Structure & Transformations · Vacancies

Equilibrium vacancy concentration: how many atoms are missing?

Drag the temperature in the first picture and watch how many lattice sites are empty. Then see how the number was measured, and why it sets how fast atoms move.

1 How many sites are empty 2 Counting them with a dilatometer 3 Diffusion and quenching Takeaways Questions References

How do you calculate the equilibrium number of vacancies?

A vacancy is a lattice site with no atom on it. The fraction of empty sites is cv = exp(Sf/k)·exp(−Ef/kT). Ef is the energy to make one vacancy. Sf is its formation entropy: extra entropy from the changed vibrations of the atoms around it. k is Boltzmann's constant and T the absolute temperature.

A: one plane; red rings are empty sites. B: cv against 1/T; squares are measured values. C: why some sites stay empty. D: how far apart vacancies are.
Metal
Fraction of sites empty, cv
One empty site in
Mean spacing between vacancies
Vacancies in a 1 mm cube
Lowest G, below a perfect crystal
cv grows, per 100 °C hotter

Try it: pick Copper and drag the temperature down to 20 °C: about 1 site in 2.9 × 1019 is empty, about 3 vacancies in a 1 mm cube. At 1075 °C it is 1 in 5,300.

How do you measure the vacancy concentration?

Simmons and Balluffi heated aluminium, then copper, and measured at the same temperatures the length of a bar and the lattice spacing by X-ray diffraction. Each vacancy adds one atom site, so the bar grows more than its lattice.

Top: bar length and lattice spacing, from 20 °C. Bottom: cv = 3(ΔL/L − Δa/a). The curves model the experiment (see More detail); squares are the published values.
Metal
Bar length ΔL/L
Lattice spacing Δa/a
Gap ΔL/L − Δa/a
cv = 3 × gap
Gap passes 1 in 105 above
Published cv

Try it: with aluminium, drag from 400 °C to 660 °C: the gap grows from 7.5 × 10−6 to 2.9 × 10−4, so cv = 3 × gap = 8.7 × 10−4. They measured 9.4 × 10−4.

How do vacancies help diffusion, and what does quenching do?

In a pure metal an atom moves by jumping into an empty neighbouring site, so it moves when a vacancy comes by. A quench (very fast cooling) keeps the vacancies the metal had when hot.

Grid: one vacancy (dashed ring) on a grid that wraps round at its edges. Bars: copper’s diffusion energy.
Vacancy jumps
Tagged atom jumps
Vacancy distance from start
Tagged atom distance from start
Quench: slow cooling follows the equilibrium curve down; a quench keeps the hot value at 20 °C.
Metal
Kept after the quench
Equilibrium at 20 °C

Try it: press Play; it pauses at 1,000 vacancy jumps. The tagged atom has moved only 7 times, and 4 of those jumps went straight back. Then quench aluminium from 600 °C: it keeps 4.8 × 108 times the equilibrium number at 20 °C.

What to take away

Few, but not zeroNear its melting point copper has about 2 empty sites per 10,000 (1.9 × 10−4 measured at 1075 °C). The same formula gives 3 × 10−20 at 20 °C.
Counted by bar against latticecv = 3(ΔL/L − Δa/a). In aluminium at 660 °C the bar outgrows its lattice by about 3 × 10−4.
Diffusion needs both energiesFor self-diffusion Q = Ef + Em. In copper, diffusion data give Q = 2.07 to 2.19 eV, and Ef = 1.17 eV is just over half of it.
More detail: the equations, the numbers and the limits

The formula. Put n vacancies among N + n sites. Then G = nEf − T(nSf + Sconfig), where Sconfig = k ln W and W = (N + n)!/(N!n!) counts the ways to place them. Setting dG/dn = 0 for a fixed number of atoms N (with Stirling's formula) gives n/(N + n) = exp(Sf/k)exp(−Ef/kT) (Porter, Easterling and Sherif, section 1.5.8). Panel C plots G per atom; its lowest point is about kTcv below zero. Many papers call Ef the formation enthalpy, Hf.

The Arrhenius plot. ln cv = Sf/k − Ef/kT is a straight line against 1/T with slope −Ef/k. Sf shifts it up or down.

Spacing and counts. The volume per atom is Ω = A/(ρNA): 0.0166 nm3 for aluminium and 0.0118 nm3 for copper (NIST densities and atomic masses). The mean spacing is (Ω/cv)1/3, the edge of a cube that holds one vacancy. Panel A is one plane; every plane has the same fraction empty.

Room temperature is an extrapolation. At 20 °C, 0.1 eV more or less in Ef changes cv 52 times. Published values differ by more than that, so treat room-temperature numbers as orders of magnitude.

