Structure & Transformations · Crystallography
Interstitial sites: octahedral and tetrahedral holes in FCC, BCC and HCP
Click a marker in the cell below to see the atoms around that hole. Then switch from FCC to BCC and watch the counts change.
1 Where the holes are 2 How big each hole is 3 Filling the holes Takeaways Questions References
Where are the octahedral and tetrahedral sites in FCC, BCC and HCP?
An interstitial site is a gap between atoms where a small atom can sit: 6 atoms surround an octahedral site, 4 surround a tetrahedral site.
Try it: FCC, Octahedral: 13 markers are drawn, but the cell owns 4 of them (1 + 12 × ¼). Press BCC: now 2 atoms share 6 octahedral and 12 tetrahedral sites.
Which is bigger, the octahedral or the tetrahedral hole?
Treat atoms as hard spheres of radius R that touch. The hole radius r is the largest sphere that fits without pushing any atom.
Try it: α-Fe with carbon: the octahedral hole (0.019 nm) is about half the tetrahedral one (0.036 nm), yet just 2 of its 6 atoms must move. Press H: hydrogen fits the tetrahedral hole with no push at all.
Why does austenite dissolve more carbon than ferrite, and which holes do ions fill?
Austenite has bigger holes than ferrite and dissolves far more carbon. The second tab fills the same FCC holes with ions to build salt crystals.
Try it: drag carbon to 1.2 wt%: c/a reaches 1.054, close to the 1.05 measured by X-rays in a 1.2 wt% C steel. Then open Ions in salts and set the ratio to 0.30: the cation is too small to touch its 6 anions, and the rule switches to tetrahedral.
What to take away
More detail: the equations, the numbers and the limits
How the page finds the holes. It searches a grid of points 1/24 of a cell edge apart. A hole centre is a point whose nearest atoms are all the same distance away, sit evenly around it and leave room. Four such atoms make a tetrahedral site; six nearest atoms in three opposite pairs make an octahedral site. Moving every site back into one cell removes the copies on faces and edges, which gives the count.
Hole sizes. r = d − R, with R half the nearest atom spacing and d the distance from the hole centre to its nearest atom. FCC: R = a√2/4; octahedral d = a/2 gives r/R = √2 − 1 = 0.414; tetrahedral d = a√3/4 gives √(3/2) − 1 = 0.225. BCC: R = a√3/4; octahedral d = a/2 gives 2/√3 − 1 = 0.155; tetrahedral d = a√5/4 gives √(5/3) − 1 = 0.291. In α-Fe that is 0.067a and 0.126a (Bhadeshia 2013). Ideal HCP (c/a = 1.633) has the FCC values; titanium (c/a = 1.588) gives 0.427 and 0.236.
Hosts and solutes. Lattice parameters at 25 °C from NBS Circular 539: α-Fe 0.28664, Cu 0.36150, Ni 0.35238, Al 0.40494 nm; Ti a = 0.2950, c = 0.4686 nm. For γ-Fe the page uses 0.3656 nm at 1000 °C. This comes from the X-ray fit of Seki and Nagata (2005), a = 0.35519 + 8.1593 × 10−6 T nm (T in kelvin, 1183 to 1550 K). Solute radii are covalent radii (Cordero and co-workers 2008). That is one choice: a dissolved atom has no single fixed size, so use the pushes to compare sites. The main result does not depend on this choice. Any carbon radius from 0.020 to 0.078 nm pushes just the 2 close atoms of the α-Fe octahedral hole.
Why carbon picks the smaller hole in α-Fe. In the octahedral hole carbon pushes 2 atoms along ⟨100⟩, the softest direction in an iron crystal at room temperature (Young's modulus E = 134 GPa). In the tetrahedral hole it pushes 4 atoms along ⟨210⟩ (E = 180 GPa). Beshers' estimate, energy ∝ sum of (push)2 × E, gives tetrahedral ÷ octahedral = 1.38 for a 0.08 nm atom (Bhadeshia 2013). Bhadeshia does not list the moduli he used; with the Rayne and Chandrasekhar elastic constants the same formula gives 1.40. On this estimate the octahedral site stays cheaper for any carbon radius from 0.063 to 0.15 nm. Bhadeshia adds that newer calculations find a large difference in chemical energy, which also favours the octahedral site.
