Materials Science & Engineering Series Part 2: Crystal Structures, Defects & Diffusion

Crystal Structures & Crystallography

If the atomic scale (Part 1) is the alphabet of materials science, then crystal structure is the grammar — the rules for how atoms arrange themselves into ordered patterns that determine mechanical strength, electrical conductivity, and thermal behavior. The vast majority of engineering metals, ceramics, and semiconductors are crystalline: their atoms are arranged in a periodic, repeating pattern that extends in three dimensions.

Analogy — The Tile Floor: Imagine tiling a floor. You can use squares, hexagons, or triangles — each shape tiles the plane differently, leaving different amounts of empty space. Atoms do the same thing in 3D: they "tile" space in different crystal structures, and the choice of tiling determines the material's properties.

import numpy as np

# Crystal structure comparison calculator
structures = {
    'Simple Cubic': {
        'atoms_per_cell': 1,
        'coordination': 6,
        'APF': np.pi / 6,  # 0.524
        'relation': 'a = 2r',
    },
    'BCC': {
        'atoms_per_cell': 2,
        'coordination': 8,
        'APF': np.pi * np.sqrt(3) / 8,  # 0.680
        'relation': 'a = 4r/√3',
    },
    'FCC': {
        'atoms_per_cell': 4,
        'coordination': 12,
        'APF': np.pi * np.sqrt(2) / 6,  # 0.740
        'relation': 'a = 2√2·r',
    },
    'HCP': {
        'atoms_per_cell': 6,
        'coordination': 12,
        'APF': np.pi * np.sqrt(2) / 6,  # 0.740
        'relation': 'a = 2r, c = 1.633a (ideal)',
    },
}

print(f"{'Structure':<16s} {'Atoms/cell':<12s} {'CN':<6s} {'APF':<8s} {'Lattice relation'}")
print("=" * 65)
for name, props in structures.items():
    print(f"{name:<16s} {props['atoms_per_cell']:<12d} {props['coordination']:<6d} "
          f"{props['APF']:.3f}   {props['relation']}")

Packing Factor & Unit Cells

The Atomic Packing Factor (APF) measures how efficiently atoms fill space — the ratio of total atom volume in a unit cell to the cell's volume. Higher APF generally means higher density, higher elastic modulus, and different deformation behavior.

import numpy as np

# Calculate theoretical density from crystal structure
def calculate_density(n_atoms, atomic_mass_amu, lattice_param_nm):
    """Calculate theoretical density in g/cm³."""
    Na = 6.022e23
    a_cm = lattice_param_nm * 1e-7
    mass_per_cell = n_atoms * atomic_mass_amu / Na
    volume = a_cm**3
    return mass_per_cell / volume

metals = {
    'Copper (FCC)':   {'n': 4, 'M': 63.55, 'a': 0.3615, 'measured': 8.96},
    'Iron-α (BCC)':   {'n': 2, 'M': 55.85, 'a': 0.2866, 'measured': 7.87},
    'Aluminum (FCC)': {'n': 4, 'M': 26.98, 'a': 0.4049, 'measured': 2.70},
    'Tungsten (BCC)': {'n': 2, 'M': 183.84,'a': 0.3165, 'measured': 19.3},
}

print(f"{'Metal':<20s} {'Calculated ρ':<14s} {'Measured ρ':<12s} {'Error %'}")
print("=" * 55)
for name, props in metals.items():
    rho_calc = calculate_density(props['n'], props['M'], props['a'])
    error = abs(rho_calc - props['measured']) / props['measured'] * 100
    print(f"{name:<20s} {rho_calc:<14.2f} {props['measured']:<12.2f} {error:.1f}%")

Miller Indices & Slip Systems

Miller indices provide standardized notation for describing directions and planes in crystal lattices. They are essential for understanding how crystals deform, where X-ray diffraction peaks appear, and how to specify crystal orientations for single-crystal components like turbine blades and semiconductor wafers.

To find Miller indices of a plane: (1) find intersections with crystallographic axes, (2) take reciprocals, (3) reduce to smallest integers. A plane intercepting at $a, \infty, \infty$ gives reciprocals (1, 0, 0) → the (100) plane.

