Question

Calculate Miller indices of crystal plane, which is cut through the crystal axes at \(3 a,-3 b,-3 c ?\)

Short answer

(1, -1, -1)

Step-by-step solution

Identify the Intercepts

Determine where the plane cuts the crystallographic axes. The given intercepts are at \(3a\) on the x-axis, \(-3b\) on the y-axis, and \(-3c\) on the z-axis.

Take Reciprocal of Intercepts

Find the reciprocals of the intercepts identified in Step 1. The reciprocals are \( \frac{1}{3a} \) for the x-axis, \( \frac{1}{-3b} \) for the y-axis, and \( \frac{1}{-3c} \) for the z-axis.

Simplify the Fractions

Simplify each reciprocal fraction. This gives us \(\frac{1}{3a} = \frac{1}{3} \cdot \frac{1}{a}\)\(\frac{1}{-3b} = -\frac{1}{3} \cdot \frac{1}{b}\)\(\frac{1}{-3c} = - \frac{1}{3} \cdot \frac{1}{c}\).

Clear the Denominator

Multiply each simplified reciprocal fraction by the lowest common multiple (LCM) to clear any fractions. Since all denominators are 3, we multiply each term by 3: \( \frac{1}{3} \rightarrow 1\)\( -\frac{1}{3} \rightarrow -1\)\( -\frac{1}{3} \rightarrow -1\).

Determine Miller Indices

Combine the values obtained in Step 4 into the form (h, k, l). For the given intercepts, the Miller indices are (1, -1, -1).

Key concepts

Crystal Axes

In crystallography, crystal axes are the reference lines used to describe the position of points within a crystal structure. These axes are typically denoted as x, y, and z. Consider them as a three-dimensional grid that helps us pinpoint specific locations within the crystal.
Each axis correlates to a crystallographic direction, which helps in identifying and categorizing different crystallographic planes. Understanding the orientation of crystal axes is crucial for calculating Miller indices, as they serve as the foundation for this process.
It's like having a map where you need longitude, latitude, and elevation to describe a point. In crystals, the axes provide a similar reference framework.

Intercepts

The concept of intercepts in crystallography refers to the points at which a crystallographic plane cuts through the crystal axes. For instance, if a plane intersects the x-axis at 3 units (3a), this is the x-intercept. Similarly, the y and z-intercepts could be -3 units (-3b) and -3 units (-3c), respectively.
These intercepts are essential for calculating the Miller indices. They give us a way to understand how a particular plane is oriented within the crystal structure.
To identify these intercepts:

• Locate where the plane cuts each axis.
• Note down these points as the initial step in calculating Miller indices.

Visualize intercepts as if a plane is slicing through a rectangular box, touching the three sides (axes) at specific points.

Reciprocals

Once we identify the intercepts, the next step is to find their reciprocals. The reciprocal of a number is simply 1 divided by that number. If the intercept is at 3a, the reciprocal is \(\frac{1}{3a}\).
Taking reciprocals transforms the coordinates to a more manageable form for final Miller index calculations.
The process involves:

• Finding the mathematical inverse of each intercept.
• This includes converting all negative intercepts to negative reciprocals.

For instance, if one intercept is -3b, its reciprocal will be \(\frac{1}{-3b}\) or -\(\frac{1}{3b}\). This step simplifies the determination of Miller indices.

Miller Indices

Miller indices are a set of three integers (h, k, l) that uniquely define a crystallographic plane within a crystal lattice. They are derived from the reciprocals of the intercepts and provide a standardized way to specify planes, which is widely used in materials science.
To determine Miller indices:

• Take the reciprocals of the intercepts.
• Simplify each reciprocal.
• Clear any fractions by multiplying with the lowest common multiple.
• The resulting values (h, k, l) are the Miller indices.

For example, with intercepts 3a, -3b, -3c, the Miller indices are calculated as (1, -1, -1). This notation helps communicate the orientation of planes clearly and efficiently.
Miller indices are invaluable for describing crystal structures, analyzing X-ray diffraction patterns, and understanding material properties.