Solubility. First-principles calculations find that dissolving carbon costs energy in ferrite (+0.74 eV) but releases it in austenite (−0.17 eV) (Jiang and Carter 2003).
Martensite. Roberts (1953): a = 0.2861 − 0.0013x and c = 0.2861 + 0.0116x nm, with x in wt% C, so c/a ≈ 1 + 0.045x. It is a straight-line fit to X-ray data. This 0.045 slope came from steels with more than about 0.6 wt% C. Newer X-ray work on steels with 0.12 to 1.24 wt% C finds a smaller slope, c/a = 1 + 0.031x (Lu and co-workers 2017). The page uses Roberts' fit from 0 to 1.4 wt%, so below 0.6 wt% it is an extension. The carbon fills one family of octahedral sites, so the cell stretches along one axis (Bhadeshia 2013).
Radius ratio rule. A cation is taken as stable when it touches all its anion neighbours: from 0.225 for a tetrahedron of touching anions, 0.414 for an octahedron and 0.732 for a cube (Pauling 1929). The examples use Shannon (1976) radii for the real coordination. The rule is a rough guide with many exceptions: it fails here for LiI, ZnO and KCl. A test of Pauling's rules on about 5,000 oxides found they predict much less well than expected (George and co-workers 2020).
Limits. Hard spheres in a rigid host, one solute atom at a time. Real hosts relax, solutes interact, and solubility limits come from free energies, not from hole size alone.
Questions people ask
How many octahedral and tetrahedral voids are in FCC?
4 octahedral and 8 tetrahedral per unit cell. The cell holds 4 atoms, so that is 1 octahedral and 2 tetrahedral voids per atom. HCP has the same numbers per atom.
Which is bigger, the octahedral or the tetrahedral void?
In FCC and HCP the octahedral void: r/R = 0.414 against 0.225 for hard spheres. In BCC it is the other way round: the tetrahedral void is 0.291 and the octahedral void 0.155.
Why does carbon occupy octahedral sites in BCC iron?
The octahedral hole is small (0.019 nm), but just 2 of its 6 iron atoms are close, along one cube edge. Carbon pushes those 2 apart along ⟨100⟩, the softest direction in iron. In the bigger tetrahedral hole (0.036 nm) it must push 4 atoms along a stiffer direction. Beshers' elastic estimate makes that about 1.4 times more costly. Newer calculations add a chemical energy difference that also favours the octahedral site (Bhadeshia 2013).
Why is carbon more soluble in austenite than ferrite?
Austenite's octahedral holes are bigger: 0.054 nm at 1000 °C, against 0.019 nm in ferrite at 25 °C. Austenite dissolves up to 2.14 wt% C at 1147 °C; ferrite at most 0.022 wt% at 727 °C, about 100 times less (Callister).
What is the radius ratio for an octahedral site?
0.414, which is √2 − 1. Four touching spheres in a square leave a gap whose centre is R√2 from each sphere centre, so the largest sphere that fits has radius (√2 − 1)R. For ions, cation to anion ratios from 0.414 to 0.732 favour 6 neighbours.
Are interstitial sites the same in HCP and FCC?
Per atom, yes: 1 octahedral and 2 tetrahedral, with the same sizes at the ideal c/a of 1.633. The layout differs. In HCP the tetrahedral sites come in pairs 0.41 atom spacings apart, and the octahedral sites stack in columns along c, 0.82 apart. In FCC the closest pairs are 0.71 and 1.0 apart.
Why does martensite become tetragonal?
A fast quench traps the carbon. It sits in the octahedral sites whose 2 close atoms lie along z, so it pushes the cell longer along z. Roberts' fit gives c/a = 1 + 0.045 × wt% C, for example 1.036 at 0.8 wt% C. That slope comes from steels above about 0.6 wt% C.
Where does hydrogen sit in iron?
