Chapter 2: Bravais Lattices and Space Groups

Building on the lattice concepts learned in Chapter 1, this chapter delves into the core of crystallography: the 14 Bravais lattices and 230 space groups. These constitute the fundamental framework for classifying all crystal structures and represent one of the most important concepts in materials science. You will also learn practical analysis techniques using the Python library "pymatgen."

Learning Objectives

By reading this chapter, you will master the following:

✅ Classify and identify the 14 Bravais lattices
✅ Understand the difference between translational symmetry and point symmetry
✅ Understand the basics of symmetry operations (rotation, reflection, inversion, rotoinversion)
✅ Master the basic concepts of space groups and how to read international symbols
✅ Retrieve and analyze space group information using pymatgen
✅ Investigate space groups of real materials and understand symmetry operations

2.1 What are Bravais Lattices?

Classification of Lattices

As we learned in Chapter 1, crystals have a periodic arrangement called a lattice. However, not all lattices have the same shape. In 1848, French physicist Auguste Bravais systematically classified lattices in three-dimensional space and proved that only 14 Bravais lattices exist.

Bravais lattices are classified based on two criteria:

Crystal system: Seven classifications based on lattice symmetry
Centering type: Positions of lattice points within the unit cell

Lattice Point Arrangement Patterns

There are four types of lattice point arrangements within unit cells:

Symbol	Name (Japanese)	Name (English)	Lattice Point Positions
P	Primitive lattice	Primitive	Corners only (8 locations)
I	Body-centered lattice	Body-centered (Innenzentriert)	Corners + body center
F	Face-centered lattice	Face-centered	Corners + center of each face (6 locations)
C/A/B	Base-centered lattice	Base-centered	Corners + center of one pair of opposite faces

Important note: These arrangements must satisfy the condition "cannot reduce the unit cell while preserving symmetry." For example, monoclinic systems do not have body-centered (I) or face-centered (F) types (because they can be reduced to primitive (P) lattices).

Existence of 14 Bravais Lattices

Combining seven crystal systems with four centering types theoretically yields 28 lattices, but in reality only 14 exist independently. This is because some combinations can be converted to other lattices.

Mathematical Background: The number 14 for Bravais lattices derives from the combination of point group symmetry and translational symmetry in three-dimensional Euclidean space. This classification is based on the theory of crystallographic point groups in group theory.

2.2 Details of the 14 Bravais Lattices

Below, Bravais lattices are classified by the seven crystal systems. Characteristics and examples of real materials are shown for each lattice.

graph TB A[14 Bravais Lattices] --> B[Triclinic
1 type] A --> C[Monoclinic
2 types] A --> D[Orthorhombic
4 types] A --> E[Tetragonal
2 types] A --> F[Hexagonal
1 type] A --> G[Trigonal
1 type] A --> H[Cubic
3 types] B --> B1[P] C --> C1[P] C --> C2[C] D --> D1[P] D --> D2[C] D --> D3[I] D --> D4[F] E --> E1[P] E --> E2[I] F --> F1[P] G --> G1[P or R] H --> H1[P
simple cubic] H --> H2[I
BCC] H --> H3[F
FCC] style B fill:#fce7f3 style C fill:#fce7f3 style D fill:#fce7f3 style E fill:#fce7f3 style F fill:#fce7f3 style G fill:#fce7f3 style H fill:#fce7f3

1. Triclinic System - 1 type

Lattice parameters: $a \neq b \neq c$, $\alpha \neq \beta \neq \gamma \neq 90°$ (all different)

P (Primitive): Lattice with lowest symmetry
Examples: CuSO₄·5H₂O (copper sulfate pentahydrate), K₂Cr₂O₇ (potassium dichromate)

2. Monoclinic System - 2 types

Lattice parameters: $a \neq b \neq c$, $\alpha = \gamma = 90° \neq \beta$

P (Primitive): Corners only
C (Base-centered): Adds lattice point at center of face perpendicular to c-axis (ab plane)
Examples: β-S (orthorhombic sulfur), gypsum (CaSO₄·2H₂O)

3. Orthorhombic System - 4 types

Lattice parameters: $a \neq b \neq c$, $\alpha = \beta = \gamma = 90°$

P (Primitive): Corners only
C (Base-centered): Lattice point at center of ab plane
I (Body-centered): Lattice point at body center
F (Face-centered): Lattice points at centers of all faces
Examples: α-S (orthorhombic sulfur, P), BaSO₄ (barite, P), U (uranium, C)

4. Tetragonal System - 2 types

Lattice parameters: $a = b \neq c$, $\alpha = \beta = \gamma = 90°$

P (Primitive): Corners only
I (Body-centered): Lattice point at body center
Examples: TiO₂ (rutile, P), In (indium, I), Sn (white tin, I)

5. Hexagonal System - 1 type

Lattice parameters: $a = b \neq c$, $\alpha = \beta = 90°$, $\gamma = 120°$

P (Primitive): Lattice points at hexagonal vertices
Examples: Graphite (C), Mg (magnesium), Zn (zinc), ice

Note: Hexagonal system crystals often have Hexagonal Close-Packed (HCP) structure. HCP is not a Bravais lattice itself but a crystal structure with atoms arranged on the lattice.

6. Trigonal (Rhombohedral) System - 1 type

Lattice parameters: $a = b = c$, $\alpha = \beta = \gamma \neq 90°$

P (or R): Rhombohedral lattice
Examples: Calcite (CaCO₃), quartz (α-SiO₂), Bi (bismuth)

Relationship with hexagonal system: Trigonal system crystals can also be described using hexagonal system settings. In crystallography, they are often represented using hexagonal axes.

7. Cubic System - 3 types

Lattice parameters: $a = b = c$, $\alpha = \beta = \gamma = 90°$

P (Simple cubic): Corners only (very few real materials)
I (Body-Centered Cubic, BCC): Lattice point at body center
F (Face-Centered Cubic, FCC): Lattice points at centers of all faces

Examples:

P (simple cubic): α-Po (polonium), CsCl-type structure (though CsCl itself has 2-atom basis)
I (BCC): Fe (α-iron, room temperature), Cr (chromium), W (tungsten), Na (sodium)
F (FCC): Al (aluminum), Cu (copper), Au (gold), Ag (silver), Ni (nickel), Pb (lead)

Summary Table of 14 Bravais Lattices

Crystal System	Lattice Parameter Conditions	Bravais Lattices	Total
Triclinic	$a \neq b \neq c$, $\alpha \neq \beta \neq \gamma$	P	1
Monoclinic	$a \neq b \neq c$, $\alpha = \gamma = 90° \neq \beta$	P, C	2
Orthorhombic	$a \neq b \neq c$, $\alpha = \beta = \gamma = 90°$	P, C, I, F	4
Tetragonal	$a = b \neq c$, $\alpha = \beta = \gamma = 90°$	P, I	2
Hexagonal	$a = b \neq c$, $\alpha = \beta = 90°, \gamma = 120°$	P	1
Trigonal	$a = b = c$, $\alpha = \beta = \gamma \neq 90°$	P (R)	1
Cubic	$a = b = c$, $\alpha = \beta = \gamma = 90°$	P, I, F	3
Total			14

2.3 Symmetry Operations

Crystal symmetry is described by symmetry operations. A symmetry operation is a transformation (rotation, reflection, etc.) that moves the crystal but leaves it indistinguishable from its original configuration.

