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
\(1\)\([2 \bar 1 \bar 1 0]\)\((0 0 0 1)\)
\(2\)\([\bar 1 2 \bar 1 0]\)\((0 0 0 1)\)
\(3\)\([\bar 1 \bar 1 2 0]\)\((0 0 0 1)\)
basal slip system

Figure 2: \(⟨1 1 \bar 2 0⟩\{0 0 0 1\}\) basal slip system#

index
slip direction
plane normal
\(4\)\([2 \bar 1 \bar 1 0]\)\((0 \bar 1 \bar 1 0)\)
\(5\)\([\bar 1 2 \bar 1 0]\)\((\bar 1 0 0 1)\)
\(6\)\([\bar 1 \bar 1 2 0]\)\((1 \bar 1 0 0)\)
prismatic slip system

Figure 3: \(⟨1 1 \bar 2 0⟩\{1 \bar 1 0 0\}\) prismatic slip system#

index
slip direction
plane normal
\(7\)\([2 \bar 1 \bar 1 0]\)\((0 1 \bar 1 1)\)
\(8\)\([\bar 1 2 \bar 1 0]\)\((\bar 1 0 1 1)\)
\(9\)\([\bar 1 \bar 1 2 0]\)\((1 \bar 1 0 1)\)
\(10\)\([1 1 \bar 2 0]\)\((\bar 1 1 0 1)\)
\(11\)\([\bar 2 1 1 0]\)\((0 \bar 1 1 1)\)
\(12\)\([1 \bar 2 1 0]\)\((1 0 \bar 1 1)\)
1st order pyramidal slip system

Figure 4: \(⟨1 1 \bar 2 0⟩\{1 0 \bar 1 1\}\) pyramidal <a> slip system#

index
slip direction
plane normal
\(13\)\([2 \bar 1 \bar 1 3]\)\((\bar 1 1 0 1)\)
\(14\)\([\bar 1 2 \bar 1 3]\)\((\bar 1 1 0 1)\)
\(15\)\([\bar 1 \bar 1 2 3]\)\((1 0 \bar 1 1)\)
\(16\)\([\bar 2 1 1 3]\)\((1 0 \bar 1 1)\)
\(17\)\([\bar 1 2 \bar 1 3]\)\((0 1 \bar 1 1)\)
\(18\)\([1 1 \bar 2 3]\)\((0 \bar 1 1 1)\)
\(19\)\([2 \bar 1 \bar 1 3]\)\((1 \bar 1 0 1)\)
\(20\)\([\bar 1 2 \bar 1 3]\)\((1 \bar 1 0 1)\)
\(21\)\([1 1 \bar 2 3]\)\((\bar 1 0 1 1)\)
\(22\)\([2 \bar 1 \bar 1 3]\)\((\bar 1 0 1 1)\)
\(23\)\([1 \bar 2 1 3]\)\((0 1 \bar 1 1)\)
\(24\)\([\bar 1 \bar 1 2 3]\)\((0 1 \bar 1 1)\)
1st order pyramidal <c+a> slip system

Figure 5: \(⟨1 1 \bar 2 3⟩\{1 0 \bar 1 1\}\) 1st order pyramidal <c+a> slip system#

index
slip direction
plane normal
\(25\)\([2 \bar 1 \bar 1 3]\)\((\bar 2 1 1 2)\)
\(26\)\([\bar 1 2 \bar 1 3]\)\((1 \bar 2 1 2)\)
\(27\)\([\bar 1 \bar 1 2 3]\)\((1 1 \bar 2 2)\)
\(28\)\([\bar 2 1 1 3]\)\((2 \bar 1 \bar 1 2)\)
\(29\)\([1 \bar 2 1 3]\)\((\bar 1 2 \bar 1 2)\)
\(30\)\([1 1 \bar 2 3]\)\((\bar 1 \bar 1 2 2)\)
2nd order pyramidal <c+a> slip system

Figure 6: \(⟨1 1 \bar 2 3⟩\{1 1 \bar 2 2\}\) 2nd order pyramidal <c+a> slip system#

