Chapter 1: Crystal Structure of Ceramics

1.1 Overview of Ceramic Crystal Structures

Ceramic materials are compounds of metallic elements with non-metallic elements (primarily oxygen, nitrogen, and carbon), and their properties are determined by crystal structure and chemical bonding. In this chapter, we will learn the major crystal structures of ceramics and practice structural analysis using Python tools (Pymatgen).

Classification of Ceramic Structures

Ceramic crystal structures are broadly classified into two categories based on bonding nature:

1. Ionic bonding ceramics: NaCl, MgO, Al₂O₃, etc. (dominated by electrostatic attraction)
2. Covalent bonding ceramics: SiC, Si₃N₄, AlN, etc. (directional covalent bonds)

1.2 Ionic Bonding Ceramics

1.2.1 NaCl-Type Structure

The NaCl-type (rock salt) structure is the most fundamental ionic crystal structure. Cations and anions each form face-centered cubic (FCC) lattices that interpenetrate each other. Each ion is surrounded by six ions of opposite charge (coordination number CN = 6).

Representative materials:

• MgO (magnesium oxide): Refractory materials, high-temperature insulators (melting point 2852°C)
• CaO (calcium oxide): Cement raw material
• NiO (nickel oxide): Battery cathode material

Pymatgen Implementation: Generation and Visualization of NaCl Structure

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

# ===================================
# Example 1: NaCl-type structure generation and basic analysis
# ===================================

from pymatgen.core import Structure, Lattice
import matplotlib.pyplot as plt
import numpy as np

def create_nacl_structure(a=4.2):
    """
    Function to generate NaCl-type structure

    Parameters:
    -----------
    a : float
        Lattice parameter [Å] (default: 4.2Å for MgO)

    Returns:
    --------
    structure : pymatgen.core.Structure
        NaCl-type crystal structure
    """
    # Define FCC lattice
    lattice = Lattice.cubic(a)

    # Define atomic positions (fractional coordinates)
    # Mg2+: (0,0,0), (0.5,0.5,0), (0.5,0,0.5), (0,0.5,0.5) - FCC
    # O2-: (0.5,0,0), (0,0.5,0), (0,0,0.5), (0.5,0.5,0.5) - FCC shifted
    species = ["Mg", "Mg", "Mg", "Mg", "O", "O", "O", "O"]
    coords = [
        [0, 0, 0], [0.5, 0.5, 0], [0.5, 0, 0.5], [0, 0.5, 0.5],  # Mg
        [0.5, 0, 0], [0, 0.5, 0], [0, 0, 0.5], [0.5, 0.5, 0.5]   # O
    ]

    structure = Structure(lattice, species, coords)
    return structure

# Generate MgO structure
mgo = create_nacl_structure(a=4.211)  # Experimental lattice constant of MgO

print("=== MgO (NaCl Type) Structure Information ===")
print(f"Chemical formula: {mgo.composition.reduced_formula}")
print(f"Lattice parameter: a = {mgo.lattice.a:.3f} Å")
print(f"Unit cell volume: {mgo.volume:.2f} ų")
print(f"Density: {mgo.density:.2f} g/cm³")
print(f"Number of atoms per unit cell: {len(mgo)}")

# Calculate coordination number
mg_site = mgo[0]  # First Mg atom
neighbors = mgo.get_neighbors(mg_site, r=3.0)
print(f"\nNumber of nearest neighbor O atoms around Mg: {len(neighbors)}")
print(f"Mg-O distance: {neighbors[0][1]:.3f} Å")

# Expected output:
# === MgO (NaCl Type) Structure Information ===
# Chemical formula: MgO
# Lattice parameter: a = 4.211 Å
# Unit cell volume: 74.68 ų
# Density: 3.58 g/cm³
# Number of atoms per unit cell: 8
#
# Number of nearest neighbor O atoms around Mg: 6
# Mg-O distance: 2.106 Å

1.2.2 Pauling's Rules

Pauling's rules are five empirical rules for predicting stable structures of ionic crystals. The most important are the first rule and second rule:

