Crystallography and X-Ray Diffraction

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Questions and Answers

What is the coordination number for an atom in a body centered cubic structure?

8 (correct)

How many total atoms are present in a body centered cubic unit cell?

2 (correct)

What is the atomic packing factor (APF) of a body centered cubic structure?

0.78

0.68 (correct)

0.74

0.52

In what stacking sequence do the layers of a face centered cubic structure occur?

ABC Signup and view all the answers

Who discovered X-rays and is noted for receiving the first Nobel Prize in Physics?

William Röntgen Signup and view all the answers

What principle relates to the diffraction of X-rays by crystals?

Bragg's Law Signup and view all the answers

Which of the following statements about the body centered cubic structure is incorrect?

Each atom has 12 nearest neighbors. Signup and view all the answers

Which of the following describes a crystalline solid?

Atoms in a repeating pattern Signup and view all the answers

Which element is an example of having a body centered cubic structure?

Iron Signup and view all the answers

What distinguishes crystalline solids from amorphous solids?

Solidification occurs gradually in crystalline solids. Signup and view all the answers

How many face atoms are present in a face centered cubic unit cell?

6 Signup and view all the answers

What additional atom exists in a body centered cubic structure aside from those at the corners?

Central atom Signup and view all the answers

In what year did Max Von Laue win the Nobel Prize for Physics?

1914 Signup and view all the answers

How many crystal lattice structures did Auguste Bravais identify?

14 Signup and view all the answers

What type of beam is NOT typically used in crystallography?

Infrared beam Signup and view all the answers

What is one of the main focuses of the science of crystallography?

The geometric form of crystalline solids Signup and view all the answers

What characterizes a single crystal?

Its atomic structure repeats periodically across its whole volume. Signup and view all the answers

What defines a polycrystal?

It is made up of many small single crystals called grains. Signup and view all the answers

Which statement describes an amorphous solid?

It is composed of randomly oriented atoms. Signup and view all the answers

What is the primary component of a crystal structure?

Space lattice plus basis. Signup and view all the answers

How is a space lattice defined?

A three-dimensional periodic arrangement of points in space. Signup and view all the answers

What do you call the atomic grouping associated with each lattice point?

Basis. Signup and view all the answers

What happens when a cell is defined in crystallography?

It is a finite representation of the infinite lattice. Signup and view all the answers

What distinguishes nano-crystalline polycrystals?

Grains that are < 10 nm in diameter. Signup and view all the answers

What is the co-ordination number of a corner atom in a face-centered cubic (FCC) structure?

12 Signup and view all the answers

How many total atoms are present in a face-centered cubic unit cell?

4 Signup and view all the answers

What is the atomic packing factor (APF) of a face-centered cubic structure?

0.74 Signup and view all the answers

In a hexagonal close packed (HCP) structure, what is the effective number of atoms per unit cell?

6 Signup and view all the answers

What is the ratio of the height of a hexagonal close packed unit cell (c) to its base edge length (a)?

√8/3 Signup and view all the answers

In a HCP structure, how is the nearest neighbor distance related to the base edge length?

2r = a Signup and view all the answers

What shape is used to determine the volume of the hexagonal close packed unit cell?

Hexagon Signup and view all the answers

What is the relationship between the atomic radius (r) and the edge length (a) in a face-centered cubic (FCC) structure?

r = a/√8 Signup and view all the answers

What is the first step in determining the Miller Indices for the intercepts 2, 1, 2?

Take the reciprocals of the intercepts Signup and view all the answers

Which of the following correctly describes the notation for Miller Indices that provides all possible integers?

Signup and view all the answers

What is the first step in determining the Miller Indices from intercepts 2, 1, and 2?

Take the reciprocals of the intercepts Signup and view all the answers

Which of the following correctly represents the Miller Indices for the reciprocals of the intercepts 2, 1, and 2?

(1, 2, 1) Signup and view all the answers

What does the notation <1,2,1> or [1,2,1] indicate in Miller Indices?

A family of possible integers Signup and view all the answers

What is the formula for the interplanar distance between adjacent parallel planes given in the content?

<code>d₁</code> = a/V(h² + k² + l²) Signup and view all the answers

In the context of Miller Indices, what does the term 'equivalent planes' refer to?

Planes with the same orientation Signup and view all the answers

What is the relationship described by the equation Cos²a' + cos²β' + cos²y' = 1?

It describes the direction cosines Signup and view all the answers

What does the term `d₁` represent in the context of the interplanar distance?

The distance from the origin to the nearest plane Signup and view all the answers

Which of the following is not a valid Miller Index representation for a plane?

{102} Signup and view all the answers

Study Notes

Crystallography: The Study of Crystalline Solids

Crystallography is the scientific study of the geometric form and physical properties of crystalline solids.
This involves using techniques such as X-ray diffraction, electron beam diffraction, and neutron beam diffraction.

Auguste Bravais and Crystal Symmetry

Auguste Bravais developed a mathematical theory of crystal symmetry based on the concept of the crystal lattice.
Bravais identified 14 unique types of crystal lattices.

Discovery of X-Rays and X-Ray Diffraction

Wilhelm Röntgen discovered X-rays in 1895.
He observed that these penetrating rays could fog photographic plates even when shielded.
Röntgen received the Nobel Prize in Physics in 1901 for his discovery.
In 1912, Max von Laue and his colleagues demonstrated that X-rays are diffracted by crystals.
Von Laue received the Nobel Prize in Physics in 1914 for his discovery.
Diffraction is a phenomenon where waves bend around obstacles or spread out when passing through a narrow opening.

Bragg's Law and X-Ray Spectroscopy

William Henry Bragg and his son William Lawrence Bragg conducted a series of experiments in 1913.
They developed the X-ray spectroscope, which allowed them to analyze the structure of crystals based on X-ray diffraction patterns.
Bragg's Law describes the relationship between the angle of incidence, wavelength of the X-rays, and the spacing between atomic planes in a crystal.
Both Braggs received the Nobel Prize in Physics in 1915 for their contributions to X-ray crystallography.

Matter States and Crystalline Structures

Solids can be classified as crystalline or amorphous.
Crystalline solids have a regular, repeating arrangement of atoms or molecules.
Amorphous solids lack a defined structure and their atoms are arranged randomly.

Types of Crystalline Solids

Crystalline solids can be further categorized as single crystals, polycrystals, or nanocrystalline.
Single crystals have a perfectly repeating structure throughout their volume.
Polycrystals consist of many small single crystals (grains) aggregated together.
Nanocrystalline materials have grains smaller than 10 nanometers.

Crystal Structures

A crystal is composed of a repeating three-dimensional pattern of atoms or molecules called a crystal structure.
Examples of crystal structures include:
- Simple cubic
- Body-centered cubic (BCC)
- Face-centered cubic (FCC)
- Hexagonal close-packed (HCP)
The arrangement of atoms within a crystal structure is crucial for its physical properties, such as hardness, conductivity, and melting point.

Key Crystallographic Concepts

Space Lattice: A three-dimensional arrangement of points representing the repeating unit of a crystal structure.
Unit Cell: The smallest repeating unit that contains all the structural information of the crystal lattice.
Lattice Parameters: The lengths of the unit cell edges and the angles between them.
Lattice Constant: The distance between two adjacent lattice points.
Atomic Packing Factor (APF): The ratio of the volume occupied by atoms within the unit cell to the total volume of the unit cell. A higher APF indicates a denser structure.
Coordination Number: The number of nearest neighbors surrounding a particular atom in a crystal structure.
Motif (Basis): An atom or group of atoms associated with each lattice point.

Body-Centered Cubic (BCC) Structure

In a BCC structure, atoms occupy all eight corners of the unit cell and one additional atom is located at the center of the cube.
Coordination number: 8
APF: 0.68
Examples: tungsten, sodium, iron, chromium.

Face-Centered Cubic (FCC) Structure

In an FCC structure, atoms occupy all eight corners of the unit cell and six additional atoms are located at the center of each face of the cube.
Coordination number: 12
APF: 0.74
Examples: copper, lead, aluminum, silver.

Hexagonal Close-Packed (HCP) Structure

The HCP structure has a hexagonal base and consists of three layers of atoms.
Coordination number: 12
The c/a ratio (height to base diameter) is √8/3.
Examples: magnesium, zinc, titanium.

Miller Indices

Miller indices are used to describe the orientation of crystal planes.
They are determined by taking the reciprocals of the intercepts of the plane with the crystallographic axes and then reducing them to the smallest integers.
The Miller indices for a plane are enclosed in parentheses, such as (1,2,1).
Angles and distances between planes can be calculated using Miller indices, which is essential for understanding crystal properties.

Interplanar Spacing

The interplanar distance, denoted by d₁, is the distance between two adjacent parallel crystal planes.
It can be calculated using the formula: d₁ = a/√h² + k² + I², where a is the lattice constant and h, k, and l are the Miller indices of the plane.
Interplanar spacing plays a significant role in X-ray diffraction, as it determines the angles at which diffracted beams occur.

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Description

Explore the fascinating field of crystallography, focusing on the study of crystalline solids and their geometric forms. Learn about key figures like Auguste Bravais, who developed the theory of crystal symmetry, and Wilhelm Röntgen, the discoverer of X-rays. Dive into the principles and techniques that make X-ray diffraction a vital tool in understanding crystal structures.