Symmetry operations are classified into two broad categories:

Point symmetry operations: At least one point remains fixed (does not move)
Translational symmetry operations: All points move

Point Symmetry Operations

1. Rotation

Operation that rotates by a specific angle around an axis. In crystallography, only 1-fold, 2-fold, 3-fold, 4-fold, and 6-fold rotation axes are allowed (5-fold and 7-fold or higher are incompatible with translational symmetry).

Rotational Symmetry	Symbol	Rotation Angle	Repetition Count
1-fold rotation	1	360°	1 (identity operation)
2-fold rotation	2	180°	2
3-fold rotation	3	120°	3
4-fold rotation	4	90°	4
6-fold rotation	6	60°	6

Crystallographic restriction: Why don't 5-fold or 7-fold rotational symmetries exist? This is because it has been mathematically proven that only 1, 2, 3, 4, and 6-fold rotational symmetries are compatible with translational symmetry (lattice periodicity). 5-fold symmetry appears only in quasicrystals.

2. Mirror Reflection

Operation having mirror symmetry with respect to a plane. Denoted by symbol m.

3. Inversion

Operation that inverts all coordinates with respect to a point (inversion center). Denoted by symbol $\bar{1}$ or i.

Mathematical expression for inversion: $(x, y, z) \rightarrow (-x, -y, -z)$

4. Rotoinversion

Operation combining rotation and inversion. Denoted by symbols $\bar{1}, \bar{2}, \bar{3}, \bar{4}, \bar{6}$.

For example, $\bar{4}$ is the operation "rotate 90° then invert."

Translational Symmetry Operations

Translational symmetry operations combine rotation or reflection with translation (parallel displacement).

1. Glide Reflection

Operation combining reflection and translation. Denoted by symbols a, b, c, n, d.

Example: a-glide is reflection with respect to a plane followed by translation in the a-axis direction by $a/2$.

2. Screw Axis

Operation combining rotation and translation. Denoted by symbols 2₁, 3₁, 4₁, 6₁, etc.

Example: 2₁ is 180° rotation followed by translation in the rotation axis direction by half the lattice vector (read as "21-screw").

Hermann-Mauguin Notation

Combinations of symmetry operations are represented by Hermann-Mauguin notation (international symbols). This is the standard method for describing point groups and space groups of crystals.

Examples:

2/m: 2-fold rotation axis and perpendicular mirror plane
4mm: 4-fold rotation axis and two mirror planes
$\bar{3}$m: 3-fold rotoinversion axis and mirror plane

2.4 Space Group Concepts

What are Space Groups?

A space group is the set of all symmetry operations (point symmetry operations + translational symmetry operations) of a crystal. In 1891, Russian crystallographer Evgraf Fedorov and German mathematician Arthur Schönflies independently proved that only 230 space groups exist in three-dimensional space.

This is a remarkable result: all crystal structures (infinitely diverse) can be classified into only 230 symmetry patterns.

Relationship Between Point Groups and Space Groups

To understand space groups, we must first understand point groups.

Point group: Set of point symmetry operations only (no translation) → 32 types exist
Space group: Set of point symmetry operations + translational symmetry operations → 230 types exist

Each space group corresponds to one point group. The 230 space groups are constructed based on the 32 point groups.

Space Group Numbers and International Symbols

The 230 space groups are assigned numbers from 1 to 230 and international symbols using Hermann-Mauguin notation.

Examples of representative space groups:

Number	International Symbol	Crystal System	Examples
1	P1	Triclinic	Lowest symmetry
2	P$\bar{1}$	Triclinic	Has inversion center
63	Cmcm	Orthorhombic	Base-centered + mirror + c-glide + mirror
139	I4/mmm	Tetragonal	Body-centered + 4-fold rotation + mirror
194	P6₃/mmc	Hexagonal	Mg, Zn, Ti (HCP structure)
225	Fm$\bar{3}$m	Cubic	Al, Cu, Au, Ag (FCC structure)
229	Im$\bar{3}$m	Cubic	Fe, Cr, W (BCC structure)

Reading International Symbols

Space group international symbols are structured as follows:

Symbol structure: [Lattice type][Symmetry element 1][Symmetry element 2][Symmetry element 3]...

Example: Fm$\bar{3}$m (Space group 225)

F: Face-centered lattice
m: Mirror plane
$\bar{3}$: 3-fold rotoinversion axis
m: Another mirror plane

This space group corresponds to FCC structure metals such as copper (Cu) and aluminum (Al).

General Positions and Special Positions

Space groups have general positions and special positions (Wyckoff positions) where atoms can be placed.

General position: Symmetry operations generate multiple equivalent positions
Special position: On symmetry elements, so fewer equivalent positions are generated

For example, the general position in Fm$\bar{3}$m (FCC) multiplies 48-fold, but special positions (e.g., (0, 0, 0)) remain 1-fold.

2.5 Analyzing Bravais Lattices and Space Groups with Python

From here, we will practically analyze Bravais lattices and space groups using the Python library pymatgen (Python Materials Genomics).

Installing pymatgen

Pymatgen is a powerful Python library for materials science. It provides many features including crystal structure manipulation, symmetry analysis, and Materials Project API integration.

# Install pymatgen
pip install pymatgen

# Optionally, install these as well
pip install matplotlib numpy pandas plotly

Notes:

Pymatgen has many dependencies, so installation may take time
Latest versions (2024 onward) have API changes
Using Materials Project API requires a free API key (described later)

Code Example 1: Information Table for 14 Bravais Lattices

First, create a table organizing basic information about the 14 Bravais lattices.

# Requirements:
# - Python 3.9+
# - pandas>=2.0.0, <2.2.0

"""
Example: First, create a table organizing basic information about the

Purpose: Demonstrate data manipulation and preprocessing
Target: Beginner to Intermediate
Execution time: 1-5 minutes
Dependencies: None
"""

import pandas as pd

# Data for 14 Bravais lattices
bravais_lattices = [
    # Triclinic
    {'Crystal System': 'Triclinic', 'Bravais Symbol': 'aP', 'Lattice Type': 'P (Primitive)',
     'Lattice Points/Unit Cell': 1, 'Symmetry': 'Lowest', 'Example': 'CuSO₄·5H₂O'},

    # Monoclinic
    {'Crystal System': 'Monoclinic', 'Bravais Symbol': 'mP', 'Lattice Type': 'P (Primitive)',
     'Lattice Points/Unit Cell': 1, 'Symmetry': 'Low', 'Example': 'Gypsum'},
    {'Crystal System': 'Monoclinic', 'Bravais Symbol': 'mC', 'Lattice Type': 'C (Base-centered)',
     'Lattice Points/Unit Cell': 2, 'Symmetry': 'Low', 'Example': 'β-S'},

