Miller indices angle brackets, a notation system used in crystallography, provides a concise representation of crystallographic planes and directions. These brackets enclose a set of three integers, denoted h, k, and l, which indicate the intercepts of the plane or direction with the crystallographic axes a, b, and c, respectively. The indices are expressed as the smallest set of integers that maintain the ratio of the true intercepts, providing a unique identifier for each plane or direction. Miller indices angle brackets are closely related to the concepts of reciprocal lattice, atomic planes, lattice planes, and crystallographic directions.
The Best Structure for Miller Indices Angle Brackets
Miller indices angle brackets are used to describe the orientation of crystallographic planes. While three different notations can be used to represent Miller indices, the angle bracket notation is widely accepted as the most unambiguous.
The angle bracket notation uses three numbers enclosed in angle brackets (< >). These numbers represent the intercepts of the plane on the crystallographic axes. For example, the plane that intercepts the x-axis at a distance of a, the y-axis at a distance of b, and the z-axis at a distance of c would be represented as .
When writing Miller indices in angle brackets, it is important to follow the following rules:
- The numbers should be written in the order of the crystallographic axes (x, y, z).
- The numbers should be written in their lowest terms.
- The numbers should be separated by spaces.
- If a number is negative, it should be preceded by a minus sign.
For example, the plane that intercepts the x-axis at a distance of -a, the y-axis at a distance of b, and the z-axis at a distance of c would be represented as <-a b c>.
The table below summarizes the rules for writing Miller indices in angle brackets:
By following these rules, you can ensure that your Miller indices are written in a clear and unambiguous way.
Question 1:
What are Miller indices angle brackets and how are they used?
Answer:
Miller indices angle brackets, denoted as <>, are a mathematical notation used to represent the direction of a plane or crystallographic axis in a crystal lattice. They are calculated by finding the reciprocals of the fractional intercepts of the plane or axis with the unit cell axes. The Miller indices are then enclosed in angle brackets to indicate that they represent a direction rather than a plane.
Question 2:
How do Miller indices angle brackets differ from Miller indices parentheses?
Answer:
Miller indices angle brackets differ from Miller indices parentheses in that:
- Angle brackets <> represent directions within the crystal lattice.
- Parentheses () represent planes within the crystal lattice.
- Angle brackets encompass three numbers, while parentheses encompass only two numbers.
Question 3:
What is the significance of the sign of the Miller indices angle brackets?
Answer:
The sign of the Miller indices angle brackets indicates the direction of the plane or axis relative to the crystal lattice’s reference frame:
- Positive indices: Plane or axis is pointing in the positive direction of the corresponding axis.
- Negative indices (indicated by an overbar): Plane or axis is pointing in the negative direction of the corresponding axis.
- Zero indices: Plane or axis is parallel to the corresponding axis.
Thanks for taking the time to learn about Miller indices angle brackets! I hope this article has helped you understand this important concept. If you have any questions, please don’t hesitate to ask. And be sure to check back later for more crystallography content.