Crystal Defects

No real crystal is perfect. All engineering materials contain defects — deviations from the perfect periodic arrangement. Far from being unwanted flaws, defects are often deliberately introduced. Steel is strong because of its defects, not despite them. The entire field of metallurgy is essentially the science of controlling defect populations.

Analogy — The Library: Imagine a library where all books are perfectly ordered. A point defect is a missing book (vacancy) or a wrong book (substitutional impurity). A line defect is an extra half-shelf inserted partway (dislocation). A planar defect is where two sections have different cataloging systems (grain boundary).

import numpy as np
import matplotlib.pyplot as plt

# Vacancy concentration vs temperature
Qv_Cu = 0.9   # Vacancy formation energy for copper (eV)
kB = 8.617e-5  # Boltzmann constant (eV/K)

T = np.linspace(300, 1356, 500)
nv_fraction = np.exp(-Qv_Cu / (kB * T))

fig, ax = plt.subplots(figsize=(8, 5))
ax.semilogy(T, nv_fraction, 'b-', linewidth=2)
ax.set_xlabel('Temperature (K)')
ax.set_ylabel('Vacancy Fraction (n_v / N)')
ax.set_title('Equilibrium Vacancy Concentration in Copper')
ax.axvline(x=1356, color='red', linestyle='--', alpha=0.7, label='Melting point')

for T_mark in [300, 800, 1356]:
    frac = np.exp(-Qv_Cu / (kB * T_mark))
    ax.plot(T_mark, frac, 'ro', markersize=8)
    ax.annotate(f'{T_mark}K: {frac:.2e}', xy=(T_mark, frac),
                xytext=(T_mark+50, frac*5), fontsize=9)

ax.legend()
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

Line Defects & Dislocations

Dislocations are the most important defect type for understanding plastic deformation. A dislocation is a line defect — an extra half-plane of atoms inserted into the crystal. When stress is applied, the dislocation moves like a wrinkle in a carpet, allowing atoms to shift positions one row at a time.

Analogy — Moving a Heavy Rug: Trying to slide a heavy rug across a floor all at once requires enormous friction. But push a wrinkle from one end to the other, and the rug moves with very little force. The wrinkle is a dislocation — it propagates through the crystal while shifting the rug by one wrinkle-width (one Burgers vector).

Strengthening Mechanisms — all impede dislocation motion:

• Work hardening: Dislocations tangle and block each other. Why bending a paperclip makes it harder
• Solid solution: Foreign atoms distort lattice and pin dislocations. Brass (Cu-Zn) > pure Cu
• Precipitation hardening: Precipitate particles block dislocations. Basis of aircraft Al alloys
• Hall-Petch: Grain boundaries halt dislocations. $\sigma_y = \sigma_0 + k/\sqrt{d}$

Planar Defects & Grain Boundaries

Most engineering metals are polycrystalline — many small crystals (grains) with different orientations, connected at grain boundaries. This thin disorder zone (1-2 atoms wide) has higher energy than the bulk and profoundly influences mechanical properties via the Hall-Petch relationship.

Phase Diagrams

A phase diagram maps which phase(s) are stable at each combination of temperature and composition — arguably the single most useful tool in materials science. It tells you what to expect when you heat, cool, or alloy a material.

Ternary Phase Diagrams

With three components, the composition space becomes a Gibbs triangle with temperature as the vertical axis. Isothermal sections (horizontal slices) are most commonly used. Ternary diagrams are critical for designing stainless steels (Fe-Cr-Ni), brazing alloys (Ag-Cu-Zn), and superalloys (Ni-Al-Cr). The same phase rule applies: $F = C - P + 2$.