    # Orthorhombic
    {'Crystal System': 'Orthorhombic', 'Bravais Symbol': 'oP', 'Lattice Type': 'P (Primitive)',
     'Lattice Points/Unit Cell': 1, 'Symmetry': 'Medium', 'Example': 'α-S'},
    {'Crystal System': 'Orthorhombic', 'Bravais Symbol': 'oC', 'Lattice Type': 'C (Base-centered)',
     'Lattice Points/Unit Cell': 2, 'Symmetry': 'Medium', 'Example': 'U'},
    {'Crystal System': 'Orthorhombic', 'Bravais Symbol': 'oI', 'Lattice Type': 'I (Body-centered)',
     'Lattice Points/Unit Cell': 2, 'Symmetry': 'Medium', 'Example': 'TiO₂ (brookite)'},
    {'Crystal System': 'Orthorhombic', 'Bravais Symbol': 'oF', 'Lattice Type': 'F (Face-centered)',
     'Lattice Points/Unit Cell': 4, 'Symmetry': 'Medium', 'Example': 'NaNO₃'},

    # Tetragonal
    {'Crystal System': 'Tetragonal', 'Bravais Symbol': 'tP', 'Lattice Type': 'P (Primitive)',
     'Lattice Points/Unit Cell': 1, 'Symmetry': 'High', 'Example': 'TiO₂ (rutile)'},
    {'Crystal System': 'Tetragonal', 'Bravais Symbol': 'tI', 'Lattice Type': 'I (Body-centered)',
     'Lattice Points/Unit Cell': 2, 'Symmetry': 'High', 'Example': 'In, Sn (white)'},

    # Hexagonal
    {'Crystal System': 'Hexagonal', 'Bravais Symbol': 'hP', 'Lattice Type': 'P (Primitive)',
     'Lattice Points/Unit Cell': 1, 'Symmetry': 'High', 'Example': 'Mg, Zn, Graphite'},

    # Trigonal
    {'Crystal System': 'Trigonal', 'Bravais Symbol': 'hR', 'Lattice Type': 'R (Rhombohedral)',
     'Lattice Points/Unit Cell': 1, 'Symmetry': 'High', 'Example': 'Calcite, Bi'},

    # Cubic
    {'Crystal System': 'Cubic', 'Bravais Symbol': 'cP', 'Lattice Type': 'P (Simple cubic)',
     'Lattice Points/Unit Cell': 1, 'Symmetry': 'Highest', 'Example': 'α-Po'},
    {'Crystal System': 'Cubic', 'Bravais Symbol': 'cI', 'Lattice Type': 'I (BCC)',
     'Lattice Points/Unit Cell': 2, 'Symmetry': 'Highest', 'Example': 'Fe, Cr, W'},
    {'Crystal System': 'Cubic', 'Bravais Symbol': 'cF', 'Lattice Type': 'F (FCC)',
     'Lattice Points/Unit Cell': 4, 'Symmetry': 'Highest', 'Example': 'Al, Cu, Au, Ag'},
]

# Convert to DataFrame
df = pd.DataFrame(bravais_lattices)

print("=" * 100)
print("Classification of 14 Bravais Lattices")
print("=" * 100)
print(df.to_string(index=False))

# Aggregation by crystal system
print("\n" + "=" * 100)
print("Number of Bravais Lattices by Crystal System")
print("=" * 100)
print(df.groupby('Crystal System').size())

print("\n【Explanation】")
print("- Bravais Symbol: Also called Pearson symbol, represented by crystal system initial + lattice type")
print("- Lattice points: Number of lattice points in unit cell (P=1, C/I=2, F=4)")
print("- Higher symmetry means stronger constraints on lattice parameters")
print("- Cubic systems (cP, cI, cF) have highest symmetry and are important in materials science")

Example output:

====================================================================================================
Classification of 14 Bravais Lattices
====================================================================================================
Crystal System Bravais Symbol           Lattice Type  Lattice Points/Unit Cell Symmetry                Example
     Triclinic             aP     P (Primitive)                         1   Lowest       CuSO₄·5H₂O
    Monoclinic             mP     P (Primitive)                         1      Low             Gypsum
    Monoclinic             mC  C (Base-centered)                         2      Low               β-S
  Orthorhombic             oP     P (Primitive)                         1   Medium               α-S
  Orthorhombic             oC  C (Base-centered)                         2   Medium                 U
  Orthorhombic             oI  I (Body-centered)                         2   Medium  TiO₂ (brookite)
  Orthorhombic             oF  F (Face-centered)                         4   Medium            NaNO₃
    Tetragonal             tP     P (Primitive)                         1     High   TiO₂ (rutile)
    Tetragonal             tI  I (Body-centered)                         2     High      In, Sn (white)
     Hexagonal             hP     P (Primitive)                         1     High   Mg, Zn, Graphite
      Trigonal             hR  R (Rhombohedral)                         1     High         Calcite, Bi
         Cubic             cP  P (Simple cubic)                         1  Highest             α-Po
         Cubic             cI         I (BCC)                         2  Highest          Fe, Cr, W
         Cubic             cF         F (FCC)                         4  Highest      Al, Cu, Au, Ag

====================================================================================================
Number of Bravais Lattices by Crystal System
====================================================================================================
Crystal System
Triclinic       1
Trigonal        1
Hexagonal       1
Monoclinic      2
Tetragonal      2
Orthorhombic    4
Cubic           3
dtype: int64

Code Example 2: Visualization of Three Cubic Bravais Lattices

Visualize the three most important cubic Bravais lattices (P, I, F) in 3D.

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0
# - plotly>=5.14.0

import numpy as np
import plotly.graph_objects as go
from plotly.subplots import make_subplots

def create_cubic_lattice_points(lattice_type='P', a=1.0):
    """
    Generate lattice point coordinates for cubic lattices

    Parameters:
    -----------
    lattice_type : str
        'P' (Primitive), 'I' (Body-centered), 'F' (Face-centered)
    a : float
        Lattice constant

    Returns:
    --------
    np.ndarray : Lattice point coordinates (N, 3)
    """
    # Corners (8 locations)
    corners = np.array([
        [0, 0, 0], [a, 0, 0], [a, a, 0], [0, a, 0],
        [0, 0, a], [a, 0, a], [a, a, a], [0, a, a]
    ])

    if lattice_type == 'P':
        return corners

    elif lattice_type == 'I':
        # Add body center
        body_center = np.array([[a/2, a/2, a/2]])
        return np.vstack([corners, body_center])

    elif lattice_type == 'F':
        # Add 6 face centers
        face_centers = np.array([
            [a/2, a/2, 0],   # bottom face
            [a/2, a/2, a],   # top face
            [a/2, 0, a/2],   # front face
            [a/2, a, a/2],   # back face
            [0, a/2, a/2],   # left face
            [a, a/2, a/2]    # right face
        ])
        return np.vstack([corners, face_centers])

    else:
        raise ValueError("lattice_type must be 'P', 'I', or 'F'")


def plot_cubic_unit_cell(ax, lattice_type='P', a=1.0):
    """
    Plot cubic lattice unit cell
    """
    # Generate lattice points
    points = create_cubic_lattice_points(lattice_type, a)

    # Plot lattice points
    ax.add_trace(go.Scatter3d(
        x=points[:, 0], y=points[:, 1], z=points[:, 2],
        mode='markers',
        marker=dict(size=8, color='red', symbol='circle'),
        name='Lattice points'
    ))