The presets. Aluminium: Ef = 0.76 eV and Sf = 2.4k, Simmons and Balluffi's fit to their data. Copper: Ef = 1.17 ± 0.11 eV with an assumed Sf = 1.5k. Kraftmakher reviews why the methods disagree. Melting points are the ITS-90 values, 660.323 and 1084.62 °C.

The experiment. To first order, 3(ΔL/L − Δa/a) is the fraction of added sites: vacancies minus interstitials. Simmons and Balluffi estimated that divacancies (two vacancies side by side) are probably under about 15% of the total at aluminium's melting point. The lattice curves are a model. Aluminium uses Hidnert and Krider's mean expansion up to 400 °C and a straight-line trend above. Copper uses a smooth fit to Suh, Ohta and Waseda's X-ray lattice spacings from 20 to 1070 °C. Only the gap comes from cv.

Diffusion. An atom can jump only into an empty neighbour site. Each neighbour site is empty with a chance of about cv, and the jump must cross the barrier Em. So D ∝ exp(−(Ef + Em)/kT) (Porter, Easterling and Sherif, section 2.3). For copper, Butrymowicz, Manning and Read recommend Q = 2.19 eV from 700 to 1060 °C. Its monovacancy part is 2.07 eV; divacancies give about 25% of D at 1060 °C. Simmons and Balluffi's Ef is about 57% of Q, and their results imply Em = 0.88 eV. Aluminium's measured Q runs from 1.25 to 1.48 eV, depending on the temperature range (Mantina et al.).

The walk and the limits. On the 19 × 11 grid the vacancy's mean squared distance after n jumps is n jump lengths squared. The tagged atom often jumps straight back, because right after a swap the vacancy is beside it. In long runs on this square grid about half of its jumps go straight back. So its mean squared distance is only about 0.47 times that of a free random walk with as many jumps. This number is the correlation factor; in a 3D FCC metal it is 0.78 (Murch and Dyre). The quench picture assumes no vacancy is lost while cooling. In a real quench some are, and the rest leave later at grain boundaries, dislocations and surfaces (Porter, Easterling and Sherif, section 5.5.3).

Questions people ask

How do you calculate the number of vacancies?

Multiply the number of atom sites per m3, N = ρNA/A (8.49 × 1028 for copper at room-temperature density), by cv. With Ef = 1.17 eV and Sf = 1.5k, copper at 1000 °C has cv = 1.05 × 10−4, about 8.9 × 1024 vacancies per m3. Textbooks often write Nv = N exp(−Qv/kT) (Callister, section 4.2), which is the same as Sf = 0.

Why are there vacancies at equilibrium?

Each vacancy costs energy, but a few vacancies can sit among many sites in a huge number of ways, which raises the entropy. So G = H − TS first falls as vacancies are added, then rises. The lowest point is at cv: for aluminium at 660 °C, 7 × 10−5 eV per atom below a perfect crystal.

What is the vacancy concentration at the melting point?

Simmons and Balluffi measured 9.4 × 10−4 for aluminium at 660 °C and 1.9 × 10−4 for copper at 1075 °C, 10 °C below its melting point. That is about 1 site in 1,000 to 1 in 5,000. Some calorimetry studies give larger values; Kraftmakher reviews the debate.

What is the vacancy formation energy of copper and aluminium?

Copper: 1.17 ± 0.11 eV (Simmons and Balluffi, assuming Sf = 1.5k) and 1.29 ± 0.02 eV (Triftshäuser and McGervey). The second comes from positron annihilation: positrons get trapped in vacancies. A reanalysis that also counts divacancies gives 1.03 eV (Neumann et al.). Aluminium: 0.76 eV (Simmons and Balluffi) and 0.67 ± 0.03 eV, compiled from measurements by Ehrhart (as listed by Mantina et al.).

What is the difference between a Schottky and a Frenkel defect?

Both keep an ionic crystal neutral. A Schottky defect is 1 missing cation plus 1 missing anion (in a compound with equal numbers of each). A Frenkel defect is a cation moved from its site to an interstitial position: 1 vacancy plus 1 interstitial (Callister, section 12.5). A metal has no charge to balance, so a single vacancy can form alone.

Why does quenching increase the vacancy concentration?

A quench gives the extra vacancies of the hot metal no time to reach grain boundaries, dislocations or the surface, so they stay. In this page's model, aluminium quenched from 600 °C keeps 4.5 × 10−4, about 5 × 108 times its equilibrium value at 20 °C. In aluminium alloys they speed up natural ageing (Porter, Easterling and Sherif, section 5.5.3).

Are there more vacancies or interstitials in a metal?

Vacancies, by far. A self-interstitial squeezes a whole extra atom in between neighbours, which costs much more energy, so at equilibrium there are far fewer of them (Callister, section 4.2). The dilatometer counts vacancies minus interstitials: 9.4 × 10−4 for aluminium at 660 °C.