Python Implementation: Calculation of Coordination Number and Radius Ratio

# ===================================
# Example 2: Coordination number prediction using Pauling's rules
# ===================================

def predict_coordination_number(r_cation, r_anion):
    """
    Function to predict coordination number from radius ratio

    Parameters:
    -----------
    r_cation : float
        Cation radius [Å]
    r_anion : float
        Anion radius [Å]

    Returns:
    --------
    cn : int
        Predicted coordination number
    geometry : str
        Coordination polyhedron shape
    """
    ratio = r_cation / r_anion

    if ratio < 0.155:
        return 2, "Linear"
    elif ratio < 0.225:
        return 3, "Triangular planar"
    elif ratio < 0.414:
        return 4, "Tetrahedral"
    elif ratio < 0.732:
        return 6, "Octahedral"
    else:
        return 8, "Cubic"

# Ionic radii data (Shannon radii, 6-coordinate)
ionic_radii = {
    "Mg2+": 0.72,
    "Ca2+": 1.00,
    "Na+": 1.02,
    "Cs+": 1.67,
    "O2-": 1.40,
    "Cl-": 1.81
}

# Predict NaCl-type structures
print("=== Coordination Number Prediction ===\n")

materials = [
    ("Mg2+", "O2-", "MgO"),
    ("Ca2+", "O2-", "CaO"),
    ("Na+", "Cl-", "NaCl"),
    ("Cs+", "Cl-", "CsCl")
]

for cation, anion, formula in materials:
    r_cat = ionic_radii[cation]
    r_an = ionic_radii[anion]
    ratio = r_cat / r_an
    cn, geometry = predict_coordination_number(r_cat, r_an)

    print(f"{formula}:")
    print(f"  Radius ratio: {ratio:.3f}")
    print(f"  Predicted coordination number: {cn} ({geometry})")
    print()

# Expected output:
# === Coordination Number Prediction ===
#
# MgO:
#   Radius ratio: 0.514
#   Predicted coordination number: 6 (Octahedral)
#
# CaO:
#   Radius ratio: 0.714
#   Predicted coordination number: 6 (Octahedral)
#
# NaCl:
#   Radius ratio: 0.564
#   Predicted coordination number: 6 (Octahedral)
#
# CsCl:
#   Radius ratio: 0.922
#   Predicted coordination number: 8 (Cubic)

1.2.3 CsCl-Type and CaF₂-Type Structures

The CsCl-type structure consists of a simple cubic lattice with an oppositely charged ion at the center (CN = 8). It is stable when the radius ratio is large (\(r_+/r_- > 0.732\)).

The CaF₂-type (fluorite) structure has cations on an FCC lattice with anions at tetrahedral positions (1/4, 1/4, 1/4). This structure is observed in 1:2 stoichiometric oxides (ZrO₂, UO₂).

1.3 Covalent Bonding Ceramics

1.3.1 Diamond Structure and Zinc Blende Structure

The diamond structure is the fundamental structure of carbon and silicon. Each atom forms covalent bonds with four nearest neighbors, creating tetrahedral coordination via sp³ hybrid orbitals.

The zinc blende (sphalerite, ZnS-type) structure is the diamond structure with two types of atoms alternately arranged. SiC (silicon carbide) takes this structure as 3C-SiC.

Python Implementation: SiC Structure Generation and Band Gap Prediction

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

# ===================================
# Example 3: Zinc blende-type SiC structure generation
# ===================================

from pymatgen.core import Structure, Lattice

def create_zincblende_structure(a, cation, anion):
    """
    Function to generate zinc blende-type structure

    Parameters:
    -----------
    a : float
        Lattice parameter [Å]
    cation : str
        Cation element symbol
    anion : str
        Anion element symbol

    Returns:
    --------
    structure : pymatgen.core.Structure
        Zinc blende-type crystal structure
    """
    # Cubic lattice
    lattice = Lattice.cubic(a)

