Crystal Structure: Chemistry of Engineering Materials

• Lecture 9 focuses on the chemistry of engineering materials, specifically the basic concepts of crystal structure.

Crystal Structure

• Crystal structure is the ordered arrangement of atoms, ions, or molecules in crystalline materials, described within the field of crystallography.
• Crystallography is the experimental science dedicated to determining the arrangement of atoms in crystalline solids.
• Strontium titanate (SrTiO3) is an oxide of strontium and titanium.
• Brighter atoms are strontium, while the darker ones are titanium, in an atomic resolution image of strontium titanate.

Atoms, Ions, and Molecules

• An atom is the smallest unit of ordinary matter with the properties of a chemical element, and can be neutral or ionized.
• Solids, liquids, gases, and plasmas are composed of neutral or ionized items.
• An ion is an atom or molecule with a non-zero net electrical charge, has an imbalance between the number of electrons and protons
• A cation is a positively charged ion.
• An anion is a negatively charged ion.
• A molecule is an electrically neutral group of two or more atoms held together by chemical bonds.

Crystals and Crystalline Solids

• A crystal or crystalline solid is a material with constituents (atoms, molecules, or ions) arranged in a highly ordered microscopic structure.
• This structure forms a crystal lattice that extends in all directions.
• Ordered structures arise from the nature of constituent particles, creating symmetric patterns repeating along the principal directions in 3D space.

Crystal Definition

• A crystal is any solid material with component atoms arranged in a definite pattern and surface regularity reflecting internal symmetry.

Unit Cell

• The smallest group of particles in a material is a unit cell, constituting the repeating pattern of the structure.
• It fully defines the symmetry and structure of the entire crystal lattice.
• The entire crystal lattice is built up by repetitive translation of the unit cell along its principal axes with repeating patterns found at the points of the Bravais lattice.

Unit Cells and Basic Structures

• A crystalline lattice involves the regular arrangement of atoms within a crystalline solid
• Particles combine in a crystalline lattice to minimize their energy.
• A unit cell is the smallest divisible unit of a crystal.
• It reproduces the entire crystal lattice when repeated in three dimensions.
• Lattice points, represented by circles, are occupied by an atom, ion, or molecule inside cubic unit cells
• Each atom is identical to other atoms in the structure.
• Different colors are for aid visualization.

Coordination Number and Packing Efficiency

• The coordination number refers to the number of atoms directly in contact with each atom.
• It indicates the number of atoms with which a given atom can strongly interact.
• Packing efficiency is the percentage of the unit cell volume occupied by spheres that is related to high coordination numbers that create greater packing efficiency.

Simple Cubic Lattice

• In a simple cubic unit cell, an atom is located at each corner, touching atoms along each edge.
• One atom is contained per cell with 1/8 of each of eight atoms.
• The length of the simple cubic is l = 2r.
• The coordination number in a simple cubic structure is 6.
• The packing efficiency is 52%.

Body-Centered Cubic Lattice

• A body-centered cubic unit cell features one atom in the center and one at each corner.
• Atoms touch along a diagonal through the cube's center, not along the edge.
• Two Atoms per unit cell= (1/8 x 8) + 1 = 2

Face-Centered Cubic Lattice

• The face-centered cubic unit cell has one atom at each corner and one in the center of each face.
• Atoms touch along the diagonal face.
• The packing efficiency is 68%
• Packing efficiency is 74%.
• Four atoms per unit cell

Density and Crystal Structure

• Chromium crystallizes in a body-centered cubic unit cell.
• To calculate the density of solid crystalline chromium, given a chromium atom radius of 125 pm :
  • l = (4r)/√3
  • V = l^3
  • for the mass, m = 2 atoms x (1 mole Cr/ / 6.023 x 10^23) x (51.996 g / 1 mole Cr)
  • Density = m/V where ρ = 7.18 g/cm^3.

Closest-Packed Structures

• Another way to envision crystal structures is in terms of atoms stacking in layers.
• The simple cubic structure is one layer of atoms in a square pattern with the second layer aligned exactly on top of the atoms beneath
• Structure has a lot of empty space

Efficient Packing

• More efficient packing is achieved by offsetting the second layer by 1/2 atom so that the atoms sit in the indentations formed by the atoms in layer below.

