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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
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Geometry
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Quartz (a speies of zeolites)
Ideal Crystal; Quartz: Internal Struture
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Zeolites have miroporous interior struture
Quartz is a speies of Zeolites
Denition of a Crystal; Fedorov, 1885
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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
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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.
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Regular system: Lattie
Regular Point Sets. Examples.
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Regular system: an orbit with 4 latties
Regular Point Sets. Examples.
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Geometry
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Regular system: generi orbit with 4 latties
Crystal: Example.
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Crystal: of 3 regular systems= of 9 latties
From Disorder to Global Order
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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
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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
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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:
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r
is a Paking Radius
Balls entered at pts of X with radius r form paking ( do
not overlap).
Delone Set:
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R
is Covering Radius
R-balls entered at points of X over spae
d
R
ρ-luster in a Delone Set
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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
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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
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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
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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
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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
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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
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Close reformulation of the last result gives a representation on
the behavior of the enumerative funtion.
Geometry
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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
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Geometry
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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
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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
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Geometry
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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
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Geometry
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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
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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
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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
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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
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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
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Geometry
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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
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Order in
Crystals:
Rigorous
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Geometry
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Äîðîãîé Þðèé ðèãîðüåâè÷ !
Yuri
Grigorievih
Reshetnyak
C äíåì ðîæäåíèÿ !