Solid-State Chemistry/Part 4/Crystal Structure & Symmetry

4.1 Crystal Structure & Symmetry

Crystalline solids are characterized by a periodic arrangement of atoms in three dimensions. Understanding crystal structure is the starting point for predicting physical, chemical, and electronic properties of materials.

7 Crystal Systems & 14 Bravais Lattices

All crystals belong to one of 7 crystal systems defined by the relationships between lattice parameters $a, b, c$ and angles $\alpha, \beta, \gamma$:

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

Unit Cells

A primitive unit cell contains exactly one lattice point and is the smallest repeating unit. A conventional unit cell may contain multiple lattice points but reflects the full symmetry of the lattice.

- SC: 1 atom per unit cell (8 corners x 1/8)
- BCC: 2 atoms per unit cell (8 x 1/8 + 1 body center)
- FCC: 4 atoms per unit cell (8 x 1/8 + 6 x 1/2)

Miller Indices

Miller indices provide a notation for crystal planes and directions:

- (hkl): A crystal plane. Procedure: take intercepts on axes, take reciprocals, clear fractions.
- [uvw]: A crystal direction, specified by the smallest integers along the axes.
- {hkl}: Family of equivalent planes (by symmetry).
- <uvw>: Family of equivalent directions.

For cubic systems, the interplanar spacing is:

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

Packing Fraction

The atomic packing fraction (APF) is the fraction of volume in the unit cell occupied by atoms, assuming hard spheres:

$$\text{APF} = \frac{n \cdot \frac{4}{3}\pi r^3}{a^3}$$

$r = a/2$

52.4%

BCC

$r = a\sqrt{3}/4$

68.0%

FCC

$r = a\sqrt{2}/4$

74.0%

Common Crystal Structures

Structure	Lattice	CN	Radius Ratio	Examples
NaCl (rock salt)	FCC	6	0.414--0.732	NaCl, MgO, FeO
CsCl	SC	8	0.732--1.0	CsCl, CsBr
Diamond	FCC	4	--	C, Si, Ge
Zinc blende	FCC	4	0.225--0.414	GaAs, ZnS, InP
Wurtzite	HCP	4	0.225--0.414	ZnO, GaN, AlN

Radius Ratios

The radius ratio $r_+/r_-$ (cation to anion) predicts the coordination number and structure type for ionic crystals:

- $r_+/r_- < 0.155$: CN = 2 (linear)
- $0.155 \leq r_+/r_- < 0.225$: CN = 3 (trigonal planar)
- $0.225 \leq r_+/r_- < 0.414$: CN = 4 (tetrahedral, zinc blende)
- $0.414 \leq r_+/r_- < 0.732$: CN = 6 (octahedral, NaCl)
- $0.732 \leq r_+/r_- < 1.0$: CN = 8 (cubic, CsCl)

Video Lectures

18. Introduction to Crystallography

Goodie Bag 6: Crystallography

19. Crystallographic Notation

Python: Cubic Crystal Structures

3D visualization of SC, BCC, and FCC unit cells with atomic positions and packing fraction calculations.

Cubic Crystal Structures Visualization

Python

Atomic positions and packing fractions for SC, BCC, and FCC

script.py106 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

# ─────────────────────────────────────────────────
# Cubic Crystal Structures: SC, BCC, FCC
# Atomic positions and packing fraction calculation
# ─────────────────────────────────────────────────

fig = plt.figure(figsize=(18, 6))
fig.patch.set_facecolor('#0a0a1a')

structures = {
    'Simple Cubic (SC)': {
        'atoms': [(0,0,0), (1,0,0), (0,1,0), (0,0,1),
                  (1,1,0), (1,0,1), (0,1,1), (1,1,1)],
        'interior': [],
        'atoms_per_cell': 1,
        'packing': np.pi / 6,
        'coord_num': 6,
        'r_over_a': 0.5,
        'color': '#22d3ee',
    },
    'BCC': {
        'atoms': [(0,0,0), (1,0,0), (0,1,0), (0,0,1),
                  (1,1,0), (1,0,1), (0,1,1), (1,1,1)],
        'interior': [(0.5, 0.5, 0.5)],
        'atoms_per_cell': 2,
        'packing': np.pi * np.sqrt(3) / 8,
        'coord_num': 8,
        'r_over_a': np.sqrt(3)/4,
        'color': '#f59e0b',
    },
    'FCC': {
        'atoms': [(0,0,0), (1,0,0), (0,1,0), (0,0,1),
                  (1,1,0), (1,0,1), (0,1,1), (1,1,1)],
        'interior': [(0.5,0.5,0), (0.5,0,0.5), (0,0.5,0.5),
                     (1,0.5,0.5), (0.5,1,0.5), (0.5,0.5,1)],
        'atoms_per_cell': 4,
        'packing': np.pi * np.sqrt(2) / 6,
        'coord_num': 12,
        'r_over_a': np.sqrt(2)/4,
        'color': '#a78bfa',
    },
}