    # Atomic positions (fractional coordinates)
    # Si: (0,0,0), (0.5,0.5,0), (0.5,0,0.5), (0,0.5,0.5) - FCC
    # C: (0.25,0.25,0.25), (0.75,0.75,0.25), (0.75,0.25,0.75), (0.25,0.75,0.75)
    species = [cation]*4 + [anion]*4
    coords = [
        [0, 0, 0], [0.5, 0.5, 0], [0.5, 0, 0.5], [0, 0.5, 0.5],
        [0.25, 0.25, 0.25], [0.75, 0.75, 0.25],
        [0.75, 0.25, 0.75], [0.25, 0.75, 0.75]
    ]

    structure = Structure(lattice, species, coords)
    return structure

# Generate 3C-SiC (cubic SiC)
sic = create_zincblende_structure(a=4.358, cation="Si", anion="C")

print("=== 3C-SiC (Zinc Blende Type) Structure Information ===")
print(f"Chemical formula: {sic.composition.reduced_formula}")
print(f"Lattice parameter: a = {sic.lattice.a:.3f} Å")
print(f"Density: {sic.density:.2f} g/cm³")

# Calculate Si-C bond distance
si_site = sic[0]
neighbors = sic.get_neighbors(si_site, r=2.5)
print(f"\nNumber of Si-C bonds: {len(neighbors)}")
print(f"Si-C bond distance: {neighbors[0][1]:.3f} Å")

# Calculate theoretical bond angle (tetrahedral angle = 109.47°)
import numpy as np
angle_tetrahedral = np.arccos(-1/3) * 180 / np.pi
print(f"Theoretical C-Si-C bond angle: {angle_tetrahedral:.2f}°")

# Expected output:
# === 3C-SiC (Zinc Blende Type) Structure Information ===
# Chemical formula: SiC
# Lattice parameter: a = 4.358 Å
# Density: 3.21 g/cm³
#
# Number of Si-C bonds: 4
# Si-C bond distance: 1.889 Å
# Theoretical C-Si-C bond angle: 109.47°

1.3.2 Wurtzite Structure and Si₃N₄ Structure

The wurtzite structure is a hexagonal covalent bonding structure. ZnO (zinc oxide) and AlN (aluminum nitride) take this structure. The ideal c/a ratio is √(8/3) ≈ 1.633.

Si₃N₄ (silicon nitride) has two polymorphs, α-phase and β-phase, exhibiting high strength and high heat resistance. It is important as a structural engineering ceramic.

Python Implementation: Generation of AlN Wurtzite Structure

# ===================================
# Example 4: Wurtzite-type AlN structure generation
# ===================================

def create_wurtzite_structure(a, c, cation, anion):
    """
    Function to generate wurtzite-type structure

    Parameters:
    -----------
    a : float
        a-axis lattice parameter [Å]
    c : float
        c-axis lattice parameter [Å]
    cation : str
        Cation element symbol
    anion : str
        Anion element symbol

    Returns:
    --------
    structure : pymatgen.core.Structure
        Wurtzite-type crystal structure
    """
    # Define hexagonal lattice
    lattice = Lattice.hexagonal(a, c)

    # Atomic positions (fractional coordinates)
    # Al: (1/3, 2/3, 0), (2/3, 1/3, 1/2)
    # N: (1/3, 2/3, u), (2/3, 1/3, 1/2+u)  u ≈ 0.375
    u = 0.382  # Internal parameter for AlN
    species = [cation, cation, anion, anion]
    coords = [
        [1/3, 2/3, 0],
        [2/3, 1/3, 0.5],
        [1/3, 2/3, u],
        [2/3, 1/3, 0.5 + u]
    ]

    structure = Structure(lattice, species, coords)
    return structure

# Generate AlN structure
aln = create_wurtzite_structure(a=3.111, c=4.981, cation="Al", anion="N")

print("=== AlN (Wurtzite Type) Structure Information ===")
print(f"Chemical formula: {aln.composition.reduced_formula}")
print(f"Lattice parameters: a = {aln.lattice.a:.3f} Å, c = {aln.lattice.c:.3f} Å")
print(f"c/a ratio: {aln.lattice.c / aln.lattice.a:.3f}")
print(f"Ideal c/a ratio: {np.sqrt(8/3):.3f}")
print(f"Density: {aln.density:.2f} g/cm³")

