Within a cubic unit cell, sketch the following directions: (a) [101] (e) [1 1 1] (b) [211] (f) [

Within a cubic unit cell, sketch the following directions:...

Key Concepts

Crystallographic Directions

Crystallographic directions refer to specific orientations or lines within a crystal lattice, described using a standardized notation. This notation, typically enclosed in square brackets, assigns indices to a vector that indicates direction relative to the unit cell axes. Understanding these directions is essential for interpreting and predicting crystal properties such as slip and diffraction patterns.

Miller Indices Notation

Miller indices provide a shorthand method for denoting crystallographic directions and planes in a lattice. For directions, indices in the form [uvw] represent the components of a vector in the crystal's coordinate system. This system of notation is key to systematically describing spatial directions in materials science and solid state physics.

Cubic Unit Cell Geometry

The cubic unit cell is a fundamental building block in crystallography, characterized by having equal edge lengths and orthogonal axes. Its high symmetry simplifies the description and sketching of directions and planes using Miller indices, making it an ideal model for teaching concepts in crystallography and materials science.

Representation of Negative Indices

Negative indices are indicated by a bar placed over a number in the Miller indices notation, signifying that the corresponding vector component is directed opposite to the positive axis. This notation is crucial for accurately describing directions in the lattice where orientation with respect to the chosen coordinate system matters.

Transcript

00:01 So, it has been asked in the question to sketch the following directions in a cubic unit cell and the directions are a 101 b 2 1 1 c 1 0 2 bar d 3 1 bar 3 e 1 bar 1 1 bar f 2 bar 1 2 g 3 1 bar 2 and h 3 0 1.

00:54 Let us consider a cubic unit cell of lattice parameter a.

01:08 Also the tail coordinates of the vectors representing each direction is fixed at the origin that is having the coordinates 0 0 0.

01:33 The representation of the directions is restricted to or is restricted within a unit cell.

01:48 So, the first part is to represent or to draw to sketch the direction 1 0 1.

01:57 So, the head coordinates in this case would be a 0 a.

02:03 The coordinate differences would be tail coordinate coordinates subtracted from the head coordinates that is a minus 0 that is a 0 minus 0 that is 0 and a minus 0 that is a.

02:30 Now the normalized coordinate differences are so a by a that is 1 0 by a that is 0 and a by a is that is 1.

02:45 Now if we take this as the unit cell and this one as the origin then these three directions represent the x y z directions.

02:58 So, let this be x this is y and this is z then the coordinate 1 this is 0 0 0 the origin this is 1 this is 0 the y is 0 and this is 1 for x for z.

03:15 So, this is the coordinate 1 0 1 bar the normalized coordinate differences and this represents this vector represents the direction 1 0 1.

03:25 Similarly for b the head coordinates are 2 a a and a the coordinate differences are 2 a minus 0 that is 2 a a minus 0 that is a and a minus 0 that is a.

03:49 So, the normalized coordinate differences would be 2 a by a that is 2 a by a that is 1 and a by a that is 1.

04:04 Now in order to restrict the representation of the direction within a unit cell the normalized coordinate differences must be less than or equal to 1.

04:37 So, in order to do so we can reduce these normalized coordinate differences to values less than or equal to 1 by dividing it with the highest or the largest common factors or the largest coordinate difference that is 2 by 2 is 1 1 by 2 is 0 .5 and 1 by 2 is 0 .5.

05:16 So, if this is the origin this is my x axis y axis and z axis then the coordinate is along x is 1 along y is 0 .5 along z is 0 .5 this is the coordinate.

05:30 So, this would be the point b having coordinates 1 half and half so along the so this would be the direction.

05:41 So, there so these are the intercepts along the three directions these are perpendicular intercepts.

05:51 So, this represent the direction 2 1 1.

05:58 So, this is these are the coordinates along the x y z axis.

06:05 Now for the third in part c we have to represent 1 0 2 bar so the head coordinates are a 0 minus 2 a the coordinate differences would be a minus 0 that is a 0 minus 0 that is 0 and minus 2 a minus 0 that is minus 2 a.

06:25 So, normalized coordinates coordinate differences would be a by a that is 1 0 by a that is 0 minus 2 a by a that is minus 2.

06:38 Now reducing the coordinates coordinate differences to a value less than or equal to 1 that is 1 by 2 is 0 .5 0 and minus 1.

06:51 So, here we can shift the origin to restrict the direction in the same unit cell because the choice of origin is arbitrary.

07:01 So, if we choose this would be our negative z axis and this is z axis this is x axis and this is y axis you have to go half of along x axis then 0 along y axis and minus 1 along z axis.

07:22 So, this would be the point half the point would be half 0 minus 1.

07:29 So, vector from the origin to this point p a represents the direction 1 0 2 bar.

07:39 Coming to part d that is 3 1 bar 3 again the head coordinates are 3 a minus a and 3 a the coordinate differences are 3 a minus 0 that is 3 a minus a minus 0 that is minus a and 3 a minus 0 that is 3 a.

08:12 Normalized coordinate differences are 3 a by a that is 3 minus a by a that is minus 1 and 3 a by 3 that 3 a by a that is 3...

Please give Ace some feedback