Descripción del proceso

(a) [101] (e)

(b) [211] (f)

(c) (g)

(d) (h) [301]

Solution

(a) For the [101] direction, it is the case that

u =1 v = 0 w = 1

lf we select the origin of the coordinate system as the position of the vector tail, then x₁ = 0a y₁ = 0b z₁ = 0c

It is now possible to determine values of x₂, y₂, and z₂ using rearranged forms of Equations 3.10a through 3.10c as follows:

Thus, the vector head is located at a, 0b, and c, and the direction vector having these head coordinates is plotted below.

[Note: even though the unit cell is cubic, which means that the unit cell edge lengths are the same (i.e., a), in order to clarify construction of the direction vector, we have chosen to use b and c to designate edge lengths along y and z axes, respectively.]

(b) For a [211] direction, it is the case that [111]

[212]

[102] [312]

[313]

x₂ uax₁(1)a0aa y₂ vby₁(0)b0b0b

z₂ wcz₁(1)c0cc

u =2 v = 1 w = 1

lf we select the origin of the coordinate system as the position of the vector tail, then x₁ = 0a y₁ = 0b z₁ = 0c

It is now possible to determine values of x₂, y₂, and z₂ using rearranged forms of Equations 3.10a through 3.10c as follows:

Thus, the vector head is located at 2a, b, and c, and the direction vector having these head coordinates is plotted below.

[Note: even though the unit cell is cubic, which means that the unit cell edge lengths are the same (i.e., a), in order to clarify construction of the direction vector, we have chosen to use b and c to designate edge lengths along y and z axes, respectively.]

(c) For the direction, it is the case that

u =1 v = 0 w = 2

lf we select the origin of the coordinate system as the position of the vector tail, then x₁ = 0a y₁ = 0b z₁ = 0c

It is now possible to determine values of x₂, y₂, and z₂ using rearranged forms of Equations 3.10a through 3.10c as follows:

x₂ uax₁(2)a0a2a y₂ vby₁(1)b0bb z₂ wcz₁(1)c0cc

[102]

x₂ uax₁(1)a0aa y₂ vby₁(0)b0b0b

Thus, the vector head is located at a, 0b, and c. However, in order to reduce the vector length, we have divided these coordinates by ½, which gives the new set of head coordinates as a/2, 0b, and c; the direction vector having these head coordinates is plotted below.

[Note: even though the unit cell is cubic, which means that the unit cell edge lengths are the same (i.e., a), in order to clarify construction of the direction vector, we have chosen to use b and c to designate edge lengths along y and z axes, respectively.]

(d) For the direction, it is the case that

u = 3 v = 1 w = 3

lf we select the origin of the coordinate system as the position of the vector tail, then x₁ = 0a y₁ = 0b z₁ = 0c

It is now possible to determine values of x₂, y₂, and z₂ using rearranged forms of Equations 3.10a through 3.10c as follows:

Thus, the vector head is located at 3a, b, and 3c. However, in order to reduce the vector length, we have divided these coordinates by 1/3, which gives the new set of head coordinates as a, b/3, and c; the direction vector having these head coordinates is plotted below.

z₂ wcz₁(2)c0c 2c

[3 13]

x₂ uax₁(3)a0a3a y₂ vby₁(1)b0b b

z₂ wcz₁(3)c0c3c

[Note: even though the unit cell is cubic, which means that the unit cell edge lengths are the same (i.e., a), in order to clarify construction of the direction vector, we have chosen to use b and c to designate edge lengths along y and z axes, respectively.]

(e) For the direction, it is the case that

u = 1 v = 1 w = 1

lf we select the origin of the coordinate system as the position of the vector tail, then x₁ = 0a y₁ = 0b z₁ = 0c

It is now possible to determine values of x₂, y₂, and z₂ using rearranged forms of Equations 3.10a through 3.10c as follows:

Thus, the vector head is located at a, b, and c, and the direction vector having these head coordinates is plotted below.

[11 1]

x₂ uax₁(1)a0a a y₂ vby₁(1)b0bb z₂ wcz₁(1)c0c c

[Note: even though the unit cell is cubic, which means that the unit cell edge lengths are the same (i.e., a), in order to clarify construction of the direction vector, we have chosen to use b and c to designate edge lengths along y and z axes, respectively.]

(f) For the direction, it is the case that

u = 2 v = 1 w = 2

lf we select the origin of the coordinate system as the position of the vector tail, then x₁ = 0a y₁ = 0b z₁ = 0c

It is now possible to determine values of x₂, y₂, and z₂ using rearranged forms of Equations 3.10a through 3.10c as follows:

Thus, the vector head is located at 2a, b, and 2c. However, in order to reduce the vector length, we have divided these coordinates by 1/2, which gives the new set of head coordinates as a, b/2, and c; the direction vector having these head coordinates is plotted below.

[Note: even though the unit cell is cubic, which means that the unit cell edge lengths are the same (i.e., a), in order to clarify construction of the direction vector, we have chosen to use b and c to designate edge lengths along y and z axes, respectively.]

(g) For the direction, it is the case that

u =3 v = 1 w = 2

lf we select the origin of the coordinate system as the position of the vector tail, then x₁ = 0a y₁ = 0b z₁ = 0c

[212]

x₂ uax₁(2)a0a 2a y₂ vby₁(1)b0bb z₂ wcz₁(2)c0c2c

[3 12]

It is now possible to determine values of x₂, y₂, and z₂ using rearranged forms of Equations 3.10a through 3.10c as follows:

Thus, the vector head is located at 3a, b, and 2c. However, in order to reduce the vector length, we have divided these coordinates by 1/3, which gives the new set of head coordinates as a, b/3, and 2c/3; the direction vector having these head coordinates is plotted below.

[Note: even though the unit cell is cubic, which means that the unit cell edge lengths are the same (i.e., a), in order to clarify construction of the direction vector, we have chosen to use b and c to designate edge lengths along y and z axes, respectively.]

(g) For the [301] direction, it is the case that

u =3 v = 0 w = 1

lf we select the origin of the coordinate system as the position of the vector tail, then x₁ = 0a y₁ = 0b z₁ = 0c

It is now possible to determine values of x₂, y₂, and z₂ using rearranged forms of Equations 3.10a through 3.10c as follows:

x₂ uax₁(3)a0a3a y₂ vby₁(1)b0b b

z₂ wcz₁(2)c0c2c

x₂ uax₁(3)a0a3a y₂ vby₁(0)b0b0b z₂ wcz₁(1)c0cc

Thus, the vector head is located at 3a, 0b, and c. However, in order to reduce the vector length, we have divided these coordinates by 1/3, which gives the new set of head coordinates as a, 0b, and c/3; the direction vector having these head coordinates is plotted below.

[Note: even though the unit cell is cubic, which means that the unit cell edge lengths are the same (i.e., a), in order to clarify construction of the direction vector, we have chosen to use b and c to designate edge lengths along y and z axes, respectively.]