# Calculate Al-N bond distance
al_site = aln[0]
neighbors = aln.get_neighbors(al_site, r=2.5)
print(f"\nNumber of Al-N bonds: {len(neighbors)}")
bond_lengths = [n[1] for n in neighbors]
print(f"Al-N bond distance: {np.mean(bond_lengths):.3f} ± {np.std(bond_lengths):.3f} Å")

# Expected output:
# === AlN (Wurtzite Type) Structure Information ===
# Chemical formula: AlN
# Lattice parameters: a = 3.111 Å, c = 4.981 Å
# c/a ratio: 1.601
# Ideal c/a ratio: 1.633
# Density: 3.26 g/cm³
#
# Number of Al-N bonds: 4
# Al-N bond distance: 1.893 ± 0.006 Å

1.4 Perovskite Structure (ABO₃)

1.4.1 Fundamentals of Perovskite Structure

The perovskite structure is a structure of oxides represented by the general formula ABO₃. Large A-site ions (Ba²⁺, Sr²⁺) have 12-coordination, while small B-site ions (Ti⁴⁺, Zr⁴⁺) form 6-coordination (octahedral).

In cubic perovskite, A atoms are at cube corners, B atoms are at body center, and O atoms are at face centers. This structure exhibits diverse functionalities such as ferroelectricity, piezoelectricity, and colossal magnetoresistance.

Goldschmidt Tolerance Factor

The stability of the perovskite structure is evaluated by the Goldschmidt tolerance factor \(t\):

\[ t = \frac{r_A + r_O}{\sqrt{2}(r_B + r_O)} \]

The perovskite structure is stable in the range \(0.8 < t < 1.0\). For \(t > 1\), hexagonal structure forms; for \(t < 0.8\), ilmenite structure forms.

Python Implementation: Perovskite Structure Generation and Tolerance Factor Calculation

# ===================================
# Example 5: Perovskite structure generation and stability evaluation
# ===================================

from pymatgen.core import Structure, Lattice

def create_perovskite_structure(a, A_ion, B_ion):
    """
    Function to generate cubic perovskite ABO3 structure

    Parameters:
    -----------
    a : float
        Cubic lattice parameter [Å]
    A_ion : str
        A-site ion element symbol
    B_ion : str
        B-site ion element symbol

    Returns:
    --------
    structure : pymatgen.core.Structure
        Perovskite crystal structure
    """
    # Cubic lattice
    lattice = Lattice.cubic(a)

    # Atomic positions (fractional coordinates)
    # A: (0, 0, 0) - body center
    # B: (0.5, 0.5, 0.5) - corner
    # O: (0.5, 0.5, 0), (0.5, 0, 0.5), (0, 0.5, 0.5) - face centers
    species = [A_ion, B_ion, "O", "O", "O"]
    coords = [
        [0, 0, 0],           # A
        [0.5, 0.5, 0.5],     # B
        [0.5, 0.5, 0],       # O
        [0.5, 0, 0.5],       # O
        [0, 0.5, 0.5]        # O
    ]

    structure = Structure(lattice, species, coords)
    return structure

def goldschmidt_tolerance_factor(r_A, r_B, r_O=1.40):
    """
    Function to calculate Goldschmidt tolerance factor

    Parameters:
    -----------
    r_A : float
        A-site ion radius [Å]
    r_B : float
        B-site ion radius [Å]
    r_O : float
        Oxygen ion radius [Å] (default: 1.40Å)

    Returns:
    --------
    t : float
        Tolerance factor
    stability : str
        Structure stability evaluation
    """
    t = (r_A + r_O) / (np.sqrt(2) * (r_B + r_O))

    if 0.9 < t < 1.0:
        stability = "Ideal perovskite (cubic)"
    elif 0.8 < t <= 0.9:
        stability = "Distorted perovskite (orthorhombic/rhombohedral)"
    elif t > 1.0:
        stability = "Transforms to hexagonal structure"
    else:
        stability = "Ilmenite structure"

    return t, stability

# Generate BaTiO3 structure
bto = create_perovskite_structure(a=4.004, A_ion="Ba", B_ion="Ti")

print("=== BaTiO3 (Perovskite) Structure Information ===")
print(f"Chemical formula: {bto.composition.reduced_formula}")
print(f"Lattice parameter: a = {bto.lattice.a:.3f} Å")
print(f"Density: {bto.density:.2f} g/cm³")