Twin systems#

\(η_1\)
\(K_1\)
\(η_2\)
\(K_2\)
\(⟨\bar 1 0 1 1⟩\)\(\{1 0 \bar 1 2\}\)\(⟨1 0 \bar 1 1⟩\)\(\{1 0 \bar 1 \bar 2\}\)
index
slip direction
plane normal
\(1\)\([1 \bar 1 0 1]\)\((\bar 1 1 0 2)\)
\(2\)\([\bar 1 0 1 1]\)\((1 0 \bar 1 2)\)
\(3\)\([0 1 \bar 1 1]\)\((0 \bar 1 1 2)\)
\(4\)\([\bar 1 1 0 1]\)\((1 \bar 1 0 2)\)
\(5\)\([1 0 \bar 1 1]\)\((\bar 1 0 1 2)\)
\(6\)\([0 \bar 1 1 1]\)\((0 1 \bar 1 2)\)
twin system

Figure 7: \(⟨\bar 1 0 1 1⟩ \{1 0 \bar 1 2\}\) T1 tensile twinning in Co, Mg, Zr, Ti, and Be; compressive twinning in Cd and Zn.#

\(η_1\)
\(K_1\)
\(η_2\)
\(K_2\)
\(⟨\bar 1 \bar 1 2 6⟩\)\(\{1 1 \bar 2 1\}\)\(⟨1 1 2 0⟩\)\(\{0 0 0 2\}\)
index
slip direction
plane normal
\(7\)\([2 \bar 1 \bar 1 6]\)\((\bar 2 1 1 1)\)
\(8\)\([\bar 1 2 \bar 1 6]\)\((1 1 \bar 2 1)\)
\(9\)\([\bar 1 \bar 1 2 6]\)\((2 \bar 1 \bar 1 1)\)
\(10\)\([\bar 2 1 1 6]\)\((\bar 1 2 \bar 1 1)\)
\(11\)\([1 \bar 2 1 6]\)\((\bar 1 0 1 2)\)
\(12\)\([1 1 \bar 2 6]\)\((\bar 1 \bar 1 2 1)\)
twin system

Figure 8: \(⟨\bar 1 \bar 1 2 6⟩ \{1 1 \bar 2 1\}\) T2 tensile twinning in Co, Re, and Zr.#

\(η_1\)
\(K_1\)
\(η_2\)
\(K_2\)
\(⟨1 0 \bar 1 \bar 2⟩\)\(\{1 0 \bar 1 1\}\)\(⟨3 0 \bar 3 2⟩\)\(\{1 0 \bar 1 \bar 3\}\)
index
slip direction
plane normal
\(13\)\([\bar 1 1 0 \bar 2]\)\((\bar 1 1 0 1)\)
\(14\)\([1 0 \bar 1 \bar 2]\)\((1 0 \bar 1 1)\)
\(15\)\([0 \bar 1 1 \bar 2]\)\((0 \bar 1 1 1)\)
\(16\)\([1 \bar 1 0 \bar 2]\)\((1 \bar 1 0 1)\)
\(17\)\([\bar 1 0 1 \bar 2]\)\((\bar 1 0 1 1)\)
\(18\)\([0 1 \bar 1 \bar 2]\)\((0 1 \bar 1 1)\)
twin system

Figure 9: \(⟨1 0 \bar 1 \bar 2⟩ \{1 0 \bar 1 1\}\) C1 compressive twinning in Mg.#

\(η_1\)
\(K_1\)
\(η_2\)
\(K_2\)
\(⟨1 1 \bar 2 \bar 3⟩\)\(\{1 1 \bar 2 2\}\)\(⟨2 2 \bar 4 3⟩\)\(\{1 1 \bar 2 \bar 4\}\)
index
slip direction
plane normal
\(19\)\([2 \bar 1 \bar 1 \bar 3]\)\((2 \bar 1 \bar 1 2)\)
\(20\)\([\bar 1 2 \bar 1 \bar 3]\)\((\bar 1 2 \bar 1 2)\)
\(21\)\([\bar 1 \bar 1 2 \bar 3]\)\((\bar 1 \bar 1 2 2)\)
\(22\)\([\bar 2 1 1 \bar 3]\)\((\bar 2 1 1 2)\)
\(23\)\([1 \bar 2 1 \bar 3]\)\((1 \bar 2 1 2)\)
\(24\)\([1 1 \bar 2 \bar 3]\)\((1 1 \bar 2 2)\)
twin system

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