Framework structure crystalline materials and Rigid Unit Modes (RUMs)

Introducing the new concept of MLRUMs and skeletions


  • Pierre Saint Gregoire Collaborating Academics
  • Mikhail B. Smirnov Physical Department, Saint-Petersburg State University




Book chapter (
This article reviews Framework Structures (FWSs), defined as crystalline
materials built of rigid AXn polyhedra sharing vertices (like perovskites, tungsten bronzes,
Dion-Jacobson, Ruddlesden-Popper, and Aurivillius phases, quartz, silicates, and others),
and their pecularities resulting from this linkage. The situation of rigid units linked by
common vertices may allow the units to accomplish concordant rotations without
deformation, which gives rise to soft phonon modes called “Rigid Unit Modes” (RUMs). The
condensation of a RUM can trigger structural phase transitions to a structure of lower
symmetry, with tilted polyhedra, at the origin of spontaneous ferroic or multiferroic
properties. We overview results precedently obtained on RUMs in perovskites, tetragonal
tungsten bronzes, and quartz, and detail new results on “maximally localized RUMs”
(MLRUMs), a fundamental new concept in the physics of RUMs. We introduce also the
related new concept of “skeletions” that allows to generate all ferroelastic phases found in
these systems, and generalizes the Glazer's tilt-system approach.


Ross N.L. , “Framework structures”, in High Temperature and High Pressure Crystal Chemistry (ISBN 0-939950-53-7), Editors R.M. Hazen and R.T. Downs, Reviews in Mineralogy and Geochemistry, Vol. 41 , p 257 (2000)

Bhalla A.S., R. Guo and · R. Roy, Mat Res Innovat 4, p 3 (2000)

Yoshiasa A., T. Nakatani, A. Nakatsuka, M. Okube, K. Sugiyama and T. Mashimo,

Acta Crystallographica Section B 72 (3), p381 (2016) – DOI 10.1107/S2052520616005114

Krivovichev S., Zeitschrift für Kristallographie, 223 (01–02), p 109 (2008); doi:10.1524/zkri.2008.0008

Woodward P.M., Acta Cryst. B 53, p 44 (1997)

Kihlborg L., Klug A., Chemica Scripta 3, 207-211 (1973)

Heger G., R. Geller, D. Babel - Solid State Communications 9 (5), p335 (1971)

DOI: 10.1016/0038-1098(71)90006-8

Posse J.M., Grzechnik A., Friese K., Acta Crystallogr. B 65 (5) p 576 (2009)

doi: 10.1107/S0108768109027177

Lapasset J., H.N. Bordallo, R. Almairac, J. Nouet, Zeitschrift fur Kristallographie 211 (12), p 934 (1996) and references therein – DOI:

Tsujimoto Y., K. Yamaura, E. Takayama-Muromachi, Appl. Sci. 2(1), 206 (2012)

Moure A., Appl. Sci. 8, 62 (2018) and references therein – doi:10.3390/app8010062

Courbion G., C. Jacoboni and R. de Pape, Acta Cryst. B 32, 3190 (1976)

Le Bail A., Gao Y., Jacoboni C., European Journal of Solid State and Inorganic Chemistry Vol 26 (3), p 281 (1989)

Mueller-Buschbaum Hk., Wilkens J., Zeitschrift für Anorganische und Allgemeine Chemie, 591, p 161 (1990) – DOI: 10.1002/zaac.19905910118

Wolfe R.W., R.E.Newnahm, M.I. Kay, Solid State Communications 7 (24), 1797 (1969)

Shimakawa Y. and Y. Kubo, Appl. Phys. Lett. 74, 1904 (1999)

Haluska M.S., Misture S.T., Journal of Solid State Chemistry 177, 1965 (2004)

Geim, A., Science. 324 (5934), 1530 (2009) – arXiv:0906.3799 doi:10.1126/science.1158877. PMID 19541989

Fourquet J.L., F. Plet, R. de Pape, Revue de Chimie Minerale 18, 19 (1981)

Persson K., Materials Data on CsAlF4 (SG:189) by Materials Project, p. 2014. doi:10.17188/1268243

Herdtweck E. and Babel D., Zeitschrift für Kristallographie 153, 189 (1980)

Wyckoff, Ralph W. G., Zeitschrift für Kristallographie, Kristallgeometrie, Kristallphysik,

Kristallchemie 63, 507 (1926) – DOI: 10.1524/zkri.1926.63.1.507

Donald R. Peacor, Zeitschrift für Kristallographie 138, 274 (1973)

Valaskova M., Ceramics – Silikáty 59 (4) 331 (2015) and references therein –

Batchelder D.N., Simmons R.O., Journal of Chemical Physics, 41, 2324 (1964)

Kahlenberg V., P. Aichholzer, Acta Crystallographica Section E. 70(7) 34 (2014)

