# Fileset [Acta Cryst-B82-120-130-MPr5(MoO4)8.pdf](https://mdr.nims.go.jp/filesets/3df95a03-d18d-4972-9928-b5daaa60fd34/download) ## Creator Vladimir A. Morozov, Dina V. Deyneko, Darya G. Filatova, [Alexei A. Belik](https://orcid.org/0000-0001-9031-2355), Olga M. Basovich, Elena G. Khaikina, Bogdan I. Lazoryak ## Rights [In Copyright](http://rightsstatements.org/vocab/InC/1.0/) ## Other metadata [Incommensurately modulated structures of the M _1/8 Pr _5/8 MoO ₄ ( M = Li, Na, K) scheelites](https://mdr.nims.go.jp/datasets/b05e3618-5ebf-449a-9eab-8d5af088a6ca) ## Fulltext tq5028 120..130electronic reprintISSN: 2052-5206journals.iucr.org/bIncommensurately modulated structures of theM 1/8Pr5/8MoO4 (M = Li, Na, K) scheelitesVladimir A. Morozov, Dina V. Deyneko, Darya G. Filatova, Alexei A. Belik,Olga M. Basovich, Elena G. Khaikina and Bogdan I. LazoryakActa Cryst. (2026). B82, 120–130IUCr JournalsCRYSTALLOGRAPHY JOURNALS ONLINEAuthor(s) of this article may load this reprint on their own web site or institutional repository and onnot-for-profit repositories in their subject area provided that this cover page is retained and a permanentlink is given from your posting to the final article on the IUCr website.For further information see https://journals.iucr.org/services/authorrights.htmlActa Cryst. (2026). B82, 120–130 V. A. Morozov et al. · Incommensurately modulated structures of Li, Na, K scheeliteshttps://journals.iucr.org/b/https://doi.org/10.1107/S2052520626000375https://journals.iucr.org/services/authorrights.htmlhttps://crossmark.crossref.org/dialog/?doi=10.1107/S2052520626000375&domain=pdf&date_stamp=2026-04-01research papers120 https://doi.org/10.1107/S2052520626000375 Acta Cryst. (2026). B82, 120–130ISSN 2052-5206Received 7 November 2025Accepted 14 January 2026Edited by T. B. Bekker, Siberian Branch ofRussian Academy of Science, Russian FederationThis article is part of a special issue on currentresearch in crystal growth and related char-acterizationKeywords: incommensurate modulation; crystalstructure; scheelite; synchrotron powder X-raydiffraction data; ordering.B-IncStrDB reference: L8zZeSPwZg1Supporting information: this article hassupporting information at journals.iucr.org/bIncommensurately modulated structures of theM1/8Pr5/8MoO4 (M = Li, Na, K) scheelitesVladimir A. Morozov,a* Dina V. Deyneko,a Darya G. Filatova,a Alexei A. Belik,bOlga M. Basovich,c Elena G. Khaikinac and Bogdan I. LazoryakaaChemistry Department, Moscow State University, 119991, Russian Federation, bResearch Center for MaterialsNanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba, Ibaraki, 305-0044,Japan, and cBaikal Institute of Nature Management, Siberian Branch, Russian Academy of Science, Ulan-Ude, 670047,Russian Federation. *Correspondence e-mail: morozov111vla@mail.ruThe effect of cation substitutions in the scheelite-type framework is investigatedto understand the ordering of M+/Pr3+ cations and vacancies in the structure.The cation deficientM1/8Pr5/8&1/4MoO4 (M = Li, Na, K) phases were synthesizedusing solid state reactions. The K:Pr ratio in the K1/8Pr5/8MoO4 sample wasdetermined by inductively coupled plasma mass spectroscopy, inductivelycoupled plasma optical emission spectrometry and total reflection X-ray fluor-escence spectroscopy. The local element distributions in K1/8Pr5/8&1/4MoO4 weredetermined by energy-dispersive X-ray spectrometry. The incommensuratelymodulated crystal structures of the scheelite-based M1/8Pr5/8MoO4 (M = Li, Na,K) phases were refined from synchrotron powder X-ray diffraction data. Theanalysis revealed that cation ordering in the M1/8Pr5/8MoO4 (M = Li, Na)structures is incomplete and is more accurately described by continuous, ratherthan discontinuous (step-like), occupational modulation functions. The occu-pancy modulation waves for Li/Na and Pr atoms demonstrate an antiphaserelation. Compared with M = Li, Na, the order of K and Pr cations in the Aposition in the K1/8Pr5/8MoO4 structure is best approximated by crenel func-tions. Refining the coordinates and lengths of three atomic domains yields thecomposition of K0.145Pr0.618MoO4. In all cases, the modulation arises fromordering of M/Pr cations and cation vacancies at the A-sublattice of the parentscheelite ABO4 structure. The distortion of PrO8 and MO8 polyhedra is prac-tically independent of the radius of the M+ cations. The refinements of theM1/8Pr5/8MoO4 (M = Li, Na, K) structures reveal that MoO42� tetrahedra inthese scheelite-type compounds demonstrate a flexible geometry. Both Mo—Obond distances and O—Mo—O bond angles vary significantly with changing thepopulation of the A site by cations with different sizes.1. IntroductionScheelite related compounds (SRC) with a general formula(A0,A00)n[(B0,B00)O4]m (A0, A00 = Ag, alkali, alkali-earth orrare-earth elements; B0, B00 = W, Mo) are a family of inorganicmaterials that have attracted widespread attention due to theirexcellent thermal stability, hydrolysis resistance, long lifespan,and low cost. Additionally, they can accommodate variousrare earth (RE) activators to achieve effective luminescence.SRC have been widely studied and applied in phosphorconverted WLED (white-light-emitting diodes), solid-statelasers, fingerprint detection, environmental lighting, medicaltreatment, and optical thermometers (Bin et al., 2019; Wu etal., 2021; Lu et al., 2018; Singh et al., 2023; Morozov et al., 2023;Huang et al., 2004; Huang et al., 2018; Zhao et al., 2013;Pusdekar et al., 2025; Meert et al., 2014; Morozov et al., 2018;Zhao et al., 2021; Zhang et al., 2024) etc.electronic reprintThe scheelite-type ABO4 structure consists of columns ofvertex sharing AO8 polyhedra and BO4 tetrahedra runningalong the c axis and forming a 3D framework. CaWO4 is thearistotype of this family and has tetragonal symmetry withspace group I41/a. The scheelite-type structures may havelower symmetries, depending on the nature of the A and Bcations or the presence of ordered vacancies. The creation of acation vacancy (&) in the scheelite-type framework andordering of the A cations and vacancies offers a new para-meter for monitoring structure and properties of SRC. The Acations and vacancies can order, frequently forming incom-mensurately modulated structures (Abakumov et al., 2014;Arakcheeva et al., 2012; Batuk et al., 2017; Morozov et al.,2012; Morozov et al., 2013; Morozov et al., 2017; Morozov etal., 2018; Morozov et al., 2023).The cationic vacancies formation and the cationic andanionic substitutions in SRC lead to the formation of (3+1)Dincommensurate modulated structures, which require a(3+n)D approach. For example, replacing the smaller Gd3+with the larger Eu3+ at the A-sublattice inCaGd2(1–x)Eu2x(MoO4)4(1–y)(WO4)4y does not affect thenature of the incommensurate modulation. However,increasing replacement of W6+ by Mo6+ switches the modu-lation from (3+1)D to (3+2)D regime (Abakumov et al., 2014;Morozov et al., 2013). At low vacancy concentration (<15%)(Arakcheeva et al., 2012; Batuk et al., 2017), cations and cationvacancies are randomly distributed in the structure, and thematerials preserve the I41/a symmetry of the parent scheelitestructure. The structure with 20% vacancies has a(3+2)D tetragonal symmetry (Batuk et al., 2017). Furtherincreasing the cation vacancies from 20% to 25% inCa0.85–1.5xGdxEu0.1&0.05+0.5xWO4 (Batuk et al., 2017) and from<15% to 17% in NaxEu3+(2–x)/3MoO4 (Arakcheeva et al.,2012), the (3+2)D incommensurately modulated structureundergoes a monoclinic distortion, forming the (3+1)Dstructure.In this contribution, we provide the structure refinement for(3+1)D-modulated monoclinic M1/8Pr5/8&1/4MoO4 (M = Li,Na, K) using synchrotron powder X-ray diffraction and revealan influence of cationic composition on ordering of M+/Pr3+cations and vacancies.2. Experimental2.1. Materials and sample preparationM1/8Pr5/8&1/4MoO4 (M = Li, Na, K) samples were synthe-sized using solid state reactions from a stoichiometric mixtureof M2MoO4 (M = Li, Na, K) and Pr2(MoO4)3 at 773–1073 Kfor 70 h in air, followed by quenching from 1073 K to roomtemperature (TR). M2MoO4 (M = Li, Na, K) samples wereprepared using solid state reactions from stoichiometricamounts of M2CO3 (99.99%) and MoO3 (99.99%) at 673–823 K for 60 h. The Pr2(MoO4)3 precursor was synthesizedusing a solid state reaction from Pr6O11 (99.99%) and MoO3 at823 K for 10 h, followed by annealing at 1123 K for 80 h.Powder X-ray diffraction (PXRD) patterns of Pr2(MoO4)3matched the Pr2(MoO4)3 XRD patterns reported by Logvi-novich et al. (2010) and did not contain reflections of anyforeign phases.2.2. CharacterizationK:Pr ratio for the bulk K1/8Pr5/8&1/4MoO4 sample wasdetermined: (1) by inductively coupled plasma mass spectro-metry (ICP-MS) using a quadrupole Agilent 7500C massspectrometer (Agilent Technologies, Japan) controlled via thePC software ChemStation (version G1834B) package (AgilentTechnologies); (2) by inductively coupled plasma opticalemission spectroscopy (ICP-OES) using an Agilent 5800 VDVICP-OES spectrometer (Agilent Technologies, USA); (3) bytotal reflection X-ray fluorescence spectroscopy (TXRF) usinga S2 Picofox spectrometer (Bruker Nano GmbH, Germany)with Mo K� radiation.The ICP-MS, ICP-OES and TXRF results were obtained bythe external standard method using single element (K and Pr)standard solutions (High-Purity Standards, USA). Threesamples weighing between 6 and 15 mg were analyzed. Amixture of concentrated nitric and hydrochloric acids (aquaregia) and deionized water (18.2 mQ cm�1) was used forsolution preparation. All samples were weighed in a vial anddissolved by aqua regia (1:1). After sample dissolution, theobtained solution was adjusted to a volume of 2 ml withdeionized water. Further, the stock solution was diluted 1000-fold for ICP-MS and tenfold for ICP-OES and TXRFmeasurements in a 10 ml vial using a manual sampler. K K1,2(3.314 keV), Pr L1 (5.035 keV)L and Ga K1,2 (9.251 keV) linesin the TXRF spectra were used for the element compositionstudy.A Jeol JSM-6490LV scanning electron microscope (SEM)equipped with an EDX spectrometer INCA X-Sight (OxfordInstruments) was used for SEM observations ofK1/8Pr5/8&1/4MoO4. Samples were coated with a thin layer ofcarbon for SEM examinations. KK, PrL, and MoL lines in theSEM-EDX spectra were used for the element compositionstudy. The oxygen content was not