[D2] CRYSTALLOGRAPHY AND METALLOGRAPHY MERGED

Comprehensive flashcards covering physical metallurgy, crystallography, crystal structures, Miller indices, X-ray diffraction, crystalline imperfections, and metallography.

What is physical metallurgy concerned with?

Physical metallurgy is concerned with the physical and mechanical properties of metals as affected by composition, structure, processing, and environmental conditions.

What two force components determine the net interatomic force $F_{\text{net}}$ between bonding atoms?

The attractive force ($F_A$), which depends on the bonding type, and the repulsive force ($F_R$), which results from the overlapping of orbital clouds.

What is the formula for the net force $F_{\text{net}}$ between two atoms?

$$F_{\text{net}} = F_A + F_R$$

What condition occurs regarding atomic forces and energy at the equilibrium bonding distance?

At the equilibrium distance, the attractive force equals the repulsive force ($F_{\text{repulsion}} = F_{\text{attraction}}$) and the potential energy is at a minimum ($E_0$).

How does the depth/steepness of a potential energy curve relate to material stiffness?

Steeper potential energy wells correspond to relatively stiff materials with higher elastic moduli, whereas shallow wells correspond to flexible materials.

What is the definition of crystallography?

Crystallography deals with the arrangement of atoms within the structures of materials and how this arrangement relates to the behavior of materials.

How did Glusker & Trueblood define a crystal?

A crystal is a solid having a regularly repeating arrangement of atoms.

How did the IUCr Commission on Aperiodic Crystals define a crystal?

A crystal is any solid with an essentially discrete diffraction diagram.

What structural difference distinguishes crystalline solids from amorphous solids?

Crystalline solids possess long-range order with a repeating atomic arrangement, whereas amorphous solids possess only short-range order with no long-range ordering.

What defines a polycrystalline solid?

A polycrystalline solid consists of many small single crystals or grains separated by grain boundaries (crystalline discontinuities).

What two primary factors affect the crystallinity of a material?

1. Size of the building block (smaller size crystallizes easier).
2. Rate of cooling (slower cooling rate crystallizes easier).

What cooling rate is used in splat cooling, and what material type is produced?

A high cooling rate of $10^6\,^{\circ}\text{C}/s$, which produces metallic glass (an amorphous structure free of crystalline defects).

In crystallography, what is a lattice?

A three-dimensional array of points coinciding with atom positions.

What is a unit cell?

The smallest unit that completely describes the crystal pattern.

What are the unit cell parameters?

Interaxial angles (angles between coordinate axes) and lattice constants (edge lengths along the major axes).

What are the lattice constant and interaxial angle relationships for the Triclinic crystal system?

$a \neq b \neq c$ and $\alpha \neq \beta \neq \gamma \neq 90^{\circ}$

What are the lattice constant and interaxial angle relationships for the Monoclinic crystal system?

$a \neq b \neq c$ and $\alpha = \gamma = 90^{\circ} \neq \beta$

What are the lattice constant and interaxial angle relationships for the Orthorhombic crystal system?

$a \neq b \neq c$ and $\alpha = \beta = \gamma = 90^{\circ}$

What are the lattice constant and interaxial angle relationships for the Tetragonal crystal system?

$a = b \neq c$ and $\alpha = \beta = \gamma = 90^{\circ}$

What are the lattice constant and interaxial angle relationships for the Hexagonal crystal system?

$a = b \neq c$ and $\alpha = \beta = 90^{\circ}, \gamma = 120^{\circ}$

What are the lattice constant and interaxial angle relationships for the Rhombohedral crystal system?

$a = b = c$ and $\alpha = \beta = \gamma \neq 90^{\circ}$

What are the lattice constant and interaxial angle relationships for the Cubic/Isometric crystal system?

$a = b = c$ and $\alpha = \beta = \gamma = 90^{\circ}$

How many Bravais lattices exist by combining the 4 basic unit cell types with the 7 crystal systems?

14 Bravais lattices (possible lattice variations).

What are the four basic unit cell types?

Primitive ($P$), Body-centred ($I$), Face-centred ($F$), and Base-centred ($C$).

In a Simple Cubic (SC) unit cell, where are the atoms situated and how do they touch?