print("=" * 60)
print("Cubic Crystal Structures")
print("=" * 60)

for idx, (name, props) in enumerate(structures.items()):
    ax = fig.add_subplot(1, 3, idx + 1, projection='3d')
    ax.set_facecolor('#0a0a1a')

# Draw unit cell edges
    edges = [
        [(0,0,0),(1,0,0)], [(0,0,0),(0,1,0)], [(0,0,0),(0,0,1)],
        [(1,0,0),(1,1,0)], [(1,0,0),(1,0,1)], [(0,1,0),(1,1,0)],
        [(0,1,0),(0,1,1)], [(0,0,1),(1,0,1)], [(0,0,1),(0,1,1)],
        [(1,1,0),(1,1,1)], [(1,0,1),(1,1,1)], [(0,1,1),(1,1,1)],
    ]
    for edge in edges:
        xs = [edge[0][0], edge[1][0]]
        ys = [edge[0][1], edge[1][1]]
        zs = [edge[0][2], edge[1][2]]
        ax.plot(xs, ys, zs, color='#475569', linewidth=0.8, alpha=0.6)

# Plot corner atoms
    for atom in props['atoms']:
        ax.scatter(*atom, color=props['color'], s=200, alpha=0.6, edgecolors='white', linewidths=0.5)

# Plot interior atoms
    for atom in props['interior']:
        ax.scatter(*atom, color=props['color'], s=300, alpha=0.9, edgecolors='white', linewidths=1)

ax.set_title(name, color='white', fontsize=13, pad=10)
    ax.set_xlim(-0.2, 1.2)
    ax.set_ylim(-0.2, 1.2)
    ax.set_zlim(-0.2, 1.2)
    ax.tick_params(colors='#64748b', labelsize=7)
    ax.xaxis.pane.fill = False
    ax.yaxis.pane.fill = False
    ax.zaxis.pane.fill = False

pf = props['packing'] * 100
    print(f"\n{name}:")
    print(f"  Atoms per unit cell: {props['atoms_per_cell']}")
    print(f"  Coordination number: {props['coord_num']}")
    print(f"  r/a = {props['r_over_a']:.4f}")
    print(f"  Packing fraction: {pf:.1f}%")

print("\n" + "=" * 60)
print("Common Crystal Structure Types")
print("=" * 60)
print("  NaCl (rock salt): FCC with basis, CN = 6")
print("  CsCl: SC with basis, CN = 8")
print("  Diamond: FCC with 2-atom basis, CN = 4")
print("  Zinc blende (ZnS): Diamond with 2 different atoms")
print("  Wurtzite: HCP with 2-atom basis, CN = 4")

plt.tight_layout()
plt.savefig('crystal_structures.png', dpi=150, bbox_inches='tight', facecolor='#0a0a1a')
print("\nPlot saved as 'crystal_structures.png'")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Miller Index & d-Spacing Calculator

Computes interplanar d-spacings and Bragg angles for all planes up to a maximum Miller index, including FCC and BCC selection rules.

