The Miller-Bravais direction indices of the vectors originating at the center of the lower basal plane.

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Answer 1

Question

Chapter 3.15, Problem 55AAP

To determine

The Miller-Bravais direction indices of the vectors originating at the center of the lower basal plane.

Expert Solution & Answer

Answer to Problem 55AAP

The Miller-Bravais Direction indices of the vector OA is (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

Explanation of Solution

For finding the Miller-Bravais direction indices we need coordinates of intercept of the plane. By using the method of indices the intercept (a1, a2, a3 and c) of the vectors is,

Coordinate of intercept of the vectors OA is (−1,−1,12,1).

Coordinate of intercept of the vectors OB is (1,1,−12,1).

Coordinate of intercept of the vectors OC is (12,−1,−1,1).

Coordinate of intercept of the vectors OD is (1,1,−12,1).

Coordinate of intercept of the vectors OE is (−1,12,−1,1).

Coordinate of intercept of the vectors OF is (−12,1,1,1).

By using these intercept we will find the Miller-Bravais direction indices.

Conclusion:

Direction vector originating at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell.

Foundations of Materials Science and Engineering, Chapter 3.15, Problem 55AAP

Figure-(1)

The given closed packing is Hexagonal closed packing and the direction vector of the planes are shown in figure-1. Here, the originating vector at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell is defined in the figure-1

Miller-Bravais direction Indices for the Direction vector are tabulated below.

Direction Vector	Co-ordinates of intercepts	Reciprocal of intercept	Direction indices
OA	(−1,−1,12,1)	(−1,−1,2,1)	(1¯ 1¯ 2 1)
OB	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OC	(12,−1,−1,1)	(2,−1,−1,1)	(2 1¯ 1¯ 1)
OD	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OE	(−1,12,−1,1)	(−1,2,−1,1)	(1¯ 2 1¯ 1)
OF	(−12,1,1,1)	(−2,1,1,1)	(2¯ 1 1 1)

The Miller-Bravais Direction indices of the vector OAis (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

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Answer 2

Question

Chapter 3.15, Problem 55AAP

To determine

The Miller-Bravais direction indices of the vectors originating at the center of the lower basal plane.

Expert Solution & Answer

Answer to Problem 55AAP

The Miller-Bravais Direction indices of the vector OA is (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

Explanation of Solution

For finding the Miller-Bravais direction indices we need coordinates of intercept of the plane. By using the method of indices the intercept (a1, a2, a3 and c) of the vectors is,

Coordinate of intercept of the vectors OA is (−1,−1,12,1).

Coordinate of intercept of the vectors OB is (1,1,−12,1).

Coordinate of intercept of the vectors OC is (12,−1,−1,1).

Coordinate of intercept of the vectors OD is (1,1,−12,1).

Coordinate of intercept of the vectors OE is (−1,12,−1,1).

Coordinate of intercept of the vectors OF is (−12,1,1,1).

By using these intercept we will find the Miller-Bravais direction indices.

Conclusion:

Direction vector originating at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell.

Foundations of Materials Science and Engineering, Chapter 3.15, Problem 55AAP

Figure-(1)

The given closed packing is Hexagonal closed packing and the direction vector of the planes are shown in figure-1. Here, the originating vector at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell is defined in the figure-1

Miller-Bravais direction Indices for the Direction vector are tabulated below.

Direction Vector	Co-ordinates of intercepts	Reciprocal of intercept	Direction indices
OA	(−1,−1,12,1)	(−1,−1,2,1)	(1¯ 1¯ 2 1)
OB	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OC	(12,−1,−1,1)	(2,−1,−1,1)	(2 1¯ 1¯ 1)
OD	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OE	(−1,12,−1,1)	(−1,2,−1,1)	(1¯ 2 1¯ 1)
OF	(−12,1,1,1)	(−2,1,1,1)	(2¯ 1 1 1)

The Miller-Bravais Direction indices of the vector OAis (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

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Answer 3

Question

Chapter 3.15, Problem 55AAP

To determine

The Miller-Bravais direction indices of the vectors originating at the center of the lower basal plane.

Expert Solution & Answer

Answer to Problem 55AAP

The Miller-Bravais Direction indices of the vector OA is (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

Explanation of Solution

For finding the Miller-Bravais direction indices we need coordinates of intercept of the plane. By using the method of indices the intercept (a1, a2, a3 and c) of the vectors is,

Coordinate of intercept of the vectors OA is (−1,−1,12,1).

Coordinate of intercept of the vectors OB is (1,1,−12,1).

Coordinate of intercept of the vectors OC is (12,−1,−1,1).

Coordinate of intercept of the vectors OD is (1,1,−12,1).