Atoms are situated at the corners of the unit cell and touch each other along the unit cell edges.

What is the coordination number of a Simple Cubic (SC) crystal structure?

6

How many equivalent atoms are contained within a Simple Cubic (SC) unit cell?

1 atom (calculated as $\frac{1}{8} \times 8 = 1$).

What is the formula relating lattice constant $a$ to atomic radius $R$ for a Simple Cubic (SC) structure?

$$a = 2R$$

What is the Atomic Packing Factor (APF) of a Simple Cubic (SC) unit cell?

0.52

In a Body-Centered Cubic (BCC) structure, where are the atoms located?

Atoms are situated at the eight corners of the unit cell with one atom in the center (body) of the cube touching all eight corner atoms.

What is the coordination number of a Body-Centered Cubic (BCC) crystal structure?

8

How many equivalent atoms are in a Body-Centered Cubic (BCC) unit cell?

2 atoms (calculated as $8 \times \frac{1}{8} + 1 = 2$).

What is the formula relating lattice constant $a$ to atomic radius $R$ in a BCC structure?

$$a = \frac{4R}{\sqrt{3}}$$

What is the Atomic Packing Factor (APF) of a Body-Centered Cubic (BCC) unit cell?

0.68

In a Face-Centered Cubic (FCC) unit cell, how are corner and face atoms shared?

Each corner atom is shared among 8 unit cells, while each face atom is shared by 2 unit cells.

What is the coordination number of a Face-Centered Cubic (FCC) structure?

12

How many equivalent atoms belong to a Face-Centered Cubic (FCC) unit cell?

4 atoms (calculated as $8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 4$).

What is the formula relating lattice constant $a$ to atomic radius $R$ in an FCC structure?

$$a = \frac{4R}{\sqrt{2}}$$

What is the Atomic Packing Factor (APF) of a Face-Centered Cubic (FCC) unit cell?

0.74

What is the spatial arrangement of atoms in a Hexagonal Close-Packed (HCP) structure?

It has 2 basal planes (hexagons) with atoms at the corners and center, plus a middle plane containing 3 atoms located between the basal planes.

What are the coordination number and atoms per unit cell for an HCP structure?

Coordination number = 12; Atoms per cell = 6.

What is the Atomic Packing Factor (APF) of an HCP structure?

0.74

Which primary metals exhibit a Body-Centered Cubic (BCC) structure?

Iron (Fe), Chromium (Cr), Molybdenum (Mo), Vanadium (V), and Tungsten (W).

Which primary metals exhibit a Face-Centered Cubic (FCC) structure?

Aluminum (Al), Gold (Au), Lead (Pb), Nickel (Ni), Silver (Ag), Copper (Cu), and Platinum (Pt).

Which primary metals exhibit a Hexagonal Close-Packed (HCP) structure?

Cadmium (Cd), Cobalt (Co), Titanium (Ti), Zinc (Zn), and Magnesium (Mg).

What is the definition of Atomic Packing Factor (APF)?

The fraction of space filled by spherical atoms, defined as the ratio of the volume occupied by atoms in a unit cell to the total volume of the unit cell.

What is the formula for theoretical density $\rho$ of a crystalline material?

$$\rho = \frac{nA}{V_C N_A}$$

What do the variables $n$, $A$, $V_C$, and $N_A$ represent in the theoretical density formula?

$n$ = number of atoms per unit cell $A$ = atomic weight $V_C$ = volume of the unit cell $N_A$ = Avogadro's number

What is the unit cell volume $V_C$ expression for FCC copper with atomic radius $R = 0.128\,\text{nm}$?

$$V_C = a^3 = \left(\frac{4R}{\sqrt{2}}\right)^3$$

What are Miller Indices?

A symbolic notation used for expressing locations, directions, and planes in crystal lattice systems.

How are crystallographic point coordinates specified?

As fractional multiples of the unit cell edge lengths along the $x$, $y$, and $z$ axes without brackets (e.g., $1, \frac{1}{2}, 0$).

What are the steps to index a crystallographic direction vector?