Hexagonal Closest Packing

• The third layer is aligned with first layer
• ABAB (or ACAC) pattern
• Coordination # = 12
• Packing efficiency = 74%
• Unit cell is hexagonal

Cubic Closest Packing

• The third layer is offset from first layer
• ABCABC pattern
• Coordination # = 12
• Packing efficiency = 74%
• Structure identical to face-centered cubic unit cell

Crystal Systems

• The goal is to quantitatively describe crystal systems.
  • (a) Shape and Size of the Unit Cell (point symmetry)
  • (b) Location of the Lattice Points (translational symmetry)
• What will be done:
  • For (a) specify the Crystal System and Lattice Parameters.
  • For (b) define the "Bravais" Lattice.

Crystal Systems and Symmetry

• Unit cells need to be able to "stack" to fill all space.
• This puts restrictions on Unit Cell Shapes, for example, cubes work, but pentagons don't.
• Different types of unit cells are classified based on their level of symmetry.
• Symmetry is the set of mathematical rules describing an object's shape.
• A sphere is the only object with perfect symmetry, with infinite planes and rotational axes, always appearing the same regardless of rotation.

Crystal: Space Group

• A crystal is a periodic arrangement of repeating "motifs" (atoms, ions)
• Symmetry of motifs is the total set of symmetry operations allowed.
• A 90-degree rotation about the center or a linear shift (translation) results in the same pattern.
• The total set of symmetry operations applicable to a pattern is the pattern's symmetry and is the Space Group.
• Space Group describes crystal symmetry and its internal structure.

Translational Symmetry

• A Space Group includes Translational and Point Symmetries. Translations executable shifting movements along a straight line at a specified distance resulting in no change.
• Translational symmetries are macroscopically not visible since translation lengths are on the order of Å.

Point Symmetries

• Point Symmetries are macroscopically visible symmetry operations where at least one point remains where it was, it can be applied to the crystal.
• These operations include Reflection in a point (inversion): Center of Symmetry, Reflection in a plane or Mirror Symmetry, Rotation about an imaginary axis: Rotational Symmetry, and Rotation-and-after-it-inversion or Roto-inversion.
• Center of Symmetry is the item that consists of parts that are also found on the opposite side.
• Mirror Symmetry can be horizontal or vertical

Rotational Symmetry

• A point around which we rotate - symmetry axis
• Figure looks the same n times in a 360° rotation with n-fold symmetry

N-fold Roto-Inversion Symmetry

• The object has transformed into itself after a rotation of 90 degrees along the axis.
• Followed by the inversion with respect to a point on the axis

Multiple symmetries

• Objects can have more than one kind of symmetry.

Basic Symmetry Elements

• The Sets of Basic Symmetry Elements for Crystals
  • 1-fold rotation (rotation through 360 degrees); symbol: none
  • 2 - fold rotation (rotation through 180 degrees); symbol: 2
  • 3 - fold rotation (rotation through 120 degrees); symbol: 3
  • 4- fold rotation (rotation through 90 degrees); symbol: 4
  • 6-fold rotation (rotation through 60 degrees); symbol: 6
• Only above rotation axes can occur for crystals.
  • Mirror plane; symbol: m
  • Center of Symmetry: p
  • 4-fold roto-inversion axis - unique element!; symbol: 4*

Crystal Classes and JCPDS Card

• Combinations of point symmetries (Rotation, Reflection, and Roto-inversion) can be made, and which result in 32 unique possibilities.
• All crystals are classified in 32 Crystal Symmetry Classes according to their symmetry content
• A JCPDS Card and Quality is important.

There are six Crystal System

• The Crystal Systems are divided among the different Crystal Systems There are consist of: -The CUBIC (also called Isometric system) -The TETRAGONAL system -The HEXAGONAL system -The ORTHORHOMBIC system -The MONOCLINIC system -The TRICLINIC system
• Every Crystal System involves a number of Crystal Classes.