    # Draw unit cell edges
    edges = [
        [0, 1], [1, 2], [2, 3], [3, 0],  # bottom
        [4, 5], [5, 6], [6, 7], [7, 4],  # top
        [0, 4], [1, 5], [2, 6], [3, 7]   # vertical
    ]

    corners = create_cubic_lattice_points('P', a)

    for edge in edges:
        ax.add_trace(go.Scatter3d(
            x=corners[edge, 0], y=corners[edge, 1], z=corners[edge, 2],
            mode='lines',
            line=dict(color='black', width=2),
            showlegend=False
        ))


# Create 3 subplots
fig = make_subplots(
    rows=1, cols=3,
    subplot_titles=('(a) Simple Cubic (P)', '(b) Body-Centered Cubic (I)', '(c) Face-Centered Cubic (F)'),
    specs=[[{'type': 'scatter3d'}, {'type': 'scatter3d'}, {'type': 'scatter3d'}]],
    horizontal_spacing=0.05
)

# Plot each lattice type
for i, lattice_type in enumerate(['P', 'I', 'F'], start=1):
    points = create_cubic_lattice_points(lattice_type, a=1.0)

    # Lattice points
    fig.add_trace(go.Scatter3d(
        x=points[:, 0], y=points[:, 1], z=points[:, 2],
        mode='markers',
        marker=dict(size=6, color='red'),
        showlegend=False
    ), row=1, col=i)

    # Unit cell edges
    edges = [
        [0, 1], [1, 2], [2, 3], [3, 0],
        [4, 5], [5, 6], [6, 7], [7, 4],
        [0, 4], [1, 5], [2, 6], [3, 7]
    ]
    corners = create_cubic_lattice_points('P', a=1.0)

    for edge in edges:
        fig.add_trace(go.Scatter3d(
            x=corners[edge, 0], y=corners[edge, 1], z=corners[edge, 2],
            mode='lines',
            line=dict(color='black', width=2),
            showlegend=False
        ), row=1, col=i)

# Layout settings
fig.update_layout(
    title_text="Three Bravais Lattices of Cubic System",
    height=500,
    showlegend=False
)

# Axis settings
for i in range(1, 4):
    fig.update_scenes(
        xaxis=dict(range=[0, 1], title='x'),
        yaxis=dict(range=[0, 1], title='y'),
        zaxis=dict(range=[0, 1], title='z'),
        aspectmode='cube',
        row=1, col=i
    )

fig.show()

# Compare lattice point counts
print("\n【Three Bravais Lattices of Cubic System】")
print("=" * 60)
for lattice_type, name, example in [('P', 'Simple Cubic', 'α-Po'),
                                      ('I', 'BCC', 'Fe, Cr, W'),
                                      ('F', 'FCC', 'Al, Cu, Au, Ag')]:
    points = create_cubic_lattice_points(lattice_type)
    print(f"{name:20s} | Lattice points: {len(points):2d} | Example: {example}")
print("=" * 60)
print("Note: Corner lattice points are shared by 8 unit cells, counted as 1/8 each")
print("    P: 8×(1/8) = 1, I: 8×(1/8) + 1 = 2, F: 8×(1/8) + 6×(1/2) = 4")

Code Example 3: Comparison of Four Orthorhombic Bravais Lattices

Visualize the differences among the four orthorhombic Bravais lattices (P, C, I, F).

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0
# - plotly>=5.14.0

import numpy as np
import plotly.graph_objects as go
from plotly.subplots import make_subplots

def create_orthorhombic_lattice_points(lattice_type='P', a=1.0, b=1.2, c=0.8):
    """
    Generate lattice point coordinates for orthorhombic lattices

    Parameters:
    -----------
    lattice_type : str
        'P', 'C', 'I', 'F'
    a, b, c : float
        Lattice constants (a ≠ b ≠ c)
    """
    # Corners (8 locations)
    corners = np.array([
        [0, 0, 0], [a, 0, 0], [a, b, 0], [0, b, 0],
        [0, 0, c], [a, 0, c], [a, b, c], [0, b, c]
    ])

    if lattice_type == 'P':
        return corners

    elif lattice_type == 'C':
        # C-centered: center of ab plane (z=0 and z=c, 2 locations)
        base_centers = np.array([
            [a/2, b/2, 0],
            [a/2, b/2, c]
        ])
        return np.vstack([corners, base_centers])

    elif lattice_type == 'I':
        # Body-centered: body center
        body_center = np.array([[a/2, b/2, c/2]])
        return np.vstack([corners, body_center])

    elif lattice_type == 'F':
        # Face-centered: centers of 6 faces
        face_centers = np.array([
            [a/2, b/2, 0],   # z=0 face
            [a/2, b/2, c],   # z=c face
            [a/2, 0, c/2],   # y=0 face
            [a/2, b, c/2],   # y=b face
            [0, b/2, c/2],   # x=0 face
            [a, b/2, c/2]    # x=a face
        ])
        return np.vstack([corners, face_centers])

    else:
        raise ValueError("lattice_type must be 'P', 'C', 'I', or 'F'")


# Create 4 subplots
fig = make_subplots(
    rows=2, cols=2,
    subplot_titles=('(a) Primitive (P)', '(b) C-centered (C)',
                    '(c) Body-centered (I)', '(d) Face-centered (F)'),
    specs=[[{'type': 'scatter3d'}, {'type': 'scatter3d'}],
           [{'type': 'scatter3d'}, {'type': 'scatter3d'}]],
    horizontal_spacing=0.05,
    vertical_spacing=0.1
)

a, b, c = 1.0, 1.3, 0.7

positions = [(1, 1), (1, 2), (2, 1), (2, 2)]
lattice_types = ['P', 'C', 'I', 'F']

for (row, col), lattice_type in zip(positions, lattice_types):
    points = create_orthorhombic_lattice_points(lattice_type, a, b, c)

    # Lattice points
    fig.add_trace(go.Scatter3d(
        x=points[:, 0], y=points[:, 1], z=points[:, 2],
        mode='markers',
        marker=dict(size=5, color='red'),
        showlegend=False
    ), row=row, col=col)

    # Unit cell edges
    edges = [
        [0, 1], [1, 2], [2, 3], [3, 0],
        [4, 5], [5, 6], [6, 7], [7, 4],
        [0, 4], [1, 5], [2, 6], [3, 7]
    ]
    corners = create_orthorhombic_lattice_points('P', a, b, c)

    for edge in edges:
        fig.add_trace(go.Scatter3d(
            x=corners[edge, 0], y=corners[edge, 1], z=corners[edge, 2],
            mode='lines',
            line=dict(color='black', width=2),
            showlegend=False
        ), row=row, col=col)

# Layout settings
fig.update_layout(
    title_text="Four Bravais Lattices of Orthorhombic System (a ≠ b ≠ c, α = β = γ = 90°)",
    height=800,
    showlegend=False
)

fig.show()

