The indices for the directions in the hexagonal lattice using three-digit and four-digit systems are to be determined. Concept Introduction: The indices for the hexagonal structure needs to be written at given direction A, B, C. For hexagonal lattice, the three-digit system uses three indices (h,k,l) and the four-digit system uses four indices (h,k,i,l)

Question

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Answer 1

Question

Chapter 3, Problem 3.56P

Interpretation Introduction

Interpretation:

The indices for the directions in the hexagonal lattice using three-digit and four-digit systems are to be determined.

Concept Introduction:

The indices for the hexagonal structure needs to be written at given direction A, B, C. For hexagonal lattice, the three-digit system uses three indices (h,k,l) and the four-digit system uses four indices (h,k,i,l)

Expert Solution & Answer

Answer to Problem 3.56P

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

Explanation of Solution

Given:

The following hexagonal lattce is given:

Essentials of Materials Science and Engineering, SI Edition, Chapter 3, Problem 3.56P

Calculation:

Miller indices for direction A using three-digit system,

Head point =1, −1, 0Tail point = 0, 0, 0Head point – Tail point = ( 1, −1, 0) = [1 1 ¯ 0]Four digit systemConvert of four digit system.[h k l] = [ 1 1 ¯ 0]Coordinates of four digit system.h=13[2h−k]=13[2+1]=1k=13[2k−h]=13[−2−1]=−1i=−13[h+k]=−13[1−1]=0l=l'=0[h k i l] = [1,−1,0,0] = [1 1 ¯ 0 0]

Miller indices for direction B,

Three-digit system

Head point = 1,1,0Tail point = 0,0,1Head point−Tail point =( 1,1,−1) =[1 1 1 ¯]Four digit systemConvert three digit into four digit system.[h k l] = [ 1 1 1 ¯]Coordinates of four digit system.h=13[2h−k]=13[2−1]=13k=13[2k−h]=13[2−1]=13i=−13[h+k]=−13[1+1]=−23l=l'=−1[h k i l] = [ 1 3, 1 3,− 2 3,−1] = [ 1 3, 1 3,− 2 3,−1]×3 = [1,1,−2,−3] = [1 1 2 ¯ 3 ¯]

Miller indices for direction C,

Three-digit system

Head point = 0,1,1Tail point = 0,0,0Head point−Tail point =( 0,1,1) =[0 1 1]Four digit systemConvert three digit into four digit system.[h k l] = [0 1 1]Coordinates of four digit system.h=13[2h−k]=13[0−1]=−13k=13[2k−h]=13[2−0]=23i=−13[h+k]=−13[0+1]=−13l=l'=1[h k i l] = [− 1 3, 2 3,− 1 3,1] = [− 1 3, 2 3,− 1 3,1]×3 = [ 1 ¯,2, 1 ¯,3]

Conclusion

Thus,

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

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Answer 2

Question

Chapter 3, Problem 3.56P

Interpretation Introduction

Interpretation:

The indices for the directions in the hexagonal lattice using three-digit and four-digit systems are to be determined.

Concept Introduction:

The indices for the hexagonal structure needs to be written at given direction A, B, C. For hexagonal lattice, the three-digit system uses three indices (h,k,l) and the four-digit system uses four indices (h,k,i,l)

Expert Solution & Answer

Answer to Problem 3.56P

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

Explanation of Solution

Given:

The following hexagonal lattce is given:

Essentials of Materials Science and Engineering, SI Edition, Chapter 3, Problem 3.56P

Calculation:

Miller indices for direction A using three-digit system,

Head point =1, −1, 0Tail point = 0, 0, 0Head point – Tail point = ( 1, −1, 0) = [1 1 ¯ 0]Four digit systemConvert of four digit system.[h k l] = [ 1 1 ¯ 0]Coordinates of four digit system.h=13[2h−k]=13[2+1]=1k=13[2k−h]=13[−2−1]=−1i=−13[h+k]=−13[1−1]=0l=l'=0[h k i l] = [1,−1,0,0] = [1 1 ¯ 0 0]

Miller indices for direction B,

Three-digit system

Head point = 1,1,0Tail point = 0,0,1Head point−Tail point =( 1,1,−1) =[1 1 1 ¯]Four digit systemConvert three digit into four digit system.[h k l] = [ 1 1 1 ¯]Coordinates of four digit system.h=13[2h−k]=13[2−1]=13k=13[2k−h]=13[2−1]=13i=−13[h+k]=−13[1+1]=−23l=l'=−1[h k i l] = [ 1 3, 1 3,− 2 3,−1] = [ 1 3, 1 3,− 2 3,−1]×3 = [1,1,−2,−3] = [1 1 2 ¯ 3 ¯]