Crystallographic Axes

• By using crystallographic axes we can define six large groups or crystal systems that all crystal forms may be placed in
• Refer to the axes in the order a, b, c
• The point of intersection of the three axes is called the AXIAL CROSS Crystallographic axes

CUBIC (or ISOMETRIC) System -1

• The three crystallographic axes are all equal in length and intersect at right angles to each other.
• a = b = c α=β=y=90°

CUBIC (or ISOMETRIC)-II

• In If you glance on the Hexoctahedron, when compared to all the other crystal systems You will understand why crystal forms in the isometric system have the highest degree of SYMMETRY

TETRAGONAL System

• Three axes, all at right angles, two of which are equal in length (a and b)
• One (c) which is different in length (shorter or longer).
• A tetragonal prism is one of the 9 forms in this crystallographic system With 7 classes of symmetry
• a = b/c α=β=y=90° If c was equal in length to a or b, then we would be in the cubic system!

HEXAGONAL system

• Four axes! Three of the axes fall in the same plane and intersect at the axial cross at 120°. These 3 axes, labeled a1, a2, and a3, are the same length.
• The fourth axis, c, may be longer or shorter than the a axes set. The c axis also passes through the intersection of the a axes set at right angle to the plane formed by the a set.

ORTHOROMBIC System

• Three axes, all at right angles, all three have different length.
• Note: If any axis was of equal length to any other, then we would be in the tetragonal system

MONOCLINIC System

• Three axes, all unequal in length, two of which (a and c) intersect at an oblique angle (not 90 degrees), the third axis (b) is perpendicular to the other two axes.

TRICLINIC System

• The three axes are all unequal in length and intersect at three different angles (any angle but 90 degrees).

Comparing Crystal Systems

• Crystal systems can be compared based on their symmetry elements

Bravais Lattices

• By means of unit cells, we managed to reduce all possible crystal structures to a relatively small number of basic unit cell geometries.
• Lattices contain how atoms can be stacked together within a given unit cell.
• Lattice points are theoretical points are arranged periodically in 3-D space, rather than actual atoms
• Again there is a limited number of possibilities, referred to as Bravais lattice.
• The French scientist August Bravais, demonstrated in 1850 that only these 14 types of unit cells are compatible with the orderly arrangements of atoms found in crystals.
• These three-dimensional configurations of points used to describe the orderly arrangement of atoms in a crystal.
  • Each point represents one or more atoms in the actual crystal, and if the points are connected by lines, a crystal lattice is formed.

Unit Cell Properties

• Unit cell properties are as follows:
  • Triclinic with no symmetry.
  • Monoclinic with one 2-fold axis.
  • Orthorhombic with three perpendicular 2-folds.
  • Tetragonal with one 4-fold axis (parallel c).
  • Trigonal with one 3-fold axis.
  • Hexagonal with one 6-fold axis.
  • Cubic with four 3-folds along space diagonal

Simple Cells

• F face cell is very important because it is the pattern for cubic closest packing

Lattice Types

• Lattice types include Primitive, Face-centered, Body-centered, Base-Centered, Rhombohedral

Trigonal Cells

• The R cell is unique to hexagonal crystals with two points that divide between diagonals
• cell in thirds,Rhombohedron can be thought of as a cube distorted along one of its diagonals.

Packing Fraction

• Packing Fraction (defines how dense (closely packed) the atoms in a lattice area.

Bragg's Law and Crystal Lattices

• Lattice constant is equal to the change in lattice plane spacing which can be defined by Braggs Law.

Defects in Crystals

• A crystal defect is an imperfection in the regular geometrical arrangement of atoms in a crystalline solid.
  • Imperfections result from deformation, rapid cooling, or high-energy radiation.
  • Defects are located at points, along lines, or on whole surfaces, they influence mechanical, electrical, and optical behavior.

Weiss and Miller Indices

• The Weiss parameters,are the ancestors of the Miller indices.
• Miller Indices are a symbolic vector representation for the orientation of an atomic plane.

Weiss Index

• unit distance of scale in x – axis
• unit distance of scale in y – axis
• unit distance of scale in z – axis

Polymorphism

• Polymorphism is the ability of a solid material to exist in more than one crystal structure, found in crystalline materials like polymers, minerals, and metals, and is related to allotropy (chemical elements).
• Glycine exhibits polymorphism, forming monoclinic and hexagonal crystals.
• Calcite and aragonite are polymorphic forms of calcium carbonate.