Lever Rule & Phase Fractions

The lever rule determines relative amounts of each phase in a two-phase region using mass balance:

import numpy as np

# Lever Rule Calculator
def lever_rule(C0, C_alpha, C_liquid):
    """Calculate phase fractions using the lever rule."""
    W_alpha = (C_liquid - C0) / (C_liquid - C_alpha)
    W_liquid = (C0 - C_alpha) / (C_liquid - C_alpha)
    return W_alpha, W_liquid

# Example: Cu-Ni system at 1250°C, 40 wt% Ni alloy
C0 = 40        # Overall composition (wt% Ni)
C_alpha = 46   # Solid phase composition
C_liquid = 35  # Liquid phase composition

W_alpha, W_liquid = lever_rule(C0, C_alpha, C_liquid)

print("Lever Rule: Cu-Ni System at 1250°C")
print("=" * 45)
print(f"Overall composition:  {C0} wt% Ni")
print(f"Solid composition:    {C_alpha} wt% Ni")
print(f"Liquid composition:   {C_liquid} wt% Ni")
print(f"\nFraction solid (α):   {W_alpha:.3f} ({W_alpha*100:.1f}%)")
print(f"Fraction liquid (L):  {W_liquid:.3f} ({W_liquid*100:.1f}%)")
print(f"Sum check: {W_alpha + W_liquid:.3f}")

Diffusion & Solidification

Diffusion — atom transport driven by concentration gradients — is the rate-controlling step in carburizing steel, sintering ceramics, semiconductor doping, and metal oxidation. Without diffusion, heat treatment would be pointless.

Analogy — The Perfume Bottle: Open perfume in a corner of a room — the scent spreads from high to low concentration. In solids, atoms do the same but much slower. At room temperature, iron atoms in steel diffuse ~1 nm/year. At 900°C, ~1 mm/hour. Temperature matters enormously.

Fick's First & Second Laws

Fick's First Law (steady-state): $J = -D \frac{dC}{dx}$ — flux proportional to concentration gradient.

Fick's Second Law (transient): $\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}$ — how concentration profile evolves over time.

For a semi-infinite solid with constant surface concentration: $\frac{C(x,t) - C_s}{C_0 - C_s} = \text{erf}\left(\frac{x}{2\sqrt{Dt}}\right)$

import numpy as np
import matplotlib.pyplot as plt
from scipy.special import erf

# Carbon diffusion during carburizing
D = 1.2e-11     # D for C in gamma-Fe at 925°C (m²/s)
C_s = 0.8       # Surface concentration (wt%)
C_0 = 0.2       # Initial concentration (wt%)

x = np.linspace(0, 3e-3, 500)  # Depth (m)
times = [1, 3, 6, 12]           # Hours

fig, ax = plt.subplots(figsize=(8, 5))
colors = ['#132440', '#16476A', '#3B9797', '#BF092F']

for t_hr, color in zip(times, colors):
    t = t_hr * 3600
    C = C_s - (C_s - C_0) * erf(x / (2 * np.sqrt(D * t)))
    ax.plot(x * 1000, C, color=color, linewidth=2, label=f't = {t_hr} hr')

ax.axhline(y=0.4, color='gray', linestyle='--', alpha=0.5, label='Target 0.4% C')
ax.set_xlabel('Depth from Surface (mm)')
ax.set_ylabel('Carbon Concentration (wt%)')
ax.set_title('Carburizing Profiles at 925°C')
ax.legend()
ax.set_ylim(0.15, 0.85)
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

Nucleation & Solidification Theory

When liquid metal cools below its melting point, it doesn't solidify instantly — it must overcome an energy barrier to form the first tiny crystals (nuclei). Nucleation is followed by growth, and their interplay determines final grain size and microstructure.

Solidification microstructure depends on cooling rate: slow cooling (sand casting) → large grains; rapid cooling (die casting, laser melting) → fine grains or amorphous structures; directional solidification → columnar or single crystals for turbine blades.

Exercises & Practice Problems

Conclusion & Next Steps

We've moved from individual atoms to organized assemblies. Crystal structure (FCC, BCC, HCP) determines available slip systems and ductility. Defects — vacancies, dislocations, grain boundaries — are deliberately manipulated to strengthen materials. Phase diagrams map equilibrium phases at any temperature/composition, while diffusion kinetics determine how fast equilibrium is reached. Solidification theory connects processing to microstructure.

These concepts bridge atomic-scale quantum mechanics (Part 1) with the specific material classes we'll study next.