Miller indices for direction C,

Three-digit system

Head point = 0,1,1Tail point = 0,0,0Head point−Tail point =( 0,1,1) =[0 1 1]Four digit systemConvert three digit into four digit system.[h k l] = [0 1 1]Coordinates of four digit system.h=13[2h−k]=13[0−1]=−13k=13[2k−h]=13[2−0]=23i=−13[h+k]=−13[0+1]=−13l=l'=1[h k i l] = [− 1 3, 2 3,− 1 3,1] = [− 1 3, 2 3,− 1 3,1]×3 = [ 1 ¯,2, 1 ¯,3]

Conclusion

Thus,

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

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Answer 3

Question

Chapter 3, Problem 3.56P

Interpretation Introduction

Interpretation:

The indices for the directions in the hexagonal lattice using three-digit and four-digit systems are to be determined.

Concept Introduction:

The indices for the hexagonal structure needs to be written at given direction A, B, C. For hexagonal lattice, the three-digit system uses three indices (h,k,l) and the four-digit system uses four indices (h,k,i,l)

Expert Solution & Answer

Answer to Problem 3.56P

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

Explanation of Solution

Given:

The following hexagonal lattce is given:

Essentials of Materials Science and Engineering, SI Edition, Chapter 3, Problem 3.56P

Calculation:

Miller indices for direction A using three-digit system,

Head point =1, −1, 0Tail point = 0, 0, 0Head point – Tail point = ( 1, −1, 0) = [1 1 ¯ 0]Four digit systemConvert of four digit system.[h k l] = [ 1 1 ¯ 0]Coordinates of four digit system.h=13[2h−k]=13[2+1]=1k=13[2k−h]=13[−2−1]=−1i=−13[h+k]=−13[1−1]=0l=l'=0[h k i l] = [1,−1,0,0] = [1 1 ¯ 0 0]

Miller indices for direction B,

Three-digit system

Head point = 1,1,0Tail point = 0,0,1Head point−Tail point =( 1,1,−1) =[1 1 1 ¯]Four digit systemConvert three digit into four digit system.[h k l] = [ 1 1 1 ¯]Coordinates of four digit system.h=13[2h−k]=13[2−1]=13k=13[2k−h]=13[2−1]=13i=−13[h+k]=−13[1+1]=−23l=l'=−1[h k i l] = [ 1 3, 1 3,− 2 3,−1] = [ 1 3, 1 3,− 2 3,−1]×3 = [1,1,−2,−3] = [1 1 2 ¯ 3 ¯]

Miller indices for direction C,

Three-digit system

Head point = 0,1,1Tail point = 0,0,0Head point−Tail point =( 0,1,1) =[0 1 1]Four digit systemConvert three digit into four digit system.[h k l] = [0 1 1]Coordinates of four digit system.h=13[2h−k]=13[0−1]=−13k=13[2k−h]=13[2−0]=23i=−13[h+k]=−13[0+1]=−13l=l'=1[h k i l] = [− 1 3, 2 3,− 1 3,1] = [− 1 3, 2 3,− 1 3,1]×3 = [ 1 ¯,2, 1 ¯,3]

Conclusion

Thus,

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

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Answer 4

Question

Chapter 3, Problem 3.56P

Interpretation Introduction

Interpretation:

The indices for the directions in the hexagonal lattice using three-digit and four-digit systems are to be determined.

Concept Introduction:

The indices for the hexagonal structure needs to be written at given direction A, B, C. For hexagonal lattice, the three-digit system uses three indices (h,k,l) and the four-digit system uses four indices (h,k,i,l)

Expert Solution & Answer

Answer to Problem 3.56P

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

Explanation of Solution

Given:

The following hexagonal lattce is given:

Essentials of Materials Science and Engineering, SI Edition, Chapter 3, Problem 3.56P

Calculation:

Miller indices for direction A using three-digit system,

Head point =1, −1, 0Tail point = 0, 0, 0Head point – Tail point = ( 1, −1, 0) = [1 1 ¯ 0]Four digit systemConvert of four digit system.[h k l] = [ 1 1 ¯ 0]Coordinates of four digit system.h=13[2h−k]=13[2+1]=1k=13[2k−h]=13[−2−1]=−1i=−13[h+k]=−13[1−1]=0l=l'=0[h k i l] = [1,−1,0,0] = [1 1 ¯ 0 0]

