Borcea-Streinu rigidity matrix of a crystal framework. Stephen Power. Lancaster
University. Rigidity and Symmetry Workshop,. Fields Institute, October, 2011 ...
Space group and point group symmetry equations for the Borcea-Streinu rigidity matrix of a crystal framework. Stephen Power Lancaster University Rigidity and Symmetry Workshop, Fields Institute, October, 2011
Outline
S.C. Power, Crystal frameworks, symmetry and affinely periodic flexes. ArXiv 2011. Bar-joint frameworks, rigidity matrices Maxwell counting formula - degrees of freedom verses constraints. Maxwell-Calladine counting formula - via ”row rank = column rank”. Symmetry - [Fowler and Guest] symmetry version, (also [Owen-Power], [Schulze]) Applications - eg [Connelly, Guest, Fowler, Schulze, Whiteley] Periodic frameworks, Infinite frameworks - [Whiteley], [Borcea-Streinu], [Malestein-Theran], [Owen-Power], [Ross], [Ross-Schulze-Whiteley]. Borcea-Streinu rigidity matrix for affinely-periodic flexes of a periodic framework.
Maxwell-Calladine counting formulae
For a 3D finite framework (G , p) ”mechanism”dimension − selfstress dimension = 3|V | − |E | − 6 [Fowler-Guest] Let σ be a spatial symmetry: mσ − sσ = 3|Vσ | − |Eσ | − fσ . [P] Crystal framework affine flex variant. Let σ be an element of the point group of C with space group representative g . Then mσ − sσ = 3|Fvσ | − |Feσ | + dim Eg − fσ , appropriately interpreted
Figure: The kagome framework.
Some 3D frameworks (implied by materials): The kagome net CKNET Sodalite CSOD
- a tetrahedral net - a tetrahedral net
Figure: The top 4-ring of the sodalite cage. Paulingite CPAU Perovskite COCT
- a tetrahedral net, with large unit cell - simplest octahedron net
Figure: Part of the hexahedron tower, a cylindrical framework. The hexahedron framework is the 3D version, CHEX . (Physical ?)
A crystal framework C is a bar-joint framework generated by a discrete translation group T and a finite ”motif”. eg. Motif (Fv , Fe ) and Unit Cell :
6
4 I 3
1 II
2
III
5
DEF: A crystal framework C is a triple (Fv , Fe , T)
Formal Definition: A crystal framework C = (Fv , Fe , T) in R3 , with full rank translation group T and finite motif (Fv , Fe ), is a countable bar-joint framework (G , p) in R3 with a periodic labelling V = {vκ,k : κ ∈ {1, . . . , t}, k ∈ Z3 },
p = (pκ,k ),
such that (i) Fv = {pκ,0 : κ ∈ {1, . . . , t}} and Fe is a finite set of framework edges, (ii) for each κ and k the point pκ,k is the translate Tk pκ,0 , (iii) the set Cv of framework vertices is the disjoint union of the sets Tk (Fv ), k ∈ Z3 , (iv) the set Ce of framework edges is the disjoint union of the sets Tk (Fe ), k ∈ Z3 .
u2
p2 p1
u1
Figure: Infinitesimal flex velocities - no first order length change. Notation for a crystal framework Joints pκ,k
Edges Velocities [pκ,k , pτ,l ] uκ,k
An infinitesimal flex of C: a vector of velocity vectors ∏ Rd u = (uκ,k ) ∈ Hv = κ,k
such that for each edge e = [pκ,k , pτ,l ] ⟨pκ,k − pτ,l , uκ,k − uτ,l ⟩ = 0. Equivalently R(C)u = 0 where R(C) is the rigidity matrix whose e th row is [. . . 0 . . . 0 ve 0 . . . 0 − ve 0 . . . 0 . . . ] where ve is edge vector pκ,k − pτ,l appearing in columns for κ, and for τ (with sign −).
Affinely periodic infinitesimal flexes I: ”periodic modulo a background affine velocity distribution:” Let Z : Rd → Rd map standard basis vectors of Rd to the period vectors a1 , . . . , ad for C. DEF The vector subspace Hvaff ⊆ Hv consists of affinely periodic velocity vectors v˜ = (˜ vκ,k ) with the form v˜κ,k = vκ − AZk,
k ∈ Zd ,
where (vκ )κ∈Fv is a finite ”motif” velocity vector in R|Fv | and A is a d × d real matrix. v˜ is encoded by (v , A). v˜ is an affinely periodic infinitesimal flex if R(C)˜ v = 0.
Affine infinitesimal flexes II: via flow flexes: A smoothly continuous ”finite” flex p(t) relative to a background affine flow t → At .... the flow-periodicity condition holds: pκ,k (t) = At Tk A−1 t pκ,0 (t),
for all
κ, k, t,
The derivative p ′ (0) is an affinely periodic infinitesimal flex relative to the derivative (matrix) A of the function t → At at t = 0.