How do vacancies affect diffusion?

An atom moves by jumping into an empty neighbouring site. Its jump rate goes as cv (the chance that a given neighbour site is empty) times the rate of crossing the barrier Em. So Q = Ef + Em. For copper, diffusion measurements give Q = 2.19 eV between 700 and 1060 °C (Butrymowicz et al.). Simmons and Balluffi’s Ef = 1.17 eV is about 57% of Q, and their results imply Em = 0.88 eV.

Related: Diffusion couples (Fick's laws and the Kirkendall effect) · Kinetics for real alloys · Nucleation, spinodal and coarsening · Creep and deformation maps (diffusion creep) · Interstitial sites in FCC, BCC and HCP (where an extra atom would sit) · Dislocations and Burgers vectors · Short-range order

References

Show the 14 references
  1. R. O. Simmons and R. W. Balluffi, Measurements of equilibrium vacancy concentrations in aluminum, Physical Review 117, 52 to 61 (1960). doi:10.1103/PhysRev.117.52
  2. R. O. Simmons and R. W. Balluffi, Measurement of equilibrium concentrations of vacancies in copper, Physical Review 129, 1533 to 1544 (1963). doi:10.1103/PhysRev.129.1533
  3. W. Triftshäuser and J. D. McGervey, Monovacancy formation energy in copper, silver, and gold by positron annihilation, Applied Physics 6, 177 to 180 (1975). doi:10.1007/BF00883748
  4. G. Neumann, V. Tölle and C. Tuijn, Monovacancies and divacancies in copper: reanalysis of experimental data, Physica B 271, 21 to 27 (1999). doi:10.1016/S0921-4526(99)00209-4
  5. Y. Kraftmakher, Equilibrium vacancies and thermophysical properties of metals, Physics Reports 299, 79 to 188 (1998). doi:10.1016/S0370-1573(97)00082-3
  6. M. Mantina, Y. Wang, R. Arroyave, L. Q. Chen, Z. K. Liu and C. Wolverton, First-principles calculation of self-diffusion coefficients, Physical Review Letters 100, 215901 (2008), Tables I and II (measured values for aluminium, including P. Ehrhart's 1991 Landolt-Börnstein compilation). doi:10.1103/PhysRevLett.100.215901
  7. D. B. Butrymowicz, J. R. Manning and M. E. Read, Diffusion in copper and copper alloys. Part I. Volume and surface self-diffusion in copper, Journal of Physical and Chemical Reference Data 2, 643 to 656 (1973). doi:10.1063/1.3253129
  8. G. E. Murch and J. C. Dyre, Correlation effects in ionic conductivity, Critical Reviews in Solid State and Materials Sciences 15, 345 to 365 (1989): correlation factor 0.7815 for vacancy diffusion in FCC. doi:10.1080/10408438908243739
  9. P. Hidnert and H. S. Krider, Thermal expansion of aluminum and some aluminum alloys, Journal of Research of the National Bureau of Standards 48, 209 to 220 (1952), RP2308, Table 2.
  10. I. K. Suh, H. Ohta and Y. Waseda, High-temperature thermal expansion of six metallic elements measured by dilatation method and X-ray diffraction, Journal of Materials Science 23, 757 to 760 (1988). doi:10.1007/BF01174717. Copper lattice parameters, 293 to 1343 K, as listed in the American Mineralogist Crystal Structure Database, entries 0013087 to 0013096.
  11. NIST Temperature Scale Database (SRD 60): ITS-90 fixed points of aluminium and copper.
  12. J. H. Hubbell and S. M. Seltzer, X-Ray Mass Attenuation Coefficients, NIST Standard Reference Database 126, Table 1. doi:10.18434/T4D01F
  13. D. A. Porter, K. E. Easterling and M. Y. Sherif, Phase Transformations in Metals and Alloys, 3rd ed., CRC Press (2009): sections 1.5.8 (equilibrium vacancy concentration), 2.3 (substitutional diffusion) and 5.5.3 (quenched-in vacancies).
  14. W. D. Callister Jr. and D. G. Rethwisch, Materials Science and Engineering: An Introduction, 10th ed., Wiley (2018): sections 4.2 (vacancies and self-interstitials) and 12.5 (imperfections in ceramics).
Cite this page: Tripathy, Manisha. “Vacancy Concentration Lab.” untethered atom, 2026, https://untetheredatom.com/phase-transformations/vacancies.
BibTeX 
@misc{tripathy2026vacancies,
  author = {Tripathy, Manisha},
  title  = {Vacancy Concentration Lab},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/phase-transformations/vacancies}},
  note   = {Interactive web tool}
}
Last updated 3 November 2026.