# 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¶

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

Figure 2: $$⟨1 1 \bar 2 0⟩\{0 0 0 1\}$$ basal slip system

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

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

 $$\textnormal{index}$$ $$\textnormal{slip direction}$$ $$\textnormal{plane normal}$$ $$7$$ $$[0 1 \bar 1 0]$$ $$(2 \bar 1 \bar 1 0)$$ $$8$$ $$[\bar 1 0 1 0]$$ $$(\bar 1 2 \bar 1 0)$$ $$9$$ $$[1 \bar 1 0 0]$$ $$(\bar 1 \bar 1 2 0)$$

Figure 4: $$⟨\bar 1 1 0 0⟩\{1 1 \bar 2 0\}$$ 2nd order prismatic compound slip system

 $$\textnormal{index}$$ $$\textnormal{slip direction}$$ $$\textnormal{plane normal}$$ $$10$$ $$[2 \bar 1 \bar 1 0]$$ $$(0 1 \bar 1 1)$$ $$11$$ $$[\bar 1 2 \bar 1 0]$$ $$(\bar 1 0 1 1)$$ $$12$$ $$[\bar 1 \bar 1 2 0]$$ $$(1 \bar 1 0 1)$$ $$13$$ $$[1 1 \bar 2 0]$$ $$(\bar 1 1 0 1)$$ $$14$$ $$[\bar 2 1 1 0]$$ $$(0 \bar 1 1 1)$$ $$15$$ $$[1 \bar 2 1 0]$$ $$(1 0 \bar 1 1)$$

Figure 5: $$⟨1 1 \bar 2 0⟩\{1 0 \bar 1 1\}$$ 1st order pyramidal slip system

 $$\textnormal{index}$$ $$\textnormal{slip direction}$$ $$\textnormal{plane normal}$$ $$16$$ $$[2 \bar 1 \bar 1 3]$$ $$(\bar 1 1 0 1)$$ $$17$$ $$[\bar 1 2 \bar 1 3]$$ $$(\bar 1 1 0 1)$$ $$18$$ $$[\bar 1 \bar 1 2 3]$$ $$(1 0 \bar 1 1)$$ $$19$$ $$[\bar 2 1 1 3]$$ $$(1 0 \bar 1 1)$$ $$20$$ $$[\bar 1 2 \bar 1 3]$$ $$(0 1 \bar 1 1)$$ $$21$$ $$[1 1 \bar 2 3]$$ $$(0 \bar 1 1 1)$$ $$22$$ $$[2 \bar 1 \bar 1 3]$$ $$(1 \bar 1 0 1)$$ $$23$$ $$[\bar 1 2 \bar 1 3]$$ $$(1 \bar 1 0 1)$$ $$24$$ $$[1 1 \bar 2 3]$$ $$(\bar 1 0 1 1)$$ $$25$$ $$[2 \bar 1 \bar 1 3]$$ $$(\bar 1 0 1 1)$$ $$26$$ $$[1 \bar 2 1 3]$$ $$(0 1 \bar 1 1)$$ $$27$$ $$[\bar 1 \bar 1 2 3]$$ $$(0 1 \bar 1 1)$$

Figure 6: $$⟨1 1 \bar 2 3⟩\{1 0 \bar 1 1\}$$ 1st order pyramidal <c+a> slip system

 $$\textnormal{index}$$ $$\textnormal{slip direction}$$ $$\textnormal{plane normal}$$ $$28$$ $$[2 \bar 1 \bar 1 3]$$ $$(\bar 2 1 1 2)$$ $$29$$ $$[\bar 1 2 \bar 1 3]$$ $$(1 \bar 2 1 2)$$ $$30$$ $$[\bar 1 \bar 1 2 3]$$ $$(1 1 \bar 2 2)$$ $$31$$ $$[\bar 2 1 1 3]$$ $$(2 \bar 1 \bar 1 2)$$ $$32$$ $$[1 \bar 2 1 3]$$ $$(\bar 1 2 \bar 1 2)$$ $$33$$ $$[1 1 \bar 2 3]$$ $$(\bar 1 \bar 1 2 2)$$

Figure 7: $$⟨1 1 \bar 2 3⟩\{1 1 \bar 2 2\}$$ 2nd order pyramidal <c+a> slip system

## Twin systems¶

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

Figure 8: $$⟨\bar 1 0 1 1⟩ \{1 0 \bar 1 2\}$$ T1 tensile twinning in Co, Mg, Zr, Ti, and Be; compressive twinning in Cd and Zn.

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

Figure 9: $$⟨\bar 1 \bar 1 2 6⟩ \{1 1 \bar 2 1\}$$ T2 tensile twinning in Co, Re, and Zr.

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

Figure 10: $$⟨1 0 \bar 1 \bar 2⟩ \{1 0 \bar 1 1\}$$ C1 compressive twinning in Mg.

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

Figure 11: $$⟨1 1 \bar 2 \bar 3⟩ \{1 1 \bar 2 2\}$$ C2 compressive twinning in Ti and Zr.