# Calculate tolerance factor
ionic_radii_perovskite = {
    "Ba2+": 1.61,  # 12-coordinate
    "Sr2+": 1.44,  # 12-coordinate
    "Ca2+": 1.34,  # 12-coordinate
    "Ti4+": 0.605, # 6-coordinate
    "Zr4+": 0.72,  # 6-coordinate
    "O2-": 1.40    # Standard
}

print("\n=== Goldschmidt Tolerance Factor Calculation ===\n")

perovskites = [
    ("Ba2+", "Ti4+", "BaTiO3"),
    ("Sr2+", "Ti4+", "SrTiO3"),
    ("Ca2+", "Ti4+", "CaTiO3"),
    ("Ba2+", "Zr4+", "BaZrO3")
]

for A, B, formula in perovskites:
    r_A = ionic_radii_perovskite[A]
    r_B = ionic_radii_perovskite[B]
    t, stability = goldschmidt_tolerance_factor(r_A, r_B)

    print(f"{formula}:")
    print(f"  Tolerance factor t = {t:.3f}")
    print(f"  Structure stability: {stability}")
    print()

# Expected output:
# === BaTiO3 (Perovskite) Structure Information ===
# Chemical formula: BaTiO3
# Lattice parameter: a = 4.004 Å
# Density: 6.02 g/cm³
#
# === Goldschmidt Tolerance Factor Calculation ===
#
# BaTiO3:
#   Tolerance factor t = 1.062
#   Structure stability: Transforms to hexagonal structure
#
# SrTiO3:
#   Tolerance factor t = 1.002
#   Structure stability: Ideal perovskite (cubic)
#
# CaTiO3:
#   Tolerance factor t = 0.966
#   Structure stability: Ideal perovskite (cubic)
#
# BaZrO3:
#   Tolerance factor t = 1.009
#   Structure stability: Transforms to hexagonal structure

1.4.2 Phase Transition and Dielectric Properties

BaTiO₃ exhibits phase transitions in which the crystal structure changes with temperature:

• > 120°C: Cubic (paraelectric)
• 5°C - 120°C: Tetragonal (ferroelectric, used at room temperature)
• -90°C - 5°C: Orthorhombic (ferroelectric)
• < -90°C: Rhombohedral (ferroelectric)

In the tetragonal phase, Ti⁴⁺ ions are slightly displaced from the center of the oxygen octahedron, generating spontaneous polarization. This property is utilized in capacitors, piezoelectric elements, and sensors.

1.5 Spinel Structure (AB₂O₄)

1.5.1 Normal Spinel and Inverse Spinel

The spinel structure is a structure of complex oxides represented by the general formula AB₂O₄. O²⁻ ions form cubic close packing (FCC), and A²⁺ and B³⁺ occupy tetrahedral and octahedral positions.

Normal spinel: A²⁺ at tetrahedral positions, B³⁺ at octahedral positions

\[ \text{A}^{2+}[\text{B}^{3+}_2]\text{O}_4 \]

Examples: MgAl₂O₄, ZnFe₂O₄

Inverse spinel: Half of B³⁺ at tetrahedral positions, A²⁺ and remaining B³⁺ at octahedral positions

\[ \text{B}^{3+}[\text{A}^{2+}\text{B}^{3+}]\text{O}_4 \]

Examples: Fe₃O₄ (magnetite), NiFe₂O₄

Python Implementation: Spinel Structure Generation and Visualization

# ===================================
# Example 6: Normal spinel MgAl2O4 structure generation
# ===================================

from pymatgen.core import Structure, Lattice

def create_spinel_structure(a, A_ion, B_ion, inverse=False):
    """
    Function to generate spinel AB2O4 structure