In BCC iron, in the tetrahedral sites (Jiang and Carter 2004). In the hard-sphere picture its covalent radius, 0.031 nm, is smaller than the 0.036 nm tetrahedral hole, so no iron atom has to move.
Related: Bravais lattices and Miller indices · Hume-Rothery rules (when an atom goes interstitial) · Diffusion couples (atoms on the move) · Strengthening mechanisms (solid solutions) · Vacancies (the other point defect) · Stacking faults and twins (FCC and HCP stacking) · Intermetallic structures · Space groups
References
Show the 14 references
- H. K. D. H. Bhadeshia, Carbon in cubic and tetragonal ferrite, Philosophical Magazine 93, 3714 to 3725 (2013). doi:10.1080/14786435.2013.775518. Hole sizes, the carbon site and Beshers' estimate (from D. N. Beshers, Journal of Applied Physics 36, 290 to 300, 1965).
- D. E. Jiang and E. A. Carter, Carbon dissolution and diffusion in ferrite and austenite from first principles, Physical Review B 67, 214103 (2003). doi:10.1103/PhysRevB.67.214103
- D. E. Jiang and E. A. Carter, Diffusion of interstitial hydrogen into and through bcc Fe from first principles, Physical Review B 70, 064102 (2004). doi:10.1103/PhysRevB.70.064102
- C. S. Roberts, Effect of carbon on the volume fractions and lattice parameters of retained austenite and martensite, JOM 5, 203 to 204 (1953), also Trans. AIME 197. doi:10.1007/BF03397477
- G. Nolze, A. Winkelmann, G. Cios and T. Tokarski, Tetragonality mapping of martensite in a high-carbon steel by EBSD, Materials Characterization 175, 111040 (2021). doi:10.1016/j.matchar.2021.111040. Quotes Roberts' relation; X-ray c/a = 1.05 at 1.2 wt% C.
- Y. Lu, H. Yu and R. D. Sisson, The effect of carbon content on the c/a ratio of as-quenched martensite in Fe-C alloys, Materials Science and Engineering A 700, 592 to 597 (2017). doi:10.1016/j.msea.2017.05.094
- W. D. Callister, Materials Science and Engineering: An Introduction, 7th edition, Wiley (2007): Section 9.18, the iron–iron carbide phase diagram (Fig. 9.24, adapted from T. B. Massalski, Binary Alloy Phase Diagrams, 2nd edition, ASM International, 1990).
- L. Pauling, The principles determining the structure of complex ionic crystals, Journal of the American Chemical Society 51, 1010 to 1026 (1929). doi:10.1021/ja01379a006
- J. George, D. Waroquiers, D. Di Stefano, G. Petretto, G.-M. Rignanese and G. Hautier, The limited predictive power of the Pauling rules, Angewandte Chemie International Edition 59, 7569 to 7575 (2020). doi:10.1002/anie.202000829
- R. D. Shannon, Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides, Acta Crystallographica A 32, 751 to 767 (1976). doi:10.1107/S0567739476001551
- B. Cordero and co-workers, Covalent radii revisited, Dalton Transactions, 2832 to 2838 (2008). doi:10.1039/b801115j
- H. E. Swanson and co-workers, Standard X-ray Diffraction Powder Patterns, NBS Circular 539: vol. I (1953) Al, Cu, Ni; vol. III (1954) Ti; vol. IV (1955) α-Fe.
- I. Seki and K. Nagata, Lattice constant of iron and austenite including its supersaturation phase of carbon, ISIJ International 45, 1789 to 1794 (2005). doi:10.2355/isijinternational.45.1789. γ-Fe lattice parameter from 1183 to 1550 K.
- J. A. Rayne and B. S. Chandrasekhar, Elastic constants of iron from 4.2 to 300 K, Physical Review 122, 1714 to 1716 (1961). doi:10.1103/PhysRev.122.1714. C11, C12, C44 = 233.1, 135.4, 117.8 GPa.
BibTeX
@misc{tripathy2026interstitialsites,
author = {Tripathy, Manisha},
title = {Interstitial Sites Lab},
year = {2026},
howpublished = {\url{https://untetheredatom.com/crystallography/interstitial-sites}},
note = {Interactive web tool}
}