# Crystallography; from hcp to fcc

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In summary, the stacking sequence of atoms in fcc lattice can be altered to form the hexagonal close packed structure by sliding the third plane by the vector r = (c/2)[100] + (c/√3)[010] + (0)[001] over the second plane. The translation vector r is found by considering the relative position of one atom from hcp from an arbitrary origin point O defined using fcc lattice.
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## Homework Statement

Three close-packed planes of atoms are stacked to form fcc lattice. The stacking sequence of the three planes can be altered to form the hexagonal close packed structure by sliding the third plane by the vector r over the second. If the planes in the fcc structure are all (111) planes, what is the translation vector r in terms of unit vectors [100], [010], and [001] of the fcc lattice?

## The Attempt at a Solution

Unit vectors [100], [010], and [001] of the fcc lattice can be expressed as a, b, and c (any arbitrary origin point O would work).
The problem can be converted to finding relative position of one atom from hcp from an arbitrary origin point O defined using fcc lattice.

From the diagram of hcp lattice, it can be seen that in fcc, there is a shift of c/2 and c/√3 along x and y axis respectively. Therefore, the translation vector r = (c/2)[100] + (c/√3)[010]

## 1. What is crystallography?

Crystallography is the scientific study of crystals, which are solid materials with a highly ordered atomic structure. It involves the analysis and description of the arrangement of atoms within a crystal, as well as the physical and chemical properties that result from this arrangement.

## 2. What is the difference between hcp and fcc crystal structures?

HCP (hexagonal close-packed) and FCC (face-centered cubic) are two common crystal structures. The main difference between them lies in the arrangement of atoms. In HCP, the atoms are arranged in a hexagonal pattern, while in FCC, they are arranged in a cubic pattern with additional atoms in the center of each face. This results in different physical properties such as density, strength, and thermal conductivity.

## 3. How do you determine the crystal structure of a material?

The crystal structure of a material can be determined through a variety of techniques, including X-ray diffraction, electron microscopy, and neutron scattering. These methods involve analyzing the diffraction patterns or scattering of radiation by the atoms in the material, which can provide information about the arrangement of the atoms.

## 4. What are some real-life applications of crystallography?

Crystallography has numerous applications in various fields, such as materials science, geology, chemistry, and biology. It is used to study the structure and properties of materials, minerals, and biomolecules, as well as to develop new materials with specific properties. It is also crucial in drug discovery and development, as it allows scientists to determine the structures of proteins and other molecules and design more effective drugs.

## 5. How does crystallography contribute to our understanding of the world around us?

Crystallography plays a vital role in expanding our knowledge of the natural world. It helps us understand the structure and properties of minerals and materials, which are essential for various industries. It also allows us to gain insights into the structure of biomolecules and their functions, leading to advancements in medicine and biotechnology. Overall, crystallography helps us understand the fundamental building blocks of our world and how they interact to create the physical and chemical properties we observe.

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