Coordinate of intercept of the vectors OE is (−1,12,−1,1).

Coordinate of intercept of the vectors OF is (−12,1,1,1).

By using these intercept we will find the Miller-Bravais direction indices.

Conclusion:

Direction vector originating at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell.

Foundations of Materials Science and Engineering, Chapter 3.15, Problem 55AAP

Figure-(1)

The given closed packing is Hexagonal closed packing and the direction vector of the planes are shown in figure-1. Here, the originating vector at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell is defined in the figure-1

Miller-Bravais direction Indices for the Direction vector are tabulated below.

Direction Vector	Co-ordinates of intercepts	Reciprocal of intercept	Direction indices
OA	(−1,−1,12,1)	(−1,−1,2,1)	(1¯ 1¯ 2 1)
OB	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OC	(12,−1,−1,1)	(2,−1,−1,1)	(2 1¯ 1¯ 1)
OD	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OE	(−1,12,−1,1)	(−1,2,−1,1)	(1¯ 2 1¯ 1)
OF	(−12,1,1,1)	(−2,1,1,1)	(2¯ 1 1 1)

The Miller-Bravais Direction indices of the vector OAis (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

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Answer 4

Question

Chapter 3.15, Problem 55AAP

To determine

The Miller-Bravais direction indices of the vectors originating at the center of the lower basal plane.

Expert Solution & Answer

Answer to Problem 55AAP

The Miller-Bravais Direction indices of the vector OA is (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

Explanation of Solution

For finding the Miller-Bravais direction indices we need coordinates of intercept of the plane. By using the method of indices the intercept (a1, a2, a3 and c) of the vectors is,

Coordinate of intercept of the vectors OA is (−1,−1,12,1).

Coordinate of intercept of the vectors OB is (1,1,−12,1).

Coordinate of intercept of the vectors OC is (12,−1,−1,1).

Coordinate of intercept of the vectors OD is (1,1,−12,1).

Coordinate of intercept of the vectors OE is (−1,12,−1,1).

Coordinate of intercept of the vectors OF is (−12,1,1,1).

By using these intercept we will find the Miller-Bravais direction indices.

Conclusion:

Direction vector originating at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell.

Foundations of Materials Science and Engineering, Chapter 3.15, Problem 55AAP

Figure-(1)

The given closed packing is Hexagonal closed packing and the direction vector of the planes are shown in figure-1. Here, the originating vector at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell is defined in the figure-1

Miller-Bravais direction Indices for the Direction vector are tabulated below.

Direction Vector	Co-ordinates of intercepts	Reciprocal of intercept	Direction indices
OA	(−1,−1,12,1)	(−1,−1,2,1)	(1¯ 1¯ 2 1)
OB	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OC	(12,−1,−1,1)	(2,−1,−1,1)	(2 1¯ 1¯ 1)
OD	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OE	(−1,12,−1,1)	(−1,2,−1,1)	(1¯ 2 1¯ 1)
OF	(−12,1,1,1)	(−2,1,1,1)	(2¯ 1 1 1)

The Miller-Bravais Direction indices of the vector OAis (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

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Answer 5

Chapter 3.15, Problem 55AAP

To determine

The Miller-Bravais direction indices of the vectors originating at the center of the lower basal plane.

Expert Solution & Answer

Answer to Problem 55AAP

The Miller-Bravais Direction indices of the vector OA is (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

Explanation of Solution

For finding the Miller-Bravais direction indices we need coordinates of intercept of the plane. By using the method of indices the intercept (a1, a2, a3 and c) of the vectors is,

Coordinate of intercept of the vectors OA is (−1,−1,12,1).

Coordinate of intercept of the vectors OB is (1,1,−12,1).

Coordinate of intercept of the vectors OC is (12,−1,−1,1).

Coordinate of intercept of the vectors OD is (1,1,−12,1).

Coordinate of intercept of the vectors OE is (−1,12,−1,1).

Coordinate of intercept of the vectors OF is (−12,1,1,1).

By using these intercept we will find the Miller-Bravais direction indices.

Conclusion:

Direction vector originating at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell.

Foundations of Materials Science and Engineering, Chapter 3.15, Problem 55AAP

Figure-(1)

The given closed packing is Hexagonal closed packing and the direction vector of the planes are shown in figure-1. Here, the originating vector at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell is defined in the figure-1

Miller-Bravais direction Indices for the Direction vector are tabulated below.

