\(\Delta H_{ij} \equiv \Delta H_{AB}^{\mathrm{mix}}(c_A = c_B = \tfrac{1}{2})\) is the
equiatomic binary mixing enthalpy. The factor of 4 absorbs the relation
\(\Omega_{AB} = 4\,\Delta H_{AB}^{\mathrm{mix}}\), which follows from
\(\Delta H^{\mathrm{reg}}(\tfrac{1}{2},\tfrac{1}{2}) = \tfrac{1}{4}\Omega_{AB}\). Each
pair is counted once (\(i < j\)). Ternary and higher-order interactions are ignored.
Within Miedema, the equiatomic chemical mixing enthalpy comes from inserting
\(c_A = c_B = \tfrac{1}{2}\) into the chemical SS expression of section 10.
\[\Delta H_{AB}^{\mathrm{mix}}(\tfrac{1}{2},\tfrac{1}{2}) \;=\;
\frac{1}{2}\,\frac{V_A^{2/3*}\,V_B^{2/3*}}{V_A^{2/3*} + V_B^{2/3*}}\;\Gamma_{AB}\]
The \(\Delta H_{ij}\) values used for \(\Delta H_{\mathrm{mix}}\) come from a fixed,
vendored Miedema pair table (Matminer-derived; 1484 of the 1485 binary pairs of the 55 supported
elements, the lone Th–U gap reported rather than zeroed). It is the same table the Python library uses, kept in lock-step by a parity
test. A manual-override field lets you substitute validated literature values per pair,
for example the integer table of Takeuchi & Inoue (2005).
Provenance and the Ω singularity
The multi-component sum applies the factor of 4 exactly once, and each
\(\Delta H_{ij}\) is the equiatomic binary value, so an equiatomic binary reduces to
\(\Delta H_{\mathrm{mix}} = \Delta H_{ij}\) with no double counting. Absolute
\(\Delta H_{\mathrm{mix}}\) values nevertheless depend on which Miedema
parametrization supplies the pair table; published compilations disagree most on
Mn, whose Miedema parameters are assigned inconsistently across
sources. Because \(\Omega = \bar T_m\,\Delta S_{\mathrm{mix}} / |\Delta H_{\mathrm{mix}}|\)
diverges as \(\Delta H_{\mathrm{mix}} \to 0\), \(\Omega\) is extremely sensitive for
near-ideal alloys (\(|\Delta H_{\mathrm{mix}}| \lesssim 1\text{--}2\;\mathrm{kJ\,mol^{-1}}\)):
a few-kJ shift in one pair value can move \(\Omega\) by an order of magnitude, so read
it qualitatively in that regime.
08The Yang–Zhang Ω parameter
Yang & Zhang (2012) proposed a single dimensionless parameter that contrasts the
entropic stabilisation of the disordered state against the enthalpic push for ordering.
\[\Omega \;=\; \frac{\bar T_{m}\,\Delta S_{\mathrm{mix}}}{\lvert\Delta H_{\mathrm{mix}}\rvert}\]
The average melting temperature is \(\bar T_m = \sum_i c_i T_{m,i}\). \(\Omega\) is the
ratio of the maximum entropic Gibbs-energy contribution
(\(T\Delta S_{\mathrm{mix}}\) evaluated at \(\bar T_m\)) to the size of the chemical
mixing enthalpy. When \(\Omega > 1\), entropy at the solidus dominates. When
\(\Omega \ll 1\), enthalpy dominates. The empirical threshold for solid-solution formation
is \(\Omega \gtrsim 1.1\), usually combined with \(\delta \lesssim 6.5\,\%\).
\(\Omega \to \infty\) when \(\Delta H_{\mathrm{mix}} \to 0\). The calculator reports this
case explicitly to avoid numerical divergence.