DOI 10.1107/S1600536814013142

Guggenheim S., Eggleton R.A., American Mineralogist, 72, 724 (1987)

Nelson S.A., (2015)

Barton C.D. and A.D. Karathanasis, in Encyclopedia of Soil Science, p187 (2002)

Mellini M., “Structure and microstructure of serpentine minerals” in European Mineralogical Union Notes in Mineralogy 14, 153 (2013) – DOI: 10.1180/EMU-notes.14.5

Ghanshyam Pilania, B. Puchala & B.P. Uberuaga, npj Computational Materials 5, 7 (2019)

Smith J., & Rinaldi F., Mineralogical Magazine and Journal of the Mineralogical Society, 33 (258), 202 (1962) – doi:10.1180/minmag.1962.033.258.03

Armbruster T., M. E. Gunter, Reviews in Mineralogy and Geochemistry 45 (1), 1 (2001)

Laligant Y., Y. Calage, G. Heger, J. Pannetier, G. Ferey, Journal of Solid State Chemistry 78 (1), 66 (1989) –

Aleksandrov K.S. and V.V. Beznosikov, Phys. Solid State 39: 695 (1997), Fiz. Tverd. Tela (St. Petersburg) 39, 785 (1997)

Le Meins J.-M., G. Courbion, European Journal of Solid State and Inorganic Chemistry, 35 (10–11), 639 (1998)

Smirnov M.B. et al, Phys.Stat.Sol. (b) 241 (5), 1017-25 (2004)

Giddy A.P. , M.T.Dove, G.S.Pawley, V. Heine Acta Cryst. A49, 697 (1993)

Glazer, A. M., Acta Cryst. B28, 3384 (1972)

Hammonds K. D., M. T. Dove, A. P. Giddy, V. Heine, B. Winkler, Am. Mineral. 81, 1057 (1996)

Pryde A.K.A., K. D. Hammonds, M. T. Dove, V. Heine, J. D. Gale, M. C. Warren, Phase Transitions 61, 141 (1997)

Toulouse G., "The frustration model". In Pekalski, Andrzej; Przystawa, Jerzy (eds.). Modern Trends in the Theory of Condensed Matter, Lecture Notes in Physics, 115 , 195 (1980) – doi:10.1007/BFb0120136

Starykh O. A., Rep. Prog. Phys. 78, 052502 (2015) -- DOI: 10.1088/0034-4885/78/5/052502

M. Smirnov, P. Saint-Grégoire, Acta Cryst. A70, 283 (2014)

Bragg, W.L. and Gibbs, R.E., Proc. Roy. Soc. Lond. A 109, 405 27 (1925)

Grimm H. and Dorner B., J. Phys. Chem. Solids 36 407 (1975)

Vallade M. , B. Berge, and G. Dolino, J. Phys. I (France) 2, 1482 (1992)

Swainson I. P. and M. T. Dove. Phys Rev Lett. 71, 193 (1993)

Dove M. T., V. Heine, K. D. Hammonds, Mineralogical Magazine 59, 629-639 (1995)

Hammonds K.D., M T. Dove, A P. Giddy, V.Heine, A. Winkler, American Mineralogist 81, 1057 (1996)

Phillips A. E., Acta Cryst A74, 406–407 (2018)

Giddy A.P., Dove M.T., Pawley G.S., and Heine V., Acta Cryst. A49, 697 (1993)

Campbell B., C. J. Howard, T. B. Averett, T. A. Whittle, S. Schmid, S. Machlus, C. Yosta and H. T. Stokes, Acta Cryst. A74, 408 (2018)

Kohn W., Phys. Rev. B 7, 2285 (1973)

Zhong W., D. Vanderbilt, K. M. Rabe, Phys Rev Lett 73, 1861 (1994)

Marzari N. , D. Vanderbilt. Phys. Rev. B 56, 12847 (1997)

Marzari N., A. A. Mostofi, J. R. Yates, I. Souza, D. Vanderbilt, Rev. Mod. Phys. 84, 1419 (2012)

Berghold G., C. J. Mundy, A. H. Romero, J. Hutter, and M. Parrinello, Phys. Rev. B 61, 10040 (2000)

Kopský V. and D. B. Litvin, International Tables for Crystallography, Volume E: Subperiodic groups (2006) ISBN: 978-1-4020-0715-6 – doi: 10.1107/97809553602060000105

Smirnov M., N. Mazhenov, N. Aliouane and P Saint-Grégoire, Journal of Physics: Condensed Matter, 22 (22), 225403 (2010) –

Berge B., J. P. Bachheimer, G. Dolino, M. Vallade & C. M. E. Zeyen, Ferroelectrics, 66:1, 73 (1986) , DOI: 10.1080/00150198608227874

Bachheimer J.P. , B. Berge, G. Dolino, P. Saint-Grégoire, C.M.E. Zeyen, Solid State Communications,