# Compare lattice point counts
print("\n【Four Bravais Lattices of Orthorhombic System】")
print("=" * 70)
print(f"{'Lattice Type':15s} | {'Points':8s} | {'Effective':12s} | Example")
print("=" * 70)
for lt, name, eff, example in [
    ('P', 'Primitive', 1, 'α-S'),
    ('C', 'C-centered', 2, 'U'),
    ('I', 'Body-centered', 2, 'TiO₂ (brookite)'),
    ('F', 'Face-centered', 4, 'NaNO₃')
]:
    points = create_orthorhombic_lattice_points(lt, a, b, c)
    print(f"{name:15s} | {len(points):8d} | {eff:12d} | {example}")
print("=" * 70)

Code Example 4: Retrieving Space Group Information with pymatgen

Now, let's perform practical space group analysis using pymatgen. First, learn how to retrieve basic space group information.

from pymatgen.symmetry.groups import SpaceGroup

# Retrieve information for representative space groups
space_groups = [
    1,    # P1 (triclinic, lowest symmetry)
    2,    # P-1 (with inversion center)
    194,  # P6₃/mmc (HCP structure)
    225,  # Fm-3m (FCC structure)
    229   # Im-3m (BCC structure)
]

print("=" * 100)
print("Detailed Information on Representative Space Groups")
print("=" * 100)

for sg_num in space_groups:
    sg = SpaceGroup.from_int_number(sg_num)

    print(f"\n【Space Group {sg_num}】")
    print(f"  International symbol:     {sg.symbol}")
    print(f"  Crystal system:       {sg.crystal_system}")
    print(f"  Point group:         {sg.point_group}")
    print(f"  Number of symmetry operations:   {len(sg.symmetry_ops)}")
    print(f"  Centrosymmetric: {sg.is_centrosymmetric} (presence of inversion center)")

    # Display some symmetry operations (first 5)
    print(f"  Symmetry operations (first 5):")
    for i, op in enumerate(sg.symmetry_ops[:5], 1):
        print(f"    {i}. {op}")

    if len(sg.symmetry_ops) > 5:
        print(f"    ... plus {len(sg.symmetry_ops) - 5} more")

print("\n" + "=" * 100)
print("【Explanation】")
print("- Number of symmetry operations varies by space group (No. 1: 1, No. 225: 192)")
print("- Centrosymmetric indicates presence of inversion center")
print("- Symmetry operations are represented by rotation matrices and translation vectors")
print("=" * 100)

Example output:

====================================================================================================
Detailed Information on Representative Space Groups
====================================================================================================

【Space Group 1】
  International symbol:     P1
  Crystal system:       triclinic
  Point group:         1
  Number of symmetry operations:   1
  Centrosymmetric: False (presence of inversion center)
  Symmetry operations (first 5):
    1. Rot:
[[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
tau
[0. 0. 0.]

【Space Group 225】
  International symbol:     Fm-3m
  Crystal system:       cubic
  Point group:         m-3m
  Number of symmetry operations:   192
  Centrosymmetric: True (presence of inversion center)
  Symmetry operations (first 5):
    1. Rot:
[[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]
tau
[0. 0. 0.]
    2. Rot:
[[-1.  0.  0.]
 [ 0. -1.  0.]
 [ 0.  0.  1.]]
tau
[0. 0. 0.]
    ... plus 187 more

Code Example 5: List of Symmetry Operations for a Specific Space Group

Examine symmetry operations of space group 225 (Fm-3m, FCC structure) in detail.

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

"""
Example: Examine symmetry operations of space group 225 (Fm-3m, FCC s

Purpose: Demonstrate core concepts and implementation patterns
Target: Beginner to Intermediate
Execution time: 10-30 seconds
Dependencies: None
"""

from pymatgen.symmetry.groups import SpaceGroup
import numpy as np

# Space group 225 (Fm-3m, FCC structure)
sg = SpaceGroup.from_int_number(225)

print("=" * 100)
print(f"List of Symmetry Operations for Space Group {sg.int_number}: {sg.symbol}")
print("=" * 100)

print(f"Crystal system: {sg.crystal_system}")
print(f"Point group: {sg.point_group}")
print(f"Total number of symmetry operations: {len(sg.symmetry_ops)}")
print(f"Inversion center: {'Present' if sg.is_centrosymmetric else 'Absent'}")

print("\n【Classification of Symmetry Operations】")

# Classify symmetry operations
pure_translations = []
rotations = []
reflections = []
inversions = []
other_ops = []

for i, op in enumerate(sg.symmetry_ops, 1):
    rotation_matrix = op.rotation_matrix
    translation_vector = op.translation_vector

    # Identity operation
    if np.allclose(rotation_matrix, np.eye(3)) and np.allclose(translation_vector, 0):
        continue  # Skip (identity operation)

    # Pure translation
    if np.allclose(rotation_matrix, np.eye(3)):
        pure_translations.append((i, op))

    # Inversion (-I)
    elif np.allclose(rotation_matrix, -np.eye(3)):
        inversions.append((i, op))

    # Other operations
    else:
        # Determine rotation/reflection from determinant
        det = np.linalg.det(rotation_matrix)
        if np.isclose(det, 1):
            rotations.append((i, op))
        elif np.isclose(det, -1):
            reflections.append((i, op))
        else:
            other_ops.append((i, op))

print(f"  Identity operation:        1")
print(f"  Pure translations:      {len(pure_translations)}")
print(f"  Rotation operations:        {len(rotations)}")
print(f"  Reflection operations:        {len(reflections)}")
print(f"  Inversion operations:        {len(inversions)}")
print(f"  Other:          {len(other_ops)}")
print(f"  Total:            {1 + len(pure_translations) + len(rotations) + len(reflections) + len(inversions) + len(other_ops)}")

# Display details of pure translations
print("\n【Pure Translation Operations (first 10)】")
for i, (num, op) in enumerate(pure_translations[:10], 1):
    print(f"  {num:3d}. Translation vector: {op.translation_vector}")

# Display details of rotation operations (first 5)
print("\n【Rotation Operations (first 5)】")
for i, (num, op) in enumerate(rotations[:5], 1):
    print(f"  {num:3d}. Rotation matrix:")
    print(f"       {op.rotation_matrix}")
    print(f"       Translation: {op.translation_vector}")

print("\n" + "=" * 100)
print("【Explanation】")
print("- Fm-3m (FCC) has the highest symmetry among cubic systems")
print("- 192 symmetry operations = 48 point symmetry operations × 4 translations (FCC lattice characteristic)")
print("- This high symmetry makes physical properties (elastic moduli, optical properties, etc.) isotropic")
print("=" * 100)

Code Example 6: Space Group Determination from Crystal Structure

Show an example where pymatgen automatically determines the space group from a known crystal structure.

from pymatgen.core import Structure, Lattice
from pymatgen.symmetry.analyzer import SpacegroupAnalyzer

# Example 1: FCC structure aluminum (Al)
# Lattice constant: a = 4.05 Å
lattice_al = Lattice.cubic(4.05)

# FCC structure: atoms at (0,0,0), (0.5,0.5,0), (0.5,0,0.5), (0,0.5,0.5)
al_structure = Structure(
    lattice_al,
    ["Al", "Al", "Al", "Al"],
    [[0, 0, 0], [0.5, 0.5, 0], [0.5, 0, 0.5], [0, 0.5, 0.5]]
)