quantified by EDX.Powder X-ray diffraction patterns of M1/8Pr5/8&1/4MoO4(M = Li, Na, K) were collected on a SIEMENS D500diffractometer equipped with a primary SiO2 monochromator(Cu K�1 radiation, � = 1.5406 Å, Bragg–Brentano geometry)and a position sensitive detector (BRAUN). PXRD data werecollected at TR in the 7�–140� 2� range with steps of 0.02�. Todetermine the unit-cell parameters, the PXRD data wererefined by Le Bail decomposition (Le Bail et al., 1988) usingthe JANA2006 software (Petřı́ček et al., 2014).Synchrotron powder X-ray diffraction data forM1/8Pr5/8&1/4MoO4 (M = Li, Na, K) were measured in a largeDebye–Scherrer camera at the BL15XU beamline of SPring-8(� = 0.5008 Å) (Tanaka et al., 2008; Tanaka et al., 2013)between 0.1� and 40.52� with a step of 0.0025� in 2�. Thesample was packed into a Lindemann glass capillary (innerdiameter 0.1 mm), which was rotated during the measurement.Details of experiments and characteristics of the final struc-ture refinements are listed in Table 1. The Rietveld analysis ofresearch papersActa Cryst. (2026). B82, 120–130 V. A. Morozov et al. � Incommensurately modulated structures of Li, Na, K scheelites 121electronic reprintincommensurately modulatedM1/8Pr5/8&1/4MoO4 (M = Li, Na,K) structures was performed using the JANA2006 software(Petřı́ček et al., 2014). Illustrations were produced with thispackage in combination with the program DIAMOND(Brandenburg, 1999).3. Results3.1. Elemental composition39K and 141Pr isotopes were used for ICP-MS measurementsto determine the K:Pr ratio for the bulk K1/8Pr5/8&1/4MoO4sample. Measurements using different isotopes yielded iden-tical results with no overlaps. The reproducibility (sr) of theICP-MS results was 0.1 for both K and Pr determination.Apparently, the high value of sr is associated with the inter-fering effect of argon plasma 38Ar+40Ar+ on the determinationof 39K and 40K isotopes, as well as with the possible overlap ofpolyatomic ions with 141Pr isotope. The formation of polya-tomic ions in plasma is probabilistic in nature, so it is difficultto take them into account in a control experiment. Pr(417.939 nm) and K (766.490 nm) lines were used for ICP-OES measurements. This method reduced the sr of Pr and Kdetermination to 0.05. The time of TXRF spectrum acquisitionwas 250 s (lifetime). To verify substrate purity, admixtureswere determined after cleaning the substrate with HNO3solution (10%). A Ga3+ ion solution with the concentration50 mg l�1 as an internal standard was used for quantitativecalculations.SEM-EDX analysis revealed K:Pr:Mo ratios inK1/8Pr5/8MoO4 of 0.12:0.61:1 (7.13 � 0.26 at% K, 35.20 �0.67 at% Pr, 57.67 � 0.87 at% Mo), which closely matchthe expected composition. Fig. S1 in supporting informationshows representative TXRF and SEM-EDX spectra forthe K1/8Pr5/8MoO4 sample. Table 2 summarizes the K:Prratio results in K1/8Pr5/8MoO4 determined by differentmethods. The determined K:Pr ratio is consistent withexpected values.research papers122 V. A. Morozov et al. � Incommensurately modulated structures of Li, Na, K scheelites Acta Cryst. (2026). B82, 120–130Table 1Crystal data, data collection and refinement of M1/8Pr5/8MoO4 (M = Li, Na, K).Li scheelite Na scheelite K scheeliteCrystal dataChemical formula Li0.125Pr0.625&1/4MoO4 Na0.125Pr0.625&1/4MoO4 K0.145Pr0.618&0.237MoO4Mr 246.5 250.9 252.7Crystal system, space group Monoclinic, I2/b(��0)00Temperature (K) 293a, b, c (Å) 5.30263 (1), 5.29792 (1), 11.75179 (2) 5.30752 (3), 5.30900 (3), 11.75478 (3) 5.31857 (1), 5.33959 (1), 11.87729 (3)� (�) 90.0184 (4) 90.0084 (9) 90.0509 (2)V (Å3) 330.14 (1) 331.22 (1) 337.30 (1)Wave vectors q = 0.568641a* + 0.835383b* q = 0.578340a* + 0.837694b* q = 0.566419a* + 0.836554b*Z 4 4 4Dcalc (g cm�3 4.959 5.031 4.976Colour Light greenData collectionDiffractometer BL15XU beamline of SPring-8Radiation type, � (Å) Synchrotron, 0.5008 Synchrotron, 0.5008 Synchrotron, 0.50082� values (�) 2�min = 3.148, 2�max = 45.518, 2�step =0.00252�min = 2.395, 2�max = 45.518, 2�step =0.0032�min = 2.69, 2�max = 42.818, 2�step =0.0025Imax 81873 17940 69838Number of points 14121 14374 13376RefinementRefinement RietveldBackground function Legendre polynomials, 15 termsNumber of reflections (all/observed) 1920/1215 1930/839 2739/2321Among themMain (all/observed) 645/464 647/348 548/4931st order satellites (all/observed) 1275/751 1283/491 1099/9352nd order satellites (all/observed) – – 1092/893No. of refined parameters/ refinedatomic parameters55/31 55/27 75/47R and Rw (%) for Bragg reflections(Rall/Robs)5.89/3.91 and 2.41/2.32 6.19/3.40 and 2.76/2.48 4.76/4.00 and 2.41/2.37Among them:Main 2.93/2.36 and 1.85/1.81 3.36/2.42 and 2.09/1.92 2.33/2.09 and 1.57/1.551st order satellites 10.27/7.03 and 3.87/3.66 10.86/6.51 and 4.63/4.08 6.73/5.84 and 3.42/3.372nd order satellites – – 8.81/7.07 and 2.99/2.93R factors and goodness of fit Rp = 0.0736, Rwp = 0.0957, Rexp =0.0709, R(F) = 0.0589, �2 = 1.35Rp = 0.1097, Rwp = 0.1461, Rexp =0.1485, R(F) = 0.0610, �2 = 0.98Rp = 0.0661, Rwp = 0.0922, Rexp =0.0654, R(F) = 0.0476, �2 = 1.41Max./min. residual density (e Å�3) 2.77/�4.7 2.35/�4.99 2.62/�3.01CSD number 2500363 2500364 2500362Computer programs: JANA2000 (Petřı́ček et al., 2014).electronic reprint3.2. Preliminary characterizationFig. 