From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of From Loal to Global Order in Crystals: Rigorous Results Yuri Geometry Days in Novosibirsk Dediated to 85th anniversary of Grig- Yuri Grigorievih Reshetnyak orievih Reshetnyak Nikolay Dolbilin Steklov Mathematis Institute September 26, 2014 Ideal Crystal; Quartz: Exterior Shape From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Quartz (a speies of zeolites) Ideal Crystal; Quartz: Internal Struture From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Zeolites have miroporous interior struture Quartz is a speies of Zeolites Denition of a Crystal; Fedorov, 1885 From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Denition (of rystal) Let Xd be spae with onstant urvature, G a rystallographi group operating in Xd , X0 ⊂ Xd a nite point subset: X0 = {x1 , . . . , xm }, G -orbit of the set X0 G · X0 = ∪m i G · xi is alled a rystal Denition (of regular systems) If X0 := {x }, an orbit G · x of a single point is alled a regular system. A regular system is a partiular ase of a rystal. Crystallographi groups From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak A subgroup G ⊂ Iso (d ) of isometries is a rystallographi group if G is a disrete subgroup of Iso (d ) the spae of orbits Xd \G is ompat. Theorem (Shoenis: d = 3; Bieberbah: ∀d > 3; Hilbert XVIII Problem) Let Xd = Rd , then a rystallographi group G ontains a subgroup T of translations of spae with a nite index h: G = T ∪ Tg2 ∪ . . . ∪ Tgh . Due to the Theorem a rystal G · X0 is the union of nite number ongruent and parallel latties G · X0 = ∪m i (T · xi ∪ T · g2 (xi ) ∪ . . . ∪ T · gh (xi )). Therefore, a rystal is periodi ("in all d dimensions"). Regular Systems: Example. From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Regular system: Lattie Regular Point Sets. Examples. From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Regular system: an orbit with 4 latties Regular Point Sets. Examples. From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Regular system: generi orbit with 4 latties Crystal: Example. From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Crystal: of 3 regular systems= of 9 latties From Disorder to Global Order From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak The inner well-ordered struture in rystal appears from amorphous solution under rystallization Under rystallization atoms try to bind to the loal arrangements with the minimally possible binding energy For any two idential atoms minimal energy of loal ongurations is attained on idential ongurations. Therefore atoms of the same speies try to bind to the pairwise idential loal patterns minimizing the energy Physists (Pauling, Feynmann) found the following postulate obvious (see Feynmann Letures on Physis, v. VII): The reurrene in rystal of loal idential arrangements implies global periodiity of a rystalline struture. However, in quasirystals (Shehtman, 1982, Nobel Prize, 2011) there is reurrene of loal arrangements but NO global order/periodiity at all. Main Goal of the Loal Approah From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Loal theory of rystalline struture was (and is) designed: orretly formulate appropriate loal onditions and from them rigorously derive "the global order". to distint whih loal arrangments do admit globally ordered extensions and whih do not. The initiator of the rst line was Boris Delone (1890-1980) Delone sets From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Denition A point set X ⊂ Rd is a Delone set the following onditions hold: (r) ∃r > 0 suh that balls Bx (r ), (entered at x ∈ X with radius r ) form a paking of spae (the balls do not overlap) (disreteness); (R) ∃R > 0 suh that balls Bx (R ), (entered at x ∈ X with radius R) do over all spae (∪Bx (R ) = Rd ) (no empty holes in X) Delone Set: From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak r is a Paking Radius Balls entered at pts of X with radius r form paking ( do not overlap). Delone Set: From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak R is Covering Radius R-balls entered at points of X over spae d R ρ-luster in a Delone Set From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Given Delone set X , x ∈ X , and a positive ρ, a ρ-luster at point x is Cx (ρ) =: {x′ ∈ X : |xx'| ≤ ρ}. Enumerative Funtion From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Given a Delone set X and number ρ > 0, set of rho-luster splits into lasses of ongruent lusters the number N (ρ) of lasses of ρ-lusters in X is a funtion of ρ and alled enumerative funtion Yuri Grigorievih Reshetnyak Assume that X is a Delone set of nite type, i.e. N (ρ) is determined and nite for any ρ > 0 Enumerative funtion and Crystals From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak N (ρ) is monotonially non-dereasing, integer-valued funtion; X is regular system ⇔ N (ρ) ≡ 1 X is rystal with m regular sets ⇔ maxρ N (ρ) = m, Symmetries of a luster From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Denote by Sx (ρ) : Sym(Cx (ρ), x) the symmetry group of a ρ-luster. Note that Sx (ρ) ⊇ Sx (ρ′ ) if ρ < ρ′ Let Mx (ρ) := |Sx (ρ)| be the order of the group of a luster Cx . The funtion Mx (ρ) is pieewise-onstant, non-inreasing, integer-valued funtion. Mx (ρ) ≥ Mx (ρ′ ) if ρ < ρ′ . Assume for two points x and x' its lusters Cx (ρ) and Cx' (ρ) are equivalent. Then the groups of the lusters are onjugate in Iso (d ) and Mx (ρ) = Mx' (ρ) Loal Criterion for rystals From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Theorem (Loal Criterion, Dolbilin, Stogrin) Delone set X is a rystal of m regular systems if (and only if) for some ρ0 > 0 two onditions hold: (1) N (ρ0 ) = N (ρ0 + 2R ) = m (2) for every i-th lass (i = 1, . . . , m) of ρ-lusters we have: Mi (ρ0 ) = Mi (ρ0 + 2R ), where Mi (ρ0 ) denotes the order of groups of lusters from i-th lass. Some Corollaries From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak From Loal Criterion an be derived that the following theorem Theorem There is suh a ρ0 that a Delone set X with parameters r , R is a rystal of m orbits if and only if N (ρ0 ) = m, where ρ0 = ρ0 (r , R , m, d ). Some Corollaries From Loal to Global Order in Crystals: Rigorous Results Close reformulation of the last result gives a representation on the behavior of the enumerative funtion. Geometry Days in Novosibirsk Dediated to 85th anniversary of Let X ⊂ R d be a Delone set with onstants (r , R ). If for some radius ρ0 the number m = N (ρ0 ) of lasses of its ρ0 -lusters satises ρ N (ρ0 ) < 0 , CR where C = C (R /r , d ). Then X is a rystal with exatly m orbits. Yuri Grigorievih Reshetnyak Theorem (Suient Conditions, Lagarias, Senehal and N.D.) Thus, if the enumerative funtion N (ρ) at the beginning grows rather slowly then it is bounded for all ρ > 0 Note that for the Penrose quasi-periodi 2D-patterns N (ρ) ∼ ρ2 Loal riterion for regular systems From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak The following riterion for regular systems follows from the loal riterion for rystals Theorem (Loal theorem, Delone, Dolbilin, Stogrin) A Delone set is a regular set if and only if for some radius ρ0 two onditions hold: N (ρ0 + 2R ) = 1 and M (ρ0 ) = M (ρ0 + 2R ) Corollaries from Loal Theorem From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Statement . Assume in X N (4R ) = 1 and a 2R-luster has no symmetry. Then X is a regular System 4R is preise, i.e. for any ε > 0 the ondition N (4R − ε) = 1 does not sue: In any dimension for any ε > 0 there exists a Delone set X (with parameter R) suh that N (4R − ε) = 1 but X is not a regular system. Moreover, among suh non-regular sets with N (4Rε ) = 1 there are suh X that funtion N (ρ) ∼ ρ2 → ∞ as ρ → ∞. Results for From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak d = 2 and d =3 Theorem (Stogrin, Dolbilin) d = 2. N (4R ) = 1 ⇒ N (ρ) ≡ 1, i.e. X is a regular system Reall that for any ε > 0 the ondition N (4R − ε) = 1 does not sue: In any dimension for any ε > 0 there exists a Delone set X (with parameter R) suh that N (4R − ε) = 1 but X is not a regular system. Theorem (Stogrin, N.D) d = 3: N (10R ) = 1 ⇒ X is a regular system Central Symmetry: Loal-Global From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Many natural rystals (e.g., sault NaCl) have loally antipodal struture. Theorem (N.D.) Let X ⊂ Rd be suh that All 2R-lusters Cx (2R ) are entrally symmetrial. Then the whole X is entrally symmetrial about any x ∈ X Central Symmetry: Loal-Global From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Theorem (N.D) Let X ⊂ Rd be suh that All 2R-lusters Xx (2R ) are entrally symmetrial. Then the whole X is entrally symmetrial about any x ∈ X Regular Systems From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Theorem Let X ⊂ Rd be suh that (1) N (2R ) = 1, (2) 2R-luster Cx (2R ) is entrally symmetrial. Then X is regular system !! Compare with N (4R − ε) = 1 is not enough !! Symmetry of 2R -lusters ⇒ Uniqueness From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Theorem (Uniqueness theorem, N.D., A.Magazinov ) Let X and Y be Delone (r , R )-sets and suh that 1) all 2R-lusters are entrally symmetrial; 2) for some x ∈ X , y ∈ Y x = y , Xx (2R ) = Yy (2R ). Then X = Y . Symmetry of 2R -lusters ⇒ Crystal From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak The uniqueness theorem easily implies the following theorems (just mentioned theorem) Theorem Let X ⊂ Rd be suh that (1) N (2R ) = 1, (2) 2R-luster Cx (2R ) is entrally symmetrial. Then X is a regular system It is important that if even N (2R ) = 1 is not required the loal symmetry implies rystalline struture: Theorem (N.D., A.Magazinov) Let X ⊂ Rd be suh that 2R-luster Cx (2R ) for ∀x ∈ X is entrally symmetrial. Then X is a rystal Some of Open Problems From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Yuri Grigorievih Reshetnyak Prove (or disprove) : d = 3, N (4R ) = 1 ⇒ N (ρ) = 1 for any ρ > 0, i.e. X is Regular system. d ≥ 4, prove: for any d ≥ 4 ∃ k (d ) > 0 suh that does not depend on r and R and N (kR ) = 1 ⇒ N (ρ) ≡ 1 for any ρ. i.e. X is a regular system The most hallenging problems: d ≥ 2, nd loal onditions of X to be a quasirystal (not a rystal). to study onditions for nuleous of rystalline and quasirystalline strutures with this or that kind of symmetry Conjeture From Loal to Global Order in Crystals: Rigorous Results Geometry Days in Novosibirsk Dediated to 85th anniversary of Äîðîãîé Þðèé ðèãîðüåâè÷ ! Yuri Grigorievih Reshetnyak C äíåì ðîæäåíèÿ !

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