Miller Indices of a Plane in an Orthorhombic Crystal

Introduction
In crystallography, Miller indices are used to describe the orientation of a plane or a direction within a crystal lattice. These indices are represented by three integers (h, k, l), known as Miller indices, which denote the reciprocals of the fractional intercepts made by the plane along the three crystallographic axes.

Given Data
- Intercepts along the three crystallographic axes: 2.93 mm, 4.47 mm, and 2.35 mm
- Corresponding vectors: 3.05 Å, 6.991 Å, and 4.90 Å

Step 1: Conversion of Intercepts to Reciprocal Values
To determine the Miller indices, we first need to convert the intercepts into reciprocal values. We can do this by taking the reciprocal of each intercept:

Reciprocal of intercept along a-axis: 1/2.93 mm = 0.3413
Reciprocal of intercept along b-axis: 1/4.47 mm = 0.2237
Reciprocal of intercept along c-axis: 1/2.35 mm = 0.4255

Step 2: Conversion of Reciprocal Values to Integer Values
Next, we need to convert the reciprocal values to integers. To do this, we multiply each reciprocal value by a common factor to obtain whole numbers. In this case, we can use 1000 as the common factor to avoid decimal values:

Integer value along a-axis: 0.3413 * 1000 = 341
Integer value along b-axis: 0.2237 * 1000 = 224
Integer value along c-axis: 0.4255 * 1000 = 426

Step 3: Finding the Least Common Multiple
The next step is to find the least common multiple (LCM) of the three integers obtained in Step 2. The LCM will be used as the common factor to obtain the final Miller indices.

LCM of 341, 224, and 426 = 38,344

Step 4: Determining the Miller Indices
To obtain the Miller indices, we divide the LCM by each of the integer values obtained in Step 2:

Miller index along a-axis: 38,344 / 341 ≈ 113
Miller index along b-axis: 38,344 / 224 ≈ 171
Miller index along c-axis: 38,344 / 426 ≈ 90

Therefore, the Miller indices of the plane are (113, 171, 90).

Explanation
The Miller indices (113, 171, 90) indicate that the plane intersects the a-axis at a distance of 1/113 of the unit cell length along this axis, the b-axis at a distance of 1/171 of the unit cell length, and the c-axis at a distance of 1/90 of the unit cell length.

These indices provide a concise representation of the orientation and position of the plane within the crystal lattice. They are widely used in crystallography to describe the symmetry and structure of crystalline materials.

Summary
- Miller