Miller Indices of a Plane in an Orthorhombic Crystal

An orthorhombic crystal is a type of crystal system characterized by three unequal axes that are mutually perpendicular to each other. The ratio of the lengths of these axes is defined by the parameters a:b:c, where a, b, and c are the lengths of the axes.

Miller Indices

Miller indices are a system of notation used to describe crystal planes and directions within a crystal lattice. They are represented by three integers (hkl) enclosed in parentheses, where h, k, and l are the reciprocals of the intercepts made by the plane on the three axes.

Intercepts on Coordinate Axes

In this case, the plane intersects the coordinate axes of the orthorhombic crystal at 3 Ångstrom, 4 Ångstrom, and 5 Ångstrom, respectively. Let's assign these intercepts to the parameters a, b, and c.

Given:
a:b:c = 1:2:5

Therefore, we can assign:
a = 3 Ångstrom
b = 4 Ångstrom
c = 5 Ångstrom

Determining Miller Indices

To determine the Miller indices of the plane, we need to find the reciprocals of the intercepts made by the plane on the three axes.

Reciprocal of the intercept on the a-axis (h):
1/a = 1/3 Ångstrom

Reciprocal of the intercept on the b-axis (k):
1/b = 1/4 Ångstrom

Reciprocal of the intercept on the c-axis (l):
1/c = 1/5 Ångstrom

Simplifying the reciprocals, we get:
h = 1/3
k = 1/4
l = 1/5

Miller Indices Representation

Now that we have the values of h, k, and l, we can represent the Miller indices of the plane as (hkl).

(hkl) = (1/3, 1/4, 1/5)

However, Miller indices are always simplified to the smallest possible integers. To achieve this, we multiply all the indices by a common factor to eliminate any fractions.

Multiplying all indices by 60 (common multiple of 3, 4, and 5), we get:
(hkl) = (20, 15, 12)

Therefore, the Miller indices of the plane that make intercepts of 3 Ångstrom, 4 Ångstrom, and 5 Ångstrom on the coordinate axes of the orthorhombic crystal with a:b:c = 1:2:5 are (20, 15, 12).

Summary
- Orthorhombic crystals have three unequal axes, represented by the parameters a, b, and c.
- Miller indices describe crystal planes and directions within a crystal lattice.
- The intercepts made by the plane on the three axes are used to determine the Miller indices.
- Reciprocals of the intercepts on the axes give the values of h, k, and l.
- The Miller indices are represented as (hkl).
- To simplify the indices, a common factor is multiplied to eliminate fractions.
- The final Miller indices for this plane are (