# Space group analysis
sga_al = SpacegroupAnalyzer(al_structure)

print("=" * 100)
print("【Example 1: Crystal Structure Analysis of Aluminum (Al)】")
print("=" * 100)
print(f"Lattice constant: a = 4.05 Å (cubic)")
print(f"Number of atoms: {len(al_structure)}")
print(f"\nSpace group number:       {sga_al.get_space_group_number()}")
print(f"Space group symbol:       {sga_al.get_space_group_symbol()}")
print(f"Point group:             {sga_al.get_point_group_symbol()}")
print(f"Crystal system:           {sga_al.get_crystal_system()}")
print(f"Lattice type:       {sga_al.get_lattice_type()}")

# Example 2: BCC structure iron (Fe)
lattice_fe = Lattice.cubic(2.87)

# BCC structure: atoms at (0,0,0) and (0.5,0.5,0.5)
fe_structure = Structure(
    lattice_fe,
    ["Fe", "Fe"],
    [[0, 0, 0], [0.5, 0.5, 0.5]]
)

sga_fe = SpacegroupAnalyzer(fe_structure)

print("\n" + "=" * 100)
print("【Example 2: Crystal Structure Analysis of Iron (Fe)】")
print("=" * 100)
print(f"Lattice constant: a = 2.87 Å (cubic)")
print(f"Number of atoms: {len(fe_structure)}")
print(f"\nSpace group number:       {sga_fe.get_space_group_number()}")
print(f"Space group symbol:       {sga_fe.get_space_group_symbol()}")
print(f"Point group:             {sga_fe.get_point_group_symbol()}")
print(f"Crystal system:           {sga_fe.get_crystal_system()}")
print(f"Lattice type:       {sga_fe.get_lattice_type()}")

# Example 3: HCP structure magnesium (Mg)
# HCP: a = 3.21 Å, c = 5.21 Å, c/a = 1.624
lattice_mg = Lattice.hexagonal(a=3.21, c=5.21)

# HCP structure: atoms at (0,0,0), (1/3, 2/3, 1/2)
mg_structure = Structure(
    lattice_mg,
    ["Mg", "Mg"],
    [[0, 0, 0], [1/3, 2/3, 0.5]]
)

sga_mg = SpacegroupAnalyzer(mg_structure)

print("\n" + "=" * 100)
print("【Example 3: Crystal Structure Analysis of Magnesium (Mg)】")
print("=" * 100)
print(f"Lattice constants: a = 3.21 Å, c = 5.21 Å (hexagonal)")
print(f"c/a ratio: {5.21/3.21:.3f}")
print(f"Number of atoms: {len(mg_structure)}")
print(f"\nSpace group number:       {sga_mg.get_space_group_number()}")
print(f"Space group symbol:       {sga_mg.get_space_group_symbol()}")
print(f"Point group:             {sga_mg.get_point_group_symbol()}")
print(f"Crystal system:           {sga_mg.get_crystal_system()}")
print(f"Lattice type:       {sga_mg.get_lattice_type()}")

print("\n" + "=" * 100)
print("【Explanation】")
print("- Al (FCC): Space group 225 Fm-3m")
print("- Fe (BCC): Space group 229 Im-3m")
print("- Mg (HCP): Space group 194 P6₃/mmc")
print("- pymatgen can automatically determine space groups from atomic coordinates")
print("=" * 100)

Code Example 7: Calculating Equivalent Positions

Calculate equivalent positions generated from a single atomic position by symmetry operations of a space group.

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

"""
Example: Calculate equivalent positions generated from a single atomi

Purpose: Demonstrate core concepts and implementation patterns
Target: Beginner to Intermediate
Execution time: 10-30 seconds
Dependencies: None
"""

from pymatgen.symmetry.groups import SpaceGroup
import numpy as np

# Space group 225 (Fm-3m, FCC structure)
sg = SpaceGroup.from_int_number(225)

print("=" * 100)
print(f"Generation of Equivalent Positions in Space Group {sg.int_number}: {sg.symbol}")
print("=" * 100)

# General position example: (x, y, z) = (0.3, 0.2, 0.1)
original_position = np.array([0.3, 0.2, 0.1])

print(f"\nOriginal atomic position: ({original_position[0]:.2f}, {original_position[1]:.2f}, {original_position[2]:.2f})")
print(f"\nEquivalent positions generated by symmetry operations:")

# Set to store equivalent positions (for duplicate removal)
equivalent_positions = set()

for i, op in enumerate(sg.symmetry_ops, 1):
    # Apply symmetry operation
    new_position = op.operate(original_position)

    # Apply periodic boundary conditions (0 ≤ coord < 1)
    new_position = new_position % 1.0

    # Convert to tuple and add to set (for duplicate checking)
    pos_tuple = tuple(np.round(new_position, 6))
    equivalent_positions.add(pos_tuple)

# Display results
print(f"\nTotal number of equivalent positions: {len(equivalent_positions)}\n")

for i, pos in enumerate(sorted(equivalent_positions), 1):
    print(f"  {i:3d}. ({pos[0]:7.4f}, {pos[1]:7.4f}, {pos[2]:7.4f})")

# Special position example: (0, 0, 0) - high symmetry point
print("\n" + "=" * 100)
print("【Special Position Example: (0, 0, 0)】")
print("=" * 100)

special_position = np.array([0.0, 0.0, 0.0])
special_positions = set()

for op in sg.symmetry_ops:
    new_position = op.operate(special_position)
    new_position = new_position % 1.0
    pos_tuple = tuple(np.round(new_position, 6))
    special_positions.add(pos_tuple)

print(f"Total number of equivalent positions: {len(special_positions)}")
print("→ Special positions (high symmetry points) generate fewer equivalent positions than general positions")

print("\n" + "=" * 100)
print("【Explanation】")
print("- General position (x, y, z): many equivalent positions generated by symmetry operations")
print("- Special position (on symmetry elements): fewer equivalent positions (e.g., (0,0,0) only 1)")
print("- General position in Fm-3m multiplies 192-fold (192 symmetry operations)")
print("- In crystal structure description, using special positions requires fewer atoms")
print("=" * 100)

Code Example 8: Retrieving Crystal Examples from Space Group Number (Conceptual)

Finally, introduce how to search for real materials with specific space groups using Materials Project API integration (requires API key).