Affine infinitesimal flexes III: direct definition of (u, A): The pair (u, A) is an affinely periodic infinitesimal flex of C if and only if for the affine flow At = I + tA and each motif edge e = [pκ,0 , pτ,δ(e) ] the bond deviation at time t |pκ,0 − pτ,δ(e) | − |(pκ,0 + tuκ ) − At Tδ(e) A−1 t (pτ,δ(e) + tuτ )| is of order t 2 as t → 0.
Borcea-Streinu rigidity matrix DEF Let C = (Fv , Fe , T). Then the affinely periodic rigidity matrix 2 R(M, Rd ) is the |Fe | × (d|Fv | + d 2 ) real matrix whose rows, labelled by the edges e = [pκ,0 , pτ,δ(e) ] of Fe , have the form [0 · · · 0 ve 0 · · · 0 − ve 0 · · · 0 δ1 ve · · · δd ve ], where ve = pκ,0 − pτ,δ(e) is the edge vector for e, distributed in the d columns for κ, where −ve appears in the columns for τ , and where δ(e) = (δ1 , . . . , δd ) is the exponent of e. If e is a reflexive edge in the sense that κ = τ then the entries in the d columns for κ are zero. 2
The notation R(M, Rd ) signals that the domain of the usual periodic rigidity matrix R(M) has been augmented by a real d 2 dimensional space. 2 Also R(M, Rd ) = [R(M) X ] with affine motion restricted variants R(M, E) with E a subspace of allowable affine velocities, eg. symmetry preserving restrictions.
THM [P, 2011] The restriction of the rigidity matrix transformation R(C) : Hv → He to the finite-dimensional space Hvaff has representing 2 matrix R(M, Rd ). Moreover, the FAE: 2
(i) (u, AZ ) ∈ ker R(M, Rd ). (matrix AZ as a long vector) (ii) u˜ ∈ ker R(C), where u˜ ∈ Hv is the vector defined by the affinely periodic extension formula u˜κ,k = uκ − AZk,
k ∈ Zd .
COR [BS] : C is affinely periodically rigid iff 2
nullity R(M, Rd ) = d(d + 1)/2 = rigid motion dimension
Capturing the Symmetry of C Some representations of the crystallographic group G(C) of C. The space group acts by permutation on the set of vertex labels: (κ, k) → g · (κ, k),
g ∈ G(C),
and acts on edge labels, (e, k) → g · (e, k). There is a induced permutation action on the motif set Fv (coset permutation): κ → g · κ where
g · (κ, k) = (g · κ, k ′ ),
which gives a finite-dimensional linear transformation repn ν on R|Fv | .
Capturing the Symmetry of C, II The spatial representation ρsp : g → Tg (to affine isometries of Rd ) satisfies pg ·(κ,k) = Tg (pκ,k ). THM [Owen-P] For the rigidity matrix transfn., R(C) : Hv → He , ρe (g )R(C) = R(C)˜ ρv (g ),
g ∈ G(C),
where ρ˜v = ρv ⊗ ρsp and where ρe and ρv are natural linear representations of G(C), associated with edges and with vertices. This allows in operator methods, group representation methods, invariant subspace methods ...
Space group symmetry equation: 2
THM [P 2011] For the BS matrix R(M, Rd ) for C, 2
2
πe (g )R(M, Rd ) = R(M, Rd )πv (g ),
g ∈ G(C),
where πe is the natural space group representation in R|Fe | , and πv is the representation in the domain space Rd|Fv | ⊕ Rd given by 2
πv (g )(u, A) = ((ν(g ) ⊗ Tg )u, Tg ATg−1 ),
g ∈ G(C),
Point group symmetry equation: Gpt (C) := G(C)/T = G(C)/Zd g ∈ G(C) gives coset g˙ ∈ Gpt (C) Special case: G(C) is separable: G(C) ≈ Zd × Gpt (C) Then there are well-defined induced representations, π˙ e , π˙ v and a similar symmetry equation COR inequality predictor of Ross, Schulze, Whiteley.
General (nonseparable) case Complication: Representation πv of G(C) on the domain space 2 Rd|Fv | ⊕ Rd does not restrict to (nor induce) a representation of the point group Gpt (C). But can obtain individual symmetry equations: Let g be a symmetry in G(C) and let Eg be the space of matrices A (for (u, A) vectors) that commute with the symmetry: gA = Ag Then π˙ e (g˙ )R(M, Eg ) = R(M, Eg )π˙ v (g˙ ), where π˙ v (g˙ ) = (ν(g˙ ) ⊗ Rg˙ ) ⊕ I , and Rg˙ is the ”local isometry” for g˙ . COR Symmetry adapted Maxwell-Calladine formula.
The hexahedron framework. CHEX No nontrivial periodic infinitesimal flexes No nontrivial RUM infinitesimal flexes No nontrivial continuous (finite) deformation (of any kind) There exists an affinely periodic infinitesimal flex.