09Compound formation enthalpy
For an ordered binary intermetallic, chemical short-range order enhances the
mixing-degree factor relative to the random-solution case. Miedema introduced a
phenomenological factor.
\[f_{AB}(c^{\,s}) \;=\; c_A^{\,s}\,c_B^{\,s}\,\bigl(1 + 8\,(c_A^{\,s}\,c_B^{\,s})^2\bigr)\]
The coefficient \(\mu = 8\) is fitted to fully ordered compounds (de Boer
et al., 1988). The chemical part of the formation enthalpy is then
\[\Delta H^{\mathrm{Mied}} \;=\; f_{AB}(c^{\,s})\,\bigl(c_A V_A^{2/3} + c_B V_B^{2/3}\bigr)\,\Gamma_{AB}\]
Zhang, Zhang, He, Jing & Sheng (2016) refined the model with an
atomic-size-difference (ASD) prefactor \(S_C\) that suppresses the formation
enthalpy of compounds with poor geometric compatibility.
\[\Delta H_{\mathrm{form}} \;=\; S_C\;\Delta H^{\mathrm{Mied}} \;+\; \Delta H_{\mathrm{trans}}\]
Two functional forms appear in practice. The composition-dependent (“original”)
form is
\(S_C = 1 - |V_B^{2/3} - V_A^{2/3}| / (c_A^{\,s}V_A^{2/3} + c_B^{\,s}V_B^{2/3})^{2}\). The
composition-independent (“recent”) form, used by default in this calculator,
is \(S_C = V_A^{2/3}\,V_B^{2/3} / [\tfrac{1}{2}(V_A^{2/3} + V_B^{2/3})]^{2}\).
\(\Delta H_{\mathrm{trans}}\) is the semiconductor-to-metal transformation enthalpy
applied to elements that are non-metallic in their ground state. Si is the main case in
this dataset, with \(\Delta H_{\mathrm{trans}} \approx 34\;\mathrm{kJ\,mol^{-1}}\)
(de Boer et al., 1988). For all transition and noble metals
\(\Delta H_{\mathrm{trans}} = 0\).
10Solid-solution formation enthalpy
For a disordered solid solution, where atoms occupy lattice sites at random, the
formation enthalpy decomposes into three additive contributions.
\[\Delta H_{\mathrm{SS}} \;=\; \Delta H_{\mathrm{chem}} + \Delta H_{\mathrm{elast}} + \Delta H_{\mathrm{struct}}\]
Chemical contribution
With \(\mu = 0\) (no chemical short-range order), the mixing-degree factor reduces to
\(f_{AB} = c_A^{\,s} c_B^{\,s}\). After symmetrisation,
\[\Delta H_{\mathrm{chem}} \;=\; c_A\,c_B\bigl(c_B^{\,s}\,V_A^{2/3} + c_A^{\,s}\,V_B^{2/3}\bigr)\,\Gamma_{AB}\]
Elastic mismatch contribution (Eshelby–Friedel)
A solute atom of molar volume \(W_A\) inserted into a matrix of molar volume \(W_B\)
produces a strain field. Its energy comes from the Eshelby (1957) inclusion problem and
was adapted to dilute alloys by Friedel (1954). For an elastically isotropic host,
\[E_{A\;\mathrm{in}\;B} \;=\; \frac{2\,K_A\,G_B\,(W_A - W_B)^{2}}{3\,K_A\,W_B + 4\,G_B\,W_A}\]
where \(K_A\) is the bulk modulus of the inclusion and \(G_B\) the shear modulus of the
matrix. The symmetric expression \(E_{B\;\mathrm{in}\;A}\) describes the reverse case.
The composition-weighted total elastic enthalpy (Niessen & Miedema, 1983) is
\[\Delta H_{\mathrm{elast}} \;=\; c_A\,c_B\bigl(c_B\,E_{A\;\mathrm{in}\;B} + c_A\,E_{B\;\mathrm{in}\;A}\bigr)\]
\(\Delta H_{\mathrm{elast}}\) is identically zero for size-matched pairs and grows
quadratically with \(\Delta W\). It is always non-negative. Elastic strain
destabilises the solid solution.
Structural contribution
When the constituent elements adopt different ground-state crystal structures, forming
a single-phase solution requires one or both elements to sit in a metastable lattice.
The energy cost comes from the lattice-stability data of Niessen & Miedema
(1983) and Loeff, Weeber & Miedema (1988). The structure-dependent energy
\(E_{\alpha}(z)\) of a transition metal is tabulated as a function of valence-electron
count \(z\), referenced to the bcc structure with \(E_{\mathrm{bcc}}(z) \equiv 0\).
