System parameters
Critical ordering temperature Tc = W/2R:
Stoichiometric (50-50) AB alloy on two sublattices, the classic β-brass (CuZn) / CsCl-type case. Below Tc, atoms sort onto the "correct" sublattice; above it, the arrangement is fully random.
Free energy vs. order parameter, at the current T
G(S,T)
Equilibrium S
Equilibrium order parameter vs. temperature
Sublattice occupancy (one random realization at this S)
A atom
B atom
α sites (correct = A) on odd rows, β sites (correct = B) on even rows
Schematic diffraction pattern: fundamental vs. superlattice reflections
Fundamental (always present)
Superlattice, intensity ∝ S²
What this is actually computing. Two-sublattice combinatorics give the configurational entropy
exactly: R·ln2 − (R/2)[(1+S)ln(1+S) + (1−S)ln(1−S)]. Combined with the point/mean-field
(Bragg-Williams) bond energy −(W/4)S², minimizing G(S,T) gives the self-consistency equation
S = tanh(WS/2RT), solved here by bisection, not asserted. Its bifurcation point is Tc = W/2R, the
same formula as the miscibility-gap consolute temperature in the other two Labs, because it is the
same mean-field mathematics applied to a different physical degree of freedom (sublattice occupancy
instead of composition). Unlike that miscibility gap, this is a second-order transition: S rises
continuously from zero at Tc, with no discontinuous jump. The point approximation used here ignores
short-range order beyond nearest-neighbor mean-field coupling; the Bethe (quasi-chemical) approximation
below refines this and predicts a somewhat different Tc plus nonzero short-range order above it.
This model is specific to a stoichiometric 50-50, two-equal-sublattice (B2/CsCl-type) system; off-stoichiometric
alloys and other ordered structures (L12, L10, D03) need different entropy and
energy expressions not implemented here. The diffraction panel is a schematic proxy for which reflections are
fundamental vs. superlattice in this specific structure type, not a general structure-factor calculator.
For indexing a real diffraction pattern, see the SAED Zone-Axis Indexer, whose own extra-spot classifier flags
candidate superlattice/ordering reflections with an honest caveat of its own: position alone can't always
distinguish a true ordering reflection from a coincidentally-positioned second-phase precipitate spot.
teaching-scale model
References
- Bragg, W.L., and E.J. Williams. The Effect of Thermal Agitation on Atomic Arrangement in Alloys. Proceedings of the Royal Society of London A 145 (1934): 699–730. doi:10.1098/rspa.1934.0132
- Bethe, H.A. Statistical Theory of Superlattices. Proceedings of the Royal Society of London A 150 (1935): 552–575. doi:10.1098/rspa.1935.0122
- Cullity, B.D., and S.R. Stock. Elements of X-Ray Diffraction, 3rd ed. Prentice Hall, 2001. Chapter 13, Order-Disorder Transformations.
- Porter, D.A., K.E. Easterling, and M.Y. Sherif. Phase Transformations in Metals and Alloys, 3rd ed. CRC Press, 2009.
- MIT OpenCourseWare. 3.014 Materials Laboratory, Order-Disorder Transitions: X-Ray Diffraction. ocw.mit.edu