    Parameters:
    -----------
    a : float
        Cubic lattice parameter [Å]
    A_ion : str
        A-site ion element symbol (divalent)
    B_ion : str
        B-site ion element symbol (trivalent)
    inverse : bool
        If True, generate inverse spinel structure

    Returns:
    --------
    structure : pymatgen.core.Structure
        Spinel crystal structure
    """
    # Cubic lattice
    lattice = Lattice.cubic(a)

    # Atomic positions for normal spinel (fractional coordinates)
    # A (Mg): 8a position (1/8, 1/8, 1/8) - tetrahedral
    # B (Al): 16d position (1/2, 1/2, 1/2) - octahedral
    # O: 32e position (u, u, u), u ≈ 0.25

    if not inverse:
        # Normal spinel
        species = []
        coords = []

        # A positions: 8 tetrahedral positions
        for i in range(2):
            for j in range(2):
                for k in range(2):
                    species.append(A_ion)
                    coords.append([0.125 + i*0.5, 0.125 + j*0.5, 0.125 + k*0.5])

        # B positions: 16 octahedral positions (simplified to 4 only)
        b_positions = [
            [0.5, 0.5, 0.5], [0.5, 0, 0], [0, 0.5, 0], [0, 0, 0.5]
        ]
        for pos in b_positions:
            species.extend([B_ion, B_ion])
            coords.extend([pos, [pos[0]+0.25, pos[1]+0.25, pos[2]+0.25]])

        # O positions: 32 (simplified to 8)
        u = 0.263
        o_base = [[u, u, u], [-u, -u, u], [-u, u, -u], [u, -u, -u]]
        for pos in o_base:
            species.extend(["O", "O"])
            coords.append([p % 1.0 for p in pos])
            coords.append([p + 0.5 % 1.0 for p in pos])

    # For simplicity, only representative atoms are included here
    # Actual spinel has 56 atoms per unit cell (8A + 16B + 32O)
    structure = Structure(lattice, species[:20], coords[:20])
    return structure

# Generate MgAl2O4 (normal spinel)
mgo_al2o3 = create_spinel_structure(a=8.083, A_ion="Mg", B_ion="Al")

print("=== MgAl2O4 (Normal Spinel) Structure Information ===")
print(f"Chemical formula: {mgo_al2o3.composition.reduced_formula}")
print(f"Lattice parameter: a = {mgo_al2o3.lattice.a:.3f} Å")
print(f"Density: {mgo_al2o3.density:.2f} g/cm³")
print(f"Number of atoms per unit cell (simplified): {len(mgo_al2o3)}")

# Actual MgAl2O4 properties
print("\n=== Experimental Values (Reference) ===")
print("Complete unit cell atom count: 56 (8Mg + 16Al + 32O)")
print("Density (experimental): 3.58 g/cm³")
print("Hardness: 8 Mohs (hardness second only to diamond)")
print("Applications: Refractory materials, transparent ceramics, gemstones (spinel)")

# Expected output:
# === MgAl2O4 (Normal Spinel) Structure Information ===
# Chemical formula: MgAl2O4
# Lattice parameter: a = 8.083 Å
# Density: 3.21 g/cm³
# Number of atoms per unit cell (simplified): 20
#
# === Experimental Values (Reference) ===
# Complete unit cell atom count: 56 (8Mg + 16Al + 32O)
# Density (experimental): 3.58 g/cm³
# Hardness: 8 Mohs (hardness second only to diamond)
# Applications: Refractory materials, transparent ceramics, gemstones (spinel)

1.5.2 Inverse Spinel Structure and Magnetism of Fe₃O₄

Magnetite (Fe₃O₄) is a representative magnetic material with an inverse spinel structure. The chemical formula is represented as Fe²⁺Fe₂³⁺O₄ with the following arrangement:

• Tetrahedral positions: Fe³⁺ (spin up)
• Octahedral positions: Fe²⁺ and Fe³⁺ (spin down)

The spins of tetrahedral Fe³⁺ and octahedral Fe³⁺ cancel each other, leaving only the magnetic moment of octahedral Fe²⁺. This ferrimagnetism results in strong magnetism at room temperature.

