Sketch within a cubic unit cell the following planes: (a) (10 1) (e) (1 1 1) (b) (2 1 1) (f) (2

Sketch within a cubic unit cell the following planes: (a)...

Key Concepts

Visualization of Crystallographic Planes

Drawing or sketching crystallographic planes involves translating the Miller index notation into a graphical representation within a unit cell. It requires understanding the spatial orientations provided by the indices and using them to identify the cutting points on the cell’s edges, thereby allowing one to accurately depict the planes’ positions and intersections in three dimensions.

Intercept Form of Planes

The intercept form is a method used to visualize crystallographic planes by relating the Miller indices to the points where the plane intersects the axes of the unit cell. The reciprocal relationship between the Miller indices and the intercepts allows one to determine the cutting points on the axes, which is useful for sketching the planes within the unit cell.

Negative Miller Indices

Negative Miller indices are indicated by a bar over the number and denote that the plane intercepts the axes in the negative direction. This notation is crucial for accurately representing the orientation and position of the plane within the unit cell, ensuring that the full symmetry and directional characteristics of the crystal structure are captured.

Miller Indices

Miller indices are a shorthand notation in crystallography used to describe the orientation of planes in a crystal lattice. They are represented as (h k l) for a plane that intercepts the crystallographic axes proportional to 1/h, 1/k, and 1/l, respectively, and are fundamental in interpreting and predicting the geometrical arrangement of atoms in crystal structures.

Cubic Unit Cell

A cubic unit cell is the simplest crystal lattice structure, characterized by equal edge lengths and all angles measuring 90°. It provides a geometric framework for visualizing and sketching the arrangement of atoms and the orientation of various crystallographic planes described by their Miller indices.

Transcript

00:01 So it has been asked to sketch the following planes within a cubic unit cell and the planes are a 101 bar b 2 1 bar 1 c 0 1 2 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 so coming to the part a that is the plane 1 0 1 bar the plane has indices along xyz directions as 1 0 minus 1 the reciprocal of the indices are 1 by 1 that is 1 1 by 0 that is infinity and minus 1 by 1 that is minus 1 its intercepts in terms of lattice parameter a of a cubic unit cell is a infinity minus a so since the z -axis is the intercept along z -axis is negative.

01:22 So which displace our origin so as to accommodate the plane in the in the cubic unit cell, so this is our this is our intercept x intercept a and then it is parallel to y -axis.

01:42 So this line is parallel to y -axis and this is our negative z intercept minus this plane is the plane which is represented by the miller indices 1 0 1 bar.

02:03 So this is the plane 1 0 1 bar coming to part b that is 2 1 bar 1 again the indices along xyz directions are 2 minus 1 and 1 the reciprocal of these indices are 1 by 2 minus 1 and minus 1 minus 1 and 1 and then the respective intercepts would be a by 2 minus a and a so now since y is negative so we displace the origin towards the y -axis to accommodate the plane in the single cubic unit cell so this is the origin displaced origin this is the z intercept x intercept this is your y intercept and this is your z intercept.

02:52 So this is the plane which is representing the this is the plane which is representing the miller indices 2 1 bar 1 coming to part c which is 0 1 2 so we'll have this plane will have indices along xyz direction as 0 1 and 2 the reciprocal of these indices are infinity 1 and 1 by 2 so their respective intercepts are infinity a and a by 2 now since all are positive.

03:30 So the origin is intact and is not needed to be displaced so this is our origin and this is our x y and z so the plane is parallel to x axis has a z intercept minus a by 2 and y intercept the plane would be this so this is the plane which is parallel to x axis and we'll have a y intercept a and z intercept minus a by z intercept a bar this plane has a miller indices 0 1 coming to part d so part d is coming to part d so 3 1 bar 3 the indices along xyz direction are 3 minus 1 3 reciprocal of these indices are 1 by 3 minus 1 and 1 by 3 and the intercepts would be a by 3 minus a and a by 3 so now since y axis is negative so we displace it along by one unit along y axis.

04:56 So this is the axis now, so this is xyz so the plane has an intercept along x axis a by 3 along y axis minus a along z axis a by 3 we connect it with straight lines.

05:11 So this plane represents the plane 3 1 bar 3 so this is the plane which represent which is represented by miller indices 3 1 bar 3 coming to part e that is 1 bar 1 1 bar so it will have indices along xyz direction as minus 1 1 minus 1 reciprocal of these indices are minus 1 1 and minus 1 intercepts would be minus a a and minus a so now it will have the origin must be displaced towards by one unit along x and one unit along z...

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