The cubic garnet is refined in \(I\bar{4}3d\) for every Ga–M–Ta composition and for the Ga–Y–Sc series, "commonly adopted for Ga-substituted LLZO" (§III.3.1, p. 59; §IV.3.1, p. 86). Only the lattice parameter and the phase fractions are reported; no atomic coordinate, site occupancy or displacement parameter is given anywhere (Tables III-S1, IV-1).

The space group is acentric because the lithium sublattice and the oxygen framework lose the inversion of \(Ia\bar{3}d\): 12a and 12b are inversion images of each other. Putting Ga on one or the other is therefore meaningful only relative to that distortion, and a laboratory pattern dominated by La and Zr (89 % of the X-ray scattering power) hardly sees it: the acentric shift of the oxygens changes the Cu Kα pattern by 0.09 %, the neutron pattern by 0.88 %.

Neutrons and single crystals put Ga on 12a in \(I\bar{4}3d\) above about 0.07–0.15 Ga per formula unit (Wagner et al. 2016, SPODI; Rettenwander et al. 2016, ILL D20); Fe does the same. But at the thesis's very composition, \(\mathrm{Ga_{0.2}Li_{6.3}La_3Zr_2O_{12}}\), synchrotron diffraction with high-field NMR finds Ga only on 24d of \(Ia\bar{3}d\) and assigns the other Ga signals to \(\mathrm{LiGaO_{2}}\) (Vema et al. 2023), and the Ga–Ta neutron study closest to the thesis refines \(Ia\bar{3}d\) (Christopher et al. 2023, ECHIDNA). At 0.2 Ga the space group is contested in the literature.

Neutron powder diffraction at high resolution (ILL D2B), refined jointly with synchrotron data (ESRF ID22) in diff-ref-struct: O1 and O2 refined independently, Ga on 12a with Li on 12a, 12b and 48e.

The two references (Ga–Ta, Ga–Sc) and the key set K12.

The acentric distortion as a number per composition (the O1–O2 splitting, the 12a/12b Li–O distances) and the Ga occupancy of 12a with its esd; whether the distortion changes with the La-site substituent and between the Ta and Sc families.

The thesis explains the high conductivity of Ga-LLZO by the \(I\bar{4}3d\) Li pathways (§I.2.3). Whether the tri-doped garnets keep, lose or strengthen that distortion is the structural fact behind the claim.

The lithium vacancy concentration is the thesis's main explanatory variable: it decides the cubic–tetragonal balance ("about 0.7 vacancies per formula unit, above the critical value of approximately 0.4 to 0.5", p. 63), the Rb series's tetragonal fraction and conductivity loss (p. 76), and the Y-series improvement (p. 76). Lithium is never measured: 15 wt% \(\mathrm{Li_{2}CO_{3}}\) excess is weighed (p. 52), the pellets are sintered 6 h at 1200 °C (p. 55), and all Li contents are nominal.

Lithium carries 0.4 % of the X-ray scattering power of the garnet. Adding 0.3 Li per formula unit changes the Cu Kα pattern by 0.52 %, about what counting statistics and a background choice change; the neutron pattern by 1.42 %, with lithium scattering in antiphase (b = −1.90 fm) to every other atom.

Li occupancies of doped LLZO have been refined from natural-Li neutron data many times (Xie et al. 2011; Thompson et al. 2015; Chen et al. 2015; Christopher et al. 2023, where Li falls from 6.66 to 4.94 per formula unit with Ga in Ga–Ta LLZO made with the same 15 % Li excess as the thesis). Meesala et al. 2019 found the tetrahedral Li occupancy rising with the number of dopants in Ta–Ba–Ga LLZO. Never for a systematic matrix at fixed processing.

Neutron powder diffraction (D2B at room temperature; D20 for the whole matrix), with the lithium content refined over 12a, 12b and 48e, constrained by ICP-OES Li analysis of the same powder.

K12 on D2B; all 38 on D20.

Refined Li per formula unit and its distribution over the tetrahedral and octahedral sites, with esd, for each composition; the measured vacancy concentration set against the nominal one (0.7 for Ga–Ta, 0.5 for Ga–Sc, reduced by Ba and Rb), against the tetragonal fraction, and against \(\sigma_\mathrm{RT}\) and \(E_\mathrm{a}\).

It turns the thesis's central argument (vacancies set phase and conductivity) from inference into measurement, composition by composition, and gives the ML chapter the structural descriptors its Perspectives ask for (p. 126).

Pellets are sintered at 1200 °C for 6 h, buried in mother powder in covered alumina boats (p. 55). Aluminium is not analysed and not mentioned as a possible dopant.