The structural enthalpy of the alloy is the difference between the most stable structure
at the composition-averaged \(\langle z\rangle = \sum_i c_i z_i\) and the
composition-weighted reference energies of the pure elements.
\[\Delta H_{\mathrm{struct}} \;=\; \min_{\alpha}\,E_{\alpha}\!\bigl(\langle z\rangle\bigr)
\;-\; \sum_i c_i\,E_{\alpha_i^{0}}(z_i)\]
\(\alpha_i^{0}\) is the equilibrium structure of pure element \(i\). The tabulated values
are interpolated linearly between integer \(z\). For the late transition metals
(\(z \geq 10\)) lattice-stability differences are taken to vanish, matching the original
Niessen–Miedema treatment.
11Amorphous-alloy formation enthalpy
An amorphous alloy has no long-range order. Its chemical contribution matches that of
a disordered solid solution (Bakker, 1998). The structural cost is replaced by a
topological cost. That cost reflects the density-of-states penalty for
non-crystalline packing (Loeff, Weeber & Miedema, 1988).
\[\Delta H_{\mathrm{AM}} \;=\; \Delta H_{\mathrm{chem}} \;+\; \Delta H_{\mathrm{topo}},\qquad
\Delta H_{\mathrm{topo}} \;=\; \beta\,\sum_i c_i\,T_{m,i}\]
with \(\beta \approx 3.5\;\mathrm{J\,mol^{-1}\,K^{-1}}\) (Bakker, 1998). The topological
term reflects an empirical observation. The enthalpy difference between an amorphous
and a crystalline metal scales linearly with melting temperature, because both are
controlled by the cohesive energy. \(\Delta H_{\mathrm{topo}}\) is always positive.
Amorphisation is energetically unfavourable without a chemical driving force.
The crossover \(\Delta H_{\mathrm{AM}} < \Delta H_{\mathrm{SS}}\) is the basic
glass-forming criterion in this picture. Senkov & Miracle (2001) added
size-distribution descriptors, but the enthalpy crossover stays the dominant indicator.
12Phase-selection criteria
The descriptors above combine into a small number of phenomenological rules. The most
widely cited are listed below.
- Yang & Zhang (2012). A single-phase solid solution is likely when
\(\Omega \geq 1.1\) and \(\delta \leq 6.5\,\%\).
- Guo & Liu (2011). Within the SS field, the structure follows VEC. Fcc
above 8.0, bcc below 6.87, mixed in between.
- Zhang et al. (2008). A complementary cut on
\(\Delta H_{\mathrm{mix}}\) between roughly \(-22\) and \(+5\,\mathrm{kJ\,mol^{-1}}\)
separates SS from compounds and miscibility gaps.
- Miedema-model crossover (Zhang et al., 2016). When
\(\Delta H_{\mathrm{form}}^{\mathrm{compound}}\) is much more negative than
\(\Delta H_{\mathrm{SS}} - T\Delta S_{\mathrm{mix}}\), an ordered intermetallic is
expected. When \(\Delta H_{\mathrm{AM}} < \Delta H_{\mathrm{SS}}\), an amorphous
phase is favoured.
- Tsai et al. (2013). In annealed alloys containing Cr and/or
V, the topologically close-packed σ phase precipitates when
\(6.88 \leq \mathrm{VEC} \leq 7.84\); outside the window it generally does not. The
rule says nothing about alloys without Cr or V, and the calculator reports it as
not applicable there.
- Sheikh et al. (2016). Among single-phase bcc refractory
HEAs, \(\mathrm{VEC} < 4.5\) marks the intrinsically ductile alloys and
\(\mathrm{VEC} \geq 4.6\) the brittle ones. Read it together with the Guo bands; it
is a screen for bcc refractory chemistry, not a universal ductility predictor.
No single parameter is reliable on its own: the canonical practice is to read
\(\delta\), \(\Delta H_{\mathrm{mix}}\), and \(\Omega\) jointly, and to consult VEC
only after a solid-solution verdict (Pickering & Jones, 2016). The calculator's
phase-selection maps draw exactly these joint windows with the current alloy plotted,
and every verdict card reports its margin so borderline calls are visible. Empirical
screens are the start of an assessment; CALPHAD or DFT remain the confirmation step.