Miller indices for direction B,

Three-digit system

Head point = 1,1,0Tail point = 0,0,1Head point−Tail point =( 1,1,−1) =[1 1 1 ¯]Four digit systemConvert three digit into four digit system.[h k l] = [ 1 1 1 ¯]Coordinates of four digit system.h=13[2h−k]=13[2−1]=13k=13[2k−h]=13[2−1]=13i=−13[h+k]=−13[1+1]=−23l=l'=−1[h k i l] = [ 1 3, 1 3,− 2 3,−1] = [ 1 3, 1 3,− 2 3,−1]×3 = [1,1,−2,−3] = [1 1 2 ¯ 3 ¯]

Miller indices for direction C,

Three-digit system

Head point = 0,1,1Tail point = 0,0,0Head point−Tail point =( 0,1,1) =[0 1 1]Four digit systemConvert three digit into four digit system.[h k l] = [0 1 1]Coordinates of four digit system.h=13[2h−k]=13[0−1]=−13k=13[2k−h]=13[2−0]=23i=−13[h+k]=−13[0+1]=−13l=l'=1[h k i l] = [− 1 3, 2 3,− 1 3,1] = [− 1 3, 2 3,− 1 3,1]×3 = [ 1 ¯,2, 1 ¯,3]

Conclusion

Thus,

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

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Answer 5

Chapter 3, Problem 3.56P

Interpretation Introduction

Interpretation:

The indices for the directions in the hexagonal lattice using three-digit and four-digit systems are to be determined.

Concept Introduction:

The indices for the hexagonal structure needs to be written at given direction A, B, C. For hexagonal lattice, the three-digit system uses three indices (h,k,l) and the four-digit system uses four indices (h,k,i,l)

Expert Solution & Answer

Answer to Problem 3.56P

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

Explanation of Solution

Given:

The following hexagonal lattce is given:

Essentials of Materials Science and Engineering, SI Edition, Chapter 3, Problem 3.56P

Calculation:

Miller indices for direction A using three-digit system,

Head point =1, −1, 0Tail point = 0, 0, 0Head point – Tail point = ( 1, −1, 0) = [1 1 ¯ 0]Four digit systemConvert of four digit system.[h k l] = [ 1 1 ¯ 0]Coordinates of four digit system.h=13[2h−k]=13[2+1]=1k=13[2k−h]=13[−2−1]=−1i=−13[h+k]=−13[1−1]=0l=l'=0[h k i l] = [1,−1,0,0] = [1 1 ¯ 0 0]

Miller indices for direction B,

Three-digit system

Head point = 1,1,0Tail point = 0,0,1Head point−Tail point =( 1,1,−1) =[1 1 1 ¯]Four digit systemConvert three digit into four digit system.[h k l] = [ 1 1 1 ¯]Coordinates of four digit system.h=13[2h−k]=13[2−1]=13k=13[2k−h]=13[2−1]=13i=−13[h+k]=−13[1+1]=−23l=l'=−1[h k i l] = [ 1 3, 1 3,− 2 3,−1] = [ 1 3, 1 3,− 2 3,−1]×3 = [1,1,−2,−3] = [1 1 2 ¯ 3 ¯]

Miller indices for direction C,

Three-digit system

Head point = 0,1,1Tail point = 0,0,0Head point−Tail point =( 0,1,1) =[0 1 1]Four digit systemConvert three digit into four digit system.[h k l] = [0 1 1]Coordinates of four digit system.h=13[2h−k]=13[0−1]=−13k=13[2k−h]=13[2−0]=23i=−13[h+k]=−13[0+1]=−13l=l'=1[h k i l] = [− 1 3, 2 3,− 1 3,1] = [− 1 3, 2 3,− 1 3,1]×3 = [ 1 ¯,2, 1 ¯,3]

Conclusion

Thus,

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

Answer 6

Chapter 3, Problem 3.56P

Interpretation Introduction

Interpretation:

The indices for the directions in the hexagonal lattice using three-digit and four-digit systems are to be determined.

