The crystallography, in detail

The crystallography of the thesis, in detail

Every structural statement of the thesis, what it rests on, and how firmly. The point is not to fault the work: the laboratory diffractometer did what it can do, and the revision of October 2026 (V4.0) made the Chapter III refinements a coherent set. The point is to know exactly which structural facts the thesis has, which it infers, and which it needs: that is what the beam time must deliver.

The data and the refinement

One instrument, one radiation. Panalytical Empyrean, Cu Kα, 2\(\theta\) 10–100°, step 0.01°, room temperature, calcined powders (§II.4.6). The upper limit, 100°, means \(d_\mathrm{min} \approx 1.01\) Å; the garnet has 207 independent reflections to 0.81 Å (the neutron range of D2B) and about 90 to 1.01 Å.
What was refined. Chapter III: all 19 Ga–M–Ta compositions in \(I\bar{4}3d\), with tetragonal LLZO (\(I4_1/acd\)), \(\mathrm{BaZrO_{3}}\) (\(Pm\bar{3}m\)) and \(\mathrm{Li_{2}ZrO_{3}}\) (\(C2/c\)) when detected; reported are the lattice parameter (to \(10^{-5}\) Å), the phase fractions (wt%), Rp, Rwp, Rexp, \(\chi^2\) and the Bragg R of the cubic phase (Table III-S1, pp. 79–80). Chapter IV: lattice parameters to two decimals (Table IV-1) and refinement plots of the Ga–Y–Sc series only (Fig. IV-2).
What was not reported. The software, the starting model, the list of refined parameters, the zero-shift or sample-displacement treatment, any estimated standard deviation, any atomic coordinate, site occupancy or displacement parameter.
The scattering. In the reference garnet La and Zr carry 89 % of the X-ray scattering power (\(\sum n f^2\)), lithium 0.4 %, oxygen 5.4 %. Whatever the refinement program was asked to do with lithium, the data could hardly answer. The contrast page has the numbers.

The structural claims and what they rest on

Claim of the thesis	What it rests on	Status, and what would settle it
The cubic garnet of every composition is \(I\bar{4}3d\) (pp. 59, 86, 123).	The literature on Ga-LLZO; the (310) reflection near 21.5°, forbidden in \(Ia\bar{3}d\) (p. 86).	Plausible, not tested. No comparison of refinements in the two groups; the acentric distortion not refined. G01, G05.
The Ga–Ta reference is a single cubic phase, a = 12.977 Å (p. 62).	Rietveld, Table III-S1.	Measured. 12.977 Å agrees with published Ga-LLZO. Three different sets of agreement factors are printed for this one powder (Table III-S1 and Fig. II-9b against Fig. III-1d).
Rb is progressively incorporated on the La site (p. 62).	Lattice expansion \(12.987 \to 13.001\) Å; Rb signal in EDS rising with x.	Supported. X-rays can measure the La-site occupancy directly (Rb, Z = 37, against La, 57). G06.
Ba enters the garnet only in a limited amount; solubility limit between x = 0.10 and 0.15 (p. 62).	Nearly constant lattice parameter; \(\mathrm{BaZrO_{3}}\) from x = 0.15; Ba–Zr co-enrichment in EDS.	Inferred. X-rays cannot see Ba on the La site (Z = 56 against 57); neutrons can (5.07 against 8.24 fm). G06.
Part of the Y occupies the octahedral Zr site in the Ta family (p. 63); in the Sc family Y is on the La site (p. 89).	A slight expansion of the cell (Ta family) or a contraction (Sc family); the rise of \(\mathrm{Li_{2}ZrO_{3}}\) with Y.	Not measurable with these data. Y and Zr differ by one electron; the \(\mathrm{Li_{2}ZrO_{3}}\) mass balance does not close (next row). Y K-edge EXAFS and the La-site occupancy from synchrotron data would settle it. G04.
The Zr displaced by Y reacts with the lithium excess to form \(\mathrm{Li_{2}ZrO_{3}}\) (p. 63); 14.2 wt% \(\mathrm{Li_{2}ZrO_{3}}\) does not limit transport (p. 76).	Phase fractions, Table III-S1.	Does not close. 14.24 wt% beside 85.76 wt% garnet is 0.92 \(\mathrm{Li_{2}ZrO_{3}}\) per garnet formula unit; Y on the Zr site can displace at most 0.20 Zr. Either the fraction is overestimated (a light monoclinic minor phase with overlapping lines) or Zr leaves the garnet for another reason. G09.
The lithium vacancy concentration controls the cubic–tetragonal balance and the conductivity (pp. 63–64, 76–78).	Nominal Li contents; the critical 0.4–0.5 vacancies of the literature.	Plausible, not measured. Lithium never measured; the 15 % \(\mathrm{Li_{2}CO_{3}}\) excess, the losses at 1200 °C, possible protonation and Al uptake all move it. G02, G03, G08.
Sc is on the Zr site; its expulsion as \(\mathrm{Sc_{2}O_{3}}\) in the Rb series reflects a destabilised Zr site (pp. 88–89).	Charge-balance reasoning; \(\mathrm{Sc_{2}O_{3}}\) reflections; Sc-rich inclusions in EDS.	Assumed. Sc could also go to a Li site; neutrons separate Sc (12.29 fm) from Li and Ga. G07.
The Ga–Sc reference is a mixture of comparable amounts of cubic and tetragonal garnet (p. 88).	Splitting of the main reflections (text).	Contradicted within the chapter. Its refinement (Fig. IV-2a) is single cubic and Table IV-1 gives a cubic cell; it is the chapter's cubic baseline. G05, G07.
The Chapter IV lattice parameters, 13.03–13.07 Å (Table IV-1).	Refinements with the earlier protocol (not redone in V4.0).	Not comparable with Chapter III (0.06–0.09 Å higher for almost the same framework); two decimals, no esd. G07.
The decrease of Eₐ to 0.32 eV in Ga–Y₀.₂₀–Ta reflects a more favourable Li migration environment in the lattice (p. 77).	Total (bulk + grain boundary) activation energy.	Inferred. Bulk and grain boundary not separated; the lithium pathway not observed. G10.

Measured: refined from the thesis's data. Inferred / assumed / supported: consistent with the data, which cannot establish it. Not measurable / does not close / contradicted: the data cannot carry it, or contradict it.

The numbers, checked

The fact base transcribes every refinement row and checks it. In fifteen of the nineteen Ga–M–Ta rows the printed \(\chi^2\) equals (Rwp/Rexp)² within rounding; in four it does not (the reference, Ba 0.03 and 0.05, and Rb 0.20, where Rp and Rexp seem exchanged). In five compositions the lattice parameter of Table III-1 is not the rounding of Table III-S1 (Rb 0.05, 0.15, 0.20; Y 0.05, 0.15; Rb 0.15 by 0.003 Å). None of this changes a trend; all of it says that the refinements should be redone, with esd, on data that can carry them.

Powders and pellets

Every phase fraction is for a powder calcined at 900 °C; every density and conductivity is for a pellet sintered at 1200 °C for 6 h in an alumina boat. Between the two, lithium is lost, the tetragonal fraction can change, \(\mathrm{Li_{2}ZrO_{3}}\) can form or react, and Al can enter from the boat. The EDS maps already show \(\mathrm{Li_{2}ZrO_{3}}\)-like regions in a pellet whose powder had none by XRD (xRb = 0.10, p. 69). The structural state of the conductivity samples is not known: G09 and G03.