1 shows parts of powder X-ray diffraction patterns ofM1/8Pr5/8&1/4MoO4 (M = Ag, Li, Na, K) phases. Similar toAg1/8Pr5/8MoO4 (Morozov et al., 2006), the PXRD patterns ofM1/8Pr5/8&1/4MoO4 (M = Li, Na, K) contain intense reflectionscorresponding to the scheelite-type structure and weakersatellite reflections. Earlier the appearance of weaker reflec-tions was observed on PXRD patterns of Na1/8Pr5/8&1/4MoO4(Dhanya et al., 2019). The unit-cell parameters were deter-mined by Le Bail decomposition (Le Bail et al., 1988) inI2/b(��0)00 superspace group using JANA2006 software(Petřı́ček et al., 2014).3.3. Refinement of M1/8Pr5/8&1/4MoO4 (M = Li, Na, K) crystalstructuresSynchrotron powder X-ray diffraction (SXRD) data wasused to refine crystal structures of theM1/8Pr5/8&1/4MoO4 (M =Li, Na, K) phases. The incommensurately modulated struc-tures were refined from powder diffraction intensities usingsuperspace group I2/b(��0)00. Fig. 2 shows the observedelectron density along the internal axis, x4, calculated in thevicinity of A position for the M1/8Pr5/8MoO4 [M = Li (a),Na (b), K (c)] structures. The continuous variation of theelectron density distribution along the x4 axis indicates thewavy behavior of the A position occupation by M/Pr andvacancies. Based on refined structures of other scheelite-related compounds (Abakumov et al., 2014; Arakcheeva et al.,2012; Batuk et al., 2017; Morozov et al., 2013; Morozov et al.,2017; Morozov et al., 2018; Morozov et al., 2023), we testeddifferent models during the Rietveld refinement. Appendix 1of the supporting information provides detailed descriptionsof the models. The nine models (i)–(ix) differ by the functionused for the occupancy of the A position and by the functionused for the displacive modulation. We constrained all A sitecations to the same coordinates, displacive modulation func-tions, and isotropic atomic displacement parameters (ADPs),and refined x04 coordinates (except x04 = 0 and x04 = 0.5) andlengths (�) of the atomic domains. Tables S1–S3 and Figs. S2and S3 list results of the Rietveld refinement of the structuresin these models.research papersActa Cryst. (2026). B82, 120–130 V. A. Morozov et al. � Incommensurately modulated structures of Li, Na, K scheelites 123Table 2K:Pr ratio in K1/8Pr5/8&1/4MoO4 determined by ICP-MS, ICP-OES, TXRFand SEM-EDX.Compound ICP-MS ICP-OES TXRF SEM-EDXK1/8Pr5/8&1/4MoO4 1:5.2� 0.2 1:5.0� 0.3 1:4.8� 0.3 1:4.9� 0.3Figure 2The vicinity of the A position in the structural model for theM1/8Pr5/8MoO4 [M = Li (a), Na (b), K (c)] structures. The x1x4, x2x4, andx3x4 sections are presented for observed electron density maps. Linesshow positive values of the electron density. The step between lines is10 Å�3 for electron density.Figure 1Parts of PXRD patterns of M1/8Pr5/8MoO4 [M = Ag (1) (Morozov et al.,2006b), Li (2), Na (3), K (4)] phases. Bragg reflections (5) for CaMoO4(JCPDS, PDF2 51-0267) are shown. The satellite reflections are indicatedwith red diamonds. Inset shows low-angle parts of the profile. Theindexation of some satellite (green) and main (black) reflections areshown.electronic reprint3.3.1. M1/8Pr5/8&1/4MoO4 (M = Li, Na) structuresComparing the refinement results of all models with one[crenel-1H (ii), crenel-1L (iii)], crenel-3H (vi), crenel-3L (vii),crenel-4H (viii), and crenel-4L (ix)] and three crenel domainsof Pr [crenel-2H (iv) and crenel-2L (v)] shows that the firstmodel (i) has better reliability factors for all reflections(Tables S1 and S2). The harmonic model (i) demonstrates thebest agreement with experimental data (Figs. S2 and S3). Italso shows the best reliability factors for main [RF(mainreflections) = 2.93% (Li) and 3.72% (Na)] and for the first-order satellites [RF(first-order satellites) = 10.27% (Li) and11.79% (Na)] (Tables S1 and S2). The bond valence sums(BVS) variations forM and Pr inM1/8Pr5/8MoO4 (M = Li, Na)structures for refinement models with the best reliabilityfactors for all reflections are shown in Figs. S4 and S5. BVSwere calculated with 3 Å cutoff. BVS for Li differ only slightlyamong the three refinement models, with ranges 0.71–0.95 for(i), 0.82–0.88 for (iv) and 0.83–0.91 for (v). The difference inBVS for Pr in Li1/8Pr5/8MoO4 structure is greater for thesemodels, but not wrong: ranges are 2.05–3.24 for model (i),2.92–3.30 (Pr1) and 2.28–2.55 (Pr2) for (iv), 3.06–3.41 (Pr1)and 2.30–2.68 (Pr2) for (v). Consequently, selecting a refine-ment model for the Li1/8Pr5/8MoO4 structure based solely onBVS variations for Li and Pr is challenging.A similar situation was observed when choosing betweenrefinement models for the Na1/8Pr5/8MoO4 structure based onBVS variations for Na and Pr. Only the crenel-4H (viii) modelmodel can be clearly rejected, as it yields erroneous BVSranges of 1.83–4.08 for Pr and 1.58–1.63 for Na. InLi1/8Pr5/8MoO4, BVSs differ slightly between the three models,ranging from 2.13 to 3.17 for (i), 2.94 to 3.33 (Pr1) and 2.20 to2.55 (Pr2) for (iv), 3.12 to 3.45 (Pr1) and 2.24 to 2.65 (Pr2) for(v). The BVS values for Na in these models show a widerspread than those for Li in the