Al\(^{3}\)⁺ from alumina is a well known involuntary dopant of LLZO and goes to the same tetrahedral Li sites as Ga\(^{3}\)⁺. In X-rays it is a light atom beside La; in neutrons its scattering length (3.45 fm) differs from both Li (−1.90 fm) and Ga (7.29 fm). 0.1 Al per formula unit changes the Cu Kα pattern by 0.71 %, the neutron pattern by 1.39 %.

Al from crucibles was what stabilised the first cubic LLZO (Geiger et al. 2011); some groups avoid it with zirconia or MgO crucibles (Christopher et al. 2023). It is assumed far more often than it is measured.

ICP-OES of sintered pellets (Al, Ga, Li), and neutron diffraction of crushed pellets with Ga and Al refined on 12a under the ICP constraint.

Sintered pellets of the references and of the best compositions (Ga–Y₀.₂₀–Ta, Ga–Rb₀.₀₃–Sc).

The Al content of the conductivity samples, or an upper bound; if Al is present, which site it shares with Ga.

Every conductivity in the thesis was measured on sintered pellets. If they contain Al, the best compositions are Ga–Al–M–Ta and Ga–Al–M–Sc, and the comparison between series must include it.

The slight lattice expansion of the Ga–Y–Ta series (\(12.983 \to 12.994\) Å) is read as "part of the Y\(^{3}\)⁺ occupies the octahedral Zr site", supported by the rise of \(\mathrm{Li_{2}ZrO_{3}}\) (p. 63); the same partial occupation is then used to explain the tetragonal fraction (p. 64), the extra lithium and the best conductivity of the thesis, 1.04 mS \(\mathrm{cm}^{-1}\) at x = 0.20 (p. 76). In the Ga–Y–Sc series the opposite is concluded: the lattice contracts and Y is on the La site (p. 89).

Y and Zr differ by one electron (Z = 39, 40): no X-ray pattern can tell them apart on the octahedral site (0.06 % change for 0.1 Y per formula unit). Neutrons cannot either: b(Y) = 7.75 fm against b(Zr) = 7.16 fm gives 0.16 %. This question is not for neutrons. What X-rays can measure is the La site: 0.1 Y on the La site changes the pattern by 1.8 % (Z = 39 against 57), so a precise La-site occupancy, with the Y content fixed by ICP, says how much Y is not there.

Y on the Zr site of LLZO has been argued only from a slight lattice expansion (Hitz et al. 2013), the thesis's own argument. No Y K-edge study of a garnet electrolyte was found; recent neutron studies of Y-containing medium-entropy garnets do not report the Y site.

Y K-edge EXAFS (17.04 keV; ESRF BM23, or BM25 SpLine through the Spanish route): coordination 6 on the Zr site, 8 on the La site, Y–O near 2.2 Å against 2.4 Å. La-site occupancy from high-resolution synchrotron diffraction (ID22). Resonant powder diffraction at the Y K edge (ID22) as a direct, site-resolved contrast.

Ga–Y–Ta at x = 0.05, 0.10, 0.20, 0.30; Ga–Y–Sc at 0.05 and 0.20; \(\mathrm{Y_{2}O_{3}}\), YSZ and a Y-garnet as references.

The fraction of Y on each site for each composition, and whether it differs between the Ta and Sc families as the thesis's two readings imply.

The best composition of the thesis rests its explanation on this site. Measured, it either becomes the thesis's strongest structural result or is replaced by a better one.

\(I\bar{4}3d\) "maintained throughout" (General Conclusions, p. 123), while tetragonal LLZO is refined at 3–21 wt% in the Rb and Y series (Table III-S1) and the Ga–Sc reference is described as half tetragonal (p. 88) but refined as single cubic (Fig. IV-2a).

The weak reflections that distinguish \(I\bar{4}3d\) from \(Ia\bar{3}d\), like (310), and the small splittings of the tetragonal cell are at the limit of laboratory resolution and counting statistics. Their intensity comes mostly from the lithium sublattice and the oxygen distortion, where neutrons are 3–9 times more sensitive.

Two causes can make a garnet \(I\bar{4}3d\): Ga (or Fe) ordering on 12a, and protonation, which neutrons showed to give an acentric \(I\bar{4}3d\) phase with H on the octahedral sites (Orera et al. 2016, Spanish neutron group). No study has separated the two in a Ga-doped garnet; aged powders cannot be read without it (G08).

High-resolution synchrotron diffraction (ID22, instrumental FWHM 0.0025°) for the splittings and the weak reflections of all 38 powders; neutrons (D2B, D20) for the intensities; refinement in both space groups compared on the same data.

All 38 on ID22; K12 on D2B.

For each composition: the space group the data support, the tetragonal fraction with esd and its detection limit (\(\approx 0.1\) wt% on ID22), the tetragonality c/a of the minor phase.

It fixes the phase state that every conductivity is compared against, and settles the Chapter IV baseline.