Miller Index d-Spacing Calculator

Fortran

Interplanar spacings and selection rules for cubic crystals

miller_index.f9095 lines

program miller_index_calculator
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: pi = 3.14159265358979_dp

integer :: h, k, l, hmax, count
  real(dp) :: a, d_hkl
  real(dp) :: lambda, theta_deg

! Lattice parameter (angstroms) - default Cu
  a = 3.615_dp   ! Cu lattice parameter in angstroms
  lambda = 1.5406_dp  ! Cu K-alpha wavelength in angstroms
  hmax = 5

write(*,'(A)') '============================================================'
  write(*,'(A)') ' Miller Index Calculator: d-spacings for Cubic Crystal'
  write(*,'(A)') '============================================================'
  write(*,'(A,F7.4,A)') ' Lattice parameter a = ', a, ' Angstroms (Cu)'
  write(*,'(A,F7.4,A)') ' X-ray wavelength  = ', lambda, ' Angstroms (Cu K-alpha)'
  write(*,'(A,I2)')      ' Max Miller index  = ', hmax
  write(*,'(A)') ''
  write(*,'(A)') ' (hkl)    h^2+k^2+l^2    d_hkl(A)    2theta(deg)'
  write(*,'(A)') ' ------   -----------     --------    -----------'

count = 0
  do h = 0, hmax
    do k = 0, h
      do l = 0, k
        if (h == 0 .and. k == 0 .and. l == 0) cycle
        d_hkl = a / sqrt(real(h*h + k*k + l*l, dp))

! Check if Bragg condition can be satisfied (sin(theta) <= 1)
        if (lambda / (2.0_dp * d_hkl) <= 1.0_dp) then
          theta_deg = asin(lambda / (2.0_dp * d_hkl)) * 180.0_dp / pi
          write(*,'(2X,A1,I1,I1,I1,A1,5X,I5,10X,F8.4,5X,F8.3)') &
            '(', h, k, l, ')', h*h+k*k+l*l, d_hkl, 2.0_dp * theta_deg
        else
          write(*,'(2X,A1,I1,I1,I1,A1,5X,I5,10X,F8.4,5X,A)') &
            '(', h, k, l, ')', h*h+k*k+l*l, d_hkl, 'not accessible'
        end if
        count = count + 1
      end do
    end do
  end do

write(*,'(A)') ''
  write(*,'(A,I3,A)') ' Total unique planes computed: ', count, ''

! FCC selection rules
  write(*,'(A)') ''
  write(*,'(A)') ' --- FCC Selection Rules ---'
  write(*,'(A)') ' Allowed: h,k,l all odd or all even'
  write(*,'(A)') ' Allowed FCC reflections for Cu:'
  do h = 0, hmax
    do k = 0, h
      do l = 0, k
        if (h == 0 .and. k == 0 .and. l == 0) cycle
        ! FCC: all odd or all even
        if ((mod(h,2)==mod(k,2)) .and. (mod(k,2)==mod(l,2))) then
          d_hkl = a / sqrt(real(h*h + k*k + l*l, dp))
          if (lambda / (2.0_dp * d_hkl) <= 1.0_dp) then
            theta_deg = asin(lambda / (2.0_dp * d_hkl)) * 180.0_dp / pi
            write(*,'(4X,A1,I1,I1,I1,A1,3X,A3,F8.4,A,3X,A7,F7.3)') &
              '(', h, k, l, ')', 'd =', d_hkl, ' A', '2th =', 2.0_dp * theta_deg
          end if
        end if
      end do
    end do
  end do

! BCC selection rules
  write(*,'(A)') ''
  write(*,'(A)') ' --- BCC Selection Rules ---'
  write(*,'(A)') ' Allowed: h+k+l = even'
  write(*,'(A,F7.4,A)') ' Allowed BCC reflections (Fe, a = ', 2.8665_dp, ' A):'
  a = 2.8665_dp  ! Fe
  do h = 0, hmax
    do k = 0, h
      do l = 0, k
        if (h == 0 .and. k == 0 .and. l == 0) cycle
        if (mod(h+k+l, 2) == 0) then
          d_hkl = a / sqrt(real(h*h + k*k + l*l, dp))
          if (lambda / (2.0_dp * d_hkl) <= 1.0_dp) then
            theta_deg = asin(lambda / (2.0_dp * d_hkl)) * 180.0_dp / pi
            write(*,'(4X,A1,I1,I1,I1,A1,3X,A3,F8.4,A,3X,A7,F7.3)') &
              '(', h, k, l, ')', 'd =', d_hkl, ' A', '2th =', 2.0_dp * theta_deg
          end if
        end if
      end do
    end do
  end do

write(*,'(A)') ''
  write(*,'(A)') 'Done.'
end program miller_index_calculator

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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