Li1/8Pr5/8MoO4, with ranges0.95–1.29 for model (i); 1.11–1.21 for (iv), and 1.12–1.24for (v).The t-plots of BVS variations for K and Pr cations inK1/8Pr5/8&1/4MoO4 structure, calculated using the harmonicmodel, are incorrect. In contrast, the t-plots for K and Prcations are similar for the crenel-2H (iv) and crenel-2L (v)models. The harmonic model for M1/8Pr5/8&1/4MoO4 (M = Li,Na) structures fits the experimental data significantly betterthan all crenel models. Therefore, model (i) with the harmo-research papers124 V. A. Morozov et al. � Incommensurately modulated structures of Li, Na, K scheelites Acta Cryst. (2026). B82, 120–130Figure 3Experimental, calculated and difference SXPD profiles after Rietveldrefinement of M1/8Pr5/8MoO4 [M = Li (a), Na (b), K (c)]. Insets show alow-angle part of the profile. Black and green bars mark the positions ofthe main and satellite reflections, respectively.Figure 4Illustration of the different model refinements performed from SXPDdata for K0.145Pr0.618&0.237MoO4: (a) wave occupation function approx-imation (harmonic model); (b) crenel-1H approximation; (c) crenel-2Happroximation. The lower-angle parts of the experimental, calculated anddifference SXPD profiles with the indexation of reflections are shown.Black and green bars mark the positions of the main and satellitereflections, respectively.electronic reprintnically occupational modulation of the A site was adopted forM1/8Pr5/8MoO4 (M = Li, Na) structure refinements.The harmonic model assumes that the occupancy factor ofthe A site is harmonically modulated and there is no orderingbetween different A cations and vacancies. Three species withdifferent scattering factors (Pr, M and vacancy) occupy thesame crystallographic position. Therefore, their influence onthe overall modulation of the scattering density at this positioncannot be determined unambiguously. Only an antiphaseoccupancy modulation of the M and Pr atoms agrees with theexperimentally determined modulation of the scatteringdensity at the A position.No satellites with an order higher than one were observedon the SXPD pattern of M1/8Pr5/8&1/4MoO4 (M = Li, Na);therefore, the displacive modulations were fitted with a first-order harmonic. Parameters of the modulation functions thatdid not exceed their standard deviations were fixed to 0 in therefinement. Identical atomic coordinates and ADPs wererefined for the M and Pr atoms. O atoms were refined withcommon ADPs. Table 1 lists the crystallographic information;Tables S4 and S5 list the atomic parameters and the coeffi-cients of the modulation functions; Tables S6 and S7 give themain interatomic distances. Fig. 3 shows the experimental,calculated, and difference SXPD profiles of theM1/8Pr5/8&1/4MoO4 [M = Li (a), Na (b)] phases after theRietveld refinement.3.3.2. K1/8Pr5/8MoO4 structureIn contrast to the PXRD patterns of M1/8Pr5/8&1/4MoO4(M = Ag, Li, Na, K) (Fig. 1) and the SXPD patterns ofM1/8Pr5/8&1/4MoO4 [M = Li, Na (Fig. 3(a), 3(b)], second-ordersatellites appear on the SXPD patterns of K1/8Pr5/8&1/4MoO4[Figs. 3(c) and 4]. The refinement of the K1/8Pr5/8&1/4MoO4structure in harmonic model (i) showed poor agreementbetween the calculated and experimental profiles (Table S3).Therefore, we discarded model (i).Comparing the refinement results of all models shows thatmodel (iv) has better reliability factors for all reflections withone [crenel-1H (ii), crenel-1L (iii), crenel-3H (vi), crenel-3L(vii), crenel-4H (viii), and crenel-4L (ix)]. Three creneldomains of Pr [crenel-2H (iv) and crenel-2L (v)] have betterreliability factors for all reflections (Table S3). The model (iv)demonstrates the best agreement with the experimental data(Fig. 4) and best reliability factors for main reflections[RF(main reflections) = 2.33% and for the first- and second-order satellites RF(first-order satellites) = 6.73% and RF(se-cond-order satellites) = 8.81%] (Table S3). The bond valencesums variations in K1/8Pr5/8&1/4MoO4 structure for refinementmodels with the best reliability factors for all reflections areshown in Fig. S6. The model with the displacive modulationfitted with harmonic functions with the second-order Fourieramplitudes for atom positions for K1/8Pr5/8&1/4MoO4 structure(Appendix 1 of the supporting information) fits the experi-mental data significantly better than the harmonic model andother crenel models.Therefore, model (iv) with the step-like occupationalmodulation and three crenel domains of Pr was adopted. Thedisplacive modulations were fitted with a second-orderharmonic.Refining the coordinates and lengths of three atomicdomains results in the composition K0.145Pr0.618&0.237MoO4.The reliability factors show good agreement between thecalculated and experimental profiles (Table 1). Fig. 3(c) showsthe experimental, calculated, and difference SXPD profiles ofthe K0.145Pr0.618&0.237MoO4 phases after the Rietveld refine-ment. Table S8 lists the atomic parameters and the coefficientsof the modulation functions; Table S9 gives the main intera-tomic distances.4. DiscussionOverview images of the Li1/8Pr5/8&1/4MoO4 structure with theoccupation of the A-sites (A = Pr1–xLix) > 5% and > 50% areshown in Fig. 5. The