1. Identify lead and starting point coordinates.
2. Subtract starting point from lead point: $(x,y,z)_{\text{lead}} - (x,y,z)_{\text{start}}$.
3. Clear fractions to obtain lowest integers.
4. Write in square brackets $[uvw]$ without commas.
5. Overbar negative integers.

How are negative integer values represented in Miller indices?

By placing a bar over the integer (e.g., $\bar{1}$).

What formula gives the cosine of angle $\theta$ between two directions $[uvw]$ and $[u'v'w']$?

$$\cos(\theta) = \frac{uu' + vv' + ww'}{\sqrt{u^2 + v^2 + w^2}\sqrt{u'^2 + v'^2 + w'^2}}$$

What are the steps to index a crystallographic plane?

1. Identify the intercepts of the plane along axes $x, y, z$ (use $\infty$ if parallel, shift origin if passing through origin).
2. Take reciprocals of intercepts.
3. Clear fractions to lowest integers.
4. Enclose in parentheses $(hkl)$.
5. Overbar negative values.

What is the mathematical condition for direction $[uvw]$ to be perpendicular to direction $[u'v'w']$?

$$uu' + vv' + ww' = 0$$

What is the mathematical condition for direction $[uvw]$ to be parallel to plane $(hkl)$?

$$hu + kv + lw = 0$$

What 4-axis coordinate notation is used for directions and planes in hexagonal crystals?

$[uvtw]$ for directions and $(hkil)$ for planes.

In hexagonal Miller-Bravais plane notation $(hkil)$, how is index $i$ calculated?

$$i = -(h + k)$$

What is the formula for Linear Density (LD)?

$$\text{LD} = \frac{\text{number of atoms entered on direction vector}}{\text{length of direction vector}}$$

What is the linear density (LD) of an FCC metal in the $[110]$ direction?

$$\text{LD} = \frac{1}{2R}$$

What is a family of directions, denoted as $\langle uvw \rangle$?

A set of crystallographic directions that have equivalent atomic spacing and identical linear densities.

What is the formula for Planar Density (PD)?

$$\text{PD} = \frac{\text{number of atoms centered on the plane}}{\text{area of plane}}$$

What is the planar density (PD) of an FCC structure on the $(100)$ plane?

$$\text{PD} = \frac{1}{4R^2}$$

What is a family of planes, denoted as $\{hkl\}$?

Planes having identical planar densities and equivalent crystallographic arrangements.

What is polymorphism?

The capability of a solid material to exist in more than one crystal structure.

What is the difference between isotropy and anisotropy?

Isotropy is when material properties are independent of crystallographic direction; anisotropy is when properties vary along different directions.

What is X-ray diffraction (XRD)?

The chief analytical technique for investigating the atomic and crystal structure of minerals and materials using X-rays.

What physical process produces diffraction in XRD?

Constructive interference of scattered X-ray wavelengths from periodically arranged atomic scattering centers.

What is Bragg's Law equation?

$$2d \sin(\theta) = n\lambda$$

In Bragg's Law $2d \sin(\theta) = n\lambda$, what do $d$, $\theta$, and $\lambda$ represent?

$d$ = interplanar spacing between crystal planes $\theta$ = angle of incidence / Bragg angle $\lambda$ = wavelength of the incident X-ray beam

Why must the extra path length traveled by scattered X-rays equal $n\lambda$ for diffraction to occur?

To ensure that the diffracted waves remain in phase and reinforce each other constructively.

What is the interplanar spacing $d_{hkl}$ formula for a cubic crystal lattice?

$$d_{hkl} = \frac{a_0}{\sqrt{h^2 + k^2 + l^2}}$$

What planes reflect X-rays in a Simple Cubic (SC) lattice?

All planes reflect (e.g., $(100)$, $(111)$, $(123)$).

What condition must be met for diffraction reflections in a Body-Centered Cubic (BCC) lattice?

The sum of Miller indices $(h + k + l)$ must be an even integer (e.g., $(110)$, $(200)$, $(220)$).

What condition must be met for diffraction reflections in a Face-Centered Cubic (FCC) lattice?

The Miller indices $h, k, l$ must be all odd or all even (e.g., $(111)$, $(200)$, $(220)$).

How are crystalline imperfections categorized by geometry?

