Solid-State Chemistry/Part 5/Point and Line Defects

5.1 Point and Line Defects

Real crystals are never perfect. Defects govern many material properties including mechanical strength, electrical conductivity, diffusion rates, and chemical reactivity. Understanding defects is essential for engineering materials with desired properties.

Point Defects

Point defects are localized disruptions involving one or a few atoms:

Vacancy

A missing atom from a lattice site. The most common point defect in metals. Concentration increases exponentially with temperature.

Interstitial

An extra atom squeezed into a space between regular lattice sites. Can be self-interstitial or foreign atom.

Substitutional

A foreign atom replacing a host atom on a regular lattice site. Governed by the Hume-Rothery rules.

Vacancy Concentration

The equilibrium vacancy concentration at temperature $T$ follows an Arrhenius relationship:

$$n_v = N\exp\!\left(-\frac{E_v}{k_BT}\right)$$

where $N$ is the total number of atomic sites and $E_v$ is the vacancy formation energy. Near the melting point of most metals, about 1 in $10^4$ sites is vacant.

Schottky and Frenkel Defects

In ionic crystals, defects must maintain charge neutrality:

Schottky Defect

A pair of cation and anion vacancies. Common in NaCl, KCl, MgO.

$$n_s = N\exp\!\left(-\frac{E_s}{2k_BT}\right)$$

Frenkel Defect

An ion displaced to an interstitial site, leaving a vacancy. Common in AgCl, AgBr.

$$n_f = \sqrt{NN_i}\exp\!\left(-\frac{E_f}{2k_BT}\right)$$

Line Defects: Dislocations

Dislocations are line defects that play a central role in plastic deformation:

Edge Dislocation

An extra half-plane of atoms inserted into the crystal. The Burgers vector$\vec{b}$ is perpendicular to the dislocation line.

Screw Dislocation

A spiral ramp around the dislocation line. The Burgers vector $\vec{b}$ is parallel to the dislocation line.

The Burgers vector $\vec{b}$ characterizes the magnitude and direction of lattice distortion. Slip occurs on close-packed planes in close-packed directions, where $|\vec{b}|$ is smallest.

Strengthening Mechanisms

All strengthening mechanisms work by impeding dislocation motion:

- Work hardening: Increasing dislocation density makes further motion more difficult (tangled dislocations block each other).
- Grain boundary strengthening: (Hall-Petch) Grain boundaries act as barriers. Finer grains give higher strength: $\sigma_y = \sigma_0 + k_y / \sqrt{d}$.
- Solid solution strengthening: Solute atoms (substitutional or interstitial) create local strain fields that interact with dislocations.
- Precipitation hardening: Second-phase particles act as obstacles that dislocations must cut through or bow around (Orowan mechanism).

Video Lectures

23. Point and Line Defects I

Goodie Bag 7: Defects

24. Point and Line Defects II

Python: Vacancy Concentration vs Temperature

Equilibrium vacancy fraction as a function of temperature for several metals with different formation energies. Includes both linear and Arrhenius plots.

Vacancy Concentration vs Temperature

Python

Equilibrium vacancy fractions for Cu, Al, Au, Fe, and W

script.py79 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
# ─────────────────────────────────────────────────
# Vacancy Concentration vs Temperature
# for different formation energies
# ─────────────────────────────────────────────────

kB = 8.617e-5  # eV/K

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))
fig.patch.set_facecolor('#0a0a1a')
for ax in (ax1, ax2):
    ax.set_facecolor('#0a0a1a')
    for spine in ax.spines.values():
        spine.set_edgecolor('#334155')

T = np.linspace(300, 1800, 500)

# Formation energies for different metals (eV)
metals = {
    'Cu (Ev=1.28 eV)': {'Ev': 1.28, 'Tm': 1358, 'N': 8.49e22, 'color': '#22d3ee'},
    'Al (Ev=0.75 eV)': {'Ev': 0.75, 'Tm': 933, 'N': 6.02e22, 'color': '#a78bfa'},
    'Au (Ev=0.94 eV)': {'Ev': 0.94, 'Tm': 1337, 'N': 5.90e22, 'color': '#f59e0b'},
    'Fe (Ev=1.49 eV)': {'Ev': 1.49, 'Tm': 1811, 'N': 8.49e22, 'color': '#ef4444'},
    'W  (Ev=3.10 eV)': {'Ev': 3.10, 'Tm': 3695, 'N': 6.32e22, 'color': '#22c55e'},
}

print("=" * 65)
print("Equilibrium Vacancy Concentration in Metals")
print("=" * 65)
print(f"\n{'Metal':<20} {'Ev(eV)':<10} {'Tm(K)':<10} {'nv/N at Tm':<15} {'nv(cm-3) at Tm'}")
print("-" * 65)

for name, props in metals.items():
    Ev = props['Ev']
    Tm = props['Tm']
    N = props['N']

# nv/N = exp(-Ev/kBT)
    ratio = np.exp(-Ev / (kB * T))
    nv = N * ratio

ax1.semilogy(T, ratio, color=props['color'], linewidth=2, label=name)

