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Question: Determine the indices for the directions A, B, C and D shown in the following cubic unit cell: –Free Chegg Question Answer

Determine the indices for the directions A, B, C and D shown in the following cubic unit cell:

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Answer

Step-by-step

Step 1 of 4

Using the stated steps, obtain the miller indices for direction A.

-
ab
-1
0
Axes
Projections
Projection in
terms of
a,b and c
Reduction to
integers
Enclosure
Not Necessary
[110]

The miller indices for direction A are \left[ {\bar 110} \right][1ˉ10] .
Explanation | Common mistakes | Hint for next step

Since direction A is not passing through the origin of the coordinate system, hence change the position of direction A to the origin of the coordinate system by moving one unit cell distance horizontally opposite to x – axisxaxis , thus the intercept will be in the negative side of the origin. Since it is shifted by one unit in negative direction, the x – {\rm{intercept}}x−intercept in term of lattice parameters is found to be \left( { – a} \right)(−a) . The z – {\rm{intercept}}z−intercept in terms of lattice parameter is found to be \left( {0c} \right)(0c) as the direction A is not passing along z – axiszaxis . The y – {\rm{intercept}}y−intercept in terms of lattice parameters is found to be \left( b \right)(b) as it is moved by one complete unit along y – axisyaxis . Then write the intercepts in terms of integers. It is not necessary for reduction in this case as intercepts are not in fraction. The three integer indices are enclosed within the parentheses as \left[ {\bar 110} \right][1ˉ10] and should not be separated by commas.

Step 2 of 4

Using the stated steps, obtain the miller indices for direction B.

Axes
Projections
Projection in
terms of
a,b and c
Reduction
to integers
Enclosure
(121)

The miller indices for direction {\rm{B}}B are \left[ {121} \right][121] .
Explanation | Common mistakes | Hint for next step

Since direction B is passing through the origin of the coordinate system, hence no translation is necessary in this case. The x – {\rm{intercept}}x−intercept in terms of lattice parameters is found to be \left( {\frac{a}{2}} \right)(2a​) as it is moved by half unit along x – axisxaxis . The y – {\rm{intercept}}y−intercept in terms of lattice parameters is found to be \left( b \right)(b) as it is moved by one complete unit along y – axisyaxis . The z – {\rm{intercept}}z−intercept in terms of lattice parameter is found to be \left( {\frac{c}{2}} \right)(2c​) as it is moved by half unit along z – axiszaxis . The reduction to integers is necessary in this case, so multiply the projections along each axis with 2 to obtain the miller indices. The three integer indices are enclosed within the parentheses as \left[ {121} \right][121] and should not be separated by commas.

Step 3 of 4

Using the stated steps, obtain the miller indices for direction C.

Axes
Projections
Projection in
terms of
a,b and c
Reduction
to integers
Enclosure

The miller indices for direction {\rm{C}}C are \left[ {0\overline 1 \overline 2 } \right][012] .
Explanation | Common mistakes | Hint for next step

Since direction C is not passing through the origin of the coordinate system, hence change the position of direction C to the origin of the coordinate system by moving one unit cell distance horizontally opposite to z – axiszaxis , thus the intercept will be in the negative side of the origin. Since it is shifted by one unit in negative direction, the z – {\rm{intercept}}z−intercept in term of lattice parameters is found to be \left( { – c} \right)(−c) . The x – {\rm{intercept}}x−intercept in terms of lattice parameter is found to be \left( {0a} \right)(0a) as the direction C is not passing along x – axisxaxis . The y – {\rm{intercept}}y−intercept in terms of lattice parameters is found to be \left( {\frac{{ – b}}{2}} \right)(2−b​) as it is moved by half unit along y – {\rm{axis}}y−axis opposite to the direction of the axis. Then write the intercepts in terms of integers. The reduction to integers is necessary in this case, so multiply the projections along each axis with 2 to obtain the miller indices. The three integer indices are enclosed within the parentheses as \left[ {0\overline 1 \overline 2 } \right][012] and should not be separated by commas.

Step 4 of 4

Using the stated steps, obtain the miller indices for direction D.

Axes
Projections
Projection in
terms of
a,b and c
Reduction
to integers
Enclosure


The miller indices for direction {\rm{D}}D are \left[ {1\overline 2 1} \right][121] .
Explanation | Common mistakes

Since direction D is not passing through the origin of the coordinate system, hence change the position of direction D to the origin of the coordinate system by moving one unit cell distance horizontally opposite to y – axisyaxis , thus the intercept will be in the negative side of the origin. Since it is shifted by one unit in negative direction, the y – {\rm{intercept}}y−intercept in term of lattice parameters is found to be \left( { – b} \right)(−b) . The x – {\rm{intercept}}x−intercept in terms of lattice parameters is found to be \left( {\frac{a}{2}} \right)(2a​) as it is moved by half unit along x – axisxaxis . The z – {\rm{intercept}}z−intercept in terms of lattice parameter is found to be \left( {\frac{c}{2}} \right)(2c​) as it is moved by half unit along z – axiszaxis . The reduction to integers is necessary in this case, so multiply the projections along each axis with 2 to obtain the miller indices. The three integer indices are enclosed within the parentheses as \left[ {1\overline 2 1} \right][121] and should not be separated by commas.

Answer

The miller indices for direction A are \left[ {\bar 110} \right][1ˉ10] .

The miller indices for direction {\rm{B}}B are \left[ {121} \right][121] .

The miller indices for direction {\rm{C}}C are \left[ {0\overline 1 \overline 2 } \right][012] .

The miller indices for direction {\rm{D}}D are \left[ {1\overline 2 1} \right][121] .


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