Crystallography

From the upper-hemisphere stereographic projection:

A filled square at the center denotes a single tetrad (4-fold rotation axis) perpendicular to the projection → one tetrad present.

Oval symbols on the perimeter represent diads (2-fold axes). There are four diads arranged symmetrically in the horizontal plane.

The symmetry corresponds to a tetragonal crystal system, where

The symmetry elements imply one horizontal mirror plane and four vertical mirror planes, giving 5 mirror planes total.

Answer: (A), (B), (C) and (D)

To determine the value of h in the Miller-Bravais Index (4̅1h0), we need to understand how indices are calculated in the hexagonal crystal system. The Miller-Bravais indices are used to describe planes in this system, which involves four indices: (hkl) and (hkil). For the four-index system:

u + v + t = 0

where:

• u = h
• v = k
• t = -(u + v)
• w = l

In our index (4̅1h0):

• u = 4̅ (representing negative 4)
• v = 1
• w = 0 (for the l in traditional Miller index)

Substitute these values into the equation for t:

t = -(u + v) = -(-4 + 1) = -(−3) = 3

Thus, h = t = 3.

The computed value for h is 3

Step 1: Understanding the hexagonal dipyramid:

A hexagonal dipyramid is a crystal form consisting of two hexagonal pyramids joined at their bases. It belongs to the hexagonal crystal system.

Step 2: Evaluating each option:

(A) A vertical six fold axis of symmetry:

• This is present in the hexagonal dipyramid
• However, this alone does NOT define the complete form
• The 6-fold axis is characteristic of all hexagonal forms, not specific to the dipyramid

(B) A horizontal mirror plane:

• This is present (perpendicular to the 6-fold axis)
• However, this alone does not define the dipyramid
• This could apply to other hexagonal forms

(C) Six vertical mirror planes at an angle of 30° with each other:

• Incorrect angle: vertical mirror planes in hexagonal system are at 60° to each other (360°/6 = 60°), not 30°

(D) A mirror plane that is perpendicular to the vertical six fold axis of symmetry:

• This is the horizontal mirror plane
• This element, combined with the 6-fold axis, specifically defines the dipyramid form
• This symmetry element creates the reflection that generates the upper and lower pyramids
• This is the defining characteristic that distinguishes the dipyramid from a single pyramid

Answer: (D) A mirror plane that is perpendicular to the vertical six fold axis of symmetry

Given:

• Miller Index: (326)

Concept:

Miller indices and Weiss parameters are reciprocally related.

If Miller indices are (hkl), then the Weiss parameters are:

$ $

Comparing with xa : xb : xc, we have:

• Coefficient of a: x = 2
• Coefficient of b: 3
• Coefficient of c: 1

Answer: x = 2