Direction Vector	Co-ordinates of intercepts	Reciprocal of intercept	Direction indices
OA	(−1,−1,12,1)	(−1,−1,2,1)	(1¯ 1¯ 2 1)
OB	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OC	(12,−1,−1,1)	(2,−1,−1,1)	(2 1¯ 1¯ 1)
OD	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OE	(−1,12,−1,1)	(−1,2,−1,1)	(1¯ 2 1¯ 1)
OF	(−12,1,1,1)	(−2,1,1,1)	(2¯ 1 1 1)

The Miller-Bravais Direction indices of the vector OAis (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

Answer 6

Chapter 3.15, Problem 55AAP

To determine

The Miller-Bravais direction indices of the vectors originating at the center of the lower basal plane.

Expert Solution & Answer

Answer to Problem 55AAP

The Miller-Bravais Direction indices of the vector OA is (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

Explanation of Solution

For finding the Miller-Bravais direction indices we need coordinates of intercept of the plane. By using the method of indices the intercept (a1, a2, a3 and c) of the vectors is,

Coordinate of intercept of the vectors OA is (−1,−1,12,1).

Coordinate of intercept of the vectors OB is (1,1,−12,1).

Coordinate of intercept of the vectors OC is (12,−1,−1,1).

Coordinate of intercept of the vectors OD is (1,1,−12,1).

Coordinate of intercept of the vectors OE is (−1,12,−1,1).

Coordinate of intercept of the vectors OF is (−12,1,1,1).

By using these intercept we will find the Miller-Bravais direction indices.

Conclusion:

Direction vector originating at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell.

Foundations of Materials Science and Engineering, Chapter 3.15, Problem 55AAP

Figure-(1)

The given closed packing is Hexagonal closed packing and the direction vector of the planes are shown in figure-1. Here, the originating vector at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell is defined in the figure-1

Miller-Bravais direction Indices for the Direction vector are tabulated below.

Direction Vector	Co-ordinates of intercepts	Reciprocal of intercept	Direction indices
OA	(−1,−1,12,1)	(−1,−1,2,1)	(1¯ 1¯ 2 1)
OB	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OC	(12,−1,−1,1)	(2,−1,−1,1)	(2 1¯ 1¯ 1)
OD	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OE	(−1,12,−1,1)	(−1,2,−1,1)	(1¯ 2 1¯ 1)
OF	(−12,1,1,1)	(−2,1,1,1)	(2¯ 1 1 1)

The Miller-Bravais Direction indices of the vector OAis (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

Answer 7

Chapter 3.15, Problem 55AAP

To determine

The Miller-Bravais direction indices of the vectors originating at the center of the lower basal plane.

Answer 8

Expert Solution & Answer

Answer to Problem 55AAP

The Miller-Bravais Direction indices of the vector OA is (1¯ 1¯ 2 1).

The Miller-Bravais Direction indices of the vector OB is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OC is (2 1¯ 1¯ 1).

The Miller-Bravais Direction indices of the vector OD is (1 1 2¯ 1).

The Miller-Bravais Direction indices of the vector OE is (1¯ 2 1¯ 1)

The Miller-Bravais Direction indices of the vector OF is (2¯ 1 1 1).

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Explanation of Solution

For finding the Miller-Bravais direction indices we need coordinates of intercept of the plane. By using the method of indices the intercept (a1, a2, a3 and c) of the vectors is,

Coordinate of intercept of the vectors OA is (−1,−1,12,1).

Coordinate of intercept of the vectors OB is (1,1,−12,1).

Coordinate of intercept of the vectors OC is (12,−1,−1,1).

Coordinate of intercept of the vectors OD is (1,1,−12,1).

Coordinate of intercept of the vectors OE is (−1,12,−1,1).

Coordinate of intercept of the vectors OF is (−12,1,1,1).

By using these intercept we will find the Miller-Bravais direction indices.

Conclusion:

Direction vector originating at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell.

Foundations of Materials Science and Engineering, Chapter 3.15, Problem 55AAP

Figure-(1)

The given closed packing is Hexagonal closed packing and the direction vector of the planes are shown in figure-1. Here, the originating vector at the centre of the lower basal plane and ending at the end point of the upper basal plane for a Hexagonal closed packed unit cell is defined in the figure-1

Miller-Bravais direction Indices for the Direction vector are tabulated below.

Direction Vector	Co-ordinates of intercepts	Reciprocal of intercept	Direction indices
OA	(−1,−1,12,1)	(−1,−1,2,1)	(1¯ 1¯ 2 1)
OB	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OC	(12,−1,−1,1)	(2,−1,−1,1)	(2 1¯ 1¯ 1)
OD	(1,1,−12,1)	(1,1,−2,1)	(1 1 2¯ 1)
OE	(−1,12,−1,1)	(−1,2,−1,1)	(1¯ 2 1¯ 1)
OF	(−12,1,1,1)	(−2,1,1,1)	(2¯ 1 1 1)