13Geometric packing parameters Λ and γ
Two descriptors condense the size-mismatch geometry beyond \(\delta\) itself. Singh
et al. (2014) observed that entropy and mismatch compete directly and
defined
\[\Lambda = \frac{\Delta S_{\mathrm{mix}}}{\delta^{2}},\]
with \(\delta\) in percent. Surveying known alloys they found \(\Lambda > 0.96\) for
single disordered solid solutions, \(0.24 \leq \Lambda \leq 0.96\) for mixtures of
solutions and compounds, and \(\Lambda < 0.24\) when compounds dominate. Two elements
with identical tabulated radii give \(\delta = 0\) and an unbounded \(\Lambda\); the
calculator reports it as such rather than truncating.
Wang et al. (2015) instead asked how much solid angle a neighbour of
average radius \(\bar r\) subtends around the smallest and the largest atom. Two
touching spheres of radii \(r_c\) and \(\bar r\) define a tangent cone with
\(\sin\theta = \bar r/(r_c + \bar r)\), so the fractional solid angle is
\[\omega(r_c) = 1 - \cos\theta
= 1 - \sqrt{\frac{(r_c + \bar r)^2 - \bar r^{\,2}}{(r_c + \bar r)^2}}.\]
The ratio \(\gamma = \omega_{\mathrm{S}}/\omega_{\mathrm{L}}\) equals 1 for equal
radii and grows with mismatch; their alloy survey places the crystalline
solid-solution ceiling at \(\gamma < 1.175\), a modern restatement of the
Hume–Rothery 15 % rule for multicomponent packing.
14The Senkov–Miracle κ criterion
The empirical windows above ignore temperature, yet solid solution and intermetallic
compete through their Gibbs energies. Senkov & Miracle (2016) compare the two
phases directly: the solution wins when
\(\Delta H_{\mathrm{mix}} - T\,\Delta S_{\mathrm{mix}} <
\Delta H_{\mathrm{IM}} - T\,\Delta S_{\mathrm{IM}}\), with the working assumption
\(\Delta S_{\mathrm{IM}} = k_2\,\Delta S_{\mathrm{mix}}\), \(k_2 = 0.6\), for the
partial order of real intermetallics. For the usual case
\(\Delta H_{\mathrm{mix}} < 0\) this rearranges to the published parameterization
\[k_1 = \frac{\Delta H_{\mathrm{IM}}}{\Delta H_{\mathrm{mix}}}
< k_1^{\mathrm{cr}}(T) = 1 +
\frac{(1 - k_2)\,T\,\Delta S_{\mathrm{mix}}}{\left|\Delta H_{\mathrm{mix}}\right|},\]
where the \((1-k_2)\) factor multiplies the temperature term; deriving the inequality
from the Gibbs comparison fixes the typography ambiguity found in some restatements.
The calculator evaluates the verdict from the direct Gibbs comparison, which stays
well defined when \(\Delta H_{\mathrm{mix}} \geq 0\), and reports \(k_1\) and
\(k_1^{\mathrm{cr}}\) whenever the ratio form is meaningful.
\(\Delta H_{\mathrm{IM}}\) uses the same most-negative-binary proxy documented for
King’s \(\Phi\), and the temperature input is shared with that rule. This is
the calculator’s only temperature-explicit phase criterion: raising the anneal
temperature visibly widens the solid-solution window.
15Elastic-strain energy \(\Delta H_{\mathrm{el}}\)
Size mismatch costs elastic energy in proportion to how stiff the misfitting atoms
are, which \(\delta\) alone cannot see. Andreoli et al. (2019) evaluate
the strain energy of squeezing each atom from its own volume \(V_i\) into the
stiffness-weighted common volume
\[\bar V = \frac{\sum_i c_i B_i V_i}{\sum_i c_i B_i},
\qquad
\Delta H_{\mathrm{el}} = \sum_i \frac{c_i\,B_i\,(V_i - \bar V)^2}{2\,V_i}\]
(their Eqs. 9–10). With \(B_i\) in GPa and \(V_i\) in cm³/mol the result is
directly in kJ/mol. Across their 235-alloy survey, fcc solid solutions form mainly
below \(6.05\;\mathrm{kJ\,mol^{-1}}\) (96.4 % of fcc SS), bcc solutions between
6.05 and 22, and everything above 22 is glass- or intermetallic-forming territory.
