Structure & Transformations · Short-range order

Short-range order in alloys: which atom sits next to which

Watch alloy atoms pick their neighbours instead of mixing at random, and see the X-ray glow this makes.

The latticeDiffractionShort vs long rangeTakeawaysQuestionsReferences

What does short-range order look like in an alloy?

Grey and gold atoms swap places; the bars count how often unlike atoms touch.

Schematic 2D model, live. Coral lines: unlike atoms touching.
α compares each ring of neighbours with a random mix.
Start from
unlike pairslike pairs
α1 (4 touching)
α2 (4 diagonal)
Unlike touching pairs: now · if random

Try it: press Ordering: coral lines fill the lattice and α1 drops below 0. Press Clustering: α1 rises above 0.

How does short-range order show up in diffraction?

Every crystal gives sharp X-ray spots (Bragg spots); short-range order adds a faint glow between them.

Glow from the lattice above; 1 = random. h, k: position in the pattern.
Lattice above
Glow halfway, (½,½)
Glow by a spot, (1/16, 0)

Try it: press Ordering: a broad glow grows at (½,½). Press Clustering: the glow moves next to the Bragg spots.

How is short-range order different from long-range order?

No, weak and strong preference for unlike neighbours, at 50% B.

Schematic 2D model. Blue: each lattice's X-ray pattern.
α1: random · short · long
Glow near (½,½): random · short · long
Real case: Cu3Au. Ordered below about 400 °C (Au corners, Cu faces); short-range order remains above.

Try it: move the right slider to −0.3: the sharp spot fades into a broad glow.

What to take away

α counts neighbours

α = 0: random. Below 0: more unlike neighbours.

Short range vs long range

Weak preference: order fades within a few atoms. Strong: one checkerboard spans the crystal.

X-rays see a glow

On this lattice, ordering: glow halfway between Bragg spots. Clustering: glow next to them.

More detail: the model, the formulas and their limits

The Warren-Cowley parameter

For neighbour shell n, αn = 1 − Pn(B|A)/cB, where Pn(B|A) is the chance of finding a B atom in shell n of an A atom (Walsh, Abu-Odeh and Asta 2023, after Cowley 1950). In a random solution Pn(B|A) = cB, so every αn = 0. The page counts the pairs on the lattice at every step and averages over recent steps. On the square lattice shell 1 is the 4 nearest sites, shell 2 the 4 diagonal sites, then (2,0), (2,1) and (2,2). α can never pass 1, and α1 can never fall below −cB/cA, because there are too few B atoms to surround every A: the floor is −1 only at cB = 0.5, −1/3 at 0.25. He et al. (2024), on the atom probe page, write α = (PAB − XB)/XB, the same number with the opposite sign.

The diffuse intensity

The short-range order part of the scattering is I(q) = N cAcB(fA − fB)2 Σ α(r) cos(2π q·r), summed over all pair vectors r (Jagodzinski and Frey, International Tables B; Stana et al. 2016). Here q = (h, k) is the position in the X-ray pattern, fA and fB are how strongly each kind of atom scatters X-rays, and the "glow" on the page is this diffuse intensity. The page prints it in Laue units, cAcB(fA − fB)2 per atom, so a random alloy gives a flat 1 (α(0) = 1, all others 0). "Every distance" uses α for every pair vector in the 32 by 32 lattice; the shell options cut the sum short. A cut-off sum can dip below 0, which a real intensity cannot; the page draws such values as 0.

The simulation

Each site holds σ = +1 (A) or −1 (B). The energy is the nearest-neighbour Ising form U = −J Σ σiσj: a like pair has energy −J and an unlike pair +J, so J < 0 favours unlike neighbours. Atoms move by Kawasaki exchange: pick a nearest-neighbour pair at random and swap it with probability min[1, exp(−ΔU/kT)] (Metropolis et al. 1953; the same moves as Gorentsveig, Fratzl and Lebowitz 1997, whose model also has a second-neighbour term that this page leaves out). Swaps keep the composition fixed. The three presets (+0.35, 0, −0.35) and the slider ranges are the page's own choice. The third picture prints the mean glow over the 3 by 3 grid points around (½,½). It calls a panel long-range order when the sum over those 9 points passes 25% of N, where N is the sum a perfect checkerboard gives; this 25% rule is the page's own choice, as is calling a panel short-range order when α1 is below −0.05. The page looks around (½,½), not only at it, because two checkerboard patches that are out of step cancel at (½,½) itself. Only the ratio J/kT matters: a stronger preference or a lower temperature have the same effect. For this model at cB = 0.5, long-range order sets in when |J|/kT passes 0.44 (Onsager's exact result kTc = 2|J|/ln(1+√2), as given by Dolfi et al. 2014; flipping every second site turns the unlike-pair model into the like-pair one). So the presets ±0.35 sit above the ordering temperature and show short-range order only, while −0.8 and −0.9 sit below it.