1. Point defects (0-dimensional)
2. Line defects (1-dimensional)
3. Planar defects (2-dimensional)
4. Volume/Bulk defects (3-dimensional)

What are the main types of point defects?

Vacancies, Self-interstitials, Schottky imperfections, Frenkel defects, and Impurity defects.

What formula calculates the equilibrium number of vacancies $N_V$ at temperature $T$?

$$N_V = N \exp\left(-\frac{Q_v}{kT}\right)$$

What is a Schottky imperfection?

A point defect in ionic compounds where a cation-anion pair is missing from the lattice.

What is a Frenkel defect?

A point defect in ionic crystals where a cation is displaced from a lattice site into an interstitial site.

What are the roles of solute and solvent in a solid solution?

Solute is the element present in minor concentration (impurity); Solvent is the host element present in greatest amount.

What are the two modes of solid solutions?

1. Substitutional solid solutions (solute replaces host atoms).
2. Interstitial solid solutions (solute fits into spaces between host atoms).

What are the four Hume-Rothery rules for complete substitutional solid solubility?

1. Atomic Size Factor: atomic radius difference $< \pm 15\%$
2. Crystal Structure: solute and solvent crystal structures must match
3. Electronegativity: electronegativity difference close to 0
4. Valences: higher solute valency increases solubility

What two factors govern the Hume-Rothery rules for interstitial solid solutions?

1. Atomic Size Factor: interstitial atom diameter must be substantially smaller than host atoms.
2. Electronegativity: solute and solvent electronegativities should be comparable/close.

What two parameters define a dislocation line defect?

The Dislocation Line and the Burgers Vector ($\mathbf{b}$).

What is the geometric relationship between the Burgers vector and dislocation line in an edge dislocation?

The Burgers vector is perpendicular to the dislocation line ($\mathbf{b} \perp \text{dislocation line}$).

What is the geometric relationship between the Burgers vector and dislocation line in a screw dislocation?

The Burgers vector is parallel to the dislocation line ($\mathbf{b} \parallel \text{dislocation line}$).

What is a mixed dislocation?

A dislocation that exhibits characteristics of both edge and screw dislocations.

What are six examples of planar defects?

1. External surfaces
2. Grain boundaries
3. Twin boundaries
4. Stacking faults
5. Phase boundaries
6. Ferromagnetic domain walls

In which crystal structures are annealing twins and mechanical twins most prevalent?

Annealing twins are most prevalent in FCC metals; Mechanical twins are most prevalent in HCP and BCC metals.

What is a stacking fault?

An interruption in the normal stacking sequence of close-packed atomic planes.

What are four examples of volume/bulk defects?

Pores, Cracks, Foreign inclusions, and Other phases.

What is metallography?

The scientific discipline of examining and determining the constitution and underlying structure of constituents in metals, alloys, and materials.

What are the five steps for metallographic specimen preparation according to ASTM E3?

1. Cutting and Sectioning
2. Rough Grinding / Planar Grinding
3. Mounting and Fine Grinding
4. Polishing (Rough & Final)
5. Etching / Contrasting

What distinct structural features are evaluated using transverse versus longitudinal sections in metallography?

Transverse sections evaluate structure variation from center to surface, coating thickness, decarburization, and depth of corrosion. Longitudinal sections evaluate inclusion content, plastic deformation (grain distortion), and microstructural banding.

When should cold castable mounting be selected over hot compression mounting?

When the specimen is too large for a mounting press or when mounting press heat would induce microstructural changes in the sample.

What grit paper progression is used for planar and fine grinding of metallographic specimens?

Planar grinding starts with P120 grit SiC, followed by fine grinding steps with P220, P500, and P1200 grit SiC papers.

What is the composition and etching function of Nital etchant?

Composition: $1\text{--}5\,\text{mL}\,\text{HNO}_3$ in $100\,\text{mL}$ ethanol ($95\%$). Function: Reveals ferrite grain boundaries and attacks pearlite in carbon steels.

What is the composition and etching function of Picral etchant?

Composition: $4\,\text{g}$ picric acid in $100\,\text{mL}$ ethanol ($95\%$). Function: Attacks ferrite-carbide boundaries, making it ideal for revealing bainite and martensite.