Ba: lattice parameter nearly constant (12.969–12.981 Å) although Ba\(^{2}\)⁺ is larger than La\(^{3}\)⁺, read as limited incorporation; \(\mathrm{BaZrO_{3}}\) from x = 0.15 (p. 62). Rb: steady expansion to 13.001 Å, read as progressive incorporation (p. 62).

Ba (Z = 56) and La (Z = 57) are indistinguishable by X-rays: 0.1 Ba per formula unit on the La site changes the Cu Kα pattern by 0.09 %. Neutrons separate them well, b(Ba) = 5.07 fm against b(La) = 8.24 fm: 1.0 % change, twelve times more. For Rb the roles reverse: X-rays see it (Z = 37, 1.9 %) better than neutrons (0.37 %).

Ba with Ta has been reported single cubic up to 0.30 Ba (Inada et al. 2016), against the thesis's limit near 0.10; with Ga, \(\mathrm{BaZrO_{3}}\) forms between grains (Tian et al. 2022). Rb on the La site of Ga-LLZO is published at 0.05 only (Wu et al. 2017); the thesis goes to 0.30, six times higher.

Neutrons (D2B) for the La-site occupancy of the Ba series; synchrotron (ID22) for the Rb series; Rb K-edge EXAFS (15.2 keV, BM23) for the local environment; the phase fraction of \(\mathrm{BaZrO_{3}}\) from ID22.

Ga–Ba–Ta at x = 0.05, 0.10, 0.20, 0.30; Ga–Rb–Ta at 0.05, 0.20; Ga–Rb–Sc at 0.03, 0.20.

The Ba solubility limit measured in the garnet (not inferred from \(\mathrm{BaZrO_{3}}\)), the Rb content of the garnet, and the lithium that compensates each.

The thesis ranks the three substituents by how much lithium each adds; that ranking needs the amount of each that is actually in the garnet.

Chapter IV was not re-refined with the Chapter III protocol: its lattice parameters (13.03–13.07 Å, two decimals, Table IV-1) are 0.06–0.09 Å above Chapter III's for an almost identical framework; \(\mathrm{Sc_{2}O_{3}}\) appears in the Rb series and \(\mathrm{Y_{2}O_{3}}\) in the Y series; Sc is assumed on the Zr site (p. 88).

The two families need the same data and the same refinement to be compared. Sc on the Zr site is seen by both probes (X-rays 3.8 %, neutrons 1.4 %, with b(Sc) = 12.29 fm the strongest scatterer of the cell); Ta by X-rays only (Z = 73; neutrons 0.03 %). Sc\(^{3}\)⁺ could also go to a Li site; the scattering length tells it apart from Li and Ga.

The direct precedent of the Sc family, Ga on Li with Sc on Zr (Buannic et al. 2017, 1.8 mS \(\mathrm{cm}^{-1}\)), used DFT, XRD, ICP, NMR and STEM, no neutrons. Sc in LLZO has never been refined from neutron data, although its contrast with Zr is the strongest of the cell.

All 18 Ga–M–Sc powders on ID22 with the Ga–M–Ta set, refined with one protocol; the Sc compositions of K12 on D2B; Sc K-edge EXAFS at BM08 LISA or BM20 (4.49 keV, below BM23's range).

All 38 (ID22); Ga–Sc, Ga–Rb₀.₀₃–Sc, Ga–Rb₀.₂₀–Sc, Ga–Y₀.₁₀–Sc (D2B).

One table of lattice parameters with esd for the 38 compositions, from one instrument and one protocol; the site of Sc.

The thesis's cross-family conclusion (the inversion of the Rb and Y rankings between families, p. 123) depends on it.

Not discussed for the bulk. Oxygen enrichment at grain boundaries is seen by EDS and attributed to \(\mathrm{Li_{2}CO_{3}}\) formed on exposure of fractured surfaces to air (p. 69).

Li⁺/H⁺ exchange in moist air is well documented for garnets and changes the lattice parameter and the Li content. X-rays do not see hydrogen; neutrons see it twice, by its negative scattering length on the Li sites and by its incoherent scattering (80 b) in the background. 0.5 H⁺ for 0.5 Li⁺ changes the Cu Kα pattern by 0.59 %, the neutron pattern by 2.3 %.

Exchange starts on the first exposure to air and Ga-LLZO behaves like Al-LLZO (Redhammer et al. 2021); octahedral 48e Li goes first (Redhammer et al. 2022); deep hydration gives an \(I\bar{4}3d\) phase (Orera et al. 2016); dry sintering avoids it (Bernuy-López et al. 2014).

Neutron background and Li-site refinement (D2B, D20); TGA–MS of the same powders; fresh samples kept under argon from synthesis to beam.

Old and fresh powder of the same composition (Ga–Ta, Ga–Y₀.₂₀–Ta); a deliberately aged sample.

The H content of the samples as kept, the rate of exchange, and the protocol that the beam-time samples must follow.