Limits

This is a 2D toy on a square lattice with one pair energy, not a model of a real alloy, which is 3D (fcc, bcc) with several pair energies. The spot positions (½,½) and (1/16, 0) belong to this square lattice; where the glow sits in a real alloy depends on its lattice and on how its order repeats (Jagodzinski and Frey). The lattice is 32 by 32 with wrap-around edges, so the α values jump about by a few hundredths and the glow values by much more. Real diffuse scattering also contains terms from atoms sitting off their ideal sites (static displacements), which the page leaves out; Butler and Cohen (1988) measured such displacements in Cu3Au.

On this site: Order-disorder lab · Short-range order in atom probe · 4D-STEM diffuse scattering · Correlative APT and TEM · Diffusion couples · 4D-STEM basics

Questions people ask

What is chemical short-range order?

It is a preference, in a solid solution (a crystal where two kinds of atoms share one lattice), for certain atoms to be close neighbours, so the arrangement of atoms on the lattice sites deviates from random. It is measured with Warren-Cowley parameters, one per neighbour shell.

What does a negative Warren-Cowley parameter mean?

A negative αn means more unlike (A-B) pairs in shell n than in a random alloy; a positive value means fewer. For nearest neighbours, a positive α1 suggests clustering and a negative α1 usually means ordering (Walsh, Abu-Odeh and Asta 2023).

How is short-range order measured?

The classic way is to measure the faint diffuse X-ray glow from a single crystal, as Cowley did for Cu3Au in 1950. Butler and Cohen measured it in absolute units over a whole volume of the X-ray pattern (reciprocal space). Owen and co-workers used X-ray total scattering (pair distribution functions) on heated powders.

What is the difference between short-range and long-range order?

Long-range order means one kind of atom sits on one set of sites across the whole crystal, as in L12 Cu3Au. Short-range order is only local. In the third picture, long-range order gives a sharp spot at (½,½) and short-range order a broad glow.

Does short-range order exist above the ordering temperature?

In Cu3Au, yes. Butler and Cohen measured the diffuse scattering at 703 K, about 35 K above the ordering temperature, and obtained short-range order parameters. Owen and co-workers saw short-range order resembling L12 develop before the transition near 400 °C.

Is a quenched solid solution random?

Not always. Owen and co-workers found a degree of short-range order even in quenched Cu3Au samples, which are usually assumed to be completely random.

References

Show the 11 references
  1. J. M. Cowley, An approximate theory of order in alloys, Physical Review 77, 669 (1950). doi:10.1103/PhysRev.77.669
  2. J. M. Cowley, X-ray measurement of order in single crystals of Cu3Au, Journal of Applied Physics 21, 24 to 30 (1950). doi:10.1063/1.1699415
  3. H. Jagodzinski and F. Frey, Disorder diffuse scattering of X-rays and neutrons, section 4.2.4.4, Disorder with three-dimensional correlations (defects, local ordering and clustering), International Tables for Crystallography, Vol. B, ch. 4.2 (2006).
  4. F. Walsh, A. Abu-Odeh and M. Asta, Reconsidering short-range order in complex concentrated alloys, MRS Bulletin 48(7), 753 to 761 (2023). doi:10.1557/s43577-023-00555-y
  5. M. Stana, B. Sepiol, R. Kozubski and M. Leitner, Chemical ordering beyond the superstructure in long-range ordered systems, New Journal of Physics 18, 113051 (2016). arXiv:1608.06870
  6. B. D. Butler and J. B. Cohen, Diffuse X-ray scattering study of Cu3Au above the order-disorder transition temperature, MRS Proceedings 138, 59 (1988). doi:10.1557/PROC-138-59
  7. L. R. Owen, H. Y. Playford, H. J. Stone and M. G. Tucker, Analysis of short-range order in Cu3Au using X-ray pair distribution functions, Acta Materialia 125, 15 to 26 (2017). doi:10.1016/j.actamat.2016.11.048
  8. L. R. Owen, H. Y. Playford, H. J. Stone and M. G. Tucker, A new approach to the analysis of short-range order in alloys using total scattering, Acta Materialia 115, 155 to 166 (2016). doi:10.1016/j.actamat.2016.05.031
  9. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines, Journal of Chemical Physics 21, 1087 to 1092 (1953), doi:10.1063/1.1699114; and V. I. Gorentsveig, P. Fratzl and J. L. Lebowitz, Kinetics of joint ordering and decomposition in binary alloys, arXiv:cond-mat/9703094 (1997).
  10. M. Dolfi et al., A model project for reproducible papers: critical temperature for the Ising model on a square lattice, arXiv:1401.2000 (2014).
  11. M. He, W. J. Davids, A. J. Breen and S. P. Ringer, Quantifying short-range order using atom probe tomography, Nature Materials 23, 1200 to 1207 (2024). doi:10.1038/s41563-024-01912-1
Cite this page: Tripathy, Manisha. “Short-Range Order Lab.” untethered atom, 2026, https://untetheredatom.com/phase-transformations/short-range-order.
BibTeX 
@misc{tripathy2026shortrangeorder,
  author = {Tripathy, Manisha},
  title  = {Short-Range Order Lab},
  year   = {2026},
  howpublished = {\url{https://untetheredatom.com/phase-transformations/short-range-order}},
  note   = {Interactive web tool}
}
Last updated 1 October 2026.