scheelite-type ABO4 (CaWO4) structureconsists of a 3D framework. Along the c axis, it containscolumns of vertex sharingAO8 polyhedra and BO4 tetrahedra.The scheelite-type structure can accommodate a high numberof vacancies at the A-cation position, leading to a cation ratiowhere (A0+A00) < (B0+B00), i.e. n < m. The A-site cationordering can occur in scheelites with or without A-site cationvacancies (Morozov et al., 2006a; Morozov et al., 2015). The Acations and vacancies can order, frequently forming incom-mensurately modulated structures (Abakumov et al., 2014;research papersActa Cryst. (2026). B82, 120–130 V. A. Morozov et al. � Incommensurately modulated structures of Li, Na, K scheelites 125Figure 5Overview of the Li1/8Pr5/8&1/4MoO4 O (7a� 6b� 1c supercell) with theoccupation of the A sites (A = Pr1–xLix) (a) > 5% and (b) > 50%. Pr/LiO8polyhedra are shown as green colour. The Mo and O atoms are shown asyellow and red spheres, respectively.electronic reprintArakcheeva et al., 2012; Batuk et al., 2017; Morozov et al.,2013; Morozov et al., 2017; Morozov et al., 2018; Morozov etal., 2023). It was suspected that the main driving force for thecompositional modulation was a tendency to optimize thebond valence balance for the A cations and oxygen atoms(Abakumov et al., 2014).Earlier, we studied the structure and the luminescenceproperties of Eu-based cation deficient scheelite-relatedphases (Arakcheeva et al., 2012; Batuk et al., 2017; Morozov etal., 2013; Morozov et al., 2017; Morozov et al., 2023). In theseEu-based scheelite-related incommensurately modulatedstructures, Eu-aggregates exist in two forms: [Eu2O14] dimersand infinite chains of EuO8 polyhedra parallel to the c axis(Arakcheeva et al., 2012; Morozov et al., 2017). A correlationwas proposed between the relative amount of [Eu3+2O14]dimers and the characteristic parameters of Eu3+-centeredluminescence of the NaxEu3+(2–x)/3&(1–x)/3MoO4 (Arakcheevaet al., 2012) and AgxEu(2–x)/3&(1–2x)/3WO4 (Morozov et al.,2017) phases. Luminescence parameters increase with thegrowing relative amount of Eu3+ dimers.Figs. 6 and 7 show the main structural parameters of theM1/8Pr5/8&1/4MoO4 (M = Li, Na) as t-plots, providing acomprehensive overview of the parameters variation indifferent unit cells of the basic structure. The occupancymodulation waves for the Li/Na and Pr atoms demonstrate anantiphase relation, peaking at t = 1/2 and t = 0, respectively.However, the maximum Pr content for the A site is observedat t = 0 in Li1/8Pr5/8&1/4MoO4 [A = 0.94Pr3+ + 0.06Li+, Fig. 6(a)]and at t = 1/2 in Na1/8Pr5/8&1/4MoO4 structure [A = 0.94Pr3+ +0.06Na+, Fig. 7(a)]. According to the SXPD data, there is noordering between the Li+/Na+ and Pr3+ cations. This agreeswith the published criterion (Abakumov et al., 2014) thatcation ordering requires an ionic size difference of�0.35 Å [inthe case of M1/8Pr5/8&1/4MoO4: �r � 0.206 Å (Li) and �r �0.054 Å (Na); r(Pr) = 1.126 Å, r(Na) = 1.18 Å, r(Li) = 0.92 Å,CN = 8 (Shannon, 1976)].The distortion of the PrO8 andMO8 polyhedra is practicallyindependent of the M radius. The Pr/M–O interatomicdistances in the AO8 polyhedra vary between 2.372–2.619 Åand 2.405–2.635 Å [Figs. 6(b), 7(b) and Tables S6–S7] for Liand Na, respectively. Notably, all eight Pr/M–O distanceschange almost synchronously, indicating ‘breathing‘ of thePrO8 andMO8 polyhedra, and not the displacement of the Pr/M cation from the center.Compared to M1/8Pr5/8&1/4MoO4 (M = Li, Na), the K+ andPr3+ cations are ordered in the K0.145Pr0.618&0.237MoO4 struc-ture. This ordering occurs because the ionic size difference of�r = 0.384 Å [r(Pr) = 1.126 Å, r(K) = 1.51 Å, CN = 8(Shannon, 1976)] satisfies the cation ordering criterion(Abakumov et al., 2014). In the K0.145Pr0.618&0.237MoO4scheelite-type structure, columns of [ . . . –(PrO8/KO8)–MoO4– . . . ] and [ . . . –&–MoO4– . . . ] run along the c axis[Fig. 8(a)]. Basically, the framework can be considered asconsisting of groups of AO8 polyhedra alternating with cationvacancies. An A-cationic subset of the K0.145Pr0.618&0.237MoO4aperiodic structure is shown in Fig. 8(b). In contrast to theNaxEu3+(2–x)/3&(1–2x/3MoO4 (Arakcheeva et al., 2012; Batuk etal., 2017; Morozov et al., 2017; Morozov et al., 2018) andAgxR(2–x)/3&(1–2x)/3WO4 (R = Eu, Sm), only extended [PrO8]nresearch papers126 V. A. Morozov et al. � Incommensurately modulated structures of Li, Na, K scheelites Acta Cryst. (2026). B82, 120–130Figure 7t-plots in the Na1/8Pr5/8&1/4MoO4 structure: variation of the Na/Pr occu-pancies (a); A—O bond length (b); Mo—O bond length (c); O—Mo—Obond angles (d); and tetrahedral distortion parameters d� (e) and�� (f).Figure 6t-plots in the Li1/8Pr5/8&1/4MoO4 structure: variation of the Li/Pr occu-pancies (a), A—O bond length (b), Mo—O bond length (c), O—Mo—Obond angles (d) and tetrahedral distortion parameters �d (e) and�� (f).electronic reprintchains exist in the K0.145Pr0.618&0.237MoO4 structure. Two typesof K fragments can be distinguished in the structure: singleKO8 polyhedra and [K2O14] dimers with K–K distances of3.94 Å within the dimers and 5.18 Å between them [Fig. 9(a)].Neighboring K fragments are separated by vacancies[Fig. 