Without it, every Li occupancy of G02 is uncertain by the unknown protonation.

14.2 wt% \(\mathrm{Li_{2}ZrO_{3}}\) in the best composition (Ga–Y₀.₂₀–Ta), explained by Zr displaced by Y (p. 63) and said not to limit transport (p. 76). All phase fractions come from calcined powders (900 °C, p. 59); all conductivities from pellets sintered at 1200 °C (p. 55).

14.24 wt% \(\mathrm{Li_{2}ZrO_{3}}\) beside 85.76 wt% garnet is 0.92 \(\mathrm{Li_{2}ZrO_{3}}\) per garnet formula unit (fact base), while Y on the Zr site can displace at most x = 0.20 Zr. The mass balance does not close: either the light monoclinic minor phase, with heavily overlapping reflections, is overestimated on laboratory data, or Zr leaves the garnet for another reason and the garnet is Zr-poor. The phase content of the pellets, the samples of every conductivity, was not measured.

Excess lithium pushes Ga out of the garnet into \(\mathrm{LiGaO_{2}}\) (Vema et al. 2023), and Ga-rich compositions give \(\mathrm{La_{2}Zr_{2}O_{7}}\) + \(\mathrm{LiGaO_{2}}\) (Rettenwander et al. 2014): with 15 % \(\mathrm{Li_{2}CO_{3}}\) excess, how much of the nominal 0.2 Ga is in the garnet is itself a question. In situ neutron studies show LLZO forming from a carbonate melt and residual phases left by \(\mathrm{Li_{2}CO_{3}}\) evaporation (Rao et al. 2015; Chen et al. 2015).

High-resolution synchrotron diffraction (ID22) of powders and of crushed pellets of the same compositions, with an internal standard for the amorphous fraction; neutron diffraction, where \(\mathrm{Li_{2}ZrO_{3}}\) scatters differently from the garnet; ICP for the Zr/La ratio.

All 38 powders and 12 sintered pellets (crushed) on ID22.

Phase fractions with esd and detection limits for powders and pellets, \(\mathrm{LiGaO_{2}}\) and \(\mathrm{La_{2}Zr_{2}O_{7}}\) looked for explicitly; how much of the nominal Ga is in the garnet; a closed mass balance (Li, Zr, La) for each composition; the evolution of the phase state from 900 °C to 1200 °C.

The conductivities can be interpreted only against the phase content of the pellets they were measured on.

Activation energies from 0.32 to 0.46 eV, total (bulk and grain boundary unresolved) for all Ta compositions (Table III-2); the decrease to 0.32 eV for Ga–Y₀.₂₀–Ta is read as "a more favorable Li⁺ migration environment within the lattice" (p. 77).

Lithium transport pathways show up as anharmonic, elongated nuclear density between 12a/12b and 48e, which neutron diffraction at high temperature maps (Fourier and maximum-entropy maps, anharmonic displacement parameters). X-rays cannot see it.

High-temperature neutron diffraction with maximum-entropy maps showed the Li pathway in LLZO (Han et al. 2012, ⁷Li sample, LANSCE); single-crystal neutron diffraction of \(\mathrm{Li_{6}La_{3}ZrTaO_{12}}\) to 873 K resolved the Li1 displacement towards 96h (Redhammer et al. 2021).

Neutron diffraction from room temperature to 900–1000 K (D2B furnace for high-Q data; D20 for continuous ramps and the tetragonal–cubic transition of the Rb and Y series).

Ga–Ta, Ga–Y₀.₂₀–Ta, Ga–Rb₀.₀₃–Sc, Ga–Ba₀.₁₀–Ta.

Li nuclear density maps at several temperatures; the bottleneck of the 12a/12b–48e path; whether the lowest-\(E_\mathrm{a}\) composition has the most open path; the tetragonal–cubic transition temperatures.

It links the measured activation energies to the structure, which is what the thesis's Chapter III conclusion claims and cannot show.

EDS maps at the micrometre scale show homogeneous Ga, Y and Rb (pp. 69, 91); local structure is not addressed.

Average-structure refinement cannot see Ga–Li or dopant–vacancy short-range order, nor the local distortion around Y or Rb. Total scattering (PDF) can, with X-rays for the heavy-atom framework and neutrons for Li–O and Li–Li.

The only total-scattering study of cubic LLZO is of the undoped compound (Tian et al. 2023, neutron and X-ray PDF with reverse Monte Carlo).

X-ray PDF (ESRF ID15A or ID22 fast-PDF), neutron PDF (ILL D4c).

Ga–Ta, Ga–Y₀.₂₀–Ta, Ga–Rb₀.₀₃–Sc.

Local distortions around the substituents; Li–Li correlations; the length scale over which the average \(I\bar{4}3d\) structure holds.

Second round: it explains what the average structure of G01–G02 will leave unexplained.