Concept Introduction:

The indices for the hexagonal structure needs to be written at given direction A, B, C. For hexagonal lattice, the three-digit system uses three indices (h,k,l) and the four-digit system uses four indices (h,k,i,l)

Expert Solution & Answer

Answer to Problem 3.56P

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

Explanation of Solution

Given:

The following hexagonal lattce is given:

Essentials of Materials Science and Engineering, SI Edition, Chapter 3, Problem 3.56P

Calculation:

Miller indices for direction A using three-digit system,

Head point =1, −1, 0Tail point = 0, 0, 0Head point – Tail point = ( 1, −1, 0) = [1 1 ¯ 0]Four digit systemConvert of four digit system.[h k l] = [ 1 1 ¯ 0]Coordinates of four digit system.h=13[2h−k]=13[2+1]=1k=13[2k−h]=13[−2−1]=−1i=−13[h+k]=−13[1−1]=0l=l'=0[h k i l] = [1,−1,0,0] = [1 1 ¯ 0 0]

Miller indices for direction B,

Three-digit system

Head point = 1,1,0Tail point = 0,0,1Head point−Tail point =( 1,1,−1) =[1 1 1 ¯]Four digit systemConvert three digit into four digit system.[h k l] = [ 1 1 1 ¯]Coordinates of four digit system.h=13[2h−k]=13[2−1]=13k=13[2k−h]=13[2−1]=13i=−13[h+k]=−13[1+1]=−23l=l'=−1[h k i l] = [ 1 3, 1 3,− 2 3,−1] = [ 1 3, 1 3,− 2 3,−1]×3 = [1,1,−2,−3] = [1 1 2 ¯ 3 ¯]

Miller indices for direction C,

Three-digit system

Head point = 0,1,1Tail point = 0,0,0Head point−Tail point =( 0,1,1) =[0 1 1]Four digit systemConvert three digit into four digit system.[h k l] = [0 1 1]Coordinates of four digit system.h=13[2h−k]=13[0−1]=−13k=13[2k−h]=13[2−0]=23i=−13[h+k]=−13[0+1]=−13l=l'=1[h k i l] = [− 1 3, 2 3,− 1 3,1] = [− 1 3, 2 3,− 1 3,1]×3 = [ 1 ¯,2, 1 ¯,3]

Conclusion

Thus,

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

Answer 7

Chapter 3, Problem 3.56P

Interpretation Introduction

Interpretation:

The indices for the directions in the hexagonal lattice using three-digit and four-digit systems are to be determined.

Concept Introduction:

The indices for the hexagonal structure needs to be written at given direction A, B, C. For hexagonal lattice, the three-digit system uses three indices (h,k,l) and the four-digit system uses four indices (h,k,i,l)

Answer 8

Expert Solution & Answer

Answer to Problem 3.56P

Miller indices for hexagonal lattice at given directions.

PlaneThree digit systemFour digit systemA[1 1 ¯ 0][1 1 ¯ 0 0]B[1 1 1 ¯][1 1 2 ¯ 3 ¯]C[0 1 1][ 1 ¯ 2 1 ¯ 3]

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Explanation of Solution

Given:

The following hexagonal lattce is given:

Essentials of Materials Science and Engineering, SI Edition, Chapter 3, Problem 3.56P

Calculation:

Miller indices for direction A using three-digit system,

Head point =1, −1, 0Tail point = 0, 0, 0Head point – Tail point = ( 1, −1, 0) = [1 1 ¯ 0]Four digit systemConvert of four digit system.[h k l] = [ 1 1 ¯ 0]Coordinates of four digit system.h=13[2h−k]=13[2+1]=1k=13[2k−h]=13[−2−1]=−1i=−13[h+k]=−13[1−1]=0l=l'=0[h k i l] = [1,−1,0,0] = [1 1 ¯ 0 0]

Miller indices for direction B,

Three-digit system

Head point = 1,1,0Tail point = 0,0,1Head point−Tail point =( 1,1,−1) =[1 1 1 ¯]Four digit systemConvert three digit into four digit system.[h k l] = [ 1 1 1 ¯]Coordinates of four digit system.h=13[2h−k]=13[2−1]=13k=13[2k−h]=13[2−1]=13i=−13[h+k]=−13[1+1]=−23l=l'=−1[h k i l] = [ 1 3, 1 3,− 2 3,−1] = [ 1 3, 1 3,− 2 3,−1]×3 = [1,1,−2,−3] = [1 1 2 ¯ 3 ¯]

Miller indices for direction C,

Three-digit system

Head point = 0,1,1Tail point = 0,0,0Head point−Tail point =( 0,1,1) =[0 1 1]Four digit systemConvert three digit into four digit system.[h k l] = [0 1 1]Coordinates of four digit system.h=13[2h−k]=13[0−1]=−13k=13[2k−h]=13[2−0]=23i=−13[h+k]=−13[0+1]=−13l=l'=1[h k i l] = [− 1 3, 2 3,− 1 3,1] = [− 1 3, 2 3,− 1 3,1]×3 = [ 1 ¯,2, 1 ¯,3]