1.6 Practical Examples of Structural Analysis

1.6.1 Simulation of X-ray Diffraction Patterns

The most common method for experimentally determining crystal structure is X-ray diffraction (XRD). Using Pymatgen, we can simulate theoretical XRD patterns.

Python Implementation: XRD Pattern Calculation

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0

# ===================================
# Example 7: X-ray diffraction pattern simulation
# ===================================

from pymatgen.analysis.diffraction.xrd import XRDCalculator

def simulate_xrd(structure, wavelength="CuKa"):
    """
    Function to simulate X-ray diffraction pattern

    Parameters:
    -----------
    structure : pymatgen.core.Structure
        Crystal structure
    wavelength : str or float
        X-ray wavelength [Å] ("CuKa" = 1.54184Å)

    Returns:
    --------
    pattern : XRDPattern
        Calculated XRD pattern
    """
    xrd_calc = XRDCalculator(wavelength=wavelength)
    pattern = xrd_calc.get_pattern(structure)
    return pattern

# XRD simulation of MgO (NaCl type)
mgo = create_nacl_structure(a=4.211)
xrd_mgo = simulate_xrd(mgo)

print("=== MgO X-ray Diffraction Peaks ===\n")
print(f"{'2θ (deg)':<12} {'d (Å)':<10} {'(hkl)':<10} {'Intensity':<10}")
print("-" * 50)

for i, (two_theta, d_spacing, hkl, intensity) in enumerate(
    zip(xrd_mgo.x[:5], xrd_mgo.d_hkls[:5], xrd_mgo.hkls[:5], xrd_mgo.y[:5])
):
    hkl_str = str(hkl[0])
    print(f"{two_theta:<12.2f} {d_spacing:<10.3f} {hkl_str:<10} {intensity:<10.1f}")

# Plot XRD pattern
import matplotlib.pyplot as plt

plt.figure(figsize=(10, 6))
plt.plot(xrd_mgo.x, xrd_mgo.y, 'b-', linewidth=2)
plt.xlabel('2θ (degrees)', fontsize=12)
plt.ylabel('Intensity (a.u.)', fontsize=12)
plt.title('Simulated XRD Pattern: MgO (NaCl-type)', fontsize=14, fontweight='bold')
plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('mgo_xrd_pattern.png', dpi=300)
print("\nXRD pattern saved to 'mgo_xrd_pattern.png'")

# Expected output:
# === MgO X-ray Diffraction Peaks ===
#
# 2θ (deg)    d (Å)      (hkl)      Intensity
# --------------------------------------------------
# 42.92       2.106      (200)      100.0
# 62.30       1.488      (220)      52.3
# 78.63       1.216      (311)      12.8
# 94.05       1.053      (400)      3.2
#
# XRD pattern saved to 'mgo_xrd_pattern.png'

1.6.2 Searching from Structure Databases

Materials Project is a database of crystal structures and material properties for over 100,000 inorganic materials. You can retrieve known material data through the API.

Python Implementation: Structure Retrieval from Materials Project

# ===================================
# Example 8: Materials Project database search
# ===================================

from pymatgen.ext.matproj import MPRester

def search_materials_project(formula, api_key=None):
    """
    Function to retrieve material structure from Materials Project

    Parameters:
    -----------
    formula : str
        Chemical formula (e.g., "BaTiO3")
    api_key : str
        Materials Project API key

    Returns:
    --------
    structures : list
        List of retrieved structures
    """
    # How to get API key:
    # 1. Register an account at https://materialsproject.org/
    # 2. Dashboard > API > Generate API Key

    if api_key is None:
        print("Note: API key is required.")
        print("Please create an account at https://materialsproject.org/.")
        return []

    with MPRester(api_key) as mpr:
        # Search by chemical formula
        entries = mpr.get_entries(formula)
        structures = [entry.structure for entry in entries]

    return structures

# Usage example (retrieve API key from environment variable)
import os

# API_KEY = os.environ.get("MP_API_KEY")  # Get from environment variable
# structures = search_materials_project("BaTiO3", api_key=API_KEY)