"""
Example of space group search using Materials Project API

Note: This code requires a Materials Project API key.
You can obtain an API key by registering for free at https://next-gen.materialsproject.org/

Installation:
pip install mp-api
"""

# The following is conceptual code (requires API key)

from mp_api.client import MPRester

def search_materials_by_space_group(space_group_number, api_key=None, max_results=5):
    """
    Search for materials with a specific space group

    Parameters:
    -----------
    space_group_number : int
        Space group number (1-230)
    api_key : str
        Materials Project API key
    max_results : int
        Maximum number of results to retrieve

    Returns:
    --------
    list : Search results (list of material ID, formula, space group symbol)
    """
    if api_key is None:
        print("Error: API key required")
        print("How to obtain: Register for free at https://next-gen.materialsproject.org/")
        return []

    with MPRester(api_key) as mpr:
        # Search by space group
        docs = mpr.materials.summary.search(
            spacegroup_number=space_group_number,
            fields=["material_id", "formula_pretty", "symmetry"],
            num_chunks=1
        )

        results = []
        for i, doc in enumerate(docs[:max_results]):
            results.append({
                'material_id': doc.material_id,
                'formula': doc.formula_pretty,
                'space_group': doc.symmetry.symbol,
                'crystal_system': doc.symmetry.crystal_system
            })

        return results


# Usage example (if you have API key)
# API_KEY = "your_api_key_here"
# results = search_materials_by_space_group(225, api_key=API_KEY, max_results=10)

# Demo output without API key
print("=" * 100)
print("【Representative Materials with Space Group 225 (Fm-3m, FCC)】")
print("=" * 100)

# Known material examples (manual list)
fcc_materials = [
    {'formula': 'Al', 'space_group': 'Fm-3m', 'crystal_system': 'cubic', 'description': 'Aluminum'},
    {'formula': 'Cu', 'space_group': 'Fm-3m', 'crystal_system': 'cubic', 'description': 'Copper'},
    {'formula': 'Au', 'space_group': 'Fm-3m', 'crystal_system': 'cubic', 'description': 'Gold'},
    {'formula': 'Ag', 'space_group': 'Fm-3m', 'crystal_system': 'cubic', 'description': 'Silver'},
    {'formula': 'Ni', 'space_group': 'Fm-3m', 'crystal_system': 'cubic', 'description': 'Nickel'},
    {'formula': 'Pt', 'space_group': 'Fm-3m', 'crystal_system': 'cubic', 'description': 'Platinum'},
    {'formula': 'Pb', 'space_group': 'Fm-3m', 'crystal_system': 'cubic', 'description': 'Lead'},
    {'formula': 'NaCl', 'space_group': 'Fm-3m', 'crystal_system': 'cubic', 'description': 'Sodium chloride'},
    {'formula': 'MgO', 'space_group': 'Fm-3m', 'crystal_system': 'cubic', 'description': 'Magnesium oxide'},
]

for i, mat in enumerate(fcc_materials, 1):
    print(f"{i:2d}. {mat['formula']:10s} | {mat['space_group']:10s} | {mat['description']}")

print("\n" + "=" * 100)
print("【Representative Materials with Space Group 229 (Im-3m, BCC)】")
print("=" * 100)

bcc_materials = [
    {'formula': 'Fe', 'space_group': 'Im-3m', 'crystal_system': 'cubic', 'description': 'Iron (α-phase, room temp)'},
    {'formula': 'Cr', 'space_group': 'Im-3m', 'crystal_system': 'cubic', 'description': 'Chromium'},
    {'formula': 'W', 'space_group': 'Im-3m', 'crystal_system': 'cubic', 'description': 'Tungsten'},
    {'formula': 'Mo', 'space_group': 'Im-3m', 'crystal_system': 'cubic', 'description': 'Molybdenum'},
    {'formula': 'V', 'space_group': 'Im-3m', 'crystal_system': 'cubic', 'description': 'Vanadium'},
    {'formula': 'Na', 'space_group': 'Im-3m', 'crystal_system': 'cubic', 'description': 'Sodium'},
    {'formula': 'K', 'space_group': 'Im-3m', 'crystal_system': 'cubic', 'description': 'Potassium'},
]

for i, mat in enumerate(bcc_materials, 1):
    print(f"{i:2d}. {mat['formula']:10s} | {mat['space_group']:10s} | {mat['description']}")

print("\n" + "=" * 100)
print("【Explanation】")
print("- Using Materials Project API (https://next-gen.materialsproject.org/),")
print("  you can search for real materials by space group number (free API key registration required)")
print("- The above examples are manually listed representative materials")
print("- FCC (No. 225) and BCC (No. 229) are the most common structures in metallic materials")
print("=" * 100)

2.6 Practice Problems

Exercise 1: Classification of 14 Bravais Lattices

Problem: Crystals with the following lattice parameters are classified into which crystal system and which Bravais lattice?

$a = b = c = 5.0$ Å, $\alpha = \beta = \gamma = 90°$, lattice points only at corners
$a = b = 3.2$ Å, $c = 5.1$ Å, $\alpha = \beta = 90°$, $\gamma = 120°$
$a = 4.0$ Å, $b = 5.0$ Å, $c = 6.0$ Å, $\alpha = \beta = \gamma = 90°$, lattice point at body center
$a = b = c = 4.05$ Å, $\alpha = \beta = \gamma = 90°$, lattice points at centers of all faces

Answers:

Cubic system, P (simple cubic) - all equal, 90°, corners only
Hexagonal system, P - $a = b \neq c$, $\gamma = 120°$
Orthorhombic system, I (body-centered) - $a \neq b \neq c$, all 90°, body center
Cubic system, F (FCC) - all equal, 90°, face-centered

Exercise 2: Packing Fraction Comparison of Three Cubic Bravais Lattices

Problem: For the three cubic Bravais lattices (P, I, F), calculate the packing fraction when atoms are treated as rigid spheres. Let lattice constant be $a$ and atomic radius be $r$.

Hint:

Packing fraction = (Volume of atoms) / (Volume of unit cell)
Volume of atom = $\frac{4}{3}\pi r^3$
Volume of unit cell = $a^3$
P: Atoms only at corners, nearest-neighbor distance = $a$
I (BCC): Atoms at corners + body center, nearest-neighbor distance = $\frac{\sqrt{3}}{2}a$
F (FCC): Atoms at corners + face centers, nearest-neighbor distance = $\frac{a}{\sqrt{2}}$

Answers:

1. Simple Cubic (P):

Atoms per unit cell: $8 \times \frac{1}{8} = 1$
Nearest-neighbor distance: $2r = a$ → $r = \frac{a}{2}$
Packing fraction: $\frac{1 \times \frac{4}{3}\pi r^3}{a^3} = \frac{\frac{4}{3}\pi (\frac{a}{2})^3}{a^3} = \frac{\pi}{6} \approx 0.524$ (52.4%)

2. BCC (I):

Atoms per unit cell: $8 \times \frac{1}{8} + 1 = 2$
Nearest-neighbor distance: $2r = \frac{\sqrt{3}}{2}a$ → $r = \frac{\sqrt{3}}{4}a$
Packing fraction: $\frac{2 \times \frac{4}{3}\pi r^3}{a^3} = \frac{2 \times \frac{4}{3}\pi (\frac{\sqrt{3}}{4}a)^3}{a^3} = \frac{\sqrt{3}\pi}{8} \approx 0.680$ (68.0%)

3. FCC (F):

Atoms per unit cell: $8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 4$
Nearest-neighbor distance: $2r = \frac{a}{\sqrt{2}}$ → $r = \frac{a}{2\sqrt{2}}$
Packing fraction: $\frac{4 \times \frac{4}{3}\pi r^3}{a^3} = \frac{4 \times \frac{4}{3}\pi (\frac{a}{2\sqrt{2}})^3}{a^3} = \frac{\pi}{3\sqrt{2}} \approx 0.740$ (74.0%)

Conclusion: FCC (74.0%) > BCC (68.0%) > Simple Cubic (52.4%)

Exercise 3: Identifying Types of Symmetry Operations

Problem: Classify the following symmetry operations by type:

90° rotation around z-axis
Reflection with respect to xy plane
Inversion with respect to origin: $(x, y, z) \rightarrow (-x, -y, -z)$
180° rotation followed by translation in rotation axis direction by half lattice vector
Reflection followed by translation parallel to reflection plane by half lattice vector

Answers:

4-fold rotation - Symbol: 4
Mirror reflection - Symbol: m
Inversion - Symbol: $\bar{1}$ or i
Screw axis - Symbol: 2₁ (2-fold screw)
Glide reflection - Symbol: a, b, c, n, d (varies by direction)

Exercise 4: Reading Hermann-Mauguin Symbols from Space Group Numbers

Problem: Explain the meaning of the following space group symbols.