HEA-Bench takes \(B_i\) and \(V_i\) from the vendored matminer Miedema table, which
differs from the paper’s elastic sources; window placements are robust to this,
third decimals are not, and the provenance panel says so.
16High-entropy oxides and sublattice entropy
Rost, Sachet, Borman et al. (2015) showed that the five-cation rock salt
(Mg,Co,Ni,Cu,Zn)O is single-phase and entropy-stabilized: it transforms reversibly
between multiphase and single-phase states with temperature, which is direct evidence
that the configurational entropy term drives the stabilization. That result opened the
high-entropy concept to ionic compounds. The structural difference from alloys is that
an oxide crystal has distinct sublattices. Cations mix on one or more cation
sublattices while oxygen occupies its own, essentially ordered, anion sublattice, so
the single-lattice entropy formula of section 02 must be generalized to a sum over
sublattices:
\[\Delta S_{\mathrm{config}} = -R \sum_{k} a_k \sum_{i} x_{ik}\,\ln x_{ik}\]
where \(a_k\) is the number of sites of sublattice \(k\) per formula unit (one A and
one B site for ABO3, two of each for A2B2O7)
and \(x_{ik}\) is the site fraction of species \(i\) on sublattice \(k\). A sublattice
carrying a single species contributes zero, so the ordered oxygen sublattice drops out.
Anion disorder and oxygen vacancies are not modelled. The calculator reports the value
per mole of formula units and per cation site, and it classifies the composition on the
most-disordered sublattice, which is the HEO literature convention: a perovskite with
five equimolar B cations is a high-entropy oxide at \(1.61\,R\) on the B sublattice
even though the value per total cation site is half that. The thresholds are the usual
ones, high entropy at or above \(1.5\,R\) and medium entropy from \(1.36\,R\) to
\(1.5\,R\).
17Oxidation states and Shannon ionic radii
Every geometric oxide descriptor depends on ionic radii, and an ionic radius is not a
single number per element. It depends on the oxidation state, the coordination number,
and for some 3d cations the spin state. The calculator uses the effective ionic radii
of Shannon (1976), on the scale fixed by \(r(\mathrm{O}^{2-},\mathrm{VI}) = 1.40\) Å,
from a vendored 94-element digitization of the published tables.
Before any radius can be looked up, each cation needs an oxidation state. The
calculator assigns them by exact charge balance against the oxygen content,
\[\sum_{k} a_k \sum_{i} x_{ik}\,q_{ik} = 2\,n_{\mathrm O}\]
where \(q_{ik}\) is the assigned state and \(n_{\mathrm O}\) the number of oxygen atoms
per formula unit. The solver tries each element's common oxidation states first and
widens to the rarer literature-documented states only if no combination balances, with
a warning. When several combinations balance, it prefers chemically uniform
assignments, for example all trivalent cations on a pyrochlore A site rather than a
mixed-valence pair. Coordination numbers follow the structure family: six-fold in rock
salt, twelve-fold A and six-fold B in perovskite, eight-fold in fluorite, and
eight-fold A with six-fold B in pyrochlore. Where Shannon distinguishes spin states the
high-spin radius is the default, and when a radius is missing at the family's
coordination number the nearest tabulated coordination is used, again with a warning.
From these radii the cation size-disorder parameter of each sublattice is the ionic
analog of the alloy \(\delta\) (Manchón-Gordón et al., 2025),
and the two-sublattice combination is
\(\delta r^{*} = \sqrt{\delta_A^2 + \delta_B^2}\).