8(b)].Fig. 9(b) shows one layer of the incommensurately modu-lated K0.145Pr0.618&0.237MoO4 structure and of the subset of theA-cations in the [001] projection. The cation order is definedby the coefficients � and � of the modulation vector q = �a* +�b* (Table 1) and the parameters of three atomic domains forthe same cation position (Pr1, Pr2 and K) (Table S8). As canbe deduced from Fig. 9(b), the ordering of Pr/K cations andvacancies in the A position is periodic along the c axis butaperiodic (modulated) in the ab plane.Fig. 10 shows a variation of the K/Pr occupancies, Pr/K—Obond lengths, and Pr–Pr distances for theK0.145Pr0.618&0.237MoO4 structure. The PrO8 and KO8 poly-hedra are significantly distorted. The Pr—O bond distancesare higher for Pr2 (2.441–2.800 Å) than for Pr1 (2.310–2.600 Å), while the K–O distances vary between 2.367 and2.685 Å [Fig. 10(b)]. The Pr–Pr distances within the [PrO8]nchains range from 3.759 to 4.147 Å [Fig. 9(c), Table S9].The Mo—O bond lengths and O—Mo—O bond angles int-plots for the K0.145Pr0.618&0.237MoO4 structure are shown inFigs. 11(a) and 11(b), respectively. The MoO4 tetrahedraexhibit minimal distortion, with Mo–O distances remainingnearly independent of the M radius. These distances rangefrom 1.739 to 1.832 Å (Table S9) for K0.145Pr0.618&0.237MoO4[r(K) = 1.51 Å, CN = 8], 1.762–1.836 Å (Table S7) forNa1/8Pr5/8&1/4MoO4 [r(Na) = 1.18 Å, CN = 8] and 1.744–1.814 Å (Table S6) for Li1/8Pr5/8&1/4MoO4 [r(Li) = 0.92 Å,CN = 8] (Shannon, 1976).The O—Mo—O bond angles inM1/8Pr5/8&1/4MoO4 (M = Li,Na, K) (Figs. 6–7 and Fig. 11) deviate significantly from theideal tetrahedral angle of 109.5o varying in the range from100.3 (8) to 124.5 (8)� for K0.145Pr0.618&0.237MoO4 [r(K) =1.51 Å, CN = 8], 104.0 (7)–119.5 (8)� for Na1/8Pr5/8&1/4MoO4[r(Na) = 1.18 Å, CN = 8], and 104.7 (4)–118.7 (5)� forLi1/8Pr5/8&1/4MoO4 [r(Li) = 0.92 Å, CN = 8] (Shannon, 1976).Usually the structure adopts changing size and charge of thecations at the A position through the deformation of theMoO4 tetrahedra, which occurs as an antiphase alternation oftwo distortion modes: stretching the Mo—O bonds or bendingthe O—Mo—O bond angles. The tetrahedral distortionparameters �d and ��, characterizing the deviations of theMo—O interatomic distances from the average value and thedeviations of the O—Mo—O bond angles from the idealtetrahedral angle were used for analysis of the distortion ofresearch papersActa Cryst. (2026). B82, 120–130 V. A. Morozov et al. � Incommensurately modulated structures of Li, Na, K scheelites 127Figure 9(a) A single [K2O14] dimer surrounded byMoO4 tetrahedra; (b) one layer(along z, 0.40 � z � 0.85) of the 12a� 11b supercell ofK0.145Pr0.618&0.237MoO4. Blue squares indicate some of the vacancies inthe structure.Figure 8(a) Overview of the 12a� 11b� 1c supercell in the ab projection (a) andA-cation subset (b) of the K0.145Pr0.618&0.237MoO4 aperiodic structure.PrO8 polyhedra are shown in green. K, Mo and O atoms are shown asblack, yellow and red spheres, respectively. Blue squares indicate some ofthe vacancies in the structure. [K2O14] dimers are marked by red ellipses.electronic reprintthe MoO4 inM1/8Pr5/8&1/4MoO4 (M = Li, Na, K) (Figs. 6, 7 and11). The parameters are defined as�d ¼ 1=4Xn¼1-4dn � dð Þ=d� �2;�� ¼ 1=6Xn¼1-6�n � �ð Þ=�� 2;where dn is the individual Mo—O bond length, d is theaverage Mo—O bond length, �n is the individual O—Mo—Obond angle and � is the ideal tetrahedral angle.The occupancy coefficients of the Li/Na and Pr positions aremodulated in antiphase [Figs. 6(a) and 7(a)]. The A–Odistances vary significantly, being smallest and largest at theunit cells with the highest and lowest occupancy of the Prpositions, respectively [Figs. 6(b) and 7(b)]. At the same time,the occupancy of the A site with the Pr3+ cations inK0.145Pr0.618&0.237MoO4 causes significant contraction of theA–O distance [Fig. 10(b)]. This tendency agrees with thesmaller ionic radius of the Pr3+ cations (r = 1.126 Å, CN = 8)compared to that of K (r = 1.51 Å, CN = 8) (Shannon, 1976).The stretching distortion of the Mo—O bonds (�d) anddistortion of the O—Mo—O angles (��) depend on thedifference between the ionic radii of the M and Pr cations(�r = |r(M)–r(A00)|). The �d and �� parameters rise withincreasing of �r: �d = 1.8–2.6� 10�4 and �� = 1.2–2.0�10�3 for Na1/8Pr5/8&1/4MoO4 (�r = 0.054 Å); �d = 0.2–2.5�10�4 and �� = 1.0–2.4� 10�3 for Li1/8Pr5/8&1/4MoO4 (�r =0.206 Å), and �d = 0–4.8� 10�4 and �� = 1.5–4.0� 10�3 forK0.145Pr0.618&0.237MoO4 (�r = 0.384 Å) (Shannon, 1976). Thesmallest stretching distortion of the Mo—O bonds (minimalvalues of �d) occurs in the unit cells with the highest Prcontent [Figs. 6(e) and 7(e)] in M1/8Pr5/8MoO4 (M = Li, Na)while the largest bending distortion of the O—Mo—O angles(largest��) occurs in the unit cells with the highest Pr contentforM = Li and the lowest Pr content forM = Na [Figs. 6(f) and7(f)]. The minimal values of �d and the largest �� areobserved for t = 1/2 (center of Pr1 crenel domain) while themaximal �d and smallest �� are found for t = �0.323 and t =�0.677 (close to boundaries of Pr1 crenel domain) [Figs. 11(c)and 11(d)].5. ConclusionWe investigated the effect of