Within a cubic unit cell, sketch the following directions: (a) [101] (b) [211] (c) [10 2] (d) [3

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

Key Concepts

Cubic Unit Cell

A cubic unit cell is the simplest repeating unit in a crystal lattice that has all sides equal and angles at 90°. It serves as the fundamental building block for the lattice structure in many crystalline materials, providing a framework for understanding and visualizing the spatial arrangement of atoms or ions.

Crystallographic Directions

Crystallographic directions describe specific orientations within a crystal lattice. They are typically expressed in square brackets and represent a vector that points from the origin of the unit cell to a specific point in the lattice, guiding the visualization and analysis of material properties and symmetry.

Index Notation

Index notation in crystallography is a standardized way of representing directions and planes in a crystal. The indices are usually enclosed in square brackets for directions, and each number correspondingly defines the component along the unit cell's axes. Understanding this notation is essential for accurately interpreting structural details in crystallographic studies.

Interpretation of Negative Indices

Negative indices, indicated with an overline above the number, represent directions opposite to the chosen positive coordinate axes. This convention allows for a complete and precise description of any direction within a crystal lattice, which is crucial for accurate analysis and representation of crystal structures.

Transcript

00:01 So this is a simple cubic unit cell and in this unit cell we have to mark these eight directions so these eight directions are as follows the first one is one zero one the second one is two one one the third one option c is one zero two bar option d is three one bar three one bar option e is one bar one bar option f is two option g is three one bar two and option h is three zero one so in this unit cell, this particular unit cell, i will take this as a sample and i will redraw all of these directions in this particular unit cell for convenience.

00:42 So you can pause the video whenever it is convenient to you to find the solution to a particular direction, that is to a particular bead of this question.

00:51 You can pause to that particular moment where i have drawn these directions.

00:55 So i will basically erase and redraw all these directions in this unit cell just to try.

01:00 Show how it is done for illustration purposes so the first direction is 1 -0 -1 so this is a simple one and in this case we will consider a coordinate axis like this is the x -axis this is the y -axis and this is the x -axis so this direction is basically nothing but a line from 0 -0 to 101 so this direction these two coordinates are the endpoints of this line segment.

01:37 So this is the 0 0, 0 coordinate and 1 .01 coordinate is x1, y, 0, z1.

01:46 So this particular coordinates.

01:47 So i will draw this with a different color.

01:50 So this is the unit cell direction, 101.

01:55 So this is 1 .01.

01:58 Now let us move to the next direction.

02:02 The next direction is 2 .1.

02:06 So this is also simple.

02:08 We will use the same coordinate axis.

02:11 So two means x is half.

02:14 So let me tell the rule as well for clarification purpose...