18Perovskite tolerance factors
For an ABO3 perovskite, Goldschmidt (1926) observed that the structure forms
when the A–O and B–O bond lengths are geometrically compatible, expressed
through the tolerance factor
\[t = \frac{r_A + r_{\mathrm O}}{\sqrt{2}\,(r_B + r_{\mathrm O})}\]
with cubic perovskites near \(t = 1\). The calculator applies the window
\(0.92 \leq t \leq 1.04\) compiled for high-entropy perovskites by
Manchón-Gordón et al. (2025); the classic textbook window is
0.9–1.0 and the bound is configurable. A second geometric condition is the
octahedral factor \(\mu = r_B / r_{\mathrm O}\), which must lie between 0.414 and 0.732
for a stable BO6 octahedron. Bartel et al. (2019) constructed an
improved one-dimensional factor,
\[\tau = \frac{r_{\mathrm O}}{r_B} - n_A\!\left(n_A - \frac{r_A/r_B}{\ln(r_A/r_B)}\right)\]
with \(n_A\) the oxidation state of the A cation, predicting a perovskite when
\(\tau < 4.18\) with about 91 % reported accuracy over 576 experimentally
characterized ABX3 compounds. It is defined for \(r_A > r_B\). For
high-entropy perovskites the radii and \(n_A\) are composition-weighted sublattice
means. Jiang et al. (2018) found that the tolerance factor, not the size
mismatch, is what separates their single-phase high-entropy perovskites from the
multiphase ones, which is why these factors are the primary perovskite screen here.
19Fluorite and pyrochlore criteria
For fluorite-structured high-entropy oxides, Spiridigliozzi, Ferone, Cioffi &
Dell'Agli (2021) found that a single descriptor separates their synthesis outcomes: the
sample standard deviation of the constituent cation radii, taken at eight-fold Shannon
coordination over the equimolar cation set,
\[\sigma = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}\,(r_i - \bar r)^2}\]
with \(\sigma > 0.095\) Å predicting a single-phase, entropy-stabilized
fluorite and lower values giving bixbyite-type or mixed phases. The implementation here
reproduces the standard deviations published by the same group (Spiridigliozzi,
Bortolotti & Dell'Agli, 2023) to every quoted digit, which pins the radius and
normalization conventions. The criterion was calibrated on equimolar five-cation
rare-earth-based oxides, so verdicts far outside that domain are extrapolations.
For A2B2O7 pyrochlores the classic geometric screen is
the plain radius ratio of the two cation sites (Subramanian, Aravamudan & Subba
Rao, 1983): a pyrochlore is expected for \(1.46 < r_A/r_B < 1.78\), a disordered
defect fluorite below the window, and no single cubic phase above it. As with the alloy
rules, all of these oxide criteria are weak empirical screens calibrated on small
historical datasets, and the calculator reports the value, the window, and the verdict
together so the margin is always visible.
20Limitations and scope
All descriptors implemented here share four structural limitations worth keeping in
mind when reading the outputs.
- Pairwise additivity. The mixing-enthalpy and Miedema solid-solution
formulae are sums over binary pairs. Ternary and higher-order interactions are
ignored. That is exact for ideal regular solutions and a controlled approximation
for sub-regular systems. It can fail when three- or four-body chemical interactions
are large. Alloys that combine Al with several transition metals are a common
example.
- Equilibrium-only. The descriptors describe the equilibrium driving force.
They say nothing about kinetics, nucleation rates, diffusion-limited segregation,
or processing-induced metastable phases. As-cast HEAs often disagree with equilibrium
predictions for that reason.
- Empirical parameters. Miedema's \(\varphi^{*}\), \(n_{ws}^{1/3}\), and
\(V^{2/3}\) values are calibrated against a fixed dataset of binary heats of
formation. The model performs best for transition-metal-rich alloys. It degrades
for systems with lanthanides, actinides, or strongly covalent partners (de Boer
et al., 1988).
- Small calibration sets for the oxide windows. The oxide formability
windows are fitted to small historical synthesis datasets: the fluorite
\(\sigma\) threshold to equimolar five-cation rare-earth oxides, the
high-entropy-perovskite \(t\) window to a compilation of a few hundred reported
compositions, and the pyrochlore radius-ratio window to the classic
pyrochlore literature. They assume full cation mixing, the stated oxygen
stoichiometry, and equilibrium synthesis, and verdicts far from those domains
are extrapolations.
For quantitative phase prediction, supplement these descriptors with a CALPHAD
free-energy minimisation (Lukas, Fries & Sundman, 2007) or first-principles
methods such as the special quasirandom structure approach within DFT (Zunger, Wei,
Ferreira & Bernard, 1990). The descriptors here remain useful as a fast,
transparent pre-screen and as a sanity check on more expensive calculations.