# Arrhenius plot: ln(nv/N) vs 1000/T
    ax2.semilogy(1000/T, ratio, color=props['color'], linewidth=2, label=name)

# Value at melting point
    ratio_Tm = np.exp(-Ev / (kB * Tm))
    nv_Tm = N * ratio_Tm
    print(f"{name:<20} {Ev:<10.2f} {Tm:<10.0f} {ratio_Tm:<15.3e} {nv_Tm:<.3e}")

ax1.set_xlabel('Temperature (K)', color='#94a3b8', fontsize=12)
ax1.set_ylabel('nv / N (vacancy fraction)', color='#94a3b8', fontsize=12)
ax1.set_title('Vacancy Fraction vs Temperature', color='white', fontsize=14)
ax1.legend(fontsize=9, facecolor='#0a0a1a', edgecolor='#334155', labelcolor='white')
ax1.tick_params(colors='#94a3b8')
ax1.grid(True, alpha=0.2)
ax1.set_ylim(1e-20, 1e-1)

ax2.set_xlabel('1000/T (K$^{-1}$)', color='#94a3b8', fontsize=12)
ax2.set_ylabel('nv / N', color='#94a3b8', fontsize=12)
ax2.set_title('Arrhenius Plot', color='white', fontsize=14)
ax2.legend(fontsize=9, facecolor='#0a0a1a', edgecolor='#334155', labelcolor='white')
ax2.tick_params(colors='#94a3b8')
ax2.grid(True, alpha=0.2)
ax2.set_ylim(1e-20, 1e-1)

print("\nNote: nv/N at room temperature (300 K):")
for name, props in metals.items():
    r300 = np.exp(-props['Ev'] / (kB * 300))
    print(f"  {name}: {r300:.2e}")

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

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Vacancy & Schottky Defect Calculator

Computes equilibrium concentrations for single vacancies in Cu, Schottky pairs in NaCl, and Frenkel defects in AgCl across a range of temperatures.

Vacancy & Schottky Defect Concentrations

Fortran

Equilibrium defect concentrations for metals and ionic crystals

vacancy_schottky.f9085 lines

program vacancy_schottky
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  real(dp), parameter :: kB = 8.617d-5  ! eV/K

real(dp) :: T, Ev, Es, N, nv, ns
  real(dp) :: temps(6)
  integer :: i, j

temps = (/ 300.0_dp, 500.0_dp, 800.0_dp, 1000.0_dp, 1200.0_dp, 1500.0_dp /)

write(*,'(A)') '============================================================'
  write(*,'(A)') ' Equilibrium Vacancy & Schottky Defect Concentrations'
  write(*,'(A)') '============================================================'

! --- Single Vacancies in Metals ---
  write(*,'(A)') ''
  write(*,'(A)') ' === Single Vacancies in Cu (Ev = 1.28 eV) ==='
  Ev = 1.28_dp
  N = 8.49d22  ! atomic sites per cm^3

write(*,'(A)') ' T(K)      nv/N           nv(cm-3)'
  write(*,'(A)') ' ----      ----           --------'

do i = 1, 6
    T = temps(i)
    nv = N * exp(-Ev / (kB * T))
    write(*,'(F7.1,3X,ES14.4,3X,ES14.4)') T, exp(-Ev / (kB * T)), nv
  end do

! --- Schottky Defects in NaCl ---
  write(*,'(A)') ''
  write(*,'(A)') ' === Schottky Defect Pairs in NaCl (Es = 2.30 eV) ==='
  Es = 2.30_dp  ! Schottky pair formation energy
  N = 2.23d22   ! ion pairs per cm^3

write(*,'(A)') ' ns = N * exp(-Es/(2*kB*T))  [note factor of 2]'
  write(*,'(A)') ''
  write(*,'(A)') ' T(K)      ns/N           ns(cm-3)       pairs/million'
  write(*,'(A)') ' ----      ----           --------       -------------'

do i = 1, 6
    T = temps(i)
    ! Schottky: ns = N * exp(-Es/(2kT))
    ns = N * exp(-Es / (2.0_dp * kB * T))
    write(*,'(F7.1,3X,ES14.4,3X,ES14.4,3X,F14.4)') &
      T, exp(-Es / (2.0_dp * kB * T)), ns, &
      exp(-Es / (2.0_dp * kB * T)) * 1.0d6
  end do

! --- Frenkel Defects in AgCl ---
  write(*,'(A)') ''
  write(*,'(A)') ' === Frenkel Defects in AgCl (Ef = 1.40 eV) ==='
  write(*,'(A)') ' nf = sqrt(N*Ni) * exp(-Ef/(2*kB*T))'
  write(*,'(A)') ' where Ni ~ N (number of interstitial sites)'
  write(*,'(A)') ''

N = 2.57d22  ! ion sites per cm^3
  Ev = 1.40_dp ! Frenkel pair formation energy

write(*,'(A)') ' T(K)      nf/N           nf(cm-3)'
  write(*,'(A)') ' ----      ----           --------'

do i = 1, 6
    T = temps(i)
    ! Frenkel: nf = sqrt(N*Ni)*exp(-Ef/(2kT)), with Ni ~ N
    nv = N * exp(-Ev / (2.0_dp * kB * T))
    write(*,'(F7.1,3X,ES14.4,3X,ES14.4)') T, exp(-Ev / (2.0_dp * kB * T)), nv
  end do

! --- Comparison at melting points ---
  write(*,'(A)') ''
  write(*,'(A)') ' === Summary: Defect Fractions near Melting Point ==='
  write(*,'(A)') ' System          Tm(K)    E(eV)    n/N at Tm'
  write(*,'(A)') ' -------         -----    -----    ---------'
  write(*,'(A,F7.0,F8.2,ES14.4)') ' Cu vacancy      ', 1358.0_dp, 1.28_dp, &
    exp(-1.28_dp / (kB * 1358.0_dp))
  write(*,'(A,F7.0,F8.2,ES14.4)') ' NaCl Schottky   ', 1074.0_dp, 2.30_dp, &
    exp(-2.30_dp / (2.0_dp * kB * 1074.0_dp))
  write(*,'(A,F7.0,F8.2,ES14.4)') ' AgCl Frenkel    ',  728.0_dp, 1.40_dp, &
    exp(-1.40_dp / (2.0_dp * kB * 728.0_dp))

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

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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