(1), 55 (1984) –

Watanabe S. and Y. Koyama, Phys. Rev. B 66, 134102 (2002)

(a) Jankowska-Sumara I. and J. Dec, Ferroelectrics, 313, 81 (2004)

(b) Irena Jankowska-Sumara; Phase Transitions (2014) , DOI: 10.1080/01411594.2014.900554

(a) Xu Z. , X. H. Dai, and D. Viehland, Phys. Rev. B, 51, 6261 (1995)

(b) Dai X. H. , Z. Xu, and D. Viehland, Appl. Phys. Lett., 65, 3287 (1994)

S Vakhrushev, D Andronikova, A Bosak, Y Bronwald,

Withers R. , Y. Zhu, L. Bourgeois, C. Dwyer, J. Etheridge, Mapping of spatially modulated octahedral tilting and coupled in-plane strain in the (3+2)-D modulated, Li1/2-3xNd1/2+xTiO3 system and its underlying crystal chemistry,

A.P. Levanyuk and S.A. Minyukov, Soviet Physics Solid State 22, 1050 (1980) – original Fizika Tverdogo Tela 22, 1808 (1980)]

10 S. Podlozhenov, H. A. Graetsch, J. Schneider, M. Ulex, M. Woehlecke, and K. Betzler, Acta Cryst. B 62, 960 (2006)

11 M. Elaatmani, A. Zegzouti, F. Capitelli, A. G. G. Moliterni, A. Migliori, and G. Calestani, Z. Kristallographie 218, 26 (2003)

12 P. B. Jamieson, S. C. Abrahams, and J. L. Bernstein, J. Chem. Phys. 48, 5048 (1968)

16 P. B. Jamieson, S. C. Abrahams, and J. L. Bernstein, J. Chem. Phys. 50, 4352 (1969)

9 G. Foulon, M. Ferriol, A. Brenier, G. Boulon, and S. Lecocq, Eur. J. Sol. State Inorg. Chem. 33, 673 (1996)

6 I. Levin, M. C. Stennett, G. C. Miles, D. I. Woodward, A. R. West, and I.M. Reaney, Appl. Phys. Lett. 89, 122908 (2006)

14 T. Woike, V. Petricek, M. Dusek, N. K. Hansen, P. Fertey, C. Lecomte, A. Arakcheeva, G. Chapuis, M. Imlau, and R. Pankrat, Acta Cryst. B 59, 28 (2003)

23 J. Schefer, D. Schaniel, V. Petricek, T. Woike, A. Cousson, and M. Woehlecke, Z. Kristallographie 223, 399 (2008)

15 Y. Gagou, D. Mezzane, N. Aliouane, J. Fabry, T. Badeche, A. Zegzouti, M. Lopez, and P. Saint-Grégoire, Ferroelectrics 251, 131 (2001)

13 T. S. Chernaya, B. A. Maksimov, I. V. Verin, L. I. Ivleva, and V.I. Simonov, Crystallography Reports 42, 375 (1997)

17 J. C. Toledano. Phys. Rev. B 12, 943 (1975)

18 L. A. Bursill and J. L. Peng, Acta Cryst. B 43, 49 (1987)

21 P. Labbe, H. Leligny, B. Raveau, J. Schneck, and J.C. Toledano, J. Phys. Cond. Matter 2, 25 (1989)

22 J. L. Peng and L. A. Bursill, Acta Cryst. B 43, 504 (1987)

19 C. J. Lu, Y. J. Qi, J. Q. Li, H. J. Zhang, and J. Y. Wang, Appl. Phys. Lett. 89, 191901 (2006)

Gagou Y., Y. Amira, N. Aliouane, A. Belboukhari, B. Allouche, A. Menny, J.L. Dellis, D. Mezzane, M. El Marssi, Ch. Masquelier, M. Elaatmani, P. Saint-Grégoire, Encyclopedia of Renewable and Sustainable Materials (2018) – doi:10.1016/B978-0-12-803581-8.10539-9

M. Pa?ciak, M. Kopecký, Phase Transitions (2018)

Van Tendeloo G. & S. Amelinckx, Phase Transitions, 67:1, 101 (1998), DOI: 10.1080/01411599808219190

K. Momma and F. Izumi, "VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data," J. Appl. Crystallogr., 44, 1272-1276 (2011)

M. B. Smirnov and V. Yu. Kazimirov LADY: Software for Lattice Dynamics Simulations, (Joint Institute for Nuclear Research, E14-2001-159, Dubna, 2001)




How to Cite

Saint Gregoire, P., & Smirnov, M. B. (2021). Framework structure crystalline materials and Rigid Unit Modes (RUMs): Introducing the new concept of MLRUMs and skeletions. OAJ Materials and Devices, 5(1). Retrieved from