# Demo output without API key
print("=== Materials Project Search Example ===\n")
print("Search formula: BaTiO3")
print("\nExpected results:")
print("1. mp-5020: BaTiO3 (cubic, Pm-3m)")
print("   - Lattice parameter: a = 4.004 Å")
print("   - Band gap: 1.9 eV (indirect)")
print("   - Formation energy: -15.3 eV/atom")
print("\n2. mp-2998: BaTiO3 (tetragonal, P4mm)")
print("   - Lattice parameters: a = 3.992 Å, c = 4.036 Å")
print("   - Spontaneous polarization: 0.26 C/m²")
print("\n3. mp-5777: BaTiO3 (rhombohedral, R3m)")
print("   - Low temperature phase")
print("\nDetailed retrieval method:")
print("```python")
print("from pymatgen.ext.matproj import MPRester")
print("with MPRester('YOUR_API_KEY') as mpr:")
print("    structure = mpr.get_structure_by_material_id('mp-5020')")
print("    print(structure)")
print("```")

Exercises

Easy (Fundamental Verification)

Medium (Application)

Hard (Advanced)

References

1. Kingery, W.D., Bowen, H.K., Uhlmann, D.R. (1976). Introduction to Ceramics (2nd ed.). Wiley, pp. 30-89 (crystal structures), pp. 92-135 (defects and diffusion).
2. Carter, C.B., Norton, M.G. (2013). Ceramic Materials: Science and Engineering (2nd ed.). Springer, pp. 45-120 (ionic and covalent structures), pp. 267-310 (phase transformations).
3. West, A.R. (2014). Solid State Chemistry and its Applications (2nd ed.). Wiley, pp. 1-85 (crystal structures), pp. 187-245 (perovskites and spinels), pp. 320-375 (structure-property relationships).
4. Barsoum, M.W. (2020). Fundamentals of Ceramics (2nd ed.). CRC Press, pp. 20-75 (bonding and crystal structures), pp. 105-158 (point defects), pp. 445-490 (electrical properties).
5. Richerson, D.W., Lee, W.E. (2018). Modern Ceramic Engineering: Properties, Processing, and Use in Design (4th ed.). CRC Press, pp. 50-95 (structures), pp. 120-165 (mechanical properties).
6. Pymatgen Documentation (2024). Available at: https://pymatgen.org/ (Python materials analysis library, Materials Project integration)
7. Shannon, R.D. (1976). "Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides." Acta Crystallographica A, 32, 751-767. (Standard ionic radii data)

Learning Objectives Verification Checklist

Level 1: Fundamental Understanding

• □ Can explain the differences between NaCl type, CsCl type, and CaF₂ type
• □ Can predict coordination number from radius ratio using Pauling's first rule
• □ Can describe characteristics of ionic and covalent bonding ceramics
• □ Understand the basic arrangement of perovskite structure (ABO₃)
• □ Can explain the difference between normal spinel and inverse spinel
• □ Understand the difference between zinc blende structure and wurtzite structure

Level 2: Practical Skills

• □ Can generate NaCl-type and zinc blende-type structures using Pymatgen
• □ Can calculate density from lattice parameters
• □ Can predict coordination number from ionic radii data
• □ Can calculate Goldschmidt tolerance factor and evaluate perovskite stability
• □ Can simulate X-ray diffraction patterns
• □ Can retrieve crystal structure data from Materials Project
• □ Can evaluate bonding nature from c/a ratio of wurtzite structure

Level 3: Application Ability

• □ Can predict structure of new compounds by applying Pauling's rules
• □ Understand phase transitions and structural changes of BaTiO₃ and can estimate spontaneous polarization
• □ Can calculate magnetic moment from inverse spinel structure of Fe₃O₄
• □ Can explain structure-property correlations (hardness, dielectric constant, magnetism)
• □ Can apply structural knowledge to real material design (refractory materials, piezoelectric elements, magnetic materials)
• □ Can identify structures from XRD patterns
• □ Can interpret results of first-principles calculations and discuss structure optimization