cationic composition on theordering of the M+/Pr3+ cations and vacancies for (3+1)D-modulated monoclinic M1/8Pr5/8&1/4MoO4 (M = Li, Na, K)structures. The M1/8Pr5/8&1/4MoO4 (M = Li, Na, K) structureswere refined using synchrotron powder X-ray diffraction.M1/8Pr5/8&1/4MoO4 (M = Li, Na, K) were synthesized usingsolid state reactions. Element contents were determined bydifferent methods. Different models during the Rietveldrefinement were tested based on earlier refined scheelite–related structures of other compounds. The cation ordering isnot complete for M1/8Pr5/8&1/4MoO4 (M = Li, Na, K) struc-tures and better described with harmonic rather than withstep–like occupational modulation functions. The occupancymodulation waves for the Li/Na and Pr atoms demonstrate anantiphase relation. In comparison withM = Li, Na the order ofthe K and Pr cations in the A position in K1/8Pr5/8&1/4MoO4structure is best approximated by crenel functions. Refiningthe coordinates and lengths of three atomic domains results inthe composition K0.145Pr0.618&0.237MoO4. In all cases themodulation arises from ordering of the M/Pr cations and thecation vacancies at the A-sublattice of the parent scheeliteresearch papers128 V. A. Morozov et al. � Incommensurately modulated structures of Li, Na, K scheelites Acta Cryst. (2026). B82, 120–130Figure 11t-plots for the K0.145Pr0.618&0.237MoO4 structure: Mo—O bond length (a);O—Mo—O bond angles (b), tetrahedral distortion parameters �d (c)and �� (d).Figure 10t-plots for the K0.145Pr0.618&0.237MoO4 structure: (a) variation of the K/Proccupancies, (b) A—O bond lengths and (c) Pr–Pr distances. Black linesshow Pr1–Pr2 distances. Green lines show Pr1–Pr1 and Pr2–Pr2 distances.electronic reprintABO4 structure. The distortion of the PrO8 and MO8 poly-hedra is practically independent of the M radius. The refine-ments of the M1/8Pr5/8&1/4MoO4 (M = Li, Na, K) structuresreveal that the MoO42� tetrahedra in scheelite-typecompounds demonstrate a flexible geometry. Both Mo—Odistances and O—Mo—O bond angles vary significantly withchanging the population of the A site by cations with differentcation size.6. Supporting informationDTXRF and EDX spectra of K1/8Pr5/8MoO4 (Fig. S1). Modelstested during the Rietveld Li1/8Pr5/8&1/4MoO4 refinementusing the structure SXPD (Appendix 1). Characteristics of thedifferent of the different model refinements model refine-ments performed of from PXPD data for Li1/8Pr5/8&1/4MoO4(Fig. S2). Characteristics of the different model refinements ofNa1/8Pr5/8&1/4MoO4 structure (Table S2). Illustration of thedifferent model refinements performed from PXPD data forNa1/8Pr5/8&1/4MoO4 (Fig. S3). t-plots of bond valence sumsvariations in Li1/8Pr5/8&1/4MoO4 structure for refinementmodels with the best Na1/8Pr5/8&1/4MoO4 reliability factors forall reflections (Fig. S4). t-plots of bond valence sums variationsin Na1/8Pr5/8&1/4MoO4 structure for refinement models withthe best reliability factors for all reflections (Fig. S5). Char-acteristics of the different model refinements ofK1/8Pr5/8&1/4MoO4 structure (Table S3). t-plots of bondvalence sums variations in K1/8Pr5/8&1/4MoO4 structure forrefinement models with the best reliability factors for allreflections (Fig. S6). Atomic coordinates, amplitudes ofFourier components for the occupational and displacivemodulation functions and isotropic atomic displacementparameters for Li1/8Pr5/8&1/4MoO4 structure (Table S4).Atomic coordinates, amplitudes of Fourier components for theoccupational and displacive modulation functions andisotropic atomic displacement parameters forNa1/8Pr5/8&1/4MoO4 structure (Table S5). Main interatomicdistances for Li1/8Pr5/8&1/4MoO4 (Table S6). Main interatomicdistances for Na1/8Pr5/8&1/4MoO4 (Table S7). Atomic coordi-nates, amplitudes of Fourier components for the occupationaland displacive modulation functions and isotropic atomicdisplacement parameters for in K0.145Pr0.618&0.237MoO4structure (Table S8). Selected interatomic distances forK0.145Pr0.618&0.237MoO4. (Table S9).AcknowledgementsThe synthesis of samples was carried out within the frameworkof the state assignment of the BINM SB RAS. The synchro-tron radiation experiments were conducted at the formerNIMS beamline (BL15XU) of SPring-8 with the approval ofthe former NIMS Synchrotron X-ray Station. We thank Dr Y.Katsuya and Dr M. Tanaka for their help at SPring-8. Theauthors are grateful to A. M. Alekseeva (Chemistry Depart-ment, Moscow State University) for the SEM-EDX study.Funding informationThe following funding is acknowledged: State assignment"Substances and materials for ensuring safety, reliability andenergy efficiency" (grant No. AAAA-A21-121011590086-0);state assignment of the BINM SB RAS; former NIMSSynchrotron X-ray Station (proposal No. 2019A4501).ReferencesAbakumov, A. M., Morozov, V. A., Tsirlin, A. A., Verbeeck, J. &Hadermann, J. (2014). Inorg. Chem. 53, 9407–9415.Arakcheeva, A., Logvinovich, D., Chapuis, G., Morozov, V., Eliseeva,S. V., Bünzli, J. G. & Pattison, P. (2012). Chem. Sci. 3, 384–390.Batuk, D., Batuk, M., Morozov, V. A., Meert, K. W., Smet, P. F.,Poelman, D., Abakumov, A. 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