P6₃/mmc (Space group 194)
Fm$\bar{3}$m (Space group 225)
Im$\bar{3}$m (Space group 229)

Answers:

1. P6₃/mmc:

P: Primitive lattice
6₃: 6-fold screw axis
/: Perpendicular symmetry element
m: Mirror plane
m: Another mirror plane
c: c-glide (glide reflection in c-axis direction)
Hexagonal system, HCP structure (Mg, Zn, Ti, etc.)

2. Fm$\bar{3}$m:

F: Face-centered lattice
m: Mirror plane
$\bar{3}$: 3-fold rotoinversion axis
m: Another mirror plane
Cubic system, FCC structure (Al, Cu, Au, Ag, etc.)

3. Im$\bar{3}$m:

I: Body-centered lattice
m: Mirror plane
$\bar{3}$: 3-fold rotoinversion axis
m: Another mirror plane
Cubic system, BCC structure (Fe, Cr, W, etc.)

Exercise 5: Investigating Space Group of Specific Crystal with pymatgen

Problem: Use pymatgen to investigate the space group of diamond (C) crystal structure. Diamond has the following characteristics:

Lattice constant: $a = 3.567$ Å (cubic)
Atomic coordinates: (0, 0, 0), (0.25, 0.25, 0.25), (0.5, 0.5, 0), (0.75, 0.75, 0.25), (0.5, 0, 0.5), (0.75, 0.25, 0.75), (0, 0.5, 0.5), (0.25, 0.75, 0.75)

Answer Code:

from pymatgen.core import Structure, Lattice
from pymatgen.symmetry.analyzer import SpacegroupAnalyzer

# Diamond structure
lattice_diamond = Lattice.cubic(3.567)

diamond_structure = Structure(
    lattice_diamond,
    ["C"] * 8,
    [
        [0, 0, 0], [0.25, 0.25, 0.25],
        [0.5, 0.5, 0], [0.75, 0.75, 0.25],
        [0.5, 0, 0.5], [0.75, 0.25, 0.75],
        [0, 0.5, 0.5], [0.25, 0.75, 0.75]
    ]
)

sga = SpacegroupAnalyzer(diamond_structure)

print("【Diamond (C) Crystal Structure Analysis】")
print(f"Space group number: {sga.get_space_group_number()}")
print(f"Space group symbol: {sga.get_space_group_symbol()}")
print(f"Point group: {sga.get_point_group_symbol()}")
print(f"Crystal system: {sga.get_crystal_system()}")
print(f"Lattice type: {sga.get_lattice_type()}")

Expected Output:

【Diamond (C) Crystal Structure Analysis】
Space group number: 227
Space group symbol: Fd-3m
Point group: m-3m
Crystal system: cubic
Lattice type: face_centered

Explanation:

Diamond is space group 227 Fd-3m (face-centered diamond cubic)
Structure with 2-atom basis on FCC lattice (diamond cubic structure)
Si and Ge also have this structure

2.7 Chapter Summary

What We Learned

14 Bravais Lattices
- Combination of 7 crystal systems and 4 centering types (P, C, I, F)
- Three cubic types (P, I, F) are most important in materials science
- Hexagonal (HCP) and cubic systems (FCC, BCC) are main structures in metallic materials
Symmetry Operations
- Point symmetry operations: rotation (1, 2, 3, 4, 6), mirror (m), inversion ($\bar{1}$), rotoinversion ($\bar{2}, \bar{3}, \bar{4}, \bar{6}$)
- Translational symmetry operations: screw, glide
- Crystallographic restriction: 5-fold and 7-fold or higher rotational symmetries do not exist
Space Groups
- 230 space groups classify all crystal structures
- Space group = point group (32 types) + translational symmetry operations
- Represented by Hermann-Mauguin notation (international symbols)
- General positions and special positions (Wyckoff positions)
Practical Use with pymatgen
- Retrieving and analyzing space group information
- Automatic space group determination from crystal structure
- Generating equivalent positions through symmetry operations
- Integration with Materials Project API (material search)

Important Points

Bravais lattices describe the "framework" of crystals
Space groups describe the "complete symmetry" of crystals
Same Bravais lattice can have different space groups due to atomic arrangement (basis)
FCC (No. 225), BCC (No. 229), HCP (No. 194) are the most important space groups
Pymatgen is a powerful tool for crystallographic analysis

To the Next Chapter

In Chapter 3, we will learn about Miller Indices and Reciprocal Lattice:

Notation of crystal planes and directions using Miller indices
Calculation of interplanar spacing
Reciprocal lattice concept and application to X-ray diffraction
Ewald sphere and Bragg condition
Calculation of reciprocal lattice vectors using pymatgen

Learning Objectives

2.1 What are Bravais Lattices?

Classification of Lattices

Lattice Point Arrangement Patterns

Existence of 14 Bravais Lattices

2.2 Details of the 14 Bravais Lattices

1. Triclinic System - 1 type

2. Monoclinic System - 2 types

3. Orthorhombic System - 4 types

4. Tetragonal System - 2 types

5. Hexagonal System - 1 type

6. Trigonal (Rhombohedral) System - 1 type

7. Cubic System - 3 types

Summary Table of 14 Bravais Lattices

2.3 Symmetry Operations

Point Symmetry Operations

1. Rotation

2. Mirror Reflection

3. Inversion

4. Rotoinversion

Translational Symmetry Operations

1. Glide Reflection

2. Screw Axis

Hermann-Mauguin Notation

2.4 Space Group Concepts

What are Space Groups?

Relationship Between Point Groups and Space Groups

Space Group Numbers and International Symbols

Reading International Symbols

General Positions and Special Positions

2.5 Analyzing Bravais Lattices and Space Groups with Python

Installing pymatgen

Code Example 1: Information Table for 14 Bravais Lattices

Code Example 2: Visualization of Three Cubic Bravais Lattices

Code Example 3: Comparison of Four Orthorhombic Bravais Lattices

Code Example 4: Retrieving Space Group Information with pymatgen

Code Example 5: List of Symmetry Operations for a Specific Space Group

Code Example 6: Space Group Determination from Crystal Structure

Code Example 7: Calculating Equivalent Positions

Code Example 8: Retrieving Crystal Examples from Space Group Number (Conceptual)

2.6 Practice Problems

2.7 Chapter Summary

What We Learned

Important Points

To the Next Chapter

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