# How to find the weight fraction of each phase in a mixture Refine every phase in Rietveld mode, read `result.qpa.phases`, and check the minor phase’s esd and correlations before you quote its fraction. A Rietveld fit with more than one phase converts the phase scales into weight fractions by the Hill-Howard relation, W_p ∝ S_p·(ZMV)_p, renormalised to sum to 1. Nothing has to be switched on. ## Read the fractions The NAC pattern that ships with rietx, measured at APS 11-BM, carries a CaF₂ impurity. This refines both phases with `mccusker_structural`: ```python import rietx as rx from rietx.examples import examples_dir data = rx.read_pattern(examples_dir() / "11BM_NAC.fxye") def site(label, species, xyz, biso): x, y, z = (rx.Parameter(value=v) for v in xyz) return rx.Atom(label=label, species=species, x=x, y=y, z=z, biso=rx.Parameter(value=biso, min=0.0, max=25.0)) def nac_and_fluorite(): structure = rx.Structure.from_cif(examples_dir() / "cod_1000236.cif") structure.phases[0].name = "NAC" structure.phases.append(rx.Phase( name="CaF2", space_group="F m -3 m", cell=rx.Cell.cubic(5.4631), atoms=[site("Ca", "Ca2+", (0, 0, 0), 0.6), site("F", "F1-", (0.25, 0.25, 0.25), 0.9)])) instrument = rx.Instrument.debye_scherrer(wavelength=0.4139090) instrument.profile.w.value = 2e-5 instrument.profile.x.value = 2e-3 instrument.background = rx.BackgroundChebyshev.with_terms(6) return rx.Refinement(structure, instrument) def show(result): for row in result.qpa.phases: print(f"{row.name:<5} {100 * row.weight_fraction:6.2f} ± {100 * row.weight_fraction_stderr:.2f} wt%") ref = nac_and_fluorite() result = ref.fit(data, plan="mccusker_structural", two_theta_limits=(2, 24)) show(result) ``` ```text NAC 96.42 ± 3.53 wt% CaF2 3.58 ± 3.53 wt% ``` Each row is a `PhaseQuantity`, with the fraction, its esd, the refined scale and the Z·M and V it was computed from. The esd is propagated through the covariance of all the scales. With two phases the fractions sum to 1, so their esds are equal. The fractions are of the crystalline phases in the model, and `result.qpa.crystalline_only` says so. An amorphous component or a crystalline phase you did not model is shared out among the phases you did, so the fractions still sum to 1. Report them as “wt % of the crystalline content”. A phase with no reflection in the fitted range has no fraction (`None`), and the others are relative to the phases seen in range ([QPA reference](../reference/qpa.html.md#a-phase-with-no-reflection-in-the-fitted-range)). ## Check the minor phase An esd of 3.53 wt % on a 3.58 wt % phase, measured at a synchrotron, says something is trading against the scale. The diagnostics name it: ```python for d in result.diagnostics: if d.code in ("HIGH_CORRELATION", "FLOOR_HIT"): print(d.code, "|", d.message) ``` ```text FLOOR_HIT | phases.1.atoms.0.biso refined to its floor (ρ=+0.021). FLOOR_HIT | phases.1.atoms.1.biso refined to its floor (ρ=+0.091). HIGH_CORRELATION | Correlated pair: phases.1.scale ~ phases.1.extinction (ρ=+0.998). Flagged in stages: extinction, roughness, closing. ``` The plan freed CaF₂’s extinction, and with a few weak peaks the extinction and the scale can scale the same intensities. Both CaF₂ Biso values went to 0, which no real crystal has. The data cannot fix these three parameters for a phase this weak, so hold them: ```python held = nac_and_fluorite() held.hold(["phases.1.extinction", "phases.1.atoms.*.biso"]) held_result = held.fit(data, plan="mccusker_structural", two_theta_limits=(2, 24)) show(held_result) ``` ```text NAC 98.63 ± 0.07 wt% CaF2 1.37 ± 0.07 wt% ``` The esd falls from 3.53 to 0.07 wt %, and `mccusker_default`, which frees no structural parameter, gives 1.38 wt % for CaF₂. Rwp rose from 0.082 to 0.089 when the three parameters were held. The free extinction lowered Rwp while it moved the fraction, so Rwp cannot choose between the two fits. The fit reports the holds as `HOLD_BLOCKED_PLAN`, because the plan tried to free paths you held. [How to keep a parameter fixed whatever the plan frees](hold.html.md) explains holds. ## Check that the fraction has one answer A weak phase’s scale can also trade against its peak width: broadened far enough, its peaks become a hump the background can share. `Refinement.profile_fraction` pins a width term at a series of values, refits everything else at each, and reports every fraction the data cannot tell from the best one: ```python profile = held.profile_fraction(data, "CaF2", axes=["phases.1.lor_strain"]) print(f"admissible {100 * profile.range_low:.2f} to {100 * profile.range_high:.2f} wt%," f" {len(profile.points)} refits") ``` ```text admissible 1.37 to 1.37 wt%, 12 refits ``` Here only the fit’s own fraction is admissible, so the fraction is determined. A point is admissible when its χ² is within 3.84 × χ²_red × f² of the lowest χ² found, f being the Bérar-Lelann factor. Every profile of a fit with an esd carries a `QPA_FRACTION_RANGE` info record, whose value is the farthest admissible fraction’s distance from the fit’s, in the esd’s 95 % half-widths. Below 1 the profile confirms the esd. Well above 1 the esd describes only the basin the fit stopped in, and you should quote the range rather than the value. Without `axes=`, it uses the phase’s width terms that were free in the last fit, and refuses if none was. ## Correct for microabsorption A strongly absorbing coarse phase hides the inside of its own particles, so its fraction comes out low. The Brindley correction needs a particle radius for every phase, which the pattern cannot give: measure it, for example from a micrograph. Set `Phase.particle_radius_um` on every phase before the fit: ```python radii = nac_and_fluorite() radii.hold(["phases.1.extinction", "phases.1.atoms.*.biso"]) radii.structure.phases[0].particle_radius_um = 5.0 radii.structure.phases[1].particle_radius_um = 2.0 corrected = radii.fit(data, plan="mccusker_structural", two_theta_limits=(2, 24)) for row in corrected.qpa.phases: print(f"{row.name:<5} µR {row.mu_r:.4f} {100 * row.weight_fraction:.2f} ->" f" {100 * row.weight_fraction_corrected:.2f} wt%") ``` ```text NAC µR 0.0021 98.63 -> 98.63 wt% CaF2 µR 0.0014 1.37 -> 1.37 wt% ``` The radii here are assumed, to show the call. At 0.414 Å both phases absorb so little that the correction changes nothing. With Cu Kα and a heavy phase it can move a fraction by several wt %. The corrected fraction is reported beside the uncorrected one, and the esd belongs to the uncorrected one: the radii’s uncertainty is not statistical. Above µR = 0.05 the correction is outside the regime Brindley derived it for, and `BRINDLEY_OUTSIDE_REGIME` names the phases. ## Write the table `Refinement.write_qpa_table` writes the fractions to a CSV with the crystalline-only caveat in it. A joint fit reports the fractions per histogram on `HistogramResult.qpa`, and a series per pattern on `SeriesEntry.qpa`. ## See also - [How to quote a parameter that ended on its bound](interval.html.md) gives an interval for a Biso on its floor. - [Intensity corrections](../theory/corrections.html.md) gives the Hill-Howard and Brindley equations.