal $⟨ c + a ⟩$ non-collinear
59	15	2. order pyramidal $⟨ c + a ⟩$ semi-collinear
60	$-$	2. order pyramidal $⟨ c + a ⟩$ /1. order pyramidal $⟨ c + a ⟩$
61	$-$	2. order pyramidal $⟨ c + a ⟩$ /1. order pyramidal $⟨ c + a ⟩$ collinear
62	$-$	2. order pyramidal $⟨ c + a ⟩$ /1. order pyramidal $⟨ c + a ⟩$
63	$-$	2. order pyramidal $⟨ c + a ⟩$ /1. order pyramidal $⟨ c + a ⟩$
64	$-$	2. order pyramidal $⟨ c + a ⟩$ /pyramidal $⟨ a ⟩$ non-collinear
65	$-$	2. order pyramidal $⟨ c + a ⟩$ /pyramidal $⟨ a ⟩$ semi-collinear
66	14	2. order pyramidal $⟨ c + a ⟩$ /prismatic non-collinear
67	13	2. order pyramidal $⟨ c + a ⟩$ /prismatic semi-collinear
68	12	2. order pyramidal $⟨ c + a ⟩$ /basal non-collinear
69	11	2. order pyramidal $⟨ c + a ⟩$ /basal semi-collinear

N. Bertin, C.N. Tomé, I.J. Beyerlein, M.R. Barnett, and L. Capolungo. On the strength of dislocation interactions and their effect on latent hardening in pure Magnesium International Journal of Plasticity, 62:72-92, 2014. doi:10.1016/j.ijplas.2014.06.010.
B. Devincre. Dislocation dynamics simulations of slip systems interactions and forest strengthening in ice single crystal Philosophical Magazine A, 93(1-3):1-12, 2013. doi:10.1080/14786435.2012.699689.